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Encyclopedia > Control theory

Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Psychological science redirects here. ... Sociology (from Latin: socius, companion; and the suffix -ology, the study of, from Greek Î»ÏŒÎ³Î¿Ï‚, lÃ³gos, knowledge [1]) is the systematic and scientific study of society, including patterns of social relationships, social action, and culture[2]. Areas studied in sociology can range from the analysis of brief contacts between anonymous... Control theory, as an extension to the field of psychoanalysis, postulates human behaviors driven by the therapeutic function of taming the threatening Otherness of oneâ€™s surroundings. ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... The Lorenz attractor is an example of a non-linear dynamical system. ... Basic Principles A controller is the brain component of a system that monitors certain input variables and adjusts other output variables to achieve the desired operation. ...

The concept of the feedback loop to control the dynamic behavior of the reference: this is negative feedback because the sensed value is subtracted from the desired value to create the error signal which is amplified by the controller.

Image File history File links Size of this preview: 800 Ã— 352 pixelsFull resolution (1135 Ã— 500 pixel, file size: 44 KB, MIME type: image/jpeg) mbbradford I, the copyright holder of this work, hereby release it into the public domain. ... Image File history File links Size of this preview: 800 Ã— 352 pixelsFull resolution (1135 Ã— 500 pixel, file size: 44 KB, MIME type: image/jpeg) mbbradford I, the copyright holder of this work, hereby release it into the public domain. ...

Control theory is

The Lorenz attractor is an example of a non-linear dynamical system. ... Engineering is the discipline of acquiring and applying knowledge of design, analysis, and/or construction of works for practical purposes. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Psychological science redirects here. ... Control theory, as an extension to the field of psychoanalysis, postulates human behaviors driven by the therapeutic function of taming the threatening Otherness of oneâ€™s surroundings. ... Criminology is the scientific study of crime as an individual and social phenomenon. ...

### An example

Consider an automobile's cruise control, which is a device designed to maintain a constant vehicle speed; the desired or reference speed, provided by the driver. The system in this case is the vehicle. The system output is the vehicle speed, and the control variable is the engine's throttle position which influences engine torque output. Cruise control (sometimes known as speed control or Autocruise) is a system to automatically control the speed of an automobile. ... In an engine, the throttle is the mechanism by which the engines power is increased or decreased. ... For other senses of this word, see torque (disambiguation). ...

A simple way to implement cruise control is to lock the throttle position when the driver engages cruise control. However, on hilly terrain, the vehicle will slow down going uphill and accelerate going downhill. In fact, any parameter different than what was assumed at design time will translate into a proportional error in the output velocity, including exact mass of the vehicle, wind resistance, and tire pressure. This type of controller is called an open-loop controller because there is no direct connection between the output of the system (the engine torque) and the actual conditions encountered; that is to say, the system does not and can not compensate for unexpected forces. An open-loop controller does not use feedback to control states or outputs of a dynamic system. ...

In a closed-loop control system, a sensor monitors the output (the vehicle's speed) and feeds the data to a computer which continuously adjusts the control input (the throttle) as necessary to keep the control error to a minimum (to maintain the desired speed). Feedback on how the system is actually performing allows the controller (vehicle's on board computer) to dynamically compensate for disturbances to the system, such as changes in slope of the ground or wind speed. An ideal feedback control system cancels out all errors, effectively mitigating the effects of any forces that may or may not arise during operation and producing a response in the system that perfectly matches the user's wishes.

## History

Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the centrifugal governor, conducted by the physicist James Clerk Maxwell in 1868 entitled On Governors[1]. This described and analyzed the phenomenon of "hunting," in which lags in the system can lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate Edward John Routh generalized the results of Maxwell for the general class of linear systems. Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877. This result is called the Routh-Hurwitz Criterion. A centrifugal governor is a specific type of governor that controls the speed of an engine by regulating the amount of fuel admitted, so as to maintain a near constant speed whatever the load or fuel supply conditions. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance â€” eponymously named Maxwells equations â€” including an important modification (extension) of the AmpÃ¨res... Year 1868 (MDCCCLXVIII) was a leap year starting on Wednesday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Monday of the 12-day slower Julian calendar). ... Edward John Routh (1831-1907) was a British mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the nineteenth century. ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... 1877 (MDCCCLXXVII) was a common year starting on Monday (see link for calendar). ... In mathematics, Routh-Hurwitz theorem permits to determine whether a given polynomial is Hurwitz stable. ...

