Minimum phase

In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
The most general causal LTI transfer function can be uniquely factored into a series of an all-pass and a minimum phase system. The system function is then the product of the two parts, and in the time domain the response of the system is the convolution of the two part responses. The difference between a minimum phase and a general transfer function is that a minimum phase system has all of the poles and zeroes of its transfer function in the left half of the s-plane representation (in discrete time, respectively, inside the unit circle of the z-plane). Since inverting a system function leads to poles turning to zeroes and vice versa, and poles on the right side (s-plane imaginary line) or outside (z-plane unit circle) of the complex plane lead to unstable systems, only the class of minimum phase systems is closed under inversion. Intuitively, the minimum phase part of a general causal system implements its amplitude response with minimum group delay, while its all pass part corrects its phase response alone to correspond with the original system function.
The analysis in terms of poles and zeroes is exact only in the case of transfer functions which can be expressed as ratios of polynomials. In the continuous time case, such systems translate into networks of conventional, idealized LCR networks. In discrete time, they conveniently translate into approximations thereof, using addition, multiplication, and unit delay. It can be shown that in both cases, system functions of rational form with increasing order can be used to efficiently approximate any other system function; thus even system functions lacking a rational form, and so possessing an infinitude of poles and/or zeroes, can in practice be implemented as efficiently as any other.
In the context of causal, stable systems, we would in theory be free to choose whether the zeroes of the system function are outside of the stable range (to the right or outside) if the closure condition wasn't an issue. However, inversion is of great practical importance, just as theoretically perfect factorizations are in their own right. (Cf. the spectral symmetric/antisymmetric decomposition as another important example, leading e.g. to Hilbert transform techniques.) Many physical systems also naturally tend towards minimum phase response, and sometimes have to be inverted using other physical systems obeying the same constraint.
Insight is given below as to why this system is called minimum-phase, and why the basic idea applies even when the system function cannot be cast into a rational form that could be implemented.
Inverse system.
A system formula_1 is invertible if we can uniquely determine its input from its output. I.e., we can find a system formula_2 such that if we apply formula_1 followed by formula_2, we obtain the identity system formula_5. (See Inverse matrix for a finite-dimensional analog). That is,
formula_6
Suppose that formula_7 is input to system formula_1 and gives output formula_9.
formula_10
Applying the inverse system formula_2 to formula_9 gives the following
formula_13
So we see that the inverse system formula_14 allows us to determine uniquely the input formula_7 from the output formula_9.
Discrete-time example.
Suppose that the system formula_1 is a discrete-time, linear, time-invariant (LTI) system described by the impulse response formula_18 for in . Additionally, suppose formula_2 has impulse response formula_20. The cascade of two LTI systems is a convolution. In this case, the above relation is the following:
formula_21
where formula_22 is the Kronecker delta or the identity system in the discrete-time case. (Changing the order of formula_23 and formula_24 is allowed because of commutativity of the convolution operation.) Note that this inverse system formula_2 need not be unique.
Minimum phase system.
When we impose the constraints of causality and stability, the inverse system is unique; and the system formula_1 and its inverse formula_2 are called minimum-phase. The causality and stability constraints in the discrete-time case are the following (for time-invariant systems where is the system's impulse response):
Causality.
formula_28
and
formula_29
Stability.
formula_30
and
formula_31
See the article on stability for the analogous conditions for the continuous-time case.
Frequency analysis.
Discrete-time frequency analysis.
Performing frequency analysis for the discrete-time case will provide some insight. The time-domain equation is the following:
formula_32
Applying the Z-transform gives the following relation in the z-domain
formula_33
From this relation, we realize that
formula_34
For simplicity, we consider only the case of a rational transfer function . Causality and stability imply that all poles of must be strictly inside the unit circle (See stability). Suppose
formula_35
where and are polynomial in . Causality and stability imply that the poles – the roots of – must be strictly inside the unit circle. We also know that
formula_36
So, causality and stability for formula_37 imply that its poles – the roots of – must be inside the unit circle. These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the unit circle.
Continuous-time frequency analysis.
Analysis for the continuous-time case proceeds in a similar manner except that we use the Laplace transform for frequency analysis. The time-domain equation is the following.
formula_38
where formula_39 is the Dirac delta function. The Dirac delta function is the identity operator in the continuous-time case because of the sifting property with any signal .
formula_40
Applying the Laplace transform gives the following relation in the s-plane.
formula_41
From this relation, we realize that
formula_42
Again, for simplicity, we consider only the case of a rational transfer function . Causality and stability imply that all poles of must be strictly inside the left-half s-plane (See stability). Suppose
formula_43
where and are polynomial in . Causality and stability imply that the poles – the roots of – must be inside the left-half s-plane. We also know that
formula_44
So, causality and stability for formula_45 imply that its poles – the roots of – must be strictly inside the left-half s-plane. These two constraints imply that both the zeros and the poles of a minimum phase system must be strictly inside the left-half s-plane.
Relationship of magnitude response to phase response.
A minimum-phase system, whether discrete-time or continuous-time, has an additional useful property that the natural logarithm of the magnitude of the frequency response (the "gain" measured in nepers which is proportional to dB) is related to the phase angle of the frequency response (measured in radians) by the Hilbert transform. That is, in the continuous-time case, let
formula_46
be the complex frequency response of system . Then, only for a minimum-phase system, the phase response of is related to the gain by
formula_47
where formula_48 denotes the Hilbert transform, and, inversely,
formula_49
Stated more compactly, let
formula_50
where formula_51 and formula_52 are real functions of a real variable. Then
formula_53
and
formula_54
The Hilbert transform operator is defined to be
formula_55
An equivalent corresponding relationship is also true for discrete-time minimum-phase systems.
Minimum phase in the time domain.
For all causal and stable systems that have the same magnitude response, the minimum phase system has its energy concentrated near the start of the impulse response. i.e., it minimizes the following function which we can think of as the delay of energy in the impulse response.
formula_56
Minimum phase as minimum group delay.
For all causal and stable systems that have the same magnitude response, the minimum phase system has the minimum group delay. The following proof illustrates this idea of minimum group delay.
Suppose we consider one zero formula_57 of the transfer function formula_58. Let's place this zero formula_57 inside the unit circle (formula_60) and see how the group delay is affected.
formula_61
Since the zero formula_57 contributes the factor formula_63 to the transfer function, the phase contributed by this term is the following.
formula_64
formula_65 contributes the following to the group delay.
formula_66
The denominator and formula_67 are invariant to reflecting the zero formula_57 outside of the unit circle, i.e., replacing formula_57 with formula_70. However, by reflecting formula_57 outside of the unit circle, we increase the magnitude of formula_72 in the numerator. Thus, having formula_57 inside the unit circle minimizes the group delay contributed by the factor formula_63. We can extend this result to the general case of more than one zero since the phase of the multiplicative factors of the form formula_75 is additive. I.e., for a transfer function with formula_76 zeros,
formula_77
So, a minimum phase system with all zeros inside the unit circle minimizes the group delay since the group delay of each individual zero is minimized.
Non-minimum phase.
Systems that are causal and stable whose inverses are causal and unstable are known as "non-minimum-phase" systems. A given non-minimum phase system will have a greater phase contribution than the minimum-phase system with the equivalent magnitude response.
Maximum phase.
A "maximum-phase" system is the opposite of a minimum phase system. A causal and stable LTI system is a "maximum-phase" system if its inverse is causal and unstable. That is,
Such a system is called a "maximum-phase system" because it has the maximum group delay of the set of systems that have the same magnitude response. In this set of equal-magnitude-response systems, the maximum phase system will have maximum energy delay.
For example, the two continuous-time LTI systems described by the transfer functions
formula_78
have equivalent magnitude responses; however, the second system has a much larger contribution to the phase shift. Hence, in this set, the second system is the maximum-phase system and the first system is the minimum-phase system. These systems are also famously known as nonminimum-phase systems that raise many stability concerns in control. One recent solution to these systems is moving the RHP zeros to the LHP using the PFCD method.
Mixed phase.
A "mixed-phase" system has some of its zeros inside the unit circle and has others outside the unit circle. Thus, its group delay is neither minimum or maximum but somewhere between the group delay of the minimum and maximum phase equivalent system.
For example, the continuous-time LTI system described by transfer function
formula_79
is stable and causal; however, it has zeros on both the left- and right-hand sides of the complex plane. Hence, it is a "mixed-phase" system. To control the transfer functions that include these systems some methods such as internal model controller (IMC), generalized Smith's predictor (GSP) and parallel feedforward control with derivative (PFCD) are proposed.
Linear phase.
A linear-phase system has constant group delay. Non-trivial linear phase or nearly linear phase systems are also mixed phase.


