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.
248:
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
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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
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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
808:
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
1975:
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
406:
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.
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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
336:
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.
228:
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.
2028:
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:
413:
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:
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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
854:
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
344:
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.
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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
158:
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
119:
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:
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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
2002:
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
1603:
3050:"Some fundamental control theory I: Controllability, observability, and duality —AND— Some fundamental control Theory II: Feedback linearization of single input nonlinear systems"
1012:
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
1819:
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
1650:
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,
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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.
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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
921:
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.
515:
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
880:(SISO) system design, except when analyzing for disturbance rejection using a second input. The system analysis is carried out in the time domain using
79:
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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
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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
1998:
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
17:
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An electromechanical timer, normally used for open-loop control based purely on a timing sequence, with no feedback from the process
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274:. A centrifugal governor was already used to regulate the velocity of windmills. Maxwell described and analyzed the phenomenon of
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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
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1849:. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold.
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and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.
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1959:, this can be obtained by directly placing the poles. Nonlinear control systems use specific theories (normally based on
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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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1981:
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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
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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
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who was one of the main contributors to nonlinear control theory and published many books on perturbation methods
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2143:, now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the
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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
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Hallion, Richard P. (1980). Sicherman, Barbara; Green, Carol Hurd; Kantrov, Ilene; Walker, Harriette (eds.).
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828:, in which the control input is the input audio signal and the output is the sound waves from the speaker.
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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".
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608:. A major subclass is systems which in addition have parameters which do not change with time, called
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analyzed system stability using differential equations in 1877, resulting in what is now known as the
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488:) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with
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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
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764:. The advantage of this technique is that it results in a simplification of the mathematics; the
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Modern performance assessments use some variation of integrated tracking error (IAE, ISA, CQI).
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2317:(LMI) in control theory. He pioneered the behavioral approach to mathematical systems theory.
1657:(the time needed by the control system to reach the desired value after a perturbation), peak
666:, which give solutions for system response and design techniques for most systems of interest.
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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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1661:(the highest value reached by the response before reaching the desired value) and others (
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507:, control algorithms, and actuators is arranged in an attempt to regulate a variable at a
154:(SP). The difference between actual and desired value of the process variable, called the
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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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282:, abstracted Maxwell's results for the general class of linear systems. Independently,
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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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3501:
Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide
3266:
2537:
2508:
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2384:
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2272:
2240:
2204:
2194:
1985:
1972:
1820:
1686:
1486:
1477:
913:
860:
843:
839:
834:(MIMO) – These are found in more complicated systems. For example, modern large
793:
663:
322:
309:
developed the theory of discontinuous automatic control systems, and applied the
294:
225:
159:
136:
3406:
2181:
in 1927. He managed to develop stable negative feedback amplifiers in the 1930s.
993:(ISS), which combines Lyapunov stability and a notion similar to BIBO stability.
683:
480:. Its name comes from the information path in the system: process inputs (e.g.,
5689:
5323:
5074:
5069:
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4113:
4036:
4021:
3988:
3947:
3781:
3699:
3573:, a set of worked-through control examples solved by several different methods.
3499:
2995:
2602:
2513:
2399:
2394:
2250:
2226:
2110:
2100:
2047:
1991:
1908:, trajectory linearization control normally take advantage of results based on
1681:
1666:
1446:
and is not BIBO stable since the pole has a modulus strictly greater than one.
975:
929:
889:
864:
785:
733:
570:
512:
485:
350:
293:
A notable application of dynamic control was in the area of crewed flight. The
112:
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5409:
5361:
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4974:
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3927:
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3429:
2769:
2300:
2294:
2290:
2220:
2184:
1956:
1662:
1490:
1481:
1017:
971:
851:
707:
516:
317:
for aircraft. Other areas of application for discontinuous controls included
283:
205:
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163:
74:
was found. You can help implement the merge by following the instructions at
5419:
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4092:
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3140:
2728:
2498:
2051:
2007:
1901:
1824:
1605:
is a fixed value strictly greater than zero, instead of simply asking that
1458:
1090:
When the appropriate conditions above are satisfied a system is said to be
825:
500:
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302:
197:
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3238:
2567:
2282:
2244:
2230:
2136:
2096:
2080:
2015:
1868:
For MIMO systems, pole placement can be performed mathematically using a
1665:, quarter-decay). Frequency domain specifications are usually related to
1507:
1032:
1028:
761:
718:
679:
627:
544:
534:
disturbance rejection (such as hills in the cruise control example above)
393:
326:
217:
209:
2867:. Cambridge, Mass.: Belknap Press of Harvard University Press. pp.
