Co-Design of Event-Triggered Scheme and H∞ Output Control for Markov Jump Systems Against Deception Attacks

This paper is concerned with the co-design problem of event-triggered scheme and H static output control of linear Markov jump systems with deception attacks. To save the previous communication resources, a mode-dependent event-triggered scheme is utilized based on system output. To describe the deception attacks, a random variable satisfying Bernoulli distribution is employed. By using a separation approach, sufficient linear matrix inequality conditions for the existence of event-triggered output controllers that ensure the stochastic stability with prescribed H_∞ are obtained. Then, a co-design algorithm is proposed to obtain the trade-off between the communication cost and H_∞ performance. Lastly, the validity of the developed method is verified by a numerical example and a practical single-link robot arm system.

SECTION I. Introduction

As a type of stochastic systems, Markov jump systems (MJSs) have been widely adopted in various fields, such as networked systems, manufacturing systems, fault diagnosis [1]–​[7]. With the rapid development of communication technique, the integration of traditional control systems and communication network has attracted much attention of researchers. In recent years, various problems of networked MJSs have been reported in [8]–​[11] and the references therein. Note that these results are obtained based on a hypothesis that the communication resources like node energy and network bandwidth are adequate. However, communication resources in practical systems are usually constrained by the limited ability and cost of equipments.

Since the advantage of saving limited communication resources, event-triggered scheme (ETS) become an effective manner to reduce communication cost without losing much system performance [12]–​[16]. By using event-triggered scheme, many outcomes of MJSs about robust control [17], quantized control [18] and finite-time control [19] have been reported. Specially, [17] studies the robust event-triggered control problem of discrete-time uncertain MJSs with nonlinear input. To decrease the waste of network resources, both event-triggered mechanism and input quantization are considered to investigate the issue of guaranteed cost finite-time control of semi-MJSs in [18]. In [19], the finite-time event-triggered $H_{\infty }$ controller is obtained to ensure the finite-time bound of T-S MJSs, where the asynchronous premise variables are taken into account. It is noted that in these event-triggered control results, communication cost and system performance, such as stability, $H_{\infty }/H_{2}$ performance, are closely related to the triggering threshold. Generally speaking, a large triggering threshold will lead to fewer triggered signal but worse system performance. Thus, a trade-off should be considered between the system performance and the communication cost (like channel bandwidth and node energy) [20], [21]. To minimize both system performance and communication cost, a data packet transmission ratio is introduced to evaluate the communication cost in [22], where a co-design iterative algorithm is developed to minimize both the $H_{\infty }$ performance and the communication cost. Based on this method, however, the data packet transmission ratio only can be obtained after executing system simulations in a given time period, which needs much computation burden.

On the other hand, the triggered data transmitted over the network is highly possible to face the threaten of cyber-attacks, which may cause system performance deduction even instability [23]–​[27]. Although the dedicated protocols for network is able to defend the attacks from external attackers to some extent [28], it is difficult to ensure the complete protection against cyber-attacks. Thus, the security issues have been researched from the perspective of control theory. In [29], a centralized security-guaranteed filter is investigated for stochastic systems subject to cyber-attacks. The problem of event-triggered output control of networked systems with state observer and denial-of-service attacks is developed in [30], in which cyber-attacks can block the transmitted data in communication channels. In [31], the resilient load frequency control issue of multiarea power systems under deception attacks is addressed based on an event-triggered communication scheme. In [32], a memory ETS is used to save network resources and a resilient event-triggered controller is designed for networked systems with randomly occurring deception attacks. An event-based security control issue of discrete-time stochastic systems subject to denial-of-service attacks and deception attacks is studied in [33]. For MJSs, a finite-time sliding-mode control issue is addressed in [34], where the intruders can inject false data into the communication signals. Reference [35] studies the static output feedback asynchronous control of MJSs with deception attacks. Note that the event-triggered communication scheme and the trade-off between communication cost and system performance for MJSs are not considered in [34], [35]. Moreover, to cope with the static output feedback controller, an inequality technique is used in [35], which could result in some design conservativeness. Thus, how to design the event-triggered static output controller for MJSs with deception attacks and obtain the trade-off between communication cost and system performance needs further investigation.

Inspired by the above discussions, this article studies the co-design issue of event-triggered scheme and $H_{\infty }$ static output control of MJSs with deception attacks. The main contributions are summarized as below:

1. To handle the nonlinearity induced by static output feedback control, a constructive separation strategy is utilized to remove the coupling among controller gain and system matrices. Compared with some existing methods, some constraints like equality constraint [36], rank constraints [37] or inequality technique [35] are not required any more.

2. A novel co-design algorithm is proposed to derive the trade-off between the communication cost and $H_{\infty }$ performance. In contrast to the method in [22], an extra step of computing the data packet transmission ratio is removed, which means that our proposed trade-off algorithm is simpler to be executed.

Notation: In this paper, the notation $X>0~( < 0)$ means $X$ is a symmetric and positive (negative) definite matrix. $\mathrm {He}(X)=X^{T}+X$ . The kernel of matrix $X$ is represented by $X^{\perp }$ . * denotes $[*]UV=V^{T}UV$ or $V^{T}U[*]=V^{T}UV$ .

SECTION II. Preliminaries

Consider the following continuous MJS: $$\begin{equation*} \begin{cases} \dot {x}(t)=A(r_{t})x(t)+B_{1}(r_{t})u(t)+B_{2}(r_{t})\omega (t) \\ z(t)=C_{1}(r_{t})x(t)+D(r_{t})\omega (t)\\ y(t)=C_{2}(r_{t})x(t) \end{cases}\tag{1}\end{equation*}$$ where $x(t)\in \mathbb {R}^{n}$ , $u(t)\in \mathbb {R}^{n}$ , $w(t)\in \mathbb {R}^{p}$ , $z(t)\in \mathbb {R}^{q}$ and $y(t)\in \mathbb {R}^{m}$ denote the state vector, control input, exogenous disturbance in $\mathcal {L}_{2}~[0\hspace {2mm}\infty$ ), performance output and measured output, respectively. $A(r_{t})$ , $B_{1}(r_{t})$ , $B_{2}(r_{t})$ , $C_{1}(r_{t})$ , $C_{2}(r_{t})$ , and $D(r_{t})$ are known system matrices. $r_{t}$ is a continuous-time Markov process denoting the system mode, which takes discrete values in a finite set $S=\{1,2,\ldots,s\}$ . The transition rates are given as follows: $$\begin{equation*} Pr\{r_{t+\Delta }=j|r_{t}=i\}= \begin{cases} \lambda _{ij}\Delta +o(\Delta),&i\neq j\\ 1+\lambda _{ii}\Delta +o(\Delta),&i=j, \end{cases}\tag{2}\end{equation*}$$ where $\Delta >0$ , $\lambda _{ij}\geqslant 0$ for $i\neq j$ and $\lambda _{ii}=-\sum \limits ^{s}_{j=1,j\neq i}\lambda _{ij}$ for each mode $i$ , $\lim \limits _{\Delta \rightarrow 0}o(\Delta)/\Delta =0$ . As a result, the transition rate matrix is denoted by $$\begin{equation*} \Lambda =\left [{ \begin{matrix} \lambda _{11}&\quad \lambda _{12}&\quad \cdots &\quad \lambda _{1s}\\ \lambda _{21}&\quad \lambda _{22}&\quad \cdots &\quad \lambda _{2s}\\ \vdots \\ \lambda _{s1}&\quad \lambda _{s2}&\quad \cdots &\quad \lambda _{ss}~\end{matrix} }\right].\tag{3}\end{equation*}$$

For $r_{t}=i\in S=\{1,2,\ldots,s\}$ , the abbreviations of system matrices of the $i$ th mode are given as $A_{i}$ , $B_{1i}$ , $B_{2i}$ , $C_{1i}$ , $C_{2i}$ and $D_{i}$ .

