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第81行： − + − ==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: − + − − − − * 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.+ − − − − − − A common closed-loop controller architecture is the [[PID controller]]. − − A common closed-loop controller architecture is the PID controller. + − + − − − ==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)。 − − − − − A simple feedback control loop+ − − 一个简单的反馈控制回路 − − − − 第205行：
第113行： − + − − − − − <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> − − 数学 c (s) e (s) / 数学 − − <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>+ − − 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}} − + − 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.]]
无编辑摘要
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，计算得到的差称为'''误差'''，决定了节气门的位置(控制)。输出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，差值增加，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。
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，计算得到的差称为'''误差'''，决定了节气门的位置(控制)。输出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，差值增加，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==
==经典控制理论==
为了克服开回路控制器的局限性，控制理论引入了反馈。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。
闭环控制器比开环控制器有以下优点:
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)
*干扰抑制（例如上述巡航控制示例中的山丘）
*当模型结构与实际过程不完全匹配且模型参数不精确时，即模型不确定，也可以保证系统性能。
*不稳定的过程可以稳定下来
*对参数变化的敏感性降低
*改进的参考跟踪性能
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。
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。
常用的闭环控制器结构是 PID 控制器。
常用的闭环控制器结构是 PID 控制器。
==闭环传递函数==
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。
[[File:simple feedback control loop2.svg|center|A simple feedback control loop]]
[[File:simple feedback control loop2.svg|center|A simple feedback control loop]]
这被称为单输入单输出 SISO 控制系统; 另一种常见的控制系统为 MIMO 即多输入多输出系统，具有多个输入/输出。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)。
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:
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:
如果我们假设控制器 ''C''、被控对象''P''和传感器 ''F''是[[线性时不变]] LMI 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拉普拉斯 Laplace 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>
: <math>U(s) = C(s) E(s)</math>
: <math>U(s) = C(s) E(s)</math>
: <math>E(s) = R(s) - F(s)Y(s).</math>
: <math>E(s) = R(s) - F(s)Y(s).</math>
用给定的 ''R''(''s'') 解出''Y''(''s'')。
: <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>
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加上反馈回路中的增益，即所谓的回路增益。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''，且<math>|F(s)| \approx 1</math>，那么''Y(s)''大约等于''R(s)''，且输出紧密跟踪参考输入。
==PID 反馈控制==
[[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.]]
[[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.]]
[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]
[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]