﻿ DoS攻击下不确定网络控制系统基于事件触发机制的鲁棒有限时间可靠控制 Robust Finite-Time Reliable Control for Uncertain Networked Control Systems under Denial of Service Attacks Based on Event Triggering Mechanism

Dynamical Systems and Control
Vol. 11  No. 03 ( 2022 ), Article ID: 53437 , 13 pages
10.12677/DSC.2022.113012

DoS攻击下不确定网络控制系统基于事件触发机制的鲁棒有限时间可靠控制

Robust Finite-Time Reliable Control for Uncertain Networked Control Systems under Denial of Service Attacks Based on Event Triggering Mechanism

Mengjie Xue, Weiqun Wang, Tianrui Li

School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing Jiangsu

Received: Jun. 16th, 2022; accepted: Jun. 28th, 2022; published: Jul. 8th, 2022

ABSTRACT

In this paper, robust finite-time stability of Networked Control Systems (NCSs) based on event triggering mechanism under actuator failures and non-periodic Denial of Service (DoS) attacks is investigated. First, a switching-like delay system model is presented under discrete event-triggered communication scheme. Then, by constructing piecewise Lyapunov-Krasovskii functional, sufficient conditions for finite-time stability of the system are derived and expressed with a set of linear matrix inequalities. Moreover, the co-design of the robust reliable controller and the event-triggered parameter is obtained. Finally, two examples are employed to demonstrate the effectiveness of the proposed method.

Keywords:Networked Control Systems, Reliable Control, Event-Triggered, Finite-Time, Denial of Service Attacks

1. 引言

2. 问题描述

$\stackrel{˙}{x}\left(t\right)=\left(A+\Delta A\left(t\right)\right)x\left(t\right)+\left(B+\Delta B\left(t\right)\right){u}^{f}\left(t\right),$ (1)

$\Delta A\left(t\right)=EH\left(t\right){F}_{1},\Delta B\left(t\right)=EH\left(t\right){F}_{2},$ (2)

${u}^{f}\left(t\right)=\Xi u\left(t\right),$ (3)

$\Xi =diag\left\{{d}_{1},{d}_{2},\cdots ,{d}_{m}\right\},$

$U=diag\left\{{u}_{1},{u}_{2},\cdots ,{u}_{m}\right\},L=diag\left\{{p}_{1},{p}_{2},\cdots ,{p}_{m}\right\},E=diag\left\{{e}_{1},{e}_{2},\cdots ,{e}_{m}\right\}$

$\Xi =U\left(I+E\right),|E|\le L\le I,$ (4)

${I}_{DOS}\left(t\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{h}_{n},{h}_{n}+{l}_{n}\right)\\ 1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }t\in \left[{h}_{n}+{l}_{n},{h}_{n+1}\right).\end{array}$ (5)

Figure 1. Structure diagram of NCSs based on event triggering mechanism

$u\left(t\right)=Kx\left({t}_{k}h\right),t\in \left[{t}_{k}h,{t}_{k+1}h\right),$ (6)

${e}_{k,j}^{\text{T}}\left({t}_{{k}_{j}}h\right)\Phi {e}_{k,j}\left({t}_{{k}_{j}}h\right)\ge \delta {x}^{\text{T}}\left({t}_{k}h\right)\Phi x\left({t}_{k}h\right),$ (7)

${t}_{k,n}h=\left\{{t}_{{k}_{j}}h\text{ }\text{ }满足\text{(7)}|{t}_{{k}_{j}}h\in {L}_{1,n-1}\right\}\cup \left\{{h}_{n}\right\},$ (8)

$u\left(t\right)=\left\{\begin{array}{l}Kx\left({t}_{k,n}h\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[{t}_{k,n}h,{t}_{k+1,n}h\right)\cap {L}_{1,n-1}\\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {L}_{2,n-1},\end{array}$ (9)

$\stackrel{˙}{x}\left(t\right)=\left\{\begin{array}{l}\left(A+\Delta A\left(t\right)\right)x\left(t\right)+\left(B+\Delta B\left(t\right)\right)\Xi Kx\left({t}_{k,n}h\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\nu }_{k,n}\cap {L}_{1,n-1}\\ \left(A+\Delta A\left(t\right)\right)x\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {L}_{2,n-1}.\end{array}$ (10)

${\nu }_{k,n}={\cup }_{l=1}^{{\lambda }_{k,n}+1}{F}_{k,n}^{l},$ (11)

${F}_{k,n}^{l}=\left[{t}_{k,n}h+\left(l-1\right)h,{t}_{k,n}h+lh\right),l=1,2,\cdots ,{\lambda }_{k,n},$

${\lambda }_{k,n}\triangleq \mathrm{sup}\left\{l\in ℕ|{t}_{k,n}h+lh<{t}_{k+1,n}h\right\},$

