Control theory - Knowledge

353:. 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 a 809:

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.

892:. 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. 888:, 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 are 248: 5738: 2330: 439: 394: 577: 1984:(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 5750: 5136: 5148: 5774: 5762: 5160: 51: 1502:, 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. 1807:. 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. 908:(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 order 1931:

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

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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

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and 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

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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 requirement is important, as no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a

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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

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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,

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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

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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.

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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

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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). Continuous, reliable control of the airplane was necessary for flights

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representation, 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

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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

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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

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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

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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.

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has 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.

1102:; 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. 1895:

typically 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.,

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theories 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 like

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are 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.

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systems 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 and

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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 the

1094:; 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 402:

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.

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deals 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

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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:

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is "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."

523:, 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. 1946:

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.

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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

674:– 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 by 1805: 3145:

Here we use tools from control and network theories to offer a mechanistic explanation for how the brain moves between cognitive states drawn from the network organization of white matter microstructure

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control 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 the

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also 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 an

1963:'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. 1522:

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

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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

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signal, 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 are

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in 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 any

1415: 1243: 1576: 2018:, 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/or 678:. 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 include 420:

is 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."

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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 in

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and 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 include

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must 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

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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 using

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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.

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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 termed

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systems, time domain is widely used to analyze real-world nonlinear systems. Although these are more difficult to solve, modern computer simulation techniques such as

1641: 278:, 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, 1468:

that 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.

1444: 1272: 1085: 824:(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 an 5198: 3184:

N. A. Sinitsyn. S. Kundu, S. Backhaus (2013). "Safe Protocols for Generating Power Pulses with Heterogeneous Populations of Thermostatically Controlled Loads".

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present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.

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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

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in the frequency domain which is much simpler to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above.

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to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called, led to the study of the

1707:. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its 4266: 2642: 1464:

Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use

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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

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Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the

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to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as

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methods tend to be able to cope with small differences between the true system and the nominal model used for design. The early methods of

185:, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the 5100: 5706: 5191: 3509: 2106: 1719: 5778: 2876: 2523: 1981: 877: 821: 397:

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

2518: 2254: 201: 274:. A centrifugal governor was already used to regulate the velocity of windmills. Maxwell described and analyzed the phenomenon of 5242: 4929: 3839: 3186: 2831: 5815: 5656: 4163: 675: 2690: 5754: 5090: 804:

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain

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In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed

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Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by

1849:. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold. 5810: 5184: 5095: 4542: 4010: 3993: 3892: 3730: 2592: 2542: 2359: 2123: 1846: 643: 492:

and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.

410: 4133: 1959:, this can be obtained by directly placing the poles. Nonlinear control systems use specific theories (normally based on 5805: 3601: 605: 541:

uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact

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Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the

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Liu, Jie; Wilson Wang; Farid Golnaraghi; Eric Kubica (2010). "A novel fuzzy framework for nonlinear system control".

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be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have

1354: 1182: 3456: 2891:"Feedback and control systems" - JJ Di Steffano, AR Stubberud, IJ Williams. Schaums outline series, McGraw-Hill 1967 208:, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of 5252: 4944: 4796: 4143: 4026: 4016: 3882: 3791: 2577: 2493: 2423: 2072: 1926: 1533: 167: 35: 287: 5666: 5638: 5275: 5164: 4077: 3942: 3877: 3834: 2597: 2303:

who was one of the main contributors to nonlinear control theory and published many books on perturbation methods

508: 151: 39: 2143:, now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the 788:

are represented as functions of time. With this model, the system being analyzed is represented by one or more

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A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion: Particularly Steady Motion

2213:, developed synchronous reinforcement learning algorithms to solve optimal control and game theoretic problems 255: 2859:

Hallion, Richard P. (1980). Sicherman, Barbara; Green, Carol Hurd; Kantrov, Ilene; Walker, Harriette (eds.).

2210: 5800: 5596: 5586: 5556: 5490: 5225: 4178: 4108: 3907: 3887: 3594: 2428: 2369: 2003: 1581: 1005: 749: 601: 5766: 921: 5694: 5591: 5571: 5566: 5495: 5220: 5152: 4844: 4647: 4501: 4391: 4246: 4193: 4118: 4062: 3973: 3952: 3646: 2755: 2409: 2404: 2335: 1897: 1869: 1863: 901: 881: 828:, in which the control input is the input audio signal and the output is the sound waves from the speaker. 805: 737: 562: 519:

in the controller restores the actual speed to the desired speed in an optimum way, with minimal delay or

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Hurwitz, A. (1964). "On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts".

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Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the

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

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

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

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Melby, Paul; et., al. (2002). "Robustness of Adaptation in Controlled Self-Adjusting Chaotic Systems".

608:. A major subclass is systems which in addition have parameters which do not change with time, called 286:

analyzed system stability using differential equations in 1877, resulting in what is now known as the

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deals explicitly with uncertainty in its approach to controller design. Controllers designed using

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

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industrial applications, as it has been shown they can guarantee closed-loop stability. These are

1291: 1115: 764:. The advantage of this technique is that it results in a simplification of the mathematics; the 5742: 5661: 5601: 5533: 5523: 5462: 5437: 5313: 5270: 5265: 4989: 4924: 4904: 4894: 4854: 4839: 4814: 4772: 4767: 4621: 4611: 4606: 4421: 4396: 4336: 4261: 4188: 4138: 3932: 3745: 3704: 3221: 3195: 3100: 3069: 3009: 2981: 2732: 2503: 2306: 2264: 2164: 2140: 2092: 2036: 2025: 1960: 1941: 1892: 1658: 1099: 963: 938: 772: 659: 655: 647: 538: 520: 1672:

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

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

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simpler mathematical model is chosen in order to simplify calculations, otherwise, the true

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with the requisite corrective behavior is required. This controller monitors the controlled

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This article is about control theory in engineering. For control theory in linguistics, see

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the controller will adjust itself consequently in order to ensure the correct performance.

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Mathematical Systems Theory I – Modelling, State Space Analysis, Stability and Robustness

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Every control system must guarantee first the stability of the closed-loop behavior. For

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Sometimes, mechanical methods are used to improve the stability of systems. For example,

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

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Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition

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representing the system's input, output and feedback are represented as functions of

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The process of determining the equations that govern the model's dynamics is called

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

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Mathematically, this means that for a causal linear system to be stable all of the

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Advanced control structures, free on-line simulators explaining the control theory

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

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Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide

834:(MIMO) – These are found in more complicated systems. For example, modern large 309:

developed the theory of discontinuous automatic control systems, and applied the

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in 1927. He managed to develop stable negative feedback amplifiers in the 1930s.

2136: 2096: 2080: 2015: 1507: 1173: 1032: 1028: 993:(ISS), which combines Lyapunov stability and a notion similar to BIBO stability. 761: 679: 627: 544: 480:. Its name comes from the information path in the system: process inputs (e.g., 326: 217: 209: 3573:, a set of worked-through control examples solved by several different methods. 3499: 2995: 1908:, trajectory linearization control normally take advantage of results based on 1446:

and is not BIBO stable since the pole has a modulus strictly greater than one.

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A notable application of dynamic control was in the area of crewed flight. The

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for aircraft. Other areas of application for discontinuous controls included

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was found. You can help implement the merge by following the instructions at

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is a fixed value strictly greater than zero, instead of simply asking that

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When the appropriate conditions above are satisfied a system is said to be

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For MIMO systems, pole placement can be performed mathematically using a

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disturbance rejection (such as hills in the cruise control example above)

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are well known among the people who have shaped modern control theory.

