Mixed H ∞ and passive resilient control for discrete-time systems with Round-Robin protocol and deception attacks

Abstract

This paper focuses on the mixed ${\mathcal{H}}_{\infty }$ and passive resilient control problem for uncertain discrete-time networked control systems subject to time delays. The measurement output is considered to be quantized by the dynamic quantizer before transmission through an unreliable communication channel, where the deception attacks may occur randomly. To further reduce the communication burden, the Round-Robin (RR) protocol is adopted to schedule sensors in a predetermined transmission order. Meanwhile, a new performance index is developed to solve restriction of the same dimension of control output and the exogenous disturbance. The S-procedure combined with a two-step uncertainties elimination method is adopted to deal with the nonlinear product terms involving multiple uncertainties. Furthermore, a decoupling method is introduced for static output feedback controller design. Sufficient conditions of the resilient controller and the dynamic quantizer are provided, which can improve the robustness of resulting closed-loop system and guarantee its stochastic stability and the specified mixed ${\mathcal{H}}_{\infty }$ and passivity performance index. Finally, an example is given to illustrate the feasibility of the theoretical results.

Introduction

Networked control systems (NCSs) have become a hot research field, which have attracted increasing attention because of their applications in manufacturing plants, remote processes, and mobile sensor networks (Qiu et al., 2013; Tong et al., 2022; Zhang et al., 2020a). Although they have many practical advantages, there might be network-induced phenomena in the networked systems, such as network-induced time delay, quantization, and packet loss. These phenomena can have an impact on the system performance. Therefore, it is practical to study the network-induced phenomena in the NCSs (Bahreini et al., 2021; Wang et al., 2020; Zou et al., 2016). These network-induced phenomena are mainly caused by the process of data transmission in the network.

In general, the limited network bandwidth has a great influence on the transmission of data. Then, some methods have been proposed to overcome the challenge. On the one hand, the signal must be quantized before transmission through the network channels. Up to now, the research on quantization has obtained abundant research results (Chen and Chang, 2022; Wang and Yu, 2018; Wu et al., 2021). In general, the quantization strategies can be divided into static quantization and dynamic quantization. As for the static quantization, the common method is a sector boundary, which regards the quantization error as norm-bounded uncertainty (Fu and Xie, 2005; Gao et al., 2008). However, static quantization is required to have the limited quantization level. Compared with the static quantization, dynamic quantization is time-varying (Brockett and Liberzon, 2000). The quantization range of dynamic quantizer can be adjusted by the quantization parameter, which can scale the quantization level. Therefore, dynamic quantization is more general and has attracted the attention of a large number of scholars. On the other hand, network communication protocol is an effective method to solve the problems of data conflicts and channel congestion caused by the limited bandwidth. During the transmission of data through network channel, special transmission rules can be designed. According to the special rules, each node in the network can be assigned transmission rights at specific instants. As the result of which, the communication burden of the network can be reduced. Thus, it is of practical significance to study the communication protocols, and there have been many research results on communication protocols. At present, RR protocol (Shen et al., 2020; Zhao et al., 2020), weighted try-once-discard protocol (Zhang et al., 2020b; Zou et al., 2016), and stochastic communication protocol (Ren et al., 2021) are the most studied communication protocols. Among them, the RR protocol is a static protocol, which evenly distributes the network usage rights to all network nodes through a preset token ring. Because of its unique characteristics, the RR protocol naturally receives attention in load balancing and security monitoring (Wang et al., 2018; Zhao et al., 2020).

During the transmission of data through network channels, not only the problem of the limited bandwidth needs attention but also the cybersecurity deserves attention. When data are transmitted through the network channels, it may be subject to malicious attacks (Liu et al., 2018; Xiong et al., 2020; Yu and Chang, 2022). Due to the attacks, the receiver may not receive data or receive an incorrect data. Generally speaking, the common attacks include denial-of-service attacks and deception attacks, among which deception attacks are one of the most dangerous network attacks. The means of deception attacks is to inject pseudo data into normal data during transmission, with the purpose of destroying normal transmission to obtain confidential information or forge data (Du et al., 2019; Yuan et al., 2020). Recently, some important works about security control against network attacks had been launched. Wang et al. (2020) studied the problem of resilient memory event-triggered control for a new NCSs model under random deception attacks. Zhao et al. (2022) proposed a dynamic detection method based on rotor speed observer/estimator to identify false data injection attacks. After the location attacks, the adaptive resilient control strategy of generator torque is realized to ensure the control target of optimal power tracking. In the work by Li and Zhao (2021), the problem of resilient adaptive dynamic surface control was researched for more general nonlinear network physical systems, and an attack compensator was constructed to mitigate the impact of sensor and actuator attacks.

