Control theory

This article is about control theory in engineering. For control theory in linguistics, seecontrol (linguistics).For control theory in psychology and sociology, seecontrol theory (sociology)andPerceptual control theory.

Control theoryis a field ofcontrol engineeringandapplied mathematicsthat deals with thecontrolofdynamical systemsin engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing anydelay,overshoot,orsteady-state errorand ensuring a level of controlstability;often with the aim to achieve a degree ofoptimality.

To do this, acontrollerwith the requisite corrective behavior is required. This controller monitors the controlledprocess variable(PV), and compares it with the reference orset point(SP). The difference between actual and desired value of the process variable, called theerrorsignal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied arecontrollabilityandobservability.Control theory is used incontrol system engineeringto design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such asrobotics.

Extensive use is usually made of a diagrammatic style known as theblock diagram.In it thetransfer function,also known as the system function or network function, is a mathematical model of the relation between the input and output based on thedifferential equationsdescribing the system.

Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described byJames Clerk Maxwell.Control theory was further advanced byEdward Routhin 1874,Charles Sturmand in 1895,Adolf Hurwitz,who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development ofPID controltheory byNicolas Minorsky. Although a major application ofmathematicalcontrol theory is incontrol systems engineering,which deals with the design ofprocess controlsystems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs - thus control theory also has applications in life sciences, computer engineering, sociology andoperations research.

History

See also:Control engineering § History

Centrifugal governorin aBoulton & Watt engineof 1788

Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of thecentrifugal governor,conducted by the physicistJames Clerk Maxwellin 1868, entitledOn Governors.A centrifugal governor was already used to regulate the velocity of windmills.Maxwell described and analyzed the phenomenon ofself-oscillation,in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate,Edward John Routh,abstracted Maxwell's results for the general class of linear systems.Independently,Adolf Hurwitzanalyzed system stability using differential equations in 1877, resulting in what is now known as theRouth–Hurwitz theorem.

A notable application of dynamic control was in the area of crewed flight. TheWright brothersmade 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). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds.

ByWorld War II,control theory was becoming an important area of research.Irmgard Flügge-Lotzdeveloped the theory of discontinuous automatic control systems, and applied thebang-bang principleto the development ofautomatic flight control equipmentfor aircraft.Other areas of application for discontinuous controls includedfire-control systems,guidance systemsandelectronics.

Sometimes, mechanical methods are used to improve the stability of systems. For example,ship stabilizersare fins mounted beneath the waterline and emerging laterally. In contemporary vessels, they may be gyroscopically controlled active fins, which have the capacity to change their angle of attack to counteract roll caused by wind or waves acting on the ship.

TheSpace Racealso depended on accurate spacecraft control, and control theory has also seen an increasing use in fields such as economics and artificial intelligence. Here, one might say that the goal is to find aninternal modelthat obeys thegood regulator theorem.So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of aregulatorinteracting with aplant.

Open-loop and closed-loop (feedback) control

This section is an excerpt fromControl loop § Open-loop and closed-loop.[edit]

Fundamentally, there are two types of control loop:open-loop control(feedforward), andclosed-loop control(feedback).

An electromechanical timer, normally used for open-loop control based purely on a timing sequence, with no feedback from the process

In open-loop control, the control action from the controller is independent of the "process output" (or "controlled process variable" ). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the switching on/off of the boiler, but the controlled variable should be the building temperature, but is not because this is open-loop control of the boiler, which does not give closed-loop control of the temperature.

In closed loop control, the control action from the controller is dependent on the process output. In the case of the boiler analogy this would include a thermostat to monitor the building temperature, and thereby feed back a signal to ensure the controller maintains the building at the temperature set on the thermostat. A closed loop controller therefore has a feedback loop which ensures the controller exerts a control action to give a process output the same as the "reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.

The definition of a closed loop control system according to theBritish Standards Institutionis "a control system possessing monitoring feedback, the deviation signal formed as a result of this feedback being used to control the action of a final control element in such a way as to tend to reduce the deviation to zero."

Likewise; "AFeedback Control Systemis a system which tends to maintain a prescribed relationship of one system variable to another by comparing functions of these variables and using the difference as a means of control. "

Classical control theory

This section is an excerpt fromClosed-loop controller.[edit]

Example of a single industrial control loop; showing continuously modulated control of process flow.

