非线性广义半马尔可夫跳跃系统的H_∞控制

H_∞ Control of Nonlinear Singular Semi-Markov Jump Systems

摘要: 探讨一类非线性广义半Markov跳跃系统的随机稳定性和H_∞控制问题。设计新型的Lyapunov-Krasovkii泛函(LKF)，用于减少冗余决策变量。同时，引入参数依赖的互凸矩阵不等式(PDRCMI)来降低保守性，保证了非线性广义Markov跳跃系统渐进稳定并满足性能，最后，通过数值算例验证了所得方法的有效性。

Abstract: The stochastic stability of a class of nonlinear singular semi-Markov jumping systems and H_∞ control problems are discussed. A new type of Lyapunov-Krasovkii functional (LKF) is designed to reduce redundant decision variables. At the same time, the parameter-dependent Convex Matrix Inequality (PDRCMI) is introduced to reduce the conservatism, which ensures the asymptotic stability and satisfies the performance of the nonlinear singular Markov jumping system, and finally, the effectiveness of the proposed method is verified by numerical examples.

1. 引言

广义系统因其广泛应用于电力的系统、航空航天工程、社会经济系统、化学过程、生物系统、网络分析、飞机控制系统、电力网络、太阳能热中央接收器、机器人机械手系统等[1]-[3]领域而引起了广泛的兴趣。几十年来，稳定性问题得到了广泛的研究[1]-[3]。然而，在这些实际系统的参数和结构中，一些现象，如子系统互连的修改、突变环境干扰、组件故障或维修等原因，可能导致系统结构，参数发生随机变化。而幸运的是，这些变化可以用马尔可夫跳跃系统(MJSs)适当地描述，并且每种操作模式都与一个动态系统关联，其中模型转换在一个马尔可夫过程的控制下。因此，学术界和工业界的大量关注都集中在广义马尔可夫跳跃系统(SMJSs)的研究上。

MJSs作为一种特殊的随机系统，由于其理论意义已经在网络系统控制、能源、制造和经济系统[1]-[6]中得到了广泛的研究。许多问题已经解决，取得良好的结果，例如，H_∞控制滤波的不确定时滞MJSs [2]，具有部分未知转移速率的MJSs的稳定性分析[3]，不定二次最优控制问题[4]，基于分散二维马尔可夫跳跃系统的故障检测[7]。

2. 新判据

2.1. 模型介绍与假设

本文采用以下记号本文考虑下述不确定广义半马尔可夫跳跃系统:

$\{\begin{array}{l}E\dot{x}\left(t\right)=\left(A\left(r_t\right)+\Delta A\left(r_t\right)\right)x\left(t\right)+\left(A_d\left(r_t\right)+\Delta A_d\left(r_t\right)\right)x\left(t-d\left(t\right)\right)+B_w\left(r_t\right)\varpi \left(t\right)\\ +B\left(r_t\right)\left(u\left(t\right)+f\left(t,x\left(t\right),r\left(t\right)\right)\right)+B_d\left(r_t\right)\left(u\left(t-d\left(t\right)\right)+f\left(t,x\left(t-d\left(t\right)\right),r\left(t\right)\right)\right)\\ z\left(t\right)=C\left(r_t\right)x\left(t\right)+C_d\left(r_t\right)x\left(t-d\left(t\right)\right)+D\left(r_t\right)w\left(t\right)\end{array}$(2.1)

其中$x\left(t\right)\in {ℝ}^n$ 为系统状态向量，$u\left(t\right)\in {ℝ}^m$ 为控制输入，$z\left(t\right)\in {ℝ}^p$ 为控制输出，$w\left(t\right)\in {ℝ}^q$ 为外部扰动，矩阵$E\in {ℝ}^{n\times n}$ 可能是奇异矩阵，$Rank\left(E\right)=r\le n$ ，$f\left(t,x\left(t\right)\right)$ 为非线性函数。$A\left(r_t\right)$ ，$A_d\left(r_t\right)$ ，$B\left(r_t\right)$ ，$B_d\left(r_t\right)$ ，$B_w\left(r_t\right)$ ，$C\left(r_t\right)$ ，$C_d\left(r_t\right)$ 和$D\left(r_t\right)$ 是适当维度的已知常数矩阵。$\Delta A\left(r_t\right)$ 和$\Delta A_d\left(r_t\right)$ 是未知时变矩阵。时间延迟$d\left(t\right)$ 满足$0\le d_1\le d\left(t\right)\le d_2$ ，$h_1\le \dot{d}\left(t\right)\le h_2$ ，$\forall t\ge 0$ ，$d_1,d_2,h_1,h_2$ 是给定常数边界，同时定义$0<d_{12}=d_2-d_1$ 。模态$\left\{r_t,t\ge 0\right\}$ 是连续时间半马尔可夫过程，该过程在有限集中取值$S=\left\{1,2,\cdots ,s\right\}$ [1]，转移速率矩阵$\Pi =\left\{{\pi }_{ij}\right\}$ 如下定义：

$\mathrm{P}\left\{r\left(t+\Delta \right)=j|r\left(t\right)=i\right\}=\{\begin{array}{l}{\pi }_{ij}\Delta +o\left(\Delta \right),i\ne j,\\ 1+{\pi }_{ii}\Delta +o\left(\Delta \right),i=j,\end{array}$

这里$\Delta >0$ ，$\underset{\Delta \to 0}{\lim }\frac{o\left(\Delta \right)}{\Delta }=0$ 及${\pi }_{ij}\left(h\right)\ge 0\left(i,j\in S,i\ne j\right)$ ，指将i从t的模态转变到$t+\Delta$ 的j模态，$\forall i\in S$ 都有${\pi }_{ij}=-{\displaystyle \sum _{j=1,j\ne i}^s{\pi }_{ij}}$ 。为了便于表达定义：

$A\left(r_t\right)=A_i,A_d\left(r_t\right)=A_{di},B\left(r_t\right)=B_i,B_d\left(r_t\right)=B_{di},B_w\left(r_t\right)=B_{wi},C\left(r_t\right)=C_i,C_d\left(r_t\right)=C_{di},D\left(r_t\right)=D_i$

$\Delta A\left(r_t\right)=\Delta A_i,\Delta A_d\left(r_t\right)=\Delta A_{di},f\left(t,x\left(t\right),r\left(t\right)\right)=f\left(t\right),f\left(t,x\left(t-d\left(t\right)\right),r\left(t\right)\right)=f\left(t-d\left(t\right)\right).$

为了简化系统的形式，我们可以将系统(2.1)重写如下：

$\{\begin{array}{l}E\dot{x}\left(t\right)=\left(A_i+\Delta A_i\right)x\left(t\right)+\left(A_{di}+\Delta A_{di}\right)x\left(t-d\left(t\right)\right)+B_i\left(u\left(t\right)+f\left(t\right)\right)\\ +B_{di}\left(u\left(t-d\left(t\right)\right)+f\left(t-d\left(t\right)\right)\right)+B_{wi}\varpi \left(t\right)\\ Z\left(t\right)=C_{i}x\left(t\right)+C_{di}x\left(t-d\left(t\right)\right)+D_{i}w\left(t\right)\end{array}$(2.2)

假设转移速率矩阵$\Pi \triangleq \left({\pi }_{ij}\right)$ 通常不确定，我们为系统(2.1)计算转移速率矩阵，如下所示：

$\left[\begin{array}{ccccc}{\widehat{\pi }}_{11}+\Delta {\pi }_{11}& ?& {\widehat{\pi }}_{13}+\Delta {\pi }_{13}& \cdots & ?\\ ?& ?& {\widehat{\pi }}_{23}+\Delta {\pi }_{23}& \cdots & {\widehat{\pi }}_{2s}+\Delta {\pi }_{2s}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ ?& {\widehat{\pi }}_{s2}+\Delta {\pi }_{s2}& ?& \cdots & {\widehat{\pi }}_{ss}+\Delta {\pi }_{ss}\end{array}\right]$ (2.3)

其中${\widehat{\pi }}_{ij}$ 和$\Delta {\pi }_{ij}\in \left[-{\delta }_{ij},{\delta }_{ij}\right]{\left({\delta }_{ij}\ge 0\right)}_{ij}\in \left[-{\delta }_{ij},{\delta }_{ij}\right]\left({\delta }_{ij}\ge 0\right)$ 分别代表估计值和不确定转移速率${\pi }_{ij}$ 的估计误差。${\widehat{\pi }}_{ij}$ 和${\delta }_{ij}$ 为已知，“?”是完全未知的。对任意$i\in S$ ，集合$U^i$ 表示$U^i=U_k^i\cup U_{uk}^i$ 。其中$U_k^i\triangleq \left\{j:{\pi }_{ij}为已知,j\in S\right\}$ ，$U_{uk}^i\triangleq \left\{j:{\pi }_{ij}为未知,j\in S\right\}$ 。此外，若$U_k^i\ne \varnothing$ ，它可以被描述为$U_k^i=\left\{k_1^i,k_2^i,\cdots ,k_{mi}^i\right\}$ ，其中$k_{mi}^i\in S^{+}$ 表示矩阵$\Pi$ 的第i排的第m个已知边界。

备注2.1无论是有界不确定的TR(BUTR)或部分未知TR(PUTR)模型均不如上述不确定转移速率的描述那么宽泛。重写了以下两个不确定的模型：

$\left[\begin{array}{cccc}{\widehat{\pi }}_{11}+{\Delta }_{11}& {\widehat{\pi }}_{12}+{\Delta }_{12}& \cdots & {\widehat{\pi }}_{1s}+{\Delta }_{1s}\\ {\widehat{\pi }}_{21}+{\Delta }_{21}& {\widehat{\pi }}_{22}+{\Delta }_{22}& \cdots & {\widehat{\pi }}_{2s}+{\Delta }_{2s}\\ ⋮& ⋮& \ddots & ⋮\\ {\widehat{\pi }}_{s1}+{\Delta }_{s1}& {\widehat{\pi }}_{s2}+{\Delta }_{s2}& \cdots & {\widehat{\pi }}_{ss}+{\Delta }_{ss}\end{array}\right]$ (2.4)

其中${\widehat{\pi }}_{ij}-{\delta }_{ij}\ge 0\left(\forall j\in S,j\ne i\right)$ ，${\widehat{\pi }}_{ii}=-{\displaystyle \sum _{j=1,j\ne i}^s{\widehat{\pi }}_{ij}}$ 和${\delta }_{ii}=-{\displaystyle \sum _{j=1,j\ne i}^s{\delta }_{ij}}$ 。

$\left[\begin{array}{ccccc}{\pi }_{11}& ?& {\pi }_{13}& \cdots & ?\\ ?& ?& {\pi }_{23}& \cdots & {\pi }_{2s}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ ?& {\pi }_{s2}& ?& \cdots & {\pi }_{ss}\end{array}\right]$ (2.5)

显然，当$U_k^i\ne \varnothing$ ，$\forall i\in S$ 时，GUTR模型(2.3)简化为BUTR模型(2.5)，若${\delta }_{ij}=0$ ，$\forall i\in S$ ，$\forall j\in U_k^i$ 则简化为PUTR模型(2.5)。显然，BUTR或PUTR模型不如GUTR模型(2.3)通用，这意味着它更实用。

假设已知TR的估计值如下定义：

假设2.1若$U_k^i=S$ ，则${\widehat{\pi }}_{ij}-{\delta }_{ij}\ge 0$ ，$\left(\forall j\in S,j\ne i\right)$ ，${\widehat{\pi }}_{ii}=-{\displaystyle \sum _{j=1,j\ne i}^s{\widehat{\pi }}_{ij}\le 0}$ ，${\delta }_{ii}={\displaystyle \sum _{j=1,j\ne i}^s{\delta }_{ij}}>0$ ；

假设2.2若$U_k^i\ne S$ 且$i\in U_k^i$ ，则${\widehat{\pi }}_{ij}-{\delta }_{ij}\ge 0$ ，$\left(\forall j\in U_k^i,j\ne i\right)$ ，${\widehat{\pi }}_{ii}+{\delta }_{ii}\le 0$ ，${\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}\le 0$ ；

假设2.3若$U_k^i\ne S$ 且$i\in U_k^i$ ，则${\widehat{\pi }}_{ij}-{\delta }_{ij}\ge 0$ ，$\left(\forall j\in U_k^i\right)$ 。

这三个假设是基于TR特征而得出的，由此可以推断它们具有合理性。

$\left(e.g.{\pi }_{ij}\ge 0\left(\forall j\in S,j\ne i\right)\text{and}{\pi }_{ii}=-{\displaystyle \sum _{j=1,j\ne i}^s{\pi }_{ij}}\right).$

假设2.4在本文中所涉及的不确定性属于有界范数的，假设如下：

$\left[\Delta A_i\Delta A_{di}\right]=M_i{F}_i\left(t\right)\left[N_i{N}_{di}\right],\forall i\in S$

已知的常数矩阵$M_i$ ，$N_i$ ，$N_{di}$ 具有适当的维度，使$F_i\left(t\right)$ 得满足$F_i^{\text{T}}\left(t\right)F_i\left(t\right)\le I$ ，$i\in S$ 。

假设2.5非线性函数$f\left(t,\xi \left(t\right)\right)$ 满足

$\{\begin{array}{l}{f}^{\text{T}}\left(t\right)f\left(t\right)\le x^{\text{T}}\left(t\right)T^{2}x\left(t\right)\\ f^{\text{T}}\left(t-d\left(t\right)\right)f\left(t-d\left(t\right)\right)\le x^{\text{T}}\left(t-d\left(t\right)\right)T^{2}x\left(t-d\left(t\right)\right)\end{array}$ 其中$T=diag\left\{{\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_s\right\}$ 。

定义2.1 [7]若$\det \left(s\widehat{E}-{\widehat{A}}_i\right)$ 对任意$r_t=i\in S$ 均不为0，则系统(2.2)正则。

1) 若对任意$r_t=i\in S$ 有$\mathrm{deg}\left(\det \left(s\widehat{E}-{\widehat{A}}_i\right)\right)=rank\left(\widehat{E}\right)$ ，则系统(2.2)被称为无脉冲。

2) 当$w\left(t\right)=0$ ，$u\left(t\right)=0$ 时且存在标量$M\left(r_0,\phi \left(t\right)\right)>0$ 使得

$\underset{t\to \infty }{\lim }\epsilon \left\{{\displaystyle {\int }_0^{\infty }{\Vert x\left(t\right)\Vert }^2\text{d}t}|r_0,x\left(t\right)=\phi \left(t\right),t\in \left[-h,0\right]\right\}\le M\left(r_0,\phi \left(t\right)\right)$ ，则(2.2)随机稳定的。

3) 当$w\left(t\right)=0$ ，$u\left(t\right)=0$ 时，若系统(2.2)是正则的、无脉冲的，随机稳定的，则称它是随机容许的。

定义2.2对于给定的标量$\gamma >0$ ，系统(2.2)是随机容许的并满足H_∞性能$\gamma$ ，如果它在$w\left(t\right)=0$ 及零初始条件下是随机容许的，非零的$w\left(t\right)\in L_2\left[0,\infty \right)$ ，满足以下条件：

$\epsilon \left\{{\displaystyle {\int }_0^{\infty }\left(z^{\text{T}}\left(t\right)z\left(t\right)-{\lambda }^2{w}^{\text{T}}\left(t\right)w\left(t\right)\right)\text{d}t}\right\}<0$ .

引理2.1设H和G是具有适当维数的实数矩阵，并且$F_i^{\text{T}}\left(t\right)F_i\left(t\right)\le I$ 。以下不等式适用于任何标量$\lambda >0$ ：

1) $H{F}_i\left(t\right)G^{\text{T}}+G{F}_i^{\text{T}}\left(t\right)H^{\text{T}}\le \lambda G{G}^{\text{T}}+{\lambda }^{-1}H{H}^{\text{T}}$ ，2) $\pm 2H^{\text{T}}G\le H^{\text{T}}H+G^{\text{T}}G$ 。

引理2.2设$C\in R^{n\times n}$ ，$B\in R^{m\times m}$ 为正定矩阵，当且仅当$A>B^{\text{T}}{C}^{-1}B$ ，则对$\forall A\in R^{m\times m}$ 有

$F=\left(\begin{array}{cc}A& B^{\text{T}}\\ B& C\end{array}\right)>0$

2.2. 主要结果

对于给定的标量$0\le d_1\le d_2$ ，$h_1$ ，$h_2$ ，${\epsilon }_1\ge 0$ ，${\epsilon }_2\ge 0$ 和$\gamma >0$ ，如果存在矩阵$P_i>0$ ，$R_{1i}>0$ ，$R_{2i}>0$ ，$R_{3i}>0$ ，$Q_1>0$ ，$Q_2>0$ ，$S_1>0$ ，$S_2>0$ ，$Z_1>0$ ，$Z_2>0$ ，$Y_1$ ，$Y_2$ 使得以下不等式对任意$i\in I$ 成立，则系统渐进稳定的并具有相应的H_∞衰减指数$\gamma$

$E^{\text{T}}{P}_i=P_i^{\text{T}}E>0,$

${\Omega }^{\left[m\right]}=\left[\begin{array}{cc}{E}^{\left[m\right]}& E_{12}\\ & E_{22}\end{array}\right]<0,m=1,2,3,4,$

${\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{1j}-Q_1\le 0},{\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{3j}-Q_2\le 0},$

${\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{1j}-Q_1\le 0},{\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{3j}-Q_2\le 0},R_{2i}-R_{3i}\le 0,{\Phi }_1>0,{\Phi }_2>0,$

$E\left(d\left(t\right),\dot{d}\left(t\right)\right)=E_{1i\mu |d\left(t\right),\dot{d}\left(t\right)}+E_2+E_3,$

$E_{1i\mu |d\left(t\right),\dot{d}\left(t\right)}=\Xi +{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}\right){\Lambda }_1+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}x\left(t\right),$