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may be defined as attempts to interfere in the processes by which the

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have mirrors composed of many separate segments each controlled by an

50: 4566: 3678: 2832:"Optimum and Quasi-Optimum Control of Third and Fourth-Order Systems" 2752:"Control Theory: History, Mathematical Achievements and Perspectives" 504: 489: 3089:"Controllability of structural brain networks (Article Number 8414)" 3065: 2863:

Notable American Women: The Modern Period: A Biographical Dictionary

1832:

a control technique by including these qualities in its properties.

850:. The shape of the entire mirror is constantly adjusted by a MIMO 177:

Extensive use is usually made of a diagrammatic style known as the

4527: 4051: 3200: 3105: 2986: 2050:

in which a set of devices and governing software is arranged in a

859:

due to turbulence in the atmosphere. Complicated systems such as

575: 392: 246: 3586: 2662:(1922). "Directional stability of automatically steered bodies". 1278:). This system is BIBO (asymptotically) stable since the pole is 748:

are converted from time functions to functions of frequency by a

3547:

Process Modeling, Simulation, and Control for Chemical Engineers

2309:

Introduced the concept of dissipativity, as a generalization of

1800:{\displaystyle m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)} 527:

do not make use of feedback, and run only in pre-arranged ways.

409:

The definition of a closed loop control system according to the

5180: 4531: 4066: 3590: 3320:

Feedback Systems: An Introduction for Scientists and Engineers

2691:"Katalog der Deutschen Nationalbibliothek (Authority control)" 2229:

co-developed the Wiener–Kolmogorov filter and coined the term

2039:

in the 1950s, and have found particular success in that field.

1680:

A control system must always have some robustness property. A

642:. These lead to a description of the system using terms like 591:

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

566:

and serves to further improve reference tracking performance.

44: 305:, control theory was becoming an important area of research. 3521:

Classical Feedback Control with Nonlinear Multi-loop Systems

3576: 2285:

approach to systems and control. Introduced the notions of

1689:

can be so complicated that a complete model is impossible.

3581: 2754:. Boletin de la Sociedad Espanola de Matematica Aplicada. 2061:, then that hierarchical control system is also a form of 867:

are simulated by a computer as large MIMO control systems.

1498:. Observability instead is related to the possibility of 1063:

axis is the real axis and the discrete Z-transform is in

618:

mathematical techniques of great generality, such as the

600:– This applies to systems made of devices which obey the 2817:

Selected Papers on Mathematical Trends in Control Theory

614:(LTI) systems. These systems are amenable to powerful 2830:

Flugge-Lotz, Irmgard; Titus, Harold A. (October 1962).

1873:

be included and incorporated in pole placement design.

2939:"Feedback for physicists: A tutorial essay on control" 719:

Analysis techniques - frequency domain and time domain

34:. For control theory in psychology and sociology, see 2247:

in control theory (invented by Laplace) in the 1950s.

1722: 1611: 1584: 1536: 1426: 1357: 1294: 1254: 1185: 1118: 1073: 1049: 876:

The scope of classical control theory is limited to

792:. Since frequency domain techniques are limited to 569:

A common closed-loop controller architecture is the

468:. A closed-loop controller uses feedback to control 378:

Fundamentally, there are two types of control loop:

5680: 5637: 5547: 5509: 5476: 5428: 5400: 5347: 5294: 5251: 5109: 5083: 4795: 4671: 4635: 4565: 4227: 4101: 4002: 3961: 3870: 3822: 3713: 3624: 2057:. When the links in the tree are implemented by a 2860: 2664:Journal of the American Society of Naval Engineers 1799: 1635: 1597: 1570: 1438: 1409: 1337: 1266: 1237: 1161: 1079: 1055: 70:, consensus to merge this with content from 3275:. Vol. 211, no. 3. pp. 186–200. 2970:"Thermodynamics of feedback controlled systems" 1410:{\displaystyle \ X(z)={\frac {1}{1-1.5z^{-1}}}} 1238:{\displaystyle \ X(z)={\frac {1}{1-0.5z^{-1}}}} 710:by approximating them by a linear system using 694:. Nonlinear systems are often analyzed using 5192: 4543: 4078: 3602: 1571:{\displaystyle Re<-{\overline {\lambda }}} 556:improved rectification of random fluctuations 8: 4093:Subfields of and cyberneticians involved in 1936:Deterministic and stochastic systems control 900:Modern control theory is carried out in the 365:Open-loop and closed-loop (feedback) control 150:(PV), and compares it with the reference or 135:; often with the aim to achieve a degree of 2095:or a combination of these methods, such as 550:reduced sensitivity to parameter variations 5199: 5185: 5177: 4677: 4550: 4536: 4528: 4085: 4071: 4063: 3609: 3595: 3587: 3317:Karl J. Åström; Richard M. Murray (2008). 2926:. Clinton, MA US: The Colonial Press, Inc. 2908:. Clinton, MA US: The Colonial Press, Inc. 2717:Proceedings of the Royal Society of London 2071:uses various AI computing approaches like 976:bounded-input bounded-output (BIBO) stable 770:that represent the system are replaced by 196:. Control theory was further advanced by 3392:(4 ed.). New Jersey: Prentice Hall. 3199: 3130: 3104: 2985: 2759: 2022:in the presence of small modeling errors. 1777: 1776: 1771: 1727: 1726: 1721: 1610: 1585: 1583: 1558: 1535: 1425: 1395: 1376: 1356: 1317: 1293: 1253: 1223: 1204: 1184: 1141: 1117: 1072: 1048: 3519:Boris J. Lurie; Paul J. Enright (2019). 3027:. Schaum's outline series. McGraw Hill. 2968:Cao, F. J.; Feito, M. (April 10, 2009). 1024:is used to obtain the transfer function. 437: 372:Control loop § Open-loop and closed-loop 2648:from the original on December 19, 2008. 2615: 1932:channels and coordinate their actions. 553:improved reference tracking performance 2715:Maxwell, J.C. (1868). "On Governors". 1598:{\displaystyle {\overline {\lambda }}} 781:Time-domain state space representation 3413:Feedback Control of Computing Systems 1285:However, if the impulse response was 744:. The input signal and the system's 27:Branch of engineering and mathematics 7: 5761: 5159: 2937:Bechhoefer, John (August 31, 2005). 2839:Stanford University Technical Report 2167:in the 1890s marks the beginning of 962:with no input can be described with 813:System interfacing - SISO & MIMO 714:, and linear techniques can be used. 5773: 5101:Systems theory in political science 3390:Feedback Control of Dynamic Systems 1967:List of the main control techniques 1676:Model identification and robustness 884:, in the complex-s domain with the 587:Linear and nonlinear control theory 298:lasting longer than a few seconds. 3281:10.1038/scientificamerican0964-186 2783:Routh, E.J.; Fuller, A.T. (1975). 2676:10.1111/j.1559-3584.1922.tb04958.x 2207:, control theorist and a professor 2191:for feedback systems in the 1930s. 2159:automatic aircraft control systems 2107:Self-organized criticality control 2006:developed by Duncan McFarlane and 1772: 906:multiple-input and multiple-output 511:(SP). An everyday example is the 315:automatic flight control equipment 243:Control engineering § History 216:. Although a major application of 25: 3577:Control Tuning and Best Practices 3433:and Anthony J. Pritchard (2005). 2524:Markov chain approximation method 1982:linear-quadratic-Gaussian control 1472:Controllability and observability 800:have made their analysis routine. 784:– In this type the values of the 732:– In this type the values of the 537:guaranteed performance even with 224:, which deals with the design of 5772: 5760: 5749: 5748: 5736: 5158: 5146: 5135: 5134: 3298:Levine, William S., ed. (1996). 3187:Energy Conversion and Management 3025:State space & linear systems 2635:Proceedings of the Royal Society 2328: 676:nonlinear differential equations 430:This section is an excerpt from 370:This section is an excerpt from 131:and ensuring a level of control 49: 5657:Computational complexity theory 4164:Cybernetics in the Soviet Union 2924:The Origins of Feedback Control 2906:The Origins of Feedback Control 2750:Fernandez-Cara, E.; Zuazua, E. 2155:discontinuous automatic control 1105:If a system in question has an 5091:Systems theory in anthropology 3388:Franklin; et al. (2002). 3371:Optimal Control and Estimation 3326:. Princeton University Press. 3218:10.1016/j.enconman.2012.11.021 1794: 1788: 1765: 1759: 1744: 1738: 1624: 1618: 1549: 1543: 1370: 1364: 1332: 1326: 1307: 1301: 1198: 1192: 1156: 1150: 1131: 1125: 1020:for continuous time, when the 878:single-input and single-output 832:Multiple-input multiple-output 1: 5096:Systems theory in archaeology 4011:Automation and Remote Control 3994:Programmable logic controller 3893:Distributed parameter systems 3731:Closed-loop transfer function 3159:Fluctuation and Noise Letters 3054:American Mathematical Monthly 2593:People in systems and control 2543:Automation and remote control 2360:Distributed parameter systems 2124:People in systems and control 2118:People in systems and control 1921:Decentralized systems control 1887:Processes in industries like 1526:or aircraft cruise control). 1016:in the open left half of the 698:on computers, for example by 606:linear differential equations 411:British Standards Institution 266:, conducted by the physicist 166:. Control theory is used in 3571:Control Tutorials for Matlab 3362:Operational Circuit Analysis 3345:. Thompson Delmar Learning. 3087:Gu Shi; et al. (2015). 2519:Youla–Kucera parametrization 2271:into stochastic control and 2179:negative feedback amplifiers 1590: 1563: 1338:{\displaystyle \ x=1.5^{n}u} 1162:{\displaystyle \ x=0.5^{n}u} 1031:for discrete time, when the 872:Classical SISO system design 450:or feedback controller is a 4052:Supervisory control (SCADA) 3364:. John Wiley and Sons, Inc. 3341:Christopher Kilian (2005). 