Aclosed-loop controlleror feedback controller is acontrol loopwhich incorporatesfeedback,in contrast to anopen-loop controllerornon-feedback controller. A closed-loop controller uses feedback to controlstatesoroutputsof adynamical system.Its name comes from the information path in the system: process inputs (e.g.,voltageapplied to anelectric motor) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured withsensorsand processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.

In the case of linearfeedbacksystems, acontrol loopincludingsensors,control algorithms, and actuators is arranged in an attempt to regulate a variable at asetpoint(SP). An everyday example is thecruise controlon 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. ThePID algorithmin the controller restores the actual speed to the desired speed in an optimum way, with minimal delay orovershoot,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 systemsdo not make use of feedback, and run only in pre-arranged ways.

Closed-loop controllers have the following advantages over open-loop controllers:

• disturbance rejection (such as hills in the cruise control example above)
• guaranteed performance even withmodeluncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
• unstableprocesses can be stabilized
• reduced sensitivity to parameter variations
• improved reference tracking performance
• improved rectification of random fluctuations

In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termedfeedforwardand serves to further improve reference tracking performance.

A common closed-loop controller architecture is thePID controller.

A basic feedback loop

Linear and nonlinear control theory

The field of control theory can be divided into two branches:

• Linear control theory– This applies to systems made of devices which obey thesuperposition principle,which means roughly that the output is proportional to the input. They are governed bylinear differential equations.A major subclass is systems which in addition have parameters which do not change with time, calledlinear time invariant(LTI) systems. These systems are amenable to powerfulfrequency domainmathematical techniques of great generality, such as theLaplace transform,Fourier transform,Z transform,Bode plot,root locus,andNyquist stability criterion.These lead to a description of the system using terms likebandwidth,frequency response,eigenvalues,gain,resonant frequencies,zeros and poles,which give solutions for system response and design techniques for most systems of interest.
• Nonlinear control theory– This covers a wider class of systems that do not obey the superposition principle, and applies to more real-world systems because all real control systems are nonlinear. These systems are often governed bynonlinear differential equations.The few mathematical techniques which have been developed to handle them are more difficult and much less general, often applying only to narrow categories of systems. These includelimit cycletheory,Poincaré maps,Lyapunov stability theorem,anddescribing functions.Nonlinear systems are often analyzed usingnumerical methodson computers, for example bysimulatingtheir operation using asimulation language.If only solutions near a stable point are of interest, nonlinear systems can often belinearizedby approximating them by a linear system usingperturbation theory,and linear techniques can be used.

Analysis techniques - frequency domain and time domain

Mathematical techniques for analyzing and designing control systems fall into two different categories:

• Frequency domain– In this type the values of thestate variables,the mathematicalvariablesrepresenting the system's input, output and feedback are represented as functions offrequency.The input signal and the system'stransfer functionare converted from time functions to functions of frequency by atransformsuch as theFourier transform,Laplace transform,orZ transform.The advantage of this technique is that it results in a simplification of the mathematics; thedifferential equationsthat represent the system are replaced byalgebraic equationsin the frequency domain which is much simpler to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above.
• Time-domain state space representation– In this type the values of thestate variablesare represented as functions of time. With this model, the system being analyzed is represented by one or moredifferential equations.Since frequency domain techniques are limited tolinearsystems, time domain is widely used to analyze real-world nonlinear systems. Although these are more difficult to solve, modern computer simulation techniques such assimulation languageshave made their analysis routine.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domainstate spacerepresentation,a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). 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 inputs and outputs, we would otherwise have to write down 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. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.

System interfacing - SISO & MIMO

Control systems can be divided into different categories depending on the number of inputs and outputs.

• Single-input single-output(SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or anaudio system,in which the control input is the input audio signal and the output is the sound waves from the speaker.
• Multiple-input multiple-output(MIMO) – These are found in more complicated systems. For example, modern largetelescopessuch as theKeckandMMThave mirrors composed of many separate segments each controlled by anactuator.The shape of the entire mirror is constantly adjusted by a MIMOactive opticscontrol system using input from multiple sensors at the focal plane, to compensate for changes in the mirror shape due to thermal expansion, contraction, stresses as it is rotated and distortion of thewavefrontdue to turbulence in the atmosphere. Complicated systems such asnuclear reactorsand humancellsare simulated by a computer as large MIMO control systems.

Classical SISO system design

The scope of classical control theory is limited tosingle-input and single-output(SISO) system design, except when analyzing for disturbance rejection using a second input. The system analysis is carried out in the time domain usingdifferential equations,in the complex-s domain with theLaplace transform,or in the frequency domain by transforming from the complex-s domain. Many systems may be assumed to have a second order and single variable system response in the time domain. A controller designed using classical theory often requires on-site tuning due to incorrect design approximations. Yet, due to the easier physical implementation of classical controller designs as compared to systems designed using modern control theory, these controllers are preferred in most industrial applications. The most common controllers designed using classical control theory arePID controllers.A less common implementation may include either or both a Lead or Lag filter. The ultimate end goal is to meet requirements typically provided in the time-domain called the step response, or at times in the frequency domain called the open-loop response. The step response characteristics applied in a specification are typically percent overshoot, settling time, etc. The open-loop response characteristics applied in a specification are typically Gain and Phase margin and bandwidth. These characteristics may be evaluated through simulation including a dynamic model of the system under control coupled with the compensation model.

Modern MIMO system design

Modern control theory is carried out in thestate space,and can deal withmultiple-input and multiple-output(MIMO) systems. This overcomes the limitations of classical control theory in more sophisticated design problems, such as fighter aircraft control, with the limitation that no frequency domain analysis is possible. In modern design, a system is represented to the greatest advantage as a set of decoupled first orderdifferential equationsdefined usingstate variables.Nonlinear,multivariable,adaptiveandrobust controltheories come under this division. Matrix methods are significantly limited for MIMO systems where linear independence cannot be assured in the relationship between inputs and outputs.Being fairly new, modern control theory has many areas yet to be explored. Scholars likeRudolf E. KálmánandAleksandr Lyapunovare well known among the people who have shaped modern control theory.

Topics in control theory

Stability

Thestabilityof a generaldynamical systemwith no input can be described withLyapunov stabilitycriteria.

• Alinear systemis calledbounded-input bounded-output (BIBO) stableif its output will stayboundedfor any bounded input.
• Stability fornonlinear systemsthat take an input isinput-to-state stability(ISS), which combines Lyapunov stability and a notion similar to BIBO stability.

For simplicity, the following descriptions focus on continuous-time and discrete-timelinear systems.

Mathematically, this means that for a causal linear system to be stable all of thepolesof itstransfer functionmust have negative-real values, i.e. the real part of each pole must be less than zero. Practically speaking, stability requires that the transfer function complex poles reside

• in the open left half of thecomplex planefor continuous time, when theLaplace transformis used to obtain the transfer function.
• inside theunit circlefor discrete time, when theZ-transformis used.

The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions. The continuous Laplace transform is inCartesian coordinateswhere the$x$axis is the real axis and the discrete Z-transform is incircular coordinateswhere the$\rho$axis is the real axis.

When the appropriate conditions above are satisfied a system is said to beasymptotically 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 amodulusequal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it ismarginally stable;in this case the system transfer function has non-repeated poles at the 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.

If a system in question has animpulse responseof

$\ x[n]=0.5^{n}u[n]$

then the Z-transform (seethis example), is given by

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

which has a pole in$z=0.5$(zeroimaginary part). This system is BIBO (asymptotically) stable since the pole isinsidethe unit circle.

However, if the impulse response was

$\ x[n]=1.5^{n}u[n]$

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 theroot locus,Bode plotsor theNyquist plots.

Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships useantiroll finsthat extend transversely from the side of the ship for perhaps 30 feet (10 m) and are continuously rotated about their axes to develop forces that oppose the roll.

Controllability and observability

Main articles:ControllabilityandObservability

Controllabilityandobservabilityare 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 is termedstabilizable.Observability instead is related to the possibility ofobserving,through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behavior 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.

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 behavior in the closed-loop system. That is, if one of theeigenvaluesof 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.

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

Control specification

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 (especiallyroboticsor aircraft cruise control).

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 asking that$Re[\lambda ]<0$.

Another typical specification is the rejection of a step disturbance; including anintegratorin 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.

Other "classical" control theory specifications regard the time-response of the closed-loop system. These include therise time(the time needed by the control system to reach the desired value after a perturbation), peakovershoot(the highest value reached by the response before reaching the desired value) and others (settling time,quarter-decay). Frequency domain specifications are usually related torobustness(see after).

