﻿ 惯性串联系统的自抗扰控制
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 自动化学报  2018, Vol. 44 Issue (3): 562-568 PDF

1. 华南理工大学自动化科学与工程学院 自主系统与网络控制教育部重点实验室 广州 510641

Active Disturbance Rejection Control of Cascade Inertia Systems
LI Xiang-Yang1, AI Wei1, TIAN Sen-Ping1
1. Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, School of Automation Science and Engineering, South China University of Technology, Guangzhou 510641
Manuscript received : August 3, 2016, accepted: March 2, 2017.
Foundation Item: Supported by National Natural Science Foundation of China (61773170, 61374104), Natural Science Foundation of Guangdong Province of China (2016A030313505), and Guangdong Science and Technology Plan (2013A011402003)
Corresponding author. LI Xiang-Yang  Associate professor at the School of Automation Science and Engineering, South China University of Technology. His research interest covers active disturbance rejection control, iterative learning control, and industrial automation. Corresponding author of this paper
Recommended by Associate Editor JI Hai-Bo
Key words: Active disturbance rejection control (ADRC)     cascade inertia systems     extended state observer (ESO)     disturbance observer (DOB)     three-degree of freedom (3-DOF) control

1 问题描述

 $(s + \sigma )^n y = f(y, y^{(1)}, \cdots, y^{(n - 1)}, t) + \\ b(y, y^{(1)}, \cdots, y^{(n - 1)}, t)u(t)$ (1)

 $\left| b \right| \ge b_{\min } > 0$ (6)

$b_0$是与$b$具有相同符号的粗略估计.由假设1和假设2, 系统$(5)$可以写成扩张状态方程形式

 $\begin{cases} \dot x_i (t) = x_{i + 1} (t), \quad i = 1, \cdots, n - 1 \\ \dot x_n (t) = x_{n + 1} (t) + b_0 u(t) \\ x_{n + 1} (t) = f({\pmb{x}}, t) + (b - b_0 )u(t) \\ \end{cases}$ (7)

 $\begin{cases} \dot {\hat x}_i (t) = \hat x_{i + 1} - a_i (\hat x_1 - x_1 ), \quad i = 1, \cdots, n - 1 \\ \qquad \vdots \\ \dot {\hat x}_n (t) = \hat x_{n + 1} - a_n (\hat x_1 - x_1 ) + b_0 u(t) \\ \dot {\hat x}_{n + 1} (t) = - a_{n + 1} (\hat x_1 - x_1 ) \\ \end{cases}$ (8)

1) 给定任意正数$t_\varepsilon$, 对于$t \in {\rm{[}}t_\varepsilon, \infty {\rm{)}}$, 下式一致收敛.

 $\mathop {\lim }\limits_{\varepsilon \to 0} \left| {x_i (t)- \hat x_i (t)} \right| = 0$ (9)

2) 系统$(7)$和系统$(8)$的解$x_i$$\hat x_i 之差的上极限满足  \mathop {\overline {\lim } }\limits_{t \to \infty } \left| {x_i (t)- \hat x_i (t)} \right| \le o(\varepsilon ^{(n + 2- i)} ) (10) 引理1的证明和说明见文献[11]和文献[12], 引理1中 {\pmb{\hat x}} 实现了对系统状态 {\pmb{x}}的估计, \hat x_{n + 1}实现了对系统总扰动x_{n + 1}的估计.据此可以构造如图 1所示的ADRC系统结构.  图 1 自抗扰控制的系统结构 Figure 1 System structure of ADRC 图 1中, ADRC的关键在于通过ESO得到对系统总扰动估计的扩展状态\hat x_{n + 1} , 并加入到系统控制输入中补偿系统(5)得到一个近似线性积分串联标准型, 实现了系统(5)的动态补偿线性化, 在此基础上通过反馈控制 u_f实现补偿后的积分串联系统的极点配置, 从而保证闭环系统的性能要求, 最终系统的控制量为  u = u_f - \frac{{\hat x_{n + 1} }}{{b_0 }} (11) 由于系统(1)在实际中大量存在, 例如过程控制中三容过程, 在ADRC的应用中, 是否可以直接采用与实际系统更接近的惯性串联模型, 而不一定是积分串联模型呢?同样采用ADRC的动态补偿线性化思想, 但是线性化后得到的不是积分串联系统, 而是与实际系统更接近的惯性串联系统, 可以减少ESO的计算量, 更接近实际系统工作时的状态变量有明显的物理意义, 其估计值可以用于实际控制系统的故障诊断, 有利于提高整个控制系统的可靠性. 2 惯性串联系统的ADRC 针对系统(1), 重新定义系统的状态变量  \begin{cases} z_1 = y \\ z_i = (s + \sigma )z_{i - 1}, \quad i = 2, \cdots, n \end{cases} (12) 其中, -\sigma 为惯性环节的极点, 则式(1)可以化为如下的惯性串联系统  \begin{cases} \dot z_i (t) = - \sigma z_i (t) + z_{i + 1} (t), \quad i = 1, \cdots, n - 1 \\ \dot z_n (t) = -\sigma z_n (t) + z_{n + 1} {\rm{(}}t) + b_0 u(t) \\ z_{n + 1} (t) = f(z, t) + (b(z, t) - b_0 )u(t) \\ \end{cases} (13) 其中, {\pmb{z}}^{\rm{T}} = [\begin{array}{*{20}c} {z_{\rm{1}} } & \cdots & {z_n } \\ \end{array}], z_{n + 1} (t)为系统的惯性串联型ES.对比式(13)和式(5), 可以看出积分串联系统(5)是惯性串联系统(13)$$\sigma =0$的特殊情况, 仿照积分串联型ESO$(8)$可以构造式$(13)$的ESO为

 $\begin{cases} \dot {\hat z}_i (t) = -\sigma \hat z_i + \hat z_{i + 1} - \alpha _i (\hat z_1 - z_1 ), \quad i = 1, \cdots, n - 1 \\ \dot {\hat z}_n (t) = - \sigma \hat z_n + \hat z_{n + 1} - \alpha _n (\hat z_1 - z_1 ) + b_0 u(t) \\ \dot {\hat z}_{n + 1} (t) = - \sigma \hat z_{n + 1} - \alpha _{n + 1} (\hat z_1 - z_1 ) \end{cases}$ (14)

 图 2 线性惯性串联系统 Figure 2 Linear inertia series system

2.1 惯性串联系统ESO

1) 给定任意正数$t_\varepsilon$, 对于$t \in {\rm{[}}t_\varepsilon, \infty {\rm{)}}$下式一致收敛.

 $\mathop {\lim }\limits_{\varepsilon \to 0} \left| {z_i (t)- \hat z_i (t)} \right| = 0$ (15)

2) 系统$(13)$和系统$(14)$的解$z_i$$\hat z_i 之差的上极限满足  \overline {\mathop {\lim }\limits_{{{t}} \to \infty } } \left| {z_i (t)- \hat z_i (t)} \right| \le o(\varepsilon ^{(n + 2- i)} ) (16) 其中, i = 1, \cdots, n + 1 , z_{n + 1} (t) = f + (b -b_0)u(t) 为系统(13)的扩张状态. 证明.根据式(4)和式(12)的两种状态变量的定义, 状态变量之间的关系为  \begin{cases} z_1 = x_1 = y \\ z_2 = { {C}}_1^0 \sigma x_1 + x_2 \\ z_3 = { {C}}_2^0 \sigma ^2 x_1 + { {C}}_2^1 \sigma x_2 + x_3 \\ \begin{array}{*{20}c} {} & \vdots & {} \\ \end{array} \\ z_n = { {C}}_{n - 1}^0 \sigma ^{n - 1} x_1 + \cdots + { {C}}_{n - 1}^{n - 2} \sigma x_{n - 1} + x_n \end{cases} (17) (17)写成向量和矩阵形式为  {\pmb{z}} = T{\pmb{x}} (18) 其中, T为可逆变换矩阵  T = \left[{\begin{array}{*{20}c} 1 & 0 & \cdots & 0 & 0 \\ {{ {C}}_1^0 \sigma ^1 } & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {{ {C}}_{n-1}^0 \sigma ^{n-1} } & {{ {C}}_{n-1}^1 \sigma ^{n - 2} } & \cdots & {{ {C}}_{n - 1}^{n - 2} \sigma } & 1 \\ \end{array}} \right]_{n \times n} (19) 同理有  \begin{cases} \hat z_1 = \hat x_1 \\ \hat z_2 = \hat x_1 { {C}}_1^0 \sigma + \hat x_2 \\ \hat z_3 = \hat x_1 { {C}}_2^0 \sigma ^2 + \hat x_2 { {C}}_2^1 \sigma + \hat x_3 \\ \begin{array}{*{20}c} {} & \vdots & {} \\ \end{array} \\ \hat z_n = \hat x_1 { {C}}_{n - 1}^0 \sigma ^{n - 1} + \cdots + \hat x_{n - 1} { {C}}_{n - 1}^{n - 2} \sigma + \hat x_n \\ \hat z_{n + 1} = \hat x_1 { {C}}_n^0 \sigma ^n + \cdots + \hat x_n { {C}}_n^{n - 1} \sigma + \hat x_{n + 1} \\ \end{cases} (20) 式(20)写成向量和矩阵形式为  \hat {\pmb{z}} = \hat T\hat {\pmb{x}} (21) 其中, \hat T为可逆变换矩阵  \hat T = \left[{\begin{array}{*{20}c} 1 & 0 & \cdots & 0 & 0 \\ {{ {C}}_1^0 \sigma ^1 } & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {{ {C}}_n^0 \sigma ^n } & {{ {C}}_n^1 \sigma ^{n-1} } & \cdots & {{ {C}}_n^{n-1} \sigma } & 1 \\ \end{array}} \right]_{(n + 1) \times (n + 1)} (22) 根据引理1, 采用线性ESO(8)对式(2)的状态进行估计, 式(9)和式(10)成立, 则有  {\pmb{z}} - \hat {\pmb{z}} = T({\pmb{x}} - \hat {\pmb{x}}) (23)  \mathop {\lim }\limits_{\varepsilon \to 0} \left| {{\pmb{z}}(t) - \hat {\pmb{z}}(t)} \right| = T\mathop {\lim }\limits_{\varepsilon \to 0} \left| {{\pmb{x}}(t) - \hat {\pmb{x}}(t)} \right| = 0 (24)  x_{n + 1} - \hat x_{n + 1} = z_{n + 1} - \hat z_{n + 1} - \sum\limits_{i = 1}^n {{ {C}}_n^i \sigma (x_i - \hat x_i )} (25) 式(25)两边取绝对值有  \left| {z_{n + 1} (t) - \hat z_{n + 1} (t)} \right| \le \left| {x_{n + 1} - \hat x_{n + 1} } \right| + \left| {\sum\limits_{i = 1}^n {{ {C}}_n^i \sigma (x_i - \hat x_i )} } \right| (26) 由式(9)和式(24)有  \mathop {\lim }\limits_{\varepsilon \to 0} \left| {z_{n + 1} (t) - \hat z_{n + 1} (t)} \right| = 0 (27) 即定理1中的式(15)成立.同理, 由式(17)、式(20)和式(25)以及高阶无穷小的概念有  \overline {\mathop {\lim }\limits_{{{t}} \to \infty } } \left| {z_i (t) - \hat z_i (t)} \right| \le o(\varepsilon ^{(n + 2 - i)} ) 即定理1中的式(16)成立. 定理1给出了惯性串联系统的线性ESO, 从\hat x_{n + 1}$$ \hat z_{n+ 1}$的关系可以看出, 采用ESO估计多了一项$\sum_{i = 1}^n { {C}}_n^i \sigma (x_i -\hat x_i)$, 这项正是在补偿控制中直接用于配置$n$重极点$\sigma$引起的.