2841:(134): 8–12. Archived from 2583:Negative feedback amplifier 2563:Controller (control theory) 2558:Control–feedback–abort loop 2375:Hierarchical control system 2343:Examples of control systems 2189:Nyquist stability criterion 2044:hierarchical control system 640:Nyquist stability criterion 547:processes can be stabilized 222:control systems engineering 5832: 5707:Films about mathematicians 4144:Computational neuroscience 4027:Industrial control systems 4017:Distributed control system 3883:Coefficient diagram method 3792:State space representation 3369:Robert F. Stengel (1994). 3269:(1964). "Control Theory". 3060:(9): 705–719 and 812–828. 2996:10.1103/PhysRevE.79.041118 2578:Mathematical system theory 2494:State space representation 2424:Coefficient diagram method 2121: 2073:artificial neural networks 1939: 1927:Distributed control system 1924: 1880: 1870:state space representation 1861: 1696: 1475: 1172:then the Z-transform (see 822:Single-input single-output 688:Lyapunov stability theorem 429: 369: 240: 168:control system engineering 36:control theory (sociology) 29: 5730: 5276:Philosophy of mathematics 5216: 5130: 4728: 4680: 4592:Coupled human–environment 3943:Perceptual control theory 3878:Artificial neural network 3835:Digital signal processing 3523:(3 ed.). CRC Press. 3498:Christophe Basso (2012). 3343:Modern Control Technology 3253:10.1016/j.fss.2010.04.009 3171:10.1142/S0219477502000919 3048:Terrell, William (1999). 2955:10.1103/RevModPhys.77.783 2943:Reviews of Modern Physics 2598:Perceptual control theory 2147:which is named after him. 2113:system dissipates energy. 2099:algorithms, to control a 1877:Nonlinear systems control 896:Modern MIMO system design 525:Open-loop control systems 256:Boulton & Watt engine 59:This is currently being 40:Perceptual control theory 5712:Recreational mathematics 4317:Charles Geoffrey Vickers 4204:Second-order cybernetics 3898:Fractional-order control 3668:Model predictive control 3545:Luyben, William (1989). 3539:For Chemical Engineering 3479:Goodwin, Graham (2001). 3023:Donald M Wiberg (1971). 2417:Topics in control theory 2380:Model predictive control 2365:Fractional-order control 2315:linear matrix inequality 2221:Wiener–Kolmogorov filter 2177:invented the concept of 2153:developed the theory of 2089:evolutionary computation 2063:networked control system 1978:Model Predictive Control 1843:model predictive control 1348:then the Z-transform is 991:input-to-state stability 978:if its output will stay 945:Topics in control theory 702:their operation using a 671:Nonlinear control theory 425:Classical control theory 72:Classical control theory 5597:Mathematical statistics 5587:Mathematical psychology 5557:Engineering mathematics 5491:Algebraic number theory 4925:Charles A. S. Hall 4179:Engineering cybernetics 4109:Artificial intelligence 3908:H-infinity loop-shaping 3888:Control reconfiguration 3415:. John Wiley and Sons. 3411:; Sujay Parekh (2004). 3407:Joseph L. Hellerstein; 3302:. New York: CRC Press. 2787:. Taylor & Francis. 2429:Control reconfiguration 2370:H-infinity loop-shaping 2211:Kyriakos G. Vamvoudakis 2004:H-infinity loop-shaping 1951:Main control strategies 1636:{\displaystyle Re<0} 1087:axis is the real axis. 602:superposition principle 466:non-feedback controller 418:Feedback Control System 189:describing the system. 5816:Management cybernetics 5743:Mathematics portal 5592:Mathematical sociology 5572:Mathematical economics 5567:Mathematical chemistry 5496:Analytic number theory 5377:Differential equations 4845:Ludwig von Bertalanffy 4502:Walter Bradford Cannon 4392:Ludwig von Bertalanffy 4247:Alfred Radcliffe-Brown 4194:Management cybernetics 4119:Biomedical cybernetics 4114:Biological cybernetics 3974:Closed-loop controller 3647:Energy-shaping control 3373:. Dover Publications. 3360:Vannevar Bush (1929). 3241:Fuzzy Sets and Systems 2729:10.1098/rspl.1867.0055 2405:State space (controls) 2336:Systems science portal 2297:for linear estimation. 1898:feedback linearization 1864:State space (controls) 1858:Linear systems control 1853:System classifications 1801: 1637: 1599: 1572: 1440: 1411: 1339: 1268: 1239: 1163: 1081: 1057: 989:that take an input is 982:for any bounded input. 910:differential equations 882:differential equations 790:differential equations 767:differential equations 581: 495:In the case of linear 448:closed-loop controller 443: 432:Closed-loop controller 398: 351:good regulator theorem 313:to the development of 259: 187:differential equations 78:and the resolution on 5722:Mathematics education 5652:Theory of computation 5372:Hypercomplex analysis 5122:Principia Cybernetica 4835:Anthony Stafford Beer 4705:Sociotechnical system 4462:Anthony Stafford Beer 4297:Ernst von Glasersfeld 3913:Hankel singular value 3857:System identification 3805:analysis & design 3481:Control System Design 3093:Nature Communications 2588:Outline of management 2474:Radial basis function 2444:Hankel singular value 1914:Differential geometry 1829:gain and phase margin 1802: 1705:system identification 1699:System identification 1697:Further information: 1693:System identification 1638: 1600: 1573: 1518:Control specification 1441: 1439:{\displaystyle z=1.5} 1412: 1340: 1269: 1267:{\displaystyle z=0.5} 1240: 1164: 1092:asymptotically stable 1082: 1080:{\displaystyle \rho } 1058: 1041:Cartesian coordinates 611:linear time invariant 597:Linear control theory 580:A basic feedback loop 579: 441: 396: 288:Routh–Hurwitz theorem 250: 220:control theory is in 32:control (linguistics) 5811:Computer engineering 5702:Informal mathematics 5582:Mathematical physics 5577:Mathematical finance 5562:Mathematical biology 5501:Diophantine geometry 4985:Mihajlo D. Mesarovic 4960:Edward Norton Lorenz 4915:Jay Wright Forrester 4720:World-systems theory 4695:Earth system science 4492:Valentin Braitenberg 4372:Jay Wright Forrester 4003:Control applications 3979:Lead-lag compensator 3830:Discrete-time signal 3431:Diederich Hinrichsen 3300:The Control Handbook 2798:Routh, E.J. (1877). 