diff --git a/samples/texts/1516894/page_1.md b/samples/texts/1516894/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..be5b560caf580843a0ffd02e3b600f03a6eab647 --- /dev/null +++ b/samples/texts/1516894/page_1.md @@ -0,0 +1,7 @@ +ADVANCED +FUNCTIONAL +MATERIALS + +THERMOELECTRICS + +In article 1705823, Hongyun So, Debbie G. Senesky, Eric Pop, and co-workers describe the ability to manipulate electrical transport separately from thermal transport by leveraging the 2-dimensional electron gas in AlGaN/GaN heterostructures. The image shows the transport of electrons and phonons in the heterostructure, which, under a thermal gradient, can be used for on-chip thermal sensing and energy harvesting within power electronics modules. \ No newline at end of file diff --git a/samples/texts/1516894/page_10.md b/samples/texts/1516894/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..b17744728cb24560eaf359c2e4b819f5458788b2 --- /dev/null +++ b/samples/texts/1516894/page_10.md @@ -0,0 +1,31 @@ +[21] Z. H. Liu, S. Arulkumaran, G. I. Ng, *Appl. Phys. Lett.* **2009**, *94*, 142105. + +[22] A. S. Yalamarthy, D. G. Senesky, *Semicond. Sci. Technol.* **2016**, *31*, 035024. + +[23] Y. K. Koh, Y. Cao, D. G. Cahill, D. Jena, *Adv. Funct. Mater.* **2009**, *19*, 610. + +[24] A. J. H. McGaughey, E. S. Landry, D. P. Sellan, C. H. Amon, *Appl. Phys. Lett.* **2011**, *99*, 131904. + +[25] A. Sztein, J. Haberstroh, J. E. Bowers, S. P. Denbaars, S. Nakamura, *J. Appl. Phys.* **2013**, *113*, 183707. + +[26] T. E. Beechem, A. E. McDonald, E. J. Fuller, A. A. Talin, C. M. Rost, J. P. Maria, J. T. Gaskins, P. E. Hopkins, A. A. Allerman, *J. Appl. Phys.* **2016**, *120*, 095104. + +[27] J. Ohta, H. Fujioka, S. Ito, M. Oshima, *Appl. Phys. Lett.* **2002**, *81*, 2373. + +[28] W. Liu, A. A. Balandin, *J. Appl. Phys.* **2005**, *97*, 073710. + +[29] J. Zou, D. Kotchetkov, A. A. Balandin, D. I. Florescu, F. H. Pollak, *J. Appl. Phys.* **2002**, *92*, 2534. + +[30] E. Ziade, J. Yang, G. Brummer, D. Nothern, T. Moustakas, A. J. Schmidt, *Appl. Phys. Lett.* **2017**, *110*, 031903. + +[31] J. Sonntag, *Phys. Rev. B* **2006**, *73*, 045126. + +[32] J. Bahk, T. Favaloro, A. Shakouri, *Annu. Rev. Heat Transfer* **2013**, *16*, 51. + +[33] S. Shimizu, M. S. Bahramy, T. Iizuka, S. Ono, K. Miwa, Y. Tokura, Y. Iwasa, *Proc. Natl. Acad. Sci. USA* **2016**, *113*, 6438. + +Adv. Funct. Mater. **2018**, *28*, 1705823 + +1705823 (9 of 9) + +© 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim \ No newline at end of file diff --git a/samples/texts/1516894/page_11.md b/samples/texts/1516894/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..22fb3bc3d5cb33eca7ae22ca1548d649be5a44f4 --- /dev/null +++ b/samples/texts/1516894/page_11.md @@ -0,0 +1,11 @@ +**ADVANCED** +**FUNCTIONAL** +**MATERIALS** + +Supporting Information + +for *Adv. Funct. Mater.*, DOI: 10.1002/adfm.201705823 + +Tuning Electrical and Thermal Transport in AlGaN/GaN Heterostructures via Buffer Layer Engineering + +Ananth Saran Yalamarthy, Hongyun So,*, Miguel Muñoz Rojo, Ateeq J. Suria, Xiaoqing Xu, Eric Pop, and Debbie G. Senesky* \ No newline at end of file diff --git a/samples/texts/1516894/page_12.md b/samples/texts/1516894/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..6556ce28c35632cdce75bd5371670ef089f50968 --- /dev/null +++ b/samples/texts/1516894/page_12.md @@ -0,0 +1,33 @@ +# Supporting Information + +## Tuning Electrical and Thermal Transport in AlGaN/GaN Heterostructures via Buffer Layer Engineering + +Ananth Saran Yalamarthy¹, Hongyun So²*, Miguel Muñoz Rojo³, Ateeq J. Suria¹, Xiaoqing Xu⁴, Eric Pop³,⁵,⁶, and Debbie G. Senesky³,⁶,⁷* + +¹Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA. + +²Department of Mechanical Engineering, Hanyang University, Seoul 04763, South Korea. + +³Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA. + +⁴Stanford Nanofabrication Facility, Stanford University, Stanford, CA 94305, USA. + +⁵Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA. + +⁶Precourt Institute for Energy, Stanford University, Stanford, CA 94305, USA. + +⁷Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA. + +*E-mail: Hongyun So (hyso@hanyang.ac.kr), Debbie G. Senesky (dsenesky@stanford.edu) + +**Table of Contents:** + +**Supplementary Note 1:** Fabrication Process + +**Supplementary Note 2:** Test Setup and Calibration + +**Supplementary Note 3:** Finite-Element Model, Measurement Process and Error Correction + +**Supplementary Note 4:** Schrödinger-Poisson Model Notes and Validation + +**Supplementary Note 5:** Circular Transfer Length Method (CTLM) \ No newline at end of file diff --git a/samples/texts/1516894/page_13.md b/samples/texts/1516894/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..b3e1fca662976671d8a53f872d0c14ae26cb5e14 --- /dev/null +++ b/samples/texts/1516894/page_13.md @@ -0,0 +1,3 @@ +**Figure S1.** Seven-mask process used to fabricate suspended AlGaN/GaN electrical and thermal measurement platform. + +**Figure S2.** (a) Schematic of grown AlGaN/GaN heterostructure, along with the different buffer layers. (b) Schrödinger-Poisson simulation of decrease in 2DEG charge density ($n_s$) with reducing the thickness of the GaN buffer layer. \ No newline at end of file diff --git a/samples/texts/1516894/page_14.md b/samples/texts/1516894/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..d8ef9f4964aa05bef00cb109e15e3443b4dd2912 --- /dev/null +++ b/samples/texts/1516894/page_14.md @@ -0,0 +1 @@ +Figure S3. (a) Schematic of test setup. We measured the resistance of the heater electrode using a DC multimeter and voltage source with a calibration current of 50 µA. For the sensor side, we used an AC lock-in amplifier with frequency of 97 Hz to minimize self-heating effects. (b) Resistance-temperature calibration for the heater and sensor lines showing linear behavior. (c) TCR for heater and sensor lines. \ No newline at end of file diff --git a/samples/texts/1516894/page_15.md b/samples/texts/1516894/page_15.md new file mode 100644 index 0000000000000000000000000000000000000000..5c7c680580d7e6e0847f9b801cb0b6e3285dcedb --- /dev/null +++ b/samples/texts/1516894/page_15.md @@ -0,0 +1 @@ +**Figure S4.** (a) Thermal resistance network with the different pathways for heat sinking when current is applied to the heater metal. (b) Simulated lateral temperature profile in the AlGaN/GaN/buffer film with ~14 mA heater current. The substrate fixed at 25°C (for the bulk GaN film). (c) Simulated vertical temperature drop across the heater/Al₂O₃/AlGaN/GaN/buffer film for the same conditions in (b). \ No newline at end of file diff --git a/samples/texts/1516894/page_16.md b/samples/texts/1516894/page_16.md new file mode 100644 index 0000000000000000000000000000000000000000..9b8881e8558a0f9cc73be5c487b2015da29ffc97 --- /dev/null +++ b/samples/texts/1516894/page_16.md @@ -0,0 +1 @@ +**Figure S5.** Thermal conductivity measurement. Panels (a)-(c) are for the heater line, while (d)-(f) are for the sensor line. These panels are for the bulk GaN sample with the substrate held at 25°C. \ No newline at end of file diff --git a/samples/texts/1516894/page_17.md b/samples/texts/1516894/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..14f6a3b58c87cbad45fb7642d71d414b593c3292 --- /dev/null +++ b/samples/texts/1516894/page_17.md @@ -0,0 +1 @@ +**Figure S6.** Seebeck coefficient measurement. Panels (a)-(c) are for the heater line, while (d) shows the Seebeck voltage measured in the 2DEG mesa. These panels are for the bulk GaN sample with the substrate held at 25°C. (e,f) Cross-section and top view showing the different electrodes for Seebeck coefficient measurement. **Scale bar of (f) 200 µm.** \ No newline at end of file diff --git a/samples/texts/1516894/page_18.md b/samples/texts/1516894/page_18.md new file mode 100644 index 0000000000000000000000000000000000000000..51d9cb76fd89d1b7f5ee6a0690a03c11893fcd09 --- /dev/null +++ b/samples/texts/1516894/page_18.md @@ -0,0 +1,5 @@ +**Figure S7.** (a) Half-symmetric finite-element simulation of experimental structure, showing sample temperature profile when current is applied through the heater with the bottom fixed at room temperature. (b) Temperature profile when no current is applied through the heater with bottom fixed at 200°C. Notice the cooling in the suspended membrane due to external convection. This effect becomes prominent at temperatures above ~100°C. (c,d) Thermal conductivity measurements predicted from finite-element model for bulk and thin GaN samples, respectively. At higher temperatures, the measured thermal conductivity is higher than the actual thermal conductivity due to the cooling losses described in Figure S7b. **Scale bars of (a), (b) 200 µm.** + +**Table S1.** Estimated thermal conductivity correction factors. + +
T (°C)Bulk GaNThin GaN
75 °C--0.055
100 °C--0.097
150 °C0.0960.19
200 °C0.170.31
250 °C0.320.62
300 °C0.631.1
\ No newline at end of file diff --git a/samples/texts/1516894/page_19.md b/samples/texts/1516894/page_19.md new file mode 100644 index 0000000000000000000000000000000000000000..25068b9f16f5184ca6a9b377ba85b3a8e54d3730 --- /dev/null +++ b/samples/texts/1516894/page_19.md @@ -0,0 +1,2 @@ +**Figure S8.** Room temperature Hall-effect Measurements. (a) Van der Pauw structure used for Hall-effect measurements. An octagonal 2DEG mesa region is used. These devices are co-fabricated with the other electrical and thermal transport test structures described in this article. (b) Extracted sheet density ($n_s$) for a range of applied bias currents ($I_B$) at room temperature. The estimates from the Schrödinger-Poisson model are shown using dashed lines. +Scale bar of (a) 200 µm. \ No newline at end of file diff --git a/samples/texts/1516894/page_2.md b/samples/texts/1516894/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..13ec323fc0988421a9a52e1b2c2821ec72860bc1 --- /dev/null +++ b/samples/texts/1516894/page_2.md @@ -0,0 +1,54 @@ +# Tuning Electrical and Thermal Transport in AlGaN/GaN Heterostructures via Buffer Layer Engineering + +Ananth Saran Yalamarthy, Hongyun So,* Miguel Muñoz Rojo, Ateeq J. Suria, Xiaoqing Xu, Eric Pop, and Debbie G. Senesky* + +Progress in wide bandgap, III-V material systems based on gallium nitride (GaN) has enabled the realization of high-power and high-frequency electronics. Since the highly conductive, 2D electron gas (2DEG) at the aluminum gallium nitride (AlGaN)/GaN interface is based on built-in polarization fields and is confined to nanoscale thicknesses, its charge carriers exhibit much higher mobilities compared to their doped counterparts. This study shows that such 2DEGs also offer the unique ability to manipulate electrical transport separately from thermal transport, through the examination of fully suspended AlGaN/GaN diaphragms of varied GaN buffer layer thickness. Notably, ≈100 nm thin GaN layers can considerably impede heat flow without electrical transport degradation. These achieve 4× improvement in the thermoelectric figure of merit ($zT$) over externally doped GaN, with state-of-the-art power factors of 4–7 mW m$^{-1}$ K$^{-2}$. The remarkable tuning behavior and thermoelectric enhancement, elucidated here for the first time in a polarization-based heterostructure, are achieved because electrons are at the heterostructured interface, while phonons are within the material system. These results highlight the potential for using 2DEGs in III-V materials for on-chip thermal sensing and energy harvesting. + +## 1. Introduction + +Over the past decade, gallium nitride on silicon (GaN-on-Si) substrates have gained widespread use as a materials platform for high-power,[1] high-frequency,[1] and extreme temperature electronics.[2] This technology is enabled by the presence of a 2D electron gas (2DEG) that is formed when a nanometer-thick layer of unintentionally doped aluminum gallium nitride + +(AlGaN), indium gallium nitride (InGaN), or aluminum indium nitride (AlInN) is deposited on an underlying GaN buffer layer.[3–5] The 2DEG, created from built-in polarization fields and surface states in the undoped III-V layers, has a high sheet density $n_s \approx 10^{13}$ cm$^{-2}$, high room-temperature mobility $\mu \approx 1500–2000$ cm$^2$ V$^{-1}$ s$^{-1}$, and has been reported to operate at temperatures as high as 1000 °C.[2] + +GaN thin films have been utilized in the design of thermoelectric devices because of their high-temperature operation and potential for on-chip energy harvesting and sensing.[6] As a result, the electro-thermal transport in various GaN-based materials is being investigated, including externally n-doped AlInN with a thermoelectric figure of merit $zT \approx 0.1$ at 25 °C,[7] and bulk InGaN with a $zT$ as high as 0.34 at ≈600 °C.[8] Recently, 2DEG heterostructures[9,10] and GaN-based superlattices have attracted great attention in the design of thermoelectric devices, as bulk doped mate- + +rials are limited by lower mobilities[11] (200–400 cm$^2$ V$^{-1}$ s$^{-1}$). For example, recent experiments showed ≈10× improvement in power factor ($S^2\sigma$) over bulk doped GaN (as high as $2 \times 10^{-3}$ Wm$^{-1}$ K$^{-2}$ using an AlGaN/GaN superlattice[10]), as well as simultaneous increase in Seebeck coefficient ($S$) and electrical conductivity ($\sigma$) for the 2DEG,[9] contrary to bulk doped materials. The primary contributor in the power factor enhancement is the improved 2DEG mobility,[10] yet, the overall + +A. S. Yalamarthy, Dr. A. J. Suria +Department of Mechanical Engineering +Stanford University +Stanford, CA 94305, USA + +Prof. H. So +Department of Mechanical Engineering +Hanyang University +Seoul 04763, South Korea +E-mail: hyso@hanyang.ac.kr + +Dr. M. Muñoz Rojo, Prof. E. Pop, Prof. D. G. Senesky +Department of Electrical Engineering +Stanford University +Stanford, CA 94305, USA +E-mail: dsenesky@stanford.edu + +The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adfm.201705823. + +Dr. X. Xu +Stanford Nanofabrication Facility +Stanford University +Stanford, CA 94305, USA + +E. Pop +Department of Materials Science and Engineering +Stanford University +Stanford, CA 94305, USA + +E. Pop, D. G. Senesky +Precourt Institute for Energy +Stanford University +Stanford, CA 94305, USA + +D. G. Senesky +Department of Aeronautics and Astronautics +Stanford University +Stanford, CA 94305, USA \ No newline at end of file diff --git a/samples/texts/1516894/page_20.md b/samples/texts/1516894/page_20.md new file mode 100644 index 0000000000000000000000000000000000000000..6bcc4866b145506e5e3cc71d7e284f1bcea0c1d7 --- /dev/null +++ b/samples/texts/1516894/page_20.md @@ -0,0 +1,5 @@ +**Figure S9.** CTLM Measurements. (a) CTLM test array for measurement of contact & sheet resistance with gap spacing varying from 20 µm to 70 µm. (b) Transfer resistance versus gap spacing for the thin and bulk GaN samples to extract sheet & contact resistance at 25°C. (c) Temperature dependent contact resistance for the thin and bulk GaN samples. (d) Temperature dependent sheet resistance for the thin and bulk GaN samples. Error bars are obtained from measurements across 4 identical CTLM test arrays. **Scale bar of (a) 500 µm.** + +**Figure S10.** (a) XRD scan to estimate the density of edge-type dislocations in the bulk GaN sample. The edge type dislocation density is estimated as ≈3×10⁹ cm⁻² following the methods described in Lee *et al.* [11] (b) Cross-plane thermal conductivity estimates of the layers in our composite film as a function of thickness. The experimental data points correspond to cross-plane measurements in GaN. [2,3] + +**Figure S11.** Room temperature Seebeck coefficients ($S$) as a function of carrier concentration ($n_v$) in GaN. The black line represents the Seebeck coefficient model. The experimental data points[4–6] and the model are at room temperature. All the data points correspond to doped, bulk films. The measured values for the 2DEG in the bulk and thin GaN sample in this work are also plotted. \ No newline at end of file diff --git a/samples/texts/1516894/page_21.md b/samples/texts/1516894/page_21.md new file mode 100644 index 0000000000000000000000000000000000000000..e32fe16d88cbecdc88aa8e5188c4b56483e472e0 --- /dev/null +++ b/samples/texts/1516894/page_21.md @@ -0,0 +1,5 @@ +**Figure S12.** Power factor of the 2DEG in the thin and bulk GaN samples. The power factor for state-of-the-art thermoelectric materials is usually between $1.5 \times 10^{-3}$ and $4 \times 10^{-3} \text{ Wm}^{-1}\text{K}^{-2}$ at room temperature.[7] + +**Supplementary Note 1: Fabrication Process** + +Figure S1 shows the seven-mask process to fabricate the fully-suspended AlGaN/GaN platform for thermal measurements. A schematic of the heterostructure showing the different buffer layers and the silicon substrate is illustrated in Figure S2a. The AlGaN/GaN/buffer heterostructure for the thin and bulk GaN samples was grown using an in-house metal organic chemical vapor deposition (MOCVD) chamber on a Si (111) substrate. In order to define the 2DEG mesa, we etched the AlGaN/GaN layers to a depth of ~100 nm using an inductive coupled plasma technique with BCl$_3$/Cl$_2$ gases as shown in Figure S1a. This was followed by the deposition of ~4 µm PECVD oxide on the backside and selectively patterned to define the Si removal region, as depicted in Figure S1b. The Ohmic contacts to the 2DEG were patterned by depositing Ti/Al/Pt/Au (20/100/40/80 nm) followed by a rapid thermal anneal (RTA) in N$_2$ ambient at 850°C for 35 seconds (Figure S1c). Next, we deposited ~47 nm of atomic-layer deposited (ALD) Al$_2$O$_3$ followed by patterning Ti/Pt (10/100 nm) heater and sensor metal lines, as shown in Figure S1d. To deposit Ti/Au (20/200 nm) bondpad metal, we opened vias in the ALD film using a 20:1 buffered oxide etch for ~2 min (Figure S1e). In order to release the AlGaN/GaN/buffer heterostructure, Si was finally etched from the backside using a deep reactive ion etching (DRIE) technique, stopping at the buffer/Si \ No newline at end of file diff --git a/samples/texts/1516894/page_22.md b/samples/texts/1516894/page_22.md new file mode 100644 index 0000000000000000000000000000000000000000..83a3c42eaacc4f26de518888b21e84c9a7b97e0d --- /dev/null +++ b/samples/texts/1516894/page_22.md @@ -0,0 +1,9 @@ +interface. X-Ray diffraction data for the AlGaN/GaN/buffer layers in available in our former work.[8] After suspension, the total thickness of the heterostructure layers was obtained as ~1.695 µm for the thin GaN heterostructure and ~2.85 µm for the bulk GaN heterostructure from the SEM cross-section images. + +**Choice of buffer layer thicknesses:** + +The thermal conductivity of the AlGaN/GaN/buffer hetero-structure is typically determined by the thermal conductivity of the GaN buffer layer. The thickness of the GaN ($t_{GaN}$) layer in the thin GaN sample was designed to lower the thermal conductivity of the buffer layer structure due to size effect, while preserving the charge density $n_s$ of the 2DEG. This effect is shown in Figure S2b, which depicts the rapid decline in $n_s$ with $t_{GaN}$ due to band bending in the AlGaN and GaN layers from the Schrödinger Poisson model. As the GaN layer thickness decreases below ~100 nm, the decline is much sharper as the strain difference in the AlGaN/GaN layers decreases, leading to the GaN layer becoming pseudomorphic with the buffer layers beneath. + +**Supplementary Note 2: Test Setup and Calibration** + +Figure S3a shows the test setup used to measure the in-plane thermal conductivity of the AlGaN/GaN hetero-structure. In order to ensure accuracy in the thermal conductivity measurements, we performed careful resistance versus temperature calibration for the Ti/Pt heater and sensor lines. For the heater line, a DC current source (Keithley 2400) and a DC voltage source (Agilent 34401) were used to measure the resistance of the Ti/Pt trace. To measure the resistance of the sensor line, we used an AC voltage lock-in amplifier (Zurich Instruments HF2LI) with a lock-in frequency of 97 Hz. AC voltage measurement across a fixed resistor (1 kΩ, ultra-low TCR of less than 1 ppm) was used to infer the AC current from the applied AC voltage. The lock in-amplifier was chosen for the sensor side to minimize self- \ No newline at end of file diff --git a/samples/texts/1516894/page_23.md b/samples/texts/1516894/page_23.md new file mode 100644 index 0000000000000000000000000000000000000000..ae4f690cbb8407cba18a4243cada40439158002e --- /dev/null +++ b/samples/texts/1516894/page_23.md @@ -0,0 +1,9 @@ +heating effects and block environmental noise. In order to calibrate the resistances of both lines, the substrate of the suspended membrane was attached to a temperature controlled chuck using high vacuum thermal grease (Apiezon, Inc.). A current value of ~100 µA was carefully chosen for the purpose of resistance calibration to avoid self-heating effects. + +Figure S3b and Figure S3c show the calibration curves of resistance (R) and temperature coefficient of resistance (TCR) till 300°C. Initially, we obtained a non-linear resistance-temperature calibration curve due to the effects of alloying in the Ti/Pt metal. In order to alleviate this problem, the entire sample was heated to ~400°C and held for ~10 minutes to anneal the Ti/Pt metals. Upon annealing, the resistance calibration curve is found to be extremely linear as can be seen in Figure S3b ($R^2 \approx 0.998$), and this is later used to extract the temperature of the heater line when heating power is applied to it to extract the thermal conductivity of the AlGaN/GaN heterostructure. The fitted slope is ~0.148 ΩK⁻¹, as can be seen in Figure S3b. Note that the plotted resistance values are obtained by averaging over 20 measurements spaced by 2 seconds at each substrate temperature. In each case, the error bar (defined as the range) for the resistance measurement is smaller than the size of the markers in Figure S3b and Figure S3c, with steadily increasing error bar as the temperature increases. Typical values for the error bars are ~1.5 mΩ at 25°C with a steady increase to ~50 mΩ at 300°C. Using the fitted slope, this implies an error of < +/- 0.5°C for the measured temperature even at 300°C, which is accounted for in the extraction of thermal conductivity. + +A similar calibration procedure was performed for the heater line in the Seebeck coefficient measurement platform for the bulk and thin GaN samples. + +**Supplementary Note 3: Finite-Element Model, Measurement Process and Error Correction** + +In order to understand the errors associated with the measurement and extract the thermal properties of the AlGaN/GaN/buffer hetero-structure, a 3-D finite-element model using a commercial software (COMSOL) was implemented. Figure S7a shows a half-symmetric \ No newline at end of file diff --git a/samples/texts/1516894/page_24.md b/samples/texts/1516894/page_24.md new file mode 100644 index 0000000000000000000000000000000000000000..858d4ed630875e845b3dc81eae11f97abf2b13b2 --- /dev/null +++ b/samples/texts/1516894/page_24.md @@ -0,0 +1,26 @@ +finite-element simulation of experimental structure, showing sample temperature profile when +current is applied through the heater with the substrate fixed 25°C. The heater and sensor lines +have a width (W) of 5 µm, and are spaced by a distance (DHS) of 75 µm. The location of the +heater and sensor resistances on the suspended membrane (RH and RS) are chosen such that +the heat transfer can be well approximated as 1-D,[9] which facilitates the extraction of the +thermal conductivity. The typical variation of temperature along the length of these resistors +is estimated to be < 0.01% from the simulation model, which supports this assumption. Figure +S4a shows a cross-section schematic of the thermal resistance network with the different +pathways for heat sinking when current is applied to the heater metal. In the absence of +external convective and radiative losses and negligible contribution of the Alumina film to the +in-plane heat conduction, the thermal resistance of the film (RF) can be estimated as: + +$$ +R_F = \frac{2(T_H - T_S)}{P_H} - 2R_{Al} - \frac{2(R_{mox} + R_{oxg})}{A_H} \quad (S1) +$$ + +where $T_H$ and $T_S$ are the heater and sensor line temperatures, $P_H$ is the input power to the heater and $R_{Al}$ is the thermal resistance of the Al$_2$O$_3$ layer, $A_H$ is the area projected area of the heater electrode (5 $\mu$m × 200 $\mu$m), $R_{mox}$ is the thermal boundary resistance of the Heater/Al$_2$O$_3$ interface and $R_{oxg}$ is the thermal boundary resistance of the Al$_2$O$_3$/GaN interface. The thermal conductivity of the film can be extracted from $R_F$ and the known film dimensions. The simulated temperature drop from the heater to the sensor in the GaN film is linear, as can be seen along the Y direction in Figure S4b. Equation S1 suggests that the thermal conductivity of the film can be measured accurately in the limit of $R_F \gg R_{Al+Interfaces}$. + +The resistance ratio is analytically estimated as: + +$$ +\frac{R_{Al+Interfaces}}{R_F} = \frac{T_{Al} k_F T_F}{k_{Al} W D_{HS}} + \frac{(R_{mox} + R_{oxg}) k_F T_F}{W D_{HS}} \quad (S2) +$$ + +Where $T_{Al}$ and $T_F$ are the thicknesses of the alumina and AlGaN/GaN/buffer film, respectively. +We used a thermal boundary resistance of 2.8×10⁻⁸ m²K W⁻¹ for $R_{mox}$. [10] Although data for \ No newline at end of file diff --git a/samples/texts/1516894/page_25.md b/samples/texts/1516894/page_25.md new file mode 100644 index 0000000000000000000000000000000000000000..eff73af43e06352d2caa65af95061309f306da26 --- /dev/null +++ b/samples/texts/1516894/page_25.md @@ -0,0 +1,3 @@ +the thermal boundary resistance across the Al₂O₃/GaN film interface is not available, we estimated Roxg ~ 1×10⁻⁸ m²KW⁻¹ based on measurements across amorphous dielectric/Si interfaces,[11] since GaN and Si have similar Debye temperatures.[12,13] At room temperature for the bulk GaN film, using TF ≈ 3 µm, TAl ≈ 47 nm, kAl ≈ 2 Wm-1K-1[10] and kF ≈ 115 Wm-1K-1, we estimate a thermal resistance ratio of ~5.7%. For the thin GaN film, since TF ≈ 2 µm and kF ≈ 45 Wm-1K-1, the thermal resistance ratio is ~1.48%, thus the error due to loss in Al₂O₃ is smaller than the bulk GaN film. Note that these values also support the assumption in Equation S1 that Alumina does not contribute to the in-plane heat conduction, since kAlTAl << kFTF. At higher temperatures, the error due to this effect is less pronounced as kAl is found to increase,[10] while kF further decreases, as can be seen in Figure 3b. The loss via the Al₂O₃ and the interfaces can also be observed via the COMSOL model, as can be seen in Figure S4c (bulk GaN film, substrate at 25°C). In the model, in addition to using kAl ≈ 2 Wm-1K-1, The temperature drops by ~0.47 K in Al₂O₃ and interfaces on either side relative to ~7.4 K in the GaN film (across DHS), giving an error of ~6.35%. + +Figure S5 shows the typical thermal conductivity measurement procedure for our films. These plots are from experiments with the bulk GaN sample. In this experiment, the substrate is held at 25°C. The sensor is maintained at the calibration current of ~100 µA (Figure S5f), while the heater current is ramped up in a half-sinusoid from its initial calibration value (Figure S5b). Before each resistance measurement, we wait for 2 seconds after the current ramp to allow the system to equilibrate. The waiting interval of 2 seconds was chosen based on an estimation of a thermal time constant of ~2 milli-seconds for the suspended membrane from COMSOL simulations. The heater & sensor temperature (converted from the resistance via the calibration curve in Figure S3b) track the current pattern, with the initial temperature equal to the substrate temperature, as seen in Figure S5c and Figure S5d. The extracted temperature difference can be used to calculate the in-plane film thermal conductivity via Equation S1, after accounting for the Al₂O₃ temperature drop and external losses, which are \ No newline at end of file diff --git a/samples/texts/1516894/page_26.md b/samples/texts/1516894/page_26.md new file mode 100644 index 0000000000000000000000000000000000000000..2087ee4551fdf5bc81dc9db7896c90a5d41f567a --- /dev/null +++ b/samples/texts/1516894/page_26.md @@ -0,0 +1,9 @@ +discussed later. In addition, we also ensured that hysteresis did not occur in our heater and sensor lines. This is clear from observing the temperature versus power lines in Figure S5a and Figure S5e. + +Figure S6 shows a typical Seebeck coefficient measurement procedure. Similar to the thermal conductivity measurement, the heater current is ramped up from its calibration value, setting up a lateral temperature gradient along the 2DEG mesa which translates to a measurable Seebeck voltage (Figure S6d). The measured Seebeck voltage includes a minor contribution from the temperature drop across the Ti/Al/Pt/Au Ohmic contacts to the 2DEG (visualized in Figure S6f). At room temperature, we measured the Seebeck voltage across the 2DEG mesa and the Ohmic metal line for an identical temperature gradient. Using this, we estimated the contribution of the Ohmic metal line to be less than 2% of the overall Seebeck voltage, and thus neglected its effect in subsequent measurements. The Seebeck voltage of the 2DEG is given as $S_{2DEG}=V_{2DEG}/(T_1-T_2)$, as depicted in Figure S6e and Figure S6f. The temperature at the contact outside the suspended region ($T_2$) is assumed to be at the substrate temperature. The temperature drop in the silicon supported region is <1% of the total temperature drop ($T_1-T_2$) (Figure S4b), thus, the contribution to the Seebeck coefficient from the supported region can be ignored. $T_1$ is related to the heater temperature $T_H$ as: + +$$ \frac{T_H - T_1}{P_H} = R_{Al} + \frac{R_F}{2} + \frac{(R_{mox} + R_{oxg})}{A_H} \quad (S3) $$ + +where $R_F$ is calculated using the measured film thermal conductivity and a length of 30 µm ($D_S$, depicted in Figure S6e) and $R_{Al}$ is calculated as discussed earlier. + +Finally, external losses from convection are significant at high temperatures which lead to errors in the thermal conductivity measurement. This can be seen in Figure S7b, which shows the temperature profile when no current is applied through the heater with bottom fixed at 200°C for the bulk GaN film. Notice the cooling in the suspended membrane due to external convection, leading a relative temperature difference between the heater and the \ No newline at end of file diff --git a/samples/texts/1516894/page_27.md b/samples/texts/1516894/page_27.md new file mode 100644 index 0000000000000000000000000000000000000000..ba8072961578e49139b373a8ddee6ce7e462fee5 --- /dev/null +++ b/samples/texts/1516894/page_27.md @@ -0,0 +1,5 @@ +sensor line. This underestimates the ΔT between the heater and the sensor line, leading a higher measured thermal conductivity than the true value. The actual temperature gradient can be expressed as $\Delta T_{\text{real}} = \Delta T_{\text{meas}} + \Delta T_{\text{corr}}$, where $\Delta T_{\text{corr}}$ is the temperature gradient from the heater to the sensor with no current applied. The prominence of this effect is seen in Figure S7c and Figure S7d, which shows the simulated true and estimated measurements of the thermal conductivity for the normal GaN and thin GaN film. The error in the measured thermal conductivity is estimated to be ~4% and ~10% for the normal and thin GaN films from simulations, using an external convection coefficient of 10 Wm⁻²K⁻¹. The correction factor $\Delta T_{\text{corr}}$ can be estimated from the natural convection coefficient ($h$). At progressively higher temperatures, a non-zero Seebeck voltage is observed when no current is applied in the heater line due to the effect shown in Figure S7b. Then, using the most recently corrected thermal conductivity value, we estimated the value of $h$ required to produce the observed non-zero Seebeck voltage using our knowledge of the Seebeck coefficient from the COMSOL model. Following this, we obtained the correction factor $\Delta T_{\text{corr}}$ for the current thermal conductivity value. Typical values for $h$ estimated using this procedure are in the range of 12-14 Wm⁻²K⁻¹, which are reasonable coefficients for natural convection. The correction factors ($\Delta T_{\text{corr}}$) we obtained for the normal GaN and thin GaN films for the different substrate are tabulated in Table S1. + +**Supplementary Note 4: Schrödinger-Poisson Model Notes and Validation** + +The models for the bulk and thin GaN heterostructures were made using a commercially available Schrödinger-Poisson device physics simulator (NextNano Inc.).[14] In both models, we set the barrier height for the GaN capping layer to 1 eV, based on the assumption that the surface is exposed to air.[15] The entire structure is simulated using a 1-D grid size of 0.5 nm, except in the region where the 2DEG quantum well forms, where we used a finer grid of 0.1 \ No newline at end of file diff --git a/samples/texts/1516894/page_28.md b/samples/texts/1516894/page_28.md new file mode 100644 index 0000000000000000000000000000000000000000..06606b9540c9541d22a9e14dd75881c2c389fb53 --- /dev/null +++ b/samples/texts/1516894/page_28.md @@ -0,0 +1,7 @@ +nm. In both cases, the simulation requires a substrate to determine the strain in the heterostructure. For the bulk GaN model, we used a thick 500 nm GaN layer as the substrate to converge the 2DEG sheet density ($n_s$). However, for the thin GaN model, the GaN layer was set to 100 nm, while the thick layer below it, $Al_{0.2}Ga_{0.8}N$, was used as the substrate to determine the strain level. A 500 nm thick $Al_{0.2}Ga_{0.8}N$ layer was sufficient to converge the charge density in the thin GaN model. A lattice temperature of 300 K was used for the both models. The source code for both models is available at https://github.com/ananthy/GaNThick. + +The Schrödinger-Poisson model is validated by comparing the sheet densities obtained from the model with the values obtained from Hall-effect measurements. We first recall that we obtained a sheet density $n_s = 1.06 \times 10^{13}$ cm⁻² and $n_s = 0.91 \times 10^{13}$ cm⁻² for the bulk and thin GaN heterostructures from the simulations, respectively. We recall that the sheet density is ~16% lower for the thin GaN sample due to the reduced 2DEG quantum well depth ($t_{2D}$), but has a similar peak volumetric charge density as the bulk GaN sample, as discussed in Figure 3. A simple, 4-contact Van der Pauw structure was used to characterize the sheet density via Hall-effect measurements. The structure is shown in Figure S8a. The conducting 2DEG area for each Hall-effect plate was 200 µm × 200 µm. Using a range of bias current levels ($I_B$), an external magnetic field ($B$) of ~1 mT, $n_s$ is related to the Hall voltage $V_H$ as + +$$n_s = \frac{I_B B}{q V_H} \quad (S5)$$ + +where $q$ is the electronic charge. The results from these experiments performed at room temperature are depicted in Figure S8b. Good match between the sheet densities obtained from the experiments and the model is observed. The values from the experiments are about 5-10% higher than those predicted from the Schrödinger-Poisson model, however, we note that the average difference in the sheet densities of the thin and bulk GaN samples from experiments (~14.3%) and the model (~16.5%) is predicted accurately, which serves as a validation for the model, in particular, the thin GaN heterostructure. The fluctuation in the \ No newline at end of file diff --git a/samples/texts/1516894/page_29.md b/samples/texts/1516894/page_29.md new file mode 100644 index 0000000000000000000000000000000000000000..4534e973161a8036d3c2aa4199315c30672fe847 --- /dev/null +++ b/samples/texts/1516894/page_29.md @@ -0,0 +1,19 @@ +experimentally observed *n*s values for different bias currents (*I*B) could arise from non- +linearity in the current-voltage characteristics and other sources such as offset voltage from +thermal effects.[16] + +**Supplementary Note 5: Circular Transfer Length Method (CTLM)** + +CTLM test structures with gap spacing (d) varying from 20-70 µm (Figure S9a) were used to measure the sheet resistance of the 2DEG and the contact resistance for the thin and bulk GaN samples. The inner radius (L) of the circular test structures was designed to be 500 µm, such that the approximation L ≫ d is valid. In the regime where L ≫ d, the total resistance RT between the inner and outer contact can be expressed as[17]: + +$$ +R_T = \frac{R_{sh}(d + 2L_T)}{2\pi d} \log \left( 1 + \frac{d}{L} \right) \quad (S4) +$$ + +where $L_T$ is the transfer length and $R_{sh}$ is the sheet resistance of the 2DEG. Here, $L_T$ is related to $R_{sh}$ and the contact resistance $\rho_c$ as: + +$$ +L_T = \sqrt{\rho_c / R_{sh}} \tag{S5} +$$ + +Figure S9b illustrates the variation of the total resistance $R_T$ with gap spacing $d$ (Equation S4) for the thin and bulk GaN samples at 25°C, which is used to extract $R_{sh}$ and $\rho_c$. Using the 2DEG thickness, $t_{2D}$, the extracted $R_{sh}$ is used to extract the 2DEG conductivity, as plotted in Figure 3c from 25 °C to 300°C. The values of $R_{sh}$ for the bulk and thin GaN sample are also shown in Figure S9d. The contact resistance values are noted to be in the right range for typical 2DEG Ohmic contacts[18] ($10^{-5}-10^{-6} \Omega \cdot \text{cm}^{-2}$) and decreasing with increasing temperature due to enhanced thermionic field emission across the GaN/AlGaN layers,[19,20] as observed in Figure S9c. A similar magnitude of decrease has also been observed in former work.[19] \ No newline at end of file diff --git a/samples/texts/1516894/page_3.md b/samples/texts/1516894/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..c5a0581f0535ab32a0281465d434a0df8b505e52 --- /dev/null +++ b/samples/texts/1516894/page_3.md @@ -0,0 +1,106 @@ +thermoelectric efficiency is limited by the high thermal conduc- +tivity[12] of its underlying GaN buffer layer. To date, researchers +have overlooked the ability to modify the GaN buffer layer to a +reduced size in order to significantly scatter the phonons in it, +while simultaneously preserving the electrical integrity of the +2DEG. Furthermore, the thermoelectric properties of material +systems where the interfacial charge is caused by polarization +fields (e.g., AlGaN/GaN) have not yet been investigated over a +wide range of temperature and buffer configurations. + +In this communication, we present this remarkable ability to +independently manipulate electrical transport separately from +thermal transport in AlGaN/GaN heterostructures. This tuning +behavior is possible as it arises from polarization fields; the elec- +trons are tightly confined at the interface, while the phonons are +in the material system.[10] Our device test platform is composed +of a fully suspended, microfabricated device architecture that +enables characterization of the in-plane thermal and electrical +transport in AlGaN/GaN heterostructures with varying buffer +layer thickness. **Figure 1** shows a conceptual schematic of the +aims of this study. Transport properties are studied over a wide +temperature range from 25 to 300 °C. Notably, we show that +≈100 nm thin GaN layers can considerably impede heat flow +without significant electrical transport degradation, and that a +large improvement (≈4x) in the thermoelectric figure of merit +over externally doped GaN is observed in 2DEG-based hetero- +structures. Furthermore, our experiments also demonstrate state- +of-the-art[13] thermoelectric power factors (4–7 × 10⁻³ Wm⁻¹ K⁻² +at room temperature) observed in the 2DEG of this material +system. The remarkable tuning behavior and thermoelectric +enhancement, elucidated here for the first time in the AlGaN/ +GaN 2DEG heterostructure, demonstrate how manipulating the +polarization fields at material interfaces can be used for thermal +sensing and energy harvesting applications. + +**Figure 1.** Conceptual schematic showing thermoelectric voltage genera- +tion via a lateral temperature gradient in the AlGaN/GaN 2DEG. The high +mobility electrons in the 2DEG can lead to high thermoelectric power +factors across the 2DEG. In this study, we explore how the thickness of +the underlying GaN and buffer layers can be designed to preserve the +thermoelectric power factor of the 2DEG but significantly reduce the in- +plane thermal conductivity. This allows for a large improvement in the +thermoelectric figure of merit in comparison with bulk doped GaN. Note +that we consider the in-plane thermal conductivity of the GaN and the +buffer (but not Si), since only these are necessary for 2DEG formation. + +## 2. Test Structures and Measurements + +Figure 2a,b shows microscope images of our two fully sus- +pended platforms for the measurement of in-plane thermal con- +ductivity of the heterostructure stack and Seebeck coefficient of +the 2DEG. A scanning electron microscope (SEM) image of a +cross-section of the suspension region is shown in Figure 2c. +These structures are based on the well-known central line heater +method used in thermal characterization.[14,15] The suspended +platform was microfabricated using a seven-mask process +(Section S1, Supporting Information), with deep reactive-ion +etching (DRIE) used as the final processing step to remove +the Si(111) below the heterostructure, as seen in Figure 2c. +Two parallel, ≈5 µm wide Ti/Pt metal lines separated by 75 µm +are used as heater and sensor thermometers, patterned on a +≈47 nm thick amorphous Al₂O₃ layer that provides electrical +isolation from the heterostructure. For Seebeck coefficient +measurement, only a heater thermometer is patterned adjacent +to a 2DEG mesa with Ohmic contacts extending to the sub- +strate, as illustrated in Figure 2b. We used an in-house metal +organic chemical vapor deposition (MOCVD) system (Aixtron, +Inc.) to deposit the AlGaN/GaN/buffer heterostructure layers on +top of p-type Si(111) substrates with resistivity of 0.1–1 Ω·cm. +Additional details about the growth process can be found in +our former work.[16] The buffer layers (AlₓGa₁-xN, 0 ≤ x ≤ 1) +are unintentionally doped below 10¹⁶ cm⁻³. Current–voltage +(I–V) measurements after etching the 2DEG mesa were below +the measurement resolution of our system (≈10 pA), which +supports the assumption of Rbuffer >> R₂₂DEG, where R is the +resistance. Thus, the buffer layers can be considered semi- +insulating. Two variants of the heterostructure with GaN thick- +nesses of 1.2 µm and ≈100–150 nm are grown and called the +“bulk GaN” (Figure 2d) and “thin GaN” (Figure 2e) samples, +respectively. The “bulk GaN” heterostructure is still a thin film +and reflects the heterostructure thicknesses that are typical for +AlGaN/GaN power devices.[17] The selection of the GaN thick- +ness in the “thin GaN” device structure is based on a trade- +off to reduce the thermal conductivity of the buffer structure +while preserving the 2DEG conductivity (Section S1, Sup- +porting Information). Forming of the 2DEG was accomplished +by depositing ≈30 nm of unintentionally doped Al₀.₂₅Ga₀.₇₅N +barrier layer on the GaN layer in both heterostructure variants. +A thin GaN capping layer of ≈3 nm was grown on top of the +AlGaN barrier layer, and a 1 nm thick AlN spacer was inserted +between the AlGaN and the GaN layers for 2DEG mobility +enhancement.[18] + +Measurement of the in-plane thermal conductivity is con- +ducted as follows. The sample is attached to a temperature- +controlled chuck (Signatone Inc.) via a vacuum-compatible +thermal grease (Apiezon Inc.) with air as the ambient. We pass +a range of DC currents through the heater metal line to induce +a temperature gradient in the heterostructure (Figure 2f) and +simultaneously measure the electrical resistance of the metal +electrodes. Typical current values are chosen to induce a max- +imum ΔT ≈ 20 K referenced to the substrate temperature, which +varies from 25 to 300 °C. The placement of the sensor electrode +was carefully designed to allow for a 1D in-plane heat transfer +approximation in the diaphragm.[14] The electrical resistance of +the electrodes was calibrated over the entire temperature range \ No newline at end of file diff --git a/samples/texts/1516894/page_30.md b/samples/texts/1516894/page_30.md new file mode 100644 index 0000000000000000000000000000000000000000..69e5e881b3bdee14d4d0ec10883f4761d5018742 --- /dev/null +++ b/samples/texts/1516894/page_30.md @@ -0,0 +1,41 @@ +**References** + +[1] H.-P. Lee, J. Perozek, L. D. Rosario, C. Bayram, *Sci. Rep.* **2016**, *6*, 37588. + +[2] Y. K. Koh, Y. Cao, D. G. Cahill, D. Jena, *Adv. Funct. Mater.* **2009**, *19*, 610. + +[3] J. Cho, Y. Li, W. E. Hoke, D. H. Altman, M. Asheghi, K. E. Goodson, *Phys. Rev. B* **2014**, *89*, 115301. + +[4] K. Nagase, S. Takado, K. Nakahara, *Phys. Status Solidi A* **2016**, 213, 1088. + +[5] E. N. Hurwitz, B. Kucukgok, A. G. Melton, Z. Liu, N. Lu, I. Ferguson, *MRS Proc.* **2012**, 1396, mrsf11-1396-o08-10. + +[6] B. Kucukgok, B. Wang, A. G. Melton, N. Lu, I. T. Ferguson, *Phys. Status Solidi* **2014**, *11*, 894. + +[7] H. Ohta, S. W. Kim, S. Kaneki, A. Yamamoto, T. Hashizume, *Adv. Sci.* **2017**, 1700696. + +[8] X. Xu, J. Zhong, H. So, A. Norvilas, C. Sommerhalter, D. G. Senesky, M. Tang, *AIP Adv.* **2016**, *6*, 115016. + +[9] C. Dames, *Annu. Rev. Heat Transf.* **2013**, *16*, 7. + +[10] A. Cappella, J.-L. Battaglia, V. Schick, A. Kusiak, A. Lamperti, C. Wiemer, B. Hay, *Adv. Eng. Mater.* **2013**, *15*, 1046. + +[11] M.-H. Bae, Z. Li, Z. Aksamija, P. N. Martin, F. Xiong, Z. Ong, I. Knezevic, E. Pop, *Nat. Commun.* **2013**, *4*, 1734. + +[12] H. R. Shanks, P. D. Maycock, P. H. Sidles, G. C. Danielson, *Phys. Rev.* **1963**, *130*, 1743. + +[13] A. Sztein, J. Haberstroh, J. E. Bowers, S. P. Denbaars, S. Nakamura, *J. Appl. Phys.* **2013**, *113*, 183707. + +[14] S. Birner, T. Zibold, T. Andlauer, T. Kubis, M. Sabathil, A. Trellakis, P. Vogl, *IEEE Trans. Electron Devices* **2007**, *54*, 2137. + +[15] S. Heikman, S. Keller, Y. Wu, J. S. Speck, S. P. Denbaars, U. K. Mishra, *Appl. Phys. Lett.* **2003**, *93*, 10114. + +[16] U. Ausserlechner, *J. Sensors* **2016**, *2016*, 5625607. + +[17] D. K. Schroder, Semiconductor Material and Device Characterization **1990**. + +[18] A. Schmid, C. Schroeter, R. Otto, M. Schuster, V. Klemm, D. Rafaja, J. Heitmann, *Appl. Phys. Lett.* **2015**, *106*, 053509. + +[19] Z. H. Liu, S. Arulkumaran, G. I. Ng, *Appl. Phys. Lett.* **2009**, *94*, 142105. + +[20] A. Fontsere, A. Pérez-Tomás, M. Placidi, P. Fernández-Martínez, N. Baron, S. Chenot, Y. Cordier, J. C. Moreno, P. M. Gammon, M. R. Jennings, *Microelectron. Eng.* **2011**, *88*, 3140. \ No newline at end of file diff --git a/samples/texts/1516894/page_4.md b/samples/texts/1516894/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..f2af4daa13b56220819e97631c80be3ad95e9376 --- /dev/null +++ b/samples/texts/1516894/page_4.md @@ -0,0 +1,11 @@ +**Figure 2.** a) Microscope image of the fully suspended AlGaN/GaN heterostructure used for in-plane thermal conductivity measurements. The separation between the heater and sensor lines is ≈75 µm. A thin ≈47 nm Al₂O₃ layer provides electrical isolation between the metal lines and the AlGaN/GaN heterostructure underneath. b) Microscope image of the suspended AlGaN/GaN heterostructure to measure the Seebeck coefficient of the 2DEG. The 2DEG mesa is contacted via Ohmic Ti/Al/Pt/Au contacts. The Seebeck 2DEG mesa is shaded white for clarity. c) Cross-section SEM image (of the A–B section in (a)) of the suspended heterostructure, with the Si substrate selectively etched out from the backside via DRIE. d) SEM image of the bulk GaN structure. The GaN thickness is ≈1.2 µm and false colored. The buffer structure, starting from the Si interface, is composed of AlN (300 nm)/Al₀.₈Ga₀.₂N (300 nm)/Al₀.₅Ga₀.₅N (400 nm)/Al₀.₂Ga₀.₈N (500 nm). Further details are in Section S1 in the Supporting Information. e) SEM image of the thin GaN structure, showing the 2DEG at the AlGaN/GaN interface and the buffer layers. The GaN thickness is ≈100–150 nm and is false colored. The other buffer layers are identical to the bulk GaN structure. f) Half-symmetric finite-element simulation of experimental structure, showing sample temperature profile when current is applied through the heater (Section S3, Supporting Information). Scale bars of (a)–(e), 200, 200, 3, 2, and 2 µm. + +using sufficiently low currents to avoid self-heating (Section S2, Supporting Information). The calibration allows us to convert the electrical resistance into corresponding temperature values using the measured temperature coefficient of resistance. From the collected temperature data, we can infer the in-plane thermal conductivity of the heterostructure given the heater power ($P_H$), after accounting for errors due to heat spreading into the Al₂O₃ and external losses (Section S3, Supporting Information) through a simple analytical model in conjunction with a three-dimensional (3D) finite-element simulation. In the model, we also included estimated values of the thermal contact resistance between the electrode, insulation, and heterostructure interfaces. Overall, the errors due the insulation are found to be less than ≈6%, while errors due to external convective and radiative losses progressively increase to ≈10% at a substrate temperature of 300 °C (Section S3, Supporting Information). + +The measurement of the Seebeck coefficient follows a similar procedure; a current passed through the heater electrode induces a temperature gradient in the diaphragm, resulting in a Seebeck voltage across the 2DEG mesa that spans the suspension and the substrate regions (Figure 2b). Using a similar calibration procedure for the heater line, the temperature drop + +across the mesa can be used to extract the Seebeck coefficient, after accounting for external losses, Ohmic contact voltage drop, and a minor temperature drop in the substrate (Section S3, Supporting Information). Note that the measured Seebeck coefficient corresponds to the 2DEG contribution exclusively since the III–V buffer layers are semi-insulating. Lastly, electrical conductivity of the 2DEG for the bulk and thin GaN samples is estimated using circular transfer length method (CTLM) structures with varying channel lengths ($d = 20$ to $70$ µm), with the aid of simulations to obtain the thickness of the 2DEG triangular potential ($t_{2D}$) well, to be discussed in the next section. + +## 3. Charge Profiles + +The thickness and charge density of the 2DEG for the bulk and thin GaN heterostructures are simulated using a commercially available Schrödinger-Poisson solver (NextNano GmbH[19]). The simulated band structures and volumetric charge density profiles for the bulk and thin GaN heterostructures are illustrated in Figure 3. All the heterostructure layers are assumed \ No newline at end of file diff --git a/samples/texts/1516894/page_5.md b/samples/texts/1516894/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..1014cfa8f4411a1b228a0449caa5ba4ef3ca9ad6 --- /dev/null +++ b/samples/texts/1516894/page_5.md @@ -0,0 +1,11 @@ +**Figure 3.** a) Schrödinger–Poisson model of the energy band diagram for the bulk GaN structure with AlGaN thickness of 30 nm. The thickness of the 2DEG region, $t_{2D}$, is shown in the region where GaN is degenerate. b) Volumetric charge density, $n_v$, versus position, depicting the approximately triangular charge profile, with $t_{2D} \approx 6.1$ nm. c) Simulated energy band diagram for the thin GaN structure with AlGaN thickness of 30 nm. Note that the GaN layer is 100 nm. d) Volumetric charge density, $n_v$, versus position, depicting the approximately triangular charge profile, with $t_{2D} \approx 4.4$ nm for the thin GaN structure. + +to be undoped, and the barrier height for the GaN capping layer is set to 1 eV.[20] For the bulk and thin GaN models, the GaN thickness and the Al0.2Ga0.8N layer thickness (first buffer layer below the GaN) were varied until 2DEG sheet density ($n_s$) convergence was observed. In both cases, the 2DEG region is visible as a triangular potential well near the AlGaN/AlN/GaN interface. From the simulation, we found $n_s = 1.06 \times 10^{13}$ cm-2 and $n_v = 0.91 \times 10^{13}$ cm-2 for the bulk and thin GaN heterostructures, respectively. It should be noted that a good match, within ≈10% of the theoretically calculated values, is observed when comparing these values with experimental data extracted from Hall-effect devices fabricated on the same platform, which supports the model (Section S4, Supporting Information). The physical thickness of the 2DEG region, $t_{2D}$, can be extracted as the region where GaN is degenerate.[9] From simulation, these thickness values were obtained to be ≈6.1 and ≈4.4 nm for the bulk and thin GaN heterostructures, respectively, which can be used to obtain the 2DEG conductivity $\sigma$ from the sheet resistance ($R_{sh}$) extracted via CTLM measurements. Finally, we note that an average 2DEG volumetric density can be estimated as $n_v = n_s/t_{2D}$ for the bulk GaN (1.73 × 1019 cm-3) and thin GaN (2.07 × 1019 cm-3) heterostructures. We note that the higher $n_v$ for the thin GaN sample reflects the smaller 2DEG quantum well thickness. + +## 4. Electrical and Thermal Property Measurements + +The measurements of $R_{sh}$ averaged over four samples up to 300 °C via CTLM measurements (Section S5, Supporting + +Information) can be combined with the 2DEG thickness $t_{2D}$ to obtain the average electrical conductivity [$\sigma = 1/(R_{sh} \times t_{2D})$] of the electrons in the 2DEG. At room temperature, we obtained $R_{sh}$ values of ≈350 and ≈500 Ω sq-1 for the bulk and thin GaN samples, respectively. We note that these values are among the lowest reported $R_{sh}$ values for AlGaN/GaN 2DEGs, which highlights the quality of our samples.[16] The average conductivity in the thin GaN sample is observed to be similar to the bulk GaN sample due to simultaneous reduction in the sheet density and quantum well depth, as seen in **Figure 4a**. Sheet densities in this temperature range are approximately constant due to negligible strain relaxation in the heterostructure layers,[21] stable piezoelectric coefficients,[22] and minimal intrinsic carrier concentration change due to the wide bandgap. Thus, the decrease of $\sigma$ at high temperatures is mainly determined by the 2DEG mobility, $\mu$. The dependence is well described by a temperature power law ≈$T^{-2.5}$ that arises from electron–optical phonon scattering, which further supports this fact. We also note that our exponent is consistent with former exponent ranges (from −2.2 to −3.4) reported in the literature.[21] + +Temperature-dependent in-plane thermal conductivity measurements for the bulk and thin GaN samples are shown in Figure 4b. Room temperature thermal conductivity dropped from ≈115 Wm-1 K-1 for the bulk GaN sample to ≈45 Wm-1 K-1 for the thin GaN sample due to phonon boundary scattering, i.e., the size effect.[15] The measurements for the bulk GaN sample follow a $T^{-1.18}$ fit. This is consistent with a similar temperature exponent observed in measurements of cross-plane thermal conductivity measurements of GaN films of thickness of ≈0.7 µm.[12,23] However, for the thin GaN sample, we note that the measured thermal conductivity values follow a $T^{-0.88}$ \ No newline at end of file diff --git a/samples/texts/1516894/page_6.md b/samples/texts/1516894/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..e33088243b6957b10c7fdad724d8243a077ec9f7 --- /dev/null +++ b/samples/texts/1516894/page_6.md @@ -0,0 +1,15 @@ +Figure 4. a) Temperature-dependent electrical conductivity of the 2DEG for the thin and bulk GaN samples, extracted via CTLM measurements (Section S5, Supporting Information). b) Temperature-dependent thermal conductivity measurements for the bulk and thin GaN samples. Fits are shown with black, dotted lines. c) XRD rocking curve scan of the (0002) lattice plane in GaN, to investigate the density of screw-type dislocations. d) Modeled in-plane thermal conductivities of the layers in our composite stack as a function of thickness using a dislocation density of $10^9$ cm⁻². e) Thermal conductivity reduction due to size effect at 25 °C. The model corresponds to the dashed lines with increasing dislocation densities ($N_{dis}$), and the data points are the experimental measurements. f) Measured Seebeck coefficient versus temperature for thin and bulk GaN samples. The model uses $\eta_v \approx 2 \times 10^{19}$ cm⁻³, which is estimated from the Schrödinger-Poisson simulation. + +fit, indicating that Umklapp scattering is less prominent for long-wavelength phonons which are suppressed due to the size effect. + +In-plane thermal conductivity data in these films are limited, with little data available on the size effect and temperature dependence.[12] Since our suspended film is a composite consisting of an AlN layer, AlₓGa₁-xN transition layers, and a GaN layer, the overall thermal conductivity (k) can be estimated as $\sum k_i t_i / \sum t_i$, where $k_i$ and $t_i$ refer to the thermal conductivities and thicknesses of individual layers. For each multilayer, we used a Boltzmann Transport Equation (BTE) model to quantify $k_i$ with layer thickness ($t_i$). Using a simple Debye approximation for the phonon dispersion with an average velocity over the acoustic phonon modes ($v_{ac}$), the in-plane thermal conductivity for each layer can be written as[24] + +$$k_i = \frac{3k_B^3 T^3}{8\pi^3 \hbar^3 v_{ac}^3} \int_0^{\theta_D/T} \int_0^{2\pi} \int_0^{\pi} c_{ph} \sin(\theta) \tau_c(x) x^2 v_g^2 d\theta d\phi dx \quad (1)$$ + +where $k_B$ is the Boltzmann constant, $\theta_D$ is the Debye temperature for the multilayer,[25] T is the temperature, $\hbar$ is the reduced Planck's constant, $c_{ph}$ is the mode-specific volu- + +metric heat capacity, evaluated as $3k_B \left(\frac{T}{\theta_D}\right)^3 \int_0^{\theta_D/T} \frac{x^4 e^x}{(e^x - 1)^2} dx$, + +and $x = \hbar\omega/k_B T$, where $\omega$ is the phonon frequency. The integration is performed over the angular directions ($\theta$ and $\phi$) using a direction-dependent group velocity $v_g = v_{ac}\sin(\theta)\cos(\phi)$. + +The total scattering time $\tau_C$ is calculated by Mathiessen's rule with contributions from Umklapp ($\tau_U$), impurity ($\tau_I$), alloy ($\tau_A$), boundary ($\tau_B$), and defect scattering ($\tau_D$), respectively. The Umklapp scattering term is evaluated via the Callaway relationship, $\tau_U = A/\omega^2$. We evaluated the constant A in the bulk limit as $2\pi^2 v_{ac} k_{\infty} / (c_{ph} \omega_D)$, where $k_{\infty}$ in the bulk thermal conductivity of the layer and $\omega_D$ is the Debye frequency. For instance, $k_{\infty}$ values of 240 and 285 Wm⁻¹ K⁻¹ are used for GaN and AlN at room temperature, respectively.[26,27] The Debye frequencies are \ No newline at end of file diff --git a/samples/texts/1516894/page_7.md b/samples/texts/1516894/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..32cc835ec33739613b5262c61aab2c29ad4843a4 --- /dev/null +++ b/samples/texts/1516894/page_7.md @@ -0,0 +1,21 @@ +extracted from the known Debye temperatures of these material layers.[25] Scattering with impurities is neglected since its effect is found to be negligible for unintentionally doped films.[26] For the underlying transition layers, all the material parameters (e.g., vac, θD, k∞) are averaged over the AlN and GaN fractions, in context of the virtual crystal model.[25] Alloy scattering severely reduces the thermal conductivity of the transition layers and is evaluated as a point defect scattering term.[28] For the sake of brevity, we skip the details, which can be found in the study by Liu and Balandin.[28] The defect scattering term (τD) included core, screw, edge, and mixed dislocations with total density Ndis, whose effect is to reduce the thermal conductivity.[29] + +Although we have a composite film (and thus, the dislocation density is expected to vary for the different layers), we estimated an average value for the composite film via X-ray diffraction (XRD) measurements. For example, Figure 4c shows symmetric (rocking curve) scans of the (0002) lattice planes in the thin and thick GaN samples. Using the full width at half maximum (denoted by β) value of the XRD scans, the screw-type dislocation density can be estimated as ≈β²/(4.35c²), where c is the lattice constant (0.5185 nm) along the c-axis direction for the GaN wurtzite crystal.[17] Using this, we calculate the screw-type dislocation density to be ≈9 × 10⁸ cm⁻² and ≈2.5 × 10⁹ cm⁻² for the thick and thin GaN samples, respectively. An estimate of the edge-type dislocation density from XRD measurements is shown in Figure S10a (Supporting Information), also on the order of 10⁹ cm⁻². Finally, the direction-dependent boundary scattering term[24] is evaluated as τB = ts/2vaccos(θ)/|t|. The modeled in-plane thermal conductivities of the individual layers in the composite stack are depicted in Figure 4d as a function of the corresponding layer thickness. + +Due to the lack of in-plane thermal conductivity data in these films, it is difficult to compare the model in Figure 4d with literature. However, the cross-plane thermal conductivity[24] of the layers can be estimated using a different vg = vaccos(θ) in Equation (1), the results of which are given in Figure S10b (Supporting Information), showing good agreement with previous measurements. Using the thicknesses of the multilayers (Figure 2), the overall in-plane thermal conductivity of the composite stack (k), where the GaN buffer layer thickness (tGaN) is varied (at room temperature), is shown in Figure 4e. In particular, we note that the model agrees with the experimental data well, and that dislocation densities in our range (10⁹–10¹⁰ cm⁻²) are estimated to have little effect on thermal conductivity of the film layers. In other words, the in-plane thermal conductivity reduction from the thick GaN to the thin GaN composite stack is expected to arise almost exclusively from the size effect. We note that this observation is consistent with former reports of the cross-plane thermal conductivity in GaN films with similar dislocation densities.[29,30] + +The measured Seebeck coefficients for the bulk and thin GaN heterostructures are shown in Figure 4f. While the Seebeck coefficients for bulk doped III–V films have been reported in the past,[8,11,25] values for a polarization-induced 2DEG have not been well studied in literature. The approximate linear increase in the Seebeck coefficient with temperature indicates a degenerate semiconductor that may be well approximated with a Cutler–Mott[31] formula for nearly free electrons. Since Rbuffer >> R2DEG, the measured values can be considered to arise exclusively from the 2DEG,[32] which is in agreement with the degenerate semiconductor characteristic. To understand the + +magnitude and the observed dependencies with temperature, we implemented a simple analytical model based on Sztein et al.[25] For a bulk doped material with fixed electron concentration, the Seebeck coefficient is analytically evaluated as[25] + +$$S = \frac{-\int (E - E_F) \sigma(E) dE}{qT \int \sigma(E) dE} \quad (2)$$ + +where σ(E) is the differential electronic conductivity, q is the fundamental charge, E is the electron energy, and EF is the Fermi energy. + +Following Sztein et al.,[25] we evaluated S as a function of doping densities in GaN. The model for the Seebeck coefficient shows good match with measurements for doped GaN films (Figure S11, Supporting Information). These doping densities only serve to “mimic” the effect of 2DEG charge density, since the origin of the 2DEG is related to built-in polarization fields as discussed earlier. The differential electronic conductivity is evaluated using a nonparabolic energy dispersion relation and typical scattering mechanisms found in III-nitrides: optical phonon, piezoelectric, deformation potential, and charged dislocation scattering.[25] The parameters and assumptions in the model follow from Sztein et al.,[25] with the notable exception that we neglected ionized impurity scattering in the evaluation of σ(E), since the 2DEG region is assumed to be undoped. In principle, the Seebeck coefficient for the 2DEG region can be evaluated by discretizing it into fine regions with approximately constant charge concentration using Equation (2) via a thickness average. For simplicity, we instead compare the average volumetric density nv for the 2DEG region that we obtained by calculating the average charge density ns/t2D against the simulated Seebeck coefficients for “mimic” doping densities. The predicted Seebeck coefficient dependence using nv (= 1.73 × 1019 cm-3) is in good agreement with the measured values for the 2DEG for the bulk GaN sample, as illustrated in Figure 4f. However, the Seebeck coefficient values for the thin GaN sample are lower than the bulk GaN sample, which could arise from a larger nv (= 2.07 × 1019 cm-3) due to reduced 2DEG thickness and differing relevance of scattering mechanisms (e.g., greater dislocation scattering) in comparison to the bulk GaN film. + +Finally, we also note that our measured values for the AlGaN/GaN 2DEG Seebeck coefficients do not show a similar enhancement in comparison to a bulk 3D Seebeck coefficient, which is unlike other systems, such as ZnO-based 2DEG.[33] This could arise from the relatively large 2DEG confinement thickness (t2D) of ≈5 nm in our structures, as opposed to much smaller confinement depths (≈1 nm) in the ZnO-based 2DEG.[33] Nonetheless, our results show that tuning the thermal transport while maintaining the electrical transport via buffer layer engineering can be achieved and is useful for the design of AlGaN/GaN devices. + +## 5. Applications + +The measured electrical, thermal, and thermoelectric properties could be important for a variety of GaN-based temperature, power electronics, heat-flux, magnetic field, and \ No newline at end of file diff --git a/samples/texts/1516894/page_8.md b/samples/texts/1516894/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..ac0950ae85c0ca684ce02c1501fa90d299bd3138 --- /dev/null +++ b/samples/texts/1516894/page_8.md @@ -0,0 +1,7 @@ +**Figure 5.