Stability radius

In mathematics, the stability radius of an object (system, function, matrix, parameter) at a given nominal point is the radius of the largest ball, centered at the nominal point, all of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:
where formula_1 denotes the nominal point, formula_2 denotes the space of all possible values of the object formula_3, and the shaded area, formula_4, represents the set of points that satisfy the stability conditions. The radius of the blue circle, shown in red, is the stability radius.
Abstract definition.
The formal definition of this concept varies, depending on the application area. The following abstract definition is quite useful
where formula_6 denotes a closed ball of radius formula_7 in formula_2 centered at formula_1.
History.
It looks like the concept was invented in the early 1960s. In the 1980s it became popular in control theory and optimization. It is widely used as a model of local robustness against small perturbations in a given nominal value of the object of interest.
Relation to Wald's maximin model.
It was shown that the stability radius model is an instance of Wald's maximin model. That is,
where
The large penalty (formula_12) is a device to force the formula_13 player not to perturb the nominal value beyond the stability radius of the system. It is an indication that the stability model is a model of local stability/robustness, rather than a global one.
Info-gap decision theory.
Info-gap decision theory is a recent non-probabilistic decision theory. It is claimed to be radically different from all current theories of decision under uncertainty. But it has been shown that its robustness model, namely
is actually a stability radius model characterized by a simple stability requirement of the form formula_15 where formula_16 denotes the decision under consideration, formula_17 denotes the parameter of interest, formula_18 denotes the estimate of the true value of formula_17 and formula_20 denotes a ball of radius formula_21 centered at formula_18.
Since stability radius models are designed to deal with small perturbations in the nominal value of a parameter, info-gap's robustness model measures the "local robustness" of decisions in the neighborhood of the estimate formula_18.
Sniedovich argues that for this reason the theory is unsuitable for the treatment of severe uncertainty characterized by a poor estimate and a vast uncertainty space.
Alternate definition.
There are cases where it is more convenient to define the stability radius slightly different. For example, in many applications in control theory the radius of stability is defined as the size of the smallest destabilizing perturbation in the nominal value of the parameter of interest. The picture is this:
More formally,
where formula_25 denotes the "distance" of formula_26 from formula_1.
Stability radius of functions.
The stability radius of a continuous function "f" (in a functional space "F") with respect to an open stability domain "D" is the distance between "f" and the set of unstable functions (with respect to "D"). We say that a function is "stable" with respect to "D" if its spectrum is in "D". Here, the notion of spectrum is defined on a case-by-case basis, as explained below.
Definition.
Formally, if we denote the set of stable functions by "S(D)" and the stability radius by "r(f,D)", then:
where "C" is a subset of "F".
Note that if "f" is already unstable (with respect to "D"), then "r(f,D)=0" (as long as "C" contains zero).
Applications.
The notion of stability radius is generally applied to special functions as polynomials (the spectrum is then the roots) and matrices (the spectrum is the eigenvalues). The case where "C" is a proper subset of "F" permits us to consider structured perturbations (e.g. for a matrix, we could only need perturbations on the last row). It is an interesting measure of robustness, for example in control theory.
Properties.
Let "f" be a (complex) polynomial of degree "n", "C=F" be the set of polynomials of degree less than (or equal to) "n" (which we identify here with the set formula_29 of coefficients). We take for "D" the open unit disk, which means we are looking for the distance between a polynomial and the set of Schur stable polynomials. Then:
where "q" contains each basis vector (e.g. formula_31 when "q" is the usual power basis). This result means that the stability radius is bound with the minimal value that "f" reaches on the unit circle.


Rubber pad forming

Rubber pad forming (RPF) is a metalworking process where sheet metal is pressed between a die and a rubber block, made of polyurethane. Under pressure, the rubber and sheet metal are driven into the die and conform to its shape, forming the part. The rubber pads can have a general purpose shape, like a membrane. Alternatively, they can be machined in the shape of die or punch.
Rubber pad forming is a deep drawing technique that is ideally suited for the production of small and medium-sized series. Deep drawing makes it possible to deform sheet metal in two directions, which offers great benefits in terms of function integration, weight reduction, cleanability and such.
The disadvantage of regular deep drawing is that expensive tools consisting of an upper and lower mold are needed. Once these tools have been made, the variable costs are low, which makes regular deep drawing very suitable for large and very large numbers of products.
Technique.
In the rubber pad forming process only a milled lower die is required on which a metal plate is placed. Afterwards, the shape of the lower die is pressed in the plate with the rubber mold. In most cases, the contour, hole patterns and the like will be cut with a 3D laser cutter.
The simplicity of the rubber press tool causes tooling costs to be around 85 to 90% lower than those of regular deep drawing while the variable costs are higher. This combination makes rubber pad pressing very suitable for smaller and medium-sized series (up to 5,000-10,000 pieces per year), even though traditional cutting, lace, welding, finishing etc. is used more due to the unfamiliarity with rubber pad forming.

380 Ton Rubber Pad Press
Rubber pad forming has been used in production lines for many years. Up to 60% of all sheet metal parts in the aerospace industry are fabricated using this process.
The most relevant applications are indeed in the aerospace field. It is frequently used in prototyping shops and for the production of kitchenware. For a decade, rubber pad pressing has developed greatly into a widely used technology for many industrial applications.
Pressing power.
Enormous pressing forces are required for the rubber presses to work. In the Netherlands there are several rubber pad presses, of which the largest one has a press force of no less than 8,000 tons with a maximum surface area of 1.10x2.20m, these presses are used for very diverse industrial applications.
Worldwide, presses are in use up to about 14,000 tonnes.
Pros and cons.
In summary, the benefits of rubber pad pressing are:
And the disadvantages:
Definition.
Rubber pad forming can be accomplished in many different ways, and as technology has advanced, so have the applications for this simple process. In general, an elastic upper die, usually made of rubber, is connected to a hydraulic press. A rigid lower die, often called a form block, provides the mold for the sheeted metal to be formed to. Because the upper (male) die can be used with separate lower (female) dies, the process is relatively cheap and flexible. The worked metal is not worn as quickly as in more conventional processes such as deep drawing, however, rubber pads exert less pressure in the same circumstances as non-elastic parts, which may lead to less definition in forming, and rubber pads wear more quickly than steel parts.
The Guerin process.
The Guerin process, also called Guerin Stamping, is a manufacturing process used in the shaping of sheet metals. It is the oldest and most basic of the production rubber-pad forming processes. It was developed in the late 1930s by Henry Guerin, an employee of the Douglas Aircraft Co. in California. Thereafter, it was used extensively by all major aircraft manufacturers to shape the many complex shapes inherent in the design of aircraft.


Lead frame

A lead frame (pronounced ) is a metal structure inside a chip package that carries signals from the die to the outside, used in DIP, QFP and other packages where connections to the chip are made on its edges.
The lead frame consists of a central die pad, where the die is placed, surrounded by leads, metal conductors leading away from the die to the outside world. The end of each lead closest to the die ends in a bond pad. Small bond wires connect the die to each bond pad. Mechanical connections fix all these parts into a rigid structure, which makes the whole lead frame easy to handle automatically.
Manufacturing.
Lead frames are manufactured by removing material from a flat plate of copper, copper-alloy, or iron-nickel alloy like alloy 42. Two processes used for this are etching (suitable for high density of leads), or stamping (suitable for low density of leads). The mechanical bending process can be applied after both techniques.
The die is glued or soldered to the die pad inside the lead frame, and then bond wires are attached between the die and the bond pads to connect the die to the leads. In a process called encapsulation, a plastic case is moulded around the lead frame and die, exposing only the leads. The leads are cut off outside the plastic body and any exposed supporting structures are cut away. The external leads are then bent to the desired shape.
Uses.
Amongst others, lead frames are used to manufacture a quad flat no-leads package (QFN), a quad flat package (QFP), or a dual in-line package (DIP).


Moldmaker

A moldmaker (mouldmaker in English-speaking countries other than the US) or molder is a skilled tradesperson who fabricates molds for use in casting metal products.
Moldmakers are generally employed in foundries, where molds are used to cast products from metals such as aluminium and cast iron.
Injection molding.
The term moldmaker may also be used to describe workers employed in fabricating dies and metal moulds for use in injection molding and die-casting, such as in the plastics, rubber or ceramics industries, in which case it is sometimes regarded as a variety of the trade of the toolmaker. The process of manufacturing molds is now often highly automated.
While much of the machining processes involved in mold making use computer-controlled equipment for the actual manufacturing of molds (particularly plastic and rubber injection and transfer). Moldmaking is still a highly skilled trade requiring expertise in manual machining, CNC machining, CNC wire EDM, CNC Ram EDM, surface grinding, hand polishing and more. Because of the high skill and intense labor involved much of the mold making in the US has been outsourced to low wage countries.
The majority of plastic and rubber parts that are in existence today are made using injection or transfer molds, requiring a mold to be manufactured by a moldmaker.