A notable application of dynamic control was in the area of manned flight. The Wright Brothers made their first successful test flights on December 17, 1903 and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Control of the airplane was necessary for safe flight. The Wright brothers, Orville (August 19, 1871â€“January 30, 1948) and Wilbur (April 16, 1867â€“May 30, 1912), were two Americans generally credited with building the worlds first successful airplane and making the first controlled, powered and heavier-than-air human flight on December 17, 1903. ... December 17 is the 351st day of the year (352nd in leap years) in the Gregorian calendar. ... Year 1903 (MCMIII) was a common year starting on Thursday (link will display calendar) of the Gregorian calendar or a common year starting on Wednesday of the 13-day slower Julian calendar. ...

## People in systems and control

A lot of active and historical figures made significant contribution to control theory, for example: People in systems and control is an alphabetical list (in two parts) of people who have made significant contributions in the fields of System analysis and Control theory. ...

Richard Ernest Bellman (1920â€“1984) was an applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics. ... In mathematics and computer science, dynamic programming is a method of solving problems exhibiting the properties of overlapping subproblems and optimal substructure (described below) that takes much less time than naive methods. ... Harold Stephen Black (1898-1983) was an cock who revolutionized the field of applied electronics by inventing the buttplug in 1927. ... A negative feedback amplifier, or more commonly simply a feedback amplifier, is an amplifier which uses a negative feedback network, generally for improving performance (gain stability, linearity, frequency response etc. ... Aleksandr Mikhailovich Lyapunov (&#1040;&#1083;&#1077;&#1082;&#1089;&#1072;&#1085;&#1076;&#1088; &#1052;&#1080;&#1093;&#1072;&#1081;&#1083;&#1086;&#1074;&#1080;&#1095; &#1051;&#1103;&#1087;&#1091;&#1085;&#1086;&#1074;) (June 6, 1857 - November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist. ... In mathematics, stability theory deals with the stability of the solutions of differential equations and dynamical systems. ... Harry Nyquist (pron. ... The Nyquist plot for . ... John R. Ragazzini John Ralph Ragazzini (1912 â€“ November 22, 1988) was an American electrical engineer and a professor of Electrical Engineering. ... Digital control is a branch of control theory that uses digital computers to act as a system. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... Norbert Wiener Norbert Wiener (November 26, 1894, Columbia, Missouri â€“ March 18, 1964, Stockholm Sweden) was an American theoretical and applied mathematician. ... For other uses, see Cybernetics (disambiguation). ...

## Classical control theory: the closed-loop controller

To avoid the problems of the open-loop controller, control theory introduces feedback. 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. velocity or torque of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is used as input to the process, closing the loop. For the superhero, see Feedback (Dark Horse Comics). ... Basic Principles A controller is the brain component of a system that monitors certain input variables and adjusts other output variables to achieve the desired operation. ... In control theory, states are what characterize a system. ... Output is the term denoting either an exit or changes which exits a system and which activate/modify a process. ... The Lorenz attractor is an example of a non-linear dynamical system. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ... For other kinds of motors, see motor. ... Not to be confused with censure, censer, or censor. ...

Closed-loop controllers have the following advantages over open-loop controllers: An open-loop controller does not use feedback to control states or outputs of a dynamic system. ...

• disturbance rejection (such as unmeasured friction in a motor)
• guaranteed performance even with model uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
• unstable processes can be stabilized
• reduced sensitivity to parameter variations
• improved reference tracking performance

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 mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... Instability in systems is generally characterized by some of the outputs or internal states growing without bounds. ...

A common closed-loop controller architecture is the PID controller. A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. ...

The output of the system y(t) is fed back to the reference value r(t), through a sensor measurement. 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). In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. ... In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. ... Distributed parameter system (as opposed to a lumped parameter system) refers to system whose state-space is infinite-dimensional. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...

If we assume the controller C and the plant P are linear and time-invariant (i.e.: elements of their transfer function C(s) and P(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations: Simple feedback control loop block diagram Created in Dia. ... For other uses, see Linear (disambiguation). ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t â‰¥ 0) that transforms it to a function F(s) (image) with a complex argument s. ...