State-space representation

In control engineering, model based fault detection and system identification a state-space representation is a mathematical model of a physical system specified as a set of input, output and variables related by first-order (not involving second derivatives) differential equations or difference equations. Such variables, called state variables, evolve over time in a way that depends on the values they have at any given instant and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables.
The "state space" or "phase space" is the geometric space in which the variables on the axes are the state variables. The state of the system can be represented as a vector, the state vector, within state space.
If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
The state-space method is characterized by significant algebraization of general system theory, which makes it possible to use Kronecker vector-matrix structures. The capacity of these structures can be efficiently applied to research systems with modulation or without it.
The state-space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With formula_1 inputs and formula_2 outputs, we would otherwise have to write down formula_3 Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions.
The state-space model can be applied in subjects such as economics, statistics, computer science and electrical engineering, and neuroscience. In econometrics, for example, state-space models can be used to decompose a time series into trend and cycle, compose individual indicators into a composite index, identify turning points of the business cycle, and estimate GDP using latent and unobserved time series. Many applications rely on the Kalman Filter or a state observer to produce estimates of the current unknown state variables using their previous observations.
State variables.
The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system, formula_4, is usually equal to the order of the system's defining differential equation, but not necessarily. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction. It is important to understand that converting a state-space realization to a transfer function form may lose some internal information about the system, and may provide a description of a system which is stable, when the state-space realization is unstable at certain points. In electric circuits, the number of state variables is often, though not always, the same as the number of energy storage elements in the circuit such as capacitors and inductors. The state variables defined must be linearly independent, i.e., no state variable can be written as a linear combination of the other state variables, or the system cannot be solved.
Linear systems.
The most general state-space representation of a linear system with formula_1 inputs, formula_2 outputs and formula_4 state variables is written in the following form:
where:
In this general formulation, all matrices are allowed to be time-variant (i.e. their elements can depend on time); however, in the common LTI case, matrices will be time invariant. The time variable formula_26 can be continuous (e.g. formula_27) or discrete (e.g. formula_28). In the latter case, the time variable formula_29 is usually used instead of formula_26. Hybrid systems allow for time domains that have both continuous and discrete parts. Depending on the assumptions made, the state-space model representation can assume the following forms:
Example: continuous-time LTI case.
Stability and natural response characteristics of a continuous-time LTI system (i.e., linear with matrices that are constant with respect to time) can be studied from the eigenvalues of the matrix formula_31. The stability of a time-invariant state-space model can be determined by looking at the system's transfer function in factored form. It will then look something like this:
The denominator of the transfer function is equal to the characteristic polynomial found by taking the determinant of formula_33,
The roots of this polynomial (the eigenvalues) are the system transfer function's poles (i.e., the singularities where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is asymptotically stable or marginally stable. An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's Lyapunov stability.
The zeros found in the numerator of formula_35 can similarly be used to determine whether the system is minimum phase.
The system may still be input–output stable (see BIBO stable) even though it is not internally stable. This may be the case if unstable poles are canceled out by zeros (i.e., if those singularities in the transfer function are removable).
Controllability.
The state controllability condition implies that it is possible – by admissible inputs – to steer the states from any initial value to any final value within some finite time window. A continuous time-invariant linear state-space model is controllable if and only if
where rank is the number of linearly independent rows in a matrix, and where "n" is the number of state variables.
Observability.
Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals (i.e., as controllability provides that an input is available that brings any initial state to any desired final state, observability provides that knowing an output trajectory provides enough information to predict the initial state of the system).
A continuous time-invariant linear state-space model is observable if and only if
Transfer function.
The "transfer function" of a continuous time-invariant linear state-space model can be derived in the following way:
First, taking the Laplace transform of
yields
Next, we simplify for formula_40, giving
and thus
Substituting for formula_40 in the output equation
giving
Assuming zero initial conditions formula_46 and a single-input single-output (SISO) system, the transfer function is defined as the ratio of output and input formula_47. For a multiple-input multiple-output (MIMO) system, however, this ratio is not defined. Therefore, assuming zero initial conditions, the transfer function matrix is derived from
using the method of equating the coefficients which yields
Consequently, formula_50 is a matrix with the dimension formula_3 which contains transfer functions for each input output combination. Due to the simplicity of this matrix notation, the state-space representation is commonly used for multiple-input, multiple-output systems. The Rosenbrock system matrix provides a bridge between the state-space representation and its transfer function.
Canonical realizations.
Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system):
Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
The coefficients can now be inserted directly into the state-space model by the following approach:
This state-space realization is called controllable canonical form because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).
The transfer function coefficients can also be used to construct another type of canonical form
This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).
Proper transfer functions.
Transfer functions which are only proper (and not strictly proper) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant.
The strictly proper transfer function can then be transformed into a canonical state-space realization using techniques shown above. The state-space realization of the constant is trivially formula_58. Together we then get a state-space realization with matrices "A", "B" and "C" determined by the strictly proper part, and matrix "D" determined by the constant.
Here is an example to clear things up a bit:
which yields the following controllable realization
Notice how the output also depends directly on the input. This is due to the formula_62 constant in the transfer function.
Feedback.
A common method for feedback is to multiply the output by a matrix "K" and setting this as the input to the system: formula_63.
Since the values of "K" are unrestricted the values can easily be negated for negative feedback.
The presence of a negative sign (the common notation) is merely a notational one and its absence has no impact on the end results.
becomes
solving the output equation for formula_68 and substituting in the state equation results in
The advantage of this is that the eigenvalues of "A" can be controlled by setting "K" appropriately through eigendecomposition of formula_71.
This assumes that the closed-loop system is controllable or that the unstable eigenvalues of "A" can be made stable through appropriate choice of "K".
Example.
For a strictly proper system "D" equals zero. Another fairly common situation is when all states are outputs, i.e. "y" = "x", which yields "C" = "I", the Identity matrix. This would then result in the simpler equations
This reduces the necessary eigendecomposition to just formula_74.
Feedback with setpoint (reference) input.
In addition to feedback, an input, formula_75, can be added such that formula_76.
becomes
solving the output equation for formula_68 and substituting in the state equation
results in
One fairly common simplification to this system is removing "D", which reduces the equations to
Moving object example.
A classical linear system is that of one-dimensional movement of an object (e.g., a cart).
Newton's laws of motion for an object moving horizontally on a plane and attached to a wall with a spring:
where
The state equation would then become
where
The controllability test is then
which has full rank for all formula_91 and formula_93. This means, that if initial state of the system is known (formula_87, formula_88, formula_89), and if the formula_91 and formula_93 are constants, then there is a force formula_108 that could move the cart into any other position in the system.
The observability test is then
which also has full rank. Therefore, this system is both controllable and observable.
Nonlinear systems.
The more general form of a state-space model can be written as two functions.
The first is the state equation and the latter is the output equation.
If the function formula_112 is a linear combination of states and inputs then the equations can be written in matrix notation like above.
The formula_90 argument to the functions can be dropped if the system is unforced (i.e., it has no inputs).
Pendulum example.
A classic nonlinear system is a simple unforced pendulum
where
The state equations are then
where
Instead, the state equation can be written in the general form
The equilibrium/stationary points of a system are when formula_127 and so the equilibrium points of a pendulum are those that satisfy
for integers "n".


Discrete system

In theoretical computer science, a discrete system is a system with a countable number of states. Discrete systems may be contrasted with continuous systems, which may also be called analog systems. A final discrete system is often modeled with a directed graph and is analyzed for correctness and complexity according to computational theory. Because discrete systems have a countable number of states, they may be described in precise mathematical models.
A computer is a finite-state machine that may be viewed as a discrete system. Because computers are often used to model not only other discrete systems but continuous systems as well, methods have been developed to represent real-world continuous systems as discrete systems. One such method involves sampling a continuous signal at discrete time intervals.