$\begin{array}{c}{E}_{1i\mu |d\left(t\right),\dot{d}\left(t\right)}=sym\left({\Lambda }_1^{\text{T}}{P}_i{\Lambda }_2\right)+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}\right){\Lambda }_1+e_1^{\text{T}}\left(R_{1i}+R_{3i}+d_1{Q}_1+d_2{Q}_2+{\Theta }_1\right)e_1\\ +e_2^{\text{T}}\left({\lambda }_{i6}^{-1}{T}^2-R_{1i}\right)e_2-e_3^{\text{T}}{R}_{2i}{e}_3+e_5^{\text{T}}\left(d_2^2{S}_1+d_{12}^2{S}_2+\frac{1}{2}{d}_1^2\left(Z_1+Z_2\right)\right)e_5\\ -{\Lambda }_6^{\text{T}}{\overline{Z}}_1{\Lambda }_6-{\Lambda }_7^{\text{T}}{\overline{Z}}_2{\Lambda }_7+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}x\left(t\right)+mt{e}_4^{\text{T}}\left(R_{2i}-R_{3i}\right)e_4+e_1^{\text{T}}{P}_i{\Theta }_2{e}_2,\end{array}$

$\begin{array}{l}\Xi =sym\left({\Lambda }_1^{\text{T}}{P}_i{\Lambda }_2\right)+e_1^{\text{T}}\left(R_{1i}+R_{3i}+d_1{Q}_1+d_2{Q}_2+{\Theta }_1\right)e_1+e_2^{\text{T}}\left({\lambda }_{i6}^{-1}{T}^2-R_{1i}\right)e_2-e_3^{\text{T}}{R}_{2i}{e}_3\\ +mt{e}_4^{\text{T}}\left(R_{2i}-R_{3i}\right)e_4+e_1^{\text{T}}{P}_i{\Theta }_2{e}_2+e_5^{\text{T}}\left(d_2^2{S}_1+d_{12}^2{S}_2+\frac{1}{2}{d}_1^2\left(Z_1+Z_2\right)\right)e_5-{\Lambda }_6^{\text{T}}{\overline{Z}}_1{\Lambda }_6-{\Lambda }_7^{\text{T}}{\overline{Z}}_2{\Lambda }_7,\end{array}$

$\begin{array}{l}{E}_2=-{\Gamma }_1^{\text{T}}{\Phi }_1{\Gamma }_1-{\Gamma }_2^{\text{T}}{\Phi }_2{\Gamma }_2,E_3=e_{zi\mu }^{\text{T}}{e}_{zi\mu }-e_{14}^{\text{T}}{\gamma }^{2}I{e}_{14},\\ E_{12}=\left[\sqrt{1-{\alpha }_1}{\Lambda }_3^{\text{T}}{Y}_1\sqrt{{\alpha }_1}{\Lambda }_4^{\text{T}}{Y}_1{}^{\text{T}}\sqrt{1-{\alpha }_2}{\Lambda }_5^{\text{T}}{Y}_2\sqrt{{\alpha }_2}{\Lambda }_4^{\text{T}}{Y}_2{}^{\text{T}}\right],\end{array}$

$\begin{array}{l}{\Phi }_1=\left[\begin{array}{cc}\left(2-{\alpha }_1\right){\overline{S}}_1+\left(1-{\alpha }_1\right){\epsilon }_{1}I& Y_1\\ & \left(1+{\alpha }_1\right){\overline{S}}_1+{\alpha }_1{\epsilon }_{1}I\end{array}\right],\\ {\Phi }_2=\left[\begin{array}{cc}\left(2-{\alpha }_2\right){\overline{S}}_2+\left(1-{\alpha }_2\right){\epsilon }_{2}I& Y_2\\ & \left(1+{\alpha }_2\right){\overline{S}}_2+{\alpha }_2{\epsilon }_{2}I\end{array}\right],\end{array}$

$\begin{array}{c}{\Theta }_1={\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}+P_i(2A_i+2B_{i}K+{\lambda }_{i1}^{-1}{N}_i^{\text{T}}{N}_i+{\lambda }_{i3}^{-1}{N}_{ki}^{\text{T}}{N}_{ki}+{\lambda }_{i1}{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}\\ +{\lambda }_{i3}{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}}+{\lambda }_{i6}{B}_{di}{B}_{di}^{\text{T}}{P}_i^{\text{T}})+{\lambda }_{i5}^{-1}{T}^2\end{array}$

$\begin{array}{l}{\Theta }_2=2A_{di}+2B_{di}{K}_{di}+{\lambda }_{i2}^{-1}{N}_{di}^{\text{T}}{N}_{di}+{\lambda }_{i4}^{-1}{N}_{kdi}^{\text{T}}{N}_{kdi}+{\lambda }_{i2}{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}+{\lambda }_{i4}{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}},\\ E_{22}=diag\left\{-{\overline{S}}_1,-{\overline{S}}_1,-{\overline{S}}_2,-{\overline{S}}_2\right\},\end{array}$

$\begin{array}{l}{\Lambda }_1={\left[\left(d\left(t\right)-d_1\right)e_6^{\text{T}}\left(d_2-d\left(t\right)\right)e_7^{\text{T}}d\left(t\right)e_8^{\text{T}}{\left(d\left(t\right)-d_1\right)}^2{e}_{10}^{\text{T}}{\left(d_2-d\left(t\right)\right)}^2{e}_{11}^{\text{T}}d{\left(t\right)}^2{e}_{12}^{\text{T}}\right]}^{\text{T}},\\ {\Gamma }_1={\left[{\Lambda }_3^{\text{T}}{\Lambda }_4^{\text{T}}\right]}^{\text{T}},{\Gamma }_2={\left[{\Lambda }_5^{\text{T}}{\Lambda }_4^{\text{T}}\right]}^{\text{T}},\end{array}$

$\begin{array}{l}{\Lambda }_2=[E^{\text{T}}{e}_2^{\text{T}}-mt{E}^{\text{T}}{e}_4^{\text{T}}mt{E}^{\text{T}}{e}_4^{\text{T}}-E^{\text{T}}{e}_3^{\text{T}}{e}_1^{\text{T}}-mt{E}^{\text{T}}{e}_4^{\text{T}}\left(d\left(t\right)-d_1\right)E^{\text{T}}{e}_1^{\text{T}}-\left(d\left(t\right)-d_1\right)e_6^{\text{T}}\\ \left(d_2-d\left(t\right)\right)E^{\text{T}}{e}_1^{\text{T}}-\left(d_2-d\left(t\right)\right)e_7^{\text{T}}{d\left(t\right)E^{\text{T}}{e}_1^{\text{T}}-d\left(t\right)e_8^{\text{T}}]}^{\text{T}},\end{array}$

${\Lambda }_3={\left[r_1^{\text{T}}{r}_2^{\text{T}}{r}_3^{\text{T}}\right]}^{\text{T}},{\Lambda }_4={\left[r_4^{\text{T}}{r}_5^{\text{T}}{r}_6^{\text{T}}\right]}^{\text{T}},{\Lambda }_5={\left[r_7^{\text{T}}{r}_8^{\text{T}}{r}_9^{\text{T}}\right]}^{\text{T}},{\Lambda }_6={\left[r_{10},r_{11}\right]}^{\text{T}},{\Lambda }_7={\left[r_{12},r_{13}\right]}^{\text{T}},$

$\begin{array}{l}{r}_1=e_1-e_4,r_2=e_1+e_4-2e_8,r_3=e_1-e_4-6e_8+12e_{12},\\ r_4=e_4-e_3,r_5=e_3+e_4-2e_7,r_6=e_3-e_4-6e_7+12e_{11},\\ r_7=e_2-e_4,r_8=e_2+e_4-2e_6,r_9=e_2-e_4-6e_6+12e_{10},\\ r_{10}=e_1-e_9,r_{11}=e_1+2e_9-6e_{13},r_{12}=e_2-e_9,r_{13}=e_2-4e_9+6e_{13},\end{array}$

${\mu }_1=\left(1-{\alpha }_1\right){\Lambda }_3^{\text{T}}{Y}_1{\overline{S}}_1^{-1}{Y}_1{\Lambda }_3+{\alpha }_1{\Lambda }_4^{\text{T}}{Y}_1{\overline{S}}_1^{-1}{Y}_1{\Lambda }_4,{\mu }_2=\left(1-{\alpha }_2\right){\Lambda }_5^{\text{T}}{Y}_2{\overline{S}}_2^{-1}{Y}_2{\Lambda }_5+{\alpha }_2{\Lambda }_6^{\text{T}}{Y}_2{\overline{S}}_2^{-1}{Y}_2{\Lambda }_6,$

${\alpha }_1=\frac{d\left(t\right)}{d_2},{\alpha }_2=\frac{d\left(t\right)-d_1}{d_{12}},mt=1-\dot{d}\left(t\right),{\overline{S}}_1=diag\left\{S_1,3S_1,5S_1\right\},{\overline{S}}_{2i}=diag\left\{S_2,3S_2,5S_2\right\},$

