Neural tangent kernel

In the study of artificial neural networks (ANNs), the neural tangent kernel (NTK) is a kernel that describes the evolution of deep artificial neural networks during their training by gradient descent. It allows ANNs to be studied using theoretical tools from kernel methods.
In general, a kernel is a positive-semidefinite symmetric function of two inputs which represents some notion of similarity between the two inputs. The NTK is a specific kernel derived from a given neural network; in general, when the neural network parameters change during training, the NTK evolves as well. However, in the limit of large layer width the NTK becomes constant, revealing a duality between training the wide neural network and kernel methods: gradient descent in the infinite-width limit is fully equivalent to kernel gradient descent with the NTK. As a result, using gradient descent to minimize least-square loss for neural networks yields the same mean estimator as ridgeless kernel regression with the NTK. This duality enables simple closed form equations describing the training dynamics, generalization, and predictions of wide neural networks.
The NTK was introduced in 2018 by Arthur Jacot, Franck Gabriel and Clément Hongler, who used it to study the convergence and generalization properties of fully connected neural networks. Later works extended the NTK results to other neural network architectures. In fact, the phenomenon behind NTK is not specific to neural networks and can be observed in generic nonlinear models, usually by a suitable scaling.
Main results (informal).
Let formula_1 denote the scalar function computed by a given neural network with parameters formula_2 on input formula_3. Then the neural tangent kernel is defined asformula_4Since it is written as a dot product between mapped inputs (with the gradient of the neural network function serving as the feature map), we are guaranteed that the NTK is symmetric and positive semi-definite. The NTK is thus a valid kernel function.
Consider a fully connected neural network whose parameters are chosen i.i.d. according to any mean-zero distribution. This random initialization of formula_2 induces a distribution over formula_1 whose statistics we will analyze, both at initialization and throughout training (gradient descent on a specified dataset). We can visualize this distribution via a neural network ensemble which is constructed by drawing many times from the initial distribution over formula_1 and training each draw according to the same training procedure.
The number of neurons in each layer is called the layer’s width. Consider taking the width of every hidden layer to infinity and training the neural network with gradient descent (with a suitably small learning rate). In this infinite-width limit, several nice properties emerge:
Applications.
Ridgeless kernel regression and kernel gradient descent.
Kernel methods are machine learning algorithms which use only pairwise relations between input points. Kernel methods do not depend on the concrete values of the inputs; they only depend on the relations between the inputs and other inputs (such as the training set). These pairwise relations are fully captured by the kernel function: a symmetric, positive-semidefinite function of two inputs which represents some notion of similarity between the two inputs. A fully equivalent condition is that there exists some feature map formula_14 such that the kernel function can be written as a dot product of the mapped inputsformula_15The properties of a kernel method depend on the choice of kernel function. (Note that formula_16 may have higher dimension than formula_17.) As a relevant example, consider linear regression. This is the task of estimating formula_18 given formula_19 samples formula_20 generated from formula_21, where each formula_22 is drawn according to some input data distribution. In this setup, formula_18 is the weight vector which defines the true function formula_24; we wish to use the training samples to develop a model formula_25 which approximates formula_18. We do this by minimizing the mean-square error between our model and the training samples:formula_27There exists an explicit solution for formula_25 which minimizes the squared error: formula_29, where formula_30 is the matrix whose columns are the training inputs, and formula_31 is the vector of training outputs. Then, the model can make predictions on new inputs: formula_32.
However, this result can be rewritten as: formula_33. Note that this dual solution is expressed solely in terms of the inner products between inputs. This motivates extending linear regression to settings in which, instead of directly taking inner products between inputs, we first transform the inputs according to a chosen feature map and then evaluate the inner products between the transformed inputs. As discussed above, this can be captured by a kernel function formula_34, since all kernel functions are inner products of feature-mapped inputs. This yields the ridgeless kernel regression estimator:formula_35If the kernel matrix formula_36 is singular, one uses the Moore-Penrose pseudoinverse. The regression equations are called "ridgeless" because they lack a ridge regularization term.
In this view, linear regression is a special case of kernel regression with the identity feature map: formula_37. Equivalently, kernel regression is simply linear regression in the feature space (i.e. the range of the feature map defined by the chosen kernel). Note that kernel regression is typically a "nonlinear" regression in the input space, which is a major strength of the algorithm.
Just as it’s possible to perform linear regression using iterative optimization algorithms such as gradient descent, one can perform kernel regression using kernel gradient descent. This is equivalent to performing gradient descent in the feature space. It’s known that if the weight vector is initialized close to zero, least-squares gradient descent converges to the minimum-norm solution, i.e., the final weight vector has the minimum Euclidean norm of all the interpolating solutions. In the same way, kernel gradient descent yields the minimum-norm solution with respect to the RKHS norm. This is an example of the implicit regularization of gradient descent.
The NTK gives a rigorous connection between the inference performed by infinite-width ANNs and that performed by kernel methods: when the loss function is the least-squares loss, the inference performed by an ANN is in expectation equal to ridgeless kernel regression with respect to the NTK. This suggests that the performance of large ANNs in the NTK parametrization can be replicated by kernel methods for suitably chosen kernels.
Overparametrization, interpolation, and generalization.
In overparametrized models, the number of tunable parameters exceeds the number of training samples. In this case, the model is able to memorize (perfectly fit) the training data. Therefore, overparametrized models interpolate the training data, achieving essentially zero training error.
Kernel regression is typically viewed as a non-parametric learning algorithm, since there are no explicit parameters to tune once a kernel function has been chosen. An alternate view is to recall that kernel regression is simply linear regression in feature space, so the “effective” number of parameters is the dimension of the feature space. Therefore, studying kernels with high-dimensional feature maps can provide insights about strongly overparametrized models.
As an example, consider the problem of generalization. According to classical statistics, memorization should cause models to fit noisy signals in the training data, harming their performance on unseen data. To mitigate this, machine learning algorithms often introduce regularization to mitigate noise-fitting tendencies. Surprisingly, modern neural networks (which tend to be strongly overparametrized) seem to generalize well, even in the absence of explicit regularization. To study the generalization properties of overparametrized neural networks, one can exploit the infinite-width duality with ridgeless kernel regression. Recent works have derived equations describing the expected generalization error of high-dimensional kernel regression; these results immediately explain the generalization of sufficiently wide neural networks trained to convergence on least-squares.
Convergence to a global minimum.
For a convex loss functional formula_38 with a global minimum, if the NTK remains positive-definite during training, the loss of the ANN formula_39 converges to that minimum as formula_40. This positive-definiteness property has been shown in a number of cases, yielding the first proofs that large-width ANNs converge to global minima during training.
Extensions and limitations.
The NTK can be studied for various ANN architectures, in particular convolutional neural networks (CNNs), recurrent neural networks (RNNs) and transformers. In such settings, the large-width limit corresponds to letting the number of parameters grow, while keeping the number of layers fixed: for CNNs, this involves letting the number of channels grow.
Individual parameters of a wide neural network in the kernel regime change negligibly during training. However, this implies that infinite-width neural networks cannot exhibit feature learning, which is widely considered to be an important property of realistic deep neural networks. This is not a generic feature of infinite-width neural networks and is largely due to a specific choice of the scaling by which the width is taken to the infinite limit; indeed several works have found alternate infinite-width scaling limits of neural networks in which there is no duality with kernel regression and feature learning occurs during training. Others introduce a "neural tangent hierarchy" to describe finite-width effects, which may drive feature learning.
Neural Tangents is a free and open-source Python library used for computing and doing inference with the infinite width NTK and neural network Gaussian process (NNGP) corresponding to various common ANN architectures. In addition, there exists a scikit-learn compatible implementation of the infinite width NTK for Gaussian processes called scikit-ntk.
Details.
When optimizing the parameters formula_41 of an ANN to minimize an empirical loss through gradient descent, the NTK governs the dynamics of the ANN output function formula_42 throughout the training.
Case 1: Scalar output.
An ANN with scalar output consists of a family of functions formula_43 parametrized by a vector of parameters formula_41.
The NTK is a kernel formula_45 defined byformula_46In the language of kernel methods, the NTK formula_47 is the kernel associated with the feature map formula_48. To see how this kernel drives the training dynamics of the ANN, consider a dataset formula_49 with scalar labels formula_50 and a loss function formula_51. Then the associated empirical loss, defined on functions formula_52, is given byformula_53When the ANN formula_54 is trained to fit the dataset (i.e. minimize formula_55) via continuous-time gradient descent, the parameters formula_56 evolve through the ordinary differential equation:
During training the ANN output function follows an evolution differential equation given in terms of the NTK:
This equation shows how the NTK drives the dynamics of formula_59 in the space of functions formula_60 during training.
Case 2: Vector output.
An ANN with vector output of size formula_61 consists in a family of functions formula_62 parametrized by a vector of parameters formula_41.
In this case, the NTK formula_64 is a matrix-valued kernel, with values in the space of formula_65 matrices, defined byformula_66Empirical risk minimization proceeds as in the scalar case, with the difference being that the loss function takes vector inputs formula_67. The training of formula_68 through continuous-time gradient descent yields the following evolution in function space driven by the NTK:formula_69This generalizes the equation shown in case 1 for scalar outputs.
Interpretation.
The NTK formula_70 represents the influence of the loss gradient formula_71 with respect to example formula_72 on the evolution of ANN output formula_73 through a gradient descent step: in the scalar case, this readsformula_74In particular, each data point formula_75 influences the evolution of the output formula_73 for each formula_3 throughout the training, in a way that is captured by the NTK formula_70.
Wide fully-connected ANNs have a deterministic NTK, which remains constant throughout training.
Consider an ANN with fully-connected layers formula_79 of widths formula_80, so that formula_81, where formula_82 is the composition of an affine transformation formula_83 with the pointwise application of a nonlinearity formula_84, where formula_2 parametrizes the maps formula_86. The parameters formula_41 are initialized randomly, in an independent, identically distributed way.
As the widths grow, the NTK's scale is affected by the exact parametrization of the formula_83's and by the parameter initialization. This motivates the so-called NTK parametrization formula_89. This parametrization ensures that if the parameters formula_41 are initialized as standard normal variables, the NTK has a finite nontrivial limit. In the large-width limit, the NTK converges to a deterministic (non-random) limit formula_91, which stays constant in time.
The NTK formula_91 is explicitly given by formula_93, where formula_94 is determined by the set of recursive equations:
where formula_96 denotes the kernel defined in terms of the Gaussian expectation:
In this formula the kernels formula_98 are the ANN's so-called activation kernels.
Wide fully connected networks are linear in their parameters throughout training.
The NTK describes the evolution of neural networks under gradient descent in function space. Dual to this perspective is an understanding of how neural networks evolve in parameter space, since the NTK is defined in terms of the gradient of the ANN's outputs with respect to its parameters. In the infinite width limit, the connection between these two perspectives becomes especially interesting. The NTK remaining constant throughout training at large widths co-occurs with the ANN being well described throughout training by its first order Taylor expansion around its parameters at initialization:


Margin (machine learning)

In machine learning the margin of a single data point is defined to be the distance from the data point to a decision boundary. Note that there are many distances and decision boundaries that may be appropriate for certain datasets and goals. A margin classifier is a classifier that explicitly utilizes the margin of each example while learning a classifier. There are theoretical justifications (based on the VC dimension) as to why maximizing the margin (under some suitable constraints) may be beneficial for machine learning and statistical inferences algorithms.
There are many hyperplanes that might classify the data. One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the two classes. So we choose the hyperplane so that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known as the "maximum-margin hyperplane" and the linear classifier it defines is known as a "maximum margin classifier"; or equivalently, the "perceptron of optimal stability."


Adaptive control

Adaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself to such changing conditions. Adaptive control is different from robust control in that it does not need "a priori" information about the bounds on these uncertain or time-varying parameters; robust control guarantees that if the changes are within given bounds the control law need not be changed, while adaptive control is concerned with control law changing itself.
Parameter estimation.
The foundation of adaptive control is parameter estimation, which is a branch of system identification. Common methods of estimation include recursive least squares and gradient descent. Both of these methods provide update laws that are used to modify estimates in real-time (i.e., as the system operates). Lyapunov stability is used to derive these update laws and show convergence criteria (typically persistent excitation; relaxation of this condition are studied in Concurrent Learning adaptive control). Projection and normalization are commonly used to improve the robustness of estimation algorithms.
Classification of adaptive control techniques.
In general, one should distinguish between:
as well as between
Direct methods are ones wherein the estimated parameters are those directly used in the adaptive controller. In contrast, indirect methods are those in which the estimated parameters are used to calculate required controller parameters. Hybrid methods rely on both estimation of parameters and direct modification of the control law.
There are several broad categories of feedback adaptive control (classification can vary):
Some special topics in adaptive control can be introduced as well:
In recent times, adaptive control has been merged with intelligent techniques such as fuzzy and neural networks to bring forth new concepts such as fuzzy adaptive control.
Applications.
When designing adaptive control systems, special consideration is necessary of convergence and robustness issues. Lyapunov stability is typically used to derive control adaptation laws and show .
Usually these methods adapt the controllers to both the process statics and dynamics. In special cases the adaptation can be limited to the static behavior alone, leading to adaptive control based on characteristic curves for the steady-states or to extremum value control, optimizing the steady state. Hence, there are several ways to apply adaptive control algorithms.
A particularly successful application of adaptive control has been adaptive flight control. This body of work has focused on guaranteeing stability of a model reference adaptive control scheme using Lyapunov arguments. Several successful flight-test demonstrations have been conducted, including fault tolerant adaptive control.


Wiener filter

In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant (LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and additive noise. The Wiener filter minimizes the mean square error between the estimated random process and the desired process.
Description.
The goal of the Wiener filter is to compute a statistical estimate of an unknown signal using a related signal as an input and filtering that known signal to produce the estimate as an output. For example, the known signal might consist of an unknown signal of interest that has been corrupted by additive noise. The Wiener filter can be used to filter out the noise from the corrupted signal to provide an estimate of the underlying signal of interest. The Wiener filter is based on a statistical approach, and a more statistical account of the theory is given in the minimum mean square error (MMSE) estimator article.
Typical deterministic filters are designed for a desired frequency response. However, the design of the Wiener filter takes a different approach. One is assumed to have knowledge of the spectral properties of the original signal and the noise, and one seeks the linear time-invariant filter whose output would come as close to the original signal as possible. Wiener filters are characterized by the following:
This filter is frequently used in the process of deconvolution; for this application, see Wiener deconvolution.
Wiener filter solutions.
Let formula_1 be an unknown signal which must be estimated from a measurement signal formula_2. Where alpha is a tunable parameter. formula_3 is known as prediction, formula_4 is known as filtering, and formula_5 is known as smoothing (see Wiener filtering chapter of for more details).
The Wiener filter problem has solutions for three possible cases: one where a noncausal filter is acceptable (requiring an infinite amount of both past and future data), the case where a causal filter is desired (using an infinite amount of past data), and the finite impulse response (FIR) case where only input data is used (i.e. the result or output is not fed back into the filter as in the IIR case). The first case is simple to solve but is not suited for real-time applications. Wiener's main accomplishment was solving the case where the causality requirement is in effect; Norman Levinson gave the FIR solution in an appendix of Wiener's book.
Noncausal solution.
where formula_7 are spectral densities. Provided that formula_8 is optimal, then the minimum mean-square error equation reduces to
and the solution formula_8 is the inverse two-sided Laplace transform of formula_11.
Causal solution.
where
This general formula is complicated and deserves a more detailed explanation. To write down the solution formula_23 in a specific case, one should follow these steps:
Finite impulse response Wiener filter for discrete series.
The causal finite impulse response (FIR) Wiener filter, instead of using some given data matrix X and output vector Y, finds optimal tap weights by using the statistics of the input and output signals. It populates the input matrix X with estimates of the auto-correlation of the input signal (T) and populates the output vector Y with estimates of the cross-correlation between the output and input signals (V).
In order to derive the coefficients of the Wiener filter, consider the signal "w"["n"] being fed to a Wiener filter of order (number of past taps) "N" and with coefficients formula_34. The output of the filter is denoted "x"["n"] which is given by the expression
The residual error is denoted "e"["n"] and is defined as "e"["n"] = "x"["n"] − "s"["n"] (see the corresponding block diagram). The Wiener filter is designed so as to minimize the mean square error (MMSE criteria) which can be stated concisely as follows:
where formula_37 denotes the expectation operator. In the general case, the coefficients formula_38 may be complex and may be derived for the case where "w"["n"] and "s"["n"] are complex as well. With a complex signal, the matrix to be solved is a Hermitian Toeplitz matrix, rather than symmetric Toeplitz matrix. For simplicity, the following considers only the case where all these quantities are real. The mean square error (MSE) may be rewritten as:
To find the vector formula_40 which minimizes the expression above, calculate its derivative with respect to each formula_41
Assuming that "w"["n"] and "s"["n"] are each stationary and jointly stationary, the sequences formula_43 and formula_44 known respectively as the autocorrelation of "w"["n"] and the cross-correlation between "w"["n"] and "s"["n"] can be defined as follows:
The derivative of the MSE may therefore be rewritten as:
Note that for real formula_47, the autocorrelation is symmetric:formula_48Letting the derivative be equal to zero results in:
which can be rewritten (using the above symmetric property) in matrix form
These equations are known as the Wiener–Hopf equations. The matrix T appearing in the equation is a symmetric Toeplitz matrix. Under suitable conditions on formula_51, these matrices are known to be positive definite and therefore non-singular yielding a unique solution to the determination of the Wiener filter coefficient vector, formula_52. Furthermore, there exists an efficient algorithm to solve such Wiener–Hopf equations known as the Levinson-Durbin algorithm so an explicit inversion of T is not required.
In some articles, the cross correlation function is defined in the opposite way:formula_53Then, the formula_54 matrix will contain formula_55; this is just a difference in notation.
Whichever notation is used, note that for real formula_56:formula_57
Relationship to the least squares filter.
The realization of the causal Wiener filter looks a lot like the solution to the least squares estimate, except in the signal processing domain. The least squares solution, for input matrix formula_58 and output vector formula_59 is
The FIR Wiener filter is related to the least mean squares filter, but minimizing the error criterion of the latter does not rely on cross-correlations or auto-correlations. Its solution converges to the Wiener filter solution.
Complex signals.
For complex signals, the derivation of the complex Wiener filter is performed by minimizing formula_61 =formula_62. This involves computing partial derivatives with respect to both the real and imaginary parts of formula_38, and requiring them both to be zero.
The resulting Wiener-Hopf equations are:
which can be rewritten in matrix form:
Note here that:formula_66
The Wiener coefficient vector is then computed as:formula_67
Applications.
The Wiener filter has a variety of applications in signal processing, image processing, control systems, and digital communications. These applications generally fall into one of four main categories:
For example, the Wiener filter can be used in image processing to remove noise from a picture. For example, using the Mathematica function:
codice_1 on the first image on the right, produces the filtered image below it.
It is commonly used to denoise audio signals, especially speech, as a preprocessor before speech recognition.
History.
The filter was proposed by Norbert Wiener during the 1940s and published in 1949. The discrete-time equivalent of Wiener's work was derived independently by Andrey Kolmogorov and published in 1941. Hence the theory is often called the "Wiener–Kolmogorov" filtering theory ("cf." Kriging). The Wiener filter was the first statistically designed filter to be proposed and subsequently gave rise to many others including the Kalman filter.