2.2 惯性串联型自抗扰控制

 ${ \varPhi }(s) = \frac{{K(s)P_d (s)}}{{1 + K(s)P_d (s)}} = \frac{{Kb_0 }}{{(s + \sigma )^n + Kb_0 }}$ (28)

 图 4 部分模型已知的惯性串联型自抗扰控制结构 Figure 4 ADRC topology for inertia series systems with partially known model

 $\begin{cases} \dot {\hat z}_i (t) = - \sigma \hat z_i + \hat z_{i + 1} - \alpha _i (\hat z_1 - z_1 ), \quad i = 1, \cdots, n - 1 \\[1mm] \dot {\hat z}_n (t) = - \sigma \hat z_n + \hat z_{n + 1} - \alpha _n (\hat z_1 - z_1 ) + f_0 (\hat z)+b_0 u(t) \\[1mm] \dot {\hat z}_{n + 1} (t) = - \sigma \hat z_{n + 1} - \alpha _{n + 1} (\hat z_1 - z_1 ) \end{cases}$ (32)

 $u = r_{n + 1} + u_f - \frac{{f_0 (\hat {\pmb{z}}) + \hat z_{n + 1} }}{{b_0 }}$ (33)

 $P(s) = P_n (s)(1 + \Delta (s))$ (34)
 图 5 基于DOB的控制的系统框图 Figure 5 Block diagram of DOB control system

 $u = u_f - \hat d$ (35)
 $y = P(u + d)$ (36)

 $y = \frac{P}{{P_n (1 - Q) + QP}}P_n u_f + \frac{{P(1 - Q)}}{{P_n (1 - Q) + QP}}P_n d$ (37)
 $\hat d = \frac{{Q(P - P_n )}}{{P_n (1 - Q) + QP}}u_f + \frac{{QP}}{{P_n (1 - Q) + QP}}d$ (38)

 $P_n (s) = P_d (s) = \frac{{b_0 }}{{(s + \sigma )^n }}$ (39)

 $Q(s) = \frac{1}{{(\varepsilon s + \varepsilon \sigma + 1)^{n + 1}}}$ (40)

 $\mathop {\lim }\limits_{\varepsilon \to 0} y = P_n u_f$ (41)
 $\mathop {\lim }\limits_{\varepsilon \to 0} \hat z_{n + 1} = b_0 \mathop {\lim }\limits_{\varepsilon \to 0} \hat d$ (42)

 $z_3 = 2.25x_1 - 3x_2 + f(t) + (b - b_0 )u$ (53)

 图 6 参考输入$r_1$和系统输出$y_1$ Figure 6 Reference input $r_1$ and system output $y_1$
 图 7 系统控制输入$u$ Figure 7 System input $u$

 图 8 系统(47)的扩张状态$z_3$及其估计${\hat z}_3$ Figure 8 ES $z_3$ and its estimation ${\hat z}_3$ of system (47)

4 结论

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