$Y(s) = P(s) U(s),!$
$U(s) = C(s) E(s),!$
$E(s) = R(s) - Y(s),!$

Solving for Y(s) in terms of R(s) gives:

$Y(s) = left( frac{P(s)C(s)}{1 + P(s)C(s)} right) R(s)$

The term $frac{P(s)C(s)}{1 + P(s)C(s)}$ is referred to as the transfer function of the system. The numerator is the forward gain from r to y, and the denominator is one plus the loop gain of the feedback loop. If $P(s)C(s) gg 1$, i.e. it has a large norm with each value of s, then Y(s) is approximately equal to R(s). This means simply setting the reference controls the output. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

## Topics in control theory

### Stability

Stability (in control theory) often means that for any bounded input over any amount of time, the output will also be bounded. This is known as BIBO stability (see also Lyapunov stability). If a system is BIBO stable then the output cannot "blow up" (i.e., become infinite) if the input remains finite. Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must satisfy some criteria depending on whether a continuous or discrete time analysis is used: In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ... In mathematics, the notion of Lyapunov stability occurs in the study of dynamical systems. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

• In continuous time, the Laplace transform is used to obtain the transfer function. A system is stable if the poles of this transfer function lie in the closed left half of the complex plane. I.e. the real part of all the poles is less than or equal to zero).

OR In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original ) with a real argument t (t â‰¥ 0) that transforms it to a function F(s) (image) with a complex argument s. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

• In discrete time the Z-transform is used. A system is stable if the poles of this transfer function lie on or inside the unit circle. I.e. the magnitude of the poles is less than or equal to one)

When the appropriate conditions above are satisfied a system is said to be asymptotically stable: the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at at complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero. In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... Illustration of a unit circle. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In the theory of dynamical systems, a linear time-invariant system is marginally stable if every eigenvalue in the systems transfer-function is non-positive, and all eigenvalues with zero real value are simple roots. ...

Differences between the two cases are not a contradiction. The Laplace transform is in Cartesian coordinates and the Z-transform is in circular coordinates, and it can be shown that Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

• the negative-real part in the Laplace domain can map onto the interior of the unit circle
• the positive-real part in the Laplace domain can map onto the exterior of the unit circle

If the system in question has an impulse response of The Impulse response from a simple audio system. ...

x[n] = 0.5nu[n]

and considering the Z-transform (see this example), it yields In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

$X(z) = frac{1}{1 - 0.5z^{-1}}$

which has a pole in z = 0.5 (zero imaginary part). This system is BIBO (asymptotically) stable since the pole is inside the unit circle.

However, if the impulse response was

x[n] = 1.5nu[n]

then the Z-transform is

$X(z) = frac{1}{1 - 1.5z^{-1}}$

which has a pole at z = 1.5 and is not BIBO stable since the pole has a modulus strictly greater than one.

Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the root locus , Bode plots or the Nyquist plots. In control theory, the root locus is the locus of the poles of a transfer function as the system gain K is varied on some interval. ... The Bode plot for a first-order Butterworth filter A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show... A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ...

### Controllability and observability

Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state it is termed Stabilizable. Observability instead is related to the possibility of "observing", through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behaviour of an unobservable state and hence cannot use it to stabilize the system. However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable. Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. ... Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. ...

From a geometrical point of view, looking at the states of each variable of the system to be controlled, every "bad" state of these variables must be controllable and observable to ensure a good behaviour in the closed-loop system. That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis. In linear algebra, a scalar &#955; is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=&#955;x. ...

Solutions to problems of uncontrollable or unobservable system include adding actuators and sensors.

### Control specifications

Several different control strategies have been devised in the past years. These vary from extremely general ones (PID controller), to others devoted to very particular classes of systems (especially robotics or aircraft cruise control). A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. ... The Shadow robot hand system holding a lightbulb. ... Flying machine redirects here. ...

A control problem can have several specifications. Stability, of course, is always present: the controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have $Re[lambda] < -overline{lambda}$, where $overline{lambda}$ is a fixed value strictly greater than zero, instead of simply ask that Re[λ] < 0.

Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included. An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus. ...

Other "classical" control theory specifications regard the time-response of the closed-loop system: these include the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). Frequency domain specifications are usually related to robustness (see after). In electronics, when approximating a voltage or current step function, rise time (also risetime) refers to the time required for a signal to change from a specified low value to a specified high value. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Robust means healthy, strong, durable, and often adaptable, innovative, flexible. ...

Modern performance assessments use some variation of integrated tracking error (IAE,ISA,CQI).