Bearing reducer

A Bearing reducer in engineering is a bearing that designates the full integration of high-precision reduction gear and high-precision radial-axial bearing in a compact unit. This transmission system allows the utilization of the bearing reducer in several technics, such as robotics and automation, machine tools, measuring equipment, navigation systems, the aircraft industry, the military and medicine field, the woodworking field, the printers branch, the machines for the textile industry and glass treatment, and the filling machines.


Triplet loss

Triplet loss is a loss function for machine learning algorithms where a reference input (called anchor) is compared to a matching input (called positive) and a non-matching input (called negative). The distance from the anchor to the positive is minimized, and the distance from the anchor to the negative input is maximized.
An early formulation equivalent to triplet loss was introduced (without the idea of using anchors) for metric learning from relative comparisons by M. Schultze and T. Joachims in 2003.
By enforcing the order of distances, triplet loss models embed in the way that a pair of samples with same labels are smaller in distance than those with different labels. Unlike t-SNE which preserves embedding orders via probability distributions, triplet loss works directly on embedded distances. Therefore, in its common implementation, it needs soft margin treatment with a slack variable formula_1 in its hinge loss-style formulation. It is often used for learning similarity for the purpose of learning embeddings, such as learning to rank, word embeddings, thought vectors, and metric learning.
Consider the task of training a neural network to recognize faces (e.g. for admission to a high security zone). A classifier trained to classify an instance would have to be retrained every time a new person is added to the face database. This can be avoided by posing the problem as a similarity learning problem instead of a classification problem. Here the network is trained (using a contrastive loss) to output a distance which is small if the image belongs to a known person and large if the image belongs to an unknown person. However, if we want to output the closest images to a given image, we want to learn a ranking and not just a similarity. A triplet loss is used in this case.
The loss function can be described by means of the Euclidean distance function
This can then be used in a cost function, that is the sum of all losses, which can then be used for minimization of the posed optimization problem


Bowl Prechamber Ignition

Bowl Prechamber Ignition, abbreviated BPI, is a combustion process designed for Otto cycle engines running on an air-fuel mixture leaner than stochiometric formula_1. Its distinguishing feature is a special type of spark plug, capable of reliably igniting very lean air-fuel mixtures. This spark plug is called "prechamber spark plug". The ignition electrodes of this spark plug are housed in a perforated enclosure, the "prechamber". At the engine's compression stroke, some fuel (usually less than 5 % of the total injected fuel) is injected into the piston bowl; this fuel is then forced through the small holes into the prechamber due to the high pressure in the cylinder near top dead centre. Inside the prechamber spark plug, the air-fuel mixture is ignitable by the ignition spark. Flame jets occurring due to the small holes in the prechamber then ignite the air-fuel mixture in the main combustion chamber, that would not catch fire using a regular spark plug.


Bilinear transform

The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa.
The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used to convert a transfer function formula_1 of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function formula_2 of a linear, shift-invariant filter in the discrete-time domain (often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the formula_3 axis, formula_4, in the s-plane to the unit circle, formula_5, in the z-plane. Other bilinear transforms can be used to warp the frequency response of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays formula_6 with first order all-pass filters.
The transform preserves stability and maps every point of the frequency response of the continuous-time filter, formula_7 to a corresponding point in the frequency response of the discrete-time filter, formula_8 although to a somewhat different frequency, as shown in the Frequency warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency.
Discrete-time approximation.
The bilinear transform is a first-order Padé approximant of the natural logarithm function that is an exact mapping of the "z"-plane to the "s"-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of
where formula_10 is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation; or, in other words, the sampling period. The above bilinear approximation can be solved for formula_11 or a similar approximation for formula_12 can be performed.
The inverse of this mapping (and its first-order bilinear approximation) is
The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, formula_1
That is
Stability and minimum-phase property preserved.
A continuous-time causal filter is stable if the poles of its transfer function fall in the left half of the complex s-plane. A discrete-time causal filter is stable if the poles of its transfer function fall inside the unit circle in the complex z-plane. The bilinear transform maps the left half of the complex s-plane to the interior of the unit circle in the z-plane. Thus, filters designed in the continuous-time domain that are stable are converted to filters in the discrete-time domain that preserve that stability.
Likewise, a continuous-time filter is minimum-phase if the zeros of its transfer function fall in the left half of the complex s-plane. A discrete-time filter is minimum-phase if the zeros of its transfer function fall inside the unit circle in the complex z-plane. Then the same mapping property assures that continuous-time filters that are minimum-phase are converted to discrete-time filters that preserve that property of being minimum-phase.
Transformation of a General LTI System.
A general LTI system has the transfer function
formula_17
The order of the transfer function is the greater of and (in practice this is most likely as the transfer function must be proper for the system to be stable). Applying the bilinear transform
formula_18
where is defined as either or otherwise if using frequency warping, gives
formula_19
Multiplying the numerator and denominator by the largest power of present, gives
formula_20
It can be seen here that after the transformation, the degree of the numerator and denominator are both .
Consider then the pole-zero form of the continuous-time transfer function
formula_21
The roots of the numerator and denominator polynomials, and , are the zeros and poles of the system. The bilinear transform is a one-to-one mapping, hence these can be transformed to the z-domain using
formula_22
yielding some of the discretized transfer function's zeros and poles and
formula_23
As described above, the degree of the numerator and denominator are now both , in other words there is now an equal number of zeros and poles. The multiplication by means the additional zeros or poles are
formula_24
Given the full set of zeros and poles, the z-domain transfer function is then
formula_25
Example.
As an example take a simple low-pass RC filter. This continuous-time filter has a transfer function
If we wish to implement this filter as a digital filter, we can apply the bilinear transform by substituting for formula_27 the formula above; after some reworking, we get the following filter representation:
The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward' coefficients used to implement a real-time digital filter.
Transformation for a general first-order continuous-time filter.
It is possible to relate the coefficients of a continuous-time, analog filter with those of a similar discrete-time digital filter created through the bilinear transform process. Transforming a general, first-order continuous-time filter with the given transfer function
using the bilinear transform (without prewarping any frequency specification) requires the substitution of
where
However, if the frequency warping compensation as described below is used in the bilinear transform, so that both analog and digital filter gain and phase agree at frequency formula_31, then
This results in a discrete-time digital filter with coefficients expressed in terms of the coefficients of the original continuous time filter:
Normally the constant term in the denominator must be normalized to 1 before deriving the corresponding difference equation. This results in
The difference equation (using the Direct form I) is
General second-order biquad transformation.
A similar process can be used for a general second-order filter with the given transfer function
This results in a discrete-time digital biquad filter with coefficients expressed in terms of the coefficients of the original continuous time filter:
Again, the constant term in the denominator is generally normalized to 1 before deriving the corresponding difference equation. This results in
The difference equation (using the Direct form I) is
Frequency warping.
To determine the frequency response of a continuous-time filter, the transfer function formula_1 is evaluated at formula_41 which is on the formula_3 axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function formula_43 is evaluated at formula_44 which is on the unit circle, formula_5. The bilinear transform maps the formula_3 axis of the "s"-plane (of which is the domain of formula_1) to the unit circle of the "z"-plane, formula_5 (which is the domain of formula_43), but it is not the same mapping formula_50 which also maps the formula_3 axis to the unit circle. When the actual frequency of formula_52 is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency, formula_53, for the continuous-time filter that this formula_52 is mapped to.
This shows that every point on the unit circle in the discrete-time filter z-plane, formula_56 is mapped to a point on the formula_57 axis on the continuous-time filter s-plane, formula_58. That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is
and the inverse mapping is
The discrete-time filter behaves at frequency formula_61 the same way that the continuous-time filter behaves at frequency formula_62. Specifically, the gain and phase shift that the discrete-time filter has at frequency formula_61 is the same gain and phase shift that the continuous-time filter has at frequency formula_64. This means that every feature, every "bump" that is visible in the frequency response of the continuous-time filter is also visible in the discrete-time filter, but at a different frequency. For low frequencies (that is, when formula_65 or formula_66), then the features are mapped to a "slightly" different frequency; formula_67.
One can see that the entire continuous frequency range
is mapped onto the fundamental frequency interval
The continuous-time filter frequency formula_70 corresponds to the discrete-time filter frequency formula_71 and the continuous-time filter frequency formula_72 correspond to the discrete-time filter frequency formula_73
One can also see that there is a nonlinear relationship between formula_53 and formula_75 This effect of the bilinear transform is called frequency warping. The continuous-time filter can be designed to compensate for this frequency warping by setting formula_59 for every frequency specification that the designer has control over (such as corner frequency or center frequency). This is called pre-warping the filter design.
It is possible, however, to compensate for the frequency warping by pre-warping a frequency specification formula_77 (usually a resonant frequency or the frequency of the most significant feature of the frequency response) of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system. When designing a digital filter as an approximation of a continuous time filter, the frequency response (both amplitude and phase) of the digital filter can be made to match the frequency response of the continuous filter at a specified frequency formula_77, as well as matching at DC, if the following transform is substituted into the continuous filter transfer function. This is a modified version of Tustin's transform shown above.
However, note that this transform becomes the original transform
as formula_81.
The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with Impulse invariance.