${F}_{k,n}^{{\lambda }_{k,n}+1}=\left[{t}_{k,n}h+{\lambda }_{k,n}h,{t}_{k+1,n}h\right).$

${L}_{1,n-1}\subseteq {\cup }_{k=0}^{k\left(n\right)}{\nu }_{k,n},$ (12)

${L}_{1,n-1}={\cup }_{k=0}^{k\left(n\right)}\left\{{\upsilon }_{k,n}\cap {L}_{1,n-1}\right\}={\cup }_{k=0}^{k\left(n\right)}{\cup }_{l=1}^{{\lambda }_{k,n}+1}\left\{{F}_{k,n}^{l}\cap {L}_{1,n-1}\right\}.$

${\varphi }_{k,n}^{l}={F}_{k,n}^{l}\cap {L}_{1,n-1}$，则 ${L}_{1,n-1}={\cup }_{k=0}^{k\left(n\right)}{\cup }_{l=1}^{{\lambda }_{k,n}+1}{\varphi }_{k,n}^{l}$

${\tau }_{k,n}\left(t\right)=\left\{\begin{array}{l}t-{t}_{k,n}h,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\varphi }_{k,n}^{1}\\ t-{t}_{k,n}h-h,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\varphi }_{k,n}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ t-{t}_{k,n}h-{\lambda }_{k,n}h,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\varphi }_{k,n}^{{\lambda }_{k,n}+1},\end{array}$

${e}_{k,n}\left(t\right)=\left\{\begin{array}{l}0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\varphi }_{k,n}^{1}\\ x\left({t}_{k,n}h\right)-x\left({t}_{k,n}h+h\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\varphi }_{k,n}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ x\left({t}_{k,n}h\right)-x\left({t}_{k,n}h+{\lambda }_{k,n}h\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\varphi }_{k,n}^{{\lambda }_{k,n}+1},\end{array}$

$x\left({t}_{k,n}h\right)=x\left(t-{\tau }_{k,n}\left(t\right)\right)+{e}_{k,n}\left(t\right),t\in {\nu }_{n,k}\cap {L}_{1,n-1}.$ (13)

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left\{\begin{array}{l}\left(A+\Delta A\left(t\right)\right)x\left(t\right)+\left(B+\Delta B\left(t\right)\right)\Xi Kx\left(t-{\tau }_{k,n}\left(t\right)\right)+\left(B+\Delta B\left(t\right)\right)\Xi K{e}_{k,n}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\nu }_{k,n}\cap {L}_{1,n-1}\\ \left(A+\Delta A\left(t\right)\right)x\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {L}_{2,n-1}\end{array}\\ x\left(t\right)=\varphi \left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[-h,0\right],\end{array}$ (14)

$\underset{\theta \in \left[-h,0\right]}{\mathrm{sup}}\left\{{x}^{\text{T}}\left(\theta \right)Rx\left(\theta \right),{\stackrel{˙}{x}}^{\text{T}}\left(\theta \right)R\stackrel{˙}{x}\left(\theta \right)\right\}\le {c}_{1}⇒{x}^{\text{T}}\left(t\right)Rx\left(t\right)<{c}_{2}\text{,}\forall t\in \left[0,{T}_{f}\right],$

$-{\tau }_{M}{\int }_{t-{\tau }_{M}}^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right)Q\stackrel{˙}{x}\left(s\right)\text{d}s\le {\left[\begin{array}{c}{x}^{\text{T}}\left(t\right)\\ {x}^{\text{T}}\left(t-\tau \left(t\right)\right)\\ {x}^{\text{T}}\left(t-{\tau }_{M}\right)\end{array}\right]}^{\text{T}}\left[\begin{array}{ccc}-Q& Q-U& -U\\ *& -2Q+U+{U}^{\text{T}}& Q-U\\ *& *& -Q\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ x\left(t-\tau \left(t\right)\right)\\ x\left(t-{\tau }_{M}\right)\end{array}\right].$

$X+UVW+{W}^{\text{T}}{V}^{\text{T}}{U}^{\text{T}}<0,$

$X+\rho U{U}^{\text{T}}+{\rho }^{-1}{W}^{\text{T}}W<0.$

3. 主要结果

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)=\left\{\begin{array}{l}Ax\left(t\right)+B\Xi Kx\left(t-{\tau }_{k,n}\left(t\right)\right)+B\Xi K{e}_{k,n}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\nu }_{k,n}\cap {L}_{1,n-1}\\ Ax\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {L}_{2,n-1}\end{array}\\ x\left(t\right)=\phi \left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in \left[-h,0\right].\end{array}$ (15)

3.1. 有限时间稳定性分析

$V\left(t\right)=\left\{\begin{array}{l}{V}_{1}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\nu }_{k,n}\cap {L}_{1,n-1}\\ {V}_{2}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {L}_{2,n-1},\end{array}$

${\Pi }_{1}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Sigma }_{1}>0,$ (16)