** a) Fully suspended platform to measure local temperature rise using the bulk GaN heterostructure. The six labeled AlGaN-GaN Seebeck 2DEG mesa regions (e.g., 2-2†, shaded white for clarity) can be used to estimate temperature rise near the heater line, which mimics a power device. Note that the temperatures are measured at the “hot” ends of the mesa regions, marked by a † symbol. b) Finite element model of temperature rise with a heater power of 25 mW using the extracted thermal properties of the bulk GaN sample. The model is for a quarter symmetry region of (a). Three locations for temperature rise measurement (2†, 3†, and 6†) are also shown. The Si substrate is assumed to be at 25 °C in this model. c) Simulated and experimentally determined temperature rise at locations. The measured temperatures are noted to be accurate within ≈15%. We used the measured Seebeck coefficient of the bulk GaN sample to extract the temperature rise. d) Measured thermoelectric figure of merit (zT) for thin and bulk GaN samples, showing steady rise with temperature. An improvement of ≈22–51% is noted for the thin GaN sample over the bulk GaN sample across the temperature range. Scale bars of (a) and (b), 500 and 100 µm. + +energy harvesting devices using III-V heterostructures. As an example, we demonstrate how the measured electrical and thermal properties could be used to detect local temperatures from on-chip heat loads using the fully suspended bulk GaN platform. Figure 5a shows an on-chip circuit with four resistive heat sources (similar to the heater and sensor metal electrodes) with six 2DEG mesa regions (spanning the suspension and substrate regions) across which the Seebeck voltage can be measured. The substrate is held at 25 °C using a temperature-controlled probe station. We then apply a heating power (25 mW) to induce a temperature gradient relative to the substrate in the suspended heterostructures. The simulated temperature profile for a quarter region using the measured thermal properties and external losses (Section S3, Supporting Information) is shown in Figure 5b. Three sample locations of the “hot” ends of the 2DEG mesa where temperature is measured are illustrated in Figure 5b. Finally, with the heater power held constant, we measured the differential Seebeck voltage for + +the six 2DEG mesa regions. Using the measured temperature-dependent Seebeck voltage for the bulk GaN film, we extracted the temperatures at the hot ends of the 2DEG mesa regions and compared them with the simulated local temperatures from the finite element model. Excellent agreement is observed (within a maximum of ≈15% error), as seen in Figure 5c, which demonstrates how the measured properties could be used for local on-chip monitoring of thermal loads in addition to validating our measurements. + +Finally, Figure 5d shows the temperature-dependent thermoelectric figure of merit ($zT$) up to ≈300 °C using the measurements of 2DEG electrical conductivity, Seebeck coefficient, and in-plane heterostructure thermal conductivity. These $zT$ values are 3 to 4× higher than in bulk doped GaN films,[10] owing to large 2DEG conductivities and thermal conductivity reduction from size effect. Further, we notice that the thin GaN film displays almost ≈22–51% higher $zT$ values in comparison to the bulk GaN film due to the large, ≈2.5× decrease of the in-plane \ No newline at end of file diff --git a/samples/texts/1516894/page_9.md b/samples/texts/1516894/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..c1fa52fa51243059a42001b069d3b640d6da3809 --- /dev/null +++ b/samples/texts/1516894/page_9.md @@ -0,0 +1,95 @@ +thermal conductivity. The trend with temperature is almost +linear, which is different from the super-linear trend[25] observed +for doped III–V films, due a greater temperature exponent for +the 2DEG mobility. Since the background doping concentra- +tion is low (<10¹⁶ cm⁻³) and the layers used are wide bandgap +materials, it is expected that the thermopower peak occurs at +much higher temperatures[25] in comparison to doped III–V films, +which warrants further investigation of these properties beyond +300°C. Apart from the thermoelectric figure of merit, the 2D +electron gases in these films also show very high power fac- +tors between 4 and 7 × 10⁻³ Wm⁻¹ K⁻² at room temperature +(Figure S12, Supporting Information), which is comparable to +the values for state-of-the-art thermoelectric materials.[13] These +observations suggest that this approach of engineering the +heterostructure layers in conjunction with polarization-based +2DEGs has promising potential for thermoelectric applications +at high temperatures. + +6. Conclusion + +In summary, we designed and implemented a suspended AlGaN/GaN heterostructure platform to investigate temperature-dependent thermal and electrical transport by tuning the GaN layer thickness. We demonstrate effective manipulation of these properties, and in the process shed light on several transport parameters that have not been previously explored in detail in former literature. Notably, we show that thin GaN layers of ≈100 nm significantly impede heat flow, but preserve the 2DEG conductivity, which could be useful for a range of GaN-based devices. We also show a simple example of how the measured properties can be used to monitor local heat fluxes on an AlGaN/GaN power device. We used a single AlGaN barrier layer in this study, which could serve as a backbone for extension to multiple, alternating GaN/AlGaN superlattice layers. Further work along these lines could include how changing the AlGaN thickness affects transport, improving the temperature-dependent mobility degradation in AlGaN/GaN 2DEGs, exploring transport at higher temperatures beyond 300 °C, and exploring thermoelectric phenomena in other III–V heterostructure families with 2DEGs using this experimental platform. + +Supporting Information + +Supporting Information is available from the Wiley Online Library or +from the author. + +Acknowledgements + +This work was supported in part by the National Science Foundation (NSF) Engineering Research Center for Power Optimization of Electro Thermal Systems (POETS) under Grant EEC-1449548, by the NSF DMREF grant 1534279, and by the research fund of Hanyang University (HY-2017). The MOCVD experiments were conducted at the MOCVD Lab of the Stanford Nanofabrication Facility (SNF), which was partly supported by the NSF as part of the National Nanotechnology Coordinated Infrastructure (NNCI) under award ECCS-1542152. The authors thank Caitlin Chapin, Hannah Alpert, and Karen Dowling for assistance with fabrication. The authors also thank Karen Dowling and + +Hannah Alpert for assistance with Hall measurements, and Thomas Heuser for assistance with the XRD measurements. The authors also acknowledge Prof. Andrew Alleyne and Pamela Tannous for useful discussions. + +Conflict of Interest + +The authors declare no conflict of interest. + +Keywords + +2DEG, AlGaN/GaN, polarization, Seebeck coefficients, thermal conductivity + +Received: October 8, 2017 + +Revised: January 22, 2018 + +Published online: March 30, 2018 + +[1] R. S. Pengelly, S. M. Wood, J. W. Milligan, S. T. Sheppard, W. L. Pribble, *IEEE Trans. Microwave Theory Tech.* **2012**, *60*, 1764. + +[2] D. Maier, M. Alomari, N. Grandjean, J.-F. Carlin, M.-A. Diforte-Poisson, C. Dua, S. Delage, E. Kohn, *IEEE Electron Device Lett.* **2012**, *33*, 985. + +[3] O. Ambacher, J. Majewski, C. Miskys, A. Link, M. Hermann, M. Eickhoff, M. Stutzmann, F. Bernardini, V. Fiorentini, V. Tilak, B. Schaff, L. F. Eastman, *J. Phys.: Condens. Matter* **2002**, *14*, 3399. + +[4] O. Ambacher, B. Foutz, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, A. J. Sierakowski, W. J. Schaff, L. F. Eastman, R. Dimitrov, A. Mitchell, M. Stutzmann, *J. Appl. Phys.* **2000**, *87*, 334. + +[5] O. Ambacher, J. Smart, J. R. Shealy, N. G. Weimann, K. Chu, M. Murphy, W. J. Schaff, L. F. Eastman, R. Dimitrov, L. Wittmer, M. Stutzmann, W. Rieger, J. Hilsenbeck, *J. Appl. Phys.* **1999**, *85*, 3222. + +[6] A. Sztein, H. Ohta, J. Sonoda, A. Ramu, J. E. Bowers, S. P. Denbaars, S. Nakamura, *Appl. Phys. Express* **2009**, *2*, 111003. + +[7] J. Zhang, H. Tong, G. Liu, J. A. Herbsommer, G. S. Huang, N. Tansu, *J. Appl. Phys.* **2011**, *109*, 053706. + +[8] A. Sztein, H. Ohta, J. E. Bowers, S. P. Denbaars, S. Nakamura, *J. Appl. Phys.* **2011**, *110*, 123709. + +[9] K. Nagase, S. Takado, K. Nakahara, *Phys. Status Solidi A* **2016**, *213*, 1088. + +[10] A. Sztein, J. E. Bowers, S. P. DenBaars, S. Nakamura, *Appl. Phys. Lett.* **2014**, *104*, 042106. + +[11] A. Sztein, J. E. Bowers, S. P. Denbaars, S. Nakamura, *J. Appl. Phys.* **2012**, *112*, 083716. + +[12] J. Cho, Z. Li, M. Asheghi, K. E. Goodson, *Annu. Rev. Heat Transfer* **2014**, *18*, 7. + +[13] H. Ohta, S. W. Kim, S. Kaneki, A. Yamamoto, T. Hashizume, *Adv. Sci.* **2017**, 1700696. + +[14] C. Dames, *Annu. Rev. Heat Transfer* **2013**, *16*, 7. + +[15] M. Asheghi, K. Kurabayashi, R. Kasnavi, K. E. Goodson, *J. Appl. Phys.* **2002**, *91*, 5079. + +[16] X. Xu, J. Zhong, H. So, A. Norvilas, C. Sommerhalter, D. G. Senesky, M. Tang, *AIP Adv.* **2016**, *6*, 115016. + +[17] H.-P. Lee, J. Perozek, L. D. Rosario, C. Bayram, *Sci. Rep.* **2016**, *6*, 37588. + +[18] I. P. Smorchkova, L. Chen, T. Mates, L. Shen, S. Heikman, B. Moran, S. Keller, S. P. DenBaars, J. S. Speck, U. K. Mishra, *J. Appl. Phys.* **2001**, *90*, 5196. + +[19] S. Birner, T. Zibold, T. Andlauer, T. Kubis, M. Sabathil, A. Trellakis, P. Vogl, *IEEE Trans. Electron Devices* **2007**, *54*, 2137. + +[20] S. Heikman, S. Keller, Y. Wu, J. S. Speck, S. P. Denbaars, U. K. Mishra, *Appl. Phys. Lett.* **2003**, *93*, 10114. + +Adv.Funct.Mater. +2018, +28, +1705823 + +**1705823** (8 of 9) + +© 2018 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim \ No newline at end of file diff --git a/samples/texts/2651514/page_1.md b/samples/texts/2651514/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..a3f9f97186fec19b474bdc6a442735405abfa088 --- /dev/null +++ b/samples/texts/2651514/page_1.md @@ -0,0 +1,19 @@ +# Random Effects Cox Models: A Poisson Modelling Approach + +Renjun Ma$^{0,1}$, Daniel Krewski$^{1,2}$ and Richard T. Burnett$^{1,3}$ + +¹Faculty of Medicine, University of Ottawa, Ottawa, Canada, K1H 8M5 + +² School of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6 + +³ Environmental Health Directorate, Health Canada, Ottawa, Canada, K1A 0L2 + +February 1, 2000 + +## Abstract + +We propose a Poisson modelling approach to random effects Cox proportional hazards models. Specifically we describe methods of statistical inference for a class of random effects Cox models which accommodate a wide range of nested random effects distributions. The orthodox BLUP approach to random effects Poisson modeling techniques enables us to study this new class of models as a single class, rather than as a collection of unrelated models. The explicit expressions for the random effects given by our approach facilitate incorporation of relatively large number of random effects. An important feature of this approach is that the principal results depend only on the first and second moments of the unobserved random effects. The application of proposed methods is illustrated through the re-analysis of data on the time to failure (tumour onset) in an animal carcinogenesis experiment previously reported by Mantel and Ciminera (1979). + +Key words: Cox model; BLUP; estimating equation; frailty; generalized linear models; random effects; Tweedie exponential dispersion model + +⁰Email address: renjun@zeus.med.uottawa.ca \ No newline at end of file diff --git a/samples/texts/2651514/page_10.md b/samples/texts/2651514/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..724ee661e3b15bbc004fd66efababf4e042487c9 --- /dev/null +++ b/samples/texts/2651514/page_10.md @@ -0,0 +1,23 @@ +## 4.3 Estimation of Dispersion Parameters + +We now discuss the situation in which the dispersion parameters are unknown. In analogy with generalized linear models, we adopt the following adjusted Pearson estimator for the dispersion parameter $\sigma^2$: + +$$ \hat{\sigma}^2 = \frac{1}{m} \sum_{i=1}^{m} \{(U_i - 1)^2 + c_i\}. $$ + +The first term is the Pearson estimator, with the second term being a bias correction term. The corresponding adjusted Pearson estimator for $\omega^2$ is: + +$$ \hat{\omega}^2 = \frac{1}{m} \sum_{i=1}^{m} \frac{1}{J_i} \sum_{j=1}^{J_i} \{(U_{ij} - \hat{U}_i)^2 + c_{ij} + c_i - 2c_i w_{ij}\}. $$ + +Again, the first term is the Pearson estimator, whereas the remaining terms are bias correction terms. These dispersion parameter estimates can also be shown to be consistent as $m \to \infty$ (Ma 1999). Unlike most other approaches in the literature, our asymptotic variance of the regression parameter estimator is not affected by the variability in the dispersion parameter estimators. + +In fact, this orthodox BLUP approach depends on the random effects only via the first and second moments of the sub-cluster random effects. It has been shown to be robust, to a certain extent, against mispecification of the random effects distributions (Ma 1999), and thus covers non-Tweedie random effects such as log-normal random effects. + +## 4.4 Computational Procedures + +Initial values for the regression parameters are taken as the regression parameter estimates obtained from standard Poisson regression techniques assuming independent responses. Initial random effects predictions $\hat{U}_i$ and $\hat{U}_{ij}$ are given by the average of the responses within cluster $i$ divided by the average of all responses and the average of the responses within sub-cluster $(i, j)$ divided by the average of all responses, respectively. The initial dispersion parameter estimates are calculated from the adjusted Pearson estimators, omitting the bias-correction terms. + +The algorithm then iterates between updating the regression parameter estimates via the Newton scoring algorithm, updating random effect predictors via the orthodox BLUP, and updating dispersion parameter estimates via the adjusted Pearson estimators. + +# 5 An Illustrative Example + +We illustrate the application of our approach to the random effects Cox model using data from an animal carcinogenesis experiment originally re- \ No newline at end of file diff --git a/samples/texts/2651514/page_11.md b/samples/texts/2651514/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..a5353482fc7d491bbd674649e2c7b67621d7b01c --- /dev/null +++ b/samples/texts/2651514/page_11.md @@ -0,0 +1,15 @@ +ported by Mantel and Ciminera (1979). This experiment involved 50 sets of three female weanling rats selected from within the same litter, with one animal assigned to a treatment group exposed to a putative carcinogen, and the remaining two serving as litter-matched controls. The time to tumour occurrence or censoring was recorded to the nearest week for each of the 150 animals employed in this study. This experiment thus involved a single binary covariate with values of 0 and 1 indicating assignment to the control or treated group, respectively. + +Because of the possibility of intra-litter correlation (Gart et al. 1986), we included a random effect for each litter. The corresponding Cox regression model assumes that, given the random effects, the hazard functions for individuals are conditionally independent, with the hazard function for individual *j* from litter *i* given by + +$$h_{ij}(t) = h_0(t)u_i \exp(x_{ij}\beta),$$ + +where $x_{ij}$ is the indicator variable, reflecting exposure to the test agent. The litter random effect $u_i$ are assumed to follow independent and identical Tweedie distributions with unity mean and dispersion parameter $\sigma^2$ described in (2). + +Parameter estimates for both the standard and random effects Cox models are shown in Table 1 where the Peto-Breslow approximation (Cox and Oakes 1984) for tied failure times was used in both analyses. The estimates of the regression parameter $\beta$ associated with the treatment effect are comparable under both models, as are the standard errors of these estimates. Based on the ratio of these estimates to their respective standard errors, the treatment effect is significant under both models. + +Table 1: Parameter estimates for the animal carcinogenesis data. + +
Parameter Estimates
Cox Modelβ ± SEσ²
Standard0.898 ± 0.317-
Random effects0.902 ± 0.3120.293
+ +Scatter plot of the litter random effects is shown in Figures 1. These 50 litters were labelled as 1, 3, ..., 99 by Mantel and Ciminera (1979) and are re-numbered as 1, 2, ..., 50 here for convenience. Litters 3, 21, 22, 25 and 37 demonstrated the lowest litter-specific relative risks, whereas litter 13 had the highest (Figure 1). Figure 2 shows that the litter random effects match the number of tumour occurrences in the corresponding litter; the higher the litter-specific relative risk, the higher the litter tumour occurrence. The one exception is litter 13, which had a higher litter-specific relative risk than \ No newline at end of file diff --git a/samples/texts/2651514/page_12.md b/samples/texts/2651514/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..ea7c90d22e1b8eb7e1543fc7d372d3c96bed8507 --- /dev/null +++ b/samples/texts/2651514/page_12.md @@ -0,0 +1,15 @@ +litter 32, although litter 32 was the only litter with tumours occurring in all three littermates. Examination of the data revealed that all three rats in litter 13 had exceptionally low tumour onset times (Figure 3). + +Figure 1, 2 and 3 are approximately here. + +# 6 Discussion + +In this paper, we have introduced a Poisson modelling approach to random effects Cox models. We have specifically focussed on Cox models with two levels of nested random effects. We may consider models with more than two levels of random effects. For such models, our method remains valid with $(i, j, k)$ replaced by higher dimensional indices. The proposed Poisson modelling approach can also be extended to random effects Cox models with time dependent covariates in the following way. Suppose that all covariates assume constant values between two distinct failure times, as reflected by the corresponding step functions for the cumulative failure times. The incorporation of such time dependent covariates can be simply achieved by replacing $\mathbf{x}_{ijk}^{(s)}$ by $\mathbf{x}_{ijk}^{(s)}(t) = \mathbf{x}_{ijk}(\tau_{sh})$ for $Y_{ijk,h}^{(s)}$ in the model. + +For the Cox model with one level of random effects ($J_i = 1, \omega^2 = 0$, with $u_{ij} = u_i$), the random effects have been previously characterized by gamma (Clayton 1991), positive stable (Hougaard 1986a, 1986b) and log-normal (McGilchrist 1993) distributions. Our framework effectively covers the gamma, log-normal and inverse Gaussian distributed random effects. + +Our Poisson approach is not limited to Cox models with the nested random effects structures. Taking $u_{ij} = v_i v_j$ for balanced designs will lead to crossed random effects. For Cox models with only time dependent subject frailties $u_i(t)$ for each subject $i$, we can employ the techniques developed for Poisson models with an AR(p) structure on the latent variable $u_i(t)$. Since the distinct failure times are not equally spaced, a specific time series structure for time dependent frailties may not be appropriate. In the Cox model specified by (1)-(3), taking the second level random effects $u_{it} = u_i(t)$ as conditional on the subject random effect $u_i$, where $t$ represents the distinct failure times in the stratum of the $i$th subject, we have correlated time dependent frailties for each subject. + +## Acknowledgements + +This research was supported by the Health Effects Institute and by a grant A8664 from the Natural Sciences and Engineering Research Council of Canada to D. Krewski. \ No newline at end of file diff --git a/samples/texts/2651514/page_13.md b/samples/texts/2651514/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..c7a5dcc4b6b69a37876b87c4cff551c817b42166 --- /dev/null +++ b/samples/texts/2651514/page_13.md @@ -0,0 +1,29 @@ +References + +Breslow, N.E. and Clayton, D.G. (1993). Approximate inference in generalized linear mixed model. *Journal of American Statistical Association* **88**, 9-25. + +Brockwell, P.J. and Davis, R.A. (1991). *Time Series: Theory and Methods* 2nd ed. New York: Springer-Verlag. + +Clayton, D.G. (1991) A Monte Carlo method for Bayesian inference in frailty models. *Biometrics* **47**, 467-485. + +Cox, D.R. and Oakes, D. (1984) *Analysis of Survival Data* New York: Chapman and Hall. + +Gart, J., Krewski, D., Lee, P., Tarone, R. and Wahrendorf, J. (1986) *Statistical Methods in Cancer Research, Vol.III: The Design and Analysis of Long-Term Animal Experiments*. Lyon: International Agency for Research on Cancer. + +Glifford, P. (1993). Discussion on the meeting on the Gibbs sampler and other Markov chain Monte Carlo methods. *Journal of Royal Statistical Society Ser. B* **55**, 53-54. + +Hougaard, P. (1986a) Survival models for heterogeneous population derived from stable distributions. *Biometrika* **73**, 387-396. + +Hougaard, P. (1986) A class of multivariate failure time distributions. *Biometrika* **73**, 671-678. + +Jørgensen, B. (1997). *The Theory of Dispersion Models*. London: Chapman and Hall. + +Lee, Y. and Nelder, J.A. (1996). Hierarchical generalized linear models. *Journal of Royal Statistical Society B* **58**, 619-678. + +Ma, R. (1999) An Orthodox BLUP Approach to Generalized Linear Mixed Models. Ph.D. Thesis. Department of Statistics, The University of British Columbia. + +Mantel, N. and Ciminera, J.L. (1979) Mantel-Haenszel analysis of litter-matched time-to-response data with modifications for recovery of interlitter information. *Cancer Research* **37**, 3863-3868. + +McGilchrist, C.A. (1993) REML estimation for survival models with frailty. *Biometrics* **49**, 221-225. + +Sastry, N. (1997) A nested frailty model for survival data, with an application to the study of child survival in northeast Brazil. *Journal of American Statistical Association* **92**, 426-435. \ No newline at end of file diff --git a/samples/texts/2651514/page_14.md b/samples/texts/2651514/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..20ffb403ab44a9d4d13d9add28d56fc147182ad4 --- /dev/null +++ b/samples/texts/2651514/page_14.md @@ -0,0 +1,5 @@ +Sargent, D.J. (1998) A general framework for random effects survival analysis in the Cox proportional hazards setting. *Biometrics* **54**, 1486-1497. + +Smith, A.F.M. and Roberts, G.O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. *Journal of Royal Statistical Society Ser. B* **55**, 3-23. + +Whitehead, J. (1980) Fitting Cox's regression model to survival data using GLIM. *Applied Statistics* **29**, 268-275. \ No newline at end of file diff --git a/samples/texts/2651514/page_2.md b/samples/texts/2651514/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..12f4cfc67e40a59fbdd8ad9519a14c68071f221e --- /dev/null +++ b/samples/texts/2651514/page_2.md @@ -0,0 +1,11 @@ +# 1 Introduction + +Although the incorporation of random effects into Cox models has gained increasing attention in analyses of event history data, these models pose considerable theoretical difficulties in the development of estimation and inference procedures (Clayton 1991). Until recently, previous research in this area has focussed mainly on survival models with one level of random effects (Sastry 1997; Sargent 1998). The frequentist approaches to nested frailty survival models have usually been restricted to piecewise constant baseline hazard functions and specific random effects distributions (Sastry 1997). On the other hand, Bayesian approaches to nested random effects Cox models are computationally intensive, and the assessment of convergence of computational techniques such as the Gibbs sampler remains an area of debate (Glifford 1993; Smith and Roberts 1993; Sargent 1998). Flexible frailty models that can be fit with reasonable computational effort are therefore needed. + +Considerable progress has been made in recent years in the area of random effects generalized linear models (Breslow and Clayton 1993; Lee and Nelder 1996; Ma 1999). The connection between the Cox and Poisson regression models has long been recognized (Whitehead 1980). In this paper, we show that random effects methods developed for use with generalized linear models can be applied by characterizing the random effects Cox model as a random effects Poisson regression model. Our approach deals with an unspecified baseline hazard function and a wide range of random effects distributions. Our approach can also handle ties and stratification in the same way as in the standard Cox model. Further, our explicit expressions for the random effects facilitate incorporation of relatively large numbers of random effects. + +The organization of the paper is as follows. We introduce the random effects Cox model and its auxiliary random effects Poisson models in Sections 2 and 3, respectively. In Section 4, we discuss the estimation of the nested random effects Cox models based on the orthodox BLUP approach to the auxiliary random effects Poisson models. An illustrative example involving animal carcinogenesis data is presented in Section 5, and potential extensions of the models are discussed in Section 6. + +# 2 Random Effects Cox Model + +In this section, we consider a Cox model with two levels of random effects. Suppose that the cohort of interest is stratified on the basis of one or more relevant covariates. Let the hazard function for individual $(i, j, k)$ from stratum $s = 1, 2, \dots, a$ at time $t$ be denoted by $h_{ijk}^{(s)}(t)$. Given the random effects, we assume that the individual hazard functions are conditionally independent \ No newline at end of file diff --git a/samples/texts/2651514/page_3.md b/samples/texts/2651514/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..8dc6c6d7c4e11893320636dea9ec9137c3acd781 --- /dev/null +++ b/samples/texts/2651514/page_3.md @@ -0,0 +1,21 @@ +with + +$$h_{ijk}^{(s)}(t) = h_{0}^{(s)}(t)u_{ij} \exp(\beta^{\top}\mathbf{x}_{ijk}^{(s)}). \quad (1)$$ + +Here, $u_{ij} > 0$ are random effects, or frailties, shared by all individuals within the same group, and $h_0^{(s)}(t)$ is the baseline hazard function for stratum s. Clearly the survival times (either failed or censored) within the same group are correlated. The random effects are traditionally assumed not to depend on the regression parameter $\beta$. Without loss of generality, we assume that the design matrix is of full rank. + +Here, we will focus on three-level hierarchical Cox models with the following nested random effects structure. Suppose the cohort is composed of $m$ independent clusters indexed by $i$. Within each cluster $i$, there are $J_i$ correlated sub-clusters indexed by $(i, j)$. Further, within each sub-cluster $(i, j)$ there are $n_{ij}$ individuals whose survival times are given by (1). One such hierarchy example was presented by Sastry (1997) where the children were clustered at both community and family levels. + +We introduce a class of models with nested random effects based on the class of Tweedie exponential dispersion model distributions denoted by $\text{Tw}_r(\mu, \sigma^2)$, where $\text{Tw}_r(\mu, \sigma^2)$ includes the normal ($r=0$), Poisson ($r=1$), gamma ($r=2$), compound Poisson ($11.1The overlapping region and switching surfaces employed in hysteresis logic.62.1Feedback connection between an LPV plant and an LPV controller.132.2An example of a two-dimensional parameter trajectory with four subregions.142.3Parameterizations of switching surfaces (dashed lines) in terms of the centers (dotted lines) and the widths of the overlapping regions for the case when N1 = 2 and N2 = 3.152.4The optimization flowchart.213.1Mass-spring-damper schematic.293.2A feedback structure for controller design.323.3Functionality of γ with respect to pS = (c1,1, c2,1) obtained by the full-search method.343.4Comparison of step responses between the optimal controller and a heuristic one with m = 60.373.5Plant variation region with 5 divisions on each axis.383.6The initial and final performance level for different number of switching surface variables while applying the descent algorithm.39 \ No newline at end of file diff --git a/samples/texts/5122728/page_100.md b/samples/texts/5122728/page_100.md new file mode 100644 index 0000000000000000000000000000000000000000..25d16a984bdb13f7f24c366c7a558b6e69c16563 --- /dev/null +++ b/samples/texts/5122728/page_100.md @@ -0,0 +1,5 @@ +by juxtaposing closed-loop performance of the switching LPV controller with optimized switching surfaces and that with heuristic switching surfaces, it was shown that optimized switching LPV controller has the better performance. + +In Chapter 4, a magnetically-actuated optical image stabilizer was used as a control application in which the mass and stiffness-value are subject to product variations. Two switching LPV controllers were designed for this system: the controller with optimized switching surfaces and the one with heuristic switching surfaces. Simulation results of implementing two switching LPV controllers showed that the closed-loop performance of the worst case product will be improved if the optimized controller is employed. Nevertheless, many products showed relatively worse responses with the optimized controller. This phenomenon coincides with our expectation that applying switching LPV controller with optimized switching surfaces can only improve the worst case L₂-gain performance of the closed-loop which relates to the worst case product. + +In Chapter 5, the proposed algorithm was applied to another control application which is air-fuel ratio control in spark ignition internal combustion engines. The dynamic of the system depends on the engine speed and air flow which form a 2-dimensional plant variation region. The switching LPV controller with optimized switching surfaces was designed using the descent algorithm. Then, simulation results of implementing such controller were compared with that of implementing a switching LPV controller having heuristic switching surfaces. A realistic plant parameters trajectory was used, and an arbitrary reference signal was input to the closed-loop. It was shown that the optimized controller could result in better tracking response in terms of the error, overshoot, and settling time. \ No newline at end of file diff --git a/samples/texts/5122728/page_101.md b/samples/texts/5122728/page_101.md new file mode 100644 index 0000000000000000000000000000000000000000..08e872a99eb75473132289a79779ce9dcd7d785c --- /dev/null +++ b/samples/texts/5122728/page_101.md @@ -0,0 +1,13 @@ +## 6.2 Contributions + +The main contributions of this thesis can be outlined as follows: + +* A numerical algorithm was developed to optimize switching surfaces in switch-ing LPV control design. By applying this algorithm, decrease in the closed-loop $L_2$-gain performance from initial selection of the switching surfaces to the convergent point is guaranteed. + +* A hybrid method which is the combination of the steepest descent method and Newton's method was proposed to be used in the descent algorithm. + +* A numerical example was developed, and the proposed algorithm was applied to design a switching LPV controller with optimized switching surfaces. The potential of applying the descent algorithm to design switching surfaces with any number of variables was successfully tested in this example. + +* The proposed algorithm was applied to design switching surfaces for a switch-ing LPV controller used in a magnetically-actuated optical image stabilizer system with production variation, and improvement in the worst case refer-ence tracking error was achieved. + +* For air-fuel ratio control in spark ignition internal combustion engine, a switch-ing LPV control with optimized switching surfaces was designed by applying the descent algorithm and improvement in the closed-loop performance of the system was obtained. \ No newline at end of file diff --git a/samples/texts/5122728/page_102.md b/samples/texts/5122728/page_102.md new file mode 100644 index 0000000000000000000000000000000000000000..e102a40f425fcf6adb7204d936ced9f990ba78c2 --- /dev/null +++ b/samples/texts/5122728/page_102.md @@ -0,0 +1,11 @@ +## 6.3 Future Work + +Optimization of switching surfaces in switching LPV control design is an effective way to improve the performance of the systems, the dynamics of which vary during the operation of them. What is done in this research has been addressed this optimization problem in a particular manner with a diverse set of premises. To put it in a nutshell, there are still lots of room for research and study in this field. Some research topics can be categorized as follows: + +* Combine the proposed descent algorithm with the genetic algorithm in order to achieve global optimum: It is important to have an algorithm which does not depend on the initial point, and the global minimum point is possible to be obtained, so that further improvement of the performance can be achieved. + +* Optimize the number of divisions on each axis of the plant variation region, so that a balance between computational time and improvement in the performance level is sought: Intuitively, increase in the number of subregions leads to improvement in the performance level. However, in high number of subregions, the computational cost increases dramatically. Thus, it is important to maintain a balance between these two factors in controller design. + +* Optimize shapes of subregions: In current research the shape of the subregions are assumed to be rectangular. However, different shapes and layouts of the subregions can be assumed, and optimization of the switching surfaces can be done while maintaining those shapes, instead of rectangular shapes. + +* Implement the switching LPV controllers with optimized switching surfaces on physical systems of the applications presented in this thesis: The simulation \ No newline at end of file diff --git a/samples/texts/5122728/page_103.md b/samples/texts/5122728/page_103.md new file mode 100644 index 0000000000000000000000000000000000000000..423e59364145f96080e0c68626a1d793a3d1d69d --- /dev/null +++ b/samples/texts/5122728/page_103.md @@ -0,0 +1 @@ +results of applying optimized controllers to the control applications show that there are potentials to improve the performance of the systems. Therefore, it is tempting to implement such controllers in physical systems to obtain experimental results and validate simulation ones. \ No newline at end of file diff --git a/samples/texts/5122728/page_104.md b/samples/texts/5122728/page_104.md new file mode 100644 index 0000000000000000000000000000000000000000..4ef2b3710dbbbbbaf3569e3754584ce47f355e0a --- /dev/null +++ b/samples/texts/5122728/page_104.md @@ -0,0 +1,13 @@ +# Bibliography + +[1] Per Andersson and Lars Eriksson. Three way catalyst control using PI-style controller with HEGO sensor feedback. In *CCSE, Norrkoping*, Sweden, 2002. + +[2] Per Andersson, Lars Eriksson, and Lars Nielsen. Modeling and architecture examples of model based engine control. In *Proceedings of the Second Conference on Computer Science and Systems Engineering*, Norrkoping, Sweden, 1999. + +[3] Pierre Apkarian and Richard J. Adams. Advanced gain-scheduling techniques for uncertain systems. *IEEE Transactions on Control Systems Technology*, 6(1):21-32, 1998. + +[4] Pierre Apkarian, Pascal Gahinets, and Greg Becker. Self-scheduled $H_\infty$ control of linear parameter-varying systems: A design example. *Automatica*, 31(9):1251-1261, 1995. + +[5] Ehsan Azadi Yazdi and Ryozo Nagamune. A parameter set division and switching gain-scheduling controllers design method for time-varying plants. *Systems & Control Letters*, 60:1016-1023, 2011. + +[6] Stephen Boyd and Qinping Yang. Structured and simultaneous Lyapunov functions for system stability problems. *International Journal of Control*, 49:2215- \ No newline at end of file diff --git a/samples/texts/5122728/page_105.md b/samples/texts/5122728/page_105.md new file mode 100644 index 0000000000000000000000000000000000000000..be491dd01100d7e771587322fdb2dc39abd53ed4 --- /dev/null +++ b/samples/texts/5122728/page_105.md @@ -0,0 +1,17 @@ +2240, 1989. + +[7] Michael S. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. *IEEE Transactions on Automatic Control*, 43(4):475-482, April 1998. + +[8] Pang-Chia Chen. The design of smooth switching control with application to V/STOL aircraft dynamics under input and output constraints. *Asian Journal of Control*, 14:439-453, 2012. + +[9] Pang-Chia Chen, Sun-Li Wu, and Hung-Shiang Chuang. The smooth switching control for TORA system via LMIs. In *2010 8th IEEE International Conference on Control and Automation (ICCA)*, pages 1338-1343, June 2010. + +[10] Raymond A. Decarlo, Michael S. Branicky, Stefan Pettersson, and Bengt Lennartson. Perspectives and results on the stability and stabilizability of hybrid systems. In *Proceedings of the IEEE*, volume 88, pages 1069-1082, July 2000. + +[11] John C. Doyle and Keith Glover. *Robust and Optimal Control*. Prentice Hall, Upper Saddle River, New Jersey, 1996. + +[12] Nazli E. Kahveci and Mrdjan J. Jankovic. 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Springer, New York, 2003. + +[21] Sungyung Lim. *Analysis and Control of Linear Parameter-Varying Systems*. dissertation, Stanford University, 1998. + +[22] Sungyung Lim, Sungyung Lim and Kam Chan, Kam Chan. Analysis of hybrid linear parameter-varying systems. In *Proceedings of the 2003 American Control Conference*, volume 6, pages 4822–4827, 2003. \ No newline at end of file diff --git a/samples/texts/5122728/page_107.md b/samples/texts/5122728/page_107.md new file mode 100644 index 0000000000000000000000000000000000000000..adcf5c4720300ae19a212a036ead0d2743de1aa5 --- /dev/null +++ b/samples/texts/5122728/page_107.md @@ -0,0 +1,15 @@ +[23] Bei Lu, Heeju Choi, Gregory D. Buckner, and Kari Tammi. Linear parameter-varying techniques for control of a magnetic bearing system. *Control Engineering Practice*, 16:1161–1172, 2008. + +[24] Bei Lu and Fen Wu. 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In *Proceedings of the 2001 American Control Conference*, pages 2867–2871, Arlington, Virginia, 2001. + +[30] Javad Mohammadpour and Carsten W. (Eds.) Scherer. *Control of Linear Parameter Varying Systems with Applications*. Springer, New York, 2012. \ No newline at end of file diff --git a/samples/texts/5122728/page_108.md b/samples/texts/5122728/page_108.md new file mode 100644 index 0000000000000000000000000000000000000000..f3875e99cb91de698f88b22b4b03eaa4c69b9e23 --- /dev/null +++ b/samples/texts/5122728/page_108.md @@ -0,0 +1,15 @@ +[31] Kenneth R. Muske, James C. Peyton Jones, and E. M. Franceschi. Adaptive analytical model-based control for SI engine air fuel ratio. *IEEE Transactions on Control Systems Technology*, 16(4):763–768, July 2008. + +[32] João P. Hespanha, Daniel Liberzon, and A. Stephen Morse. Hysteresis-based switching algorithms for supervisory control of uncertain systems. *Automatica*, 39(2):263–272, February 2003. + +[33] João P. Hespanha and A. Stephen Morse. 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3.7Computational time of convergence of the descent algorithm for different number of switching surface variables, having used the hybrid method and the steepest descent method for the DS step.40
3.8Computational time of convergence of the descent algorithm for different number of switching surface variables, having used the steepest descent method for the DS step.40
4.1Physical layout of the lens-shifting image stabilizer [37].45
4.2Block diagram of the plant and actuator.46
4.3Block diagram of the closed-loop system.47
4.4Actual control input computation.50
4.5A feedback structure for controller design.53
4.6Functionality of γ with respect to pS = (c1,1, c2,1) obtained by the full-search method.55
4.7Convergence path of the descent algorithm having employed the hybrid method or the steepest descent method.56
4.8Error envelope comparison between the cases having an optimal controller and a heuristic one.59
5.1Plant variation region.64
5.2Block diagram of the closed-loop system.67
5.3Generalized plant.68
5.4The plant variation region with a partition.71
5.5Comparison of equivalent air-fuel ratio reference tracking between controllers with the optimized switching surface and the heuristic one.75
\ No newline at end of file diff --git a/samples/texts/5122728/page_110.md b/samples/texts/5122728/page_110.md new file mode 100644 index 0000000000000000000000000000000000000000..881377af87204c71fd23cabbf906a088e904fcf3 --- /dev/null +++ b/samples/texts/5122728/page_110.md @@ -0,0 +1,16 @@ +[47] Ehsan Azadi Yazdi, Mohammad Sepasi, Farrokh Sassani, and Ryozo Naga- +mune. Automated multiple robust track-following control system design in hard +disk drives. *IEEE Transactions on Control Systems Technology*, 19(4):920–928, +2011. + +[48] Hui Ye, Anthony N. Michel, and Ling Hou. Stability theory for hybrid dynamical systems. *IEEE Transactions on Automatic Control*, 43(4):461–474, April 1998. + +[49] Yildiray Yildiz, Anuradha M. Annaswamy, Diana Yanakiev, and Ilya Kolmanovsky. Adaptive air fuel ratio control for internal combustion engines. In *Proceedings of 2008 American Control Conference*, Seattle, Washington, June 2008. + +[50] Yildiray Yildiz, Anuradha M. Annaswamy, Diana Yanakiev, and Ilya Kol- +manovsky. Spark ignition engine fuel-to-air ratio control: An adaptive control +approach. *Control Engineering Practice*, 18(12):1369–1378, 2010. + +[51] Feng Zhang, Karolos M. Grigoriadis, Matthew A. Franchek, and Imad H. Makki. Transient lean burn air-fuel ratio control using input shaping method combined with linear parameter-varying control. In *Proceedings of 2006 American Control Conference*, Minneapolis, Minnesota, June 2006. + +[52] Feng Zhang, Karolos M. Grigoriadis, Matthew A. Franchek, and Imad H. Makki. Linear parameter-varying lean burn air-fuel ratio control for a spark ignition engine. *Journal of Dynamic Systems, Measurement, and Control*, 129(4):404–414, 2007. \ No newline at end of file diff --git a/samples/texts/5122728/page_111.md b/samples/texts/5122728/page_111.md new file mode 100644 index 0000000000000000000000000000000000000000..a9c2ed61a589aae70d7ee60709f617cc175b603a --- /dev/null +++ b/samples/texts/5122728/page_111.md @@ -0,0 +1,7 @@ +# Plant Model Derivations of Spark Ignition Internal Combustion Engines + +As explained in [36], a parameter dependent first-order plus dead time (FOPDT) model is used in this thesis for modelling the IC engine. This model is simple enough to design controller for and accurate enough to represent the overall dynamics of the IC engine. The input to the system is the fuel mass flow $\dot{m}_{fuel}$ which is a control input to manipulate the air-fuel ratio(AFR), denoted by $\psi$, as the output of the system. The FOPDT model is + +$$ \frac{\d\psi}{\d t} = \frac{g}{sT+1} e^{-s\alpha}, \quad (A.1) $$ + +where $g$ is the steady state gain of the system, $T$ is the time constant of the first-order term, and $\alpha$ is the pure delay. These parameters are functions of $N$ and $\dot{m}_{air}$, the varying parameters setting the dynamics of the plant. The mentioned \ No newline at end of file diff --git a/samples/texts/5122728/page_112.md b/samples/texts/5122728/page_112.md new file mode 100644 index 0000000000000000000000000000000000000000..cb8bdddfef6637fcb6bf60759fd99d0a324c4675 --- /dev/null +++ b/samples/texts/5122728/page_112.md @@ -0,0 +1,19 @@ +functionalities explained in [36] can be seen in below: + +$$g(\theta) = \frac{R_{st}}{\dot{m}_{air}}, \qquad (A.2)$$ + +where $R_{st}$ is the stoichiometric mass ratio of air over fuel which is almost 14.7. It +can be seen that steady state gain $g$ is just function of $\dot{m}_{air}$, not $N$. Also, we have + +$$T(\theta) \approx 120 \cdot \left(1 - \frac{1}{n_{cyl}}\right) \cdot \frac{1}{N}, \qquad (A.3)$$ + +where $n_{cyl}$ is the number of cylinders in the engine which is equal to 4 in our case, +and + +$$\alpha(\theta) = \frac{180}{N} + \frac{5.33}{\dot{m}_{air}}, \qquad (A.4)$$ + +shows that the pure delay is function of both the engine speed and the air flow [36]. + +The delay in FOPDT model is estimated by a Pade approximation with a second-order denominator and first-order numerator. Then, the plant transfer function is approximated by + +$$\frac{\psi}{\dot{m}_{fuel}} = \frac{g(\theta)}{sT(\theta) + 1} \cdot \frac{6 - 2s\alpha(\theta)}{6 + 4s\alpha(\theta) + (s\alpha(\theta))^2}. \qquad (A.5)$$ \ No newline at end of file diff --git a/samples/texts/5122728/page_113.md b/samples/texts/5122728/page_113.md new file mode 100644 index 0000000000000000000000000000000000000000..2aad3de56e565ae366d43208edcfd0d61b9c9ce2 --- /dev/null +++ b/samples/texts/5122728/page_113.md @@ -0,0 +1,7 @@ +# Appendix B + +## A Feedforward Term Added to the Control Input in Air-Fuel Ratio Control of Internal Combustion Engines + +After designing switching LPV controllers for air-fuel ratio control in internal combustion engines, it has been seen that the simulation performance of the system will be improved if a feedforward term is added to the output of the switching LPV controller to make the control input to the engine. In other words, *ṁfuel*-command to the engine has two components: a base-fuelling term obtained by a feedforward controller and a correction term which is the output of the switching LPV controller, as shown in Fig. B.1. + +To set the base-fuelling term, we make three assumptions as follows: \ No newline at end of file diff --git a/samples/texts/5122728/page_114.md b/samples/texts/5122728/page_114.md new file mode 100644 index 0000000000000000000000000000000000000000..2c1c937007f92dfcd4a0ce1b53061a5023e4f262 --- /dev/null +++ b/samples/texts/5122728/page_114.md @@ -0,0 +1,19 @@ +Figure B.1: The block diagram of the control system with a feedforward controller. + +* The output of the switching LPV controller in Fig. B.1 is set to be zero, and the feedforward controller is used exclusively in the system to control the air-fuel ratio. + +* There is no delay in the system. + +* By applying the feedforward controller, the output of the plant, i.e., the air-fuel ratio $\psi$, will be equal to 1. + +Having set these assumptions, we have + +$$ \dot{m}_{fuel} = \dot{m}_{fuel\_forward}, \quad (B.1) $$ + +and the transfer function of the engine block in Fig. B.1 with no delay is then expressed by + +$$ \frac{\d\psi}{\d t} = \frac{g(\theta)}{sT(\theta) + 1}. \quad (B.2) $$ + +This system can be realized in a state-space form as + +$$ P(\theta): \begin{cases} \dot{x} = \frac{-1}{T(\theta)} x + \dot{m}_{fuel} \\ \psi = \frac{g(\theta)}{T(\theta)} x, \end{cases} \quad (B.3) $$ \ No newline at end of file diff --git a/samples/texts/5122728/page_115.md b/samples/texts/5122728/page_115.md new file mode 100644 index 0000000000000000000000000000000000000000..ff802d5fa3c05bb9ccf6afd6b5a1aedcf617e885 --- /dev/null +++ b/samples/texts/5122728/page_115.md @@ -0,0 +1,29 @@ +in which $x \in \mathbb{R}$ is the state vector of the plant. By taking derivative of the output equation, we have + +$$ \dot{\psi} = \frac{\dot{g}(\theta)T(\theta) - g(\theta)\dot{T}(\theta)}{T^2(\theta)} \cdot x + \frac{g(\theta)}{T(\theta)} \cdot \dot{x}. \quad (B.4) $$ + +As mentioned before, we would like to input $\dot{m}_{fuel}$ such that the output of the plant is equal to 1. Therefore + +$$ \psi = 1, \quad (B.5) $$ + +and + +$$ \dot{\psi} = 0. \quad (B.6) $$ + +By substituting (B.5) in the output equation in (B.3), we have + +$$ x = \frac{T(\theta)}{g(\theta)}. \quad (B.7) $$ + +Now, by plugging (B.7), (B.6), and (B.3) in (B.4), the equation can be expressed by + +$$ 0 = \frac{\dot{g}(\theta)T(\theta) - g(\theta)\dot{T}(\theta)}{T^2(\theta)} \cdot \frac{T(\theta)}{g(\theta)} + \frac{g(\theta)}{T(\theta)} \cdot \left( \frac{-1}{g(\theta)} + \dot{m}_{fuel} \right). \quad (B.8) $$ + +By simplification, $\dot{m}_{fuel}$ for the base-fuelling is obtained as follows: + +$$ \dot{m}_{fuel} = \frac{1}{g(\theta)} + \frac{g(\theta)\dot{T}(\theta) - \dot{g}(\theta)T(\theta)}{g^2(\theta)}. \quad (B.9) $$ + +From (B.1), the relation between input and output of the feedforward controller in Fig. B.1 can be written as + +$$ \dot{m}_{\text{fuel-forward}} = \frac{1}{g(\theta)} + \frac{g(\theta)\dot{T}(\theta) - \dot{g}(\theta)T(\theta)}{g^2(\theta)}, \quad (B.10) $$ + +where the functionalities of $g$ and $T$ with respect to the varying parameter $\theta$ are available. In all the simulations of Chapter 5, this feedforward term has been added to the output of the designed switching LPV controllers, so as to build up the final control input to the plant. \ No newline at end of file diff --git a/samples/texts/5122728/page_12.md b/samples/texts/5122728/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..f14f31bc52d24c4a8cb3a4c9059b830ce0e3b990 --- /dev/null +++ b/samples/texts/5122728/page_12.md @@ -0,0 +1,19 @@ +5.6 Error comparison of equivalent air-fuel ratio reference tracking, averaged in every 10 seconds time interval, between controllers with the optimized switching surface and the heuristic one. . . . . . . . . . . 76 + +5.7 Air-flow trajectory and the lower and upper bounds on its variations. . . . . . 77 + +5.8 Engine speed trajectory and the lower and upper bounds on its variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 + +5.9 The operating point of the engine in the plant variation region. . . . 78 + +5.10 Air-flow variation rate trajectory and the lower and upper bounds on its variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 + +5.11 Engine Speed variation rate trajectory and the lower and upper bounds on its variations. . . . . . . . . . . . . . . . . . . . . . . . . . 80 + +5.12 Comparison of switching variable on air-flow axis between controllers with the optimized switching surface and the heuristic one. . . . . . 81 + +5.13 Comparison of switching variable on engine speed axis between controllers with the optimized switching surface and the heuristic one. . 81 + +5.14 Comparison of control input to the plant between controllers with the optimized switching surface and the heuristic one. . . . . . . . . . 82 + +B.1 The block diagram of the control system with a feedforward controller. 100 \ No newline at end of file diff --git a/samples/texts/5122728/page_13.md b/samples/texts/5122728/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..51f45be704ce39314fcaf1116fd55c81c559fe61 --- /dev/null +++ b/samples/texts/5122728/page_13.md @@ -0,0 +1,19 @@ +# Acknowledgements + +Foremost, I would like to express my deep gratitude to my supervisor and in- +structor, Dr. Ryozo Nagamune, for his enlightening guidance throughout the entire +research process. He has been a tremendous mentor for me with the useful comments +and remarks. His assiduity towards research will inspire me for my lifetime. Further- +more, I would like to thank the rest of my thesis committee, Dr. Farrokh Sassani +and Dr. Mu Chiao, for their great academic support of my thesis. My gratitude +also goes to the members of Control Engineering Laboratory at the University of +British Columbia, especially Mr. Masih Hanifzadegan for his insightful comments +about my thesis. I would also like to thank my friends, Mr. Mahdi Nematimehr, +Mr. Sina Amini Niaki, Mr. Ashkan Babaei, Mr. Amirhossein Hadi Hossein Abadi, +and Mr. Reza Nickmanesh for their supports and encouragements. + +MOEIN JAVADIAN + +*The University of British Columbia* + +*February 2014* \ No newline at end of file diff --git a/samples/texts/5122728/page_14.md b/samples/texts/5122728/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..c2b0487d796cd22ee28c18e7055b05bbd5ed874a --- /dev/null +++ b/samples/texts/5122728/page_14.md @@ -0,0 +1,2 @@ +This thesis is dedicated to my Parents, +for their priceless support and endless love \ No newline at end of file diff --git a/samples/texts/5122728/page_15.md b/samples/texts/5122728/page_15.md new file mode 100644 index 0000000000000000000000000000000000000000..3128547c4f244e8f67f7f29eab10c682c1f5b25f --- /dev/null +++ b/samples/texts/5122728/page_15.md @@ -0,0 +1,9 @@ +# Chapter 1 + +# Introduction + +## 1.1 Motivation + +A controller design procedure usually involves trial and error. Various characteristics of a controller to be designed are selected heuristically and required to be modified multiple times. The controller structure, a state-space order of the controller, some tuning parameters of a specific type of controller, and weighing parameters used in some sort of control design methods are just a few examples of such characteristics. An appropriate selection of these characteristics to maintain a stability and a satisfactory performance is always a challenging task in the control design procedure which requires trial and error. To facilitate such procedure and to obtain the best controller in the search region of the heuristically-selected parameters, it is usually desired to automate the parameters selection. + +Similarly, in switching linear parameter-varying (LPV) control design method which is used in a wide range of control applications, many premises are made beforehand and different parameters are heuristically selected. Nevertheless, optimal \ No newline at end of file diff --git a/samples/texts/5122728/page_16.md b/samples/texts/5122728/page_16.md new file mode 100644 index 0000000000000000000000000000000000000000..37fc96e2cac16127ad56466645acf191ca63daa6 --- /dev/null +++ b/samples/texts/5122728/page_16.md @@ -0,0 +1,13 @@ +selections instead of heuristic selections can be extremely beneficial. Therefore, a systematic approach to switching LPV control design method is sought, so as to shortcut the design process and to further improve the performance of the closed-loop system containing a switching LPV controller. + +## 1.2 Literature Review + +The genesis of switching LPV control design method through the literature is discussed in this section. Before addressing this topic, LPV models and LPV control design method are explained and related research endeavours are mentioned. + +### 1.2.1 LPV Model + +LPV models are known to be useful in representing nonlinear systems with varying operating conditions, as well as time-varying systems with measurable parameters. Different methods used to express nonlinear plants as LPV systems can be found in [17] and [39]. In LPV models, one or more of the plant state-space matrices are functions of a vector, $\theta$, called the scheduling parameter, which is assumed to be measurable in real-time. The scheduling parameter changes within bounds, called the plant variation region. Change in the vector $\theta$ causes variation not only in the input and output of the plant but also in the dynamics of the linear-represented system expressed by the state-space matrices. + +### 1.2.2 LPV-Based Control + +For LPV systems, it is desired to design gain-scheduled LPV controllers rather than time-invariant ones, since LPV controllers can relatively increase the stability margin and improve the performance of the system. Like any other LPV systems, the \ No newline at end of file diff --git a/samples/texts/5122728/page_17.md b/samples/texts/5122728/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..c78fa36e4c090a9862750792d99fdbf86d31a83c --- /dev/null +++ b/samples/texts/5122728/page_17.md @@ -0,0 +1,5 @@ +dynamics of LPV controllers depend on the scheduling parameter, and the control input is updated by measuring the scheduling parameter in real-time. The gain-scheduling design techniques are first introduced in [4] where a single Lyapunov function defined over the entire plant variation region is used to formulate the LPV synthesis condition as a linear matrix inequality (LMI) optimization problem. The existence of a single LPV controller, the state-space matrices of which satisfy the formulated LMI, guarantees the stability of the closed-loop system. Also, the performance of the system is optimized by minimization of the bound of the closed-loop $L_2$-gain performance. This problem is a convex optimization which can be solved using available LMI techniques. + +Although it is required in [4] that the state-space matrices of the plant are affine functions of the scheduling parameter, this necessity is removed in further development of the method in [3] by finitely sampling the plant variation region. In this approach, LMIs generated by formulating the problem for each sampled point are solved together to compute the optimal LPV controller state-space matrices. However, this controller satisfying LMI conditions for all the sampled points are required to be analyzed for other points in the plant variation region, so that the stability and performance for the whole region are checked. If the controller is not satisfactory, more sampling in the plant variation region is required for control design. In addition to the aforementioned development of LPV controller design in [3], unlike [4], a limit can be enforced for the variation rate of the scheduling parameter resulting in a less conservative controller. + +A novel control design approach is also proposed in [45] in which parameter-dependent Lyapunov functions are used to design LPV controllers for linear frac- \ No newline at end of file diff --git a/samples/texts/5122728/page_18.md b/samples/texts/5122728/page_18.md new file mode 100644 index 0000000000000000000000000000000000000000..88ec8aacce7388434b80a7fb4a65598f15e135c4 --- /dev/null +++ b/samples/texts/5122728/page_18.md @@ -0,0 +1,27 @@ +tional transformation (LFT) systems with guaranteed closed-loop stability and per- +formance. Finally, control of LPV systems has been thoroughly and comprehensively +studied in [30]. + +The aforementioned methods have been successfully applied to real engineering +problems, e.g., automotive engines [13, 36], aircraft systems [28], and active magnetic +bearing systems [23], to name a few. + +1.2.3 Switching LPV Control + +In the design of LPV controllers, or any other type of robust controllers, as the +range of plant variations increases, it is often the case that we have to compro- +mise the overall control performance, or that the stabilizing controller cannot be +found, due to the conservatism inherent to robust controller design methods. With +large plant variations, one approach to the design of a satisfactory controller is the +switching LPV control system design in [24]. In this approach, the region of pa- +rameter variations is divided into a given set of subregions, and for each subregion, +one LPV controller is designed so that it performs well while the parameter varies +within the subregion. The LPV controller switches to another LPV controller when +the parameter moves into another subregion, with performance guarantee. These +switched systems are considered as a combination of continuous systems and discrete +switching events which is described in [20]. + +Stability analysis of switched systems has come a long way primarily studied +in [10], [7], [35], and [20]. As shown in [6] and [20], the existence of a common +Lyapunov function for a group of stable LTI systems guarantees stability of switched +systems for any switching arrangements. Nevertheless, satisfying such a stability \ No newline at end of file diff --git a/samples/texts/5122728/page_19.md b/samples/texts/5122728/page_19.md new file mode 100644 index 0000000000000000000000000000000000000000..f46b880b31ba45620cfd0c0b36d893127cad322f --- /dev/null +++ b/samples/texts/5122728/page_19.md @@ -0,0 +1,3 @@ +condition results in a too conservative controller when a specific switching logic is applied. However, for a particular class of switching signals, multiple Lyapunov functions can be employed for stability analysis. Multiple Lyapunov functions can be defined either as piecewise continuous ones studied in [34], [44], [18], and [38], or as discontinuous ones described in [48] and [7]. Nevertheless, most of these research have considered each subsystem having LTI dynamics. In references [21], [22], and [33], this field of research has been further developed to the stability analysis of switched LPV systems. + +Switching logic plays an important role in the stability characteristics of switched systems. Diverse switching rules can be found in the literature. In references [44], [18], and [38], the state-space is partitioned and a construction method of conic switching logic is developed for the stability of a family of LTI systems. This method is further improved in [32] where a hysteresis switching logic is employed for conic sets. For this logic, it is assumed that every two adjacent sets have an overlapping region engendering two switching surfaces in between as shown in Fig. 1.1. The active controller is switched when one of the switching surfaces is hit. A min-projection strategy is the basis for another type of state-dependent switching logic in [27] and [35]. Also, switching with average dwell time studied in [33] is an alternative method based on which switches occur in an adequately slow manner, so that the effects of transient responses after switches are suppressed and stability is guaranteed. In [24], both hysteresis switching logic and switching with average dwell time are employed to develop the switching LPV control design of LPV systems using multiple parameter-dependent Lyapunov functions. \ No newline at end of file diff --git a/samples/texts/5122728/page_2.md b/samples/texts/5122728/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..87aeab56aeb519877ed4960b56ee33b9d3a0dc3b --- /dev/null +++ b/samples/texts/5122728/page_2.md @@ -0,0 +1,5 @@ +Abstract + +This thesis proposes an algorithm to design switching surfaces for the switching linear parameter-varying (LPV) controller with hysteresis switching. The switching surfaces are sought for to optimize the bound of the closed-loop $L_2$-gain performance. An optimization problem is formulated with respect to parameters characterizing Lyapunov matrix variables, local controller matrix variables, and locations of the switching surfaces. Since the problem turns out to be non-convex in terms of these characterizing parameters, a numerical algorithm is given to guarantee the decrease of the cost function value after each iteration, which consists of two steps: direction selection and line search. A *hybrid method* which is a combination of the *steepest descent method* and *Newton's method* is employed in the direction selection step to decide the orientation of proceeding. A numerical algorithm is used to compute the most appropriate length of the proceeding along the selected direction which generates the most decrease in the cost function. + +To demonstrate the efficiency and usefulness of the proposed algorithm, it will be applied to three examples in control applications: a tracking problem for a mass-spring-damper system, a vibration suppression problem for a magnetically-actuated \ No newline at end of file diff --git a/samples/texts/5122728/page_20.md b/samples/texts/5122728/page_20.md new file mode 100644 index 0000000000000000000000000000000000000000..ebe91905ca73725ce40fa534f569f482b209b5ec --- /dev/null +++ b/samples/texts/5122728/page_20.md @@ -0,0 +1,7 @@ +Figure 1.1: The overlapping region and switching surfaces employed in hysteresis logic. + +The switching LPV control design method has been successfully applied to different applications, such as ball-screw drive servo control [15], aircraft systems control [25], and air-fuel ratio control in automotive engines [36]. + +### 1.2.4 Smooth Switching LPV Control + +A deficiency inherent to general switching LPV controllers described in [24] is that the smooth transition is not guaranteed while a switch event occurs. In other words, a jump in the control input is generally expected since the state-space matrices of the active controller would be changed abruptly when controller switching occurs. This nonsmooth transition might lead to signal saturations or damage the system in some cases. In [9] and [8], a smooth switching LPV control design method has been proposed which can be applied to the plants with one-dimensional parameter variation regions. Besides, state feedback is required in order to use this method. An alternative method is developed in [14] which can be applied to control design for systems having output feedback and one or two-dimensional plant variation region. \ No newline at end of file diff --git a/samples/texts/5122728/page_21.md b/samples/texts/5122728/page_21.md new file mode 100644 index 0000000000000000000000000000000000000000..1448c98c3ecd7719d9dd90adeb91b03d4dad9e54 --- /dev/null +++ b/samples/texts/5122728/page_21.md @@ -0,0 +1,11 @@ +### 1.2.5 Switching LPV Control with Optimized Switching Surfaces + +Another shortcoming of the method in [24] is that the set of subregions are assumed to be given a priori. Although the subregions may be selected by trial and error, optimization-based methods will be desirable for finding satisfactory subregions systematically. Methods have been developed to determine the subregions automatically based on the nonsmooth optimization technique and applied to various applications in [46, 5, 47]. In these methods, the subregions are designed sequentially such that the performance of the system over the entire plant variation region is optimized. However, this optimization does not guarantee achieving the global optimum. + +## 1.3 Research Objectives + +As described above, various works have been done in regard to switching LPV control design to address stability analysis as well as smoothness of the transient response while switching the controller occurs. In addition, research has been recently done to design the set of subregions in the plant variation region rather than to prespecify such subregions set. Nevertheless, the nature of the problem is such that different approaches to optimized partition design are expectable, and more room to improve the performance of the system is still remained to be explored. This thesis addresses the partition design problem in a different manner from how [46, 5, 47] do and presents an alternative method to optimized switching LPV control design. + +In this approach: + +* An algorithm is proposed to design both subregions of the parameter variation \ No newline at end of file diff --git a/samples/texts/5122728/page_22.md b/samples/texts/5122728/page_22.md new file mode 100644 index 0000000000000000000000000000000000000000..f83463d683985e1db3ba1c1f7b8d12c5e25f4da0 --- /dev/null +++ b/samples/texts/5122728/page_22.md @@ -0,0 +1,14 @@ +region and LPV controllers associated with the subregions, leading to the +switching LPV controller with hysteresis switching. + +* The proposed method optimizes the *size* of subregions such that the performance of the system over the entire plant variation region is optimized. However, the number of subregions as well as the shape of each subregion is still prespecified. + +* The subregions are characterized by two kinds of parameters, i.e., the centres and the widths of overlapping regions between adjacent subregions. + +* To design a switching LPV controller that optimizes the bound of the closed-loop $L_2$-gain performance, an optimization problem is formulated by means of these characterizing parameters, as well as matrix variables for switching LPV controller design. + +* Since the problem is non-convex, we will provide an iterative algorithm to guarantee the decrease of the cost function value after each iteration, which consists of a direction selection step and line search step. + +* A *hybrid method* is proposed for the direction selection step which is a combination of *the steepest descent method* and *Newton's method* explained, for example, in [26]. + +After proposing the algorithm, it is applied to three control applications including LPV systems: a tracking problem for a mass-spring-damper system, a vibration suppression problem for a magnetically-actuated optical image stabilizer, and an air-fuel ratio control problem for automotive engines. Simulation results of implementing the controllers are also provided and compared with that of implementing \ No newline at end of file diff --git a/samples/texts/5122728/page_23.md b/samples/texts/5122728/page_23.md new file mode 100644 index 0000000000000000000000000000000000000000..86f021e13306d668797595ceb83504c44018cd36 --- /dev/null +++ b/samples/texts/5122728/page_23.md @@ -0,0 +1,11 @@ +non-optimized controllers. + +## 1.4 Organization of Thesis + +The rest of this thesis is organized as follows. The numerical iterative descent algorithm to optimize the switching surfaces in switching LPV control design is presented in Chapter 2. Before the algorithm is described, formulation of an LPV plant is provided, and switching LPV control design method with fixed partition is reviewed. Then, switching surface design method is formulated in the form of an optimization problem, to solve of which the proposed algorithm is employed. + +In Chapter 3, the proposed algorithm is applied to a numerical