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

Model identification and robustness

A control system must always have some robustness property. Arobust controlleris such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This requirement is important, as 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 truesystem dynamicscan be so complicated that a complete model is impossible.

System identification

Further information:System identification

The process of determining the equations that govern the model's dynamics is calledsystem identification.This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically itstransfer functionor 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 amass-spring-dampersystem we know that$m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)$.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 a physical system with true parameter values away from nominal.

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 (single input single output) control system can be performed in the frequency domain, considering the system's transfer function and usingNyquistandBode diagrams.Topics includegain and phase marginand amplitude margin. For MIMO (multi-input multi output) 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 their attention to a control technique by including these qualities in its properties.

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 damage or break actuators or other subsystems. Specific control techniques are available to solve the problem:model predictive control(see later), andanti-wind up systems.The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold.

System classifications

Linear systems control

Main article:State space (controls)

For MIMO systems, pole placement can be performed mathematically using astate space representationof 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.

Nonlinear systems control

Main article:Nonlinear control

Processes in industries likeroboticsand theaerospace industrytypically have strong nonlinear 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 nonlinear systems. These, e.g.,feedback linearization,backstepping,sliding mode control,trajectory linearization control normally take advantage of results based onLyapunov's theory.Differential geometryhas been widely used as a tool for generalizing well-known linear control concepts to the nonlinear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states.

Decentralized systems control

Main article:Distributed control system

When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions.

Deterministic and stochastic systems control

Main article:Stochastic control

A stochastic control problem is one in which the evolution of the state variables is subjected to random shocks from outside the system. A deterministic control problem is not subject to external random shocks.

Main control strategies

Every control system must guarantee first the stability of the closed-loop behavior. Forlinear systems,this can be obtained by directly placing the poles. Nonlinear control systems use specific theories (normally based onAleksandr 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.

List of the main control techniques

• Optimal controlis 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 areModel Predictive Control(MPC) andlinear-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 inprocess control.
• Robust controldeals explicitly with uncertainty in its approach to controller design. Controllers designed usingrobust controlmethods tend to be able to cope with small differences between the true system and the nominal model used for design.The early methods ofBodeand others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness. Examples of modern robust control techniques includeH-infinity loop-shapingdeveloped by Duncan McFarlane andKeith Glover,Sliding mode control(SMC) developed byVadim Utkin,and safe protocols designed for control of large heterogeneous populations of electric loads in Smart Power Grid applications.Robust methods aim to achieve robust performance and/orstabilityin the presence of small modeling errors.
• Stochastic controldeals with control design with uncertainty in the model. In typical stochastic control problems, it is assumed that there exist random noise and disturbances in the model and the controller, and the control design must take into account these random deviations.
• Adaptive controluses 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 theaerospace industryin the 1950s, and have found particular success in that field.
• Ahierarchical control systemis a type ofcontrol systemin which a set of devices and governing software is arranged in ahierarchicaltree.When the links in the tree are implemented by acomputer network,then that hierarchical control system is also a form ofnetworked control system.
• Intelligent controluses various AI computing approaches likeartificial neural networks,Bayesian probability,fuzzy logic,machine learning,evolutionary computationandgenetic algorithmsor a combination of these methods, such asneuro-fuzzyalgorithms, to control adynamic system.
• Self-organized criticality controlmay be defined as attempts to interfere in the processes by which theself-organizedsystem dissipates energy.

People in systems and control

Main article:People in systems and control

Many active and historical figures made significant contribution to control theory including