${\Pi }_{2}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Sigma }_{2}>0,$ (17)

${\Sigma }_{1}=\left[\begin{array}{cc}{Z}_{1}& {S}_{1}\\ *& {Z}_{1}\end{array}\right],\text{}{\Sigma }_{2}=\left[\begin{array}{cc}{Z}_{2}& {S}_{2}\\ *& {Z}_{2}\end{array}\right],\text{}{\Pi }_{1}=\left[\begin{array}{ccc}\stackrel{˜}{\Upsilon }& \phi & \psi \\ *& -{\rho }^{-1}I& 0\\ *& 0& -\rho I\end{array}\right],\text{}\stackrel{˜}{\Upsilon }=\left[\begin{array}{cc}\stackrel{˜}{\Theta }& h{\stackrel{˜}{\Gamma }}_{1}^{\text{T}}{Z}_{1}\\ *& -{Z}_{1}\end{array}\right],\text{}{\stackrel{˜}{\Gamma }}_{1}^{\text{T}}=\left[\begin{array}{c}{A}^{\text{T}}\\ {\left(BUK\right)}^{\text{T}}\\ 0\\ {\left(BUK\right)}^{\text{T}}\end{array}\right],$

$\phi =\left[\begin{array}{c}{P}_{1}B\\ 0\\ 0\\ \begin{array}{l}\text{}0\\ h{Z}_{1}B\end{array}\end{array}\right],\text{}\psi =\left[\begin{array}{c}0\\ {K}^{\text{T}}{U}^{\text{T}}\\ 0\\ \begin{array}{l}{K}^{\text{T}}{U}^{\text{T}}\\ \text{}0\end{array}\end{array}\right],\text{}\stackrel{˜}{\Theta }=\left[\begin{array}{cccc}{\stackrel{˜}{\Theta }}_{11}& {\stackrel{˜}{\Theta }}_{12}& {\stackrel{˜}{\Theta }}_{13}& {\stackrel{˜}{\Theta }}_{14}\\ *& {\stackrel{˜}{\Theta }}_{22}& {\stackrel{˜}{\Theta }}_{23}& {\stackrel{˜}{\Theta }}_{24}\\ *& *& {\stackrel{˜}{\Theta }}_{33}& 0\\ *& *& *& \text{}{\stackrel{˜}{\Theta }}_{44}\end{array}\right],\text{}{\Pi }_{2}=\left[\begin{array}{cccc}{\Omega }_{11}& {\Omega }_{12}& {\Omega }_{13}& h{A}^{\text{T}}{Z}_{2}\\ *& {\Omega }_{22}& {\Omega }_{23}& 0\\ *& *& {\Omega }_{33}& 0\\ *& *& *& -{Z}_{2}\end{array}\right],$

$\begin{array}{l}{\stackrel{˜}{\Theta }}_{11}={A}^{\text{T}}{P}_{1}+{P}_{1}A+{Q}_{1}+2{\alpha }_{1}{P}_{1}-{\text{e}}^{-{\alpha }_{1}h}{Z}_{1},\text{}{\stackrel{˜}{\Theta }}_{12}={P}_{1}BUK-{\text{e}}^{-{\alpha }_{1}h}\left({S}_{1}-{Z}_{1}\right),\text{}{\stackrel{˜}{\Theta }}_{13}={\text{e}}^{-{\alpha }_{1}h}{S}_{1},\text{}{\stackrel{˜}{\Theta }}_{14}={P}_{1}BUK,\\ {\stackrel{˜}{\Theta }}_{22}=\delta \text{ }\Phi -{\text{e}}^{-{\alpha }_{1}h}\left(2{Z}_{1}-{S}_{1}-{S}_{1}^{\text{T}}\right),\text{}{\stackrel{˜}{\Theta }}_{23}=-{\text{e}}^{-{\alpha }_{1}h}\left({S}_{1}-{Z}_{1}\right),\text{}{\stackrel{˜}{\Theta }}_{24}=\delta \text{ }\Phi ,\text{}{\stackrel{˜}{\Theta }}_{33}=-{\text{e}}^{-{\alpha }_{1}h}\left({Q}_{1}+{Z}_{1}\right),\text{}{\stackrel{˜}{\Theta }}_{44}=\delta \text{ }\Phi -\Phi ,\\ {\Omega }_{11}={A}^{\text{T}}{P}_{2}+{P}_{2}A+{Q}_{2}-{\alpha }_{2}{P}_{2}-{Z}_{2},\text{}{\Omega }_{12}={Z}_{2}-{S}_{2},\text{}{\Omega }_{13}={S}_{2},\text{}{\Omega }_{22}={S}_{2}+{S}_{2}^{\text{T}}-2{Z}_{2},\text{}{\Omega }_{23}={Z}_{2}-{S}_{2},\\ {\Omega }_{33}=-{\text{e}}^{{\alpha }_{2}h}{Q}_{2}-{Z}_{2},\end{array}$