example which is control of a simple mass-spring-damper system. Diverse aspects of applying the numerical algorithm are investigated, and both strengths and weaknesses of the proposed optimization approach are explained. + +In Chapters 4 and 5, the numerical algorithm is applied to design switching LPV controllers with optimized switching surfaces for two control applications: magnetically-actuated optical image stabilization in cameras and air-fuel ratio control in spark ignition internal combustion engines. For each application, control design results are provided, and simulation results of applying controllers with optimized switching surfaces are compared with that of applying controllers with heuristic switching surfaces. + +Conclusion of the thesis and a summary of results are provided in Chapter 6. The contributions of this research are also briefly discussed and possible future works are mentioned. \ No newline at end of file diff --git a/samples/texts/5122728/page_24.md b/samples/texts/5122728/page_24.md new file mode 100644 index 0000000000000000000000000000000000000000..b396c0258f08ef7ab9266e1007d37d7e8f308598 --- /dev/null +++ b/samples/texts/5122728/page_24.md @@ -0,0 +1,5 @@ +# Chapter 2 + +## An Optimization Approach to Switching LPV Control Design + +Design objectives in parameter-varying control problems consist of guaranteeing stability and improving performance over the entire plant variation region. Performance of a control system can be characterized by, e.g., speed of response, tracking error, control input magnitude, disturbance rejection, and noise rejection. All these performance components using some weighting functions, introduced later in this thesis, can be quantified to closed-loop $L_2$-gain performance of the system, called performance level $\gamma$. It can be shown that there is an inverse relation between performance of the system and the performance level $\gamma$. Namely, the less $\gamma$ it is, the better performance the system has. Moreover, the existence of $\gamma$ itself, which correlates to existence of a set of common Lyapunov functions, guarantees the stability. In other words, the control design objective is to design a controller so that a $\gamma$-value is found to have a stable system, and its value is minimized to have a satisfactory performance. \ No newline at end of file diff --git a/samples/texts/5122728/page_25.md b/samples/texts/5122728/page_25.md new file mode 100644 index 0000000000000000000000000000000000000000..47ad14ccc3bf59efa2d4870bcf151112ebaee921 --- /dev/null +++ b/samples/texts/5122728/page_25.md @@ -0,0 +1,17 @@ +To further minimize $\gamma$-value and improve the performance, a numerical algorithm is proposed in this section to optimize switching surfaces in switching LPV control design. The formulation of switching LPV control design method with optimized switching surfaces is introduced before presenting the algorithm. + +## 2.1 Problem Formulation + +### 2.1.1 Description of an LPV Plant + +Consider the following continuous-time LPV plant: + +$$ \begin{bmatrix} \dot{x}(t) \\ z(t) \\ y(t) \end{bmatrix} = G(\theta(t)) \begin{bmatrix} x(t) \\ w(t) \\ u(t) \end{bmatrix}, \quad (2.1) $$ + +where the vector $x$ is the state, $z$ is the controlled variable, $y$ is the measurement, $w$ is the exogenous signal, and $u$ is the control input. A matrix-valued function $G$ is defined for a parameter vector $\theta \in \mathbb{R}^{n_{\theta}}$ by + +$$ G(\theta) := \begin{bmatrix} A(\theta) & B_1(\theta) & B_2(\theta) \\ C_1(\theta) & D_{11}(\theta) & D_{12}(\theta) \\ C_2(\theta) & D_{21}(\theta) & 0 \end{bmatrix}, \quad (2.2) $$ + +where the matrices in (2.2) have dimensions which are compatible with the lengths of vectors in (2.1). + +Hereafter, to simplify the notation and the explanation of our proposed method, we will assume that the dimension of the parameter vector $\theta$ is two, i.e., $n_{\theta} = 2$. However, all the results are valid even for cases $n_{\theta} > 2$. The parameter vector $\theta$ which varies in time is assumed to be measurable in real-time, to be used for \ No newline at end of file diff --git a/samples/texts/5122728/page_26.md b/samples/texts/5122728/page_26.md new file mode 100644 index 0000000000000000000000000000000000000000..1bf3c06a407cf7560ea22a33d5913e1c5f807e3b --- /dev/null +++ b/samples/texts/5122728/page_26.md @@ -0,0 +1,32 @@ +the gain-scheduling control purpose, and to move within a given normalized square +region and given rates of variations as $\theta(t) \in \Theta$ and $\dot{\theta}(t) \in \Omega$, where + +$$ +\Theta := \{ \theta \in \mathbb{R}^2 : \theta_k \in [-1, 1], k = 1, 2 \}, \quad (2.3) +$$ + +$$ +\Omega := \{ \omega \in \mathbb{R}^2 : \omega_k \in [\underline{\omega}_i, \overline{\omega}_i], k = 1, 2 \}, \quad (2.4) +$$ + +To guarantee the existence of the output feedback stabilizing controller, we also +assume the stabilizability and detectability of the plant for each $\theta$ meeting the +above conditions. + +**2.1.2 Description of a Switching LPV Controller with Hysteresis Switching** + +As shown in Figure 2.1, the LPV plant in (2.1) is connected with a switching LPV +controller expressed by + +$$ +\begin{bmatrix} \dot{x}_K(t) \\ u(t) \end{bmatrix} = K(\theta(t)) \begin{bmatrix} x_K(t) \\ y(t) \end{bmatrix}, \qquad (2.5) +$$ + +where the vector $x_K$ is the controller state, and the system matrix of the controller, +involving a piecewise-constant switching signal $\sigma$, is given by + +$$ +K(\theta(t)) := \begin{bmatrix} A_K^{(\sigma(t))}(\theta(t)) & B_K^{(\sigma(t))}(\theta(t)) \\ C_K^{(\sigma(t))}(\theta(t)) & D_K^{(\sigma(t))}(\theta(t)) \end{bmatrix}. \quad (2.6) +$$ + +The switching signal $\sigma$ specifies one active controller among a set of controllers at each time instant. The switching rule in this thesis is the hysteresis switching rule presented in [25, 24], and will be briefly reviewed with a simple example next. \ No newline at end of file diff --git a/samples/texts/5122728/page_27.md b/samples/texts/5122728/page_27.md new file mode 100644 index 0000000000000000000000000000000000000000..f526787b6ec2a19fef3956f665c2fac38f2265f4 --- /dev/null +++ b/samples/texts/5122728/page_27.md @@ -0,0 +1,25 @@ +Figure 2.1: Feedback connection between an LPV plant and an LPV controller. + +Suppose that the normalized square region (2.3) is divided into four overlapping +rectangular subregions (see Figure 2.2): + +$$ +\begin{align*} +\Theta^{(1,1)} &:= \{\theta \in \mathbb{R}^2 : \theta_1 \in [-0.1, 1], \theta_2 \in [-1, 0.1]\}, \\ +\Theta^{(1,2)} &:= \{\theta \in \mathbb{R}^2 : \theta_1 \in [-0.1, 1], \theta_2 \in [-0.1, 1]\}, \\ +\Theta^{(2,1)} &:= \{\theta \in \mathbb{R}^2 : \theta_1 \in [-1, 0.1], \theta_2 \in [-1, 0.1]\}, \\ +\Theta^{(2,2)} &:= \{\theta \in \mathbb{R}^2 : \theta_1 \in [-1, 0.1], \theta_2 \in [-1, 0.1]\}, +\end{align*} +$$ + +and that system matrices of the LPV controller for each subregion $\Theta^{(i,j)}$, $i=1,2$, +$j=1,2$, are given by + +$$ +K^{(i,j)}(\theta) := \begin{bmatrix} A_K^{(i,j)}(\theta) & B_K^{(i,j)}(\theta) \\ C_K^{(i,j)}(\theta) & D_K^{(i,j)}(\theta) \end{bmatrix}. \quad (2.7) +$$ + +In Figure 2.2, the switching surfaces are indicated with dashed lines. With the hys- +teresis switching rule, the controller switches when the varying parameter $\theta$ crosses +the switching surfaces which are also the boundary of the subregion from which $\theta$ +moves into an adjacent region. For an example of a trajectory of $\theta$ provided in \ No newline at end of file diff --git a/samples/texts/5122728/page_28.md b/samples/texts/5122728/page_28.md new file mode 100644 index 0000000000000000000000000000000000000000..f9b3a2e89536f6b882592ba53d40e0f523d26905 --- /dev/null +++ b/samples/texts/5122728/page_28.md @@ -0,0 +1,17 @@ +Figure 2.2: An example of a two-dimensional parameter trajectory with four subregions. + +Figure 2.2, the switching signal $\sigma$ in (2.6) changes its value at points marked with ‘x’ as $(1,1) \rightarrow (1,2) \rightarrow (2,2) \rightarrow (1,2) \rightarrow (1,1) \rightarrow (2,1)$. + +### 2.1.3 Switching Surface Design Problem + +We will tackle the following design problem in this chapter. Given an LPV plant (2.1)-(2.4), and the number of divisions in each axis of the square region $\Theta$, denoted by $N_1$ and $N_2$, design both the subregions + +$$ \Theta^{(i,j)}, i = 1, \dots, N_1, j = 1, \dots, N_2, \qquad (2.8) $$ + +and the associated LPV controllers + +$$ K^{(i,j)}, i = 1, \dots, N_1, j = 1, \dots, N_2, \qquad (2.9) $$ + +such that the bound $\gamma$ of the closed-loop $L_2$-gain from $w$ to $z$ is minimized, namely, + +$$ \min \gamma \text{ subject to } \|z\|_2 < \gamma \|w\|_2, \qquad (2.10) $$ \ No newline at end of file diff --git a/samples/texts/5122728/page_29.md b/samples/texts/5122728/page_29.md new file mode 100644 index 0000000000000000000000000000000000000000..eb85d69df4dfdd0857f3cd91ec63da30b104651a --- /dev/null +++ b/samples/texts/5122728/page_29.md @@ -0,0 +1,7 @@ +for any trajectory of $\theta$ with (2.3) and (2.4), and with the hysteresis switching LPV controller (2.6). + +We remark that the problem formulated above is different from the one in [24], in that the subregions $\Theta^{(i,j)}$ are prespecified in that paper, while they are variables to be sought in our formulation. + +If we consider only regularly aligned rectangular subregions, as shown in Fig. 2.2, then the design problem of subregions is equivalent to that of switching surfaces. The switching surfaces for each coordinate $\theta_k$, $k = 1, 2$, are characterized by the pairs of centers and widths, denoted by $\{(c_{k,n_k}, w_{k,n_k}) : n_k = 1, \dots, N_k - 1\}$, of the overlapping regions. This characterization is illustrated in Figure 2.3. We will utilize these characterizing parameters to solve the formulated problem. + +Figure 2.3: Parameterizations of switching surfaces (dashed lines) in terms of the centers (dotted lines) and the widths of the overlapping regions for the case when $N_1 = 2$ and $N_2 = 3$. \ No newline at end of file diff --git a/samples/texts/5122728/page_3.md b/samples/texts/5122728/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..b4f97d3d7b76275639f5adfc123d48f8951c6186 --- /dev/null +++ b/samples/texts/5122728/page_3.md @@ -0,0 +1,6 @@ +optical image stabilizer, and an air-fuel-ratio control problem for automotive en- +gines. In these examples, it will be shown that the proposed optimization approach +to the design of the switching surfaces and the switching LPV controller is superior +to heuristic approaches in closed-loop performances, at the price of higher compu- +tational costs. Additionally, it will be shown that the algorithm can be applied to +the general *n*-parameters case. \ No newline at end of file diff --git a/samples/texts/5122728/page_30.md b/samples/texts/5122728/page_30.md new file mode 100644 index 0000000000000000000000000000000000000000..2c2a0c6e67ad6093a7d54a77a27e364dee682555 --- /dev/null +++ b/samples/texts/5122728/page_30.md @@ -0,0 +1,24 @@ +## 2.2 Review of LPV Switching Controller Design + +Here, we review the LPV switching controller design method developed in [24], and set the notation to propose an algorithm for switching surface design in the next section. + +The $L_2$-gain performance in (5.21) holds as far as the vector $\theta$ changes within the region $\Theta$ and within the region of rate of variations $\Omega$, if there exist a constant matrix $Y^\dagger$ and matrices which depend on $\theta$ by means of scalar-valued functions $h_m$, $m = 1, \dots, M^\ddag$, expressed as + +$$ +\begin{align} +\hat{A}_K^{(i,j)}(\theta) &:= \hat{A}_0^{(i,j)} + \sum_{m=1}^M h_m(\theta) \hat{A}_m^{(i,j)}, \notag \\ +\hat{B}_K^{(i,j)}(\theta) &:= \hat{B}_0^{(i,j)} + \sum_{m=1}^M h_m(\theta) \hat{B}_m^{(i,j)}, \notag \\ +\hat{C}_K^{(i,j)}(\theta) &:= \hat{C}_0^{(i,j)} + \sum_{m=1}^M h_m(\theta) \hat{C}_m^{(i,j)}, \tag{2.11} \\ +\hat{D}_K^{(i,j)}(\theta) &:= \hat{D}_0^{(i,j)} + \sum_{m=1}^M h_m(\theta) \hat{D}_m^{(i,j)}, \notag \\ +X^{(i,j)}(\theta) &:= X_0^{(i,j)} + \sum_{m=1}^M h_m(\theta) X_m^{(i,j)}, \notag \\ +& i = 1, \dots, N_1, \quad j = 1, \dots, N_2, \notag +\end{align} +$$ + +that have appropriate matrix dimensions§ and fulfill the following matrix inequalities. (The dependency of the matrices in (2.12)–(2.14) on $\theta$ and $\dot{\theta}$ is omitted for brevity.) + +†For practical validity, we have to assume that either *X* or *Y* is a constant matrix. Here, we assume *Y* to be constant, but formula are analogous for a constant *X*. See [3, 36]. + +‡The functions $h_m$, $m = 1, \dots, M$, are chosen based on the functionality of the plant $G(\theta)$ in (2.2). See [3, Section IV]. + +§See [42] for the dimensions of these matrices. \ No newline at end of file diff --git a/samples/texts/5122728/page_31.md b/samples/texts/5122728/page_31.md new file mode 100644 index 0000000000000000000000000000000000000000..b8d40a9f7a24de34b4db2dc352045032065c01a6 --- /dev/null +++ b/samples/texts/5122728/page_31.md @@ -0,0 +1,59 @@ +1. For each vector $\theta \in \Theta^{(i,j)}$ and each vector $\dot{\theta} \in \Omega$, + +$$ +\begin{bmatrix} +M_{11} & \star & \star & \star \\ +M_{21} & M_{22} & \star & \star \\ +M_{31} & M_{32} & -\gamma I & \star \\ +M_{41} & M_{42} & M_{43} & -\gamma I +\end{bmatrix} < 0, \qquad (2.12) +$$ + +$$ +-\left[ \begin{array}{cc} X^{(i,j)} & I \\ I & Y \end{array} \right] < 0, \qquad (2.13) +$$ + +where the entry $\star$ represents a matrix which makes the whole matrix symmet- +ric, and block matrices are defined by + +$$ +M_{11} := \dot{X}^{(i,j)} + X^{(i,j)} A + \hat{B}_K^{(i,j)} C_2 + (\star), +$$ + +$$ +M_{21} := [\hat{A}_K^{(i,j)}]^T + A + B_2 \hat{D}_K^{(i,j)} C_2, +$$ + +$$ +M_{22} := -\dot{Y} + AY + B_2 \hat{C}_K^{(i,j)} + (\star), +$$ + +$$ +M_{31} := (X^{(i,j)} B_1 + \hat{B}_K^{(i,j)} D_{21})^T, +$$ + +$$ +M_{32} := (B_1 + B_2 \hat{D}_K^{(i,j)}) D_{21})^T, +$$ + +$$ +M_{41} := C_1 + D_{12} \hat{D}_K^{(i,j)} C_2, +$$ + +$$ +M_{42} := C_1 Y + D_{12} \hat{C}_K^{(i,j)}, +$$ + +$$ +M_{43} := D_{11} + D_{12} \hat{D}_K^{(i,j)} D_{21}. +$$ + +2. For each vector $\theta$ on each switching surface associated with the case when $\theta$ +leaves the subregion $\Theta^{(i,j)}$ and enters its adjacent subregion $\Theta^{(k,l)}$, + +$$ +X^{(i,j)} - X^{(k,l)} < 0. \tag{2.14} +$$ + +Once matrices in (2.11) satisfying the matrix inequalities (2.12)–(2.14) are found, +the controller parameters for a subregion $\Theta^{(i,j)}$ are obtained, with $N^{(i,j)} := Y^{-1}$ – \ No newline at end of file diff --git a/samples/texts/5122728/page_32.md b/samples/texts/5122728/page_32.md new file mode 100644 index 0000000000000000000000000000000000000000..2c4e924d98c29a421096bfa99214f5f5bc4633b4 --- /dev/null +++ b/samples/texts/5122728/page_32.md @@ -0,0 +1,23 @@ +$X^{(i,j)}$, by + +$$ +\begin{align*} +A_K^{(i,j)} &:= [N^{(i,j)}]^{-1} [\hat{A}_K^{(i,j)} - X^{(i,j)} \{A - B_2 \hat{D}_K^{(i,j)} C_2\} Y \\ +&\phantom{:=} - \hat{B}_K^{(i,j)} C_2 Y - X^{(i,j)} B_2 \hat{C}_K^{(i,j)}] Y, \\ +B_K^{(i,j)} &:= [N^{(i,j)}]^{-1} [\hat{B}_K^{(i,j)} - X^{(i,j)} B_2 \hat{D}_K^{(i,j)}], \\ +C_K^{(i,j)} &:= [\hat{C}_K^{(i,j)} - \hat{D}_K^{(i,j)} C_2 Y] Y^T, \\ +D_K^{(i,j)} &:= \hat{D}_K^{(i,j)}(\theta), +\end{align*} +$$ + +For controller design, matrix inequalities (2.12)–(2.14) must hold at infinitely many points $\theta$, and extreme points of $\dot{\theta}$, i.e., $\omega_k$ and $\bar{\omega}_k$ in (2.4), due to the linearity of the matrix inequality (2.12) with respect to $\dot{\theta}$. In order to make the design practically feasible, a common technique is the gridding of the parameter region, leading to a finite number of matrix inequalities. + +We can gather all of the inequalities into one matrix inequality, expressed in the form of $M(\gamma, p_K, p_S) < 0$. Here, the matrix $M$ is a block diagonal matrix consisting of left-hand sides of the inequalities (2.12)–(2.14) evaluated at gridded points, and $p_K$ and $p_S$ are vectors of optimization variables related to, respectively, the controller design $(\hat{A}_K^{(i,j)}, \hat{B}_K^{(i,j)}, \hat{C}_K^{(i,j)}, \hat{D}_K^{(i,j)}, X^{(i,j)}, Y, i = 1, \dots, N_1, j = 1, \dots, N_2)$ and the switching surface design $(\{(c_{k,n_k}, w_{k,n_k}) : n_k = 1, \dots, N_k - 1\}, k = 1, 2)$. + +The optimization problem + +$$ +\min \gamma \quad \text{subject to} \quad M(\gamma, \mathbf{p}_K, \mathbf{p}_S) < 0 \tag{2.15} +$$ + +is non-convex in general, due to the coupling between optimization variables $\mathbf{p}_K$ and $\mathbf{p}_S$ in the matrix $M$. However, with a fixed vector $\mathbf{p}_S = \mathbf{p}_S^\star$, this problem reduces \ No newline at end of file diff --git a/samples/texts/5122728/page_33.md b/samples/texts/5122728/page_33.md new file mode 100644 index 0000000000000000000000000000000000000000..3aa8efff804103e95ace53e745a46c37269e8d1c --- /dev/null +++ b/samples/texts/5122728/page_33.md @@ -0,0 +1,17 @@ +to + +$$ \min \gamma \quad \text{subject to} \quad M(\gamma, \boldsymbol{p}_K, \boldsymbol{p}_S^*) < 0, \qquad (2.16) $$ + +which is exactly same as that considered in [24], and thus solvable via numerically efficient convex optimization techniques. By solving (2.16), the corresponding minimizers $p_K^*$ and $\gamma^*$ are obtained. One of the contribution of this thesis is to propose an algorithm for the adjustment of the vector $p_S^*$, i.e., the switching surface, so as to further improve the $L_2$-gain bound $\gamma^*$. + +## 2.3 An Optimization Approach to Switching Surface Design + +An optimization algorithm presented in this thesis utilizes numerical methods to suggest such $p_S^*$, so that the performance level $\gamma^*$ is minimized. The iterative descent algorithm is explained in this section. For ease of explanation of the algorithm, we introduce a function $f$ which relates $\gamma^*$ to $p_S^*$ as follows: + +$$ \gamma^* = f(p_S^*). \qquad (2.17) $$ + +Note that the explicit form of $f$ is not available. However, for any input to the function $f$, it is possible to find the output by solving a convex optimization problem (2.16). + +### 2.3.1 The Descent Algorithm's Main Structure + +The main structure of the algorithm can be seen in Fig. 2.4 and described as follows: \ No newline at end of file diff --git a/samples/texts/5122728/page_34.md b/samples/texts/5122728/page_34.md new file mode 100644 index 0000000000000000000000000000000000000000..0d59b5891770ec0c0b32ecef2ac93f4b58a2e01c --- /dev/null +++ b/samples/texts/5122728/page_34.md @@ -0,0 +1,29 @@ +1. Initialize a switching surfaces vector + +$$ \boldsymbol{p}_{S,\text{old}} := \boldsymbol{p}_{S,\text{init}}, \quad (2.18) $$ + +and compute + +$$ \gamma_{\text{old}} := f(\boldsymbol{p}_{S,\text{old}}), \quad (2.19) $$ + +by solving the minimization problem (2.16) at given $\boldsymbol{p}_{S,\text{old}}$. + +2. Find a new vector + +$$ \boldsymbol{p}_{S,\text{new}} := \boldsymbol{p}_{S,\text{old}} + \delta \boldsymbol{p}_S, \quad (2.20) $$ + +and compute + +$$ \gamma_{\text{new}} := f(\boldsymbol{p}_{S,\text{new}}), \quad (2.21) $$ + +by solving the minimization problem (2.16) at given $\boldsymbol{p}_{S,\text{new}}$. In (2.20), we have + +$$ \delta \boldsymbol{p}_S := \epsilon \cdot d, \quad (2.22) $$ + +where $d$ is a direction vector numerically obtained by applying the idea behind *the steepest descent method* and *Newton's method* explained in [26]. Also, $\epsilon$ in (2.22) is numerically set by line search. In the next subsections, the direction selection and line search steps will be described. + +3. Test whether the $\gamma$-value has decreased by comparing $\gamma_{new}$ with the $\gamma$-value in the previous iteration, $\gamma_{old}$. Assume $\delta$ is a sufficiently small positive value. + +• If $\gamma_{old} - \gamma_{new} < \delta$, then terminate the algorithm. The optimizers are $\gamma^* := \gamma_{old}$ and $\boldsymbol{p}_S^* := \boldsymbol{p}_{S,\text{old}}$. + +• Otherwise, set $\gamma_{old} := \gamma_{new}$, $\boldsymbol{p}_{S,\text{old}} := \boldsymbol{p}_{S,\text{new}}$, and go back to Step 2. \ No newline at end of file diff --git a/samples/texts/5122728/page_35.md b/samples/texts/5122728/page_35.md new file mode 100644 index 0000000000000000000000000000000000000000..b1b944be0f1cf969ef15d898b89f1dfe9fa772df --- /dev/null +++ b/samples/texts/5122728/page_35.md @@ -0,0 +1,5 @@ +Figure 2.4: The optimization flowchart. + +The algorithm is terminated at Step 3 and the optimizers, $\gamma^*$ and $p_S^*$, are obtained. Meanwhile, the optimized controller parameter vector denoted by $p_K^*$ is computed by solving the optimization problem (2.16) at given $p_S^*$. Having obtained all the design parameters, controller reconstruction is done as explained in Section 2.2. + +Regarding the algorithm presented above, we give two remarks. First, the proposed algorithm guarantees that the $\gamma$-value decreases every time when the algorithm comes to Step 2. In other words, the algorithm is a descent algorithm. Secondly, since the problem being solved is non-convex, the final solution obtained by this algorithm may depend on the initial selection of the switching surfaces in Step 1, which can be heuristic. If the final value of $\gamma$ is not satisfactory, we have to either \ No newline at end of file diff --git a/samples/texts/5122728/page_36.md b/samples/texts/5122728/page_36.md new file mode 100644 index 0000000000000000000000000000000000000000..4a29d4ecb3239e1443cae5dbdb99021efe6e0069 --- /dev/null +++ b/samples/texts/5122728/page_36.md @@ -0,0 +1,9 @@ +change the initial switching surfaces, or increase the number of divisions $N_1$ and $N_2$ on each axis. + +### 2.3.2 Direction Selection Computations + +There are different numerical methods for the direction selection (DS) step, two of which are the steepest descent method and Newton's method used in this thesis [26]. The steepest descent method utilizes a first order approximation of the cost function to select a descending direction. The order of approximation of this method brings about a low convergence rate. However, selection of a descending direction is guaranteed. On the other hand, Newton's method employs a second order approximation which makes convergence rate be comparably faster. The disadvantage inherent to Newton's method is that it needs Hessian matrix of the objective function to be positive definite to ensure a descending direction [26]. + +A hybrid method is employed in this thesis to use the advantages of both methods while covering the disadvantages of each. It starts with Newton's method and check eigenvalues of Hessian matrix to see whether they are positive or not. If the Hessian matrix is positive definite, the direction suggested by Newton's method is adopted. Otherwise, the method is switched to the steepest descent method, so that the descending direction is guaranteed. In the next chapters of this thesis, the hybrid method is applied for DS step of the algorithm. However, the results of applying either of above methods exclusively are also presented and compared with that of the hybrid method. + +In the following, computation steps of both the steepest descent method and Newton's method are described. \ No newline at end of file diff --git a/samples/texts/5122728/page_37.md b/samples/texts/5122728/page_37.md new file mode 100644 index 0000000000000000000000000000000000000000..293d6e63f6515a91d1e4c41ae4302eb50361d046 --- /dev/null +++ b/samples/texts/5122728/page_37.md @@ -0,0 +1,29 @@ +# DS Based on the Steepest Descent Method + +Based on the steepest descent method, we do first order approximation to decide +the direction to proceed. Therefore, it is assumed that + +$$d = -\nabla f(\mathbf{p}_S)^T, \quad (2.23)$$ + +where $\nabla f(\mathbf{p}_S)$ is the gradient of the function $f$ evaluated at $\mathbf{p}_S$. We have + +$$\nabla f(\mathbf{p}_S) = \left[ \frac{\partial f}{\partial \mathbf{p}_{S_1}} \quad \frac{\partial f}{\partial \mathbf{p}_{S_2}} \quad \dots \quad \frac{\partial f}{\partial \mathbf{p}_{S_n}} \right], \quad (2.24)$$ + +where $\mathbf{p}_{S_i}$ is the $i_{th}$ element and $n$ is the number of elements in the vector $\mathbf{p}_S$. For obtaining the $i_{th}$ element of the gradient vector $\nabla f(\mathbf{p}_S)$, first we need to perturb the $i_{th}$ element of $\mathbf{p}_S$ in forward and backward while other elements are set to be fixed, i.e., + +$$\mathbf{p}_{S,b} := [\mathbf{p}_{S_1} \ \mathbf{p}_{S_2} \ \dots \ \mathbf{p}_{S_i} - h \ \dots \ \mathbf{p}_{S_n}],$$ + +$$\mathbf{p}_{S,f} := [\mathbf{p}_{S_1} \ \mathbf{p}_{S_2} \ \dots \ \mathbf{p}_{S_i} + h \ \dots \ \mathbf{p}_{S_n}], \quad (2.25)$$ + +where indices $b$ and $f$ stand for backward step and forward step around $\mathbf{p}_S$, respectively. Then we compute + +$$\frac{\partial f}{\partial \mathbf{p}_{S_i}} = \frac{f(\mathbf{p}_{S,f}) - f(\mathbf{p}_{S,b})}{2h}, \quad (2.26)$$ + +by solving the minimization problem (2.16) at given $\mathbf{p}_{S,f}$ and $\mathbf{p}_{S,b}$. We iterate above computations for every element of $\mathbf{p}_S$ to obtain all the elements of $\nabla f(\mathbf{p}_S)$. + +# DS Based on Newton's Method + +As it is explained before, Newton's method utilizes second order approximation +for setting the direction, $d$. Based on this method it is assumed that + +$$d = -\mathbf{H}(\mathbf{p}_S)^{-1} \nabla f(\mathbf{p}_S)^T. \quad (2.27)$$ \ No newline at end of file diff --git a/samples/texts/5122728/page_38.md b/samples/texts/5122728/page_38.md new file mode 100644 index 0000000000000000000000000000000000000000..4df4b6c842cffca0efaab8c47851fd3280fca803 --- /dev/null +++ b/samples/texts/5122728/page_38.md @@ -0,0 +1,25 @@ +where $\nabla f(\mathbf{p}_S)$ is the gradient of the function $f$ and computed same as what is shown above. Also, $\mathbf{H}(\mathbf{p}_S)$ is the Hessian of the function $f$ evaluated at $\mathbf{p}_S$. Each element of $\mathbf{H}(\mathbf{p}_S)$ is expressed by + +$$ \mathbf{H}_{ij}(\mathbf{p}_S) = \frac{\partial^2 f}{\partial \mathbf{p}_{S_i} \partial \mathbf{p}_{S_j}}, \quad i,j = 1, \dots, n, \qquad (2.28) $$ + +where $\mathbf{p}_{S_i}$ is the $i_{th}$ element of $\mathbf{p}_S$ and $n$ is the number of elements in the vector $\mathbf{p}_S$. + +For computing $\mathbf{H}_{ij}(\mathbf{p}_S)$, we need to perturb the first-order partial derivative of $f$. + +First assume + +$$ \begin{align*} \mathbf{p}_{S,bb} &:= [\mathbf{p}_{S_1} \ \mathbf{p}_{S_2} \ \dots \ \mathbf{p}_{S_i} - h \ \dots \ \mathbf{p}_{S_j} - h \ \dots \ \mathbf{p}_{S_n}], \\ \mathbf{p}_{S,bf} &:= [\mathbf{p}_{S_1} \ \mathbf{p}_{S_2} \ \dots \ \mathbf{p}_{S_i} - h \ \dots \ \mathbf{p}_{S_j} + h \ \dots \ \mathbf{p}_{S_n}], \\ \mathbf{p}_{S,fb} &:= [\mathbf{p}_{S_1} \ \mathbf{p}_{S_2} \ \dots \ \mathbf{p}_{S_i} + h \ \dots \ \mathbf{p}_{S_j} - h \ \dots \ \mathbf{p}_{S_n}], \\ \mathbf{p}_{S,ff} &:= [\mathbf{p}_{S_1} \ \mathbf{p}_{S_2} \ \dots \ \mathbf{p}_{S_i} + h \ \dots \ \mathbf{p}_{S_j} + h \ \dots \ \mathbf{p}_{S_n}]. \end{align*} $$ + +Then we compute + +$$ \frac{\partial^2 f}{\partial \boldsymbol{p}_{S_i} \partial \boldsymbol{p}_{S_j}} = \frac{\left( \frac{f(\boldsymbol{p}_{S,ff}) - f(\boldsymbol{p}_{S,bf})}{2h} \right) - \left( \frac{f(\boldsymbol{p}_{S,bf}) - f(\boldsymbol{p}_{S,bb})}{2h} \right)}{2h}, \qquad (2.29) $$ + +by solving the minimization problem (2.16) at given $\mathbf{p}_{S,ff}$, $\mathbf{p}_{S,bf}$, $\mathbf{p}_{S,bf}$, and $\mathbf{p}_{S,bb}$. + +We need to iterate above computations for obtaining every element of $\mathbf{H}(\mathbf{p}_S)$. + +### 2.3.3 Line Search + +There are different methods for doing line search, such as the bisection technique and Armijo's method [26]. However, applying these methods on this particular problem brings about additional charge to total computational time of the algorithm. + +In other words, although searching along the previously selected direction to find the best point on the line is reasonable at the first glance, it requires solving the \ No newline at end of file diff --git a/samples/texts/5122728/page_39.md b/samples/texts/5122728/page_39.md new file mode 100644 index 0000000000000000000000000000000000000000..8fffc86ef2f60a01c11ef19ec9cbd49cf032764d --- /dev/null +++ b/samples/texts/5122728/page_39.md @@ -0,0 +1,45 @@ +minimization problem 2.16 numerously. Therefore, using a technique with less com- +putation for the line search is beneficial. For that matter, an alternative algorithm +is proposed to set the step size $\epsilon$ as described below: + +1. Initialize + +$$ +\epsilon := \bar{\epsilon}, \tag{2.30} +$$ + +where $\bar{\epsilon}$ is a sufficiently large positive pre-specified value, so that sufficiently +large step size to proceed is guaranteed in the first iteration. + +2. Compute + +$$ +\delta \boldsymbol{p}_S := \epsilon \cdot d, +\quad (2.31) +$$ + +using $d$ obtained by the DS done before. Then set a candidate for the new +switching surface vector: + +$$ +\boldsymbol{p}_{S,cand} := \boldsymbol{p}_{S,old} + \delta \boldsymbol{p}_{S}. \tag{2.32} +$$ + +Next, Compute + +$$ +\gamma_{cand} := f(\mathbf{p}_{S,cand}), \tag{2.33} +$$ + +3. Test whether the $\gamma$-value has sufficiently decreased by comparing $\gamma_{cand}$ with $\gamma_{old}$. Assume $\hat{\delta}$ and $\hat{\epsilon}$ are sufficiently small positive values. + +• If $\frac{\gamma_{old}-\gamma_{cand}}{\gamma_{old}} < \hat{\delta}$ or $\epsilon \le \hat{\epsilon}$, then terminate the line search algorithm. $\epsilon$ is the desired step size. + +• Otherwise, choose a smaller $\epsilon$-value by setting + +$$ +\epsilon := \frac{\epsilon}{2}, +\quad (2.34) +$$ + +and go to step 2. \ No newline at end of file diff --git a/samples/texts/5122728/page_4.md b/samples/texts/5122728/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..46fa7075082fdc697e9cf086997bd8457571b019 --- /dev/null +++ b/samples/texts/5122728/page_4.md @@ -0,0 +1,11 @@ +Preface + +This thesis is an original intellectual property of the author, Moein Javadian, +working under the supervision of Dr. Ryozo Nagamune. This work, which presents +a numerical optimization algorithm to switching surface design for switching linear +parameter-varying control, has been completed in the Control Engineering Labora- +tory at the University of British Columbia. + +A version of results in Chapters 2 and 3 has been accepted for publication in The +2014 American Control Conference. Results in Chapter 4 and 5 will be submitted +for publication. \ No newline at end of file diff --git a/samples/texts/5122728/page_40.md b/samples/texts/5122728/page_40.md new file mode 100644 index 0000000000000000000000000000000000000000..8eb0ad2e4e129a5a33e6a0ef56c0b4743ca12f90 --- /dev/null +++ b/samples/texts/5122728/page_40.md @@ -0,0 +1,5 @@ +In other words, we generally expect that $\gamma$-value will decrease because of the direction that has been chosen. Therefore, if the sufficient decrease in $\gamma$-value, i.e., $\delta$, is not the case, the step size is too large that the first-order approximation is not accurate enough. That is why we need to choose smaller $\epsilon$ to make smaller step size. Also, if the $\epsilon$-value, which has been initialized with an adequately large value, has become too small, without resulting in sufficient decrease in the $\gamma$-value in between, the algorithm is terminated as well. + +## 2.4 Summary + +The numerical algorithm was developed and explained in this chapter. Formula tion of an LPV plant, as well as a switching LPV controller with hysteresis switching logic was first introduced. It was shown how the plant variation region could be di vided into a set of overlapping subregions by means of a set of switching surfaces. The switching LPV controller design problem with fixed switching surfaces was then reviewed. The closed-loop $L_2$-gain performance of the system was set as a cost function to be minimized in this problem. The resulting minimizers would be the switching LPV controller parameters. Afterwards, optimization of the switching surfaces was formulated as optimization of centres and widths of the overlapping regions. This optimization problem, which is non-convex in general, was rewritten in the form of minimization of a cost function, the explicit form of which is not avail able. However, obtaining the function at each point, i.e., a specific set of switching surfaces, is possible by solving a convex optimization problem. This feature was widely used in the proposed algorithm to numerically minimize the cost function. \ No newline at end of file diff --git a/samples/texts/5122728/page_41.md b/samples/texts/5122728/page_41.md new file mode 100644 index 0000000000000000000000000000000000000000..f78bc888eb16a0b1a1db48153967718f7168b6e3 --- /dev/null +++ b/samples/texts/5122728/page_41.md @@ -0,0 +1,3 @@ +Next, the main structure of the optimization algorithm was provided. This algorithm is iterative and descent in which a decrease in the cost function in each iteration is guaranteed. However, a local minimum point is obtained depending on the initial condition, and finding the global minimum point is not guaranteed. Each iteration consists of two steps: a direction selection (DS) step and a line search step. + +In DS step, the orientation of proceedings to the next point in the variables space is selected by means of *the hybrid method* which is a combination of *the steepest descent method* and *Newton's method*. The idea behind *the hybrid method* is to use the relatively better accuracy of *Newton's method* while preserving a guaranteed descent direction obtained by *the steepest descent method*. The first and second order approximations of the cost function corresponding respectively to *the steepest descent method* and *Newton's method* are computed numerically by solving a series of convex optimization problems. In the line search step, the size of the proceeding along the proposed direction in the DS step is determined using an iterative algorithm. These two steps are repeated until sufficiently large decrease in the cost function is no more the case, resulting in termination of the optimization algorithm. \ No newline at end of file diff --git a/samples/texts/5122728/page_42.md b/samples/texts/5122728/page_42.md new file mode 100644 index 0000000000000000000000000000000000000000..e134ac430d9a06ec464c52a3205c3b0ef9c896c7 --- /dev/null +++ b/samples/texts/5122728/page_42.md @@ -0,0 +1,5 @@ +# Chapter 3 + +## A Numerical Example + +The proposed algorithm in Chapter 2 is applied to design a switching LPV controller with optimized switching surfaces for a mass-spring-damper system. It will be also shown how computationally it costs to design such a controller via the descent algorithm rather than a *full search method*, described later. Moreover, a time domain response of the closed-loop having the optimized switching LPV controller is compared with that having a switching LPV controller with a trivial partition. \ No newline at end of file diff --git a/samples/texts/5122728/page_43.md b/samples/texts/5122728/page_43.md new file mode 100644 index 0000000000000000000000000000000000000000..395894688f15bb5d9cb3e960ce95c3d217f2db2d --- /dev/null +++ b/samples/texts/5122728/page_43.md @@ -0,0 +1,15 @@ +## 3.1 Problem Statement + +### 3.1.1 Plant Model + +Consider a mass-spring-damper system shown in Fig. 3.1 and represented in a state-space form as + +$$ P: \begin{cases} \dot{x}(t) = \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} x(t) + \underbrace{\begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix}}_{B_p} u(t), \\ y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix} x(t). \end{cases} \quad (3.1) $$ + +Here, scalars $m$, $k$, and $b$ are respectively mass, spring, and damping coefficients, $x$ is a state vector consisting of the position and the velocity of the mass, $u$ is the input force, and $y$ is the position measurement. + +### 3.1.2 Plant Parameter Variation + +The dynamics of this system is subjected to change due to change in the mass and stiffness. To cover different scenarios, we consider two different sources of plant variation. It is assumed that the mass changes due to product variation, meaning + +Figure 3.1: Mass-spring-damper schematic. \ No newline at end of file diff --git a/samples/texts/5122728/page_44.md b/samples/texts/5122728/page_44.md new file mode 100644 index 0000000000000000000000000000000000000000..299f0c276bc1ae30cde481dfd90cd0425fc3ee90 --- /dev/null +++ b/samples/texts/5122728/page_44.md @@ -0,0 +1,41 @@ +that it remains constant for each product during the operation of the system. The +range of product variation of the mass is set as follows: + +$$ +m \in [20, 100]. \tag{3.2} +$$ + +Besides, it is assumed that the spring is nonlinear, and the stiffness-value depends on the length of the spring, so it varies while the system operates. As explained in [41], the force-displacement relations for a hard spring are given by + +$$ +F = c_1 y + c_2 y^3, \quad c_1, c_2 > 0. \tag{3.3} +$$ + +Then, the stiffness-displacement equation is obtained by dividing above equation by $y$ as follows: + +$$ +k = c_1 + c_2 y^2. \tag{3.4} +$$ + +In this example $c_1$ and $c_2$-values are given by + +$$ +c_1 = 10, \quad c_2 = 50. \tag{3.5} +$$ + +The displacement $y$ is assumed to be within the range of $[-1.34\ 1.34]$. Therefore, +the allowable range of the stiffness is obtained as + +$$ +k \in [10, 100], \tag{3.6} +$$ + +where $k = 10$ and $k = 100$ correspond to $y = 0$ and $y = \pm 1.34$, respectively. Also, +the maximum stiffness variation rate is assumed to be 100 standard unit per second +either increasing or decreasing. Moreover, the damping coefficient $b$ is assumed to +be fixed and equal to 5. Having defined the plant parameters, we introduce a gain- +scheduling parameter $\theta$ which is a function of the varying parameters and given by + +$$ +\theta = \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix} = \begin{bmatrix} 1/m \\ k \end{bmatrix}, \qquad (3.7) +$$ \ No newline at end of file diff --git a/samples/texts/5122728/page_45.md b/samples/texts/5122728/page_45.md new file mode 100644 index 0000000000000000000000000000000000000000..f52b75de8e0f95a731e38ca6fa68126254dc2c50 --- /dev/null +++ b/samples/texts/5122728/page_45.md @@ -0,0 +1,22 @@ +where $\theta_1$ is equal to the inverse of the mass for controller reconstruction purposes since the mass appears in the denominator in (3.1). The variation range and variation's rate range can be expressed by + +$$ +\begin{align} +\Theta &:= \{\theta \in \mathbb{R}^2 : \theta_1 \in [1/100, 1/20], \theta_2 \in [10, 100]\}, \tag{3.8} \\ +\Omega &:= \{\omega \in \mathbb{R}^2 : \omega_1 = 0, \omega_2 \in [-100, 100]\}. +\end{align} +$$ + +## 3.2 Control Objectives + +### 3.2.1 Weighting Functions + +For the system $P$, we form a feedback structure shown in Fig. 3.2, where $r$ is the reference signal, $e$, $e_w$, $u$, and $u_w$ are respectively the error, the weighted error, the control input, and the weighted control input, $W_e$ and $W_u$ are weighting functions, and $K(\theta)$ is a switching LPV controller. In this example, we assume that $W_u$ is a scalar constant gain, while $W_e$ is a dynamical system with a state-space realization as + +$$ W_e: \begin{cases} \dot{x}_e(t) = W_{eA}x_e(t) + W_{eB}e(t), \\ e_w(t) = W_{eC}x_e(t) + W_{eD}e(t), \end{cases} \quad (3.9) $$ + +where the vector $x_e$ is the state vector of the system $W_e$, and matrices in (3.9) have compatible dimensions. + +### 3.2.2 A Generalized Plant + +The controller $K(\theta)$ is designed such that the $L_2$-gain from $r$ to $e_w$ and $u_w$ is to be minimized. This $L_2$-gain minimization corresponds to reference tracking and control input energy minimization. For this purpose, we obtain the input-output \ No newline at end of file diff --git a/samples/texts/5122728/page_46.md b/samples/texts/5122728/page_46.md new file mode 100644 index 0000000000000000000000000000000000000000..a54511a6d09c6c39ef12f0dde65207a0ddef59f0 --- /dev/null +++ b/samples/texts/5122728/page_46.md @@ -0,0 +1,13 @@ +Figure 3.2: A feedback structure for controller design. + +relation for the generalized plant $G(\theta)$ in Fig. 2.1 from Fig. 3.2 as + +$$ \begin{bmatrix} \begin{bmatrix} \dot{x} \\ \dot{x}_e \\ e_w \\ u_w \\ e \end{bmatrix} \\ \begin{bmatrix} x \\ x_e \\ r \\ u \end{bmatrix} \end{bmatrix} = G(\theta) \begin{bmatrix} \begin{bmatrix} e \\ u \end{bmatrix} \\ \begin{bmatrix} 0 & 0 & 0 & B_p \\ W_{eB}C_p & W_{eA} & W_{eB} & 0 \\ -W_{eD}C_p & W_{eC} & W_{eD} & 0 \\ 0 & 0 & 0 & W_u \\ -C_p & 0 & 1 & 0 \end{bmatrix} \end{bmatrix}, \quad (3.10) $$ + +where the matrix-valued function $G(\theta)$ is given by + +$$ G(\theta) = \begin{bmatrix} A_p & 0 & 0 & B_p \\ W_{eB}C_p & W_{eA} & W_{eB} & 0 \\ -W_{eD}C_p & W_{eC} & W_{eD} & 0 \\ 0 & 0 & 0 & W_u \\ -C_p & 0 & 1 & 0 \end{bmatrix}. \quad (3.11) $$ + +In addition, we select the weighting functions as + +$$ \begin{bmatrix} W_{eA} & W_{eB} \\ W_{eC} & W_{eD} \end{bmatrix} = \begin{bmatrix} -0.0499 & 3.2 \\ 3.12 & 0.05 \end{bmatrix}, \quad W_u = 0.01. \quad (3.12) $$ \ No newline at end of file diff --git a/samples/texts/5122728/page_47.md b/samples/texts/5122728/page_47.md new file mode 100644 index 0000000000000000000000000000000000000000..84de225df9e2c9681ecc7190c84a7de775439a2c --- /dev/null +++ b/samples/texts/5122728/page_47.md @@ -0,0 +1,9 @@ +## 3.3 Controller Design + +In this section, control design has been done for two cases to properly address different aspects of the numerical algorithm. In the first case, 2 switching surface variables are assumed to be designed. Different methods used to find the optimized partition are applied, and simulation results of implementing switching LPV controllers with optimized switching surfaces and heuristic ones are compared to each other. For the second case, applicability of the proposed algorithm to the *n*-variables case is investigated. The relations between the number of switching surface variables and both the performance level and computational time of convergence are discussed. + +### 3.3.1 Case One: 2 Switching Surface Variables + +For the LPV generalized plant $G(\theta)$, we design a switching LPV controller with $N_1 = 2$ and $N_2 = 2$, i.e., four subregions, including the center lines of the two overlapping regions $c_{1,1}$ and $c_{2,1}$. For simplicity, the width vector $w$ of the overlapping regions are fixed to be 0.1 times the variation range of each gain-scheduling parameter. The results of applying different methods to obtain the switching LPV controller with optimized switching surfaces are explained below and summarized in Table 3.1. + +First, we applied the full-search method to find the optimal $p_S = (c_{1,1}, c_{2,1})$ by gridding the $p_S$-space. To do the full search, we designed the switching LPV controller that minimizes $\gamma$-value for each gridded point by solving (2.16). The result of the full-search method is shown in Fig. 3.3. As it can be observed in this figure, the minimum performance level $\gamma^* = 10.06$ is achieved at $p_S^* = (0.038, 23)$. Although \ No newline at end of file diff --git a/samples/texts/5122728/page_48.md b/samples/texts/5122728/page_48.md new file mode 100644 index 0000000000000000000000000000000000000000..572605903fabc5ad2f8cc0086a674e058215fee6 --- /dev/null +++ b/samples/texts/5122728/page_48.md @@ -0,0 +1,7 @@ +Figure 3.3: Functionality of $\gamma$ with respect to $p_S = (c_{1,1}, c_{2,1})$ obtained by the full-search method. + +it is guaranteed that this method finds the optimal $p_S$, up to the resolution of +the gridding, its main disadvantage is that the computational complexity increases +exponentially as the dimension of $\theta$ does. + +Secondly, we applied the descent algorithm proposed in Section 2.3. Having employed *the hybrid method* for the DS step and having started at $p_{s,init} = (0.03, 55)$, the middle point of the plant variation region, the algorithm converged to the same pair of $(\gamma^*, p_s^*)$ with much less computational time in comparison to *the full-search method*. The advantages of this method will be more remarkable if plant has two varying parameters and more divisions are desired in each axis. Although depending on the initial point, the algorithm could have converged to the other local minimum point, one could try different initial points to find the global minimum while still spending much less time compared to *the full search method*. \ No newline at end of file diff --git a/samples/texts/5122728/page_49.md b/samples/texts/5122728/page_49.md new file mode 100644 index 0000000000000000000000000000000000000000..4e663d09b41f15f5b4b23b451e2571165411627a --- /dev/null +++ b/samples/texts/5122728/page_49.md @@ -0,0 +1,7 @@ +Table 3.1: Results of applying different numerical methods to optimize the switching surface + +
MethodInitial PointFinal PointComp. Time (min)
Full SearchN/Aγ* = 10.06
ps* = (0.038, 23)		228
Descent Algorithm
(Hybrid)		γinit = 10.24
ps,init = (0.03, 55)		γ* = 10.06
ps* = (0.038, 23)		9
Descent Algorithm
(Steepest Descent)		γinit = 10.24
ps,init = (0.03, 55)		γ* = 10.06
ps* = (0.038, 23)		15
Descent Algorithm
(Newton's)		γinit = 10.24
ps,init = (0.03, 55)		N/AN/A
+ +Thirdly, both *the steepest descent method* and *Newton's method* were used singly for the DS step, instead of *the hybrid method*. It was seen that having the same initial point, applying *Newton's method* did not lead to convergence to any local minimum point, since Hessian matrix is negative definite around the initial point trying to maximize the cost function. On the other hand, applying *the steepest descent method* singly for the DS step led to convergence to $(\gamma^*, p_s^*)$. However, *the steepest descent method* utilizing first order approximation took one and half times longer than *the hybrid method* to converge, since the latter one employs second order approximation at some iterations in which Hessian matrix is positive definite. + +In order to meaningfully compare the computational time of applying different methods, computations were done with the same number of gridded points of the plant variation region, equal to 2500, as well as the same 3.20 GHz CPU computer with 32.0 GB RAM and MATLAB version 8.1.0.604 (R2013a). \ No newline at end of file diff --git a/samples/texts/5122728/page_5.md b/samples/texts/5122728/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..2d030f1f2bb66eba5d060543086ac1e1887b88cf --- /dev/null +++ b/samples/texts/5122728/page_5.md @@ -0,0 +1,3 @@ +# Table of Contents + +
Abstractii
Prefaceiv
Table of Contentsv
List of Tablesix
List of Figuresx
Acknowledgementsxiii
Dedicationxiv
1 Introduction1
1.1 Motivation1
1.2 Literature Review2
1.2.1 LPV Model2
1.2.2 LPV-Based Control2
1.2.3 Switching LPV Control4
1.2.4 Smooth Switching LPV Control.6
1.2.5 Switching LPV Control with Optimized Switching Surfaces7
\ No newline at end of file diff --git a/samples/texts/5122728/page_50.md b/samples/texts/5122728/page_50.md new file mode 100644 index 0000000000000000000000000000000000000000..6e395504578e1571ebaec71495c7c6618b4924ca --- /dev/null +++ b/samples/texts/5122728/page_50.md @@ -0,0 +1,9 @@ +Simulation Results + +It is important to note that improvement of the worst case $L_2$-gain, $\gamma$, does not necessarily correlate to improvement of the time domain simulation results of a specific plant parameter trajectory. However, advancement of $\gamma$ could increase the possibility of advancement in time domain responses for trajectories that are often the case in each particular example. Therefore, it is inherent to the controller design procedure to go back and forth and readjust the switching surfaces as well as weighting functions, so that improvement in time domain responses for required trajectories is guaranteed. + +Nevertheless, for the system explained above, improvement of the step response for different mass-values in the product variation range specified in (3.2) was observed. In this section, the step responses of two switching LPV controllers for just a sample mass-value are compared to each other in Fig. 3.4. First controller is the optimal one, the switching surfaces of which are placed at $p_S^* = (0.038, 23)$. The other one is designed with a heuristic partition where the switching surfaces are placed at the middle of each plant variation region's axis, $p_S = (0.03, 55)$. As can be seen in Fig. 3.4, the optimized controller generates step response with smaller overshoot and less oscillations. + +### 3.3.2 Case Two: *n* Switching Surface Variables + +In the second case study, more than two number of switching surface variables are required to be designed. This work was done by making different assumptions on the number of divisions on each axis of the plant variation region. These assumptions are summarized in Table 3.2. As seen in this table, the widths of the overlapping \ No newline at end of file diff --git a/samples/texts/5122728/page_51.md b/samples/texts/5122728/page_51.md new file mode 100644 index 0000000000000000000000000000000000000000..39d63ebf1402ac050614e43b53399dd421761935 --- /dev/null +++ b/samples/texts/5122728/page_51.md @@ -0,0 +1,10 @@ +Figure 3.4: Comparison of step responses between the optimal controller and a heuristic one with $m = 60$. + +regions have been assumed to be fixed for simplicity. However, varying width could +have also been assumed. For the case that number of variables is eight, a schematic +layout of the plant variation region with assumed switching surfaces are shown in +Fig. 3.5. + +Table 3.2: Assumptions on the switching surfaces + +
N1N2WidthNumber of Variables
22Fixed2
23Fixed3
33Fixed4
34Fixed5
44Fixed6
54Fixed7
55Fixed8
\ No newline at end of file diff --git a/samples/texts/5122728/page_52.md b/samples/texts/5122728/page_52.md new file mode 100644 index 0000000000000000000000000000000000000000..93f769908956e50df5a7ca94b88a41831cdd7741 --- /dev/null +++ b/samples/texts/5122728/page_52.md @@ -0,0 +1,5 @@ +Figure 3.5: Plant variation region with 5 divisions on each axis. + +A switching LPV controller with optimized switching surfaces is designed for each set of assumptions in Table 3.2 by applying the descent algorithm. Employing both *the hybrid method* and *steepest descent method* have resulted in the same final switching surfaces $p_s^*$ and performance level $\gamma^*$. However, applying *Newton's method* singly for the DS step of the algorithm does not lead to convergence to any local minimum point. The initial and final performance levels are shown in Fig. 3.6. It can be seen that $\gamma$-value decreases when the number of switching surface variables increases—an observation which coincides with our expectations. + +Fig. 3.7 shows the time of convergence for each case. It can be seen that *the steepest descent method* results in smaller computational time for high number of variables than *the hybrid method*, different from two-variables case. The computational time of applying *the hybrid method* increases dramatically by increasing the number of variables since Hessian matrix is not positive definite and useless in most \ No newline at end of file diff --git a/samples/texts/5122728/page_53.md b/samples/texts/5122728/page_53.md new file mode 100644 index 0000000000000000000000000000000000000000..daeebafce6f736491ddb96495ee269200db231dd --- /dev/null +++ b/samples/texts/5122728/page_53.md @@ -0,0 +1,5 @@ +Figure 3.6: The initial and final performance level for different number of switching surface variables while applying the descent algorithm. + +of iterations. Therefore, computing second order approximation of the function does not help the algorithm to speed up converging, while it charges the algorithm in terms of computational time. + +Convergence computational time of applying *the steepest descent method* has been replotted singly in Fig. 3.8, so that the trend can be seen more clearly. Looking at these three figures demonstrates that increasing number of switching surface variables can burden the design process computationally, and it should be evaluated that if such computation is beneficial. In other words, it is important to maintain a balance between the computational time and improvement in the performance level by choosing an appropriate number of switching surface variables which can be an interesting topic for future research in this field. \ No newline at end of file diff --git a/samples/texts/5122728/page_54.md b/samples/texts/5122728/page_54.md new file mode 100644 index 0000000000000000000000000000000000000000..41f9b2a7f547f01a96de658457f2182077965409 --- /dev/null +++ b/samples/texts/5122728/page_54.md @@ -0,0 +1,3 @@ +Figure 3.7: Computational time of convergence of the descent algorithm for different number of switching surface variables, having used the *hybrid method* and the *steepest descent method* for the DS step. + +Figure 3.8: Computational time of convergence of the descent algorithm for different number of switching surface variables, having used the *steepest descent method* for the DS step. \ No newline at end of file diff --git a/samples/texts/5122728/page_55.md b/samples/texts/5122728/page_55.md new file mode 100644 index 0000000000000000000000000000000000000000..884a360eb4256b01d7f64758fd4e473f54562345 --- /dev/null +++ b/samples/texts/5122728/page_55.md @@ -0,0 +1,5 @@ +## 3.4 Summary + +The proposed numerical algorithm in this thesis was then applied to a numerical example in this chapter. A simple mass-spring-damper system was used as a plant for which controllers were designed. It was assumed that the plant's mass is subject to product variation, and the stiffness-value of the spring changes during the operation of the system as a result of nonlinearity in the spring. These two parameters form a plant variation region with specified variation range and variation rate range. This problem was assumed as reference tracking one in which the position of the mass is required to track a reference signal by means of applying a force generated by a controller. + +For controller design, two cases were studied. First, two divisions on each axis of the plant variation region were assumed, and the widths of the overlapping regions were set to be fixed. Therefore, 2 switching surface variables, i.e., centres of the overlapping regions, are sought to be designed. Results of applying different methods to find the switching LPV controller with optimized switching surfaces were obtained in this chapter. It was shown that applying the proposed algorithm in Chapter 2 is superior than using *the full search method* in terms of computational cost. Moreover, it was seen that using *Newton's method* singly for the DS step of the descent algorithm does not lead to convergence to a local minimum point since Hessian matrix of the cost function at some iterations are not positive definite. However, the algorithm perfectly worked while *the hybrid method* or *the steepest descent method* were used for the DS step of the descent algorithm. Afterwards, closed-loop performance of the switching LPV controller with optimized switching surfaces and that with heuristic switching surfaces were juxtaposed to each other. \ No newline at end of file diff --git a/samples/texts/5122728/page_56.md b/samples/texts/5122728/page_56.md new file mode 100644 index 0000000000000000000000000000000000000000..adcc155465c4f51b68c0fde49ae5d3e00c7c0f7b --- /dev/null +++ b/samples/texts/5122728/page_56.md @@ -0,0 +1,3 @@ +Although advancement in bounds of the closed-loop $L_2$-gain performance is only guaranteed, it was shown time-domain performance of the system with a specific plant parameters trajectory is more likely to be improved if optimized switching surfaces are employed. + +In the second case, the control design problem with more than 2 switching surface variables were studied by dividing the plant variation region into more than 4 subregions. Final performance levels having different number of variables were obtained and compared to each other. Also, computational time of convergence of the descent algorithm for the different cases were plotted. It was seen that in high number of variables, the computational time increases dramatically, while the final performance level is not remarkably decreased. Therefore, cost benefit analysis suggests that lots of divisions on the axes of the plant variation region is not always favourable. \ No newline at end of file diff --git a/samples/texts/5122728/page_57.md b/samples/texts/5122728/page_57.md new file mode 100644 index 0000000000000000000000000000000000000000..4279f955e2b0d4e1b0bf724e3d9f1f45cce7ef42 --- /dev/null +++ b/samples/texts/5122728/page_57.md @@ -0,0 +1,9 @@ +# Chapter 4 + +## Application in +Magnetically-Actuated Optical +Image Stabilizer + +Image blurriness as a result of involuntary hand movement has always been an issue for the users while taking photos. Especially at low-light conditions, this phenomenon is intensified because the shutter speed is slow. In photography, the effective length of time when a camera's shutter is open is called shutter speed. In order to avoid this annoying blurriness and stabilize the image, there are different methods widely used in industry which are categorized into software-based and hardware-based technologies. + +Software-based image stabilization methods are mainly applied for lower-end cameras [19], while hardware-based technologies are used for higher-end ones. Although hardware-based mechanisms are more effective in stabilizing the images, most mo- \ No newline at end of file diff --git a/samples/texts/5122728/page_58.md b/samples/texts/5122728/page_58.md new file mode 100644 index 0000000000000000000000000000000000000000..e2afceadaee003b55bb1c1c03589227fb1648e31 --- /dev/null +++ b/samples/texts/5122728/page_58.md @@ -0,0 +1,9 @@ +bile cameras are not equipped with them because of size constraints. However, a hardware-based method is designed recently for miniaturized applications in which a lens-shifting mechanism is used to stabilize the image [37]. The idea is that when the whole camera including the micro lens moves due to hand shake, a controller system tries to compensate for the hand movement and move back the lens to its right position in order to restore the image in a good quality with no blurriness. This mechanism utilizing a magnetic actuator is used for not only compensation for undesired vibrations but also auto focusing purposes. + +This device called optical image stabilizer (OIS) is built in micro scale. Production variation in micro-scale devices is considerable and causes huge effect on the dynamics of the system from one product to another. In micro-scale OISs, the change in dynamics of the plant is caused by variation in the equivalent mass and stiffness of the system as a result of imprecise cutting. This variation in dynamics is considered in this chapter for designing a switching LPV controller to stabilize the image. + +## 4.1 Problem Statement + +### 4.1.1 Physical System + +The physical system described in [37] consists of a sufficiently flexible platform at the center of which a micro lens is placed. See Fig. 4.1. Four magnets are attached to the platform. After some air gap, there are four shielded electromagnetic coils under each magnet. Depending on the currents in the electromagnetic coils and depending on the air gap between each magnet and coil, four vertical forces will be applied to the whole platform. Due to the flexibility and elasticity of the platform, \ No newline at end of file diff --git a/samples/texts/5122728/page_59.md b/samples/texts/5122728/page_59.md new file mode 100644 index 0000000000000000000000000000000000000000..a914c615fda2fb7aaaf9ebd4c01796674c1d85a4 --- /dev/null +++ b/samples/texts/5122728/page_59.md @@ -0,0 +1,5 @@ +it will work like a spring at each side connected to the chassis of the camera. The superposition of the forces applied to the platform will generate some movements, namely vertical motion along the z axis and rotational motions about the x and the y axes. More technically, the plant including a platform, a lens, and magnets has outputs of $(z, \theta_x, \theta_y)$ measured using strain gauge sensors, and inputs of $(F, T_x, T_y)$. The set of the output defines the position and attitude of the whole platform with three degrees of freedom in the space, and the set of the input consists of the net vertical force and torques around the x and the y axes. These torques are generated because of the eccentricity of four forces applied by interaction of four pairs of coil and magnet. + +Figure 4.1: Physical layout of the lens-shifting image stabilizer [37]. + +The inputs to the plant are the outputs of an actuator consisting of coils, magnets, and some electric circuits. The inputs to the actuator are four currents going through the coils, and the air gap between each pair of coil and magnet. This air gap can be determined by vertical and angular positions of the platform which are the outputs of the plant, so the air gap can not be manipulated directly. Therefore, the four currents are the only inputs to the actuator which can be determined by a controller. \ No newline at end of file diff --git a/samples/texts/5122728/page_6.md b/samples/texts/5122728/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..5ebc06c0df6a4e59a52f829ba8a29d2d846e5c2f --- /dev/null +++ b/samples/texts/5122728/page_6.md @@ -0,0 +1,119 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
1.3Research Objectives7
1.4Organization of Thesis9
2An Optimization Approach to Switching LPV Control Design10
2.1Problem Formulation.11
2.1.1Description of an LPV Plant11
2.1.2Description of a Switching LPV Controller with Hysteresis
Switching		12
2.1.3Switching Surface Design Problem14
2.2Review of LPV Switching Controller Design16
2.3An Optimization Approach to Switching Surface Design19
2.3.1The Descent Algorithm's Main Structure19
2.3.2Direction Selection Computations.22
2.3.3Line Search24
2.4Summary26
3A Numerical Example.28
3.1Problem Statement.29
3.1.1Plant Model.29
3.1.2Plant Parameter Variation.29
3.2Control Objectives31
3.2.1Weighting Functions31
3.2.2A Generalized Plant31
3.3Controller Design.33
3.3.1Case One: 2 Switching Surface Variables.33
3.3.2Case Two: n Switching Surface Variables36
\ No newline at end of file diff --git a/samples/texts/5122728/page_60.md b/samples/texts/5122728/page_60.md new file mode 100644 index 0000000000000000000000000000000000000000..ec0eece44a256cd7d19436e36b74af0078425c58 --- /dev/null +++ b/samples/texts/5122728/page_60.md @@ -0,0 +1,7 @@ +A typical block diagram of the plant and the actuator can be seen in Fig. 4.2. + +Figure 4.2: Block diagram of the plant and actuator. + +### 4.1.2 Control Problem + +Now that the plant, actuator, and sensor are explained, it is desired to find a controller which inputs such currents to the actuator that the outputs of the plant, i.e., $(z, \theta_x, \theta_y)$, track a reference signal which is obtained based on the hand movement. First, the entire hand movement, including voluntary and involuntary ones, are measured using some gyroscopes or accelerometers. Then the measured movements are passed through a high pass filter in order to ignore zero frequency disturbance, since we do not want to reject intentional hand movements. In other words, by applying a control system we intend to reject the involuntary hand shakes which have high frequencies. Therefore, we pass a part of the signal which has such frequencies and reject the rest. Afterwards, the oscillatory signal is multiplied by -1, and eventually, the reference signal is built. This multiplication is done because we would like to track negative involuntary hand shake in order to compensate for that. The block diagram of the closed-loop system can be seen in Fig. 4.3. \ No newline at end of file diff --git a/samples/texts/5122728/page_61.md b/samples/texts/5122728/page_61.md new file mode 100644 index 0000000000000000000000000000000000000000..6f056326531d2641f74c4ab2587e9843fbf8e45a --- /dev/null +++ b/samples/texts/5122728/page_61.md @@ -0,0 +1,9 @@ +Figure 4.3: Block diagram of the closed-loop system. + +Now, if the controller enforces the output of the plant to track the obtained reference signal perfectly, the overall movement of the micro lens will be almost zero and the image will be clear and in good quality. Moreover, there is a bound for the currents going through the coils which should be considered in the control design process. Additionally, due to the product variation, the dynamics of the plant changes from one device to another, and designing a single controller for all the devices might not be satisfying. Therefore, an optimized switching LPV controller is designed in this chapter to achieve the design objectives explained above. + +### 4.1.3 Plant Model + +After applying Newton's 2nd law and doing realization, the state space form of the plant dynamics is obtained in [37] which has the form + +$$ P: \begin{cases} \dot{x}_p = A_p x_p + B_p u_F \\ y = C_p x_p + D_p u_F \end{cases}, \quad (4.1) $$ \ No newline at end of file diff --git a/samples/texts/5122728/page_62.md b/samples/texts/5122728/page_62.md new file mode 100644 index 0000000000000000000000000000000000000000..e6b7f2afcad9b8da7bf0e21d4e3097fa2fff74cc --- /dev/null +++ b/samples/texts/5122728/page_62.md @@ -0,0 +1,13 @@ +where + +$$x_p = \begin{bmatrix} z \\ \dot{z} \\ \theta_x \\ \dot{\theta}_x \\ \theta_y \\ \dot{\theta}_y \end{bmatrix}, \quad u_F = \begin{bmatrix} F \\ T_x \\ T_y \end{bmatrix}, \quad y = \begin{bmatrix} z \\ \theta_x \\ \theta_y \end{bmatrix}, \qquad (4.2)$$ + +and + +$$A_p = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ \frac{-k_{eq}}{M_{eq}} & \frac{-b_{eq}}{M_{eq}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{-k_x}{I_x} & \frac{-b_x}{I_x} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & \frac{-k_y}{I_y} & \frac{-b_y}{I_y} \end{bmatrix}, \quad B_p = \begin{bmatrix} 0 & 0 & 0 \\ \frac{1}{M_{eq}} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & \frac{1}{I_x} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{1}{I_y} \end{bmatrix},$$ + +$$C_p = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}, \quad D_p = \mathbf{0} \qquad (4.3)$$ + +In this state space form, $M_{eq}$ is the equivalent mass of the platform, $F$ is the total vertical force, $k_{eq}$ is the equivalent translational stiffness and $b_{eq}$ is the equivalent damping coefficient. Also, $I_x$ and $I_y$ are the mass moments of inertia, $T_x$ and $T_y$ are the equivalent torques, $k_x$ and $k_y$ are the equivalent rotational stiffness, and $b_x$ and $b_y$ are the rotational damping constants about each axis. + +It can be seen that the matrices $A_P$ and $B_P$ are block diagonal which provides us with the situation to deal with each degree of freedom independently in controller \ No newline at end of file diff --git a/samples/texts/5122728/page_63.md b/samples/texts/5122728/page_63.md new file mode 100644 index 0000000000000000000000000000000000000000..9c037278eb99dac0d458e988292db9afbfd761c9 --- /dev/null +++ b/samples/texts/5122728/page_63.md @@ -0,0 +1,30 @@ +design. Therefore, the model can be broken down into three subsystems for con- +troller design purposes, and each controller is designed for each subsystem. This +approach to the problem needs to assume the force vector $u_F$ as the control input to +the plant, while the actual control input for implementation purposes is the currents +going through the coils. To deal with this issue, we need to have the input-output +equation of the actuator. The simplified version of this equation obtained in [37] +can be expressed as + +$$ +\begin{bmatrix} F \\ T_x \\ T_y \end{bmatrix} = M \cdot \begin{bmatrix} i_1 \\ i_2 \\ i_3 \\ i_4 \end{bmatrix}. \qquad (4.4) +$$ + +We have + +$$ +M = \alpha \begin{bmatrix} 1 & 1 & 1 & 1 \\ -R & -R & R & R \\ -R & R & -R & R \end{bmatrix}, \quad (4.5) +$$ + +where $\alpha$ and $R$ are constants shown in Table 4.1. Therefore, the force vector $u_F$ is +considered as the control input to be obtained by designing the controller, and the +actual control input, i.e., the currents to the coils, is computed as follows: + +$$ +\begin{bmatrix} i_1 \\ i_2 \\ i_3 \\ i_4 \end{bmatrix} = M^T (M M^T)^{-1} \begin{bmatrix} F \\ T_x \\ T_y \end{bmatrix}. \quad (4.6) +$$ + +See Fig. 4.4. Note that if we dealt with the currents as the control input in the +control design process, the combination of the plant and the actuator as the new- +defined plant would not have block diagonal $A_p$ matrix and would not be broken +down into three subsystems. \ No newline at end of file diff --git a/samples/texts/5122728/page_64.md b/samples/texts/5122728/page_64.md new file mode 100644 index 0000000000000000000000000000000000000000..8d21701b44af7b9e309bb40ead6c3a4b7fc9dbb7 --- /dev/null +++ b/samples/texts/5122728/page_64.md @@ -0,0 +1,194 @@ +Figure 4.4: Actual control input computation. + +Table 4.1 shows all the nominal values of the parameters of the plant and actuator +which are obtained in [37]. + +Table 4.1: System parameters [37]. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ Parameter + + Value + + Unit +
