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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback}}+ − − − − {{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}} − − − − '''Control theory''' deals with the control of continuously operating [[dynamical system]]s in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without ''delay or overshoot'' and ensuring control [[Stability theory|stability]]. Control theory is subfield of [[mathematics]], [[computer science]]<ref>{{Cite web|url=https://portal.dnb.de/opac.htm?method=simpleSearch&cqlMode=true&query=nid=4032317-1|title=Katalog der Deutschen Nationalbibliothek (Authority control)|last=GND|website=portal.dnb.de|url-status=live|archive-url=|archive-date=|access-date=2020-04-26}}</ref> and [[control engineering]]. − − Control theory deals with the control of continuously operating dynamical systems in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability. Control theory is subfield of mathematics, computer science and control engineering. − − 控制理论研究工程过程和机器中连续运行的动态系统的控制。目标是建立一个控制模型来控制这样的系统，使用控制行动在一个最佳的方式没有延迟或超调，并确保控制稳定性。控制理论是数学、计算机科学和控制工程的一个分支。 − − − − − − To do this, a ''[[Controller (control theory)|controller]]'' with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or 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]]. This is the basis for the advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries. This is ''feedback control'', which is usually ''continuous'' and involves taking measurements using a [[sensor]] and making calculated adjustments to keep the measured variable within a set range by means of a "final control element", such as a [[control valve]].<ref>Bennett, Stuart (1992). A history of control engineering, 1930-1955. IET. p. 48. {{ISBN|978-0-86341-299-8}}.</ref> − − 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 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. This is the basis for the advanced type of automation that revolutionized manufacturing, aircraft, communications and other industries. This is feedback control, which is usually continuous and involves taking measurements using a sensor and making calculated adjustments to keep the measured variable within a set range by means of a "final control element", such as a control valve. − − 要做到这一点，需要一个具有必要纠正行为的控制器。这个控制器监视受控程序变数，并将其与参考点或设定点进行比较。程序变数的实际值和期望值之间的差，称为误差信号，或者 SP-PV 误差，作为反馈来产生一个控制动作，使被控制的程序变数达到设定值。研究的其他方面还有可控性和可观测性。这是先进的自动化类型的基础，革命性的制造业，飞机，通信和其他行业。这就是反馈控制，它通常是连续的，涉及使用传感器进行测量，并通过控制阀等”最终控制元件”进行计算调整，使测量的变量保持在一定范围内。 − − − − − − Extensive use is usually made of a diagrammatic style known as the [[block diagram]]. In it the [[transfer function]], also known as the system function or network function, is a mathematical model of the relation between the input and output based on the [[differential equation]]s describing the system. − − Extensive use is usually made of a diagrammatic style known as the block diagram. In it the transfer function, also known as the system function or network function, is a mathematical model of the relation between the input and output based on the differential equations describing the system. − − 广泛使用的通常是一种被称为框图的图解风格。其中的传递函数，也称为系统函数或网络函数，是基于描述系统的微分方程的输入输出关系的数学模型。 − − − − − − Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by [[James Clerk Maxwell]].<ref>{{cite journal |first=J. C. |last=Maxwell |authorlink=James Clerk Maxwell |title=On Governors |date=1868 |journal=Proceedings of the Royal Society |volume=100 |issue= |pages= |url=https://upload.wikimedia.org/wikipedia/commons/b/b1/On_Governors.pdf}}</ref> Control theory was further advanced by [[Edward Routh]] in 1874, [[Jacques Charles François Sturm|Charles Sturm]] and in 1895, [[Adolf Hurwitz]], who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of [[PID control]] theory by [[Nicolas Minorsky]].<ref>{{cite journal |last=Minorsky |first=Nicolas |authorlink=Nicolas Minorsky |title=Directional stability of automatically steered bodies |journal=Journal of the American Society of Naval Engineers |year=1922 |volume=34 |pages=280–309 |issue=2 |ref=harv |doi=10.1111/j.1559-3584.1922.tb04958.x}}</ref> − − Control theory dates from the 19th century, when the theoretical basis for the operation of governors was first described by James Clerk Maxwell. Control theory was further advanced by Edward Routh in 1874, Charles Sturm and in 1895, Adolf Hurwitz, who all contributed to the establishment of control stability criteria; and from 1922 onwards, the development of PID control theory by Nicolas Minorsky. − − 控制理论可以追溯到19世纪，当时詹姆斯·克拉克·麦克斯韦首次描述了统治者运作的理论基础。1874年 Edward Routh，Charles Sturm 和1895年 Adolf Hurwitz 进一步提出了控制理论，他们都为建立控制稳定性标准做出了贡献; 从1922年开始，尼古拉斯·米诺尔斯基发展了 PID 控制理论。 − − Although a major application of control theory is in control systems engineering, which deals with the design of [[process control]] systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs. − − Although a major application of control theory is in control systems engineering, which deals with the design of process control systems for industry, other applications range far beyond this. As the general theory of feedback systems, control theory is useful wherever feedback occurs. − − 虽然控制理论的一个主要应用是在控制系统工程，其中涉及工业过程控制系统的设计，其他应用范围远远超出这一范围。作为反馈系统的一般理论，控制理论在反馈发生的任何地方都是有用的。 − − − − − − ==History== − − ==History== − − 历史 − − [[File:Boulton and Watt centrifugal governor-MJ.jpg|thumb|right|[[Centrifugal governor]] in a [[Boulton & Watt engine]] of 1788]] − − [[Centrifugal governor in a Boulton & Watt engine of 1788]] − − [1788年在 Boulton & Watt 发动机上的离心式调速器] − − − − − − 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=free}}<!--| accessdate = 2008-04-14--></ref> A centrifugal governor was already used to regulate the velocity of windmills.<ref>[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.302.5633&rep=rep1&type=pdf Control Theory: History, Mathematical Achievements and Perspectives | E. Fernandez-Cara1 and E. Zuazua]</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. A centrifugal governor was already used to regulate the velocity of windmills. 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 − − 虽然各种类型的控制系统可以追溯到古代，一个更加正式的领域分析开始于离心式调速器的动力学分析，由物理学家詹姆斯·克拉克·麦克斯韦在1868年进行，题为统治者。离心式调速器已经被用来调节风车的速度。麦克斯韦描述并分析了自激振荡现象，其中系统的滞后可能导致系统的过度补偿和不稳定行为。这引起了人们对这个话题的浓厚兴趣，在这期间，Maxwell 的同学，爱德华·约翰·劳思，抽象出了 Maxwell 关于线性系统的一般类别的结果。 1975{ cite book − − | author = Routh, E.J. − − | author = Routh, E.J. − − 作者: 劳斯 e.j。 − − |author2=Fuller, A.T. − − |author2=Fuller, A.T. − − 2 Fuller，a.t. − − | year = 1975 − − | year = 1975 − − 1975年 − − | title = Stability of motion − − | title = Stability of motion − − 运动的稳定性 − − | publisher = Taylor & Francis − − | publisher = Taylor & Francis − − 出版商泰勒弗朗西斯集团 − − | isbn = − − | isbn = − − | isbn − − }}</ref> Independently, [[Adolf Hurwitz]] analyzed system stability using differential equations in 1877, resulting in what is now known as the [[Routh–Hurwitz theorem]].<ref name=Routh1877>{{cite book − − }}</ref> Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the Routh–Hurwitz theorem.<ref name=Routh1877>{{cite book − − 阿道夫 · 赫尔维茨在1877年独立地用微分方程分析了系统的稳定性，导致了现在被称为劳斯-赫尔维茨定理的结果。 1877{ cite book − − | author = Routh, E.J. − − | author = Routh, E.J. − − 作者: 劳斯 e.j。 − − | year = 1877 − − | year = 1877 − − 1877年 − − | title = A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion: Particularly Steady Motion − − | title = A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion: Particularly Steady Motion − − | 题目: 论述给定运动状态的稳定性，特别是稳定运动: 特别是稳定运动 − − | url = https://archive.org/details/atreatiseonstab00routgoog − − | url = https://archive.org/details/atreatiseonstab00routgoog − − Https://archive.org/details/atreatiseonstab00routgoog − − | publisher = Macmillan and co. − − | publisher = Macmillan and co. − − 出版商麦克米伦出版公司。 − − | isbn = − − | isbn = − − | isbn − − }}</ref><ref name=Hurwitz1964>{{cite journal − − }}</ref><ref name=Hurwitz1964>{{cite journal − − 1964{ cite journal − − | author = Hurwitz, A. − − | author = Hurwitz, A. − − 作者: Hurwitz，a。 − − | year = 1964 − − | year = 1964 − − 1964年 − − | title = On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts − − | title = On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts − − 关于方程只有带负实部根的条件 − − | journal = Selected Papers on Mathematical Trends in Control Theory − − | journal = Selected Papers on Mathematical Trends in Control Theory − − 控制理论中的数学趋势论文选 − − }}</ref> − − }}</ref> − − {} / ref − − − − − − A notable application of dynamic control was in the area of manned flight. The [[Wright brothers]] made their first successful test flights on December 17, 1903 and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds. − − A notable application of dynamic control was in the area of manned flight. The Wright brothers made their first successful test flights on December 17, 1903 and were distinguished by their ability to control their flights for substantial periods (more so than the ability to produce lift from an airfoil, which was known). Continuous, reliable control of the airplane was necessary for flights lasting longer than a few seconds. − − 动态控制的一个显著应用是在载人飞行领域。1903年12月17日，莱特兄弟进行了首次成功的试飞，他们的杰出之处在于能够在相当长的时间内控制飞行(比已知的用机翼产生升力的能力还要强)。持续、可靠地控制飞机对于持续时间超过几秒钟的飞行是必要的。 − − − − − − By [[World War II]], control theory was becoming an important area of research. [[Irmgard Flügge-Lotz]] developed the theory of discontinuous automatic control systems, and applied the [[bang–bang control|bang-bang principle]] to the development of [[autopilot|automatic flight control equipment]] for aircraft.<ref>{{cite journal|last1=Flugge-Lotz|first1=Irmgard|last2=Titus|first2=Harold A.|title=Optimum and Quasi-Optimum Control of Third and Fourth-Order Systems|journal=Stanford University Technical Report|date=October 1962|issue=134|pages=8–12|url=http://www.dtic.mil/dtic/tr/fulltext/u2/621137.pdf}}</ref><ref>{{cite book|last1=Hallion|first1=Richard P.|editor1-last=Sicherman|editor1-first=Barbara|editor2-last=Green|editor2-first=Carol Hurd|editor3-last=Kantrov|editor3-first=Ilene|editor4-last=Walker|editor4-first=Harriette|title=Notable American Women: The Modern Period: A Biographical Dictionary|url=https://archive.org/details/notableamericanw00sich|url-access=registration|date=1980|publisher=Belknap Press of Harvard University Press|location=Cambridge, Mass.|isbn=9781849722704|pages=[https://archive.org/details/notableamericanw00sich/page/241 241–242]}}</ref> Other areas of application for discontinuous controls included [[fire-control system]]s, [[guidance system]]s and [[electronics]]. − − By World War II, control theory was becoming an important area of research. Irmgard Flügge-Lotz developed the theory of discontinuous automatic control systems, and applied the bang-bang principle to the development of automatic flight control equipment for aircraft. Other areas of application for discontinuous controls included fire-control systems, guidance systems and electronics. − − 到了第二次世界大战，控制理论成为一个重要的研究领域。Irmgard fl gge-lotz 发展了非连续自动控制系统理论，并将 bang-bang 原理应用于飞机自动飞行控制装置的开发。不连续控制的其他应用领域包括火控系统、制导系统和电子学。 − − − − − − Sometimes, mechanical methods are used to improve the stability of systems. For example, [[Stabilizer (ship)|ship stabilizers]] 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. − − Sometimes, mechanical methods are used to improve the stability of systems. For example, ship stabilizers 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. − − 有时采用机械方法来提高系统的稳定性。例如，船舶稳定器是安装在吃水线以下和横向出现的鳍。在现代船舶中，它们可能是陀螺控制的活动鳍，这种鳍有能力改变攻角，以抵消作用在船上的风浪造成的横摇。 − − − − − − The [[Space Race]] 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 [[internal model (motor control)|internal model]] that obeys the [[Good regulator|good regulator theorem]]. So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of a [[controller (control theory)|regulator]] interacting with a [[plant (control theory)|plant]]. − − The Space Race 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 internal model that obeys the good regulator theorem. So, for example, in economics, the more accurately a (stock or commodities) trading model represents the actions of the market, the more easily it can control that market (and extract "useful work" (profits) from it). In AI, an example might be a chatbot modelling the discourse state of humans: the more accurately it can model the human state (e.g. on a telephone voice-support hotline), the better it can manipulate the human (e.g. into performing the corrective actions to resolve the problem that caused the phone call to the help-line). These last two examples take the narrow historical interpretation of control theory as a set of differential equations modeling and regulating kinetic motion, and broaden it into a vast generalization of a regulator interacting with a plant. − − 太空竞赛也依赖于精确的航天器控制，控制理论在经济学和人工智能等领域的应用也越来越多。在这里，有人可能会说，目标是找到一个内部模型，遵守良好的调节器定理。因此，举例来说，在经济学中，一个(股票或商品)交易模型越准确地代表了市场的行为，它就越容易控制市场(并从中提取“有用的工作”(利润))。在人工智能中，一个例子可能是一个聊天机器人模拟人类的话语状态: 它模拟人类状态越精确(例如:。在电话语音支持热线) ，它可以更好地操纵人类(例如:。进行纠正措施，以解决问题，造成电话呼叫的帮助热线)。最后两个例子将控制理论狭隘的历史解释视为一组微分方程建模和调节动力学运动，并将其扩展为调节器与植物相互作用的广义推广。 − − − − − − ==Open-loop and closed-loop (feedback) control== − − ==Open-loop and closed-loop (feedback) control== − − 开环和闭环(反馈)控制 − − [[File:Feedback loop with descriptions.svg|thumb|right|400px|A [[block diagram]] of a [[negative feedback]] [[control system]] using a [[feedback loop]] to control the process variable by comparing it with a desired value, and applying the difference as an error signal to generate a control output to reduce or eliminate the error.]] − − A [[block diagram of a negative feedback control system using a feedback loop to control the process variable by comparing it with a desired value, and applying the difference as an error signal to generate a control output to reduce or eliminate the error.]] − − 一个[负反馈控制系统的框图，使用反馈回路通过比较期望值来控制程序变数，并将差值作为一个错误信号来产生控制输出，以减少或消除错误] − − [[File:Industrial control loop.jpg|thumb|400px|Example of a single industrial control loop; showing continuously modulated control of process flow.]] − − Example of a single industrial control loop; showing continuously modulated control of process flow. − − 单个工业控制回路的例子; 显示过程流程的连续调制控制。 − − Fundamentally, there are two types of control loops: open loop control and closed loop (feedback) control. − − Fundamentally, there are two types of control loops: open loop control and closed loop (feedback) control. − − 基本上，有两种类型的控制回路: 开环控制和闭环(反馈)控制。 − − − − − − In open loop control, the control action from the controller is independent of the "process output" (or "controlled process variable" - PV). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the timed switching on/off of the boiler, the process variable is the building temperature, but neither is linked. − − In open loop control, the control action from the controller is independent of the "process output" (or "controlled process variable" - PV). A good example of this is a central heating boiler controlled only by a timer, so that heat is applied for a constant time, regardless of the temperature of the building. The control action is the timed switching on/off of the boiler, the process variable is the building temperature, but neither is linked. − − 在开环控制中，来自控制器的控制动作独立于“过程输出”(或“受控程序变数”-PV)。这方面的一个很好的例子是一个集中供热锅炉只由一个定时器控制，这样加热就是一个恒定的时间，不管建筑物的温度如何。控制动作是锅炉的定时开关，程序变数是建筑物的温度，但两者都没有联系。 − − − − − − In closed loop control, the control action from the controller is dependent on feedback from the process in the form of the value of the process variable (PV). In the case of the boiler analogy, a closed loop would include a thermostat to compare the building temperature (PV) with the temperature set on the thermostat (the set point - SP). This generates a controller output to maintain the building at the desired temperature by switching the boiler on and off. A closed loop controller, therefore, has a feedback loop which ensures the controller exerts a control action to manipulate the process variable to be the same as the "Reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers.<ref>"Feedback and control systems" - JJ Di Steffano, AR Stubberud, IJ Williams. Schaums outline series, McGraw-Hill 1967</ref> − − In closed loop control, the control action from the controller is dependent on feedback from the process in the form of the value of the process variable (PV). In the case of the boiler analogy, a closed loop would include a thermostat to compare the building temperature (PV) with the temperature set on the thermostat (the set point - SP). This generates a controller output to maintain the building at the desired temperature by switching the boiler on and off. A closed loop controller, therefore, has a feedback loop which ensures the controller exerts a control action to manipulate the process variable to be the same as the "Reference input" or "set point". For this reason, closed loop controllers are also called feedback controllers. − − 在闭环控制中，来自控制器的控制动作依赖于来自过程的反馈，其形式为程序变数值。在类似于锅炉的情况下，一个闭合回路将包括一个恒温器，用于比较建筑物温度(PV)和恒温器上的温度设置(设置点 -SP)。这会产生一个控制器输出，通过开关锅炉使建筑物保持在所需的温度。一个闭环控制器，因此，有一个反馈回路，确保控制器施加一个控制动作来操纵程序变数，使其与“参考输入”或“设定点”相同。因此，闭环控制器也称为反馈控制器。 − − − − − − The definition of a closed loop control system according to the British Standard Institution 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." <ref>{{cite book|title= The Origins of Feedback Control|last=Mayr|first= Otto| author-link= Otto Mayr |year= 1970 − − The definition of a closed loop control system according to the British Standard Institution 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." <ref>{{cite book|title= The Origins of Feedback Control|last=Mayr|first= Otto| author-link= Otto Mayr |year= 1970 − − 根据英国标准制度，闭环控制系统的定义是“一个具有监控反馈的控制系统，这种反馈产生的偏差信号被用来控制最终控制元件的动作，从而使偏差趋于零。”反馈控制的起源 | 去年5月 | 第一任 Otto | 作者链接 Otto Mayr | 1970年 − − |publisher =The Colonial Press, Inc.|location= Clinton, MA USA|isbn= |pages=}}</ref> − − |publisher =The Colonial Press, Inc.|location= Clinton, MA USA|isbn= |pages=}}</ref> − − | 出版商 The Colonial Press，inc. | location Clinton，MA USA | isbn | pages } / ref − − − − − − Likewise; "A ''Feedback Control System'' 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."<ref>{{cite book|title= The Origins of Feedback Control|last=Mayr|first= Otto| author-link= Otto Mayr |year= 1969|publisher =The Colonial Press, Inc.