$\begin{array}{l}{\overline{Z}}_1=diag\left\{2Z_1,4Z_1\right\},{\overline{Z}}_2=diag\left\{2Z_2,4Z_2\right\},\\ e_i=\left[0_{\left(2n+k\right)\times \left(i-1\right)\left(2n+k\right)}{I}_{\left(2n+k\right)}{0}_{\left(2n+k\right)\times \left(13-i\right)\left(2n+k\right)}{0}_{\left(2n+k\right)\times \left(m+q+l\right)}\right],i=1,\cdots ,14.\end{array}$

证明：首先证明系统是正则的、无脉冲的，存在M，N满足

$MEN=\left[\begin{array}{cc}{I}_r& 0\\ 0& 0\end{array}\right],M{A}_{i}N=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],M^{-\text{T}}{P}_{i}N=\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right],$

且$E^{\text{T}}{P}_i=P_i^{\text{T}}E$ ，可知$P_{12}=0$ 。将左右乘以N和$N^{\text{T}}$ 有${\Omega }_{i11}<0$ ，则$A_{22}^{\text{T}}{P}_{22}+P_{22}^{\text{T}}{A}_{22}<0$ ，故$A_{22}$ 是非奇异的。因此$\det \left(A_{22}\right)\ne 0$ 和$\det \left(\det \left(sE-A_i\right)\right)=r=rank\left(E\right)$ ，则$\left(E,A_i\right)$ 是正则的、无脉冲的。同样地，进行左右分别乘以${\left[\begin{array}{ccc}I& \cdots & I\end{array}\right]}_{7\times 1}^{\text{T}}$ 和${\left[\begin{array}{ccc}I& \cdots & I\end{array}\right]}_{7\times 1}^{}$ 有${\left(A_i+A_{di}\right)}^{\text{T}}{P}_i+P_i^{\text{T}}\left(A_i+A_{di}\right)+{\displaystyle \sum _{j=1}^s{\pi }_{ij}{E}^{\text{T}}{P}_{j}E<0}$ 可知$\left(E,A_i+A_{di}\right)$ 是正则，无脉冲的。根据[2]中定义1系统是正则，无脉冲的。接下来，证明系统是渐进稳定，构造LKF函数如下。

$V\left(x_t\right)={\displaystyle \sum _{k=1}^5{V}_k},$

$V_1\left(x_t\right)=x^{\text{T}}\left(t\right)E^{\text{T}}{P}_{i}x\left(t\right)+{\eta }^{\text{T}}\left(t\right)P_i\eta \left(t\right),$

$V_2\left(x_t\right)={\displaystyle {\int }_{t-d_1}^t{x}^{\text{T}}\left(s\right)R_{1i}}x\left(s\right)\text{d}s+{\displaystyle {\int }_{-d_1}^0{\displaystyle {\int }_{t-d_1}^t{x}^{\text{T}}\left(s\right)Q_1}x\left(s\right)\text{d}s}\text{d}\theta ,$

$V_3\left(x_t\right)={\displaystyle {\int }_{t-d_2}^{t-d\left(t\right)}{x}^{\text{T}}\left(s\right)R_{2i}}x\left(s\right)\text{d}s+{\displaystyle {\int }_{t-d\left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{3i}}x\left(s\right)\text{d}s+{\displaystyle {\int }_{-d_2}^0{\displaystyle {\int }_{t+\theta }^t{x}^{\text{T}}\left(s\right)Q_2}x\left(s\right)\text{d}s\text{d}\theta },$

$V_4\left(x_t\right)=d_2{\displaystyle {\int }_{-d_2}^0{\displaystyle {\int }_{t+\theta }^t{\dot{x}}^{\text{T}}\left(s\right)E^{\text{T}}{S}_{1}E}\dot{x}\left(s\right)\text{d}s}\text{d}\theta +d_{12}{\displaystyle {\int }_{-d_2}^{-d_1}{\displaystyle {\int }_{t+\theta }^t{\dot{x}}^{\text{T}}\left(s\right)E^{\text{T}}{S}_{2}E}\dot{x}\left(s\right)\text{d}s}\text{d}\theta ,$

$V_5\left(x_t\right)={\displaystyle {\int }_{t-d_1}^t{\displaystyle {\int }_u^t{\displaystyle {\int }_{\theta }^t{\dot{x}}^{\text{T}}\left(s\right)E^{\text{T}}{Z}_{1}E}}\dot{x}\left(s\right)\text{d}s\text{d}\theta }\text{d}u+{\displaystyle {\int }_{t-d_1}^t{\displaystyle {\int }_{t-d_1}^u{\displaystyle {\int }_{\theta }^t{\dot{x}}^{\text{T}}\left(s\right)E^{\text{T}}{Z}_{2}E\dot{x}\left(s\right)\text{d}s}}\text{d}\theta }\text{d}u,$

$\begin{array}{l}\eta \left(t\right)=[{\displaystyle {\int }_{t-d\left(t\right)}^{t-d_1}{x}^{\text{T}}\left(t\right)E^{\text{T}}\text{d}s}{\displaystyle {\int }_{t-d_2}^{t-d\left(t\right)}{x}^{\text{T}}\left(t\right)E^{\text{T}}\text{d}s}{\displaystyle {\int }_{t-d\left(t\right)}^t{x}^{\text{T}}\left(t\right)E^{\text{T}}\text{d}s}{\displaystyle {\int }_{t-d\left(t\right)}^{t-d_1}{\displaystyle {\int }_{\theta }^t{x}^{\text{T}}\left(t\right)E^{\text{T}}}\text{d}}s\text{d}\theta \\ {{\displaystyle {\int }_{t-d_2}^{t-d\left(t\right)}{\displaystyle {\int }_{\theta }^t{x}^{\text{T}}\left(t\right)E^{\text{T}}\text{d}s}\text{d}\theta }{\displaystyle {\int }_{t-d\left(t\right)}^t{\displaystyle {\int }_{\theta }^t{x}^T\left(t\right)E^{\text{T}}}\text{d}s}\text{d}\theta ]}^{\text{T}}.\end{array}$

L为弱无穷小算子，有：

$\begin{array}{c}L{V}_1\left(x_t\right)=2{\Psi }^{\text{T}}\left(t\right){\Lambda }_1^{\text{T}}{P}_i{\Lambda }_2\Psi \left(t\right)+2x^{\text{T}}\left(t\right)E{P}_i\dot{x}\left(t\right)+{\Psi }^{\text{T}}\left(t\right){\Lambda }_1^{\text{T}}{\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}{\Lambda }_1\Psi \left(t\right)+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}x\left(t\right)\\ ={\Psi }^{\text{T}}\left(t\right)\left[sym\left({\Lambda }_1^{\text{T}}{P}_i{\Lambda }_2\right)+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}\right){\Lambda }_1\right]\Psi \left(t\right)+x^{\text{T}}\left(t\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}x\left(t\right)+sym\left[x^{\text{T}}\left(t\right)E^{\text{T}}{P}_i\dot{x}\left(t\right)\right]\\ ={\Psi }^{\text{T}}\left(t\right)\left[sym\left({\Lambda }_1^{\text{T}}{P}_i{\Lambda }_2\right)+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}\right){\Lambda }_1\right]\Psi \left(t\right)+x^{\text{T}}\left(t\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}x\left(t\right)+2x^{\text{T}}\left(t\right)P_i{A}_{i}x\left(t\right)\\ +2x^{\text{T}}\left(t\right)P_i\Delta A_{i}x\left(t\right)+2x^{\text{T}}\left(t\right)P_i{B}_i\Delta K_{i}x\left(t\right)+2x^{\text{T}}\left(t\right)P_i{A}_{di}x\left(t-d\left(t\right)\right)+2x^{\text{T}}\left(t\right)P_i\Delta A_{di}x\left(t-d\left(t\right)\right)\\ +2x^{\text{T}}\left(t\right)P_i{B}_i{K}_{i}x\left(t\right)+2x^{\text{T}}\left(t\right)P_i{B}_{di}{K}_{di}x\left(t-d\left(t\right)\right)+2x^{\text{T}}\left(t\right)P_i{B}_{di}\Delta K_{di}x\left(t-d\left(t\right)\right)\\ +2x^{\text{T}}\left(t\right)P_i{B}_{i}f\left(t\right)+2x^{\text{T}}\left(t\right)P_i{B}_{di}f\left(t-d\left(t\right)\right),\end{array}$

$L{V}_2\left(x_t\right)={\Psi }^{\text{T}}\left(t\right)\left(e_1^{\text{T}}\left(R_{1i}+d_1{Q}_1\right)e_1-e_2^{\text{T}}{R}_{1i}{e}_2\right)\Psi \left(t\right)+{\displaystyle {\int }_{t-d_1}^t{x}^{\text{T}}\left(s\right)\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{1j}-Q_1}\right)x\left(s\right)\text{d}s},$