Quadratically constrained quadratic program

In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has the form
where "P"0, …, "P""m" are "n"-by-"n" matrices and "x" ∈ R"n" is the optimization variable.
If "P"0, …, "P""m" are all positive semidefinite, then the problem is convex. If these matrices are neither positive nor negative semidefinite, the problem is non-convex. If "P"1, … ,"P""m" are all zero, then the constraints are in fact linear and the problem is a quadratic program.
Hardness.
Solving the general case is an NP-hard problem. To see this, note that the two constraints "x"1("x"1 − 1) ≤ 0 and "x"1("x"1 − 1) ≥ 0 are equivalent to the constraint "x"1("x"1 − 1) = 0, which is in turn equivalent to the constraint "x"1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since 0–1 integer programming is NP-hard in general, QCQP is also NP-hard.
Relaxation.
There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available.
Nonconvex QCQPs with non-positive off-diagonal elements can be exactly solved by the SDP or SOCP relaxations, and there are polynomial-time-checkable sufficient conditions for SDP relaxations of general QCQPs to be exact. Moreover, it was shown that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables.
Semidefinite programming.
When "P"0, …, "P""m" are all positive-definite matrices, the problem is convex and can be readily solved using interior point methods, as done with semidefinite programming.
Example.
Max Cut is a problem in graph theory, which is NP-hard. Given a graph, the problem is to divide the vertices in two sets, so that as many edges as possible go from one set to the other. Max Cut can be formulated as a QCQP, and SDP relaxation of the dual provides good lower bounds.


SEMCI

Single Entry Multiple Company Interface (SEMCI) is a computer system based on service-oriented architecture (SOA) that is used to submit the same information to multiple companies. SEMCI is use by insurance agents to insurance quotes from several insurance companies at once.
SEMCI is an interface which connects the software of the agent to the software of multiple companies so that an inquiry can go to all the companies at once. Previously, the agent would need to send the inquiry to each company individually.


Predictive control of switching power converters

Predictive controllers rely on optimum control systems theory and aim to solve a cost function minimization problem. Predictive controllers are relatively easy to numerically implement but electronic power converters are non-linear time-varying dynamic systems, so a different approach to predictive must be taken.
Principles of non-linear predictive optimum control.
The first step to designing a predictive controller is to derive a detailed direct dynamic model (including non-linearities) of the switching power converter. This model must contain enough detail of the converter dynamics to allow, from initial conditions, a forecast in real time and with negligible error, of the future behavior of the converter.
Sliding mode control of switching power converters chooses a vector to reach sliding mode as fast as possible (high switching frequency).
It would be better to choose a vector to ensure zero error at the end of the sampling period Δt.To find such a vector, a previous calculation can be made (prediction);
The converter has a finite number of vectors (states) and is usually non-linear: one way is to try all vectors to find the one that minimizes the control errors, prior to the application of that vector to the converter.


Factory automation infrastructure

Factory automation infrastructure describes the process of incorporating automation into the manufacturing environment and processing of input goods into final products.
The manufacturing environment is defined by its ability to manufacture and/or assemble goods by machines, integrated assembly lines, and robotic arms. Automated environments are also defined by their coordination with (and usually their systematic integration with) the required automatic equipment to form a complete system.
Factory automation intends to decrease risks associated with laborious and dangerous work faced by human workers. This system is essentially a solution for the automation and manufacturing of a particular production process of an intended output and/or final/end product.
Automation.
Automation has produced sophisticated parts with similar or higher output qualities with minor quality fluctuation. It also can help cut overall manufacturing costs and create safer working environments for workers.
The use of automation in manufacturing started by using technologies such as pneumatic and hydraulic systems in applications where their mechanical advantages could be used to raise output quality and efficiency in production. Complex and highly integrated systems have since evolved, composed of many different technologies and innovative procedures controlled under High Language programming environments with sophisticated operation drivers. These drivers often are running languages that support 6, 7, and 8-axis controls for sophisticated robotics.
Robotic arm.
A robotic arm is a type of mechanical arm, usually programmable, with functions similar to a human arm; the arm may be the total of the mechanism or may be part of a more complex robot. The links of such a manipulator are connected by joints allowing either rotational motion (such as in an articulated robot) or transnational (linear) displacement. The links of the manipulator can be considered to form a kinematic chain. The terminus of the kinematic chain of the manipulator is called the end effector and is analogous to the human hand.
Advantages and disadvantages.
The main advantages of automation are:
The following methods are often employed to improve productivity, quality, or robustness.
The main disadvantages of automation are:
External links.
kinematic chain.


Identical-machines scheduling

Identical-machines scheduling is an optimization problem in computer science and operations research. We are given "n" jobs "J"1, "J"2, ..., "Jn" of varying processing times, which need to be scheduled on "m" identical machines, such that a certain objective function is optimized, for example, the makespan is minimized.
Identical machine scheduling is a special case of uniform machine scheduling, which is itself a special case of optimal job scheduling. In the general case, the processing time of each job may be different on different machines; in the case of identical machine scheduling, the processing time of each job is the same on each machine. Therefore, identical machine scheduling is equivalent to multiway number partitioning. A special case of identical machine scheduling is single-machine scheduling.
In the standard three-field notation for optimal job scheduling problems, the identical-machines variant is denoted by P in the first field. For example, " P||formula_1" is an identical machine scheduling problem with no constraints, where the goal is to minimize the maximum completion time.
In some variants of the problem, instead of minimizing the "maximum" completion time, it is desired to minimize the "average" completion time (averaged over all "n" jobs); it is denoted by P||formula_2. More generally, when some jobs are more important than others, it may be desired to minimize a "weighted average" of the completion time, where each job has a different weight. This is denoted by P||formula_3.
Algorithms.
Minimizing average and weighted-average completion time.
Minimizing the "average" completion time (P||formula_2) can be done in polynomial time. The SPT algorithm (Shortest Processing Time First), sorts the jobs by their length, shortest first, and then assigns them to the processor with the earliest end time so far. It runs in time O("n" log "n"), and minimizes the average completion time on identical machines, P||formula_2.
Minimizing the "weighted average" completion time is NP-hard even on identical machines, by reduction from the knapsack problem. It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the partition problem.
Sahni presents an exponential-time algorithm and a polynomial-time approximation scheme for solving both these NP-hard problems on identical machines:
Minimizing the maximum completion time (makespan).
Minimizing the "maximum" completion time (P||formula_1) is NP-hard even for "identical" machines, by reduction from the partition problem. Many exact and approximation algorithms are known.
Graham proved that:
Coffman, Garey and Johnson presented a different algorithm called multifit algorithm, using techniques from bin packing, which has an approximation factor of 13/11≈1.182.
Huang and Lu presented a simple polynomial-time algorithm that attains an 11/9≈1.222 approximation in time O("m" log "m" + "n"), through the more general problem of "maximin-share allocation of chores".
Sahni presented a PTAS that attains (1+ε)OPT in time formula_9. It is an FPTAS if "m" is fixed. For m=2, the run-time improves to formula_10. The algorithm uses a technique called "interval partitioning".
Hochbaum and Shmoys presented several approximation algorithms for any number of identical machines (even when the number of machines is not fixed):
Leung improved the run-time of this algorithm to formula_14.
Maximizing the minimum completion time.
Maximizing the minimum completion time (P||formula_15) is applicable when the "jobs" are actually spare parts that are required to keep the machines running, and they have different life-times. The goal is to keep machines running for as long as possible. The LPT algorithm attains at least formula_16 of the optimum.
Woeginger presented a PTAS that attains an approximation factor of formula_17 in time formula_18, where formula_19 a huge constant that is exponential in the required approximation factor ε. The algorithm uses Lenstra's algorithm for integer linear programming.
General objective functions.
Alon, Azar, Woeginger and Yadid consider a more general objective function. Given a positive real function "f", which depends only on the completion times "Ci", they consider the objectives of minimizing formula_20, minimizing formula_21, maximizing formula_20, and maximizing formula_23. They prove that, if "f" is non-negative, convex, and satisfies a strong continuity assumption that they call "F*", then both minimization problems have a PTAS. Similarly, if "f" is non-negative, concave, and satisfies F*, then both maximization problems have a PTAS. In both cases, the run-time of the PTAS is O("n"), but with constants that are exponential in 1/"ε."