2531:Other related topics 2454:Lead-lag compensator 2133:Pierre-Simon Laplace 2077:Bayesian probability 2012:Sliding mode control 1906:sliding mode control 1847:anti-wind up systems 1720: 1716:system we know that 1609: 1582: 1534: 1424: 1420:which has a pole at 1355: 1292: 1252: 1248:which has a pole in 1183: 1116: 1071: 1065:circular coordinates 1047: 904:, and can deal with 798:simulation languages 692:describing functions 660:resonant frequencies 461:open-loop controller 458:, in contrast to an 319:fire-control systems 264:centrifugal governor 252:Centrifugal governor 111:that deals with the 5806:Control engineering 5717:Mathematics and art 5627:Operations research 5382:Functional analysis 4860:Kenneth E. Boulding 4517:William Grey Walter 4457:Sergei P. Kurdyumov 4417:N. Katherine Hayles 4199:Medical cybernetics 4159:Conversation theory 3969:Embedded controller 3938:Minor loop feedback 3871:Advanced techniques 3663:Intelligent control 3272:Scientific American 3210:2013ECM....67..297S 3115:2015NatCo...6.8414G 2804:. Macmillan and co. 2785:Stability of motion 2573:Intelligent control 2553:Control engineering 2464:Multi-loop feedback 2459:Minor loop feedback 2355:Deadbeat controller 2269:viscosity solutions 2259:bang-bang principle 2199:dynamic programming 2151:Irmgard Flugge-Lotz 2069:Intelligent control 2014:(SMC) developed by 773:algebraic equations 736:, the mathematical 712:perturbation theory 704:simulation language 454:which incorporates 387:closed-loop control 384:(feedforward), and 357:interacting with a 311:bang-bang principle 307:Irmgard Flügge-Lotz 268:James Clerk Maxwell 231:operations research 194:James Clerk Maxwell 109:applied mathematics 105:control engineering 85:Process started in 5662:Numerical analysis 5271:Mathematical logic 5266:Information theory 4990:James Grier Miller 4945:Faina M. Kirillova 4905:Heinz von Foerster 4895:Edsger W. Dijkstra 4855:Alexander Bogdanov 4840:Richard E. Bellman 4815:William Ross Ashby 4497:William Ross Ashby 4422:Natalia Bekhtereva 4397:Maleyka Abbaszadeh 4337:Heinz von Foerster 4262:Buckminster Fuller 4189:Information theory 4139:Catastrophe theory 3933:Lyapunov stability 3746:Frequency response 3705:Stochastic control 3123:10.1038/ncomms9414 2848:on April 27, 2019. 2504:Transient response 2484:Signal-flow graphs 2265:Pierre-Louis Lions 2165:Alexander Lyapunov 2157:and applied it to 2141:probability theory 2093:genetic algorithms 2037:aerospace industry 2026:Stochastic control 1961:Aleksandr Lyapunov 1942:Stochastic control 1893:aerospace industry 1797: 1714:mass-spring-damper 1633: 1595: 1568: 1436: 1407: 1335: 1264: 1235: 1159: 1077: 1053: 964:Lyapunov stability 939:Aleksandr Lyapunov 648:frequency response 582: 444: 399: 270:in 1868, entitled 260: 129:steady-state error 5788: 5787: 5387:Harmonic analysis 5174: 5173: 5055:Manuela M. Veloso 4970:Humberto Maturana 4910:Stephanie Forrest 4880:C. West Churchman 4805:Russell L. Ackoff 4791: 4790: 4663:Positive feedback 4658:Negative feedback 4525: 4524: 4447:Ranulph Glanville 4362:Jakob von Uexküll 4342:Humberto Maturana 4302:Francis Heylighen 4060: 4059: 3984:Numerical control 3814:Transfer function 3787:Signal-flow graph 3771:Positive feedback 3756:Negative feedback 3751:Laplace transform 3741:Fourier transform 3714:System properties 3695:Real-time control 3685:Nonlinear control 3556:978-0-07-039159-8 3530:978-1-1385-4114-6 3490:978-0-13-958653-8 3483:. Prentice Hall. 3471:978-0-387-98489-6 3444:978-3-540-44125-0 3422:978-0-471-26637-2 3399:978-0-13-032393-4 3380:978-0-486-68200-6 3352:978-1-4018-5806-3 3333:978-0-691-13576-2 3309:978-0-8493-8570-4 3247:(21): 2746–2759. 3034:978-0-07-070096-3 2974:Physical Review E 2660:Minorsky, Nicolas 2489:Stable polynomial 2311:Lyapunov function 2255:maximum principle 2237:John R. Ragazzini 2219:co-developed the 2217:Andrey Kolmogorov 2145:Laplace transform 1910:Lyapunov's theory 1883:Nonlinear control 1845:(see later), and 1827:. Topics include 1785: 1735: 1709:transfer function 1682:robust controller 1593: 1566: 1405: 1360: 1297: 1282:the unit circle. 1233: 1188: 1121: 1100:marginally stable 1056:{\displaystyle x} 1022:Laplace transform 1010:transfer function 987:nonlinear systems 886:Laplace transform 758:Laplace transform 754:Fourier transform 746:transfer function 696:numerical methods 624:Fourier transform 620:Laplace transform 381:open-loop control 280:Edward John Routh 183:transfer function 117:dynamical systems 96: 95: 91: 16:(Redirected from 5823: 5776: 5775: 5764: 5763: 5752: 5751: 5741: 5740: 5672:Computer algebra 5647:Computer science 5367:Complex analysis 5201: 5194: 5187: 5178: 5162: 5161: 5150: 5138: 5137: 5050:Francisco Varela 4782:Systems thinking 4715:Urban metabolism 4678: 4552: 4545: 4538: 4529: 4512:Warren McCulloch 4487:Valentin Turchin 4437:Pyotr Grigorenko 4382:John N. Warfield 4307:Francisco Varela 4267:Charles François 4237:Alexander Lerner 4214:Sociocybernetics 4134:Neurocybernetics 4087: 4080: 4073: 4064: 3923:Krener's theorem 3632:Adaptive control 3611: 3604: 3597: 3588: 3560: 3534: 3515: 3504:. Artech House. 3494: 3475: 3463: 3448: 3426: 3403: 3384: 3365: 3356: 3337: 3325: 3313: 3285: 3284: 3263: 3257: 3256: 3236: 3230: 3229: 3203: 3181: 3175: 3174: 3165:(4): L285–L292. 