Control theory: Difference between revisions
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{{short description|Branch of engineering and mathematics}} | {{short description|Branch of engineering and mathematics}} | ||
{{About|control theory in engineering|control theory in linguistics| | {{About|control theory in engineering|control theory in linguistics|Control (linguistics)|control theory in psychology and sociology|Control theory (sociology)|and|Perceptual control theory}} | ||
{{Use mdy dates|date=July 2016}} | {{Use mdy dates|date=July 2016}} | ||
'''Control theory''' is a field of [[control engineering]] and [[applied mathematics]] that deals with the [[control system|control]] of [[dynamical system]]s. The | '''Control theory''' is a field of [[control engineering]] and [[applied mathematics]] that deals with the [[control system|control]] of [[dynamical system]]s. The aim is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control [[Stability theory|stability]]; often with the aim to achieve a degree of [[Optimal control|optimality]]. | ||
To do this, a '''controller''' with the requisite corrective behavior is required. This controller monitors the controlled [[process variable]] (PV), and compares it with the reference or [[Setpoint (control system)|set point]] (SP). The difference between actual and desired value of the process variable, called the ''error'' 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 [[controllability]] and [[observability]]. Control theory is used in [[control system engineering]] to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as [[robotics]]. | To do this, a '''controller''' with the requisite corrective behavior is required. This controller monitors the controlled [[process variable]] (PV), and compares it with the reference or [[Setpoint (control system)|set point]] (SP). The difference between actual and desired value of the process variable, called the ''error'' 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 [[controllability]] and [[observability]]. Control theory is used in [[control system engineering]] to design automation that have revolutionized manufacturing, aircraft, communications and other industries, and created new fields such as [[robotics]]. | ||
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[[File:Boulton and Watt centrifugal governor-MJ.jpg|thumb|right|[[Centrifugal governor]] in a [[Boulton & Watt engine]] of 1788]] | [[File:Boulton and Watt centrifugal governor-MJ.jpg|thumb|right|[[Centrifugal governor]] in a [[Boulton & Watt engine]] of 1788]] | ||
Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the [[centrifugal governor]], conducted by the physicist [[James Clerk Maxwell]] in 1868, entitled ''On Governors''.<ref name="Maxwell1867">{{cite journal|author=Maxwell, J.C.|year=1868|title=On Governors|journal=Proceedings of the Royal Society of London|volume=16|pages=270–283|doi=10.1098/rspl.1867.0055|jstor=112510|doi-access=}}<!--| access-date = 2008-04-14--></ref> A centrifugal governor was already used to regulate the velocity of windmills.<ref>{{cite web| title = Control Theory: History, Mathematical Achievements and Perspectives|url=https://citeseerx.ist.psu.edu/doc/10.1.1.302.5633|author1=Fernandez-Cara, E. |author2=Zuazua, E. |publisher=Boletin de la Sociedad Espanola de Matematica Aplicada|citeseerx=10.1.1.302.5633 |issn=1575-9822}}</ref> Maxwell described and analyzed the phenomenon of [[self-oscillation]], in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate, [[Edward John Routh]], abstracted Maxwell's results for the general class of linear systems.<ref name=Routh1975>{{cite book | Although control systems of various types date back to antiquity, a more formal analysis of the field began with a dynamics analysis of the [[centrifugal governor]], conducted by the physicist [[James Clerk Maxwell]] in 1868, entitled ''On Governors''.<ref name="Maxwell1867">{{cite journal|author=Maxwell, J.C.|year=1868|title=On Governors|journal=Proceedings of the Royal Society of London|volume=16|issue=16 |pages=270–283|doi=10.1098/rspl.1867.0055|jstor=112510|doi-access=}}<!--| access-date = 2008-04-14--></ref> A centrifugal governor was already used to regulate the velocity of windmills.<ref>{{cite web| title = Control Theory: History, Mathematical Achievements and Perspectives|url=https://citeseerx.ist.psu.edu/doc/10.1.1.302.5633|author1=Fernandez-Cara, E. |author2=Zuazua, E. |publisher=Boletin de la Sociedad Espanola de Matematica Aplicada|citeseerx=10.1.1.302.5633 |issn=1575-9822}}</ref> Maxwell described and analyzed the phenomenon of [[self-oscillation]], in which lags in the system may lead to overcompensation and unstable behavior. This generated a flurry of interest in the topic, during which Maxwell's classmate, [[Edward John Routh]], abstracted Maxwell's results for the general class of linear systems.<ref name=Routh1975>{{cite book | ||
