Bayesian interpretation of kernel regularization

Within bayesian statistics for machine learning, kernel methods arise from the assumption of an inner product space or similarity structure on inputs. For some such methods, such as support vector machines (SVMs), the original formulation and its regularization were not Bayesian in nature. It is helpful to understand them from a Bayesian perspective. Because the kernels are not necessarily positive semidefinite, the underlying structure may not be inner product spaces, but instead more general reproducing kernel Hilbert spaces. In Bayesian probability kernel methods are a key component of Gaussian processes, where the kernel function is known as the covariance function. Kernel methods have traditionally been used in supervised learning problems where the "input space" is usually a "space of vectors" while the "output space" is a "space of scalars". More recently these methods have been extended to problems that deal with multiple outputs such as in multi-task learning.
A mathematical equivalence between the regularization and the Bayesian point of view is easily proved in cases where the reproducing kernel Hilbert space is "finite-dimensional". The infinite-dimensional case raises subtle mathematical issues; we will consider here the finite-dimensional case. We start with a brief review of the main ideas underlying kernel methods for scalar learning, and briefly introduce the concepts of regularization and Gaussian processes. We then show how both points of view arrive at essentially equivalent estimators, and show the connection that ties them together.
The supervised learning problem.
The classical supervised learning problem requires estimating the output for some new input point formula_1 by learning a scalar-valued estimator formula_2 on the basis of a training set formula_3 consisting of formula_4 input-output pairs, formula_5. Given a symmetric and positive bivariate function formula_6 called a "kernel", one of the most popular estimators in machine learning is given by
where formula_7 is the kernel matrix with entries formula_8, formula_9, and formula_10. We will see how this estimator can be derived both from a regularization and a Bayesian perspective.
A regularization perspective.
The main assumption in the regularization perspective is that the set of functions formula_11 is assumed to belong to a reproducing kernel Hilbert space formula_12.
Reproducing kernel Hilbert space.
A reproducing kernel Hilbert space (RKHS) formula_12 is a Hilbert space of functions defined by a symmetric, positive-definite function formula_14 called the "reproducing kernel" such that the function formula_15 belongs to formula_12 for all formula_17. There are three main properties make an RKHS appealing:
1. The "reproducing property", which gives name to the space,
where formula_19 is the inner product in formula_12.
2. Functions in an RKHS are in the closure of the linear combination of the kernel at given points,
This allows the construction in a unified framework of both linear and generalized linear models.
3. The squared norm in an RKHS can be written as
and could be viewed as measuring the "complexity" of the function.
The regularized functional.
The estimator is derived as the minimizer of the regularized functional
where formula_23 and formula_24 is the norm in formula_12. The first term in this functional, which measures the average of the squares of the errors between the formula_26 and the formula_27, is called the "empirical risk" and represents the cost we pay by predicting formula_26 for the true value formula_27. The second term in the functional is the squared norm in a RKHS multiplied by a weight formula_30 and serves the purpose of stabilizing the problem as well as of adding a trade-off between fitting and complexity of the estimator. The weight formula_30, called the "regularizer", determines the degree to which instability and complexity of the estimator should be penalized (higher penalty for increasing value of formula_30).
Derivation of the estimator.
The explicit form of the estimator in equation  is derived in two steps. First, the representer theorem states that the minimizer of the functional  can always be written as a linear combination of the kernels centered at the training-set points,
for some formula_33. The explicit form of the coefficients formula_34 can be found by substituting for formula_35 in the functional . For a function of the form in equation , we have that
We can rewrite the functional  as
This functional is convex in formula_38 and therefore we can find its minimum by setting the gradient with respect to formula_38 to zero,
Substituting this expression for the coefficients in equation , we obtain the estimator stated previously in equation ,
A Bayesian perspective.
The notion of a kernel plays a crucial role in Bayesian probability as the covariance function of a stochastic process called the "Gaussian process".
A review of Bayesian probability.
As part of the Bayesian framework, the Gaussian process specifies the "prior distribution" that describes the prior beliefs about the properties of the function being modeled. These beliefs are updated after taking into account observational data by means of a "likelihood function" that relates the prior beliefs to the observations. Taken together, the prior and likelihood lead to an updated distribution called the "posterior distribution" that is customarily used for predicting test cases.
The Gaussian process.
A Gaussian process (GP) is a stochastic process in which any finite number of random variables that are sampled follow a joint Normal distribution. The mean vector and covariance matrix of the Gaussian distribution completely specify the GP. GPs are usually used as a priori distribution for functions, and as such the mean vector and covariance matrix can be viewed as functions, where the covariance function is also called the "kernel" of the GP. Let a function formula_42 follow a Gaussian process with mean function formula_43 and kernel function formula_44,
In terms of the underlying Gaussian distribution, we have that for any finite set formula_46 if we let formula_47 then
where formula_49 is the mean vector and formula_50 is the covariance matrix of the multivariate Gaussian distribution.
Derivation of the estimator.
In a regression context, the likelihood function is usually assumed to be a Gaussian distribution and the observations to be independent and identically distributed (iid),
This assumption corresponds to the observations being corrupted with zero-mean Gaussian noise with variance formula_52. The iid assumption makes it possible to factorize the likelihood function over the data points given the set of inputs formula_53 and the variance of the noise formula_52, and thus the posterior distribution can be computed analytically. For a test input vector formula_1, given the training data formula_56, the posterior distribution is given by
where formula_58 denotes the set of parameters which include the variance of the noise formula_52 and any parameters from the covariance function formula_44 and where
The connection between regularization and Bayes.
A connection between regularization theory and Bayesian theory can only be achieved in the case of "finite dimensional RKHS". Under this assumption, regularization theory and Bayesian theory are connected through Gaussian process prediction.
In the finite dimensional case, every RKHS can be described in terms of a feature map formula_62 such that
Functions in the RKHS with kernel formula_64 can be then be written as
and we also have that
We can now build a Gaussian process by assuming formula_67 to be distributed according to a multivariate Gaussian distribution with zero mean and identity covariance matrix,
If we assume a Gaussian likelihood we have
where formula_70. The resulting posterior distribution is the given by
We can see that a "maximum posterior (MAP)" estimate is equivalent to the minimization problem defining Tikhonov regularization, where in the Bayesian case the regularization parameter is related to the noise variance.
From a philosophical perspective, the loss function in a regularization setting plays a different role than the likelihood function in the Bayesian setting. Whereas the loss function measures the error that is incurred when predicting formula_72 in place of formula_73, the likelihood function measures how likely the observations are from the model that was assumed to be true in the generative process. From a mathematical perspective, however, the formulations of the regularization and Bayesian frameworks make the loss function and the likelihood function to have the same mathematical role of promoting the inference of functions formula_42 that approximate the labels formula_73 as much as possible.


Representer theorem

For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer formula_1 of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.
Formal statement.
The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola:
Theorem: Consider a positive-definite real-valued kernel formula_2 on a non-empty set formula_3 with a corresponding reproducing kernel Hilbert space formula_4. Let there be given
which together define the following regularized empirical risk functional on formula_4:
Then, any minimizer of the empirical risk
admits a representation of the form:
where formula_12 for all formula_13.
Proof:
Define a mapping
(so that formula_15 is itself a map formula_16). Since formula_17 is a reproducing kernel, then
where formula_19 is the inner product on formula_4.
Given any formula_21, one can use orthogonal projection to decompose any formula_22 into a sum of two functions, one lying in formula_23, and the other lying in the orthogonal complement:
where formula_25 for all formula_26.
The above orthogonal decomposition and the reproducing property together show that applying formula_27 to any training point formula_28 produces
which we observe is independent of formula_30. Consequently, the value of the error function formula_31 in (*) is likewise independent of formula_30. For the second term (the regularization term), since formula_30 is orthogonal to formula_34 and formula_35 is strictly monotonic, we have
Therefore setting formula_37 does not affect the first term of (*), while it strictly decreases the second term. Consequently, any minimizer formula_1 in (*) must have formula_37, i.e., it must be of the form
which is the desired result.
Generalizations.
The Theorem stated above is a particular example of a family of results that are collectively referred to as "representer theorems"; here we describe several such.
The first statement of a representer theorem was due to Kimeldorf and Wahba for the special case in which
for formula_42. Schölkopf, Herbrich, and Smola generalized this result by relaxing the assumption of the squared-loss cost and allowing the regularizer to be any strictly monotonically increasing function formula_43 of the Hilbert space norm.
It is possible to generalize further by augmenting the regularized empirical risk functional through the addition of unpenalized offset terms. For example, Schölkopf, Herbrich, and Smola also consider the minimization
i.e., we consider functions of the form formula_45, where formula_22 and formula_47 is an unpenalized function lying in the span of a finite set of real-valued functions formula_48. Under the assumption that the formula_49 matrix formula_50 has rank formula_51, they show that the minimizer formula_52 in formula_53
admits a representation of the form
where formula_55 and the formula_56 are all uniquely determined.
The conditions under which a representer theorem exists were investigated by Argyriou, Micchelli, and Pontil, who proved the following:
Theorem: Let formula_3 be a nonempty set, formula_17 a positive-definite real-valued kernel on formula_59 with corresponding reproducing kernel Hilbert space formula_4, and let formula_61 be a differentiable regularization function. Then given a training sample formula_62 and an arbitrary error function formula_63, a minimizer
of the regularized empirical risk admits a representation of the form
where formula_12 for all formula_13, if and only if there exists a nondecreasing function formula_68 for which
Effectively, this result provides a necessary and sufficient condition on a differentiable regularizer formula_70 under which the corresponding regularized empirical risk minimization formula_71 will have a representer theorem. In particular, this shows that a broad class of regularized risk minimizations (much broader than those originally considered by Kimeldorf and Wahba) have representer theorems.
Applications.
Representer theorems are useful from a practical standpoint because they dramatically simplify the regularized empirical risk minimization problem formula_71. In most interesting applications, the search domain formula_4 for the minimization will be an infinite-dimensional subspace of formula_74, and therefore the search (as written) does not admit implementation on finite-memory and finite-precision computers. In contrast, the representation of formula_75 afforded by a representer theorem reduces the original (infinite-dimensional) minimization problem to a search for the optimal formula_76-dimensional vector of coefficients formula_77; formula_78 can then be obtained by applying any standard function minimization algorithm. Consequently, representer theorems provide the theoretical basis for the reduction of the general machine learning problem to algorithms that can actually be implemented on computers in practice.
The following provides an example of how to solve for the minimizer whose existence is guaranteed by the representer theorem. This method works for any positive definite kernel formula_79, and allows us to transform a complicated (possibly infinite dimensional) optimization problem into a simple linear system that can be solved numerically.
Assume that we are using a least squares error function
formula_80
and a regularization function formula_81
for some formula_42. By the representer theorem, the minimizer
formula_83
has the form
formula_84
for some formula_85. Noting that
formula_86
we see that formula_87 has the form
formula_88
where formula_89 and formula_90. This can be factored out and simplified to
formula_91
Since formula_92 is positive definite, there is indeed a single global minimum for this expression. Let formula_93 and note that formula_94 is convex. Then formula_87, the global minimum, can be solved by setting formula_96. Recalling that all positive definite matrices are invertible, we see that
formula_97
so the minimizer may be found via a linear solve.