3154: 3148: 3147: 3134: 3108: 3084: 3078: 3077: 3045: 3039: 3038: 3020: 3014: 3013: 3006: 3000: 2999: 2989: 2965: 2959: 2958: 2934: 2928: 2927: 2916: 2910: 2909: 2898: 2892: 2889: 2883: 2882: 2866: 2856: 2850: 2849: 2847: 2836: 2827: 2821: 2820: 2812: 2806: 2805: 2795: 2789: 2788: 2780: 2774: 2773: 2763: 2747: 2741: 2740: 2712: 2706: 2705: 2703: 2701: 2686: 2680: 2679: 2656: 2650: 2649: 2647: 2632: 2620: 2469:Positive systems 2449:Krener's theorem 2338: 2333: 2332: 2331: 2293:. Developed the 2279:Rudolf E. Kálmán 2169:stability theory 2085:machine learning 2059:computer network 2033:Adaptive control 1806: 1804: 1803: 1798: 1787: 1786: 1778: 1775: 1737: 1736: 1728: 1642: 1640: 1639: 1634: 1604: 1602: 1601: 1596: 1594: 1586: 1577: 1575: 1574: 1569: 1567: 1559: 1445: 1443: 1442: 1437: 1416: 1414: 1413: 1408: 1406: 1404: 1403: 1402: 1377: 1358: 1344: 1342: 1341: 1336: 1322: 1321: 1295: 1273: 1271: 1270: 1265: 1244: 1242: 1241: 1236: 1234: 1232: 1231: 1230: 1205: 1186: 1168: 1166: 1165: 1160: 1146: 1145: 1119: 1107:impulse response 1086: 1084: 1083: 1078: 1062: 1060: 1059: 1054: 960:dynamical system 935:Rudolf E. Kálmán 861:nuclear reactors 729:Frequency domain 616:frequency domain 478:dynamical system 334:ship stabilizers 323:guidance systems 276:self-oscillation 214:Nicolas Minorsky 148:process variable 88: 83: 53: 45: 21: 5831: 5830: 5826: 5825: 5824: 5822: 5821: 5820: 5791: 5790: 5789: 5784: 5735: 5726: 5676: 5633: 5612:Systems science 5543: 5539:Homotopy theory 5505: 5472: 5424: 5396: 5343: 5290: 5261:Category theory 5247: 5212: 5205: 5175: 5170: 5126: 5105: 5079: 5020:Anatol Rapoport 5005:Talcott Parsons 4980:Donella Meadows 4955:Allenna Leonard 4875:Mary Cartwright 4870:Kathleen Carley 4830:Gregory Bateson 4825:Béla H. Bánáthy 4787: 4724: 4673: 4667: 4653:Limiting factor 4648:Leverage points 4631: 4569: 4561: 4559:Systems science 4556: 4526: 4521: 4477:Talcott Parsons 4467:Stuart Kauffman 4367:Jason Jixuan Hu 4352:Igor Aleksander 4332:Gregory Bateson 4327:Gordon S. Brown 4312:Frederic Vester 4292:Erich von Holst 4252:Allenna Leonard 4242:Alexey Lyapunov 4223: 4169:Decision theory 4097: 4091: 4061: 4056: 4042:Process control 4022:Electric motors 3998: 3957: 3866: 3823:Digital control 3818: 3809:System dynamics 3736:Controllability 3709: 3690:Optimal control 3642:Digital control 3620: 3615: 3567: 3557: 3549:. McGraw Hill. 3544: 3531: 3518: 3512: 3497: 3491: 3478: 3472: 3461: 3453:Sontag, Eduardo 3451: 3445: 3429: 3423: 3409:Dawn M. Tilbury 3406: 3400: 3387: 3381: 3368: 3359: 3353: 3340: 3334: 3323: 3316: 3310: 3297: 3294: 3292:Further reading 3289: 3288: 3267:Richard Bellman 3265: 3264: 3260: 3238: 3237: 3233: 3183: 3182: 3178: 3156: 3155: 3151: 3086: 3085: 3081: 3066:10.2307/2589614 3047: 3046: 3042: 3035: 3022: 3021: 3017: 3008: 3007: 3003: 2967: 2966: 2962: 2936: 2935: 2931: 2918: 2917: 2913: 2900: 2899: 2895: 2890: 2886: 2879: 2858: 2857: 2853: 2845: 2834: 2829: 2828: 2824: 2814: 2813: 2809: 2797: 2796: 2792: 2782: 2781: 2777: 2761:10.1.1.302.5633 2749: 2748: 2744: 2714: 2713: 2709: 2699: 2697: 2688: 2687: 2683: 2658: 2657: 2653: 2645: 2630: 2622: 2621: 2617: 2612: 2607: 2538:Adaptive system 2528: 2509:Transient state 2414: 2390:Process control 2385:Optimal control 2334: 2329: 2327: 2324: 2287:controllability 2273:optimal control 2253:introduced the 2243:and the use of 2241:digital control 2205:Warren E. Dixon 2195:Richard Bellman 2175:Harold S. Black 2139:in his work on 2126: 2120: 1986:process control 1973:Optimal control 1953: 1944: 1938: 1929: 1923: 1885: 1879: 1866: 1860: 1855: 1718: 1717: 1701: 1687:system dynamics 1678: 1607: 1606: 1580: 1579: 1532: 1531: 1520: 1487:Controllability 1484: 1478:Controllability 1476:Main articles: 1474: 1422: 1421: 1391: 1381: 1353: 1352: 1313: 1290: 1289: 1250: 1249: 1219: 1209: 1181: 1180: 1176:), is given by 1137: 1114: 1113: 1069: 1068: 1045: 1044: 952: 947: 914:state variables 898: 890:PID controllers 874: 815: 786:state variables 734:state variables 721: 664:zeros and poles 589: 584: 583: 435: 427: 422: 421: 375: 367: 349:that obeys the 295:Wright brothers 245: 239: 226:process control 160:controllability 92: 86: 65: 54: 43: 28: 23: 22: 15: 12: 11: 5: 5829: 5827: 5819: 5818: 5813: 5808: 5803: 5801:Control theory 5793: 5792: 5786: 5785: 5783: 5782: 5770: 5758: 5746: 5731: 5728: 5727: 5725: 5724: 5719: 5714: 5709: 5704: 5699: 5698: 5697: 5690:Mathematicians 5686: 5684: 5682:Related topics 5678: 5677: 5675: 5674: 5669: 5664: 5659: 5654: 5649: 5643: 5641: 5635: 5634: 5632: 5631: 5630: 5629: 5624: 5619: 5617:Control theory 5609: 5604: 5599: 5594: 5589: 5584: 5579: 5574: 5569: 5564: 5559: 5553: 5551: 5545: 5544: 5542: 5541: 5536: 5531: 5526: 5521: 5515: 5513: 5507: 5506: 5504: 5503: 5498: 5493: 5488: 5482: 5480: 5474: 5473: 5471: 5470: 5465: 5460: 5455: 5450: 5445: 5440: 5434: 5432: 5426: 5425: 5423: 5422: 5417: 5412: 5406: 5404: 5398: 5397: 5395: 5394: 5392:Measure theory 5389: 5384: 5379: 5374: 5369: 5364: 5359: 5353: 5351: 5345: 5344: 5342: 5341: 5336: 5331: 5326: 5321: 5316: 5311: 5306: 5300: 5298: 5292: 5291: 5289: 5288: 5283: 5278: 5273: 5268: 5263: 5257: 5255: 5249: 5248: 5246: 5245: 5240: 5235: 5234: 5233: 5228: 5217: 5214: 5213: 5206: 5204: 5203: 5196: 5189: 5181: 5172: 5171: 5169: 5168: 5156: 5144: 5131: 5128: 5127: 5125: 5124: 5119: 5113: 5111: 5107: 5106: 5104: 5103: 5098: 5093: 5087: 5085: 5081: 5080: 5078: 5077: 5075:Anthony Wilden 5072: 5070:Jennifer Wilby 5067: 5065:Norbert Wiener 5062: 5057: 5052: 5047: 5042: 5037: 5035:Claude Shannon 5032: 5027: 5022: 5017: 5012: 5010:Ilya Prigogine 5007: 5002: 5000:Howard T. Odum 4997: 4995:Radhika Nagpal 4992: 4987: 4982: 4977: 4972: 4967: 4965:Niklas Luhmann 4962: 4957: 4952: 4947: 4942: 4937: 4932: 4927: 4922: 4917: 4912: 4907: 4902: 4897: 4892: 4890:George Dantzig 4887: 4885:Manfred Clynes 4882: 4877: 4872: 4867: 4862: 4857: 4852: 4850:Margaret Boden 4847: 4842: 4837: 4832: 4827: 4822: 4817: 4812: 4810:Victor Aladjev 4807: 4801: 4799: 4793: 4792: 4789: 4788: 4786: 4785: 4775: 4770: 4765: 4760: 4755: 4750: 4745: 4740: 4735: 4729: 4726: 4725: 4723: 4722: 4717: 4712: 4707: 4702: 4700:Living systems 4697: 4692: 4687: 4685:Control theory 4681: 4675: 4669: 4668: 4666: 4665: 4660: 4655: 4650: 4645: 4639: 4637: 4633: 4632: 4630: 4629: 4624: 4619: 4614: 4609: 4604: 4599: 4594: 4589: 4584: 4579: 4573: 4571: 4563: 4562: 4557: 4555: 4554: 4547: 4540: 4532: 4523: 4522: 4520: 4519: 4514: 4509: 4504: 4499: 4494: 4489: 4484: 4479: 4474: 4472:Stuart Umpleby 4469: 4464: 4459: 4454: 4449: 4444: 4439: 4434: 4432:Norbert Wiener 4429: 4427:Niklas Luhmann 4424: 4419: 4414: 4409: 4404: 4402:Manfred Clynes 4399: 4394: 4389: 4384: 4379: 4377:Jennifer Wilby 4374: 4369: 4364: 4359: 4354: 4349: 4347:I. A. Richards 4344: 4339: 4334: 4329: 4324: 4319: 4314: 4309: 4304: 4299: 4294: 4289: 4284: 4282:Claude Bernard 4279: 4277:Margaret Boden 4274: 4272:Genevieve Bell 4269: 4264: 4259: 4257:Anthony Wilden 4254: 4249: 4244: 4239: 4233: 4231: 4229:Cyberneticians 4225: 4224: 4222: 4221: 4216: 4211: 4209:Cybersemiotics 