The decision on sampled data $y(i_{k}h)$ to be sent or not is determined by the following event-detector $$\begin{align*}&\hspace {-.5pc}t_{k+1}h=t_{k}h+\min _{l}\{lh|e^{T}(i_{k}h)\Phi _{i} e(i_{k}h) \\&\qquad\qquad\qquad\qquad\qquad\qquad\quad\! \displaystyle {\geq \delta _{i} y^{T}(t_{k}h) \Phi _{i} y(t_{k}h), \}} \tag{4}\end{align*}$$ where $e(i_{k}h)$ represent the error term $y(i_{k}h)-y(t_{k}h)$ and it is reset as zero at the triggering instants, $h$ denotes the sampling period, $t_{k}h$ is the latest triggering time when $y(t_{k}h)$ is successfully sent to the controller, $i_{k}h$ satisfies $i_{k}h=t_{k}h+lh$ ($l\in \mathbb {N}$ ) and $y(i_{k}h)$ is the current sampled output, $\Phi _{i}$ is a positive weighting matrices to be designed, $\delta _{i}$ is the triggering threshold parameter belongs to $[0,~1$ ).

It is assumed that no data packet loss occurs during the data transmission and the sum of communication delay, computation delay, and waiting delay is expressed as $\tau _{t_{k}}$ . With the modified event-detector and considering the effect of a zero-order-hold (ZOH), $y(t_{k}h)$ to the controller is held until a new trigger happens. Incorporating network induced delay $\tau _{t_{k}}$ , the signal $\hat {y}(t)$ received by controller is $$\begin{equation*} \hat {y}(t)=y(t_{k}h), t\in \Omega _{l}\triangleq [t_{k}h+\tau _{t_{k}}, t_{k+1}h+\tau _{t_{k}+1}).\tag{5}\end{equation*}$$ Set $\tau (t)\triangleq t-i_{k}h$ , obviously, $\tau (t)$ is a piecewise-linear function and meets $0\le \tau (t)\le h+\bar {\tau }\triangleq \tau _{m}$ , where $\bar \tau ={sup}_{l}\{\tau _{t_{k}+1}\}$ is the upper bound of network delay. Consequently, (5) can be expressed as: $$\begin{equation*} \hat {y}(t)=y(t-\tau (t))-e(i_{k}h),\hspace {4mm} t\in \Omega _{l}.\tag{6}\end{equation*}$$

In this paper, the deception attack is assumed can be modelled by a Bernoulli variable, $\theta (t)\in \{0,~1\}$ with $Prob\{\theta (t)=1\}=\theta _{1}$ and $Prob\{\theta (t)=0\}=\theta _{2}$ . Furthermore, the deception attack modifies $\hat y(t)$ into $f(\hat y(t))$ , where $f(\hat y(t))$ satisfies the following assumption.

Assumption 1:

The deception attack function $f(\cdot)$ is assumed to satisfy the following sector bound condition: $$\begin{equation*} (f(x)-L_{1}x)^{T}(f(x)-L_{2}x)\leq 0,\tag{7}\end{equation*}$$ where $L_{1}$ and $L_{2}$ are two known constant matrices and satisfy $L_{2}-L_{1}\geq 0$ .

The proposed controller law is given as $$\begin{align*}&\hspace {-.5pc}u_{a}(t)= (1-\theta (t))(K_{i}y(t-\tau (t))-K_{i}e(i_{k}h)) \\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \displaystyle {+\,\theta (t)K_{i}f(\hat y(t)),} \tag{8}\end{align*}$$ where $K_{i}$ denotes the controller gains to be designed.

Remark 1:

The data transmitted over the communication network is highly possible to be attacked by external hackers. Randomly occurring deception attacks may affect the system performance. In this paper, the stochastic process of deception attacks is modelled by a random variable $\theta (t)$ satisfying Bernoulli distribution where $\theta (t)=0$ and $\theta (t)=1$ are used to represent the data communication is attacked or not, respectively. Examples of deception attack functions $f(\cdot)$ that satisfy Assumption 1 are saturation function $f(x(t))=sat(x(t))$ , time-varying percentage function $f(x(t))=(I+\Delta (t))x(t)$ with $|\Delta (t)| < 1$ , etc.

Combing (1) and (8), for $t\in \Omega \triangleq \cup \Omega _{j}$ , one has the closed-loop system as: $$\begin{equation*} \begin{cases} \dot {x}(t)=A_{i}x(t)+(1-\theta (t))\Big (B_{1i}K_{i}C_{2i}x(t-\tau (t))\\ - B_{1i}K_{i}e(i_{k}h)\Big)+\theta (t)B_{1i}K_{i}f(\hat y(t)) +B_{2i}\omega (t) \\ z(t)=C_{1i}x(t)+D_{i}\omega (t) \end{cases}\tag{9}\end{equation*}$$

The objective of this paper is to design the controller in (8) with the event-triggered scheme (4) such that system (9) is stochastically stable with the required $H_{\infty }$ performance index $\gamma$ .

Some definitions and technical lemmas are introduced to help to achieve the above objective.

Definition 1:

[2] The system (1) with $\omega (t)\equiv 0$ is said to to be stochastically stable if $$\begin{equation*} \mathbb {E}\left \{{\int ^{\infty }_{0}\|x(t)\|^{2}|x_{0},r_{0} }\right \} < \infty\tag{10}\end{equation*}$$ under any initial conditions $x_{0}$ and $r_{0}$ .

Definition 2:

[3] Given a scalar $\gamma > 0$ , the system (1) is stochastically stable with $H_{\infty }$ performance $\gamma > 0$ if the following inequality $$\begin{equation*} \mathbb {E}\left \{{\int _{0}^{\infty }z^{T}(t)z(t)dt}\right \} < \gamma ^{2}\mathbb {E}\left \{{\int _{0}^{\infty }\omega ^{T}(t)\omega (t)dt}\right \}\tag{11}\end{equation*}$$ holds for all nonzero $\omega (t)\in \mathcal {L}_{2}$ under the zero initial condition $x_{0}=0$ .

Lemma 1:

[38] $\eta ^{T}(t)\mathcal {P}\eta (t)+g(\eta (t)) < 0$ , $\eta (t)\neq 0, \forall \mathcal {N}\eta (t)=0$ , where $\mathcal {P}=\mathcal {P}^{T}$ , $\mathcal {N}\in \mathbb {R}^{m\times n}$ and $g(\eta (t))$ is a scalar function, if there exists $\mathcal {U}\in \mathbb {R}^{n\times m}$ such that $\eta ^{T}(t)\left ({\mathcal {P}+\textit {He}(\mathcal {UN})}\right)\eta (t)+g(\eta (t)) < 0$ .