$\begin{array}{c}L{V}_3\left(x_t\right)={\Psi }^{\text{T}}\left(t\right)\left(e_1^{\text{T}}\left(R_{3i}+d_2{Q}_2\right)e_1+mt{e}_4^{\text{T}}\left(R_{2i}-R_{3i}\right)e_4-e_3^{\text{T}}{R}_{2i}{e}_3\right)\Psi \left(t\right)\\ +{\displaystyle {\int }_{t-d_2}^{t-d\left(t\right)}{x}^{\text{T}}\left(s\right)\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}\left(R_{2j}-R_{3j}\right)}\right)x^{\text{T}}\left(s\right)\text{d}s}+{\displaystyle {\int }_{t-d_2}^t{x}^{\text{T}}\left(s\right)\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{3j}-Q_2}\right)x\left(s\right)\text{d}s},\end{array}$

$\begin{array}{c}L{V}_4\left(x_t\right)={\Psi }^{\text{T}}\left(t\right)\left(e_5^{\text{T}}\left(d_2^2{S}_1+d_{12}^2{S}_2\right)e_5\right)\Psi \left(t\right)-d_2{\displaystyle {\int }_{t-d_2}^t{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)S_{1}E\dot{x}\left(s\right)\text{d}s}\\ -d_{12}{\displaystyle {\int }_{t-d_2}^{t-d_1}{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)S_{2}E\dot{x}\left(s\right)\text{d}s},\end{array}$

$\begin{array}{c}L{V}_5\left(x_t\right)={\Psi }^{\text{T}}\left(t\right)\left(\frac{d_1^2}{2}{e}_5^{\text{T}}\left(Z_1+Z_2\right)e_5\right)\Psi \left(t\right)-{\displaystyle {\int }_{t-d_1}^t{\displaystyle {\int }_{\theta }^t{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)Z_{1}E\dot{x}\left(s\right)\text{d}s}}\text{d}\theta \\ -{\displaystyle {\int }_{t-d_1}^t{\displaystyle {\int }_{t-d_1}^{\theta }{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)Z_{2}E\dot{x}\left(s\right)\text{d}s}}\text{d}\theta ,\end{array}$

$\Psi \left(t\right)={\left[{\psi }_1^{\text{T}}\left(t\right){\psi }_2^{\text{T}}\left(t\right){\psi }_3^{\text{T}}\left(t\right)w^{\text{T}}\left(t\right)\right]}^{\text{T}},$

${\psi }_1\left(t\right)={\left[x^{\text{T}}\left(t\right)x^{\text{T}}\left(t-d_1\right)x^{\text{T}}\left(t-d_2\right)x^{\text{T}}\left(t-d\left(t\right)\right){\dot{x}}^{\text{T}}\left(t\right)E^{\text{T}}\right]}^{\text{T}},$

${\psi }_2\left(t\right)=[\frac{1}{d\left(t\right)-d_1}{\displaystyle {\int }_{t-d\left(t\right)}^{t-d_1}{x}^{\text{T}}\left(s\right)E^{\text{T}}}\text{d}s\frac{1}{d_2-d\left(t\right)}{\displaystyle {\int }_{t-d_2}^{t-d\left(t\right)}{x}^{\text{T}}\left(s\right)E^{\text{T}}\text{d}s}\frac{1}{d\left(t\right)}{\displaystyle {\int }_{t-d\left(t\right)}^t{x}^{\text{T}}\left(s\right)E^{\text{T}}\text{d}s}{\frac{1}{d_1}{\displaystyle {\int }_{t-d_1}^t{x}^{\text{T}}\left(s\right)E^{\text{T}}\text{d}s}]}^{\text{T}},$

$\begin{array}{l}{\psi }_3\left(t\right)=[\frac{1}{{\left(d\left(t\right)-d_1\right)}^2}{\displaystyle {\int }_{t-d\left(t\right)}^{t-d_1}{\displaystyle {\int }_{\theta }^t{x}^{\text{T}}\left(s\right)E^{\text{T}}}}\text{d}s\text{d}\theta \frac{1}{{\left(d_2-d\left(t\right)\right)}^2}{\displaystyle {\int }_{t-d_2}^{t-d\left(t\right)}{\displaystyle {\int }_{\theta }^t{x}^{\text{T}}\left(s\right)E^{\text{T}}\text{d}s\text{d}\theta }}\\ \frac{1}{{\left(d\left(t\right)\right)}^2}{\displaystyle {\int }_{t-d\left(t\right)}^t{\displaystyle {\int }_{\theta }^t{x}^{\text{T}}\left(s\right)E^{\text{T}}}}\text{d}s\text{d}\theta {\frac{1}{d_1^2}{\displaystyle {\int }_{t-d_1}^t{\displaystyle {\int }_{\theta }^t{x}^{\text{T}}\left(s\right)E^{\text{T}}\text{d}s}}]}^{\text{T}}.\end{array}$

由[8]中引理2，4和假设1，2

$2x^{\text{T}}\left(t\right)P_i\Delta A_{i}x\left(t\right)=2x^{\text{T}}\left(t\right)P_i{M}_i{F}_i\left(t\right)N_{i}x\left(t\right)\le {\lambda }_{i1}{x}^{\text{T}}\left(t\right)P_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}x\left(t\right)+{\lambda }_{i1}^{-1}{x}^{\text{T}}\left(t\right)N_i^{\text{T}}{N}_{i}x\left(t\right),$

$\begin{array}{c}2x^{\text{T}}\left(t\right)P_i\Delta A_{di}x\left(t-d\left(t\right)\right)=2x^{\text{T}}\left(t\right)P{M}_i{F}_i\left(t\right)N_{di}x\left(t-d\left(t\right)\right)\\ \le {\lambda }_{i2}{x}^{\text{T}}\left(t\right)P_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{x}_1\left(t-d\left(t\right)\right)+{\lambda }_{i2}^{-1}{x}^{\text{T}}\left(t\right)N_{di}^{\text{T}}{N}_{di}x\left(t-d\left(t\right)\right),\end{array}$

$2x^{\text{T}}\left(t\right)P_i{B}_i\Delta K_{i}x\left(t\right)=2x^{\text{T}}\left(t\right)P_i{B}_i{M}_i{F}_i\left(t\right)N_{ki}x\left(t\right)\le {\lambda }_{i3}{x}^{\text{T}}\left(t\right)P_i{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}}x\left(t\right)+{\lambda }_{i3}^{-1}{x}^{\text{T}}\left(t\right)N_{ki}^{\text{T}}{N}_{ki}x\left(t\right),$

$2x^{\text{T}}\left(t\right)P_i{B}_i\Delta K_{kdi}x\left(t-d\left(t\right)\right)\le {\lambda }_{i4}{x}^{\text{T}}\left(t\right)P_i{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}}x\left(t-d\left(t\right)\right)+{\lambda }_{i4}^{-1}{x}^{\text{T}}\left(t\right)N_{kdi}^{\text{T}}{N}_{kdi}x\left(t-d\left(t\right)\right),$

$2x^{\text{T}}\left(t\right)P_i{B}_{i}f\left(t\right)\le {\lambda }_{i5}{x}^{\text{T}}\left(t\right)P_i{B}_i{B}_i^{\text{T}}{P}_i^{\text{T}}x\left(t\right)+{\lambda }_{i5}^{-1}{f}^{\text{T}}\left(t\right)f\left(t\right)\le {\lambda }_{i5}{x}^{\text{T}}\left(t\right)P_i{B}_i{B}_i^{\text{T}}{P}_i^{\text{T}}x\left(t\right)+{\lambda }_{i5}^{-1}{x}^{\text{T}}\left(t\right)T^{2}x\left(t\right),$

$\begin{array}{c}2x^{\text{T}}\left(t\right)P_i{B}_{di}f\left(t-d\left(t\right)\right)\le {\lambda }_{i6}{x}^{\text{T}}\left(t\right)P_i{B}_{di}{B}_{di}^{\text{T}}{P}_i^{\text{T}}x\left(t\right)+{\lambda }_{i6}^{-1}{f}^{\text{T}}\left(t-d\left(t\right)\right)f\left(t-d\left(t\right)\right)\\ \le {\lambda }_{i6}{x}^{\text{T}}\left(t\right)P_i{B}_{di}{B}_{di}^T{P}_i^{T}x\left(t\right)+{\lambda }_{i6}^{-1}{x}^{\text{T}}\left(t-d\left(t\right)\right)T^{2}x\left(t-d\left(t\right)\right).\end{array}$