Disco Corporation

 is a Japanese precision tools maker, especially for the semiconductor production industry.
The company makes dicing saws and laser saws to cut semiconductor silicon wafers and other materials; grinders to process silicon and compound semiconductor wafers to ultra-thin levels; polishing machines to remove the grinding damage layer from the wafer back-side and to increase chip strength.
History.
The company was founded as Daiichi-Seitosho in May 1937, as an industrial abrasive wheel manufacturer.
After World War II Japan faced a construction boom which also helped DISCO to boost its sales. The company's grinder discs were in high demand from utility companies, which needed them to manufacture watt-meters.
In December 1968 the company developed and released an ultra-thin resinoid cutting wheel, "Microncut". The wheel contained diamond powder and as a result it was capable of making sharp, precision cuts as demanded in the semiconductor manufacturing process. There were no cutting machines available in the market on which ultra-thin precision wheels could be mounted and run, DISCO decided to develop its own machine in 1975. The cutting machine, DAD-2h, received instant recognition from semiconductor companies, including Texas Instruments.
The company adopted the name of DISCO Corporation in May 1977, was listed with the Japan Securities Dealers' Association in October 1989, and entered the First Section of the Tokyo Stock Exchange in December 1999.


RoboCup Small Size League

The RoboCup Small Size League (SSL) is one of the RoboCup soccer leagues.
Old Format.
Two teams of six robots which are limited to an 18 cm diameter and 15 cm height play soccer with an orange golf ball. They are identified and tracked by four overhead cameras connected to an off-field computer. The field size is 9metersx6meters. Then robots' and balls' status including their position and id are sent to teams' computers. Their AI software sends commands to the robots based on the vision data.
New Format.
As of 2018, there are now two divisions, the B division that continued to use the same parameters as before (6 robots, 9x6 meter field, etc) and the A division that changed a lot of parameters. This change was brought to improve the tournament experience by making the more experienced teams face each other while letting the newer teams improve their algorithms with more balanced face offs.


Pyragas method

In the mathematics of chaotic dynamical systems, in the Pyragas method of stabilizing a periodic orbit, an appropriate continuous controlling signal is injected into the system, whose intensity is nearly zero as the system evolves close to the desired periodic orbit but increases when it drifts away from the desired orbit. Both the Pyragas and OGY (Ott, Grebogi and Yorke) methods are part of a general class of methods called "closed loop" or "feedback" methods which can be applied based on knowledge of the system obtained through solely observing the behavior of the system as a whole over a suitable period of time.
The method was proposed by Lithuanian physicist .


Continuous-flow manufacturing

Continuous-flow manufacturing, or repetitive-flow manufacturing, is an approach to discrete manufacturing that contrasts with batch production. It is associated with a just-in-time and kanban production approach, and calls for an ongoing examination and improvement efforts which ultimately requires integration of all elements of the production system. The goal is an optimally balanced production line with little waste, the lowest possible cost, on-time and defect-free production.
This strategy is typically applied in discrete manufacturing as an attempt to handle production volumes comprising discrete units of product in a flow which is more naturally found in process manufacturing. The basic fact is that in most cases, discrete units of a solid product cannot be handled in the same way as continuous quantities of liquid, gas or powder.
Discrete manufacturing is more likely to be performed in batches of product units that are routed from process to process in the factory. Each process may add value to the batch during a run-time or work-time. There is usually some time spent waiting for the process during a queue-time or wait-time. The larger the batch, the longer each unit has to wait for the rest of the batch to be completed, before it can go forward to the next process. This queue-time is waste, "Muda", and represents time lost that is not value-added in the eyes of the customer. This waste is one of the most important elements targeted for reduction and elimination in lean manufacturing.
Reducing the batch size in discrete manufacturing is therefore a desirable goal: it improves the speed of response to the customer, whilst improving the ratio of value-added to non value-added work. However, it should be balanced against the finite capacity of resources at the value-adding processes. Capacity is consumed by changeover whenever a process is required to perform work on a different part or product model than the preceding one. Time consumed in changeover is also considered waste, and it reduces the amount of resource capacity that is available to perform value-adding work. Reducing batch sizes can also increase handling time, risk and complexity in planning and controlling production.
The paradigm aim is to achieve single-piece flow where a single discrete unit of product flows from process to process. In effect, the batch quantity is one. If there is no change in part or product model, then this objective needs to be balanced against the additional handling time, and the work-centres that perform the process will typically have to be arranged in close proximity to one another in a flow-line. This is often a characteristic of Repetitive-flow manufacturing and most manual assembly work is performed this way in the modern factory.
If there is a change in part or product model, then the process engineer should also consider to balance the changeover time with run-time. If the changeover time is long, as it might be on a machine, batch size reduction is typically preceded with setup reduction techniques such as Single-Minute Exchange of Die.
One methodology for Repetitive-flow manufacturing is Demand Flow Technology which combines the principles of Repetitive-flow and demand-driven manufacturing. The production planning and control is linked to a pull signal that is triggered from a customer order or consumption of finished goods stock. A pull signal can also link a process to the down-stream, and synchronize the flow to the demand of the customer.


Hirschberg–Sinclair algorithm

The Hirschberg–Sinclair algorithm is a distributed algorithm designed for leader election problem in a synchronous ring network. It is named after its inventors, Dan Hirschberg and J. B. Sinclair.
The algorithm requires the use of unique IDs (UID) for each process. The algorithm works in phases and sends its UID out in both directions. The message goes out a distance of 2Phase Number hops and then the message heads back to the originating process. While the messages are heading "out" each receiving process will compare the incoming UID to its own. If the UID is greater than its own UID then it will continue the message on. Otherwise if the UID is less than its own UID, it will not pass the information on. At the end of a phase, a process can determine if it will send out messages in the next round by if it received both of its incoming messages. Phases continue until a process receives both of its out messages, from both of its neighbors. At this time the process knows it is the largest UID in the ring and declares itself the leader.


Automation Studio

Automation Studio is a circuit design, simulation and project documentation software for fluid power systems and electrical projects conceived by Famic Technologies Inc.. It is used for CAD, maintenance, and training purposes. Mainly used by engineers, trainers, and service and maintenance personnel. Automation Studio can be applied in the design, training and troubleshooting of hydraulics, pneumatics, HMI, and electrical control systems.
Two versions of the software exist:
The educational version of Automation Studio is a limited features version used by engineering and technical schools to train students who are future engineers or technicians. The software is designed for schools that teach technical subjects such as industrial technologies, mechatronics, electromechanical technologies, electrical & electronics, automation, and maintenance. Modeling and simulation are used to illustrate theoretical aspects.
Libraries.
Automation Studio has various symbol libraries. All libraries follow standards such as ISO, IEC, JIC and NEMA.
Libraries features.
Automation Studio is used as a design and simulation tool in the fields of hydraulics, pneumatics, electrical and automation.
Automation Studio Hydraulics.
Automation Studio Hydraulics’ functions are used for hydraulic system engineering purposes. Automation Studio Hydraulics includes a specific symbol library and uses modeling techniques such the Bernoulli's law and the gradient method.
Automation Studio Hydraulics is the main aspect of Automation Studio: it is used to conceive and to test hydraulic systems while taking into account thermal parameters. It displays inside views of the elements in the schematics. The Automation Studio library includes additional elements such as commands and control devices (PID controller, CAN bus, and servo-direction).
Fluid power is one of the central elements in such simulation.
Automation Studio Pneumatics.
Automation Studio Pneumatics is similar to Automation Studio Hydraulics, but the simulation is done for air rather than fluids. This library, like Automation Studio Hydraulics, is used to design and test models.
Thus, the simulation elements that are used are not the same as those in the hydraulics library.
Automation Studio Electrotechnical.
The electrotechnical module in Automation Studio is used for design, simulation, validation, documentation and troubleshooting of electrical diagrams. It includes multi-line and one-line representation according to the users' choice. The different aspects of the IEC and NEMA international standards are respected: components’ identification, symbols, ratings, port names, … etc.
The electrotechnical module works simultaneously with the fluid power technologies which allows the users to design and simulate complete systems.