### Model identification and robustness

Main article: System identification

A control system must always have some robustness property. A robust controller is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This specification is important: no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise the true system dynamics can be so complicated that a complete model is impossible. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. ... In robust control, the word robust means that the controller or regulator should work well (e. ...

#### System identification

The process of determining the equations that govern the model's dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations: for example, in the case of a mass-spring-damper system we know that $m ddot{{x}}(t) = - K x(t) - Beta dot{x}(t)$. Even assuming that a "complete" model is used in designing the controller, all the parameters included in these equations (called "nominal parameters") are never known with absolute precision; the control system will have to behave correctly even when connected to physical system with true parameter values away from nominal. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. ...

Some advanced control techniques include an "on-line" identification process (see later). The parameters of the model are calculated ("identified") while the controller itself is running: in this way, if a drastic variation of the parameters ensues (for example, if the robot's arm releases a weight), the controller will adjust itself consequently in order to ensure the correct performance.

#### Analysis

Analysis of the robustness of a SISO control system can be performed in the frequency domain, considering the system's transfer function and using Nyquist and Bode diagrams. Topics include phase margin and amplitude margin. For MIMO and, in general, more complicated control systems one must consider the theoretical results devised for each control technique (see next section): i.e., if particular robustness qualities are needed, the engineer must shift his attention to a control technique including them in its properties. A Nyquist plot is used in signal processing for assessing the stability of a system with feedback. ... The Bode plot for a first-order Butterworth filter A Bode plot, named for Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot: A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show... In electronic amplifiers, phase margin is the difference, measured in degrees, between the phase angle of the amplifiers output signal and -360Â°. In feedback amplifiers, the phase margin is measured at the frequency at which the open loop voltage gain of the amplifier and the closed loop voltage gain...

#### Constraints

A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system: for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even break actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control (see later), and anti-wind up systems. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold. Model Predictive Control, or MPC, is an advanced method of process control that has been in use in the process industries such as chemical plants and oil refineries since the 1980s. ...

## Main control strategies

Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be obtained by directly placing the poles. Non-linear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen. Here a summary list of the main control techniques is shown: A linear system is a model of a system based on some kind of linear operator. ... Aleksandr Mikhailovich Lyapunov (ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ€ ÐœÐ¸Ñ…Ð°Ð¹Ð»Ð¾Ð²Ð¸Ñ‡ Ð›ÑÐ¿ÑƒÐ½Ð¾Ð²) (June 6, 1857 â€“ November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist. ...

### PID controllers

Main article: PID controller

The PID controller is probably the most-used feedback control design, being the simplest one. "PID" means Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control signal. If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and tracking error e(t) = r(t) − y(t), a PID controller has the general form A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. ... A proportional-integral-derivative controller (PID controller) is a generic control loop feedback mechanism widely used in industrial control systems. ...

$u(t) = K_P e(t) + K_I int e(t)dt + K_D dot{e}(t)$

The desired closed loop dynamics is obtained by adjusting the three parameters KP, KI and KD, 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. Process control is a statistics and engineering discipline that deals with architectures, mechanisms, and algorithms for controlling the output of a specific process. ...

### Direct pole placement

For MIMO systems, pole placement can be performed mathematically using a State space representation of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design. In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. ... In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. ...

### Optimal control

Main article: Optimal control

Optimal control is a particular control technique in which the control signal optimizes a certain "cost index": for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are Model Predictive Control (MPC) and Linear-Quadratic-Gaussian control (LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important feature in many industrial processes. However, the "optimal control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely used control technique in process control. Optimal control theory, a generalization of the calculus of variations, is a mathematical optimization method for deriving control policies. ... Model Predictive Control, or MPC, is an advanced method of process control that has been in use in the process industries such as chemical plants and oil refineries since the 1980s. ... Linear quadratic Gaussian (LQG) control is a method of designing feedback control laws for linear systems with additive Gaussian noise processes that minimize a given quadratic cost functional. ... Process control is a statistics and engineering discipline that deals with architectures, mechanisms, and algorithms for controlling the output of a specific process. ...

Adaptive control uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the aerospace industry in the 1950s, and have found particular success in that field. When the parameters of a system are slowly time-varying or uncertain, we need a control law that adapts itself under such conditions to give reliable performance. ... Aerospace engineering is the branch of engineering concerning aircraft, spacecraft and related topics. ... The 1950s decade refers to the years 1950 to 1959 inclusive. ...