4206: 4201: 4196: 4191: 4186: 4181: 4176: 4171: 4166: 4161: 4156: 4154:Control theory 4151: 4146: 4141: 4136: 4131: 4126: 4121: 4116: 4111: 4105: 4103: 4099: 4098: 4092: 4090: 4089: 4082: 4075: 4067: 4058: 4057: 4055: 4054: 4049: 4044: 4039: 4037:Motion control 4034: 4029: 4024: 4019: 4014: 4006: 4004: 4000: 3999: 3997: 3996: 3991: 3989:PID controller 3986: 3981: 3976: 3971: 3965: 3963: 3959: 3958: 3956: 3955: 3953:Vector control 3950: 3948:State observer 3945: 3940: 3935: 3930: 3925: 3920: 3915: 3910: 3905: 3900: 3895: 3890: 3885: 3880: 3874: 3872: 3868: 3867: 3865: 3864: 3859: 3854: 3848: 3842: 3837: 3832: 3826: 3824: 3820: 3819: 3817: 3816: 3811: 3806: 3800: 3794: 3789: 3784: 3782:Servomechanism 3779: 3773: 3768: 3763: 3758: 3753: 3748: 3743: 3738: 3733: 3728: 3723: 3717: 3715: 3711: 3710: 3708: 3707: 3702: 3700:Robust control 3697: 3692: 3687: 3682: 3676: 3670: 3665: 3660: 3654: 3649: 3644: 3639: 3637:Control theory 3634: 3628: 3626: 3622: 3621: 3618:Control theory 3616: 3614: 3613: 3606: 3599: 3591: 3585: 3584: 3579: 3574: 3566: 3565:External links 3563: 3562: 3561: 3555: 3541: 3540: 3536: 3535: 3529: 3516: 3511:978-1608075577 3510: 3495: 3489: 3476: 3470: 3449: 3443: 3427: 3421: 3404: 3398: 3385: 3379: 3366: 3357: 3351: 3338: 3332: 3314: 3308: 3293: 3290: 3287: 3286: 3258: 3231: 3176: 3149: 3079: 3040: 3033: 3015: 3001: 2960: 2949:(3): 783–836. 2929: 2911: 2893: 2884: 2877: 2851: 2822: 2807: 2790: 2775: 2742: 2707: 2681: 2670:(2): 280–309. 2651: 2628:"On Governors" 2624:Maxwell, J. C. 2614: 2613: 2611: 2608: 2606: 2605: 2603:Systems theory 2600: 2595: 2590: 2585: 2580: 2575: 2570: 2565: 2560: 2555: 2550: 2545: 2540: 2534: 2533: 2532: 2527: 2526: 2521: 2516: 2514:Underactuation 2511: 2506: 2501: 2496: 2491: 2486: 2481: 2476: 2471: 2466: 2461: 2456: 2451: 2446: 2441: 2436: 2431: 2426: 2420: 2419: 2418: 2413: 2412: 2410:Vector control 2407: 2402: 2400:Servomechanism 2397: 2395:Robust control 2392: 2387: 2382: 2377: 2372: 2367: 2362: 2357: 2352: 2346: 2345: 2344: 2340: 2339: 2323: 2320: 2319: 2318: 2307:Jan C. Willems 2304: 2298: 2281:pioneered the 2276: 2262: 2251:Lev Pontryagin 2248: 2234: 2227:Norbert Wiener 2224: 2214: 2208: 2202: 2192: 2187:developed the 2182: 2172: 2162: 2148: 2122:Main article: 2119: 2116: 2115: 2114: 2111:self-organized 2104: 2101:dynamic system 2066: 2048:control system 2040: 2030: 2023: 1996:robust control 1992:Robust control 1989: 1969: 1968: 1957:linear systems 1952: 1949: 1940:Main article: 1937: 1934: 1925:Main article: 1922: 1919: 1881:Main article: 1878: 1875: 1862:Main article: 1859: 1856: 1854: 1851: 1838: 1837: 1817: 1816: 1796: 1793: 1790: 1784: 1781: 1774: 1770: 1767: 1764: 1761: 1758: 1755: 1752: 1749: 1746: 1743: 1740: 1734: 1731: 1725: 1695: 1694: 1677: 1674: 1632: 1629: 1626: 1623: 1620: 1617: 1614: 1592: 1589: 1565: 1562: 1557: 1554: 1551: 1548: 1545: 1542: 1539: 1519: 1516: 1473: 1470: 1435: 1432: 1429: 1418: 1417: 1401: 1398: 1394: 1390: 1387: 1384: 1380: 1375: 1372: 1369: 1366: 1363: 1346: 1345: 1334: 1331: 1328: 1325: 1320: 1316: 1312: 1309: 1306: 1303: 1300: 1276:imaginary part 1263: 1260: 1257: 1246: 1245: 1229: 1226: 1222: 1218: 1215: 1212: 1208: 1203: 1200: 1197: 1194: 1191: 1170: 1169: 1158: 1155: 1152: 1149: 1144: 1140: 1136: 1133: 1130: 1127: 1124: 1076: 1052: 1037: 1036: 1025: 999:linear systems 995: 994: 985:Stability for 983: 951: 948: 946: 943: 930:robust control 912:defined using 897: 894: 873: 870: 869: 868: 829: 814: 811: 802: 801: 777: 720: 717: 716: 715: 667: 588: 585: 571:PID controller 558: 557: 554: 551: 548: 542: 535: 513:cruise control 486:electric motor 484:applied to an 436: 428: 426: 423: 376: 368: 366: 363: 347:internal model 238: 235: 142:To do this, a 103:is a field of 101:Control theory 94: 93: 80:the discussion 57: 55: 48: 26: 24: 18:Control Theory 14: 13: 10: 9: 6: 4: 3: 2: 5828: 5817: 5814: 5812: 5809: 5807: 5804: 5802: 5799: 5798: 5796: 5781: 5780: 5771: 5769: 5768: 5759: 5757: 5756: 5747: 5745: 5744: 5739: 5733: 5732: 5729: 5723: 5720: 5718: 5715: 5713: 5710: 5708: 5705: 5703: 5700: 5696: 5693: 5692: 5691: 5688: 5687: 5685: 5683: 5679: 5673: 5670: 5668: 5665: 5663: 5660: 5658: 5655: 5653: 5650: 5648: 5645: 5644: 5642: 5640: 5639:Computational 5636: 5628: 5625: 5623: 5620: 5618: 5615: 5614: 5613: 5610: 5608: 5605: 5603: 5600: 5598: 5595: 5593: 5590: 5588: 5585: 5583: 5580: 5578: 5575: 5573: 5570: 5568: 5565: 5563: 5560: 5558: 5555: 5554: 5552: 5550: 5546: 5540: 5537: 5535: 5532: 5530: 5527: 5525: 5522: 5520: 5517: 5516: 5514: 5512: 5508: 5502: 5499: 5497: 5494: 5492: 5489: 5487: 5484: 5483: 5481: 5479: 5478:Number theory 5475: 5469: 5466: 5464: 5461: 5459: 5456: 5454: 5451: 5449: 5446: 5444: 5441: 5439: 5436: 5435: 5433: 5431: 5427: 5421: 5418: 5416: 5413: 5411: 5410:Combinatorics 5408: 5407: 5405: 5403: 5399: 5393: 5390: 5388: 5385: 5383: 5380: 5378: 5375: 5373: 5370: 5368: 5365: 5363: 5362:Real analysis 5360: 5358: 5355: 5354: 5352: 5350: 5346: 5340: 5337: 5335: 5332: 5330: 5327: 5325: 5322: 5320: 5317: 5315: 5312: 5310: 5307: 5305: 5302: 5301: 5299: 5297: 5293: 5287: 5284: 5282: 5279: 5277: 5274: 5272: 5269: 5267: 5264: 5262: 5259: 5258: 5256: 5254: 5250: 5244: 5241: 5239: 5236: 5232: 5229: 5227: 5224: 5223: 5222: 5219: 5218: 5215: 5210: 5202: 5197: 5195: 5190: 5188: 5183: 5182: 5179: 5167: 5166: 5157: 5155: 5154: 5149: 5145: 5143: 5142: 5133: 5132: 5129: 5123: 5120: 5118: 5115: 5114: 5112: 5110:Organizations 5108: 5102: 5099: 5097: 5094: 5092: 5089: 5088: 5086: 5082: 5076: 5073: 5071: 5068: 5066: 5063: 5061: 5060:Kevin Warwick 5058: 5056: 5053: 5051: 5048: 5046: 5043: 5041: 5038: 5036: 5033: 5031: 5028: 5026: 5023: 5021: 5018: 5016: 5013: 5011: 5008: 5006: 5003: 5001: 4998: 4996: 4993: 4991: 4988: 4986: 4983: 4981: 4978: 4976: 4975:Margaret Mead 4973: 4971: 4968: 4966: 4963: 4961: 4958: 4956: 4953: 4951: 4948: 4946: 4943: 4941: 4938: 4936: 4935:Lydia Kavraki 4933: 4931: 4928: 4926: 4923: 4921: 4920:Barbara Grosz 4918: 4916: 4913: 4911: 4908: 4906: 4903: 4901: 4898: 4896: 4893: 4891: 4888: 4886: 4883: 4881: 4878: 4876: 4873: 4871: 4868: 4866: 4863: 4861: 4858: 4856: 4853: 4851: 4848: 4846: 4843: 4841: 4838: 4836: 4833: 4831: 