Not only can the network-induced phenomena have an influence on the performance and stability of NCSs but also the uncertainty and external disturbance may affect them (Xu et al., 2021). Due to the sensor aging and other factors, there always are uncertainties in networked systems. The robust control method can be used to eliminate the impact of uncertainties on system performance, including ${\mathcal{H}}_{\infty }$ control, ${\mathcal{L}}_2-{\mathcal{L}}_{\infty }$ control, and induced ${\mathcal{L}}_{\infty }$ control. One of the differences between these control strategies is that ${\mathcal{H}}_{\infty }$ control has requirements for the energy of external disturbance, which requires energy-bounded external disturbances. In addition, passive control (Fu and Ma, 2016; Wu et al., 2012) is also the common control strategy, which requires that the energy supplied from the outside is greater than the energy consumed internally. Furthermore, a control strategy known as mixed ${\mathcal{H}}_{\infty }$ and passive control has been exploited to minimize the degree of disturbances in the system. Li et al. (2018) solved the problem that alarm-based hybrid controller was designed for a delayed Markovian jump system with mixed ${\mathcal{H}}_{\infty }$ and passivity performance. Zheng et al. (2018) designed the passivity-based ${\mathcal{H}}_{\infty }$ controller for nonlinear system approximated by means of the Takagi–Sugeno fuzzy model. As far as the authors know, under the joint influence of quantization, deception attacks, and the RR protocol, the problem of mixed ${\mathcal{H}}_{\infty }$ and passive resilient control for uncertain discrete time-delay systems has not been solved in the unified framework.

Motivated by the observations above, the aim of this paper is devoted to solve the mixed ${\mathcal{H}}_{\infty }$ and passive control problem for uncertain time-delay systems with dynamic quantization and the RR protocol subject to deception attacks. In view of the actual environmental factors, networks inevitably suffer from noise and device limitations, so the system parameter uncertainty and the controller gain perturbation are also included in the scope of consideration. The main highlights are outlined below:

1. A unified framework of the mixed ${\mathcal{H}}_{\infty }$ and passive resilient control problem for uncertain discrete time-delay systems is established, which involves dynamic quantization, RR protocol, and deception attacks.

2. A new performance analysis index is proposed to handle the same-dimensional constraint of control output and external disturbance.

3. Sufficient conditions for the design of the mode-dependent resilient controller and the dynamic quantizer are presented to ensure the stochastic stability and specified mixed ${\mathcal{H}}_{\infty }$ and passivity performance index of the resulting closed-loop system.

The paper is organized as follows. The system description, communication network with quantization, deception attacks, and the RR protocol, and the closed-loop system are provided in section “Problem formulation.” Section “Main results” gives the main results. Section “Simulation results” presents a simulation result to show the feasibility of the proposed method. The conclusion is drawn in section “Conclusion.”

Notations: Throughout this paper, standard notations are used here. ${\mathbb{R}}^n$ means the n-dimensional Euclidean space. $\mathbb{E}\{o\}$ denotes the expected value of the stochastic variable o. $\mathrm{Prob}\{·\}$ stands for the probability of the event “·.” ${\tau }_{\min }(\mathbb{T})$ is the smallest eigenvalue of matrix $\mathbb{T}$ . $l_2[0,\infty )$ is the space of square integrable vectors. For simplicity, using the notation $\mathrm{He}(L)=L+L^T$ and $\mathrm{Ha}(L)=L^{T}L$ , respectively. In symmetric block matrices, * is used as an ellipsis for terms induced by symmetry.