Conley's fundamental theorem of dynamical systems or Conley's decomposition theorem states that every flow of a dynamical system with compact phase portrait admits a decomposition into a chain-recurrent part and a gradient-like flow part. Due to the concise yet complete description of many dynamical systems, Conley's theorem is also known as the fundamental theorem of dynamical systems. Conley's fundamental theorem has been extended to systems with non-compact phase portraits and also to hybrid dynamical systems.
Complete Lyapunov functions.
Conley's decomposition is characterized by a function known as complete Lyapunov function. Unlike traditional Lyapunov functions that are used to assert the stability of an equilibrium point (or a fixed point) and can be defined only on the basin of attraction of the corresponding attractor, complete Lyapunov functions must be defined on the whole phase-portrait.
In the particular case of an autonomous differential equation defined on a compact set "X", a complete Lyapunov function "V" from "X" to R is a real-valued function on "X" satisfying:
Conley's theorem states that a continuous complete Lyapunov function exists for any differential equation on a compact metric space. Similar result hold for discrete-time dynamical systems.


Control (optimal control theory)

In optimal control theory, a control is a variable chosen by the controller or agent to manipulate state variables, similar to an actual control valve. Unlike the state variable, it does not have a predetermined equation of motion. The goal of optimal control theory is to find some sequence of controls (within an admissible set) to achieve an optimal path for the state variables (with respect to a loss function).
A control given as a function of time only is referred to as an "open-loop control". In contrast, a control that gives optimal solution during some remainder period as a function of the state variable at the beginning of the period is called a "closed-loop control".


Diamond norm

In quantum information, the diamond norm, also known as completely bounded trace norm, is a norm on the space of quantum operations, or more generally on any linear map that acts on complex matrices. Its main application is to measure the "single use distinguishability" of two quantum channels. If an agent is randomly given one of two quantum channels, permitted to pass one state through the unknown channel, and then measures the state in an attempt to determine which operation they were given, then their maximal probability of success is determined by the diamond norm of the difference of the two channels.
Although the diamond norm can be efficiently computed via semidefinite programming, it is in general difficult to obtain analytical expressions and those are known only for a few particular cases.
Definition.
The diamond norm is the trace norm of the output of a trivial extension of a linear map, maximized over all possible inputs with trace norm at most one. More precisely, let formula_1 be a linear transformation, where formula_2 denotes the formula_3 complex matrices, let formula_4 be the identity map on formula_3 matrices, and formula_6. Then the diamond norm of formula_7 is given by
where formula_9 denotes the trace norm.
The diamond norm induces the diamond distance, which in the particular case of completely positive, trace non-increasing maps formula_10 is given by
where the maximization is done over all density matrices formula_12 of dimension formula_13.
Discrimination of quantum channels.
In the task of single-shot discrimination of quantum channels, an agent is given one of the channels formula_10 with probabilities "p" and "1-p", respectively, and attempts to guess which channel they received by preparing a state formula_12, passing it through the unknown channel, and making a measurement on the resulting state. The maximal probability that the agent guesses correctly is given by
Semidefinite programming formulation.
The diamond norm can be efficiently calculated via semidefinite programming. Let formula_17 be a linear map, as before, and formula_18 its Choi state, defined as
The diamond norm of formula_7 is then given by the solution of the following semidefinite programming problem:
where formula_22 are Hermitian matrices and formula_23 is the usual spectral norm.


PhotoModeler

PhotoModeler is a software application that performs image-based modeling and close range photogrammetry – producing 3D models and measurements from photography. The software is used for close-range, aerial and uav photogrammetry.
Close Range Photogrammetry (CRP) can mean photographs taken from the ground with a handheld camera, or taken from a UAV/drone at a relatively low altitude. PhotoModeler and CRP are used for performing measurement and modeling in agriculture, archaeology, architecture, biology, engineering, fabrication, film production, forensics, mining, stockpile volumes, etc.
How Photomodeler works.
1) Take photos from different angles: Ensure that many images are taken to capture the entire object. The amount of required images varies accordingly to the size and complexity of the object being captured.
2) Load images into Photomodeler software: Upload the images by any means on to the computer.
3) Chose the method: Chose one amongst the four following options.
4) Review, measure, and export
Applications.
Some of the applications of PhotoModeler are:
Notes and references.
Notes


Netsh

In computing, codice_1, or network shell, is a command-line utility included in Microsoft's Windows NT line of operating systems beginning with Windows 2000. It allows local or remote configuration of network devices such as the interface.
Overview.
A common use of codice_1 is to reset the TCP/IP stack to default, known-good parameters, a task that in Windows 98 required reinstallation of the TCP/IP adapter.
netsh, among many other things, also allows the user to change the IP address on their machine.
Starting from Windows Vista, one can also edit wireless settings (for example, SSID) using codice_1.
codice_1 can also be used to read information from the IPv6 stack.
The command codice_5 can be used to reset TCP/IP problems when communicating with a networked device.


Reconfigurable manufacturing system

A reconfigurable manufacturing system (RMS) is one designed at the outset for rapid change in its structure, as well as its hardware and software components, in order to quickly adjust its production capacity and functionality within a part family in response to sudden market changes or intrinsic system change.
From 1996 to 2007 Yoram Koren received an NSF grant of $32.5 million to develop the RMS science-base and its software and hardware tools, which were implemented in the automotive, aerospace, and engine factories.
The term reconfigurability in manufacturing was likely coined by Kusiak and Lee.
The RMS, as well as one of its components—the reconfigurable machine tool (RMT)—were invented in 1998 in the Engineering Research Center for Reconfigurable Manufacturing Systems (ERC/RMS) at the University of Michigan College of Engineering. The RMS goal is summarized by the statement: "Exactly the capacity and functionality needed, exactly when needed".
Ideal reconfigurable manufacturing systems possess six core RMS characteristics: modularity, integrability, customized flexibility, scalability, convertibility, and diagnosability. A typical RMS will have several of these characteristics, though not necessarily all. When possessing these characteristics, RMS increases the speed of responsiveness of manufacturing systems to unpredicted events, such as sudden market demand changes or unexpected machine failures.. The RMS facilitates a quick production launch of new products, and allows for adjustment of production quantities that might unexpectedly vary. The ideal reconfigurable system provides exactly the functionality and production capacity needed, and can be economically adjusted exactly when needed. These systems are designed and operated according to Yoram Koren's RMS principles.
The components of RMS are CNC machines, reconfigurable machine tools, reconfigurable inspection machines and material transport systems (such as gantries and conveyors) that connect the machines to form the system. Different arrangements and configurations of these machines will affect the system's productivity. A collection of mathematical tools, which are defined as the RMS science base, may be utilized to maximize system productivity with the smallest possible number of machines.
Rationale for RMS.
Globalization has created a new landscape for industry, one of fierce competition, short windows of market opportunity, and frequent changes in product demand. This change presents both a threat and an opportunity. To capitalize on the opportunity, industry needs to possess manufacturing systems that can produce a wide range of products within a product family. That range must meet the requirements of multiple countries and various cultures, not just one regional market. A design for the right mix of products must be coupled with the technical capabilities that allow for quick changeover of product mix and quantities that might vary dramatically, even on a monthly basis. Reconfigurable manufacturing systems have these capabilities.
RMS System Architecture and Operation.
The system architecture of a typical RMS is shown below.
The system is composed of stages: 10, 20, 30, 40, etc. Each stage consists of identical machines, such as CNC milling machines, or RMT machines. The system produces one product, for example, an automotive engine block or a cylinder head. The manufactured product moves on the horizontal conveyor. Then Gantry-10 grips the product and brings it to one of CNC-10. When CNC-10 finishes the processing, Gantry-10 moves it back to the conveyor. The conveyor moves the product to Gantry-20, which grips the product and load it on the RMT-20, and so on. Inspection machines are placed at several stages, and at the end of the manufacturing system.
RMS is defined as a “system designed at the outset for rapid changes in its structure.” In practice this feature is implemented by designing an open space with an access to the gantry at each stage. These spaces enable matching rapidly higher market demand by adding machines in these spaces, which increases production rate to match the demand.
The product may move during its production in many production paths. Three paths are shown in the figure. Although the CNC machines at each stage are identical, in practice there are small variations in the precision of identical machines, which create accumulated errors in the manufactured product. The magnitude of the error depends on the path in which the product moved; each path has its own “stream-of-variations” (a term coined by Y. Koren).
RMS characteristics.
Ideal reconfigurable manufacturing systems possess six core characteristics: modularity, integrability, customized flexibility, scalability, convertibility, and diagnosability. These characteristics, which were introduced by professor Yoram Koren in 1995, apply to the design of whole manufacturing systems, as well as to some of its components: reconfigurable machines, their controllers, and system control software.
Modularity refers to the modules that reconfigurable manufacturing systems consist of"." At the system level the machines are modules. At the machine level the axes of motion are modules (see the RMT Figure). The system control may be composed of control modules. Modules are easier to maintain and update.
Integrability is the ability to rapidly integrate modules by mechanical, informational, and control interfaces that enable module integration and communication.  At the system level the machines are the modules that are integrated via material transport systems (such as conveyors and gantries) to form a reconfigurable manufacturing system.
Customization allows the design of system flexibility just around a product family, obtaining thereby customized-flexibility, as opposed to the general flexibility of FMS. Customization allows a reduction in the investment cost without sacrificing performance.
Convertibility is the ability to easily transform the functionality of existing systems, machines, or controls to suit new production requirements. Examples included changing a machine in the system to another type of machine to respond to  a new required functionality, or  switching spindles on a milling machine (e.g., from low-torque high-speed spindle for aluminum to high-torque low-speed spindle for titanium).
Scalability is the ability to easily change production capacity by adding (or reducing) manufacturing resources. Scalability of a manufacturing system is increased by adding machines to expand the system production rate to match a sudden market growth. Adding machines requires extending the reach of the station gantries.
Diagnosability is the ability to automatically detect and diagnose the source of the manufactured product quality or precision defects. This automatic diagnosis  allows rapid correction of the defects. The RMS must be designed with product inspection machines embedded at optimal locations in the system.
RMS principles.
Reconfigurable manufacturing systems operate according to a set of basic principles formulated by professor Yoram Koren and are called Koren's RMS principles. The more of these principles applicable to a given manufacturing system, the more reconfigurable is that system. The RMS principles are:
RMS and FMS.
Reconfigurable manufacturing systems (RMS) and flexible manufacturing systems (FMS) have different goals. FMS aims at increasing the variety of parts produced. RMS aims at increasing the speed of responsiveness to market changes and customer's demand. RMS is also flexible, but only to a limited extent—its flexibility is confined to only that necessary to produce a part family. This is the "customized flexibility" or the customization characteristic, which is not the general flexibility that FMS offers. The customized flexibility enables higher production rates. Other important advantages of RMS are rapid scalability to the desired volume, and convertibility, which are obtained within reasonable cost to manufacturers. The best application of FMS is found in production of small sets of products [see Wikipedia].
RMS science base.
The RMS technology is based on a systematic approach to the design and operation of reconfigurable manufacturing systems. The approach consists of key elements, the compilation of which is called the RMS science base. These elements are summarized below.