Linear dynamical system

Linear dynamical systems are dynamical systems whose evolution functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point.
Introduction.
In a linear dynamical system, the variation of a state vector
(an formula_1-dimensional vector denoted formula_2) equals a constant matrix
(denoted formula_3) multiplied by
formula_2. This variation can take two forms: either
as a flow, in which formula_2 varies
continuously with time
or as a mapping, in which
formula_2 varies in discrete steps
These equations are linear in the following sense: if
formula_9 and formula_10
are two valid solutions, then so is any linear combination
of the two solutions, e.g.,
formula_11
where formula_12 and formula_13
are any two scalars. The matrix formula_3
need not be symmetric.
Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.
Solution of linear dynamical systems.
If the initial vector formula_15
is aligned with a right eigenvector formula_16 of
the matrix formula_3, the dynamics are simple
where formula_19 is the corresponding eigenvalue;
the solution of this equation is
as may be confirmed by substitution.
If formula_3 is diagonalizable, then any vector in an formula_1-dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted formula_23) of the matrix formula_3.
Therefore, the general solution for formula_9 is
a linear combination of the individual solutions for the right
eigenvectors
Similar considerations apply to the discrete mappings.
Classification in two dimensions.
The roots of the characteristic polynomial det(A - λI) are the eigenvalues of A. The sign and relation of these roots, formula_28, to each other may be used to determine the stability of the dynamical system
For a 2-dimensional system, the characteristic polynomial is of the form formula_30 where formula_31 is the trace and formula_32 is the determinant of A. Thus the two roots are in the form:
and formula_35 and formula_36. Thus if formula_37 then the eigenvalues are of opposite sign, and the fixed point is a saddle. If formula_38 then the eigenvalues are of the same sign. Therefore, if formula_39 both are positive and the point is unstable, and if formula_40 then both are negative and the point is stable. The discriminant will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).


Automatic server discovery

Automatic server discovery is a software licensing feature that allows client applications to find license servers automatically on the network, thus eliminating the need for end users to manually configure server information and allowing system administrators to perform their tasks more easily and efficiently. If you have, for example, over 70 machines then the configuration process will take a long time if it is done manually.
Automatic server discovery often uses Multicast UDP to send broadcasts, to which available license servers respond with information about their network location. When a licensed server is discovered, the information is locally cached on the client machine, so automatic server discovery does not have to be performed at each application startup.
The newest version of NTP also supports Automatic server discovery. There are three schemes provided by NTPv4:
Automatic server discovery typically works only on local networks, and will not work on WAN or VPN connections.


Sampled data system

In systems science, a sampled-data system is a control system in which a continuous-time plant is controlled with a digital device. Under periodic sampling, the sampled-data system is time-varying but also periodic; thus, it may be modeled by a simplified discrete-time system obtained by discretizing the plant. However, this discrete model does not capture the inter-sample behavior of the real system, which may be critical in a number of applications.
The analysis of sampled-data systems incorporating full-time information leads to challenging control problems with a rich mathematical structure. Many of these problems have only been solved recently.


Class kappa-ell function

In control theory, it is often required to check if a nonautonomous system is stable or not. To cope with this it is necessary to use some special comparison functions. Class formula_1 functions belong to this family:
Definition: A continuous function formula_2 is said to belong to class formula_1 if:


Signal processing

Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing "signals", such as sound, images, potential fields, seismic signals, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal.
History.
According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s.
In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was published in the "Bell System Technical Journal". The paper laid the groundwork for later development of information communication systems and the processing of signals for transmission.
Signal processing matured and flourished in the 1960s and 1970s, and digital signal processing became widely used with specialized digital signal processor chips in the 1980s.
Definition Signal.
A signal is a function formula_1, where this function is either
Categories.
Analog.
Analog signal processing is for signals that have not been digitized, as in most 20th-century radio, telephone, and television systems. This involves linear electronic circuits as well as nonlinear ones. The former are, for instance, passive filters, active filters, additive mixers, integrators, and delay lines. Nonlinear circuits include compandors, multipliers (frequency mixers, voltage-controlled amplifiers), voltage-controlled filters, voltage-controlled oscillators, and phase-locked loops.
Continuous time.
Continuous-time signal processing is for signals that vary with the change of continuous domain (without considering some individual interrupted points).
The methods of signal processing include time domain, frequency domain, and complex frequency domain. This technology mainly discusses the modeling of a linear time-invariant continuous system, integral of the system's zero-state response, setting up system function and the continuous time filtering of deterministic signals
Discrete time.
Discrete-time signal processing is for sampled signals, defined only at discrete points in time, and as such are quantized in time, but not in magnitude.
"Analog discrete-time signal processing" is a technology based on electronic devices such as sample and hold circuits, analog time-division multiplexers, analog delay lines and analog feedback shift registers. This technology was a predecessor of digital signal processing (see below), and is still used in advanced processing of gigahertz signals.
The concept of discrete-time signal processing also refers to a theoretical discipline that establishes a mathematical basis for digital signal processing, without taking quantization error into consideration.
Digital.
Digital signal processing is the processing of digitized discrete-time sampled signals. Processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors (DSP chips). Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, multiplication and addition. Other typical operations supported by the hardware are circular buffers and lookup tables. Examples of algorithms are the fast Fourier transform (FFT), finite impulse response (FIR) filter, Infinite impulse response (IIR) filter, and adaptive filters such as the Wiener and Kalman filters.
Nonlinear.
Nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatiotemporal domains. Nonlinear systems can produce highly complex behaviors including bifurcations, chaos, harmonics, and subharmonics which cannot be produced or analyzed using linear methods.
Polynomial signal processing is a type of non-linear signal processing, where polynomial systems may be interpreted as conceptually straightforward extensions of linear systems to the non-linear case.
Statistical.
Statistical signal processing is an approach which treats signals as stochastic processes, utilizing their statistical properties to perform signal processing tasks. Statistical techniques are widely used in signal processing applications. For example, one can model the probability distribution of noise incurred when photographing an image, and construct techniques based on this model to reduce the noise in the resulting image.
Application fields.
In communication systems, signal processing may occur at:


Hierarchical control system

A hierarchical control system (HCS) is a form of control system in which a set of devices and governing software is arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that hierarchical control system is also a form of networked control system.
Overview.
A human-built system with complex behavior is often organized as a hierarchy. For example, a command hierarchy has among its notable features the organizational chart of superiors, subordinates, and lines of organizational communication. Hierarchical control systems are organized similarly to divide the decision making responsibility.
Each element of the hierarchy is a linked node in the tree. Commands, tasks and goals to be achieved flow down the tree from superior nodes to subordinate nodes, whereas sensations and command results flow up the tree from subordinate to superior nodes. Nodes may also exchange messages with their siblings. The two distinguishing features of a hierarchical control system are related to its layers.
Besides artificial systems, an animal's control systems are proposed to be organized as a hierarchy. In perceptual control theory, which postulates that an organism's behavior is a means of controlling its perceptions, the organism's control systems are suggested to be organized in a hierarchical pattern as their perceptions are constructed so.
Control system structure.
The accompanying diagram is a general hierarchical model which shows functional manufacturing levels using computerised control of an industrial control system.
Referring to the diagram;
Applications.
Manufacturing, robotics and vehicles.
Among the robotic paradigms is the hierarchical paradigm in which a robot operates in a top-down fashion, heavy on planning, especially motion planning. Computer-aided production engineering has been a research focus at NIST since the 1980s. Its Automated Manufacturing Research Facility was used to develop a five layer production control model. In the early 1990s DARPA sponsored research to develop distributed (i.e. networked) intelligent control systems for applications such as military command and control systems. NIST built on earlier research to develop its Real-Time Control System (RCS) and Real-time Control System Software which is a generic hierarchical control system that has been used to operate a manufacturing cell, a robot crane, and an automated vehicle.
In November 2007, DARPA held the Urban Challenge. The winning entry, Tartan Racing employed a hierarchical control system, with layered mission planning, motion planning, behavior generation, perception, world modelling, and mechatronics.
Artificial intelligence.
Subsumption architecture is a methodology for developing artificial intelligence that is heavily associated with behavior based robotics. This architecture is a way of decomposing complicated intelligent behavior into many "simple" behavior modules, which are in turn organized into layers. Each layer implements a particular goal of the software agent (i.e. system as a whole), and higher layers are increasingly more abstract. Each layer's goal subsumes that of the underlying layers, e.g. the decision to move forward by the eat-food layer takes into account the decision of the lowest obstacle-avoidance layer. Behavior need not be planned by a superior layer, rather behaviors may be triggered by sensory inputs and so are only active under circumstances where they might be appropriate.
Reinforcement learning has been used to acquire behavior in a hierarchical control system in which each node can learn to improve its behavior with experience.
James Albus, while at NIST, developed a theory for intelligent system design named the Reference Model Architecture (RMA), which is a hierarchical control system inspired by RCS. Albus defines each node to contain these components.
At its lowest levels, the RMA can be implemented as a subsumption architecture, in which the world model is mapped directly to the controlled process or real world, avoiding the need for a mathematical abstraction, and in which time-constrained reactive planning can be implemented as a finite state machine. Higher levels of the RMA however, may have sophisticated mathematical world models and behavior implemented by automated planning and scheduling. Planning is required when certain behaviors cannot be triggered by current sensations, but rather by predicted or anticipated sensations, especially those that come about as result of the node's actions.


Real-time Control System Software

The Real-time Control System (RCS) is a software system developed by NIST based on the Real-time Control System Reference Model Architecture, that implements a generic Hierarchical control system. The RCS Software Library is an archive of free C++, Java and Ada code, scripts, tools, makefiles, and documentation developed to aid programmers of software to be used in real-time control systems (especially those using the Reference Model Architecture for Intelligent Systems Design).
Introduction.
RCS has been used in automated manufacturing, robotics, and automated vehicle research at NIST. The software consists of a C++ library and GUI and configuration tools written in a variety of software languages. The Software Library is offering the following RCS tools:


Riveting machine

A riveting machine is used to automatically set (squeeze) rivets in order to join materials together. The riveting machine offers greater consistency, productivity, and lower cost when compared to manual riveting.
Types.
Automatic feed riveting machines include a hopper and feed track which automatically delivers and presents the rivet to the setting tools which overcomes the need for the operator to position the rivet. The downward force required to deform the rivet with an automatic riveting machine is created by a motor and flywheel combination, pneumatic cylinder, or hydraulic cylinder. Manual feed riveting machines usually have a mechanical lever to deliver the setting force from a foot pedal or hand lever.
Riveting machines can be sub-divided into two broad groups — impact riveting machines and orbital (or radial) riveting machines.
Impact riveting.
Impact riveting machines set the rivet by driving the rivet downwards, through the materials to be joined and on into a forming tool (known as a rollset). This action causes the end of the rivet to roll over in the rollset which causes the end of the rivet to flare out and thus join the materials together. Impact riveting machines are very fast and a cycle time of 0.5 seconds is typical.
Orbital riveting.
Orbital riveting machines have a spinning forming tool (known as a peen) which is gradually lowered into the rivet which spreads the material of the rivet into a desired shape depending upon the design of the tool. Orbital forming machines offer the user more control over the riveting cycle but the trade off is in cycle time which can be 2 or 3 seconds.
There are different types of riveting machines. Each type of machine has unique features and benefits. The orbital riveting process is different from impact riveting and spiralform riveting. Orbital riveting requires less downward force than impact or spiral riveting. Also, orbital riveting tooling typically lasts longer.
Orbital riveting machines are used in a wide range of applications including brake linings for commercial vehicles, aircraft, and locomotives, textile and leather goods, metal brackets, window and door furniture, latches and even mobile phones. Many materials can be riveted together using orbital riveting machines including delicate and brittle materials, and sensitive electrical or electronic components.
The orbital riveting process uses a forming tool mounted at a 3 or 6° angle. The forming tool contacts the material and then presses it while rotating until the final form is achieved. The final form often has height and/or diameter specifications.
Pneumatic orbital riveting machines typically provide downward force in the range. Hydraulic orbital riveting machines typically provide downward force in the range.
Radial (Spiralform) riveting.
Radial riveting is subtly different from orbital forming. In most cases however, where high-quality joints are demanded, the radial riveting technology is the appropriate procedure due to the low cyle time, the little force needed and the high quality results obtained.
The riveting peen describes a rose-petal path. The rivet is deformed in three directions. Radially outwards, radially inwards and overlying also tangentially.
Excellent surface structure of the closing head: With the Radial riveting process, the tool itself does not rotate. The friction between tool and work-piece is thus at a minimum. The result is an excellent surface structure.
Low workpiece loading: Even bakelite or ceramic parts can be riveted. Lateral forces are negligible. Clamping is usually unnecessary.
Rollerform riveting.
Rollerforming is a subset of orbital forming. Rollerforming uses the same powerhead as orbital forming but instead of a peen has multiple wheels that circle the workpiece and combine two similar or non-similar materials together with a seamless and smooth gentle bonding via downward pressure as the rollers move downward or inward on the piece.
Automatic drilling and riveting machine.
These machines take the automation one step farther by clamping the material and drilling or countersinking the hole in addition to riveting. They are commonly used in the aerospace industry because of the large number of holes and rivets required to assemble the aircraft skin.
Applications.
Riveting machines are used in a wide range of applications including brake linings for commercial vehicles, aircraft, and locomotives, textile and leather goods, metal brackets, window and door furniture, latches and even mobile phones. Many materials can be riveted together using riveting machines including delicate and brittle materials, and sensitive electrical or electronic components.


Deneb Robotics

Deneb Robotics was a company founded by Scott Walter, Jay Harrison, Nathan Yoffa, and Rakesh Mahajan in 1985. The company pioneered graphics-based 3D factory simulation software for simulation and digital manufacturing tools. The application area reached from concept development to shop floor implementation, including off-line programming. Deneb Robotics is known for their IGRIP, Quest, Ultra and VirtualNC software packages.
The company was acquired by Dassault Systèmes in 1997 and is branded DELMIA. Major companies such as Boeing, and General Motors used Deneb’s suite of tools to optimize their design and manufacturing processes.


Robust fuzzy programming

Robust fuzzy programming (ROFP) is a powerful mathematical optimization approach to deal with optimization problems under uncertainty. This approach is firstly introduced at 2012 by Pishvaee, Razmi & Torabi in the Journal of Fuzzy Sets and Systems. ROFP enables the decision makers to be benefited from the capabilities of both fuzzy mathematical programming and robust optimization approaches. At 2016 Pishvaee and Fazli put a significant step forward by extending the ROFP approach to handle flexibility of constraints and goals. ROFP is able to achieve a "robust solution" for an optimization problem under uncertainty.
Definition of robust solution.
Robust solution is defined as a solution which has "both "feasibility robustness" and "optimality robustness"; Feasibility robustness means that the solution should remain feasible for (almost) all possible values of uncertain parameters and flexibility degrees of constraints and optimality robustness means that the value of objective function for the solution should remain close to optimal value or have minimum (undesirable) deviation from the optimal value for (almost) all possible values of uncertain parameters and flexibility degrees on target value of goals".
Classification of ROFP methods.
As fuzzy mathematical programming is categorized into "Possibilistic programming" and "Flexible programming", ROFP also can be classified into:
The first category is used to deal with imprecise input parameters in optimization problems while the second one is employed to cope with flexible constraints and goals. Also, the last category is capable to handle both uncertain parameters and flexibility in goals and constraints.
From another point of view, it can be said that different ROFP models developed in the literature can be classified in three categories according to degree of conservatism against uncertainty. These categories include:
Hard worst case ROFP has the most conservative nature among ROFP methods since it provides maximum safety or immunity against uncertainty. Ignoring the chance of infeasibility, this method immunizes the solution for being infeasible for all possible values of uncertain parameters. Regarding the optimality robustness, this method minimizes the worst possible value of objective function (min-max logic). On the other hand, Soft worst case ROFP method behaves similar to hard worst case method regarding optimality robustness, however does not satisfy the constraints in their extreme worst case. Lastly, realistic method establishes a reasonable trade-off between the robustness, the cost of robustness and other objectives such as improving the average system performance (cost-benefit logic).
Applications.
ROFP is successfully implemented in different practical application areas such as the following ones.


