具有欺骗攻击的中立型半马尔可夫随机系统的自适应滑模控制

Adaptive SMC for Neutral Semi-Markov Stochastic Systems with Deception Attack

摘要: 本文研究了在欺骗攻击模式下的中立型半马尔可夫随机系统的滑模控制问题。首先，本文提出了一种能自适应调整阈值的动态事件触发机制，该方法可有效地节省网络通信资源；其次，设计了一种考虑网络延迟的滑模面，并建立了基于收敛因子的滑模控制律，该方法可保证系统在有限时间内能够到达预定的滑模面，然后应用线性矩阵不等式方法，得到保证滑模控制系统限时稳定的充分条件，最后，通过一个数值算例验证结果的有效性。

Abstract: This paper studies the sliding mode control problem of neutral semi-Markov stochastic systems under fraud attack patterns. Firstly, a dynamic event-triggering mechanism capable of adaptively adjusting the threshold is proposed, which can effectively save network communication resources. Secondly, a sliding mode surface considering network delay was designed, and a sliding mode control law based on convergence factors was established. This method can ensure that the system can reach the predetermined sliding mode surface within a finite time. Then, the linear matrix inequality method was applied to obtain sufficient conditions to ensure the time-limited stability of the sliding mode control system. Finally, the validity of the results was verified through a numerical example.

1. 引言

半马尔可夫系统是一个复杂的动态系统，该系统的模态在不断切换并服从一个与驻留时间有关的半马尔科夫过程，这使得模型能描述更广泛的现实现象。近几十年来越来越多的学者投入对中立型半马尔科夫系统的研究之中，并取得了许多重要成果[1]-[3]，文献[1]研究了受分数布朗运动影响的不确定延迟中立型半马尔可夫跳变系统的随机$H_{\infty }$ 控制，得到了使系统稳定的控制器增益。文献[3]基于事件触发方案和隐马尔可夫模型，设计了异步滤波器并推导了滤波误差系统为耗散随机有限时间有界的充分条件。然而在现实生活中，系统常常遭受到来自外界的网络攻击，常见的如Dos攻击[4]、欺诈攻击[5]等，为应对此类问题，学者们提出了多种控制策略。文献[6]提出了一种基于攻击参数的自适应事件触发机制，该机制可根据网络状态和DoS攻击的严重程度动态调整每个节点的阈值参数，通过建立的协同设计准则，得到了自适应动态事件触发机制和同步控制器的增益矩阵。文献[7]讨论了在存在拒绝服务攻击的情况下，马尔可夫跳变系统的滑模控制问题，并通过仅使用成功传输的状态信号设计了与模式相关的滑模控制器，以保证系统在有限时间内到达所设计的滑模面。

滑模控制[9]方法因其鲁棒性已成为一种有效的控制方法，并应用于各种控制系统。文献[10]研究了具有马尔可夫跳变参数和时变延迟的不确定中立型随机系统的$H_{\infty }$ 滑模控制问题。通过设计合适的滑模控制律，保证系统状态在有限时间内能到达滑模面。随着网络化马尔科夫系统研究的不断深入，需要传输到滑模控制器的数据信息越来越多，因此，学者们在马尔科夫系统滑模控制的研究中引入事件触发来减小通信负担。但是，文献[12] [13]在设计事件触发器时，触发阈值是固定的，无法动态调整以适应动态系统，这不利于通信资源的有效利用，在[14] [15]中，提出的自适应事件触发方案的事件触发阈值可以通过一种新的自适应法则进行调整，以适应动态系统。基于此，本文设计了一种自适应事件触发方案，使阈值在动态调整适应动态系统的同时达到节约网络资源的目的。

本文基于Lyapunov方法，结合线性矩阵不等式等方法，探讨了具有欺骗攻击的中立型半马尔可夫随机系统的自适应滑模控制问题。主要贡献如下：(1) 提出了一种新型自适应事件触发方案，比起传统的事件触发方案，该事件触发方案在节省网络资源方面更有效。(2) 设计了一种考虑网络延迟的滑模面，构建了基于收敛因子的滑模控制律，推导得到了保证滑模控制系统实现限时稳定的充分条件。

2. 问题描述

2.1. 系统简介

考虑具有时变时滞的中立型半马尔可夫随机系统

$\{\begin{array}{l}d\left[x\left(t\right)-C\left(r_t\right)x\left(t-\tau \left(t\right)\right)\right]=\left[A\left(r_t\right)x\left(t\right)+B\left(r_t\right)u^{\aleph }\left(t\right)+D\left(r_t\right)x\left(t-h\left(t\right)\right)\right]\text{d}t\\ +\left[A_1\left(r_t\right)x\left(t\right)+D_1\left(r_t\right)x\left(t-h\left(t\right)\right)\right]\text{d}w\left(t\right);\\ x\left(t\right)=\chi \left(t\right),\forall t\in \left[-\rho ,0\right].\end{array}$ (1)

其中$x(t)\in R^n$ 为系统的状态向量，$u^{\aleph }(t)\in R^m$ 为控制输入，$A(r_t),A_1(r_t),B(r_t),C(r_t),D(r_t),D_1(r_t)$ 为适当维数的系统矩阵，$r_t$ 为系统模态，$w(t)$ 为标准布朗运动且满足$E\{dw(t)\}=0$ ，$\tau (t),h(t)$ 分别表示中立型时滞和时变时滞，且满足条件：

$\{\begin{array}{l}0<\tau (t)<\tau ,0<\dot{\tau }(t)<{\tau }_d<1,\\ 0<h(t)<h,0<\dot{h}(t)<h_d<1.\end{array}$

$\left\{r_t,t\ge 0\right\}$ 是取自有限集$S=\left\{1,2,\cdots ,S\right\}$ 的齐次半马尔可夫过程，其模态转移概率为

$\mathrm{Pr}\left(r\left(t+\Delta \right)=j|r\left(t\right)=i\right)=\{\begin{array}{l}\mathrm{Pr}\left\{\hslash \le h+\Delta ,{\overline{r}}_{k+1}=j|\hslash >h\right\}={\pi }_{ij}\left(h\right)\Delta +o\left(\Delta \right),i\ne j;\\ \mathrm{Pr}\left\{\hslash >h+\Delta |\hslash >h,{\overline{r}}_k=i\right\}=1+{\pi }_{ii}\left(h\right)\Delta +o\left(\Delta \right),i=j.\end{array}$ (2)

这里$\Delta >0$ 是时间增量，$\underset{\Delta \to 0}{\lim }\frac{o\left(\Delta \right)}{\Delta }=0$ 。当$i\ne j$ 时，${\pi }_{ij}\left(h\right)$ 表示系统从第$i$ 模态跳跃到第$j$ 模态的转移率，且${\pi }_{ii}\left(h\right)=-{\displaystyle \sum _{j=1,j\ne i}^S{\pi }_{ij}\left(h\right)}$ ，其转移率矩阵为