Costate equation

The costate equation is related to the state equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order differential equations
where the right-hand side is the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables.
Interpretation.
The costate variables formula_2 can be interpreted as Lagrange multipliers associated with the state equations. The state equations represent constraints of the minimization problem, and the costate variables represent the marginal cost of violating those constraints; in economic terms the costate variables are the shadow prices.
Solution.
The state equation is subject to an initial condition and is solved forwards in time. The costate equation must satisfy a transversality condition and is solved backwards in time, from the final time towards the beginning. For more details see Pontryagin's maximum principle.


Laser-hybrid welding

Laser-hybrid welding is a type of welding process that combines the principles of laser beam welding and arc welding.
The combination of laser light and an electrical arc into an amalgamated welding process has existed since the 1970s, but has only recently been used in industrial applications. There are three main types of hybrid welding process, depending on the arc used: TIG, plasma arc or MIG augmented laser welding. While TIG-augmented laser welding was the first to be researched, MIG is the first to go into industry and is commonly known as hybrid laser welding.
Whereas in the early days laser sources still had to prove their suitability for industrial use, today they are standard equipment in many manufacturing enterprises.
The combination of laser welding with another weld process is called a "hybrid welding process". This means that a laser beam and an electrical arc act simultaneously in one welding zone, influencing and supporting each other.
Laser.
Laser welding not only requires high laser power but also a high quality beam to obtain the desired "deep-weld effect". The resulting higher quality of beam can be exploited either to obtain a smaller focus diameter or a larger focal distance. A variety of laser types are used for this process, in particular where the laser light can be transmitted via a water-cooled glass fiber. The beam is projected onto the workpiece by collimating and focusing optics. Carbon dioxide laser can also be used where the beam is transmitted via lens or mirrors.
Laser-hybrid process.
For welding metallic objects, the laser beam is focused to obtain intensities of more than 1 MW/cm2. When the laser beam hits the surface of the material, this spot is heated up to vaporization temperature, and a vapor cavity is formed in the weld metal due to the escaping metal vapor. This is known as a keyhole. The extraordinary feature of the weld seam is its high depth-to-width ratio. The energy-flow density of the freely burning arc is slightly more than 100 kW/cm2. Unlike a dual process where two separate weld processes act in succession, hybrid welding may be viewed as a combination of both weld processes acting simultaneously in one and the same process zone. Depending on the kind of arc or laser process used, and depending on the process parameters, the two systems will influence each other in different ways.
The combination of the laser process and the arc process results in an increase in both weld penetration depth and welding speed (as compared to each process alone). The metal vapor escaping from the vapor cavity acts upon the arc plasma. Absorption of the laser radiation in the processing plasma remains negligible. Depending on the ratio of the two power inputs, the character of the overall process may be mainly determined either by the laser or by the arc.
Absorption of the laser radiation is substantially influenced by the temperature of the workpiece surface. Before the laser welding process can start, the initial reflectance must be overcome, especially on aluminum surfaces. This can be achieved by preheating the material. In the hybrid process, the arc heats the metal, helping the laser beam to couple in. After the vaporisation temperature has been reached, the vapor cavity is formed, and nearly all radiation energy can be put into the workpiece. The energy required for this is thus determined by the temperature-dependent absorption and by the amount of energy lost by conduction into the rest of the workpiece. In laser-hybrid welding, using MIG, vaporisation takes place not only from the surface of the workpiece but also from the filler wire, so that more metal vapor is available to facilitate the absorption of the laser radiation.
Fatigue behavior.
Over the years a great deal of research has been done to understand fatigue behavior, particularly for new techniques like laser-hybrid welding, but knowledge is still limited. Laser-hybrid welding is an advanced welding technology that creates narrow deep welds and offers greater freedom to control the weld surface geometry. Therefore, fatigue analysis and life prediction of hybrid weld joints has become more important and is the subject of ongoing research.


Minimal realization

In control theory, given any transfer function, any state-space model that is both controllable and observable and has the same input-output behaviour as the transfer function is said to be a minimal realization of the transfer function. The realization is called "minimal" because it describes the system with the minimum number of states.
The minimum number of state variables required to describe a system equals the order of the differential equation; more state variables than the minimum can be defined. For example, a second order system can be defined by two or more state variables, with two being the minimal realization.
Gilbert's realization.
Given a matrix transfer function, it is possible to directly construct a minimal state-space realization by using Gilbert's method (also known as Gilbert's realization).


Step detection

In statistics and signal processing, step detection (also known as step smoothing, step filtering, shift detection, jump detection or edge detection) is the process of finding abrupt changes (steps, jumps, shifts) in the mean level of a time series or signal. It is usually considered as a special case of the statistical method known as change detection or change point detection. Often, the step is small and the time series is corrupted by some kind of noise, and this makes the problem challenging because the step may be hidden by the noise. Therefore, statistical and/or signal processing algorithms are often required.
The step detection problem occurs in multiple scientific and engineering contexts, for example in statistical process control (the control chart being the most directly related method), in exploration geophysics (where the problem is to segment a well-log recording into stratigraphic zones), in genetics (the problem of separating microarray data into similar copy-number regimes), and in biophysics (detecting state transitions in a molecular machine as recorded in time-position traces). For 2D signals, the related problem of edge detection has been studied intensively for image processing.
Algorithms.
When the step detection must be performed as and when the data arrives, then online algorithms are usually used,
and it becomes a special case of sequential analysis.
Such algorithms include the classical CUSUM method applied to changes in mean.
By contrast, "offline" algorithms are applied to the data potentially long after it has been received. Most offline algorithms for step detection in digital data can be categorised as "top-down", "bottom-up", "sliding window", or "global" methods.
Top-down.
These algorithms start with the assumption that there are no steps and introduce possible candidate steps one at a time, testing each candidate to find the one that minimizes some criteria (such as the least-squares fit of the estimated, underlying piecewise constant signal). An example is the "stepwise jump placement" algorithm, first studied in geophysical problems, that has found recent uses in modern biophysics.
Bottom-up.
Bottom-up algorithms take the "opposite" approach to top-down methods, first assuming that there is a step in between every sample in the digital signal, and then successively merging steps based on some criteria tested for every candidate merge.
Sliding window.
By considering a small "window" of the signal, these algorithms look for evidence of a step occurring within the window. The window "slides" across the time series, one time step at a time. The evidence for a step is tested by statistical procedures, for example, by use of the two-sample Student's t-test. Alternatively, a nonlinear filter such as the median filter is applied to the signal. Filters such as these attempt to remove the noise whilst preserving the abrupt steps.
Global.
Global algorithms consider the entire signal in one go, and attempt to find the steps in the signal by some kind of optimization procedure. Algorithms include wavelet methods, and total variation denoising which uses methods from convex optimization. Where the steps can be modelled as a Markov chain, then Hidden Markov Models are also often used (a popular approach in the biophysics community). When there are only a few unique values of the mean, then k-means clustering can also be used.
Linear versus nonlinear signal processing methods for step detection.
Because steps and (independent) noise have theoretically infinite bandwidth and so overlap in the Fourier basis, signal processing approaches to step detection generally do not use classical smoothing techniques such as the low pass filter. Instead, most algorithms are explicitly nonlinear or time-varying.
Step detection and piecewise constant signals.
Because the aim of step detection is to find a series of instantaneous jumps in the mean of a signal, the wanted, underlying, mean signal is piecewise constant. For this reason, step detection can be profitably viewed as the problem of recovering a piecewise constant signal corrupted by noise. There are two complementary models for piecewise constant signals: as 0-degree splines with a few knots, or as level sets with a few unique levels. Many algorithms for step detection are therefore best understood as either 0-degree spline fitting, or level set recovery, methods.
Step detection as level set recovery.
When there are only a few unique values of the mean, clustering techniques such as k-means clustering or mean-shift are appropriate. These techniques are best understood as methods for finding a level set description of the underlying piecewise constant signal.
Step detection as 0-degree spline fitting.
Many algorithms explicitly fit 0-degree splines to the noisy signal in order to detect steps (including stepwise jump placement methods), but there are other popular algorithms that can also be seen to be spline fitting methods after some transformation, for example total variation denoising.
Generalized step detection by piecewise constant denoising.
All the algorithms mentioned above have certain advantages and disadvantages in particular circumstances, yet, a surprisingly large number of these step detection algorithms are special cases of a more general algorithm. This algorithm involves the minimization of a global functional:
Here, "x""i" for "i" = 1, ..., "N" is the discrete-time input signal of length "N", and "m""i" is the signal output from the algorithm. The goal is to minimize "H"["m"] with respect to the output signal "m". The form of the function formula_1 determines the particular algorithm. For example, choosing:
where "I"("S") = 0 if the condition "S" is false, and one otherwise, obtains the total variation denoising algorithm with regularization parameter formula_3. Similarly:
leads to the mean shift algorithm, when using an adaptive step size Euler integrator initialized with the input signal "x". Here "W" > 0 is a parameter that determines the support of the mean shift kernel. Another example is:
leading to the bilateral filter, where formula_6 is the tonal kernel parameter, and "W" is the spatial kernel support. Yet another special case is:
specifying a group of algorithms that attempt to greedily fit 0-degree splines to the signal. Here, formula_8 is defined as zero if "x" = 0, and one otherwise.
Many of the functionals in equation  defined by the particular choice of formula_1 are convex: they can be minimized using methods from convex optimization. Still others are non-convex but a range of algorithms for minimizing these functionals have been devised.
Step detection using the Potts model.
A classical variational method for step detection is the Potts model. It is given by the non-convex optimization problem
The term formula_11 penalizes the number of jumps and the term formula_12 measures fidelity to data "x". The parameter γ > 0 controls the tradeoff between regularity and data fidelity. Since the minimizer formula_13 is piecewise constant the steps are given by the non-zero locations of the gradient formula_14.
For formula_15 and formula_16 there are fast algorithms which give an exact solution of the Potts problem in formula_17.