Hybrid system

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both "flow" (described by a differential equation) and "jump" (described by a state machine or automaton). Often, the term "hybrid dynamical system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena.
In general, the "state" of a hybrid system is defined by the values of the "continuous variables" and a discrete "mode". The state changes either continuously, according to a flow condition, or discretely according to a "control graph". Continuous flow is permitted as long as so-called "invariants" hold, while discrete transitions can occur as soon as given "jump conditions" are satisfied. Discrete transitions may be associated with "events".
Examples.
Hybrid systems have been used to model several cyber-physical systems, including physical systems with "impact", logic-dynamic controllers, and even Internet congestion.
Bouncing ball.
A canonical example of a hybrid system is the bouncing ball, a physical system with impact. Here, the ball (thought of as a point-mass) is dropped from an initial height and bounces off the ground, dissipating its energy with each bounce. The ball exhibits continuous dynamics between each bounce; however, as the ball impacts the ground, its velocity undergoes a discrete change modeled after an inelastic collision. A mathematical description of the bouncing ball follows. Let formula_1 be the height of the ball and formula_2 be the velocity of the ball. A hybrid system describing the ball is as follows:
When formula_3, flow is governed by
formula_4,
where formula_5 is the acceleration due to gravity. These equations state that when the ball is above ground, it is being drawn to the ground by gravity.
When formula_6, jumps are governed by
formula_7,
where formula_8 is a dissipation factor. This is saying that when the height of the ball is zero (it has impacted the ground), its velocity is reversed and decreased by a factor of formula_9. Effectively, this describes the nature of the inelastic collision.
The bouncing ball is an especially interesting hybrid system, as it exhibits Zeno behavior. Zeno behavior has a strict mathematical definition, but can be described informally as the system making an "infinite" number of jumps in a "finite" amount of time. In this example, each time the ball bounces it loses energy, making the subsequent jumps (impacts with the ground) closer and closer together in time.
It is noteworthy that the dynamical model is complete if and only if one adds the contact force between the ground and the ball. Indeed, without forces, one cannot properly define the bouncing ball and the model is, from a mechanical point of view, meaningless. The simplest contact model that represents the interactions between the ball and the ground, is the complementarity relation between the force and the distance (the gap) between the ball and the ground. This is written as
formula_10
Such a contact model does not incorporate magnetic forces, nor gluing effects. When the complementarity relations are in, one can continue to integrate the system after the impacts have accumulated and vanished: the equilibrium of the system is well-defined as the static equilibrium of the ball on the ground, under the action of gravity compensated by the contact force formula_11. One also notices from basic convex analysis that the complementarity relation can equivalently be rewritten as the inclusion into a normal cone, so that the bouncing ball dynamics is a differential inclusion into a normal cone to a convex set. See Chapters 1, 2 and 3 in Acary-Brogliato's book cited below (Springer LNACM 35, 2008). See also the other references on non-smooth mechanics.
Hybrid systems verification.
There are approaches to automatically proving properties of hybrid systems (e.g., some of the tools mentioned below). Common techniques for proving safety of hybrid systems are computation of reachable sets, abstraction refinement, and barrier certificates.
Most verification tasks are undecidable, making general verification algorithms impossible. Instead, the tools are analyzed for their capabilities on benchmark problems. A possible theoretical characterization of this is algorithms that succeed with hybrid systems verification in all robust cases implying that many problems for hybrid systems, while undecidable, are at least quasi-decidable.
Other modeling approaches.
Two basic hybrid system modeling approaches can be classified, an implicit and an explicit one. The explicit approach is often represented by a hybrid automaton, a hybrid program or a hybrid Petri net. The implicit approach is often represented by guarded equations to result in systems of differential algebraic equations (DAEs) where the active equations may change, for example by means of a hybrid bond graph.
As a unified simulation approach for hybrid system analysis, there is a method based on DEVS formalism in which integrators for differential equations are quantized into atomic DEVS models. These methods generate traces of system behaviors in discrete event system manner which are different from discrete time systems. Detailed of this approach can be found in references [Kofman2004] [CF2006] [Nutaro2010] and the software tool PowerDEVS.


Closed-loop controller

A closed-loop controller or feedback controller is a control loop which incorporates feedback, in contrast to an "open-loop controller" or "non-feedback controller".
A closed-loop controller uses feedback to control states or outputs of a dynamical system. Its name comes from the information path in the system: process inputs (e.g., voltage applied to an electric motor) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.
In the case of linear feedback systems, a control loop including sensors, control algorithms, and actuators is arranged in an attempt to regulate a variable at a setpoint (SP). An everyday example is the cruise control on a road vehicle; where external influences such as hills would cause speed changes, and the driver has the ability to alter the desired set speed. The PID algorithm in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay or overshoot, by controlling the power output of the vehicle's engine.
Control systems that include some sensing of the results they are trying to achieve are making use of feedback and can adapt to varying circumstances to some extent. Open-loop control systems do not make use of feedback, and run only in pre-arranged ways.
Closed-loop controllers have the following advantages over open-loop controllers:
In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed "feedforward" and serves to further improve reference tracking performance.
A common closed-loop controller architecture is the PID controller.
Closed-loop transfer function.
The output of the system "y"("t") is fed back through a sensor measurement "F" to a comparison with the reference value "r"("t"). The controller "C" then takes the error "e" (difference) between the reference and the output to change the inputs "u" to the system under control "P". This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.
This is called a single-input-single-output ("SISO") control system; "MIMO" (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).
If we assume the controller "C", the plant "P", and the sensor "F" are linear and time-invariant (i.e., elements of their transfer function "C"("s"), "P"("s"), and "F"("s") do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:
Solving for "Y"("s") in terms of "R"("s") gives
The expression formula_5 is referred to as the "closed-loop transfer function" of the system. The numerator is the forward (open-loop) gain from "r" to "y", and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If formula_6, i.e., it has a large norm with each value of "s", and if formula_7, then "Y"("s") is approximately equal to "R"("s") and the output closely tracks the reference input.
PID feedback control.
A proportional–integral–derivative controller (PID controller) is a control loop feedback mechanism control technique widely used in control systems.
A PID controller continuously calculates an "error value" as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms. "PID" is an initialism for "Proportional-Integral-Derivative", referring to the three terms operating on the error signal to produce a control signal.
The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and later in industrial process computers.
The PID controller is probably the most-used feedback control design.
If is the control signal sent to the system, is the measured output and is the desired output, and is the tracking error, a PID controller has the general form
The desired closed loop dynamics is obtained by adjusting the three parameters , and , often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in process control). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered.
Applying Laplace transformation results in the transformed PID controller equation
with the PID controller transfer function
As an example of tuning a PID controller in the closed-loop system , consider a 1st order plant given by
where and are some constants. The plant output is fed back through
where is also a constant. Now if we set formula_14, and formula_15, we can express the PID controller transfer function in series form as
Plugging , and into the closed-loop transfer function , we find that by setting
. With this tuning in this example, the system output follows the reference input exactly.
However, in practice, a pure differentiator is neither physically realizable nor desirable due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead.


Open-loop controller

In control theory, an open-loop controller, also called a non-feedback controller, is a control loop part of a control system in which the control action is independent of the "process output", which is the process variable that is being controlled. It does not use feedback to determine if its output has achieved the desired goal of the input command or process setpoint.
There are many open-loop controls, such as on/off switching of valves, machinery, lights, motors or heaters, where the control result is known to be approximately sufficient under normal conditions without the need for feedback. The advantage of using open-loop control in these cases is the reduction in component count and complexity. However, an open-loop system cannot correct any errors that it makes or correct for outside disturbances, and cannot engage in machine learning, unlike a closed-loop control system.
Applications.
An open-loop controller is often used in simple processes because of its simplicity and low cost, especially in systems where feedback is not critical. A typical example would be an older model domestic clothes dryer, for which the length of time is entirely dependent on the judgement of the human operator, with no automatic feedback of the dryness of the clothes.
For example, an irrigation sprinkler system, programmed to turn on at set times could be an example of an open-loop system if it does not measure soil moisture as a form of feedback. Even if rain is pouring down on the lawn, the sprinkler system would activate on schedule, wasting water.
Another example is a stepper motor used for control of position. Sending it a stream of electrical pulses causes it to rotate by exactly that many steps, hence the name. If the motor was always assumed to perform each movement correctly, without positional feedback, it would be open-loop control. However, if there is a position encoder, or sensors to indicate the start or finish positions, then that is closed-loop control, such as in many inkjet printers. The drawback of open-loop control of steppers is that if the machine load is too high, or the motor attempts to move too quickly, then steps may be skipped. The controller has no means of detecting this and so the machine continues to run slightly out of adjustment until reset. For this reason, more complex robots and machine tools instead use servomotors rather than stepper motors, which incorporate encoders and closed-loop controllers.
However, open-loop control is very useful and economic for well-defined systems where the relationship between input and the resultant state can be reliably modeled by a mathematical formula. For example, determining the voltage to be fed to an electric motor that drives a constant load, in order to achieve a desired speed would be a good application. But if the load were not predictable and became excessive, the motor's speed might vary as a function of the load not just the voltage, and an open-loop controller would be insufficient to ensure repeatable control of the velocity.
An example of this is a conveyor system that is required to travel at a constant speed. For a constant voltage, the conveyor will move at a different speed depending on the load on the motor (represented here by the weight of objects on the conveyor). In order for the conveyor to run at a constant speed, the voltage of the motor must be adjusted depending on the load. In this case, a closed-loop control system would be necessary.
Thus there are many open-loop controls, such as switching valves, lights, motors or heaters on and off, where the result is known to be approximately sufficient without the need for feedback.
Combination with feedback control.
A feed back control system, such as a PID controller, can be improved by combining the feedback (or closed-loop control) of a PID controller with feed-forward (or open-loop) control. Knowledge about the system (such as the desired acceleration and inertia) can be fed forward and combined with the PID output to improve the overall system performance. The feed-forward value alone can often provide the major portion of the controller output. The PID controller primarily has to compensate whatever difference or "error" remains between the setpoint (SP) and the system response to the open-loop control. Since the feed-forward output is not affected by the process feedback, it can never cause the control system to oscillate, thus improving the system response without affecting stability. Feed forward can be based on the setpoint and on extra measured disturbances. Setpoint weighting is a simple form of feed forward.
For example, in most motion control systems, in order to accelerate a mechanical load under control, more force is required from the actuator. If a velocity loop PID controller is being used to control the speed of the load and command the force being applied by the actuator, then it is beneficial to take the desired instantaneous acceleration, scale that value appropriately and add it to the output of the PID velocity loop controller. This means that whenever the load is being accelerated or decelerated, a proportional amount of force is commanded from the actuator regardless of the feedback value. The PID loop in this situation uses the feedback information to change the combined output to reduce the remaining difference between the process setpoint and the feedback value. Working together, the combined open-loop feed-forward controller and closed-loop PID controller can provide a more responsive control system in some situations.