$\begin{array}{c}L{V}_1\left(x_t\right)\le {\Psi }^{\text{T}}\left(t\right)\left[sym\left({\Lambda }_1^{\text{T}}{P}_i{\Lambda }_2\right)+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}\right){\Lambda }_1\right]\Psi \left(t\right)+x^{\text{T}}\left(t\right)[{\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}+P_i(2A_i+2B_{i}K+{\lambda }_{i1}^{-1}{N}_i^{\text{T}}{N}_i\\ \stackrel{}{}+{\lambda }_{i3}^{-1}{N}_{ki}^{\text{T}}{N}_{ki}+{\lambda }_{i1}{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}+{\lambda }_{i3}{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}}+{\lambda }_{i6}{B}_{di}{B}_{di}^{\text{T}}{P}_i^{\text{T}})+{\lambda }_{i5}^{-1}{T}^2]x\left(t\right)\\ +x^{\text{T}}\left(t\right)P_i(2A_{di}+2B_{di}{K}_{di}+{\lambda }_{i2}^{-1}{N}_{di}^{\text{T}}{N}_{di}+{\lambda }_{i4}^{-1}{N}_{kdi}^{\text{T}}{N}_{kdi}+{\lambda }_{i2}{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}\\ +{\lambda }_{i4}{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}})x\left(t-d\left(t\right)\right)+{\lambda }_{i6}^{-1}{x}^{\text{T}}\left(t-d\left(t\right)\right)T^{2}x\left(t-d\left(t\right)\right)\\ ={\Psi }^{\text{T}}\left(t\right)\left[sym\left({\Lambda }_1^{\text{T}}{P}_i{\Lambda }_2\right)+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}\right){\Lambda }_1+e_1^{\text{T}}{\Theta }_1{e}_1+e_1^{\text{T}}{P}_i{\Theta }_2{e}_2+e_2^{\text{T}}{\lambda }_{i6}^{-1}{T}^2{e}_2\right]\Psi \left(t\right),\end{array}$

${\Theta }_1={\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}+P_i(2A_i+2B_{i}K+{\lambda }_{i1}^{-1}{N}_i^{\text{T}}{N}_i+{\lambda }_{i3}^{-1}{N}_{ki}^{\text{T}}{N}_{ki}+{\lambda }_{i1}{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}+{\lambda }_{i3}{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}}+{\lambda }_{i6}{B}_{di}{B}_{di}^{\text{T}}{P}_i^{\text{T}})+{\lambda }_{i5}^{-1}{T}^2,$

${\Theta }_2=2A_{di}+2B_{di}{K}_{di}+{\lambda }_{i2}^{-1}{N}_{di}^T{N}_{di}+{\lambda }_{i4}^{-1}{N}_{kdi}^{\text{T}}{N}_{kdi}+{\lambda }_{i2}{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}+{\lambda }_{i4}{B}_i{M}_i{M}_i^{\text{T}}{P}_i^{\text{T}}{B}_i^{\text{T}},$

[9]结论1和[10]引理3，有

$\begin{array}{l}-d_2{\displaystyle {\int }_{t-d_2}^t{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)S_1}E\dot{x}\left(s\right)\text{d}s=-d_2{\displaystyle {\int }_{t-d\left(t\right)}^t{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)S_1}E\dot{x}\left(s\right)\text{d}s-d_2{\displaystyle {\int }_{t-d_2}^{t-d\left(t\right)}{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)S_{2}E\dot{x}\left(s\right)\text{d}s}\\ \le {\Psi }^{\text{T}}\left(t\right)\left[-\frac{1}{{\alpha }_1}\left(r_1^{\text{T}}{S}_1{r}_1+3r_2^{\text{T}}{S}_1{r}_2+5r_3^{\text{T}}{S}_1{r}_3\right)-\frac{1}{1-{\alpha }_1}\left(r_4^{\text{T}}{S}_1{r}_4+3r_5^{\text{T}}{S}_1{r}_5+5r_6^{\text{T}}{S}_1{r}_6\right)\right]\Psi \left(t\right)\\ \le {\Psi }^{\text{T}}\left(t\right){\Gamma }_1^{\text{T}}\left[\begin{array}{cc}{\overline{S}}_1+\left(1-{\alpha }_1\right)\left({\overline{S}}_1+{\epsilon }_{1}I-Y_1{\overline{S}}_1^{-1}{Y}_1^{\text{T}}\right)& Y_1\\ & {\overline{S}}_1+{\alpha }_1\left({\overline{S}}_1+{\epsilon }_{1}I-Y_1{}^{\text{T}}{\overline{S}}_1^{-1}{Y}_1\right)\end{array}\right]{\Gamma }_1\Psi \left(t\right)\\ ={\Psi }^{\text{T}}\left(t\right)\left({\mu }_1-{\Gamma }_1^{\text{T}}{\Phi }_1{\Gamma }_1\right){\Psi }^{\text{T}}\left(t\right)\end{array}$

$-d_{12}{\displaystyle {\int }_{t-d_2}^{t-d_1}E{\dot{x}}^{\text{T}}\left(s\right)S_{2}E\dot{x}\left(s\right)}\le {\Psi }^{\text{T}}\left(t\right)\left({\mu }_2-{\Gamma }_2^{\text{T}}{\Phi }_2{\Gamma }_2\right){\Psi }^{\text{T}}\left(t\right)$

$\begin{array}{l}{\mu }_1=\left(1-{\alpha }_1\right){\Lambda }_3^{\text{T}}{Y}_1{\overline{S}}_1^{-1}{Y}_1{\Lambda }_3+{\alpha }_1{\Lambda }_4^{\text{T}}{Y}_1{\overline{S}}_1^{-1}{Y}_1{\Lambda }_4,\\ {\mu }_2=\left(1-{\alpha }_2\right){\Lambda }_5^{\text{T}}{Y}_2{\overline{S}}_2^{-1}{Y}_2{\Lambda }_5+{\alpha }_2{\Lambda }_6^{\text{T}}{Y}_2{\overline{S}}_2^{-1}{Y}_2{\Lambda }_6,{\overline{S}}_n=diag\left\{S_n,3S_n,5S_n\right\},\end{array}$

根据[11]中(9)，(10)，我们得到

$\begin{array}{l}-{\displaystyle {\int }_{t-d_1}^t{\displaystyle {\int }_{\theta }^t{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)Z_1}}E\dot{x}\left(s\right)\text{d}s\text{d}\theta \le -{\Psi }^{\text{T}}\left(t\right)\left[2r_{10}^{\text{T}}{Z}_1{r}_{10}+4r_{11}^{\text{T}}{Z}_1{r}_{11}\right]\Psi \left(t\right)={\Psi }^{\text{T}}\left(t\right)\left[-\left({\Lambda }_6^{\text{T}}{\overline{Z}}_1{\Lambda }_6\right)\right]\Psi \left(t\right)\\ -{\displaystyle {\int }_{t-d_1}^t{\displaystyle {\int }_{t-d_1}^{\theta }{E}^{\text{T}}{\dot{x}}^{\text{T}}\left(s\right)Z_2}}E\dot{x}\left(s\right)\text{d}s\text{d}\theta \le -{\Psi }^{\text{T}}\left(t\right)\left[2r_{12}^{\text{T}}{Z}_2{r}_{12}+4r_{13}^{\text{T}}{Z}_2{r}_{13}\right]\Psi \left(t\right)={\Psi }^{\text{T}}\left(t\right)\left[-\left({\Lambda }_7^{\text{T}}{\overline{Z}}_2{\Lambda }_7\right)\right]\Psi \left(t\right).\end{array}$

通过上述，我们可以得到

$LV\left(x_t\right)\le {\xi }^{\text{T}}\left(t\right){\Omega }_0\xi \left(t\right),$

其中${\Omega }_0={E_{1i\mu }|}_{d\left(t\right),\dot{d}\left(t\right)}+E_2+{\eta }_1+{\eta }_2$ ，由Schur补，我们有，

${\Omega }_0={E_{1i\mu }|}_{d\left(t\right),\dot{d}\left(t\right)}+E_2+{\eta }_1+{\eta }_2<0,$

$LV\left(x_t\right)<0,$

这意味着$LV\left(x_t\right)\le -\varsigma x^{\text{T}}\left(t\right)x\left(t\right)$ ，使用Dynkin公式，所有$i\in I$ ，$T>0$ ，遵循$i\in I$ ，$T>0$ ，

$E\left\{V\left(x\left(T\right),r\left(T\right)\right)\right\}-V\left(x\left(0\right),r\left(0\right)\right)\le -\varsigma E\left\{{\displaystyle {\int }_0^T{x}^{\text{T}}\left(s\right)x\left(s\right)|\left(x\left(0\right),r\left(0\right)\right)}\right\},$

此外，让$T\to \infty$ ，$E\left\{{\displaystyle {\int }_0^T{x}^{\text{T}}\left(s\right)x\left(s\right)|\left(x\left(0\right),r\left(0\right)\right)}\right\}\le \frac{1}{\varsigma }V\left(x\left(0\right),r\left(0\right)\right)<\infty$ 。

根据定义2.1，系统(2.2)是随机稳定的。接下来我们考虑H_∞性能函数J，

$\begin{array}{c}J={\displaystyle {\int }_0^{\infty }{z}^{\text{T}}\left(t\right)z\left(t\right)-{\gamma }^2{w}^{\text{T}}\left(t\right)w\left(t\right)\text{d}t}\le {\displaystyle {\int }_0^{\infty }LV\left(x_t\right)+z^{\text{T}}\left(t\right)z\left(t\right)-{\gamma }^2{w}^{\text{T}}\left(t\right)w\left(t\right)\text{d}t}\\ ={\displaystyle {\int }_0^{\infty }{\Psi }^{\text{T}}\left(t\right)\Omega \Psi \left(t\right)\text{d}t},\end{array}$