| author = Routh, E.J. | | author = Routh, E.J. | ||
|author2=Fuller, A.T. | |author2=Fuller, A.T. | ||
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* ''[[Frequency domain]]'' – In this type the values of the [[state variable]]s, the mathematical [[variable (mathematics)|variables]] representing the system's input, output and feedback are represented as functions of [[frequency]]. The input signal and the system's [[transfer function]] are converted from time functions to functions of frequency by a [[transform (mathematics)|transform]] such as the [[Fourier transform]], [[Laplace transform]], or [[Z transform]]. The advantage of this technique is that it results in a simplification of the mathematics; the ''[[differential equation]]s'' that represent the system are replaced by ''[[algebraic equation]]s'' in the frequency domain which is much simpler to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above. | * ''[[Frequency domain]]'' – In this type the values of the [[state variable]]s, the mathematical [[variable (mathematics)|variables]] representing the system's input, output and feedback are represented as functions of [[frequency]]. The input signal and the system's [[transfer function]] are converted from time functions to functions of frequency by a [[transform (mathematics)|transform]] such as the [[Fourier transform]], [[Laplace transform]], or [[Z transform]]. The advantage of this technique is that it results in a simplification of the mathematics; the ''[[differential equation]]s'' that represent the system are replaced by ''[[algebraic equation]]s'' in the frequency domain which is much simpler to solve. However, frequency domain techniques can only be used with linear systems, as mentioned above. | ||
* ''[[Time-domain state space representation]]'' – In this type the values of the [[state variable]]s are represented as functions of time. With this model, the system being analyzed is represented by one or more [[differential equation]]s. Since frequency domain techniques are limited to [[linear function|linear]] 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 [[simulation language]]s have made their analysis routine. | * ''[[Time-domain state space representation]]'' – In this type the values of the [[state variable]]s are represented as functions of time. With this model, the system being analyzed is represented by one or more [[differential equation]]s. Since frequency domain techniques are limited to [[linear function|linear]] 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 [[simulation language]]s have made their analysis routine. | ||
In contrast to the frequency-domain analysis of the classical control theory, modern control theory utilizes the time-domain [[state space (controls)|state space]] representation,{{citation needed|date=December 2022}} a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.<ref>{{cite book|title=State space & linear systems|series=Schaum's outline series |publisher=McGraw Hill|author=Donald M Wiberg|year=1971 |isbn=978-0-07-070096-3}}</ref><ref>{{cite journal|author=Terrell, William|title=Some fundamental control theory I: Controllability, observability, and duality —AND— Some fundamental control Theory II: Feedback linearization of single input nonlinear systems|journal=American Mathematical Monthly|volume=106|issue=9|year=1999|pages=705–719 and 812–828|url=http://www.maa.org/programs/maa-awards/writing-awards/some-fundamental-control-theory-i-controllability-observability-and-duality-and-some-fundamental|doi=10.2307/2589614|jstor=2589614}}</ref> | In contrast to the frequency-domain analysis of the classical control theory, modern control theory utilizes the time-domain [[state space (controls)|state space]] representation,{{citation needed|date=December 2022}} a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs, and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state-space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a point within that space.