$V\left(t\right)\le \left\{\begin{array}{l}{\text{e}}^{-{\alpha }_{1}\left(t-{h}_{n-1}\right)}{V}_{1}\left({h}_{n-1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {\nu }_{k,n}\cap {L}_{1,n-1}\\ {\text{e}}^{{\alpha }_{2}\left(t-{h}_{n-1}-{l}_{n-1}\right)}{V}_{2}\left({h}_{n-1}+{l}_{n-1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\in {L}_{2,n-1}.\end{array}$ (18)

(i) $\forall t\in {\nu }_{k,n}\cap {L}_{1,n-1}$，误差 ${e}_{k,n}\left(t\right)$ 满足

${e}_{k,n}^{\text{T}}\left(t\right)\Phi {e}_{k,n}\left(t\right)\le \delta {\left[x\left(t-{\tau }_{k,n}\left(t\right)\right)+{e}_{k,n}\left(t\right)\right]}^{\text{T}}\Phi \left[x\left(t-{\tau }_{k,n}\left(t\right)\right)+{e}_{k,n}\left(t\right)\right].$ (19)

${V}_{1}\left(t\right)$ 沿着系统(17)的轨迹关于t求导可得

$\begin{array}{c}{\stackrel{˙}{V}}_{1}\left(t\right)=-{\alpha }_{1}V\left(t\right)+{\alpha }_{1}{x}^{\text{T}}\left(t\right){P}_{1}x\left(t\right)+2{x}^{\text{T}}\left(t\right){P}_{1}\stackrel{˙}{x}\left(t\right)-{x}^{\text{T}}\left(t-h\right){\text{e}}^{-{\alpha }_{1}h}{P}_{1}x\left(t-h\right)\\ \text{\hspace{0.17em}}\text{ }\text{ }+{h}^{2}{\stackrel{˙}{x}}^{\text{T}}\left(t\right){Z}_{1}\stackrel{˙}{x}\left(t\right)-h{\int }_{t-h}^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right){\text{e}}^{-{\alpha }_{1}\left(t-s\right)}{Z}_{1}\stackrel{˙}{x}\left(s\right)\text{d}s\\ \le -{\alpha }_{1}V\left(t\right)+{\alpha }_{1}{x}^{\text{T}}\left(t\right){P}_{1}x\left(t\right)+2{x}^{\text{T}}\left(t\right){P}_{1}\stackrel{˙}{x}\left(t\right)-{x}^{\text{T}}\left(t-h\right){\text{e}}^{-{\alpha }_{1}h}{P}_{1}x\left(t-h\right)\\ \text{\hspace{0.17em}}\text{ }\text{ }+{h}^{2}{\stackrel{˙}{x}}^{\text{T}}\left(t\right){Z}_{1}\stackrel{˙}{x}\left(t\right)-h{\text{e}}^{-{\alpha }_{1}h}{\int }_{t-h}^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right){Z}_{1}\stackrel{˙}{x}\left(s\right)\text{d}s.\end{array}$ (20)

$-h{\int }_{t-h}^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right){Z}_{1}\stackrel{˙}{x}\left(s\right)\text{d}s\le -{\varsigma }^{\text{T}}\left(t\right)\left[\begin{array}{ccc}{Z}_{1}& {S}_{1}-{Z}_{1}& -{S}_{1}\\ *& 2{Z}_{1}-{S}_{1}-{S}_{1}^{\text{T}}& {S}_{1}-{Z}_{1}\\ *& *& {Z}_{1}\end{array}\right]\varsigma \left(t\right).$ (21)

${\stackrel{˙}{V}}_{1}\left(t\right)\le -{\alpha }_{1}V\left(t\right)+{\xi }^{\text{T}}\left(t\right)\left(\Theta +{h}^{2}{\Gamma }_{1}^{\text{T}}{Z}_{1}{\Gamma }_{1}\right)\xi \left(t\right),$ (22)

$\xi \left(t\right)={\left[{x}^{\text{T}}\left(t\right),{x}^{\text{T}}\left(t-{\tau }_{k,n}\left(t\right)\right),{x}^{\text{T}}\left(t-h\right),{e}_{k,n}^{\text{T}}\left(t\right)\right]}^{\text{T}},$

$\Theta =\left[\begin{array}{cccc}{\Theta }_{11}& {\Theta }_{12}& {\Theta }_{13}& {\Theta }_{14}\\ *& {\Theta }_{22}& {\Theta }_{23}& {\Theta }_{24}\\ *& *& {\Theta }_{33}& 0\\ *& *& *& {\Theta }_{44}\end{array}\right],\text{\hspace{0.17em}}{\Gamma }_{1}^{\text{T}}=\left[\begin{array}{c}{A}^{\text{T}}\\ {\left(B\Xi K\right)}^{\text{T}}\\ 0\\ {\left(B\Xi K\right)}^{\text{T}}\end{array}\right],$