Distributed artificial intelligence

Distributed Artificial Intelligence (DAI) also called Decentralized Artificial Intelligence is a subfield of artificial intelligence research dedicated to the development of distributed solutions for problems. DAI is closely related to and a predecessor of the field of multi-agent systems.
Multi-agent systems and distributed problem solving are the two main DAI approaches. There are numerous applications and tools.
Definition.
Distributed Artificial Intelligence (DAI) is an approach to solving complex learning, planning, and decision-making problems. It is embarrassingly parallel, thus able to exploit large scale computation and spatial distribution of computing resources. These properties allow it to solve problems that require the processing of very large data sets. DAI systems consist of autonomous learning processing nodes (agents), that are distributed, often at a very large scale. DAI nodes can act independently, and partial solutions are integrated by communication between nodes, often asynchronously. By virtue of their scale, DAI systems are robust and elastic, and by necessity, loosely coupled. Furthermore, DAI systems are built to be adaptive to changes in the problem definition or underlying data sets due to the scale and difficulty in redeployment.
DAI systems do not require all the relevant data to be aggregated in a single location, in contrast to monolithic or centralized Artificial Intelligence systems which have tightly coupled and geographically close processing nodes. Therefore, DAI systems often operate on sub-samples or hashed impressions of very large datasets. In addition, the source dataset may change or be updated during the course of the execution of a DAI system.
Development.
In 1975 distributed artificial intelligence emerged as a subfield of artificial intelligence that dealt with interactions of intelligent agents. Distributed artificial intelligence systems were conceived as a group of intelligent entities, called agents, that interacted by cooperation, by coexistence or by competition. DAI is categorized into multi-agent systems and distributed problem solving. In multi-agent systems the main focus is how agents coordinate their knowledge and activities. For distributed problem solving the major focus is how the problem is decomposed and the solutions are synthesized.
Goals.
The objectives of Distributed Artificial Intelligence are to solve the reasoning, planning, learning and perception problems of artificial intelligence, especially if they require large data, by distributing the problem to autonomous processing nodes (agents). To reach the objective, DAI requires:
There are many reasons for wanting to distribute intelligence or cope with multi-agent systems. Mainstream problems in DAI research include the following:
Approaches.
Two types of DAI has emerged:
DAI can apply a bottom-up approach to AI, similar to the subsumption architecture as well as the traditional top-down
approach of AI. In addition, DAI can also be a vehicle for emergence.
Challenges.
The challenges in Distributed AI are:
Applications and tools.
Areas where DAI have been applied are:
DAI integration in tools has included:
Agents.
Systems: Agents and multi-agents.
Notion of Agents: Agents can be described as distinct entities with standard boundaries and interfaces designed for problem solving.
Notion of Multi-Agents: Multi-Agent system is defined as a network of agents which are loosely coupled working as a single entity like society for problem solving that an individual agent cannot solve.
Software agents.
The key concept used in DPS and MABS is the abstraction called software agents. An agent is a virtual (or physical) entity that has an understanding of its environment and acts upon it. An agent is usually able to communicate with other agents in the same system to achieve a common goal, that one agent alone could not achieve. This communication system uses an agent communication language.
A first classification that is useful is to divide agents into:
Well-recognized agent architectures that describe how an agent is internally structured are:


Spiral optimization algorithm

In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature.
The first SPO algorithm was proposed for two-dimensional unconstrained optimization
based on two-dimensional spiral models. This was extended to n-dimensional problems by generalizing the two-dimensional spiral model to an n-dimensional spiral model.
There are effective settings for the SPO algorithm: the periodic descent direction setting
and the convergence setting.
Metaphor.
The motivation for focusing on spiral phenomena was due to the insight that the dynamics that generate logarithmic spirals share the diversification and intensification behavior. The diversification behavior can work for a global search (exploration) and the intensification behavior enables an intensive search around a current found good solution (exploitation).
Algorithm.
The SPO algorithm is a multipoint search algorithm that has no objective function gradient, which uses multiple spiral models that can be described as deterministic dynamical systems. As search points follow logarithmic
spiral trajectories towards the common center, defined as the current best point, better solutions can be found and the common center can be updated.
The general SPO algorithm for a minimization problem under the maximum iteration formula_1 (termination criterion) is as follows:
 0) Set the number of search points formula_2 and the maximum iteration number formula_1.
 1) Place the initial search points formula_4 and determine the center formula_5, formula_6,and then set formula_7.
 2) Decide the step rate formula_8 by a rule.
 3) Update the search points: formula_9
 4) Update the center: formula_10 where formula_11.
 5) Set formula_12. If formula_13 is satisfied then terminate and output formula_14. Otherwise, return to Step 2).
Setting.
The search performance depends on setting the composite rotation matrix formula_15, the step rate formula_8, and the initial points formula_17.
The following settings are new and effective.
Setting 1 (Periodic Descent Direction Setting).
This setting is an effective setting for high dimensional problems under the maximum iteration formula_1. The conditions on formula_15 and formula_17 together ensure that the spiral models generate descent directions periodically. The condition of formula_8 works to utilize the periodic descent directions under the search termination formula_1.
formula_31
where formula_32. Note that this condition is almost all satisfied by a random placing and thus no check is actually fine.
Setting 2 (Convergence Setting).
This setting ensures that the SPO algorithm converges to a stationary point under the maximum iteration formula_38. The settings of formula_15 and the initial points formula_17 are the same with the above Setting 1. The setting of formula_8 is as follows.
Extended works.
Many extended studies have been conducted on the SPO due to its simple structure and concept; these studies have helped improve its global search performance and proposed novel applications.


Wanfeng Auto Holding Group

Wanfeng Auto Holding Group Co., Ltd., headquartered in Xinchang County, China, is a parts and equipment manufacturer for the automotive industry, aerospace industry, military industry, and energy saving.
It manufactures lightweight metal parts: aluminium wheels and magnesium alloy materials, and provides environmental protection coatings, hybrid vehicle and electric vehicle assembly and powertrain systems, technical services and control systems.
It is involved in aircraft manufacturing and research & development, airport construction and management and general aviation operations: aerial sightseeing, air sports and flight training services.
It supplies industrial automatic gating systems, low pressure and gravity die cast machines and auxiliary equipment, with consulting and maintenance and repair services, and provides automation for auto and heavy industries and industrial robot intelligent equipment.
It provides private equity, hedge fund and finance leasing services.
It designs, develops, and constructs real estate properties and services high-rise buildings, multi-story residences, villas, office buildings, community and commercial buildings and government properties.
It is amongst the Chinese Private Enterprises Top 500 (#123) and China Auto Industry Top 30 (#18).
It is listed on the Shenzhen Stock Exchange and is a part of the SZSE 200 Index.


Infinite-dimensional optimization

In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body. Such a problem is an infinite-dimensional optimization problem, because, a continuous quantity cannot be determined by a finite number of certain degrees of freedom.
Examples.
Infinite-dimensional optimization problems can be more challenging than finite-dimensional ones. Typically one needs to employ methods from partial differential equations to solve such problems.
Several disciplines which study infinite-dimensional optimization problems are calculus of variations, optimal control and shape optimization.


Empowerment (artificial intelligence)

Empowerment in the field of artificial intelligence formalises and quantifies (via information theory) the potential an agent perceives that it has to influence its environment. An agent which follows an empowerment maximising policy, acts to maximise future options (typically up to some limited horizon). Empowerment can be used as a (pseudo) utility function that depends only on information gathered from the local environment to guide action, rather than seeking an externally imposed goal, thus is a form of intrinsic motivation.
The empowerment formalism depends on a probabilistic model commonly used in artificial intelligence. An autonomous agent operates in the world by taking in sensory information and acting to change its state, or that of the environment, in a cycle of perceiving and acting known as the perception-action loop. Agent state and actions are modelled by random variables (formula_1) and time (formula_2). The choice of action depends on the current state, and the future state depends on the choice of action, thus the perception-action loop unrolled in time forms a causal bayesian network.
Definition.
Empowerment (formula_3) is defined as the channel capacity (formula_4) of the actuation channel of the agent, and is formalised as the maximal possible information flow between the actions of the agent and the effect of those actions some time later. Empowerment can be thought of as the future potential of the agent to affect its environment, as measured by its sensors.
formula_5
In a discrete time model, Empowerment can be computed for a given number of cycles into the future, which is referred to in the literature as 'n-step' empowerment.
formula_6
The unit of empowerment depends on the logarithm base. Base 2 is commonly used in which case the unit is bits.
Contextual Empowerment.
In general the choice of action (action distribution) that maximises empowerment varies from state to state. Knowing the empowerment of an agent in a specific state is useful, for example to construct an empowerment maximising policy. State-specific empowerment can be found using the more general formalism for 'contextual empowerment'. formula_4 is a random variable describing the context (e.g. state).
formula_8
Application.
Empowerment maximisation can be used as a pseudo-utility function to enable agents to exhibit intelligent behaviour without requiring the definition of external goals, for example balancing a pole in a cart-pole balancing scenario where no indication of the task is provided to the agent.
Empowerment has been applied in studies of collective behaviour and in continuous domains. As is the case with Bayesian methods in general, computation of empowerment becomes computationally expensive as the number of actions and time horizon extends, but approaches to improve efficiency have led to usage in real-time control. Empowerment has been used for intrinsically motivated reinforcement learning agents playing video games, and in the control of underwater vehicles.