$\Pi \left(h\right)={\left[{\pi }_{ij}\left(h\right)\right]}_{S\times S}=\left[\begin{array}{cccc}{\pi }_{11}\left(h\right)& {\pi }_{12}\left(h\right)& \cdots & {\pi }_{1N}\left(h\right)\\ {\pi }_{21}\left(h\right)& {\pi }_{22}\left(h\right)& \cdots & {\pi }_{2N}\left(h\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\pi }_{N1}\left(h\right)& {\pi }_{N2}\left(h\right)& \cdots & {\pi }_{NN}\left(h\right)\end{array}\right].$

考虑到系统会遭受到网络欺诈攻击，本文根据所受到的欺诈攻击模态的控制输入具体设计为

$u^{\aleph }\left(t\right)=u\left(t\right)+\alpha \left(t\right)H_{r\left(t\right)}\left(t\right)\zeta \left(x\left(t\right),r\left(t\right),t\right).$ (3)

其中$H_{r\left(t\right)}\left(t\right)$ 是用来表征攻击特征的未知加权矩阵，满足$\Vert H_{r\left(t\right)}\left(t\right)\Vert \le {\upsilon }_{r\left(t\right)}$ ，${\upsilon }_{r\left(t\right)}$ 是一个常数，$\zeta \left(x\left(t\right),r\left(t\right)\right)$ 表示网络攻击所注入的非线性函数，满足$\Vert \zeta \left(x\left(t\right),r\left(t\right)\right)\Vert \le \vartheta \left(x\left(t\right),r\left(t\right)\right)$ ，$\vartheta \left(x\left(t\right),r\left(t\right)\right)$ 是一个先验函数。随机变量$\alpha \left(t\right)$ 服从伯努利分布序列，且满足$\mathrm{Pr}\left\{\alpha =1\right\}={\alpha }^{*}$ ，$\mathrm{Pr}\left\{\alpha =0\right\}=1-{\alpha }^{*}$ ，${\alpha }^{*}\in \left(0,1\right]$ 。当系统遭受网络攻击时${\alpha }^{*}=1$ ，系统未遭受网络攻击时${\alpha }^{*}=0$ 。

2.2. 自适应事件触发

为了节省网络资源，减少不必要的数据传输，本文建立了自适应事件触发方案，以$h$ 为采样周期，采样时间序列为${\Im }_1=\left\{\beta h|\beta \in N^{+}\right\}$ ，受事件触发影响的触发时间序列为${\Im }_2=\left\{t_{k}h|t_k\in N^{+}\right\},t_0=0$ ，$d_k\in \left[0,d_M\right)$ 为网络传输引起的延迟，将自适应事件触发机制定义为

$e^{\text{T}}\left(t\right){\Lambda }_{i}e\left(t\right)\ge \eta \left(t\right)x^{\text{T}}\left(t_{k}h\right){\Lambda }_{i}x\left(t_{k}h\right)$ (4)

下一个切换时刻为

$t_{k+1}=t_k+\underset{\rho \ge 1}{\inf }\left\{\rho h|e^{\text{T}}\left(t\right){\Lambda }_{i}e\left(t\right)\ge \eta \left(t\right)x^{\text{T}}\left(t_{k}h+\rho h\right){\Lambda }_{i}x\left(t_{k}h+\rho h\right)\right\}$ (5)

这里$t\in \left[t_{k}h+d_k,t_{k+1}h+d_{k+1}\right)$ ，${\Lambda }_i>0$ 为自适应触发矩阵，$t_{k}h+\rho h$ 和$t_{k}h$ 分别为最新采样时间和最近触发时间，$e\left(t\right)=x\left(t_{k}h+\rho h\right)-x\left(t_{k}h\right)$ ，$x\left(t_{k}h\right)$ 为最新传输数据，$x\left(t_{k}h+\rho h\right)$ 为当前采样数据。动态阈值$\eta \left(t\right)$ 满足

$\dot{\eta }\left(t\right)=\frac{1}{\eta \left(t\right)}\left(\frac{1}{\eta \left(t\right)}-{\eta }_0\right)e^{\text{T}}\left(t\right){\Lambda }_{i}e\left(t\right)$(6)

其中，$\eta \left(0\right)={\eta }_0\ge 0$ 为一个预定的常数。对当$t\in \left[t_{k}h+d_k,t_{k+1}h+d_{k+1}\right]$ 时，网络延迟$d\left(t\right)=t-t_{k}h-\rho h$ ，$0\le d_k\le d\left(t\right)\le d_M$ ，$\dot{d}\left(t\right)\le d_{\ell }\le 1$ ，可以得到$x\left(t_{k}h\right)=x\left(t-d\left(t\right)\right)-e\left(t\right)$ 。

注1：事件触发参数$\eta \left(t\right)$ 可以通过(6)中描述的自适应律进行动态调整。自适应律依赖于最新触发的数据与当前数据之间的误差，当误差$e\left(t\right)$ 趋于0时，$\dot{\eta }\left(t\right)\to 0$ ，此时事件触发阈值趋于常数。此外，如果选择事件触发参数$\eta \left(t\right)$ 为0初值，则$\dot{\eta }\left(t\right)=0$ ，所提出的自适应事件触发方案将转化为传统的事件触发方案[16] [17]，触发条件如下

$t_{k+1}=t_k+\underset{\rho \ge 1}{\inf }\left\{\rho h|e^{\text{T}}\left(t\right){\Lambda }_{i}e\left(t\right)\ge \overline{\eta }{x}^{\text{T}}\left(t_{k}h+\rho h\right){\Lambda }_{i}x\left(t_{k}h+\rho h\right)\right\}$(7)

其中，$\overline{\eta }\in \left(0,1\right)$ 是一个预定的常数。与传统的事件触发方案相比，本文提出的自适应事件触发方案在节省网络资源方面更加有效。

为了方便，当$r_t=i$ 时，记$A\left(r_t\right)=A_i,A_1\left(r_t\right)=A_{1i},B\left(r_t\right)=B_i,C\left(r_t\right)=C_i,D\left(r_t\right)=D_i,D_1\left(r_t\right)=D_{1i}$ 。

2.3. 基于事件的滑模控制

基于上述的动态事件触发机制，本文设计如下的滑模面

$\{\begin{array}{l}s\left(t\right)=G_i\left[x\left(t\right)-C\left(r_t\right)x\left(t-\tau \left(t\right)\right)\right]-G_i\delta \left(t\right);\\ d\delta \left(t\right)=G_i\left(A\left(r_t\right)x\left(t\right)+D\left(r_t\right)x\left(t-h\left(t\right)\right)\right)+G_{i}B\left(r_t\right)K_i\left(x\left(t-d\left(t\right)\right)-e\left(t\right)\right).\end{array}$ (8)