<ref>{{cite book|title=State space & linear systems|series=Schaum's outline series |publisher=McGraw Hill|author=Donald M Wiberg|year=1971 |isbn=978-0-07-070096-3}}</ref><ref>{{cite journal|author=Terrell, William|title=Some fundamental control theory I: Controllability, observability, and duality —AND— Some fundamental control Theory II: Feedback linearization of single input nonlinear systems|journal=American Mathematical Monthly|volume=106|issue=9|year=1999|pages=705–719 and 812–828|url=http://www.maa.org/programs/maa-awards/writing-awards/some-fundamental-control-theory-i-controllability-observability-and-duality-and-some-fundamental|doi=10.2307/2589614|jstor=2589614|archive-url=http://web.archive.org/web/20130910191729/http://www.maa.org/programs/maa-awards/writing-awards/some-fundamental-control-theory-i-controllability-observability-and-duality-and-some-fundamental|archive-date=2013-09-10}}</ref> | ||
==System interfacing== | ==System interfacing== | ||
Control systems can be divided into different categories depending on the number of inputs and outputs. | Control systems can be divided into different categories depending on the number of inputs and outputs. | ||
* [[Single-input single-output system|Single-input single-output]] (SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an [[audio | * [[Single-input single-output system|Single-input single-output]] (SISO) – This is the simplest and most common type, in which one output is controlled by one control signal. Examples are the cruise control example above, or an [[audio power amplifier]], in which the control input is the input [[audio signal]] and the output drives the [[loudspeaker]]. | ||
* [[Multiple-input multiple-output system|Multiple-input multiple-output]] (MIMO) – These are found in more complicated systems. For example, modern large [[telescope]]s such as the [[Keck telescopes|Keck]] and [[MMT Observatory|MMT]] have mirrors composed of many separate segments each controlled by an [[actuator]]. The shape of the entire mirror is constantly adjusted by a MIMO [[active optics]] 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 [[wavefront]] due to turbulence in the atmosphere. Complicated systems such as [[nuclear reactor]]s and human [[cell (biology)|cells]] are simulated by a computer as large MIMO control systems. | * [[Multiple-input multiple-output system|Multiple-input multiple-output]] (MIMO) – These are found in more complicated systems. For example, modern large [[telescope]]s such as the [[Keck telescopes|Keck]] and [[MMT Observatory|MMT]] have mirrors composed of many separate segments each controlled by an [[actuator]]. The shape of the entire mirror is constantly adjusted by a MIMO [[active optics]] 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 [[wavefront]] due to turbulence in the atmosphere. Complicated systems such as [[nuclear reactor]]s and human [[cell (biology)|cells]] are simulated by a computer as large MIMO control systems. | ||
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* [[Richard Bellman]] developed [[dynamic programming]] in the 1940s.<ref>{{cite magazine |author=Richard Bellman |date=1964 |title=Control Theory |doi=10.1038/scientificamerican0964-186 |magazine=[[Scientific American]] |volume=211 |issue=3 |pages=186–200|author-link=Richard Bellman }}</ref> | * [[Richard Bellman]] developed [[dynamic programming]] in the 1940s.<ref>{{cite magazine |author=Richard Bellman |date=1964 |title=Control Theory |doi=10.1038/scientificamerican0964-186 |magazine=[[Scientific American]] |volume=211 |issue=3 |pages=186–200|author-link=Richard Bellman }}</ref> | ||
* [[Warren E. Dixon]], control theorist and a professor | * [[Warren E. Dixon]], control theorist and a professor | ||
* [[Andrey Kolmogorov]] co-developed the [[Wiener filter|Wiener–Kolmogorov filter]] in 1941. | * [[Andrey Kolmogorov]] co-developed the [[Wiener filter|Wiener–Kolmogorov filter]] in 1941. | ||
* [[Norbert Wiener]] co-developed the Wiener–Kolmogorov filter and coined the term [[cybernetics]] in the 1940s. | * [[Norbert Wiener]] co-developed the Wiener–Kolmogorov filter and coined the term [[cybernetics]] in the 1940s. | ||