GloMoSim

Global Mobile Information System Simulator (GloMoSim) is a network protocol simulation software that simulates wireless and wired network systems.
GloMoSim is designed using the parallel discrete event simulation capability provided by "Parsec", a parallel programming language. GloMoSim currently supports protocols for a purely wireless network.
It uses the Parsec compiler to compile the simulation protocols.
Parsec.
Parsec is a C-based simulation language, developed by the Parallel Computing Laboratory at UCLA, for sequential and parallel execution of discrete-event simulation models.
Development.
GloMoSim is no longer under active development


Integral sliding mode

In 1996, V. Utkin and J. Shi proposed an improved sliding control method named integral sliding mode control (ISMC). In contrast with conventional sliding mode control, the system motion under integral sliding mode has a dimension equal to that of the state space. In ISMC, the system trajectory always starts from the sliding surface.
Accordingly, the reaching phase is eliminated, and robustness in the whole state space is promised.
Control scheme.
For a system formula_1, formula_2 bounded uncertainty.
Mathews and DeCarlo [1] suggested to select an integral sliding surface as
 formula_3
In this case there exists a unit or discontinuous sliding mode controller compensating uncertainty formula_2.
Utkin and Shi [2] have remarked that, if formula_5 is guaranteed, the reaching phase is eliminated.
In the case, when unmatched uncertainties occur formula_6 should be selected as formula_7
where formula_8 is a pseudo inverse matrix [3-5].
References.
1.G.P. Matthews, R.A. DeCarlo, Decentralized tracking for a class of interconnected
nonlinear systems using variable structure control. Automatica 24,
187–193 (1988)
2. V.I. Utkin, J. Shi, Integral sliding mode in systems operating under uncertainty
conditions, in Proceedings of the 35th IEEE-CDC, Kobe, Japan, 1996
3. Y. Shtessel, C. Edwards, L. Fridman, A. Levant. Sliding Mode Control and Observation, Series: Control Engineering, Birkhauser: Basel, 2014, ISBN 978-0-81764-8923.
4. L. Fridman, A. Poznyak, F.J. Bejarano. Robust Ouput LQ Optimal Control via Integral Sliding Modes. Birkhäuser Basel, 2014, ISBN 978-0-8176-4961-6.
5. Rubagotti, M.; Estrada, A.; Castaños, F.; Ferrara, A., L. Fridman. Integral Sliding Mode Control for Nonlinear Systems With Matched and Unmatched Perturbations IEEE Transactions on Automatic Control, 2011, Vol. 56, 11, pp. 2699-2704


Terminal sliding mode

In the early 1990s, a new type of sliding mode control, named terminal sliding modes (TSM) was invented at the Jet Propulsion Laboratory (JPL) by Venkataraman and Gulati. TSM is robust non-linear control approach.
The main idea of terminal sliding mode control evolved out of seminal work on terminal attractors done by Zak in the JPL, and is evoked by the concept of terminal attractors which guarantee finite time convergence of the states. While, in normal sliding mode, asymptotic stability is promised which leads to the convergence of the states to the origin.
But this convergence may only be guaranteed within infinite time. In TSM, a nonlinear term is introduced in the sliding surface design so that the manifold is formulated as an attractor. After the sliding surface is intercepted, the trajectory is attracted within the manifold and converges to the origin following a power rule.
There are some variations of the TSM including: Non-singular TSM, Fast TSM,
Terminal sliding mode also has been widely applied to nonlinear process control, for example, rigid robot control etc.. Several open questions still remain on the mathematical treatment of the system's behavior at the origin since it is non-Lipschitz.
Control Scheme.
Consider a continuous nonlinear system in canonical form
formula_1 ...
formula_2
formula_3
where formula_4 is the state vector, formula_5 is the control
input, formula_6 and formula_7 are nonlinear functions in formula_8.
Then a sequence of terminal sliding surfaces can be designed as follows:
formula_9
formula_10 ...
formula_11 where formula_12 and formula_13 . formula_14 are positive odd numbers and formula_15.
References.
Venkataraman, S., Gulati, S., Control of Nonlinear Systems Using Terminal Sliding Modes
J. Dyn. Sys., Meas., Control, Sept 1993, Volume 115, Issue 3.


Witsenhausen's counterexample

Witsenhausen's counterexample, shown in the figure below, is a deceptively simple toy problem in decentralized stochastic control. It was formulated by Hans Witsenhausen in 1968. It is a counterexample to a natural conjecture that one can generalize a key result of centralized linear–quadratic–Gaussian control systems—that in a system with linear dynamics, Gaussian disturbance, and quadratic cost, affine (linear) control laws are optimal—to decentralized systems. Witsenhausen constructed a two-stage linear quadratic Gaussian system where two decisions are made by decision makers with decentralized information and showed that for this system, there exist nonlinear control laws that outperform all linear laws. The problem of finding the optimal control law remains unsolved.
Statement of the counterexample.
The statement of the counterexample is simple: two controllers attempt to control the system by attempting to bring the state close to zero in exactly two time steps. The first controller observes the initial state formula_1 There is a cost on the input formula_2 of the first controller, and a cost on the state formula_3 after the input of the second controller. The input formula_4 of the second controller is free, but it is based on noisy observations formula_5 of the state formula_6 after the first controller's input. The second controller cannot communicate with the first controller and thus cannot observe either the original state formula_7 or the input formula_2 of the first controller. Thus the system dynamics are
with the second controller's observation equation
The objective is to minimize an expected cost function,
where the expectation is taken over the randomness in the initial state formula_7 and the observation noise formula_14, which are distributed independently. The observation noise formula_14 is assumed to be distributed in a Gaussian manner, while the distribution of the initial state value formula_7 differs depending on the particular version of the problem.
The problem is to find control functions
that give at least as good a value of the objective function as do any other pair of control functions. Witsenhausen showed that the optimal functions formula_18 and formula_19 cannot be linear.
Specific results of Witsenhausen.
Witsenhausen obtained the following results:
The significance of the problem.
The counterexample lies at the intersection of control theory and information theory. Due to its hardness, the problem of finding the optimal control law has also received attention from the theoretical computer science community. The importance of the problem was reflected upon in the 47th IEEE Conference on Decision and Control (CDC) 2008, Cancun, Mexico, where an entire session was dedicated to understanding the counterexample 40 years after it was first formulated.
The problem is of conceptual significance in decentralized control because it shows that it is important for the controllers to communicate with each other implicitly in order to minimize the cost. This suggests that control actions in decentralized control may have a dual role: those of control and communication.
The hardness of the problem.
The hardness of the problem is attributed to the fact that information of the second controller depends on the decisions of the first controller. Variations considered by Tamer Basar show that the hardness is also because of the structure of the performance index and the coupling of different decision variables. It has also been shown that problems of the spirit of Witsenhausen's counterexample become simpler if the transmission delay along an external channel that connects the controllers is smaller than the propagation delay in the problem. However, this result requires the channels to be perfect and instantaneous, and hence is of limited applicability. In practical situations, the channel is always imperfect, and thus one can not assume that decentralized control problems are simple in presence of external channels.
A justification of the failure of attempts that discretize the problem came from the computer science literature: Christos Papadimitriou and John Tsitsiklis showed that the discrete version of the counterexample is NP-complete.
Attempts at obtaining a solution.
A number of numerical attempts have been made to solve the counterexample. Focusing on a particular choice of problem parameters formula_25, researchers have obtained strategies by discretization and using neural networks. Further research (notably, the work of Yu-Chi Ho, and the work of Li, Marden and Shamma) has obtained slightly improved costs for the same parameter choice. The best known numerical results for a variety of parameters, including the one mentioned previously, are obtained by a local search algorithm proposed by S.-H. Tseng and A. Tang in 2017. The first provably approximately optimal strategies appeared in 2010 (Grover, Park, Sahai) where information theory is used to understand the communication in the counterexample. The optimal solution of the counterexample is still an open problem.