其中$G_i$ 非奇异，$K_i$ 为控制器增益矩阵，$B\left(r_t\right)K_i$ 是一个Hurwitz矩阵。

结合(1)和(8)可以推导出

$\begin{array}{c}E\left\{ds\left(t\right)\right\}=E\{G_i\left[A\left(r_t\right)x\left(t\right)+B\left(r_t\right)u^{\aleph }\left(t\right)+D\left(r_t\right)x\left(t-h\left(t\right)\right)\right]\\ -\left[G_i{A}_{r_t}x\left(t\right)+G_i{B}_{r_t}{K}_i\left(x\left(t-d\left(t\right)\right)-e\left(t\right)\right)\right]\}\\ =E\left[G_{i}B\left(r_t\right)u^{\aleph }\left(t\right)-G_{i}B\left(r_t\right)K_i\left(x\left(t-d\left(t\right)\right)-e\left(t\right)\right)\right].\end{array}$ (9)

考虑滑模控制理论，建立等效控制器为

$u_{eq}=K_i\left(x\left(t-d\left(t\right)\right)-e\left(t\right)\right)-\alpha \left(t\right)H\left(r_t\right)\zeta \left(x\left(t\right),r\left(t\right)\right).$ (10)

将(10)和(3)式带入(1)式，可以得到系统(1)的等效闭环系统

$\{\begin{array}{l}d\left[x\left(t\right)-C_{r_t}x\left(t-\tau \left(t\right)\right)\right]=\left[A\left(r_t\right)x\left(t\right)+B\left(r_t\right)K_i\left(x\left(t-d\left(t\right)\right)-e\left(t\right)\right)+D\left(r_t\right)x\left(t-h\left(t\right)\right)\right]\text{d}t\\ +\left[A_1\left(r_t\right)x\left(t\right)+D_1\left(r_t\right)x\left(t-h\left(t\right)\right)\right]\text{d}w\left(t\right),\\ x\left(t\right)=\chi \left(t\right),\forall t\in \left[-\rho ,0\right].\end{array}$ (11)

2.4. 引理与定义

定义1 [18]对于系统(11)，若任意给定的正标量$c_1,c_2,T$ ，正定矩阵$U$ ，如果状态变量满足关系

$\underset{t_0-{\rho }^{*}\le s\le t_0}{\sup }\left\{x^{\text{T}}\left(s\right)Ux\left(s\right),{\dot{x}}^{\text{T}}\left(s\right)U\dot{x}\left(s\right)\right\}\le c_1,\forall t\in \left[t_0,T\right],$(12)

其中${\rho }^{*}=\max \left(\tau ,h,d_M\right)$ ，有

$x^{\text{T}}\left(t\right)Ux\left(t\right)<c_2.$ (13)

则称系统(11)对于${\Gamma }_i=\left(c_1,c_2,T,U\right)$ 是限时稳定的。

引理1 [19]对于任意正定矩阵$M$ ，标量$h_2>h_1\ge 0$ ，向量函数$\nu :\left[h_2,h_1\right]\mapsto R^n$ ，有下列不等式成立：

$\begin{array}{l}-\left(h_2-h_1\right)\left({\displaystyle {\int }_{t-h_2}^{t-h_1}{\nu }^{\text{T}}\left(s\right)M\nu \left(s\right)\text{d}s}\right)\\ \le -{\left({\displaystyle {\int }_{t-h_2}^{t-h_1}\nu \left(s\right)\text{d}s}\right)}^{\text{T}}M\left({\displaystyle {\int }_{t-h_2}^{t-h_1}\nu \left(s\right)\text{d}s}\right).\end{array}$ (14)

引理2 [20]对给定的对称常数矩阵${\sum }_1,{\sum }_2,{\sum }_3$ 若${\sum }_1+{\sum }_3^{\text{T}}{\sum }_2^{-1}{\sum }_3<0$ 成立，当且仅当

$\left[\begin{array}{cc}{\sum }_1& {\sum }_3^{\text{T}}\\ {\sum }_3& -{\sum }_2\end{array}\right]<0$ 或$\left[\begin{array}{cc}-{\sum }_2& {\sum }_3\\ {\sum }_3^T& {\sum }_1\end{array}\right]<0.$ (15)

3. 主要结果及其证明

3.1. 可达性分析

本文设计如下滑模控制律

$u\left(t\right)=K_i\left(x\left(t_k\right)-\alpha {\text{e}}^{-\iota t}sign\left(B_i^{\text{T}}{G}_i^{\text{T}}s\left(t\right)\right)-\beta \left(t\right)sign\left(B_i^{\text{T}}{G}_i^{\text{T}}s\left(t\right)\right)\right),\forall t\in \left[t_k,t_{k+1}\right].$ (16)

其中$\beta \left(t\right)=\beta +{\alpha }^{*}{\varsigma }_i\zeta \left(x\left(t\right),i\right)$ ，$\beta >0$ 是一个正标量，${\text{e}}^{-\iota t}$ 是引入的收敛因子。

下面将证明在控制律(16)下，系统(11)将在有限时间内到达所设计的滑模面。

定理3.1：在滑模控制律(16)作用下的系统(11)的状态轨迹将在有限时间内收敛到滑模面(8)

证明：构造正定函数

$V\left(t\right)=\frac{1}{2}{s}^{\text{T}}\left(t\right)s\left(t\right).$ (17)

结合(9)式、(16)式和(17)式，计算$LV\left(t\right)$ ，可以得到

$\begin{array}{c}LV\left(t\right)=s^{\text{T}}\left(t\right)\left(G_i{B}_i\left(u^{\aleph }\left(t\right)+\alpha \left(t\right)H\left(r_t\right)\zeta \left(x\left(t\right),r\left(t\right)\right)\right)-G_i{B}_i{K}_{i}x\left(t_{k}h\right)\right)\\ \le s^{\text{T}}\left(t\right)G_i{B}_i\left[\alpha \left(t\right)H\left(r_t\right)\zeta \left(x\left(t\right),r\left(t\right)\right)-\left(\beta +{\alpha }^{*}{\varsigma }_i\zeta \left(x\left(t\right),i\right)\right)sign\left(B_i^{\text{T}}{G}_i^{\text{T}}s\left(t\right)\right)\right]\\ \le -\left(\beta +\alpha {\text{e}}^{-\iota t}\right)\left|B_i^{\text{T}}{G}_i^{\text{T}}s\left(t\right)\right|<0.\end{array}$ (18)

由(18)知$V\left(t\right)$ 单调不增，即系统(11)的轨线将在有限时间内到达预定的滑模面(8)，证毕。

3.2. 滑动模态有限时间稳定分析

下面主要通过Lyapunov方法和矩阵不等式技巧，研究系统(11)的限时稳定问题。

定理3.2：当矩阵$C_i$ 的谱半径小于1时，若存在适当维数的矩阵$P_i>0$ ，$R_1>0$ ，$R_2>0$ ，$R_3>0$ ，$N_{\text{1}}>0$ ，$N_{\text{2}}>0$ ，$N_{\text{3}}>0$ ，标量${\gamma }_i>0$ ，${ƛ}_1>0$ ，${ƛ}_2>0$ ，${\lambda }_1>0$ ，${\lambda }_2>0$ ，${\lambda }_3>0$ ，${\lambda }_4>0$ ，${\lambda }_5>0$ ，${\lambda }_6>0$ ，以及非退化矩阵$U$ ，使得下面线性矩阵不等式成立

$\left[\begin{array}{ccccccccc}{\varphi }_{11}& {\varphi }_{12}& {\varphi }_{1\text{3}}& {\varphi }_{14}& 0& 0& 0& {\varphi }_{18}& A_{1i}^{\text{T}}{P}_i\\ *& {\varphi }_{22}& 0& 0& 0& 0& 0& 0& C_{1i}^{\text{T}}{A}_{1i}^{\text{T}}{P}_i\\ *& *& {\varphi }_{33}& 0& 0& 0& 0& 0& D_{1i}^{\text{T}}{P}_i\\ *& *& *& {\varphi }_{44}& 0& 0& 0& {\varphi }_{\text{48}}& 0\\ *& *& *& *& {\varphi }_{55}& 0& 0& 0& 0\\ *& *& *& *& *& {\varphi }_{66}& 0& 0& 0\\ *& *& *& *& *& *& {\varphi }_{77}& 0& 0\\ *& *& *& *& *& *& *& {\varphi }_{88}& 0\\ *& *& *& *& *& *& *& *& -P_i\end{array}\right]<0;$ (19)

${\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{1j}}\le {\gamma }_i{R}_{1i};{\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{2j}}\le {\gamma }_i{R}_{2i};{\displaystyle \sum _{j=1}^N{\pi }_{ij}{R}_{3j}}\le {\gamma }_i{R}_{3i};$ (20)

$\begin{array}{l}{ƛ}_{1}I<{\widehat{P}}_i<{ƛ}_{2}I;0<{\widehat{R}}_1\le {\lambda }_{1}I;0<{\widehat{R}}_2\le {\lambda }_{2}I;0<{\widehat{R}}_3\le {\lambda }_{3}I;\\ 0<{\widehat{N}}_1\le {\lambda }_{4}I;0<{\widehat{N}}_2\le {\lambda }_{5}I;0<{\widehat{N}}_3\le {\lambda }_{6}I.\end{array}$ (21)

其中

$\begin{array}{l}{\widehat{P}}_i=U^{-\frac{1}{2}}{P}_i{U}^{-\frac{1}{2}};{\widehat{R}}_1=U^{-\frac{1}{2}}{R}_1{U}^{-\frac{1}{2}};{\widehat{R}}_2=U^{-\frac{1}{2}}{R}_2{U}^{-\frac{1}{2}};{\widehat{R}}_3=U^{-\frac{1}{2}}{R}_3{U}^{-\frac{1}{2}};\\ {\widehat{N}}_1=U^{-\frac{1}{2}}{N}_1{U}^{-\frac{1}{2}};{\widehat{N}}_2=U^{-\frac{1}{2}}{N}_2{U}^{-\frac{1}{2}};{\widehat{N}}_3=U^{-\frac{1}{2}}{N}_3{U}^{-\frac{1}{2}};\\ {\varphi }_{11}=P_i{A}_i+A_i^{\text{T}}{P}_i+{\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}-{\gamma }_i{P}_i+R_1+R_2+R_3+h^2{N}_1+{\tau }^2{N}_2+d_M^2{N}_3;\\ {\varphi }_{12}=P_i{A}_i{C}_i+\left(R_1+R_2+R_3\right)C_i+h^2{N}_1+{\tau }^5{N}_2+d_M^2{N}_3;\\ {\varphi }_{13}=P_i{D}_i;{\varphi }_{14}=P_i{B}_i{K}_i;{\varphi }_{18}=-P_i{B}_i{K}_i;\\ {\varphi }_{22}=C_i^{\text{T}}\left(R_1+R_2+R_3\right)C_i-\left(1-{\tau }_d\right)R_2+h^2{N}_1+{\tau }^5{N}_2+d_M^2{N}_3;\\ {\varphi }_{33}=-\left(1-h_d\right)R_1;{\varphi }_{44}={\Lambda }_i-\left(1-d_{\ell }\right)R_{3i};{\varphi }_{48}=-{\Lambda }_i;\\ {\varphi }_{55}=-N_1;{\varphi }_{66}=-N_2;{\varphi }_{77}=-N_3;{\varphi }_{88}=-{\Lambda }_i-{\eta }_0{\Lambda }_i.\end{array}$

则系统(11)关于$\left(c_1,c_2,T,U\right)$ 是限时稳定的。

证明：定义算子$\phi \left(x_t,i\right)=x\left(t\right)-C_{i}x\left(t-\tau \left(t\right)\right)$ 。构造如下的Lyapunov泛函

$V\left(x_t,t,r_t\right)=V_1\left(x_t,t,i\right)+V_2\left(x_t,t,i\right)+V_3\left(x_t,t,i\right)+V_4\left(x_t,t,i\right).$ (22)

其中

$\begin{array}{l}{V}_1\left(x_t,t,i\right)={\phi }^{\text{T}}\left(x_t,i\right)P_i\phi \left(x_t,i\right);\\ V_2\left(x_t,t,i\right)={\displaystyle {\int }_{t-h\left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{1}x\left(s\right)\text{d}s}+{\displaystyle {\int }_{t-\tau \left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{2}x\left(s\right)\text{d}s}+{\displaystyle {\int }_{t-d\left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{3}x\left(s\right)\text{d}s};\\ V_3\left(x_t,t,i\right)=h{\displaystyle {\int }_{-h}^0{\displaystyle {\int }_{t+\theta }^t{x}^{\text{T}}\left(s\right)N_{1}x\left(s\right)\text{d}s\text{d}\theta }}+\tau {\displaystyle {\int }_{-\tau }^0{\displaystyle {\int }_{t+\theta }^t{x}^{\text{T}}\left(s\right)N_{2}x\left(s\right)\text{d}s\text{d}\theta }}+d_M{\displaystyle {\int }_{-d_M}^0{\displaystyle {\int }_{t+\theta }^t{x}^{\text{T}}\left(s\right)N_{3}x\left(s\right)\text{d}s\text{d}\theta }};\\ V_4\left(x_t,t,i\right)=\frac{1}{2}{\eta }^2\left(t\right).\end{array}$