5606:
5485:
5280:
5044:
4899:
3122:
3073:
2547:
2478:
2438:
2349:
1647:
1450:
941:
are well known among the people who have shaped modern control theory.
699:
651:
635:
576:
341:
2736:
2109:
may be defined as attempts to interfere in the processes by which the
846:
have mirrors composed of many separate segments each controlled by an
4709:
4596:
4173:
3720:
3183:
2919:
2901:
2832:"Optimum and Quasi-Optimum Control of Third and Fourth-Order Systems"
2752:"Control Theory: History, Mathematical Achievements and Perspectives"
2158:
1654:
1454:
856:
835:
741:
631:
314:
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.
5510:
5429:
5356:
4046:
2433:
1888:
1523:
850:. The shape of the entire mirror is constantly adjusted by a MIMO
847:
496:
455:
177:
Extensive use is usually made of a diagrammatic style known as the
171:
4527:
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
5295:
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
481:
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
4566:
3678:
3316:
1935:
504:
489:
364:
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.
4051:
3518:
305:, control theory was becoming an important area of research.
3521:
Classical Feedback Control with Nonlinear Multi-loop Systems
50:
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
2749:
614:(LTI) systems. These systems are amenable to powerful
2830:
Flugge-Lotz, Irmgard; Titus, Harold A. (October 1962).
2782:
1873:
be included and incorporated in pole placement design.
812:
2939:"Feedback for physicists: A tutorial essay on control"
1675:
719:
Analysis techniques - frequency domain and time domain
586:
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:
2325:
1471:
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:
2057:. When the links in the tree are implemented by a
3340:
2860:
2664:Journal of the American Society of Naval Engineers
1799:
1635:
1597:
1570:
1438:
1409:
1337:
1266:
1237:
1161:
1079:
1055:
3368:
2885:
70:, consensus to merge this with content from
5792:
3497:
3047:
2829:
2117:
1920:
3275:. Vol. 211, no. 3. pp. 186–200.
3265:
3022:
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}}}}
871:
710:by approximating them by a linear system using
694:. Nonlinear systems are often analyzed using
5192:
4543:
4078:
3602:
3359:
2714:
1571:{\displaystyle Re<-{\overline {\lambda }}}
556:improved rectification of random fluctuations
4093:Subfields of and cyberneticians involved in
3080:
1936:Deterministic and stochastic systems control
1876:
900:Modern control theory is carried out in the
895:
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
5206:
2814:
2797:
2095:or a combination of these methods, such as
944:
550:reduced sensitivity to parameter variations
424:
5199:
5185:
4550:
4536:
4085:
4071:
3609:
3595:
3317:Karl J. Åström; Richard M. Murray (2008).
2936:
2926:. Clinton, MA US: The Colonial Press, Inc.
2908:. Clinton, MA US: The Colonial Press, Inc.
2808:
2717:Proceedings of the Royal Society of London
2708:
2071:uses various AI computing approaches like
1950:
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.
1857:
1852:
3519:Boris J. Lurie; Paul J. Enright (2019).
3387:
3156:
3027:. Schaum's outline series. McGraw Hill.
2968:Cao, F. J.; Feito, M. (April 10, 2009).
2967:
2791:
2776:
2658:
1517:
1024:is used to obtain the transfer function.
575:
437:
392:
372:Control loop § Open-loop and closed-loop
246:
3478:
2858:
2648:from the original on December 19, 2008.
2622:
1932:channels and coordinate their actions.
553:improved reference tracking performance
14:
5793:
3544:
3451:
3297:
3086:
2715:Maxwell, J.C. (1868). "On Governors".
