Guaranteed Cost Output Feedback Control for Nonlinear Systems via Networks with Adaptive Event-Triggered SCP and Hybrid Attacks

Abstract

This paper addresses the problem of guaranteed cost output feedback control for a class of networked interval type-2 Takagi–Sugeno (IT2 T-S) fuzzy systems with adaptive event-triggered stochastic communication protocol (AETSCP) scheduling and hybrid attacks. A novel AETSCP scheduling is designed to judge whether or not data are triggered as well as to determine which node transmits data to the controller. Meanwhile, the security problem of hybrid attacks with respect to denial-of-service (DoS) attacks and deception attacks on the system is considered. The quadratic boundedness (QB) technique is employed to depict the closed-loop stability of the concerned networked control systems (NCSs). Two adequate theorems are given based on Lyapunov stability theory for designing the observer-based and dynamic output feedback-based controllers, which guarantee the stability and robust performance of the required system. In the end, a simulation example of the mass-spring-damping system is provided to confirm the effectiveness of the presented control strategy.

Appendices

Appendix A

Proof

Consider the performance index function $J_h^1={\mathcal {E}}\{ \sum _{h=0}^{\infty }(\Vert {\tilde{x}}_{h_t} \Vert _{{\mathscr {Q}}}^2+\Vert u_h \Vert _{{\mathscr {R}}}^2 ) \}$, where ${\mathscr {Q}}$ and ${\mathscr {R}}$ are positive-definite matrices of appropriate dimensions. For the closed-loop system (17), we construct the Lyapunov function $V(\aleph _h^1)=\Vert \aleph _h^1 \Vert _{\digamma _1}^2+\epsilon _h$, where $\digamma _1=\textrm{diag}\{ P_1,\varrho N_e,S_1,M_1 \}$. If NCS (17) is quadratically stable, then ${\mathcal {E}}\{V(\aleph _h^1)\}\ge 1 \Longrightarrow {\mathcal {E}}\{ V(\aleph _h^1) \}-{\mathcal {E}}\{ V(\aleph _{h+1}^1 \}\ge \frac{1}{\gamma }{\mathcal {E}}\{ \Vert {\tilde{x}}_{h_t} \Vert _{{\mathscr {Q}}}^2+\Vert u_h \Vert _{{\mathscr {R}}}^2 \}$ holds, where $\gamma$ is the upper bound of $J_h^1$. Since $\Vert \xi _h \Vert _H^2\le 1$, by the S-procedure method, there exists a scalar $\phi _1\in (0,1)$, which makes

$$\begin{aligned}&{\mathcal {E}}\{ V(\aleph _h^1) \}-{\mathcal {E}}\{ V(\aleph _{h+1}^1 \}-\frac{1}{\gamma }{\mathcal {E}}\{ \Vert {\tilde{x}}_{h_t} \Vert _{{\mathscr {Q}}}^2 + \Vert u_h \Vert _{{\mathscr {R}}}^2 \} \nonumber \\&\quad -\phi _1({\mathcal {E}}\{ V(\aleph _h^1) \}-\Vert \xi _h \Vert _H^2)\ge 0 \end{aligned}$$

(29)

Besides, by (3) and (4), it is possible to obtain the inequality

$$\begin{aligned} \Delta \epsilon _h=\epsilon _{h+1}-\epsilon _{h}\le \Vert {\hat{x}}_h+{\bar{e}}_h \Vert _{\Psi }^2-{\tilde{\epsilon }}\Vert {\bar{e}}_h \Vert _{\Psi }^2 \end{aligned}$$

(30)

Substituting (17) into (29), and combining (30), a new matrix inequality can be acquired by using the Schur complement. And then, since there is a non-convex optimization problem in this new matrix inequality, it can be handled by the cone complement linearization algorithm in Lemma 1, i.e., by making $P^{-1}={\bar{P}}$, $S^{-1}={\bar{S}}$, $M^{-1}={\bar{M}}$, $\Psi ^{-1}={\bar{\Psi }}$, such that $P{\bar{P}}=I$, $S{\bar{S}}=I$, $M{\bar{M}}=I$, $\Psi {\bar{\Psi }}=I$. Next, using a congruence transformation on the matrix inequality via $\textrm{diag}\{ I,I,I,I,I,I,I,I,I,I,I,\varrho I,I,I,I,I,I,I,I,I,I,I,I \}$, (19) can be proved. Finally, we give the upper bound $\gamma$ of $J_h^1$ such that $\gamma {\mathcal {E}}\{V(\aleph _0^1)\}\le \gamma$. Again using the Schur complement, (20) is confirmed. $\square$

Appendix B

Proof

Similarly, consider the performance index function $J_h^2={\mathcal {E}}\{ \sum _{h=0}^{\infty }(\Vert {\tilde{y}}_{h_t} \Vert _{{\mathscr {Q}}}^2+\Vert u_h \Vert _{{\mathscr {R}}}^2 ) \}$. For the closed-loop system (23), we construct the Lyapunov function $V(\aleph _h^2)=\Vert \aleph _h^2 \Vert _{\digamma _2}^2+\epsilon _h$, where $\digamma _2=\textrm{diag}\{ P_2,S_2,M_2 \}$. If NCS (23) is quadratically stable, then there exists a scalar $\phi _2\in (0,1)$ satisfying the following condition:

$$\begin{aligned}&{\mathcal {E}}\{ V(\aleph _h^2) \}-{\mathcal {E}}\{ V(\aleph _{h+1}^2) \}-\frac{1}{\gamma }{\mathcal {E}}\{ \Vert {\tilde{y}}_{h_t} \Vert _{{\mathscr {Q}}}^2 + \Vert u_h \Vert _{{\mathscr {R}}}^2 \} \nonumber \\&\quad -\phi _2({\mathcal {E}}\{ V(\aleph _h^2) \}-\Vert \xi _h \Vert _H^2)\ge 0 \end{aligned}$$

(31)

In addition, $\Delta \epsilon _h \le \Vert y_h+{\bar{e}}_h \Vert _{\Psi }^2-{\tilde{\epsilon }}_h\Vert {\bar{e}}_h \Vert _{\Psi }^2$. By applying the Schur complement, the new matrix inequality is obtained. Define

$$\begin{aligned} P_2=\begin{bmatrix} W &{} X^\textrm{T} \\ X &{} R \end{bmatrix},\; P_2^{-1}=\begin{bmatrix} Y &{} K^\textrm{T} \\ K &{} G \end{bmatrix} \end{aligned}$$

with $X=-W$. Make ${\mathcal {T}}_1=\begin{bmatrix} I &{} Y \\ 0 &{} K \end{bmatrix}$ and ${\mathcal {T}}_2=\begin{bmatrix} I &{} W \\ 0 &{} X \end{bmatrix}$. By pre- and post-multiplying $\textrm{diag}\{{\mathcal {T}}_1^\textrm{T},I,I,I,I,I,{\mathcal {T}}_2^\textrm{T},I,I,I,I,I,I,I,I,I,I,I,I,I,I\}$ and its transpose for both the left and right sides of the matrix inequality, (25) is deduced. The rest of the proof procedure is similar to Appendix A. $\square$