$\begin{array}{l}{\Theta }_{11}={A}^{\text{T}}{P}_{1}+{P}_{1}A+{Q}_{1}+{\alpha }_{1}{P}_{1}-{\text{e}}^{-{\alpha }_{1}h}{Z}_{1},\text{\hspace{0.17em}}{\Theta }_{12}={P}_{1}B\Xi K-{\text{e}}^{-{\alpha }_{1}h}\left({S}_{1}-{Z}_{1}\right),\text{\hspace{0.17em}}{\Theta }_{13}={\text{e}}^{-{\alpha }_{1}h}{S}_{1},\\ {\Theta }_{14}={P}_{1}B\Xi K,\text{\hspace{0.17em}}{\Theta }_{22}=\delta \text{ }\Phi -{\text{e}}^{-{\alpha }_{1}h}\left(2{Z}_{1}-{S}_{1}-{S}_{1}^{\text{T}}\right),\text{\hspace{0.17em}}{\Theta }_{23}=-{\text{e}}^{-{\alpha }_{1}h}\left({S}_{1}-{Z}_{1}\right),\text{\hspace{0.17em}}{\Theta }_{24}=\delta \text{ }\Phi ,\\ {\Theta }_{33}=-{\text{e}}^{-{\alpha }_{1}h}\left({Q}_{1}+{Z}_{1}\right),\text{\hspace{0.17em}}{\Theta }_{44}=\delta \text{ }\Phi -\Phi .\end{array}$

$\stackrel{˜}{\Upsilon }+\rho \phi {\phi }^{\text{T}}+{\rho }^{-1}{\psi }^{\text{T}}\psi <0.$ (23)

$\Xi =U\left(I+E\right)$，可得

$\Upsilon =\left[\begin{array}{cc}\Theta & h{\Gamma }_{1}^{\text{T}}{Z}_{1}\\ *& -{Z}_{1}\end{array}\right]<0.$ (24)

${\stackrel{˙}{V}}_{1}\left(t\right)\le -{\alpha }_{1}V\left(t\right).$ (25)

${V}_{1}\left(t\right)<{\text{e}}^{-{\alpha }_{1}\left(t-{h}_{n-1}\right)}{V}_{1}\left({h}_{n-1}\right).$

(ii) $\forall t\in {L}_{2,n-1}$，类似(i)的证明，可得式(17)成立保证下列不等式成立

${\stackrel{˙}{V}}_{2}\left(t\right)\le {\alpha }_{2}V\left(t\right).$ (26)

${V}_{2}\left(t\right)<{\text{e}}^{{\alpha }_{2}\left(t-{h}_{n-1}-{l}_{n-1}\right)}{V}_{2}\left({h}_{n-1}+{l}_{n-1}\right).$

${P}_{1}\le {\mu }_{2}{P}_{2},$ (27)

${P}_{2}\le {\mu }_{1}{\text{e}}^{\left({\alpha }_{1}+{\alpha }_{2}\right)h}{P}_{1},$ (28)

${Q}_{i}\le {\mu }_{3-i}{Q}_{3-i},$ (29)

${Z}_{i}\le {\mu }_{3-i}{Z}_{3-i},$ (30)

$\left({\lambda }_{2}+{\lambda }_{3}h\right){c}_{1}<\frac{{c}_{2}}{M},$ (31)

${V}_{1}\left({h}_{n-1}\right)\le {\mu }_{2}{V}_{2}\left({h}_{n-1}^{-}\right),$ (32)

${V}_{2}\left({h}_{n-1}+{l}_{n-1}\right)\le {\mu }_{1}{\text{e}}^{\left({\alpha }_{1}+{\alpha }_{2}\right)h}{V}_{1}\left[{\left({h}_{n-1}+{l}_{n-1}\right)}^{-}\right].$ (33)

$t\in \left[0,{T}_{f}\right)$，存在 $n\in ℕ$ 使得 $t\in {L}_{1,n-1}$$t\in {L}_{2,n-1}$

