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:

$\{\begin{array}{l}E\stackrel{\dot{}}{x}\left(t\right)=\left(A\left(r_t\right)+\Delta A\left(r_t\right)\right)x\left(t\right)+\left(A_d\left(r_t\right)+\Delta A_d\left(r_t\right)\right)x\left(t-d\left(t\right)\right)+B_w\left(r_t\right)\varpi \left(t\right)\\ +B\left(r_t\right)\left(u\left(t\right)+f\left(t,x\left(t\right),r\left(t\right)\right)\right)+B_d\left(r_t\right)\left(u\left(t-d\left(t\right)\right)+f\left(t,x\left(t-d\left(t\right)\right),r\left(t\right)\right)\right)\\ z\left(t\right)=C\left(r_t\right)x\left(t\right)+C_d\left(r_t\right)x\left(t-d\left(t\right)\right)+D\left(r_t\right)w\left(t\right)\end{array}$(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 $0\le d_1\le d\left(t\right)\le d_2$， $h_1\le \stackrel{\dot{}}{d}\left(t\right)\le h_2$， $\forall t\ge 0$， $d_1,d_2,h_1,h_2$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 ${\pi }_{ij}=-{\displaystyle \underset{j=1,j\ne i}{\overset{s}{\sum }}{\pi }_{ij}}$。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：

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

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

其中 ${\hat{\pi }}_{ij}$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。 ${\hat{\pi }}_{ij}$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：

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

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

$\left[\begin{array}{ccccc}{\pi }_{11}& ?& {\pi }_{13}& \cdots & ?\\ ?& ?& {\pi }_{23}& \cdots & {\pi }_{2s}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ ?& {\pi }_{s2}& ?& \cdots & {\pi }_{ss}\end{array}\right]$(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 ${\hat{\pi }}_{ij}-{\delta }_{ij}\ge 0$， $\left(\forall j\in S,j\ne i\right)$， ${\hat{\pi }}_{ii}=-{\displaystyle \underset{j=1,j\ne i}{\overset{s}{\sum }}{\hat{\pi }}_{ij}\le 0}$， ${\delta }_{ii}={\displaystyle \underset{j=1,j\ne i}{\overset{s}{\sum }}{\delta }_{ij}}>0$；

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

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

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。

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

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 $\det \left(s\hat{E}-{\hat{A}}_i\right)$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 $\mathrm{deg}\left(\det \left(s\hat{E}-{\hat{A}}_i\right)\right)=rank\left(\hat{E}\right)$，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

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

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 $\gamma$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ${\left(A_i+A_{di}\right)}^{\text{T}}{P}_i+P_i^{\text{T}}\left(A_i+A_{di}\right)+{\displaystyle \underset{j=1}{\overset{s}{\sum }}{\pi }_{ij}{E}^{\text{T}}{P}_{j}E<0}$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。

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

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

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

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

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

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

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

L为弱无穷小算子，有：

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

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

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

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

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