Festo

Festo is a German automation company based in Esslingen am Neckar, Germany. Festo produces and sells pneumatic and electrical control systems and drive technology for factories and process automation. Festo Didactic also offers industrial education and consultation services and is one of the sponsors and partners of the WorldSkills Mechatronics Competitions. Sales subsidiaries, distribution centres and factories of Festo are located in 61 countries worldwide. The company was named after its founders Albert Fezer and Gottlieb Stoll.
Animal robots.
Festo is known for making moving robots that move like animals, such as the seagull-like SmartBird, jellyfish, butterflies and the BionicKangaroo. In 2018 they also added a flying fox and a rolling spider to the list. Festo calls their Bionic Flying Fox an “ultra-lightweight flying object with intelligent kinematics.”


STANKIN

The Moscow State University of Technology "STANKIN" (MSUT "STANKIN") (Russian: Московский Государственный Технологический Университет "СТАНКИН" (МГТУ "СТАНКИН")), previously the Moscow Machine and Tool Institute (Russian: Московский станкоинструментальный институт, tr. "Moskovsky stankoinstrumental'ny institut"), the name of which is still preserved in the acronym STANKIN (Russian: СТАНКИН), is a Russian technical higher education institution founded in 1930. Today STANKIN trains specialists in machinery, robotics, CNC's, electronics, automation and control systems, economics of enterprises, informatics and measurement systems.
History.
The university was founded as the Moscow Machine Tool Institute in 1930 to provide the machine tool industry with qualified specialists.
Today STANKIN is a scientific industrial complex with the Technological Design Institute of Informatic of the Russian Academy of Sciences (RSA). There is a network of scientific, educational and industrial centers. It has relations with universities and firms from Austria, Brazil, Germany, Hungary, Italy, China, USA, South Korea and other countries.
There are more than 600 professors and scientists working at Stankin today. Stankin has its own newspaper "Stankinovskiy vestnik" and a peer-reviewed journal with an international editorial board "Vestnik MGTU STANKIN" from 2009 "(English: MSUT STANKIN Messenger)" included in Web of Science databases.
Education.
Studies and corresponding research cover the areas of automation of technological processes and manufactures; engineering ecology and security in machine building; information and marketing in machine building; information systems;
quality of production and management; computerization of computation durability of machine building construction; computer modeling in instrumental technics; computer control system in production and business; tool engineering and computer modeling;
laser technology; presses and metal treatment technology; metal cutting machines and tools; design and computer modeling of plastic deformation system; robotics and mechatronic systems; system production of automated technological machinery; computer-aided design systems; metrological production system; computer control machine tools; technological information of automated manufacturing technology; technology and business in no-waste production; technology and management in instrumental manufacturing; mechanical engineering; physics of high concentrated energy; economy and control of production.


Root locus analysis

In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function in the complex "s"-plane as a function of a gain parameter (see pole–zero plot).
Evans also invented in 1948 an analog computer to compute root loci, called a "Spirule" (after "spiral" and "slide rule"); it found wide use before the advent of digital computers.
Uses.
In addition to determining the stability of the system, the root locus can be used to design the damping ratio ("ζ") and natural frequency ("ω""n") of a feedback system. Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arccosine whose center points coincide with the origin. By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency, a gain "K" can be calculated and implemented in the controller. More elaborate techniques of controller design using the root locus are available in most control textbooks: for instance, lag, lead, PI, PD and PID controllers can be designed approximately with this technique.
The definition of the damping ratio and natural frequency presumes that the overall feedback system is well approximated by a second order system; i.e. the system has a dominant pair of poles. This is often not the case, so it is good practice to simulate the final design to check if the project goals are satisfied.
Definition.
The root locus of a feedback system is the graphical representation in the complex "s"-plane of the possible locations of its closed-loop poles for varying values of a certain system parameter. The points that are part of the root locus satisfy the angle condition. The value of the parameter for a certain point of the root locus can be obtained using the magnitude condition.
Suppose there is a feedback system with input signal formula_1 and output signal formula_2. The forward path transfer function is formula_3; the feedback path transfer function is formula_4.
For this system, the closed-loop transfer function is given by
Thus, the closed-loop poles of the closed-loop transfer function are the roots of the characteristic equation formula_6. The roots of this equation may be found wherever formula_7.
In systems without pure delay, the product formula_8 is a rational polynomial function and may be expressed as
where formula_10 are the formula_11 zeros, formula_12 are the formula_13 poles, and formula_14 is a scalar gain. Typically, a root locus diagram will indicate the transfer function's pole locations for varying values of the parameter formula_14. A root locus plot will be all those points in the "s"-plane where formula_7 for any value of formula_14.
The factoring of formula_14 and the use of simple monomials means the evaluation of the rational polynomial can be done with vector techniques that add or subtract angles and multiply or divide magnitudes. The vector formulation arises from the fact that each monomial term formula_19 in the factored formula_8 represents the vector from formula_21 to formula_22 in the s-plane. The polynomial can be evaluated by considering the magnitudes and angles of each of these vectors.
According to vector mathematics, the angle of the result of the rational polynomial is the sum of all the angles in the numerator minus the sum of all the angles in the denominator. So to test whether a point in the "s"-plane is on the root locus, only the angles to all the open loop poles and zeros need be considered. This is known as the angle condition.
Similarly, the magnitude of the result of the rational polynomial is the product of all the magnitudes in the numerator divided by the product of all the magnitudes in the denominator. It turns out that the calculation of the magnitude is not needed to determine if a point in the s-plane is part of the root locus because formula_14 varies and can take an arbitrary real value. For each point of the root locus a value of formula_14 can be calculated. This is known as the magnitude condition.
The root locus only gives the location of closed loop poles as the gain formula_14 is varied. The value of formula_14 does not affect the location of the zeros. The open-loop zeros are the same as the closed-loop zeros.
Angle condition.
A point formula_22 of the complex "s"-plane satisfies the angle condition if
which is the same as saying that
that is, the sum of the angles from the open-loop zeros to the point formula_22 (measured per zero w.r.t. a horizontal running through that zero) minus the angles from the open-loop poles to the point formula_22 (measured per pole w.r.t. a horizontal running through that pole) has to be equal to formula_32, or 180 degrees. Note that these interpretations should not be mistaken for the angle differences between the point formula_22 and the zeros/poles.
Magnitude condition.
A value of formula_14 satisfies the magnitude condition for a given formula_22 point of the root locus if
which is the same as saying that
Sketching root locus.
Using a few basic rules, the root locus method can plot the overall shape of the path (locus) traversed by the roots as the value of formula_14 varies. The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of formula_14. The rules are the following:
Let "P" be the number of poles and "Z" be the number of zeros:
The asymptotes intersect the real axis at formula_41 (which is called the centroid) and depart at angle formula_42 given by:
where formula_45 is the sum of all the locations of the poles, formula_46 is the sum of all the locations of the explicit zeros and formula_47 denotes that we are only interested in the real part.
The breakaway points are located at the roots of the following equation:
Once you solve for "z", the real roots give you the breakaway/reentry points. Complex roots correspond to a lack of breakaway/reentry.
Plotting root locus.
Given the general closed-loop denominator rational polynomial
the characteristic equation can be simplified to
The solutions of formula_22 to this equation are the root loci of the closed-loop transfer function.
Example.
Given
we will have the characteristic equation
The following MATLAB code will plot the root locus of the closed-loop transfer function as formula_14 varies using the described manual method as well as the codice_1 built-in function:
% Manual method
K_array = (0:0.1:220).'; % .' is a transpose. Looking up in Matlab documentation.
NK = length(K_array);
x_array = zeros(NK, 3);
y_array = zeros(NK, 3);
for nK = 1:NK
 K = K_array(nK);
 C = [1, 3, (5 + K), (1 + 3*K)];
 r = roots(C).';
 x_array(nK,:) = real(r);
 y_array(nK,:) = imag(r);
end
figure;
plot(x_array, y_array);
grid on;
% Built-in method
sys = tf([1, 3], [1, 3, 5, 1]);
figure;
rlocus(sys);
"z"-plane versus "s"-plane.
The root locus method can also be used for the analysis of sampled data systems by computing the root locus in the "z"-plane, the discrete counterpart of the "s"-plane. The equation maps continuous "s"-plane poles (not zeros) into the "z"-domain, where is the sampling period. The stable, left half "s"-plane maps into the interior of the unit circle of the "z"-plane, with the "s"-plane origin equating to "|z|" = 1 (because "e"0 = 1). A diagonal line of constant damping in the "s"-plane maps around a spiral from (1,0) in the "z" plane as it curves in toward the origin. The Nyquist aliasing criteria is expressed graphically in the "z"-plane by the "x"-axis, where . The line of constant damping just described spirals in indefinitely but in sampled data systems, frequency content is aliased down to lower frequencies by integral multiples of the Nyquist frequency. That is, the sampled response appears as a lower frequency and better damped as well since the root in the "z"-plane maps equally well to the first loop of a different, better damped spiral curve of constant damping. Many other interesting and relevant mapping properties can be described, not least that "z"-plane controllers, having the property that they may be directly implemented from the "z"-plane transfer function (zero/pole ratio of polynomials), can be imagined graphically on a "z"-plane plot of the open loop transfer function, and immediately analyzed utilizing root locus.
Since root locus is a graphical angle technique, root locus rules work the same in the and planes.
The idea of a root locus can be applied to many systems where a single parameter is varied. For example, it is useful to sweep any system parameter for which the exact value is uncertain in order to determine its behavior.