这里$P_i>0$ ，$R_1>0$ ，$R_2>0$ ，$R_3>0$ ，$N_1>0$ ，$N_2>0$ ，$N_3>0$ 为适当维数的正定矩阵，分别对$V_i\left(x_t,t,r_t\right)$ $\left(i=1,2,3,4\right)$ 求伊藤算子，有

$\begin{array}{c}L{V}_1\left(x_t,t,i\right)=2{\phi }^{\text{T}}\left(x_t,i\right)P_i\left[A_{i}x\left(t\right)+B_i{u}^{\aleph }\left(t\right)+D_{i}x\left(t-h\left(t\right)\right)\right]+{\displaystyle \sum _{j=1}^N{\pi }_{ij}}{\phi }^{\text{T}}\left(x_t,i\right)P_j\phi \left(x_t,i\right)\\ +{\left[A_{1i}x\left(t\right)+D_{1i}x\left(t-h\left(t\right)\right)\right]}^{\text{T}}{P}_i\left[A_{1i}x\left(t\right)+D_{1i}x\left(t-h\left(t\right)\right)\right]\\ ={\phi }^{\text{T}}\left(x_t,i\right)\left[P_i{A}_i+A_i^{\text{T}}{P}_i+{\displaystyle \sum _{j=1}^N{\pi }_{ij}{P}_j}\right]\phi \left(x_t,i\right)\\ +2{\phi }^{\text{T}}\left(x_t,i\right)P_i[A_i{C}_{i}x\left(t-\tau \left(t\right)\right)+B_i{K}_i\left(x\left(t-d\left(t\right)\right)-e\left(t\right)\right)+D_{i}x\left(t-h\left(t\right)\right)]\\ +{\left[A_{1i}\phi \left(x_t,i\right)+A_{1i}{C}_{i}x\left(t-\tau \left(t\right)\right)+D_{1i}x\left(t-h\left(t\right)\right)\right]}^{\text{T}}\\ \times P_i\left[A_{1i}\phi \left(x_t,i\right)+A_{1i}{C}_{i}x\left(t-\tau \left(t\right)\right)+D_{1i}x\left(t-h\left(t\right)\right)\right].\end{array}$ (23)

$\begin{array}{c}L{V}_2\left(x_t,t,i\right)\le {\phi }^{\text{T}}\left(x_t,i\right)\left[R_1+R_2+R_3\right]\phi \left(x_t,i\right)+2{\phi }^{\text{T}}\left(x_t,i\right)\left[R_1+R_2+R_3\right]C_{i}x\left(t-\tau \left(t\right)\right)\\ +x^{\text{T}}\left(t-\tau \left(t\right)\right)\left[C_i^{\text{T}}\left(R_1+R_2+R_3\right)C_i-\left(1-{\tau }_d\right)R_2\right]x\left(t-\tau \left(t\right)\right)\\ -\left(1-h_d\right)x^{\text{T}}\left(t-h\left(t\right)\right)R_{1}x\left(t-\tau \left(t\right)\right)-\left(1-d_{\ell }\right)x^{\text{T}}\left(t-d\left(t\right)\right)R_{3}x\left(t-d\left(t\right)\right)\\ +{\displaystyle {\int }_{t-h\left(t\right)}^t{x}^{\text{T}}\left(s\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}}{R}_{1j}x\left(s\right)\text{d}s}+{\displaystyle {\int }_{t-\tau \left(t\right)}^t{x}^{\text{T}}\left(s\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}}{R}_{2j}x\left(s\right)\text{d}s}\\ +{\displaystyle {\int }_{t-d\left(t\right)}^t{x}^{\text{T}}\left(s\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}}{R}_{3j}x\left(s\right)\text{d}s}.\end{array}$ (24)

根据引理1，有

$\begin{array}{c}L{V}_3\left(x_t,t,i\right)\le x^{\text{T}}\left(t\right)\left[h^2{N}_1+{\tau }^2{N}_2+d_M^2{N}_3\right]x\left(t\right)-{\left({\displaystyle {\int }_{t-h}^{t}x\left(s\right)\text{d}s}\right)}^{\text{T}}{N}_1\left({\displaystyle {\int }_{t-h}^{t}x\left(s\right)\text{d}s}\right)\\ -{\left({\displaystyle {\int }_{t-\tau }^{t}x\left(s\right)\text{d}s}\right)}^{\text{T}}{N}_2\left({\displaystyle {\int }_{t-\tau }^{t}x\left(s\right)\text{d}s}\right)-{\left({\displaystyle {\int }_{t-d_M}^{t}x\left(s\right)\text{d}s}\right)}^{\text{T}}{N}_3\left({\displaystyle {\int }_{t-d_M}^{t}x\left(s\right)\text{d}s}\right)\\ ={\phi }^{\text{T}}\left(x_t,i\right)\left[h^2{N}_1+{\tau }^2{N}_2+d_M^2{N}_3\right]\phi \left(x_t,i\right)\\ +2{\phi }^{\text{T}}\left(x_t,i\right)\left[h^2{N}_1+{\tau }^2{N}_2+d_M^2{N}_3\right]C_{i}x\left(t-\tau \left(t\right)\right)\\ +x\left(t-\tau \left(t\right)\right)C_i^{\text{T}}\left[h^2{N}_1+{\tau }^2{N}_2+d_M^2{N}_3\right]C_{i}x\left(t-\tau \left(t\right)\right)\\ -{\left({\displaystyle {\int }_{t-h}^{t}x\left(s\right)\text{d}s}\right)}^{\text{T}}{N}_1\left({\displaystyle {\int }_{t-h}^{t}x\left(s\right)\text{d}s}\right)-{\left({\displaystyle {\int }_{t-\tau }^{t}x\left(s\right)\text{d}s}\right)}^{\text{T}}{N}_2\left({\displaystyle {\int }_{t-\tau }^{t}x\left(s\right)\text{d}s}\right)\\ -{\left({\displaystyle {\int }_{t-d_M}^{t}x\left(s\right)\text{d}s}\right)}^{\text{T}}{N}_3\left({\displaystyle {\int }_{t-d_M}^{t}x\left(s\right)\text{d}s}\right).\end{array}$ (25)

结合(6)式可以得到

$\begin{array}{c}L{V}_4\left(x_t,t,i\right)={\eta }^{\text{T}}\left(t\right)\dot{\eta }\left(t\right)\\ =\left[\frac{1}{\eta \left(t\right)}-{\eta }_0\right]e^{\text{T}}\left(t\right){\Lambda }_{i}e\left(t\right)\\ \le x^{\text{T}}\left(t_{k}h\right){\Lambda }_{i}x\left(t_{k}h\right)-{\eta }_0{e}^T\left(t\right){\Lambda }_{i}e\left(t\right)\\ ={\left[x\left(t-d\left(t\right)\right)-e\left(t\right)\right]}^{\text{T}}{\Lambda }_i\left[x\left(t-d\left(t\right)\right)-e\left(t\right)\right]-{\eta }_0{e}^T\left(t\right){\Lambda }_{i}e\left(t\right).\end{array}$(26)