|location= Clinton, MA USA|isbn= |pages=}}</ref> − − Likewise; "A Feedback Control System 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." − − 同样，“反馈控制系统是一个倾向于通过比较一个系统变量的功能和使用差异作为一种控制手段来维持一个系统变量与另一个系统变量的规定关系的系统。” − − − − − − ===Other examples=== − − ===Other examples=== − − 其他例子 − − An example of a control system is a car's [[cruise control]], which is a device designed to maintain vehicle speed at a constant ''desired'' or ''reference'' speed provided by the driver. The ''controller'' is the cruise control, the ''plant'' is the car, and the ''system'' is the car and the cruise control. The system output is the car's speed, and the control itself is the engine's [[throttle]] position which determines how much power the engine delivers. − − An example of a control system is a car's cruise control, which is a device designed to maintain vehicle speed at a constant desired or reference speed provided by the driver. The controller is the cruise control, the plant is the car, and the system is the car and the cruise control. The system output is the car's speed, and the control itself is the engine's throttle position which determines how much power the engine delivers. − − 控制系统的一个例子是汽车的巡航控制，巡航控制是一种设计来保持汽车速度在司机提供的恒定期望或参考速度。控制器是巡航控制，被控对象是汽车，系统是汽车和巡航控制。系统输出是汽车的速度，控制本身是发动机的油门位置，这决定了发动机提供多少动力。 − − − − − − A primitive way to implement cruise control is simply to lock the throttle position when the driver engages cruise control. However, if the cruise control is engaged on a stretch of non-flat road, then the car will travel slower going uphill and faster when going downhill. This type of controller is called an ''[[open-loop controller]]'' because there is no [[feedback]]; no measurement of the system output (the car's speed) is used to alter the control (the throttle position.) As a result, the controller cannot compensate for changes acting on the car, like a change in the slope of the road. − − A primitive way to implement cruise control is simply to lock the throttle position when the driver engages cruise control. However, if the cruise control is engaged on a stretch of non-flat road, then the car will travel slower going uphill and faster when going downhill. This type of controller is called an open-loop controller because there is no feedback; no measurement of the system output (the car's speed) is used to alter the control (the throttle position.) As a result, the controller cannot compensate for changes acting on the car, like a change in the slope of the road. − − 实现巡航控制的一个原始方法是在驾驶员启动巡航控制时锁定油门位置。不过，如果巡航控制系统在一段不平坦的道路上启动，车辆上山时行驶速度会较慢，下山时则较快。这种类型的控制器被称为开回路控制器控制器，因为没有反馈; 没有测量系统输出(汽车的速度)来改变控制(油门位置)因此，控制器不能补偿作用在汽车上的变化，比如道路坡度的变化。 − − − − − − In a ''[[closed-loop controller|closed-loop control system]]'', data from a sensor monitoring the car's speed (the system output) enters a controller which continuously compares the quantity representing the speed with the reference quantity representing the desired speed. The difference, called the error, determines the throttle position (the control). The result is to match the car's speed to the reference speed (maintain the desired system output). Now, when the car goes uphill, the difference between the input (the sensed speed) and the reference continuously determines the throttle position. As the sensed speed drops below the reference, the difference increases, the throttle opens, and engine power increases, speeding up the vehicle. In this way, the controller dynamically counteracts changes to the car's speed. The central idea of these control systems is the ''feedback loop'', the controller affects the system output, which in turn is measured and fed back to the controller. − − In a closed-loop control system, data from a sensor monitoring the car's speed (the system output) enters a controller which continuously compares the quantity representing the speed with the reference quantity representing the desired speed. The difference, called the error, determines the throttle position (the control). The result is to match the car's speed to the reference speed (maintain the desired system output). Now, when the car goes uphill, the difference between the input (the sensed speed) and the reference continuously determines the throttle position. As the sensed speed drops below the reference, the difference increases, the throttle opens, and engine power increases, speeding up the vehicle. In this way, the controller dynamically counteracts changes to the car's speed. The central idea of these control systems is the feedback loop, the controller affects the system output, which in turn is measured and fed back to the controller. − − 在闭环控制系统中，来自监测汽车速度(系统输出)的传感器的数据进入控制器，控制器连续比较代表速度的量和代表期望速度的参考量。这个差异，称为错误，决定了节气门的位置(控制)。结果是匹配的汽车的速度参考速度(保持所需的系统输出)。现在，当汽车上坡时，输入(感知速度)和参考速度之间的差异不断地决定油门位置。当感觉到的速度低于参考，差异增加，油门打开，发动机功率增加，加速车辆。这样，控制器动态地抵消汽车速度的变化。这些控制系统的中心思想是反馈回路，控制器影响系统的输出，反过来测量并反馈给控制器。 − − − − − − ==Classical control theory== − − ==Classical control theory== − − 经典控制理论 − − {{main|Classical control theory}} − − − − − − − − To overcome the limitations of the [[open-loop controller]], control theory introduces [[feedback]]. − − To overcome the limitations of the open-loop controller, control theory introduces feedback. − − 为了克服开回路控制器的局限性，控制理论引入了反馈。 − − A [[closed-loop controller]] uses feedback to control [[state (controls)|states]] or [[Negative feedback#Overview|outputs]] of a [[dynamical system]]. Its name comes from the information path in the system: process inputs (e.g., [[voltage]] applied to an [[electric motor]]) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with [[sensor]]s and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop. − − A closed-loop controller uses feedback to control states or outputs of a dynamical system. Its name comes from the information path in the system: process inputs (e.g., voltage applied to an electric motor) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop. − − 闭环控制器利用反馈来控制动力系统的状态或输出。它的名字来源于系统中的信息路径: 过程输入(例如，电动机的电压)对过程输出(例如，电动机的速度或扭矩)有影响，用传感器测量并由控制器处理; 结果(控制信号)作为过程的输入被“反馈” ，关闭循环。 − − − − − − Closed-loop controllers have the following advantages over [[open-loop controller]]s: − − Closed-loop controllers have the following advantages over open-loop controllers: − − 闭环控制器比开环控制器有以下优点: − − * disturbance rejection (such as hills in the cruise control example above) − − − − * guaranteed performance even with [[mathematical model|model]] uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact − − − − * [[instability|unstable]] processes can be stabilized − − − − * reduced sensitivity to parameter variations − − − − * improved reference tracking performance − − − − − − − − In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed [[feed forward (control)|feedforward]] and serves to further improve reference tracking performance. − − In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed feedforward and serves to further improve reference tracking performance. − − 在一些系统中，闭环控制和开环控制同时使用。在这样的系统中，开环控制被称为前馈，用于进一步改善参考跟踪性能。 − − − − − − A common closed-loop controller architecture is the [[PID controller]]. − − A common closed-loop controller architecture is the PID controller. − − 常用的闭环控制器结构是 PID 控制器。 − − − − − − ==Closed-loop transfer function== − − ==Closed-loop transfer function== − − 闭回路传递函数 − − {{details|closed-loop transfer function}} − − − − The output of the system ''y(t)'' is fed back through a sensor measurement ''F'' to a comparison with the reference value ''r(t)''. The controller ''C'' then takes the error ''e'' (difference) between the reference and the output to change the inputs ''u'' to the system under control ''P''. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller. − − The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller. − − 系统的输出 y (t)通过传感器测量 f 反馈给参考值 r (t)进行比较。然后，控制器 c 利用参考和输出之间的误差 e (差)来改变输入 u 到控制 p 下的系统。 如图所示。这种控制器是一种闭环控制器或反馈控制器。 − − − − − − This is called a single-input-single-output (''SISO'') control system; ''MIMO'' (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through [[coordinate vector|vectors]] instead of simple [[scalar (mathematics)|scalar]] values. For some [[distributed parameter systems]] the vectors may be infinite-[[Dimension (vector space)|dimensional]] (typically functions). − − This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions). − − 这被称为单输入单输出(SISO)控制系统; MIMO (即多输入多输出)系统，具有多个输入 / 输出，是常见的。在这种情况下，变量通过向量表示，而不是简单的标量值。对于一些分布参数系统，向量可能是无限维的(典型的函数)。 − − − − − − [[File:simple feedback control loop2.svg|center|A simple feedback control loop]] − − A simple feedback control loop − − 一个简单的反馈控制回路 − − − − − − If we assume the controller ''C'', the plant ''P'', and the sensor ''F'' are [[linear]] and [[time-invariant]] (i.e., elements of their [[transfer function]] ''C(s)'', ''P(s)'', and ''F(s)'' do not depend on time), the systems above can be analysed using the [[Laplace transform]] on the variables. This gives the following relations: − − If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations: − − 如果我们假设控制器 c、被控对象 p 和传感器 f 是线性和时不变的(即，它们的传递函数 c (s)、 p (s)和 f (s)的元素不依赖于时间) ，那么上述系统可以用变量的拉普拉斯变换来分析。这就产生了以下关系: − − − − − − : <math>Y(s) = P(s) U(s)</math> − − <math>Y(s) = P(s) U(s)</math> − − 数学 y (s) p (s) u / 数学 − − : <math>U(s) = C(s) E(s)</math> − − <math>U(s) = C(s) E(s)</math> − − 数学 c (s) e (s) / 数学 − − : <math>E(s) = R(s) - F(s)Y(s).</math> − − <math>E(s) = R(s) - F(s)Y(s).</math> − − 数学 e (s) r (s)-f (s) y (s) / 数学 − − − − − − Solving for ''Y''(''s'') in terms of ''R''(''s'') gives − − Solving for Y(s) in terms of R(s) gives − − 用 r (s)解 y (s)给出 − − − − − − : <math>Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).</math> − − <math>Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).</math> − − 数学 y (s)左( frac { p (s) c (s)}{1 + p (s) c (s) f (s)}右) r (s) h (s) r (s) . / math − − − − − − The expression <math>H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}</math> is referred to as the ''closed-loop transfer function'' of the system. The numerator is the forward (open-loop) gain from ''r'' to ''y'', and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If <math>|P(s)C(s)| \gg 1</math>, i.e., it has a large [[norm (mathematics)|norm]] with each value of ''s'', and if <math>|F(s)| \approx 1</math>, then ''Y(s)'' is approximately equal to ''R(s)'' and the output closely tracks the reference input. − − The expression <math>H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}</math> is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If <math>|P(s)C(s)| \gg 1</math>, i.e., it has a large norm with each value of s, and if <math>|F(s)| \approx 1</math>, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. − − 表达式 math h (s) frac { p (s) c (s)}{1 + f (s) p (s) c (s)} / math 称为系统的闭回路传递函数。分子是从 r 到 y 的正向(开环)增益，分母是1加上反馈环路中的增益，即所谓的环路增益。如果 math | p (s) c (s) | gg 1 / math，也就是说，它有一个很大的范数，每个值都是 s，如果 math | f (s) | 大约1 / math，那么 y (s)大约等于 r (s) ，输出密切跟踪参考输入。 − − − − − − ==PID feedback control== − − ==PID feedback control== − − Pid 反馈控制 − − {{main |PID Controller}} − − − − [[File:PID en.svg|right|thumb|400x400px|A [[block diagram]] of a PID controller in a feedback loop, r(''t'') is the desired process value or "set point", and y(''t'') is the measured process value.]] − − A [[block diagram of a PID controller in a feedback loop, r(t) is the desired process value or "set point", and y(t) is the measured process value.]] − − [反馈回路中 PID 控制器的框图，r (t)是期望的过程值或“设定点” ，y (t)是测量的过程值] − − A '''proportional–integral–derivative controller ''' ('''PID controller''') is a [[control loop]] [[feedback mechanism]] control technique widely used in control systems. − − A proportional–integral–derivative controller (PID controller) is a control loop feedback mechanism control technique widely used in control systems. − − 比例-积分-微分控制器(PID 控制器)是一种广泛应用于控制系统的控制回路反馈机制控制技术。 − − − − − − A PID controller continuously calculates an ''error value'' <math>e(t)</math> as the difference between a desired [[Setpoint (control system)|setpoint]] and a measured [[process variable]] and applies a correction based on [[Proportional control|proportional]], [[integral]], and [[derivative]] terms. ''PID'' is an initialism for ''Proportional-Integral-Derivative'', referring to the three terms operating on the error signal to produce a control signal. − − A PID controller continuously calculates an error value <math>e(t)</math> as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms. PID is an initialism for Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control signal. − − 一个 PID 控制器连续计算一个误差值 math e (t) / math 作为期望设定点和被测程序变数之间的差值，并根据比例、积分和微分项进行修正。Pid 是比例积分微分的一种初始化方法，指的是对误差信号进行三项操作来产生控制信号。 − − − − − − The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and latterly in industrial process computers. − − The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and latterly in industrial process computers. − − 理论的理解和应用可以追溯到20世纪20年代，它们在几乎所有的模拟控制系统中得到实现; 最初在机械控制器中，后来在工业过程计算机中使用离散电子学。 − − The [[PID controller]] is probably the most-used feedback control design. − − The PID controller is probably the most-used feedback control design. − − Pid 控制器可能是最常用的反馈控制设计。 − − − − − − If ''u(t)'' is the control signal sent to the system, ''y(t)'' is the measured output and ''r(t)'' is the desired output, and <math>e(t)=r(t)- y(t)</math> is the tracking error, a PID controller has the general form − − If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and <math>e(t)=r(t)- y(t)</math> is the tracking error, a PID controller has the general form − − 如果 u (t)是发送到系统的控制信号，y (t)是测量输出，r (t)是期望输出，math e (t) r (t)-y (t) / math 是跟踪误差，则 PID 控制器具有通用形式 − − − − − − :<math>u(t) = K_P e(t) + K_I \int e(\tau)\text{d}\tau + K_D \frac{\text{d}e(t)}{\text{d}t}.</math> − − <math>u(t) = K_P e(t) + K_I \int e(\tau)\text{d}\tau + K_D \frac{\text{d}e(t)}{\text{d}t}.</math> − − Math u (t) k p e (t) + k i int e ( tau) text { d } tau + k d frac { text { d } e (t)}{ text { d } t } / math − − − − − − The desired closed loop dynamics is obtained by adjusting the three parameters <math> K_P</math>, <math> K_I</math> and <math> K_D</math>, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in [[process control]]). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if [[MIMO]] systems are considered. − − The desired closed loop dynamics is obtained by adjusting the three parameters <math> K_P</math>, <math> K_I</math> and <math> K_D</math>, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in process control). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered. − − 期望闭环动力学是通过调整三个参数数学 k p / math、数学 k i / math 和数学 k d / math 得到的，通常是通过“调整”迭代得到的，不需要具体的对象模型知识。稳定性往往可以确保只使用比例的条款。积分项允许抑制阶跃扰动(在过程控制中通常是一个引人注目的规范)。导数项用于提供响应的阻尼或整形。Pid 控制器是最成熟的一类控制系统: 然而，他们不能用于几个更复杂的情况，特别是如果 MIMO 系统考虑。 − − − − − − Applying [[Laplace transform|Laplace transformation]] results in the transformed PID controller equation − − Applying Laplace transformation results in the transformed PID controller equation − − 应用拉普拉斯变换得到变换后的 PID 控制器方程 − − − − − − :<math>u(s) = K_P e(s) + K_I \frac{1}{s} e(s) + K_D s e(s)</math> − − <math>u(s) = K_P e(s) + K_I \frac{1}{s} e(s) + K_D s e(s)</math> − − 数学 u (s) k p e (s) + k i frac {1}{ s } e (s) + k d s e (s) / math − − :<math>u(s) = \left(K_P + K_I \frac{1}{s} + K_D s\right) e(s)</math> − − <math>u(s) = \left(K_P + K_I \frac{1}{s} + K_D s\right) e(s)</math> − − 数学 u (s)左(k p + k i frac {1}{ s } + k d s 右) e (s) / math − − − − − − with the PID controller transfer function − − with the PID controller transfer function − − 具有 PID 控制器的传递函数 − − :<math>C(s) = \left(K_P + K_I \frac{1}{s} + K_D s\right).</math> − − <math>C(s) = \left(K_P + K_I \frac{1}{s} + K_D s\right).</math> − − 数学 c (s)左(k p + k i frac {1}{ s } + k d s 右) . / math − − − − − − As an example of tuning a PID controller in the closed-loop system <math>H(s)</math>, consider a 1st order plant given by − − As an example of tuning a PID controller in the closed-loop system <math>H(s)</math>, consider a 1st order plant given by − − 作为闭环系统数学 h (s) / math 中 PID 控制器整定的一个例子，考虑一个一阶被控对象 − − − − − − :<math>P(s) = \frac{A}{1 + sT_P}</math> − − <math>P(s) = \frac{A}{1 + sT_P}</math> − − 数学 p (s) frac { a }{1 + sT p } / math − − − − − − where <math>A</math> and <math>T_P</math> are some constants. The plant output is fed back through − − where <math>A</math> and <math>T_P</math> are some constants. The plant output is fed back through − − 其中数学 a / 数学和数学 t p / 数学是一些常数。植物的产量通过 − − − − − − :<math>F(s) = \frac{1}{1 + sT_F}</math> − − <math>F(s) = \frac{1}{1 + sT_F}</math> − − 数学 f (s) frac {1}{1 + sT f } / math − − − − − − where <math>T_F</math> is also a constant. Now if we set <math>K_P=K\left(1+\frac{T_D}{T_I}\right)</math>, <math>K_D=KT_D</math>, and <math>K_I=\frac{K}{T_I}</math>, we can express the PID controller transfer function in series form as − − where <math>T_F</math> is also a constant. Now if we set <math>K_P=K\left(1+\frac{T_D}{T_I}\right)</math>, <math>K_D=KT_D</math>, and <math>K_I=\frac{K}{T_I}</math>, we can express the PID controller transfer function in series form as − − 数学 t / 数学也是一个常数。现在如果我们设置数学 kpk 左(1 + + frac { t }{ t i }右) / math，math k d KT d / math，和 math k i frac { t i } / math，我们可以将 PID 控制器传递函数表示成串行形式如下 − − − − − − :<math>C(s) = K \left(1 + \frac{1}{sT_I}\right)(1 + sT_D)</math> − − <math>C(s) = K \left(1 + \frac{1}{sT_I}\right)(1 + sT_D)</math> − − 数学 c (s) k 左(1 + frac {1}{ sT i }右)(1 + sT d) / 数学 − − − − − − Plugging <math>P(s)</math>, <math>F(s)</math>, and <math>C(s)</math> into the closed-loop transfer function <math>H(s)</math>, we find that by setting − − Plugging <math>P(s)</math>, <math>F(s)</math>, and <math>C(s)</math> into the closed-loop transfer function <math>H(s)</math>, we find that by setting − − 把数学 p (s) / 数学，数学 f (s) / 数学，数学 c (s) / 数学输入到闭回路传递函数数学 h (s) / 数学中，我们通过设置 − − − − − − :<math>K = \frac{1}{A}, T_I = T_F, T_D = T_P</math> − − <math>K = \frac{1}{A}, T_I = T_F, T_D = T_P</math> − − 数学 k frac {1}{ a } ，t i t f，t d t p / math − − − − − − <math>H(s) = 1</math>. With this tuning in this example, the system output follows the reference input exactly. − − <math>H(s) = 1</math>. With this tuning in this example, the system output follows the reference input exactly. − − 数学 h (s)1 / 数学。通过本例中的这个调优，系统输出精确地跟随参考输入。 − − − − − − However, in practice, a pure differentiator is neither physically realizable nor desirable<ref>{{cite journal |last1=Ang |first1=K.H. |last2=Chong |first2=G.C.Y. |last3=Li |first3=Y. |date=2005 |title=PID control system analysis, design, and technology |journal=IEEE Transactions on Control Systems Technology |volume=13 |issue=4 |pages=559–576|doi=10.1109/TCST.2005.847331 }}</ref> due to amplification of noise and resonant modes in the system. Therefore, a [[Lead–lag compensator|phase-lead compensator]] type approach or a differentiator with low-pass roll-off are used instead. − − However, in practice, a pure differentiator is neither physically realizable nor desirable due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead. − − 然而，在实践中，由于噪声和谐振模式的放大，纯微分器既不是物理上可实现的，也不是理想的。