Quantitative feedback theory

In control theory, quantitative feedback theory (QFT), developed by Isaac Horowitz (Horowitz, 1963; Horowitz and Sidi, 1972), is a frequency domain technique utilising the Nichols chart (NC) in order to achieve a desired robust design over a specified region of plant uncertainty. Desired time-domain responses are translated into frequency domain tolerances, which lead to bounds (or constraints) on the loop transmission function. The design process is highly transparent, allowing a designer to see what trade-offs are necessary to achieve a desired performance level.
Plant templates.
Usually any system can be represented by its Transfer Function (Laplace in continuous time domain), after getting the model of a system.
As a result of experimental measurement, values of coefficients in the Transfer Function have a range of uncertainty. Therefore, in QFT every parameter of this function is included into an interval of possible values, and the system may be represented by a family of plants rather than by a standalone expression.
formula_1
A frequency analysis is performed for a finite number of representative frequencies and a set of "templates" are obtained in the NC diagram which encloses the behaviour of the open loop system at each frequency.
Frequency bounds.
Usually system performance is described as robustness to instability (phase and gain margins), rejection to input and output noise disturbances and reference tracking. In the QFT design methodology these requirements on the system are represented as frequency constraints, conditions that the compensated system loop (controller and plant) could not break.
With these considerations and the selection of the same set of frequencies used for the templates, the frequency constraints for the behaviour of the system loop are computed and represented on the Nichols Chart (NC) as curves.
To achieve the problem requirements, a set of rules on the Open Loop Transfer Function, for the nominal plant formula_2 may be found. That means the nominal loop is not allowed to have its frequency value below the constraint for the same frequency, and at high frequencies the loop should not cross the "Ultra High Frequency Boundary" (UHFB), which has an oval shape in the center of the NC.
Loop shaping.
The controller design is undertaken on the NC considering the frequency constraints and the "nominal loop" formula_3 of the system. At this point, the designer begins to introduce controller functions (formula_4) and tune their parameters, a process called Loop Shaping, until the best possible controller is reached without violation of the frequency constraints.
The experience of the designer is an important factor in finding a satisfactory controller that not only complies with the frequency restrictions but with the possible realization, complexity, and quality.
For this stage there currently exist different CAD ("Computer Aided Design") packages to make the controller tuning easier.
Prefilter design.
Finally, the QFT design may be completed with a pre-filter (formula_5) design when it is required. In the case of tracking conditions a shaping on the Bode diagram may be used. Post design analysis is then performed to ensure the system response is satisfactory according with the problem requirements.
The QFT design methodology was originally developed for "Single-Input Single-Output" (SISO) and "Linear Time Invariant Systems" (LTI), with the design process being as described above. However, it has since been extended to weakly nonlinear systems, time varying systems, distributed parameter systems, multi-input multi-output (MIMO) systems (Horowitz, 1991), discrete systems (these using the Z-Transform as transfer function), and non minimum phase systems. The development of CAD tools has been an important, more recent development, which simplifies and automates much of the design procedure (Borghesani et al., 1994).
Traditionally, the pre-filter is designed by using the Bode-diagram magnitude information. The use of both phase and magnitude information for the design of pre-filter was first discussed in (Boje, 2003) for SISO systems. The method was then developed to MIMO problems in (Alavi et al., 2007).


Grinding dresser

A grinding dresser or wheel dresser is a tool to dress (slightly trim) the surface of a grinding wheel. Grinding dressers are used to return a wheel to its original round shape (to true it up), to expose fresh grains for renewed cutting action (including cleaning away clogged areas), or to make a different profile (cross-sectional shape) on the wheel's edge. Utilizing pre-determined dressing parameters will allow the wheel to be conditioned for optimum grinding performance while truing and restoring the form simultaneously.
Purpose.
The objective of "dressing the wheel" is to:
Types.
Also an abrasive wheel type that has a small "grinding wheel" in a holder that is held against the spinning grinding wheel to dress and clean the face of the grinding wheel.
Grinding complex shapes.
Four types of dressers are used to dress the wheels of CNC grinders used for grinding complex shapes. This type of dresser is mainly in use on CNC grinding machine tools to automatically dress the grinding wheel via computer control in specialist areas requiring complex shapes such as grinding bearing raceways.
Wheel Conditioning
Proper wheel conditioning is a critical part of any grinding process. The condition of the wheel will determine the wheel's ability to meet the part finish requirements and metal removal rate capabilities. The grains in the wheel can be sheared to create a smooth condition or fractured to create a course open condition utilizing dressing parameters.
Skate sharpening.
Grinders used for sharpening skate blades typically have one or more thin grinding wheels mounted on vertical spindles, with a single-diamond dresser mounted on a gymbal with a horizontal axis level with the centerline of the wheel, so it can swing above and below the plane of the wheel, producing a convex grinding surface of a predetermined radius. This allows the blade to be sharpened to a specified hollow, typically very deep for hockey skates, very shallow for skates used for school figures, and moderate for skates used for freestyle skating.


Rapid casting

Rapid casting is an integration of investment casting with rapid prototyping/3D printing. In this technique disposable patterns that are used for forming molds are created with 3D printing techniques like fused deposition modeling, stereolithography or any other 3D printing technique.


System Information (Windows)

System Information (codice_1) is a system profiler included with Microsoft Windows that displays diagnostic and troubleshooting information related to the operating system, hardware and software. It has been bundled with Windows since Windows NT 4.0.
It compiles technical information on the overall system, hardware resources (including memory, I/O, etc.), physical hardware components (CD-ROM, sound, network, etc.), and the Windows environment as well (drivers, environment variables, services, etc.). It can export this information in the plain text format or in files with a .nfo extension, which can be used to diagnose problems. In addition, System Information can be used to gather technical information on a remote computer on the same network.


Magnetorheological finishing

Magnetorheological finishing (MRF) is a precision surface finishing technology. Optical surfaces are polished in a computer-controlled magnetorheological (MR) finishing slurry. Unlike conventional rigid lap polishing, the MR fluid's shape and stiffness can be magnetically manipulated and controlled in real time. The optic's final surface form and finishing results are predicted through the use of computer algorithms.
Literature.
 W.I. Kordonski (2014). "Magnetorheological Fluid-Based High Precision Finishing Technology." Magnetorheology: Advances and Applications, Norman M. Wereley, Ed., RSC Smart Materials, Cambridge, UK, Chapter 11, 261–277.
DOI:10.1039/9781849737548-00261
 S.D. Jacobs, W.I. Kordonski, I.V. Prokhorov, D. Golini, G.R. Gorodkin, T.D. Strafford (2002). "Deterministic Magnetorheological Finishing." US Patent: US5449313A
 Shorey et al. "Experiments and Observations Regarding the Mechanisms of Glass Removal in Magnetorheological Finishing", abstract and full text (pdf)
 Chunlin Miao, et al., "Shear stress in magnetorheological finishing for glasses," Applied Optics 48, 2585-2594 (2009)
 Chunlin Miao, et al., "Process parameter effects on material removal in magnetorheological finishing of borosilicate glass," Applied Optics 49, 1951-1963 (2010)


Thread control block

Thread Control Block (TCB) is a data structure in the operating system kernel which contains thread-specific information needed to manage it. The TCB is "the manifestation of a thread in an operating system."
An example of information contained within a TCB is:
The Thread Control Block acts as a library of information about the threads in a system. Specific information is stored in the thread control block highlighting important information about each process.


Network Virtualization using Generic Routing Encapsulation (NVGRE) is a network virtualization technology that attempts to alleviate the scalability problems associated with large cloud computing deployments. It uses Generic Routing Encapsulation (GRE) to tunnel layer 2 packets over layer 3 networks.
Its principal backer is Microsoft.