Elmer G. Gilbert

Elmer Grant Gilbert was an American aerospace engineer and a Professor Emeritus of Aerospace Engineering at the University of Michigan. He received his Ph.D. in Instrumentation Engineering from Michigan in 1957.
He was a member of the National Academy of Engineering and a recipient of the 1994 IEEE Control Systems Award (the citation reads: "for pioneering and innovative contributions to linear state space theory and its applications, especially realization and decoupling, as well as to control algorithms") and the 1996 Richard E. Bellman Control Heritage Award from the American Automatic Control Council.


John C. Lozier

John C. Lozier is a noted American control engineer. He was responsible for the control of the Telstar ground-tracking antenna installed near Andover, Maine, and the Brittany Peninsula in France. This equipment, involving real-time computer control, enabled the first transatlantic TV operation in 1962.
Lozier served as the president of the American Automatic Control Council from 1960 to 1962 and as the president of the International Federation of Automatic Control from 1972 to 1975.
He received numerous awards for his contribution to the field of control theory, including the Richard E. Bellman Control Heritage Award in 1987.


Pneumatic web guides

A Web is a term used in the Converting Industry that refers to continuous rolls of thin, flat materials like paper, film and plastic. Web guiding systems use a sensor to monitor the position of a web as it enters a production process for lateral tracking. Each type of web guide sensor has an actuator to shift the running web mechanically back on course whenever the sensor detects movement away from the set path. Actuators may be pneumatic or hydraulic cylinders, or some kind of electromechanical device. Because the web may be fragile — particularly at its edge — non-contact sensors are used.
Sensors developed for Web Guiding applications in the Converting Industry may be pneumatic, photoelectric, ultrasonic, or infrared. The system’s controls must process the output signals from the sensors in to a form that can drive an actuator. Many controls today are electronic, typically using an amplifier to convert signals from the sensor, then commanding a special servo motor incorporating a lead or ball screw for guiding actuation. Some electromechanical guiding systems also utilize computers.
Pneumatic web guide systems are typically easier to install, operate and are less expensive than more complex hydraulic and electronic systems. Pneumatic servo controllers are considered explosion-proof, especially in dusty or contaminated environments.


Subspace identification method

In mathematics, specifically in control theory, subspace identification (SID) aims at identifying linear time invariant (LTI) state space models from input-output data. SID does not require that the user parametrizes the system matrices before solving a parametric optimization problem and, as a consequence, SID methods do not suffer from problems related to local minima that often lead to unsatisfactory identification results.
History.
SID methods are rooted in the work by the German mathematician Leopold Kronecker (1823–1891). Kronecker showed that a power series can be written as a rational function when the rank of the Hankel operator that has the power series as its symbol is finite. The rank determines the order of the polynomials of the rational function.
In the 1960s the work of Kronecker inspired a number of researchers in the area of Systems and Control, like Ho and Kalman, Silverman and Youla and Tissi, to store the Markov parameters of an LTI system into a finite dimensional Hankel matrix and derive from this matrix an (A,B,C) realization of the LTI system. The key observation was that when the Hankel matrix is properly dimensioned versus the order of the LTI system, the rank of the Hankel matrix is the order of the LTI system and the SVD of the Hankel matrix provides a basis of the column space observability matrix and row space of the controllability matrix of the LTI system. Knowledge of this key spaces allows to estimate the system matrices via linear least squares.
An extension to the stochastic realization problem where we have knowledge only of the Auto-correlation (covariance) function of the output of an LTI system driven by white noise, was derived by researchers like Akaike.
A second generation of SID methods attempted to make SID methods directly operate on input-output measurements of the LTI system in the decade 1985–1995. One such generalization was presented under the name of the Eigensystem Realization Algorithm (ERA) made use of specific input-output measurements considering the impulse inputs. It has been used for modal analysis of flexible structures, like bridges, space structures, etc. These methods have demonstrated to work in practice for resonant structures they did not work well for other type of systems and an input different from an impulse. A new impulse to the development of SID methods was made for operating directly on generic input-output data and avoiding to first explicitly compute the Markov parameters or estimating the samples of covariance functions prior to realizing the system matrices. Pioneers that contributed to these breakthroughs were Van Overschee and De Moor – introducing the N4SID approach, Verhaegen – introducing the MOESP approach and Larimore – presenting ST in the framework of Canonical Variate Analysis (CVA)


Miyota (watch movement manufacturer)

Miyota is a brand of mechanical and quartz watch movements manufactured by Citizen Watch Manufacturing Co., Ltd. (CWMJ), a subsidiary of Citizen Watch.
In 1959 Miyota Precision Co., Ltd. was established in the town of Miyota, Nagano Prefecture, Japan as an assembly factory for wristwatches. The company renamed Miyota Co., Ltd. in 1991 and Citizen Miyota Co., Ltd. in 2005 and was merged with Citizen Finetech Co., Ltd. to form Citizen Finetech Miyota Co., Ltd. in 2008 and merged with Citizen Seimitsu Co., Ltd. to form Citizen Finedevice Co., Ltd. in 2015. The company produces watch parts such as crystal oscillators and bearing jewels, but its watch movement manufacturing business has been transferred to Citizen Watch Manufacturing Co., Ltd.
In 2016 a large movement assembly factory (CWMJ Miyota Saku Factory) in Saku, Nagano Prefecture was opened. Most watch brands do not make their own movements in-house, but rather use standard watch movements manufactured by specialized companies like Miyota and ETA SA.
Products.
Mechanical movements.
Miyota produces various 'standard' and 'premium' grade mechanical movements for automatic wristwatches.
The "Miyota 8215" is an entry level non-hacking twenty-one jewel three-hand with date automatic wristwatch movement with a uni-directional winding system (left rotation) with an accuracy of -20 to +40 seconds per day, and a power reserve of over 40 hours. It allows hand winding. The date window may be placed at the 3 o'clock position (cal. 8215-33A) or at 6 o'clock (cal. 8215-36A). The diameter of the movement is 26 mm and the thickness is 5.67 mm. It beats at 21,600 BPH or 3 Hz (6 half-cycles per second). The movement has a 49° lift angle.
The "Miyota 9015" is a more sophisticated hacking twenty-four jewel three-hand with date automatic wristwatch movement with a uni-directional winding system (left rotation). The 9015 automatic movement has a beat rate of 28,800 BPH (= 4 Hz), 51° lift angle, 24 jewels and features (automatic) winding, a ≥42 hours power reserve, manual winding and a hack function (stopping the movement of the second hand). The static accuracy rating is −10 to +30 seconds per day (23±2 °C).
Electric movements.
Miyota also produces various electric driven movements for quartz watches. These are categorized in standard, slim, small, multi-function, chronograph, and small second chronograph movements.These movements are used by a variety of watch brands such as Bulova, MVMT and Skagen.
Usage.
Miyota movements are used by many watch makers besides Citizen for some of their watches, among them Xeric, Invicta, Boldr, Bulova, Bernhardt, Casio, Corgeut, and Timex.


Open Agent Architecture

Open Agent Architecture, or OAA for short, is a framework for integrating a community of heterogeneous software agents in a distributed environment. It is also a research project of the SRI International Artificial Intelligence Center.
Roughly, the architecture is that a central "blackboard" server holds a list of tasks while a group of agents executes these tasks based on their specific capabilities.
Agents working in the structure of an OAA framework are built to universal communication and functional standards and are based on the Interagent Communication Language. The language is platform-independent and allows agents to collaborate by delegating and receiving work requests.
Open Agent Architecture was first proposed in the late 1990s and was later used as a foundation for the DARPA-funded CALO artificial intelligence project.


Gain scheduling

In control theory, gain scheduling is an approach to control of nonlinear systems that uses a family of linear controllers, each of which provides satisfactory control for a different operating point of the system.
One or more observable variables, called the "scheduling variables", are used to determine what operating region the system is currently in and to enable the appropriate linear controller. For example, in an aircraft flight control system, the altitude and Mach number might be the scheduling variables, with different linear controller parameters available (and automatically plugged into the controller) for various combinations of these two variables.
A relatively large scope state of the art about gain scheduling has been published in (Survey of Gain-Scheduling Analysis & Design, D.J.Leith, WE.Leithead).