Software independence

The term "software independence" (SI) was coined by Dr. Ron Rivest and NIST researcher John Wack. A software independent voting machine is one whose tabulation record does not rely solely on software. The goal of an SI system is to definitively determine whether all votes were recorded legitimately or in error.
The technical definition of SI is:
"A voting system is software-independent if an undetected change or error in its software cannot cause an undetectable change or error in an election outcome."
SI has been redefined as a global property for a tabulation of votes rather than of each individual vote, aiming to detect rather than prevent error and fraud through human processes.
TGDC Resolution.
The Election Assistance Commission's Technical Guidelines Development Committee adopted an SI resolution for the next iteration of the Voluntary Voting System Guidelines (VVSG):
"Election officials and vendors have appropriately responded to the growing complexity of voting systems by adding more stringent access controls, encryption, testing, and physical security to election procedures and systems. The TGDC has considered current threats to voting systems and, at this time, finds that security concerns do not warrant replacing deployed voting systems where EAC Best Practices are used."
"To provide auditability and proactively address the increasing difficulty of protecting against all prospective threats, the TGDC directs STS to write requirements for the next version of the VVSG requiring the next generation of voting systems to be software independent. The TGDC directs STS and HFP to draft usability and accessibility requirements to ensure that all voters can verify the independent voting record."
"The TGDC further directs STS and Core Requirements and Testing Subcommittees (CRT) to draft requirements to ensure that systems that produce independently verifiable voting records are reliable and provide adequate support for audits."
Example systems.
Examples of software-independent voting systems are optical scan voting systems and direct recording electronic voting computers (DRE) with a voter verified paper audit trail.


Winnow (algorithm)

The winnow algorithm is a technique from machine learning for learning a linear classifier from labeled examples. It is very similar to the perceptron algorithm. However, the perceptron algorithm uses an additive weight-update scheme, while Winnow uses a multiplicative scheme that allows it to perform much better when many dimensions are irrelevant (hence its name winnow). It is a simple algorithm that scales well to high-dimensional data. During training, Winnow is shown a sequence of positive and negative examples. From these it learns a decision hyperplane that can then be used to label novel examples as positive or negative. The algorithm can also be used in the online learning setting, where the learning and the classification phase are not clearly separated.
Algorithm.
The basic algorithm, Winnow1, is as follows. The instance space is formula_1, that is, each instance is described as a set of Boolean-valued features. The algorithm maintains non-negative weights formula_2 for formula_3, which are initially set to 1, one weight for each feature. When the learner is given an example formula_4, it applies the typical prediction rule for linear classifiers:
Here formula_6 is a real number that is called the "threshold". Together with the weights, the threshold defines a dividing hyperplane in the instance space. Good bounds are obtained if formula_7 (see below).
For each example with which it is presented, the learner applies the following update rule:
A typical value for is 2.
There are many variations to this basic approach. "Winnow2" is similar except that in the demotion step the weights are divided by instead of being set to 0. "Balanced Winnow" maintains two sets of weights, and thus two hyperplanes. This can then be generalized for multi-label classification.
Mistake bounds.
In certain circumstances, it can be shown that the number of mistakes Winnow makes as it learns has an upper bound that is independent of the number of instances with which it is presented. If the Winnow1 algorithm uses formula_14 and formula_15 on a target function that is a formula_16-literal monotone disjunction given by formula_17, then for any sequence of instances the total number of mistakes is bounded by:
formula_18.


Hole drilling method

The hole drilling method is a method for measuring residual stresses, in a material. Residual stress occurs in a material in the absence of external loads. Residual stress interacts with the applied loading on the material to affect the overall strength, fatigue, and corrosion performance of the material. Residual stresses are measured through experiments. The hole drilling method is one of the most used methods for residual stress measurement.
The hole drilling method can measure macroscopic residual stresses near the material surface. The principle is based on drilling of a small hole into the material. When the material containing residual stress is removed the remaining material reaches a new equilibrium state. The new equilibrium state has associated deformations around the drilled hole. The deformations are related to the residual stress in the volume of material that was removed through drilling. The deformations around the hole are measured during the experiment using strain gauges or optical methods. The original residual stress in the material is calculated from the measured deformations. The hole drilling method is popular for its simplicity and it is suitable for a wide range of applications and materials.
Key advantages of the hole drilling method include rapid preparation, versatility of the technique for different materials, and reliability. Conversely, the hole drilling method is limited in depth of analysis and specimen geometry, and is at least semi-destructive.
History and development.
The idea of measuring the residual stress by drilling a hole and registering the change of the hole diameter was first proposed by Mathar in 1934. In 1966 Rendler and Vignis introduced a systematic and repeatable procedure of hole drilling to measure the residual stress. In the following period the method was further developed in terms of drilling techniques, measuring the relieved deformations, and the residual stress evaluation itself. A very important milestone is the use of finite element method to compute the calibration coefficients and to evaluate the residual stresses from the measured relieved deformations (Schajer, 1981). That allowed especially the evaluation of residual stresses which are not constant along the depth. It also brought further possibilities of using the method, e.g., for inhomogeneous materials, coatings, etc. The measurement and evaluation procedure is standardised by the norm ASTM E837 of the American Society for Testing and Materials which also contributed to the popularity of the method. The hole drilling is currently one of the most widespread methods of measuring the residual stress. Modern computational methods are used for the evaluation. The method is being developed especially in terms of drilling techniques and the possibilities of measuring the deformations.
Fundamental principles.
The hole drilling method of measuring the residual stresses is based on drilling a small hole in the material surface. This relieves the residual stresses and the associated deformations around the hole. The relieved deformations are measured in at least three independent directions around the hole. The original residual stress in the material is then evaluated based on the measured deformations and using the so-called calibration coefficients. The hole is made by a cylindrical end mill or by alternative techniques. Deformations are most often measured using strain gauges (strain gauge rosettes).
The biaxial stress in the surface plane can be measured. The method is often referred to as semi-destructive thanks to the small material damage. The method is relatively simple, fast, the measuring device is usually portable. Disadvantages include the destructive character of the technique, limited resolution, and a lower accuracy of the evaluation in the case of nonuniform stresses or inhomogeneous material properties.
The so-called calibration coefficients play an important role in the residual stress evaluation. They are used to convert the relieved deformations to the original residual stress in the material. The coefficients can be theoretically derived for a through hole and a homogeneous stress. Then they depend only on the material properties, hole radius, and the distance from the hole. In the vast majority of practical applications, however, the preconditions for using the theoretically derived coefficients are not met, e.g., the integral deformation over the tensometer area is not included, the hole is blind instead of through, etc. Therefore, coefficients taking into account the practical aspects of measuring are used. They are mostly determined by a numerical computation using the finite element method. They express the relation between the relieved deformations and the residual stresses, taking into account the hole size, hole depth, shape of the tensometric rosette, material, and other parameters.
The evaluation of the residual stresses depends on the method used to calculate them from the measured relieved deformations. All the evaluation methods are built on the basic principles. They differ in the preconditions for use, the accuracy requirements on the calibration coefficients, or the possibility to take additional influences into account. In general, the hole is made in successive steps and the relieved deformations are measured after each step.
Evaluation methods for the residual stress.
Several methods have been developed for the evaluation of residual stresses from the relieved deformations. The fundamental method is the "equivalent uniform stress method". The coefficients for particular hole diameter, rosette type, and hole depth are published in the norm ASTM E837. The method is suitable for a constant or little changing stress along the depth. It can be used as a guideline for non-constant stresses, however, the method may give highly distorted results.
The most general method is the "integral method". It calculates the influence of the relieved stress in the given depth which, however, changes with the total depth of the hole. The calibration coefficients are expressed as matrices. The evaluation leads to a system of equations whose solution is a vector of residual stresses in particular depths. A numerical simulation is required to get the calibration coefficients. The integral method and its coefficients are defined in the norm ASTM E837.
There are other evaluation methods that have lower demands on the calibration coefficients and on the evaluation process itself. These include "the average stress method" and "the incremental strain method". Both the methods are based on the assumption that the change in deformation is caused solely by the relieved stress on the drilled increment. They are suitable only if there are small changes in the stress profiles. Both the methods give numerically correct results for uniform stresses.
"The power series method" and "the spline method" are other modifications of the integral method. They both take into account both the distance of the stress effect from the surface and the total hole depth. Contrary to the integral method, the resulting stress values are approximated by a polynomial or a spline. The power series method is very stable but it cannot capture rapidly changing stress values. The spline method is more stable and less susceptible to errors than the integral method. It can capture the actual stress values better than the power series method. The main disadvantage are the complicated mathematical calculations needed to solve a system of nonlinear equations.
Using the hole drilling method.
The hole drilling method finds its use in many industrial areas dealing with material production and processing. The most important technologies include heat treatment, mechanical and thermal surface finishing, machining, welding, coating, or manufacturing composites. Despite its relative universality, the method requires these fundamental preconditions to be met: the possibility to drill the material, the possibility to apply the tensometric rosettes (or other means of measuring the deformations), and the knowledge of the material properties. Additional conditions can affect the accuracy and repeatability of the measuring. These include especially the size and shape of the sample, distance of the measured area from the edges, homogeneity of the material, presence of residual stress gradients, etc. Hole drilling can be performed in the laboratory or as a field measurement, making it ideal for measuring actual stresses in large components that cannot be moved.