+ M + + eq + + + 1.49 × 10 + + -4 + + + kg +
+ k + + eq + + + 89.09 + + N.m + + -1 + +
+ b + + eq + + + 2.30 × 10 + + -3 + + + N.m + + -1 + + .s + + -1 + +
+ I + + x + + = I + + y + + + 9.40 × 10 + + -10 + + + kg.m + + 2 + +
+ k + + x + + = k + + y + + + 2.40 × 10 + + -3 + + + N.rad + + -1 + +
+ b + + x + + = b + + y + + + 3.89 × 10 + + -8 + + + N.rad + + -1 + + .s + + -1 + +
+ α + + 0.105 + + N.A + + -1 + +
+ R + + 2.5 × 10 + + -3 + + + m +
+ +In the rest of this chapter, we assume one of the subsystems, i.e., the one which +models rotation of the platform around x-axis, as the plant for which we are going +to design controller. The state space form of this system can be given by + +$$ +P_x : \begin{cases} \dot{x}_{px} = A_{px} x_{px} + B_{px} T_x \\ \theta_x = C_{px} x_{px} + D_{px} T_x \end{cases}, \tag{4.7} +$$ \ No newline at end of file diff --git a/samples/texts/5122728/page_65.md b/samples/texts/5122728/page_65.md new file mode 100644 index 0000000000000000000000000000000000000000..979940004af047ae94647b01aab3ff200e1c9991 --- /dev/null +++ b/samples/texts/5122728/page_65.md @@ -0,0 +1,17 @@ +where + +$$x_{px} = \begin{bmatrix} \theta_x \\ \dot{\theta}_x \end{bmatrix}, \quad (4.8)$$ + +and + +$$A_{px} = \begin{bmatrix} 0 & 1 \\ -\frac{k_x}{I_x} & -\frac{b_x}{I_x} \end{bmatrix}, \quad B_{px} = \begin{bmatrix} 0 \\ \frac{1}{I_x} \end{bmatrix},$$ + +$$C_{px} = \begin{bmatrix} 1 & 0 \end{bmatrix}, \quad D_{px} = 0. \quad (4.9)$$ + +### 4.1.4 Plant Parameter Variation + +As it is explained before, the change in dynamics of the system in (4.7) is caused by product variation which leads to variation in the equivalent mass moments of inertia, $I_x$ and the equivalent rotational stiffness $k_x$ of the system as a result of imprecise cutting. These two parameters, $I_x$ and $k_x$, form a plant variation region in which the bounds of the variation are defined based on manufacturing tolerances. These manufacturing tolerances are assumed to be 75% of the nominal parameter values, $\hat{I}_x$ and $\hat{k}_x$. Moreover, once the product is manufactured, it is assumed that the parameters of the plant are not varying during the operation of the system. The plant variation vector $\theta$ is defined as + +$$\theta = \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix} = \begin{bmatrix} 1/I_x \\ k_x \end{bmatrix}, \quad (4.10)$$ + +where $\theta_1$ is assumed to be equal to the inverse of the plant's moment of inertia due to controller reconstruction purposes. Also, the plant variation region $\Theta$ and \ No newline at end of file diff --git a/samples/texts/5122728/page_66.md b/samples/texts/5122728/page_66.md new file mode 100644 index 0000000000000000000000000000000000000000..4126e91e7e8573a4cf0feb9ae460dd9211b2cbb5 --- /dev/null +++ b/samples/texts/5122728/page_66.md @@ -0,0 +1,25 @@ +variation rate region $\Theta_r$ are expressed as + +$$ \Theta := \left\{ \theta \in \mathbb{R}^2 : \begin{array}{l} \theta_1 \in [0.6079, 4.2553] \, 1/(g.mm^2), \\ \theta_2 \in [0.6000, 4.2000] \, g.mm^2/ms^2 \end{array} \right\}, \quad (4.11) $$ + +$$ \Omega := \{\omega \in \mathbb{R}^2 : \omega_1 = \omega_2 = 0\}. $$ + +## 4.2 Control Objectives + +As mentioned in the previous sections, this is a reference tracking control problem. + +Moreover, there is a constraint on the control input to be applied. To formulate the problem in an appropriate manner so that these objectives are sought, weighting functions and the generalized plant are introduced in this section. + +### 4.2.1 Weighting Functions + +The block diagram of the control problem can be seen in Fig. 4.5 where $r_x$ is the reference signal for rotation tracking around x-axis, $e_x$, $\hat{e}_x$, $T_x$, and $\hat{T}_x$ are respectively the error, the weighted error, the control input, and the weighted control input. Also, $K(\theta)$ is a switching LPV controller to be designed. $W_e$ is a weighting function expressed by + +$$ W_e : \begin{cases} \dot{x}_e(t) = W_{eA}x_e(t) + W_{eB}e(t), \\ e_w(t) = W_{eC}x_e(t) + W_{eD}e(t), \end{cases} \quad (4.12) $$ + +where $x_e$ is the state vector of the system and + +$$ \begin{bmatrix} W_{eA} & W_{eB} \\ W_{eC} & W_{eD} \end{bmatrix} = \begin{bmatrix} -0.000866 & 8 \\ 10.83 & 0.5 \end{bmatrix}. \quad (4.13) $$ + +Moreover, $W_u$ is assumed to be a static gain given by + +$$ W_u = 0.01. \quad (4.14) $$ \ No newline at end of file diff --git a/samples/texts/5122728/page_67.md b/samples/texts/5122728/page_67.md new file mode 100644 index 0000000000000000000000000000000000000000..43ed189368fb450bc60e24294c286ea7b2cd6346 --- /dev/null +++ b/samples/texts/5122728/page_67.md @@ -0,0 +1,9 @@ +The selection procedure of a weighting function has been explained in [11]. The procedure for the current example can be briefly expressed in two steps. First, the shape of the weighting function in frequency-domain is selected such that it covers the FRF measurements of the hand movements. By this selection, the weighting function penalizes the performance level more around frequencies that have relatively high amplitudes. Next, parameters representing exact form of the functions are selected by trial and error such that the final performance of the system is satisfactory. + +Figure 4.5: A feedback structure for controller design. + +### 4.2.2 A Generalized Plant + +Having defined the weighting functions, we can introduce the control objective as +to minimize the error and the control input by minimization of the $L_2$-gain from $r$ +to $\hat{e}_x$ and $\hat{T}_x$. Based on this control objective, the generalized plant in (2.1) relating \ No newline at end of file diff --git a/samples/texts/5122728/page_68.md b/samples/texts/5122728/page_68.md new file mode 100644 index 0000000000000000000000000000000000000000..b4728b1537ff45d650e2bdaaaa9ad2ef070788fc --- /dev/null +++ b/samples/texts/5122728/page_68.md @@ -0,0 +1,37 @@ +the inputs and outputs as + +$$ +\begin{bmatrix} +\dot{x}_{px} \\ +\dot{x}_e \\ +\hat{e}_x \\ +\hat{T}_x \\ +e +\end{bmatrix} += +G(\theta) +\begin{bmatrix} +[x_{px}] \\ +[x_e] \\ +[r_x] \\ +[T_x] +\end{bmatrix}, +\quad (4.15) +$$ + +is obtained from Fig. 4.5 as follows: + +$$ +G(\theta) = \begin{bmatrix} +A_{p_x} & 0 & 0 & B_{p_x} \\ +W_{eB}C_{p_x} & W_{eA} & W_{eB} & 0 \\ +-W_{eD}D_{p_x} & W_{eC} & W_{eD} & 0 \\ +0 & 0 & 0 & W_u \\ +-C_{p_x} & 0 & 1 & 0 +\end{bmatrix}. +\quad (4.16) +$$ + +4.3 Controller Design + +Using different methods explained in Chapter 2, switching LPV controllers with $N_1 = 2$ and $N_2 = 2$ are designed for the LPV generalized plant described above. Like the numerical example in Chapter 3, having two divisions in each axis leads to four number of subregions in the plant variation region. The width of the overlapping regions are assumed to be zero since this is a time invariant system, and it does not need to have hysteresis switching logic. In other words, no switching occurs during the operation of the system. Having made the assumptions on the number of subregions and the width of the overlapping regions, we introduce $\mathbf{p}_S = (c_{1,1}, c_{2,1})$ as the switching surface variable vector. \ No newline at end of file diff --git a/samples/texts/5122728/page_69.md b/samples/texts/5122728/page_69.md new file mode 100644 index 0000000000000000000000000000000000000000..30414279e280e7df0ff444319abc1532da4c4409 --- /dev/null +++ b/samples/texts/5122728/page_69.md @@ -0,0 +1,5 @@ +Like what was done for the numerical example in Chapter 3, first, *the full search method* was applied by gridding the $p_S$-space to obtain the minimum point. It can be seen in Fig. 4.6 that the optimal $\gamma^* = 6.76$ was obtained at $p_S^* = (2.00, 1.95)$. + +Figure 4.6: Functionality of $\gamma$ with respect to $p_S = (c_{1,1}, c_{2,1})$ obtained by the full-search method. + +Secondly, the descent algorithm developed in Chapter 3 employing the hybrid method for the DS step was applied. Starting from $p_{S,init} = (2.43, 2.40)$, i.e., the middle point of the plant variation region, the algorithm converged to $\gamma^* = 6.82$ and $p_S^* = (2.18, 2.00)$, as shown in Fig. 4.7. It can be seen that the final point obtained by applying the descent algorithm is close to the minimum point obtained by the full search method. However, it is not the exact same point. The reason for that is because of the discontinuity in the first derivative of the function $f$ in (2.17) which can be observed in Fig. 4.6. In other words, the algorithm gets stuck in a discontinuous-first-derivative point where the numerically-computed derivatives do \ No newline at end of file diff --git a/samples/texts/5122728/page_7.md b/samples/texts/5122728/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..976a253717b9549fb7871d921a86942d223df816 --- /dev/null +++ b/samples/texts/5122728/page_7.md @@ -0,0 +1,124 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
3.4Summary41
4Application in Magnetically-Actuated Optical Image Stabilizer43
4.1Problem Statement44
4.1.1Physical System44
4.1.2Control Problem46
4.1.3Plant Model.47
4.1.4Plant Parameter Variation.51
4.2Control Objectives52
4.2.1Weighting Functions52
4.2.2A Generalized Plant53
4.3Controller Design54
4.4Simulation Results58
4.5Summary58
5Application in Spark Ignition Internal Combustion Engines61
5.1Problem Statement.63
5.1.1Plant Parameters Variation63
5.1.2Plant Model.65
5.2Control Objectives66
5.2.1Weighting Functions66
5.2.2A Generalized Plant67
5.3Controller Design.70
5.3.1Partition Design Assumptions.70
5.3.2Optimized Switching LPV Controller Design71
5.4Simulation Results74
\ No newline at end of file diff --git a/samples/texts/5122728/page_70.md b/samples/texts/5122728/page_70.md new file mode 100644 index 0000000000000000000000000000000000000000..d6945738034d6f2de24a5e67a4e0877a8d34b5ab --- /dev/null +++ b/samples/texts/5122728/page_70.md @@ -0,0 +1,15 @@ +not suggest necessarily a descending direction. Thus, the algorithm terminates at +that point which is not the local minimum point. + +In fact, the descent algorithm perfectly works with an objective function $f$ the +first derivative of which is continuous, and it exactly converges to the minimum +point. Nevertheless, since the algorithm is descent, it guarantees decrease in the +cost function and finding a point with smaller performance level $\gamma$ compared to that +of achieved by a trivial choice of the switching surface. Moreover, computational +cost of using the descent algorithm over *the full search method* is remarkably small. +See Table 4.2. + +Figure 4.7: Convergence path of the descent algorithm having employed *the hybrid method* or *the steepest descent method*. + +Thirdly, we used *the steepest descent method* and *Newton's method* singly for +the DS step of the descent algorithm. It was seen that *Newton's method* was not \ No newline at end of file diff --git a/samples/texts/5122728/page_71.md b/samples/texts/5122728/page_71.md new file mode 100644 index 0000000000000000000000000000000000000000..72be31626f2e5fb87227e14328ce09a7486fab66 --- /dev/null +++ b/samples/texts/5122728/page_71.md @@ -0,0 +1,7 @@ +Table 4.2: Results of applying different numerical methods to optimize the switching surface + +
MethodInitial PointFinal PointComp. Time (min)
Full SearchN/Aγ* = 6.76
ps* = (2.00, 1.95)		114
Descent Algorithm
(Hybrid)		γinit = 6.96
ps,init = (2.43, 2.40)		γ* = 6.82
ps* = (2.18, 2.00)		5
Descent Algorithm
(Steepest Descent)		γinit = 6.96
ps,init = (2.43, 2.40)		γ* = 6.82
ps* = (2.18, 2.00)		3
Descent Algorithm
(Newton's)		γinit = 6.96
ps,init = (2.43, 2.40)		N/AN/A
+ +applicable since Hessian matrix of the objective function *f* is negative definite all the time, as the convexity of the cost function, seen in Fig. 4.6, demonstrates such result. Due to the same reason, applying *the steepest descent method* led to the same convergence path as applying *the hybrid method* did, as shown in Fig. 4.7. In other words, second order approximation of the objective function was never used in any iteration of the algorithm employing *the hybrid method* since Hessian matrix of *f* was always negative definite. However, computational time of applying *the steepest descent method* for the DS step was smaller than that of applying *the hybrid method* because at each iteration, Hessian matrix of *f* should be computed additionally in *the hybrid method*. See Table 4.2. + +Also, in order to meaningfully compare the time, computations were done with the same number of gridded points of the plant variation region, equal to 2500, as well as the same 3.20 GHz CPU computer with 32.0 GB RAM and MATLAB version 8.1.0.604 (R2013a). \ No newline at end of file diff --git a/samples/texts/5122728/page_72.md b/samples/texts/5122728/page_72.md new file mode 100644 index 0000000000000000000000000000000000000000..41d90981f66f5ea9720032a4dc36257302363f36 --- /dev/null +++ b/samples/texts/5122728/page_72.md @@ -0,0 +1,9 @@ +## 4.4 Simulation Results + +Simulation results of implementing two switching LPV controllers are shown in Fig. 4.8. One of the controllers has the optimized switching surface obtained by applying the descent algorithm, and the other one has a heuristic switching surface assumed to be the one placed at the middle of the plant variation region. The reference signal to be tracked is a sinusoidal signal with the frequency of 20 Hz and the amplitude of $10^{-3}$ rad which is the working condition of the actual system. The error signal which is the difference between the attitude of the platform $\theta_x$ and the reference signal $r$ is plotted in the figure below. The simulation was done for different plants covering the whole product variation region. As a result, two error signal envelopes each corresponding to one of the switching LPV controllers were obtained. + +As seen in this figure, the worst case error has been improved by applying the optimized switching LPV controller. However, for many other plants in the product variation region, the amplitude of the error signal has increased compared to the case having a heuristic switching surface. This is not surprising since as explained before, improvement of the performance level $\gamma$ guarantees improvement of just *the worst case L2-gain* from $r$ to $\hat{e}_x$ and $\hat{T}_x$. + +## 4.5 Summary + +In this chapter, a magnetically-actuated optical image stabilizer was used as a control application. The micro-scale image stabilizer system studied in this thesis uses a hardware-based mechanism to suppress vibrations produced by involuntary \ No newline at end of file diff --git a/samples/texts/5122728/page_73.md b/samples/texts/5122728/page_73.md new file mode 100644 index 0000000000000000000000000000000000000000..d905f51e33138bc46c09dbe7a090c4410333bc1e --- /dev/null +++ b/samples/texts/5122728/page_73.md @@ -0,0 +1,5 @@ +Figure 4.8: Error envelope comparison between the cases having an optimal controller and a heuristic one. + +hand-movements, so that the image becomes clear with no blurriness. After reviewing the mechanical system and the operation of the image stabilizer, it was shown how the plant is modelled as a mass-spring-damper system. The mass and stiffness-value of a system in micro-scale forms are subject to product variations. Therefore, the plant variation region was defined for this system as a result of such product variations. The control objective in this control design problem is that the orientation of the platform containing a lens at the centre tracks a reference signal, which is the negative of the hand-movements. + +For designing a switching LPV controller, the plant variation region was divided into 4 rectangular subregions with fixed widths of the overlapping regions. Thus, 2 switching surface variables, i.e., the centres of the overlapping regions, were remained to be designed. The switching LPV controller with optimized switching surfaces was \ No newline at end of file diff --git a/samples/texts/5122728/page_74.md b/samples/texts/5122728/page_74.md new file mode 100644 index 0000000000000000000000000000000000000000..9617f0fb5c46db2551bd0ec230afb1ea0f2a43ea --- /dev/null +++ b/samples/texts/5122728/page_74.md @@ -0,0 +1,17 @@ +designed using the descent algorithm presented in Chapter 2. Different methods for +the DS step of the algorithm were employed and convergence computational time of +applying each of them was computed and compared with one another. Also, *the full* +*search method* was applied to validate that the final point obtained by the descent +algorithm is actually a local minimum or sufficiently close to a local minimum. +Besides, it was shown how fast is the algorithm compared to *the full search method*. + +Finally, simulation results of implementing two switching LPV controllers, one +with the optimized switching surfaces and the other one with heuristic switching +surfaces, were juxtaposed together. For each controller, simulation was done by +sampling the product variation region, and an envelope of response was obtained. By +using the switching LPV controller with optimized switching surfaces, improvement +in the closed-loop performance of the worst case product was observed. Nevertheless, +many products showed relatively worse responses. This phenomenon coincides with +our expectation that applying switching LPV controller with optimized switching +surfaces can only improve the worst case L2-gain performance of the closed-loop +which relates to the worst case product. \ No newline at end of file diff --git a/samples/texts/5122728/page_75.md b/samples/texts/5122728/page_75.md new file mode 100644 index 0000000000000000000000000000000000000000..c1431d7fd6d03bda9cde033271b8d8364b17f031 --- /dev/null +++ b/samples/texts/5122728/page_75.md @@ -0,0 +1,8 @@ +# Chapter 5 + +## Application in Spark Ignition +Internal Combustion Engines + +One of the major concerns of using Internal Combustion Engines (ICE) is to reduce hazardous emissions to health and environment. One of the technologies used in modern automotive engines for that matter is Three-Way Catalytic Converter (TWC). The TWC is located in the exhaust line of the engine, and through some chemical reactions, improves the quality of exhaust gas. Some typical reactions in TWC are reducing nitrogen oxides to nitrogen and oxygen, oxidizing hydrocarbons to carbon dioxide and water, and oxidizing carbon monoxide to carbon dioxide [16]. In general, TWC can be assumed as a buffer to small fluctuations of air and fuel flow. This happens by storing oxygen on the catalytic surface when excess oxygen exists in exhaust gas and releasing oxygen when deficiency in oxygen occurs. + +As explained in [36], TWC requires at least two conditions to work properly: First, it is important that oxygen content on the catalytic layer should be as near as \ No newline at end of file diff --git a/samples/texts/5122728/page_76.md b/samples/texts/5122728/page_76.md new file mode 100644 index 0000000000000000000000000000000000000000..d910ed796f225beda28f98e0d298ba910efe954d --- /dev/null +++ b/samples/texts/5122728/page_76.md @@ -0,0 +1,7 @@ +possible to half of its capacity, and second, exhaust gas chemical composition should be near stoichiometric composition. Therefore, it is important to have air-fuel ratio as close as possible to stoichiometric air-fuel ratio. When the driver pushes the throttle pedal, the throttle valve is opened and allows more air into the engine's cylinder. To compensate for the excess amount of air, Engine Control Unit (ECU) changes the duration of the fuel injection pulses. + +In modern control systems, Universal Exhaust Gas Oxygen (UEGO) sensor has been used to compensate for drawbacks in conventional control systems in which static maps and feedforward controllers are used to determine the amount of injected fuel. In modern systems, air-fuel ratio in the exhaust gas is measured and based on the measurement, a feedback is sent to ECU [2]. Simultaneous use of feedback and feedforward controller can perform fast and accurate air-fuel ratio control in engine. + +In the normally aspirated, port injected, Spark Ignition (SI) Internal Combustion (IC) engine, air flow is controlled by driver using throttle pedal. In intake manifold, air flow is divided to separate stream for each cylinder. Afterwards, fuel is injected to each stream and air-fuel mixture enters cylinders. After combustion process in each cylinder, exhaust gases are combined in exhaust manifold and go into the TWC. Measurement of air-fuel ratio takes place before and after TWC using UEGO sensors. + +There are three remarks about the dynamics of an IC engine: In the first place, there is no rational transfer function for an IC engine since its discrete strokes cause a pure delay in the system. In the second place, since the engine dynamics is not time invariant, the pure delay could varies notably with various parameters such as \ No newline at end of file diff --git a/samples/texts/5122728/page_77.md b/samples/texts/5122728/page_77.md new file mode 100644 index 0000000000000000000000000000000000000000..e8a28c1feb8786419103255cfccb7f0353e1d449 --- /dev/null +++ b/samples/texts/5122728/page_77.md @@ -0,0 +1,11 @@ +engine speed and air flow. Finally, as far as the air-fuel ratio is a fraction of the IC engine's inputs, i.e., air and fuel, the system could only be expressed in a nonlinear form instead of a simple linear transfer function. + +Different control design methods have been used so far in order to control air-fuel ratio in SI IC engines, e.g., PI control [1], $H_\infty$ Robust Control [29], Kalman Filter [31, 43] Adaptive Control [12, 40, 49, 50], LPV Control [51, 52], and switching LPV control [36]. + +In [36], the strengths and weaknesses of applying each control design method mentioned above are discussed, and a switching LPV controller with a heuristic selection of switching surfaces is designed. In current research, an extension to the research done in [36] is presented by designing a switching LPV controller with optimized switching surfaces using the descent algorithm. + +## 5.1 Problem Statement + +### 5.1.1 Plant Parameters Variation + +As explained before, the dynamics of the IC engine change due to change in engine speed $N$ and air flow $\dot{m}_{air}$. As shown in Fig. 5.1, these two parameters form a plant variation region in which the engine speed is assumed to change from 800 rpm to 6000 rpm because of the limits on the operation of the engine. Also, the air flow is assumed to change from 10% to 100%. The red dashed line in Fig. 5.1 shows an example of the varying parameters trajectory in the plant variation region. Additionally, the maximum rate of change of the varying parameters are assumed to be 6000 rmp/s and 100% per second for the engine speed and air flow, respectively. \ No newline at end of file diff --git a/samples/texts/5122728/page_78.md b/samples/texts/5122728/page_78.md new file mode 100644 index 0000000000000000000000000000000000000000..7d343e33fe0ac956b39a379d36792bda1e861981 --- /dev/null +++ b/samples/texts/5122728/page_78.md @@ -0,0 +1,15 @@ +Figure 5.1: Plant variation region. + +For controller reconstruction purposes, the scheduling parameter vector, $\theta$ is defined as + +$$ \theta = \begin{bmatrix} \theta_1 \\ \theta_2 \end{bmatrix} = \begin{bmatrix} 1/\dot{m}_{air} \\ 1/N \end{bmatrix}. \quad (5.1) $$ + +We also have + +$$ \Theta := \left\{ \theta \in \mathbb{R}^2 : \theta_1 \in \left[ \frac{1}{100\%}, \frac{1}{10\%} \right], \theta_2 \in \left[ \frac{1}{6000 \text{ rpm}}, \frac{1}{800 \text{ rpm}} \right] \right\}, $$ + +$$ \Omega := \left\{ \omega \in \mathbb{R}^2 : \omega_1 \in \left[ \frac{-100\%/s}{(10\%)^2}, \frac{100\%/s}{(10\%)^2} \right], \omega_2 \in \left[ \frac{-6000 \text{ rpm/s}}{(800 \text{ rpm})^2}, \frac{6000 \text{ rpm/s}}{(800 \text{ rpm})^2} \right] \right\}, \quad (5.2) $$ + +where $\Theta$ and $\Omega$ are the plant variation region and variation rate region, respectively. + +In the next subsection, the functionality of the transfer function model and a state space form of the IC engine with respect to the scheduling parameter $\theta$ will be shown. \ No newline at end of file diff --git a/samples/texts/5122728/page_79.md b/samples/texts/5122728/page_79.md new file mode 100644 index 0000000000000000000000000000000000000000..1d76d9e66a35058d86199335a61deca58f12322a --- /dev/null +++ b/samples/texts/5122728/page_79.md @@ -0,0 +1,23 @@ +### 5.1.2 Plant Model + +Below is the transfer function obtained in [36] to model the IC engine: + +$$ \frac{\psi}{\dot{m}_{fuel}} = \frac{g(\theta)}{sT(\theta) + 1} \cdot \frac{6 - 2s\alpha(\theta)}{6 + 4s\alpha(\theta) + (s\alpha(\theta))^2}, \quad (5.3) $$ + +where the input to the system is the fuel mass flow $\dot{m}_{fuel}$ which is a control input to manipulate the equivalent air-fuel ratio, denoted by $\psi$, as the output of the system. Also, as modelled in [36] and shown in Appendix A, we have + +$$ g(\theta) = \frac{14.7}{\dot{m}_{air}}, \quad (5.4) $$ + +$$ T(\theta) \approx \frac{90}{N}, \quad (5.5) $$ + +$$ \alpha(\theta) = \frac{180}{N} + \frac{5.33}{\dot{m}_{air}}, \quad (5.6) $$ + +where $g$, $T$, and $\alpha$ are parameters of the modelled transfer function and depend on the varying parameters, $\dot{m}_{air}$ and $N$. After realization, a state space form of the plant is given by + +$$ P(\theta): \begin{cases} \dot{x}_p = A_p(\theta)x_p + B_p(\theta)\dot{m}_{fuel} \\ \psi = C_p(\theta)x_p + D_p\dot{m}_{fuel} \end{cases} \quad (5.7) $$ + +where $x \in \mathbb{R}^3$ is the state vector of the plant and + +$$ A_p(\theta) = \begin{bmatrix} \frac{-1}{T(\theta)} & \frac{6}{\alpha(\theta)^2} & \frac{-2}{\alpha(\theta)} \\ 0 & 0 & 1 \\ 0 & \frac{-6}{\alpha(\theta)^2} & \frac{-4}{\alpha(\theta)} \end{bmatrix}, \quad B_p(\theta) = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, $$ + +$$ C_p(\theta) = \begin{bmatrix} \frac{g(\theta)}{T(\theta)} & 0 & 0 \end{bmatrix}, \quad D_p = 0. \quad (5.8) $$ \ No newline at end of file diff --git a/samples/texts/5122728/page_8.md b/samples/texts/5122728/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..5faeba7aeea69911d478c61a3eba4faf1deba03e --- /dev/null +++ b/samples/texts/5122728/page_8.md @@ -0,0 +1 @@ +
5.5 Summary83
6 Conclusion84
6.1 Summary84
6.2 Contributions87
6.3 Future Work88
Bibliography90
Appendix A Plant Model Derivations of Spark Ignition Internal Combustion Engines97
Appendix B A Feedforward Term Added to the Control Input in Air-Fuel Ratio Control of Internal Combustion Engines99
\ No newline at end of file diff --git a/samples/texts/5122728/page_80.md b/samples/texts/5122728/page_80.md new file mode 100644 index 0000000000000000000000000000000000000000..6fbf2963f84bf349d24b82ff6d4ac24e738c2b84 --- /dev/null +++ b/samples/texts/5122728/page_80.md @@ -0,0 +1,11 @@ +In the next section we will show how to design a controller for the above varying plant. + +## 5.2 Control Objectives + +In this problem, we would like to design an optimized switching LPV controller, $K(\theta)$, such that disturbance $d$ is rejected, and the air-fuel ratio $\psi$ tracks a reference signal which is not necessarily stoichiometric. In fact, it may happen that during warm up or while a correction in oxygen storage level of TWC is needed, the air-fuel ratio mixture is required to be rich or lean. So, if the air-fuel ratio tracks this reference well, the overall performance of the IC engine will be improved. Since this is a reference tracking problem, an integrator is used to help removing tracking error. In addition, there is a limitation for the control input, i.e., $\dot{m}_{fuel}$, which should be considered in the controller design process. In order to achieve the control objectives described above, weighting functions and a generalized plant are introduced in the following. + +### 5.2.1 Weighting Functions + +A schematic structure of the closed-loop system is shown in Fig. 5.2. In order to make the performance channels, $e_w$ and $u_w$, two weighting functions $W_e$ and $W_u$ are employed for the error signal $e$ and control input $u$, respectively, as shown in Fig. 5.2. The weighting functions are realized into the state space forms as follows: + +$$ W_e: \begin{cases} \dot{x}_e = W_{eA}x_e + W_{eB}e \\ e_w = W_{eC}x_e + W_{eD}e \end{cases}, \quad (5.9) $$ \ No newline at end of file diff --git a/samples/texts/5122728/page_81.md b/samples/texts/5122728/page_81.md new file mode 100644 index 0000000000000000000000000000000000000000..a51cc74f6a303774c6bdaf59050d964d5e7b2d09 --- /dev/null +++ b/samples/texts/5122728/page_81.md @@ -0,0 +1,17 @@ +Figure 5.2: Block diagram of the closed-loop system. + +and + +$$ W_u : \begin{cases} \dot{x}_u = W_{uA}x_u + W_{uB}u \\ u_w = W_{uC}x_u + W_{uD}u \end{cases}, \qquad (5.10) $$ + +where $x_e$ and $x_u$ are the state vectors. The weighting functions used in this thesis are first order with same parameters set in [36] and given by + +$$ \begin{bmatrix} W_{eA} & W_{eB} \\ W_{eC} & W_{eD} \end{bmatrix} = \begin{bmatrix} -502.5 & 10^{-5} \\ 5.02 \times 10^5 & 10^{-5} \end{bmatrix}, \qquad (5.11) $$ + +and + +$$ \begin{bmatrix} W_{uA} & W_{uB} \\ W_{uC} & W_{uD} \end{bmatrix} = \begin{bmatrix} -9.95e \times 10^5 & 1 \\ -9.949e \times 10^5 & 1 \end{bmatrix}. \qquad (5.12) $$ + +### 5.2.2 A Generalized Plant + +Having set the weighting functions, we define a generalized plant which is used in controller design process. The generalized plant with varying parameters in Fig. 5.3 \ No newline at end of file diff --git a/samples/texts/5122728/page_82.md b/samples/texts/5122728/page_82.md new file mode 100644 index 0000000000000000000000000000000000000000..5e428147f5cc32878aa9d8b4dcdfd58141b1789e --- /dev/null +++ b/samples/texts/5122728/page_82.md @@ -0,0 +1,11 @@ +has the state space realization as follows: + +$$G(\theta) : \begin{cases} \dot{x} = A(\theta)x + B_1(\theta)w + B_2(\theta)u \\ z = C_1(\theta)x + D_{11}(\theta)w + D_{12}(\theta)u \\ y = C_2(\theta)x + D_{21}(\theta)w + D_{22}(\theta)u \end{cases}, \quad (5.13)$$ + +where + +$$x = \begin{bmatrix} x_p \\ x_e \\ x_u \\ x_i \end{bmatrix}, \quad (5.14)$$ + +is the state vector of the generalized plant. $x_i$ is the state of the integrator added to the model. This integrator is actually a part of the controller. However, it is aggregated to the generalized plant for control design purposes. $y$, the output of the integrator, is sent to a controller to be designed, and $u$ is the control input to the generalized plant. + +Figure 5.3: Generalized plant. \ No newline at end of file diff --git a/samples/texts/5122728/page_83.md b/samples/texts/5122728/page_83.md new file mode 100644 index 0000000000000000000000000000000000000000..efedd1fc52dd496a11a3233a546077a35dcc531d --- /dev/null +++ b/samples/texts/5122728/page_83.md @@ -0,0 +1,17 @@ +Also, we have + +$$w = \begin{bmatrix} d \\ r \end{bmatrix}, \quad z = \begin{bmatrix} e_w \\ u_w \end{bmatrix}, \qquad (5.15)$$ + +which are the exogenous input and the performance channels vector, respectively. *d* is the disturbance added to the output of the IC engine plant, and *r* is the reference signal to be tracked. After doing algebraic calculations, the matrices in (5.13) are obtained as follows: + +$$A(\theta) = \begin{bmatrix} A_p(\theta) & 0 & 0 & 0 \\ -W_{eB}C_p(\theta) & W_{eA} & 0 & 0 \\ 0 & 0 & W_{uA} & 0 \\ -C_p(\theta) & 0 & 0 & 0 \end{bmatrix}$$ + +$$B_1(\theta) = \begin{bmatrix} 0 & 0 \\ -W_{eB} & W_{eB} \\ 0 & 0 \\ -1 & 1 \end{bmatrix}, \quad B_2(\theta) = \begin{bmatrix} B_p(\theta) \\ 0 \\ W_{uB} \\ 0 \end{bmatrix}$$ + +$$C_1(\theta) = \begin{bmatrix} 0 & W_{eC} & 0 & 0 \\ 0 & 0 & W_{uC} & 0 \end{bmatrix}, \quad C_2(\theta) = \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix} \qquad (5.16)$$ + +$$D_{11}(\theta) = \begin{bmatrix} -W_{eD} & W_{eD} \\ 0 & 0 \end{bmatrix}, \quad D_{12}(\theta) = \begin{bmatrix} 0 \\ W_{uD} \end{bmatrix}$$ + +$$D_{21}(\theta) = \begin{bmatrix} 0 & 0 \end{bmatrix}, \quad D_{22}(\theta) = 0.$$ + +As it can be seen, the matrices of the generalized plant are functions of $\theta$ in general due to the functionality of the IC engine model with respect to $\theta$. \ No newline at end of file diff --git a/samples/texts/5122728/page_84.md b/samples/texts/5122728/page_84.md new file mode 100644 index 0000000000000000000000000000000000000000..c349a59cc15b754d1a4943e0c035b30e71c15fab --- /dev/null +++ b/samples/texts/5122728/page_84.md @@ -0,0 +1,23 @@ +## 5.3 Controller Design + +### 5.3.1 Partition Design Assumptions + +As explained in Chapter 2, in order to design a switching LPV controller, we need to design a partition for the plant variation region as well as a couple of controllers each suitable for a subregion. To design the partition, we need to make some assumptions beforehand. + +* The number of divisions on each axis of the plant variation region in Fig. 5.1 is assumed to be 2, which means that we have 4 subregions in total. + +* The shape of each subregion is taken to be rectangular. + +* The width of the overlapping region between each two subregions are assumed to be fixed and is given by + +$$w_{or} = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = 0.1 \cdot L, \qquad (5.17)$$ + +where $L$ is a vector containing the variation length of the gain-scheduling parameter $\theta$. + +Fig. 5.4 shows an example of a partition on the plant variation region with the above assumptions. Having made these assumptions, we would like to design the subregions + +$$\Theta^{(i,j)}, \quad i = 1, 2, \quad j = 1, 2, \qquad (5.18)$$ + +which means designing the centre position of the overlapping regions. Therefore, the switching surface vector $p_S$ in (2.15) is given by + +$$p_S = \begin{bmatrix} c_{1,1} \\ c_{2,1} \end{bmatrix}, \qquad (5.19)$$ \ No newline at end of file diff --git a/samples/texts/5122728/page_85.md b/samples/texts/5122728/page_85.md new file mode 100644 index 0000000000000000000000000000000000000000..3113fa965869897a6206c6d6e40d48426ab16c98 --- /dev/null +++ b/samples/texts/5122728/page_85.md @@ -0,0 +1,17 @@ +where $c_1$ and $c_2$ are parameters to be designed. + +Figure 5.4: The plant variation region with a partition. + +### 5.3.2 Optimized Switching LPV Controller Design + +As well as designing subregions, we would like to design LPV controllers + +$$K^{(i,j)}, \quad i = 1, 2, \quad j = 1, 2, \tag{5.20}$$ + +each of which is associated with a subregion in (5.18), in order to minimize the bound $\gamma$ of the closed-loop $L_2$-gain from the input $w$ to the performance channel $z$, i.e., + +$$\min \gamma \text{ subject to } \|z\|_2 < \gamma \|w\|_2, \tag{5.21}$$ + +for any possible trajectory of $\theta$ within the region $\Theta$ and the variation rate $\dot{\theta}$ within the region $\Omega$ set in (5.2). Each LPV controller in (5.20) has the form + +$$K^{(i,j)}(\theta) : \begin{cases} \dot{x}_K = A_K^{(i,j)}(\theta)x_K + B_K^{(i,j)}(\theta)y \\ u = C_K^{(i,j)}(\theta)x_K + D_K^{(i,j)}(\theta)y \end{cases}, \quad i=1,2, \ j=1,2. \tag{5.22}$$ \ No newline at end of file diff --git a/samples/texts/5122728/page_86.md b/samples/texts/5122728/page_86.md new file mode 100644 index 0000000000000000000000000000000000000000..428e0308bff1d13612f59ea42cad256dcf24d722 --- /dev/null +++ b/samples/texts/5122728/page_86.md @@ -0,0 +1,40 @@ +As explained in Chapter 2, the controller parameters are reconstructed using the +following equations: + +$$ +\begin{align*} +D_K^{(i,j)}(\theta) &:= \hat{D}_K^{(i,j)}(\theta), \\ +C_K^{(i,j)}(\theta) &:= [\hat{C}_K^{(i,j)}(\theta) - D_K^{(i,j)}(\theta)C_2(\theta)Y]^T M^T, \\ +B_K^{(i,j)}(\theta) &:= [N^{(i,j)}(\theta)]^{-1} [\hat{B}_K^{(i,j)}(\theta) - X^{(i,j)}(\theta)B_2(\theta)D_K^{(i,j)}(\theta)], \\ +A_K^{(i,j)}(\theta) &:= [N^{(i,j)}(\theta)]^{-1} \left[ \begin{aligned}[t] +& \hat{A}_K^{(i,j)}(\theta) - N^{(i,j)}(\theta)B_K^{(i,j)}(\theta)C_2(\theta)Y \\ +& \quad - X^{(i,j)}(\theta)B_2(\theta)C_K^{(i,j)}(\theta)M^T \\ +& \quad - X^{(i,j)}(\theta) \{ A(\theta) + B_2(\theta)D_K^{(i,j)}(\theta)C_2(\theta) \} Y +\end{aligned} \right] / M^T, \\ +& i=1,2, \quad j=1,2, +\end{align*} +$$ + +in which $Y$ is assumed to be fixed, and $\hat{A}_K^{(i,j)}(\theta)$, $\hat{B}_K^{(i,j)}(\theta)$, $\hat{C}_K^{(i,j)}(\theta)$, $\hat{D}_K^{(i,j)}(\theta)$, and $X^{(i,j)}$ are all assumed to be linear functions of the scheduling parameter $\theta$, expressed + +as + +$$ +\begin{align} +\hat{A}_{K}^{(i,j)}(\theta) &:= \hat{A}_{0}^{(i,j)} + \frac{1}{\dot{m}_{fuel}} \hat{A}_{1}^{(i,j)} + \frac{1}{N} \hat{A}_{2}^{(i,j)}, \nonumber \\ +\hat{B}_{K}^{(i,j)}(\theta) &:= \hat{A}_{0}^{(i,j)} + \frac{1}{\dot{m}_{fuel}} \hat{B}_{1}^{(i,j)} + \frac{1}{N} \hat{B}_{2}^{(i,j)}, \nonumber \\ +\hat{C}_{K}^{(i,j)}(\theta) &:= \hat{C}_{0}^{(i,j)} + \frac{1}{\dot{m}_{fuel}} \hat{C}_{1}^{(i,j)} + \frac{1}{N} \hat{C}_{2}^{(i,j)}, \tag{5.23} \\ +\hat{D}_{K}^{(i,j)}(\theta) &:= \hat{D}_{0}^{(i,j)} + \frac{1}{\dot{m}_{fuel}} \hat{D}_{1}^{(i,j)} + \frac{1}{N} \hat{D}_{2}^{(i,j)}, \nonumber \\ +X_{K}^{(i,j)}(\theta) &:= X_{0}^{(i,j)} + \frac{1}{\dot{m}_{fuel}} X_{1}^{(i,j)} + \frac{1}{N} X_{2}^{(i,j)}, \nonumber \\ +& i = 1, 2, \quad j = 1, 2, \nonumber +\end{align} +$$ + +The elements of these matrices are stacked together in the controller parameters vector $\mathbf{p}_K$ and subjected to be designed. + +Now that the variables $\boldsymbol{p}_K$ and $\boldsymbol{p}_S$ are introduced for the IC engine problem, the +optimization problem (2.15) repeated as + +$$ +\min \gamma \text{ subject to } M(\gamma, \boldsymbol{p}_K, \boldsymbol{p}_S) < 0, \qquad (5.24) +$$ \ No newline at end of file diff --git a/samples/texts/5122728/page_87.md b/samples/texts/5122728/page_87.md new file mode 100644 index 0000000000000000000000000000000000000000..33bb83e4c3ee82e160502ea2b555bb16a48c27ed --- /dev/null +++ b/samples/texts/5122728/page_87.md @@ -0,0 +1,14 @@ +and transformed to the minimization problem of function *f* in (2.17) repeated as + +$$ +\gamma = f(\boldsymbol{p}_S), \tag{5.25} +$$ + +is numerically solved using the descent algorithm presented in Section 2.3. First, *the hybrid method* was employed for the DS step of the algorithm. Starting from $\boldsymbol{p}_{S,init} = (5.50 \times 10^{-2}, 7.08 \times 10^{-4})$, i.e., the middle point of the plant variation set $\Theta$, and corresponding $\gamma_{init} = 3.30$, the algorithm converged to $\boldsymbol{p}_{S,init}^* = (9.09 \times 10^{-2}, 3.50 \times 10^{-4})$ and $\gamma^* = 0.818$. + +Secondly, both *the steepest descent method* and *Newton's method* were applied singly for the DS step. Having used *the steepest descent method* and started from the middle point of the plant variation set, the algorithm converged to the same point as above but in shorter computational time. The reason for such result is that in most of iterations, Hessian matrix of *f* was not positive definite, so that the second order approximation of *f* was not used, while calculation of Hessian matrix in each iteration charged the computational cost of the algorithm. For the same reason, applying *Newton's method* singly for the DS step did not lead to convergence to any point. The results mentioned above have been summarized in Table 5.1. In order to meaningfully compare the time, computations were done with the same number of gridded points of the plant variation region, equal to 2500, as well as the same 3.20 GHz CPU computer with 32.0 GB RAM and MATLAB version 8.1.0.604 (R2013a). + +Different from what was done for the numerical example in Chapter 3 and the OIS +example in Chapter 4, *the full search method* was not applied to find the optimized +switching surface for the IC engine example. This is because of the fact that applying +*the full search method* would take weeks to find the optimized point for the IC \ No newline at end of file diff --git a/samples/texts/5122728/page_88.md b/samples/texts/5122728/page_88.md new file mode 100644 index 0000000000000000000000000000000000000000..eb275bd59354289d04b42df3a4eca45f8b19ace9 --- /dev/null +++ b/samples/texts/5122728/page_88.md @@ -0,0 +1,9 @@ +Table 5.1: Results of applying different numerical methods for the DS step of the descent algorithm to optimize the switching surface + +
MethodInitial PointFinal PointComp. Time (min)
Hybridγinit = 3.30
ps,init = [ 5.50 × 10-2
7.08 × 10-4 ]		γ* = 0.818
ps,init = [ 9.09 × 10-2
3.50 × 10-4 ]		259
Steepest Descentγinit = 3.30
ps,init = [ 5.50 × 10-2
7.08 × 10-4 ]		γ* = 0.818
ps,init = [ 9.09 × 10-2
3.50 × 10-4 ]		148
Newton'sγinit = 3.30
ps,init = [ 5.50 × 10-2
7.08 × 10-4 ]		N/AN/A
+ +engine problem which is much longer than what took for other examples. The reason for such computational time difference is that the generalized plant in the current example has 6 states which is double the number of states of the generalized plants in previous examples. Therefore, computation of $\gamma$ for each gridded point would take remarkably longer time. This evidence elucidates and emphasizes the advantage of using the descent algorithm over *the full search method*, which is not applicable to some examples. + +## 5.4 Simulation Results + +Two switching LPV controllers have been designed and implemented in the system. One of the controllers has the optimized switching surfaces and the other one has the switching surfaces heuristically placed at the middle of the plant variation region, such that the whole region is divided into 4 equal subregions. In the simulation \ No newline at end of file diff --git a/samples/texts/5122728/page_89.md b/samples/texts/5122728/page_89.md new file mode 100644 index 0000000000000000000000000000000000000000..615bf6893c701ea9695600c5909e87ac50378007 --- /dev/null +++ b/samples/texts/5122728/page_89.md @@ -0,0 +1,12 @@ +tion, a feedforward term is also added to the system to modify the control input to +the plant so as to improve the tracking performance of the system. The logic behind +such modification has been explained in Appendix B. In Fig. 5.5, reference tracking +simulations of two controllers with a square-wave reference signal are juxtaposed +together. As seen in this figure, the optimized controller produced a response with +smaller overshoots and undershoots and improved settling time. It was consequently +seen that the absolute value of the area between the equivalent air-fuel ratio and the +reference signal, i.e., the error, was decreased by 17%. To clearly show the difference +between the performance of two controllers, the error signal averaged in every 10 +seconds time interval has been shown in Fig. 5.6. + +Figure 5.5: Comparison of equivalent air-fuel ratio reference tracking between controllers with the optimized switching surface and the heuristic one. \ No newline at end of file diff --git a/samples/texts/5122728/page_9.md b/samples/texts/5122728/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..d83e2bbf9d518a7118505f1f523d143845474745 --- /dev/null +++ b/samples/texts/5122728/page_9.md @@ -0,0 +1,3 @@ +# List of Tables + +
3.1Results of applying different numerical methods to optimize the switching surface.35
3.2Assumptions on the switching surfaces37
4.1System parameters [37].50
4.2Results of applying different numerical methods to optimize the switching surface.57
5.1Results of applying different numerical methods for the DS step of the descent algorithm to optimize the switching surface74
\ No newline at end of file diff --git a/samples/texts/5122728/page_90.md b/samples/texts/5122728/page_90.md new file mode 100644 index 0000000000000000000000000000000000000000..20c8d22aca4dc70643104baf40b49d7b803505a0 --- /dev/null +++ b/samples/texts/5122728/page_90.md @@ -0,0 +1 @@ +Figure 5.6: Error comparison of equivalent air-fuel ratio reference tracking, averaged in every 10 seconds time interval, between controllers with the optimized switching surface and the heuristic one. \ No newline at end of file diff --git a/samples/texts/5122728/page_91.md b/samples/texts/5122728/page_91.md new file mode 100644 index 0000000000000000000000000000000000000000..4912522ceaad7b4c5ed120487f55b1583808a991 --- /dev/null +++ b/samples/texts/5122728/page_91.md @@ -0,0 +1,3 @@ +Fig. 5.7 and Fig. 5.8 show realistic profiles of the air flow and engine speed in [36] representing operating conditions of the engine. Also, assumed upper bounds and lower bounds on their variations are shown. As seen in these figures, the profile of the air flow and engine speed are limited to these bounds as expected. Fig. 5.9 represents the resulting trajectory of the air flow and engine speed within the plant variation region. + +Figure 5.7: Air-flow trajectory and the lower and upper bounds on its variations. \ No newline at end of file diff --git a/samples/texts/5122728/page_92.md b/samples/texts/5122728/page_92.md new file mode 100644 index 0000000000000000000000000000000000000000..bf688c1257cdedf7edbc9d13c07d1faac4ce6356 --- /dev/null +++ b/samples/texts/5122728/page_92.md @@ -0,0 +1,3 @@ +Figure 5.8: Engine speed trajectory and the lower and upper bounds on its variations. + +Figure 5.9: The operating point of the engine in the plant variation region. \ No newline at end of file diff --git a/samples/texts/5122728/page_94.md b/samples/texts/5122728/page_94.md new file mode 100644 index 0000000000000000000000000000000000000000..8b15823996a55d2f572286168015c02b338130d8 --- /dev/null +++ b/samples/texts/5122728/page_94.md @@ -0,0 +1 @@ +Figure 5.11: Engine Speed variation rate trajectory and the lower and upper bounds on its variations. \ No newline at end of file diff --git a/samples/texts/5122728/page_95.md b/samples/texts/5122728/page_95.md new file mode 100644 index 0000000000000000000000000000000000000000..51ae7eac7e813c59afbc3ca8be00701b791fd506 --- /dev/null +++ b/samples/texts/5122728/page_95.md @@ -0,0 +1,5 @@ +Fig. 5.12 and Fig. 5.13 show the resulting changes in the switching variables on the air flow axis and the engine speed axis, respectively. The combination of these two switching signals determines the active controller in the Fig. 5.4 at each time. + +Figure 5.12: Comparison of switching variable on air-flow axis between controllers with the optimized switching surface and the heuristic one. + +Figure 5.13: Comparison of switching variable on engine speed axis between controllers with the optimized switching surface and the heuristic one. \ No newline at end of file diff --git a/samples/texts/5122728/page_96.md b/samples/texts/5122728/page_96.md new file mode 100644 index 0000000000000000000000000000000000000000..f4a4bb4b4329b30ef9b33e4d94360462be79dfb4 --- /dev/null +++ b/samples/texts/5122728/page_96.md @@ -0,0 +1,3 @@ +As explained before, the fuel flow is the control input to the plant. The two profiles of the fuel flow generated by two controllers have been shown in Fig. 5.14. Both signals are within the allowable region. As seen in this figure, little difference in the control input can make a huge impact in the performance of the system. The transient responses of the control input at the times of switching events are not large enough to be observed in this figure. Albeit abrupt changes in the control input signal still exist, they are negligible compared to the absolute value of the control input. Moreover, a low pass filter has been employed just after the controller for performance improvement purposes which has faded instant variations in the control input plotted here. + +Figure 5.14: Comparison of control input to the plant between controllers with the optimized switching surface and the heuristic one. \ No newline at end of file diff --git a/samples/texts/5122728/page_97.md b/samples/texts/5122728/page_97.md new file mode 100644 index 0000000000000000000000000000000000000000..4bb0878a9743bdfccc354770b1e4acf49299a009 --- /dev/null +++ b/samples/texts/5122728/page_97.md @@ -0,0 +1,5 @@ +## 5.5 Summary + +In this chapter, the proposed algorithm was applied to air-fuel ratio control in spark ignition internal combustion engines. At the beginning of this chapter, the importance of air-fuel ratio control in the engines was discussed, and different control approaches have been used so far in the literature were mentioned. By reviewing the modelling of an IC engine, it was shown that the dynamic of the system depends on the engine speed and air flow which form a 2 dimensional plant variation region. For switching LPV controller design, the plant variation region was divided into 4 rectangular subregions with fixed widths of the overlapping regions, like what was done for the image stabilizer example. As a result, the centres of the overlapping regions are the switching surface variables. Reference tracking and disturbance rejection are design objectives in this control design problem. + +The switching LPV controller with optimized switching surfaces was designed using the descent algorithm. For the DS step of the algorithm, different methods were employed and corresponding results were compared. It was observed that the algorithm with *the steepest descent method* converges faster than that with *the hybrid method*, and applying *Newton's method* led not to converge to a local minimum point. At the end of the chapter, simulation results of implementing the switching LPV controller with optimized switching surfaces were compared with that with heuristic switching surfaces. A realistic plant parameters trajectory was used and an arbitrary reference signal was input to the closed-loop. It was shown that the optimized controller could result in better tracking response in terms of the error, overshoot, and settling time. \ No newline at end of file diff --git a/samples/texts/5122728/page_98.md b/samples/texts/5122728/page_98.md new file mode 100644 index 0000000000000000000000000000000000000000..d287fdbf0e02c3ce1e8ec42b5740029c3a7d7d90 --- /dev/null +++ b/samples/texts/5122728/page_98.md @@ -0,0 +1,11 @@ +# Chapter 6 + +## Conclusion + +In conclusion, applying the descent algorithm in this thesis to design switching surfaces for switching LPV control has been shown to be beneficial for some reasons. First of all, the computational time of getting the solution by applying the proposed method is remarkably smaller than that of applying *the full search method*. Secondly, it was shown that implementing switching LPV controllers with optimized switching surfaces improves time-domain simulation performances in different control applications. However, performance improvement of the worst-case can be only guaranteed by employing optimized switching surfaces. + +In this chapter, a summary of what was explained in this thesis, as well as the contributions of my work and some future works in this area has been provided. + +### 6.1 Summary + +In this thesis, a numerical algorithm was developed to design switching surfaces for switching LPV controllers. Time-varying systems with measurable parameters, \ No newline at end of file diff --git a/samples/texts/5122728/page_99.md b/samples/texts/5122728/page_99.md new file mode 100644 index 0000000000000000000000000000000000000000..871f0013f6cab58a091440c1bf927e3a758945fe --- /dev/null +++ b/samples/texts/5122728/page_99.md @@ -0,0 +1,5 @@ +as well as nonlinear systems, represented in the form of linear models with varying operating conditions, were considered for controller designing. It was shown how stability and performance of the system can benefit from switching LPV controllers with optimized switching surfaces rather than that with heuristic ones. The algorithm proposed was applied to three examples in control applications including LPV systems: a tracking problem for a mass-spring-damper system, a vibration suppression problem for a magnetically-actuated optical image stabilizer, and an air-fuel ratio control problem for automotive engines. + +The numerical algorithm to switching LPV control design with optimized switching surfaces was developed and explained in Chapter 2 of the thesis. It was shown that optimization of the switching surfaces can be formulated as optimization of centres and widths of the overlapping regions. The cost function to be minimized is the closed-loop $L_2$-gain performance of the system, $\gamma$. The proposed algorithm is iterative and descent in which a decrease in the cost function in each iteration is guaranteed. However, a local minimum point is obtained depending on the initial condition, and finding the global minimum point is not guaranteed. A *hybrid method* which is a combination of the *steepest descent method* and *Newton's method* is employed in this algorithm. + +The proposed numerical algorithm in this thesis was then applied to a numerical example in Chapter 3. A simple mass-spring-damper system with parameter variations was used as a plant for which controllers were designed. Switching LPV controllers with different premises on the number of subregions were designed for this system. It was shown that applying the proposed algorithm in Chapter 2 is superior than using *the full search method* in terms of computational cost. Also, \ No newline at end of file diff --git a/samples/texts/6518189/page_1.md b/samples/texts/6518189/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..905859ea59ee1054cda46c096126506719bc6a4e --- /dev/null +++ b/samples/texts/6518189/page_1.md @@ -0,0 +1,22 @@ +# An energy method for computing the use of fossil fuel energy + +## Un método energético para calcular el uso de energía de combustibles fósiles + +Timur B. Temukuyev + +Russian Presidential Academy of National Economy and Public Administration under the President of +the Russian Federation. Moscow, Russia. + +energoconsul@mail.ru + +*(recibido/received: 28-octubre-2020; aceptado/accepted: 15-enero-2021)* + +### ABSTRACT + +An energy method for computing the use of fossil fuel energy has been considered in the article. On the world market, the fuel price depends on supply and demand and involves no energy costs for fuel production. An energy analysis of economic activity was suggested by Charles Hall, an American scientist, who introduced a notion of Energy Returned on Energy Invested, as a ratio between returned and invested energy, into scientific discourse. No account has been taken of invested energy depreciation in this method. All losses are fully incorporated, when the ratio between beneficially used energy in all process flow chains from fuel deposit exploration to energy utilisation, and the considered amount of natural fuel primary energy is taken as the coefficient of beneficial primary energy use (CBPEU). When CBPEU is determined, allowance is made for all potential energy losses; the depreciation degree of energy, contained in the fuel, from its deposit to a consumer, is defined. When energy of renewable sources is utilised, a coefficient of renewable sources energy conversion, defined as the ratio between energy delivered by a power unit throughout the entire operation period, and invested energy taking into account CBPEU over the same period, will represent an objective criterion of power unit efficiency. + +**Keywords:** coefficient of beneficial primary energy use; fuel reprocessing; fuel transportation; energy breeding gain. + +### RESUMEN + +En el artículo se ha considerado un método energético para calcular el uso de energía de combustibles fósiles. En el mercado mundial, el precio del combustible depende de la oferta y la demanda y no implica costos de energía para la producción de combustible. Charles Hall, un científico estadounidense, sugirió un análisis energético de la actividad económica, quien introdujo una noción de energía devuelta sobre energía invertida, como una relación entre energía devuelta e invertida, en el discurso científico. No se ha tenido en cuenta la depreciación de la energía invertida en este método. Todas las pérdidas se incorporan por completo, cuando la relación entre la energía beneficiosa usada en todas las cadenas de flujo del proceso desde la exploración de depósitos de combustible hasta la utilización de energía, y la cantidad considerada de energía primaria de combustible natural se toma como el coeficiente de uso de beneficioso de energía primaria (CBPEU). Cuando se determina el CBPEU, se tienen en cuenta todas las pérdidas de energía potenciales; Se define el grado de depreciación de la energía contenida en el combustible, desde su depósito hasta el consumidor. Cuando se utiliza energía de fuentes renovables, un coeficiente de conversión de energía de fuentes renovables, definido como la rela- \ No newline at end of file diff --git a/samples/texts/6518189/page_10.md b/samples/texts/6518189/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..cd4ed2515831bbc239362f2626abb9ccfe2ebcf6 --- /dev/null +++ b/samples/texts/6518189/page_10.md @@ -0,0 +1,45 @@ +$Q_{pec}^{NPP}$ – costs of primary energy obtained from an external source over the whole operation period. They are determined as follows: + +$$Q_{pec}^{NPP} = Q_{eq} + Q_{cap}^{NPP} + Q_{op}^{NPP} + Q_{oth}^{NPP}, \quad (37)$$ + +where $Q_{eq}$ – primary energy expended on extraction and preparation, transportation and storage, milling of uranium ore; + +$Q_{cap}^{NPP}$ – primary energy expended on construction of power plant and repository, assembly and disassembly of NPP equipment; + +$Q_{op}^{NPP}$ – operational primary energy costs for power plant and repository, including costs for extraction, transportation, ore preparation, and storage of radioactive production waste; + +$Q_{oth}^{NPP}$ – other primary energy costs. + +Energy breeding gain largely depends on uranium content in ore and trouble-free NPP operation. + +## 5. DISCUSSIONS + +In the energy analysis of economic activity suggested by Charles Hall, there are two spelling variants and two values of EROEI abbreviation in English in the modern sense: if it is EROEI (energy returned on energy invested), it implies the ratio between obtained and invested energy; when it is EROI (energy return on investment), this is the ratio between obtained energy and investments. These ratios can accordingly be written as the following formulas: + +$$Er = \frac{E_2}{E_1}, \qquad (38)$$ + +and + +$$Ei = \frac{E_2}{C}, \quad \text{J/rub.,} \qquad (39)$$ + +where $Er$ – EROEI; + +$E_2$ – energy obtained from fuel or a device transforming the Earth's, solar, etc. energy, J; + +$E_1$ – energy expended to extract (produce) energy $E_2$, J; + +$Ei$ – EROI; + +$C$ – means spent to extract (produce) energy $E_2$, rub. + +Any predator as all living organisms, with no exceptions for salmon as well, is unable to expend more energy than food provides to him, i.e. all processes here occur according to the second thermodynamic law. Any living organism can be considered as a usual heat engine, functioning with less than 1 efficiency factor, since no organism can extract all the energy from food.# + +In a general case, the efficiency factor for predator over a fixed period will be defined by formula: + +$$\eta = \frac{E_{1(\tau)}}{E_{2(\tau)}}, \qquad (26)$$ + +where $E_{1(\tau)}$ – energy expended by predator over time $\tau$; + +$E_{2(\tau)}$ – energy obtained by predator from food over time $\tau$. + +Predator, while eating food, can move, grow, and breed but is unable to generate energy. Hence, a concept of EROEI, when extended from predator to fuel extraction, should be considered differently. If in the first case the energy expended by predator is always less than the energy, he obtained by eating food, then in the second case, when it comes to extracting fuel, the energy obtained from fuel, is always greater than the energy expended on its extraction. If the total energy expended on fuel extrac- \ No newline at end of file diff --git a/samples/texts/6518189/page_11.md b/samples/texts/6518189/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..773b98a81ba10d84e570b3f6f476592898a371a2 --- /dev/null +++ b/samples/texts/6518189/page_11.md @@ -0,0 +1,26 @@ +tion is equal to the energy contained in the extracted fuel, it makes no sense to extract fuel at a given field in terms of energy. In fact, EROEI thus interpreted is nothing but CRSEC. So, when utilising fossil fuel, CRSEC can be defined by formula: + +$$ \pi = \frac{Q_2}{Q_1}, \qquad (40) $$ + +where $Q_2$ – amount of all energy obtained at the field, J; +$Q_1$ – amount of energy expended over the period of carrying out all works at the field, J. + +It should be noted that $\pi$ is greater than 1, since $Q_2$ is not related to $Q_1$ through the second thermodynamic law, as the fuel extracted at the field, is an energy source, i.e. an energy carrier. And if $\pi$ is less or equal to 1, this technological process becomes meaningless in terms of energy, regardless of whether it relates to helioplant or fossil fuel deposit. + +In Russia, researchers under the guidance of Safronov A.F. address the problem of computing EROEI. They, in particular, present EROEI computation data for a number of energy resources, mostly with respect to American conditions obtained by Hall and revised by Richard Heinberg as of 2009. Thus, EROEI for the worldwide petroleum extraction amounts to 19, for natural gas – 10, coal – 50, bituminous sands – 5.2-5.8, shale oil – 1.5-1.4, nuclear energy – 1.1-1.5, hydropower – 11-267, wind energy – 18, photovoltaics – 3.75-10, ethanol sugar cane – 0.8-1.7 (8-10 in Brazil), corn ethanol – 1.1-1.8, biodiesel – 1.9-9 (Safronov, 2010). + +As fossil fuels are extracted, the value of their EROEI decreases due to various reasons, since prolific and accessible deposits are usually developed first. This trend is also seen in Russia, for which the numerical value of EROEI was determined for three years: 31.7 in 2005, 29.9 in 2007 and 29.5 in 2008, based on the data on direct joint energy expenditure when extracting oil and gas (Safronov, 2010). + +For particular EROEI cases, numerical values of $Er$ will largely depend on how accurately $E_1$ was determined. Since during the development of some fields the energy obtained at other fields is utilised, inaccurately defined energy cost value may create an illusion about the field energy cost effectiveness. For example, energy costs associated with works on oil and gas extraction objects are provisionally divided into groups: capital, current, closure (Safronov, 2010). The respective work stages are construction, operation and closure of objects within a field. + +EROEI will decrease with distance from a borehole and be defined by formula: + +$$ Er_i = \mu_{2i} \cdot Er_{uc}, \qquad (41) $$ + +At the point of the process cycle from extracting to using heat and electrical energy, where $Er_i$ will become equal to 1, the energy value of this energy carrier ceases to exist. When exporting energy carriers, $Er_i$ can be defined to the country frontiers. + +EROEI has not any significance for a seller of energy carriers, since their price is established by the market, where cheap fuel sets the pace. However, a seller's profit will be defined by EROEI value, i.e. the higher the costs, the lower the benefit. + +Since both $\mu_{1i}$ and $\mu_{2i}$ are always less than 1, the value of $Er_i$, computed taking CBPEU into account, will be considerably lower than that obtained by formula (24). Allowing for CBPEU, formula (24) will be written as follows: + +$$ Er = \frac{E_2}{\frac{E_1}{\mu_{1i}}} \text{ or } Er = \frac{\mu_{1i} \cdot E_2}{E_1}. \qquad (42) $$ \ No newline at end of file diff --git a/samples/texts/6518189/page_12.md b/samples/texts/6518189/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..d9e7f7eebcba39a7afb45d367e6b885c733ed260 --- /dev/null +++ b/samples/texts/6518189/page_12.md @@ -0,0 +1,31 @@ +Time factor should be taken into account when computing EROEI for hydropower plants, helioplants, stations that use geothermal energy, etc. I.e., the ratio shall be taken between the energy obtained or supposed to be obtained over the whole operation period of a power unit or a facility, and the energy expended over the entire period: from commencement of works to complete disposition of a unit or facilities. CBPEU should be taken into consideration in both cases (Safronov, 2011). + +There is no way to fully account for all energy losses in the absence of the corresponding data. Energy consumption during main process stages must be allowed for. + +To gain a full knowledge of the energy value of energy resources when computing EROEI, the data on total energy costs need be used, with allowance for their depreciation while moving along the process flow, i.e. from fuel deposit to a consumer, defined by CBPEU. + +It is important to change over to such calculation methods, which would allow determination of energy cost effectiveness of using a given fuel. + +## 6. CONCLUSIONS + +A complex method for determining CBPEU of power units will make it possible to reveal those particular processes, where energy losses are considerable, and where it is essential to enhance the quality of energy use first. To attain the stated objective, various economic measures need to be taken, which will result in increasing CBPEU, i.e. in improving a system capability without increasing fuel consumption, only at the expense of decreasing energy losses in irreversible processes occurred in the system. + +Determination of energy breeding gain taking primary energy costs into consideration will enable comprehensive evaluation of a project's implementation potential regardless of its estimated parameters. + +## REFERENCES + +Boltzmann, L. (1970). Articles and speeches. Moscow: Nauka. + +Capellán-Pérez, I., Miguel, L.J., and de Castro, C. (2019). Dynamic Energy Return on Energy Investment (EROI) and material requirements in scenarios of global transition to renewable energies. Energy Strategy Reviews, 26, 100399. + +Celi, L., Della Volpe, C., Pardi, L., and Siboni, S. (2018). A new approach to calculating the «corporate» EROI. Biophysical Economics and Resource Quality, 3, 15. + +Cherevatskyi, D. (2017). EROI the Ukrainian coal. *Ekonomicheskiy vestnik Donbassa (Economic Bulletin of Donbass)*, 4 (50), 20-31. + +De Castro, C., and Capellán-Pérez, I. (2020). Standard, Point of Use, and Extended Energy Return on Energy Invested (EROI) from Comprehensive Material Requirements of Present Global Wind, Solar, and Hydro Power Technologies. *Energies*, 13 (12), 3036. + +Fedorovsky, N.M. (1935). *Classification of mineral resources according to their energy indices*. Moscow-Leningrad. + +Fersman, A.E. (1937). *Geochemistry*. Volume III. Leningrad: Khimteoret. + +Fiapshev, A., Kilchukova, O., and Khamokov, M. (2017). Biogas unit for agricultural enterprises. *Energy security and energy saving*, 2, 27-29. \ No newline at end of file diff --git a/samples/texts/6518189/page_13.md b/samples/texts/6518189/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..f388b283f4ce0f2d72a8b3bcc4a6c9d94efb2361 --- /dev/null +++ b/samples/texts/6518189/page_13.md @@ -0,0 +1,33 @@ +Fiapshev, A., Kilchukova, O., Shekikhachev, Y., Khamokov, M., and Khazhmetov, L. (2018). Mathematical model of thermal processes in a biogas plant. In: ICRE 2018 International Scientific Conference 'Investment, Construction, Real Estate: New Technologies and Special-Purpose Development Priorities. MATEC Web of Conferences, 212, 010032, 1-13. https://www.matec-conferences.org + +Global Warming of 1.5°C. (2018). Special Report. Intergovernmental Panel on Climate Change, Geneva, Switzerland. https://www.ipcc.ch/sr15/ + +Hall, A.S. (2017). Will EROI be the primary determinant of our economic future? The view of the natural scientist versus the economist. *Joule*, **1**, 635–638 + +Hall, Ch. (2008). Why EROI matters. *The Oil Drum*. http://www.theoildrum.com/node/3786 + +Järvensivu, P., Toivanen, T., Vadén, T., Lähde, V., Antti Majava, A., Jussi, T. (2018). Governance of Economic Transition. Global Sustainable Development Report 2019. https://bios.fi/bios-governance_of_economic_transition.pdf + +King Hubbert, M. (1956). *Nuclear energy and the fossil fuels. Drilling and Production Practice.* Washington: American Petroleum Institute + +Kuznetsov, P.G. (1994). System of nutrition: common sense against genocide. *Journal of interregional statesmanship*, **5**, 182-184 + +Marx, K. (1955-1981). Written works. Volume. 35. Moscow: Politizdat + +Moriarty, P. and Honnery, D. (2019). Energy Accounting for a Renewable Energy Future. *Energies*, **12** (22), 4280 + +Podolinsky, S.A. (1880). *Human work and its relation to energy distribution*. Vol. IV-V. Moscow: Slovo + +Raugei, M., Sgouridis, S., Murphy, D., Fthenakis, V. (2017). Energy Return on Energy Invested (EROEI) for photovoltaic solar systems in regions of moderate insolation: A comprehensive response. *Energy Policy*, **102**, 377-384 + +Safronov, A.F. (2010). EROEI as an indicator of the effectiveness of energy resources extraction and production. *Drilling and Oil*, **12**, 48-51 + +Safronov, A.F. (2011). Methodology for calculating EROEI by the example of developing Sredneviluy gas-condensate field. *Oil and Gas Engineering*, **6**, 197-209. http://www.ogbus.ru + +Temukuyev, T.B. (2014). On the method of calculating EROEI allowing for coefficient of beneficial energy use. *Economic sciences*, **3** (112), 62-66. + +Timiryazev, K.A. (1948). Sun, Life, and Chlorophyll. Selected works in 4 volumes. Public lectures, speeches, and scientific research. Moscow: Ogiz-selhozgiz. + +Umov, N.A. (1916). Physico-chemical model of living matter. Collected works. Vol. III. Moscow: Typolithography of I.N. Kushnerev and K. http://eritage.ru/ras/view/publication/general.html?id=46906367 + +Vernadsky, V.I. (1928). On objectives and organization of the USSR AS applied research work. Leningrad: Publishing House of the USSR Academy of Science \ No newline at end of file diff --git a/samples/texts/6518189/page_2.md b/samples/texts/6518189/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..745d7058937b58412f1a9943f72f7452353c32f8 --- /dev/null +++ b/samples/texts/6518189/page_2.md @@ -0,0 +1,35 @@ +ción entre la energía entregada por una unidad de potencia durante todo el período de operación y la energía invertida teniendo en cuenta el CBPEU durante el mismo período, representará un criterio objetivo. de eficiencia de la unidad de potencia. + +**Palabra clave**: coeficiente de uso de beneficioso de energía primaria; reprocesamiento de combustible; transporte de combustible; ganancia de generación de energía. + +# 1. INTRODUCTION + +On the worldwide market, the price of fossil fuel, as the price of any product, mostly depends on its quality, supply, and demand. Since market competition principles underlie the international pricing policy, the goods production costs, which are generally defined by one of the world currencies or national currency unit, constitute solely a seller's problem. + +If the processes of producing and selling goods are considered in terms of energy, evaluative computations will become more complex. In conditions of the existing international commerce system, the energy computation method that can be implemented only within a particular country, where there are unified laws and regulations, may basically be of interest only to government institutions, who define the technology-related policy. + +# 2. MATERIALS AND METHODS + +A deposit most commonly contains two types of fuel, for instance, coal and methane, natural gas and gas condensate, petroleum, and associated petroleum gas. Here, only one component is principal for deposit developers. + +The total amount of primary energy within the deposit of fossil fuel found from carried out geological exploration, will be determined by formula, J: + +$$ \Sigma Q_o = \sum_{i=1}^{n} B_i \cdot Q_{Hi}^p, \quad (1) $$ + +where $B_i$ – estimated reserves of the $i$th fuel, kg, m³; +$Q_{Hi}^p$ – higher heating value of the $i$th fuel, J/kg, J/m³; +$n$– number of components. + +The stage of geological exploration involves the initiation of technical processes that result in reducing the fuel energy value. + +It is geological exploration energy costs $\Sigma Q_{g.s}$, J first. They can be deducted from the total amount of primary energy or taken into account as other losses, on conversion to the design unit of main fuel component, J/kg, J/m³, by formula: + +$$ Q_{g.s} = \frac{\Sigma Q_{g.s.}}{\sum_{i=1}^{m} B_i}, \quad (2) $$ + +where **B** – main component of fuels. + +When computing, the heating value of the main fuel component will decrease by the corresponding value. + +If there are two in-place components, and only one is used, its reduced heating value will be greater than the intrinsic; it will be defined by the following formula: + +$$ Q_{\Pi}^{o} = \frac{\Sigma Q_{o}}{B}, \quad (3) $$ \ No newline at end of file diff --git a/samples/texts/6518189/page_3.md b/samples/texts/6518189/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..d62af723c29652e1003d591d39c7876480becc43 --- /dev/null +++ b/samples/texts/6518189/page_3.md @@ -0,0 +1,47 @@ +where B – design amount of main natural fuel situated deep in the earth, kg, m³. + +Unused energy of extracted subterranean fuel should be classified as extraction losses. It is determined as the difference between heating value of natural fuel composition and higher heating value of operating composition of the main fuel component: + +$$ \Delta Q_2 = Q_{\Pi}^{0} - Q_{H}^{p}. \quad (4) $$ + +Associated petroleum gas and flammable gas condensate are typically not used. + +If the Earth's interior contains only one component without combustible admixtures, which are lost during fuel extraction, then $Q_{\Pi}^{0} = Q_{H}^{p}$, and formula (1) will be as follows: + +$$ \Sigma Q_o = B \cdot Q_H^p, \quad (5) $$ + +Energy efficiency indicator: + +1. EROEI (energy returned on energy invested) is the ratio between obtained and invested energy: + +2. + +$$ Er = \frac{E_2}{E_1}, \quad (6) $$ + +where *Er* – EROEI; + +*E₂* – energy obtained from fuel or a device transforming the Earth’s, solar, etc. energy, J; + +*E₁* – energy expended to extract (produce) energy *E₂*, J. + +2. EROI (energy return on investment) is the ratio between obtained energy and investments, J/rub.: + +$$ Ei = \frac{E_2}{C}, \quad (7) $$ + +where *Ei* – EROI; + +*E₂* – the same as in formula (6); + +*C* – means spent to extract (produce) energy *E₂*, rub. + +Let the coefficient of beneficial primary energy use (CBPEU) be the ratio between beneficially used energy at a certain stage of the process flow from exploring fuel deposit to utilising energy *Q*ᵢ (in mass or volume unit equivalent) and the considered amount of natural fuel primary energy *Q*Π0: + +$$ \mu_{oi} = \frac{Q_i}{Q_H^o}. \quad (8) $$ + +When determining CBPEU, not only energy loss during operation, but also energy losses on the unit creation, assembly, subsequent dismantling, etc., i.e. all possible energy losses should be taken into account. Actually, these are the computations of how subterranean fuel energy is depreciated until it reaches consumers in the form of electricity or heat and is used by them. + +With such a method of estimation, CBPEU of the entire system from fuel extraction to energy use will be defined by the following formula: + +$$ \mu_{o1} = \frac{Q_1}{Q_{\Pi}^0}, \quad (9) $$ + +where *Q*₁ – beneficially used energy, J/kg, J/m³. \ No newline at end of file diff --git a/samples/texts/6518189/page_4.md b/samples/texts/6518189/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..0e55513c3bbe884d079b309402d70c780e7074f3 --- /dev/null +++ b/samples/texts/6518189/page_4.md @@ -0,0 +1,66 @@ +In general, total primary energy costs, J/kg, J/m³, for a particular stage, can be presented as follows: + +$$ \Sigma Q_{ci} = \sum_{i=1}^{n} Q_{ci}^{c} + \sum_{i=1}^{n} Q_{ci}^{ut} + \sum_{i=1}^{n} Q_{ci}^{op} + \sum_{i=1}^{n} Q_{ci}^{sal} + \sum_{i=1}^{n} Q_{ci}^{oth}, \quad (10) $$ + +where $\sum_{i=1}^{n} Q_{ci}^{c}$ – total primary energy expended on the object capital construction, equipment assem- +bly and disassembly; + +$\sum_{i=1}^{n} Q_{ci}^{ut}$ – total primary energy expended on the equipment fabrication and utilisation; + +$\sum_{i=1}^{n} Q_{ci}^{op}$ – total operational primary energy costs; + +$\sum_{i=1}^{n} Q_{ci}^{sal}$ – total primary energy costs related to people's work and their salary paid; + +$\sum_{i=1}^{n} Q_{ci}^{oth}$ – other total primary energy costs. + +Only operational costs will be defined rather easily; the costs related to people's work are very com- +plicated to calculate, since it is fairly difficult to transform them into heat or energy. It will require the +development of a special methodology. + +Technological system from fuel extraction to electrical and heat energy use can be divided into several +processes: extraction, reprocessing, and transportation of fuel, generation, transportation, distribution, +consumption of electrical and heat energy. When needed, any of these processes can be divided into +parts. + +Let the total energy costs be denoted at: + +$\Sigma Q_{c2}$ – fuel extraction and preparation; + +$\Sigma Q_{c3}$ – fuel transportation and storage; + +$\Sigma Q_{c4}$ – fuel reprocessing; + +$\Sigma Q_{c5}$ – conversion into other types of energy; + +$\Sigma Q_{c6}$ – transmission of electrical and heat energy; + +$\Sigma Q_{c7}$ – distribution of electrical and heat energy; + +$\Sigma Q_{c8}$ – use (consumption) of electrical and heat energy. + +Then the absolute amount of the fuel energy available will accordingly be determined by formulas +(11-13). For fuel after: + +extraction and preparation + +$$ Q_2 = Q_n^o - \sum Q_{c2}; \qquad (11) $$ + +transportation and storage + +$$ Q_3 = Q_n^o - (\sum Q_{c2} + \sum Q_{c3}); \qquad (12) $$ + +reprocessing + +$$ Q_4 = Q_n^o - (\sum Q_{c2} + \sum Q_{c3} + \sum Q_{c4}). \qquad (13) $$ + +The fuel as it is with such available energy $Q_4$ is delivered for conversion into other types of energy – +electrical and heat. The absolute amount of primary energy allowing for previous costs will be deter- +mined by formulas (14 – 17): + +after conversion into other types of energy + +$$ Q_5 = Q_n^o - (\sum Q_{c2} + \sum Q_{c3} + \sum Q_{c4} + \sum Q_{c5}); \qquad (14) $$ + +after transportation by trunk transmission lines (pipelines) + +$$ Q_6 = Q_n^o - (\sum Q_{c2} + \sum Q_{c3} + \sum Q_{c4} + \sum Q_{c5} + \sum Q_{c6}); \qquad (15) $$ \ No newline at end of file diff --git a/samples/texts/6518189/page_5.md b/samples/texts/6518189/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..ba5a535f00afb5b1c9c0f02f61a8f4fa1ad547e3 --- /dev/null +++ b/samples/texts/6518189/page_5.md @@ -0,0 +1,29 @@ +after transportation by distribution electrical transmission lines (pipelines) + +$$Q_7 = Q_n^0 - (\sum Q_{c2} + \sum Q_{c3} + \sum Q_{c4} + \sum Q_{c5} + \sum Q_{c6} + \sum Q_{c7}); \quad (16)$$ + +beneficially used + +$$Q_1 = Q_n^0 - (\sum Q_{c2} + \sum Q_{c3} + \sum Q_{c4} + \sum Q_{c5} + \sum Q_{c6} + \sum Q_{c7} + \sum Q_{c8}). \quad (17)$$ + +To obtain dimensionless values $q_i$, denoting respective shares of losses, both parts of equation (17) should be divided by $Q_n^0$, then formula (17) will be written as: + +$$\begin{gathered} q_{01} = 1 - (q_2 + q_3 + q_4 + q_5 + q_6 + q_7 + q_8) \\ \mu_{01} = 1 - (q_2 + q_3 + q_4 + q_5 + q_6 + q_7 + q_8), \end{gathered} \quad (18)$$ + +or + +i.e. CBPEU for a consumer of electrical (heat) energy is obtained. + +Formulas (11 – 16) can be rewritten correspondingly. + +CBPEU is an indicator that reflects the effectiveness of using energy resources at all stages of the process flow, from fuel extraction to energy consumer inclusive. If needed, it can be calculated for several consecutive stages of energy consumption. + +### 3. LITERATURE REVIEW + +In the late XIX century, S.A. Podolinsky, the Russian scientist, was the first to examine life problems in the energy-related context (Podolinsky, 1880). Commenting on his work, F. Engels noted what problems scientists can face when examining this matter (Marx, 1955-1981). In 1886, L. Boltzmann proposed a thermodynamic analysis of life phenomena (Boltzmann, 1970). Russian academics had their own ideas and suggestions. In 1901, N.A. Umov put forward an idea of the third law of thermodynamics, which would determine the specifics of energy processes in life phenomena (Umov, 1916). K.A. Timiryazev (1948) analysed specific thermodynamic functions of chlorophyll apparatus in plants V.I. Vernadsky (1928) offered to introduce some general unit 'to quantitatively compare all natural productive forces. N.M. Fedorovsky (1935) suggested that mineral resources be classified based on energy principle. A.E. Fersman (1937) employed energy method in his research studies. The issues of economic activity energy analysis interested P.G. Kuznetsov (1994). + +In 1956, King Hubbert, an American scientist, deduced a formula for extracting petroleum in USA. The extraction first rises then remains unchanged for a while, and then starts to be down. At the first and second stages petroleum is cheap, and at the third stage its price begins to raise (King Hubbert, 1956). + +An American biologist Charles Hall proposed a theory of economic activity energy analysis. He introduced a concept of Energy Returned On Energy Invested (EROEI) into scientific use, asserting that predators cannot expend more energy than they receive while hunting. Hall further transferred this idea to petroleum extraction. He divided the amount of energy contained in extracted petroleum by the amount of energy expended on its extraction. Hall, through comparing such indicators of various fields, determined the most perspective of them in terms of energy (Hall, 2008). + +In Russia, researchers under the guidance of A.F. Safronov computed EROEI (ratio between obtained and invested energy) of a specific gas condensate field (Safronov, 2010, 2011), and examined its influence on the CBPEU value (Temukuyev, 2014); the EROI value for coal was determined in Ukraine (Cherevatskyi, 2017). \ No newline at end of file diff --git a/samples/texts/6518189/page_6.md b/samples/texts/6518189/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..ffe5f1b8c58527aabdb7d7f9b2db8a16a98845e5 --- /dev/null +++ b/samples/texts/6518189/page_6.md @@ -0,0 +1,42 @@ +To date, closer attention has been given to using alternative energy sources, what is largely due to their increased efficiency. Hence, the research on increasing the efficiency of using unconventional power sources has become highly relevant. The use of such unconventional sources as energy obtained by employing biotechnologies can be considered among the most appropriate trends for economies (Fiap-shev *et al.*, 2017, 2018). Rather representative is comparison between EROI for wind and solar photo-voltaic power systems (Raugei *et al.*, 2017). As time passes, the attitude towards alternative fuel types changes (IPCC, 2018). It becomes apparent that in future, the EROI values of fossils of most renewable energy sources will decrease (Järvensivu *et al.*, 2018), and the future energy balance can change significantly (Moriarty & Honnery, 2019). A new methodology for estimating EROI (Capellán-Pérez *et al.*, 2019) and standard (De Castro and Capellán-Pérez, 2020) were developed. It is not certain that EROI will be the main decisive factor in the future (Hall, 2017), at the same time a new approach to calculating “corporate” EROI is under review (Celi *et al.*, 2018). + +# 4. RESULTS + +## 4.1. Method for determining a coefficient of renewable sources energy conversion (CRSEC) + +When utilising the energy of renewable resources, CRSEC should be taken as an objective power plant (power unit) efficiency criterion, defined by formula: + +$$ \pi = \frac{Q_1}{Q_{pec}}, \qquad (19) $$ + +where $Q_1$ – energy, supplied by power plant (power unit) over the entire operation period; + +$Q_{pec}$ – primary energy costs (imported energy allowing for CBPEU), obtained from an external source over the entire period of its operation. + +They are defined as follows: + +$$ Q_{pec} = Q_{eq} + Q_{cap} + Q_{op} + Q_{oth}, \qquad (20) $$ + +where $Q_{eq}$ – primary energy costs to manufacture the equipment; + +$Q_{cap}$ – primary energy expended on capital construction, object equipment assembly and disassembly; + +$Q_{op}$ – operational primary energy costs; + +$Q_{oth}$ – other total primary energy costs. + +Other costs should also include labour costs. + +If total energy costs $Q_i$ are to the power unit operation time $\tau$, then specific primary energy losses +$q_i = \frac{q_i}{\tau}$, and $\pi = \frac{q_i}{q_{pec}}$. + +Power unit efficiency factor will be defined by formula: + +$$ \eta = \frac{Q_{un}}{Q_{max}}, \qquad (21) $$ + +where $Q_{un}$ – energy supplied by the power unit to an external consumer; + +$Q_{max}$ – maximum energy theoretically obtainable when there is an ideal power unit over the same period. + +In terms of energy, the system use is justified, when $\pi > 1$, otherwise if $\pi < 1$, it serves no purpose to invest in its development. + +Let the above suggested method be considered for specific energy sources. \ No newline at end of file diff --git a/samples/texts/6518189/page_7.md b/samples/texts/6518189/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..f39d3881e1f2db12570ac07d07d463a448d2c890 --- /dev/null +++ b/samples/texts/6518189/page_7.md @@ -0,0 +1,51 @@ +## 4.2 Hydroelectric power plants (HPPs) + +The interval from the start of operation to overhaul should be taken as the design HPP operation period, and during further operation the time should be counted from the overhaul. In addition, determination of CRSEC during subsequent operation should involve overhaul costs rather than primary costs. For HPPs, with certain data correction, CRSEC can also be determined using formula (19), which will take the following form: + +$$ \pi^{HPP} = \frac{Q_1^{HPP}}{Q_{pec}^{HPP}}, \qquad (22) $$ + +where $Q_1^{HPP}$ – energy supplied by HPP over the design period; + +$Q_{pec}^{HPP}$ – costs of primary energy obtained from an external source over the design period. + +They are determined according to formula: + +$$ Q_{pec}^{HPP} = Q_{eq}^{HPP} + Q_{cap}^{HPP} + Q_{op}^{HPP} + Q_{oth}^{HPP}, \qquad (23) $$ + +where $Q_{eq}^{HPP}$ – primary energy costs to manufacture the equipment; + +$Q_{cap}^{HPP}$ – primary energy costs on capital construction, object equipment assembly and disassembly; + +$Q_{op}^{HPP}$ – primary energy operational costs; + +$Q_{oth}^{HPP}$ – other primary energy costs. + +## 4.3. Solar energy + +For helioplants, CRSEC is defined by formula: + +$$ \pi^{Hel} = \frac{Q_1^{Hel}}{Q_{pec}^{Hel}}, \qquad (24) $$ + +where $Q_1^{Hel}$ – energy supplied by helioplant over the entire period of its operation; + +$Q_{pec}^{Hel}$ – costs of primary energy obtained from an external source over the entire period of its operation. They are defined as follows: + +$$ Q_{pec}^{Hel} = Q_{eq}^{Hel} + Q_{cap}^{Hel} + Q_{op}^{Hel} + Q_{oth}^{Hel}, \qquad (25) $$ + +where $Q_{eq}^{Hel}$ – primary energy expended on manufacturing the equipment; + +$Q_{cap}^{Hel}$ – primary energy expended on construction, assembly and disassembly of helioplant; + +$Q_{op}^{Hel}$ – operational primary energy costs; + +$Q_{oth}^{Hel}$ – other primary energy costs. + +## 4.4. Wind energy + +For wind turbines, CRSEC is determined using the same method as described in the previous cases: + +$$ \pi^W = \frac{W}{Q_{pec}^W}, \qquad (26) $$ + +where $Q_1^W$ – energy obtained from wind turbine over its entire operation period; + +$Q_{pec}^W$ – costs of primary energy from an external source for wind turbine manufacture, construction, and operation. They are determined according to expression: \ No newline at end of file diff --git a/samples/texts/6518189/page_8.md b/samples/texts/6518189/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..dadf26167f91254b7ac62c8d1278614274680b60 --- /dev/null +++ b/samples/texts/6518189/page_8.md @@ -0,0 +1,51 @@ +$$Q_{pec}^{W} = Q_{eq}^{W} + Q_{cap}^{W} + Q_{op}^{W} + Q_{oth}^{W}, \quad (27)$$ + +where $Q_{eq}^W$ – primary energy expended on manufacturing the equipment; + +$Q_{cap}^W$ – primary energy costs for equipment assembly and disassembly; + +$Q_{op}^W$ – operational primary energy costs; + +$Q_{oth}^W$ – other primary energy costs. + +### 4.5. Geothermal energy + +For geothermal power plants, CRSEC is determined using the same method as described in the previous cases: + +$$\pi^{Geo} = \frac{Q_1^{Geo}}{Q_{pec}^{Geo}}, \quad (28)$$ + +where $Q_1^{Geo}$ – energy obtained from geothermal power station over its entire operation period; + +$Q_{pec}^{Geo}$ – costs of primary energy obtained from an external source over the entire period of geo-thermal power station operation. They are defined as follows: + +$$Q_{pec}^{Geo} = Q_{eq}^{Geo} + Q_{cap}^{Geo} + Q_{op}^{Geo} + Q_{oth}^{Geo}, \quad (29)$$ + +where $Q_{eq}^{Geo}$ – primary energy expended on manufacturing the equipment; + +$Q_{cap}^{Geo}$ – primary energy costs for drilling a borehole, equipment assembly and disassembly; + +$Q_{op}^{Geo}$ – operational primary energy costs; + +$Q_{oth}^{Geo}$ – other primary energy costs. + +It is difficult to immediately and fully switch over to an energy method for evaluating the geothermal energy cost, but even a stepwise transition can provide a clear impression in a first approximation of the degree of effectiveness, which a particular system has. + +### 4.6. Waste recycling + +For waste recycling, CRSEC is determined using the same methods as those described in the previous cases: + +$$\pi^{waste} = \frac{Q_1^{waste}}{Q_{pec}^{waste}}, \quad (30)$$ + +where $Q_1^{waste}$ – energy obtained from recycled waste over the entire period of recycling facility operation (when calculating it, the energy should also be taken into account, expended on removing metal, glass, and other materials from domestic waste); + +$Q_{pec}^{waste}$ – actual costs of primary energy obtained from an external source over its entire operation period. They are defined as follows: + +$$Q_{pec}^{waste} = Q_{eq}^{waste} + Q_{cap}^{waste} + Q_{op}^{waste} + Q_{oth}^{waste}, \quad (31)$$ + +where $Q_{eq}^{waste}$ – primary energy expended on manufacturing the equipment; + +$Q_{cap}^{waste}$ – primary energy costs for construction, assembly and disassembly of recycling facility; + +$Q_{op}^{waste}$ – operational primary energy costs; + +$Q_{oth}^{waste}$ – other primary energy costs. \ No newline at end of file diff --git a/samples/texts/6518189/page_9.md b/samples/texts/6518189/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..3d76be8e9cd24d30e0a8e362fc976277aa4ece72 --- /dev/null +++ b/samples/texts/6518189/page_9.md @@ -0,0 +1,41 @@ +## 4.7. Biofuel production + +The formula to determine CRSEC as applied to biofuel production per 1 ha of land over a design period of 1 year is as follows: + +$$ \pi^{bio} = \frac{q_{1}^{bio}}{q_{pec}^{bio}}, \quad (32) $$ + +where $q_1^{bio}$ – specific energy obtained from biofuel over the design period of production, J/(ha year); +$q_{pec}^{bio}$ – costs of specific primary energy to produce biofuel over the design period, J/(ha year); + +Actual energy costs are defined by the following formula: + +$$ q_{pec}^{bio} = q_{eq}^{bio} + q_{cap}^{bio} + q_{op}^{bio} + q_{oth}^{bio}, \quad (33) $$ + +where $q_{eq}^{bio}$ – specific primary energy expended on manufacturing the equipment; +$q_{cap}^{bio}$ – costs of specific primary energy for constructing the object to reprocess biomass, assemble and dismantle its equipment; +$q_{op}^{bio}$ – operational costs of specific primary energy for machinery, fertilizers, pesticides, etc. over the design period; +$q_{oth}^{bio}$ – other specific primary energy costs. + +Other costs should also include labour costs. + +For heat pump installation (HPI) CRSEC is determined using the same method as described in the previous cases: + +$$ \pi^{HPI} = \frac{Q_{1}^{HPI}}{Q_{pec}^{HPI}}, \quad (34) $$ + +where $Q_1^{HPI}$ – energy delivered to HPI consumer over the entire period of HPI operation; +$Q_{pec}^{HPI}$ – costs of primary energy obtained from an external source over the entire operation period. They are determined as follows: + +$$ Q_{pec}^{HPI} = Q_{eq}^{HPI} + Q_{cap}^{HPI} + Q_{op}^{HPI} + Q_{oth}^{HPI}, \quad (35) $$ + +where $Q_{eq}^{HPI}$ – primary energy expended on manufacturing the equipment; +$Q_{cap}^{HPI}$ – costs of primary energy over the entire operation period from assembling to disassembling HPI; +$Q_{op}^{HPI}$ – operational primary energy costs; +$Q_{oth}^{HPI}$ – other primary energy costs. + +## 4.8. Nuclear power plants (NPPs) + +For NPPs that utilise limited reserves of nuclear fuel, it is possible to define an energy breeding gain through formula (19): + +$$ \pi^{NPP} = \frac{Q_{1}^{NPP}}{Q_{pec}^{NPP}}, \quad (36) $$ + +where $Q_1^{NPP}$ – energy, obtained from NPP over the entire period of its operation; \ No newline at end of file diff --git a/samples/texts/7407615/page_1.md b/samples/texts/7407615/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..1f0af2db2937858100506985c49f2e160ec6aa50 --- /dev/null +++ b/samples/texts/7407615/page_1.md @@ -0,0 +1,18 @@ +Searching for Signatures of Quantum Gravity in Quantum Gases. + +Simon A. Haine¹, * + +¹Department of Quantum Science, Research School of Physics and Engineering, +The Australian National University, Canberra, Australia + +We explore the possibility of testing the quantum nature of the gravitational field with an ensemble of ultra-cold atoms. The use of many microscopic particles may circumvent some of the experimental obstacles encountered in recent proposals involving a pair of particles with mesoscopic mass. We employ multi-parameter estimation techniques, including the quantum and classical Fisher information to provide a criteria for the observability of the quantum effects, and compare to other recently proposed schemes. Crucially, we find that by preparing the appropriate initial state, interactions mediated via a quantum-valued gravitational field provide a signature that is distinct from classical gravitational interactions. We find that a test with ultra-cold atoms would be challenging, but not implausible with moderate improvements to current experimental techniques. + +While a full theoretical quantum treatment of quantum gravity remains elusive, *effective* quantum field theories of gravity have been developed, by treating the quantum field as a perturbation from the classical solution [1–3]. Low-energy laboratory-scale experiments are well within this perturbative regime [3]. However, whether or not the gravitational field displays quantum properties, that is, if a quantum treatment is required at all, is a question that demands experimental resolution [4], with several proposed theories stating that the gravitational field may be fundamentally classical [5–8]. There have been several recent proposals for laboratory scale experiments to discriminate between these theories and a theory with a quantised gravitational field [3, 7–13], and experiments that rule out classes of semi-classical gravity [14, 15]. Recently, it has been argued that if the gravitational interaction between two quantum systems can create entanglement, then the gravitational field must act as a quantum mediator [7, 8, 16–18]. This principle is the basis of recent experimental proposals for testing this phenomenon by searching for gravitational induced entanglement between two freely-falling micron-sized test masses [19, 20], and more recently, optomechanical systems [21, 22]. While these proposals are feasible with a modest improvement in current experimental capabilities, there are still significant experimental challenges associated with them. As such, it is worth exploring if the differences that other systems provide can prove advantageous. + +In this paper, we investigate the gravitational interaction amongst an ensemble of ultra-cold atoms. Ultra-cold atoms potentially provide a number of advantages over other systems when performing ultra-precise measurements. Of particular relevance is that the response of an atom to electromagnetic and gravitational fields is fundamentally locked to physical constants, which provides a long-term stability which no macroscopic system can. This property differentiates atoms from quantum systems formed from a particular macroscopic objects, + +such as a nano-diamond or a micro-mechanical oscillator, where the mass or spring-constant may have a slight drift over the duration required to collect sufficient experimental statistics. This is of particular importance when searching for a weak effect that will require a large number of repetitions, and is the reason why atoms are the system of choice for atomic clocks and ultra-precise gravimeters [23, 24]. + +Bose et al. [19] and Marletto and Vedral [20] have previously shown that quantum entanglement between two systems which is generated by the gravitational field is indirect evidence of quantum gravity. Here, we take a slightly different approach, which is to consider how the effect of a quantum valued gravitational field would differ from a classical valued field. We note that in our approach, the presence of a quantum-valued field results in a Hamiltonian that can generate entanglement, while the Hamiltonian resulting from a classical-valued gravitational field cannot, implying logical consistency of our approach with the entanglement witness approaches. We find that if the gravitational interaction is mediated by a quantum-valued gravitational field, it presents different signatures to that of a classical-valued gravitational field. By carefully engineering the many-particle quantum state of the system, we can maximise the effect of the observability of these signatures. Other than the use of many particles, an important difference between this work and the work of Bose et al. [19] and Marletto and Vedral [20] is that instead of searching for an entanglement witness, we formulate the experiment as a multi-parameter estimation problem, which, in principle allows us to search for arbitrarily weak interactions by performing many repetitions of the experiment. Crucially, we show that classical theories of gravity, and decoherence effects, produce signatures in the measured probability distribution that are distinct from what an interaction mediated by a quantum-valued gravitational field predicts. + +* simon.a.haine@gmail.com \ No newline at end of file diff --git a/samples/texts/7407615/page_2.md b/samples/texts/7407615/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..c2a90792bb4548473d23833fb56c258666382018 --- /dev/null +++ b/samples/texts/7407615/page_2.md @@ -0,0 +1,43 @@ +# I. GRAVITATIONAL INTERACTION FOR AN ENSEMBLE OF TWO-LEVEL ATOMS + +We assume an ensemble of ultracold bosonic atoms with two electronic states $|a\rangle$ and $|b\rangle$. Following [19, 20], rather than attempting to present a quantum description of the gravitational field, we describe the gravitational interactions between the particles directly. The contribution to the Hamiltonian due to the gravitational interaction energy is given by + +$$ \hat{H}_G = -\frac{1}{2} G m^2 \iint : \hat{n}(\mathbf{r}) \hat{n}(\mathbf{r}') : \frac{1}{|\mathbf{r}-\mathbf{r}'|} d^3\mathbf{r} d^3\mathbf{r}', \quad (1) $$ + +where *m* is the mass of each individual particle, and *G* is Newton's constant for universal gravitation. The operator $\hat{n}$ is given by + +$$ \hat{n}(\mathbf{r}) = \hat{\psi}_a^\dagger(\mathbf{r})\hat{\psi}_a(\mathbf{r}) + \hat{\psi}_b^\dagger(\mathbf{r})\hat{\psi}_b(\mathbf{r}), \quad (2) $$ + +and represents the total particle density, and $\hat{\psi}_j(\mathbf{r})$ is the standard bosonic field operator annihilating a particle from state $|j\rangle$ at point $\mathbf{r}$, obeying the standard commutation relations + +$$ [\hat{\psi}_i(\mathbf{r}), \hat{\psi}_j^\dagger(\mathbf{r}')] = \delta_{ij}\delta(\mathbf{r}-\mathbf{r}'), \quad (3) $$ + +and ‘::’ denotes normal ordering. We have ignored the post-newtonian correction and first quantum correction, as for our system they are of orders $10^{-40}$ and $10^{-57}$ smaller [3]. It is important to note that in writing Eq. (1), we have implicitly assumed that the gravitational field can be quantum valued. Assuming that the interaction between the atoms is mediated by a gravitational field (i.e., not a direct non-local action at a distance), this field must be quantum valued. Specifically, Eq. (1) can be re-written as + +$$ \hat{H}_G = \frac{m}{2} \int : \hat{n}(\mathbf{r}) \hat{\Phi}_G(\mathbf{r}) : d^3\mathbf{r}, \quad (4) $$ + +where + +$$ \hat{\Phi}_G(\mathbf{r}) = -Gm \int \hat{n}(\mathbf{r}') \frac{1}{|\mathbf{r}-\mathbf{r}'|} d^3\mathbf{r}', \quad (5) $$ + +is the (operator valued) gravitational potential due to source masses at location $\mathbf{r}'$, and the factor of $\frac{1}{2}$ removes double-counting. That is, the gravitational potential is quantum valued as a result of the quantum uncertainty in the location of the source masses. + +Consider, in contrast, a classical-valued gravitational potential $\Phi_C(\mathbf{r})$, originating from some classical source. Placing our quantum test-particles in this field results in the gravitational potential + +$$ \hat{H}_C = m \int \hat{n}(\mathbf{r}) \Phi_C(\mathbf{r}) d^3\mathbf{r}. \quad (6) $$ + +Clearly, this gravitational interaction energy can also be quantum valued, but only due to the quantum-ness of the + +FIG. 1. Scheme for probing the gravitational interaction with ultra-cold atoms. (a): An ensemble of two level atoms are all prepared in state $|a\rangle$. (b): A state preparation unitary operation $\hat{U}_0$ distributes the atoms between two modes, and may create non-trivial quantum correlations. (c): The two modes are spatially separated and evolve for time $t$ under the gravitational interaction $\mathcal{I}_G$. (d): A second unitary operation $\hat{U}_2$ interferes the two modes, before the number in each mode is measured (e). + +location of the particles, rather than the quantum-valued nature of the gravitational source (and therefore field) as in Eq. (1). A theory of gravity that does not allow for a quantum valued field would then be described by a Hamiltonian such as Eq. (6), with the value of $\Phi_C(\mathbf{r})$ depending on how a classical valued field arises from a quantum distribution of source masses. For example, in a semi-classical theory of gravity [5, 12, 25, 26], + +$$ \Phi_C(\mathbf{r}) = \langle \hat{\Phi}_G(\mathbf{r}) \rangle. \quad (7) $$ + +However, we do not restrict ourself to this case, and simply consider any classical gravitational field, without making any assumptions on how the value of $\Phi_C(\mathbf{r})$ is determined from the quantum distribution of source masses. Our task is then to devise an experiment that can distinguish between the effects of Eq. (1) and Eq. (6). + +It has been shown that quantum entanglement can only be generated via a quantum field [19, 20, 27, 28]. We note that the Hamiltonian Eq. (1) is capable of generating entanglement between spatially separated particles, while Eq. (6) can only perform local operations. Therefore, although we are not explicitly searching for the presence of entanglement as the signature of quantum gravity, distinguishing between the effects of two Hamiltonians, one which is capable of generating entanglement, and one which is not, is logically equivalent. + +# II. SCHEME TO SEARCH FOR SIGNATURES OF QUANTUM GRAVITY WITH ULTRA-COLD ATOMS + +Our scheme is based on a standard Mach-Zehnder interferometer, and is illustrated in Fig. 1. Each atom is initially prepared in state $|a\rangle$, before being placed in an equal coherent superposition of $|a\rangle$ and $|b\rangle$ at time \ No newline at end of file diff --git a/samples/texts/7407615/page_3.md b/samples/texts/7407615/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..8da6298507a8fb865886df0c097d8d9ef4783cdd --- /dev/null +++ b/samples/texts/7407615/page_3.md @@ -0,0 +1,53 @@ +$t=0$ via a $\pi/2$ pulse implemented via a radio-frequency transition. The two components are then spatially separated via a state dependent potential. Alternatively, if a two-photon Raman transition is used instead of a radio-frequency transition, the need for a potential is removed as the two components will separate due to the momentum transfer of the Raman transition [29, 30], which would remove any systematic error due to slight uncertainties in the behaviour of the potential. After evolving under the effects of the gravitational interaction for time $t$, the two components are then brought back together and recombined by a second $\pi/2$ pulse [31], and the number in each component is measured. Quantitatively, we describe the evolution as follows: The state $|\Psi_i\rangle = \hat{U}_0|\Psi_0\rangle$ is prepared by splitting the initial state into two components. We also allow for the possibility that $\hat{U}_0$ may contribute additional operations during this step, such as creating entanglement between the particles. The atoms interact gravitationally, via the interaction $\mathcal{I}_G[|\Psi_i\rangle]$, which may be a unitary mapping, or perhaps some operation that causes decoherence, before a second unitary operation $\hat{U}_2$ is applied, and a measurement $\hat{M}$ is made. + +If the atoms in state $|j\rangle$ remain localised in spatial mode $u_j(\mathbf{r}, t)$, then its reasonable to make the approximation + +$$ \hat{\psi}_j(\mathbf{r}) = \hat{j} u_j(\mathbf{r}, t) \quad (8) $$ + +with + +$$ [\hat{j}, \hat{k}^\dagger] = \delta_{jk}, \quad [\hat{j}^\dagger, \hat{k}^\dagger] = [\hat{j}, \hat{k}] = 0 \quad (9) $$ + +Inserting this into Eq. (1) gives + +$$ \hat{H} = \kappa_{aa}(t)\hat{a}^{\dagger}\hat{a} + \kappa_{bb}(t)\hat{b}^{\dagger}\hat{b} + 2\kappa_{ab}(t)\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b} \quad (10) $$ + +where + +$$ \kappa_{ij} = -\frac{1}{2} G m^2 \iint |u_i(\mathbf{r})|^2 |u_j(\mathbf{r}')|^2 \frac{1}{|\mathbf{r} - \mathbf{r}'|} d^3\mathbf{r} d^3\mathbf{r}'. \quad (11) $$ + +Meanwhile, making the same substitution in Eq. (6) gives + +$$ \hat{H}_C = v_a \hat{a}^\dagger \hat{a} + v_b \hat{b}^\dagger \hat{b} \quad (12) $$ + +where + +$$ v_j = m \int |u_j(\mathbf{r})|^2 \Phi_c(\mathbf{r}) d^3\mathbf{r}. \quad (13) $$ + +If the wavepackets remain spatially separated by significantly more than their spatial extent for the majority of the evolution, $\kappa_{ab} \ll \kappa_{aa}, \kappa_{bb}$. This means that the gravitational potential energy of the system will be larger when all of the atoms are in one mode compared to the case when they are evenly distributed across both modes (and therefore occupying a larger volume). Setting $\kappa_{aa} = \kappa_{bb} = \kappa_0$, $\kappa_{ab} = 0$, the unitary operator for the evolution given by Eq. (10) is + +$$ \hat{U}_Q = \exp(i\alpha\hat{A}). \quad (14) $$ + +where $\alpha = \kappa_0 t / \hbar$, and $\hat{A} = \hat{a}^\dagger\hat{a}\hat{a} + \hat{b}^\dagger\hat{b}\hat{b}$. In this case, the gravitational interaction is simply described by $\mathcal{I}_G[|\Psi_i\rangle] = U_Q|\Psi_i\rangle$. + +Meanwhile, the unitary evolution operator resulting from Eq. (10) is + +$$ \hat{U}_C = \exp \left[ i(\beta \hat{J}_z + \gamma \hat{N}_+) \right] \quad (15) $$ + +where $\beta = (v_b - v_a)t$, $\gamma = -\frac{1}{2}(v_a + v_b)t$, $\hat{N}_+ = \hat{a}^\dagger\hat{a} + \hat{b}^\dagger\hat{b}$, and $\hat{J}_z = \frac{1}{2}(\hat{a}^\dagger\hat{a} - \hat{b}^\dagger\hat{b})$, and $\mathcal{I}_G[|\Psi_i\rangle] = U_C|\Psi_i\rangle$. If we assume a mean-field gravity model, then $\Phi(\mathbf{r}) = \langle\hat{\Phi}(\mathbf{r})\rangle$, and we can estimate the expected values of $v_a$ and $v_b$. However, we do not restrict ourself to this case, and treat these as unknown parameters to allow for the possibility of other models of classical gravity. Our task of demonstrating the quantum valued nature of the gravitational field can now be summarised as inferring a non-zero value of $\alpha$, while distinguishing this from the effects of a possibly non-zero, unknown value of $\beta$ and $\gamma$. Fortunately, we can ignore $\gamma$, as for a fixed total number of particles, this parameter results only in a global phase shift to the state, which is undetectable. We therefore estimate the smallest value of $\alpha$ that can be detected in the presence of the unknown nuisance parameter $\beta$. We can place a bound on our ability to determine the value of $\alpha$ by introducing the quantum Fisher covariance matrix [32], with elements + +$$ F'_{i,j} = 2 (\langle\partial_i\Psi|\partial_j\Psi\rangle + \langle\partial_j\Psi|\partial_i\Psi\rangle - 2\langle\Psi|\partial_i\Psi\rangle\langle\partial_j\Psi|\Psi\rangle). \quad (16) $$ + +for {$i,j = \{\alpha,\beta\}$}. In general, the quantum Fisher covariance matrix resulting from the state + +$$ |\Psi\rangle = \hat{U}_Q \hat{U}_C |\Psi_i\rangle, \quad (17) $$ + +contains off-diagonal terms, and therefore our ability to distinguish the presence of $\alpha$ from the possible presence of $\beta$ is hindered. However, by carefully choosing our initial state, we can find a diagonal quantum Fisher covariance matrix with large diagonal terms. Introducing the collective spin operators $\hat{J}_k = \frac{1}{2}\mathbf{a}^\dagger\sigma_k\mathbf{a}$, where $\mathbf{a} = (\hat{a}, \hat{b})^T$ and $\sigma_k$ is the $k$th Pauli matrix, and denoting $|m_k\rangle$ as the eigenstate of $\hat{J}_k$ with eigenvalue $m_k$, we find that the state + +$$ |\Psi_i\rangle = \frac{1}{\sqrt{2}} [ |0_z\rangle + \frac{1}{\sqrt{2}} (|(\mathrm{N}/2)_z\rangle + |(-\mathrm{N}/2)_z\rangle ) ] , \quad (18) $$ + +leads to a quantum Fisher covariance matrix with elements $F_{\alpha,\alpha} = N^4/4$, $F_{\beta,\beta} = 2N^2$, and $F_{\alpha,\beta} = F_{\beta,\alpha} = 0$. This state is shown in Fig. 2. The lack off-diagonal terms means that the matrix is trivial to invert, and we should be able to infer the presence of a non-zero $\alpha$ as long as + +$$ \alpha \gtrsim \frac{1}{\sqrt{kF_{\alpha,\alpha}}} = \frac{2}{\sqrt{kN^2}}, \quad (19) $$ + +where $k$ is the number of repetitions of the experiment. We note that as this state is a superposition of maximal and minimal eigenstates of $\hat{A}$, it is the state that maximises $F_{\alpha,\alpha}$ for $N$ particles. \ No newline at end of file diff --git a/samples/texts/7407615/page_4.md b/samples/texts/7407615/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..8b470edcba0b88ee9560b0b64b648cc3ad7e215b --- /dev/null +++ b/samples/texts/7407615/page_4.md @@ -0,0 +1,52 @@ +FIG. 2: The Husimi-Q function $Q(\theta, \phi)$ for the state given in Eq. (18) (a) and Eq. (21) (b). The Husimi-Q function is defined as $Q(\theta, \phi) = |\langle\xi(\theta, \phi)|\Psi\rangle|^2$, where $\|\xi(\theta, \phi)\rangle = \exp(i\phi J_z) \exp(i\theta J_x)(N/2)_z$ is a coherent spin state. The projection into the $\hat{J}_x$, $\hat{J}_y$ and $\hat{J}_z$ basis are also shown for Eq. (18) (c, e, g) and Eq. (21) (d, f, h) respectively. + +The question remains how to engineer this state, and +to find a measurement scheme that saturates the Quan- +tum Cramer-Rao bound (QCRB) [32–35]. While stan- +dard atom-optics techniques cannot easily produce Eq. +(18), we can create a state that performs nearly as well +via the well-studied one-axis twisting (OAT) technique +[36–43]. In particular, we envisage a controllable one-axis +twisting interaction via an optically induced nonlinearity +as demonstrated in [40, 44–46]. Starting with all the par- +ticles in mode $|a\rangle$, ie, $|\Psi_0\rangle = (\hat{a}^\dagger)^N / \sqrt{N!} |0\rangle = |(N/2)_z\rangle$, +and applying $\pi/2$ rotation about the $\hat{J}_x$ axis, followed +by a one-axis twisting interaction for period $\tau$, followed +by a second $\pi/2$ rotation results in the state preparation +unitary + +$$ +\hat{U}_0 = \exp(i\hat{J}_x \frac{\pi}{2}) \exp(i\chi\tau \hat{J}_z^2) \exp(i\hat{J}_x \frac{\pi}{2}). \quad (20) +$$ + +Setting $\chi\tau = \frac{\pi}{4}$ gives the state shown in Fig. 2 b, which is + +$$ +|\Psi_i\rangle = \hat{U}_0|\Psi_0\rangle = N e^{i\frac{N}{2}\pi} \left(|\eta_x\rangle + e^{-i\frac{3\pi}{4}}|\eta_z\rangle\right), \quad (21) +$$ + +where + +$$ +|\eta_x\rangle = \frac{1}{\sqrt{2}} (|(N/2)_x\rangle + |(-N/2)_x\rangle), \quad (22a) +$$ + +$$ +|\eta_z\rangle = \frac{1}{\sqrt{2}} \left( |(N/2)_z\rangle - e^{i\frac{N}{2}\pi} |(-N/2)_z\rangle \right), \quad (22b) +$$ + +and the normalisation factor $\mathcal{N} \approx \frac{1}{\sqrt{2}}$ as a result of +$|\langle\eta_x|\eta_z\rangle| \ll 1$ for $N \gg 1$. This state has $F_{\alpha,\alpha} = N^4/4 - O(N^3)$, $F_{\beta,\beta} = 2(N^2+N)$, and $F_{\alpha,\beta} = 0$. However, the task remains to find a measurement that saturates the QCRB. That is, the obtainable sensitivity from a particular measurement is given by the diagonal terms of the covariance matrix, which can be calculated from the inverse of the *classical* Fisher information matrix + +$$ +\Delta\alpha^2 = [\mathcal{F}^{-1}]_{\alpha,\alpha}, \qquad (23) +$$ + +where $\mathcal{F}$ is the classical Fisher information matrix with elements + +$$ +\mathcal{F}_{ij} = \sum_m \frac{\partial_i P_m \partial_j P_m}{P_m}. \qquad (24) +$$ + +Following [47–49], we can saturate the QCRB by apply- +ing the unitary operator $\hat{U}_2 = \hat{U}_0^\dagger$ after the interrogation \ No newline at end of file diff --git a/samples/texts/7407615/page_5.md b/samples/texts/7407615/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..aa4c97e714401de0f4d9e7c6c3755e3de8057bcf --- /dev/null +++ b/samples/texts/7407615/page_5.md @@ -0,0 +1,31 @@ +FIG. 3. (a) The projection into the $\hat{J}_z$ basis, $P(J_z)$ for the state $|\Psi_f\rangle = \hat{U}_0^\dagger \hat{U}_Q \hat{U}_C \hat{U}_0 |\Psi_0\rangle$, for $\alpha = 0, \beta = 0$. (b) $\partial_\alpha P(J_z)$ for $\alpha = 10^{-8} \ll 1/\sqrt{F_{\alpha,\alpha}}$. (c) $\partial_\beta P(J_z)$ for $\beta = 10^{-8} \ll 1/\sqrt{F_{\beta,\beta}}$. $N = 100$ for all frames. + +time, and then measuring in a basis that projects into $|\Psi_0\rangle$, which can be achieved by measuring in the $\hat{J}_z$ basis. This results in a classical Fisher covariance matrix with elements $F_{ij} = F_{ji}$, and therefore we can achieve the sensitivity given by Eq. (19). To illustrate how the effect of two potential models of gravity can be distinguished, figure 3 shows $\partial_k P_{m_z}$ for $k = \alpha, \beta$. A small, non-zero value of $\alpha$ results in transfer of probability from $P_{N/2}$ to $P_{-N/2}$, while a small, non-zero value of $\beta$ results in transfer to a distribution of values around $m_z \approx 0$. + +### III. EFFECTS OF DE-COHERENCE + +Another potential effect that our measurement must be able to discriminate is the possibility that the gravitational field causes decoherence [6–8, 15, 50]. That is, superpositions of different gravitational source configurations suffer de-phasing. We must also allow for the possibility of other unaccounted sources of decoherence, as this could cause a signature in the detection signal that is falsely attributed to a non-zero value of $\alpha$. We account for the possibility of decoherence (due to gravitational or other effects) phenomenologically by introducing an additional parameter or unknown magnitude $\delta_\Gamma$, such that the effect of the de-phasing interaction on our initial state is $\mathcal{I}_G[\hat{\rho}_i] = \exp(\delta_\Gamma \mathcal{L}[\Gamma])\hat{\rho}_i \equiv \hat{\rho}$, where + +$$ \mathcal{L}[\hat{\Gamma}] \hat{\rho}_i = \hat{\Gamma} \hat{\rho}_i \hat{\Gamma} - \frac{1}{2} (\hat{\Gamma}^2 \hat{\rho}_i + \hat{\rho} \hat{\Gamma}^2), \quad (25) $$ + +such that + +$$ \hat{\rho} = \sum_n \sum_m \exp(-\delta_\Gamma (\lambda_n - \lambda_m)^2) |n\rangle\langle n| \hat{\rho}_i |m\rangle\langle m|, \quad (26) $$ + +where $\hat{\rho}_i = |\Psi_i\rangle\langle\Psi_i|$, and $\hat{\Gamma}|n\rangle = \lambda_n|n\rangle$ [51–53]. By setting $\hat{\Gamma} = \hat{A}$, the eigenstates represent states with well-defined gravitational interaction energy. That is, off-diagonal terms representing superpositions of states with a large difference in their gravitational interaction energy will suffer greater decoherence. Another possibility is that it is just superpositions of the source potential, rather than the total interaction energy that decohere, which could be accounted for by setting $\hat{\Gamma} = \hat{J}_z$. This choice of $\hat{\Gamma}$ would also describe decoherence due to the presence of a fluctuating field interacting with a non-vanishing difference in the magnetic dipole moment of the two atomic species. + +While it is challenging to compute the full quantum Fisher covariance matrix including $\delta_A$ and $\delta_{J_z}$, it is straight-forward to numerically compute the classical Fisher covariance matrix. Unfortunately, the state preparation and measurement procedure presented in the previous section ($\hat{U}_2 = \hat{U}_0^\dagger$, followed by measurement in the $\hat{J}_z$ basis) leads to a non-diagonal Fisher covariance matrix for the parameters $\alpha, \beta, \delta_A$ and $\delta_{J_z}$. In fact, setting $\hat{\Gamma} = \hat{A}$, results in $\partial_{\delta_A} P(J_z)$ with nearly identical profile to $\partial_\alpha P(J_z)$. Fortunately, an alternate choice of $\hat{U}_2$ remediates the situation. Specifically, choosing $\hat{U}_2 = \hat{U}_0$ (rather than $\hat{U}_0^\dagger$) and computing the Fisher covariance matrix numerically results in values of $\mathcal{F}_{\alpha,k} \ll \mathcal{F}_{\alpha,\alpha}$ for $k \neq \alpha$, and as such can be inverted to find $[\mathcal{F}]_{\alpha,\alpha}^{-1} \approx F_{\alpha,\alpha}$. In figure 4 we show $P(J_z)$ and $\partial_k P(J_z)$ for $k = \alpha, \beta, \delta_A$, and $\delta_{J_z}$. Again, we see that $\beta, \delta_A$, and $\delta_{J_z}$ give signatures that are easy to distinguish from $\hat{U}_Q$. An additional benefit of using $\hat{U}_2 = \hat{U}_0$ (rather than $\hat{U}_2 = \hat{U}_0^\dagger$) is that the final state has a maximally bi-modal probability distribution and is therefore optimally robust to detection noise such that our inference of $\alpha$ saturates the noisy quantum Cramer-Rao bound, as introduced in [54]. + +### IV. EXPERIMENTAL CONSIDERATIONS + +We can estimate the magnitude of $\alpha$ by assuming $u_{a(b)}(\mathbf{r})$ are Gaussian wave-packets of the form + +$$ |u_{a(b)}(\mathbf{r})|^2 = \frac{1}{\sigma^3 \pi^{3/2}} \exp \left[ -\frac{(\mathbf{r} - (+)x_0 \hat{\mathbf{x}})^2}{\sigma^2} \right]. \quad (27) $$ + +Inserting this into Eq. (11) gives + +$$ \alpha = \frac{tGm^2}{\hbar\sigma\sqrt{\pi}}, \quad (28) $$ + +where we have assumed that $x_0 \gg \sigma$ such that we can ignore the contribution from $\kappa_{ab}$. Using Eq. (19), the minimum number of particles required to observe signatures of quantum gravity is then + +$$ N = \sqrt{\frac{2\hbar\sigma\sqrt{\pi}}{\sqrt{k}Gm^2t}}. \quad (29) $$ \ No newline at end of file diff --git a/samples/texts/7407615/page_6.md b/samples/texts/7407615/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..0888908612408f3f7d93bddf3dc677e4a8948b83 --- /dev/null +++ b/samples/texts/7407615/page_6.md @@ -0,0 +1,11 @@ +FIG. 4. (a) The projection into the $\hat{J}_z$ basis, $P(J_z)$ for the state $|\Psi_f\rangle = U_0U_QU_CU_0|\Psi_0\rangle$, for $\alpha=0, \beta=0$. (b) $\partial_\alpha P(J_z)$ for $\alpha = 10^{-8} \ll 1/\sqrt{F_{\alpha,\alpha}}$. (c) $\partial_\beta P(J_z)$ for $\beta = 10^{-8} \ll 1/\sqrt{F_{\beta,\beta}}$. (d) $\partial_{\delta_A} P(J_z)$ for $\hat{\Gamma} = \hat{A}$, and $\delta_A = 10^{-8}$. (e) $\partial_{\delta_{J_z}} P(J_z)$ for $\hat{\Gamma} = \hat{J}_z$, and $\delta_{J_z} = 10^{-8}$ ($N=100$ for all frames). + +For realistic experimental parameters, ($\sigma = 50 \mu m$, $t = 1s$, $k = 10^5$) and considering $^{87}$Rb, the required number of particles is $N \approx 5 \times 10^9$, which is 2 orders of magnitude more than are used in current state-of-the-art precision sensing experiments. These parameters result in an atomic density of $\sim 4 \times 10^{13} \text{ cm}^{-3}$, which is considerably less than the density in the precision atom-interferometry experiment reported in [55]. If we consider more massive atoms such as Ytterbium, and the possibility of employing di-atomic molecules, then the required number of particles is reduced further. Additionally, increasing the interrogation time ($t$), or the number of repetitions ($k$) are also not inconceivable. The number of repetitions $k = 10^5$ equates to approximately 1 day of continuous operation (with $t = 1$s), or 10 days including 10s of state preparation time. This duration of data collection is not uncommon in atomic squeezing experiments, such as [38], for example. Although the preparation of exotic quantum states such as Eq. (21) has not been demonstrated + +in large ensembles of particles, the field is currently in a state of rapid progress [56–59]. + +Obviously, when searching for such minuscule signals, it is important to consider sources of systematic error. Linear shifts in energy such as Zeeman and Stark shifts due to fluctuating electric and magnetic fields, as well as fluctuations in any confining potentials could potentially be orders of magnitude larger than the gravitational signature. However, these shifts are all single-particle effects, and as our multi-parameter analysis reveals, with the appropriate choice of quantum state, these effects de-couple from the gravitational signature that we are searching for. That is, these effects would couple into the value of $\beta$ or $\delta_{J_z}$, which cause a change in the measured probability distribution that is entirely distinguishable from $\alpha$. A potentially much more serious issue is imperfect knowledge of the inter-particle interactions due to short-range van der Waals interaction. For ultra-cold quantum gases, these interactions are dominated by s-wave scattering, which can be manipulated via a Feshbach resonance [60], or compensated for by an optically-induced atom-atom interaction [58]. However, any imperfect knowledge of the magnitude of this interaction will contribute to a term in Eq. (10) that has the same form as the gravitational interaction. That is, they contribute to the coefficient of $\hat{A}$ and therefore are indistinguishable from a change in $\alpha$. To successfully perform this experiment at the precision required to observe gravitational effects, the electromagnetic interaction will need to be measured to much higher precision than the current state-of-the-art [61]. Interestingly, the scheme provided in this paper may be capable of a very sensitive measurement of the scattering-length, and thus removing this source of systematic error. Alternatively, as this coefficient has different scaling with $\sigma$ compared to $\alpha$. That is, as gravity is a long-range force, $\alpha \propto \sigma^{-1}$. However, the magnitude of s-wave interactions that generally dominate ultra-cold atomic gases scale as $\sigma^{-3}$, so by varying the value of $\sigma$ it may be possible to isolate the two contributions. That is, doubling $\sigma$ reduces $\alpha$ by a factor of 2, but will decrease the effect of the atomic interactions by a factor of 8. + +V. DISCUSSION + +It is worth discussing how a scheme involving the use of many particles differs from the schemes proposed by Bose et al. [19], and Marletto and Vedral [20], and how these differences might prove advantageous. Firstly, the use of many particles allows for a far greater space of possible quantum states, and therefore a much greater capacity to tailor the specific state to the particular requirements. Formulating the problem in the language of multi-parameter estimation, rather than searching for an entanglement witness, means that, in principle, arbitrary small signals can be found by performing more repetitions of the experiment to increase precision, assum- \ No newline at end of file diff --git a/samples/texts/7407615/page_7.md b/samples/texts/7407615/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..6ad7c68ef074e1c0a7ce1bba0cbabad8bda2c985 --- /dev/null +++ b/samples/texts/7407615/page_7.md @@ -0,0 +1,43 @@ +ing long-term stability can be achieved. Additionally, the freedom in the choice of the quantum state means that we can engineer our quantum state such that it is maximally sensitive to the parameter of interest (in our case, the gravitational interaction), and ensuring that this parameter provides a signature that is distinct from other effects. It is interesting to consider the scaling of the scheme with the number of particles, $N$. Increasing $N$ not only increases the magnitude of gravitational interaction, it also increases the ability of the quantum state to perceive a small change. Although we are not searching for entanglement, and therefore not directly demonstrating entanglement mediated through the gravitational field, a Hamiltonian of the form Eq. (10) can create entanglement, so proving that the gravitational field obeys a Hamiltonian of this form is evidence that the gravitational field can transmit entanglement. It was recently argued by Hall and Reginatto that, depending on how quantum-classical interactions are modelled, the generation of entanglement through a gravitational interaction is *not* definitive proof of non-classical gravity [62, 63]. It's possible that the scheme considered here could perhaps differentiate between a broader class of classical gravitational models than the schemes presented in Bose et al. [19] and Marletto and Vedral [20], however, further investigation is required. + +It is worth discussing how this proposal differs from the experimental results discussed in Altamirano et al. [15]. By considering recent high-precision atom-interferometry experiments, they show that the observed results are inconsistent with gravity interacting as a local operations + +and classical communication (LOCC) channel, as proposed by Kafri, Taylor and Milburn [7, 8], which should cause far greater de-coherence. This decoherence is the result of a single atom with quantum degrees of freedom interacting with the earth's gravitational field, and does not directly show that the gravitational field can generate entanglement between two spatially separated masses. The experiment that we propose would provide evidence that the gravitational field can exist in a quantum superposition. + +In summary, we have investigated a scheme to search for quantum signatures of gravity in ultra-cold quantum gases. The use of atoms rather than quantum systems formed from particular macroscopic objects provide much greater long-term stability, a factor which will be important when searching for such small signatures. Additionally, the use of many microscopic particles and multi-parameter estimation may provide additional advantages over the previously proposed schemes. Although it appears that the observation of quantum signatures of gravitation in ultra-cold quantum gases is beyond the ability of current experiments, improvements in particle number, and knowledge of the residual electromagnetic forces between the atoms may enable the detection of effects of quantum gravity. + +## ACKNOWLEDGEMENTS + +The author acknowledges fruitful discussions with Joe Hope, Antigonie Bradshaw, Matthew Blacker, Ruvi Lecamwasam, John Close, Nick Robins, and Michael Hall. + +[1] John F. Donoghue, “General relativity as an effective field theory: The leading quantum corrections,” Phys. Rev. 