4828: 4826: 4823: 4821: 4820:Ruzena Bajcsy 4818: 4816: 4813: 4811: 4808: 4806: 4803: 4802: 4800: 4798: 4794: 4783: 4779: 4776: 4774: 4771: 4769: 4766: 4764: 4761: 4759: 4756: 4754: 4751: 4749: 4746: 4744: 4741: 4739: 4736: 4734: 4731: 4730: 4727: 4721: 4718: 4716: 4713: 4711: 4708: 4706: 4703: 4701: 4698: 4696: 4693: 4691: 4688: 4686: 4683: 4682: 4679: 4676: 4670: 4664: 4661: 4659: 4656: 4654: 4651: 4649: 4646: 4644: 4643:Doubling time 4641: 4640: 4638: 4634: 4628: 4625: 4623: 4620: 4618: 4615: 4613: 4610: 4608: 4605: 4603: 4600: 4598: 4595: 4593: 4590: 4588: 4585: 4583: 4580: 4578: 4575: 4574: 4572: 4568: 4564: 4560: 4553: 4548: 4546: 4541: 4539: 4534: 4533: 4530: 4518: 4515: 4513: 4510: 4508: 4505: 4503: 4500: 4498: 4495: 4493: 4490: 4488: 4485: 4483: 4482:Ulla Mitzdorf 4480: 4478: 4475: 4473: 4470: 4468: 4465: 4463: 4460: 4458: 4455: 4453: 4452:Robert Trappl 4450: 4448: 4445: 4443: 4440: 4438: 4435: 4433: 4430: 4428: 4425: 4423: 4420: 4418: 4415: 4413: 4410: 4408: 4407:Margaret Mead 4405: 4403: 4400: 4398: 4395: 4393: 4390: 4388: 4387:Kevin Warwick 4385: 4383: 4380: 4378: 4375: 4373: 4370: 4368: 4365: 4363: 4360: 4358: 4357:Jacque Fresco 4355: 4353: 4350: 4348: 4345: 4343: 4340: 4338: 4335: 4333: 4330: 4328: 4325: 4323: 4320: 4318: 4315: 4313: 4310: 4308: 4305: 4303: 4300: 4298: 4295: 4293: 4290: 4288: 4285: 4283: 4280: 4278: 4275: 4273: 4270: 4268: 4265: 4263: 4260: 4258: 4255: 4253: 4250: 4248: 4245: 4243: 4240: 4238: 4235: 4234: 4232: 4230: 4226: 4220: 4217: 4215: 4212: 4210: 4207: 4205: 4202: 4200: 4197: 4195: 4192: 4190: 4187: 4185: 4182: 4180: 4177: 4175: 4172: 4170: 4167: 4165: 4162: 4160: 4157: 4155: 4152: 4150: 4149:Connectionism 4147: 4145: 4142: 4140: 4137: 4135: 4132: 4130: 4127: 4125: 4122: 4120: 4117: 4115: 4112: 4110: 4107: 4106: 4104: 4100: 4096: 4088: 4083: 4081: 4076: 4074: 4069: 4068: 4065: 4053: 4050: 4048: 4045: 4043: 4040: 4038: 4035: 4033: 4030: 4028: 4025: 4023: 4020: 4018: 4015: 4013: 4012: 4008: 4007: 4005: 4001: 3995: 3992: 3990: 3987: 3985: 3982: 3980: 3977: 3975: 3972: 3970: 3967: 3966: 3964: 3960: 3954: 3951: 3949: 3946: 3944: 3941: 3939: 3936: 3934: 3931: 3929: 3928:Least squares 3926: 3924: 3921: 3919: 3918:Kalman filter 3916: 3914: 3911: 3909: 3906: 3904: 3901: 3899: 3896: 3894: 3891: 3889: 3886: 3884: 3881: 3879: 3876: 3875: 3873: 3869: 3863: 3860: 3858: 3855: 3852: 3849: 3846: 3843: 3841: 3838: 3836: 3833: 3831: 3828: 3827: 3825: 3821: 3815: 3812: 3810: 3807: 3804: 3801: 3798: 3795: 3793: 3790: 3788: 3785: 3783: 3780: 3777: 3774: 3772: 3769: 3767: 3764: 3762: 3761:Observability 3759: 3757: 3754: 3752: 3749: 3747: 3744: 3742: 3739: 3737: 3734: 3732: 3729: 3727: 3726:Block diagram 3724: 3722: 3719: 3718: 3716: 3712: 3706: 3703: 3701: 3698: 3696: 3693: 3691: 3688: 3686: 3683: 3680: 3677: 3674: 3673:Multivariable 3671: 3669: 3666: 3664: 3661: 3658: 3655: 3653: 3652:Fuzzy control 3650: 3648: 3645: 3643: 3640: 3638: 3635: 3633: 3630: 3629: 3627: 3623: 3619: 3612: 3607: 3605: 3600: 3598: 3593: 3592: 3589: 3583: 3580: 3578: 3575: 3572: 3569: 3568: 3564: 3558: 3552: 3548: 3543: 3542: 3538: 3537: 3532: 3526: 3522: 3517: 3513: 3507: 3503: 3502: 3496: 3492: 3486: 3482: 3477: 3473: 3467: 3460: 3459: 3454: 3450: 3446: 3440: 3436: 3432: 3428: 3424: 3418: 3414: 3410: 3405: 3401: 3395: 3391: 3386: 3382: 3376: 3372: 3367: 3363: 3358: 3354: 3348: 3344: 3339: 3335: 3329: 3322: 3321: 3315: 3311: 3305: 3301: 3296: 3295: 3291: 3282: 3278: 3274: 3273: 3268: 3262: 3259: 3254: 3250: 3246: 3242: 3235: 3232: 3227: 3223: 3219: 3215: 3211: 3207: 3202: 3197: 3193: 3189: 3188: 3180: 3177: 3172: 3168: 3164: 3160: 3153: 3150: 3146: 3142: 3138: 3133: 3128: 3124: 3120: 3116: 3112: 3107: 3102: 3098: 3094: 3090: 3083: 3080: 3075: 3071: 3067: 3063: 3059: 3055: 3051: 3044: 3041: 3036: 3030: 3026: 3019: 3016: 3011: 3005: 3002: 2997: 2993: 2988: 2983: 2980:(4): 041118. 2979: 2975: 2971: 2964: 2961: 2956: 2952: 2948: 2944: 2940: 2933: 2930: 2925: 2921: 2915: 2912: 2907: 2903: 2897: 2894: 2888: 2885: 2880: 2878:9781849722704 2874: 2870: 2865: 2864: 2855: 2852: 2844: 2840: 2833: 2826: 2823: 2818: 2811: 2808: 2803: 2802: 2794: 2791: 2786: 2779: 2776: 2771: 2767: 2762: 2757: 2753: 2746: 2743: 2738: 2734: 2730: 2726: 2722: 2718: 2711: 2708: 2696: 2695:portal.dnb.de 2692: 2685: 2682: 2677: 2673: 2669: 2665: 2661: 2655: 2652: 2644: 2640: 2636: 2629: 2625: 2619: 2616: 2609: 2604: 2601: 2599: 2596: 2594: 2591: 2589: 2586: 2584: 2581: 2579: 2576: 2574: 2571: 2569: 2566: 2564: 2561: 2559: 2556: 2554: 2551: 2549: 2546: 2544: 2541: 2539: 2536: 2535: 2530: 2529: 2525: 2522: 2520: 2517: 2515: 2512: 2510: 2507: 2505: 2502: 2500: 2497: 2495: 2492: 2490: 2487: 2485: 2482: 2480: 2477: 2475: 2472: 2470: 2467: 2465: 2462: 2460: 2457: 2455: 2452: 2450: 2447: 2445: 2442: 2440: 2437: 2435: 2432: 2430: 2427: 2425: 2422: 2421: 2416: 2415: 2411: 2408: 2406: 2403: 2401: 2398: 2396: 2393: 2391: 2388: 2386: 2383: 2381: 2378: 2376: 2373: 2371: 2368: 2366: 2363: 2361: 2358: 2356: 2353: 2351: 2348: 2347: 2342: 2341: 2337: 2326: 2321: 2316: 2312: 2308: 2305: 2302: 2301:Ali H. 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563:feedforward 499:systems, a 327:electronics 210:PID control 5795:Categories 5607:Statistics 5486:Arithmetic 5448:Arithmetic 5314:Elementary 5281:Set theory 5045:Eric Trist 4900:Fred Emery 4797:Scientists 4773:Psychology 4768:Philosophy 4597:Ecological 4582:Biological 3776:Root locus 2920:Mayr, Otto 2902:Mayr, Otto 2610:References 2548:Bond graph 2479:Root locus 2439:H infinity 2350:Automation 2267:developed 2197:developed 1980:(MPC) and 1667:robustness 1648:integrator 1455:Bode plots 1451:root locus 1067:where the 1043:where the 974:is called 966:criteria. 863:and human 836:telescopes 708:linearized 700:simulating 636:root locus 503:including 342:Space Race 241:See also: 212:theory by 144:controller 137:optimality 87:March 2023 5534:Geometric 5524:Algebraic 5463:Euclidean 5438:Algebraic 5334:Universal 4710:Systemics 4174:Emergence 4102:Subfields 3845:Real time 3797:Stability 3721:Bode plot 3201:1211.0248 3106:1406.5197 2987:0805.4824 2770:1575-9822 2756:CiteSeerX 2700:April 26, 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