# 不确定性T-S模糊时滞系统的容错控制研究Fault-Tolerant Control for Uncertain T-S Fuzzy Time-Delay Systems

• 全文下载: PDF(567KB)    PP.109-119   DOI: 10.12677/AAM.2020.91014
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This paper is concerned with fault-tolerant control for a class of nonlinear continuous systems with uncertainties and delay terms. Firstly, the T-S fuzzy model is constructed to approximate the original system more accurately, and the Lyapunov function is constructed, the robustness of the closed-loop system with actuator faults is guaranteed by the designed fuzzy controller. The value of gain matrix and the feasible conditions for the closed-loop system are proposed by solving linear matrix inequalities. Finally, the effectiveness of the method is verified by numerical simulation.

1. 引言

T-S模糊模型是一种用来描述复杂系统，有效解决线性与非线性控制系统之间差距的模糊模型，即用模糊规则来描述非线性系统，进而达到对非线性系统建模的目的。利用T-S模糊控制方法来解决系统中的非线性和不确定问题也得到了广泛的关注和研究，并取得了一系列成果。

2. 系统描述

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left({A}_{i}+\Delta {A}_{i}\right)x\left(t\right)+\left({A}_{di}+\Delta {A}_{di}\right)x\left(t-d\right)+\left({B}_{i}+\Delta {B}_{i}\right)u\left(t\right)+w\left(t\right),\\ z\left(t\right)={C}_{i}x\left(t\right),\\ x\left(t\right)=\phi \left(t\right),t\in \left[-d,0\right],i=1,2,\cdots ,r.\end{array}$ (1)

$\left[\Delta {A}_{i}\left(t\right),\Delta {B}_{i}\left(t\right),\Delta {A}_{di}\left(t\right)\right]=M{F}_{i}\left(t\right)\left[{E}_{1i},{E}_{2i},{E}_{di}\right]$(2)

${F}_{i}\left(t\right){F}_{i}^{\text{T}}\left(t\right)\le I$(3)

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)={\sum }_{i=1}^{r}{b}_{i}\left(\theta \left(t\right)\right)\left[\left({A}_{i}+\Delta {A}_{i}\right)x\left(t\right)+\left({A}_{di}+\Delta {A}_{di}\right)x\left(t-d\right)+\left({B}_{i}+\Delta {B}_{i}\right)u\left(t\right)+w\left(t\right)\right]\\ z\left(t\right)={\sum }_{i=1}^{r}{b}_{i}\left(\theta \left(t\right)\right){C}_{i}x\left(t\right)\end{array}$ (4)

$u\left(t\right)={\sum }_{i=1}^{r}{b}_{i}\left(\theta \left(t\right)\right){K}_{i}x\left(t\right),i=1,2,\cdots ,r$ (5)

$\begin{array}{c}\stackrel{˙}{x}\left(t\right)={\sum }_{i=1}^{r}{\sum }_{j=1}^{r}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[\left({A}_{i}+\Delta {A}_{i}\left(t\right)+{B}_{i}{K}_{j}+\Delta {B}_{i}\left(t\right){K}_{j}\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)x\left(t-d\right)+w\left(t\right)\right]\end{array}$ (6)

$L=diag\left({l}_{1},{l}_{2},\cdots ,{l}_{n}\right)$

${l}_{i}=\left\{\begin{array}{l}0,第i个执行器失效\\ 1,第i个执行器正常\end{array}$

$\begin{array}{c}\stackrel{˙}{x}\left(t\right)={\sum }_{i=1}^{r}{\sum }_{j=1}^{r}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[\left({A}_{i}+\Delta {A}_{i}\left(t\right)+{B}_{i}L{K}_{j}+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)x\left(t-d\right)+w\left(t\right)\right]\end{array}$ (7)

① 当干扰 $w\left(t\right)=0$ 时，证明闭环系统(7)是渐进稳定的；

② 当干扰 $w\left(t\right)\ne 0$ 时，证明对任意干扰衰减指标 $\gamma$ ，若系统控制输出 $z\left(t\right)$ 满足 ${‖z\left(t\right)‖}_{2}\le \gamma {‖w\left(t\right)‖}_{2}$ ，闭环系统(7)是渐进稳定的。

3. 主要结果

${x}_{1}^{\text{T}}Y{x}_{2}+{x}_{2}^{\text{T}}{Y}^{\text{T}}{x}_{1}\le {x}_{1}^{\text{T}}Y{R}^{-1}{Y}^{\text{T}}{x}_{1}+{x}_{2}^{\text{T}}R{x}_{2}$