• Pierre-Simon Laplaceinvented theZ-transformin his work onprobability theory,now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of theLaplace transformwhich is named after him.
• Irmgard Flugge-Lotzdeveloped the theory ofdiscontinuous automatic controland applied it toautomatic aircraft control systems.
• Alexander Lyapunovin the 1890s marks the beginning ofstability theory.
• Harold S. Blackinvented the concept ofnegative feedback amplifiersin 1927. He managed to develop stable negative feedback amplifiers in the 1930s.
• Harry Nyquistdeveloped theNyquist stability criterionfor feedback systems in the 1930s.
• Richard Bellmandevelopeddynamic programmingin the 1940s.
• Warren E. Dixon,control theorist and a professor
• Kyriakos G. Vamvoudakis,developed synchronous reinforcement learning algorithms to solve optimal control and game theoretic problems
• Andrey Kolmogorovco-developed theWiener–Kolmogorov filterin 1941.
• Norbert Wienerco-developed the Wiener–Kolmogorov filter and coined the termcyberneticsin the 1940s.
• John R. Ragazziniintroduceddigital controland the use ofZ-transformin control theory (invented by Laplace) in the 1950s.
• Lev Pontryaginintroduced themaximum principleand thebang-bang principle.
• Pierre-Louis Lionsdevelopedviscosity solutionsinto stochastic control andoptimal controlmethods.
• Rudolf E. Kálmánpioneered thestate-spaceapproach to systems and control. Introduced the notions ofcontrollabilityandobservability.Developed theKalman filterfor linear estimation.
• Ali H. Nayfehwho was one of the main contributors to nonlinear control theory and published many books on perturbation methods
• Jan C. WillemsIntroduced the concept of dissipativity, as a generalization ofLyapunov functionto input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called, led to the study of thelinear matrix inequality(LMI) in control theory. He pioneered the behavioral approach to mathematical systems theory.

See also

Examples of control systems

• Automation
• Deadbeat controller
• Distributed parameter systems
• Fractional-order control
• H-infinity loop-shaping
• Hierarchical control system
• Model predictive control
• Optimal control
• Process control
• Robust control
• Servomechanism
• State space (controls)
• Vector control

Topics in control theory

• Coefficient diagram method
• Control reconfiguration
• Feedback
• H infinity
• Hankel singular value
• Krener's theorem
• Lead-lag compensator
• Minor loop feedback
• Multi-loop feedback
• Positive systems
• Radial basis function
• Root locus
• Signal-flow graphs
• Stable polynomial
• State space representation
• Steady state
• Transient response
• Transient state
• Underactuation
• Youla–Kucera parametrization
• Markov chain approximation method

Other related topics

• Adaptive system
• Automation and remote control
• Bond graph
• Control engineering
• Control–feedback–abort loop
• Controller (control theory)
• Cybernetics
• Intelligent control
• Mathematical system theory
• Negative feedback amplifier
• Outline of management
• People in systems and control
• Perceptual control theory
• Systems theory

Further reading

• Levine, William S., ed. (1996).The Control Handbook.New York: CRC Press.ISBN978-0-8493-8570-4.
• Karl J. Åström; Richard M. Murray (2008).Feedback Systems: An Introduction for Scientists and Engineers(PDF).Princeton University Press.ISBN978-0-691-13576-2.
• Christopher Kilian (2005).Modern Control Technology.Thompson Delmar Learning.ISBN978-1-4018-5806-3.
• Vannevar Bush (1929).Operational Circuit Analysis.John Wiley and Sons, Inc.
• Robert F. Stengel (1994).Optimal Control and Estimation.Dover Publications.ISBN978-0-486-68200-6.
• Franklin; et al. (2002).Feedback Control of Dynamic Systems(4 ed.). New Jersey: Prentice Hall.ISBN978-0-13-032393-4.
• Joseph L. Hellerstein;Dawn M. Tilbury;Sujay Parekh (2004).Feedback Control of Computing Systems.John Wiley and Sons.ISBN978-0-471-26637-2.
• Diederich Hinrichsenand Anthony J. Pritchard (2005).Mathematical Systems Theory I – Modelling, State Space Analysis, Stability and Robustness.Springer.ISBN978-3-540-44125-0.
• Sontag, Eduardo(1998).Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition(PDF).Springer.ISBN978-0-387-98489-6.
• Goodwin, Graham (2001).Control System Design.Prentice Hall.ISBN978-0-13-958653-8.
• Christophe Basso (2012).Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide.Artech House.ISBN978-1608075577.
• Boris J. Lurie; Paul J. Enright (2019).Classical Feedback Control with Nonlinear Multi-loop Systems(3 ed.). CRC Press.ISBN978-1-1385-4114-6.

For Chemical Engineering

• Luyben, William (1989).Process Modeling, Simulation, and Control for Chemical Engineers.McGraw Hill.ISBN978-0-07-039159-8.

External links

• Control Tutorials for Matlab,a set of worked-through control examples solved by several different methods.
• Control Tuning and Best Practices
• Advanced control structures, free on-line simulators explaining the control theory