Connectionist expert system

Connectionist expert systems are artificial neural network (ANN) based expert systems where the ANN generates inferencing rules e.g., fuzzy-multi layer perceptron where linguistic and natural form of inputs are used. Apart from that, rough set theory may be used for encoding knowledge in the weights better and also genetic algorithms may be used to optimize the search solutions better. Symbolic reasoning methods may also be incorporated (see hybrid intelligent system). (Also see expert system, neural network, clinical decision support system.)


Zero dynamics

In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems.
History.
The idea was introduced thirty years ago as the nonlinear approach to the concept of transmission of zeros. The original purpose of introducing the concept was to develop an asymptotic stabilization with a set of guaranteed regions of attraction (semi-global stabilizability), to make the overall system stable.
Initial working.
Given the internal dynamics of any system, zero dynamics refers to the control action chosen in which the output variables of the system are kept identically zero. While, various systems have an equally distinctive set of zeros, such as decoupling zeros, invariant zeros, and transmission zeros. Thus, the reason for developing this concept was to control the non-minimum phase and nonlinear systems effectively.
Applications.
The concept is widely utilized in SISO mechanical systems, whereby applying a few heuristic approaches, zeros can be identified for various linear systems. Zero dynamics adds an essential feature to the overall system’s analysis and the design of the controllers. Mainly its behavior plays a significant role in measuring the performance limitations of specific feedback systems. In a Single Input Single Output system, the zero dynamics can be identified by using junction structure patterns. In other words, using concepts like bond graph models can help to point out the potential direction of the SISO systems.
Apart from its application in nonlinear standardized systems, similar controlled results can be obtained by using zero dynamics on nonlinear discrete-time systems. In this scenario, the application of zero dynamics can be an interesting tool to measure the performance of nonlinear digital design systems (nonlinear discrete-time systems).
Before the advent of zero dynamics, the problem of acquiring non-interacting control systems by using internal stability was not specifically discussed. However, with the asymptotic stability present within the zero dynamics of a system, static feedback can be ensured. Such results make zero dynamics an interesting tool to guarantee the internal stability of non-interacting control systems.


Separation principle in stochastic control

The separation principle is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system
with a state process formula_2, an output process formula_3 and a control formula_4, where formula_5 is a vector-valued Wiener process, formula_6 is a zero-mean Gaussian random vector independent of formula_5, formula_8, and formula_9, formula_10, formula_11, formula_12, formula_13 are matrix-valued functions which generally are taken to be continuous of bounded variation. Moreover, formula_14 is nonsingular on some interval formula_15. The problem is to design an output feedback law formula_16 which maps the observed process formula_3 to the control input formula_4 in a nonanticipatory manner so as to minimize the functional
where formula_20 denotes expected value, prime (formula_21) denotes transpose. and formula_22 and formula_23 are continuous matrix functions of bounded variation, formula_24 is positive semi-definite and formula_25 is positive definite for all formula_26. Under suitable conditions, which need to be properly stated, the optimal policy formula_27 can be chosen in the form
where formula_29 is the linear least-squares estimate of the state vector formula_30 obtained from the Kalman filter
where formula_32 is the gain of the optimal linear-quadratic regulator obtained by taking formula_33 and formula_6 deterministic, and where formula_35 is the Kalman gain. There is also a non-Gaussian version of this problem (to be discussed below) where the Wiener process formula_5 is replaced by a more general square-integrable martingale with possible jumps. In this case, the Kalman filter needs to be replaced by a nonlinear filter providing an estimate of the (strict sense) conditional mean
where
is the "filtration" generated by the output process; i.e., the family of increasing sigma fields representing the data as it is produced.
In the early literature on the separation principle it was common to allow as admissible controls formula_4 all processes that are "adapted" to the filtration formula_40 in the class of deterministically well-posed control laws that minimizes formula_41, and it is given by
where formula_32 is the deterministic control gain and formula_44 is given by the linear (distributed) filter
where formula_46 is the innovation process
and the gain formula_2 is as defined in page 120 in Lindquist.


IEC 63110

IEC 63110 is an international standard defining a protocol for the management of electric vehicles charging and discharging infrastructures, which is currently under development. IEC 63110 is one of the International Electrotechnical Commission's group of standards for electric road vehicles and electric industrial trucks, and is the responsibility of Joint Working Group 11 (JWG11) of IEC Technical Committee 69 (TC69).
Standard documents.
IEC 63110 consists of the following parts, detailed in separate IEC 63110 standard documents:


Level luffing crane

A level-luffing crane is a crane mechanism where the hook remains at the same level while luffing: moving the jib up and down, so as to move the hook inwards and outwards relative to the base.
Usually the description is only applied to those with a luffing jib that have some "additional" mechanism applied to keep the hook level when luffing.
Level-luffing is most important when careful movement of a load near ground level is required, such as in construction or shipbuilding. This partially explains the popularity of fixed horizontal jibs in these fields.
Toplis cable luffing.
An early form of level-luffing gear was the "Toplis" design, invented by a Stothert & Pitt engineer in 1914.
The crane jibs luffs as for a conventional crane, with the end of the jib rising and falling. The crane's hook is kept level by automatically paying out enough extra cable to compensate for this. This is also a purely mechanical linkage, arranged by the reeving of the hoist cables to the jib over a number of pulleys at the crane's apex above the cab, so that luffing the jib upwards allows more free cable and lowers the hook to compensate.
Horse-head jibs.
The usual mechanism for level-luffing in modern cranes is to add an additional "horse head" section to the top of the jib. By careful design of the geometry, this keeps level merely by the linked action of the pivots.
Powered level-luffing.
As cranes and their control systems became more sophisticated, it became possible to control the level of luffing directly, by winching the hoist cable in and out as needed. The first of these systems used mechanical clutches between luffing and hoist drums, giving simplicity and a "near level" result.
Later systems have used modern electronic controls and quickly reversible motors with good slow-speed control to the hoist winch motors, so as to give a positioning accuracy of inches. Some early systems used controllable hydraulic gearboxes to achieve the same result, but these added complexity and cost and so were only popular where high accuracy was needed, such as for shipbuilding.
Luffing cabs.
Luffing mechanisms have also been applied to the driver's cab being mounted on its own jib, following the movement of the crane's main jib. These are used for tasks such as ship unloading, where the view from the driver's cab is greatly improved by cantilevering it forwards and over the ship.