由(23)~(26)式有

$\begin{array}{c}LV\left(x_t,t,i\right)\le {\xi }^{\text{T}}\left(t\right){\psi }_i\xi \left(t\right)+{\displaystyle {\int }_{t-h\left(t\right)}^t{x}^{\text{T}}\left(s\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}}{R}_{1j}x\left(s\right)\text{d}s}\\ +{\displaystyle {\int }_{t-\tau \left(t\right)}^t{x}^{\text{T}}\left(s\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}}{R}_{2j}x\left(s\right)\text{d}s}+{\displaystyle {\int }_{t-d_M}^t{x}^{\text{T}}\left(s\right){\displaystyle \sum _{j=1}^N{\pi }_{ij}}{R}_{3j}x\left(s\right)\text{d}s}\\ +{\left[A_{1i}\phi \left(x_t,i\right)+A_{1i}{C}_{i}x\left(t-\tau \left(t\right)\right)+D_{1i}x\left(t-h\left(t\right)\right)\right]}^{\text{T}}\\ \times P_i\left[A_{1i}\phi \left(x_t,i\right)+A_{1i}{C}_{i}x\left(t-\tau \left(t\right)\right)+D_{1i}x\left(t-h\left(t\right)\right)\right].\end{array}$ (27)

其中

$\begin{array}{l}\xi (t)=[\begin{array}{cccc}{\phi }^{\text{T}}\left(x_t,i\right)& x^{\text{T}}\left(t-\tau \left(t\right)\right)& x^{\text{T}}\left(t-h\left(t\right)\right)& x^{\text{T}}\left(t-d\left(t\right)\right)\end{array}\\ {\begin{array}{cccc}{\displaystyle {\int }_{t-h\left(t\right)}^t{x}^{\text{T}}\left(s\right)\text{d}s}& {\displaystyle {\int }_{t-\tau \left(t\right)}^t{x}^{\text{T}}\left(s\right)\text{d}s}& x^{\text{T}}\left(t-h\left(t\right)\right)& e^{\text{T}}\left(t\right)\end{array}]}^{\text{T}}.\end{array}$

${\psi }_i=\left[\begin{array}{cccccccc}{\varphi }_{11}& {\varphi }_{12}& {\varphi }_{1\text{3}}& {\varphi }_{14}& 0& 0& 0& {\varphi }_{18}\\ *& {\varphi }_{22}& 0& 0& 0& 0& 0& 0\\ *& *& {\varphi }_{33}& 0& 0& 0& 0& 0\\ *& *& *& {\varphi }_{44}& 0& 0& 0& {\varphi }_{\text{48}}\\ *& *& *& *& {\varphi }_{55}& 0& 0& 0\\ *& *& *& *& *& {\varphi }_{66}& 0& 0\\ *& *& *& *& *& *& {\varphi }_{77}& 0\\ *& *& *& *& *& *& *& {\varphi }_{88}\end{array}\right].$

由条件(20)可得

$\begin{array}{c}LV\left(x_t,t,i\right)\le {\xi }^{\text{T}}\left(t\right){\psi }_i\xi \left(t\right)+{\gamma }_i{\displaystyle {\int }_{t-h\left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{1i}x\left(s\right)\text{d}s}\\ +{\gamma }_i{\displaystyle {\int }_{t-\tau \left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{2i}x\left(s\right)\text{d}s}+{\gamma }_i{\displaystyle {\int }_{t-d\left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{3i}x\left(s\right)\text{d}s}\\ +{\left[A_{1i}\phi \left(x_t,i\right)+A_{1i}{C}_{i}x\left(t-\tau \left(t\right)\right)+D_{1i}x\left(t-h\left(t\right)\right)\right]}^{\text{T}}\\ \cdot P_i\left[A_{1i}\phi \left(x_t,i\right)+A_{1i}{C}_{i}x\left(t-\tau \left(t\right)\right)+D_{1i}x\left(t-h\left(t\right)\right)\right].\end{array}$ (28)

通过引理2和(19)式，可以得到

$\begin{array}{c}LV\left(x_t,t,i\right)\le {\gamma }_i{\phi }^{\text{T}}\left(x_t,i\right)P_i\phi \left(x_t,i\right)+{\gamma }_i{\displaystyle {\int }_{t-h\left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{1i}x\left(s\right)\text{d}s}\\ +{\gamma }_i{\displaystyle {\int }_{t-\tau \left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{2i}x\left(s\right)\text{d}s}+{\gamma }_i{\displaystyle {\int }_{t-\eta \left(t\right)}^t{x}^{\text{T}}\left(s\right)R_{3i}x\left(s\right)\text{d}s}\\ \le {\gamma }_{i}V\left(x_t,t,i\right).\end{array}$ (29)

对(29)式的两边在$t_{\text{0}}$ 到$t$ 上取积分，对$\forall t\ge t_0$ 有

$V\left(x_t,t,i\right)\le \left(V\left(x_0,t_0,i\right)\right){\text{e}}^{{\gamma }_i\left(t-t_0\right)}.$ (30)

令${\widehat{ƛ}}_1=\underset{i\in S}{\min }\left({\lambda }_{\min }\left(P_i\right)\right)$ ，${\widehat{ƛ}}_2=\underset{i\in S}{\max }\left({\lambda }_{\max }\left(U\right)\right)$ ，${\rho }_{C_i}=\rho \left(C_i\right)$ 为$C_i$ 的谱半径，${\rho }_U=\rho \left(U\right)$ 为$U$ 的谱半径，有

$\begin{array}{c}E\left(V\left(x_t,t,i\right)\right)\ge E\left({\phi }^{\text{T}}\left(x_t,i\right)P_i\phi \left(x_t,i\right)\right)\\ \ge E\left({\phi }^{\text{T}}\left(x_t,i\right)U^{\frac{1}{2}}{\widehat{P}}_i{U}^{\frac{1}{2}}\phi \left(x_t,i\right)\right)\\ \ge {ƛ}_{1}E\left({\phi }^{\text{T}}\left(x_t,i\right)U\phi \left(x_t,i\right)\right).\end{array}$(31)

其中${\widehat{P}}_i=U^{-\frac{1}{2}}{P}_i{U}^{-\frac{1}{2}}$ ，${ƛ}_{1}I<{\widehat{P}}_i<{ƛ}_{2}I$ 。