$\begin{array}{c}{V}_{1}\left(t\right)\le {\text{e}}^{-{\alpha }_{1}\left(t-{h}_{n-1}\right)}{V}_{1}\left({h}_{n-1}\right)\\ \le {\mu }_{2}{\text{e}}^{-{\alpha }_{1}\left(t-{h}_{n-1}\right)}{V}_{2}\left({h}_{n-1}^{-}\right)\\ \le {\mu }_{2}{\text{e}}^{-{\alpha }_{1}\left(t-{h}_{n-1}\right)}{\text{e}}^{{\alpha }_{2}\left({h}_{n-1}-{h}_{n-2}-{l}_{n-2}\right)}{V}_{2}\left({h}_{n-2}+{l}_{n-2}\right)\\ \le {\mu }_{1}{\mu }_{2}{\text{e}}^{-{\alpha }_{1}\left(t-{h}_{n-1}\right)}{\text{e}}^{{\alpha }_{2}\left({h}_{n-1}-{h}_{n-2}-{l}_{n-2}\right)}{\text{e}}^{\left({\alpha }_{1}+{\alpha }_{2}\right)h}{V}_{2}\left[{\left({h}_{n-2}+{l}_{n-2}\right)}^{-}\right]\\ ⋮\\ \le {\text{e}}^{n\left(t\right)\left({\alpha }_{1}+{\alpha }_{2}\right)h+n\left(t\right)\mathrm{ln}\left({\mu }_{1}{\mu }_{2}\right)}{V}_{1}\left(0\right){\text{e}}^{d},\end{array}$ (34)

${V}_{1}\left(t\right)\le {\text{e}}^{{g}_{1}}{V}_{1}\left(0\right),$ (35)

${V}_{2}\left(t\right)\le \frac{1}{{\mu }_{2}}{\text{e}}^{{g}_{2}}{V}_{1}\left(0\right),$ (36)

$V\left(t\right)\ge {\lambda }_{1}{x}^{\text{T}}\left(t\right)Rx\left(t\right),\text{\hspace{0.17em}}{V}_{1}\left(0\right)\le \left({\lambda }_{2}+{\lambda }_{3}h\right){c}_{1},$ (37)

$M=\mathrm{max}\left\{\frac{{\text{e}}^{{g}_{1}}}{{\lambda }_{1}},\frac{{\text{e}}^{{g}_{2}}}{{\lambda }_{1}{\mu }_{2}}\right\}$，结合(31)和(37)，可得

${x}^{\text{T}}\left(t\right)Rx\left(t\right)\le M{V}_{1}\left(0\right)\le M\left({\lambda }_{2}+{\lambda }_{3}h\right){c}_{1}<{c}_{2}.$

3.2. 鲁棒可靠性控制器设计

${\stackrel{^}{\Sigma }}_{1}>0,\text{\hspace{0.17em}}{\stackrel{^}{\Pi }}_{1}<0,$ (38)

${\stackrel{^}{\Sigma }}_{2}>0,\text{\hspace{0.17em}}{\stackrel{^}{\Pi }}_{2}<0,$ (39)

$\left[\begin{array}{cc}-{\mu }_{2}{X}_{2}& {X}_{2}^{\text{T}}\\ *& -{X}_{1}\end{array}\right]\le 0,$ (40)

$\left[\begin{array}{cc}-{\mu }_{1}{\text{e}}^{\left({\alpha }_{1}+{\alpha }_{2}\right)h}{X}_{1}& {X}_{1}^{\text{T}}\\ *& -{X}_{2}\end{array}\right]\le 0,$ (41)

$\left[\begin{array}{cc}-{\mu }_{3-i}{\stackrel{^}{Q}}_{3-i}& {X}_{3-i}^{\text{T}}\\ *& {\stackrel{^}{Q}}_{i}-2{X}_{i}\end{array}\right]\le 0,$ (42)

$\left[\begin{array}{cc}-{\mu }_{3-i}{\stackrel{^}{Z}}_{3-i}& {X}_{3-i}^{\text{T}}\\ *& {\stackrel{^}{Z}}_{i}-2{X}_{i}\end{array}\right]\le 0,$ (43)

${\stackrel{^}{\Sigma }}_{1}=\left[\begin{array}{cc}{\stackrel{^}{Z}}_{1}& {\stackrel{^}{S}}_{1}\\ *& {\stackrel{^}{Z}}_{1}\end{array}\right],\text{}{\stackrel{^}{\Sigma }}_{2}=\left[\begin{array}{cc}{\stackrel{^}{Z}}_{2}& {\stackrel{^}{S}}_{2}\\ *& {\stackrel{^}{Z}}_{2}\end{array}\right],\text{}{\stackrel{^}{\Pi }}_{1}=\left[\begin{array}{ccc}\stackrel{^}{\Upsilon }& \stackrel{^}{\phi }& \stackrel{^}{\psi }\\ *& -{\rho }^{-1}I& 0\\ *& *& -\rho I\end{array}\right],\text{}\stackrel{^}{\Upsilon }=\left[\begin{array}{cc}\stackrel{^}{\Theta }& h{\stackrel{^}{\Gamma }}_{1}^{\text{T}}{Z}_{1}\\ *& -{Z}_{1}\end{array}\right],$