Lemma 2:

[39]: For given $\mu \in (0, 1)$ , $R \in \mathbb {R}^{p\times p}$ , $M_{1}\in \mathbb {R}^{p\times q}, M_{2}\in \mathbb {R}^{p\times q}$ . Define the following function $\Im (\mu,R)$ : $$\begin{equation*} \Im (\mu,R)=\cfrac {1}{\mu } \hspace {2mm}\vartheta ^{T}M_{1}^{T}RM_{1}\vartheta +\cfrac { 1}{1-\mu }\hspace {2mm}\vartheta ^{T}M_{2}^{T}RM_{2}\vartheta.\tag{12}\end{equation*}$$ Then, if there exists $X\in \mathbb {R}^{p\times p}$ satisfying $\left [{ {\begin{array}{cccccccccccccccccccc} R & X\\ * & R\end{array}} }\right]>0$ , the following inequality holds $$\begin{equation*} \min _{\mu \in (0,1)}\Im (\mu,R)\geqslant \left [{ {\begin{array}{cccccccccccccccccccc} M_{1}\vartheta \\ M _{2}\vartheta \end{array}} }\right]^{T} \left [{ {\begin{array}{cccccccccccccccccccc} R &\quad X\\ * &\quad R\end{array}} }\right] \left [{ {\begin{array}{cccccccccccccccccccc} M_{1}\vartheta \\ M _{2}\vartheta \end{array}} }\right].\tag{13}\end{equation*}$$

SECTION III. Main Results

A. Stability Analysis and Control Synthesis

In this section, sufficient conditions for the required stability and performance are given in Theorem 1 and a unified event-triggered filter design method is proposed in Theorem 2.

Theorem 1:

For given parameters $\tau _{m}$ , $\delta _{i}$ , controller gain $K_{i}$ , system (9) under the event-triggered scheme (4) is stochastically stable with a required $H_\infty$ performance $\gamma$ if there exist matrices $P_{i}>0$ , $R>0$ , $S>0$ , $\Phi _{i}>0$ , and matrices $G_{i}$ , $F_{i}$ , $X$ such that $$\begin{align*}&\hspace {-.5pc}\Pi _{i}+\Sigma _{i} < 0, \tag{14}\\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad \displaystyle {\left [{ {\begin{array}{cccccccccccccccccccc} R &\quad X\\ * &\quad R\end{array}} }\right]>0,} \tag{15}\end{align*}$$ hold, where $$\begin{align*} \Pi _{i}=&\left [{ {\begin{array}{cccccccccccccccccccc} \Pi _{i}^{11} &~\Pi _{i}^{12} &~\Pi _{i}^{13} &~0 &~\Pi _{i}^{15} &~\Pi _{i}^{16} &~ \Pi _{i}^{17} &~0 \\ * &~\Pi _{i}^{22} &~\Pi _{i}^{23} &~0 &~\Pi _{i}^{25} &~\Pi _{i}^{26} &~\Pi _{i}^{27} &~C^{T}_{1i} \\ * &~* &~\Pi _{i}^{33} &~0 &~\Pi _{i}^{35} &~0 &~0 &~0 \\ * &~* &~* &\,\,-S &~0 &~0 &~0 &~0 \\ * &~* &~* &~* &~\Pi _{i}^{55} &~0 &~0 &~0 \\ * &~* &~* &~* &~* &~0 &~0 &~0 \\ * &~* &~* &~* &~* &~* &\,\,-\gamma ^{2}I &~D^{T}_{i}\\ * &~* &~* &~* &~* &~* &~* &\,\,-I \end{array}} }\right],\\ \Sigma _{i}=&-\Sigma _{1}-\rho \Sigma _{2i},\quad \Sigma _{2i}=\mathscr {E}^{T}_{2i} \mathfrak {F}\mathscr {E}_{2i},\\ \Sigma _{1}=&\mathscr E_{1}^{T} \begin{bmatrix} R &\quad X\\ * &\quad R \end{bmatrix} \mathscr E_{1},\quad \mathfrak {F}=\left [{ {\begin{array}{cccccccccccccccccccc} \mathfrak {F}_{1} &\quad \mathfrak {F}_{2}\\ \mathfrak {F}^{T}_{2} &\quad I\\ \end{array}} }\right],\\ \mathfrak {F}_{1}=&\cfrac {L^{T}_{1}L_{2}+L^{T}_{2}L_{1}}{2},\quad \mathfrak {F}_{2}=\cfrac {L_{1}+L_{2}}{2},\\ \mathscr {E}_{1}=&\left [{ {\begin{array}{cccccccccccccccccccc} 0 &\quad I &\quad -I &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ 0 &\quad 0 &\quad I &\quad -I &\quad 0 &\quad 0 &\quad 0 &\quad 0 \end{array}} }\right],\\ \mathscr {E}_{2i}=&\begin{bmatrix}0 &\quad 0 &\quad C_{2i} &\quad 0 &\quad -I &\quad 0 &\quad 0 &\quad 0\\ 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad I &\quad 0 &\quad 0\end{bmatrix},\\ \Pi _{i}^{11}=&\tau ^{2}_{m}R-\mathrm He(G_{i}), \quad \Pi _{i}^{12}=P_{i}+G_{i}A_{i}-F^{T}_{i},\\ \Pi _{i}^{13}=&\theta _{2}G_{i} B_{1i}K_{i}C_{2i}, \quad \Pi _{i}^{15}=-\theta _{2}G_{i}B_{1i}K_{i},\\ \Pi _{i}^{16}=&\theta _{1}G_{i}B_{1i}K_{i}, \quad \Pi _{i}^{17}= G_{i}B_{2i},\\ \Pi _{i}^{23}=&\theta _{2}F_{i}B_{1i}K_{i}C_{2i}, \quad \Pi _{i}^{22}=\!S\!+\!\mathrm {He}(F_{i}A_{i})+\sum ^{s}_{j=1}\pi _{ij}P_{j},\\ \Pi _{i}^{25}=&-\theta _{2}F_{i}B_{1i}K_{i}, \quad \Pi _{i}^{26}= \theta _{1}F_{i}B_{1i}K_{i},\\ \Pi _{i}^{27}=&F_{i}B_{2i}, \quad \Pi _{i}^{33}= \delta _{i} C^{T}_{2i}\Phi _{i} C_{2i},\\ \Pi _{i}^{35}=&-\delta _{i}C^{T}_{2i}\Phi _{i}, \quad \Pi _{i}^{55}= (\delta _{i}-1)\Phi _{i}.\end{align*}$$

Proof:

Choose the Lyapunov-Krasovskii functional as below $$\begin{align*}&\hspace {-.5pc}V(t)=x^{T}(t) P_{i} x(t) + \int ^{t}_{t-\tau _{m}}x^{T}(v)Sx(v)dv \\&\qquad\qquad\qquad\qquad \displaystyle { +\,\tau _{m}\int ^{0}_{-\tau _{m}}\int ^{t}_{t+u}\dot {x}^{T}(v)R\dot {x}(v)dv du.} \tag{16}\end{align*}$$

Defining $\varsigma$ as the weak infinitesimal operator of the Markov process $x(t),r_{t}$ , one can get $$\begin{align*} \mathbb {E}\{ \varsigma V(t)\}=&2x^{T}(t) P_{i} \dot x(t)+x^{T}(t) \sum ^{s}_{j=1}\pi _{ij}P_{j} x(t) \\&+\,x^{T}(t)Sx(t)-x^{T}(t-\tau _{m})Sx(t-\tau _{m}) \\&+\,\tau ^{2}_{m}\dot { x}^{T}(t) R \dot {x}(t) -\tau _{m}\int ^{t}_{t-\tau _{m}}\dot {x}^{T}(v)R\dot {x}(v)dv.\tag{17}\end{align*}$$