$S<0$

${S}_{11}<0,{S}_{22}-{S}_{12}^{\text{T}}{S}_{11}^{-1}{S}_{12}<0$

${S}_{22}<0,{S}_{11}-{S}_{12}{S}_{22}^{-1}{S}_{12}^{\text{T}}<0$

$\left[\begin{array}{ccccc}\Psi & XR& {A}_{di}& X{E}_{1i}^{\text{T}}+{Y}_{j}^{\text{T}}L{E}_{2i}^{\text{T}}& M\\ *& -R& 0& 0& 0\\ *& *& -R& {E}_{di}^{\text{T}}& 0\\ *& *& *& -\alpha I& 0\\ *& *& *& *& -{\alpha }^{-1}I\end{array}\right]<0$ (8)

$\Psi =X{A}_{i}^{\text{T}}+{A}_{i}X+{Y}_{j}^{\text{T}}L{B}_{i}^{\text{T}}+{B}_{i}L{Y}_{j}$

$V\left(x\left(t\right)\right)={x}^{\text{T}}\left(t\right)Px\left(t\right)+\underset{t-d}{\overset{t}{\int }}{x}^{\text{T}}\left(s\right)Rx\left(s\right)\text{d}s$

$\begin{array}{c}\stackrel{˙}{V}\left(x\left(t\right)\right)={\stackrel{˙}{x}}^{\text{T}}\left(t\right)px\left(t\right)+{x}^{\text{T}}\left(t\right)p\stackrel{˙}{x}\left(t\right)+{x}^{\text{T}}\left(t\right)Rx\left(t\right)-{x}^{\text{T}}\left(t-d\right)Rx\left(t-d\right)\\ =\underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[{x}^{\text{T}}\left(t\right){\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}Px\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\text{T}}\left(t-d\right){\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}Px\left(t\right)+{x}^{\text{T}}\left(t\right)P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\text{T}}\left(t\right)P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)x\left(t-d\right)+{x}^{\text{T}}\left(t\right)Rx\left(t\right)-{x}^{\text{T}}\left(t-d\right)Rx\left(t-d\right)\right]\end{array}$

$\begin{array}{l}\le \underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[{x}^{\text{T}}\left(t\right)\left({\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}P\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{x}^{\text{T}}\left(t\right)P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right){R}^{-1}{\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}Px\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{x}^{\text{T}}\left(t-d\right)Rx\left(t-d\right)+{x}^{\text{T}}\left(t\right)Rx\left(t\right)-{x}^{\text{T}}\left(t-d\right)Rx\left(t-d\right)\right]\end{array}$

$\begin{array}{l}=\underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[{x}^{\text{T}}\left(t\right)\left({\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}P\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{x}^{\text{T}}\left(t\right)P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right){R}^{-1}{\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}Px\left(t\right)+{x}^{\text{T}}\left(t\right)Rx\left(t\right)\right]\end{array}$

$\begin{array}{l}=\underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[{x}^{\text{T}}\left(t\right)\left({\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}P\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+R+P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right){R}^{-1}{\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}P\right)x\left(t\right)\right]\end{array}$

$\begin{array}{l}\left({\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}P\\ +P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)\\ \text{ }+R+P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right){R}^{-1}{\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}P\right)<0\end{array}$

${\prod }_{1}={\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}P+P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)$

$\left[\Delta {A}_{i}\left(t\right),\Delta {B}_{i}\left(t\right),\Delta {A}_{di}\left(t\right)\right]=M{F}_{i}\left(t\right)\left[{E}_{1i},{E}_{2i},{E}_{di}\right]$ 带入(9)可得：

$\begin{array}{l}\left[\begin{array}{ccc}{\Pi }_{\text{1}}& R& P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)\\ *& -R& 0\\ *& *& -R\end{array}\right]\\ =\left[\begin{array}{ccc}{\left({A}_{i}+{B}_{i}L{K}_{j}\right)}^{\text{T}}P+P\left({A}_{i}+{B}_{i}L{K}_{j}\right)& R& P{A}_{di}\\ *& -R& 0\\ *& *& -R\end{array}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[\begin{array}{c}PM\\ 0\\ 0\end{array}\right]{F}_{i}\left(t\right)\left[\begin{array}{ccc}{E}_{1i}+{E}_{2i}L{K}_{j}& 0& {E}_{di}\end{array}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[\begin{array}{c}{\left({E}_{1i}+{E}_{2i}L{K}_{j}\right)}^{\text{T}}\\ 0\\ {E}_{di}^{T}\end{array}\right]{F}_{i}^{\text{T}}\left(t\right)\left[\begin{array}{ccc}{M}^{\text{T}}P& 0& 0\end{array}\right]\end{array}$

$\begin{array}{l}\left[\begin{array}{ccc}{\left({A}_{i}+{B}_{i}L{K}_{j}\right)}^{\text{T}}P+P\left({A}_{i}+{B}_{i}L{K}_{j}\right)& R& P{A}_{di}\\ *& -R& 0\\ *& *& -R\end{array}\right]\\ \text{ }+\alpha \left[\begin{array}{c}PM\\ 0\\ 0\end{array}\right]\left[\begin{array}{ccc}{M}^{\text{T}}P& 0& 0\end{array}\right]+{\alpha }^{-1}\left[\begin{array}{c}{\left({E}_{1i}+{E}_{2i}L{K}_{j}\right)}^{\text{T}}\\ 0\\ {E}_{di}^{\text{T}}\end{array}\right]\left[\begin{array}{ccc}{E}_{1i}+{E}_{2i}L{K}_{j}& 0& {E}_{di}\end{array}\right]\\ =\left[\begin{array}{ccc}{\Pi }_{2}& R& P{A}_{di}+{\alpha }^{-1}{\left({E}_{1i}+{E}_{2i}L{K}_{j}\right)}^{\text{T}}{E}_{di}\\ *& -R& 0\\ *& *& -R+{\alpha }^{-1}{E}_{di}^{\text{T}}{E}_{di}\end{array}\right]<0\end{array}$ (10)

${\prod }_{2}={\left({A}_{i}+{B}_{i}L{K}_{j}\right)}^{\text{T}}P+P\left({A}_{i}+{B}_{i}L{K}_{j}\right)+\alpha PM{M}^{\text{T}}P+{\alpha }^{-1}{\left({E}_{1i}+{E}_{2i}L{K}_{j}\right)}^{\text{T}}\left({E}_{1i}+{E}_{2i}L{K}_{j}\right)$

$\left[\begin{array}{ccccc}{\prod }_{3}& R& P{A}_{di}& {\left({E}_{1i}+{E}_{2i}L{K}_{j}\right)}^{\text{T}}& PM\\ *& -R& 0& 0& 0\\ *& *& -R& {E}_{di}^{\text{T}}& 0\\ *& *& *& -\alpha I& 0\\ *& *& *& *& -{\alpha }^{-1}I\end{array}\right]<0$ (11)

${\prod }_{3}={\left({A}_{i}+{B}_{i}L{K}_{j}\right)}^{\text{T}}P+P\left({A}_{i}+{B}_{i}L{K}_{j}\right)$

$\left[\begin{array}{ccccc}\Psi & XR& {A}_{di}& X{E}_{1i}^{\text{T}}+{Y}_{j}^{\text{T}}L{E}_{2i}^{\text{T}}& M\\ *& -R& 0& 0& 0\\ *& *& -R& {E}_{di}^{\text{T}}& 0\\ *& *& *& -\alpha I& 0\\ *& *& *& *& -{\alpha }^{-1}I\end{array}\right]<0$

$\left[\begin{array}{ccccccc}\Psi & I& XR& {A}_{di}& X{C}_{i}& X{E}_{1i}+{Y}_{j}^{\text{T}}L{E}_{2i}& M\\ \text{*}& -{\gamma }^{2}I& 0& 0& 0& 0& 0\\ \text{*}& \text{*}& -R& 0& 0& 0& 0\\ \text{*}& \text{*}& \text{*}& -R& 0& {E}_{di}^{\text{T}}& 0\\ \text{*}& \text{*}& \text{*}& \text{*}& -I& 0& 0\\ \text{*}& \text{*}& \text{*}& \text{*}& \text{*}& -\alpha I& 0\\ \text{*}& \text{*}& \text{*}& \text{*}& \text{*}& \text{*}& -{\alpha }^{-1}I\end{array}\right]<0$ (12)

$\Psi =X{A}_{i}^{\text{T}}+{A}_{i}X+{Y}_{j}^{\text{T}}L{B}_{i}^{\text{T}}+{B}_{i}L{Y}_{j}$

$J={\int }_{0}^{\infty }\left[{z}^{\text{T}}\left(t\right)z\left(t\right)-{\gamma }^{2}{w}^{\text{T}}\left(t\right)w\left(t\right)\right]\text{d}t$ (13)

$\begin{array}{c}J={\int }_{0}^{\infty }\left[{z}^{\text{T}}\left(t\right)z\left(t\right)-{\gamma }^{2}{w}^{\text{T}}\left(t\right)w\left(t\right)\right]\text{d}t\\ ={\int }_{0}^{\infty }\left[{z}^{\text{T}}\left(t\right)z\left(t\right)-{\gamma }^{2}{w}^{\text{T}}\left(t\right)w\left(t\right)+\frac{\text{d}}{\text{d}t}V\left(x\left(t\right)\right)\right]\text{d}t-V\left(x\left(t\right)\right)\\ \le {\int }_{0}^{\infty }\left[{z}^{\text{T}}\left(t\right)z\left(t\right)-{\gamma }^{2}{w}^{\text{T}}\left(t\right)w\left(t\right)+\frac{\text{d}}{\text{d}t}V\left(x\left(t\right)\right)\right]\text{d}t\end{array}$ (14)

$\begin{array}{l}{z}^{\text{T}}\left(t\right)z\left(t\right)-{\gamma }^{2}{w}^{\text{T}}\left(t\right)w\left(t\right)+\frac{\text{d}}{\text{d}t}V\left(x\left(t\right)\right)\\ =\underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[{x}^{\text{T}}\left(t\right){C}_{i}^{\text{T}}{C}_{i}x\left(t\right)-{\gamma }^{2}{w}^{\text{T}}\left(t\right)w\left(t\right)+{x}^{\text{T}}\left(t\right)\left({A}_{i}+{B}_{i}L{K}_{j}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\Delta {A}_{i}\left(t\right)+{\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}Px\left(t\right)+{x}^{\text{T}}\left(t-d\right){\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}Px\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{x}^{\text{T}}\left(t\right)P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)x\left(t\right)+{w}^{\text{T}}\left(t\right)Px\left(t\right)+{x}^{\text{T}}\left(t\right)Pw\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{x}^{\text{T}}\left(t\right)P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)x\left(t-d\right)+{x}^{\text{T}}\left(t\right)Rx\left(t\right)-{x}^{\text{T}}\left(t-d\right)Rx\left(t-d\right)\right]\end{array}$

$\begin{array}{l}=\underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[{x}^{\text{T}}\left(t\right)\left({\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}P\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)+{C}_{i}^{\text{T}}{C}_{i}+R\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right){R}^{-1}{\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}P\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{w}^{\text{T}}\left(t\right)Px\left(t\right)+{x}^{\text{T}}\left(t\right)Pw\left(t\right)-{\gamma }^{2}{w}^{\text{T}}\left(t\right)w\left(t\right)\right]\\ =\underset{i=1}{\overset{r}{\sum }}\underset{j=1}{\overset{r}{\sum }}{b}_{i}\left(\theta \left(t\right)\right){b}_{j}\left(\theta \left(t\right)\right)\left[\begin{array}{cc}{x}^{\text{T}}\left(t\right)& {w}^{\text{T}}\left(t\right)\end{array}\right]\left[\begin{array}{cc}\Gamma & P\\ P& -{\gamma }^{2}I\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ w\left(t\right)\end{array}\right]<0\end{array}$ (15)

$\begin{array}{l}\Gamma ={\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)}^{\text{T}}P+P\left({A}_{i}+{B}_{i}L{K}_{j}+\Delta {A}_{i}\left(t\right)+\Delta {B}_{i}\left(t\right)L{K}_{j}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{C}_{i}^{\text{T}}{C}_{i}+R+P\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right){R}^{-1}{\left({A}_{di}+\Delta {A}_{di}\left(t\right)\right)}^{\text{T}}P\end{array}$

4. 数值仿真

$\begin{array}{l}{\stackrel{˙}{x}}_{1}\left(t\right)=0.2\mathrm{sin}\left(t\right){x}_{1}\left(t\right)+\left(0.25+\mathrm{sin}\left(t\right)\right){x}_{1}\left(t-1\right)-1.5{x}_{2}\left(t\right)+0.1{x}_{1}\left(t\right){x}_{2}\left(t\right)-u\left(t\right)+w\left(t\right)\\ {\stackrel{˙}{x}}_{2}\left(t\right)={x}_{1}\left(t\right)-\left(3+0.2\mathrm{cos}\left(t\right)\right){x}_{2}\left(t\right)+0.1{x}_{2}\left(t-1\right)+0.3\mathrm{cos}\left(t\right){x}_{2}\left(t-1\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(0.1+0.02\mathrm{cos}\left(t\right)\right)u\left(t\right)+w\left(t\right)\\ z\left(t\right)=0.1{x}_{1}\left(t\right)\end{array}$ (16)