我们有$J<0$ 。H_∞性能已验证。根据GUTR矩阵的定义，我们要探讨以下三种情况下的上述不等式。

情况I $i\notin U_k^i$ ，$U_k^i=\left\{k_1^i,\cdots ,k_{m_i}^i\right\}$ ，存在正定矩阵，$V_{ij}\in R^{n\times n}\left(i\notin U_k^i,j\in U_k^i\right)$ 及$l\in U_{uk}^i$ 有

$\left[\begin{array}{cccccc}{E}^{\left[m\right]}& E_{12}& e_{zi\mu }^{\text{T}}& {\Pi }_1^{\text{T}}E\left(P_{k_1^i}-F_i\right)& \cdots & {\Pi }_1^{\text{T}}E\left(P_{i{k}_{mi}^i}-F_i\right)\\ *& E_{22}& 0& 0& \cdots & 0\\ *& *& -I& 0& \cdots & 0\\ *& *& *& -V_{i{k}_1^i}& \cdots & 0\\ *& *& *& *& \ddots & ⋮\\ *& *& *& *& *& -V_{i{k}_{mi}^i}\end{array}\right]<0,$

证明 $i\notin U_k^i$ ，应该注意的是，在这种情况下${\displaystyle \sum _{j\in U_k^i,j\ne i}{\pi }_{ij}}=-{\pi }_{ii}-{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}$ ，${\pi }_{ij}\ge 0$ ，且必须有$l\in U_k^i$ ，$l\ne j$ ，

使$E^{\text{T}}{\overline{P}}_{l}E-E^{\text{T}}{\overline{P}}_{j}E\ge 0$，定义

$\begin{array}{c}\Omega =\Xi +E_2+E_3+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}+{\pi }_{ii}{P}_i+{\displaystyle \sum _{j\in U_k^i,j\ne i}{\pi }_{ij}{P}_j}\right){\Lambda }_1\\ +x^{\text{T}}\left(t\right)E^{\text{T}}\left({\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}+{\pi }_{ii}{P}_i+{\displaystyle \sum _{j\in U_k^i,j\ne i}{\pi }_{ij}{P}_j}\right)x\left(t\right),\end{array}$ (2.6)

$\begin{array}{c}\Omega =\Xi +E_2+E_3+{\Lambda }_1^{\text{T}}{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}{\Lambda }_1+{\Lambda }_1^{\text{T}}{\pi }_{ii}{P}_i{\Lambda }_1+{\Lambda }_1^{\text{T}}{\displaystyle \sum _{j\in U_k^i,j\ne i}{\pi }_{ij}{P}_j}{\Lambda }_1+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_{j}x\left(t\right)}\\ +x^{\text{T}}\left(t\right)E^{\text{T}}{\pi }_{ii}{P}_{i}x\left(t\right)+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j\in U_k^i,j\ne i}{\pi }_{ij}{P}_j}x\left(t\right)\\ \le \Xi +E_2+E_3+{\Lambda }_1^{\text{T}}{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}{\Lambda }_1+{\Lambda }_1^{\text{T}}{\pi }_{ii}{P}_i{\Lambda }_1+{\Lambda }_1^{\text{T}}\left(-{\pi }_{ii}-{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}\right)P_l{\Lambda }_1+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_{j}x\left(t\right)}\\ +x^{\text{T}}\left(t\right)E^{\text{T}}{\pi }_{ii}{P}_{i}x\left(t\right)+x^{\text{T}}\left(t\right)E^{\text{T}}\left(-{\pi }_{ii}-{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}\right)P_{l}x\left(t\right)\end{array}$

$\begin{array}{l}=\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{x}^{\text{T}}\left(t\right)E^{\text{T}}\left(P_j-P_l\right)x\left(t\right)\\ =\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}\left({\widehat{\pi }}_{ij}+{\pi }_{ij}\right){\Lambda }_1^{\text{T}}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\left({\widehat{\pi }}_{ij}+{\pi }_{ij}\right)x^{\text{T}}}\left(t\right)E^{\text{T}}\left(P_j-P_l\right)x\left(t\right)\\ =\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\text{+}{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{x}^{\text{T}}\left(t\right)E^{\text{T}}\left(P_j-P_l\right)x\left(t\right)\\ +{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{x}^{\text{T}}\left(t\right)E^{\text{T}}\left(P_j-P_l\right)x\left(t\right).\end{array}$(2.7)

此外，由于引理2.1，可以得到

$\begin{array}{l}{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\\ ={\displaystyle \sum _{j\in U_k^i}\left[\frac{1}{2}{\pi }_{ij}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+\frac{1}{2}{\pi }_{ij}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\left[\frac{1}{2}{\pi }_{ij}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)+\frac{1}{2}{\pi }_{ij}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]}\\ \le {\displaystyle \sum _{j\in U_k^i}\left[\frac{{\delta }_{ij1}^2}{4}{V}_{ij1}+{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1{V}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\left[\frac{{\delta }_{ij2}^2}{4}{V}_{ij2}+E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)V_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]},\end{array}$ (2.8)

从(2.6)~(2.8)中，我们有

$\begin{array}{l}\Omega \le \Xi +E_2+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij1}^2}{4}{V}_{ij1}}+{\displaystyle \sum _{j\in U_k^i}\left[{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1{V}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij2}^2}{4}{V}_{ij2}}+{\displaystyle \sum _{j\in U_k^i}\left[E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)V_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]}\\ +{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\end{array}$(2.9)

如果，

$\begin{array}{l}\Xi +E_2+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij1}^2}{4}{V}_{ij1}}+{\displaystyle \sum _{j\in U_k^i}\left[{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1{V}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij2}^2}{4}{V}_{ij2}}\\ +{\displaystyle \sum _{j\in U_k^i}\left[E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)V_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]}+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)<0\end{array}$(2.10)

则$\Omega <0$ ，由引理2.2有(2.10)成立。可以看出

$LV\left(x_t\right)\le {\gamma }^2{w}^{\text{T}}\left(t\right)w\left(t\right)-z^{\text{T}}\left(t\right)z\left(t\right).$

由定义2.2，系统(2.2)在H_∞扰动水平为零时，是随机允许的$\gamma$ 。

情况II $i\in U_k^i$ ，$U_k^i\ne \varnothing$ ，存在正定矩阵，$G_{ij}\in R^{n\times n}\left(i,j\in U_k^i,l\in U_{uk}^i\right)$ 有

$\left[\begin{array}{cccccc}{E}^{\left[m\right]}& E_{12}& e_{zi\mu }^{\text{T}}& {\Pi }_1^{\text{T}}E\left(P_{k_1^i}-F_i\right)& \cdots & {\Pi }_1^{\text{T}}E\left(P_{i{k}_{mi}^i}-F_i\right)\\ *& E_{22}& 0& 0& \cdots & 0\\ *& *& -I& 0& \cdots & 0\\ *& *& *& -G_{i{k}_1^i}& \cdots & 0\\ *& *& *& *& \ddots & ⋮\\ *& *& *& *& *& -G_{i{k}_{mi}^i}\end{array}\right]<0,$

证明 $i\in U_k^i$ 且$U_{uk}^i\ne \varnothing$ ，必须有$l\in U_{uk}^i$ ，$E^{\text{T}}P\left(l\right)E-E^{\text{T}}P\left(j\right)E\ge 0$ ，$j\in U_{uk}^i$ 。因为它认为

$\Omega =\Xi +E_2+E_3+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}+{\pi }_{ii}{P}_i+{\displaystyle \sum _{j\in U_{uk}^i}{\pi }_{ij}{P}_j}\right){\Lambda }_1+x^{\text{T}}\left(t\right)E^{\text{T}}\left({\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}+{\pi }_{ii}{P}_i+{\displaystyle \sum _{j\in U_{uk}^i}{\pi }_{ij}{P}_j}\right)x\left(t\right),$

然后我们有

$\begin{array}{l}\Omega \le \Xi +E_2+E_3+{\Lambda }_1^{\text{T}}\left({\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}+{\displaystyle \sum _{j\in U_{uk}^i}{\pi }_{ij}{P}_l}\right){\Lambda }_1+E^{\text{T}}{x}^{\text{T}}\left(t\right)\left({\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}{P}_j}+{\displaystyle \sum _{j\in U_{uk}^i}{\pi }_{ij}{P}_l}\right)x\left(t\right)\\ =\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{\Lambda }_1^{\text{T}}{P}_j{\Lambda }_1-{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{\Lambda }_1^{\text{T}}{P}_l{\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{x}^{\text{T}}\left(t\right)E^{\text{T}}{P}_{j}x\left(t\right)-{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{x}^{\text{T}}\left(t\right)E^{\text{T}}{P}_{l}x\left(t\right)\\ =\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}\left({\widehat{\pi }}_{ij}+{\pi }_{ij}\right){\Lambda }_1^{\text{T}}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\left({\widehat{\pi }}_{ij}+{\pi }_{ij}\right)x^{\text{T}}}\left(t\right)E^{\text{T}}\left(P_j-P_l\right)x\left(t\right)\\ =\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{x}^{\text{T}}\left(t\right)E^{\text{T}}\left(P_j-P_l\right)x\left(t\right)\\ +{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{x}^{\text{T}}\left(t\right)E^{\text{T}}\left(P_j-P_l\right)x\left(t\right),\end{array}$