Define the following augmented vector $$\begin{align*}&\hspace {-.5pc}\zeta (t)=\left [{ {\begin{array}{cccccccccccccccccccc} \dot {x}^{T}(t) &\quad x^{T}(t) &\quad x^{T}(t-\tau (t)) &\quad x^{T}(t-\tau _{m}) \end{array}} }\right.\\&\qquad\qquad\qquad\qquad \displaystyle {\left.{ {\begin{array}{cccccccccccccccccccc} e^{T}(i_{k}h) &\quad f^{T}(\hat y(t)) &\quad \omega ^{T}(t) &\quad z^{T}(t) \end{array}} }\right]^{T}.} \end{align*}$$

By adopting Jensen inequality and Lemma 2 to the integral terms in (17), it produces $$\begin{align*} -\tau _{m}\int ^{t}_{t-\tau _{m}}\dot {x}^{T}(v)R\dot {x}(v)dv \le -\zeta ^{T}(t)\mathscr {E}_{1}^{T} \left [{ {\begin{array}{cccccccccccccccccccc}R &\quad X\\ * &\quad R\end{array}} }\right]\mathscr {E}_{1}\zeta (t). \\\tag{18}\end{align*}$$

Combing (18) and (17), one has $$\begin{equation*} \mathbb {E}\{ \varsigma V(t)\}\leq \zeta ^{T}(t)\left ({\Psi _{i}-\mathscr {E}^{T}_{1} \left [{ {\begin{array}{cccccccccccccccccccc}R &\quad X\\ * &\quad R \end{array}} }\right] \mathscr {E}_{1} }\right)\zeta (t),\tag{19}\end{equation*}$$ where $$\begin{align*} \Psi _{i}=&\left [{ {\begin{array}{cccccccccccccccccccc} \Psi ^{11}_{i} &\quad P_{i} &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ * &\quad \Psi ^{22}_{i} &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ * &\quad * &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ * &\quad * &\quad * &\quad -S &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ * &\quad * &\quad * &\quad * &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ * &\quad * &\quad * &\quad * &\quad * &\quad 0 &\quad 0 &\quad 0 \\ * &\quad * &\quad * &\quad * &\quad * &\quad * &\quad 0 &\quad 0 \\ * &\quad * &\quad * &\quad * &\quad * &\quad * &\quad * &\quad 0 \\ \end{array}} }\right],\\ \Psi ^{11}_{i}=&\tau _{m}^{2}R,\quad \Psi ^{22}_{i}=S+\sum ^{s}_{j=1}\pi _{ij}P_{j}.\end{align*}$$

Considering the $H_\infty$ performance, the following inequality is required to be satisfied: $$\begin{equation*} \mathbb {E}\Big \{\varsigma V(t) + z^{T}(t)z(t)-\gamma ^{2}w^{T}(t)w(t)\Big \} < 0.\tag{20}\end{equation*}$$

By adding and subtracting $e^{T}(i_{k}h)\Phi _{i} e(i_{k}h)$ to and from (20), it gives $$\begin{align*}&\hspace {-.5pc}\mathbb {E}\Big \{\varsigma V(t) + z^{T}(t)z(t)-\gamma ^{2}w^{T}(t)w(t) \\&\qquad\quad \displaystyle {+\,e^{T}(i_{k}h)\Phi _{i} e(i_{k}h)- e^{T}(i_{k}h)\Phi _{i} e(i_{k}h)\Big \} < 0.} \tag{21}\end{align*}$$

According to (4), at the triggering instant, $y(i_{k}h)=y(t_{k}h)$ , which implies $e(i_{k}h)=0$ and between the triggering instants, we have $e^{T}(i_{k}h)\Phi _{i} e(i_{k}h) < \delta _{i}y^{T}(t_{k}h)\Phi _{i} y(t_{k}h)$ . Therefore, $$\begin{equation*} e^{T}(i_{k}h)\Phi _{i} e(i_{k}h) \leq \delta _{i}y^{T}(t_{k}h)\Phi _{i} y(t_{k}h).\tag{22}\end{equation*}$$ By using (22), (21) becomes $$\begin{align*}&\hspace {-.5pc}\mathbb {E}\Big \{\varsigma V(t) + z^{T}(t)z(t)-\gamma ^{2}w^{T}(t)w(t) \\&\qquad\; \displaystyle {+\,\delta _{i}y^{T}(t_{k}h)\Phi _{i} y(t_{k}h)- e^{T}(i_{k}h)\Phi _{i} e(i_{k}h)\Big \} < 0,} \tag{23}\end{align*}$$ which is equivalent to $$\begin{equation*} \zeta ^{T}(t)\left ({\hat \Psi _{i}-\mathscr {E}^{T}_{1} \left [{ {\begin{array}{cccccccccccccccccccc}R &\quad X\\ * &\quad R \end{array}} }\right] \mathscr {E}_{1} }\right) \zeta (t) < 0,\tag{24}\end{equation*}$$ where $$\begin{align*} \hat \Psi _{i}=&\left [{ {\begin{array}{cccccccccccccccccccc} \Psi _{i}^{11} &~P_{i} &~0 &~0 &~0 &~0 &~0 &~0 \\ * &~\Psi _{i}^{22} &~0 &~0 &~0 &~0 &~0 &~C^{T}_{1i} \\ * &~* &~\hat \Psi _{i}^{33} &~0 &\hat \Psi _{i}^{35} &~0 &~0 &~0 \\ * &~* &~* &\,\,-S &~0 &~0 &~0 &~0 \\ * &~* &~* &~* &~\hat \Psi _{i}^{55} &~0 &~0 &~0 \\ * &~* &~* &~* &~* &~0 &~0 &~0 \\ * &~* &~* &~* &~* &~* &\,\,-\gamma ^{2}I &~D_{i}^{T} \\ * &~* &~* &~* &~* &~* &~* &\,\,-I \end{array}} }\right],\\ \hat \Psi _{i}^{33}=&\delta _{i} C^{T}_{2i}\Phi _{i} C_{2i},~~\hat \Psi _{i}^{35}=-\delta _{i} C^{T}_{2i}\Phi _{i},~~\hat \Psi _{i}^{55}=(\delta _{i}-1) \Phi _{i}.\end{align*}$$

According to Assumption 1, the deception attacks are handled as $$\begin{equation*} \left [{ {\begin{array}{cccccccccccccccccccc} * \end{array}} }\right] \left [{ {\begin{array}{cccccccccccccccccccc} \mathfrak {F}_{1} & \quad \mathfrak {F}_{2}\\ \mathfrak {F}^{T}_{2} &\quad I\\ \end{array}} }\right] \left [{ {\begin{array}{cccccccccccccccccccc} y(t-\tau (t))-e(i_{k}h) \\ f(\hat y(t)) \end{array}} }\right] \leq 0,\tag{25}\end{equation*}$$ which is equivalent to $$\begin{equation*} -\rho \zeta ^{T}(t) \mathscr {E}^{T}_{2i} \mathfrak {F}\mathscr {E}_{2i} \zeta (t)\geq 0,\tag{26}\end{equation*}$$ for any $\rho > 0$ , where $\mathfrak {F}$ , $\mathfrak {F}_{1}$ and $\mathfrak {F}_{2}$ are defined in Theorem 1.