Electroforming

Electroforming is a metal forming process in which parts are fabricated through electrodeposition on a model, known in the industry as a mandrel. Conductive (metallic) mandrels are treated to create a mechanical parting layer, or are chemically passivated to limit electroform adhesion to the mandrel and thereby allow its subsequent separation. Non-conductive (glass, silicon, plastic) mandrels require the deposition of a conductive layer prior to electrodeposition. Such layers can be deposited chemically, or using vacuum deposition techniques (e.g., gold sputtering). The outer surface of the mandrel forms the inner surface of the form.
The process involves passing direct current through an electrolyte containing salts of the metal being electroformed. The anode is the solid metal being electroformed, and the cathode is the mandrel, onto which the electroform gets plated (deposited). The process continues until the required electroform thickness is achieved. The mandrel is then either separated intact, melted away, or chemically dissolved.
The surface of the finished part that was in intimate contact with the mandrel is replicated in fine detail with respect to the original, and is not subject to the shrinkage that would normally be experienced in a foundry cast metal object, or the tool marks of a milled part. The solution side of the part is less well defined, and that loss of definition increases with thickness of the deposit. In extreme cases, where a thickness of several millimetres is required, there is preferential build-up of material on sharp outside edges and corners. This tendency can be reduced by shielding, or a process known as periodic reverse, where the electroforming current is reversed for short periods and the excess is preferentially dissolved electrochemically. The finished form can either be the finished part, or can be used in a subsequent process to produce a positive of the original mandrel shape, such as with vinyl records or CD and DVD stamper manufacture.
In recent years, due to its ability to replicate a mandrel surface with practically no loss of fidelity, electroforming has taken on new importance in the fabrication of micro and nano-scale metallic devices and in producing precision injection molds with micro- and nano-scale features for production of non-metallic micro-molded objects.
Process.
In the basic electroforming process, an electrolytic bath is used to deposit nickel or other electroformable metal onto a conductive surface of a model (mandrel). Once the deposited material has been built up to the desired thickness, the electroform is parted from the substrate. This process allows precise replication of the mandrel surface texture and geometry at low unit cost with high repeatability and excellent process control.
If the mandrel is made of a non-conductive material it can be coated with a thin conductive layer.
Advantages and disadvantages.
The main advantage of electroforming is that it accurately replicates the external shape of the mandrel. Generally, machining a cavity accurately is more challenging than machining a convex shape, however the opposite holds true for electroforming because the mandrel's exterior can be accurately machined and then used to electroform a precision cavity.
Compared to other basic metal forming processes (casting, forging, stamping, deep drawing, machining and fabricating) electroforming is very effective when requirements call for extreme tolerances, complexity or light weight. The precision and resolution inherent in the photo-lithographically produced conductive patterned substrate, allows finer geometries to be produced to tighter tolerances while maintaining superior edge definition with a near optical finish. Electroformed metal can be extremely pure, with superior properties over wrought metal due to its refined crystal structure. Multiple layers of electroformed metals can be bonded together, or to different substrate materials to produce complex structures with "grown-on" flanges and bosses.
Tolerances of 1.5 to 3 nanometres have been reported.
A wide variety of shapes and sizes can be made by electroforming, the principal limitation being the need to part the product from the mandrel. Since the fabrication of a product requires only a single model or mandrel, low production quantities can be made economically.


Exploration-exploitation dilemma

The exploration-exploitation dilemma, also known as the explore-exploit tradeoff, is a fundamental concept in decision-making that arises in many domains. It is depicted as the balancing act between two opposing strategies. Exploitation involves choosing the best-known option based on past experiences, while exploration involves trying out new options that may lead to better outcomes in the future. Finding the optimal balance between these two strategies is a crucial challenge in many decision-making situations, where the goal is to maximize long-term benefits.
Application in machine learning.
In the context of machine learning, the exploration-exploitation tradeoff is often encountered in reinforcement learning, a type of machine learning that involves training agents to make decisions based on feedback from the environment. The agent must decide whether to exploit the current best-known policy or explore new policies to improve its performance. Various algorithms have been developed to address this challenge, such as epsilon-greedy, Thompson sampling, and the upper confidence bound.


Pencil milling

Pencil milling is a cleanup toolpath generated by computer-aided manufacturing (CAM) programs to machine internal corners and fillets with smaller radius tools to remove the remaining material that are inaccessible with larger tools used for previous roughing, semi-finishing, and finishing toolpaths. The name comes from the way that a pencil could naturally be drawn along these corners. It is sometimes called a rolling ball toolpath.
Often a constant step-over passes are derived from single pencil pass to create parallel pencil passes that are very good for cleaning up corners and fillets where excess material remains from a bigger cutter.
Generating pencil toolpaths.
There are several alternative algorithms for generating pencil passes. The method most commonly published in the academic literature involves creating tool surface offsets of the model surfaces and intersecting them to find the common line where the cutter would be in contact with two surfaces at once. An example of this implementation uses the ZMap method described by Park, et al.
The industrial method, used in commercial CAM software, differs substantially and works by detecting double-contact points and linking them up into a chain to form a toolpath. A double-contact point is a pair of cutter locations displaced by a tiny distance horizontally, but with a large difference in height or sudden change in contact point. These positions can be located very precisely by binary subdivision, where a cutter location created between a pair of close cutter locations will almost always be continuous with one or the other side.


Dual control theory

Dual control theory is a branch of control theory that deals with the control of systems whose characteristics are initially unknown. It is called "dual" because in controlling such a system the controller's objectives are twofold:
These two objectives may be partly in conflict.
In the context of reinforcement learning, this is known as the exploration-exploitation trade-off (e.g. Multi-armed bandit#Empirical motivation).
Dual control theory was developed by Alexander Aronovich Fel'dbaum in 1960. He showed that in principle the optimal solution can be found by dynamic programming, but this is often impractical; as a result a number of methods for designing sub-optimal dual controllers have been devised.
Example.
To use an analogy: if you are driving a new car you want to get to your destination cheaply and smoothly, but you also want to see how well the car accelerates, brakes and steers so as to get a better feel for how to drive it, so you will do some test manoeuvers for this purpose. Similarly a dual controller will inject a so-called probing (or exploration) signal into the system that may detract from short-term performance but will improve control in the future.


Outline of machines

Machine – mechanical system that provides the useful application of power to achieve movement. A machine consists of a power source, or engine, and a mechanism or transmission for the controlled use of this power. The combination of force and movement, known as power, is an important characteristic of a machine.
Machine theory.
The mathematical tools for the analysis of movement in machines:
Machine elements.
Movement in a machine is controlled by mechanism elements that the shape forces and movement and structural elements that support these mechanisms.
General machine-related concepts.
Mechanical components.
Airfoil.
Airfoil