Process capability

The process capability is a measurable property of a process to the specification, expressed as a process capability index (e.g., Cpk or Cpm) or as a process performance index (e.g., Ppk or Ppm). The output of this measurement is often illustrated by a histogram and calculations that predict how many parts will be produced out of specification (OOS).
Two parts of process capability are: 1) measure the variability of the output of a process, and 2) compare that variability with a proposed specification or product tolerance.
Capabilities
The input of a process usually has at least one or more measurable characteristics that are used to specify outputs. These can be analyzed statistically; where the output data shows a normal distribution the process can be described by the process mean (average) and the standard deviation.
A process needs to be established with appropriate process controls in place. A control chart analysis is used to determine whether the process is "in statistical control" If the process is not in statistical control then capability has no meaning. Therefore, the process capability involves only common cause variation and not special cause variation.
A batch of data needs to be obtained from the measured output of the process. The more data that is included the more precise the result, however an estimate can be achieved with as few as 17 data points. This should include the normal variety of production conditions, materials, and people in the process. With a manufactured product, it is common to include at least three different production runs, including start-ups.
The process mean (average) and standard deviation are calculated. With a normal distribution, the "tails" can extend well beyond plus and minus three standard deviations, but this interval should contain about 99.73% of production output. Therefore, for a normal distribution of data the process capability is often described as the relationship between six standard deviations and the required specification.
Capability study.
The output of a process is expected to meet customer requirements, specifications, or engineering tolerances. Engineers can conduct a process capability study to determine the extent to which the process can meet these expectations.
The ability of a process to meet specifications can be expressed as a single number using a process capability index or it can be assessed using control charts. Either case requires running the process to obtain enough measurable output so that engineering is confident that the process is stable and so that the process mean and variability can be reliably estimated. Statistical process control defines techniques to properly differentiate between stable processes, processes that are drifting (experiencing a long-term change in the mean of the output), and processes that are growing more variable. Process capability indices are only meaningful for processes that are stable (in a state of statistical control).


GraphEdit

GraphEdit is a utility which is part of the Microsoft DirectShow SDK. It is a visual tool for building and testing filter graphs for DirectShow. Filters are displayed as boxes, with a text caption showing the name of the filter. Pins appear as small squares along the edge of the filter. Input pins are shown on the left side of the filter, and output pins are on the right side of the filter. A pin connection appears as an arrow connecting the output pin to the input pin. Connection mediatypes can be viewed as "properties" on pins and connections. GraphEdit can automatically build a filter graph that plays a file.


Trailer connectors in Australia

A number of standards prevail in Australia for trailer connectors, the electrical connectors between vehicles and the trailers they tow that provide a means of control trailer lamps, and in one case, trailer brakes, and also sometimes, manufacturer-specific non-standard functions.
The Australian market generally uses its own version of the European connectors, as well as its uniquely own contacts.
The only connector used on the Australian market that is fully ISO standard conformant is the 7-pin ABS / EBS plug.
Since Australia has vehicles from both the North American market and the European market there is a mixture of 12V and 24V.
7-pin trailer connector (AS 4735) for heavy duty vehicles.
This connector is based on both SAE J560 and ISO 1185 and is providing either 12V, 7 x 40A or 24V, 7 x 20A. The voltage varies from vehicle to vehicle.
Round trailer connectors Type 1.
These connectors are based on ISO 1724 in 5-pin and 7-pin versions, but with some difference in the wiring.
Round 7-pin trailer connector Type 1 (AS 2513).
In this pinout for an ISO 1724 connector, the "position light" pin is used for electric brakes (Pin 5, 58R), which means that if you connect a trailer with electric brakes to a towing vehicle wired according to ISO 1724 and turn on the position lights the trailer will be braking. Pin 2 (54G) is in the Australian wiring standard the reversing light, which is a minor problem.
Round 5-pin trailer connector Type 1.
This 5-pin connector has been superseded by the 7-pin (AS 2513), but can be found on older vehicles. Note that pins 1 and 4 are missing. Pin placement is identical to the 7-pin ISO 1724 with the absence of these pins. This means that you can connect a trailer with a 5-pin connector to a 7-pin socket or the other way around, but since the pins are wired in a different way the result may be far from what was expected.
Rectangular trailer connectors Type 3.
The image of the 7 and 12 pin flat plugs are from the cable entry view (and possibly the images of the round connectors too). Please see reference [2] (VSB1 section 14) for front image. Reference [1] Narva wiring diagrams also have the diagrams cable entry view. If you want the front view images in colour see
Some trailer manufacturers will wire non-standard features through non-standard pins. Auxiliary power, breakaway sense or hydraulic brake pump should never be wired to pin 2, even if the trailer does not feature reverse lights.


Sensor hub

A sensor hub is a microcontroller unit/coprocessor/DSP set that helps to integrate data from different sensors and process them. This technology can help off-load these jobs from a product's main central processing unit, thus saving battery consumption and providing a performance improvement.
Intel has the Intel Integrated Sensor Hub. Starting from Cherrytrail and Haswell, many Intel processors offers on package sensor hub. The Samsung Galaxy Note II is the first smart phone with a sensor hub, which was launched in 2012.
Examples.
Some devices with Snapdragon 800 series chips, including HTC One (M8), Sony Xperia Z1, LG G2, etc., have a sensor hub, the Qualcomm Snapdragon Sensor Core, and all HiSilicon Kirin 920 devices have sensor hub embedded in the chipset with its successor Kirin 925 integrated an i3 chip with same function into it. Some other devices that are not using these chips but with a sensor hub integrated are listed below:


Pseudospectral optimal control

Pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems. It combines pseudospectral (PS) theory with optimal control theory to produce PS optimal control theory. PS optimal control theory has been used in ground and flight systems in military and industrial applications. The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum control.
Overview.
There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo pseudospectral method, the Bellman pseudospectral method, the flat pseudospectral method and many others. Solving an optimal control problem requires the approximation of three types of mathematical objects: the integration in the cost function, the differential equation of the control system, and the state-control constraints. An ideal approximation method should be efficient for all three approximation tasks. A method that is efficient for one of them, for instance an efficient ODE solver, may not be an efficient method for the other two objects. These requirements make PS methods ideal because they are efficient for the approximation of all three mathematical objects. In a pseudospectral method, the continuous functions are approximated at a set of carefully selected quadrature nodes. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, Legendre and Chebyshev polynomials are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with a small number of points. For instance, the interpolating polynomial of any smooth function (Cformula_1) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, faster than any polynomial rate.
Details.
A basic pseudospectral method for optimal control is based on the covector mapping principle. Other pseudospectral optimal control techniques, such as the Bellman pseudospectral method, rely on node-clustering at the initial time to produce optimal controls. The node clusterings occur at all Gaussian points.
Moreover, their structure can be highly exploited to make them more computationally efficient, as ad-hoc scaling and Jacobian computation methods, involving dual number theory have been developed.
In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to formula_2. In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems.


DIDO (software)

DIDO  is a MATLAB optimal control toolbox for solving general-purpose optimal control problems. It is widely used in academia, industry, and NASA. Hailed as a breakthrough software, DIDO is based on the pseudospectral optimal control theory of Ross and Fahroo. The latest enhancements to DIDO are described in Ross.
Usage.
DIDO utilizes trademarked expressions and objects that facilitate a user to quickly formulate and solve optimal control problems. Rapidity in formulation is achieved through a set of DIDO expressions which are based on variables commonly used in optimal control theory. For example, the state, control and time variables are formatted as:
The entire problem is codified using the key words, cost, dynamics, events and path:
A user runs DIDO using the one-line command:
codice_1,
where the object defined by codice_2 allows a user to choose various options. In addition to the cost value and the primal solution, DIDO automatically outputs all the dual variables that are necessary to verify and validate a computational solution. The output codice_3 is computed by an application of the covector mapping principle.
Theory.
DIDO implements a spectral algorithm based on pseudospectral optimal control theory founded by Ross and his associates. The covector mapping principle of Ross and Fahroo eliminates the curse of sensitivity associated in solving for the costates in optimal control problems. DIDO generates spectrally accurate solutions whose extremality can be verified using Pontryagin's Minimum Principle. Because no knowledge of pseudospectral methods is necessary to use it, DIDO is often used as a fundamental mathematical tool for solving optimal control problems. That is, a solution obtained from DIDO is treated as a candidate solution for the application of Pontryagin's minimum principle as a necessary condition for optimality.
Applications.
DIDO is used world wide in academia, industry and government laboratories. Thanks to NASA, DIDO was flight-proven in 2006. On November 5, 2006, NASA used DIDO to maneuver the International Space Station to perform the zero-propellant maneuver.
Since this flight demonstration, DIDO was used for the International Space Station and other NASA spacecraft. It is also used in other industries. Most recently, DIDO has been used to solve traveling salesman type problems in aerospace engineering.
MATLAB optimal control toolbox.
DIDO is primarily available as a stand-alone MATLAB optimal control toolbox. That is, it does not require any third-party software like SNOPT or IPOPT or other nonlinear programming solvers. In fact, it does not even require the MATLAB Optimization Toolbox.
The MATLAB/DIDO toolbox does not require a "guess" to run the algorithm. This and other distinguishing features have made DIDO a popular tool to solve optimal control problems.
The MATLAB optimal control toolbox has been used to solve problems in aerospace, robotics and search theory.
History.
The optimal control toolbox is named after Dido, the legendary founder and first queen of Carthage who is famous in mathematics for her remarkable solution to a constrained optimal control problem even before the invention of calculus. Invented by Ross, DIDO was first produced in 2001. The software is widely cited and has many firsts to its credit:
Versions.
The early versions, widely adopted in academia, have undergone significant changes since 2007. The latest version of DIDO, available from Elissar Global, does not require a "guess" to start the problem and eliminates much of the minutia of coding by simplifying the input-output structure. Low-cost student versions and discounted academic versions are also available from Elissar Global.