Process optimization

Process optimization is the discipline of adjusting a process so as to optimize (make the best or most effective use of) some specified set of parameters without violating some constraint. The most common goals are minimizing cost and maximizing throughput and/or efficiency. This is one of the major quantitative tools in industrial decision making.
When optimizing a process, the goal is to maximize one or more of the process specifications, while keeping all others within their constraints. This can be done by using a process mining tool, discovering the critical activities and bottlenecks, and acting only on them.
Areas.
Fundamentally, there are three parameters that can be adjusted to affect optimal performance. They are:
The first step is to verify that the existing equipment is being used to its fullest advantage by examining operating data to identify equipment bottlenecks.
Operating procedures may vary widely from person-to-person or from shift-to-shift. Automation of the plant can help significantly. But automation will be of no help if the operators take control and run the plant in manual.
In a typical processing plant, such as a chemical plant or oil refinery, there are hundreds or even thousands of control loops. Each control loop is responsible for controlling one part of the process, such as maintaining a temperature, level, or flow.
If the control loop is not properly designed and tuned, the process runs below its optimum. The process will be more expensive to operate, and equipment will wear out prematurely. For each control loop to run optimally, identification of sensor, valve, and tuning problems is important. It has been well documented that over 35% of control loops typically have problems.
The process of continuously monitoring and optimizing the entire plant is sometimes called performance supervision.


Partially observable Markov decision process

A partially observable Markov decision process (POMDP) is a generalization of a Markov decision process (MDP). A POMDP models an agent decision process in which it is assumed that the system dynamics are determined by an MDP, but the agent cannot directly observe the underlying state. Instead, it must maintain a sensor model (the probability distribution of different observations given the underlying state) and the underlying MDP. Unlike the policy function in MDP which maps the underlying states to the actions, POMDP's policy is a mapping from the history of observations (or belief states) to the actions.
The POMDP framework is general enough to model a variety of real-world sequential decision processes. Applications include robot navigation problems, machine maintenance, and planning under uncertainty in general. The general framework of Markov decision processes with imperfect information was described by Karl Johan Åström in 1965 in the case of a discrete state space, and it was further studied in the operations research community where the acronym POMDP was coined. It was later adapted for problems in artificial intelligence and automated planning by Leslie P. Kaelbling and Michael L. Littman.
An exact solution to a POMDP yields the optimal action for each possible belief over the world states. The optimal action maximizes the expected reward (or minimizes the cost) of the agent over a possibly infinite horizon. The sequence of optimal actions is known as the optimal policy of the agent for interacting with its environment.
Definition.
Formal definition.
A discrete-time POMDP models the relationship between an agent and its environment. Formally, a POMDP is a 7-tuple formula_1, where
At each time period, the environment is in some state formula_9. The agent takes an action formula_10, which causes the environment to transition to state formula_11 with probability formula_12. At the same time, the agent receives an observation formula_13 which depends on the new state of the environment, formula_11, and on the just taken action, formula_15, with probability formula_16 (or sometimes formula_17 depending on the sensor model). Finally, the agent receives a reward formula_18 equal to formula_19. Then the process repeats. The goal is for the agent to choose actions at each time step that maximize its expected future discounted reward: formula_20, where formula_21 is the reward earned at time formula_22. The discount factor formula_23 determines how much immediate rewards are favored over more distant rewards. When formula_24 the agent only cares about which action will yield the largest expected immediate reward; when formula_25 the agent cares about maximizing the expected sum of future rewards.
Discussion.
Because the agent does not directly observe the environment's state, the agent must make decisions under uncertainty of the true environment state. However, by interacting with the environment and receiving observations, the agent may update its belief in the true state by updating the probability distribution of the current state. A consequence of this property is that the optimal behavior may often include (information gathering) actions that are taken purely because they improve the agent's estimate of the current state, thereby allowing it to make better decisions in the future.
It is instructive to compare the above definition with the definition of a Markov decision process. An MDP does not include the observation set, because the agent always knows with certainty the environment's current state. Alternatively, an MDP can be reformulated as a POMDP by setting the observation set to be equal to the set of states and defining the observation conditional probabilities to deterministically select the observation that corresponds to the true state.
Belief update.
After having taken the action formula_15 and observing formula_27, an agent needs to update its belief in the state the environment may (or not) be in. Since the state is Markovian (by assumption), maintaining a belief over the states solely requires knowledge of the previous belief state, the action taken, and the current observation. The operation is denoted formula_28. Below we describe how this belief update is computed.
After reaching formula_11, the agent observes formula_13 with probability formula_31. Let formula_32 be a probability distribution over the state space formula_2. formula_34 denotes the probability that the environment is in state formula_35. Given formula_34, then after taking action formula_15 and observing formula_27,
where formula_40 is a normalizing constant with formula_41.
Belief MDP.
A Markovian belief state allows a POMDP to be formulated as a Markov decision process where every belief is a state. The resulting "belief MDP" will thus be defined on a continuous state space (even if the "originating" POMDP has a finite number of states: there are infinite belief states (in formula_42) because there are an infinite number of probability distributions over the states (of formula_2)).
Formally, the belief MDP is defined as a tuple formula_44 where
Of these, formula_47 and formula_18 need to be derived from the original POMDP. formula_47 is
formula_54
where formula_55 is the value derived in the previous section and
formula_56
The belief MDP reward function (formula_18) is the expected reward from the POMDP reward function over the belief state distribution:
formula_58.
The belief MDP is not partially observable anymore, since at any given time the agent knows its belief, and by extension the state of the belief MDP.
Policy and value function.
Unlike the "originating" POMDP (where each action is available from only one state), in the corresponding Belief MDP all belief states allow all actions, since you (almost) always have "some" probability of believing you are in any (originating) state. As such, formula_59 specifies an action formula_60 for any belief formula_32.
Here it is assumed the objective is to maximize the expected total discounted reward over an infinite horizon. When formula_62 defines a cost, the objective becomes the minimization of the expected cost.
The expected reward for policy formula_59 starting from belief formula_64 is defined as
where formula_66 is the discount factor. The optimal policy formula_67 is obtained by optimizing the long-term reward.
where formula_64 is the initial belief.
The optimal policy, denoted by formula_67, yields the highest expected reward value for each belief state, compactly represented by the optimal value function formula_71. This value function is solution to the Bellman optimality equation:
For finite-horizon POMDPs, the optimal value function is piecewise-linear and convex. It can be represented as a finite set of vectors. In the infinite-horizon formulation, a finite vector set can approximate formula_71 arbitrarily closely, whose shape remains convex. Value iteration applies dynamic programming update to gradually improve on the value until convergence to an formula_74-optimal value function, and preserves its piecewise linearity and convexity. By improving the value, the policy is implicitly improved. Another dynamic programming technique called policy iteration explicitly represents and improves the policy instead.
Approximate POMDP solutions.
In practice, POMDPs are often computationally intractable to solve exactly, so computer scientists have developed methods that approximate solutions for POMDPs.
Grid-based algorithms comprise one approximate solution technique. In this approach, the value function is computed for a set of points in the belief space, and interpolation is used to determine the optimal action to take for other belief states that are encountered which are not in the set of grid points. More recent work makes use of sampling techniques, generalization techniques and exploitation of problem structure, and has extended POMDP solving into large domains with millions of states. For example, adaptive grids and point-based methods sample random reachable belief points to constrain the planning to relevant areas in the belief space.
Dimensionality reduction using PCA has also been explored.
Another line of approximate solution techniques for solving POMDPs relies on using (a subset of) the history of previous observations, actions and rewards up to the current time step as a pseudo-state. Usual techniques for solving MDPs based on these pseudo-states can then be used (e.g. Q-learning). Ideally the pseudo-states should contain the most important information from the whole history (to reduce bias) while being as compressed as possible (to reduce overfitting).
POMDP theory.
Planning in POMDP is undecidable in general. However, some settings have been identified to be decidable (see Table 2 in, reproduced below). Different objectives have been considered. Büchi objectives are defined by Büchi automata. Reachability is an example of a Büchi condition (for instance, reaching a good state in which all robots are home). coBüchi objectives correspond to traces that do not satisfy a given Büchi condition (for instance, not reaching a bad state in which some robot died). Parity objectives are defined via parity games; they enable to define complex objectives such that reaching a good state every 10 timesteps. The objective can be satisfied:
We also consider the finite memory case in which the agent is a finite-state machine, and the general case in which the agent has an infinite memory.
Applications.
POMDPs can be used to model many kinds of real-world problems. Notable applications include the use of a POMDP in management of patients with ischemic heart disease, assistive technology for persons with dementia, the conservation of the critically endangered and difficult to detect Sumatran tigers and aircraft collision avoidance.