Centerless grinding

Centerless grinding is a machining process that uses abrasive cutting to remove material from a workpiece. Centerless grinding differs from centered grinding operations in that no spindle or fixture is used to locate and secure the workpiece; the workpiece is secured between two rotary grinding wheels, and the speed of their rotation relative to each other determines the rate at which material is removed from the workpiece.
Centerless grinding is typically used in preference to other grinding processes for operations where many parts must be processed in a short time.
Working principle.
In centerless grinding, the workpiece is held between two wheels, rotating in the same direction at different speeds, and a work-holding platform. One wheel, known as the grinding wheel (stationary wheel in the diagram), is on a fixed axis and rotates such that the force applied to the workpiece is directed downward, against the work-holding platform. This wheel usually performs the grinding action by having a higher tangential speed than the workpiece at the point of contact. The other wheel, known as the regulating wheel (moving wheel in the diagram), is movable. This wheel is positioned to apply lateral pressure to the workpiece, and usually has either a very rough or rubber-bonded abrasive to trap the workpiece.
The speed of the two wheels relative to each other provides the grinding action and determines the rate at which material is removed from the workpiece. During operation the workpiece turns with the regulating wheel, with the same linear velocity at the point of contact and (ideally) no slipping. The grinding wheel turns faster, slipping past the surface of the workpiece at the point of contact and removing chips of material as it passes.
Types.
There are three forms of centerless grinding, differentiated primarily by the method used to feed the workpiece through the machine.
Through-feed.
In through-feed centerless grinding, the workpiece is fed through the grinding wheels completely, entering on one side and exiting on the opposite. The regulating wheel in through-feed grinding is canted away from the plane of the grinding wheel in such a way as to provide an axial force component, feeding the workpiece through between the two wheels. Through-feed grinding can be very efficient because it does not require a separate feed mechanism; however, it can only be used for parts with a simple cylindrical shape.
End-feed.
In end-feed centerless grinding, the workpiece is fed axially into the machine on one side and comes to rest against an end stop; the grinding operation is performed, and then the workpiece is fed in the opposite direction to exit the machine. End-feed grinding is best for tapered workpieces.
In-feed.
In-feed centerless grinding is used to grind workpieces with relatively complex shapes, such as an hourglass shape. Before the process begins, the workpiece is loaded manually into the grinding machine and the regulating wheel is moved into place. The complexity of the part shapes and grinding wheel shapes required to grind them accurately prevent the workpiece from being fed axially through the machine.
Equipment.
Centerless grinding uses purpose-built centerless grinding machines. Such a machine will always include the grinding wheel, regulating wheel, and some means of supporting a workpiece. Modern machines may involve computer numerical control to allow automation and improve precision. Grinding wheels are interchangeable, to allow for different grits and shapes. Machines designed to accommodate through-feed grinding operations will allow the angle of the regulating wheel to be adjusted, to accommodate parts of different sizes.


Circle criterion

In nonlinear control and stability theory, the circle criterion is a stability criterion for nonlinear time-varying systems. It can be viewed as a generalization of the Nyquist stability criterion for linear time-invariant (LTI) systems.
Overview.
Consider a linear system subject to non-linear feedback, i.e. a non linear element formula_1 is present in the feedback loop. Assume that the element satisfies a sector condition formula_2, and (to keep things simple) that the open loop system is stable. Then the closed loop system is globally asymptotically stable if the Nyquist locus does not penetrate the circle having as diameter the segment formula_3 located on the "x"-axis.
General description.
Consider the nonlinear system
Suppose that
Then formula_10 such that for any solution of the system the following relation holds:
Condition 3 is also known as the "frequency condition". Condition 1 the "sector condition".


ISO 13399

ISO 13399 (Cutting tool data representation and exchange) is an international technical standard by ISO (the International Organization for Standardization) for the computer-interpretable representation and exchange of industrial product data about cutting tools and toolholders. The objective is to provide a mechanism capable of describing product data regarding cutting tools, independent from any particular system. The nature of this description makes it suitable not only for neutral file exchange (free of proprietary format constraints), but also as a basis for implementing and sharing product databases and archiving, regarding cutting tools.
Typically ISO 13399 can be used to exchange data between computer-aided design (CAD), computer-aided manufacturing (CAM), computer-aided engineering (CAE), tool management software, product data management (PDM/EDM), manufacturing resource planning (MRP) or enterprise resource planning (ERP), and other computer-aided technologies (CAx) and systems.
The usage of the ISO 13399 standard will simplify the exchange of data for cutting tools. Expected results are lower cost for managing the information about tools and a more accurate and efficient usage of manufacturing resources. The ISO 13399 has been developed with contributions from AB Sandvik Coromant, the Royal Institute of Technology in Stockholm, Kennametal Inc, and Ferroday Ltd.
ISO 13399 is developed and maintained by the ISO technical committee TC 29, Small tools, sub-committee WG34. Like other ISO and IEC standards ISO 13399 is copyright by ISO and is not freely available. Other standards developed and maintained by ISO TC29/WG34 are:
Structure.
ISO 13399 is divided into several parts:
ISO 13399 defines a data model for cutting tool information using the EXPRESS modelling language. Application data according to this data model can be exchanged either by a STEP-File, STEP-XML or via shared database access using SDAI.
The dictionary (reference data library) of ISO 13399 currently uses PLIB (ISO 13584, IEC 61360).
See also.
List of ISO standards 12000–13999#ISO_13000_–_ISO_13999


Electrochemical grinding

Electrochemical grinding is a process that removes electrically conductive material by grinding with a negatively charged abrasive grinding wheel, an electrolyte fluid, and a positively charged workpiece. Materials removed from the workpiece stay in the electrolyte fluid. Electrochemical grinding is similar to electrochemical machining but uses a wheel instead of a tool shaped like the contour of the workpiece.
Process.
The electrochemical grinding process combines traditional electrochemical machining and grinding processes to remove material from a workpiece. A grinding wheel is used as a cutting tool as a cathode and the workpiece is an anode. During the process, electrolytic fluid, typically sodium nitrate, is pumped into the space between the workpiece and the grinding wheel. Other electrolytes used include sodium hydroxide, sodium carbonate, and sodium chloride. This electrolytic fluid will cause electrochemical reactions to occur at the workpiece surface which oxidize the surface, thereby removing material. As a consequence of the oxidation which occurs, layers of oxide films will form on the workpiece surface, and these need to be removed by the grinding wheel. A couple schematics of the process are provided below.
Abrasive materials, either diamond or aluminum oxide, are bonded to the grinding wheel, which allows the wheel to remove the oxide layers on the workpiece surface by abrasive action. Appropriate materials used for electrolyte fluid and the grinding wheel abrasives are summarized in the table below.
Most material removal is by the electrochemical reactions which occur at the workpiece surface. Five percent or less of the material removal is carried out by the abrasive action of the grinding wheel. The fact that most material is not removed by abrasive action helps increase the life of the grinding wheel; that is, the tool will take a long time to wear down. The electrolytic fluid serves another useful purpose - it flushes out leftover material in between the grinding wheel and workpiece. The abrasive particles bonded to the grinding wheel will help to electrically insulate the space between the grinding wheel and workpiece. An equation giving the material removal rate for an electrochemical grinding process is provided in and is stated here as:
MRR = GI/ρF
where ρ is the workpiece density, G is the total mass of the workpiece, I is the current supplied, MRR is the material removal rate, and F is Faraday's constant.
Some of the main factors which govern the performance of an electrochemical grinding process include current supplied, rotation speed of the grinding wheel, the workpiece feed rate, the type of electrolyte used, electrolyte feed rate, and the workpiece's chemical properties. By changing these parameters, one can alter the material removal rate. Increasing the supplied current, rotation speed of the wheel, electrolyte feed rate, or the workpiece feed rate will lead to an increase in material removal rate (MRR), while decreasing these properties will do the opposite. If the workpiece is more reactive to the electrolyte used, then the material removal rate will increase. The grinding wheel is usually rotated with a surface speed of 1200–2000 m/min and supplied currents are around 1000A.
The accuracy of parts made by electrochemical grinding is strongly dictated by the chemical properties of the workpiece and electrolytic fluid used. If the workpiece is very reactive to the electrolyte, and if too much electrolyte is pumped into the space between the grinding wheel and workpiece, it may be difficult to control the material removal, which can lead to loss of accuracy. Also, accuracy may be reduced if the workpiece feed rate is too high.
The wheels are metal disks with abrasive particles embedded. Copper, brass, and nickel are the most commonly used materials; aluminum oxide is typically used as an abrasive when grinding steel. A thin layer of diamond particles will be used when grinding carbides or steels harder than 65 Rc.
An electrolytic spindle with carbon brushes, acting as a commutator, holds the wheel. The spindle receives a negative charge from the DC power supply, which gives the workpiece a positive charge. The electrolytic fluid is applied where the work contacts the tool by a nozzle similar to that which supplies coolant in conventional grinding. The fluid works with the wheel to form electrochemical cells that oxidize the surface of the workpiece. As the wheel carries away the oxide, fresh metal is exposed. Removing the oxidized fluid may only require a pressure of 20 psi or less, causing much less distortion than mechanical grinding. The wheel is subject to little wear, reducing the need for truing and dressing.
Applications.
Electrochemical grinding is often used for hard materials where conventional machining is difficult and time-consuming, such as stainless steel and some exotic metals. For materials with hardness greater than 65 HRC, ECG can have a material removal rate 10 times that of conventional machining. Because ECG involves little abrasion, it is often used for processes where the surface of the part is needs to be free of burrs, scratches, and residual stresses. Because of these properties, electrochemical grinding has a number of useful applications.
Advantages and disadvantages.
One of the key advantages of electrochemical grinding is the minimal wear that the grinding wheel tool experiences. This is because the majority of the material is removed by the electrochemical reaction that occurs between the cathode and anode. The only time that abrasive grinding actually occurs is in removing the film that develops on the surface of the workpiece. Another advantage of electrochemical grinding is that it can be used to machine hard materials. Hard materials pose a difficulty to other types of machining due to the tool wear that is associated with machining hard materials. It may come as a bit of a surprise that electrochemical grinding can remove material from a hard surface and experience minimal wear. Because most material is removed through electrochemical reactions, the workpiece does not experience heat damage like it would in a conventional grinding process.
Electrochemical grinding also has a few disadvantages as well. The system consists of the anode workpiece and the cathode grinding wheel. In order to create those conditions both the workpiece and the grinding wheel must be conductive. This limits the types of workpiece materials that are suitable for electrochemical grinding. Another disadvantage of electrochemical grinding is that it is only applicable to surface grinding. It is not possible to apply electrochemical grinding to workpieces that have cavities, due to the grinding wheels inability to remove the film deposit with in the cavity. One other disadvantage is the electrolytic fluid can cause corrosion at the workpiece and grinding wheel surfaces. Lastly, electrochemical grinding is more complicated than traditional machining methods. This will require more experienced personnel to operate the machinery, which will lead to higher production cost. Another disadvantage is that chemical used during grinding process need to be properly disposed of depending on the environmental regulation.


