﻿ 一类不确定非线性系统的重复学习控制
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 自动化学报  2018, Vol. 44 Issue (10): 1854-1863 PDF

1. 浙江工业大学信息工程学院 杭州 310023

Repetitive Learning Control for a Class of Uncertain Nonlinear Systems
LI He1, SUN Ming-Xuan1, ZHANG Jing1
1. College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023
Manuscript received : December 4, 2016, accepted: August 17, 2017.
Foundation Item: Supported by National Natural Science Foundation of China (61174034, 61573320, 61374103)
Corresponding author. SUN Ming-Xuan  Professor at the College of Information Engineering, Zhejiang University of Technology. His main research interest is learning control. Corresponding author of this paper.
Abstract: This paper presents a repetitive learning control method to handle the nonparametric uncertain problem for a class of uncertain nonlinear systems performing a given repetitive task over a finite time interval. The learning laws with dead-zone modification are adopted to estimate the desired control input and bound functions, which avoids the divergency of estimates due to the ceaseless positive accumulation and facilitates the implementation of the controller with less knowledge about the system dynamics. The repetitive learning controller is designed in terms of Lyapunov synthesis, so as to guarantee the boundedness of all closed-loop signals while ensuring the tracking error to converge to the pre-specified neighbourhood. Numerical results for an inverted pendulum system and the AC motor experiment are conducted to testify the effectiveness of the proposed learning control scheme.
Key words: Repetitive learning control     nonparametric uncertainties     dead-zone modification     Lyapunov approach

1 问题描述

 \begin{align} \label{eq.sys} \left\{ \begin{array}{l} \dot{x}_{i} = x_{i+1}, ~~i= 1, 2, \cdots, n-1 \\ \dot{x}_{n} = f({\pmb x}, t) + g({\pmb x}, t)u~~~~~~~~~~\\ \end{array} \right.\end{align} (1)

 \begin{align} \label{sys2} \left\{ \begin{array}{l} \dot{x}_{k, i} = x_{k, i+1}, ~~i= 1, 2, \cdots, n-1 \\ \dot{x}_{k, n} = f({\pmb x}_k, t) + g({\pmb x}_k, t)u_k~~~~~~~~~~\\ \end{array} \right.\end{align} (2)

 \begin{align} \label{dek} \dot {\pmb e}_{k} =\,&A {\pmb e}_k + {\pmb b}[f({\pmb x}_k, t) + g({\pmb x}_k, t)u_k+\nonumber\\ &\lambda^{\rm T}{\pmb e}_k-\dot x_{d, n}] \end{align} (3)

 \begin{align*} A &=& \left[\begin{array}{cccccc} 0& 1& 0& \cdots& 0& 0\\ 0& 0& 1& \cdots& 0& 0\\ \vdots& \vdots& \vdots&\ddots& \vdots&\vdots\\ 0& 0& 0& \cdots& 0& 1\\ -\lambda_1& -\lambda_2& -\lambda_3& \cdots& -\lambda_{n-1}& -\lambda_{n}\\ \end{array} \right]\end{align*}
 \begin{align*} {\pmb b}=[0~ 0~ \cdots~ 0~ 1]^{\rm T} \end{align*}

 \begin{align} u_d = \frac{1}{g({\pmb x}_d, t)}\dot x_{d, n} -\frac{f({\pmb x}_d, t)}{g({\pmb x}_d, t)}\end{align} (4)

 \begin{align} \dot x_{d, n} = f({\pmb x}_d, t)+g({\pmb x}_d, t)u_d\end{align} (5)

$f_k=f({\pmb x}_k, t), g_k=g({\pmb x}_k, t), f_d=f({\pmb x}_d, t), g_d=g({\pmb x}_d, t)$, 整理误差动态特性为

 $\dot {\pmb e}_{k} = A{\pmb e}_k + {\pmb b}[f_k+g_ku_k-\nonumber\\ \ \ \ g_du_d-f_d+{\pmb\lambda}^{\rm T}{\pmb e}_k]$ (6)

 \begin{align} \label{dot ek} \dot {\pmb e}_{k} =\,&A {\pmb e}_k + {\pmb b} [f_k-f_d + (g_k-g_d)\hat u_k +\nonumber\\ &g_d(\hat u_k-u_d)+ g_k (u_k - \hat u_k) +{\pmb \lambda}^{\rm T}{\pmb e}_k] \end{align} (7)

2 RLC的设计与分析

 \begin{align} \label{dk} d_k(t) = \left\{ \begin{array}{l} 1, ~~|{\pmb e}_k(t)|>\epsilon\\ 0, ~~|{\pmb e}_k(t)|\le\epsilon\\ \end{array} \right.\end{align} (8)

 \begin{align} \label{dot vk4} \dot V_k =\,&-\frac{1}{2}d_k{\pmb e}_k^{\rm T}Q{\pmb e}_k + d_k{\pmb b}^{\rm T}P{\pmb e}_k[f_k-f_d +\nonumber\\&(g_k-g_d)\hat u_k + g_d(\hat u_k -u_d)+\nonumber\\ &g_k (u_k- \hat u_k) +{\pmb\lambda}^{\rm T} {\pmb e}_k] \end{align} (9)

 \begin{align} \label{dot vk5} \dot{V}_k &\le -\frac{1}{2}d_k{\pmb e}_k^TQe_k + \bigg( d_kl_f|{\pmb b}^TP{\pmb e}_k||{\pmb e}_k| + \nonumber\\ &d_kl_g\hat u_k |{\pmb b}^TP{\pmb e}_k||{\pmb e}_k| +d_k {\pmb \lambda}^T|{\pmb b}^TP{\pmb e}_k||{\pmb e}_k|\bigg)+ \nonumber\\ & d_k{\pmb b}^TP{\pmb e}_kg_d(\hat u_k - u_d) + d_k{\pmb b}^TP{\pmb e}_kg_k(u_k-\hat u_k) \end{align} (10)

 \begin{align} \label{ukd} u_k =\,&\hat u_k - \frac{1}{g_{\rm min}}\frac{9}{4\lambda_Q} \hat l_{f, k} {\pmb b}^{\rm T}P{\pmb e}_k- \nonumber\\&\frac{1}{g_{\rm min}}\frac{9}{4\lambda_Q} \hat l_{g, k}\hat u_k^2 {\pmb b}^{\rm T}P{\pmb e}_k- \frac{1}{g_{\rm min}}\frac{9}{4\lambda_Q}({\pmb \lambda}^{T})^2 {\pmb b}^{\rm T}P{\pmb e}_k\end{align} (11)

 $\hat u_{k} = \hat u_{k-1} - d_k\gamma_1{\pmb b}^{\rm T}P{\pmb e}_k\label{hat ukd}$ (12)
 $\hat l_{f, k} = \hat l_{f, k-1} + d_k\gamma_2\frac{9}{4\lambda_Q}({\pmb b}^{\rm T}P{\pmb e}_k)^2\label{lfk}$ (13)
 $\hat l_{g, k} = \hat l_{g, k-1} + d_k\gamma_3\frac{9}{4\lambda_Q}\hat u_k^2({\pmb b}^{\rm T}P{\pmb e}_k)^2\label{lgk}$ (14)

 \begin{align} \label{dot vk6} \dot V_k \le& -\frac{1}{6}d_k{\pmb e}_k^{\rm T}Q{\pmb e}_k - d_k{\pmb b}^{\rm T}P{\pmb e}_kg_d\tilde u_k+\nonumber\\ &d_k\frac{9}{4\lambda_Q}\tilde l_{f, k}({\pmb b}^{\rm T}P{\pmb e}_k)^2 + d_k\frac{9}{4\lambda_Q}\tilde l_{g, k}\hat u_k^2({\pmb b}^{\rm T}P{\pmb e}_k)^2 \end{align} (15)

 $\tilde u_k^2-\tilde u_{k-1}^2= -(\hat u_k - \hat u_{k-1})^2- 2(\hat u_k - \hat u_{k-1})\tilde u_k$ (16)
 $\tilde l_{f, k}^2-\tilde l_{f, k-1}^2=\nonumber\\ \ \ \ -(\hat l_{f, k} - \hat l_{f, k-1})^2-2(\hat l_{f, k} - \hat l_{f, k-1}) \tilde l_{f, k}\label{tilde lf}$ (17)
 $\tilde l_{g, k}^2-\tilde l_{g, k-1}^2=\nonumber\\ \ \ \ -(\hat l_{g, k} - \hat l_{g, k-1})^2-2(\hat l_{g, k} - \hat l_{g, k-1})\tilde l_{g, k}\label{tilde lg}$ (18)

 \begin{align} \label{LKT} L_K(T) \le L_0(T)-\frac{\lambda_Q}{6}\sum\limits_{j=1}^K\int_0^T d_j\|{\pmb e}_j\|^2 {\rm d}s\end{align} (27)

3 仿真与实验

3.1 仿真算例

 \begin{align*} f({\pmb x}, t)=\, &\dfrac{g_0 {\rm sin}(x_1)-\dfrac{m l x_2^2 {\rm sin}(2x_1)}{2(M+m)}}{l\left(\dfrac{4}{3}-\dfrac{m {\rm cos}^2(x_1)}{M+m}\right)}\nonumber\\ g({\pmb x}, t)=&\dfrac{\dfrac{{\rm cos}(x_1)}{M+m}}{l\left(\dfrac{4}{3}- \dfrac{m {\rm cos}^2(x_1)}{M+m}\right)} \end{align*}

 \begin{align*} \begin{aligned} A=\left[ \begin{array}{cc} 0&1\\ -10 &-1 \end{array} \right], \quad&Q=\left[ \begin{array}{cc} 10&0 \\ 0&5 \end{array} \right] \end{aligned} \end{align*}

 \begin{align*} P=\left[ \begin{array}{cc} 30.5&0.5 \\ 0.5&3 \end{array} \right] \end{align*}

 图 1 误差性能指标$J_k$ Figure 1 Error performance index $J_k$
 图 2 第28次迭代的控制输入$u_k$ Figure 2 The control input $u_k$ at the 28th iteration
 图 3 参考输入估计$\hat u_k$ Figure 3 Estimate $\hat u_k$
 图 4 界函数估计$\hat l_{f, \, k}$ Figure 4 Estimate of the bound function $\hat l_{f, \, k}$
 图 5 界函数估计$\hat l_{g, \, k}$ Figure 5 Estimate of the bound function $\hat l_{g, \, k}$
3.2 实验结果

 \begin{align} \label{syxd} x_d(t)=75^{\circ}\sin(4\pi t) \end{align} (28)

 \begin{align} \label{aq} A=\left[ \begin{array}{cc} 0& 1\\ -1&-5\\ \end{array} \right], ~~~~ Q=\left[ \begin{array}{cc} 0.01& 0\\ 0& 0.01\\ \end{array} \right]\end{align} (29)

 \begin{align} \label{p} P=\left[ \begin{array}{cc} 0.027& 0.005\\ 0.005& 0.002\\ \end{array} \right]\end{align} (30)

 图 6 误差性能指标$J_k$ Figure 6 Error performance index $J_k$
 图 7 位置跟踪误差$e_1$ Figure 7 Position tracking error $e_1$
 图 8 控制输入$u_k$ Figure 8 Control input $u_k$
 图 9 参考输入的估计$\hat u_k$ Figure 9 Control input $\hat u_k$
 图 10 界函数估计$\hat l_{f, \, k}$ Figure 10 Estimate of the bound function $\hat l_{f, \, k}$
 图 11 界函数估计$\hat l_{g, \, k}$ Figure 11 Estimate of the bound function $\hat l_{g, \, k}$

 \begin{align} \label{unon} u_k =\,&\hat u_k - \frac{1}{g_{\rm min}}\frac{9}{4\lambda_Q} l_{f}^2 ({\pmb b}^{\rm T}P{\pmb e}_k)^2-\nonumber\\& \frac{1}{g_{{\rm min}}}\frac{9}{4\lambda_Q} l_{g}^2\hat u_k^2 ({\pmb b}^{\rm T}P{\pmb e}_k)^2- \nonumber\\ &\frac{1}{g_{{\rm min}}}\frac{9}{4\lambda_Q}({\pmb \lambda}^{T})^2 ({\pmb b}^{\rm T}P{\pmb e}_k)^2\end{align} (31)

4 结论

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