421
个编辑
更改
跳到导航
跳到搜索
第139行：
第139行： + − + − − − 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. − + − − − − 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 − − − − − − 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 − − − − − − <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> − − 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 第233行：
第161行： − <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> − − 数学 p (s) frac { a }{1 + sT p } / math − − − − 第265行：
第169行： − 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>+ − − 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> − − 数学 c (s) k 左(1 + frac {1}{ sT i }右)(1 + sT d) / 数学 − − − 第305行：
第182行： − 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>+ − − 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. − − − − 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:+
无编辑摘要
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。
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。
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。指的是对误差信号进行比例积分微分的计算与操作来产生对应的控制信号。
该理论的生成和应用可以追溯到20世纪20年代，几乎在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。
Pid 控制器可能是最常用的反馈控制设计。
Pid 控制器可能是最常用的反馈控制设计。
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>
期望闭环动力学是通过调整三个参数<math> K_P</math>、<math> K_D</math>和<math> K_I</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来获得。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，他们难以用于更复杂的情况，特别是如果 MIMO 系统考虑。
应用拉普拉斯变换得到变换后的 PID 控制器方程
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>
:<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>
具有 PID 控制器的传递函数
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>
作为闭环系统<math>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>
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
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<math>A</math>和<math>T_P</math>是一些常数。系统的输出通过
:<math>F(s) = \frac{1}{1 + sT_F}</math>
:<math>F(s) = \frac{1}{1 + sT_F}</math>
<math>T_F</math>也是一个常数。现在如果我们设置<math>K_P=K\left(1+\frac{T_D}{T_I}\right)</math>，<math>K_D=KT_D</math>，和<math>K_I=\frac{K}{T_I}</math>，我们可以将 PID 控制器传递函数表示成如下形式
:<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>
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<math>P(s)</math>， <math>F(s)</math>，<math>C(s)</math>输入到闭回路传递函数<math>H(s)</math>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>
<math>H(s) = 1</math>。通过本例中的这个调优，系统输出精确地跟随参考输入。
<math>H(s) = 1</math>. With this tuning in this example, the system output follows the reference input exactly.
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<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> 。
==线性控制和非线性控制==
控制理论可分为两个分支:
控制理论可分为两个分支: