不确定多状态变时滞广义Lurie切换系统的鲁棒无源控制

Robust Passive Control of Uncertain Singular Lurie Switched Systems with Multiple Time-Varying Delays

摘要: 本文在基于Lyapunov-Krasovskii泛函、Schur补引理及矩阵不等式方法下，讨论了一类具有不确定性多状态变时滞广义Lurie切换系统，在切换状态反馈策略下，得到了使不确定多状态变时滞Lurie切换广义系统能够渐近稳定且严格无源存在的充分性判据，为系统的综合提供了可行性判据。设计了具有记忆的反馈控制器，为系统的稳定性分析及控制器的综合提供了更多的自由度，最后的结果转化为线性矩阵不等式给出，通过数值仿真，验证了定理的有效性和实用性。

Abstract: This paper considers the issues of the switching state strategy memory feedback for a class of uncertain Singular Lurie switched systems with multiple time-varying delays. Under the circumstances of a state-based switching law, and based on the Lyapunov-Krasovskii functional techniques, Schur complement lemma, and linear matrix inequality, the sufficient conditions are given for the asymptotic stability and strictly passive of uncertain switched singular systems with multiple time-varying delays. Meanwhile, robust passive controllers for each subsystem under the switching law are designed. It is believed that controller with memory should provide more freedom for robust stability analysis and synthesis controller for the systems. In the final, the results are given in the form of linear matrix inequalities (LMIs). And the validity of the conclusion is shown by numerical examples.

1. 引言

切换系统是由一族子系统和描述它们之间联系的切换规则组成的混杂系统 [1]。切换系统在实际工程系统中应用很广泛，如通讯系统、电力系统、航空航天系统、化工系统、经济系统、交通系统、网络控制系统等 [2] [3] [4]，近年来，切换系统已经成为广大学者的研究热点 [5] - [16]，而带有记忆的控制器的引入更加符合实际，且为控制的可行性提供更好的自由度，因而对此类问题的研究更具有实际应用及理论价值。文献 [17] 对线性切换系统的稳定性分析进展进行了综述。文献 [18] 利用广义Lyapunov方法研究了一类具有多状态不确定性的切换广义系统的鲁棒无源控制问题。文献 [19] 利用Lyapunov-Krasovskii泛函研究了一类时变时滞的不确定性Lurie动态输出反馈 $H_{}$ 控制问题。文献 [20] 研究了切换广义系统的 $H_{\infty }$ 静态输出反馈和动态输出反馈问题。文献 [21] 通过构造Lyapunov-Krasovskii泛函，研究了一类带有多时滞摄动下的非线性切换系统的稳定性分析和切换律的设计问题。文献 [22] 通过引入新颖的Lyapunov-Krasovskii泛函，分析了非线性切换延迟系统的问题。A. V. Platonov在文献 [23] 中研究了一类非线性切换系统的稳定性问题。文献 [24] 通过构造状态采样数据控制器，考虑了一类切换非线性系统的异步切换下的全局镇定问题。在文献 [25] 中王荣豪等人对具有异步切换的非线性切换系统的有限时间稳定性和稳定性问题进行了研究。

现有的文献中，很少有涉及到多状态变时滞广义Lurie切换系统反馈问题，为此本文将研究具有参数不确定性时变时滞多状态Lurie切换广义系统的无源控制问题，通过运用Lyapunov-Krasovskii泛函、线性矩阵不等式以及文献 [26] [27] 所提供的不等式，进一步考虑了带有记忆的鲁棒无源 控制的综合问题。考虑带有记忆的控制器可以使系统的稳定性分析及控制综合拥有了更多的自由度及更好的合理性。并利用线性矩阵不等式形式得到了Lurie切换广义系统渐近稳定且严格无源的充分条件。

2. 系统的描述和准备

考虑以下含参数不确定性的带有记忆的变时滞广义Lurie切换系统：

$\{\begin{array}{l}E\stackrel{\dot{}}{x}\left(t\right)=\left(A_{\sigma \left(t\right)}+\Delta A_{\sigma \left(t\right)}\left(t\right)\right)x\left(t\right)+\left(A_{1\sigma \left(t\right)}+\Delta A_{1\sigma \left(t\right)}\left(t\right)\right)x\left(t-d_1\left(t\right)\right)\\ +\left(A_{2\sigma \left(t\right)}+\Delta A_{2\sigma \left(t\right)}\left(t\right)\right)x\left(t-d_2\left(t\right)\right)+B_{1\sigma \left(t\right)}u\left(t\right)+B_{2\sigma \left(t\right)}\omega \left(t\right)+Df\left(\mu \left(t\right)\right)\\ z\left(t\right)=C_{\sigma \left(t\right)}x\left(t\right)+C_{1\sigma \left(t\right)}u\left(t\right)+C_{2\sigma \left(t\right)}\omega \left(t\right)\\ \mu \left(t\right)=Cx\left(t\right)\\ x\left(t\right)=\phi \left(t\right)\in C\left(\left[-d,0\right],R^n\right)\end{array}$ (1)

其中， $x\left(t\right)\in R^n$ 为系统的被控对象状态变量， $u\left(t\right)\in R^p$ 为控制输入状态变量， $z\left(t\right)\in R^m$ 为控制输出变量， $\sigma (\cdot ):\left[0,\infty \right]\to \left\{1,2,\cdots ,N\right\}=\bar{N}$ 为分段常值切换信号，且 $\sigma \left(t\right)=i$ 表示第i个子系统在t时刻被激活， $\omega \left(t\right)\in R^q$ 为外部干扰，且系数矩阵 $A_{\sigma \left(t\right)}\in R^{n\times n}$， $A_{1\sigma \left(t\right)}\in R^{n\times n}$， $A_{2\sigma \left(t\right)}\in R^{n\times n}$， $B_{1\sigma \left(t\right)}\in R^{n\times p}$， $B_{2\sigma \left(t\right)}\in R^{n\times q}$， $C_{\sigma \left(t\right)}\in R^{m\times n}$， $C_{1\sigma \left(t\right)}\in R^{m\times p}$， $C_{2\sigma \left(t\right)}\in R^{m\times q}$， $C=\left[C_1,C_2,\cdots ,C_l\right]\in R^{l\times n}$， $D\in R^{n\times l}$,d为正实数， $d_1\left(t\right)$， $d_2\left(t\right)$ 表示时变的状态时滞，并且时滞变化率满足 $0<{\stackrel{\dot{}}{d}}_1\left(t\right)\le h_1<1$， $0<{\stackrel{\dot{}}{d}}_2\left(t\right)\le h_2<1$。 $\Delta A_{\sigma \left(t\right)}$， $\Delta A_{1\sigma \left(t\right)}$， $\Delta A_{2\sigma \left(t\right)}$ 为参数不确定性矩阵，且满足关系式 $\left[\Delta A_{\sigma \left(t\right)}\left(t\right),\Delta A_{1\sigma \left(t\right)}\left(t\right),\Delta A_{2\sigma \left(t\right)}\left(t\right)\right]=N_{\sigma \left(t\right)}{F}_{\sigma \left(t\right)}\left(t\right)\left[G_{\sigma \left(t\right)},G_{1\sigma \left(t\right)},G_{2\sigma \left(t\right)}\right]$。 $N_{\sigma \left(t\right)}\in R^{n\times n}$， $G_{\sigma \left(t\right)}\in R^{n\times n}$， $G_{1\sigma \left(t\right)}\in R^{n\times n}$， $G_{2\sigma \left(t\right)}\in R^{n\times n}$， $F_{\sigma \left(t\right)}\left(t\right)$ 是具有Lebesgue可测的未知矩阵，满足 $F_{\sigma \left(t\right)}^{\text{T}}\left(t\right)F_{\sigma \left(t\right)}\left(t\right)\le I$ ； $f\left(\mu \right)$ 表示非线性向量，且位于有限的霍尔维茨角域内，

$f_j(·)\in K_{\left[0,k_j\right]}=\left\{f_j\left({\mu }_j\right)|f_j\left(0\right)=0,0<{\mu }_j{f}_j\left({\mu }_j\right)\le {\lambda }_j{\mu }_j^2,\left({\mu }_j\ne 0\right)\right\},j=1,2,\cdots ,l$ (2)

其中， ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_l$ 为常数，记 $ƛ=diag\left\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_l\right\}$， $\mu \left(t\right)={\left({\mu }_1\left(t\right),{\mu }_2\left(t\right),\cdots ,{\mu }_n\left(t\right)\right)}^{\text{T}}\in R^n$， $f\left(\mu \right)={\left(f_1\left({\mu }_1\right),f_2\left({\mu }_2\right),\cdots f_l\left({\mu }_l\right)\right)}^{\text{T}}\in R^l$

设计带有记忆的输出反馈控制器为 $u\left(t\right)=K_{\sigma \left(t\right)}x\left(t\right)+K_{1\sigma \left(t\right)}x\left(t-d_1\left(t\right)\right)+K_{2\sigma \left(t\right)}x\left(t-d_2\left(t\right)\right)$ 且 $K_{\sigma \left(t\right)}\in R^{p\times n}$， $K_{1\sigma \left(t\right)}\in R^{p\times n}$， $K_{2\sigma \left(t\right)}\in R^{p\times n}$。

系统(1)的闭环系统：

$\{\begin{array}{l}E\stackrel{\dot{}}{x}\left(t\right)=\left(A_{\sigma \left(t\right)}+N_{\sigma \left(t\right)}{F}_{\sigma \left(t\right)}\left(t\right)G_{\sigma \left(t\right)}+B_{1\sigma \left(t\right)}{K}_{\sigma \left(t\right)}\right)x\left(t\right)\\ +\left(A_{1\sigma \left(t\right)}+N_{\sigma \left(t\right)}{F}_{\sigma \left(t\right)}\left(t\right)G_{1\sigma \left(t\right)}+B_{1\sigma \left(t\right)}{K}_{1\sigma \left(t\right)}\right)x\left(t-d_1\left(t\right)\right)\\ +\left(A_{2\sigma \left(t\right)}+N_{\sigma \left(t\right)}{F}_{\sigma \left(t\right)}\left(t\right)G_{2\sigma \left(t\right)}+B_{1\sigma \left(t\right)}{K}_{2\sigma \left(t\right)}\right)x\left(t-d_2\left(t\right)\right)+B_{2\sigma \left(t\right)}\omega \left(t\right)+Df\left(\mu \left(t\right)\right)\\ z\left(t\right)=\left(C_{\sigma \left(t\right)}+C_{1\sigma \left(t\right)}{K}_{\sigma \left(t\right)}\right)x\left(t\right)+C_{1\sigma \left(t\right)}{K}_{1\sigma \left(t\right)}x\left(t-d_1\left(t\right)\right)+C_{1\sigma \left(t\right)}{K}_{2\sigma \left(t\right)}x\left(t-d_2\left(t\right)\right)+C_{2\sigma \left(t\right)}\omega \left(t\right)\\ x\left(t\right)=\phi \left(t\right),\phi \in C\left[-d,0\right]\end{array}$ (3)

记 ${\bar{A}}_{\sigma \left(t\right)}=A_{\sigma \left(t\right)}+N_{\sigma \left(t\right)}{F}_{\sigma \left(t\right)}\left(t\right)G_{\sigma \left(t\right)}$， ${\bar{A}}_{1\sigma \left(t\right)}=A_{1\sigma \left(t\right)}+N_{\sigma \left(t\right)}{F}_{\sigma \left(t\right)}\left(t\right)G_{1\sigma \left(t\right)}$， ${\bar{A}}_{2\sigma \left(t\right)}=A_{2\sigma \left(t\right)}+N_{\sigma \left(t\right)}{F}_{\sigma \left(t\right)}\left(t\right)G_{2\sigma \left(t\right)}$ 则系统(3)变为：

$\{\begin{array}{l}E\stackrel{\dot{}}{x}\left(t\right)=\left({\bar{A}}_{\sigma \left(t\right)}+B_{1\sigma \left(t\right)}{K}_{\sigma \left(t\right)}\right)x\left(t\right)+\left({\bar{A}}_{1\sigma \left(t\right)}+B_{1\sigma \left(t\right)}{K}_{1\sigma \left(t\right)}x\left(t-d_1\left(t\right)\right)\right)\\ +\left({\bar{A}}_{2\sigma \left(t\right)}+B_{1\sigma \left(t\right)}{K}_{2\sigma \left(t\right)}x\left(t-d_2\left(t\right)\right)\right)+B_{2\sigma \left(t\right)}\omega \left(t\right)+Df\left(\mu \left(t\right)\right)\\ z\left(t\right)=\left(C_{\sigma \left(t\right)}+C_{1\sigma \left(t\right)}{K}_{\sigma \left(t\right)}\right)x\left(t\right)+C_{1\sigma \left(t\right)}{K}_{1\sigma \left(t\right)}x\left(t-d_1\left(t\right)\right)+C_{1\sigma \left(t\right)}{K}_{2\sigma \left(t\right)}x\left(t-d_2\left(t\right)\right)+C_{2\sigma \left(t\right)}\omega \left(t\right)\\ x\left(t\right)=\phi \left(t\right),\phi \in C\left[-d,0\right]\end{array}$ (4)

定义1：使得广义Lurie不确定切换系统(1)具有鲁棒无源的。如果：

1) 当外部扰动输入 $\omega \left(t\right)\equiv 0$ 时，闭环系统是渐近稳定的。

2) 在系统初始条件 $\phi \left(t\right)\equiv 0,t\in \left[-d,0\right]$ 时，对于所有非零的 $\omega$，使得无源不等式成立 $\stackrel{\dot{}}{V}\left(x\left(t\right),t\right)\le {\omega }^{\text{T}}\left(t\right)z\left(t\right)$， $\omega \in L_2\left[0,\infty \right]$ ；若无源不等式换为严格不等式，则称广义Lurie切换不确定性系统(1)是严格无源的。

在分析和推导的过程中，我们会用到以下的引理。

引理1 [26] ：给定适当维数的矩阵Y，D和E，其中Y是对称的，则 $Y+DFE+E^{\text{T}}{F}^{\text{T}}{D}^{\text{T}}<0$，对所有满足 $F^{\text{T}}F\le I$ 的矩阵成立，当且仅当存在一个常数 $\epsilon >0$，使得 $Y+\epsilon D{D}^{\text{T}}+{\epsilon }^{-1}{E}^{\text{T}}E<0$。

引理2 [27] (Schur补引理)：对给定的对称矩阵 $S=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right]$，其中 $S_{11}\in R^{r\times r}$，以下三个条件是等价的：

(I) $S<0$ ；

(II) $S_{11}<0,S_{22}-S_{21}{S}_{11}^{-1}{S}_{12}<0$ ；

(III) $S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{21}<0$。

引理3 [27] ：对于任意适当维数的矩阵X，Y，且存在一个常数 $\epsilon >0$，有

$X^{\text{T}}Y+Y^{\text{T}}X\le \epsilon X^{\text{T}}X+\frac{1}{\epsilon }{Y}^{\text{T}}Y$

3. 带记忆的鲁棒无源性分析

定理1：对于闭环系统(4)如果 $C_{2i}+C_{2i}^{\text{T}}>0$， $E^{\text{T}}P=P^{\text{T}}E\ge 0$ 存在可逆矩阵 $P\in P^{n\times n}$， $K_{\sigma \left(t\right)}\in R^{p\times n}$， $K_{1\sigma \left(t\right)}\in R^{p\times n}$， $K_{2\sigma \left(t\right)}\in R^{p\times n}$ 和矩阵 $Q_1=Q_1^{\text{T}}>0$， $Q_2=Q_2^{\text{T}}>0$，且 $Q_1{}^{n\times n},Q_2\in R^{n\times n}$ 正实数 $h_1<1,h_2<1$ 使得下面不等式成立：

$\left[\begin{array}{cccc}\Xi & P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}& P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}& P{B}_{2i}-C_i^{\text{T}}-K_i^{\text{T}}{C}_i^{\text{T}}\\ *& -\left(1-h_1\right)Q_1& 0& -K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\\ *& *& -\left(1-h_2\right)Q_2& -K_{1i}^{\text{T}}{C}_{2i}^{\text{T}}\\ *& *& *& -C_{2i}-C_{2i}^{\text{T}}\end{array}\right]<0$ (5)

其中 $\Xi =P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P$，*表示关于对角线的对称矩阵部分，则此时Lurie切换广义系统(1)在任意切换规则下对所有扰动是渐进稳定且严格无源的。

证明：先证在外部扰动 $\omega \left(t\right)\equiv 0$ 时，闭环系统(4)是渐近稳定的。

构造Lyapunov-Krasovskii泛函如下

$\begin{array}{l}{V}_1\left(x\left(t\right),t\right)=x^{\text{T}}\left(t\right)E^{\text{T}}Px\left(t\right)+{\displaystyle {\int }_{t-d_1\left(t\right)}^t{x}^{\text{T}}\left(s\right)Q_{1}x\left(s\right)\text{d}s}\\ +{\displaystyle {\int }_{t-d_2\left(t\right)}^t{x}^{\text{T}}\left(s\right)Q_{2}x\left(s\right)\text{d}s}\end{array}$ (6)

其中 $P\in R^{n\times n}$， $Q_1\in R^{n\times n}$， $Q_2\in R^{n\times n}$。记 $\sigma \left(t\right)=i,\left(i\in \bar{N}\right)$，及 $\xi \left(t\right)={\left(x^{\text{T}}\left(t\right),x^{\text{T}}\left(t-d_1\left(t\right)\right),x^{\text{T}}\left(t-d_2\left(t\right)\right)\right)}^{\text{T}}$， $\eta \left(t\right)={\left(x^{\text{T}}\left(t\right),x^{\text{T}}\left(t-d_1\left(t\right)\right),x^{\text{T}}\left(t-d_2\left(t\right)\right),{\omega }^{\text{T}}\left(t\right)\right)}^{\text{T}}$

对 $V_1\left(x\left(t\right),t\right)$ 沿闭环系统(4)求导，可以得到

$\begin{array}{l}{{\stackrel{\dot{}}{V}}_1\left(x\left(t\right),t\right)|}_{\left(4\right)}\\ ={\stackrel{\dot{}}{x}}^{\text{T}}\left(t\right)E^{\text{T}}Px\left(t\right)+x^{\text{T}}\left(t\right)E^{\text{T}}P\stackrel{\dot{}}{x}\left(t\right)+x^{\text{T}}\left(t\right)Q_{1}x\left(t\right)-\left(1-{\stackrel{\dot{}}{d}}_1\left(t\right)\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)\\ +x^{\text{T}}\left(t\right)Q_{2}x\left(t\right)-\left(1-{\stackrel{\dot{}}{d}}_2\left(t\right)\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x\left(t-d_2\left(t\right)\right)\\ ={\left(E\stackrel{\dot{}}{x}\left(t\right)\right)}^{\text{T}}Px\left(t\right)+x^{\text{T}}\left(t\right)P^{\text{T}}\left(E\stackrel{\dot{}}{x}\left(t\right)\right)+x^{\text{T}}\left(t\right)Q_{1}x\left(t\right)-\left(1-{\stackrel{\dot{}}{d}}_1\left(t\right)\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)\\ +x^{\text{T}}\left(t\right)Q_{2}x\left(t\right)-\left(1-{\stackrel{\dot{}}{d}}_2\left(t\right)\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x\left(t-d_2(t)\right)\end{array}$

$\begin{array}{l}={\left[\left({\bar{A}}_i+B_{1i}{K}_i\right)x\left(t\right)+\left({\bar{A}}_{1i}+B_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)+\left({\bar{A}}_{2i}+B_{1i}{K}_{2i}\right)x\left(t-d_2\left(t\right)\right)+B_{2i}\omega \left(t\right)+Df\left(\mu \left(t\right)\right)\right]}^{\text{T}}Px\left(t\right)\\ +x^{\text{T}}\left(t\right)P\left[\left({\bar{A}}_i+B_{1i}{K}_i\right)x\left(t\right)+\left({\bar{A}}_{1i}+B_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)+\left({\bar{A}}_{2i}+B_{1i}{K}_{2i}\right)x\left(t-d_2\left(t\right)\right)+B_{2i}\omega \left(t\right)+Df\left(\mu \left(t\right)\right)\right]\\ +x^{\text{T}}\left(t\right)Q_{1}x\left(t\right)-\left(1-{\stackrel{\dot{}}{d}}_1\left(t\right)\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)+x^{\text{T}}\left(t\right)Q_{2}x\left(t\right)-\left(1-{\stackrel{\dot{}}{d}}_2\left(t\right)\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x\left(t-d_2\left(t\right)\right)\\ =f^{\text{T}}\left(\mu \left(t\right)\right)D^{\text{T}}Px\left(t\right)+{\omega }^{\text{T}}\left(t\right)B_{2i}^{\text{T}}Px\left(t\right)+x^{\text{T}}\left(t-d_2\left(t\right)\right)\left(K_{2i}^{\text{T}}{B}_{1i}^{\text{T}}+{\bar{A}}_{2i}^{\text{T}}\right)Px(t)\end{array}$

$\begin{array}{l}+x^{\text{T}}\left(t-d_1\left(t\right)\right)\left(K_{1i}^{\text{T}}{B}_{1i}^{\text{T}}+{\bar{A}}_{1i}^{\text{T}}\right)Px\left(t\right)+x^{\text{T}}\left(t\right)\left(K_i^{\text{T}}{B}_{1i}^{\text{T}}+{\bar{A}}_i^{\text{T}}\right)Px\left(t\right)\\ +x^{\text{T}}\left(t\right)\left(P{\bar{A}}_i+B_{1i}{K}_i\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{1i}+B_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)\\ +x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{2i}+B_{1i}{K}_{2i}\right)x\left(t-d_2\left(t\right)\right)+x^{\text{T}}\left(t\right)P{B}_{2i}\omega \left(t\right)+x^{\text{T}}\left(t\right)PDf\left(\mu \left(t\right)\right)\\ +x^{\text{T}}\left(t\right)Q_{1}x\left(t\right)-\left(1-{\stackrel{\dot{}}{d}}_1\left(t\right)\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)+x^{\text{T}}\left(t\right)Q_{2}x\left(t\right)\\ -\left(1-{\stackrel{\dot{}}{d}}_2\left(t\right)\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x(t-d_2(\; t\; ))\end{array}$

$\begin{array}{l}=x^{\text{T}}\left(t\right)\left(P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{K}_i+K_i^{\text{T}}{B}_{1i}^{\text{T}}P+Q_1+Q_2\right)x\left(t\right)+x^{\text{T}}\left(t-d_1\left(t\right)\right)\left({\bar{A}}_{1i}^{\text{T}}P+K_{1i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)\\ +x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)+x^{\text{T}}\left(t-d_2\left(t\right)\right)\left({\bar{A}}_{2i}^{\text{T}}P+K_{2i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)\\ +x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}\right)x\left(t-d_1\left(t\right)\right)-\left(1-{\stackrel{\dot{}}{d}}_1\left(t\right)\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)\\ -\left(1-{\stackrel{\dot{}}{d}}_2\left(t\right)\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x\left(t-d_2\left(t\right)\right)+x^{\text{T}}\left(t\right)P{B}_{2i}\omega \left(t\right)\\ +{\omega }^{\text{T}}\left(t\right)B_{2i}^{\text{T}}Px\left(t\right)+x^{\text{T}}\left(t\right)PDf\left(\mu \left(t\right)\right)+f^{\text{T}}\left(\mu \left(t\right)\right)D^{\text{T}}Px\left(t\right)\end{array}$ (7)

由(2)式霍尔维茨角域可以等价表示为 $f_j\left(\mu \left(t\right)\right)\left(f_j\left(\mu \left(t\right)\right)-{\lambda }_j{c}_{j}x\left(t\right)\right)\le 0,j=1,2,\cdots ,l$，可以导出

$f^{\text{T}}\left(\mu \left(t\right)\right)f\left(\mu \left(t\right)\right)\le x^{\text{T}}\left(t\right)C^{\text{T}}{ƛ}^{\text{T}}ƛCx\left(t\right)$ (8)

所以由引理3及(8)式可以得到

$\begin{array}{l}{x}^{\text{T}}\left(t\right)PDf\left(\mu \left(t\right)\right)+f^{\text{T}}\left(\mu \left(t\right)\right)D^{\text{T}}Px\left(t\right)\\ \le f^{\text{T}}\left(\mu \left(t\right)\right)f\left(\mu \left(t\right)\right)+x^{\text{T}}\left(t\right)PD{D}^{\text{T}}Px\left(t\right)\\ \le x^{\text{T}}\left(t\right)C^{\text{T}}{ƛ}^{\text{T}}ƛCx\left(t\right)+x^{\text{T}}\left(t\right)PD{D}^{\text{T}}Px\left(t\right)\end{array}$ (9)

所以由 $0<{\stackrel{\dot{}}{d}}_1\left(t\right)\le h_1<1$， $0<{\stackrel{\dot{}}{d}}_2\left(t\right)\le h_2<1$ 以及(9)式可以得到

$\begin{array}{c}{{\stackrel{\dot{}}{V}}_1\left(x\left(t\right),t\right)|}_{\left(4\right)}\le x^{\text{T}}\left(t\right)\left(P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{K}_i+K_i^{\text{T}}{B}_{1i}^{\text{T}}P+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P\right)x\left(t\right)\\ +x^{\text{T}}\left(t-d_1\left(t\right)\right)\left({\bar{A}}_{1i}^{\text{T}}P+K_{1i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)\\ +x^{\text{T}}\left(t-d_2\left(t\right)\right)\left({\bar{A}}_{2i}^{\text{T}}P+K_{2i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}\right)x\left(t-d_1\left(t\right)\right)\\ -\left(1-h_1\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)-\left(1-h_2\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x\left(t-d_2\left(t\right)\right)\\ +x^{\text{T}}\left(t\right)P{B}_{2i}\omega \left(t\right)+{\omega }^{\text{T}}\left(t\right)B_{2i}^{\text{T}}Px\left(t\right)\end{array}$ (10)

由引理3可得

$x^{\text{T}}\left(t\right)P{B}_{1i}{K}_{i}x\left(t\right)+x^{\text{T}}\left(t\right)K_i^{\text{T}}{B}_{1i}^{\text{T}}Px\left(t\right)\le x^{\text{T}}\left(t\right)P{B}_{1i}{B}_{1i}^{\text{T}}Px\left(t\right)+x^{\text{T}}\left(t\right)K_i^{\text{T}}{K}_{i}x\left(t\right)$ (11)

将(11)式代入(10)式得

$\begin{array}{c}{\stackrel{\dot{}}{V}\left(x\left(t\right),t\right)|}_{\left(4\right)}\le x^{\text{T}}\left(t\right)\left(P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P\right)x\left(t\right)\\ +x^{\text{T}}\left(t-d_1\left(t\right)\right)\left({\bar{A}}_{1i}^{\text{T}}P+K_{1i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)\\ +x^{\text{T}}\left(t-d_2\left(t\right)\right)\left({\bar{A}}_{2i}^{\text{T}}P+K_{2i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}\right)x\left(t-d_1\left(t\right)\right)\\ -\left(1-h_1\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)-\left(1-h_2\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x\left(t-d_2\left(t\right)\right)\\ +x^{\text{T}}\left(t\right)P{B}_{2i}\omega \left(t\right)+{\omega }^{\text{T}}\left(t\right)B_{2i}^{\text{T}}Px\left(t\right)\end{array}$ (12)

记

$\xi \left(t\right)={\left(x^{\text{T}}\left(t\right),x^{\text{T}}\left(t-d_1\left(t\right)\right),x^{\text{T}}\left(t-d_2\left(t\right)\right)\right)}^{\text{T}}$

所以

${{\stackrel{\dot{}}{V}}_1\left(x\left(t\right),t\right)|}_{\left(4\right)}\le {\xi }^{\text{T}}\left(t\right)H_1\xi (t)$

其中

$H_1=\left[\begin{array}{ccc}\Xi & P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}& P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}\\ *& -\left(1-h_1\right)Q_1& 0\\ *& *& -\left(1-h_2\right)Q_2\end{array}\right]$

$\Xi =P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P$

由(5)式及Schur补引理知， $H_1<0$ 所以 ${\stackrel{\dot{}}{V}}_1\left(x\left(t\right),t\right)<0$，通过Lyapunov稳定性理论得到闭环(4)是渐近稳定的。

下证闭环系统(4)满足的无源性。

$\begin{array}{l}{\stackrel{\dot{}}{V}}_1\left(x\left(t\right),t\right)-{\omega }^{\text{T}}\left(t\right)z\left(t\right)\\ ={\stackrel{\dot{}}{V}}_1\left(x\left(t\right),t\right)-{\omega }^{\text{T}}\left(t\right)z\left(t\right)-z^{\text{T}}\left(t\right)\omega \left(t\right)+z^{\text{T}}\left(t\right)\omega \left(t\right)\\ =x^{\text{T}}\left(t\right)\left(P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P\right)x\left(t\right)\\ +x^{\text{T}}\left(t-d_1\left(t\right)\right)\left({\bar{A}}_{1i}^{\text{T}}P+K_{1i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)\\ +x^{\text{T}}\left(t-d_2\left(t\right)\right)\left({\bar{A}}_{2i}^{\text{T}}P+K_{2i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}\right)x(t-d_2(t))\end{array}$

$\begin{array}{l}-\left(1-h_1\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)-\left(1-h_2\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x\left(t-d_2\left(t\right)\right)\\ +x^{\text{T}}\left(t\right)P{B}_{2i}\omega \left(t\right)+{\omega }^{\text{T}}\left(t\right)B_{2i}^{\text{T}}Px\left(t\right)-{\omega }^{\text{T}}\left(t\right)(\left(C_i+C_i{K}_i\right)x\left(t\right)+C_{1i}{K}_{1i}x\left(t-d_1\left(t\right)\right)\\ +C_{1i}{K}_{2i}x\left(t-d_2\left(t\right)\right)+C_{2i}\omega \left(t\right))-(\left(C_i+C_i{K}_i\right)x\left(t\right)+C_{1i}{K}_{1i}x\left(t-d_1\left(t\right)\right)\\ +{C_{1i}{K}_{2i}x\left(t-d_2\left(t\right)\right)+C_{2i}\omega \left(t\right))}^{\text{T}}\omega \left(t\right)+z^{\text{T}}\left(t\right)\omega (t)\end{array}$

$\begin{array}{l}=x^{\text{T}}\left(t\right)\left(P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P\right)x\left(t\right)\\ +x^{\text{T}}\left(t-d_1\left(t\right)\right)\left({\bar{A}}_{1i}^{\text{T}}P+K_{1i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}\right)x\left(t-d_1\left(t\right)\right)\\ +x^{\text{T}}\left(t-d_2\left(t\right)\right)\left({\bar{A}}_{2i}^{\text{T}}P+K_{2i}^{\text{T}}{B}_{1i}^{\text{T}}P\right)x\left(t\right)+x^{\text{T}}\left(t\right)\left(P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}\right)x\left(t-d_2\left(t\right)\right)\\ -\left(1-h_1\right)x^{\text{T}}\left(t-d_1\left(t\right)\right)Q_{1}x\left(t-d_1\left(t\right)\right)-\left(1-h_2\right)x^{\text{T}}\left(t-d_2\left(t\right)\right)Q_{2}x(t-d_2(t))\end{array}$

$\begin{array}{l}+x^{\text{T}}\left(t\right)\left(P{B}_{2i}-K_i^{\text{T}}{C}_i^{\text{T}}-C_i^{\text{T}}\right)\omega \left(t\right)+{\omega }^{\text{T}}\left(t\right)\left(B_{2i}^{\text{T}}P-C_i-C_i{K}_i\right)x\left(t\right)\\ -x^{\text{T}}\left(t-d_1\left(t\right)\right)K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\omega \left(t\right)-x^{\text{T}}\left(t-d_2\left(t\right)\right)K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\omega \left(t\right)\\ -{\omega }^{\text{T}}\left(t\right)C_{1i}{K}_{1i}x\left(t-d_1\left(t\right)\right)-{\omega }^{\text{T}}\left(t\right)C_{1i}{K}_{2i}x\left(t-d_2\left(t\right)\right)\\ +{\omega }^{\text{T}}\left(t\right)\left(-C_{2i}-C_{2i}^{\text{T}}\right)\omega \left(t\right)\\ ={\eta }^{\text{T}}\left(t\right)H_2\eta \left(t\right)+z^{\text{T}}\left(t\right)\omega (t)\end{array}$

其中

$H_2=\left[\begin{array}{cccc}\Xi & P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}& P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}& P{B}_{2i}-C_i^{\text{T}}-K_i^{\text{T}}{C}_i^{\text{T}}\\ *& -\left(1-h_1\right)Q_1& 0& -K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\\ *& *& -\left(1-h_2\right)Q_2& -K_{1i}^{\text{T}}{C}_{2i}^{\text{T}}\\ *& *& *& -C_{2i}-C_{2i}^{\text{T}}\end{array}\right]$

$\Xi =P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P$，

$\eta \left(t\right)={\left(x^{\text{T}}\left(t\right),x^{\text{T}}\left(t-d_1\left(t\right)\right),x^{\text{T}}\left(t-d_2\left(t\right)\right),{\omega }^{\text{T}}\left(t\right)\right)}^{\text{T}}$

而根据矩阵不等式(5)可以得到 $H_2<0$ 则 ${\stackrel{\dot{}}{V}}_1\left(x\right)-2{\omega }^{\text{T}}\left(t\right)z\left(t\right)<0$ 取 $V\left(x\right)=\frac{1}{2}{V}_1\left(x\right)$，此时满足定义1，所以Lurie切换广义系统(1)是渐进稳定且严格无源的。

4. 带有记忆的无源控制器设计

定理2：如果 $C_{2i}+C_{2i}^{\text{T}}>0$， $E^{\text{T}}P=P^{\text{T}}E\ge 0$ 对于给定标量 $\epsilon >0$， $0<h_1<1$， $0<h_2<1$ 系统(1)是具有鲁棒无源可切换镇定的。如果存在非奇异矩阵 $M_i\in R^{p\times n}\left(i=1,2,\cdots \right)$， $X^{\text{T}}=X>0$， $S_1\in R^{n\times n}$， $S_2\in R^{n\times n}$ 有下列矩阵不等式成立：

$\left[\begin{array}{cccccccccc}{A}_{i}X+X^{\text{T}}{A}_i^{\text{T}}+S_1+S_2& A_{1i}X+B_{1i}{M}_{1i}& A_{2i}X+B_{1i}{M}_{2i}& B_{1i}& B_{2i}-M_i^{\text{T}}{C}_i^{\text{T}}-X^{\text{T}}{C}_i^{\text{T}}& M_i^{\text{T}}& {\epsilon }^{-1}{N}_i& \epsilon X^{\text{T}}{G}_i^{\text{T}}& X^{\text{T}}{C}^{\text{T}}{ƛ}^{\text{T}}& D\\ *& -\left(1-h_1\right)S_1& 0& 0& -M_{1i}^{\text{T}}{C}_{1i}^{\text{T}}& 0& 0& \epsilon X^{\text{T}}{G}_{1i}^{\text{T}}& 0& 0\\ *& *& -\left(1-h_2\right)S_2& 0& -M_{1i}^{\text{T}}{C}_{2i}^{\text{T}}& 0& 0& \epsilon X^{\text{T}}{G}_{2i}^{\text{T}}& 0& 0\\ *& *& *& -I& 0& 0& 0& 0& 0& 0\\ *& *& *& *& -C_{2i}-C_{2i}^{\text{T}}& 0& 0& 0& 0& 0\\ *& *& *& *& *& -I& 0& 0& 0& 0\\ *& *& *& *& *& *& -I& 0& 0& 0\\ *& *& *& *& *& *& *& -I& 0& 0\\ *& *& *& *& *& *& *& *& -I& 0\\ *& *& *& *& *& *& *& *& *& -I\end{array}\right]<0$ (13)

使得控制增益矩阵为 $K_i=M_i{X}_i^{-1}$， $u\left(t\right)=K_{\sigma \left(t\right)}x\left(t\right)+K_{1\sigma \left(t\right)}x\left(t-d_1\left(t\right)\right)+K_{2\sigma \left(t\right)}x\left(t-d_2\left(t\right)\right)$ 且 $K_{\sigma \left(t\right)}\in R^{p\times n}$， $K_{1\sigma \left(t\right)}\in R^{p\times n}$， $K_{2\sigma \left(t\right)}\in R^{p\times n}$ 是系统(1)的鲁棒无源记忆反馈控制器。

证明：由定理1知，

$\left[\begin{array}{cccc}\Xi & P{\bar{A}}_{1i}+P{B}_{1i}{K}_{1i}& P{\bar{A}}_{2i}+P{B}_{1i}{K}_{2i}& P{B}_{2i}-C_i^{\text{T}}-K_i^{\text{T}}{C}_i^{\text{T}}\\ *& -\left(1-h_1\right)Q_1& 0& -K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\\ *& *& -\left(1-h_2\right)Q_2& -K_{1i}^{\text{T}}{C}_{2i}^{\text{T}}\\ *& *& *& -C_{2i}-C_{2i}^{\text{T}}\end{array}\right]<0$ (14)

$\Xi =P{\bar{A}}_i+{\bar{A}}_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P$

将不确定项 $\left[\Delta A_i,\Delta A_{1i},\Delta A_{2i}\right]=N_i{F}_i\left(t\right)\left[G_i,G_{1i},G_{2i}\right]$， ${\bar{A}}_i=A_i+N_i{F}_i\left(t\right)G_i$， ${\bar{A}}_{1i}=A_{1i}+N_i{F}_i\left(t\right)G_{1i}$， ${\bar{A}}_{2i}=A_{2i}+N_i{F}_i\left(t\right)G_{2i}$ 代入(14)式得

$\left[\begin{array}{cccc}{\Xi }_1+P{N}_i{F}_i\left(t\right)G_i+G_i^{\text{T}}{F}_i^{\text{T}}\left(t\right)N_i^{\text{T}}P& P{A}_{1i}+P{N}_i{F}_i\left(t\right)G_{1i}+P{B}_{1i}{K}_{1i}& P{A}_{2i}+P{N}_i{F}_i\left(t\right)G_{2i}+P{B}_{1i}{K}_{2i}& P{B}_{2i}-C_i^{\text{T}}-K_i^{\text{T}}{C}_i^{\text{T}}\\ \text{*}& -\left(1-h_1\right)Q_1& 0& -K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\\ *& *& -\left(1-h_2\right)Q_2& -K_{1i}^{\text{T}}{C}_{2i}^{\text{T}}\\ *& *& *& -C_{2i}-C_{2i}^{\text{T}}\end{array}\right]<0$ (15)

此时

${\Xi }_1=P{A}_i+A_i^{\text{T}}P+P{B}_{1i}{B}_{1i}^{\text{T}}P+K_i^{\text{T}}{K}_i+Q_1+Q_2+C^{\text{T}}{ƛ}^{\text{T}}ƛC+PD{D}^{\text{T}}P$

记

$H_3=\left[\begin{array}{cccc}{\Xi }_1& P{A}_{1i}+P{B}_{1i}{K}_{1i}& P{A}_{2i}+P{B}_{1i}{K}_{2i}& P{B}_{2i}-C_i^{\text{T}}-K_i^{\text{T}}{C}_i^{\text{T}}\\ \text{*}& -\left(1-h_1\right)Q_1& 0& -K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\\ *& *& -\left(1-h_2\right)Q_2& -K_{1i}^{\text{T}}{C}_{2i}^{\text{T}}\\ *& *& *& -C_{2i}-C_{2i}^{\text{T}}\end{array}\right]$

则式(15)等价为

$H_3+\left[\begin{array}{c}{G}_i^{\text{T}}\\ G_{1i}^{\text{T}}\\ G_{2i}^{\text{T}}\\ 0\end{array}\right]F_i^{\text{T}}\left(t\right)\left[\begin{array}{cccc}{N}_i^{\text{T}}P& 0& 0& 0\end{array}\right]+\left[\begin{array}{c}P{N}_i\\ 0\\ 0\\ 0\end{array}\right]F_i\left(t\right)\left[\begin{array}{cccc}{G}_1& G_{1i}& G_{2i}& 0\end{array}\right]<0$ (16)

由引理1可知，要使不等式(16)成立，当且仅存在一个常数 $\epsilon >0$

$H_3+{\epsilon }^2{\left[\begin{array}{cccc}{G}_i& G_{1i}& G_{2i}& 0\end{array}\right]}^{\text{T}}\left[\begin{array}{cccc}{G}_i& G_{1i}& G_{2i}& 0\end{array}\right]+{\epsilon }^{-2}\left[\begin{array}{c}P{N}_i\\ 0\\ 0\\ 0\end{array}\right]{\left[\begin{array}{c}P{N}_i\\ 0\\ 0\\ 0\end{array}\right]}^{\text{T}}<0$ (17)

整理(17)式可得

$\left[\begin{array}{cccc}{\Xi }_1+{\epsilon }^2{G}_i^{\text{T}}{G}_i+{\epsilon }^{-2}P{N}_i{N}_i^{\text{T}}P& P{A}_{1i}+P{B}_{1i}{K}_{1i}+{\epsilon }^2{G}_i^{\text{T}}{G}_{1i}& P{A}_{2i}+P{B}_{1i}{K}_{2i}+{\epsilon }^2{G}_i^{\text{T}}{G}_{2i}& P{B}_{2i}-C_i^{\text{T}}-K_i^{\text{T}}{C}_i^{\text{T}}\\ *& -\left(1-h_1\right)Q_1+{\epsilon }^2{G}_{1i}^{\text{T}}{G}_{1i}& {\epsilon }^2{G}_{1i}^{\text{T}}{G}_{2i}& -K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}\\ *& *& -\left(1-h_2\right)Q_2+{\epsilon }^2{G}_{2i}^{\text{T}}{G}_{2i}& -K_{1i}^{\text{T}}{C}_{2i}^{\text{T}}\\ *& *& *& -C_{2i}-C_{2i}^{\text{T}}\end{array}\right]<0$ (18)

由Schur补引理知(18)式等价为

$\left[\begin{array}{cccccccccc}P{A}_i+A_i^{\text{T}}P+Q_1+Q_2& P{A}_{1i}+P{B}_{1i}{K}_{1i}& P{A}_{2i}+P{B}_{1i}{K}_{2i}& P{B}_{1i}& P{B}_{2i}-K_i^{\text{T}}{C}_i^{\text{T}}-C_i^{\text{T}}& K_i^{\text{T}}& {\epsilon }^{-1}P{N}_i& \epsilon G_i^{\text{T}}& C^{\text{T}}{ƛ}^{\text{T}}& PD\\ *& -\left(1-h_1\right)Q_1& 0& 0& -K_{1i}^{\text{T}}{C}_{1i}^{\text{T}}& 0& 0& \epsilon G_{1i}^{\text{T}}& 0& 0\\ *& *& -\left(1-h_2\right)Q_2& 0& -K_{1i}^{\text{T}}{C}_{2i}^{\text{T}}& 0& 0& \epsilon G_{2i}^{\text{T}}& 0& 0\\ *& *& *& -I& 0& 0& 0& 0& 0& 0\\ *& *& *& *& -C_{2i}-C_{2i}^{\text{T}}& 0& 0& 0& 0& 0\\ *& *& *& *& *& -I& 0& 0& 0& 0\\ *& *& *& *& *& *& -I& 0& 0& 0\\ *& *& *& *& *& *& *& -I& 0& 0\\ *& *& *& *& *& *& *& *& -I& 0\\ *& *& *& *& *& *& *& *& *& -I\end{array}\right]<0$ (19)

记 $X=P^{-1}$， $S_1=P^{-\text{T}}{Q}_1{P}^{-1}$， $S_2=P^{-\text{T}}{Q}_2{P}^{-1}$， $K_i=M_i{X}_i^{-1}$，其中 $M_i\in R^{p\times n}$

对不等式(19)左乘 $diag\left\{P^{-\text{T}},P^{-\text{T}},P^{-\text{T}},I,I,I,I,I,I,I\right\}$ 右乘 $diag\left\{P^{-1},P^{-1},P^{-1},I,I,I,I,I,I,I\right\}$ 得到等价的线性化不等式(13)，定理2证毕。

5. 数值仿真

考虑以下系统

$\{\begin{array}{l}E\stackrel{\dot{}}{x}\left(t\right)=\left(A_{\sigma \left(t\right)}+\Delta A_{\sigma \left(t\right)}\left(t\right)\right)x\left(t\right)+\left(A_{1\sigma \left(t\right)}+\Delta A_{1\sigma \left(t\right)}\left(t\right)\right)x\left(t-d_1\left(t\right)\right)\\ +\left(A_{2\sigma \left(t\right)}+\Delta A_{2\sigma \left(t\right)}\left(t\right)\right)x\left(t-d_2\left(t\right)\right)+B_{1\sigma \left(t\right)}u\left(t\right)+B_{2\sigma \left(t\right)}\omega \left(t\right)+Df\left(\mu \left(t\right)\right)\\ z\left(t\right)=C_{\sigma \left(t\right)}x\left(t\right)+C_{1\sigma \left(t\right)}u\left(t\right)+C_{2\sigma \left(t\right)}\omega \left(t\right)\\ \mu \left(t\right)=Cx\left(t\right)\\ x\left(t\right)=\phi \left(t\right)\in C\left(\left[-d,0\right],R^n\right)\end{array}$

系数矩阵如下：

$A_1=\left[\begin{array}{cc}-4.5& -0.5\\ 1& -3.5\end{array}\right]$， $A_2=\left[\begin{array}{cc}-4.5& -0.5\\ 1& -3.5\end{array}\right]$， $A_{11}=\left[\begin{array}{cc}-2& 0.3\\ -0.3& 0.1\end{array}\right]$， $A_{12}=\left[\begin{array}{cc}0.1& 0.1\\ -0.1& 0.1\end{array}\right]$， $A_{21}=\left[\begin{array}{cc}-2& 0.4\\ -0.3& 0.1\end{array}\right]$，

$A_{22}=\left[\begin{array}{cc}0.1& 0.3\\ -0.1& 0.1\end{array}\right]$， $B_{11}=\left[\begin{array}{cc}-0.1& 0.1\\ 0.1& 0.5\end{array}\right]$， $B_{12}=\left[\begin{array}{cc}0.9& 0.1\\ -0.2& 0\end{array}\right]$， $B_{21}=\left[\begin{array}{cc}-0.1& -0.1\\ 0.1& 0.5\end{array}\right]$， $B_{22}=\left[\begin{array}{cc}0.8& 0.1\\ -0.2& 0\end{array}\right]$，

$C_1=\left[\begin{array}{cc}-0.1& 0.2\\ 0.1& 0.2\end{array}\right]$， $C_2=\left[\begin{array}{cc}-0.3& 0.13\\ -0.1& 0.1\end{array}\right]$， $C_{11}=\left[\begin{array}{cc}-0.1& 0.1\\ 0.1& 0.2\end{array}\right]$， $C_{12}=\left[\begin{array}{cc}0.1& 0.1\\ -0.1& 8\end{array}\right]$， $C_{21}=\left[\begin{array}{cc}-0.1& 0.3\\ 0.1& 0.2\end{array}\right]$，

$C_{22}=\left[\begin{array}{cc}-0.4& 0.13\\ -0.1& 0.1\end{array}\right]$， $D_1=\left[\begin{array}{cc}-0.4& -0.5\\ 0.1& -2\end{array}\right]$， $D_2=\left[\begin{array}{cc}-0.4& -0.5\\ 0.4& -2\end{array}\right]$， $G_1=\left[\begin{array}{cc}1.6& -1\\ 0.2& 0\end{array}\right]$， $G_2=\left[\begin{array}{cc}1.57& -1\\ 0.2& 0\end{array}\right]$，

$G_{11}=\left[\begin{array}{cc}0& -0.1\\ -0.1& 0.2\end{array}\right]$， $G_{12}=\left[\begin{array}{cc}-9& -0.1\\ -0.1& 0.2\end{array}\right]$， $C_{21}=\left[\begin{array}{cc}-0.1& 0.1\\ 0.2& 0.2\end{array}\right]$， $C_{22}=\left[\begin{array}{cc}0.13& 0.1\\ -0.1& 8\end{array}\right]$， $N_1=\left[\begin{array}{cc}0.1& -0.1\\ 0.3& 0.1\end{array}\right]$，

$N_2=\left[\begin{array}{cc}0.15& -0.1\\ 0.3& 0.1\end{array}\right]$， ${ƛ}_1=\left[\begin{array}{cc}0.5& 0.1\\ 0& 1\end{array}\right]$， ${ƛ}_2=\left[\begin{array}{cc}0.5& 0.1\\ 0.1& 1\end{array}\right]$

给定 $\epsilon =0.2$， $h_1=0.1$， $h_2=0.01$ 由(13)式及MATLAB求出相应的可行解为：

$X_1=\left[\begin{array}{cc}1.3653& 2.7334\\ 2.7334& 17.1041\end{array}\right]$， $X_2=\left[\begin{array}{cc}0.8294& 1.4702\\ 1.4702& 13.6065\end{array}\right]$， $M_1=\left[\begin{array}{cc}-0.2012& 0.0792\\ -0.2325& 0.1076\end{array}\right]$，

$M_2=\left[\begin{array}{cc}-0.2064& 0.0587\\ -0.0066& 0.0080\end{array}\right]$， $M_{11}=\left[\begin{array}{cc}-0.4291& -0.0081\\ -0.5273& -0.0399\end{array}\right]$， $M_{12}=\left[\begin{array}{cc}-8.7714& -22.0217\\ -11.8122& -22.7827\end{array}\right]$，

$M_{21}=\left[\begin{array}{cc}-0.9305& 0.0042\\ -5.6686& 0.0657\end{array}\right]$， $M_{22}=\left[\begin{array}{cc}20.2108& 41.4120\\ -17.5970& -32.7827\end{array}\right]$， $S_1=\left[\begin{array}{cc}3.5131& 6.2150\\ 6.2150& 34.0292\end{array}\right]$，

$S_2=\left[\begin{array}{cc}7.2766& 15.5920\\ 15.5920& 37.2564\end{array}\right]$， $S_{11}=\left[\begin{array}{cc}1.4766& 2.1657\\ 2.1657& 22.7182\end{array}\right]$， $S_{22}=\left[\begin{array}{cc}5.1256& 9.8429\\ 9.8429& 24.5751\end{array}\right]$，

$P_1=\left[\begin{array}{cc}1.0771& -0.1721\\ -0.1721& 0.0860\end{array}\right]$， $P_2=\left[\begin{array}{cc}1.4913& -0.1611\\ -0.1611& 0.0909\end{array}\right]$， $Q_1=\left[\begin{array}{cc}2.7794& -0.3953\\ -0.3953& 0.1717\end{array}\right]$，

$Q_2=\left[\begin{array}{cc}3.7641& 0.0054\\ 0.0054& 0.0295\end{array}\right]$， $Q_{11}=\left[\begin{array}{cc}2.8330& -0.3378\\ -0.3378& 0.1626\end{array}\right]$， $Q_{22}=\left[\begin{array}{cc}7.3067& -0.0017\\ -0.0017& 0.0478\end{array}\right]$

控制器增益矩阵 $K_1=\left[\begin{array}{cc}-0.2304& 0.0415\\ -0.2689& 0.0493\end{array}\right]$， $K_2=\left[\begin{array}{cc}-0.3173& 0.0386\\ -0.0112& 0.0018\end{array}\right]$， $K_{11}=\left[\begin{array}{cc}-0.4608& 0.0732\\ -0.5610& 0.0873\end{array}\right]$，

$K_{12}=\left[\begin{array}{cc}-5.6569& -0.3835\\ -8.8012& 0.0745\end{array}\right]$， $K_{21}=\left[\begin{array}{cc}-1.3883& 0.1503\\ -8.4643& 0.9194\end{array}\right]$， $K_{22}=\left[\begin{array}{cc}23.4675& 0.5078\\ -20.9601& -0.1445\end{array}\right]$

两子系统的状态轨线对应的仿真曲线图如图1，图2。

Figure 1. The state trajectory of subsystem 1

图1. 子系统1的状态轨线曲线图

Figure 2. The state trajectory of subsystem 2

图2. 子系统2的状态轨线曲线图

通过子系统的仿真曲线图，我们可以很明显的得到：在选取的切换信号 $\sigma \left(t\right)$ 作用下，可以保证系统(1)为整体稳定的。

6. 结论

本文综合利用Lyapunov-Krasovskii泛函和矩阵不等式方法，研究了一类不确定性多状态变时滞广义Lurie切换系统及带有记忆的反馈控制器的鲁棒无源 控制问题，通过设计具有记忆的反馈控制器，保证了闭环系统的鲁棒渐近稳定，同时给出了在严格无源要求下可行解存在的充分性判据，最后的结果运用线性矩阵不等式的形式给出，通过数值仿真例子验证了所得结果的可行性。

基金项目

国家自然科学基金资助项目(11561053)；内蒙古师范大学十百千人才工程项目(PCRY-2-2012-K-037)；内蒙古师范大学研究生科研创新基金资助项目(CXJJS18070)、(CXJJS19096)；内蒙古自治区研究生教育创新计划资助项目(S2018111966Z)；内蒙古自然科学基金项目(2019BS01002)；内蒙古高等学校科研研究项目(NJZY19022)。