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Hall, "Entanglement of quantum fields via classical gravity," arXiv:1809.04989 (2018). \ No newline at end of file diff --git a/samples/texts/7426999/page_45.md b/samples/texts/7426999/page_45.md new file mode 100644 index 0000000000000000000000000000000000000000..cb97161eca066df04805c0e095545f6c2b619bf3 --- /dev/null +++ b/samples/texts/7426999/page_45.md @@ -0,0 +1,7 @@ +for transmission is called C-band. 1 Tb/s is the maximum capacity in C-band using 10 Gb/s DWDM Long-haul transmission network, which uses Non Return-to-Zero (NRZ)-On-Off Keying (OOK) modulation for transmission. + +To realize still higher speeds of transmission, data rate on individual channels have to be increased from 10 Gb/s to 100 Gb/s [6]. Increasing the symbol rate of 10 Gb/s OOK transmissions is simply not a viable solution because of dispersion effects in the optical fiber. Chromatic Dispersion (CD) causes Inter Symbol Interference (ISI) at very high symbol rates and hence severely impacts single-carrier transmission. At the receiver, the complexity of time-domain equalizer increases significantly with increased symbol rate. Also, Polarization Mode Dispersion (PMD) effects are more severe at very high data rates. Compensation of PMD is done by using bulky rotators which is not flexible. Hence, compensation of both of these effects is challenging and solutions are not cost-effective. So, the Intensity Modulation-Direct Detection (IM-DD) NRZ-OOK system cannot be scaled to 100 Gb/s data rates per channel. + +To realize higher speeds, Coherent Detection (CoD) has been reintroduced into optical communication system. Direct Detection (DD) was preferred over CoD because of its simplicity in complexity and cost. CoD has come back into prominence due to advancements in VLSI circuits. CoD [7][8][9] offers additional advantages of higher detection sensitivity, higher symbol rates, use of dual polarization and more importantly the amplitude and phase information is conserved when crossing from optical to electrical domain. This opens up the possibility of Electronic Dispersion Compensation (EDC) using Digital Signal Processing (DSP) algorithms, which are low cost, powerful and reprogrammable. This has led to development of Coherent Optical Dual Polarization QPSK (CO-DP-QPSK) systems which can work at 100 Gb/s. These systems use dual polarization and two bits per symbol to essentially deliver four times the bit rate that allows the DSP to operate at four times the lower frequency. Since it uses single-carrier scheme, it requires Finite Impulse Response (FIR) filter for equalization. Also, the CO-DP-QPSK adopted in the 100 Gb/s standard uses blind Channel Estimation (CE), which increases estimation complexity [16]. With the use of Quadrature Phase Shift Keying (QPSK), the use of Digital-to-Analog Converter (DAC) can be avoided now [16]. But, if in future, higher modulation format is adapted, DAC will have to be used and this will increase transmitter and equalizer complexity [16]. + +In the same time, Coherent Optical-OFDM (CO-OFDM)[17][18] has been proposed as a possible candidate for transmission for 100 Gb/s/400 Gb/s data rate and beyond. CO-OFDM as the name indicates combines the technique of coherent detection (CoD) and multi-carrier modulation of Orthogonal Frequency Division Multiplexing (OFDM) [19] to counter the optical channel. OFDM is inherently immune to CD due to presence of Cyclic Prefix (CP) and with the usage of Training Symbols (TS), the equalizer complexity can be reduced significantly to a one-tap equalizer. Also, OFDM offers all the flexibility advantages of allocation of power per sub-carrier (bit-power loading), pilot sub-carrier locations based on channel conditions. \ No newline at end of file diff --git a/samples/texts/7426999/page_48.md b/samples/texts/7426999/page_48.md new file mode 100644 index 0000000000000000000000000000000000000000..35f31f1b280ee19e1eb0ed73a6a5ffee6d2071e0 --- /dev/null +++ b/samples/texts/7426999/page_48.md @@ -0,0 +1,9 @@ +5. Validation of the proposed algorithms/architecture is done in a practical set-up using offline and online experiments. Offline experiments are conducted in Optical Laboratory setup and online experiments using real-time FPGA platform developed as part of FUI 100GFLEX project. + +Figure 1.3 shows the possible power saving opportunities at different stages of VLSI design flow. Resource savings also follows a similar trend. As can be observed, low-complexity algorithm and architecture saves resources at block level like savings of multipliers, adders which down the flow results in significant savings compared to savings obtained at Register Transfer (RT) or Logic Level. This approach is taken in this Thesis to optimize the resource consumption in the search of parallel CO-OFDM transceiver algorithms/architectures and it is expressed using high-level synthesis (HLS) language of CatapultC [21]. + +FIGURE 1.3: Power Savings Possible at each stage in Top down VLSI Design Flow + +## 1.3 Organization of the Thesis + +The thesis is organized as follows. In Chapter 2, single-mode optical fiber is introduced. Different linear and non-linear phenomena that cause dispersion and its effects on transmitted signal is explained. An end-to-end introduction of a typical CO-OFDM system in case of single and dual-polarization transceiver is done. Commonly used algorithms in CO-OFDM systems are given. Then, need for Multi-Band approach is explained based \ No newline at end of file diff --git a/samples/texts/7426999/page_49.md b/samples/texts/7426999/page_49.md new file mode 100644 index 0000000000000000000000000000000000000000..0844cee749dc06b1f78945cf73220d761471838f --- /dev/null +++ b/samples/texts/7426999/page_49.md @@ -0,0 +1,23 @@ +on present day DAC/ADC bandwidth and precision available. Literature survey of CO- +OFDM offline and online experiments are detailed. Complexity of algorithms (Number +of operations) used in both transmitter and receiver is calculated. This gives the state-of- +the-art complexity of CO-OFDM systems and provides motivation for reducing complexity +from the top, which means starting from low-complexity algorithms to scalable parallel +architectures and fixed-point exploration for reduction in resources required. + +In Chapter 3, low-complexity algorithms for coarse time synchronization in a disper- +sive channel is explored. A novel hierarchical low-complexity synchronization algorithm +is proposed which provides low Mean Square Error (MSE) performance similar to cross- +correlation algorithms. Complexity of the algorithm is compared with previously proposed +algorithms. A novel parallel scalable architecture is proposed for coarse time synchroniza- +tion, which provides high throughput. Proposed real-time architecture is ideal for CO- +OFDM system that can receive multiple sample input per cycle and need to match the +input rate. Complexity analysis of the proposed architecture is performed and compared +with previously proposed real-time architectures for CO-OFDM systems. + +In Chapter 4, design of MB-CO-OFDM system given a target rate is shown considering dispersion parameters of the optical channel. An End-to-end parallel transceiver architecture is proposed which can generate and process multiple samples per cycle and easily scale to higher parallel inputs/outputs. Parallel architecture of each block is detailed and savings in resources at the architectural level are shown due to usage of efficient architecture. The architecture exploration is done using CatapultC [21] which is a High Level Synthesis (HLS) tool which accepts input in C and outputs Verilog/VHDL. Fixed-point analysis of the signal processing chain is done which helps in reduction of area for achieving a particular value of bit error rate (BER) at a particular value of Optical Signal-to-Noise Ratio (OSNR). Resources consumed on Xilinx FPGA are reported and break-up of resources consumed for each block is given and compared with previous architectures in terms of scalability and performance. + +In Chapter 5, CO-OFDM experiments performed using Arbitrary Waveform Generator (AWG) as transmitter and Digital Storage Oscilloscope (DSO) as receiver are explained. The transition from electrical back-to-back experiment (B2B) to Optical B2B arrangement experiment with optical fiber is explored. BER curves are given for each configuration as a function of SNR. Performance characterization is then done with different LASER used for both transmitter and receiver. Matlab is used for generating and decoding data. Experimental curves of BER are compared with theoretical BER curves for QPSK to validate the setup. Performance of the algorithm and architecture of proposed real-time synchronization algorithm is then explored in the real-time FPGA platform developed as part of 100GFLEX FUI project. The proposed architecture is integrated into this system and performance analysis done with synchronous and asynchronous sampling configurations. + +Chapter 6 concludes the thesis by outlining the major contributions done with respect +to reduction of computational complexity. Proposals to reach speeds higher than 100 Gb/s \ No newline at end of file diff --git a/samples/texts/7426999/page_5.md b/samples/texts/7426999/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..69f501163cd78ad7f8c6cc26dd1900c2c33dda3e --- /dev/null +++ b/samples/texts/7426999/page_5.md @@ -0,0 +1,18 @@ +# Acknowledgements + +I wish to express profound gratitude to my thesis director Prof. Olivier Sentieys for guiding me throughout the time span of this work. I am grateful to him for his expert advice and time for thesis discussions and begin available for questions/clarifications at all times. The meetings which I had helped with many aspects of work. His suggestions and comments improved the quality of the thesis report. + +I also wish to express my thanks to co-director Mr. Laurent Bramerie for his guidance in the topics related to Optical Communication Systems. His evaluation of ideas from optical systems point of view helped in validation of the work. I am very thankful for many discussions on optical experiments. + +I am thankful to my colleague Rémi Pallas for bringing up the real-time FPGA development system and then integrating my architecture implementation on it. It was a very hard job and he was a very good experience working with him for all the three years. I also acknowledge Arnaud Carer for his help in setting up real-time FPGA platform and discussions regarding implementation. + +Thanks are also due to faculty members, senior researchers and rest of my colleagues at ENSSAT with whom I mutually shared ideas and had discussions. I am thankful especially to Mme. Nathalie Caradec for her French classes. I would also like to acknowledge the assistance of administrative staff of ENSSAT due to which I had a pleasant work environment. Further I wish to recollect with fondness, the memorable association I developed with my friends Hai, Jérémy, Karthik, Nhan, Rémi, Rengarajan, Stéphane, Vaibhav, Vinh and Vivek. + +This thesis has been made possible thanks to the funding by 100GFLEX project and facilities extended by IRISA/INRIA, including travel assistance to attend conferences, for which I remain grateful. + +I wish to express my deep sense of gratitude to my parents for their encouragement in all phases of my academic and professional career that in the first instance enabled me take up doctoral studies. They sharing my goal of acquiring a doctorate only added to my inspiration to complete the doctoral program successfully. I am also grateful to the rest of my family members and friends whose constant support and words of encouragement enabled me to focus on my work. + +Last, but not the least, I thank the members of the jury for agreeing to make a critical assessment of the dissertation and suggesting improvements to the thesis that enhanced its quality. + +Pramod UDUPA +Lannion, France \ No newline at end of file diff --git a/samples/texts/7426999/page_59.md b/samples/texts/7426999/page_59.md new file mode 100644 index 0000000000000000000000000000000000000000..040c7eeb6ed0410ce1814fe7ae0908d0cf85320e --- /dev/null +++ b/samples/texts/7426999/page_59.md @@ -0,0 +1,21 @@ +* Sensitive to timing offset - Loss of timing synchronization causes ISI and ICI. Without timing synchronization, other offsets cannot be efficiently estimated and compensated. Symbol synchronization and frame synchronization are essentially the same in case of OFDM. + +* Sensitive to Frequency and Phase Offsets - CFO at the receiver causes loss of orthogonality and causes symbol rotation. Phase offset [24] [25] causes rotation of constellation. + +Figure 2.3 shows a single band of single polarization/dual polarization CO-OFDM system. Each of the four blocks are explained in the subsections below. + +FIGURE 2.3: Single band of a single/dual polarization CO-OFDM system + +### 2.4.3 Digital Transmitter + +Figure 2.4 shows the internal blocks of the single polarization digital OFDM transmitter block. In case of dual polarization, the digital transmitter block is repeated used for both polarizations. + +FIGURE 2.4: Digital OFDM Transmitter, S/P - Serial-to-Parallel, P/S - Parallel-to-Serial + +* Mapper - It maps bits to symbols. Typical mapping schemes range from BPSK, QAM, 16-QAM, 64-QAM. It is followed by a serial to parallel converter block before the IFFT. + +* IFFT - It modulates complex data from frequency domain to time domain. It is the most complex block in the transmitter chain. + +$$x[n] = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X[k]e^{-j2\pi kn/N} \quad (2.4)$$ + +where $x[n]$ is the time-domain signal, $X[k]$ is the frequency domain signal and $N$ is the size of IFFT. \ No newline at end of file diff --git a/samples/texts/7426999/page_61.md b/samples/texts/7426999/page_61.md new file mode 100644 index 0000000000000000000000000000000000000000..081f95d1c0196eaa5dce58390351f59a94f6d7a5 --- /dev/null +++ b/samples/texts/7426999/page_61.md @@ -0,0 +1,13 @@ +FIGURE 2.7: Resolution vs. Sampling Rate for fastest DAC available. GSa/s - Giga Samples/second. + +* DAC - DAC converts digital output of IFFT to analogue output. Present day DACs bandwidth and resolution lag behind the requirements for 100 Gb/s single-band CO-OFDM system. A survey of the fastest DAC available in the market is shown in Figure 2.7. Fastest DAC available has a sampling rate of around 34 GSamples/s with a resolution of 6 bits. To reduce the constraints on DAC/ADCs, multi-band CO-OFDM has been proposed to achieve a total data rate of 100 Gb/s in case of 100 Gb Ethernet. + +* Low Pass Filter (LPF) - It filters the output signal with a cut-off frequency near the Nyquist frequency of the DAC sampling frequency. + +* RF driver - This amplifies the electrical signal after low pass filtering and output modulates optical carrier in MZ Modulator. + +* MZM - The carrier frequency supplied by External Cavity LASER (ECL) module is modulated by the I/Q electrical signal. + +* Variable Optical Amplifier (VOA) - The real and imaginary signals are combined and amplified by the optical amplifier. The output signal is fed to Polarizer in case of dual-polarization system. + +* Polarization Beam Combiner (PBC) - The modulated signal amplitude is controlled by VOA and the two polarizations are combined and input to single mode optical fiber channel in case of dual polarization CO-OFDM system. \ No newline at end of file diff --git a/samples/texts/7426999/page_62.md b/samples/texts/7426999/page_62.md new file mode 100644 index 0000000000000000000000000000000000000000..5199cb5525ac21b329b2c2308882f79f02206fc4 --- /dev/null +++ b/samples/texts/7426999/page_62.md @@ -0,0 +1,15 @@ +### 2.4.5 Optical-to-RF Down Converter + +FIGURE 2.8: Optical-to-RF Down Converter. **BPF** - Band Pass Filter, **ECL** - External Cavity LASER, **LO** - Local Oscillator, **PBS** - Polarization Beam Splitter, **ADC** - Analog-to-Digital Converter, **IX** - Real Part of X-Polarization, **QX** - Imaginary Part of X-Polarization, **IY** - Real Part of Y-Polarization, **QY** - Imaginary Part of Y-Polarization. + +Figure 2.8 shows the front end of the optical receiver for receiving either single/dual polarized optical signal. It shows direct down conversion architecture, where conversion from optical to analogue is direct without any intermediate RF frequency. The Band Pass Filter (BPF) selects the band for processing. The filtered signal is down converted by using LASER frequency which is tuned to center frequency of the band. The optical signal's amplitude and phase information is detected by balanced photodiode circuit and then sampled by ADC and converted to digital domain. The bandwidth of ADC is the limiting factor. Oversampling by a large factor is not possible due to this limitation. Generally, for CO-OFDM systems, an oversampling factor of 1.2 is used. A survey is done of the fastest available ADC in the market as shown in Figure 2.9. The fastest available ADC has sampling rate of 56 Gb/s with resolution of 8 bits. + +### 2.4.6 Digital OFDM Receiver + +Figure 2.10 shows the digital part of the receiver. The major components are: + +* Coarse Time Synchronization - It detects start of OFDM frame by detecting start of training symbol. Estimation of fractional carrier frequency offset (CFO) is done. + +* Fractional Frequency Synchronization - Using the fractional CFO is estimated. CFO compensation is done by multiplying input signal with estimated CFO. It receives integer CFO from integer CFO estimation block which is present after FFT block. + +* Remove CP - After CFO compensation, cyclic prefix (CP) is removed and *N* samples are fed into FFT block. \ No newline at end of file diff --git a/samples/texts/7426999/page_64.md b/samples/texts/7426999/page_64.md new file mode 100644 index 0000000000000000000000000000000000000000..2a0c43e830e2d7ac66b55e1abd29c28515e3f3f7 --- /dev/null +++ b/samples/texts/7426999/page_64.md @@ -0,0 +1,15 @@ +* Common Phase Error Estimation - Phase Error in the OFDM symbol caused due to LASER's rapid variations phase is estimated using pilot symbols dedicated in every OFDM symbol. Compensation is done by multiplication by exponential multiplication. + +* Demapper - It converts received complex data to symbols of used constellation. + +Since the data converters (DAC, ADC) form the bottleneck with respect to sampling frequency and also resolution, multi-band CO-OFDM (MB-CO-OFDM) is proposed to reduce the pressure on the signal converters. Also, MB-CO-OFDM helps in the realization of architectures on FPGA since the maximum frequency attained on an FPGA is order of magnitude lesser than that of DAC/ADC. MB-CO-OFDM divides the total optical bandwidth into smaller electrical bandwidths which can be handled by DAC/ADC and target rate of 100 Gb/s is attained by the use of multiple bands working in parallel. In this thesis, all the designs are for single-band single-polarization CO-OFDM block. Dual-Polarization is indicated when it is applicable. Then, the total target data rate of 100 Gb/s is achieved by using dual polarization multiple bands. + +## 2.5 Complexity Analysis of the System + +CoD in CO-OFDM increases the cost of the system compared to IM-DD system by increasing the number of optical/analogue components required. Table 2.3 gives the number of components for transceiver for CO-OFDM, CO-QPSK and IM-DD system. From the table, it can be seen that there is a significant increase of resources for CoD systems and where this increase compensates for bandwidth increase, it is beneficial to use. Hence, long-haul core and submarine networks are better candidates at present for adoption of CoD detections schemes compared to metro and access networks. + +TABLE 2.3: Cost of Optical Transceiver for CO-OFDM, CO-QPSK and IM-DD Systems + +
SystemLASEROptical ModulatorPhoto-DiodeDAC/ADCPBS PBC
DP-CO-OFDM4444/41/1
SP-CO-OFDM2222/21/1
DP-CO-QPSK4440/41/1
SP-CO-QPSK2220/21/1
IM-DD1110/10/0
+ +For finding the increase in complexity in the digital part, a full complexity analysis is done. The analysis is done at two levels. First, the algorithmic complexity of algorithms \ No newline at end of file diff --git a/samples/texts/7426999/page_67.md b/samples/texts/7426999/page_67.md new file mode 100644 index 0000000000000000000000000000000000000000..2519167a6d7f2c5009f514aa781712801c252d00 --- /dev/null +++ b/samples/texts/7426999/page_67.md @@ -0,0 +1,11 @@ +CO-OFDM receiver architecture on FPGA, Kaneda et al. [14] and Chen et al. [32]. All architectural complexity comparisons are done with these two papers wherever it is relevant. + +### 2.5.3 Time/Frequency Synchronization + +Algorithmic complexity for some of the algorithms which can be used for coarse time synchronization are tabulated in Table 2.8, which shows number of operations required for calculation of single timing metric point. The algorithms Schmidl-Cox, Minn-Bhargava and Shi-Serpedin have an iterative form of equation for correlation operation, which makes number of operations independent of size of IFFT/FFT (N). The algorithms of Park, Choi and Zhou do not have iterative form for the correlation operation, hence number of operations depends on size of N. + +TABLE 2.8: Algorithmic Complexity of Coarse Time Synchronization Algorithms. Calculations count only correlation operation and not the energy calculation. + +
AlgorithmReal MultiplicationsReal Additions
Schmidl-Cox [33]88
Minn-Bhargava(L = 4) [34]1212
Shi [35]5457
Park [36]2(N + 2)2(N + 1)
Choi [37]2N2(N - 1)
Zhou [38]2N2(N - 1)
+ +Although the algorithmic complexity of auto-correlation algorithms is better for high-parallel outputs compared to cross-correlation algorithms, it does not translate into hardware savings and scalability in architectural space. Direct parallelization of auto-correlation equation results in huge amount of resources and implementation of iterative equation assumes operating frequency of digital circuit is the same as ADC sampling frequency which is not, in the case of CO-OFDM receiver. In case of CO-OFDM receiver, ADC frequency is of the order of GHz and FPGA operating frequency is around 250-300 MHz. Table 2.9 shows the proposals for parallel coarse time synchronization used in real-time CO-OFDM systems. Only two proposals are available, Kaneda et al. and Chen et al. Kaneda et al. proposed for R = 16-parallel output, while Chen et al. proposed for R = 8-parallel output. Their proposal is extended for R = 2, 4, 8, 16-parallel output to show the trend. The amount of resources shown is required for computation of only auto-correlation part of the timing metric. They do not include energy calculation and division by energy of training symbol. Also, both the proposals do not provide fractional CFO estimation. Table 2.9 indicates that further improvement in scalable parallel architectures for coarse timing synchronization is required. \ No newline at end of file diff --git a/samples/texts/7426999/page_68.md b/samples/texts/7426999/page_68.md new file mode 100644 index 0000000000000000000000000000000000000000..3bb786f0f8a1245c2f8e1a7f9b311bda2fd982ab --- /dev/null +++ b/samples/texts/7426999/page_68.md @@ -0,0 +1,21 @@ +TABLE 2.9: Architectural Complexity of Coarse Time Synchronization Algorithms + +
AuthorFFT Size (N)Training Sym. Size (M)Algorithm UsedReal MultipliersReal Adders
2-PARALLEL INPUT/OUTPUT
Kaneda et al.12832Auto-Corr.868
Chen et al.128128Cross-Corr.0508
4-PARALLEL INPUT/OUTPUT
Kaneda et al.12832Auto-Corr.16136
Chen et al.128128Cross-Corr.01016
8-PARALLEL INPUT/OUTPUT
Kaneda et al.12832Auto-Corr.32272
Chen et al. [32]128128Cross-Corr.02032
16-PARALLEL INPUT/OUTPUT
Kaneda et al. [14]12832Auto-Corr.64544
+ +Total CFO can be expressed as + +$$ \hat{\epsilon}_{total} = \hat{\epsilon}_{frac} + \hat{\epsilon}_{int} \quad (2.6) $$ + +$$ \hat{\epsilon}_{total} = \frac{\hat{\phi}L}{\pi N} + \frac{2z}{N} \quad (2.7) $$ + +where $\hat{\epsilon}_{frac}$ is the fractional CFO, $\hat{\epsilon}_{int}$ is the integer CFO, $L$ is the number of repeating parts of training symbol, $N$ is the size of IFFT/FFT. Fractional CFO Estimation is done with the help of training symbol used for coarse synchronization, given by + +$$ \hat{\epsilon}_{frac} = \frac{2}{\pi} \cdot \left[ P[\hat{\eta}_{start}] \right] \quad (2.8) $$ + +where $\hat{\epsilon}_{frac}$ is the fractional CFO estimate $\hat{\eta}_{start}$ is the estimate of start of OFDM symbol, $P[\eta_{start}]$ is the auto-correlation function at the index of $\hat{\eta}_{start}$. $\hat{\epsilon}_{frac}$ gives CFO in terms of sub-carrier spacing ($\frac{B}{N}$), $B$ is the bandwidth of the OFDM signal, $N$ is the size of IFFT/FFT. The range of CFO estimation of Schmidl-Cox algorithm is $\pm 1$, and for Minn-Bhargava algorithm ($L=4$) is $\pm 2$. Algorithmic complexity of CFO Estimation is arc tangent calculation and architectural complexity is LUT implementation of arc tangent function. No real-time proposals for fractional/integer CFO estimation are proposed in literature. + +### 2.5.4 CFO Compensation + +Compensation of CFO is done by exponential multiplication by using the estimated CFO, + +$$ x_c[n] = x[n] \cdot e^{-j2\pi n(\hat{\epsilon}_{frac}+\hat{\epsilon}_{int})/N} \quad (2.9) $$ \ No newline at end of file diff --git a/samples/texts/7426999/page_69.md b/samples/texts/7426999/page_69.md new file mode 100644 index 0000000000000000000000000000000000000000..98b8afb20bd59f5e7b558ad4a7796333f5bb6f27 --- /dev/null +++ b/samples/texts/7426999/page_69.md @@ -0,0 +1,21 @@ +where $x_c[n]$ is the CFO compensated signal, $x[n]$ is the input signal, $\hat{\epsilon}_{int}$ is the CFO integer estimated, $n$ is the time index $n \in [0, N - 1]$, $N$ is the size of IFFT/FFT. For $N$ samples in a single OFDM symbol, algorithmic complexity is $4N$ real multiplications and $2N$ real additions. Architectural complexity for $R$-parallel output is $5R$ real multipliers and $3R$ real adders, where $R$ is the number of parallel outputs. + +### 2.5.5 FFT + +The FFT converts received data from time domain to frequency domain. It is the most computationally intensive block in the receiver signal processing chain. Algorithmic complexity is shown in Table 2.4 and architectural complexity is shown in Table 2.5. Kaneda et al. used in-built Altera FFT for $N = 128$ size FFT, which uses 24 real multipliers and totally 384 multipliers were used for decoding 16-parallel inputs. + +### 2.5.6 Integer CFO Estimation + +Because of large variation in LASER's frequency, integer CFO is introduced. No previous real-time architecture proposals for Integer CFO Estimation in literature. There are two methods for estimating integer CFO. + +* Method 1 - The method uses either two symbols [33] or one symbol [39]. In case of two symbols/one symbol, the even sub-carriers in the symbols are related by a fixed phase factor. The timing metric for detection of integer CFO is + +$$B_1(g) = \frac{|\sum_{k \in X} x_{1,k+2g}^* v_k^* x_{2,k+2g}|^2}{2(\sum_{k \in X} |x_{2,k}|^2)^2} \quad (2.10)$$ + +where integer $g$ spans the range of possible frequency offsets, $X = -W, .., 2, ..., W$ is the set of indices for even frequency components, $W$ is the number of even frequencies with the PN sequence. The index corresponding to the maximum value of $B_1(g)$ gives the integer CFO. Algorithmic/Architectural Complexity is given in Table 2.10. + +* Method 2 - The method uses cross-correlation with known sequence to estimate integer CFO. The timing metric is given by + +$$B_2(g) = \frac{\left| \sum_{k \in X} x_{k+2g} y_{k+2g}^* \right|^2}{\sum_{k \in X} |x_{k+2g}|^2} \quad (2.11)$$ + +where $y$ is the known sequence, $x$ is the received sequence, integer $g$ spans the range of possible frequency offsets, $X = -W, .., 2, ..., W$ is the set of indices for even frequency components. The index corresponding to maximum value of $B_2(g)$ gives the integer CFO estimate. Algorithmic/Architectural Complexity is given in Table 2.10. \ No newline at end of file diff --git a/samples/texts/7426999/page_72.md b/samples/texts/7426999/page_72.md new file mode 100644 index 0000000000000000000000000000000000000000..b71b8bbbea7db6f51ad0ac72d9b102afc7686018 --- /dev/null +++ b/samples/texts/7426999/page_72.md @@ -0,0 +1,15 @@ +TABLE 2.11: Algorithmic/Architectural Complexity for Channel Estimation and Equalization. R - number of parallel outputs. + +
AlgorithmReal MultiplicationsReal AdditionsReal MultipliersReal Adders
Least-Squares Single Pol.8N3N7R3R
Least-Squares Double Pol.32N12N28R12R
NLMS Single Pol.11N8N5R6R
NLMS Dual Pol.22N16N10R12R
Channel Equalization Single Pol.4N2N8R4R
Channel Equalization Double Pol.8N4N16R8R
+ +the phase error calculated. The method is as follows + +$$e = \frac{1}{N_p} \sum_{m=0}^{N_p-1} r^*[m] \cdot c[m] \quad (2.25)$$ + +$$\phi_{err} = \angle e \quad (2.26)$$ + +where *e* is the complex error vector, *N**p* is the number of pilots in one OFDM symbol, *r*[*m*] is the received signal, *c*[*m*] is the reference pilot symbol. The CPE compensation is done by multiplying the input signal by $e^{\phi_{err}}$. Algorithmic/Architectural complexity is given in Table 2.12. Kaneda et al. simplified CPE estimation by using LUT implementation, which avoided multipliers. + +TABLE 2.12: Algorithmic/Architectural Complexity for CPE Estimation and Compensation. R - number of parallel outputs. + +
AlgorithmReal MultiplicationsReal AdditionsReal MultipliersReal Adders
CPE Estimation4Np2Np + (Np − 1)4R4R
CPE Compensation4N2N4R2R
\ No newline at end of file diff --git a/samples/texts/7426999/page_73.md b/samples/texts/7426999/page_73.md new file mode 100644 index 0000000000000000000000000000000000000000..d8824edb411b8a54c0068f9d60969c38e1946fa3 --- /dev/null +++ b/samples/texts/7426999/page_73.md @@ -0,0 +1,11 @@ +### 2.5.9 Demapper + +Demapper - It maps incoming complex symbols to symbols of a constellation. It involves comparisons with reference symbols of the constellation and calculation of distance. In case of QPSK de-mapping, it can be reduced to checking positive/negative sign and mapped to one of QPSK symbol. + +## 2.6 Observations + +From the algorithmic and architectural complexity calculations, it can be seen that IFFT/ FFT are the major resource hungry blocks. With the adoption of multi-band CO-OFDM to reach the total target data rate of 100 Gb/s, resource savings obtained from one polarization single-band is multiplied by the total number of sub-bands used. Hence, effort targeted towards resource optimization by low-complexity algorithm/architecture goes a long way in reduction of digital computational complexity of CO-OFDM. From the survey, it has been found that there is no low-complexity parallel architecture proposed for time synchronization. Also, no proposal for efficient integer CFO estimation. Channel Estimation is one more block which occupies significant area and hence needs to be optimized. Hence this thesis directs its efforts towards low-complexity scalable algorithms/architectures for single-band single-polarization CO-OFDM system. The only resource shared from transition from single-polarization to dual-polarization is channel estimation. Other than that, all the blocks are replicated in both polarizations. + +## 2.7 Conclusions + +State-of-the-art survey of real-time CO-OFDM systems show that transmitters supporting large rates have been built. But, there also optimization of IFFT architecture and scalability has not been explored. In case of real-time CO-OFDM receiver, only two major publications have appeared which explore the complexity of the system. End-to-end parallel architectures have not been explored and scalability also is an interesting option. Chapter 3 explores timing synchronization from a low-complexity algorithmic and architecture standpoint and Chapter 4 explores end-to-end parallel architectures for the complete CO-OFDM receiver. Chapter 5 details the experiments performed using proposed low-complexity algorithms and performance characterized. \ No newline at end of file diff --git a/samples/texts/7426999/page_76.md b/samples/texts/7426999/page_76.md new file mode 100644 index 0000000000000000000000000000000000000000..398cff8200e9404b3cef68f17344e98df5fcc7fe --- /dev/null +++ b/samples/texts/7426999/page_76.md @@ -0,0 +1,21 @@ +timing metric point calculation. This leads to low throughput in output timing metric computation and hence can cause delay in synchronization. There is a need for a low complexity synchronizer which can quickly synchronize with incoming signal in a highly dispersive channel. Low complexity leads to lower resource count and low power which is required both for Wireless and Optical systems. + +## 3.3 Proposed Hierarchical Low-Complexity Synchronizer for Wireless OFDM Systems + +### 3.3.1 OFDM System Description + +The transmitted baseband OFDM samples can be written in terms of IFFT equation as + +$$x[n] = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} X[k] \cdot e^{j2\pi nk/N} \quad (3.1)$$ + +where $N$ is the number of sub-carriers and $X[k]$ the complex information carrying symbol in frequency domain. The sampled signal at the receiver can be written as + +$$r[n] = s[n - \eta] \cdot e^{j(2\pi\epsilon n/N + \phi)} + w[n] \quad (3.2)$$ + +where $\eta$ is the integer timing offset, $\epsilon$ the CFO and $\phi$ the phase offset. $w[n]$ is the additive white Gaussian noise (AWGN) and $s[n]$ in multipath channel is given by + +$$s[n] = \sum_{m=0}^{L_h-1} h[m] \cdot x[n - \tau_m] \quad (3.3)$$ + +where $h$ is the sampled channel response (complex channel coefficients) at the receiver. $L_h$ is the number of channel paths and $\tau_m$ is the path delay corresponding to the $m^{th}$ channel path. The channel is assumed static for the duration of the OFDM symbol. + +To achieve synchronization in ISI channel at low complexity, a new synchronization method is proposed based on proposal of a new training symbol. The training symbol has low PAPR and can support both delay correlation and conjugate symmetry correlation operations. The training symbol is generated using CAZAC sequence, which have very low PAPR and possess impulse-like auto-correlation properties and constant cross-correlation property. \ No newline at end of file diff --git a/samples/texts/7426999/page_77.md b/samples/texts/7426999/page_77.md new file mode 100644 index 0000000000000000000000000000000000000000..7555e517543516ab89c71f27618ae651ccf45dfe --- /dev/null +++ b/samples/texts/7426999/page_77.md @@ -0,0 +1,25 @@ +### 3.3.2 Proposed Hierarchical Method + +The training sequence is based on modified Chu (CAZAC) [10] sequence, which have smaller alphabet size than Chu sequence. The modified Chu sequence [10] is given by + +$$ a_k^{(r)} = \begin{cases} \exp\left(i \frac{2\pi}{N_s} \left\lfloor \frac{rk^2}{2} \right\rfloor\right), & \text{for } N_s \text{ even} \\ \exp\left(i \frac{2r\pi k(k+1)}{N_s}\right), & \text{for } N_s \text{ odd} \end{cases} \quad (3.4) $$ + +$$ a_k^{(r)} = \begin{cases} \exp\left(i \frac{2\pi}{N_s} \left\lfloor \frac{rk^2}{2} \right\rfloor\right), & \text{for } N_s \text{ even} \\ \exp\left(i \frac{2r\pi k(k+1)}{N_s}\right), & \text{for } N_s \text{ odd} \end{cases} \quad (3.5) $$ + +where $0 \le k < N_s$, $\gcd(r, N_s) = 1$ and $\lfloor a \rfloor$ denotes the integer part of $a$. Here $r = 1$ is used. The alphabet size is $N_s$ for modified Chu sequence compared to $2N_s$ for Chu sequence. + +The training symbol proposed [11] is + +$$ [C C C - C], C = [A B], B = A^*[{-n}] $$ + +The construction of repeating part *C* is done starting from $a_k^{(r)}$, which is considered to be in frequency domain. The size of *C* is $N/8$. The sequence generated is given to IFFT. At the input of IFFT, the zero frequency sub-carrier is switched off and as well as high-frequency sub-carriers, just like in LTE standard generation of primary synchronization signal (PSS) [47]. The output of IFFT is considered as *C* and is repeated to generate the proposed training sequence. The part **A** is constructed by taking IFFT of the generated modified Chu sequence [12] of size $N_s = N/8$. Then **B** is constructed from **A** by time-reversal and conjugation operation. The sign pattern $[1 1 1 -1]$ is designed to ensure steep roll-off for initial estimation algorithm. The length of a single part is designed to be greater than maximum delay spread of the multipath channel. The initial timing metric proposed for the training symbol is delay based auto-correlation method that involves using repeating pattern for delay based correlation. The timing metric for coarse initial estimate is: + +$$ TM_{init}[n] = \left( \frac{L}{L-1} \cdot \frac{|P_{init}[n]|}{R_{init}[n]} \right)^2 \quad (3.6) $$ + +where $P_{init}$ is the auto-correlation function, $R_{init}$ is the energy calculation function, $TM_{init}$ is the timing metric function and $L$ is the number of repeating parts ($L = 4$) in the proposed training symbol. The term $\frac{L}{L-1}$ is used to normalized for maximum value of 1 at the correct starting point. The expressions for $P_{init}$ and $R_{init}$ are + +$$ P_{init}[n] = \sum_{k=0}^{L-2} u[k] \sum_{m=0}^{M-1} r^*[n + kM + m] \cdot r[n + (k+1)M + m] \quad (3.7a) $$ + +$$ R_{init}[n] = \sum_{k=0}^{L-1} \sum_{m=0}^{M-1} |r[n + kM + m]|^2 \quad (3.7b) $$ + +where $u[k] = p[k] \cdot p[k+1]$, $p[k]$ contains the sign pattern of $[1 1 1 -1]$, $k = 0, 1, ..., (L-1)$ and $M = N/L$. The time index corresponding to the maximum value gives the initial \ No newline at end of file diff --git a/samples/texts/7426999/page_8.md b/samples/texts/7426999/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..848528335cd12fd8a4c4a204f00923c993224bd6 --- /dev/null +++ b/samples/texts/7426999/page_8.md @@ -0,0 +1,7 @@ +Abstract + +Coherent Optical-OFDM (CO-OFDM) communication system is built on most advanced techniques for detection, modulation and dispersion compensation viz., coherent detection, orthogonal multi-carrier modulation (OFDM) and electronic dispersion compensation (EDC). The re-emergence of coherent detection in optical communication systems was made possible by the advancement in very high rate digital circuits. Coherent detection (CoD) has higher sensitivity for signal detection compared to direct detection (DD) methods. It enables use of dual-polarization transmission and it preserves phase information of optical signal and passes it to electrical domain. The use of OFDM modulation provides significant flexibility and efficient use of allocated bandwidth. Due to availability of phase information in digital domain, low cost digital signal processing (DSP) processors can be used for dispersion compensation in digital domain, which makes the solution flexible and re-configurable. But, the introduction of CO-OFDM system in place of older intensity modulation-direct detection (IM-DD) system significantly increases the cost of the system, i.e. higher number of optical components and higher amount of electronic resources are required for reception of the signal. Due to increase of resources both in optical and electronic domain, it is justifiable for only long-range transmission distances. The choice of algorithm, architecture and fixed-point optimization play a significant role in reduction of electronic resources required for realization of CO-OFDM systems. + +In this thesis, low-complexity algorithms and architectures for CO-OFDM systems are explored. First, low-complexity algorithms for estimation of timing and carrier frequency offset (CFO) in dispersive channel are studied. A novel low-complexity timing synchronization algorithm, which can withstand large amount of dispersive delay, is proposed and compared with previous proposals. Then, the problem of realization of low-complexity parallel architecture is studied. A generalized scalable parallel architecture, which can be used to realize any auto-correlation algorithm, is proposed. It is then extended to handle multiple parallel samples from ADC and provide outputs, which can match the input ADC rate. The scalability of the architecture for higher number of parallel outputs and different kinds of auto-correlation algorithms is explored. + +An algorithm-architecture approach is then applied to the entire CO-OFDM transceiver chain. At the transmitter side, radix-2$^2$ algorithm for IFFT is chosen and parallel Multipath Delay Commutator (MDC) Feed-forward (FF) architecture is designed which consumes lesser resources compared to MDC FF architectures of radix-2/4. At the receiver side, efficient algorithm for Integer CFO estimation is adopted and efficiently realized without the use of complex multipliers. Reduction in complexity is achieved due to efficient architectures for timing synchronization, FFT and Integer CFO estimation. Fixed-point analysis for the entire transceiver chain is done to find fixed-point sensitive blocks, which affect bit error rate (BER) significantly. The algorithms proposed are validated using optical experiments by the help of arbitrary waveform generator (AWG) at the transmitter and digital storage oscilloscope (DSO) and Matlab at the receiver. BER plots are used to show the validity of the system built. Hardware implementation of the proposed synchronization algorithm is validated using real-time FPGA platform. \ No newline at end of file diff --git a/samples/texts/7426999/page_80.md b/samples/texts/7426999/page_80.md new file mode 100644 index 0000000000000000000000000000000000000000..aa9806f5c74f0b1418ce83db7a6b674447c4d68b --- /dev/null +++ b/samples/texts/7426999/page_80.md @@ -0,0 +1,33 @@ +determined using the probability distribution of the noise component in $Q[n]$. The steps are as follows: + +* The sequence $Q[n]$ is passed through Lloyd-Max [13] quantization algorithm using three levels of quantization. + +* The lowest quantization level and its cluster are considered as noise here. It is observed that this cluster follows a lognormal distribution. Mean ($\mu_n$) and variance ($\sigma_n^2$) of noise cluster are calculated first. The corresponding mean $\mu$ and $\sigma$ for lognormal distribution is + +$$ \mu = \log \left( \frac{\mu_n^2}{\sqrt{\sigma_n^2 + \mu_n^2}} \right) \qquad (3.16a) $$ + +$$ \sigma = \sqrt{\log \left( \frac{\sigma_n^2}{\mu_n^2} + 1 \right)} \qquad (3.16b) $$ + +* A constant false alarm rate (CFAR) of "α" is used for calculation of threshold. The equation for threshold is derived by integrating probability distribution function (pdf) of the noise distribution with limits [$\beta, \infty$]. + +$$ \beta = e^{(\sqrt{2}\cdot\sigma\cdot\operatorname{erf}^{-1}(1-2\cdot\alpha)+\mu)} \qquad (3.17) $$ + +A constant false alarm rate is used across all SNR values. A windowed summation is performed after discarding the noise values using the threshold ($\beta$) calculated. + +$$ E_p(n) = \sum_{k=0}^{S_w-1} Q(\hat{\eta}_{fine} - n + k) \qquad (3.18) $$ + +where $S_w$ is the length of summation window and $J_m$ is the search window for signal component. Then the first arrival path is given by + +$$ \hat{\eta}_{first} = \arg\max_{n} E_p(n) : n = 0, 1, \dots, J_m \qquad (3.19) $$ + +Finally, + +$$ \hat{\eta}_{final} = \hat{\eta}_{init} - \hat{\eta}_{first} \qquad (3.20) $$ + +This value indicates final estimate of the starting index of the OFDM symbol. + +### 3.3.3 Carrier Frequency Offset (CFO) Estimation + +Using estimated $\hat{\eta}_{final}$ as the start of the training symbol, the negative sign in the training symbol is inverted to get [$C C C C$]. There exists some interference due to minus sign in the channel response. The CFO estimate is calculated using the formula + +$$ \hat{\epsilon}_{frac} = \frac{2}{\pi} \left| P[\eta_{final}] \right| \qquad (3.21) $$ \ No newline at end of file diff --git a/samples/texts/7426999/page_81.md b/samples/texts/7426999/page_81.md new file mode 100644 index 0000000000000000000000000000000000000000..6550026bd5c1e00d7f97b832f0d682129cfa936a --- /dev/null +++ b/samples/texts/7426999/page_81.md @@ -0,0 +1,15 @@ +where $P[\eta_{final}]$ is the autocorrelation among the four parts of the training symbol. The calculation of $P[\eta_{final}]$ is done using (3.7a), difference being sign pattern of [1 1 1 1]. It gives fractional and integer CFO estimation. The integer CFO estimation range is equal to $\pm \frac{L}{2}$ sub-carrier spacing. The CFO estimation range of $\hat{\epsilon}_{frac}$ for proposed TS is $\pm 2$ sub-carrier spacing. + +## 3.4 Simulation Results + +### 3.4.1 Parameters + +The performance of all synchronization algorithms has been investigated by using intensive Monte-Carlo simulations ($10^5$ runs). Algorithms using auto-correlation and cross-correlation techniques are compared here. The OFDM system parameters are shown in Table 3.1. The channel used here is a frequency selective channel (ISI channel) with an exponential Power Delay Profile (PDP) and ratio of first to last Rayleigh fading tap is set to 20 dB. A uniformly distributed random phase component is multiplied to every path during each simulation run. The channel has 16 taps with equal tap spacing of four samples. The windowing parameters used for Choi's synchronization method are $J = 41, S = 48$. + +TABLE 3.1: Simulation Parameters + +
ParametersValue
IFFT/FFT Size (N)1024
Number of sub-carriers1024
Length of Cyclic Prefix (Ncyp)102
OFDM Symbol Length (Nsym)1126
Window Size Sw (samples)40
Distance Jm (samples)36
Constant False alarm rate (α)0.01
Number of simulation runs105
Number of channel taps16
Channel Tap Spacing (samples)4
Ratio between first tap to last tap (in dB)20
Carrier Frequency Offset (CFO) (ε)0.75
+ +### 3.4.2 Mean Square Error (MSE) of Timing Estimate + +Figure 3.2 shows MSE of timing estimation in the ISI channel for various synchronization methods. The delay correlation based methods (Schmidl, Minn, Shi) have higher MSE compared to methods using only conjugate symmetry correlation (Park, Choi). Park's and Choi's methods estimate strongest channel path of the received multipath channel using conjugate symmetry correlation method. But only conjugate symmetry correlation does not give low MSE. Choi uses a windowed summation method for identifying first \ No newline at end of file diff --git a/samples/texts/7426999/page_82.md b/samples/texts/7426999/page_82.md new file mode 100644 index 0000000000000000000000000000000000000000..cb9935f381b3055ac1c7b47a85442acf415e66fd --- /dev/null +++ b/samples/texts/7426999/page_82.md @@ -0,0 +1,11 @@ +path, which reduces the MSE compared to Park's method. Proposed method uses conjugate symmetry and windowed summation similar to Choi's to get low MSE in estimation which is better than Park and comparable to Choi at significantly lower computational complexity as shown in Figure 3.2. + +FIGURE 3.2: MSE of Timing Estimation versus SNR in ISI channel + +### 3.4.3 Mean Square Error (MSE) of CFO Estimate + +Figure 3.3 shows the MSE of CFO estimation for different SNR values. A CFO of 0.75 was used during simulation. No CFO estimation was proposed in Choi and Park's methods. The Cramér-Rao bound for variance in estimation of frequency offset [45] is given by + +$$CRB(\hat{\epsilon}) = \frac{1}{2\pi^2} \frac{3(SNR)^{-1}}{N(1 - 1/N^2)} \quad (3.22)$$ + +where *N* is the size of FFT. Schmidl's CFO estimator comes closest to lower bound at all SNR values. Minn's CFO estimator uses algorithm of Morelli [45], which is computationally more complex compared to Schmidl's. Shi's CFO estimation algorithm hits a floor after SNR of 15 dB. Proposed CFO estimator is similar to Schmidl's and gives estimates very close to Schmidl's at medium to high SNR values because of more accurate estimation of the starting index compared to Schmidl's algorithm which does not suffer from interference from any negative sign in training symbol. The CFO estimator has a range of ± 2 sub-carrier spacing compared to Schmidl's which has a range of ± 1 sub-carrier spacing. \ No newline at end of file diff --git a/samples/texts/7426999/page_83.md b/samples/texts/7426999/page_83.md new file mode 100644 index 0000000000000000000000000000000000000000..f9989350c7e36ee79c2b606bce68dcd8a1ee7180 --- /dev/null +++ b/samples/texts/7426999/page_83.md @@ -0,0 +1,9 @@ +FIGURE 3.3: MSE of CFO Estimation versus SNR in ISI channel + +### 3.4.4 Complexity of Calculations + +The number of real operations required is given as a function of $N$ in Table 3.2. Since the algorithms of Schmidl, Minn and Shi can be written in iterative form, the number of operations is fixed per timing metric calculation. But for algorithms of Park, Choi and Zhou where the numerator part cannot be written in the form of iterative formula, the number of operations becomes a function of $N$. Since the proposed algorithm has initial iterative part and fine non-iterative part, the fine estimation algorithm is a function of $N$. But, the non-iterative part works only for calculation of $(2N_{cyp} + 1)$ samples, unlike cross-correlation algorithms which work on every OFDM symbol ($N_{sym}$). Real operations of cross-correlation algorithms and proposed fine step is $\approx 2N$, but since proposed fine step operates on lesser number of samples. For calculation of timing metric over one OFDM symbol ($N_{sym}$), the reduction in complexity is + +$$ \begin{aligned} \textit{Complexity Reduction} &= \frac{(N_{sym} - (2N_{cyp} + 1)) \cdot 2N}{N_{sym} \cdot 2N} \\ &= \frac{N_{sym} - (2N_{cyp} + 1)}{N_{sym}} \end{aligned} \qquad (3.23) $$ + +This results in approximately 80% reduction in computations using numerical values ($N_{sym} = 1126, N_{cyp} = 102$) of simulation compared to Choi, Park and Zhou, while MSE performance is very close to Choi's MSE. In case of Minn, to perform the fine timing estimation, Maximum Likelihood (ML) channel estimation is done first which requires too \ No newline at end of file diff --git a/samples/texts/7426999/page_84.md b/samples/texts/7426999/page_84.md new file mode 100644 index 0000000000000000000000000000000000000000..c73c0e759ddd5e5cdcf264321b7bcf20c07dedeb --- /dev/null +++ b/samples/texts/7426999/page_84.md @@ -0,0 +1,17 @@ +TABLE 3.2: Number of Real Operations for calculation of a single timing metric point + +
AlgorithmReal MultiplicationReal AdditionDivision
Schmidl-Cox15131
Minn(L = 4)31291
Shi59611
Park(2N + 11)(2N + 7)1
Choi(2N + 7)(2N + 3)1
Zhou(2N + 22)(2N + 16)1
Proposed coarse step (L = 4)31291
Proposed fine step(2N + 3)(2N - 1)1
+ +much complexity. In case of Schmidl, the MSE is very high although it is computationally efficient. So, in terms of computational complexity, the proposed algorithm is significantly better than Choi, Park and Zhou and in terms of MSE, significantly better than Schmidl, Minn and Shi's methods. Proposed method is specially useful for MIMO systems where each antenna calculates timing synchronization and hence it needs to be computationally and resource efficient. Thus, the proposed algorithm provides a very good trade-off between computational complexity and MSE of timing and CFO estimation in a frequency selective channel. + +## 3.5 Hierarchical Synchronizer Proposed for CO-OFDM System + +The proposed hierarchical low-complexity synchronizer for Wireless channel OFDM systems has three parts for achieving low value of MSE, namely + +* Auto-correlation Operation ($TM_{init}$, Eq. 3.6) + +* Conjugate Symmetric Correlation Operation ($TM_{fine}$, Eq. 3.9) + +* Windowed Summation ($E_p$, Eq. 3.18) + +In case of SMF optical channel, the dispersion value is not high and more stable compared to multi-path effect of the wireless channel which can have large delay. Hence, the windowed summation step in the proposed synchronizer can be eliminated and only a 2-step procedure is necessary. This modified algorithm is used for synchronization in SMF optical channel. For comparison purposes, only auto-correlation algorithms (Schmidl-Cox, Minn-Bhargava, Shi-Serpedin) are compared. Cross-correlation algorithms (Choi, Park) are not compared since they require large amount of resources and do not provide output every cycle. Also, cross-correlation algorithms do not provide CFO estimation. The hierarchical \ No newline at end of file diff --git a/samples/texts/7426999/page_85.md b/samples/texts/7426999/page_85.md new file mode 100644 index 0000000000000000000000000000000000000000..1112009ca31fb54c54c888762e39e5c535bd3a6b --- /dev/null +++ b/samples/texts/7426999/page_85.md @@ -0,0 +1,25 @@ +synchronization steps used to calculate starting point of OFDM symbol $\eta_{final} = \eta_{init} - \eta_{fine}$ are: + +* Auto-Correlation Operation - Eq. 3.6 is used without change to calculate $\eta_{init}$. + +* Conjugate Symmetric Correlation Operation - Eq. 3.9 is modified to reduce the complexity and normalization is done by using the energy of the symbol. + +$$ \eta_{fine} = \arg\max_n (TM_{fine}^{lc}[n]) \quad (3.24) $$ + +$$ TM_{fine}^{lc}[n] = \frac{|P_{fine}^{lc}[n]|^2}{R_{fine}^2} \quad (3.25) $$ + +$$ P_{fine}^{lc}[n] = \sum_{k=0}^{\frac{N}{4}-1} r[n-k-1] \cdot r[n+k] \quad (3.26) $$ + +$$ R_{fine}[n] = \sum_{k=0}^{\frac{N}{4}-1} |r[n+k]|^2 \quad (3.27) $$ + +The CFO estimation ($\hat{\epsilon}_{frac}$) is done using Eq. 3.21. + +## 3.6 Simulation Results + +### 3.6.1 Parameters + +The performance of all synchronization algorithms has been investigated by using intensive Monte-Carlo simulations (10⁴ runs). The optical fiber channel simulated was standard single mode fiber (SSMF). The SSMF was simulated in Matlab using Optilux [48] simulator. Optilux Simulator helps in simulation of each component of the optical system with realistic parameter values and non-idealities modeled. Simulation parameters are given in Table 3.3. Simulation is done first with just fractional CFO and then with integer CFO included to see the sensitivity of the algorithms to large CFO value. + +### 3.6.2 MSE of Timing Estimate + +Figure 3.4 shows the plot of MSE of timing estimation in SSMF channel with CFO = 0.75 sub-carrier spacing. It can be observed for the proposed algorithm that there is a degradation in performance at low OSNR due to reduced computational complexity in calculation of $P_{fine}^{lc}$. Proposed algorithm's MSE curve shows improvement with increasing OSNR. Figure 3.5 shows the plot of MSE of timing estimation vs OSNR with CFO = 4.75. Performance of all the algorithms is similar to the case of only fractional CFO, showing proposed algorithm incurs no performance penalty in case of large values of CFO. \ No newline at end of file diff --git a/samples/texts/7426999/page_86.md b/samples/texts/7426999/page_86.md new file mode 100644 index 0000000000000000000000000000000000000000..df253d59d97e5b35e2253fb253787e50ff3a5a6e --- /dev/null +++ b/samples/texts/7426999/page_86.md @@ -0,0 +1,7 @@ +TABLE 3.3: Simulation Parameters for CO-OFDM System Simulation + +
OFDM ParametersValue
IFFT/FFT Size (N)256
Number of sub-carriers256
Length of Cyclic Prefix (Ncyp)8
OFDM Symbol Length (Nsym)264
Number of simulation runs104
DAC Sampling Frequency (Fs,DAC)5 GHz
+ +
SSMF Channel ParametersValue
Length of Fiber (LF)1000 km
Fiber Attenuation (αdB)0.2 dB/km
Fiber Effective Area (Aeff)80 μm2
Lambda (λ)1550 nm
Chromatic Dispersion (CD)17 ps/(nm − km)
Polarization Mode Dispersion (PMD)0.05 ps/√km
Fiber Slope0.089 ps/(nm2 − km)
Differential Group Delay (DGD)12.5
Carrier Frequency Offset (CFO)(ε)0.75 and 4.75
+ +FIGURE 3.4: MSE of Timing Estimation vs. OSNR in SSMF channel with CFO = 0.75 \ No newline at end of file diff --git a/samples/texts/7426999/page_87.md b/samples/texts/7426999/page_87.md new file mode 100644 index 0000000000000000000000000000000000000000..a9190dc31b8e525a64c98fdd53d61f4e6447afec --- /dev/null +++ b/samples/texts/7426999/page_87.md @@ -0,0 +1,11 @@ +FIGURE 3.5: MSE of Timing Estimation vs. OSNR in SSMF channel with CFO = 4.75 + +### 3.6.3 MSE of CFO Estimate + +Figure 3.6 shows the MSE of CFO estimation versus OSNR for CFO = 0.75. The MSE of CFO estimation is close to Schmidl-Cox algorithm. + +## 3.7 Need for Parallel Timing Synchronization Architecture + +In this section, architectures for timing synchronization algorithms are explored in the context of CO-OFDM systems. In a typical CO-OFDM system, the input serial rate is of the order of Gb/s. The Gb/s serial input rate is provided by ADC, which operates at a sampling rate ($F_s$) of GHz. When input serial data rate is given to deserializer block of FPGA, it converts the high rate to lower rate which is a multiple of serial rate. The deserializer block now provides multiple samples per cycle at a lower frequency to FPGA, whose maximum frequency ($F_{clk}$) of operation can reach around 400 MHz. The ratio of $\frac{F_s}{F_{clk}}$ has ratio of 4, 8 or 16 [14], which makes it necessary for the OFDM receiver blocks to either process multiple parallel samples per cycle or to have multiple blocks which process single sample per cycle. The latter idea is resource intensive and does not scale well. The approach of CO-OFDM receiver blocks which can process multiple parallel samples per cycle and match the input is a scalable option which also offers resource savings by sharing among multiple parallel blocks. + +In CO-OFDM system, the timing synchronization block is the first block in the receiver chain, which detects start of OFDM symbol. It also provides fractional CFO estimation for \ No newline at end of file diff --git a/samples/texts/7426999/page_88.md b/samples/texts/7426999/page_88.md new file mode 100644 index 0000000000000000000000000000000000000000..381e847bc49bd6044235971de8ffa9757636b4bf --- /dev/null +++ b/samples/texts/7426999/page_88.md @@ -0,0 +1,9 @@ +FIGURE 3.6: MSE of CFO Estimation vs. OSNR in SSMF channel for CFO = 0.75 + +the CFO compensation block. The block operates continuously to detect training symbols repeatedly sent at the beginning of each frame and tracks CFO variations. The total processing rate of the timing synchronization block has to match the input serial rate to avoid large memory for storing incoming data. In the present literature, there has been only two proposals for real-time parallel processing timing synchronization architecture. The details of the previous proposals are given below. Both proposals are based on using sample level parallelism to provide multiple outputs to match the input rate. + +• Kaneda et al. [14] proposed an architecture by directly parallelizing the non-iterative auto-correlation equation for Schmidl-Cox algorithm, using training symbol was of the form [AA]: + +$$P_{sc}^{kan}[n] = \sum_{k=0}^{M_{sc}/R} \sum_{m=0}^{R(k+1)-1} r^*[n+m] \cdot r[n+m+M_{sc}] \quad (3.28)$$ + +where $M_{sc}$ is the length of repeating part of the training symbol (A), R is the number of parallel inputs and outputs, r is the input data stream, $P_{sc}^{kan}$ is the auto-correlation function. Figure 3.7 shows the parallel architecture resulting from Eq. 3.28. For the case of R = 16 and $M_{sc}$ = 32, it requires 64 real multipliers and 544 real adders. Hence, it is not an efficient parallel realization of the algorithm. The ratio of $F_s/F_{clk}$ was 16, but only one pipeline was realized and the training symbol was duplicated 16 times. This resulted in reduced spectrum efficiency and less accurate estimation \ No newline at end of file diff --git a/samples/texts/7426999/page_89.md b/samples/texts/7426999/page_89.md new file mode 100644 index 0000000000000000000000000000000000000000..20e85ca568fee3dc9b33f04e73dacbf9ced1c94a --- /dev/null +++ b/samples/texts/7426999/page_89.md @@ -0,0 +1,13 @@ +of starting point. The fractional CFO estimation was significantly reduced due to +duplication of training symbol. + +FIGURE 3.7: Parallel Architecture proposed by Kaneda et. al for Schmidl-Cox Algorithm + +* Chen et al. [32] proposed an 8-parallel architecture which uses cross-correlation operation, and training symbol of the form $[A A A - A]$. The length of $A$ was $M_{mb} = 32$. Although, complex multipliers were avoided, the number of adders required was large and it does not produce output every cycle unlike auto-correlation. Cross-correlation with known training symbol in presence of large CFO value results in shifted peaks, which reduces the accuracy of start point estimation. For $R = 8$-parallel architecture proposed by Chen et al. 2032 adders were required, and does not provide fractional CFO estimation. + +Architectural complexity of the two architectures is shown in Table 2.9 for different par- +allel inputs/outputs. Previously proposed sample-level parallel architectures consume too +much of resources and do not scale efficiently. Due to complexity of architecture, they +do not provide fractional CFO estimation. The two proposals indicate that further effi- +ciency improvement of symbol synchronization in parallel processing is required [14]. In the +next sections, block-parallel architectures are proposed for acceleration of auto-correlation \ No newline at end of file diff --git a/samples/texts/7426999/page_9.md b/samples/texts/7426999/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..ba71df865542d47f6d046736be73902ba52cb947 --- /dev/null +++ b/samples/texts/7426999/page_9.md @@ -0,0 +1,3 @@ +# Contents + +
Acknowledgementsi
Résuméiii
Abstractv
Contentsx
List of Figuresx
List of Tablesxiv
List of Abbreviationsxvii
0 Résumé étendu1
0.1 Système de communications optiques OFDM à détection cohérente1
0.2 Contexte du travail2
0.3 Algorithme de synchronisation temporelle à faible complexité pour les systèmes OFDM3
0.4 Synchronisation temporelle hiérarchique à faible complexité pour les systèmes CO-OFDM6
0.5 Architecture parallèle pour l'auto-corrélation8
0.5.1 Architecture parallèle partielle(PSBP)9
0.5.2 Architecture parallèle complète (FSBP)10
0.6 Architecture parallèle pour les systèmes CO-OFDM11
0.6.1 Emetteur11
0.6.2 Récepteur11
0.7 Experimentations13
0.8 Conclusion16
1 Introduction19
1.1 Context of the Work22
1.2 Contributions23
1.3 Organization of the Thesis.24
2 CO-OFDM Transceiver System27
2.1 Introduction.27
2.2 Single-Mode Optical Fiber (SMF).27
\ No newline at end of file diff --git a/samples/texts/7426999/page_91.md b/samples/texts/7426999/page_91.md new file mode 100644 index 0000000000000000000000000000000000000000..003e1d130be6746da69b691b50034c206a23aa71 --- /dev/null +++ b/samples/texts/7426999/page_91.md @@ -0,0 +1,23 @@ +equations are as follows: + +$$P_{sc}[n] = \sum_{m=0}^{M_{sc}-1} r^*[n+m] \cdot r[n+m+M_{sc}] \quad (3.29)$$ + +$$P_{sc}[n] = P_{sc}[n-1] + r^*[n+M_{sc}] \cdot r[n+2M_{sc}] - r^*[n] \cdot r[n+M_{sc}] \quad (3.30)$$ + +where Eq. 3.29 is non-iterative equation and Eq. 3.30 the iterative equation. $P_{sc}$ is auto-correlation function, $M_{sc} = N/2$ is the size of repeating symbol (A). It can be observed that non-iterative correlation is time-consuming and does not produce outputs every cycle, while iterative equation can produce outputs every clock cycle, but depends on availability of past sample auto-correlation value. Since the non-iterative equation only depends on inputs, it can be used to calculate auto-correlation value which can be fed into iterative equation computation. Block-level parallelism [49] uses this idea and applies to multiple parallel blocks working in this fashion. Block size is important to increase sharing of resources. Observation of auto-correlation operation indicates its dependency of samples delayed by $M_{sc}$ samples. This dependency can be used to decide the size of block to ensure maximum resource sharing among multiple parallel blocks. If the non-iterative and iterative equations are written for $R=4$-parallel computation separated by $M_{sc}$ samples apart, it is given by + +$$P_{sc}[n] = \sum_{m=0}^{M_{sc}-1} r^*[n+m] \cdot r[n+m+M_{sc}] \quad (3.31)$$ + +$$P_{sc}[n+M_{sc}] = \sum_{m=0}^{M_{sc}-1} r^*[n+m+M_{sc}] \cdot r[n+m+2M_{sc}] \quad (3.32)$$ + +$$P_{sc}[n+2M_{sc}] = \sum_{m=0}^{M_{sc}-1} r^*[n+m+2M_{sc}] \cdot r[n+m+3M_{sc}] \quad (3.33)$$ + +$$P_{sc}[n+3M_{sc}] = \sum_{m=0}^{M_{sc}-1} r^*[n+m+3M_{sc}] \cdot r[n+m+4M_{sc}] \quad (3.34)$$ + +$$\begin{aligned} P_{sc}[n+1] &= P_{sc}[n] + r^*[n+M_{sc}] \cdot r[n+2M_{sc}] \\ &\quad - r^*[n] \cdot r[n+M_{sc}] \end{aligned} \quad (3.35)$$ + +$$\begin{aligned} P_{sc}[n+M_{sc}+1] &= P_{sc}[n+M_{sc}] + r^*[n+2M_{sc}] \cdot r[n+3M_{sc}] \\ &\quad - r^*[n+M_{sc}] \cdot r[n+2M_{sc}] \end{aligned} \quad (3.36)$$ + +$$\begin{aligned} P_{sc}[n+2M_{sc}+1] &= P_{sc}[n+2M_{sc}] + r^*[n+3M_{sc}] \cdot r[n+4M_{sc}] \\ &\quad - r^*[n+2M] \cdot r[n+3M_{sc}] \end{aligned} \quad (3.37)$$ + +$$\begin{aligned} P_{sc}[n+3M_{sc}+1] &= P_{sc}[n+3M_{sc}] + r^*[n+4M_{sc}] \cdot r[n+5M_{sc}] \\ &\quad - r^*[n+3M] \cdot r[n+4M_{sc}] \end{aligned} \quad (3.38)$$ \ No newline at end of file diff --git a/samples/texts/7426999/page_92.md b/samples/texts/7426999/page_92.md new file mode 100644 index 0000000000000000000000000000000000000000..1136ce13ba19ff49708aa8f674e48a1bb839b758 --- /dev/null +++ b/samples/texts/7426999/page_92.md @@ -0,0 +1,50 @@ +It can be observed that, in the block-level parallel non-iterative equation computation the data retrieved from memory can be shared across computations and memory access is regular. In case of block-level parallel iterative equation, multiplier outputs can be shared which leads to significant decrease in multipliers used. Due to selection of block size of $M_{sc}$, this sharing has been made possible. Consider now the auto-correlation equations of Minn-Bhargava algorithm for training symbol pattern [A A A - A], + +$$ +\begin{align} +P_{mb}[n] &= \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+kM_{mb}] \nonumber \\ +&\quad \cdot r[n+m+(k+1)M_{mb}] \tag{3.39} \\ +P_{mb}[n] &= P_{mb}[n-1] + r^*[n] \cdot r[n+M_{mb}] - r^*[n+3M_{mb}] \cdot r[n+4M_{mb}] \nonumber \\ +&\quad + 2r^*[n+2M_{mb}] \cdot r[n+3M_{mb}] \tag{3.40} +\end{align} +$$ + +where Eq. 3.39 is the non-iterative equation and Eq. 3.40 is the iterative equation, $P_{mb}$ is the auto-correlation function, $M_{mb}$ is the size of the repeating part (A), $M_{mb} = N/4$. If the non-iterative and iterative equations are written for $R=4$-parallel computation using block-size of $M_{mb}$, it is given by + +$$ P_{mb}[n] = \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+kM_{mb}] \quad (3.41) $$ + +$$ P_{mb}[n + M_{mb}] = \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+(k+1)M_{mb}] \quad (3.42) $$ + +$$ P_{mb}[n + 2M_{mb}] = \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+(k+2)M_{mb}] \quad (3.43) $$ + +$$ P_{mb}[n + 3M_{mb}] = \sum_{k=0}^{L-2} p[k] \cdot p[k+1] \sum_{m=0}^{M_{mb}-1} r^*[n+m+(k+3)M_{mb}] \quad (3.44) $$ + +$$ +\begin{align} +P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\ +&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \tag{3.45} +\end{align} +$$ + +$$ +\begin{align} +P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\ +&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \tag{3.46} +\end{align} +$$ + +$$ +\begin{align} +P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\ +&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \tag{3.47} +\end{align} +$$ + +$$ +\begin{align} +P_{mb}[n+1] &= P_{mb}[n] - r^*[n] \cdot r[n + M_{mb}] - r^*[n + 3M_{mb}] \cdot r[n + 4M_{mb}] \nonumber \\ +&\quad + 2r^*[n + 2M_{mb}] \cdot r[n + 3M_{mb}] \tag{3.48} +\end{align} +$$ + +It can be observed that due to selection of block size $M_{mb}$, which is the distance of auto-correlation, significant sharing of resources can be obtained. Based on sharing of \ No newline at end of file diff --git a/samples/texts/7426999/page_93.md b/samples/texts/7426999/page_93.md new file mode 100644 index 0000000000000000000000000000000000000000..23662275f66e86356e5250eb0c33caa49d997217 --- /dev/null +++ b/samples/texts/7426999/page_93.md @@ -0,0 +1,21 @@ +resources for non-iterative and iterative equations, two architectures are proposed which can support block-parallel computation. + +## 3.9 Partial-Streaming Block-Parallel (PSBP) Architecture + +Proposed architecture uses the shares resources completely between non-iterative and iterative equations. The architecture has two modes of operation, one for non-iterative mode which is done for initial auto-correlation point calculation, and the other one for iterative mode which is subsequent points in the block. Consider the computation of $R M_{sc}$ auto-correlation outputs using $R$ parallel PSBP architecture. Dividing the total computation work into $R$ parallel blocks, each block computes $\frac{R M_{sc}}{R}$ outputs. The order of computation is as follows: + +* Initially $R$ initial auto-correlation points are calculated in non-iterative mode. It requires $M_{sc}$ cycles in case of SCA and $M_{mb}$ in case of MBA for completing the operation. + +* The remaining ($M_{sc} - 1$) in case of SCA or ($M_{mb} - 1$) in case of MBA is computed in iterative mode. It can output ($M_{sc} - 1$) outputs in the same number of cycles and gives a throughput of $R$ samples/cycle. + +After the computation of $R M_{sc}$ outputs, the same process is repeated for the calculation of next set of $R M_{sc}$ outputs. The architecture consumes $(2M_{sc} - 1)$ cycles for computing $M_{sc}$ auto-correlation outputs per block. The architecture is called partial streaming because, it does not produce output every cycle and has a delay when computing initial auto-correlation output using non-iterative equation. The outputs are not fully streaming, but it uses minimum set of resources for computation. The following subsections depict the architecture for SCA and MBA. + +### 3.9.1 Proposed PSBP architecture for Schmidl-Cox algorithm (SCA) + +Figure 3.9 shows the $R = 4$-parallel PSBP architecture for auto-correlation calculation in case of SCA. Energy calculation of SCA written in both non-iterative and iterative form are given by: + +$$R_{sc}[n] = \sum_{m=0}^{M_{sc}-1} |r[n + m + M_{sc}]|^2 \quad (3.49)$$ + +$$R_{sc}[n] = R_{sc}[n-1] + |r[n + 2M_{sc}]|^2 - |r[n]|^2 \quad (3.50)$$ + +Figure 3.10 shows the $R = 4$-parallel PSBP architecture for energy calculation in case of SCA. Architectural complexity of the proposed PSBP architecture as a function of $R$ parallel inputs/outputs is shown in Table 3.4. Resource requirement scales only as a function of $R$-parallel input/output and is independent of $M_{sc}$. It is compared with Kaneda's \ No newline at end of file diff --git a/samples/texts/7426999/page_94.md b/samples/texts/7426999/page_94.md new file mode 100644 index 0000000000000000000000000000000000000000..cbb3a97019e84e5d63ba74d9802703e4a28ea97a --- /dev/null +++ b/samples/texts/7426999/page_94.md @@ -0,0 +1,3 @@ +FIGURE 3.9: Proposed *R* = 4-Parallel PSBP Architecture for *P*sc calculation in case of SCA. iter_flag = 0 indicates non-iterative computation mode, while iter_flag = 1 indicates iterative computation mode. + +FIGURE 3.10: Proposed *R* = 4-Parallel PSBP Architecture for *R*sc calculation in case of SCA. iter_flag = 0 indicates non-iterative computation mode, while iter_flag = 1 indicates iterative computation mode. \ No newline at end of file diff --git a/samples/texts/7426999/page_95.md b/samples/texts/7426999/page_95.md new file mode 100644 index 0000000000000000000000000000000000000000..3efc66fc02153a58861169c76d01d248451b9889 --- /dev/null +++ b/samples/texts/7426999/page_95.md @@ -0,0 +1,21 @@ +TABLE 3.4: Architectural Complexity calculation as a function of R-parallel input/output for SCA + +Proposed architecture for Schmidl-Cox + +
AlgorithmReal MultipliersReal Adders
Psc4(R + 1)2(3R + 1)
Rsc2(R + 1)3R + 1
+ +Kaneda's architecture for Schmidl-Cox + +
AlgorithmReal MultipliersReal Adders
Psc4R2R(Msc/2 + 1)
+ +proposals for SCA. It can be seen that proposed PSBP architecture requires four more multipliers compared to Kaneda's proposal, but this difference is fixed and independent of $R$. Number of adders required by Kaneda's architecture is large due to its dependence on size of training symbol ($M_{sc}$). For $M_{sc} = 32$, adder savings of proposed architecture for $R = 16$-parallel architecture is around 82%. Similar savings can be obtained for the architecture of $R_{sc}$ compared to Kaneda's method of sample-level parallelization. + +### 3.9.2 Proposed PSBP architecture of Minn-Bhargava algorithm (MBA) + +Figure 3.11 shows the $R = 4$-parallel PSBP architecture for auto-correlation calculation in case of MBA. Energy calculation of MBA algorithm is given by + +$$R_{mb}[n] = \sum_{k=1}^{L-2} \sum_{m=0}^{M_{mb}-1} |r[n+m+kM_{mb}]|^2 \quad (3.51)$$ + +$$R_{mb}[n] = R_{mb}[n-1] + |r[n+4M_{mb}]|^2 - |r[n+M_{mb}]|^2 \quad (3.52)$$ + +Figure 3.12 shows the $R = 4$-parallel PSBP architecture for energy calculation in case of MBA. Table 3.5 shows the architectural complexity as a function of R-parallel input/output for Minn-Bhargava algorithm. It is again compared with Kaneda's proposal for Schmidl-Cox. Proposed architecture requires twelve more real multipliers compared to Kaneda's proposal and it is fixed independent of $R$. This is because Minn-Bhargava auto-correlation ($P_{mb}$) has inherently higher number of computations compared to Schmidl-Cox auto-correlation ($P_{sc}$). But, there are savings in area required compared to Kaneda's proposal. For $R = 16$-parallel output, adder savings for proposed architecture is around 81%. Similar savings can be obtained for the architecture of $R_{mb}$ compared to Kaneda's method of sample-level parallelization. \ No newline at end of file diff --git a/samples/texts/7426999/page_96.md b/samples/texts/7426999/page_96.md new file mode 100644 index 0000000000000000000000000000000000000000..c05f79ef80dc11090f542e937dbb5854aa4fbf7f --- /dev/null +++ b/samples/texts/7426999/page_96.md @@ -0,0 +1,3 @@ +FIGURE 3.11: Proposed PSPB Architecture for calculation of $P_{mb}$ in case of MBA. $iter_flag = 0$ indicates non-iterative computation mode, while $iter_flag = 1$ indicates iterative computation mode. + +FIGURE 3.12: Proposed R = 4-Parallel PSBP Architecture for $R_{mb}$ calculation in case of MBA. $iter_flag = 0$ indicates non-iterative computation mode, while $iter_flag = 1$ indicates iterative computation mode. \ No newline at end of file diff --git a/samples/texts/7426999/page_97.md b/samples/texts/7426999/page_97.md new file mode 100644 index 0000000000000000000000000000000000000000..04d8a7ea41c3b2ceb7ae9d440e9c4f0ca65d87ff --- /dev/null +++ b/samples/texts/7426999/page_97.md @@ -0,0 +1,15 @@ +TABLE 3.5: Architectural Complexity calculation as a function of *R*-parallel input/output for MBA + +Proposed architecture for Minn-Bhargava + +
AlgorithmReal MultipliersReal Adders
Pmb4(R + 3)2(3R + 3)
Rmb2(R + 3)(3R + 3)
+ +Kaneda's architecture for Schmidl-Cox + +
AlgorithmReal MultipliersReal Adders
Psc4R2R(Msc/2 + 1)