Shop floor

The shop floor is the production area, such as in a factory or another working space and is the floor where workers produce goods. The term "shop floor" refers to the area of a factory where production takes place. The shop floor excludes the area used or designated for administrative activities.
Shop stewards and Shop Stewards Movement.
A shop steward is an employee of a company or organization who, as a labor union member and official, represents and defends the interests of their coworkers. During the First World War, the Shop Stewards Movement brought together shop stewards from across the United Kingdom. It began with the Clyde Workers Committee, Britain's first shop stewards committee, which organized in response to the imprisonment of three of their members in 1915.
Shop floor control.
Systems for managing the various components of the manufacturing process are known as shop floor control (SFC) systems.
Shop floor control is one of the functions of manufacturing control; it is the process of monitoring the production activities as they happen, such as when the product is being processed, assembled, inspected, etc. It is also concerned with the shop floor inventories—short and excessive inventories that may cause losses.
Integrated shop floor management.
The manufacturing industry is significantly impacted by technological advances such as the Internet, the Web, and intelligent agents. Changing shop floor environments and customer needs are sufficed with new kinds of shop floor control systems that are web-based shop floor control systems, also called e-shop floor or i-shop floor.


SNOPT

SNOPT, for Sparse Nonlinear OPTimizer, is a software package for solving large-scale nonlinear optimization problems written by Philip Gill, Walter Murray and Michael Saunders. SNOPT is mainly written in Fortran, but interfaces to C, C++, Python and MATLAB are available.
It employs a sparse sequential quadratic programming (SQP) algorithm with limited-memory quasi-Newton approximations to the Hessian of the Lagrangian. It is especially effective for nonlinear problems with functions and gradients that are expensive to evaluate. The functions should be smooth but need not be convex.
SNOPT is used in several trajectory optimization software packages, including Copernicus, AeroSpace Trajectory Optimization and Software (ASTOS), General Mission Analysis Tool, and Optimal Trajectories by Implicit Simulation (OTIS). It is also available in the Astrogator module of Systems Tool Kit.
SNOPT is supported in the AIMMS, AMPL, APMonitor, General Algebraic Modeling System (GAMS), and TOMLAB modeling systems.


Exact algorithm

In computer science and operations research, exact algorithms are algorithms that always solve an optimization problem to optimality.
Unless P = NP, an exact algorithm for an NP-hard optimization problem cannot run in worst-case polynomial time. There has been extensive research on finding exact algorithms whose running time is exponential with a low base.


Inferential programming

In most computer programming, a programmer keeps a program's intended results in mind and painstakingly constructs a program to achieve those results. Inferential programming refers to (still mostly hypothetical) techniques and technologies enabling the inverse. This would allow describing an intended result to a computer, using a metaphor such as a fitness function, a test specification, or a logical specification, and then the computer, on its own, would construct a program needed to meet the supplied criteria.
During the 1980s, approaches to achieve inferential programming mostly involved techniques for logical inference. Today the term is sometimes used in connection with evolutionary computation techniques that enable a computer to evolve a solution in response to a problem posed as a fitness or reward function.
In July 2022, GitHub Copilot was released, which is an example of inferential programming.


Superplastic forming and diffusion bonding (SPF/DB) is a technique allowing the manufacture of complex-shaped hollow metallic parts. It combines Superplastic forming (SPF) with a second element "Diffusion Bonding" to create the completed structures.
Principle.
Two metal sheets are welded together at their edges, then heated within the confines of a female mould tool.
When the part is hot, an inert gas is injected between the two sheets ; the part becomes hollow to the form of the mould. Parts may be welded in other areas than the edges to give an internal structure as the sheets are blown.


Multi-agent planning

In computer science multi-agent planning involves coordinating the resources and activities of multiple "agents".
NASA says, "multiagent planning is concerned with planning by (and for) multiple agents. It can involve agents planning for a common goal, an agent coordinating the plans (plan merging) or planning of others, or agents refining their own plans while negotiating over tasks or resources. The topic also involves how agents can do this in real time while executing plans (distributed continual planning). Multiagent scheduling differs from multiagent planning the same way planning and scheduling differ: in scheduling often the tasks that need to be performed are already decided, and in practice, scheduling tends to focus on algorithms for specific problem domains".


Manufacturing Message Specification

Manufacturing Message Specification (MMS) is an international standard (ISO 9506) dealing with messaging systems for transferring real time process data and supervisory control information between networked devices or computer applications. The standard is developed and maintained by the ISO Technical Committee 184 (TC184). MMS defines the following
MMS original communication stack.
MMS was standardized in 1990 under two separate standards as
This version of MMS used seven layers of OSI network protocols as its communication stack:
MMS stack over TCP/IP.
Because the Open Systems Interconnection protocols are challenging to implement, the original MMS stack never became popular. In 1999, Boeing created a new version of MMS using Internet protocols instead of the bottom four layers of the original stack plus RFC 1006 ("ISO Transport over TCP") in the transport layer. The top three layers use the same OSI protocols as before.
In terms of the seven-layer OSI model, the new MMS stack looks like this:
With the new stack, MMS has become a globally accepted standard.


Arcam

Arcam AB manufactures electron beam melting (EBM) systems for use in additive manufacturing, which create solid parts from metal powders. Arcam also produces metal powder through AP&C and medical implants through DiSanto Technologies.
Arcam AB was founded by innovator Ralf Larson and financier Jarl Assmundson, in 1997.
Arcam AB was a publicly traded company listed on the Stockholm Stock Exchange under ARCM but was also commonly quoted as OTC stock under AMAVF. Arcam AB corporate headquarters are in Mölndal, Sweden. EBM has applications in the medical, aerospace and automotive industries.
In September 2016, General Electric announced its plans for acquisition of Arcam AB.


Petri net unfoldings

Analysis of Petri nets can be performed by means of constructing either reachable state spaces (or reachable markings) or via the process of graph-based unfolding. The prefix of a Petri net unfolding, which is an acyclic Petri net graph, contains the same information about the properties of the Petri net as the reachability graph, plus it contains information about sequence, concurrency and conflict relations between Petri net transitions and Petri net places. The advantages of the use of unfolding in practice are typically associated with the fact that the unfolding prefix is much more compact than the reachability graph of the Petri net being analysed.
Petri net unfoldings were originally introduced by Ken McMillan. Later they were studied by several authors, who improved the original criterion for producing the prefix of the unfolding in terms of its compactness and hence efficient analysis.
There are applications of Petri net unfoldings in the analysis and synthesis of concurrent systems and asynchronous circuits. The latter is normally achieved through the use of Signal transition graphs (STGs).


Ross–Fahroo pseudospectral method

Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control. Examples of the Ross–Fahroo pseudospectral methods are the pseudospectral knotting method, the flat pseudospectral method, the Legendre-Gauss-Radau pseudospectral method and pseudospectral methods for infinite-horizon optimal control.
Overview.
The Ross–Fahroo methods are based on shifted Gaussian pseudospectral node points. The shifts are obtained by means of a linear or nonlinear transformation while the Gaussian pseudospectral points are chosen from a collection of Gauss-Lobatto or Gauss-Radau distribution arising from Legendre or Chebyshev polynomials. The Gauss-Lobatto pseudospectral points are used for finite-horizon optimal control problems while the Gauss-Radau pseudospectral points are used for infinite-horizon optimal control problems.
Mathematical applications.
The Ross–Fahroo methods are founded on the Ross–Fahroo lemma; they can be applied to optimal control problems governed by differential equations, differential-algebraic equations, differential inclusions, and differentially-flat systems. They can also be applied to infinite-horizon optimal control problems by a simple domain transformation technique.
 The Ross–Fahroo pseudospectral methods also form the foundations for the Bellman pseudospectral method.
Flight applications and awards.
The Ross–Fahroo methods have been implemented in many practical applications and laboratories around the world. In 2006, NASA used the Ross–Fahroo method to implement the "zero propellant maneuver" on board the International Space Station.
In recognition of all these advances, the AIAA presented Ross and Fahroo, the 2010 Mechanics and Control of Flight Award, for "... changing the landscape of flight mechanics." Ross was also elected AAS Fellow for "his pioneering contributions to pseudospectral optimal control."
Distinctive features.
A remarkable feature of the Ross–Fahroo methods is that it does away with the prior notions of "direct" and "indirect" methods. That is, through a collection of theorems put forth by Ross and Fahroo,
they showed that it was possible to design pseudospectral methods for optimal control that were equivalent in both the direct and indirect forms. This implied that one could use their methods as simply as a "direct" method while automatically generating accurate duals as in "indirect" methods. This revolutionized solving optimal control problems leading to widespread use of the Ross–Fahroo techniques.
Software implementation.
The Ross–Fahroo methods are implemented in the MATLAB optimal control solver, DIDO.