因此，相位超前补偿器类型的方法或微分器与低通滚降被用来代替。 − − − − − − ==Linear and nonlinear control theory== − − ==Linear and nonlinear control theory== − − 线性和非线性控制 − − The field of control theory can be divided into two branches: − − The field of control theory can be divided into two branches: − − 控制理论可分为两个分支: − − * ''[[Linear control theory]]'' – This applies to systems made of devices which obey the [[superposition principle]], which means roughly that the output is proportional to the input. They are governed by [[linear equation|linear]] [[differential equation]]s. A major subclass is systems which in addition have parameters which do not change with time, called ''[[linear time invariant]]'' (LTI) systems. These systems are amenable to powerful [[frequency domain]] mathematical techniques of great generality, such as the [[Laplace transform]], [[Fourier transform]], [[Z transform]], [[Bode plot]], [[root locus]], and [[Nyquist stability criterion]]. These lead to a description of the system using terms like [[Bandwidth (signal processing)|bandwidth]], [[frequency response]], [[eigenvalue]]s, [[gain (electronics)|gain]], [[resonant frequency|resonant frequencies]], [[zeros and poles]], which give solutions for system response and design techniques for most systems of interest. − − − − * ''[[Nonlinear control theory]]'' – This covers a wider class of systems that do not obey the superposition principle, and applies to more real-world systems because all real control systems are nonlinear. These systems are often governed by [[nonlinear differential equation]]s. 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 [[limit cycle]] theory, [[Poincaré map]]s, [[Lyapunov function|Lyapunov stability theorem]], and [[describing function]]s. Nonlinear systems are often analyzed using [[numerical method]]s on computers, for example by [[simulation|simulating]] their operation using a [[simulation language]]. If only solutions near a stable point are of interest, nonlinear systems can often be [[linearization|linearized]] by approximating them by a linear system using [[perturbation theory]], and linear techniques can be used.<ref>[http://www.mathworks.com/help/toolbox/simulink/slref/trim.html trim point]</ref> − − − − − − − − ==Analysis techniques - frequency domain and time domain== − − ==Analysis techniques - frequency domain and time domain== − − 分析技术. 频域和时域 − − Mathematical techniques for analyzing and designing control systems fall into two different categories: − − Mathematical techniques for analyzing and designing control systems fall into two different categories: − − 分析和设计控制系统的数学技术可分为两类: − − * ''[[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. − − − − 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, 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|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 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 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. − − 与经典控制理论的频域分析不同，现代控制理论利用时域状态空间，一个物理系统的数学模型，作为一组输入、输出和状态变量，由一阶微分方程相关联。为了从输入、输出和状态的数量中抽象出来，变量被表示为向量，微分方程和代数方程被写成矩阵形式(后者只有当动力系统是线性的时候才可能)。状态空间模型(也称为“时域方法”)提供了一种方便而紧凑的方式来建模和分析具有多个输入和输出的系统。对于输入和输出，否则我们将不得不写下拉普拉斯转换来对系统的所有信息进行编码。与频域方法不同，状态空间表示的使用并不局限于具有线性分量和零初始条件的系统。“状态空间”是指其轴为状态变量的空间。系统的状态可以表示为该空间中的一个点。 − − − − − − ==System interfacing - SISO & MIMO== − − ==System interfacing - SISO & MIMO== − − 系统接口 -SISO & MIMO − − Control systems can be divided into different categories depending on the number of inputs and outputs. − − − − 根据输入和输出的数量，控制系统可以分为不同的类别。 − − * [[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 system]], in which the control input is the input audio signal and the output is the sound waves from the speaker. − − − − * [[MIMO|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. − − − − − − − − ==Topics in control theory== − − ==Topics in control theory== − − 控制理论主题 − − − − − − ===Stability=== − − ===Stability=== − − 稳定性 − − − − − − The ''stability'' of a general [[dynamical system]] with no input can be described with [[Lyapunov stability]] criteria. − − The stability of a general dynamical system with no input can be described with Lyapunov stability criteria. − − 一个没有输入的一般动力系统的稳定性可以用李雅普诺夫稳定性标准来描述。 − − *A [[linear system]] is called [[BIBO stability|bounded-input bounded-output (BIBO) stable]] if its output will stay [[bounded function|bounded]] for any bounded input. − − − − *Stability for [[nonlinear system]]s that take an input is [[input-to-state stability]] (ISS), which combines Lyapunov stability and a notion similar to BIBO stability. − − − − − − − − For simplicity, the following descriptions focus on continuous-time and discrete-time '''linear systems'''. − − For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems. − − 为简单起见，下面的描述集中于连续时间和离散时间线性系统。 − − − − − − Mathematically, this means that for a causal linear system to be stable all of the [[Pole (complex analysis)|poles]] of its [[transfer function]] 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 − − Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function 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 − − 在数学上，这意味着一个因果线性系统要稳定，其传递函数的所有极点都必须有负实值，即。每个极点的真实部分必须小于零。实际上，稳定性要求存在传递函数复极点 − − * in the open left half of the [[complex plane]] for continuous time, when the [[Laplace transform]] is used to obtain the transfer function. − − − − * inside the [[unit circle]] for discrete time, when the [[Z-transform]] is used. − − − − The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions. The continuous Laplace transform is in [[Cartesian coordinates]] where the <math>x</math> axis is the real axis and the discrete Z-transform is in [[circular coordinates]] where the <math>\rho</math> axis is the real axis. − − 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 Cartesian coordinates where the <math>x</math> axis is the real axis and the discrete Z-transform is in circular coordinates where the <math>\rho</math> axis is the real axis. − − 这两种情况之间的区别仅仅是由于传统的绘制连续时间和离散时间传递函数的方法。连续的拉普拉斯变换是在笛卡尔坐标系中，其中数学 x / math 轴是实轴，离散 z 变换是在圆坐标系中，其中数学 rho / math 轴是实轴。 − − − − − − When the appropriate conditions above are satisfied a system is said to be [[asymptotic stability|asymptotically stable]]; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is [[marginal stability|marginally stable]]; in this case the system transfer function has non-repeated poles at the complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero. − − When the appropriate conditions above are satisfied a system is said to be asymptotically stable; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable; in this case the system transfer function has non-repeated poles at 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. − − 当满足上述适当条件时，系统被称为渐近稳定的，渐近稳定控制系统的变量总是从它们的初始值减少，并且不表现永久的振荡。当极点的实部精确等于零(在连续时间情况下)或模量等于一(在离散时间情况下)时，就会发生永久振荡。如果一个简单稳定的系统响应既不随时间衰减也不随时间增长，并且没有振荡，那么它是边际稳定的; 在这种情况下，系统传递函数在复平面原点处具有非重复极点(即:。在连续时间情况下，它们的实数和复数分量为零。当实部等于零的极点的虚部不等于零时，就存在振荡。 − − − − − − If a system in question has an [[impulse response]] of − − If a system in question has an impulse response of − − 如果一个系统的脉冲响应为 − − − − − − :<math>\ x[n] = 0.5^n u[n]</math> − − <math>\ x[n] = 0.5^n u[n]</math> − − 数学[ n ]0.5 ^ u [ n ] / math − − − − − − then the Z-transform (see [[Z-transform#Example 2 (causal ROC)|this example]]), is given by − − then the Z-transform (see this example), is given by − − 然后 z 变换(参见本例)由 − − − − − − :<math>\ X(z) = \frac{1}{1 - 0.5z^{-1}}</math> − − <math>\ X(z) = \frac{1}{1 - 0.5z^{-1}}</math> − − 数学 x (z) frac {1}{1-0.5 z ^ {1} / math − − − − − − which has a pole in <math>z = 0.5</math> (zero [[imaginary number|imaginary part]]). This system is BIBO (asymptotically) stable since the pole is ''inside'' the unit circle. − − which has a pole in <math>z = 0.5</math> (zero imaginary part). This system is BIBO (asymptotically) stable since the pole is inside the unit circle. − − 在数学 z0.5 / math (零虚部)中有一个极点。这个系统是 BIBO (渐近)稳定的，因为极点在单位圆内。 − − − − − − However, if the impulse response was − − However, if the impulse response was − − 然而，如果冲动反应是 − − − − − − :<math>\ x[n] = 1.5^n u[n]</math> − − <math>\ x[n] = 1.5^n u[n]</math> − − 数学[ n ]1.5 ^ u [ n ] − − − − − − then the Z-transform is − − then the Z-transform is − − 那么 z 变换是 − − − − − − :<math>\ X(z) = \frac{1}{1 - 1.5z^{-1}}</math> − − <math>\ X(z) = \frac{1}{1 - 1.5z^{-1}}</math> − − 1-1.5 z ^ {-1} / math − − − − − − which has a pole at <math>z = 1.5</math> and is not BIBO stable since the pole has a modulus strictly greater than one. − − which has a pole at <math>z = 1.5</math> and is not BIBO stable since the pole has a modulus strictly greater than one. − − 它的极点在数学 z1.5 / math 上，不是 BIBO 稳定的，因为极点的模严格大于1。 − − − − − − Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the [[root locus]], [[Bode plot]]s or the [[Nyquist plot]]s. − − Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the root locus, Bode plots or the Nyquist plots. − − 有许多工具可用来分析系统的极点。这些包括图形系统，如根轨迹，波德图或奈奎斯特图。 − − − − − − Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use [[Ship stability#Stabilizer fins|antiroll fins]] 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. − − Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use antiroll fins 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. − − 机械的改变可以使设备(和控制系统)更加稳定。水手们增加压舱物以改善船只的稳定性。游轮使用从船侧横向延伸约30英尺(10米)的防侧翻鳍，并不断地绕轴旋转，以产生阻止侧翻的力。 − − − − − − ===Controllability and observability=== − − ===Controllability and observability=== − − 可控性和可观察性 − − {{Main| Controllability|Observability}} − − − − − − − − [[Controllability]] and [[observability]] are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed ''stabilizable''. Observability instead is related to the possibility of ''observing'', through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the 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. − − Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed stabilizable. Observability instead is related to the possibility of observing, through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the 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. − − 能控性和能观性是系统分析中的主要问题，它直接影响到系统的可控性和稳定性。可控性与使用适当的控制信号使系统进入特定状态的可能性有关。如果一个状态是不可控的，那么没有信号将永远能够控制该状态。如果一个状态是不可控的，但它的动力学是稳定的，那么这个状态称为可稳定的。相反，可观测性与通过输出测量观测系统状态的可能性有关。如果一个状态不可观测，控制器将永远不能确定一个不可观测状态的行为，因此不能使用它来稳定系统。然而，类似于上面的稳定性条件，如果一个状态不能被观察到，它仍然可能是可检测的。 − − − − − − 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 [[eigenvalues]] of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis. − − 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 eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis. − − 从几何的角度来看，看待控制系统的每个变量的状态，这些变量的每个“坏”状态必须是可控和可观测的，以确保在闭环系统中的良好行为。也就是说，如果系统的特征值之一不是可控和可观测的，这部分动态将保持不变的闭环系统。如果这样一个特征值不稳定，这个特征值的动力学将出现在闭环系统中，因此将是不稳定的。在状态空间表示的传递函数实现中，不可观测的极点并不存在，这就是为什么在动力系统分析中有时倾向于后者的原因。 − − − − − − Solutions to problems of an uncontrollable or unobservable system include adding actuators and sensors. − − Solutions to problems of an uncontrollable or unobservable system include adding actuators and sensors. − − 解决不可控或不可观测系统问题的方法包括增加执行器和传感器。 − − − − − − ===Control specification=== − − ===Control specification=== − − 控制规格 − − Several different control strategies have been devised in the past years. These vary from extremely general ones ([[PID controller]]), to others devoted to very particular classes of systems (especially [[robotics]] or aircraft cruise control). − − Several different control strategies have been devised in the past years. These vary from extremely general ones (PID controller), to others devoted to very particular classes of systems (especially robotics or aircraft cruise control). − − 在过去的几年里，人们设计了几种不同的控制策略。这些不同从非常一般的(PID 控制器) ，其他致力于非常特殊类型的系统(特别是机器人或飞机巡航控制)。 − − − − − − A control problem can have several specifications. Stability, of course, is always present. The controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have <math>Re[\lambda] < -\overline{\lambda}</math>, where <math>\overline{\lambda}</math> is a fixed value strictly greater than zero, instead of simply asking that <math>Re[\lambda]<0</math>. − − A control problem can have several specifications. Stability, of course, is always present. The controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have <math>Re[\lambda] < -\overline{\lambda}</math>, where <math>\overline{\lambda}</math> is a fixed value strictly greater than zero, instead of simply asking that <math>Re[\lambda]<0</math>. − − 一个控制问题可以有几个规范。当然，稳定总是存在的。控制器必须确保闭环系统是稳定的，而不管开环的稳定性。控制器的选择不当甚至会恶化开环系统的稳定性，这是通常必须避免的。有时需要在闭环中获得特定的动力学:。极点有数学 Re [ lambda ]- overline { lambda } / math，其中 math overline { lambda } / math 是一个严格大于零的固定值，而不是简单地问数学 Re [ lambda ]0 / math。 − − − − − − Another typical specification is the rejection of a step disturbance; including an [[integrator]] in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included. − − Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included. − − 另一个典型的特性是抑制阶跃扰动，包括开环链中的积分器(即。直接在系统控制之前)很容易做到这一点。其他类型的扰动需要包括不同类型的子系统。 − − − − − − Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the [[rise time]] (the time needed by the control system to reach the desired value after a perturbation), peak [[overshoot (signal)|overshoot]] (the highest value reached by the response before reaching the desired value) and others ([[settling time]], quarter-decay). Frequency domain specifications are usually related to [[robust control|robustness]] (see after). − − Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). Frequency domain specifications are usually related to robustness (see after). − − 其他“经典”控制理论规范考虑了闭环系统的时间响应。其中包括上升时间(控制系统在扰动后达到期望值所需的时间)、峰值超调(响应达到期望值之前达到的最高值)和其他(稳定时间、四分之一衰减)。频域规范通常与健壮性有关(见后)。 − − − − − − Modern performance assessments use some variation of integrated tracking error (IAE, ISA, CQI). − − Modern performance assessments use some variation of integrated tracking error (IAE, ISA, CQI). − − 现代的性能评估使用一些综合跟踪误差的变体(IAE，ISA，CQI)。 − − − − − − ===Model identification and robustness=== − − ===Model identification and robustness=== − − 模型辨识与鲁棒性 − − A control system must always have some robustness property. A [[robust control]]ler 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 simpler mathematical model is chosen in order to simplify calculations, otherwise, the true system dynamics can be so complicated that a complete model is impossible. − − A control system must always have some robustness property. A robust controller is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This requirement is important, as no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise, the true system dynamics can be so complicated that a complete model is impossible. − − 控制系统必须具有一定的鲁棒性。一个鲁棒控制器是这样的，其性质不会改变很多，如果应用到一个系统略有不同的数学之一用于其综合。这一要求很重要，因为没有任何一个真正的物理系统能够像用于数学表示的一系列微分方程那样真正地表现出来。为了简化计算，通常会选择一个更简单的数学模型，否则，真正的系统动力学可能会非常复杂，以至于不可能建立一个完整的模型。 − − − − − − ;System identification − − System identification − − 系统辨识 − − {{details|System identification}} − − − − The process of determining the equations that govern the model's dynamics is called [[system identification]]. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its [[transfer function]] or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations, for example, in the case of a [[Damping#Example: mass–spring–damper|mass-spring-damper]] system we know that <math> m \ddot{{x}}(t) = - K x(t) - \Beta \dot{x}(t)</math>. 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. − − The process of determining the equations that govern the model's dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations, for example, in the case of a mass-spring-damper system we know that <math> m \ddot(t) = - K x(t) - \Beta \dot{x}(t)</math>. 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. − − 确定支配模型动力学的方程式的过程被称为系统辨识。这可以离线完成: 例如，执行一系列的措施，从中计算一个近似的数学模型，通常是它的传递函数或矩阵。然而，这种对输出的识别不能考虑不可观测的动态性。有时候模型直接从已知的物理方程出发建立，例如，在质量-弹簧-阻尼系统的情况下，我们知道 math m-dot (t)-kx (t)- Beta-dot { x }(t) / math。即使假设在设计控制器时使用”完整”模型，这些方程中包括的所有参数(称为”名义参数”)也从来不是绝对精确的; 即使控制系统连接到远离名义参数值的真实参数值的物理系统，也必须正确地工作。 − − − − − − Some advanced control techniques include an "on-line" identification process (see later). The parameters of the model are calculated ("identified") while the controller itself is running. In this way, if a drastic variation of the parameters ensues, for example, if the robot's arm releases a weight, the controller will adjust itself consequently in order to ensure the correct performance. − − Some advanced control techniques include an "on-line" identification process (see later). The parameters of the model are calculated ("identified") while the controller itself is running. In this way, if a drastic variation of the parameters ensues, for example, if the robot's arm releases a weight, the controller will adjust itself consequently in order to ensure the correct performance. − − 一些先进的控制技术包括“在线”识别过程(见后文)。当控制器本身运行时，模型的参数被计算(“辨识”)。通过这种方式，如果参数发生剧烈变化，例如，如果机器人的手臂释放一个重量，控制器将自我调整，以确保正确的性能。 − − − − − − ;Analysis − − Analysis − − 分析 − − 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 [[Nyquist diagram|Nyquist]] and [[Bode diagram]]s. Topics include [[Bode plot#Gain margin and phase margin|gain and phase margin]] 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 his attention to a control technique by including them in its properties. − − 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 Nyquist and Bode diagrams. Topics include gain and phase margin 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 his attention to a control technique by including them in its properties. − − 考虑系统的传递函数，利用奈奎斯特图和伯德图，可以在频域上对单输入单输出控制系统的鲁棒性进行分析。主题包括增益和相位裕度和振幅裕度。对于 MIMO (多输入多输出)和一般更复杂的控制系统，必须考虑为每种控制技术设计的理论结果(见下一节)。也就是说，如果需要特定的鲁棒性质，工程师必须将注意力转移到控制技术上，将其包含在控制技术的属性中。 − − − − − − ;Constraints − − Constraints − − 约束 − − A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system, for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even damage or break actuators or other subsystems. Specific control techniques are available to solve the problem: [[model predictive control]] (see later), and [[anti-wind up system (control)|anti-wind up systems]]. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold. − − A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system, for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even damage or break actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control (see later), and anti-wind up systems. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold. − − 一个特别的鲁棒性问题是控制系统在存在输入和状态约束的情况下正确执行的要求。在物理世界中，每个信号都是有限的。控制器可能会发出物理系统无法跟踪的控制信号，例如，试图以过高的速度旋转阀门。这可能导致闭环系统的不良行为，甚至损坏或破坏执行机构或其他子系统。特定的控制技术可以解决这个问题: 模型预估计控制控制系统(见后面) ，和反上风系统。后者包括一个额外的控制块，确保控制信号永远不超过给定的阈值。 − − − − − − ==System classifications== − − ==System classifications== − − 系统分类 − − − − − − ===Linear systems control=== − − ===Linear systems control=== − − 线性系统控制 − − {{Main|State space (controls)}} − − − − For MIMO systems, pole placement can be performed mathematically using a [[State space (controls)|state space representation]] of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design. − − For MIMO systems, pole placement can be performed mathematically using a state space representation of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design. − − 对于多输入多输出系统，极点配置可以通过使用开环系统的状态空间并计算反馈矩阵在期望位置的极点配置来实现。在复杂系统中，这可能需要计算机辅助计算能力，并且不能总是确保健壮性。此外，所有的系统状态一般不能测量，因此观测器必须包括在极点配置设计中。 − − − − − − ===Nonlinear systems control=== − − ===Nonlinear systems control=== − − 非线性系统控制 − − {{Main|Nonlinear control}} − − − − Processes in industries like [[robotics]] and the [[aerospace industry]] 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., [[feedback linearization]], [[backstepping]], [[sliding mode control]], trajectory linearization control normally take advantage of results based on [[Lyapunov's theory]]. [[Differential geometry]] has been widely used as a tool for generalizing well-known linear control concepts to the non-linear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states.<ref name=Shi_Gu_et_al>{{cite journal |author1 = Gu Shi|year = 2015 |title = Controllability of structural brain networks (Article Number 8414) |journal = Nature Communications |volume = 6 |quote = 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. |lay-url = http://www.nature.com/ncomms/2015/151001/ncomms9414/full/ncomms9414.html |doi = 10.1038/ncomms9414|issue = 6 |display-authors=etal|arxiv = 1406.5197|bibcode = 2015NatCo...6E8414G |pmid = 26423222 |pmc = 4600713 |page = 8414}}</ref> − − Processes in industries like robotics and the aerospace industry 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., feedback linearization, backstepping, sliding mode control, trajectory linearization control normally take advantage of results based on Lyapunov's theory. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the non-linear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states. − − 机器人技术和航空航天工业等行业的流程通常具有强烈的非线性动力学。在控制理论中，有时可以将这类系统线性化并应用线性技术，但在许多情况下，有必要从头设计允许控制非线性系统的理论。这些控制方法，例如，回授线性化控制、反推控制、滑动模式控制控制、轨迹线性化控制通常利用了基于李亚普诺夫理论的结果。微分几何已经被广泛地用作一种工具，将众所周知的线性控制概念推广到非线性情况，以及展示使它成为一个更具挑战性的问题的微妙之处。控制理论也被用来解释指导认知状态的神经机制。 − − − − − − ===Decentralized systems control=== − − ===Decentralized systems control=== − − 分散式系统控制 − − {{Main|Distributed control system}} − − − − When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions. − − When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions. − − 当系统由多个控制器控制时，问题是分散控制问题。地方分权在许多方面都很有帮助，例如，它帮助控制系统在更大的经纬度范围内运行。分散控制系统中的智能体可以利用通信信道进行交互并协调其行为。 − − − − − − ===Deterministic and stochastic systems control=== − − ===Deterministic and stochastic systems control=== − − 确定性和随机性系统控制 − − {{Main|Stochastic control}} − − − − − − − − A stochastic control problem is one in which the evolution of the state variables is subjected to random shocks from outside the system. A deterministic control problem is not subject to external random shocks. − − 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. − − 统计控制问题是指状态变量的演化受到系统外部随机冲击的问题。确定性控制问题不受外部随机冲击的影响。 − − − − − − ==Main control strategies== − − ==Main control strategies== − − 主要控制策略 − − Every control system must guarantee first the stability of the closed-loop behavior. For [[linear system]]s, this can be obtained by directly placing the poles. Non-linear control systems use specific theories (normally based on [[Aleksandr Lyapunov]]'s Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen. − − Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be obtained by directly placing the poles. Non-linear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen. − − 每个控制系统首先必须保证闭环行为的稳定性。对于线性系统，这可以通过直接放置极点来实现。非线性控制系统使用特定的理论(通常基于亚历山大·李亚普诺夫理论)来确保系统的稳定性，而不考虑系统的内部动态。实现不同规格的可能性不同于所考虑的模型和所选择的控制策略。 − − − − − − ;List of the main control techniques − − List of the main control techniques − − 主要控制技术清单 − − − − − − *[[Adaptive control]] uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the [[aerospace industry]] in the 1950s, and have found particular success in that field. − − − − *A [[hierarchical control system]] is a type of [[control system]] in which a set of devices and governing software is arranged in a [[hierarchical]] [[tree (data structure)|tree]]. When the links in the tree are implemented by a [[computer network]], then that hierarchical control system is also a form of [[networked control system]]. − − − − *[[Intelligent control]] uses various AI computing approaches like [[artificial neural networks]], [[Bayesian probability]], [[fuzzy logic]],<ref>{{cite journal | title=A novel fuzzy framework for nonlinear system control| journal=Fuzzy Sets and Systems | year=2010 | last=Liu |first1=Jie |author2=Wilson Wang |author3=Farid Golnaraghi |author4=Eric Kubica | volume=161 | issue=21 | pages=2746–2759 | doi=10.1016/j.fss.2010.04.009}}</ref> [[machine learning]], [[evolutionary computation]] and [[genetic algorithms]] or a combination of these methods, such as [[neuro-fuzzy]] algorithms, to control a [[dynamic system]]. − − − − *[[Optimal control]] is a particular control technique in which the control signal optimizes a certain "cost index": for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are [[Model Predictive Control]] (MPC) and [[linear-quadratic-Gaussian control]] (LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important feature in many industrial processes. However, the "optimal control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely used control technique in [[process control]]. − − − − *[[Robust control]] deals explicitly with uncertainty in its approach to controller design. Controllers designed using ''robust control'' methods tend to be able to cope with small differences between the true system and the nominal model used for design.<ref>{{cite journal|last1=Melby|first1=Paul|last2=et.|first2=al.|title=Robustness of Adaptation in Controlled Self-Adjusting Chaotic Systems |journal=Fluctuation and Noise Letters |volume=02|issue=4|pages=L285–L292|date=2002|doi=10.1142/S0219477502000919}}</ref> The early methods of [[Hendrik Wade Bode|Bode]] 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 [[H-infinity loop-shaping]] developed by [[Duncan McFarlane]] and [[Keith Glover]], [[Sliding mode control]] (SMC) developed by [[Vadim Utkin]], and safe protocols designed for control of large heterogeneous populations of electric loads in Smart Power Grid applications.<ref name='TCL1'>{{cite journal|title=Safe Protocols for Generating Power Pulses with Heterogeneous Populations of Thermostatically Controlled Loads |author=N. A. Sinitsyn. S. Kundu, S. Backhaus |journal=[[Energy Conversion and Management]]|volume=67|year=2013|pages=297–308|arxiv=1211.0248|doi=10.1016/j.enconman.2012.11.021}}</ref> Robust methods aim to achieve robust performance and/or [[Stability theory|stability]] in the presence of small modeling errors. − − − − *[[Stochastic control]] 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 deviations. − − − − *[[Energy-shaping control]] view the plant and the controller as energy-transformation devices. The control strategy is formulated in terms of interconnection (in a power-preserving manner) in order to achieve a desired behavior. − − − − *[[Self-organized criticality control]] may be defined as attempts to interfere in the processes by which the [[self-organized]] system dissipates energy. − − − − − − − − ==People in systems and control== − − ==People in systems and control== − − 系统和控制中的人 − − {{Main|People in systems and control}} − − − − Many active and historical figures made significant contribution to control theory including − − Many active and historical figures made significant contribution to control theory including − − 许多活跃的历史人物对防治理论作出了重要贡献，其中包括 − − * [[Pierre-Simon Laplace]] invented the [[Z-transform]] in his work on [[probability theory]], now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the [[Laplace transform]] which is named after him. − − − − * [[Irmgard Flugge-Lotz]] developed the theory of [[bang-bang control|discontinuous automatic control]] and applied it to [[autopilot|automatic aircraft control systems]]. − − − − * [[Alexander Lyapunov]] in the 1890s marks the beginning of [[stability theory]]. − − − − * [[Harold Stephen Black|Harold S. Black]] invented the concept of [[negative feedback amplifier]]s in 1927. He managed to develop stable negative feedback amplifiers in the 1930s. − − − − * [[Harry Nyquist]] developed the [[Nyquist stability criterion]] for feedback systems in the 1930s. − − − − * [[Richard Bellman]] developed [[dynamic programming]] since the 1940s.