$\begin{array}{c}E\left\{x^{\text{T}}\left(t\right)Ux\left(t\right)\right\}\le E\left({\rho }_U{c}_3{\Vert \phi \left(x_t,i\right)\Vert }^2\right){\text{e}}^{{\gamma }_i\left(t-t_0\right)}\\ \le \frac{{\rho }_U{c}_3}{{\widehat{ƛ}}_1}E\left\{{\phi }^{\text{T}}\left(x_t,i\right)P_i\phi \left(x_t,i\right)\right\}{\text{e}}^{{\gamma }_i\left(t-t_0\right)}\\ \le \frac{{\widehat{ƛ}}_2{c}_3}{{\widehat{ƛ}}_1}\frac{1}{{ƛ}_1}E\left\{{\phi }^{\text{T}}\left(x_t,i\right)U^{\frac{1}{2}}{\widehat{P}}_i{U}^{\frac{1}{2}}\phi \left(x_t,i\right)\right\}{\text{e}}^{{\gamma }_i\left(t-t_0\right)}\\ \le \frac{{\widehat{ƛ}}_2{c}_3}{{\widehat{ƛ}}_1{ƛ}_1}E\left(V\left(x_t,t,i\right)\right){\text{e}}^{{\gamma }_i\left(t-t_0\right)}.\end{array}$(32)

其中$c_3=\frac{1}{1-\min \left({\rho }_{C_i}\right)}$ ，令${\widehat{\lambda }}_1=\underset{i\in S}{\max }\left\{{\lambda }_{\max }\left({\widehat{R}}_1\right)\right\}$ ，${\widehat{\lambda }}_2=\underset{i\in S}{\max }\left\{{\lambda }_{\max }\left({\widehat{R}}_2\right)\right\}$ ，${\widehat{\lambda }}_3=\underset{i\in S}{\max }\left\{{\lambda }_{\max }\left({\widehat{R}}_3\right)\right\}$ ， ${\widehat{\lambda }}_4=\underset{i\in S}{\max }\left\{{\lambda }_{\max }\left({\widehat{N}}_1\right)\right\}$ ，${\widehat{\lambda }}_5=\underset{i\in S}{\max }\left\{{\lambda }_{\max }\left({\widehat{N}}_2\right)\right\}$ ，${\widehat{\lambda }}_6=\underset{i\in S}{\max }\left\{{\lambda }_{\max }\left({\widehat{N}}_3\right)\right\}$ 当$\underset{t_0-{\rho }^{*}<s<t_0}{\sup }\left(x^{\text{T}}\left(s\right)Ux\left(s\right)\right)<c_1$ ，$\forall t\in \left(0,T\right)$ 时，根据(22)式和(32)式，有下列不等式成立

$\begin{array}{c}E\left\{x^{\text{T}}\left(t\right)Ux\left(t\right)\right\}\le \frac{{\widehat{ƛ}}_3{c}_3}{{\widehat{ƛ}}_1{ƛ}_1}E\{\left(1+{\rho }_{C_i}\right)\underset{t_0-{\rho }^{*}<s<t_0}{\sup }{x}^{\text{T}}\left(t\right)P_{i}x\left(x\right)+h\underset{t_0-h<s<t_0}{\sup }{x}^{\text{T}}\left(t\right)R_{1}x\left(x\right)\\ +\tau \underset{t_0-\tau <s<t_0}{\sup }{x}^{\text{T}}\left(t\right)R_{2}x\left(x\right)\}+d_M\underset{t_0-d_M<s<t_0}{\sup }{x}^{\text{T}}\left(t\right)R_{3}x\left(x\right)\\ +\frac{h^3}{2}\underset{t_0-h<s<t_0}{\sup }{x}^{\text{T}}\left(t\right)N_{1}x\left(x\right)\}+\frac{{\tau }^3}{2}\underset{t_0-\tau <s<t_0}{\sup }{x}^{\text{T}}\left(t\right)N_{2}x\left(x\right)\\ +\frac{d_M^3}{2}\underset{t_0-d_M<s<t_0}{\sup }{x}^{\text{T}}\left(t\right)N_{3}x\left(x\right)+\frac{1}{2}{\eta }_0^2\}{\text{e}}^{{\gamma }_i\left(t-t_0\right)}\\ \le \frac{{\widehat{ƛ}}_2{c}_3}{{\widehat{ƛ}}_1{ƛ}_1}\{[\left(1+{\rho }_{C_i}\right){ƛ}_2+h{\widehat{\lambda }}_1+\tau {\widehat{\lambda }}_2+d_M{\widehat{\lambda }}_3+\frac{h^3}{2}{\widehat{\lambda }}_4+\frac{{\tau }^3}{2}{\widehat{\lambda }}_5\\ +\frac{d_M^3}{2}{\widehat{\lambda }}_6]\underset{t_0-{\rho }^{*}<s<t_0}{\sup }E\left\{x^{\text{T}}\left(s\right)Ux\left(s\right)\right\}+\frac{1}{2}{\eta }_0^2\}{\text{e}}^{{\gamma }_i\left(t-t_0\right)}\\ \le \frac{{\widehat{ƛ}}_2{c}_3}{{\widehat{ƛ}}_1{ƛ}_1}\{[\left(1+{\rho }_{C_i}\right){ƛ}_2+h_M{\lambda }_1+{\tau }_M{\lambda }_2+d_M{\lambda }_3+\frac{h^3}{2}{\lambda }_4+\frac{{\tau }^3}{2}{\lambda }_5\\ +\frac{d_M^3}{2}{\lambda }_6]\underset{t_0-{\rho }^{*}<s<t_0}{\sup }E\left\{x^{\text{T}}\left(s\right)Ux\left(s\right)\right\}+\frac{1}{2}{\eta }_0^2\}{\text{e}}^{\gamma \left(T-t_0\right)}\\ \le \frac{{\widehat{ƛ}}_2{c}_3}{{\widehat{ƛ}}_1{ƛ}_1}\left({\lambda }^{*}{c}_1+\frac{1}{2}{\eta }_0^2\right){\text{e}}^{\gamma \left(T-t_0\right)}<c_2.\end{array}$(33)

其中$0<{\widehat{R}}_1\le {\lambda }_{1}I,0<{\widehat{R}}_2\le {\lambda }_{2}I,0<{\widehat{R}}_3\le {\lambda }_{3}I,0<{\widehat{N}}_1\le {\lambda }_{4}I,0<{\widehat{N}}_2\le {\lambda }_{5}I,0<{\widehat{N}}_3\le {\lambda }_{6}I$ ，

${\lambda }^{*}=\left(1+{\rho }_{C_i}\right){ƛ}_2+h{\lambda }_1+\tau {\lambda }_2+d_M{\lambda }_3+\frac{h^3}{2}{\lambda }_4+\frac{{\tau }^3}{2}{\lambda }_5+\frac{d_M^3}{2}{\lambda }_6,\gamma =\max \left({\gamma }_i\right).$