1598:{\displaystyle {\overline {\lambda }}}
781:Time-domain state space representation
5180:
4531:
4066:
3590:
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
5761:
5159:
3177:
2937:Bechhoefer, John (August 31, 2005).
2918:
2900:
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.
44:
5773:
5101:Systems theory in political science
4557:
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.
24:
3616:
3291:
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:
5827:
3577:Control Tuning and Best Practices
3564:
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
3259:
3232:
3150:
3041:
3016:
3002:
2961:
2930:
2924:The Origins of Feedback Control
2912:
2906:The Origins of Feedback Control
2894:
2852:
2750:Fernandez-Cara, E.; Zuazua, E.
2688:
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
2823:
2743:
2682:
2652:
2616:
1794:
1788:
1765:
1759:
1744:
1738:
1624:
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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
13:
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
2609:
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
949:
872:Classical SISO system design
450:or feedback controller is a
7:
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
2321:
2189:Nyquist stability criterion
2044:hierarchical control system
640:Nyquist stability criterion
547:processes can be stabilized
222:control systems engineering
18:Controller (control theory)
10:
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:
236:
168:control system engineering
36:control theory (sociology)
29:
5730:
5680:
5637:
5547:
5509:
5476:
5428:
5400:
5347:
5294:
5276:Philosophy of mathematics
5251:
5216:
5130:
5109:
5083:
4795:
4728:
4680:
4671:
4635:
4592:Coupled human–environment
4565:
4227:
4101:
4002:
3961:
3943:Perceptual control theory
3878:Artificial neural network
3870:
3835:Digital signal processing
3822:
3713:
3624:
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 FlĂĽgge-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 UexkĂĽll
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:
5177:
5162:
5161:
5150:
5138:
5137:
5050:Francisco Varela
4782:Systems thinking
4715:Urban metabolism
4678:
4677:
4552:
4545:
4538:
4529:
4528:
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:
4063:
3923:Krener's theorem
3632:Adaptive control
3611:
3604:
3597:
3588:
3587:
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.
3531:
3512:
3491:
3472:
3461:
3453:Sontag, Eduardo
3445:
3423:
3409:Dawn M. Tilbury
3400:
3381:
3353:
3334:
3323:
3310:
3294:
3292:Further reading
3289:
3288:
3267:Richard Bellman
3264:
3260:
3237:
3233:
3182:
3178:
3155:
3151:
3085:
3081:
3066:10.2307/2589614
3046:
3042:
3035:
3021:
3017:
3008:
3007:
3003:
2966:
2962:
2935:
2931:
2917:
2913:
2899:
2895:
2890:
2886:
2879:
2857:
2853:
2845:
2834:
2828:
2824:
2813:
2809:
2796:
2792:
2781:
2777:
2761:10.1.1.302.5633
2748:
2744:
2713:
2709:
2699:
2697:
2687:
2683:
2657:
2653:
2645:
2630:
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:
1777:
1776:
1771:
1727:
1726:
1721:
1718:
1717:
1701:
1687:system dynamics
1678:
1610:
1607:
1606:
1585:
1583:
1580:
1579:
1558:
1535:
1532:
1531:
1520:
1487:Controllability
1484:
1478:Controllability
1476:Main articles:
1474:
1425:
1422:
1421:
1395:
1391:
1381:
1376:
1356:
1353:
1352:
1317:
1313:
1293:
1290:
1289:
1253:
1250:
1249:
1223:
1219:
1209:
1204:
1184:
1181:
1180:
1176:), is given by
1141:
1137:
1117:
1114:
1113:
1072:
1069:
1068:
1048:
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:
5819:
5818:
5813:
5808:
5803:
5801:Control theory
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:
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:
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:
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
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:
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2122:Main article:
2119:
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2111:self-organized
2104:
2101:dynamic system
2066:
2048:control system
2040:
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2023:
1996:robust control
1992:Robust control
1989:
1969:
1968:
1957:linear systems
1952:
1949:
1940:Main article:
1937:
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1925:Main article:
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1881:Main article:
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1862:Main article:
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999:linear systems
995:
994:
985:Stability for
983:
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948:
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930:robust control
912:defined using
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829:
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588:
585:
571:PID controller
558:
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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:
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48:
26:
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6:
4:
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5639:Computational
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5478:Number theory
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5410:Combinatorics
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5362:Real analysis
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5110:Organizations
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5060:Kevin Warwick
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4975:Margaret Mead
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4935:Lydia Kavraki
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4920:Barbara Grosz
4918:
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4820:Ruzena Bajcsy
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4659:
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4646:
4644:
4643:Doubling time
4641:
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4483:
4482:Ulla Mitzdorf
4480:
4478:
4475:
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4460:
4458:
4455:
4453:
4452:Robert Trappl
4450:
4448:
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4430:
4428:
4425:
4423:
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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:
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4149:Connectionism
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3954:
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3941:
3939:
3936:
3934:
3931:
3929:
3928:Least squares
3926:
3924:
3921:
3919:
3918:Kalman filter
3916:
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3762:
3761:Observability
3759:
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3752:
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3737:
3734:
3732:
3729:
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3726:Block diagram
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3674:
3673:Multivariable
3671:
3669:
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3661:
3658:
3655:
3653:
3652:Fuzzy control
3650:
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2980:(4): 041118.