### Intelligent control

Main article: Intelligent control

Intelligent control use various AI computing approaches like neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation and genetic algorithms to control a dynamic system All control techniques that use various soft computing approaches like neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation and genetic algorithms can be put into the class of intelligent control. ... A neural network is an interconnected group of neurons. ... Bayesian probability is an interpretation of probability suggested by Bayesian theory, which holds that the concept of probability can be defined as the degree to which a person believes a proposition. ... Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... As a broad subfield of artificial intelligence, machine learning is concerned with the design and development of algorithms and techniques that allow computers to learn. At a general level, there are two types of learning: inductive, and deductive. ... In computer science evolutionary computation is a subfield of artificial intelligence (more particularly computational intelligence) involving combinatorial optimization problems. ... A genetic algorithm (GA) is an algorithm used to find approximate solutions to difficult-to-solve problems through application of the principles of evolutionary biology to computer science. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...

### Non-linear control systems

Main article: Non-linear control

Processes in industries like robotics and the aerospace industry typically have strong non-linear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques: but in many cases it can be necessary to devise from scratch theories permitting control of non-linear systems. These normally take advantage of results based on Lyapunov's theory. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the non-linear case, as well as showing the subtleties that make it a more challenging problem. Non-linear control is a sub-division of control engineering which deals with the control of non-linear systems. ... The Shadow robot hand system holding a lightbulb. ... Aerospace engineering is the branch of engineering concerning aircraft, spacecraft and related topics. ... Lyapunov theory is a collection of results regarding stability of dynamical systems, named after a Russian mathematician Aleksandr Mikhailovich Lyapunov. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

 Examples of control systems Automation Deadbeat Controller Distributed parameter systems Fractional order control H-infinity loop-shaping PID controller Model predictive control Process control Robust control Servomechanism State space (controls) Topics in control theory Coefficient diagram method Control reconfiguration Feedback H infinity Hankel singular value Lead-lag compensator Radial basis function Robotic unicycle Root locus Stable polynomial Underactuation Other related topics

## References

1. ^ J. C. Maxwell, "On Governers," Proc. R Soc. London Vol 16, 270-283 (1968) Reprinted

## Literature

• Vannevar Bush (1929). Operational Circuit Analysis. John Wiley and Sons, Inc..
• Robert F. Stengel (1994). Optimal Control and Estimation. Dover Publications. ISBN 0-486-68200-5, ISBN-13: 978-0-486-68200-6.
• Franklin et al. (2002). Feedback Control of Dynamic Systems, 4, New Jersey: Prentice Hall. ISBN 0-13-032393-4.
• Joseph L. Hellerstein, Dawn M. Tilbury, and Sujay Parekh (2004). Feedback Control of Computing Systems. John Wiley and Sons. ISBN 0-47-126637-X, ISBN-13: 978-0-471-26637-2.
• Diederich Hinrichsen and Anthony J. Pritchard (2005). Mathematical Systems Theory I - Modelling, State Space Analysis, Stability and Robustness. Springer. ISBN 0-978-3-540-44125-0.
• Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer. ISBN 0-387-984895.

// Diederich Hinrichsen (born 17. ... Eduardo Sontag (b. ...

• Andrei, Neculai (2005). "Modern Control Theory - A historical Perspective". Retrieved on 2007-10-10.

Results from FactBites:

 Control Theory (824 words) When you press the button on a light controlled crossing, you initiate a sequence of light changes, which enable you to cross the road safely, and when you relax in the comfort of your centrally heated home, spare a thought for the thermostat, which works constantly for you maintaining a comfortable temperature. Experts in the field of control theory would like us to refer to this as open loop control, for reasons that will become obvious later, but I prefer the more descriptive term of “sequencer”. One point to note here is that the light timer control illustrates the use of a sequencer based on events (on/off) linked to a particular time and day.
 Robert Agnew’s General Strain Theory (4719 words) General strain theory has defined measurements of strain, the major types of strain, the links between strain and crime, coping strategies to strain, the determinants of delinquent or nondelinquent behavior, and policy recommendations that are based on this theory. While control theory rests on the premise that the breakdown of society frees the individual to commit crime, strain theory is focused on the pressure that is placed on the individual to commit crime (Agnew, 1992:49). Control theory, however, is based on the absence of significant relationships with nondeviant others, and social learning theory is based on positive relationships with deviant others (Agnew, 1992:49).
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