Based on (26) and S-procedure [38], (24) is further ensured by $$\begin{equation*} \zeta ^{T}(t)\left ({\hat \Psi _{i}-\mathscr {E}^{T}_{1} \left [{ {\begin{array}{cccccccccccccccccccc}R &\quad X\\ * &\quad R \end{array}} }\right] \mathscr {E}_{1}- \rho \mathscr {E}^{T}_{2i} \mathfrak {F}\mathscr {E}_{2i} }\right) \zeta (t) < 0.\tag{27}\end{equation*}$$

To be consistent with Lemma 1, we rewrite the closed-loop system (9) as $$\begin{equation*} \mathbb {E}\{\mathcal {H}_{i} \zeta (t)\}=\mathscr {H}_{i}\zeta (t)=0,\tag{28}\end{equation*}$$ where $$\begin{align*} \mathcal {H}_{i}=&\left [{ {\begin{array}{cccccccccccccccccccc} -I &\quad A_{i} &\quad (1-\theta (t))B_{1i}K_{i}C_{2i} &\quad 0 \end{array}} }\right.\\&\qquad \left.{ {\begin{array}{cccccccccccccccccccc} -(1-\theta (t))B_{1i}K_{i} &\quad \theta (t)B_{1i}K_{i} &\quad B_{2i} &\quad 0 \end{array}} }\right],\\ \mathscr {H}_{i}=&\left [{ {\begin{array}{cccccccccccccccccccc} -I &\quad A_{i} & \quad \theta _{2}B_{1i}K_{i}C_{2i} &\quad 0 \end{array}} }\right.\\&\qquad {\left.{ {\begin{array}{cccccccccccccccccccc} -\theta _{2}B_{1i}K_{i} &\quad \theta _{1} B_{1i}K_{i} &\quad B_{2i} &\quad 0 \end{array}} }\right]}.\end{align*}$$

Applying Lemma 1 to (27) gives $$\begin{equation*} \zeta ^{T}(t)\left ({\hat \Psi _{i}+\Sigma _{i}+He\left ({\mathscr {M}_{i}\mathscr {H}_{i}}\right) }\right) \zeta (t) < 0,\tag{29}\end{equation*}$$ where $$\begin{equation*} \mathscr {M}_{i}=\left [{ {\begin{array}{cccccccccccccccccccc} G^{T}_{i} &\quad F^{T}_{i} &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0\end{array}} }\right]^{T}.\end{equation*}$$

According to the given condition (14), (29) is ensured.

Then, integrating both sides of (23) over $[0,~\infty)$ yields $$\begin{align*}&\hspace {-.5pc}\mathbb {E}\Bigg \{\int ^{\infty }_{0}\varsigma V(t)dt + \int ^{\infty }_{0}\left ({z^{T}(t)z(t) -\gamma ^{2} \omega ^{T}(t)\omega (t)}\right)dt \\& \displaystyle {+\, \int ^{\infty }_{0}\left ({{\delta _{i}}y^{T}(t_{k}h)\Phi _{i} y(t_{k}h)- e^{T}(i_{k}h)\Phi _{i} e(i_{k}h)}\right)dt\Bigg \} < 0.} \\\tag{30}\end{align*}$$

Based on ${\delta _{i}}y^{T}(t_{k}h)\Phi _{i} y(t_{k}h)- e^{T}(i_{k}h)\Phi _{i} e(i_{k}h)>0$ , the initial condition $x(0) = 0$ , $\mathbb {E}\{V(\infty)\} \geq 0$ and (30), we have $$\begin{align*}&\hspace {-1.2pc}\mathbb {E}\left \{{\int ^{\infty }_{0}z^{T}(t)z(t)dt}\right \}-\gamma ^{2}\mathbb {E}\left \{{\int ^{\infty }_{0}\omega ^{T}(t)\omega (t)dt}\right \} \\\leq&\mathbb {E}\Bigg \{ V(0)-V(\infty) \\&-\,\int ^{\infty }_{0}\left ({{\delta _{i}}y^{T}(t_{k}h)\Phi _{i} y(t_{k}h)- e^{T}(i_{k}h)\Phi _{i} e(i_{k}h)}\right)dt\Bigg \} < 0, \\\tag{31}\end{align*}$$ which guarantees the required $H_{\infty }$ index $\gamma$ .

Remark 2:

To cope with the time-varying communication delay, a combination method of Jensen inequality and Lemma 2 is utilized, which is less conservative than the existing result [40] based on Jensen inequality. Moreover, with the help of some improved inequalities, such as Wirtinger inequality [41] and Bessel-Legendre inequality [42], the design conservativeness can further be reduced. When the improved inequalities are used, more variable matrices will be introduced and the size of obtained conditions in Theorem 1 will get larger, which could lead to higher computation and complexity.

Theorem 2:

For given parameters $\tau _{m}$ , $b_{1}$ , $b_{2}$ , $\mu$ , $\delta _{i}$ , system (9) under the event-triggered scheme (4) is stochastically stable with a required $H_\infty$ index $\gamma$ if there exist matrices $P_{i}>0$ , $R>0$ , $S>0$ , $\Phi _{i}>0$ , and matrices $G_{i}$ , $F_{i}$ , $X$ , $Z_{i}$ , and $M_{i}$ such that (15) and $$\begin{equation*} \Xi _{i}+\Theta _{i} < 0,\tag{32}\end{equation*}$$ where $$\begin{align*} \Xi _{i} !=&\left [{\! {\begin{array}{cccccccccccccccccccc} \Xi _{i}^{11} &\, \Xi _{i}^{12} &\, \Xi _{i}^{13} &\, 0 &\, \Xi _{i}^{15} &\, \Xi _{i}^{16} &\, \Xi _{i}^{17} &\, 0 &\, \Xi _{i}^{19}\\ * &\, \Xi _{i}^{22} &\, \Xi _{i}^{23} &\, 0 &\Xi _{i}^{25} &\, \Xi _{i}^{26} &\, \Xi _{i}^{27} &\, C^{T}_{1i} &\, \Xi _{i}^{29} \\ * &\, * &\, \Xi _{i}^{33} &\, 0 &\Xi _{i}^{35} &\, 0 &\, 0 &\, 0 &\Xi _{i}^{39} \\ * &\, * &\, * &\, -S &\, 0 &\, 0 &\, 0 &\, 0 &\, 0 \\ * &\, * &\, * &\, * &\, \Xi _{i}^{55} &\, 0 &\, 0 &\, 0 &\, \Xi _{i}^{59}\\ * &\, * &\, * &\, * &\, * &\, 0 &\, 0 &\, 0 &\, 0\\ * &\, * &\, * &\, * &\, * &\, * &\, -\gamma ^{2}I &\, D^{T}_{i} &\, 0\\ * &\, * &\, * &\, * &\, * &\, * &\, * &\, -I &\, 0 \\ * &\, * &\, * &\, * &\, * &\, * &\, * &\, * &\, \Xi _{i}^{99} \end{array}} \!}\right],\\ \Theta _{i}=&-\Theta _{1}-\rho \Theta _{2i},~~\Theta _{1}= \hat {\mathscr {E}}_{1}^{T} \begin{bmatrix} R &\quad X\\ * &\quad R \end{bmatrix} \hat {\mathscr {E}}_{1}, \\ \Theta _{2i}=&\hat {\mathscr {E}}^{T}_{2i} \mathfrak {F}\hat {\mathscr {E}}_{2i},~~\hat {\mathscr {E}}_{1}=\left [{ {\begin{array}{cccccccccccccccccccc} \mathscr {E}_{1} &\quad 0 \end{array}} }\right],~\hat {\mathscr {E}}_{2i}=\left [{{\begin{array}{cccccccccccccccccccc}{\mathscr {E}}_{2i} &\quad 0 \end{array}} }\right],\\ \Xi _{i}^{22}=&S+He(F_{i}A_{i})+\sum ^{s}_{j=1}\pi _{ij}P_{j},\\ \Xi _{i}^{11}=&\tau ^{2}_{m}R-\mathrm {He}(G_{i}), \quad \Xi _{i}^{12}=P_{i}+G_{i}A_{i}-F_{i}^{T},\\ \Xi _{i}^{13}=&b_{1}\theta _{2}B_{1i}M_{i}C_{2i}, \quad \Xi _{i}^{15}=-b_{1}\theta _{2}B_{1i}M_{i},\\ \Xi _{i}^{16}=&b_{1}\theta _{1}B_{1i}M_{i}, \quad \Xi _{i}^{17}=G_{i}B_{2i},\\ \Xi _{i}^{19}=&G_{i}B_{1i}-b_{1}B_{1i}Z_{i}, \quad \Xi _{i}^{23}=b_{2}\theta _{2}B_{1i}M_{i}C_{2i},\\ \Xi _{i}^{25}=&-b_{2}\theta _{2}B_{1i}M_{i}, \quad \Xi _{i}^{26}=b_{2}\theta _{1}B_{1i}M_{i},\\ \Xi _{i}^{27}=&F_{i}B_{2i}, \quad \Xi _{i}^{29}=F_{i}B_{1i}-b_{2}B_{1i}Z_{i},\\ \Xi _{i}^{33}=&\delta _{i} C^{T}_{2i}\Phi _{i} C_{2i}, \quad \Xi _{i}^{35}=-\delta _{i} C^{T}_{2i}\Phi _{i},\\ \Xi _{i}^{39}=&-\mu (M_{i}C_{2i})^{T}, \quad \Xi _{i}^{55}=(\delta _{i}-1)\Phi _{i},\\ \Xi _{i}^{59}=&\mu M_{i}^{T}, \quad \Xi _{i}^{99}=-\mu He(Z_{i}).\\{}\end{align*}$$

Then, the controller gains are solved as $K_{i}=Z^{-1}_{i}M_{i}$ .

Proof:

The condition (14) in Theorem 1 is equivalent to $$\begin{equation*} {\mathscr {Y}_{i}^\perp }^{T}\left [{ {\begin{array}{cccccccccccccccccccc} \Pi _{i}+\Sigma _{i} &\quad 0 \\ * &\quad 0 \end{array}} }\right]\mathscr {Y}_{i}^\perp < 0\tag{33}\end{equation*}$$ where $$\begin{align*} \mathscr {Y}_{i}^\perp=&\left [{ {\begin{array}{cccccccccccccccccccc} \mathscr {Y}^\perp _{1i} \\ \mathscr {Y}^\perp _{2i} \end{array}} }\right],~~\mathscr {Y}^\perp _{1i}=diag\{\underbrace {I,\cdots,I}_{8}\},\\ \mathscr {Y}^\perp _{2i}=&\left [{ {\begin{array}{cccccccccccccccccccc} 0 &\quad 0 &\quad \theta _{2}K_{i}C_{2i} &\quad 0 &\quad -\theta _{2}K_{i} &\quad \theta _{1}K_{i} &\quad 0 &\quad 0 \end{array}} }\right].\end{align*}$$

By applying Lemma 1 to (33), it yields $$\begin{equation*} \mathscr {P}_{i}+He(\mathscr {X}_{i}\mathscr {Y}_{i}) < 0\tag{34}\end{equation*}$$ where $$\begin{align*} \mathscr {Y}_{i}=&\left [{ {\begin{array}{cccccccccccccccccccc} 0 &~0 &~\theta _{2}K_{i}C_{2i} &~ 0 &\,\,-\theta _{2}K_{i} &~\theta _{1}K_{i} &~0 &~0 &\,\,-I \end{array}} }\right],\\ \mathscr {X}_{i}=&\left [{ {\begin{array}{cccccccccccccccccccc} \mathscr {X}^{T}_{1i} &\mathscr {X}^{T}_{2i} &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad 0 &\quad \mu Z^{T}_{i} \end{array}} }\right]^{T},\\ \mathscr {X}_{1i}=&b_{1}B_{1i}Z_{i}-G_{i}B_{1i},~\mathscr {X}_{2i}=b_{2}B_{1i}Z_{i}-F_{i}B_{1i}.\end{align*}$$

Then, (34) can be ensured by (32).

Remark 3:

To derive the static output feedback controller, some existing methods such as equality constraint, iterative algorithms and rank assumptions are usually needed. According to the separation approach based on Lemma 1, the coupling among the Lyapunov matrix $P_{i}$ and system matrices are removed in Theorem 2, which is easy to be implemented without extra constraints or iterative computation.

B. The Co-Design Issue

As discussed in [21], the packet transmission rate ($\varrho$ ) defined as $\varrho =\frac {\mathscr {N}_{e}}{\mathscr {N}_{s}}$ ($\mathscr {N}_{e}$ means the number of transmitted data and $\mathscr {N}_{s}$ is the number of sampled data) can represent the communication cost, which is tightly related to the triggering threshold $\delta _{i}$ . Specifically, the smaller $\delta _{i}$ is, the larger $\varrho$ is and the more sampled signals are transmitted, which needs to consume more communication cost, such as network node energy, communication bandwidth. On the other hand, when $\delta _{i}$ increases, $\varrho$ is reduced and fewer control signals are transmitted to update controllers, which may cause large value of $H_{\infty }$ index $\gamma$ . Note that to show the communication cost, the value of $\varrho$ is computed after the system is operated, while $\delta _{i}$ is known in advance. Based on this fact, this paper defines ${\delta }_{i}$ as an index to evaluate the communication cost. In practical situations, the $H_{\infty }$ index $\gamma$ is expected to be as small as possible to minimize the impacts from external disturbance, while the index of communication cost ${\delta }_{i}$ is expected to be as small as possible to save the network resource.

It is hard to achieve a maximum ${\delta }_{i}$ and a minimum $\gamma$ simultaneously via solving Theorem 2. In terms of the inverse proportion relationship between communication cost $\delta _{i}$ and $H_{\infty }$ performance $\gamma$ , the following algorithm to perform the trade-off between the communication cost and $H_{\infty }$ performance is presented as:

Algorithm 1: Let $\delta _{i}$ and $\gamma$ mean the communication cost and $H_{\infty }$ performance of system (9), respectively. Then, the maximum $\delta _{i}$ and minimum $\gamma$ are derived simultaneously if the following optimization issue is feasible $$\begin{align*}&\hspace {-.5pc}\min \psi ~~s.t.~~\sum ^{s}_{i=1}\alpha _{i}(1-\delta _{i}) + \alpha _{0}\gamma ^{2}-\psi < 0, \\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\; \displaystyle {and\,\,\,\,(15),(32),} \tag{35}\end{align*}$$ where $\alpha _{i}$ and $\alpha _{0}$ are weighting coefficient such that $\sum ^{s}_{i=0}\alpha _{i}=1$ , $\psi >0$ is the co-performance of communication cost and $H_{\infty }$ performance to be minimized.