${M}_{11}\left(x{}_{1}\right)=\frac{{x}_{1}\left(t\right)-{N}_{2}}{{N}_{1}-{N}_{2}},{M}_{12}\left({x}_{1}\right)=\frac{{N}_{1}-{x}_{1}\left(t\right)}{{N}_{1}-{N}_{2}}$

R1: If ${x}_{1}$ is ${M}_{11}$, then

$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left({A}_{1}+\Delta {A}_{1}\right)x\left(t\right)+\left({A}_{d1}+\Delta {A}_{d1}\right)x\left(t-d\right)+\left({B}_{1}+\Delta {B}_{1}\right)u\left(t\right)+w\left(t\right)\\ z\left(t\right)={C}_{1}{x}_{1}\left(t\right)\end{array}$

R2: If ${x}_{1}$ is ${M}_{12}$, then

$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left({A}_{2}+\Delta {A}_{2}\right)x\left(t\right)+\left({A}_{d2}+\Delta {A}_{d2}\right)x\left(t-d\right)+\left({B}_{2}+\Delta {B}_{2}\right)u\left(t\right)+w\left(t\right)\\ z\left(t\right)={C}_{2}{x}_{1}\left(t\right)\end{array}$

${A}_{1}=\left[\begin{array}{cc}-0.2& 0.9\\ 0.4& -0.5\end{array}\right];{A}_{2}=\left[\begin{array}{cc}-2& 3\\ 4& 1\end{array}\right];{A}_{d1}=\left[\begin{array}{cc}0.4& -0.6\\ -0.5& -1.2\end{array}\right];{A}_{d2}=\left[\begin{array}{cc}-0.5& -1\\ -1.2& 1.6\end{array}\right];$

${B}_{1}=\left[\begin{array}{cc}0.3& 0.4\\ -0.2& 0.8\end{array}\right];{B}_{2}=\left[\begin{array}{cc}0.3& 0.4\\ -0.2& 0.2\end{array}\right];M=\left[\begin{array}{cc}0.2& 0\\ 0& 0.2\end{array}\right];$

${E}_{11}=\left[\begin{array}{cc}0.01& -0.21\\ 0& -0.21\end{array}\right];{E}_{12}=\left[\begin{array}{cc}0.01& -0.21\\ 0& -0.21\end{array}\right];{E}_{d1}=\left[\begin{array}{cc}0.01& -0.21\\ 0& -0.21\end{array}\right];{E}_{d2}=\left[\begin{array}{cc}0.01& -0.21\\ 0& -0.21\end{array}\right];$

${E}_{21}=\left[\begin{array}{cc}0.01& -0.21\\ 0& -0.21\end{array}\right];{E}_{22}=\left[\begin{array}{cc}0.01& -0.21\\ 0& -0.21\end{array}\right];{C}_{1}=\left[\begin{array}{cc}-1& 0.3\\ 0.8& -4\end{array}\right];{C}_{2}=\left[\begin{array}{cc}2& -0.25\\ -3& -0.8\end{array}\right];$

${L}_{1}=diag\left\{0,1\right\}$ ，即第一个执行器失效，当 ${L}_{2}=diag\left\{1,0\right\}$ ，即第二个执行器失效，取 $\alpha =1.2;\gamma =10;{N}_{1}=5;{N}_{2}=-5$ ，根据定理2，利用LMI工具箱求解可得不等式(12)是可行的，并求得增益矩阵的值为：

${K}_{1}=\left[\begin{array}{cc}-60.6962& 93.3755\\ 4.0830& -11.4038\end{array}\right],{K}_{2}=\left[\begin{array}{cc}-2.0163& -0.3153\\ 0.7722& -2.0768\end{array}\right]$

Figure 1. The state response curves with $w\left(t\right)=0$

Figure 2. The state response curves with $w\left(t\right)=\mathrm{sin}\left(t\right){e}^{-1}$

Figure 3. The State response curves of actuator failure

5. 总结

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