Robust control

In control theory, robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some (typically compact) set. Robust methods aim to achieve robust performance and/or stability in the presence of bounded modelling errors.
The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness, prompting research to improve them. This was the start of the theory of robust control, which took shape in the 1980s and 1990s and is still active today.
In contrast with an adaptive control policy, a robust control policy is static, rather than adapting to measurements of variations, the controller is designed to work assuming that certain variables will be unknown but
bounded.
Criteria for robustness.
Informally, a controller designed for a particular set of parameters is said to be robust if it also works well under a different set of assumptions. High-gain feedback is a simple example of a robust control method; with sufficiently high gain, the effect of any parameter variations will be negligible. From the closed-loop transfer function perspective, high open-loop gain leads to substantial disturbance rejection in the face of system parameter uncertainty. Other examples of robust control include sliding mode and terminal sliding mode control.
The major obstacle to achieving high loop gains is the need to maintain system closed-loop stability. Loop shaping which allows stable closed-loop operation can be a technical challenge.
Robust control systems often incorporate advanced topologies which include multiple feedback loops and feed-forward paths. The control laws may be represented by high order transfer functions required to simultaneously accomplish desired disturbance rejection performance with the robust closed-loop operation.
High-gain feedback is the principle that allows simplified models of operational amplifiers and emitter-degenerated bipolar transistors to be used in a variety of different settings. This idea was already well understood by Bode and Black in 1927.
The modern theory of robust control.
The theory of robust control system began in the late 1970s and early 1980s and soon developed a number of techniques for dealing with bounded system uncertainty.
Probably the most important example of a robust control technique is H-infinity loop-shaping, which was developed by Duncan McFarlane and Keith Glover of Cambridge University; this method minimizes the sensitivity of a system over its frequency spectrum, and this guarantees that the system will not greatly deviate from expected trajectories when disturbances enter the system.
An emerging area of robust control from application point of view is sliding mode control (SMC), which is a variation of variable structure control (VSC). The robustness properties of SMC with respect to matched uncertainty as well as the simplicity in design attracted a variety of applications.
While robust control has been traditionally dealt with along deterministic approaches, in the last two decades this approach has been criticized on the basis that it is too rigid to describe real uncertainty, while it often also leads to over conservative solutions. Probabilistic robust control has been introduced as an alternative, see e.g. that interprets robust control within the so-called scenario optimization theory.
Another example is loop transfer recovery (LQG/LTR), which was developed to overcome the robustness problems of linear-quadratic-Gaussian control (LQG) control.
Other robust techniques includes quantitative feedback theory (QFT), passivity based control, Lyapunov based control, etc.
When system behavior varies considerably in normal operation, multiple control laws may have to be devised. Each distinct control law addresses a specific system behavior mode. An example is a computer hard disk drive. Separate robust control system modes are designed in order to address the rapid magnetic head traversal operation, known as the seek, a transitional settle operation as the magnetic head approaches its destination, and a track following mode during which the disk drive performs its data access operation.
One of the challenges is to design a control system that addresses these diverse system operating modes and enables smooth transition from one mode to the next as quickly as possible.
Such state machine-driven composite control system is an extension of the gain scheduling idea where the entire control strategy changes based upon changes in system behavior.


Die cutting (web)

Die cutting is the general process of using a die to shear webs of low-strength materials, such as rubber, fibre, foil, cloth, paper, corrugated fibreboard, chipboard, paperboard, plastics, pressure-sensitive adhesive tapes, foam, and sheet metal. In the metalworking and leather industries, the process is known as clicking and the machine may be referred to as a "clicking machine". When a "dinking die" or "dinking machine" is used, the process is known as dinking. Commonly produced items using this process include gaskets, labels, tokens, corrugated boxes, and envelopes.
Die cutting started as a process of cutting leather for the shoe industry in the mid-19th century. It is now sophisticated enough to cut through just one layer of a laminate, so it is now used on labels, postage stamps, and other stickers; this type of die cutting is known as "".
Die cutting can be done on either flatbed or rotary presses. Rotary die cutting is often done inline with printing. The primary difference between rotary die cutting and flatbed die cutting is that the flatbed is not as fast but the tools are cheaper. This process lends itself to smaller production runs where it is not as easy to absorb the added cost of a rotary die.
Rotary die cutting.
Rotary die cutting is die cutting using a cylindrical die on a rotary press and may be known as a rotary die cutter or RDC. A long sheet or web of material will be fed through the press into an area known as a "station" which holds a rotary tool that will cut out shapes, make perforations or creases, or even cut the sheet or web into smaller parts. A series of gears will force the die to rotate at the same speed as the rest of the press, ensuring that any cuts the die makes line up with the printing on the material. The machines used for this process can incorporate multiple "stations" that die cut a particular shape in the material. In each of these stations lie one or more of these geared tools or printing cylinders, and some machines use automatic eye registration to make sure the cuts and/or printing are lined up with one another when lower tolerances are required.
Dies used in rotary die cutting are either solid engraved dies, adjustable dies, or magnetic plate tooling. Engraved dies have a much higher tolerance and are machined out of a solid steel bar normally made out of tool steel. Adjustable dies have removable blades that can be easily replaced with other blades, either due to wear or to cut a different material, while magnetic plate tooling has a cylinder that has magnets placed in it, and an engraved metal plate is attached or wrapped around the base cylinder holding onto it by the force of the magnets.
Dinking.
Dinking is a manufacturing process. Dinking uses special dies called dinking dies, which are hollow cutters. The edges of the dies are usually beveled about 20° and sharpened. The material is punched through into a wood or soft metal block in order to not dull the edges. The die may be pressed into the material with a hammer or a mechanical press.


State variable

A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of any external forces affecting the system. Models that consist of coupled first-order differential equations are said to be in state-variable form.
Control systems engineering.
In control engineering and other areas of science and engineering, state variables are used to represent the states of a general system. The set of possible combinations of state variable values is called the state space of the system. The equations relating the current state of a system to its most recent input and past states are called the state equations, and the equations expressing the values of the output variables in terms of the state variables and inputs are called the output equations. As shown below, the state equations and output equations for a linear time invariant system can be expressed using coefficient matrices: "A", "B", "C", and "D"
where "N", "L" and "M" are the dimensions of the vectors describing the state, input and output, respectively.
Discrete-time systems.
The state vector (vector of state variables) representing the current state of a discrete-time system (i.e. digital system) is formula_2, where "n" is the discrete point in time at which the system is being evaluated. The discrete-time state equations are
which describes the next state of the system ("x"["n"+1]) with respect to current state and inputs "u"["n"] of the system. The output equations are
which describes the output "y"["n"] with respect to current states and inputs "u"["n"] to the system.
Continuous time systems.
The state vector representing the current state of a continuous-time system (i.e. analog system) is formula_5, and the continuous-time state equations giving the evolution of the state vector are
which describes the continuous rate of change formula_7 of the state of the system with respect to current state "x"("t") and inputs "u"("t") of the system. The output equations are
which describes the output "y"("t") with respect to current states "x"("t") and inputs "u"("t") to the system.


Caratheodory-π solution

A Carathéodory- solution is a generalized solution to an ordinary differential equation. The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory. Its practicality was demonstrated in 2008 by Ross et al. in a laboratory implementation of the concept. The concept is most useful for implementing feedback controls, particularly those generated by an application of Ross' pseudospectral optimal control theory.
Mathematical background.
A Carathéodory- solution addresses the fundamental problem of defining a solution to a differential equation,
when "g"("x","t") is not differentiable with respect to "x". Such problems arise quite naturally in defining the meaning of a solution to a controlled differential equation,
when the control, "u", is given by a feedback law,
where the function "k"("x","t") may be non-smooth with respect to "x". Non-smooth feedback controls arise quite often in the study of optimal feedback controls and have been the subject of extensive study going back to the 1960s.
Ross' concept.
An ordinary differential equation,
is equivalent to a controlled differential equation,
with feedback control,
formula_6. Then, given an initial value problem, Ross partitions the time interval formula_7 to a grid, formula_8 with formula_9. From formula_10 to formula_11, generate a control trajectory,
to the controlled differential equation,
A Carathéodory solution exists for the above equation because formula_14 has discontinuities at most in "t", the independent variable. At formula_15, set formula_16 and restart
the system with formula_17,
Continuing in this manner, the Carathéodory segments are stitched together to form a Carathéodory- solution.
Engineering applications.
A Carathéodory- solution can be applied towards the practical stabilization of a control system. It has been used to stabilize an inverted pendulum, control and optimize the motion of robots, slew and control the NPSAT1 spacecraft and produce guidance commands for low-thrust space missions.


Double integrator

In systems and control theory, the double integrator is a canonical example of a second-order control system. It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input formula_1.
Differential equations.
The differential equations which represent a double integrator are:
where both formula_4
Let us now represent this in state space form with the vector formula_5
In this representation, it is clear that the control input formula_1 is the second derivative of the output formula_8. In the scalar form, the control input is the second derivative of the output formula_9.
State space representation.
The normalized state space model of a double integrator takes the form
According to this model, the input formula_1 is the second derivative of the output formula_13, hence the name double integrator.
Transfer function representation.
Taking the Laplace transform of the state space input-output equation, we see that the transfer function of the double integrator is given by
Using the differential equations dependent on formula_15 and formula_16, and the state space representation:


Viability theory

Viability theory is an area of mathematics that studies the evolution of dynamical systems under constraints on the system state. It was developed to formalize problems arising in the study of various natural and social phenomena, and has close ties to the theories of optimal control and set-valued analysis.
Motivation.
Many systems, organizations, and networks arising in biology and the social sciences do not evolve in a deterministic way, nor even in a stochastic way. Rather they evolve with a Darwinian flavor, driven by random fluctuations but yet constrained to remain "viable" by their environment. Viability theory started in 1976 by translating mathematically the title of the book Chance and Necessity by Jacques Monod to the differential inclusion formula_1 for chance and
formula_2 for necessity. The differential inclusion is a type of “evolutionary engine” (called an evolutionary system associating with any initial state x a subset of evolutions starting at x. The system is said to be deterministic if this set is made of one and only one evolution and contingent otherwise.
Necessity is the requirement that at each instant, the evolution is "viable" (remains) in the "environment" K described by "viability constraints", a word encompassing polysemous concepts as "stability, confinement, homeostasis, adaptation", etc., expressing the idea that some variables must obey some constraints (representing physical, social, biological and economic constraints, etc.) that can never be violated. So, viability theory starts as the confrontation of evolutionary systems governing evolutions and viability constraints that such evolutions must obey. They share common features:
Viability theory thus designs and develops mathematical and algorithmic methods for investigating the "adaptation to viability constraints" of evolutions governed by complex systems under uncertainty that are found in many domains involving living beings, from biological evolution to economics, from environmental sciences to financial markets, from control theory and robotics to cognitive sciences. It needed to forge a differential calculus of set-valued maps (set-valued analysis), differential inclusions and differential calculus in metric spaces (mutational analysis).
Viability kernel.
The basic problem of viability theory is to find the "viability kernel" of an environment, the subset of initial states in the environment such that there exists at least one evolution "viable" in the environment, in the sense that at each time, the state of the evolution remains confined to the environment. The second question is then to provide the regulation map selecting such viable evolutions starting from the viability kernel. The viability kernel may be equal to the environment, in which case the environment is called viable under the evolutionary system, and the empty set, in which case it is called a repellor, because all evolutions eventually violate the constraints.
The viability kernel assumes that some kind of "decision maker" controls or regulates evolutions of the system. If not, the next problem looks at the "tychastic kernel" (from tyche, meaning chance in Greek) or "invariance kernel", the subset of initial states in the environment such that all evolutions are "viable" in the environment, an alternative way to stochastic differential equations encapsulating the concept of "insurance" against uncertainty, providing a way of eradicating it instead of evaluating it.


Impulse vector

An impulse vector is a mathematical tool to graphically design and analyze input shapers that could suppress residual vibration. The impulse vector can be applied for both undamped and underdamped systems, and for both positive and negative impulses in a unified way. The impulse vector makes it easy to obtain impulse time and magnitude of the input shaper graphically.
A vector concept for an input shaper was first introduced by W. Singhose for undamped systems with positive impulses, and an impulse vector was first introduced by C.-G. Kang to generalize Singhose idea to underdamped systems with positive and negative impulses.
Definition.
For a vibratory second-order system formula_1 with undamped natural frequency formula_2 and damping ratio formula_3, the magnitude formula_4 and angle formula_5 of an impulse vector formula_6 corresponding to an impulse function formula_7, formula_8 is defined in a 2-dimensional polar coordinate system as
where formula_11 implies the magnitude of an impulse function, formula_12 implies the time location of the impulse function, and formula_13 implies damped natural frequency formula_14. For a positive impulse function with formula_15, the initial point of the impulse vector is located at the origin of the polar coordinate system, while for a negative impulse function with formula_16, the terminal point of the impulse vector is located at the origin. □
In this definition, the magnitude formula_4 is the product of formula_11 and a scaling factor for damping during time interval formula_12, which represents the magnitude formula_11 before being damped; the angle formula_5 is the product of the impulse time and damped natural frequency. formula_22 represents the Dirac delta function with impulse time at formula_23.
Note that an impulse function is a purely mathematical quantity, while the impulse vector includes a physical quantity (that is, formula_2 and formula_3 of a second-order system) as well as a mathematical impulse function. Representing more than two impulse vectors in the same polar coordinate system makes an "impulse vector diagram". The impulse vector diagram is a graphical representation of an impulse sequence.
Consider two impulse vectors formula_26 and formula_27 in the figure on the right-hand side, in which formula_26 is an impulse vector with magnitude formula_29 and angle formula_30 corresponding to a positive impulse with formula_31, and formula_27 is an impulse vector with magnitude formula_33 and angle formula_34 corresponding to a negative impulse with formula_35. Since the two time-responses corresponding to formula_26 and formula_27 are exactly same after the final impulse time formula_38 as shown in the figure, the two impulse vectors formula_26 and formula_27 can be regarded as the same vector for vector addition and subtraction. Impulse vectors satisfy the commutative and associative laws, as well as the distributive law for scalar multiplication.
The magnitude of the impulse vector determines the magnitude of the impulse, and the angle of the impulse vector determines the time location of the impulse. One rotation, formula_41 angle, on an impulse vector diagram corresponds to one (damped) period of the corresponding impulse response.
If it is an undamped system (formula_42), the magnitude and angle of the impulse vector become formula_43 and formula_44.
Properties.
Property 1: Resultant of two impulse vectors..
The impulse response of a second-order system corresponding to the resultant of two impulse vectors is same as the time response of the system with a two-impulse input corresponding to two impulse vectors after the final impulse time regardless of whether the system is undamped or underdamped. □
Property 2: Zero resultant of impulse vectors..
If the resultant of impulse vectors is zero, the time response of a second-order system for the input of the impulse sequence corresponding to the impulse vectors becomes zero also after the final impulse time regardless of whether the system is undamped or underdamped. □
Consider an underdamped second-order system with the transfer function formula_45. This system has formula_46 and formula_47. For given impulse vectors formula_26 and formula_27 as shown in the figure, the resultant can be represented in two ways, formula_50 and formula_51, in which formula_50 corresponds to a negative impulse with formula_53 and formula_54, and formula_51 corresponds to a positive impulse with formula_56 and formula_57.
The resultants formula_50, formula_51 can be found as follows.
Note that formula_63. The impulse responses formula_64 and formula_65 corresponding to formula_50 and formula_51 are exactly same with formula_68 after each impulse time location as shown in green lines of the figure (b).
Now, place an impulse vector formula_69 on the impulse vector diagram to cancel the resultant formula_70 as shown in the figure. The impulse vector formula_69 is given by
When the impulse sequence corresponding to three impulse vectors formula_73 and formula_69 is applied to a second-order system as an input, the resulting time response causes no residual vibration after the final impulse time formula_75 as shown in the red line of the bottom figure (b). Of course, another canceling vector formula_76 can exist, which is the impulse vector with the same magnitude as formula_69 but with an opposite arrow direction. However, this canceling vector has a longer impulse time that can be as much as a half period compared to formula_69.
Applications: Design of input shapers using impulse vectors.
ZVD"n" shaper.
Using impulse vectors, we can redesign known input shapers such as zero vibration (ZV), zero vibration and derivative (ZVD), and ZVD"n" shapers.
The ZV shaper is composed of two impulse vectors, in which the first impulse vector is located at 0°, and the second impulse vector with the same magnitude is located at 180° for formula_79. Then from the impulse vector diagram of the ZV shaper on the right-hand side,
Therefore, formula_82.
Since formula_83 (normalization constraint) must be hold, and formula_84,
Therefoere, formula_86.
Thus, the ZV shaper formula_87 is given by
formula_89
The ZVD shaper is composed of three impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at formula_90 rad, and the third vector at formula_41 rad, and the magnitude ratio is formula_92. Then formula_93. From the impulse vector diagram,
Therefore, formula_95.
Also from the impulse vector diagram,
Since formula_97 must be hold,
Therefore, formula_99.
Thus, the ZVD shaper formula_100 is given by
formula_89
The ZVD2 shaper is composed of four impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at formula_90 rad, the third vector at formula_41 rad, and the fourth vector at formula_105 rad, and the magnitude ratio is formula_106. Then formula_107. From the impulse vector diagram,
Therefore, formula_109.
Also, from the impulse vector diagram,
Since formula_111 must be hold,
Therefore, formula_113.
Thus, the ZVD2 shaper formula_114 is given by
formula_89
Similarly, the ZVD3 shaper with five impulse vectors can be obtained, in which the first vector is located at 0 rad, the second vector at formula_90 rad, third vector at formula_41 rad, the fourth vector at formula_105 rad, and the fifth vector at formula_120 rad, and the magnitude ratio is formula_121. In general, for the ZVD"n" shaper, "i"-th impulse vector is located at formula_122 rad, and the magnitude ratio is formula_123 where formula_124 implies a mathematical combination.
ETM shaper.
Now, consider "equal shaping-time and magnitudes" (ETM) shapers, with the same magnitude of impulse vectors and with the same angle between impulse vectors. The ETM"n" shaper satisfies the conditions
Thus, the resultant of the impulse vectors of the ETM"n" shaper becomes always zero for all formula_128. One merit of the ETM"n" shaper is that, unlike the ZVD"n" or extra insensitive (EI) shapers, the shaping time is always one (damped) period of the time response even if "n" increases.
The ETM4 shaper with four impulse vectors is obtained from the above conditions together with impulse vector definitions as
The ETM5 shaper with five impulse vectors is obtained similarly as
In the same way, the ETM"n" shaper with formula_133 can be obtained easily. In general, ETM shapers are less sensitive to modeling errors than ZVD"n" shapers in a large positive error range. Note that the ZVD shaper is an ETM3 shaper with formula_134.
NMe shaper.
Moreover, impulse vectors can be applied to design input shapers with negative impulses. Consider a "negative equal-magnitude" (NMe) shaper, in which the magnitudes of three impulse vectors are formula_135, and the angles are formula_136. Then the resultant of three impulse vectors becomes zero, and thus the residual vibration is suppressed. Impulse time formula_137 of the NMe shaper are obtained as formula_138, and impulse magnitudes are obtained easily by solving the simultaneous equations
The resulting NMe shaper formula_141 is
The NMe shaper has faster rise time than the ZVD shaper, but it is more sensitive to modeling error than the ZVD shaper. Note that the NMe shaper is the same with the UM shaper if the system is undamped (formula_42).
Figure (a) in the right side shows a typical block diagram of an input-shaping control system, and figure (b) shows residual vibration suppressions in unit-step responses by ZV, ZVD, ETM4 and NMe shapers.
Refer to the reference for sensitivity curves of the above input shapers, which represent the robustness to modeling errors in formula_2 and formula_3.