Remark 4:

If we set $\alpha _{i}=0$ ($i=1, {\dots },s$ ), (35) becomes the standard $H_{\infty }$ performance optimization problem, which implies that the optimization of communication cost is not considered in the controller design.

Algorithm 2: Define $\psi ^{*}$ as the minimized value of $\psi$ . For given $\tau _{m}$ , $\mu$ , $b_{1}$ , $b_{2}$ , $\alpha _{0}$ , $\alpha _{i}$ , and determine $\Phi _{i}$ , $K_{i}$ , $\delta _{i}$ ($i=1, {\dots },s$ ) and $\gamma$ such that $\psi$ is minimized.

Step 1. Set $j=0$ . Choose a sufficiently small initial $\delta _{i}(0)>0$ and a suitable $\sigma$ as the increment of $\delta _{i}(j)$ .

Step 2. For given scalars $\tau _{m}$ , $\mu$ , $b_{1}$ , $b_{2}$ , $\alpha$ , execute Algorithm 1. If it is feasible, obtain $\psi ^{*}=\psi (0)$ and go to Step 3, else reset $\tau _{m}$ , $\mu$ , $b_{1}$ , $b_{2}$ , $\alpha _{0}$ , $\alpha _{i}$ , and repeat Step 2, end.

Step 3.

For $i=1:s$

While $\delta _{i}(j) < 1$

(i) Execute Algorithm 1. If it is feasible, set $\psi *=\psi *$ for $\psi *\leq \psi (j)$ , and $\psi *=\psi (j)$ for $\psi *> \psi (j)$ , and update $\Phi _{i}(j)$ , $K_{i}(j)$ , $\delta _{i}(j)$ and $\gamma (j)$ , end, else go to Step 4, end.

(ii) Update $\delta _{i}(j+1)=\delta _{i}(j)+\sigma$ and set $j=j+1$ .

End

End

Step 4. Output $\psi ^{*}$ , $\delta _{i}(j)$ , $\gamma (j)$ , $K_{i}(j)$ , $\Phi _{i}(j)$ and exit.

Remark 5:

In the existing results [21], [22], a packet transmission rate is used to represent the communication cost, and a co-design iterative algorithm is proposed to minimize both the $H_{\infty }$ performance and the communication cost. However, in this method, the data packet transmission rate must be obtained after the simulation of system operations, which leads to much computation burden. In our proposed co-design algorithms, the triggering parameter $\delta _{i}$ is utilized to denote the communication cost. Compared to the method in [22], the extra step of the simulation of system operations is removed and more computation burden can be saved by our method.

Remark 6:

Based on Algorithm 1 and 2, the trade-off between the communication cost and $H_{\infty }$ performance can be made by choosing the weighting coefficients of $\alpha _{i}$ and $\alpha _{0}$ . If better $H_{\infty }$ performance is expected, a larger $\alpha _{0}$ can be chosen. And a larger $\alpha _{i}$ should be selected if less communication cost is required.

Remark 7:

If the weighting matrix $\Phi _{i}$ is given in advance, the trade-off between communication cost and $H_{\infty }$ performance can be made via Algorithm 1 directly without executing Algorithm 2, which can reduce the computation burden at the sacrifice of more design conservativeness.

SECTION IV. Numerical Example

Example 1:

To illustrate the system security against deception attacks, the system parameters and deception attacks are considered as below: $$\begin{align*} A_{1}=&\left [{ {\begin{array}{cccccccccccccccccccc} -0.5 &\quad 0.1 \\ 0.2 &\quad 0.1 \end{array}} }\right],~B_{11}=\left [{ {\begin{array}{cccccccccccccccccccc} 1 \\ 2 \end{array}} }\right],~B_{21}=\left [{ {\begin{array}{cccccccccccccccccccc} 0.3 \\ 0 \end{array}} }\right],\\ C_{11}=&\left [{ {\begin{array}{cccccccccccccccccccc} 1 &\quad 1 \end{array}} }\right],~C_{21}=\left [{ {\begin{array}{cccccccccccccccccccc} 1 &\quad 2 \end{array}} }\right],~D_{1}=0.1,\\ A_{2}=&\left [{ {\begin{array}{cccccccccccccccccccc} -1 &\quad 0.2 \\ 0.1 &\quad -0.2 \end{array}} }\right],~B_{12}=\left [{ {\begin{array}{cccccccccccccccccccc} 1 \\ 2 \end{array}} }\right],~B_{22}=\left [{ {\begin{array}{cccccccccccccccccccc} 0 \\ 0.3 \end{array}} }\right],\\ C_{12}=&\left [{ {\begin{array}{cccccccccccccccccccc} 1 &\quad 1 \end{array}} }\right],~C_{22}=\left [{ {\begin{array}{cccccccccccccccccccc} 0.5 &\quad 2 \end{array}} }\right],~D_{2}=0.2.\end{align*}$$ The triggered signals are damaged by the deception attacks $f(\cdot)=tanh(\cdot)$ , which satisfies Assumption 1 with $L_{1}=0$ and $L_{2}=0.4$ . The transition rate matrix is chosen as $\Lambda =\begin{bmatrix} -8 & 8 \\ 5 & -5 \end{bmatrix}$ .

Let $\tau _{m}=0.3$ , $h=0.01$ , $\delta _{1}=\delta _{2}=0.1$ , $\mu =b_{1}=b_{2}=1$ , $\rho =10$ , $\theta _{1}=0.4$ . By solving the conditions in Theorem 2, the optimal $H_{\infty }$ performance index is derived as $\gamma =0.9584$ , the controller gains and triggering matrices are obtained as $$\begin{align*} K_{1}=&-0.2124,~\Phi _{1}=1.7917,\\ K_{2}=&-0.2613,~\Phi _{2}=3.1129.\end{align*}$$

In the simulation, the zero initial condition $x(0)=0$ and the exogenous disturbance $\omega (t)=sin(\pi t)$ for $0 < t < 4s$ (otherwise, $\omega (t)=0$ ) are taken into account. Then, the curves of state responses with the controller designed without considering deception attacks and the controller designed by Theorem 2 are depicted in FIGURE 1 and FIGURE 2, respectively. The numbers of triggering events in FIGURE 1 and FIGURE 2 are 40 and 56, respectively. In terms of FIGURE 1, one observes that the system stability is damaged by the deception attacks. However, when the deception attacks are considered in controller design in this paper, FIGURE 2 shows that the state responses of the system can be stabilized even deception attacks happen.

FIGURE 1.

The curves of the state $x(t)$ and release instants with the controller without considering deception attacks.

FIGURE 2.

The curves of the state $x(t)$ and release instants with the controller obtained by Theorem 2.

When the initial condition is selected as $x(0)=[{1;-0.5}]$ and the disturbance is taken as $\omega (t)=0$ , the corresponding curves of the state $x(t)$ and triggering instants are drawn in FIGURE 3. The number of triggering events in FIGURE 3 is 34. From FIGURE 3, it is shown that the state of the system with $x(0)=[{1;-0.5}]$ can be ensured to be stochastically stable under the deception attacks.