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2301:Ali H. Nayfeh
2299:
2296:
2295:Kalman filter
2292:
2291:observability
2288:
2284:
2280:
2277:
2274:
2270:
2266:
2263:
2260:
2256:
2252:
2249:
2246:
2242:
2238:
2235:
2233:in the 1940s.
2232:
2228:
2225:
2222:
2218:
2215:
2212:
2209:
2206:
2203:
2201:in the 1940s.
2200:
2196:
2193:
2190:
2186:
2185:Harry Nyquist
2183:
2180:
2176:
2173:
2170:
2166:
2163:
2160:
2156:
2152:
2149:
2146:
2142:
2138:
2135:invented the
2134:
2131:
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2112:
2108:
2105:
2102:
2098:
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2090:
2086:
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2078:
2074:
2070:
2067:
2064:
2060:
2056:
2053:
2049:
2046:is a type of
2045:
2041:
2038:
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2027:
2024:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
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1825:Bode diagrams
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1688:
1683:
1673:
1670:
1669:(see after).
1668:
1664:
1663:settling time
1660:
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1627:
1621:
1615:
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1587:
1560:
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1527:
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1512:
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1503:
1501:
1497:
1492:
1491:observability
1488:
1483:
1482:Observability
1479:
1469:
1467:
1466:antiroll fins
1462:
1460:
1459:Nyquist plots
1456:
1452:
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1142:
1138:
1134:
1128:
1122:
1112:
1111:
1110:
1108:
1103:
1101:
1097:
1093:
1088:
1074:
1066:
1050:
1042:
1034:
1030:
1026:
1023:
1019:
1018:complex plane
1015:
1014:
1013:
1011:
1007:
1002:
1000:
992:
988:
984:
981:
977:
973:
972:linear system
969:
968:
967:
965:
961:
958:of a general
957:
942:
940:
936:
931:
927:
923:
922:multivariable
919:
915:
911:
907:
903:
893:
891:
887:
883:
879:
866:
862:
858:
853:
852:active optics
849:
845:
841:
837:
833:
830:
827:
823:
820:
819:
818:
810:
807:
799:
795:
791:
787:
783:
782:
778:
775:
774:
769:
768:
763:
759:
755:
751:
747:
743:
739:
735:
731:
730:
726:
725:
724:
713:
709:
705:
701:
697:
693:
689:
685:
684:Poincaré maps
681:
677:
673:
672:
668:
665:
661:
657:
653:
649:
645:
641:
637:
633:
629:
625:
621:
617:
613:
612:
607:
603:
599:
598:
594:
593:
592:
578:
574:
572:
567:
565:
564:
555:
552:
549:
546:
543:
540:
536:
533:
532:
531:
528:
526:
522:
518:
517:PID algorithm
514:
510:
506:
502:
498:
493:
491:
487:
483:
479:
475:
471:
467:
463:
462:
457:
453:
449:
440:
433:
419:
416:Likewise; "A
415:
412:
407:
403:
395:
391:
389:
388:
383:
382:
373:
362:
360:
356:
352:
348:
343:
338:
335:
330:
328:
324:
320:
316:
312:
308:
304:
299:
296:
291:
289:
285:
284:Adolf Hurwitz
281:
277:
273:
269:
265:
257:
253:
249:
244:
234:
232:
227:
223:
219:
215:
211:
207:
206:Adolf Hurwitz
204:and in 1895,
203:
202:Charles Sturm
199:
195:
190:
188:
184:
180:
179:block diagram
175:
173:
169:
165:
164:observability
161:
157:
153:
149:
145:
140:
138:
134:
130:
126:
122:
118:
114:
110:
106:
102:
98:
90:
81:
77:
73:
69:
64:
62:
56:
52:
47:
46:
41:
37:
33:
19:
5777:
5765:
5753:
5734:
5667:Optimization
5616:
5529:Differential
5453:Differential
5420:Order theory
5415:Graph theory
5319:Group theory
5163:
5151:
5139:
5084:Applications
5040:Katia Sycara
4940:James J. Kay
4930:Mike Jackson
4865:Murray Bowen
4763:Pharmacology
4758:Neuroscience
4684:
4507:Walter Pitts
4412:Marian Mazur
4287:Cliff Joslyn
4153:
4129:Biosemiotics
4032:Mechatronics
4009:
3840:Quantization
3803:Steady state
3636:
3617:
3546:
3520:
3500:
3480:
3464:. Springer.
3457:
3437:. Springer.
3434:
3412:
3389:
3370:
3361:
3342:
3319:
3299:
3270:
3261:
3244:
3240:
3234:
3191:
3185:
3179:
3162:
3158:
3152:
3144:
3096:
3092:
3082:
3057:
3053:
3043:
3024:
3018:
3010:"trim point"
3004:
2977:
2973:
2963:
2946:
2942:
2932:
2923:
2914:
2905:
2896:
2887:
2862:
2854:
2843:the original
2838:
2825:
2816:
2810:
2800:
2793:
2784:
2778:
2745:
2720:
2716:
2710:
2698:. Retrieved
2694:
2684:
2667:
2663:
2654:
2638:
2634:
2618:
2499:Steady state
2127:
2052:hierarchical
2008:Keith Glover
1995:
1954:
1945:
1930:
1902:backstepping
1886:
1867:
1839:
1818:
1809:
1702:
1679:
1671:
1652:
1645:
1528:
1521:
1513:
1504:
1499:
1496:stabilizable
1495:
1485:
1463:
1448:
1419:
1347:
1284:
1279:
1247:
1174:this example
1171:
1104:
1089:
1038:
1003:
998:
996:
955:
953:
899:
875:
838:such as the
826:audio system
816:
803:
779:
771:
765:
752:such as the
727:
722:
669:
609:
595:
590:
568:
561:
559:
529:
501:control loop
494:
465:
459:
452:control loop
445:
417:
408:
404:
400:
390:(feedback).
385:
379:
377:
339:
331:
303:World War II
300:
292:
272:On Governors
271:
261:
218:mathematical
198:Edward Routh
191:
181:. In it the
176:
155:
143:
141:
128:
124:
120:
100:
99:
97:
84:
76:Help:Merging
68:a discussion
58:
5779:WikiProject
5622:Game theory
5602:Probability
5339:Homological
5329:Multilinear
5309:Commutative
5286:Type theory
5253:Foundations
5209:mathematics
5030:Peter Senge
5025:John Seddon
5015:Qian Xuesen
4950:George Klir
4753:Engineering
4690:Cybernetics
4672:Theoretical
4622:Recommender
4612:Multi-agent
4607:Information
4442:Qian Xuesen
4322:Gordon Pask
4219:Synergetics
4184:Homeostasis
4124:Biorobotics
4095:cybernetics
3962:Controllers
3903:Fuzzy logic
3862:Z-transform
3766:Performance
3194:: 297–308.
3099:(6): 8414.
2723:: 270–283.
2568:Cybernetics
2283:state-space
2245:Z-transform
2239:introduced
2231:cybernetics
2137:Z-transform
2097:neuro-fuzzy
2081:fuzzy logic
2029:deviations.