${\stackrel{^}{\Gamma }}_{1}^{\text{T}}=\left[\begin{array}{c}X{A}^{\text{T}}\\ {\left(BUY\right)}^{\text{T}}\\ 0\\ {\left(BUY\right)}^{\text{T}}\end{array}\right],\stackrel{^}{\phi }=\left[\begin{array}{c}B\\ 0\\ 0\\ \begin{array}{l}\text{}0\\ hB\end{array}\end{array}\right],\stackrel{^}{\psi }=\left[\begin{array}{c}\begin{array}{l}\text{}0\\ {Y}^{\text{T}}{U}^{\text{T}}\end{array}\\ 0\\ {Y}^{\text{T}}{U}^{\text{T}}\\ 0\end{array}\right],\stackrel{^}{\Theta }=\left[\begin{array}{cccc}{\stackrel{^}{\Theta }}_{11}& {\stackrel{^}{\Theta }}_{12}& {\stackrel{^}{\Theta }}_{13}& {\stackrel{^}{\Theta }}_{14}\\ *& {\stackrel{^}{\Theta }}_{22}& {\stackrel{^}{\Theta }}_{23}& {\stackrel{^}{\Theta }}_{24}\\ *& *& {\stackrel{^}{\Theta }}_{33}& 0\\ *& *& *& \text{}{\stackrel{^}{\Theta }}_{44}\end{array}\right],{\stackrel{^}{\Pi }}_{2}=\left[\begin{array}{cccc}{\stackrel{^}{\Omega }}_{11}& {\stackrel{^}{\Omega }}_{12}& {\stackrel{^}{\Omega }}_{13}& h{X}_{2}{A}^{\text{T}}\\ *& {\stackrel{^}{\Omega }}_{22}& {\stackrel{^}{\Omega }}_{23}& 0\\ *& *& {\stackrel{^}{\Omega }}_{33}& 0\\ *& *& *& {\stackrel{^}{Z}}_{2}-2{X}_{2}\end{array}\right],$

$\begin{array}{l}{\stackrel{^}{\Theta }}_{11}={X}_{1}{A}^{\text{T}}+A{X}_{1}+{\stackrel{^}{Q}}_{1}+{\alpha }_{1}{X}_{1}-{\text{e}}^{-{\alpha }_{1}h}{\stackrel{^}{Z}}_{1},\text{\hspace{0.17em}}{\stackrel{^}{\Theta }}_{12}=BUY-{\text{e}}^{-{\alpha }_{1}h}\left({\stackrel{^}{S}}_{1}-{\stackrel{^}{Z}}_{1}\right),\text{\hspace{0.17em}}{\stackrel{^}{\Theta }}_{13}={\text{e}}^{-{\alpha }_{1}h}{\stackrel{^}{S}}_{1},\text{\hspace{0.17em}}{\stackrel{^}{\Theta }}_{14}=BUY,\\ {\stackrel{^}{\Theta }}_{22}=\delta \text{ }\stackrel{^}{\Phi }-{\text{e}}^{-{\alpha }_{1}h}\left(2{\stackrel{^}{Z}}_{1}-{\stackrel{^}{S}}_{1}-{\stackrel{^}{S}}_{1}^{\text{T}}\right),\text{\hspace{0.17em}}{\stackrel{^}{\Theta }}_{23}=-{\text{e}}^{-{\alpha }_{1}h}\left({\stackrel{^}{S}}_{1}-{\stackrel{^}{Z}}_{1}\right),\text{\hspace{0.17em}}{\stackrel{^}{\Theta }}_{24}=\delta \text{ }\stackrel{^}{\Phi },\text{\hspace{0.17em}}{\stackrel{^}{\Theta }}_{33}=-{\text{e}}^{-{\alpha }_{1}h}\left({\stackrel{^}{Q}}_{1}+{\stackrel{^}{Z}}_{1}\right),\\ {\stackrel{^}{\Theta }}_{44}=\delta \text{ }\stackrel{^}{\Phi }-\stackrel{^}{\Phi },\text{\hspace{0.17em}}{\stackrel{^}{\Omega }}_{11}={X}_{2}{A}^{\text{T}}+A{X}_{2}+{\stackrel{^}{Q}}_{2}-{\alpha }_{2}{X}_{2}-{\stackrel{^}{Z}}_{2},\text{\hspace{0.17em}}{\stackrel{^}{\Omega }}_{12}={\stackrel{^}{Z}}_{2}-{\stackrel{^}{S}}_{2},\text{\hspace{0.17em}}{\stackrel{^}{\Omega }}_{13}={\stackrel{^}{S}}_{2},\text{\hspace{0.17em}}{\stackrel{^}{\Omega }}_{22}={\stackrel{^}{S}}_{2}+{\stackrel{^}{S}}_{2}^{\text{T}}-2{\stackrel{^}{Z}}_{2},\\ {\stackrel{^}{\Omega }}_{23}={\stackrel{^}{Z}}_{2}-{\stackrel{^}{S}}_{2},\text{\hspace{0.17em}}{\stackrel{^}{\Omega }}_{33}=-{\text{e}}^{{\alpha }_{2}h}{\stackrel{^}{Q}}_{2}-{\stackrel{^}{Z}}_{2},\end{array}$