FIGURE 3.

The curves of the state $x(t)$ and release instants with $\omega (t)\equiv 0$ and $x(0)=[{1;-0.5}]$ .

In addition, by choosing $h=0.01$ , $\mu =b_{1}=b_{2}=1$ , $\rho =10$ , $\theta _{1}=0.4$ and the same $H_{\infty }$ performance index $\gamma =1$ , the maximum allowable upper bound of communication delay $\bar {\tau }$ obtained by Theorem 2 and the method based on Jensen inequality in [40] are obtained in TABLE 1.

TABLE 1 $\bar{\tau}$ Under the Same $\gamma=1$

From this table, one observes that the maximum allowable upper bound of communication delay $\bar {\tau }$ obtained by our method is larger than the value obtained by method in [40]. Moreover, $\bar {\tau }$ will decrease as the increase of $\delta _{1}$ and $\delta _{2}$ .

Example 2:

A signal-link robot arm system [43] is considered to illustrate the effectiveness of the proposed approach. The dynamics of the robot arm is governed by $$\begin{equation*} \ddot {\nu }(t)=-\frac {Mg\mathcal {L}}{\mathcal {J}}sin(\nu (t))-\frac {\mathcal {U}}{\mathcal {J}}\dot {\nu }(t) +\frac {u(t)}{\mathcal {J}}+\frac {\mathcal {L}}{\mathcal {J}}\omega (t),\tag{36}\end{equation*}$$ where ${\nu }(t)$ , $M$ , $g$ , $\mathcal {L}$ , $\mathcal {J}$ and $\mathcal {U}$ denote the angle position of the robot arm, mass of the payload, acceleration gravity, arm length, inertia moment and uncertain coefficient of viscous friction, respectively. The parameters $g=9.8$ , $\mathcal {L}=0.6$ and $\mathcal {U}=2$ are considered. A linearized system model with two different modes of the parameters $M$ and $\mathcal {J}$ is obtained as: $$\begin{align*} \dot {x}(t)=\begin{bmatrix} 0 &\quad I \\ -g\mathcal {L} &\quad -\dfrac {\mathcal {U}}{\mathcal {J}_{i}} \end{bmatrix}x(t) +\begin{bmatrix} 0 \\ \dfrac {1}{\mathcal {J}_{i}} \end{bmatrix} u(t) +\begin{bmatrix} 0 \\ \dfrac {\mathcal {L}}{\mathcal {J}_{i}} \end{bmatrix} \omega (t), \\\tag{37}\end{align*}$$ where $x(t)=\begin{bmatrix} {\nu }(t) & \dot {\nu }(t) \end{bmatrix}^{T}$ , ${\mathcal {J}_{i}}$ based on the mode $i$ are chosen as ${\mathcal {J}_{1}}=0.25$ and ${\mathcal {J}_{2}}=0.5$ .

The performance output matrices, system output matrices and the transition rate matrix are given as: $$\begin{align*} C_{11}=&C_{12}=\begin{bmatrix} 1 &\quad 1 \end{bmatrix},~~C_{21}=C_{22}=\begin{bmatrix} 0.1 &\quad 0.5 \end{bmatrix},\\ D_{1}=&D_{2}=0.2,~~\Lambda =\begin{bmatrix} -8 &\quad 8 \\ 5 &\quad -5 \end{bmatrix}.\end{align*}$$

Firstly, let $\tau _{m}=0.2$ , $\mu =100$ , $b_{1}=b_{2}=1$ , $\rho =10$ , $\theta _{1}=0.4$ . The same exogenous deception attacks in Example 1 is considered. For different triggering thresholds $\delta _{1}=\delta _{2}$ , Theorem 2 is solved to obtain the optimal $H_{\infty }$ index $\gamma$ , which is given in TABLE 2. This table verifies the fact that the larger triggering threshold is, the larger $H_{\infty }$ index $\gamma$ is, which implies less communication cost is consumed and worse disturbance attenuation level is derived.

TABLE 2 The $H_{\infty}$ Performance Index $\gamma$

Secondly, Algorithm 2 is implemented to show the trade-off among the communication cost $\delta _{i}$ and the $H_{\infty }$ performance $\gamma$ . By choosing different weights of $\delta _{i}$ and $\gamma$ , the optimal co-performance index $\psi ^{*}$ is achieved in TABLE 3 by Algorithm 2 with initial condition $\psi ^{*}(0)=1$ and increase step $\sigma =0.001$ . From this table, once the weights of $\delta _{i}$ and $\gamma$ are given, the optimal co-performance with the largest triggering thresholds and the smallest $H_{\infty }$ values can be derived. Moreover, different weights could lead to different optimal co-performance.

TABLE 3 The Co-Performance Index $\psi^{*}$

Thirdly, the controller gains, triggering matrices and triggering thresholds under two cases are derived as:

Case 1 ($\alpha _{1}=\alpha _{2}=0.1$ ): $$\begin{align*} K_{1}=&-0.1291,~\Phi _{1}=1.4689,\\ K_{2}=&-0.5401,~\Phi _{2}=1.5063;\end{align*}$$

Case 2 ($\alpha _{1}=\alpha _{2}=0.3$ ): $$\begin{align*} K_{1}=&-0.1734,~\Phi _{1}=2.1506,\\ K_{2}=&-0.1394,~\Phi _{2}=2.1407.\end{align*}$$

The simulation results under the zero initial state and the disturbance $\omega (t)=e^{-t}$ for $0 < t < 5s$ (otherwise, $\omega (t)=0$ ) are provided in FIGURE 4–​FIGURE 7. The numbers of the transmitted control signals under Case 1 and Case 2 are 113 and 69, respectively. These figures demonstrate that the larger weights ($\alpha _{1}$ and $\alpha _{2}$ ) of the communication cost are chosen, the larger triggering thresholds are obtained and the fewer control signals are transmitted. Therefore, the trade-off between the communication cost and the system performance should be made by choosing different weights $\alpha _{1}$ and $\alpha _{2}$ in Algorithm 1. To be specific, if the system designers want to obtain better system performance, they need to decrease $\alpha _{1}$ and $\alpha _{2}$ . If the goal is to save more communication cost, they need to choose larger $\alpha _{1}$ and $\alpha _{2}$ .

FIGURE 4.

The curves of system state $x(t)$ under Case 1.

FIGURE 5.

The release time intervals under Case 1.

FIGURE 6.

The curves of system state $x(t)$ under Case 2.

FIGURE 7.

The release time intervals under Case 2.

SECTION V. Conclusion

This paper has studied the event-triggered $H_{\infty }$ static output control of MJSs with deception attacks. A novel integral-based ETS with weighting function of the state of system has been proposed to implement the triggering condition. To model the random deception attacks, a Bernoulli variable is introduced. By using an event-triggered scheme, the amount of unnecessary data communication can be avoided. In terms of a constructive separation approach, sufficient conditions for the design of a static output controller that guarantees the stochastic stability with an $H_{\infty }$ index have been presented via LMIs. Lastly, a numerical example and an application to single-link robot arm have been simulated to illustrate the advantage of the developed approach. In the future work, the proposed event-triggered static output control method will be extended to the reliable control problem of fuzzy MJSs [44] and resilient control problem of nonhomogeneous MJSs [45].