Cutting tool (machining)

In the context of machining, a cutting tool or cutter is typically a hardened metal tool that is used to cut, shape, and remove material from a workpiece by means of machining tools as well as abrasive tools by way of shear deformation. The majority of these tools are designed exclusively for metals.
There are several different types of single-edge cutting tools that are made from a variety of hardened metal alloys that are ground to a specific shape in order to perform a specific part of the turning process resulting in a finished machined part. Single-edge cutting tools are used mainly in the turning operations performed by a lathe in which they vary in size as well as alloy composition depending on the size and the type of material being turned. These cutting tools are held stationary by what is known as a "tool post", which is what manipulates the tools to cut the material into the desired shape. Single-edge cutting tools are also the means of cutting material performed by shaping machines and planing machines, which remove material by means of one cutting edge.
Milling and drilling tools are often multipoint tools. Drilling is exclusively used to make holes in a workpiece. All drill bits have two cutting edges that are ground into two equally tapered angles which cuts through the material by applying downward rotational force. Endmills or milling bits, which also cut material by rotational force. Although these tools are not made to put holes in a workpiece. They cut by horizontal shear deformation in which the workpiece is brought into the tool as it's rotating. This is known as the tool path which is determined by the axis of the table that is holding the workpiece in place. This table is designed to accept a variety of vises and clamping tools so that it can move into the cutter at various angles and directions while the workpiece remains still. There are several different types of endmills that perform a certain type of milling action.
Grinding stones are tools that contain several different cutting edges which encompasses the entirety of the stone. Unlike metallic cutting tools, these grinding stones never go dull. In fact the formation of cutting edges of metallic cutting tools are achieved by the use of grinding wheels and other hard abrasives. There are several different types of grinding stone wheels that are used to grind several different types of metals. Although these stones are not metal, they need to be harder than the metal that they grind. In contrast to the grinding stone, if the hardness of the metal exceeds that of the stone, the metal will cut the stone. This is not ideal. Each grain of abrasive functions as a microscopic single-point cutting edge (although of high negative rake angle), and shears a tiny chip.
Cutting tool materials must be harder than the material which is to be cut, and the tool must be able to withstand the heat and force generated in the metal-cutting process. Also, the tool must have a specific geometry, with clearance angles designed so that the cutting edge can contact the workpiece without the rest of the tool dragging on the workpiece surface. The angle of the cutting face is also important, as is the flute width, number of flutes or teeth, and margin size. In order to have a long working life, all of the above must be optimized, plus the speeds and feeds at which the tool is run.
Types.
Linear cutting tools include tool bits (single-point cutting tools) and broaches. Rotary cutting tools include drill bits, countersinks and counterbores, taps and dies, reamers, and cold saw blades. Other cutting tools, such as bandsaw blades, hacksaw blades, and fly cutters, combine aspects of linear and rotary motion
Cutting tools with inserts (indexable tools).
Cutting tools are often designed with inserts or replaceable tips (tipped tools). In these, the cutting edge consists of a separate piece of material, either brazed, welded or clamped on to the tool body. Common materials for tips include cemented carbide, polycrystalline diamond, and cubic boron nitride. Tools using inserts include milling cutters (endmills, fly cutters), tool bits, and saw blades.
Tool setup.
The detailed instructions of how to combine the tool assembly out of basic holder, tool and insert can be stored in a tool management solution.
Cutting edge.
The cutting edge of a cutting tool is a very important for the performance of the cutting process. The main features of the cutting edge are:
The measurement of the cutting edge is performed using a tactile instrument or an instrument using focus variation. To quantify a cutting edge the following parameters are used:
One of the most important cutting edge parameters is the K factor. It specifies the form of the cutting edge. 1 means a symmetric cutting edge. If the value is smaller than 1 the form is called a waterfall. If the value is larger than 1 it is called a trumpet. Depending on the material being cut, feed rate and other factors, a cutting tool with the optimum K factor should be used.


Control system security

Control system security, or industrial control system (ICS) cybersecurity, is the prevention of (intentional or unintentional) interference with the proper operation of industrial automation and control systems. These control systems manage essential services including electricity, petroleum production, water, transportation, manufacturing, and communications. They rely on computers, networks, operating systems, applications, and programmable controllers, each of which could contain security vulnerabilities. The 2010 discovery of the Stuxnet worm demonstrated the vulnerability of these systems to cyber incidents. The United States and other governments have passed cyber-security regulations requiring enhanced protection for control systems operating critical infrastructure.
Control system security is known by several other names such as "SCADA security", "PCN security", "Industrial network security", "Industrial control system (ICS) Cybersecurity", "Operational Technology (OT) Security, Industrial automation and control systems" and "Control System Cyber Security".
Risks.
Insecurity of, or vulnerabilities inherent in industrial automation and control systems (IACS) can lead to severe consequences in categories such as safety, loss of life, personal injury, environmental impact, lost production, equipment damage, information theft, and company image.
Guidance to assess, evaluate and mitigate these potential risks is provided through the application of many Governmental, regulatory, industry documents and Global Standards, addressed below.
Vulnerability of control systems.
Industrial Control Systems (ICS) have become far more vulnerable to security incidents due to the following trends that have occurred over the last 10 to 15 years.
The cyber threats and attack strategies on automation systems are changing rapidly. Regulation of industrial control systems for security is rare and is a slow moving process. The United States, for example, only does so for the nuclear power and the chemical industries.
Government efforts.
The U.S. Government Computer Emergency Readiness Team (US-CERT) originally instituted a control systems security program (CSSP) now the National Cybersecurity and Communications Integration Center (NCCIC) Industrial Control Systems, which has made available a large set of free National Institute of Standards and Technology (NIST) standards documents regarding control system security. The U.S. Government Joint Capability Technology Demonstration (JCTD) known as MOSIACS (More Situational Awareness for Industrial Control Systems) is the initial demonstration of cybersecurity defensive capability for critical infrastructure control systems. MOSAICS addresses the Department of Defense (DOD) operational need for cyber defense capabilities to defend critical infrastructure control systems from cyber attack, such as power, water and wastewater, and safety controls, affect the physical environment. The MOSAICS JCTD prototype will be shared with commercial industry through Industry Days for further research and development, an approach intended to lead to an innovative, game-changing capabilities for cybersecurity for critical infrastructure control systems.
Industrial Cybersecurity Standards.
The international standard for cybersecurity in industrial automation is the IEC 62443. In addition, multiple national organizations such as the NIST and NERC in the USA released guidelines and requirements for cybersecurity in control systems.
IEC 62443.
The IEC 62443 cybersecurity standard defines processes, techniques and requirements for Industrial Automation and Control Systems (IACS). Its documents are the result of the IEC standards creation process where all national committees involved agree upon a common standard. The IEC 62443 was influenced by and is partly based on the ANSI/ISA-99 series of standards and the VDI/VDE 2182 guidelines.
 All IEC 62443 standards and technical reports are organized into four general categories called "General", "Policies and Procedures", "System" and "Component".
NERC.
The most widely recognized and latest NERC security standard is NERC 1300, which is a modification/update of NERC 1200. The latest version of NERC 1300 is called CIP-002-3 through CIP-009-3, with CIP referring to Critical Infrastructure Protection. These standards are used to secure bulk electric systems although NERC has created standards within other areas. The bulk electric system standards also provide network security administration while still supporting best-practice industry processes.
NIST.
The NIST Cybersecurity Framework (NIST CSF) provides a high-level taxonomy of cybersecurity outcomes and a methodology to assess and manage those outcomes. It is intended to help private sector organizations that provide critical infrastructure with guidance on how to protect it.
NIST Special Publication 800-82 Rev. 2 "Guide to Industrial Control System (ICS) Security" describes how to secure multiple types of Industrial Control Systems against cyber attacks while considering the performance, reliability, and safety requirements specific to ICS.
Control system security certifications.
Certifications for control system security have been established by several global Certification Bodies. Most of the schemes are based on the IEC 62443 and describe test methods, surveillance audit policy, public documentation policies, and other specific aspects of their program.