<ref>{{cite magazine |author=Richard Bellman |date=1964 |title=Control Theory |url=http://www.nature.com/scientificamerican/journal/v211/n3/pdf/scientificamerican0964-186.pdf |magazine=[[Scientific American]] |volume=211 |issue=3 |pages=186–200|author-link=Richard Bellman }}</ref> − − − − * [[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. − − − − * [[John R. Ragazzini]] introduced [[digital control]] and the use of [[Z-transform]] in control theory (invented by Laplace) in the 1950s. − − − − * [[Lev Pontryagin]] introduced the [[Pontryagin's minimum principle|maximum principle]] and the [[Bang-bang control|bang-bang principle]]. − − − − * [[Pierre-Louis Lions]] developed [[viscosity solutions]] into stochastic control and [[optimal control]] methods. − − − − * [[Rudolf Kalman]] pioneered the [[state-space]] approach to systems and control. Introduced the notions of [[controllability]] and [[observability]]. Developed the [[Kalman filter]] for linear estimation. − − − − * [[Ali H. Nayfeh]] who was one of the main contributors to Non-Linear Control Theory and published many books on Perturbation Methods − − − − * [[Jan Camiel Willems|Jan C. Willems]] Introduced the concept of dissipativity, as a generalization of [[Lyapunov function]] 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 [[linear matrix inequality]] (LMI) in control theory. He pioneered the behavioral approach to mathematical systems theory. − − − − − − − − ==See also== − − ==See also== − − 参见 − − {{Portal|Systems science}} − − − − ;Examples of control systems − − Examples of control systems − − 控制系统的例子 − − {{colbegin}} − − − − * [[Automation]] − − − − * [[Deadbeat controller]] − − − − * [[Distributed parameter systems]] − − − − * [[Fractional-order control]] − − − − * [[H-infinity loop-shaping]] − − − − * [[Hierarchical control system]] − − − − * [[Model predictive control]] − − − − * [[Optimal control]] − − − − * [[PID controller]] − − − − * [[Process control]] − − − − * [[Robust control]] − − − − * [[Servomechanism]] − − − − * [[State space (controls)]] − − − − * [[Vector control (motor)|Vector control]] − − − − {{colend}} − − − − ;Topics in control theory − − Topics in control theory − − 控制理论主题 − − {{colbegin}} − − − − * [[Coefficient diagram method]] − − − − * [[Control reconfiguration]] − − − − * [[Cut-insertion theorem]] − − − − * [[Feedback]] − − − − * [[H infinity]] − − − − * [[Hankel singular value]] − − − − * [[Krener's theorem]] − − − − * [[Lead-lag compensator]] − − − − * [[Minor loop feedback]] − − − − * [[Minor loop feedback|Multi-loop feedback]] − − − − * [[Positive systems]] − − − − * [[Radial basis function]] − − − − * [[Root locus]] − − − − * [[Signal-flow graph]]s − − − − * [[Stable polynomial]] − − − − * [[State space representation]] − − − − * [[Steady state]] − − − − * [[Transient response]] − − − − * [[Transient state]] − − − − * [[Underactuation]] − − − − * [[Youla–Kucera parametrization]] − − − − * [[Markov chain approximation method]] − − − − {{colend}} − − − − ;Other related topics − − Other related topics − − 其他相关话题 − − {{colbegin}} − − − − * [[Adaptive system]] − − − − * [[Automation and remote control]] − − − − * [[Bond graph]] − − − − * [[Control engineering]] − − − − * [[Control–feedback–abort loop]] − − − − * [[Controller (control theory)]] − − − − * [[Cybernetics]] − − − − * [[Intelligent control]] − − − − * [[Mathematical system theory]] − − − − * [[Negative feedback amplifier]] − − − − * [[People in systems and control]] − − − − * [[Perceptual control theory]] − − − − * [[Systems theory]] − − − − * [[Time scale calculus]] − − − − {{colend}} − − − − − − − − ==References== − − ==References== − − 参考资料 − − {{Reflist}} − − − − − − − − == Further reading == − − == Further reading == − − 进一步阅读 − − * {{cite book − − − − | editor-last = Levine − − | editor-last = Levine − − | 编辑-最后一个 Levine − − | editor-first = William S. − − | editor-first = William S. − − | 编辑-第一个威廉 s。 − − | title = The Control Handbook − − | title = The Control Handbook − − 控制手册 − − | publisher = CRC Press − − | publisher = CRC Press − − 出版商 CRC 出版社 − − | place = New York − − | place = New York − − 纽约 − − | year = 1996 − − | year = 1996 − − 1996年 − − | isbn = 978-0-8493-8570-4 − − | isbn = 978-0-8493-8570-4 − − | isbn 978-0-8493-8570-4 − − }} − − }} − − }} − − * {{cite book |author1=Karl J. Åström |author2=Richard M. Murray | year = 2008 | title = Feedback Systems: An Introduction for Scientists and Engineers.| publisher = Princeton University Press | url = http://www.cds.caltech.edu/~murray/books/AM08/pdf/am08-complete_28Sep12.pdf | isbn = 978-0-691-13576-2 }} − − − − * {{cite book | author= Christopher Kilian | title= Modern Control Technology | publisher= Thompson Delmar Learning | year= 2005 | isbn=978-1-4018-5806-3 }} − − − − * {{cite book | author= Vannevar Bush | title= Operational Circuit Analysis | publisher= John Wiley and Sons, Inc. | year= 1929 }} − − − − *{{cite book | author= Robert F. Stengel | title= Optimal Control and Estimation | publisher= Dover Publications | year= 1994 | isbn=978-0-486-68200-6 }} − − − − * {{cite book |last=Franklin |title=Feedback Control of Dynamic Systems |origyear= |accessdate= |edition=4 |year=2002 |publisher=Prentice Hall |location=New Jersey |language= |isbn=978-0-13-032393-4 |doi = |pages= |chapter= |url= |quote = |display-authors=etal }} − − − − * {{cite book |author1=Joseph L. Hellerstein |author2=Dawn M. Tilbury |author3=Sujay Parekh | title= Feedback Control of Computing Systems | publisher= John Wiley and Sons | year= 2004 | isbn=978-0-471-26637-2}} − − − − *{{cite book | author= [[Diederich Hinrichsen]] and Anthony J. Pritchard | title= Mathematical Systems Theory I – Modelling, State Space Analysis, Stability and Robustness | publisher= Springer | year= 2005 | isbn=978-3-540-44125-0 }} − − − − *{{cite journal | author = Andrei, Neculai | title = Modern Control Theory – A historical Perspective | version = | year = 2005 | url = http://camo.ici.ro/neculai/history.pdf | accessdate = 2007-10-10 }} − − − − *{{cite book | last = Sontag | first = Eduardo | authorlink = Eduardo D. Sontag | year = 1998 | title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition | publisher = Springer | url = http://www.sontaglab.org/FTPDIR/sontag_mathematical_control_theory_springer98.pdf | isbn = 978-0-387-98489-6 }} − − − − * {{cite book | last = Goodwin | first = Graham | year = 2001 | title = Control System Design | publisher = Prentice Hall | isbn = 978-0-13-958653-8 }} − − − − * {{cite book | author= Christophe Basso | year = 2012 | title = Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide.| publisher = Artech House | url = http://cbasso.pagesperso-orange.fr/Spice.htm | isbn = 978-1608075577 }} − − − − <!-- * {{cite book | author = Briat, Corentin | year = 2015 | title = Linear Parameter-Varying and Time-Delay Systems. Analysis, Observation, Filtering & Control | publisher = Springer Verlag Heidelberg | isbn = 978-3-662-44049-0}}--> − − <!-- * --> − − <!-- * --> − − * {{cite book |author1=Boris J. Lurie |author2=Paul J. Enright |title=Classical Feedback Control with Nonlinear Multi-loop Systems |origyear= |accessdate= |edition=3 |year=2019 |publisher=CRC Press |isbn=978-1-1385-4114-6 }} − − − − ; For Chemical Engineering − − For Chemical Engineering − − 化学工程 − − * {{cite book | last = Luyben | first = William | year = 1989 | title = Process Modeling, Simulation, and Control for Chemical Engineers | publisher = McGraw Hill | isbn = 978-0-07-039159-8 }} − − − − − − − − == External links == − − == External links == − − 外部链接 − − {{Wikibooks|Control Systems}} − − − − {{Commons category|Control theory}} − − − − * [http://www.engin.umich.edu/class/ctms/ Control Tutorials for Matlab], a set of worked-through control examples solved by several different methods. − − − − * [https://controlguru.com/ Control Tuning and Best Practices] − − − − * [https://www.pidlab.com/ Advanced control structures, free on-line simulators explaining the control theory] − − − − * "[http://delivery.acm.org/10.1145/2820000/2814734/p50-li.pdf Applying control theory to manage flash erasures/lifespan]" {{dead link|date=January 2016}} − − − − * [https://cbasso.pagesperso-orange.fr/Downloads/PPTs/Chris%20Basso%20APEC%20seminar%202012.pdf The Dark Side of Loop Control Theory], a professional seminar taught at APEC in 2012 (Orlando, FL). − − − − {{Control theory}} − − − − {{Cybernetics}} − − − − {{Systems}} − − − − {{Areas of mathematics}} − − − − − − − − {{Authority control}} − − − − {{DEFAULTSORT:Control Theory}} − − − − [[Category:Control theory| ]] − − − − [[Category:Computer science]] − − Category:Computer science − − 类别: 计算机科学 − − [[Category:Cybernetics]] − − Category:Cybernetics − − 类别: 控制论 − − [[Category:Control engineering]] − − Category:Control engineering − − 类别: 控制工程 − − [[Category:Computational mathematics]] − − Category:Computational mathematics − − 类别: 计算数学 − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Control theory]]. Its edit history can be viewed at [[控制理论/edithistory]]</small></noinclude> − − [[Category:待整理页面]]
以“=重定向控制理论 Control theory”替换内容
=重定向[[控制理论 Control theory]]
Control systems can be divided into different categories depending on the number of inputs and outputs.