+ +### 3.9.3 Comparison of Architectural Complexity + +Architectural Complexity as a function of *R*-parallel output for PSBP architecture and Kaneda's architecture in terms of number of real multipliers and adders required for realization of $P_{sc}/P_{mb}$. Figure 3.13 shows the number of real multipliers used. It can be observed difference of extra multipliers remains constant for all values of *R*. Figure 3.14 shows the number of real adders used. The gains in adders for PSBP architecture increases for higher values of *R* and reaches around 81% at *R* = 16. + +FIGURE 3.13: Multiplier requirement as a function of *R*-parallel output for PSBP and Kaneda's architecture, *M* = 32 \ No newline at end of file diff --git a/samples/texts/7426999/page_98.md b/samples/texts/7426999/page_98.md new file mode 100644 index 0000000000000000000000000000000000000000..9b17f4af9e10e9592b3e2996d9c467ddfdd3213c --- /dev/null +++ b/samples/texts/7426999/page_98.md @@ -0,0 +1,9 @@ +FIGURE 3.14: Adder requirement as a function of *R*-parallel output for PSBP and Kaneda's architecture, $M = 32$ + +## 3.10 Full-Streaming Block-Parallel (FSBP) Architecture + +The PSBP architecture in Section 3.9 takes $M_{sc}/M_{mb}$ cycles for initial point computation and this delay is encountered every time a new block computation needs to be started. This behaviour results in increase of memory for storage of OFDM symbols and hence is not completely a real-time architecture which can process multiple parallel inputs. In this section, modification of PSBP architecture is done to make the architecture fully streaming and produce $M_{sc}/M_{mb}$ auto-correlation output of in $M_{sc}/M_{mb}$ cycles respectively. The modification proposed is addition of separate block which can do initial point computation in parallel with iterative block so that auto-correlation output can be produced every cycle. Initial computation block computes auto-correlation output using non-iterative equation every $M_{sc}/M_{mb}$ samples apart and it is given to iterative block periodically to start computation on next set of blocks. + +### 3.10.1 Proposed FSBP architecture for SCA + +$R = 4$-Parallel Initial Computation block for auto-correlation and energy calculation of SCA are shown in Figure 3.15 and Figure 3.16. It takes $M_{sc}/R$ cycles for computation of output. The increase in resources is proportional to *R*-parallel input. After an initial latency of $M_{sc}$ cycles, the initial computation block feeds PSBP block *R* outputs continuously for \ No newline at end of file diff --git a/samples/texts/7426999/page_99.md b/samples/texts/7426999/page_99.md new file mode 100644 index 0000000000000000000000000000000000000000..9bff0475feadb005a76f6e8fec3d093aa5691327 --- /dev/null +++ b/samples/texts/7426999/page_99.md @@ -0,0 +1,122 @@ +iterative computation at regular intervals. In this case, the PSBP architecture is used in it- +erative mode only (iter_flag=1). This makes the FSBP architecture produce outputs every +cycle and therefore operate in real-time. Table 3.6 calculates the architectural complexity +of Psc for FSBP architecture. + +FIGURE 3.15: R = 4-Parallel Initial Point Auto-Correlation Computation Block for SCA + +FIGURE 3.16: R = 4-Parallel Initial Point Energy Computation Block for SCA + +TABLE 3.6: Architectural Complexity of $P_{sc}$ for FSBP Architecture as a function of $R$-parallel input/output + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ Algorithm + + Real Multipliers + + Real Adders +
+ P + + sc + + (Initial Point) + + 4R + + 4R +
+ P + + sc + + (Iterative Point) + + 4(R + 1) + + 2(3R + 1) +
+ P + + sc + + (Total) + + 8R + 4 + + 10R + 2 +
+ R + + sc + + (Initial Point) + + 2R + + 4R +
+ R + + sc + + (Iterative Point) + + 2(R + 1) + + (3R + 1) +
+ R + + sc + + (Total) + + 4R + 2 + + 7R + 1 +
+ +**3.10.2 Proposed FSBP architecture for MBA** + +$R = 4$-Parallel Initial Computation Auto-Correlation and Energy Calculation for MBA are shown in Figure 3.17 and Figure 3.18. Table 3.7 calculates the architectural complexity of $P_{mb}$ calculation. \ No newline at end of file diff --git a/samples/texts/7526122/page_1.md b/samples/texts/7526122/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..7affad8090582316fe3bb6a990f28efe187af562 --- /dev/null +++ b/samples/texts/7526122/page_1.md @@ -0,0 +1,36 @@ +# Theory for Cyclotron Harmonic Radiation from Plasmas + +KARL-BIRGER PERSSON AND E. G. JOHNSON, JR. + +National Bureau of Standards, Boulder, Colorado + +AND + +D. A. UHLENBROCK + +Department of Mathematics, University of Wisconsin, Madison, Wisconsin +(Received 5 September 1967; final manuscript received 7 December 1967) + +A highly simplified hydrodynamic model for the cause of cyclotron harmonics radiation in a magnetized, abnormal-glow, helium plasma is proposed. The high-velocity electrons are treated as sources for the charge density waves in the plasma. Electromagnetic radiation results from interaction of these wake waves with the statistical fluctuations (the granular structure in the ion density). This radiation contains the cyclotron harmonics. Some numerical curves of the radiation spectra are shown for a number of plasma parameters. + +## INTRODUCTION + +The present paper derives a theory which gives a reason for the incoherent electromagnetic noise that is emitted in the neighborhood of the electron cyclotron frequency and its harmonics under the circumstance that the plasma here considered is an abnormal-glow helium plasma¹ located in a uniform magnetic field. A following paper describes the experimental arrangements necessary to observe the emitted spectrum and compares this and other theories with the observations.² + +The radiation spectra discussed here have been subjected to rather intense investigations, both theoretical as well as experimental, for the past ten years without leading to positive identification of the mechanism responsible for the radiation. The confusing situation probably exists because there could be several different physical mechanisms involved and because most plasma configurations used will not allow a simple theoretical description.³ To compound the difficulties the proposed theories are so incomplete that they can not be compared with the experimental observations. A reasonably complete summary and bibliography on this subject is given by Bekefi.⁴ As he points out, in order to understand the emission of electromagnetic radiation from magnetized plasmas, it is necessary to determine the coupling between longitudinal and transverse electromagnetic waves and to calculate the absolute radiative intensities. We address ourselves to these points. + +Because the abnormal negative glow plasma is sufficiently spatially uniform, we have a situation + +that permits possible quantitative agreement between theory and experiment. Since we chose to observe only the radiation from the field-free part of the plasma, it was necessary to do some modifications to the discharge tube described in Ref. 1. These modifications are shown in Fig. 1. The discharge tube is divided into parts I and II by the mesh anode. As a consequence, the space I which includes the cathode and the cathode fall range is completely shielded by this fine metal mesh structure. Because the anode is sufficiently transparent, the high-energy electrons generated in the cathode fall range enter the space II and generate a substantial and essentially spatially uniform plasma. The magnetic field is applied parallel to the tube axis. This configuration permits observation of the emitted radiation both parallel and perpendicular to the applied magnetic field. The theory under discussion attempts to explain the cause of radiation emitted in the direction parallel to the magnetic field. This case is chosen because additional information about the radiation process can be gained by distinguishing between left- and right-hand circularly polarized radiation. + +The order of magnitude of some of the parameters which characterize the abnormal negative-glow plasma for helium at a pressure of 0.5 Torr follow: the electron density is $10^{12}$ cm$^{-3}$, the electron-beam density is $10^6$ cm$^{-3}$, the temperature of the cold electrons is 0.05 eV, the energy of the beam electrons is 3000 eV, the collision frequency of the cold electrons is $10^9$ sec$^{-1}$, the collision frequency of the beam electrons is $10^8$ sec$^{-1}$, and the Debye length is $10^{-4}$ cm. Because the Debye length is two orders of magnitude smaller than the average distance between electrons in the beam, we can rule out possible collective phenomena involving the beam. + +Briefly, this theory assumes that each individual + +¹ K. B. Persson, J. Appl. Phys. **36**, 10 (1965). + +² H. W. Wassink, Phys. Fluids **11**, 629 (1968). + +³ V. L. Ginsburg and V. V. Zhelezniakov, Soviet Astron. **2**, 653 (1958). + +⁴ G. Bekefi, *Radiation Processes in Plasmas* (John Wiley & Sons, Inc., New York, 1966), Chap. 7. \ No newline at end of file diff --git a/samples/texts/7526122/page_10.md b/samples/texts/7526122/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..1d6518a93fb180c62b278c4d807f89d22782d1c1 --- /dev/null +++ b/samples/texts/7526122/page_10.md @@ -0,0 +1,17 @@ +## SUMMARY AND DISCUSSION + +Many attempts have been made in the past to explain the basic mechanism causing the observed harmonic electron cyclotron radiation. Since Bekefi discusses them in his recent book,⁴ we will not do so here. It is common to these theories that the radiation losses within the plasma are carried exclusively by the longitudinal waves and that these waves, when they hit the boundaries or some other nonuniformity in the plasma, are converted into electromagnetic waves in a manner presently inaccessible to theory. These theories have neither explained the shape of the spectrum nor accounted for the intensity of the radiation. + +Many different mechanisms can give radiation at the harmonics of the electron cyclotron frequency. To distinguish between these mechanisms, it is presently necessary to develop the corresponding theories to a sufficient degree, so that the shape of the spectrum, its intensity, and some other characteristic features can be compared with the experimental data. The model presented in this paper is not necessarily the final explanation for the observed spectrum, but it satisfies the criteria mentioned above and is sufficiently close to the experimental data, in particular the data obtained from the abnormal negative glow plasma, both in the shape of the spectrum and the intensity, to merit further sophistication. + +The model proposed in this paper has common elements with theories suggested in the past; nevertheless, it is conceptually different. It requires hot electrons in a relatively cool background plasma. These hot electrons may be supplied by the high-velocity end of the electron-velocity distribution. For the sake of simplicity, this model considers a hot electron which pursue's a cyclotron orbit with a radius much larger than the Debye length. This electron excites plasma oscillations along its orbit which are primarily confined to a Debye length of the orbit. The amplitude of the space charge wave packet resonates when the hybrid frequency ($\omega_c^2 + \omega_s^2$)† is an integer times the cyclotron frequency. + +We showed that this wave packet has cyclotron harmonics in it; however we also found that this packet does not radiate at the harmonics if the background plasma is perfectly uniform. Because the small volume of plasma excited by the source electron contains so few ions there are significant thermal fluctuations. These nonuniformities allow electromagnetic radiation to occur. Once we had the mechanism for the radiation, we calculated its numerical form by standard methods. We note that the asymmetry between the ± polarization is due to the differences in the radiative absorption of the plasma. + +Although subject to many simplifications, this theory results in data that are compatible with the experimental information. A following paper discusses this point more thoroughly.² When a comparison is made with experimental data, the following points should be considered: It is likely that the influence of the boundary condition has been exaggerated in the present calculation. We have assumed a step profile for the plasma at one end and a perfectly reflecting boundary at the other. This is not realized in practice. The influence of the plasma frequency through the index of refraction is therefore stronger than it should be. The plasma frequency appearing in the formula above has been viewed as a fixed parameter common for all points in the plasma. This again is not strictly true in practice due to the presence of macroscopic non-uniformities. The use of two groups of electrons instead of a smooth distribution function is an idealization that will influence somewhat the shape of the radiation spectra. However, the elimination of these theoretical simplifications is expected to change the spectra in minor ways and not its major characteristics. + +## ACKNOWLEDGMENTS + +The authors acknowledge the accurate programming effort of Marlene Pratto. + +This work partially supported by Defense Atomic Support Agency. \ No newline at end of file diff --git a/samples/texts/7526122/page_2.md b/samples/texts/7526122/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..fb197850465f1c4f459e20a9738e02f1a9c36616 --- /dev/null +++ b/samples/texts/7526122/page_2.md @@ -0,0 +1,15 @@ +FIG. 1. A block diagram of the plasma tube. + +beam electron or any other fast electron in the plasma acts independently of any other electron of the same type during its Coulombic interaction with the cold background plasma. Because the fast electron has velocities at supersonic speeds, there are excited longitudinal space waves in the background plasma which, when the collision frequencies are small, result in a space-charge wake wave that has a time structure harmonically distributed relative to the cyclotron frequency. When the background plasma is perfectly uniform, these longitudinal waves are not coupled to any transverse electromagnetic modes. As a consequence there is no electromagnetic radiation from the induced wave packet. When the exciting electron orbits in a spacially nonuniform plasma, the coupling is operative and harmonic radiation ensues. Because the plasma has a finite number of particles and because it has an amorphous granular structure on the microscopic scale, we find that there can be nonuniformities in the ion density. Crudely, we can vision each ion and the associated free electron in the plasma as an oscillator with a characteristic frequency equal to the plasma frequency. When a fast electron in a cyclotron orbit periodically excites a small number of these randomly situated oscillators, we find cyclotron harmonics. The number of oscillators is controlled by the cyclotron orbit size of the fast electron and by the fact that the only oscillators that are driven are those that are located within about a Debye radius of this orbit. To give some numerical idea of the number of oscillators we would expect on the average, we consider that the fast electrons of energy 1 eV are produced by the ionizing collision of the beam electron with the neutrals. From the typical numbers associated with the plasma parameters in a helium negative-glow plasma, we find the number of ions along the cyclotron orbit is $10^4$. This is a sufficiently large number to validate the use of the hydrodynamic equations for the basic electromagnetic processes. At the same time this number is + +sufficiently small that variations in the actual number of oscillators permit strong coupling between the longitudinal and radiative electromagnetic modes. Please note that the number and location of the ions in the cyclotron orbit are stationary at the time scale of the motion of the fast electron. + +In addition to the above mechanism for efficient coupling of the longitudinal fields to the radiative fields, there can be generated a harmonics emission spectra if there are spacial time-independent ion-density nonuniformities at a scale of less than the radius of the significant cyclotron orbits. + +Ginsburg and Zheleziakov³ have proposed that fluctuations, primarily electron density, provide efficient scattering centers for the propagating longitudinal waves and efficient coupling to radiative electromagnetic waves. This model probably describes some of the observed microwave emission from hot plasmas; however, the mathematics associated with this model is currently unable to produce numbers that will allow quantitative comparison with available laboratory measurements. We differ from their analysis by considering the electromagnetic radiation induced by a longitudinal wake bounded to the generating electron. In its present form, our model is primarily applicable to the cold plasma which is experimentally well represented by the abnormal negative glow in helium.¹ It does not invoke fluctuations of the electron gas and it does not depend on the presence of propagating longitudinal waves as is the case for the model suggested by Ginzburg and Zheleziakov. + +## THEORY I. THE ELEMENTARY RADIATOR + +This theory is concerned with the emission of microwave radiation by a three-component (ions, electrons, and neutral atoms) nonrelativistic cold plasma with a homogeneous, static magnetic field $B_0$. The actual geometry of the plasma is a finite-length cylinder of radius $R$ with the cylinder axis parallel to $B_0$. However, except for the final steps of the calculation the finite size of the plasma will be ignored in the equations and an infinite plasma is assumed. + +The basic equations are derived from the Boltzmann-Vlasov equations for the three components by introducing collision terms with empirical effective collision frequencies and taking zeroth- and first-order moments with respect to the velocity. The moment hierarchy is truncated by replacing the square term in the velocities by a diagonal pressure tensor with ideal gas law dependence on the temperature. \ No newline at end of file diff --git a/samples/texts/7526122/page_3.md b/samples/texts/7526122/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..92be22d2241e4af2bc31a8f8097fc9981ef1a853 --- /dev/null +++ b/samples/texts/7526122/page_3.md @@ -0,0 +1,59 @@ +Since the ion mass is very large compared to the electron mass, only the limiting equations for infinite ionic (and atomic) mass are considered. They imply that the ionic current and charge densities $J^{(+)}$ and $q^{(+)}$ are constant in time. Thus $J^{(+)}$ will vanish if it is assumed to vanish at some initial time, as will be done here. Likewise the equations for the neutral component decouple from those for the electron component and can be ignored. + +As a result the effects of the positive and neutral components are represented in the equations for the electron component only in terms of an effective electron-neutral collision frequency $\nu$ and the neutralizing, possibly spatially varying ion-charge density $q^{(+)} = q_0^{(+)}(1+f)$, where $f$ gives the spatial deviation from the average charge density $q_0^{(+)}$. Suggested causes for this $f$ are variations in the production mechanisms for the background plasma or the intrinsic fluctuations in the ion density which are being sampled in such a way that they are significant and hence not negligible. We assume that these fluctuations are essentially direct current compared to the high-frequency processes involved in the production of microwave radiation. + +The externally injected fast (keV) beam electrons and the energetic secondaries resulting from initial ionizations are represented in terms of their charge and current-density $q^{(*)}$ and $J^{(*)}$ as the source in Maxwell's equations and are not included in the Boltzmann distribution functions. + +Thus with $q$, $J$ as the electron charge and current densities, $\kappa = |e|/m$ the charge to mass ratio of an electron, $\nu^2$ the squared electron sound speed, $\omega_c \equiv \kappa |B_0|$ and $r_c \equiv v_1/\omega_c$ the cyclotron frequency and radius of a source electron with transverse speed $\nu$, and the loss rate $\nu_1$, the basic equations including the usual Maxwell's equations (in mks units) are as follows: + +Moment equations + +$$ \frac{\partial q}{\partial t} + \nabla \cdot J = 0, \quad (1) $$ + +$$ \left(\frac{\partial}{\partial t} + \nu_1 - \kappa B \times \right) J + \nu^2 \nabla q + \kappa E q = 0; \quad (2) $$ + +Maxwell's equations + +$$ \nabla \cdot E = \epsilon_0^{-1}(q + q^{(+)} + q^{(*)}), \quad (3) $$ + +$$ \nabla \cdot B = 0, \quad (4) $$ + +$$ \nabla \times E = -\frac{\partial}{\partial t} B, \quad (5) $$ + +$$ \nabla \times B = \epsilon_0 \mu_0 \frac{\partial}{\partial t} E + \mu_0 (J + J^{(*)}); \quad (6) $$ + +Source equations + +$$ r^{(*)}(t) = r_0 + [r_c \cos \omega_c(t - t_0), $$ + +$$ -r_c \sin \omega_c(t - t_0), z_0 + v_s t], \quad (7) $$ + +$$ T_{(t_{-}, t_{+})}(t) = \Theta(t_{+} - t) - \Theta(t - t_{-}), \quad (8) $$ + +where $\Theta$ is the step function, and $t_+$ and $t_-$ are the creation and destruction times: + +$$ q^{(*)}(r, t) = e \delta[r - r^{(*)}(t)] T_{[t_{-}, t_{+}]}(t), \quad e < 0, \quad (9) $$ + +$$ J^{(*)}(r, t) = q^{(*)}(r, t) \frac{d}{dt} r^{(*)}. \quad (10) $$ + +Here $r^{(*)}$ represents the position of a beam or secondary source electron and $T_{[t_{-}, t_{+}]}$ simulates the effect of collisions on them. + +Equation (2) is the only nonlinear equation and is linearized by replacing $B$ with $B_0$ and $q$ with $-q^{(+)}$, which is found by requiring approximate local neutrality $q + q^{(+)} \cong 0$ of the background plasma. Here $E=0$ to this order in the linearization. Also $\nu^2$ is taken as a constant. + +To make the system of Eqs. (1)-(10) more manageable, Maxwell's equations are re-expressed in terms of the scalar and vector potentials $\Phi$ and $A$: + +$$ E = -\nabla\Phi - \frac{\partial}{\partial t}A, \quad (11) $$ + +$$ B = \nabla \times A + B_0, \quad (12) $$ + +$$ 0 = \nabla \cdot A + \frac{1}{c^2} \frac{\partial}{\partial t} \Phi \text{ (Lorentz condition)}, \quad (13) $$ + +$$ \left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2\right)(\epsilon_0 \Phi) = q + q^{(+)} + q^{(*)}, \quad (14) $$ + +$$ \left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2\right)\left(\frac{1}{\mu_0} A\right) = J + J^{(*)}. \quad (15) $$ + +When a preferred (right-handed, orthogonal) coordinate system with coordinates $(x, y, z)$ and the $z$ axis parallel along $B_0$ is introduced, the vectors or operators (like $J$ or $\nabla$) are usefully resolved into longitudinal and transverse components, $J_1 = J_s$, and $J_\perp = (J_+, J_-)$, where in turn $J_\perp$ is conveniently characterized in terms of $J_\pm = J_s \pm iJ_\nu$. The time dependence is treated by a Fourier integral transformation according to the generic formula: + +$$ f(t) = \int_{-\infty}^{+\infty} d\omega e^{-i\omega t} f(\omega) \quad (16) $$ + +such that a real valued $f(t)$ satisfies the identity $\tilde{f}(\omega)*\tilde{f}(-\omega)$ with respect to complex conjugation. Finally, dimensionless variables and fields labeled \ No newline at end of file diff --git a/samples/texts/7526122/page_4.md b/samples/texts/7526122/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..c6a0c1508b2c6b88f905c6582e926240ef4fd941 --- /dev/null +++ b/samples/texts/7526122/page_4.md @@ -0,0 +1,147 @@ +with an asterisk are introduced and defined as +follows: + +$$ +\begin{align*} +r &= \frac{c}{\omega_p} r^*, & t &= \frac{1}{\omega_p} t^*, & \Phi &= \frac{e\omega_p}{\epsilon_0 c} \Phi^*, \\ +A &= e\omega_p \mu_0 A^*, & J &= \frac{e\omega_p^3}{c^2} J^*, & q &= \frac{e\omega_p^3}{c^3} q^*, +\end{align*} +$$ + +where $\omega_p^2 = \epsilon_0^{-1}\kappa q_0^{(+)}$, the squared (average) plasma frequency. + +Using the notation + +$$ +\omega_v = \omega + i\nu_l +$$ + +and + +$$ +\Omega = +\begin{bmatrix} +\Omega_+ & 0 & 0 \\ +0 & \Omega_- & 0 \\ +0 & 0 & \Omega_0 +\end{bmatrix} +\\[1em] += \begin{bmatrix} +\omega(\omega_r + \omega_o) & 0 & 0 \\ +0 & \omega(\omega_r - \omega_o) & 0 \\ +0 & 0 & \omega\omega_r +\end{bmatrix} +\\[1em] += \begin{bmatrix} +(\Omega_\perp) & 0 \\ +0 & \Omega_0 +\end{bmatrix} +\tag{17} +$$ + +as well as leaving out the asterisk which labels the dimensionless variables and fields, one finds after the Fourier transformation the following set of equations: + +$$ +i\omega\tilde{q} = \nabla_{\perp} \cdot \mathcal{J}_{\perp} + \nabla_{\parallel} \cdot \mathcal{J}_{\parallel}, \quad (18) +$$ + +$$ +i\omega\tilde{\Phi} = \nabla_{\perp} \cdot \tilde{A}_{\perp} + \nabla_{\parallel} \cdot \tilde{A}_{\parallel}, \quad (19) +$$ + +$$ +(\nabla_{\perp}^2 + \nabla_{\parallel}^2 + \omega^2)\tilde{\Phi} = -(\tilde{q} + \tilde{q}^{(*)} + \tilde{q}^{(+)}) , \quad (20) +$$ + +$$ +(\nabla_{\perp}^2 + \nabla_{\parallel}^2 + \omega^2) \tilde{A}_{\perp} = -(J_{\perp} + J_{\perp}^{(*)}), \quad (21) +$$ + +$$ +(\nabla_{\perp}^2 + \nabla_{\parallel}^2 + \omega^2) \tilde{A}_{\parallel} = -J_{\parallel}, \quad (22) +$$ + +$$ +\begin{align} +& \Omega_{\perp} \mathcal{J}_{\perp} + v^2 \nabla_{\perp} (\dot{i}\omega \tilde{q}) \nonumber \\ +& \qquad + (1+f) \nabla_{\perp} (\dot{i}\omega \tilde{\Phi}) + \omega^2 \tilde{A}_{\perp} = 0, \tag{23} +\end{align} +$$ + +$$ +\Omega_{||} J_{||} + v^2 \nabla_s (i\omega q) \\ ++ (1+f) \nabla_s (i\omega \Phi) + \omega^2 A_s = 0. \quad (24) +$$ + +Since in Eqs. (7)-(10) the motion of a given source electron is assumed helical, the corresponding variables are best described in terms of a cylindrical coordinate system $(r, \varphi, z)$ whose $z$ axis coincides with the axis of the cylinder generated by the helix and whose origin is at $r_0$. Only at the end, when the incoherent contributions of the various source electrons to the radiation are summed subject to the varying initial conditions and orbit parameters, will + +such a choice of coordinates be impossible. This procedure is adequate because Eqs. (18)-(24) are linear in the variables. + +With + +$$ +\begin{align*} +& \delta[r - r^{(*)}(t)] \\ +&= \frac{1}{r} \delta(r - r_c) \delta_{pr}[\varphi - \omega_c(t - t_0)] \delta(z - z_0 - v_s t) \\ +&= (2\pi)^{-2} \sum_{l=-\infty}^{\infty} \int_{-\infty}^{+\infty} dk_z \frac{1}{r} \delta(r - r_c) \\ +&\quad \cdot \exp \{il[\varphi - \omega_c(t - t_0)] + ik_z(z - z_0 - v_s t)\}, +\end{align*} +\tag{25} +$$ + +one finds + +$$ +q^{(s)}(r, \varphi, z, \omega) = \sum_{l=-\infty}^{\infty} e^{i\ell\varphi} \int_{-\infty}^{\infty} dk_z e^{ik_z s} q_l^{(s)}(r, k_z, \omega), \quad (26) +$$ + +$$ +q_l^{(*)}(r, k_z, \omega) = -(2\pi)^{-3} r_c^{-1} \delta(r-r_c) [\exp[-i(k_z z_0 - l\omega_c t_0)] \\ +\qquad \cdot (\exp(i\omega_l t_l) - \exp(i\omega_l t_r))], +$$ + +with + +$$ +\omega_l = \omega - l\omega_c - k_z v_z +$$ + +and + +$$ +J_{\pm}^{(*)}(r, \varphi, z, \omega) = \pm i r_c \omega_c e^{\pm i \varphi_c z_0} q^{(*)}(r, \varphi, z, \omega \pm \omega_c) +$$ + +$$ += \sum_{l=-\infty}^{\infty} e^{i(l\pm 1)\varphi} \int_{-\infty}^{\infty} dk_z e^{ik_z s} J_{\pm l}^{(*)}(r, k_z, \omega), \quad (27) +$$ + +where + +$J_{z,z'}^{(s)}(r, k_z, w) = \pm ir_c w_c q_z^{(s)}(r, k_z, w)$ + +and + +$J_{z,z'}^{(s)}(r, c, z, w) = v_z c^{(s)}(r, c, z, w).$ + +The quantity relevant for the description of radiation propagating parallel to B₀ is + +$$ +\bar{A}_{\perp}(z, w) = \int_{0}^{\infty} r dr \int_{0}^{2\pi} d\varphi \bar{A}_{\perp}(r, c, z, w). \quad (28) +$$ + +With a similar notation for the other quantities of Eqs. (18)-(25) it is essential to note that for a spatially homogeneous plasma (with $\nabla_{||}f = 0$) the longitudinal and transverse equations decouple in the following manner: + +Longitudinal response: + +$$ +i\omega q = \nabla_s J_s, \quad (29) +$$ + +$$ +i\omega\Phi = \nabla_t A_t, +$$ + +$$ +(\nabla_s^2 + \omega^2)\Phi = -(\tilde{q} + \tilde{q}^{(+)}) + (\tilde{q}^{(s)}), \quad (31) +$$ \ No newline at end of file diff --git a/samples/texts/7526122/page_5.md b/samples/texts/7526122/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..edf3d19dcf5603a391e39b03e18a1cc1a97c77f6 --- /dev/null +++ b/samples/texts/7526122/page_5.md @@ -0,0 +1,71 @@ +$$ (\nabla_{\parallel}^2 + \omega^2) \vec{A}_{\parallel} = -\vec{J}_{\parallel}, \quad (32) $$ + +$$ \Omega_{\parallel} \vec{J}_{\parallel} + v^2 \nabla_{\parallel}(i\omega\vec{q}) + \nabla_{\parallel}(i\omega\vec{\Phi}) + \omega^2 \vec{A}_{\perp}. \quad (33) $$ + +Transverse response: + +$$ (\nabla_{\parallel}^2 + \omega^2) \vec{A}_{\perp} = -(\vec{J}_{\perp} + \vec{J}_{\perp}^{(z)}), \quad (34) $$ + +$$ \Omega_{\perp} J_{\perp} + \omega^2 A_{\perp} = 0. \quad (35) $$ + +Thus $A_{\perp}$ is determined from Eqs. (34) and (35) with the corresponding source $\vec{J}_{\perp}^{(z)}$. As in the case of a nonrelativistic electron gyrating in a vacuum there is no radiation in the longitudinal direction at the cyclotron harmonics except at the fundamental frequency. + +The situation changes when $\nabla_{\perp} f$ does not vanish identically. In that case the term $f \cdot \nabla_{\perp} \Phi$ in Eq. (23) couples the longitudinal and transverse systems, so that harmonic radiation can result. The resulting system of equations leads to unwieldy convolution equations when Fourier integral transforms are applied. Instead we neglect $f \ll 1$ in Eq. (24) and in the $\vec{A}_{\perp}$ term of Eq. (23) and keep it only in the critical $\nabla_{\perp} \Phi$ term of Eq. (23), which produces the coupling. + +Hence by combining Eqs. (21) and (23) and forming the Fourier transform in $x, y$ with $k_x = k_y = 0$, it follows that + +$$ [\omega^2 - \Omega_{\perp}(\nabla_{\perp}^2 + \omega^2)] \vec{A}_{\perp} \\ = \Omega_{\perp} \vec{J}_{\perp}^{(*)} + i\omega \vec{\Phi}(\nabla_{\perp} f). \quad (36) $$ + +Here the operator on the left side of Eq. (36) corresponds to the well-known Appleton-Hartree dispersion law for the special case of longitudinal propagation. + +A first-order perturbation treatment will determine $\vec{\Phi}$ to zeroth order in $f$ from Eqs. (18)-(25) with $f \equiv 0$. We substitute the result into Eq. (36) to find $\vec{A}_{\perp}$ to first order in $f$. + +Under these $f \equiv 0$ assumptions, the longitudinal source velocity $v_s$ appearing in Eqs. (26) and (27) leads to a Doppler shift by $v_s k_x$ and an inessential $\vec{J}_s^{(*)}$ current. For simplicity, we ignore the shift by setting $v_s = 0$. As a consequence the helical motion becomes a circular motion and the source electron only "samples" $f$ at $r_0$ close to the cyclotron orbit. This sampling process is most sensitive to the $\varphi$ dependence of $f$, which leads us to adopt a model of the form + +$$ f(r, \varphi, z) = f(\varphi) = \sum_{l=-\infty}^{+\infty} f_l e^{il\varphi}, \quad (37) $$ + +with constants $f_l$, satisfying $f_l^* = f_{-l}$, $f_0 = 0$. These constants, reflecting the inhomogenity of the plasma, will, in general, vary for different source electrons. + +A later statistical treatment will take this into account, when the contributions from different source electrons are compounded. With the notation + +$$ \vec{\Phi}(r, \varphi, z, \omega) = \sum_{l=-\infty}^{\infty} e^{il\varphi} \int_{-\infty}^{+\infty} dk_z e^{ik_z z} \Phi_l(r, k_z, \omega), \quad (38) $$ + +the term of interest in Eq. (36) is given by + +$$ (\nabla_{\perp} f) \vec{\Phi}(z, \omega) = \int_{-\infty}^{\infty} dk_z e^{ik_z z} \sum_{l=-\infty}^{\infty} \pm 2\pi(l \pm 1) f_l z_l \\ \cdot \int_0^{\infty} dr \, \Phi_l(r, k_z, \omega). \quad (39) $$ + +In order to determine $\int_0^\infty dr \, \Phi_l$ to zeroth-order in $f$ from Eqs. (18)-(25), one can practically neglect the radiation terms. Thus Eqs. (18), (20), (23), and (24) with $f=0$, when transformed as in Eq. (38), give a determined system from which a fourth-order ordinary differential equation in the $r$ variable for $\Phi_l$ is obtained by elimination: + +$$ (a_2 D_t^2 + a_1 D_t + a_0) \Phi_t = (b_1 D_t + b_0) q_t^{(*)}, \quad (40) $$ + +with + +$$ D_t = \left( r^{-1} \frac{\partial}{\partial r} - r^2 \frac{\partial}{\partial r} - l^2 r^{-2} \right), $$ + +$a_2 \equiv v^2$, the velocity of sound squared. + +$$ +\begin{align} +a_1 &\equiv \chi \Omega_0 - 1 - v^2 k_z^2, \\ +a_0 &\equiv \chi k_z^2 (1 - \Omega_0 + v^2 k_z^2), \\ +\chi &\equiv 1 - \left( \frac{\omega_c}{\omega_v} \right)^2, \\ +b_1 &\equiv -v^2, \\ +b_0 &\equiv \chi (v^2 k_z^2 - \Omega_0). +\end{align} +\quad (41) $$ + +In a cold plasma $v^2 \to 0$, and screening distances are correspondingly short. Since $a_2 = v^2$, the limit $v^2 \to 0$ is a singular perturbation limit for Eq. (40). + +The limiting second-order equation is + +$$ (c_0 - D_t)\phi_t = c_0 q_t^{(*)}, \quad (42) $$ + +where + +$$ c_0 = \lim_{r_* \to 0} \left(-\frac{a_0}{a_1}\right) = \chi k_z^2 (1 - \Omega_0) (1 - \chi \Omega_0)^{-1}, $$ + +$$ c_0' = \lim_{r_* \to 0} \left(-\frac{b_0}{a_1}\right) = -\chi \Omega_0 (1 - \chi \Omega_0)^{-1}. $$ + +The corresponding homogeneous equation to Eq. (42) is Bessel's equation with a complex scale factor for the argument. The regular solution at $r=0$ behaves as $r^{1/2}$ as $r \to 0$. + +Short of solving the differential Eqs. (40) or (42) and evaluating the integral $\int_0^\infty dr \, r^\alpha \Phi_l$ numerically, \ No newline at end of file diff --git a/samples/texts/7526122/page_6.md b/samples/texts/7526122/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..d004cafef8c8339be9b7b1097f4d873c51ad571f --- /dev/null +++ b/samples/texts/7526122/page_6.md @@ -0,0 +1,131 @@ +the approximation of replacing $r$ by $r_c$ in $D_\ell$ and +integrating the resulting Eq. (42) gives + +$$ +\begin{align*} +& \left(\frac{\partial}{\partial r} \Phi_t\right)(0) + \frac{1}{r_c} \Phi_t(0) + \left(c_0 + \frac{\ell^2}{r_c^2}\right) \int_0^\infty dr \, \phi_t \\ +& \qquad = c'_0 \int_0^\infty dr \, q_t^{(*)}, +\end{align*} +$$ + +or since for $\ell^2 \ge 1$ $(\partial_r \Phi_t)(0) = 0$ and $\Phi_t(0) = 0$, +we have + +$$ +\int_0^\infty dr \, \Phi_t = \left(c_0 + \frac{\ell^2}{r_c^2}\right)^{-1} c'_0 \int_0^\infty dr \, q_t^{(*)}, \quad \ell^2 > 1. \quad (43) +$$ + +According to Eq. (26) this implies + +$$ +\int_0^\infty dr \, \Phi_t = \frac{i c'_0 \exp[-i(k_x z_0 - \ell \omega_c t_n)] [\exp(i\omega_c t_+) - \exp(i\omega_c t_-)]}{8\pi^3 r_c \omega_c (c_0 + \ell^2/r_c^2)}. +$$ + +By ignoring the $k_.$ dependence coming from $c_0$ +(i.e., setting $c_0 = 0$) Eq. (39) then gives + +$$ +-\frac{i\omega}{\Omega_{\pm}} (\overline{\Phi\nabla_{\perp}f}) = \delta(z - z_0) B'_{\pm}, +$$ + +with + +$$ +B'_{\pm} = \sum_{t=-\infty}^{\infty} f_{\pm 1}^{*} \frac{\pm(\ell \pm 1)r_c\omega(\omega_r \mp \omega_c)e^{i(\omega_r - \ell\omega_c)t_+} + e^{i(\omega - \ell\omega_c)t_-}}{2\pi\ell^2(\omega - \ell\omega_c)[\omega(\omega_r^2 - \omega_c^2) - \omega_r]} = \sum_{t=-\infty}^{\infty} f_{\pm 1} H_{\pm t} Y_t, \quad (44) +$$ + +where + +$$ +Y_t = e^{i\ell t\omega_c z_0} \frac{e^{i(\omega - t\ell\omega_c)t_+} - e^{i(\omega - t\ell\omega_c)t_-}}{(\omega - t\ell\omega_c)}. +$$ + +and + +$$ +H_o = 0 +$$ + +$$ +H_{\pm t} = \frac{\pm(\ell \pm 1)r_c\omega(\omega_r \mp \omega_c)}{\ell^2[\omega(\omega_r^2 - \omega_c^2) - \omega_v]} +$$ + +By straightforward integration it follows from Eq. +(27) that + +$$ +\bar{J}_{\pm}^{(a)} = \delta(z - z_0) J_{\pm}^{(a)}, +$$ + +with + +$$ +\begin{align} +J_{\pm}^{(a)} &= \mp \frac{r_c \omega_c}{(\omega \pm \omega_c)} e^{\mp i \omega_c t_0} (e^{i (\omega \pm \omega_c) t_+} - e^{i (\omega \pm \omega_c) t_-}) \tag{45} \\ +&= Y_{\mp 1} H_{\pm (a)}, \nonumber +\end{align} +$$ + +where + +$$ +H_{\pm a} = \mp r_c \omega_c. +$$ + +Within the plasma, the $z$ dependence of $A_\pm$ is +governed by Eq. (36), thus + +$$ +(\nabla_{||}^2 + k_{\pm}^2) \bar{A}_{\pm} = B_{\pm} \cdot \delta(z - z_0). +$$ + +where + +$k_{\pm}^2 = \omega^2(1 - \Omega_{\pm}^{-1})$, and $B_{\pm} = -J_{\pm}^{(a)} + B'_{\pm}$. + +If the length of the plasma column in the *z* direction +is denoted by *L* and it is placed at the interval +[0, *L*) on the *z* axis, then Eq. (46) applies to the + +range $0 \le z \le L$, while an analogous equation +which holds for $z \ge L$ has $k_\pm^2$ replaced by $\omega^2$ and +$B_\pm = 0$. The additional conditions which uniquely +determine the solution are + +$$ +\begin{gather} +\bar{A}_{\pm}(0, \omega) = 0 \quad (\text{total reflection at } z=0), \nonumber \\ +\bar{A}_{\pm}(L-0, \omega) = \bar{A}_{\pm}(L+0, \omega), \quad (\text{continuity at } z=L), \tag{47} \\ +\frac{\partial}{\partial z}\bar{A}_{\pm}(L-0, \omega) = -\frac{\partial}{\partial z}\bar{A}_{\pm}(L+0, \omega), \quad 0 < z_0 < L, \\ +\text{with } \bar{A}_{\pm}(z, \omega) \text{ outgoing wave for } z > L. \nonumber \\ +\text{One easily finds outside the plasma that} +\end{gather} +$$ + +$$ +\bar{A}_{\pm}(z, \omega) = \frac{B_{\pm} e^{i\omega(z-L)} \sin(k_{\pm}z_0)}{k_{\pm} \cos(k_{\pm}L) - i\omega \sin(k_{\pm}L)}, \quad (48) +$$ + +and consequently, + +$$ +\bar{A}_{\pm}(L, \omega) = \frac{B_{\pm} \sin(k_{\pm}z_0)}{k_{\pm} \cos(k_{\pm}L) - i\omega \sin(k_{\pm}L)}, \quad (49) +$$ + +$$ +\left( \frac{\partial}{\partial z} A_{+} \right) (L, w) = i w \bar{A}_{+} (L, w). +$$ + +Finally, in order to approximately account for the +finite transverse size of the plasma, the far field +limit is evaluated in the formula (Green's formula +for the Helmholtz equation), + +$$ +A_{\pm}(r, w) = -\frac{1}{4\pi} \int dS' \left\{ + \begin{aligned}[t] + & \frac{e^{i\omega r'}}{r'} \left[ \frac{\partial}{\partial z'} A_{\pm}(r', w) \right] \\ + & - \frac{\partial}{\partial z'} \left( e^{i\omega r'} \right) A_{\pm}(r', w) + \end{aligned} +\right\} (50) +$$ \ No newline at end of file diff --git a/samples/texts/7526122/page_7.md b/samples/texts/7526122/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..b42dc3eab549d48824b8d1eef6acad19a2e1c537 --- /dev/null +++ b/samples/texts/7526122/page_7.md @@ -0,0 +1,65 @@ +with $r' = |r - r'|$, $r = (0, 0, z)$, $r' = (x', y', L)$, and $dS' = dx' dy'$, the element of plasma surfaces at $z = L$. + +For $z \gg L$ and $z \ll R_0$, it follows from Eq. (50) that + +$$ +\begin{align} +A_{\pm}(r, \omega) &= -\frac{1}{2\pi} \frac{e^{i\omega(z-L)}}{(z-L)} \left(\frac{\partial}{\partial z}\bar{A}_{\pm}\right)(L, \omega) \nonumber \\ +&= \frac{\omega B_{\pm}}{2\pi i(z-L)} \frac{e^{i\omega(z-L)} \sin(k_{\pm}z_0)}{[k_{\pm} \cos(k_{\pm}L) - i\omega \sin(k_{\pm}L)]}. \tag{51} +\end{align} +$$ + +THEORY II. THE POWER SPECTRUM + +The results of the previous section will now be used to calculate the power spectrum seen by a radiometer with the receiving area "a" which only receives radiation propagating parallel to the magnetic field. The calculation is done by first forming the Poynting vector at the receiving plane of the radiometer, multiplying it by the receiving area "a", and then integrating it over a time Δt, the integration time of the radiometer. The time Δt, is assumed to be large compared with T = t+ − t−, the time between two consecutive collisions of the source electron. The resulting radiated energy is a function of the time T. This energy is then averaged over the distribution of times T by using the exponential distribution law with the characteristic time τ. This operation will eventually result in the usual Lorentz line shape around each harmonic. The characteristic time is in dimensionless units written as τ = ωpν2-1 where ν2 is the collision frequency of the source electrons. The energy loss of the source electron is neglected in this process. To remind the reader, ν1 is the collision frequency of the background electrons. The power radiated by a typical source electron and associated space-charge wave packet is now obtained by multiplying the average energy radiated between two consecutive collisions by the collision frequency τ-1 for the source electron. The resulting average radiated power is a function of zo, the position of a source electron. By assuming that the source electrons are uniformly distributed throughout the plasma and by integrating over the volume of the plasma seen by the radiometer, the dependence on zo is removed, and we obtain the total radiated power seen by the radiometer. Multiplying this power with the integration time of the radiometer then gives the response of the radiometer to the observed spectrum. Specifically the following conditions are enforced: + +(1) Only plus polarization is accepted by the radiometer; that is, only $\omega = -|\omega|$ is accepted. + +(2) The radiometer accepts energy only in a narrow frequency band $\Delta\omega$ which is sufficiently small so that the integration over the frequency can be replaced by the integrand times $\Delta\omega$. + +(3) The radiometer integrates the received power over a time $\Delta t$ which is very large in comparison with the time $\tau$. + +(4) The characteristic collision $\tau$ is assumed to be sufficiently long so that interference between different "l" values can be neglected. + +Executing the operations mentioned above under the stated assumptions, one finds that the power $P_*$ received by the radiometer from a typical source electron and associated space-charge wave packet located in the plane $z_0$ can be written as + +$$ +P_* = \frac{\omega^2 \alpha \Delta \omega}{2\pi^2 \tau} FF^* \left\{ \frac{H_{+(a)} H_{+(a)}^*}{(1/\tau)^2 + (\omega + \omega_c)^2} + \sum_{t=-\infty}^{\infty} \frac{f_{t+1} f_{t+1}^* H_{+(t)} H_{+(t)}^*}{(1/\tau)^2 + (\omega - l\omega_c)^2} \right\}, \quad (52) +$$ + +where + +$$ +F = \frac{\sin(k_{+}z_{0})}{k_{+}\cos(k_{+}L) - i\omega\sin(k_{+}L)}, +$$ + +and where $L$, the length of the plasma, has been neglected in comparison with the distance $z$ between the plasma and the radiometer. The asterisk appearing in the formula above labels a conjugate complex quantity. + +In order to facilitate numerical calculations and comparison with experimental data it is convenient to introduce the following ratios: + +$$ +x_c = \frac{\omega_c}{|\omega|}, \quad x_p = |\omega|^{-1}, \quad \gamma_1 = \frac{\nu_1}{|\omega|}, \\ +\gamma_2 = \frac{1}{\tau |\omega|}, \quad \varphi_0 = |\omega| L. +$$ + +Remembering that the radiometer accepts only $\omega < 0$ and changing the notation accordingly, we can write the index of refraction $n = k_+/|\omega|$ explicitly as + +$$ +n = n_r + i n_i = \left(1 - \frac{x_p^2}{1 - i\gamma_1 - x_c}\right)^{\frac{1}{2}}. \quad (53) +$$ + +We define further for convenience the expressions + +$$ +Z = \frac{x_p^2 (1 - i\gamma_1 + x_c)}{x_c [(1 - i\gamma_1)^2 - x_c^2 - (1 - i\gamma_1)x_p^2]}, \quad (54) +$$ + +$$ +Z_a = \frac{1}{n \cos(n\varphi_0) + i \sin(n\varphi_0)}, \quad (55) +$$ + +and + +$$ +C_1 \equiv \frac{av_1^2 \omega \gamma_2 \Delta \omega}{2\pi z^2}, \qquad (56) +$$ \ No newline at end of file diff --git a/samples/texts/7526122/page_8.md b/samples/texts/7526122/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..22dcb6933331c5b318ad4489fd26be1ef371d49b --- /dev/null +++ b/samples/texts/7526122/page_8.md @@ -0,0 +1,45 @@ +The power received by the radiometer as expressed by formula (52) can then be written as + +$$P_s = C_1 P_1 \sin \left( n\varphi_0 \frac{z_0}{L} \right) \sin \left( n^* \varphi_0 \frac{z_0}{L} \right), \quad (57)$$ + +with $P_1$ defined as + +$$P_1 = Z_o Z_s^* \left( \frac{1}{\gamma_2^2 + (1-x_o)^2} + ZZ_s^* \sum_{l=-\infty}^{\infty} \frac{(\ell+1)^2}{\ell^4} \frac{f_{\ell+1} f_{\ell+1}^*}{\gamma_2^2 + (1+\ell x_o)^2} \right). \quad (58)$$ + +The total power $P$ received by the radiometer is obtained by multiplying $P$, with the density $n_s$ of source electrons in the plasma and by integrating over the volume of the plasma seen by the radiometer. If this is a tube with the radius $R$, one finds the total power $P$ received by the radiometer to be + +$$P = CP_1 \left( \frac{\sinh(2\varphi_0 n_t)}{n_t} - \frac{\sin(2\varphi_0 n_r)}{n_r} \right), \quad (59)$$ + +where + +$$C = \frac{a R^2 \gamma_2 n_t v_\perp^2 \Delta\omega}{8 z^3 (2\pi)^2}.$$ + +The formulas above are written in terms of the dimensionless parameters defined earlier. Translating the formulas back to the mks system, one finds that formula (59) is applicable provided the coefficient $C$ is now written as + +$$C = \left( \frac{e^2}{8\epsilon_0 c^2} \right) \left( \frac{aR^2}{(2\pi)^2 z^2} \right) (\nu_2 n_t v_\perp^2) \frac{\Delta\omega}{|\omega|} \quad (60)$$ + +and provided + +$$x_e = \frac{\omega_c}{|\omega|}, \quad x_p = \frac{\omega_p}{|\omega|}, \quad \gamma_1 = \frac{\nu_1}{|\omega|} \\ \gamma_2 = \frac{\nu_2}{|\omega|}, \quad \varphi_0 = \frac{|\omega|}{c} L.$$ + +It is obvious from formula (58) that the space charge wave packet generated by the source electron does not radiate at the harmonics of the electron cyclotron frequency unless the coefficients $f_\ell$ are different from zero. These coefficients are a measure of the nonuniformities of the plasma along the orbit of a typical source electron. It is instructive to consider the effect on the radiated spectrum by two radically different classes of nonuniformities. + +The first class is represented by a step in the electron density. A source electron traversing this step will see it at two points along the orbit. Assuming that all locations of the orbit with respect to the step in the electron density are equally + +probable, it is easily shown by averaging over the position that + +$$\langle f_l f_{l'}^\dagger \rangle = \frac{f_l^2}{2\pi^2 (\ell^2 - \ell)} , \quad (61)$$ + +where $f_l$ is the relative height of the step discontinuity. + +The second class of discontinuities is perhaps best illustrated in terms of the crude model of microscopic oscillators described in the introduction. The heavy mass of the ion relative to that of the electron indicates that the statistics of these oscillators is intimately related to the statistics of the location of the ions and less so to the corresponding statistics of the electrons. In the limit of cold and weakly ionized plasmas we can safely make the assumption that the ions are randomly distributed. In this case it is easily shown that the corresponding statistics gives + +$$\langle f_l f_{l'}^\dagger \rangle = \frac{1}{N(2\pi)^2}, \quad (62)$$ + +where $N$ is the average number of oscillator effectively interacting with the source electron. Note that the coefficients given above are independent of the $\ell$ value. It is possible that a more detailed analysis of the statistics will show some kind of structure which will make the coefficient functions of $\ell$ and hence will show up in the emission spectrum. The suggested models for arriving at values for $\langle f_l f_{l'}^\dagger \rangle$ are strongly idealized and do not exist in the plasma in those forms. However, it can be expected that they give the correct descriptions for low $\ell$ values. The upper limit is crudely given by $\ell_{max} = 2\pi r_c / d$, where $r_c$ is the radius of cyclotron orbit of the source electron and $d$ is the effective width of a realistic step or pulse in the electron density. A measure for $\ell_{max}$ is obtained by using the Debye length for $d$ giving + +$$\ell_{max} = 2\pi \left( \frac{v_\perp}{v} \right) \left( \frac{\omega_p}{\omega_c} \right), \quad (63)$$ + +where $v_\perp$ is the velocity of the source electron perpendicular to the magnetic field and $v$ is the thermal velocity of electrons in the cool background plasma. Radiation at high harmonics of the electron cyclotron frequency can, therefore, be seen in plasmas with hot source electrons in a cold background plasma. + +The model suggested here for the cyclotron harmonic radiation is based on the assumption that the response of the cold background plasma to the hot source electron is adequately described by the \ No newline at end of file diff --git a/samples/texts/7526122/page_9.md b/samples/texts/7526122/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..9f016ae5aa82ccab0a6ff7d10c5bd9fdb3e4b840 --- /dev/null +++ b/samples/texts/7526122/page_9.md @@ -0,0 +1,15 @@ +plasma hydrodynamic equations. The introduction of "pulses" of electron density along the orbit of the source electron may seem contradictory to this assumption. However, if one considers the rather small volume of plasma or the corresponding small number of ions that are sampled by a typical source electron, it is no longer surprising that the cyclotron harmonic radiation can be directly attributable to the fine structure of the plasma. The source electrons sample a volume in the form of a toroidal tube $2\pi^2 r_c \Lambda_D^2$, where $\Lambda_D$ is the Debye length. The average number $N$ of ions contained in this volume can be written as + +$$N = \frac{\pi \omega_p v}{2 \omega_c} N_D,$$ + +where $N_D$ is the number of ions per Debye sphere. This number $N$ is in the range $10$ to $10^3$ in the helium abnormal negative glow plasma. It is a sufficiently small number so that statistical fluctuations in the number density along the orbit of the source electron must be considered. As the thermal velocity of the ions is sufficiently small relative to the velocity of a typical source electron, the ion configuration along the orbit of the source electron remains fixed during the sampling time of the source electron. With regard to the present calculation the ion configuration is stationary as long as the ratio $(v_2/\omega_p)^2$ is much larger than the electron to ion mass ratio. The lack of $\ell$ dependence of Eq. (62) is most easily understood as being due to the thermal fluctuations of the ion gas. + +## SOME NUMERICAL RESULTS + +The results from the calculations of some spectra based on the model are illustrated in Fig. 2. For the sake of convenience in this calculation we have divided the plasma into groups of electrons; the group of cold electrons that constitute the background plasma and a group of hot electrons that act as source electrons. Furthermore we have assumed that the model is applicable to both groups of electrons and that the collision frequencies, densities, and energies of the groups are such that the coefficient $C$ [Eq. (60)] is the same for the two groups. In the calculation of the radiation from the cold electron group we have also included bremsstrahlung which appropriately is accounted for by adding a term equal to unity inside the bracket of Eq. (58). The normalized collision frequencies for this group of electrons have been set as $\gamma_1 = \gamma_2 = 0.25$. The influence of bremsstrahlung from the hot electron + +group was considered insignificant and has been neglected. The normalized collision frequencies for this group of electrons were set as $\gamma_1 = 0.25$ and $\gamma_2 = 0.027$. The parameter $\varphi_0$ was set equal to 15 corresponding to plasma length $L$ of approximately 10 cm at an observation frequency of 10 GHz. We used finally the stochastic model for the fluctuations and set $\langle f_i f_i^* \rangle = 2.5 \times 10^{-4}$. The calculation has been carried out for five different values of the ratio $\omega_p/\omega$. + +FIG. 2. Typical theoretical emission spectra in the neighborhood of the electron cyclotron frequency. + +The figure displays both the right -and left-hand circularly polarized radiation as functions of the ratio $\omega_c/\omega$ with the amplitude proportional to the square root of the power in arbitrary units. The curves are normalized such that the amplitude at the fundamental of the electron cyclotron frequency $\omega_c/\omega$ is $10^2$ and displaced 20 units relative to each other. The spectra are obviously asymmetric with respect to polarization, and this asymmetry is a strong function of the ratio $\omega_p/\omega$. The radiation at the fundamental electron cyclotron frequency appears essentially only as plus polarization, as would be the case if the electrons alone were responsible for the radiation. Radiation at the harmonics of the electron cyclotron frequency have both plus and minus polarization. The minus polarization of the harmonic radiation becomes very strong when the ratio $\omega_p/\omega$ approaches unity. This is partially due to a weak asymmetry in the source function $H_t$ but primarily due to the asymmetric index of refraction coupled with reflections at the boundary as well as due to a resonance in the space-charge wave packet occurring at $\omega^2 = \omega_c^2 + \omega_p^2$ as shown by the factor $ZZ^*$ in expression (58). \ No newline at end of file diff --git a/samples/texts/823239/page_1.md b/samples/texts/823239/page_1.md new file mode 100644 index 0000000000000000000000000000000000000000..1d753d9049b39f446c9de9d4b2762f07aafd7da3 --- /dev/null +++ b/samples/texts/823239/page_1.md @@ -0,0 +1,20 @@ +A new generalization of the Pareto distribution +and its applications + +Ehab M. Almetwally¹, Hanan A. Haj Ahmad² + +ABSTRACT + +This paper introduces a new generalization of the Pareto distribution using the Marshall-Olkin generator and the method of alpha power transformation. This new model has several desirable properties appropriate for modelling right skewed data. The Authors demonstrate how the hazard rate function and moments are obtained. Moreover, an estimation for the new model parameters is provided, through the application of the maximum likelihood and maximum product spacings methods, as well as the Bayesian estimation. Approximate confidence intervals are obtained by means of an asymptotic property of the maximum likelihood and maximum product spacings methods, while the Bayes credible intervals are found by using the Monte Carlo Markov Chain method under different loss functions. A simulation analysis is conducted to compare the estimation methods. Finally, the application of the proposed new distribution to three real-data examples is presented and its goodness-of-fit is demonstrated. In addition, comparisons to other models are made in order to prove the efficiency of the distribution in question. + +**Key words:** Marshall-Olkin distribution, alpha power transformation, maximum likelihood estimator, maximum product spacings, Bayes estimation, simulation. + +# 1. Introduction + +Marshall-Olkin (MO) is a well-known distribution, which was generated by Marshall and Olkin (1997). The basic idea in this generator is to add a parameter through which the new distribution will be more flexible and will have many good properties. Many authors used MO to generate new lifetime models, for example Jose and Alice (2001, 2005), Ghitany et al. (2005), Ghitany and Kotz (2007), Jose and Uma (2009), Haj Ahmad et al. (2017), Bdair and Haj Ahmad (2019) and Ahmad and Almetwally (2020). The method of alpha power transformation (APT) class is + +¹ Faculty of Business Administration, Delta University of Science and Technology, Egypt. +E-mail: ehabxp_2009@hotmail.com. + +² Department of Basic Science, Preparatory Year Deanship, King Faisal University, Hofuf, Al-Ahsa, 31982, +Saudi Arabia, E-mail: hhajahmed@kfu.edu.sa, hananahm1@yahoo.com. \ No newline at end of file diff --git a/samples/texts/823239/page_10.md b/samples/texts/823239/page_10.md new file mode 100644 index 0000000000000000000000000000000000000000..8816a5415f1ea910dd6c89220577c9d46c9e937b --- /dev/null +++ b/samples/texts/823239/page_10.md @@ -0,0 +1,46 @@ +(3) Repeat step 2 (T)-times and obtain $(\alpha_1, \theta_1, \lambda_1), \alpha_2, \theta_2, \lambda_2), \dots, (\alpha_T, \theta_T, \lambda_T)$ + +(4) After obtaining the posterior sample, the Bayes estimates of $\alpha$, $\theta$ and $\lambda$ with respect to quadratic loss function are: + +$$ +\begin{align*} +\hat{\alpha}^{MC} &= [E_{\pi}(\alpha/x)] \approx \left(\frac{1}{T-T_0}\right) \sum_{\substack{i=1 \\ T-T_0}}^{T-T_0} \alpha_i \\ +\hat{\theta}^{MC} &= [E_{\pi}(\theta/x)] \approx \left(\frac{1}{T-T_0}\right) \sum_{i=1}^{i=T-T_0} \theta_i \\ +\hat{\lambda}^{MC} &= [E_{\pi}(\lambda/x)] \approx \left(\frac{1}{T-T_0}\right) \sum_{i=1}^{i=T-T_0} \lambda_i +\end{align*} +$$ + +where $T_0$ is the burn-in-period of Markov Chain. + +The Bayes estimates of the unknown parameters $\alpha$, $\theta$ and $\lambda$ under the LINEX loss function can be calculated through the following equation: + +$$ +\gamma_j = -\frac{1}{v} \ln \left( \sum_{i=1}^{L} \frac{e^{-v\gamma^{(i)}}}{L} \right), +$$ + +where v reflects the direction and degree of asymmetry, L is number of periods in the MCMC. + +5. Interval Estimation Methods + +In this section we consider three methods of approximate confidence intervals for +the parameters of MOAPP distribution. Numerical analysis via simulation is used for +comparisons between these methods in Section 6. + +5.1. Asymptotic confidence Interval for (MLE) + +When the sample size is large enough, the normal approximation of the MLE can +be used to construct asymptotic confidence intervals for the parameters α, θ and λ. The +asymptotic normality of MLE can be stated as $\sqrt{n}(ŷ - γ) → ^{d} N_3(0, I^{-1}(γ))$, where +γ=(α,θ,λ) is a vector of parameters, →^d denotes convergence in distribution and I(γ) is +the Fisher information matrix + +$$ +I(\gamma) = - \begin{bmatrix} E(\ell_{\alpha\alpha}) & E(\ell_{\alpha\theta}) & E(\ell_{\alpha\lambda}) \\ E(\ell_{\theta\alpha}) & E(\ell_{\theta\theta}) & E(\ell_{\theta\lambda}) \\ E(\ell_{\lambda\alpha}) & E(\ell_{\lambda\theta}) & E(\ell_{\lambda\lambda}) \end{bmatrix} +$$ + +The expected values of the second derivatives can be found by using some +integration techniques. Therefore, the (1- ζ) 100% approximate CIs for α, θ and λ are + +$$ +\hat{\alpha} \pm z_{\frac{\zeta}{2}} \sqrt{v_{11}}, \hat{\theta} \pm z_{\frac{\zeta}{2}} \sqrt{v_{22}}, \hat{\lambda} \pm z_{\frac{\zeta}{2}} \sqrt{v_{33}}, +$$ \ No newline at end of file diff --git a/samples/texts/823239/page_11.md b/samples/texts/823239/page_11.md new file mode 100644 index 0000000000000000000000000000000000000000..e88e3dd7e0fdf5e612de7fe4f2ac294e038de8b9 --- /dev/null +++ b/samples/texts/823239/page_11.md @@ -0,0 +1,25 @@ +respectively, where $v_{11}$, $v_{22}$, $v_{33}$ are the entries in the main diagonal of the Fisher matrix $I^{-1}(\gamma)$, and $z_{\frac{\zeta}{2}}$ is the $(\frac{\zeta}{2})$ 100% lower percentile of the standard normal distribution. + +## 5.2. Asymptotic Confidence Interval for (MPS) + +In this section, we propose the asymptotic confidence intervals using MPS method. As it was mentioned by Cheng and Amin [1979], Ghosh and Jammalamadaka [2001] and Anatolyev and Kosenok [2005], the MPS method also shows asymptotic properties like the maximum likelihood estimator and is asymptotically equivalent to MLE. Therefore, we may propose the asymptotic confidence intervals using MPS. The exact distribution of the MPS cannot be obtained explicitly. Therefore, the asymptotic properties of MPS similar to that of MLE can be used to construct the confidence intervals. + +$$J(\gamma) = - \begin{bmatrix} E(M_{\alpha\alpha}) & E(M_{\alpha\theta}) & E(M_{\alpha\lambda}) \\ E(M_{\theta\alpha}) & E(M_{\theta\theta}) & E(M_{\theta\lambda}) \\ E(M_{\lambda\alpha}) & E(M_{\lambda\theta}) & E(M_{\lambda\lambda}) \end{bmatrix}$$ + +The first derivatives of the product of spacing, i.e. the function M with respect to parameters $\alpha$, $\theta$ and $\lambda$, are given by Equation (13), second derivative can be found numerically and hence one can obtain the $(1-\zeta)$ 100% asymptotic confidence intervals based on MPS as follows: + +$$\hat{\alpha}_{MP} \pm z_{\frac{\zeta}{2}} \sqrt{\omega_{11}}, \hat{\theta}_{MP} \pm z_{\frac{\zeta}{2}} \sqrt{\omega_{22}}, \hat{\lambda}_{MP} \pm z_{\frac{\zeta}{2}} \sqrt{\omega_{33}},$$ + +where $\omega_{11}$, $\omega_{22}$ and $\omega_{33}$ are the diagonal entries of the Fisher matrix $J^{-1}(\gamma)$. + +## 5.3. Credible intervals + +Using MCMC techniques in Section (4), the Bayes credible intervals of the parameter $\alpha$, $\theta$ and $\lambda$ can be obtained as follows: + +(1) Arrange $\alpha_i$, $\theta_i$ and $\lambda_i$; in ascending order as follow $\alpha_{[1]}, \alpha_{[2]}, \dots, \alpha_{[T]}$, $\theta_{[1]}, \theta_{[2]}, \dots, \theta_{[T]}$ and $\lambda_{[1]}, \lambda_{[2]}, \dots, \lambda_{[T]}$ + +(2) A two-sided (1- $\zeta$) 100% credible intervals for the unknown parameters $\alpha$, $\theta$ and $\lambda$ are given by + +$$ (\alpha[T_{\frac{\zeta}{2}}], \alpha[T_{1-\frac{\zeta}{2}}]), (\theta[T_{\frac{\zeta}{2}}], \theta[T_{1-\frac{\zeta}{2}}]), (\lambda[T_{\frac{\zeta}{2}}], \lambda[T_{1-\frac{\zeta}{2}}]), $$ + +respectively, where [x] denoted the largest integer less than or equal to x. \ No newline at end of file diff --git a/samples/texts/823239/page_12.md b/samples/texts/823239/page_12.md new file mode 100644 index 0000000000000000000000000000000000000000..62ffed0c4d2d49c017be622caa881c516cbc5995 --- /dev/null +++ b/samples/texts/823239/page_12.md @@ -0,0 +1,22 @@ +# 6. Simulation Study and Data Analysis + +## 6.1. Simulation Study + +In this section, we consider some experimental results that are produced to see the effectiveness of different point and interval estimation methods. We mainly compare different point estimates in terms of mean squared errors (MSE) and bias values. Efficiency of confidence intervals is compared in terms of average interval length (AIL). Based on the generated data, we compute MLE and MPS estimates using the Newton-Raphson method. Further, we compute Bayes estimates using a Monte Carlo simulation and the MH algorithm under both squared error and LINEX loss functions with v=1.5 by using R language. + +We start by building our model with generating all simulation controls. In this stage, we must do the following steps in sequence: + +Step 1: Suppose the following values for the parameter vector of MOAPP distribution $γ=(α,θ,λ)$, case 1=(0.5, 0.5, 1.5), case 2=(1.5, 0.5, 1.5), case 3=(3,0.5,1.5), case 4=(0.5,1.5, 1.5), case 5 =(1.5,1.5,1.5) and case 6=(3,1.5,1.5), case 7=(0.5,3,1.5), case 8=(1.5,3,1.5), case 9=(3,3,1.5). + +Step 2: Choose sample sizes n =30, 70 and 200. + +Step 3: Generate the sample random values of MOAPP distribution by using quantile function $X = \left(1 - \frac{1}{\ln(\alpha)} \ln \left(1 + \frac{\theta U(\alpha-1)}{1 - U(1-\theta)}\right)\right)^{-1/\lambda}$, where U is a uniform distribution (0, 1). + +Step 4: Solve differential equations for each estimation methods, to obtain the estimators of the parameters for MOAPP distribution, so we calculate α, θ, and λ. + +Step 5: Repeat this experiment (L-1) times. In each experiment use the same values of the parameters. It is certain that the values of generating random samples are varying from experiment to experiment even though the sample size (n) does not change. +Finally, we have L-values of bias and MSE. We compute the average biases and average MSE's over 10,000 runs. This number of runs will give the accuracy in the order ±0.01 (see Karian and Dudewicz (1998)). The bias of estimator is equal to $\hat{γ} - γ$, where $\hat{γ}$ is the estimated value of γ, and the mean squared error (MSE) of the estimator is $\text{MSE}=\text{Mean} (\hat{γ} - γ)^2$. + +The simulated results of point estimation methods are presented in Tables (1) to (3), where the MSE and the bias are given in each cell and it can be pointed out that the MPS and Bayesian methods for estimating the unknown parameters of MOAPP distribution are better than the MLE method, where the MSE value is considered for comparison. We summarize the cases as follows: + +1. For $0 < \alpha < 1$, the best point estimation method for estimating $\alpha$ is the Bayesian method under LINEX loss function, while for $\alpha > 1$ the best estimation method is the MPS and Bayesian under the SE loss function. \ No newline at end of file diff --git a/samples/texts/823239/page_13.md b/samples/texts/823239/page_13.md new file mode 100644 index 0000000000000000000000000000000000000000..20012a962e9c8cde86930e7ef746aaed5a015a14 --- /dev/null +++ b/samples/texts/823239/page_13.md @@ -0,0 +1,223 @@ +2. For $0 < \theta < 1$, the best point estimation method for $\theta$ is the Bayesian method under LINEX loss function, while for $\theta > 1$ the best estimation method is the MPS and Bayesian under the SE loss function. + +3. For all values of $\lambda$ the Bayesian under the SE loss function is the best estimation method. + +4. The average bias and MSE decrease as the sample size increases. It verifies the consistency properties of all the estimators. + +5. The MLE overestimates $\alpha$, $\theta$ and $\lambda$ for almost all cases except for case 3, where the MLE underestimates $\alpha$. It is also noticed that when the sample size n=200, the MLE underestimate $\alpha$ for cases 1, 3, 6 and underestimate $\theta$ for case 7, see Table (3). + +6. MPS and Bayesian estimation sometimes overestimate the parameters and sometimes underestimate them. + +Figure 3 shows the three dimension plots of MSE with different parameters values + +For confidence interval estimation of MOAPP parameters $\alpha$, $\theta$ and $\lambda$, we observe +the 95% confidence intervals (L,U) where L represents the lower bound and U is the +upper bound of this interval. Three confidence intervals are considered in simulation +analysis, i.e. asymptotic confidence intervals of MLE and MPS, also the credible +intervals of Bayesian method under SE and LINEX loss functions. The comparison is +conducted depending on the average interval length (AIL), hence the smaller the AIL +is the better confidence estimate we observe. The results are reported in Tables (4) to +(6) below. + +**Table 1.** Bias and MSE for *α*, *θ*, and *λ*, with n=30 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
λθαn=30MLEMPSSELINEX (v = 1.5)
BiasMSEBiasMSEBiasMSEBiasMSE
1.50.50.5α0.10720.1866-0.01600.08180.10510.09940.03100.0661
θ0.32120.49460.05400.20120.05170.05360.00710.0395
λ0.31340.9250-0.25850.6378-0.08580.1772-0.18930.1832
1.5α0.09750.44820.00920.0306-0.22420.3509-0.35520.4101
θ0.23410.3511-0.01250.12990.10580.08160.05590.0549
λ0.27860.6504-0.16770.4781-0.09660.1989-0.19190.2103
3α-0.07921.1131-0.00320.0421-0.32870.5870-0.52140.8308
θ0.29250.4977-0.00300.14250.10490.08940.05610.0620
λ0.23500.4980-0.16240.3824-0.05540.1675-0.14360.1748
1.5α0.32910.69170.02650.19050.13690.10790.06420.0693
θ0.28130.7987-0.05240.3616-0.17090.2413-0.27380.2666
λ0.12740.4280-0.24900.4257-0.11730.1662-0.20640.1787
1.51.51.5α0.41861.2902-0.00080.1538-0.14720.2932-0.27160.3255
θ0.33911.1591-0.09600.5198-0.13570.2656-0.24410.2761
λ0.13510.2503-0.14530.2163-0.17460.1315-0.24880.1643
333">αα
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-- \ No newline at end of file diff --git a/samples/texts/823239/page_14.md b/samples/texts/823239/page_14.md new file mode 100644 index 0000000000000000000000000000000000000000..6f9566c321213f0bbb6e9da8cf6f0e48d4fa8f36 --- /dev/null +++ b/samples/texts/823239/page_14.md @@ -0,0 +1,279 @@ +**Table 1. Bias and MSE for α, θ, and λ, with n=30 (cont.)** + +
λθαn=30MLEMPSSELINEX (ν = 1.5)
BiasMSEBiasMSEBiasMSEBiasMSE
1.530.5α0.46891.20820.03550.29510.14490.14380.06650.0872
θ0.16130.9882-0.04980.3326-0.31640.4939-0.49150.7154
λ0.08970.2667-0.21460.2902-0.08690.0977-0.15820.1124
1.5α0.66313.0228-0.18910.6353-0.13090.3296-0.26440.3409
θ0.45552.2871-0.07470.5972-0.36350.5986-0.55340.8297
λ0.08440.1606-0.16120.1698-0.14070.0853-0.19670.1038
+ +**Table 2. Bias and MSE for α, θ, and λ, with n=70** + +<
λθαn=70MLEMPSSELINEX (ν = 1.5)
BiasMSEBiasMSEBiasMSEBiasMSE
1.50.5α0.03360.1130-0.07390.07760.02500.0670-0.00410.0572
θ0.20130.25530.07250.12610.04000.03590.02150.0304
λ0.11530.3558-0.21040.3237-0.06390.1000-0.09900.1095
1.5α0.06230.25400.01380.0160-0.04760.1202-0.09440.1319
θ0.08850.0804-0.02610.04910.04560.02960.02890.0255
λ0.12870.2330-0.10760.2087-0.02930.0771-0.06180.0782
3α-0.02490.61910.01280.0235-0.04720.1116-0.08700.1191
θ0.11540.1198-0.01970.05180.01610.02950.00050.0265
λ0.11130.1867-0.09770.1677-0.03380.0770-0.06670.0822
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + MLE + MPS + SE + LINEX + --> + + + + + + +
λ
θ
α
n=20(cont.)		MLE
Bias
MSE
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
- 1e-4 (cont.)