Step response

The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a dynamical system using an evolution parameter.
From a practical standpoint, knowing how the system responds to a sudden input is important because large and possibly fast deviations from the long term steady state may have extreme effects on the component itself and on other portions of the overall system dependent on this component. In addition, the overall system cannot act until the component's output settles down to some vicinity of its final state, delaying the overall system response. Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.
Formal mathematical description.
This section provides a formal mathematical definition of step response in terms of the abstract mathematical concept of a dynamical system formula_1: all notations and assumptions required for the following description are listed here.
Nonlinear dynamical system.
For a general dynamical system, the step response is defined as follows:
It is the evolution function when the control inputs (or source term, or forcing inputs) are Heaviside functions: the notation emphasizes this concept showing "H"("t") as a subscript.
Linear dynamical system.
For a linear time-invariant (LTI) black box, let formula_9 for notational convenience: the step response can be obtained by convolution of the Heaviside step function control and the impulse response "h"("t") of the system itself
which for an LTI system is equivalent to just integrating the latter. Conversely, for an LTI system, the derivative of the step response yields the impulse response:
However, these simple relations are not true for a non-linear or time-variant system.
Time domain versus frequency domain.
Instead of frequency response, system performance may be specified in terms of parameters describing time-dependence of response. The step response can be described by the following quantities related to its "time behavior",
In the case of linear dynamic systems, much can be inferred about the system from these characteristics. Below the step response of a simple two-pole amplifier is presented, and some of these terms are illustrated.
In LTI systems, the function that has the steepest slew rate that doesn't create overshoot or ringing is the Gaussian function. This is because it is the only function whose Fourier transform has the same shape.
Feedback amplifiers.
This section describes the step response of a simple negative feedback amplifier shown in Figure 1. The feedback amplifier consists of a main open-loop amplifier of gain "A"OL and a feedback loop governed by a feedback factor β. This feedback amplifier is analyzed to determine how its step response depends upon the time constants governing the response of the main amplifier, and upon the amount of feedback used.
A negative-feedback amplifier has gain given by (see negative feedback amplifier):
where "A"OL = open-loop gain, "A"FB = closed-loop gain (the gain with negative feedback present) and "β" = feedback factor.
With one dominant pole.
In many cases, the forward amplifier can be sufficiently well modeled in terms of a single dominant pole of time constant τ, that it, as an open-loop gain given by:
with zero-frequency gain "A"0 and angular frequency ω = 2π"f". This forward amplifier has unit step response
an exponential approach from 0 toward the new equilibrium value of "A"0.
The one-pole amplifier's transfer function leads to the closed-loop gain:
This closed-loop gain is of the same form as the open-loop gain: a one-pole filter. Its step response is of the same form: an exponential decay toward the new equilibrium value. But the time constant of the closed-loop step function is "τ" / (1 + "β" "A"0), so it is faster than the forward amplifier's response by a factor of 1 + "β" "A"0:
As the feedback factor "β" is increased, the step response will get faster, until the original assumption of one dominant pole is no longer accurate. If there is a second pole, then as the closed-loop time constant approaches the time constant of the second pole, a two-pole analysis is needed.
Two-pole amplifiers.
In the case that the open-loop gain has two poles (two time constants, "τ"1, "τ"2), the step response is a bit more complicated. The open-loop gain is given by:
with zero-frequency gain "A"0 and angular frequency "ω" = 2"πf".
Analysis.
The two-pole amplifier's transfer function leads to the closed-loop gain:
The time dependence of the amplifier is easy to discover by switching variables to "s" = "j"ω, whereupon the gain becomes:
The poles of this expression (that is, the zeros of the denominator) occur at:
which shows for large enough values of "βA"0 the square root becomes the square root of a negative number, that is the square root becomes imaginary, and the pole positions are complex conjugate numbers, either "s"+ or "s"−; see Figure 2:
with
and
Using polar coordinates with the magnitude of the radius to the roots given by |"s"| (Figure 2):
and the angular coordinate φ is given by:
Tables of Laplace transforms show that the time response of such a system is composed of combinations of the two functions:
which is to say, the solutions are damped oscillations in time. In particular, the unit step response of the system is:
which simplifies to
when "A"0 tends to infinity and the feedback factor "β" is one.
Notice that the damping of the response is set by ρ, that is, by the time constants of the open-loop amplifier. In contrast, the frequency of oscillation is set by μ, that is, by the feedback parameter through β"A"0. Because ρ is a sum of reciprocals of time constants, it is interesting to notice that ρ is dominated by the "shorter" of the two.
Results.
Figure 3 shows the time response to a unit step input for three values of the parameter μ. It can be seen that the frequency of oscillation increases with μ, but the oscillations are contained between the two asymptotes set by the exponentials [ 1 − exp(−"ρt") ] and [ 1 + exp(−ρt) ]. These asymptotes are determined by ρ and therefore by the time constants of the open-loop amplifier, independent of feedback.
The phenomenon of oscillation about the final value is called ringing. The overshoot is the maximum swing above final value, and clearly increases with μ. Likewise, the undershoot is the minimum swing below final value, again increasing with μ. The settling time is the time for departures from final value to sink below some specified level, say 10% of final value.
The dependence of settling time upon μ is not obvious, and the approximation of a two-pole system probably is not accurate enough to make any real-world conclusions about feedback dependence of settling time. However, the asymptotes [ 1 − exp(−"ρt") ] and [ 1 + exp (−"ρt") ] clearly impact settling time, and they are controlled by the time constants of the open-loop amplifier, particularly the shorter of the two time constants. That suggests that a specification on settling time must be met by appropriate design of the open-loop amplifier.
The two major conclusions from this analysis are:
As an aside, it may be noted that real-world departures from this linear two-pole model occur due to two major complications: first, real amplifiers have more than two poles, as well as zeros; and second, real amplifiers are nonlinear, so their step response changes with signal amplitude.
Control of overshoot.
How overshoot may be controlled by appropriate parameter choices is discussed next.
Using the equations above, the amount of overshoot can be found by differentiating the step response and finding its maximum value. The result for maximum step response "S"max is:
The final value of the step response is 1, so the exponential is the actual overshoot itself. It is clear the overshoot is zero if "μ" = 0, which is the condition:
This quadratic is solved for the ratio of time constants by setting "x" = ("τ"1 / "τ"2)1/2 with the result
Because β "A"0 ≫ 1, the 1 in the square root can be dropped, and the result is
In words, the first time constant must be much larger than the second. To be more adventurous than a design allowing for no overshoot we can introduce a factor "α" in the above relation:
and let α be set by the amount of overshoot that is acceptable.
Figure 4 illustrates the procedure. Comparing the top panel (α = 4) with the lower panel (α = 0.5) shows lower values for α increase the rate of response, but increase overshoot. The case α = 2 (center panel) is the "maximally flat" design that shows no peaking in the Bode gain vs. frequency plot. That design has the rule of thumb built-in safety margin to deal with non-ideal realities like multiple poles (or zeros), nonlinearity (signal amplitude dependence) and manufacturing variations, any of which can lead to too much overshoot. The adjustment of the pole separation (that is, setting α) is the subject of frequency compensation, and one such method is pole splitting.
Control of settling time.
The amplitude of ringing in the step response in Figure 3 is governed by the damping factor exp(−"ρt"). That is, if we specify some acceptable step response deviation from final value, say Δ, that is:
this condition is satisfied regardless of the value of β "A"OL provided the time is longer than the settling time, say "t"S, given by:
where the τ1 ≫ τ2 is applicable because of the overshoot control condition, which makes "τ"1 = "αβA"OL τ2. Often the settling time condition is referred to by saying the settling period is inversely proportional to the unity gain bandwidth, because 1/(2"π" "τ"2) is close to this bandwidth for an amplifier with typical dominant pole compensation. However, this result is more precise than this rule of thumb. As an example of this formula, if the settling time condition is "t"S = 8 "τ"2.
In general, control of overshoot sets the time constant ratio, and settling time "t"S sets τ2.
System Identification using the Step Response: System with two real poles.
This method uses significant points of the step response. There is no need to guess tangents to the measured Signal. The equations are derived using numerical simulations, determining some significant ratios and fitting parameters of nonlinear equations. See also.
Here the steps:
Phase margin.
Next, the choice of pole ratio "τ"1/"τ"2 is related to the phase margin of the feedback amplifier. The procedure outlined in the Bode plot article is followed. Figure 5 is the Bode gain plot for the two-pole amplifier in the range of frequencies up to the second pole position. The assumption behind Figure 5 is that the frequency "f"0 dB lies between the lowest pole at "f"1 = 1/(2πτ1) and the second pole at "f"2 = 1/(2πτ2). As indicated in Figure 5, this condition is satisfied for values of α ≥ 1.
Using Figure 5 the frequency (denoted by "f"0 dB) is found where the loop gain β"A"0 satisfies the unity gain or 0 dB condition, as defined by:
The slope of the downward leg of the gain plot is (20 dB/decade); for every factor of ten increase in frequency, the gain drops by the same factor:
The phase margin is the departure of the phase at "f"0 dB from −180°. Thus, the margin is:
Because "f"0 dB / "f"1 = "βA"0 ≫ 1, the term in "f"1 is 90°. That makes the phase margin:
In particular, for case "α" = 1, "φ"m = 45°, and for "α" = 2, "φ"m = 63.4°. Sansen recommends "α" = 3, "φ"m = 71.6° as a "good safety position to start with".
If α is increased by shortening "τ"2, the settling time "t"S also is shortened. If "α" is increased by lengthening "τ"1, the settling time "t"S is little altered. More commonly, both "τ"1 "and" "τ"2 change, for example if the technique of pole splitting is used.
As an aside, for an amplifier with more than two poles, the diagram of Figure 5 still may be made to fit the Bode plots by making "f"2 a fitting parameter, referred to as an "equivalent second pole" position.