Maria (reachability analyzer)

Maria: The Modular Reachability Analyzer is a reachability analyzer for concurrent systems that uses Algebraic System Nets (a high-level variant of Petri nets) as its modelling formalism.


Roll forming

Roll forming, also spelled roll-forming or rollforming, is a type of rolling involving the continuous bending of a long strip of sheet metal (typically coiled steel) into a desired cross-section. The strip passes through sets of rolls mounted on consecutive stands, each set performing only an incremental part of the bend, until the desired cross-section (profile) is obtained. Roll forming is ideal for producing constant-profile parts with long lengths and in large quantities.
Overview.
A variety of cross-section profiles can be produced, but each profile requires a carefully crafted set of roll tools. Design of the rolls starts with a "flower pattern", which is the sequence of profile cross-sections, one profile for each stand of rolls. The roll contours are then derived from the flower pattern profiles. Because of the high cost of the roll sets, computer simulation is often used to develop or validate the roll designs and optimize the forming process to minimize the number of stands and material stresses in the final product.
Roll-formed sections may have advantages over extrusions of a similar shapes. Roll formed parts may be much lighter, with thinner walls possible than in the extrusion process, and stronger, having been work hardened in a cold state. Parts can be made having a finish or already painted. In addition, the roll forming process is more rapid and takes less energy than extrusion.
Roll forming machines are available that produce shapes of different sizes and material thicknesses using the same rolls. Variations in size are achieved by making the distances between the rolls variable by manual adjustment or computerized controls, allowing for rapid changeover. These specialized mills are prevalent in the light gauge framing industry where metal studs and tracks of standardized profiles and thicknesses are used. For example, a single mill may be able to produce metal studs of different web (e.g. 3-5/8" to 14 inches), flange (e.g. 1-3/8" to 2-1/2") and lip (e.g. 3/8" to 5/8") dimensions, from different gauges (e.g. 20 to 12 GA) of galvanized steel sheet.
Roll forming lines can be set up with multiple configurations to punch and cut off parts in a continuous operation. For cutting a part to length, the lines can be set up to use a pre-cut die where a single blank runs through the roll mill, or a post-cut die where the profile is cut off after the roll forming process. Features may be added in a hole, notch, embossment, or shear form by punching in a roll forming line. These part features can be done in a pre-punch application (before roll forming starts), in a mid-line punching application (in the middle of a roll forming line/process) or a post punching application (after roll forming is done). Some roll forming lines incorporate only one of the above punch or cut off applications, others incorporate some or all of the applications in one line.
Process.
Roll forming is, among the manufacturing processes, one of the simplest. It typically begins with a large coil of sheet metal, between and in width, and and thick, supported on an uncoiler. The strip is fed through an entry guide to properly align the material as it passes through the rolls of the mill, each set of rolls forming a bend until the material reaches its desired shape. Roll sets are typically mounted one over the other on a pair of horizontal parallel shafts supported by a stand(s). Side rolls and cluster rolls may also be used to provide greater precision and flexibility and to limit stresses on the material. The shaped strips can be cut to length ahead of a roll forming mill, between mills, or at the end of the roll forming line.
Geometric possibilities.
The geometric possibilities can be very broad and even include enclosed shapes as long as the cross-section is uniform. Typical sheet thicknesses range from to , but they can exceed that. Length is almost unaffected by the rolling process. The part widths typically are not smaller than however they can exceed . The primary limitation is profile depth, which is generally limited to less than and rarely larger than due to roll-imparted stresses and surface speed differentials that increase with depth.
Production rates.
The production rate depends greatly on the material thickness and the bend radius; it is however also affected by the number of required stations or steps. For bend radii of 50 times the material thickness of a low carbon steel thick can range from through eight stations to through 12 stations or through 22 stations.
The time for one product to take shape can be represented by a simple function: , where is the length of the piece being formed, is the number of forming stands, is the distance between stands, and is the velocity of the strip through the rolls.
In general, roll forming lines can run from or higher, depending on the application. In some cases the limiting factor is the punching or cut-off applications.
Other considerations.
While dealing with manufacturing, Things to consider are, for example, lubrication, the effect of the process on material properties, cost, and of course safety.
Lubrication provides an essential barrier between the roll dies and the work-piece surface. It helps reducing the tool wear and allows things to move along faster. This table shows the different kinds of lubricants, their application, and the ideal metals to use them on.
The effects of the process on the material's properties are minimal. The physical and chemical properties virtually don't change, but the process may cause work-hardening, micro-cracks, or thinning at bends when discussing the mechanical properties of the material.
The cost of roll forming is relatively low. When calculating the cost of the process things such as setup time, equipment and tool costs, load/unload time, direct labor rate, overhead rate, and the amortization of equipment and tooling must be considered.Safety is also a bit of an issue with this process. The main hazards that need to be taken into consideration are dealing with moving work-pieces (up to ), high pressure rolls, or sharp, sheared metal edges.


Networked control system

A networked control system (NCS) is a control system wherein the control loops are closed through a communication network. The defining feature of an NCS is that control and feedback signals are exchanged among the system's components in the form of information packages through a network.
Overview.
The functionality of a typical NCS is established by the use of four basic elements:
The most important feature of an NCS is that it connects cyberspace to physical space enabling the execution of several tasks from long distance. In addition, NCSs eliminate unnecessary wiring reducing the complexity and the overall cost in designing and implementing the control systems. They can also be easily modified or upgraded by adding sensors, actuators, and controllers to them with relatively low cost and no major change in their structure. Furthermore, featuring efficient sharing of data between their controllers, NCSs are able to easily fuse global information to make intelligent decisions over large physical spaces.
Their potential applications are numerous and cover a wide range of industries, such as space and terrestrial exploration, access in hazardous environments, factory automation, remote diagnostics and troubleshooting, experimental facilities, domestic robots, aircraft, automobiles, manufacturing plant monitoring, nursing homes and tele-operations. While the potential applications of NCSs are numerous, the proven applications are few, and the real opportunity in the area of NCSs is in developing real-world applications that realize the area's potential.
Problems and solutions.
Advent and development of the Internet combined with the advantages provided by NCS attracted the interest of researchers around the globe. Along with the advantages, several challenges also emerged giving rise to many important research topics. New control strategies, kinematics of the actuators in the systems, reliability and security of communications, bandwidth allocation, development of data communication protocols, corresponding fault detection and fault tolerant control strategies, real-time information collection and efficient processing of sensors data are some of the relative topics studied in depth.
The insertion of the communication network in the feedback control loop makes the analysis and design of an NCS complex, since it imposes additional time delays in control loops or possibility of packages loss. Depending on the application, time-delays could impose severe degradation on the system performance.
To alleviate the time-delay effect, Y. Tipsuwan and M-Y. Chow, in ADAC Lab at North Carolina State University, proposed the gain scheduler middleware (GSM) methodology and applied it in iSpace. S. Munir and W.J. Book (Georgia Institute of Technology) used a Smith predictor, a Kalman filter and an energy regulator to perform teleoperation through the Internet.
K.C. Lee, S. Lee and H.H. Lee used a genetic algorithm to design a controller used in a NCS. Many other researchers provided solutions using concepts from several control areas such as robust control, optimal stochastic control, model predictive control, fuzzy logic etc.
A most critical and important issue surrounding the design of distributed NCSs with the successively increasing complexity is to meet the requirements on system reliability and dependability, while guaranteeing a high system performance over a wide operating range. This makes network based fault detection and diagnosis techniques, which are essential to monitor the system performance, receive more and more attention.


Supervisory control theory

The supervisory control theory (SCT), also known as the Ramadge–Wonham framework (RW framework), is a method for automatically synthesizing supervisors that restrict the behavior of a plant such that as much as possible of the given specifications are fulfilled. The plant is assumed to spontaneously generate events. The events are in either one of the following two categories controllable or uncontrollable. The supervisor observes the string of events generated by the plant and might prevent the plant from generating a subset of the controllable events. However, the supervisor has no means of forcing the plant to generate an event.
In its original formulation the SCT considered the plant and the specification to be modeled by formal languages, not necessarily regular languages generated by finite automata as was done in most subsequent work.


Hybrid bond graph

A hybrid bond graph is a graphical description of a physical dynamic system with discontinuities (i.e., a hybrid dynamical system). Similar to
a regular bond graph, it is an energy-based technique. However, it allows instantaneous switching of the junction structure, which may violate the principle of continuity of power (Mosterman and Biswas, 1998).