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
+ 5 e -6
Cont. +
Bias
Cont.		MSE
Cont.		Bias
Cont.		MSE
Cont.		Bias
Cont.		MSE
Cont.		Bias
Cont.		MSE
Cont.		Bias
Cont.		MSE
Cont.
α
Cont.		- 2 e -5
Cont.		- 2 e -5
Cont.		- 2 e -5
Cont.		- 2 e -5
Cont.		- 2 e -5
Cont.		- 2 e -5
Cont.		- 2 e -5
Cont.		- 2 e -5
Cont.
θ
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.
- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.		- 8 e -8
Cont.
λ
Cont.		- 9 e -9
Cont.		- 9 e -9
Cont.		- 9 e -9
Cont.		- 9 e -9
Cont.		- 9 e -9
Cont.		- 9 e -9
Cont.		- 9 e -9
Cont.		- 9 e -9
Cont.
θ
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.
- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.		- 7 e -7
Cont.
α
Cont.		+ 3 e +3
Cont.		+ 3 e +3
Cont.		+ 3 e +3
Cont.		+ 3 e +3
Cont.		+ 3 e +3
Cont.		+ 3 e +3
Cont.		+ 3 e +3
Cont.		+ 3 e +3
Cont.
θ
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.
+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.		+ 6 e +6
Cont.
+ +**Table **3** Bias and MSE for *α*, *θ*, and *λ*, with n=2 + +\n \n \n \n \n \n \n \n \n \n \n \n \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n \\\\\n \\\\\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\nd\n&\\text{MSE}\\\\\d\\text{MLE}\\text{Bias}\\text{MSE}\\\\\d\\text{MPS}\\text{Bias}\\text{MSE}\\\\\d\\text{SE}\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MSE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MSE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MLE})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{MPS})\\text{Bias}\\text{MSE}\\\\\d\\text{LINEX}(v=\\text{SE})\\text{Bias}\\text{MSE}\\\\\d\\textbf{}(v=)}(x) = \\frac{x}{x} = \\frac{x}{x} = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x = x 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diff --git a/samples/texts/823239/page_15.md b/samples/texts/823239/page_15.md new file mode 100644 index 0000000000000000000000000000000000000000..9b3f4860e1d1036c0908b6aef0c74341dc191134 --- /dev/null +++ b/samples/texts/823239/page_15.md @@ -0,0 +1,15 @@ +**Table 3.** Bias and MSE for $\alpha, \theta$, and $\lambda$, with $n=200$ (cont.) + +
                                                                                                                                                               
λθαn=200MLEMPSSELINEX (ν = 1.5)
BiasMSEBiasMSEBiasMSEBiasMSE
1.51.5α0.08110.1085-0.03880.0462-0.00070.0105-0.00510.0104
θ0.10620.28930.03130.0781-0.00980.0133-0.01530.0133
λ0.00150.0617-0.08280.0750-0.00650.0107-0.01100.0110
α0.20530.7433-0.00430.0249-0.01060.0184-0.01640.0189
θ0.05140.1709-0.03820.0765-0.00620.0175-0.01200.0177
λ0.02440.0320-0.03220.0321-0.00680.0093-0.01110.0093
3α-0.02391.11260.00350.0187-0.01470.0171-0.02050.0176
θ0.17090.3354-0.03480.1077-0.02270.0146-0.02850.0151
λ0.02720.0289-0.03020.0276-0.01670.0074-0.02070.0077
3α0.09650.1246-0.01380.0489-0.00110.0108-0.00560.0106
θ-0.00130.24400.00440.0374-0.01330.0136-0.01910.0139
λ0.01640.0375-0.04980.0434-0.00050.0076-0.00470.0076
α0.08650.2790-0.04940.1157-0.01200.0160-0.01770.0163
θ0.07940.2092-0.03200.1043-0.04160.0195-0.04840.0209
λ0.02659 2/3%- 9 2/3%
+ +From Tables (4) to (6) we notice that the (AIL) of the credible intervals under SE and LINEX are smaller than the (AIL) of MLE and MPS in most cases except for some restricted ones. + +We can summarize the analysis of the confidence interval estimation in the following points: + +1. For $α < 1$, the best interval estimate for $\alpha$ is the Bayesian credible interval under SE and LINEX loss functions, while for $\alpha > 1$ the best interval estimation is the asymptotic interval under the MPS method except for cases 8 and 11, where the Bayesian credible interval under LINEX has the smallest AIL. + +2. For $\theta < 3$, the best interval estimate for $\theta$ is the Bayesian credible interval under LINEX loss function, while for $\theta \ge 3$, the best interval estimation is the asymptotic interval under the MPS method and the Bayesian credible interval under the SE loss function. + +3. Bayesian credible interval under the LINEX loss function has the smallest AIL for estimating $\lambda$, and hence it can be considered as the best confidence interval of $\lambda$. For the case 5, the Bayesian credible interval under the SE loss function is preferable to estimate $\lambda$. + +4. AIC decreases as the sample size increases. \ No newline at end of file diff --git a/samples/texts/823239/page_16.md b/samples/texts/823239/page_16.md new file mode 100644 index 0000000000000000000000000000000000000000..e080ed9a4f1683360ff2eedebcc412a91e98a422 --- /dev/null +++ b/samples/texts/823239/page_16.md @@ -0,0 +1,469 @@ +**Table 4.** 95% confidence intervals and Average Interval length for $\alpha$, $\theta$, and $\lambda$, with n=30 + +
λθα30MLEMPSSELINEX(v = 1.5)
LUAILLUAILLUAILLUAIL
1.50.5α-0.21331.42781.6411-0.07591.04391.11980.02101.18921.16830.02941.03261.0033
-0.40562.04802.4536-0.31911.42711.74620.10820.99520.88700.11690.89720.7803
0.03033.59653.5662-0.24032.72332.96360.60442.22401.61960.55642.06501.5086
θ-0.33321.80142.1346-0.21881.19381.41260.08451.12701.04250.10881.00300.8942
0.29443.26272.96830.01682.64772.63090.54802.25881.71080.48972.12641.6367
0.85784.98394.12602.59433.39920.80491.31154.03122.71961.00973.94752.9379
λ-0.46632.05142.5176-0.24311.23721.48030.05491.15491.10000.07931.03290.9535
-0.39104.50354.8945-0.18113.02243.20350.45962.20881.74920.39282.07971.6869
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/samples/texts/823239/page_17.md b/samples/texts/823239/page_17.md new file mode 100644 index 0000000000000000000000000000000000000000..5da3a38a2d05788656b79a8984333e6303c2a83b --- /dev/null +++ b/samples/texts/823239/page_17.md @@ -0,0 +1,51 @@ +**Table 5.** 95% confidence intervals and Average Interval length for $\alpha$, $\theta$, and $\lambda$, with n=70 + +
MLEMPSSELINEX(v = 1.5)
LUAILLUAILLUAILLUAIL
1.5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 + α -0.6687
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α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ +
α =-0-9-.-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9-..-9 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +
α =--.-.<--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<--.--.<-->--. +
α =---.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-..---. +
α =---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---..---.. +
α = --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- + +
α = -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- +
α =---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. +
α =------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ -----. +
α =------ ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----------. +
α =------ ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- --------------. +
α =------ -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- -------- ----------------. +
α =------ ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- --------------------. +
α =------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------------------. +
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α =------ ------------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- --------------------------------------------------------------------------------------. +
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    • + + + + + + +
    λθαn=70MLEMPSSELINEX (ν = 1.5)
    LUAILLUAILLUAILLUAIL
    1.50.5α-0.12211.18941.3115-0.10050.95271.05320.01881.03111.01240.02620.96560.9394
    θ-0.20761.61011.8178-0.10931.25431.36360.17600.90400.72800.18170.86130.6796
    λ0.46762.76302.29540.25312.32602.07290.82752.04481.21730.78052.02151.2410
    1.5α0.58172.54301.96131.26701.76060.49360.77752.12721.34970.71642.09481.3784
    θ0.06011.11691.05680.04240.90540.86300.21950.87170.65220.22040.83730.6170
    λ0.71642.54101.82450.52172.26311.74130.92812.01331.08520.90251.97401.0715
    3α1.43294.51733.08442.71303.31260.59962.30293.60271.29982.25683.56921.3124
    θ-0.02461.25531.27980.03560.92500.88930.18000.85210.67210.18080.82010.6392
    λ0.79252.43011.63760.62252.18211.55960.92492.00751.08260.88561.98101.0954
    1.50.5α-0.38421.72322.1074-0.16481.10791.27270.10260.95350.85090.10470.90070.7960
    θ0.48292.80482.32180.69372.31501.62130.8930
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    |i.
    1,5.
    α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ α θ λ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    n=70 αn=70 θn=70 λn=70 αn=70 θn=70 λn=70 αn=70 θn=70 λn=70 αn=70 θn=70 λn=70 αn=70 θn=70 λ
    αθλαθλαθλαθλαθλ
    -0,1221                   -                 -                 -                 -                 -                 -                 - 
                                                                                                                                                                                                (v = 1,5)—(v = 1,5) α⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬⹬>
    " alt="This image does not contain any tables."}] \ No newline at end of file diff --git a/samples/texts/823239/page_18.md b/samples/texts/823239/page_18.md new file mode 100644 index 0000000000000000000000000000000000000000..624cd498fdbfb38fc0a50e32dc1b41d2a407e1d6 --- /dev/null +++ b/samples/texts/823239/page_18.md @@ -0,0 +1,4 @@ +**Table 6.** 95% confidence intervals and Average Interval length for $\alpha, \theta$, and $\lambda$, with $n=200$ + +
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  • + diff --git a/samples/texts/823239/page_19.md b/samples/texts/823239/page_19.md new file mode 100644 index 0000000000000000000000000000000000000000..284f73afb22d5a46cd935b5ccd68601765817868 --- /dev/null +++ b/samples/texts/823239/page_19.md @@ -0,0 +1,5 @@ +**Figure 3.** MSE for MLE, MPS and Bayes estimation under SE and LINEX Loss Functions for n=120. + +## 6.2. Data Analysis + +In this section we take three different examples of real-life data set. The MLEs estimates of the parameters are reported in Tables (9), (10) and (11), then the MOAPP model is compared with other special case models like Pareto type 1, generalized Pareto (GP), and alpha power Pareto (APP). This comparison was conducted using Kolmogorov-Smirnov (KS) distance (D) between the fitted and the empirical distribution functions and the corresponding p-values. Also, Akaike information criterion (AIC) such that $AIC=-2 L(\gamma)+2p$, where p is the number of parameters in the model and L is the maximized value of the likelihood function for the model. Given a set of candidate models for the data, the preferred model is the one with the minimum AIC value. Bayesian information criterion (BIC) is also used for comparison between models, where BIC can be defined as: $BIC=-2 L(\gamma)+p \ln(n)$, where n is the sample size. As a model selection criterion, the researcher must choose the model with the minimum BIC value. The MLEs of $\alpha, \theta, \text{and } \lambda$ are computed numerically using the \ No newline at end of file diff --git a/samples/texts/823239/page_2.md b/samples/texts/823239/page_2.md new file mode 100644 index 0000000000000000000000000000000000000000..f5d457c86061547b8f2bd0258b6d57e95a664759 --- /dev/null +++ b/samples/texts/823239/page_2.md @@ -0,0 +1,39 @@ +a procedure which makes the lifetime distribution more applicable and rich towards +real data analysis. It was first introduced by Mahdavi and Kundu (2017). A new +generalization appeared in the literature by doing combination between MO and APT, +this was first studied by Nassar et al. (2019), and the new family is called “G-family +(MOAP-G). It was noticed that the MOAP-G family is analytically tractable and +efficient for real data analysis. + +The cumulative distribution function (CDF) of MOAP-G random variable X is of the +form + +$$ +F_{\text{MOAP}}(x; \alpha, \theta) = \begin{cases} \frac{\alpha^{G(x)-1}}{(\alpha-1)[\theta+(1-\theta)(\alpha-1)^{-1}(\alpha^{G(x)-1})]} & , \alpha > 0, \alpha \neq 1 \\ G(x) & , \alpha = 1 \end{cases} \quad (1) +$$ + +The corresponding probability density function (pdf) + +$$ +f_{\text{MOAP}}(x; \alpha, \theta) = \begin{cases} \frac{\theta \log(\alpha) \alpha^{G(x)} g(x)}{(\alpha-1)[\theta + \frac{1-\theta}{\alpha-1}(\alpha^{G(x)}-1)]^2} & , \alpha > 0, \alpha \neq 1 \\ g(x) & , \alpha = 1 \end{cases} \quad (2) +$$ + +where $G(x)$ is the baseline distribution. + +In this paper we will consider Pareto distribution with shape parameter $\lambda$ as a +baseline distribution, where the pdf and cdf are respectively as follows: + +$$ +g(x) = \frac{\lambda}{x^{\lambda+1}}, x \geq 1 +\quad (3) +$$ + +$$ +G(x) = 1 - \frac{1}{x^{\lambda}}, x \geq 1 \tag{4} +$$ + +The new generated distribution, namely Marshall-Olkin Alpha Power Pareto (MOAPP), is a lifetime model with three parameters. This distribution has several desirable properties and acts well for modelling right skewed data, it has upside-down bathtub hazard rate and attractive time series representation by which many statistical computations can be easily handled. Real data examples show that MOAPP behaves better than many other generalized Pareto distributions. + +The main purpose of this paper is to introduce MOAPP distribution and study some of its statistical properties, which are useful in data modelling. We use statistical inference such as maximum likelihood, maximum product spacings and Bayes estimation methods to perform point estimation. We construct confidence intervals for the unknown parameter as well. A simulation study is conducted to check the performance of the different estimation methods applied in this work This is done by comparing the bias and the mean square error (MSE) for point estimation methods and by using interval length for interval estimation. Finally, we present numerical examples that illustrate the model efficiency. + +The rest of this paper is organized as follows: In Section 2 we introduce MOAPP distribution with some of its properties. Classical point estimation methods for the unknown parameters are discussed in Section 3, while in Section 4 the Bayesian \ No newline at end of file diff --git a/samples/texts/823239/page_20.md b/samples/texts/823239/page_20.md new file mode 100644 index 0000000000000000000000000000000000000000..49e98fe84692b3d026b15b495b6f5fae30441e7e --- /dev/null +++ b/samples/texts/823239/page_20.md @@ -0,0 +1,21 @@ +function optimal in R statistical package. The values of the KS statistic with p-values, AIC and BIC are reported in Tables (7), (8) and (9). + +The first example is from Lawless (1982). The data set consists of failure times or censoring times for 36 appliances subjected to an automated life test. Failures are mainly classified into 18 different modes, although among 33 observed failures only 7 modes are present and only model 6 and 9 appear more than once. We are mainly interested in the failure mode 9. The data are given below: + +**Data 1:** 1167, 1925, 1990, 2223, 2400, 2471, 2551, 2568, 2694, 3034, 3112, 3214, 3478, 3504, 4329, 176976, 7846. + +Table 7. MLE estimation with KS, p-values and different model goodness of fit criterion for data 1 + +
    • λθαn=200MLEMPSSELINEX (ν = 1.5)
    LUAILLUAILLUAILLUAIL
    1.50.5α-0.04741.04441.0918-0.04230.86130.90360.28130.72050.43920.27730.71420.4369
    θ0.07581.13331.05750.11711.02570.90860.35060.66770.31700.34910.66250.3134
    λ0.86232.18221.31980.71082.01551.30461.27411.69370.41961.26831.68980.4216
    1.5α0.93432.09411.15981.35161.67200.32041.22351.74450.52101.21601.73920.5232
    θ0.24820.82240.57420.24210.72070.47860.36020.63560.27540.35820.63130.2731
    λ1.04102.06471.02370.94611.95101.00491.28321.70860.42541.27771.70490.4272
    3α1.87394.04792.17392.80743.21740.40992.77273.20010.42732.76593.19630.4304
    θ0.21690.87500.65810.24250.72680.48430.35270.64170.28900.35080.63770.2869
    λ1.08362.01040.92681.00311.90640.90321.27941.70980.43041.27511.70440.4293
    1.50.5α-0.04511.20721.25240.04660.87580.82920.29820.70040.40220.29510.69460.3996
    θ0.57212.64022.06810.98672.07601.08931.2644
    α̂MLEθ̂MLEλ̂MLEDP-valueAICBIC
    P--0.122310.578436.34E-06385.42783.86E+02
    GP-3341.0320.6092730.330890.03686334.5826336.2491
    APP78.74852-0.2570250.484670.00034367.75673.69E+02
    MOAPP9.00E+074.60E+072.575360.20880.3941324.6243327.124
    + +The second example represents survival times of guinea pigs injected with different amount of tubercle bacilli studied by Bjerkedal [1960]. Guinea pigs are subject to high susceptibility of human tuberculosis, which is one of the causes for choosing this species. + +Table 8. MLE estimation with KS, p-values and different model goodness of fit criterion for data 2 + +
    α̂MLEθ̂MLEλ̂MLEDP-valueAICBIC
    P--0.1998050.510932.2E-161098.511.10E+03
    GP-237.97880.3934780.231420.000895879.3132883.8665
    APP152.982-0.4463340.401471.67E-101009.9781014.531
    MOAPP112100322998.13.0118370.068370.8894857.2212864.0512
    + +The third example is from Almetwally et al. (2019). The data set consists of economic data of 31 observations subjected to a GDP growth of Egypt. The data are given below. + +Table 9. MLE estimation with KS, p-values and different model goodness of fit criterion for data 3 + +
    α̂MLEθ̂MLEλ̂MLEDP-valueAICBIC
    P--0.64730.39560.0001186.7626188.1966
    GP-6.8839-0.66070.29100.0080149.8744152.7423
    APP90.2664-1.36790.25010.0340160.3021163.1700
    MOAPP8.5373153.59463.78570.07260.9927139.4962143.7982
    \ No newline at end of file diff --git a/samples/texts/823239/page_21.md b/samples/texts/823239/page_21.md new file mode 100644 index 0000000000000000000000000000000000000000..fb46c22e39386acb7bb0464d7afb1e9b77fc97f6 --- /dev/null +++ b/samples/texts/823239/page_21.md @@ -0,0 +1,5 @@ +When comparing the values of KS statistics of MOAPP and other sub models like Pareto type 1, GP and APP for the two data examples above, we obtain the minimum KS for MOAPP with highest p-values. Also, it can be noticed that the values of AIC and BIC take their minimum values when the distribution is MOAPP. Therefore, this indicates that the MOAPP distribution fits the two sets of data very well and is better than other distributions. This also emphasizes the need of new distributions in managing real-life data. So, in general we can say that the new distribution is superior according to other sub models. + +## 7. Conclusions + +In this study we have considered MOAPP distribution which has three unknown parameters. This new distribution proved to be more flexible and more appropriate for monotone and right skewed lifetime data, also its hazard rate function can be either a decreasing or upside-down bathtub curve. We estimate the parameters of MOAPP using MLE, MPS and Bayesian method under SE and LINEX loss functions. It is not possible to compare different methods theoretically, so we have used some simulations to compare different estimators. We have compared different estimators mainly with respect to biases and mean squared errors. Confidence intervals are obtained and are compared numerically in terms of interval lengths. The best method for estimating $\alpha$ and $\theta$ is the Bayesian method under the LINEX and SE loss functions depending on the values of $\alpha$ and $\theta$, it is also noticed that the MPS method acts better for estimating $\alpha$ and $\theta$ than the MLE method. The Bayesian method under the SE loss function is the best appropriate method for estimating $\lambda$. Confidence intervals under the MPS method and the Bayesian credible interval are preferable to confidence intervals under the MLE method. Therefore, we recommend the use of the MPS and Bayes estimation methods for practical purposes. The flexibility of this distribution was illustrated in some applications to real data sets, where the new model proves to better fit data than some other sub models. \ No newline at end of file diff --git a/samples/texts/823239/page_22.md b/samples/texts/823239/page_22.md new file mode 100644 index 0000000000000000000000000000000000000000..2bd84c8323456f943297c2350d2430ef791d5b42 --- /dev/null +++ b/samples/texts/823239/page_22.md @@ -0,0 +1,23 @@ +## References + +AHMAD, H. H., ALMETWALLY, E., (2020). Marshall-Olkin Generalized Pareto Distribution: Bayesian and Non Bayesian Estimation. Pakistan Journal of Statistics and Operation Research, 16(1), pp. 21–33; DOI: 10.18187/pjsor.v16i1.2935. + +ALMETWALLY, E. M., ALMONGY, H. M. and EL-SHERPIENY, E. A., (2019). Adaptive Type-II Progressive Censoring Schemes based on Maximum Product Spacing with Application of Generalized Rayleigh Distribution. Journal of Data Science, 17(4), pp. 767–793. + +ALMETWALLY, E. M., ALMONGY, H. M., (2019b). Estimation method for new Weibull-Pareto distribution: Simulation and application. Journal of Data Science, 17(3), pp. 610–630. + +ALMETWALLY, E. M., ALMONGY, H. M., (2019a). Maximum Product Spacing and Bayesian Method for Parameter Estimation for Generalized Power Weibull Distribution under Censoring Scheme. Journal of Data Science, 17(2), pp. 407–444. + +ALMETWALLY, E. M., ALMONGY, H. M. and EL SAYED MUBARAK, A., (2018). Bayesian and Maximum Likelihood Estimation for the Weibull Generalized Exponential Distribution Parameters Using Progressive Censoring Schemes. Pakistan Journal of Statistics and Operation Research, 14(4), pp. 853–868. + +ANATOLYEV, S., KOSENOK, G., (2005). An Alternative to Maximum Likelihood Based on Spacings, Econometric Theory 21(02), pp. 472–476. + +BDAIR, O., HAJ AHMAD, H., (2019). Estimation of The Marshall-Olkin Pareto Distribution Parameters: Comparative Study, Revista Investigacion Operacional, 41(2), forthcoming. + +BJERKEDAL, T., (1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli, Am J Hyg, 72(1), pp. 130–148. + +CHENG R. C. H.; AMIN, N. A. K., (1979). product-of-spacings estimation with applications to the lognormal distribution, University of Wales IST, Math Report 79-1. + +CHENG, R. C. H., AMIN, N. A. K., (1983). Estimating parameters in continuous univariate distributions with a shifted origin, J. Roy. Statist. Soc. Ser. B, 45, pp. 394–403. + +GHITANY, M. E., (2005). Marshall-Olkin extended Pareto distribution and its application, International Journal of Applied Mathematics, 18, No. 1, pp. 17–31. \ No newline at end of file diff --git a/samples/texts/823239/page_23.md b/samples/texts/823239/page_23.md new file mode 100644 index 0000000000000000000000000000000000000000..0045a7c1543632824475650ebb7058ec3192b983 --- /dev/null +++ b/samples/texts/823239/page_23.md @@ -0,0 +1,27 @@ +GHITANY, M., E., KOTZ, S., (2007). Reliability Properties of Extended Linear Failure-Rate Distributions, Probability in the Engineering and informational Sciences 21, pp. 441–450. + +GHOSH, K., JAMMALAMADAKA, S. R., (2001). A general estimation method using spacings. Journal of Statistical Planning and Inference, 93, pp. 71–82. + +HAJ AHMAD, H., BDAIR, O., AHSANULLAH, M., (2017). On Marshall-Olkin Extended Weibull Distribution, Journal of Statistical Theory and Applications, Vol. 16, No. 1, pp 1–17. + +JOSE, K. K., ALICE, T., (2001). Marshall-Olkin Generalized Weibull distributions and applications, STARS: int. Journal, 2,1, pp. 1–8. + +JOSE, K. K., ALICE, T., (2005). Marshall-Olkin Family of Distributions: Applications in Time series modelling and Reliability, J.C Publications, Palakkad. + +JOSE, K. K., UMA, P., (2009). On Marshall-Olkin Mittag-Leffler distributions and processes, Far East Journal of Theoretical Statistics, 28, pp. 189–199. + +KARIAN, Z. A., DUDEWICZ, E. J., (1998). Modern statistical, systems, and GPSS simulation, CRC press. + +KARANDIKAR, R. L., (2006). On the Markov chain Monte Carlo (MCMC) method. Sadhana, 31(2), pp. 81–104. + +LAWLESS, J. F., (1982). Statistical Models and Methods for Lifetime Data. John Wiley & Sons, New York. + +MAHDAVI, A., KUNDU, D., (2017). A new method for generating distributions with an application to exponential distribution. Communications in Statistics-Theory and Methods, 46(13), pp. 6543–6557. + +MARSHALL, A. W., OLKIN, I., (1997). A New Method for Adding a Parameter to a Family of Distributions with Application to the Exponential and Weibull Families, Biometrika, 84(3), pp. 641–652. + +METROPOLIS, N., ROSENBLUTH, A. W., ROSENBLUTH, M. N., TELLER, A. H. and TELLER, E., (1953). Equation of state calculations by fast computing machines. The journal of chemical physics, 21(6), pp. 1087–1092. + +NASSAR, M., KUMAR, D., SANKU DEY, S., GAUSS M. CORDEIRO, G. and AFIFY, A. Z., (2019). The Marshall-Olkin alpha power family of distributions with applications, Journal of Computational and Applied Mathematics, 351, pp. 41–53. + +ROBERT, C., CASELLA, G., (2013). Monte Carlo statistical methods. Springer Science & Business Media. \ No newline at end of file diff --git a/samples/texts/823239/page_24.md b/samples/texts/823239/page_24.md new file mode 100644 index 0000000000000000000000000000000000000000..3651c92832e1dc9f400d49d3d3d131098df6bb29 --- /dev/null +++ b/samples/texts/823239/page_24.md @@ -0,0 +1,3 @@ +SINGH, U., SINGH, S. K. and SINGH, R. K., (2014). A comparative study of traditional estimation methods and maximum product spacing method in generalized inverted exponential distribution. Journal of Statistics Applications & Probability, 3(2), p. 153. + +KUMAR SINGH, R., KUMAR SINGH, S. and SINGH, U., (2016). Maximum product spacings method for the estimation of parameters of generalized inverted exponential distribution under Progressive Type II Censoring. Journal of Statistics and Management Systems, 19(2), pp. 219–245. \ No newline at end of file diff --git a/samples/texts/823239/page_3.md b/samples/texts/823239/page_3.md new file mode 100644 index 0000000000000000000000000000000000000000..10c3390c80de07587c5c3e293a92ca2d14793edc --- /dev/null +++ b/samples/texts/823239/page_3.md @@ -0,0 +1,27 @@ +estimation method is considered. In Section 5 interval estimation methods are presented. In Section 6 a simulation study and real-life data analysis are conducted and finally conclusions are given in Section 7. + +## 2. Probability Density Function + +Let X be a continuous random variable with Marshall-Olkin Alpha Power Pareto distribution (MOAPP), then using Eqs. (1) and (2) and assuming that the baseline distribution $G(x)$ is Pareto distribution given in Eqs. (3) and (4), we obtain the pdf and CDF of (MOAPP) respectively as + +$$ f_{\text{MOAPP}}(x; \alpha, \theta, \lambda) = \frac{\theta^{\lambda}(\log \alpha)^{\alpha-1}x^{-\lambda}}{(\alpha-1)x^{\lambda+1}[\theta+(1-\theta)(\alpha-1)^{-1}(\alpha^{1-x}-1)]^2}, \quad x \ge 1, \alpha \ne 1 \quad (5) $$ + +$$ F_{\text{MOAPP}}(x; \alpha, \theta, \lambda) = \frac{\alpha^{1-x^{-\lambda}}}{(\alpha-1)[\theta+(1-\theta)(\alpha-1)^{-1}(\alpha^{1-x^{-\lambda}}-1)]}, \quad x \ge 1, \alpha \ne 1, \quad (6) $$ + +In the following subsection we investigate some important properties of MOAPP distribution such as: monotonicity, hazard rate function, series representation, moments and quantiles. + +### 2.1. Monotonicity of MOAPP Distribution + +The monotonicity of MOAPP distribution is necessary to be investigated for data modelling, many areas such as medical, industrial, engineering and reliability researches need data modelling for prediction of future values and estimation of some unknown or missing variables; hence, in this section we study the monotonicity of MOAPP distribution. We consider the pdf of MOAPP distribution in Eq. (5), and study the monotonicity of this pdf by using the logarithmic function of its pdf. The following lemma illustrates the behaviour of MOAPP distribution for different parameter values, and Figure (1) shows these cases. + +**Lemma 1** + +The pdf of MOAPP distribution is either decreasing when $0 < \theta < 1$, or upside-down bathtub curve that attains its maximum at some point $x_0 \in [1, \infty)$ when $\theta > 1$ and $\lambda > 1$ + +**Proof** + +Consider the pdf of MOAPP density in Eq. (5), then the derivative of the logarithmic function of pdf with respect to x is + +$$ \frac{d\text{Log } f_{\text{MOAPP}}(x; \alpha, \theta, \lambda)}{dx} = \lambda\text{Log}(\alpha)x^{-\lambda-1} - \frac{\lambda+1}{x} - 2 \frac{\lambda\text{Log}(\alpha)^{\frac{(1-\theta)}{\alpha-1}} \alpha^{1-x^{-\lambda}} x^{-\lambda-1}}{[\theta+\frac{(1-\theta)}{\alpha-1}(\alpha^{1-x^{-\lambda}}-1)]} $$ + +$$ \frac{d\text{Log } f_{\text{MOAPP}}(x; \alpha, \theta, \lambda)}{dx} = \frac{s(x)(\lambda\text{Log}(\alpha) - (\lambda+1)x^{\lambda}) - 2\lambda\text{Log}(\alpha)}{s(x)x^{\lambda+1}} \quad (7) $$ \ No newline at end of file diff --git a/samples/texts/823239/page_4.md b/samples/texts/823239/page_4.md new file mode 100644 index 0000000000000000000000000000000000000000..f074144fd43445ff230439a799137cd057c00f47 --- /dev/null +++ b/samples/texts/823239/page_4.md @@ -0,0 +1,29 @@ +where $S(x) = [\frac{(\alpha-1)}{(1-\theta)} \theta \alpha^{x-\lambda-1} + (1-\alpha^{x-\lambda-1})]$. Equating (7) to zero, we obtain the cases: + +1. If $0 < \theta < 1$ then $S(x)$ is positive and since $\lambda \text{Log}(\alpha) < (\lambda+1)x^\lambda$ then the numerator of equation (7) is negative hence the derivative of the logarithmic function of MOAPP is negative, which indicates that the pdf of MOAPP is a decreasing function. + +2. If $\theta > 1$ and $\lambda > 1$ then by using Bolzano theorem on the interval $[1, \infty)$ there exist a root $x_0 \in [1, \infty)$ of $\text{Log } f_{\text{MOAPP}}$ hence $f_{\text{MOAPP}}$ attains its maximum at $x_0$. + +## 2.2. Hazard Rate Function + +The hazard rate function or failure rate is important in survival analysis and reliability theory. The hazard rate function for MOAPP distribution is of the form + +$$h(x; \alpha, \theta, \lambda) = \frac{\lambda \text{Log}(\alpha) x^{-(\lambda+1)}}{(\alpha^{x-\lambda}-1)(\theta+(1-\theta)(\alpha-1)^{-1}(\alpha^{1-x-\lambda}-1))}, x \geq 1 \quad (8)$$ + +In order to determine the shape of $h(x;\alpha,\theta, \lambda)$ it is quite enough to determine the shape of $\log h(x;\alpha,\theta, \lambda)$, as shown in the following lemma. + +**Lemma 2** + +The hazard rate of MOAPP distribution is either decreasing or upside down curve where the curve is skewed to the right. + +**Proof:** + +We consider a logarithmic function of the hazard rate given in Eq. (8) and take the first derivative with respect to x so that: + +$$\frac{d \log h(x; \alpha, \theta, \lambda)}{dx} = \frac{-(\lambda + 1)(\alpha^{x-\lambda} - 1)w(x) + \lambda \log(\alpha) x^{-\lambda}[w(x)\alpha^{x-\lambda} - (1-\theta)\alpha(1-\alpha^{x-\lambda})]}{x(\alpha^{x-\lambda} - 1)w(x)}$$ + +where $w(x) = \alpha(\theta(1 - \alpha^{-x-\lambda}) + \alpha^{-x-\lambda}) - 1$. The hazard rate curve may take several shapes according to different parameter values, so we summarize these cases by: + +1. If $\theta > 1$ and $\alpha > 1$ then $\alpha^{-x-\lambda} < 1$ and hence $w(x) < 0$, then $\log h(x; \alpha, \theta, \lambda)$ attains its maximum at a certain point $h_0 \in (1, \infty)$ so the hazard rate function is increasing on the interval $(1, h_0)$ and is decreasing $(h_0, \infty)$. + +2. If $0 < \theta < 1$ and $0 < \alpha < 1$ then $\alpha^{-x-\lambda} > 1$ hence $w(x) > 0$ and $\log h(x; \alpha, \theta, \lambda)$ is decreasing for all values of x, which indicates a decreasing hazard rate where $h(1)=\frac{\lambda \text{Log}(\alpha)}{(\alpha-1)\theta}$, and $h(\infty)=0$. \ No newline at end of file diff --git a/samples/texts/823239/page_5.md b/samples/texts/823239/page_5.md new file mode 100644 index 0000000000000000000000000000000000000000..ecc2e800fa317763b3aa9535bb5fba609088029a --- /dev/null +++ b/samples/texts/823239/page_5.md @@ -0,0 +1,23 @@ +Figure 2 illustrated the shape of the hazard rate function for some selected parameters' values. + +Figure 1. pdf of MOAPP under different values of the parameters + +Figure 2. Hazard rate function of MOAPP under different values of the parameters + +## 2.3. Moments + +In order to obtain the moments for MOAPP distribution we use series representation for the pdf that is given in Eq. (5). The generalized binomial expansion will be used for this purpose, hence the MOAPP density can be rewritten as: + +$$f_{\text{MOAPP}}(x) = \sum_{m=0}^{\infty} p_m \Omega_{m+1}(Y_m) \quad (9)$$ + +where $p_m = \left\{ +\begin{array}{ll} +\sum_{k=0}^{\infty} \sum_{j=0}^{k} (-1)^j (k+1)\theta(1-\theta)^k \binom{k}{j} \alpha^{k-j} \frac{(\log \alpha)^{m+1} (j+1)^m}{(\alpha-1)^{k+1} (m+1)!}, & 0 < \theta < 1 \\ +\\ +\sum_{k=0}^{\infty} \sum_{j=0}^{k} (-1)^j (k+1)(1-\theta^{-1})^k \binom{k}{j} \frac{(\log \alpha)^{m+1} (k+1-j)^m}{\theta^{(\alpha-1)^{k+1}(m+1)!}}, & \theta > 1 +\end{array} +\right.$$ + +and $\Omega_{m+1}(Y_m) = \frac{\lambda(m+1)}{y^{\lambda+1}} (1 - \frac{1}{y^\lambda})^m$, $y \ge 1$, which is the exponentiated-Pareto distribution with two shape parameters $(m+1, \lambda)$. + +Eq. (9) represents the MOAPP family density as a linear combination of exponentiated-Pareto density, hence some mathematical properties can be determined from this representation. \ No newline at end of file diff --git a/samples/texts/823239/page_6.md b/samples/texts/823239/page_6.md new file mode 100644 index 0000000000000000000000000000000000000000..c71c04bd33f727dc2478739b0d5cfbb66679a11f --- /dev/null +++ b/samples/texts/823239/page_6.md @@ -0,0 +1,29 @@ +The $r^{th}$ moment MOAPP distribution can be computed from + +$$E(X^r) = \sum_{m=0}^{\infty} p_m E(Y_m^r),$$ + +where $E(Y_m^r) = \int_1^\infty y^r \Omega_{m+1}(y) dy = (m+1)B(m+1, 1-\frac{r}{\lambda})$, and $B(\alpha, \beta)$ is beta function. + +## 2.4. Quantile function + +By inverting Equation (6), we have the quantile of MOAPP distribution as follows: + +$$x_q = \left( 1 - \frac{1}{\ln(\alpha)} \ln \left( 1 + \frac{\theta q (\alpha - 1)}{1 - q(1 - \theta)} \right) \right)^{-1/\lambda} ; 0 < q < 1 \qquad (10)$$ + +# 3. Classical Point Estimation Methods + +In this section we discuss two different classical point estimation methods, namely maximum likelihood estimation and maximum product spacing. Simulation analysis will take place in Section 6 in order to compare between the efficiency of these two methods. + +## 3.1. Maximum Likelihood Estimation + +The maximum likelihood estimation (MLE) is used in inferential statistics since it has many attractive properties, such as invariance, consistency and normal approximation properties. It depends basically on maximizing the likelihood function of MOAPP distribution. Let $X_1, X_2, ..., X_n$ be a random sample from MOAPP distribution, then the log likelihood function for the vector of parameters $\gamma=(\alpha, \theta, \lambda)$ can be expressed by + +$$\ell(\gamma) = n\text{Log}[\theta\lambda\text{Log}(\alpha)] + (n - \sum_{i=1}^{n} x_i^{-\lambda})\text{Log}(\alpha) - n\text{Log}(\alpha - 1) - \\ (\lambda + 1)\sum_{i=1}^{n} \text{Log}x_i - 2\sum_{i=1}^{n} \text{Log}[\theta + \frac{(1-\theta)}{(\alpha-1)}(\alpha^{1-x_i^{-\lambda}} - 1)] \quad (11)$$ + +In order to obtain the MLE of the parameters $\alpha, \theta$ and $\lambda$ it is necessary to find the derivative of equation (11) with respect to $\alpha, \theta$ and $\lambda$ respectively. + +$$\frac{\partial \ell(\gamma)}{\partial \alpha} = \frac{n+\text{Log}(\alpha)(n-\sum_{i=1}^{n} x_i)}{\alpha \text{Log}(\alpha)} - \frac{n}{\alpha-1} - 2 \sum_{i=1}^{n} \frac{(1-\theta)\alpha^{-x_i^{-\lambda}}[(1-\alpha)x_i^{-\lambda}+\alpha x_i^{-\lambda}-1]}{(\alpha-1)^2[\theta+\frac{(1-\theta)}{\alpha-1}(\alpha^{1-x_i^{-\lambda}}-1)]}$$ + +$$\frac{\partial \ell(\gamma)}{\partial \theta} = \frac{n}{\theta} - 2 \sum_{i=1}^{n} \frac{1-(\alpha-1)^{-1}(\alpha^{1-x_i^{-\lambda}}-1)}{[\theta+\frac{(1-\theta)}{\alpha-1}(\alpha^{1-x_i^{-\lambda}}-1)]}$$ + +$$\frac{\partial \ell(\gamma)}{\partial \lambda} = \frac{n}{\lambda} + 2 \sum_{i=1}^{n} \frac{(1-\theta)(\alpha-1)^{-1}\alpha^{1-x_i^{-\lambda}}x_i^{-\lambda}\text{Log}(x_i)\text{Log}(\alpha)}{[\theta+\frac{(1-\theta)}{\alpha-1}(\alpha^{1-x_i^{-\lambda}}-1)]}$$ \ No newline at end of file diff --git a/samples/texts/823239/page_7.md b/samples/texts/823239/page_7.md new file mode 100644 index 0000000000000000000000000000000000000000..f32dbe8091f2953661e13025b5526e140a178601 --- /dev/null +++ b/samples/texts/823239/page_7.md @@ -0,0 +1,31 @@ +The solution for the above normal equations is not in an explicit form, hence the MLEs can be obtained numerically by using Newton or Newton-Raphson methods. + +## 3.2. Maximum Product Spacings + +The Maximum Product Spacings (MPS) method is a new point estimation method that is considered as an alternative to MLE, see Cheng and Amin (1983). This method was recently used by many authors, see, for example Singh et al. (2014), Singh et al. (2016), and Almetwally and Almongy (2019a, b). It was observed that MPS acts better than MLE in many cases. The MPS is defined as: + +$$M = \left( \prod_{i=1}^{n+1} D_i \right)^{\frac{1}{n+1}},$$ + +where M is defined as the geometric mean of the product spacings function $D_i$ such that + +$$ +\begin{aligned} +D_1 &= F(x_1) \\ +D_i &= F(x_i) - F(x_{i-1}); i = 2, \dots, n \\ +D_{n+1} &= 1 - F(x_n) +\end{aligned} + $$ + +It is easy to see that $\sum_{i=1}^{n+1} D_i = 1$. The MPS method is based on the observed ordered sample $x_1 < \cdots < x_n$ from MOAPP distribution, hence the product spacings function is + +$$ M(\gamma) = \left\{ \frac{\alpha^{1-x_1^{-\lambda}}}{(\alpha-1)u(x_1)} \left(1 - \frac{\alpha^{1-x_n^{-\lambda}}}{(\alpha-1)u(x_n)}\right) \prod_{i=2}^{n} \left[ \frac{\alpha^{1-x_i^{-\lambda}}}{(\alpha-1)u(x_i)} - \frac{\alpha^{1-x_{i-1}^{-\lambda}}}{(\alpha-1)u(x_{i-1})} \right] \right\}^{\frac{1}{n+1}}, $$ + +where $u(x_i) = \theta + (1-\theta)(\alpha-1)^{-1}(\alpha^{1-x_i^{-\lambda}} - 1)$. + +The natural logarithm of the product spacings function is + +$$ \ln M(\gamma) = \frac{1}{n+1} \left\{ \ln \left( \alpha^{1-x_1^{-\lambda}} - 1 \right) - \ln \left( (\alpha-1)u(x_1) \right) + \ln \left( 1 - \frac{\alpha^{1-x_n^{-\lambda}}-1}{(\alpha-1)u(x_n)} \right) + \sum_{i=2}^{n} \ln \left[ \frac{\alpha^{1-x_i^{-\lambda}}-1}{(\alpha-1)u(x_i)} - \frac{\alpha^{1-x_{i-1}^{-\lambda}}-1}{(\alpha-1)u(x_{i-1})} \right] \right\}. \quad (12) $$ + +To obtain the normal equations for the unknown parameters, we differentiate Eq. (12) partially with respect to the vector parameter $\gamma$ and equate them to zero. + +$$ \frac{d \ln M(\gamma)}{d\alpha} = \frac{1}{n+1} \left\{ \frac{(1-x_1)^{-\lambda}\alpha^{-x_1^{-\lambda}}}{\alpha^{1-x_1^{-\lambda}}-1} - \frac{u(x_1)+(\alpha-1)\mu_{\alpha}(x_1)}{(\alpha-1)u(x_1)} + \frac{(\alpha-1)u(x_n)(1-x_n^{-\lambda})\alpha^{-x_n^{-\lambda}}}{((\alpha-1)u(x_n))^2 - ((\alpha-1)u(x_n)\alpha^{1-x_n^{-\lambda}}-1)} + \frac{(\alpha-1)u(x_i)(1-x_i^{-\lambda})\alpha^{-x_i^{-\lambda}}}{((\alpha-1)u(x_i))^2 - ((\alpha-1)u(x_i)\alpha^{1-x_i^{-\lambda}}-1)} - \frac{u(x_i)+(\alpha-1)\mu_{\alpha}(x_i)+u(x_i)}{(\alpha-1)u(x_i)\alpha^{1-x_i^{-\lambda}}-1} \right\} $$ \ No newline at end of file diff --git a/samples/texts/823239/page_8.md b/samples/texts/823239/page_8.md new file mode 100644 index 0000000000000000000000000000000000000000..3ce25eb950197c753f5202d12c40bfee56ab9432 --- /dev/null +++ b/samples/texts/823239/page_8.md @@ -0,0 +1,55 @@ +$$ +\begin{equation} +\begin{split} +\frac{d \ln M(\gamma)}{d\theta} &= \frac{1}{n+1} \left\{ + \begin{aligned}[t] + & \frac{u_{\theta}(x_1)}{u(x_1)} + \frac{(\alpha^{1-x_n}-1)u_{\theta}(x_n)}{u(x_n)((\alpha-1)u(x_n)-(\alpha^{1-x_n}-1))} + \\ + & \qquad \frac{\left(\frac{\alpha^{1-x_i}-1}{(u(x_i))^2}\right)^2 + \frac{\left(\frac{\alpha^{1-x_i-1}}{(u(x_i-1))^2}\right)^2}{\left[\frac{\alpha^{1-x_i}-1}{u(x_i)} - \frac{\alpha^{1-x_i-1}}{u(x_i-1)}\right]^{-\lambda}}}{(u(x_i))^2} + \end{aligned} +\right\}, +\end{split} +\end{equation} +$$ + +$$ +\begin{equation} +\begin{split} +\frac{d \ln M(\gamma)}{d \lambda} &= \frac{1}{n+1} \left\{ + \begin{aligned}[t] + & \frac{\operatorname{Log}(\alpha x_1) x_1^{-\lambda}}{1-\alpha x_1^{-\lambda-1}} - \frac{(\alpha-1) u_{\lambda}(x_1)}{(\alpha-1) u(x_1)} - \frac{u(x_n) \alpha^{1-x_n} \lambda^{-\lambda} \operatorname{Log}(\alpha x_n) x_n^{-\lambda} - (\alpha^{1-x_n} \lambda^{-1}) u_{\lambda}(x_n)}{u(x_n)((\alpha-1)u(x_n)-(\alpha^{1-x_n}-1))} + \\ + & \qquad \frac{u(x_i) \alpha^{1-x_i} \lambda^{-\lambda} \operatorname{Log}(\alpha x_i) x_i^{-\lambda} - (\alpha^{1-x_i-1}) u_{\lambda}(x_i)}{\operatorname{Log}((\alpha-1)u(x_i)-(\alpha^{1-x_i}-1))} \\ + & \qquad - \frac{u(x_{i-1}) \alpha^{1-x_{i-1}} \lambda^{-\lambda} \operatorname{Log}(\alpha x_{i-1}) x_{i-1}^{-\lambda} - (\alpha^{1-x_{i-1}-1}) u_{\lambda}(x_{i-1})}{u(x_{i-1})((\alpha-1)u(x_{i-1})-(\alpha^{1-x_{i-1}}-1))} \\ + & \qquad + \frac{\frac{\alpha^{1-x_i}-1}{(\alpha-1)u(x_i)} - \frac{\alpha^{1-x_i-1}}{(\alpha-1)u(x_{i-1})}}{\frac{\alpha^{1-x_i}-1}{(\alpha-1)u(x_i)} - \frac{\alpha^{1-x_i-1}}{(\alpha-1)u(x_{i-1})}} + \end{aligned} +\right\}, +\end{split} +\tag{13} +\end{equation} +$$ + +where $u_\alpha$, $u_\theta$ and $u_\lambda$ represent the partial derivative of $u(x_i)$ with respect to $\alpha$, $\theta$ and $\lambda$ respectively. The estimators of $\gamma$ can be obtained by solving the above system of nonlinear equations numerically, so the MPS of $\alpha$, $\theta$ and $\lambda$ are denoted by $\hat{\alpha}_{MP}$, $\hat{\theta}_{MP}$ and $\hat{\lambda}_{MP}$ respectively. + +4. Bayesian Estimation Method + +In this section we consider the non-classical method of estimation that is Bayes +estimates for the unknown parameters α, θ and λ of MOAPP distribution. +The quadratic loss and LINEX loss functions are the assumed loss functions. + +In Bayesian method, all parameters are random variables with a certain distribution +called prior distribution. If prior information is not available which is usually the case, +we need to select a prior distribution. Since the selection of prior distribution plays an +important role in estimation of the parameters, our choice for the prior of α,θ and λ are +the independent gamma distributions, which are G(a₁, b₁), G(a₂, b₂) and G(a₃, b₃) +respectively. Thus, the suggested prior for α, θ and λ can be written as: + +$$ +\pi_1(\alpha) \propto \alpha^{a_1-1} e^{-b_1\alpha}, \quad \pi_2(\theta) \propto \theta^{a_2-1} e^{-b_2\theta}, \quad \pi_3(\lambda) \propto \lambda^{a_3-1} e^{-b_3\lambda}, +$$ + +respectively, where a₁, a₂, a₃, b₁, b₂ and b₃ are the hyper parameters of prior distributions. + +The joint prior of α, θ and λ is + +$$ +k(\alpha, \theta, \lambda) \propto \alpha^{a_1-1} \theta^{a_2-1} \lambda^{a_3-1} e^{-b_1 \alpha - b_2 \theta - b_3 \lambda}, \quad \alpha, \theta, \lambda, a_1, a_2, a_3, b_1, b_2, b_3 > 0. +$$ \ No newline at end of file diff --git a/samples/texts/823239/page_9.md b/samples/texts/823239/page_9.md new file mode 100644 index 0000000000000000000000000000000000000000..d5a9c61d1746c27285401da1eff9bf7f0689b9af --- /dev/null +++ b/samples/texts/823239/page_9.md @@ -0,0 +1,37 @@ +The joint posterior of $\alpha, \theta$ and $\lambda$ is given by + +$$p(\alpha, \theta, \lambda | \underline{x}) \propto L(\underline{x} | \alpha, \theta, \lambda) k(\alpha, \theta, \lambda),$$ + +where $L(\underline{x}/\alpha, \theta, \lambda)$ is the likelihood function of MOAPP distribution. When substituting the likelihood function $L(\underline{x}/\alpha, \theta, \lambda)$ and the joint prior $k(\alpha, \theta, \lambda)$ in the above equation, the joint posterior will be: + +$$ +\begin{aligned} +p\left(\alpha, \theta, \frac{\lambda}{\underline{x}}\right) &\propto \alpha^{n-\sum_{i=1}^{n} x_i^{-\lambda} + a_1 - 1} \theta^{n+a_2-1} \lambda^{n+a_s-1} e^{-b_1} \alpha^{-b_2} \theta^{-b_s} \lambda \\ +&\qquad \prod_{i=1}^{n} \frac{1}{\alpha-1} \frac{\operatorname{Log}\alpha}{x_i^{\lambda+1}} \left( \theta + \frac{(1-\theta)}{\alpha-1} (\alpha^{1-x_i^{-\lambda}} - 1) \right)^{-2} \\ +p(\alpha, \theta, \lambda | \underline{x}) &\propto G_{\alpha \setminus \lambda} (n - \sum_{i=1}^{n} x_i^{-\lambda} + a_1, b_1) G_{\theta}(n + a_2, b_2) G_{\lambda}(n + a_3, b_3) e^{\phi(\alpha, \theta, \lambda)}, +\end{aligned} +$$ + +where $\phi(\alpha, \theta, \lambda) = \sum_{i=1}^{n} \ln \frac{1}{\alpha-1} \frac{\operatorname{Log}\alpha}{x_i^{\lambda+1}} - 2\ln \left( \theta + \frac{(1-\theta)}{\alpha-1} (\alpha^{1-x_i^{-\lambda}} - 1) \right)$ + +In the case of quadratic loss function, Bayes estimate is the posterior mean, the determination of posterior mean for the purpose of obtaining Bayes estimation of the parameters $\alpha, \theta$ and $\lambda$, is not easy to obtain unless we use numerical approximation methods. + +In the literature, there are several approximation methods available to solve this kind of problem. Here, we consider Monte Carlo Markov Chain (MCMC) approximation method, see Karandikar (2006). This approximation method reduces the ratio of integrals into a whole and produces a single numerical result. + +A wide variety of MCMC schemes are available. An important sub-class of MCMC methods are Gibbs sampling and more general Metropolis within Gibbs samplers. Indeed, the MCMC samples may be used to completely summarize the posterior uncertainty about the parameters $\alpha, \theta$ and $\lambda$, through a kernel estimate of the posterior distribution. This is also true of any function of the parameters. + +Therefore, to generate samples from MOAPP distribution, we use the Metropolis-Hastings method (Metropolis et al. (1953) with normal proposal distribution). For details regarding the implementation of the Metropolis-Hasting algorithm, the readers may refer to Robert and Casella (2013) and Almetwally et al. (2018). The full conditional posterior densities of $\alpha, \theta$ and $\lambda$ and the data are given by: + +$$ +\begin{align} +\pi(\alpha/\theta, \lambda; \underline{x}) &\propto G_{\alpha \setminus \lambda} \left( n - \sum_{i=1}^{n} x_i^{-\lambda} + a_1, b_1 \right) e^{\phi(\alpha, \theta, \lambda)} \\ +\pi(\theta/\alpha, \lambda; \underline{x}) &= G_{\theta}(n + a_2, b_2) e^{-2\ln\left(\theta + \frac{(1-\theta)}{\alpha-1}(\alpha^{1-x_i^{-\lambda}} - 1)\right)} \\ +\pi(\lambda/\alpha, \theta; \underline{x}) &= G_{\lambda}(n + a_3, b_3) e^{\phi(\alpha, \theta, \lambda)} +\end{align} +$$ + +To apply the Gibbs technique we need the following algorithm: + +(1) Start with initial values ($\alpha_0, \theta_0, \lambda_0$) + +(2) Use M-H algorithm to generate posterior sample for $\alpha, \theta$ and $\lambda$ from Eq. (14) \ No newline at end of file