${\stackrel{^}{\Sigma }}_{1}>0,\text{\hspace{0.17em}}{\stackrel{^}{\Psi }}_{1}<0,$ (44)

${\stackrel{^}{\Sigma }}_{2}>0,\text{\hspace{0.17em}}{\stackrel{^}{\Psi }}_{2}<0,$ (45)

${\stackrel{^}{\Psi }}_{1}=\left[\begin{array}{ccc}{\stackrel{^}{\Pi }}_{1}& {N}_{1}& {M}_{1}\\ *& -{\nu }_{1}^{-1}I& 0\\ *& *& -{\nu }_{1}I\end{array}\right],\text{\hspace{0.17em}}{\stackrel{^}{\Psi }}_{2}=\left[\begin{array}{ccc}{\stackrel{^}{\Pi }}_{2}& {N}_{2}& {M}_{2}\\ *& -{\nu }_{2}^{-1}I& 0\\ *& *& -{\nu }_{2}I\end{array}\right],$

$\begin{array}{l}{N}_{1}^{\text{T}}=\left[{F}_{1}{X}_{1},{F}_{2}UY,0,{F}_{2}UY,0,{F}_{2},0\right],\text{\hspace{0.17em}}{N}_{2}^{\text{T}}=\left[{F}_{1}{X}_{2},0,0,0\right],\\ {M}_{1}^{\text{T}}=\left[{E}^{\text{T}},0,0,0,h{E}^{\text{T}},0,0\right],\text{\hspace{0.17em}}{M}_{2}^{\text{T}}=\left[{E}^{\text{T}},0,0,h{E}^{\text{T}}\right],\end{array}$

${\stackrel{^}{\Pi }}_{1}+{N}_{1}{H}^{\text{T}}\left(t\right){M}_{1}^{\text{T}}+{M}_{1}H\left(t\right){N}_{1}^{\text{T}}<0,$ (46)

${\stackrel{^}{\Pi }}_{2}+{N}_{2}{H}^{\text{T}}\left(t\right){M}_{2}^{\text{T}}+{M}_{2}H\left(t\right){N}_{2}^{\text{T}}<0.$ (47)

${\stackrel{^}{\Pi }}_{1}+{\nu }_{1}{N}_{1}{N}_{1}^{\text{T}}+{\nu }_{1}^{-1}{M}_{1}{M}_{1}^{\text{T}}<0,$ (48)

${\stackrel{^}{\Pi }}_{2}+{\nu }_{2}{N}_{2}{N}_{2}^{\text{T}}+{\nu }_{2}^{-1}{M}_{2}{M}_{2}^{\text{T}}<0.$ (49)

4. 数值仿真

$A=\left[\begin{array}{cc}0.1& -0.9\\ 1.1& -0.4\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=\left[\begin{array}{cc}-0.1& 0.3\\ -0.4& 0.2\end{array}\right],$

$\Xi =diag\left\{{d}_{1},{d}_{2}\right\},$

${c}_{1}=1$${c}_{2}=20$${T}_{f}=10$$R=I$$\delta =0.15$${\alpha }_{1}=0.05$${\alpha }_{2}=0.3$$h=0.02$${\mu }_{1}={\mu }_{2}=1.01$$\rho =1$${l}_{\mathrm{min}}=1.5$${b}_{\mathrm{max}}=0.2$。根据定理3，可得事件触发矩阵和可靠性控制器增益如下

$\Phi =\left[\begin{array}{cc}2.4099& -0.8564\\ -0.8564& 1.3069\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}K=\left[\begin{array}{cc}-5.0894& 5.7539\\ -5.2299& 1.7458\end{array}\right].$

$A=\left[\begin{array}{cc}-0.5& -1\\ 0.7& 0.45\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=\left[\begin{array}{cc}0.5& -0.2\\ -0.3& -0.7\end{array}\right],$ (50)

$E=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{1}=\left[\begin{array}{cc}0.02& 0\\ 0& 0.2\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}_{2}=\left[\begin{array}{cc}0.1& 0\\ 0& 0.01\end{array}\right].$ (51)

$\Phi =\left[\begin{array}{cc}0.3995& 0.4398\\ 0.4398& 0.9713\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}K=\left[\begin{array}{cc}-1.2305& -0.2816\\ 0.8515& 1.4221\end{array}\right].$

Figure 2. The state responses, the trajectory of ${x}^{\text{T}}\left(t\right)Rx\left(t\right)$ and the event trigger time of the system (1) with the parameters (50), (51)

5. 结论

Robust Finite-Time Reliable Control for Uncertain Networked Control Systems under Denial of Service Attacks Based on Event Triggering Mechanism[J]. 动力系统与控制, 2022, 11(03): 104-116. https://doi.org/10.12677/DSC.2022.113012

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