此外，由于引理2.2，我们可以得到

$\begin{array}{l}{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}{\pi }_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\\ ={\displaystyle \sum _{j\in U_k^i}\left[\frac{1}{2}{\pi }_{ij}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+\frac{1}{2}{\pi }_{ij}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\left[\frac{1}{2}{\pi }_{ij}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)+\frac{1}{2}{\pi }_{ij}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]}\\ \le {\displaystyle \sum _{j\in U_k^i}\left[\frac{{\delta }_{ij1}^2}{4}{G}_{ij1}+{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1{G}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\left[\frac{{\delta }_{ij2}^2}{4}{G}_{ij2}+E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)G_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]},\end{array}$ (2.11)

另外，我们有

$\begin{array}{l}\Omega \le \Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij1}^2}{4}{G}_{ij1}}\\ +{\displaystyle \sum _{j\in U_k^i}\left[{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1{G}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij2}^2}{4}{G}_{ij2}}\\ +{\displaystyle \sum _{j\in U_k^i}\left[E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)G_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]},\end{array}$(2.12)

如果

$\begin{array}{l}\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\left[{\Lambda }_1^{\text{T}}\left(P_j-P_l\right)E^{\text{T}}{\Lambda }_1{G}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij1}^2}{4}{G}_{ij1}}+{\displaystyle \sum _{j\in U_k^i}\left[E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)G_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij2}^2}{4}{G}_{ij2}}+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)<0,\end{array}$(2.13)

因此$\Omega <0$ ，由引理2.2有(2.13)成立。可以看出$LV\left(x_t\right)\le {\gamma }^2{w}^{\text{T}}\left(t\right)w\left(t\right)-z^{\text{T}}\left(t\right)z\left(t\right)$ 。由定义2.2，系统(2.2)在H_∞扰动水平为零时是随机允许的$\gamma$ 。

情况III $i\in U_k^i$ ，$U_{uk}^i=\varnothing$ ，存在正定矩阵，$L_{ij}\in R^{n\times n}\left(i,j\in U_k^i\right)$ 有

$\left[\begin{array}{cccccc}{E}^{\left[m\right]}& E_{12}& e_{zi\mu }^{\text{T}}& {\Pi }_1^{\text{T}}E\left(P_{k_1^i}-F_i\right)& \cdots & {\Pi }_1^{\text{T}}E\left(P_{i{k}_{mi}^i}-F_i\right)\\ *& E_{22}& 0& 0& \cdots & 0\\ *& *& -I& 0& \cdots & 0\\ *& *& *& -L_{i{k}_1^i}& \cdots & 0\\ *& *& *& *& \ddots & ⋮\\ *& *& *& *& *& -L_{i{k}_{mi}^i}\end{array}\right]<0,$

证明 $i\in U_k^i$ ，$U_{uk}^i=\varnothing$ 。同样，它认为

$\begin{array}{l}\Omega =\Xi +E_2+E_3+{\Lambda }_1^{\text{T}}{\displaystyle \sum _{j=1,j\ne i}^s{\pi }_{ij}}{P}_j{\Lambda }_1+{\Lambda }_1^{\text{T}}{\pi }_{ii}{P}_i{\Lambda }_1+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j=1,j\ne i}^s{\pi }_{ij}}{P}_{j}x\left(t\right)+x^{\text{T}}\left(t\right)E^{\text{T}}{\pi }_{ii}{P}_{i}x\left(t\right)\\ =\Xi +E_2+E_3+{\Lambda }_1^{\text{T}}{\displaystyle \sum _{j=1,j\ne i}^s{\pi }_{ij}}\left(P_j-P_i\right){\Lambda }_1+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j=1,j\ne i}^s{\pi }_{ij}}\left(P_j-P_i\right)x\left(t\right)\\ =\Xi +E_2+E_3+{\Lambda }_1^{\text{T}}{\displaystyle \sum _{j=1,j\ne i}^s\left({\widehat{\pi }}_{ij}+{\pi }_{ij}\right)}\left(P_j-P_i\right){\Lambda }_1+x^{\text{T}}\left(t\right)E^{\text{T}}{\displaystyle \sum _{j=1,j\ne i}^s\left({\widehat{\pi }}_{ij}+{\pi }_{ij}\right)}\left(P_j-P_i\right)x\left(t\right)\\ \le \Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_i\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij1}^2}{4}{L}_{ij1}}+{\displaystyle \sum _{j\in U_k^i}\left[{\Lambda }_1^{\text{T}}\left(P_j-P_i\right){\Lambda }_1{L}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_i\right){\Lambda }_1\right]}\\ +{\displaystyle \sum _{j\in U_k^i}\left[E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_i\right)x\left(t\right)L_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_i\right)x\left(t\right)\right]}+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_i\right)x\left(t\right)+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij2}^2}{4}{L}_{ij2},}\end{array}$

如果

$\begin{array}{l}\Xi +E_2+E_3+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij1}^2}{4}{L}_{ij1}}+{\displaystyle \sum _{j\in U_k^i}{\widehat{\pi }}_{ij}}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)+{\displaystyle \sum _{j\in U_k^i}\frac{{\delta }_{ij2}^2}{4}{L}_{ij2}}\\ +{\displaystyle \sum _{j\in U_k^i}\left[{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1{L}_{ij1}^{-1}{\Lambda }_1^{\text{T}}\left(P_j-P_l\right){\Lambda }_1\right]}+{\displaystyle \sum _{j\in U_k^i}\left[E^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)L_{ij2}^{-1}{E}^{\text{T}}{x}^{\text{T}}\left(t\right)\left(P_j-P_l\right)x\left(t\right)\right]}<0\end{array}$(2.14)

则$\Omega <0$ ，引理2.2，(2.14)成立可以看出

$LV\left(x_t\right)\le {\gamma }^2{w}^{\text{T}}\left(t\right)w\left(t\right)-z^{\text{T}}\left(t\right)z\left(t\right).$

由定义2.2可知，系统(2.2)在扰动水平为H_∞时为零时是随机允许的$\gamma$ 。考虑具有恒定延迟$h_2$ 的SMJs，其中的参数与[8]中的参数相同，数值示例说明本文比以前方法有效更通用见表1：

Table 1. Upper bound contrast

表1. 上界对比

比较结果	d
[8]	1.2362
本文$\epsilon =0$	1.4731
本文$\epsilon =2$	1.5836

$\begin{array}{l}E=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],A=\left[\begin{array}{cc}0.4972& 0\\ 0& -0.9541\end{array}\right],A_d=\left[\begin{array}{cc}-1.4972& 1.5415\\ 0& 0.5449\end{array}\right],\\ M=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],N=\left[\begin{array}{cc}2& 1\\ -1& -1\end{array}\right],MEN=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\end{array}$

此外，我们选择

$\begin{array}{l}\Sigma =\left[\begin{array}{cc}0.1& 0.4\\ 0.2& 0.5\end{array}\right],B=\left[\begin{array}{c}-1\\ 0\end{array}\right],B_d=\left[\begin{array}{c}-2\\ 1\end{array}\right],B_w=\left[\begin{array}{c}1\\ 2.5\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],\\ C_d=\left[\begin{array}{cc}0.3& -0.3\end{array}\right],D_w=\left[\begin{array}{cc}1& 1.5\\ 3& 2\end{array}\right],H_R=4,\gamma =\mathrm{0.1.}\end{array}$

为了验证所提出的方法绘制了$x\left(0\right)={\left[\begin{array}{cc}0& 0\end{array}\right]}^{\text{T}}$ 和$w\left(t\right)={\left[\begin{array}{cc}1& 0\end{array}\right]}^{\text{T}}\sin {\text{e}}^{-0.2t}$ 可以看出SNJs状态响应收敛于零见图1。

Figure 1. $x\left(t\right)$ status response with external input $w\left(t\right)$

图1. 具有外部输入$w\left(t\right)$ 的$x\left(t\right)$ 状态响应

备注2.2：与一般的LKF相比，乘以更多的积分项，利用了更多的时变延迟信息，在增强的LKF中减少保守性。(2.21)和(2.22)中的条件可以通过使用[10]引理3得到更紧的界限。因此，本文定理的保守性比Wang等[12]、Fu和Ma [13]更低。

3. 结论

对于给定的标量本章探讨一类非线性广义Markov跳跃系统的随机稳定性和H_∞控制问题。设计新型的Lyapunov-Krasovkii泛函(LKF)，构造适当的Lyapunov-Krasovskii泛函，应用依赖参数相关的互复凸矩阵不等式(PDRCMI)、改进的Wirtinger不等式，Chen提出的广义积分不等式，通过LMI，得到保守性较低的准则。同时，引入参数依赖的互凸矩阵不等式(PDRCMI)来降低保守性，保证了非线性广义Markov跳跃系统渐进稳定并满足性能。此外，将自适应方法应用于MJSs具有现实意义，值得进一步探索。值得一提的是，我们的理论结果也可以在未来更复杂的系统。