2016:Vadim Utkin
1836:Constraints
1508:eigenvalues
1033:Z-transform
1029:unit circle
1027:inside the
902:state space
806:state space
762:Z transform
680:limit cycle
652:eigenvalues
628:Z transform
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,
2020:stability
1783:˙
1769:−
1751:−
1733:¨
1659:overshoot
1655:rise time
1622:λ
1591:¯
1588:λ
1564:¯
1561:λ
1556:−
1547:λ
1500:observing
1397:−
1386:−
1225:−
1214:−
1075:ρ
956:stability
950:Stability
918:Nonlinear
857:wavefront
750:transform
742:frequency
738:variables
644:bandwidth
632:Bode plot
521:overshoot
355:regulator
200:in 1874,
152:set point
133:stability
125:overshoot
5755:Category
5511:Topology
5458:Discrete
5443:Analytic
5430:Geometry
5402:Discrete
5357:Calculus
5349:Analysis
5304:Abstract
5243:Glossary
5226:Timeline
5141:Category
4743:Dynamics
4733:Analysis
4636:Concepts
4602:Economic
4047:Robotics
3847:software
3625:Branches
3455:(1998).
3226:32067734
3141:26423222
2922:(1969).
2904:(1970).
2643:Archived
2626:(1868).
2434:Feedback
2322:See also
2275:methods.
2257:and the
2223:in 1941.
1891:and the
1889:robotics
1815:Analysis
1578:, where
1524:robotics
1035:is used.
926:adaptive
848:actuator
682:theory,
545:unstable
509:setpoint
497:feedback
456:feedback
172:robotics
5767:Commons
5549:Applied
5519:General
5296:Algebra
5221:History
5165:Commons
4748:Ecology
4738:Biology
4617:Nervous
4587:Complex
3851:Sampled
3681:control
3675:control
3659:control
3206:Bibcode
3132:4600713
3111:Bibcode
3074:2589614
2869:241–242
1821:Nyquist
1457:or the
1096:modulus
1008:of its
980:bounded
505:sensors
490:sensors
482:voltage
474:outputs
258:of 1788
237:History
113:control
5468:Finite
5324:Linear
5231:Future
5207:Major
5153:Portal
4778:Theory
4674:fields
4627:Social
4567:System
3799:theory
3679:Neural
3657:Hybrid
3553:
3527:
3508:
3487:
3468:
3441:
3419:
3396:
3377:
3349:
3330:
3306:
3224:
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3072:
3031:
2875:
2768:
2758:
2737:112510
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1296:
1280:inside
1274:(zero
1187:
1120:
794:linear
690:, and
638:, and
470:states
66:After
61:merged
5695:lists
5238:Lists
5211:areas
4570:types
3778:ethod
3462:(PDF)
3324:(PDF)
3222:S2CID
3196:arXiv
3101:arXiv
3070:JSTOR
2982:arXiv
2846:(PDF)
2835:(PDF)
2733:JSTOR
2689:GND.
2646:(PDF)
2631:(PDF)
1006:poles
865:cells
760:, or
539:model
476:of a
359:plant
254:in a
156:error
127:, or
121:delay
5117:List
3853:data
3551:ISBN
3525:ISBN
3506:ISBN
3485:ISBN
3466:ISBN
3439:ISBN
3417:ISBN
3394:ISBN
3375:ISBN
3347:ISBN
3328:ISBN
3304:ISBN
3137:PMID
3029:ISBN
2873:ISBN
2766:ISSN
2702:2020
2289:and
2091:and
2055:tree
2000:Bode
1823:and
1628:<
1553:<
1489:and
1480:and
954:The
937:and
928:and
842:and
840:Keck
656:gain
340:The
325:and
174:.
162:and
107:and
38:and
4577:Art
3277:doi
3249:doi
3245:161
3214:doi
3167:doi
3127:PMC
3119:doi
3062:doi
3058:106
2992:doi
2951:doi
2725:doi
2672:doi
2639:100
1434:1.5
1389:1.5
1315:1.5
1262:0.5
1217:0.5
1139:0.5
1109:of
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472:or
464:or
301:By
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