Programming station

A programming station is a terminal or computer that allows a machine operator to control a machine remotely, rather than being on the factory or shop floor. The programming station usually provides all the functionality including management and diagnostics that are found on the main control station.


Out-of-band control

Out-of-band control is a characteristic of network protocols with which data control is regulated. Out-of-band control passes control data on a separate connection from main data. Protocols such as FTP use out-of-band control.
FTP sends its control information, which includes user identification, password, and put/get commands, on one connection, and sends data files on a separate parallel connection. Because it uses a separate connection for the control information, FTP uses out-of-band control.


Intelligent pump

An intelligent pump is a pump that has the ability to regulate and control flow or pressure. Typical advantages are energy savings, lifetime improvements and system cost reductions. Intelligent pumps are used in boilers and systems, temperature control, water treatment, industrial water supply, wash and clean, machining and desalination.


Automatic lubrication system

An automatic lubrication system (ALS), sometimes referred to as a centralized lubrication system (CLS), is a system that delivers controlled amounts of lubricant to multiple locations on a machine while the machine is operating. Even though these systems are usually fully automated, a system that requires a manual pump or button activation is still identified as a centralized lubrication system. The system can be classified into two different categories that can share a lot of the same components.
Oil systems: Oil systems primary use is for stationary manufacturing equipment such as CNC milling
Grease systems: Grease primary use is on mobile units such as trucks, mining or construction equipment.
Automatic lubrication systems are key aspects in maintenance and reliability programs. They supply lube points with metered amounts of grease or oil from a central location. The pump supplies the system with the chosen lubricant and is fed from a reservoir that is easily accessible. Depending on the application, the reservoir ranges in size and can be as small as 2 liters all the way up to an intermediate bulk container or even a bulk tank. The options are almost limitless and are application-specific. These systems have the option to be monitored remotely with feedback and can be tied directly into your plant's PLC. So whether you're running an excavator, driving a ready-mix truck, operating a crusher, or making steel, you can rest assured that your assets are being properly lubricated at all times.
Reasons for Automatic Lubrication Systems.
Automatic lubrication systems or centralized lubrication system s are designed to apply lubricant in precise, metered amounts over short, frequent time intervals. Time and human resource constraints and often the physical location on the machine often makes it impractical to manually lubricate the points. As a result, production cycles, machine availability, and manpower availability dictate the intervals at which machinery is lubricated, which is not optimal for the point requiring lubrication. Automatic lubrication systems are installed on machinery to circumvent these issues.
Benefits.
Auto lube systems have many advantages over traditional methods of manual lubrication:
Components.
A typical system consists of controller/timer, pump w/reservoir, supply line, metering valves, and feed lines. Regardless of the manufacturer or type of system, all automatic lubrication systems share these 5 main components:
Types.
There are several different types of automatic lubrication systems including:
The 4 most commonly used Automatic Lubrication System types are:
Single line progressive.
A single line progressive system uses lubricant flow to cycle individual metering valves and valve assemblies. The valves consist of dispensing pistons moving back and forth in a specific bore. Each piston depends on flow from the previous piston to shift and displace lubricant. If one piston doesn't shift, none of the following pistons will shift. Valve output is not adjustable.
Operation begins when the controller/timer sends a signal to the pump to start the lube event. The pump then feeds lubricant into the supply line which connects to the primary metering valve, for either a preprogrammed amount of time or number of times as monitored through a designated piston cycle switch. Lubricant is fed to the multiple lubrication points one after another via secondary progressive metering valves sized for each series of lubrication points, and then directly to each point via the feed lines.
Single line parallel.
The first single-line parallel system for industry was introduced in 1937 by Lincoln Engineering (now known as Lincoln Industrial) in the United States.
A single line parallel system can service a single machine, different zones on a single machine or even several separate machines and is ideal when the volume of lubricant varies for each point. In this type of system, a central pump station automatically delivers lubricant through a single supply line to multiple branches of injectors. Each injector serves a single lubrication point, operates independently and may be individually adjusted to deliver the desired amount of lubricant.
Operation begins when the controller/timer sends a signal to the pump starting the lube cycle. The pump begins pumping lubricant to build up pressure in the supply line connecting the pump to the injectors. Once the required pressure is reached, the lube injectors dispense a predetermined amount of lubricant to the lubrication points via feed lines.
Once the entire system reaches the required pressure, a pressure switch sends a signal to the controller indicating that grease has cycled through to all the distribution points. The pump shuts off. Pressure is vented out of the system and grease in the line is redirected back to the pump reservoir, until the normal system pressure level is restored.
Dual line parallel.
A dual line parallel system is similar to the single line parallel system in that it uses hydraulic pressure to cycle adjustable valves to dispense measured shots of lubricant. It has 2 main supply lines which are alternatively used as pressure / vent lines. The advantage of a two-line system is that it can handle hundreds of lubrication points from a single pump station over several thousand feet using significantly smaller tubing or pipe.
Operation begins when the controller/timer sends a signal to the pump to start the lubrication cycle. The pump begins pumping lubricant to build up pressure in the first (the pressure) supply line while simultaneously venting the second (vent) return line. Once the required pressure is reached, a predetermined amount of lubricant is dispensed by the metering devices to half of the lubrication points via feed lines.
Once the pressure switch monitoring main supply line pressure indicates a preset pressure in the line has been reached, the system is hydraulically closed. The controller shuts off the pump and signals a changeover valve to redirect lubricant to the second main supply line.
The next time the controller activates the system, the second main line now becomes the pressure line while the first line becomes the vent line. The second line is pressurized and the entire process is repeated lubricating the remaining lube points.
Multi point direct lubricator
When the controller in the pump or external controller activates the drive motor, a set of cams turns and activates individual injectors or pump elements to dispense a fixed amount of lubricant to each individual lubrication point. Systems are easy to design, direct pump to lube point without added accessories and easy to troubleshoot.


Routh–Hurwitz theorem

In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a mathematical test, the Routh-Hurwitz stability criterion, to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz.
Notations.
Let "f"("z") be a polynomial (with complex coefficients) of degree "n" with no roots on the imaginary axis (i.e. the line "Z" = "ic" where "i" is the imaginary unit and "c" is a real number). Let us define formula_1 (a polynomial of degree "n") and formula_2 (a nonzero polynomial of degree strictly less than "n") by formula_3, respectively the real and imaginary parts of "f" on the imaginary line.
Furthermore, let us denote by:
Statement.
With the notations introduced above, the Routh–Hurwitz theorem states that:
From the first equality we can for instance conclude that when the variation of the argument of "f"("iy") is positive, then "f"("z") will have more roots to the left of the imaginary axis than to its right.
The equality "p" − "q" = "w"(+∞) − "w"(−∞) can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is "p" + "q" and the "w" from the right member is the number of variations of a Sturm chain (while "w" refers to a generalized Sturm chain in the present theorem).
Routh–Hurwitz stability criterion.
We can easily determine a stability criterion using this theorem as it is trivial that "f"("z") is Hurwitz-stable iff "p" − "q" = "n". We thus obtain conditions on the coefficients of "f"("z") by imposing "w"(+∞) = "n" and "w"(−∞) = 0.