则由定义1可以得到系统(11)是限时稳定的，定理3.2得证。

4. 数值算例

本节将通过数值算例来验证上一节所提方法的有效性。考虑遭受欺骗攻击的两种模态的中立型半马尔可夫随机系统

$\{\begin{array}{l}\text{d}\left[x\left(t\right)-C\left(r_t\right)x\left(t-\tau \left(t\right)\right)\right]=\left[A\left(r_t\right)x\left(t\right)+B\left(r_t\right)u^{\aleph }\left(t\right)+D\left(r_t\right)x\left(t-h\left(t\right)\right)\right]\text{d}t\\ +\left[A_1\left(r_t\right)x\left(t\right)+D_1\left(r_t\right)x\left(t-h\left(t\right)\right)\right]\text{d}w\left(t\right);\\ x\left(t\right)=\chi \left(t\right),\forall t\in \left[-\rho ,0\right].\end{array}$ (34)

模式1参数如下：

$A_1=\left[\begin{array}{cc}-2& 0.5\\ 2& -0.4\end{array}\right]$ , $B_1=\left[\begin{array}{c}-0.2\\ 1\end{array}\right]$ , $C_1=\left[\begin{array}{cc}-0.15& -0.06\\ 0.5& -0.5\end{array}\right]$ , $D_1=\left[\begin{array}{cc}-2& 1\\ 0& 0.1\end{array}\right]$ ,

$A_{11}\left[\begin{array}{cc}-0.3& 0\\ 0.2& -0.1\end{array}\right]$ , $D_{11}=\left[\begin{array}{cc}-0.1& 0.1\\ 0& -0.1\end{array}\right]$ , $G_1=\left[\begin{array}{cc}0.01& 0.01\end{array}\right]$ .

模式2参数如下：

$A_2=\left[\begin{array}{cc}-1& 1\\ 1& -0.4\end{array}\right]$ , $B_1=\left[\begin{array}{c}-0.2\\ 1\end{array}\right]$ , $C_2=\left[\begin{array}{cc}0.25& -0.35\\ 0.05& -0.65\end{array}\right]$ , $D_2=\left[\begin{array}{cc}-0.07& -0.1\\ 0& -0.05\end{array}\right]$ , $A_{12}=\left[\begin{array}{cc}-0.3& 0\\ 0.2& -0.1\end{array}\right]$ ,$D_{12}=\left[\begin{array}{cc}-0.1& 0\\ 0& -0.1\end{array}\right]$ , $G_2=\left[\begin{array}{cc}0.01& 0.01\end{array}\right]$ .

中立型时滞$\tau \left(t\right)=0.3+0.1\sin \left(t\right)$ ，时变时滞$h\left(t\right)=0.3+0.1\cos \left(t\right)$ 。考虑到半马尔科夫系统各模态的驻留时间服从威布尔分布，选择转移概率矩阵为：

${\pi }_{ij}=\left[\begin{array}{cc}-2.7082& 2.7082\\ 3.6763& -3.6763\end{array}\right]\text{.}$

选择控制增益矩阵：

$K_1=\left[\begin{array}{cc}1& 2\end{array}\right],K_2=\left[\begin{array}{cc}1.5& 2.5\end{array}\right].$

考虑系统受到欺骗攻击的影响，其中

$H_1\left(t\right)=0.1\sin \left(t\right),\zeta \left(x\left(t\right),1,t\right)=0.1\sin \left(t\right)\sqrt{x^2+1};$

$H_2\left(t\right)=0.1\cos \left(t\right),\zeta \left(x\left(t\right),2,t\right)=0.1\sqrt{x^2+1}.$

此外，选择$h_d=0.5$ ，${\tau }_d=0.5$ ，$d_M=0.5$ ，$\alpha =0.2$ ，${\eta }_0=0.1$ ，$\iota =100$ ，采样周期$h=0.1$ ，仿真时间为10秒，仿真结果如图1~4所示，通过图1(a)可以清晰地看到遭受欺骗攻击时基于事件的开环系统是发散的，图1(b)展示了闭环系统状态$x\left(t\right)$ 的演变情况，可以清晰地看到在滑模控制器作用下遭受欺骗攻击的闭环系统的状态$x\left(t\right)$ 收敛到0。图2(a)展示了滑模面$s\left(t\right)$ 的演变情况，可以明显地看到在设计的滑模控制律下，系统在有限时间内是能到达预定的滑模面的。图2(b)展示控制输入$u\left(t\right)$ 的演变情况。图3(a)展示了系统模态跳变情况。图3(b)展示了自适应阈值$\eta \left(t\right)$ 的变化情况。图4(a)和图4(b)分别展示自适应事件触发协议下的释放瞬间和触发间隔和传统触发方案下的释放瞬间和释放间隔，在自适应事件触发方案下，平均传输数据包(采样100次)为27%，而传统事件触发方案下，平均传输数据包(采样100次)为39%，即本文提出的自适应触发方案能够有效的减少触发次数，达到节省网络资源的目的。

Figure 1. Open-loop System and close-loop System State Response of $x\left(t\right)$

图1. 开环系统与闭环系统状态$x\left(t\right)$

Figure 2. Sliding surface $s\left(t\right)$ and The control input $u\left(t\right)$

图2. 滑模面$s\left(t\right)$ 和控制输入$u\left(t\right)$

Figure 3. The curves of mode and adaptive triggered threshold $\eta \left(t\right)$

图3. 模态变化和自适应触发阈值$\eta \left(t\right)$

Figure 4. The release instant and release interval and the release instant and release interval under the traditional triggering scheme

图4. 自适应触发方案和传统触发方案下的释放瞬间和释放间隔

综上，所有的仿真结果都验证了该方法的有效性，事件触发机方案、控制器设计以及滑模控制方案都能达到理想的控制性能。

5. 结语

基于Lyapunov稳定性理论和线性矩阵不等式技术研究了具有网络攻击的中立型半马尔科夫随机系统的自适应滑模控制问题。首先，提出了一个自适应动态事件触发方案，以节省网络通信资源；然后构造了一个考虑网络延迟的滑模面，通过Lyapunov泛函和积分不等式技巧，得到了滑模面有限时间的可达性和滑模控制律下系统限时稳定的充分条件。

基金项目

贵州民族大学基金科研项目(JZMUZK [2023] YB12)；贵州省基础研究计划(自然科学)青年引导项目[2024] 210；贵州省基础研究计划(自然科学)面上项目MS [2025] 217；贵州省高等学校光通讯系统中孤子的数学理论与计算协作创新团队(黔教技[2023] 062号)。

NOTES

^*通讯作者。