Problem formulation

System description

Consider a class of uncertain discrete time-delay systems described by the equation of state

$$\begin{array}{c}\multicolumn{1}{c}{x(k+1)=(A+\mathrm{\Delta }A)x(k)+(A_0+\mathrm{\Delta }{A}_0)x(k-d)}\\ \multicolumn{1}{c}{+\mathrm{Bu}(k)+\mathrm{Fw}(k)}\\ \multicolumn{1}{c}{z(k)=\mathrm{Dx}(k)+\mathrm{Hw}(k)}\\ \multicolumn{1}{c}{y(k)=\mathrm{Cx}(k)}\\ \multicolumn{1}{c}{x(k)=x_0,k\le 0}\end{array}$$

(1)

where $x(k)\in {\mathbb{R}}^n$ is the state, $y(k)\in {\mathbb{R}}^m$ is the measurement output, $u(k)\in {\mathbb{R}}^l$ is the control input, $z(k)\in {\mathbb{R}}^p$ is control output, $w(k)\in {\mathbb{R}}^f$ is the exogenous disturbance signal belonging to $l_2[0,\infty )$ , A, $A_0$ , B, F, C, D, and H are known real matrices with suitable dimensions. The positive integer d represents a constant time delay and $x_0$ denotes an initial condition. $\mathrm{\Delta }A$ and $\mathrm{\Delta }{A}_0$ are the uncertain parameters of the system. In this paper, the uncertainty matrices $\mathrm{\Delta }A$ and $\mathrm{\Delta }{A}_0$ can be described in the following form

$$\left[\mathrm{\Delta }A\mathrm{\Delta }{A}_0\right]=\mathcal{M}{\mathcal{F}}_1(k)\left[{\mathcal{N}}_1{\mathcal{N}}_2\right]$$

(2)

where $\mathcal{M}$ , ${\mathcal{N}}_1$ , and ${\mathcal{N}}_2$ are known constant matrices with appropriate dimensions and ${\mathcal{F}}_1(k)$ satisfied ${\mathcal{F}}_1^T(k){\mathcal{F}}_1(k)\le I$ .

Communication network

In general, for network communication channels with limited bandwidth, it is inevitable that the signal is usually quantized before transmission, and the RR protocol is considered to avoid data conflict in the process of information transmission. With the use of commercial operating systems, the possibility of network systems being attacked increases greatly. Most deception attacks are directed to fraudulently inject signals from the transmitter to the processor. When it happens, the network system becomes weak, which leads to the decline of system performance. The system architecture is given in Figure 1 to incorporate these dynamic properties. In the coming part, the detailed description of the measurement output affected by dynamic quantization, random deception attacks, and RR protocol will be presented.

Figure 1.

Structure of NCSs.

First, the dynamic quantizer with general form in Brockett and Liberzon (2000) is adopted for quantization of the measurement output, whose property is introduced as follows

$$\begin{array}{c}\parallel q(\vartheta )-\vartheta \parallel \le {\mathrm{\Delta }}_{\vartheta },\mathrm{if}\parallel \vartheta \parallel \le M_{\vartheta }\hfill \\ \multicolumn{1}{c}{\parallel q(\vartheta )-\vartheta \parallel >{\mathrm{\Delta }}_{\vartheta },\mathrm{if}\parallel \vartheta \parallel >M_{\vartheta }}\end{array}$$

(3)

where ${\mathrm{\Delta }}_{\vartheta }$ is used for denoting the quantization error bound, and $M_{\vartheta }$ is used to stand for quantization range. Therefore, the following one-parameter family of quantizer is considered to quantize the measurement output

$$\overline{y}(k)=q_{\mu (k)}(y(k))=\mu (k)q\left(\frac{y(k)}{\mu (k)}\right),\mu (k)>0$$

(4)

where $\overline{y}(k)$ denotes the quantized measurement output, and $\mu (k)$ is the dynamic parameter of $q_{\mu (k)}(·)$ . Let $\nu (k)=\mu (k)(q(y(k)/\mu (k))-(y(k)/\mu (k))$ be the quantization error. Through equations (3) and (4), it is easy to be seen $\parallel \nu (k)\parallel \le \mu (k){\mathrm{\Delta }}_y$ only if $\parallel y(k)\parallel \le \mu (k)M_y$ holds.

In this paper, it is assumed that there are $\mathrm{\Pi }$ sensor nodes exchange information with each other through a shared network, then the measurement output can be described as $y(k)\stackrel{\mathrm{\Delta }}{=}[y_1^T(k)y_2^T(k)\dots y_{\mathrm{\Pi }}^T(k){]}^T\in {\mathbb{R}}^m$ , where $y_i(k)=C_{i}x(k)\in {ℝ}^{m_i}$ $(i=1,\dots ,\mathrm{\Pi })$ with $m=\sum _{i=1}^{\mathrm{\Pi }}{m}_i$ . Therefore, the quantized measurement output before transmission is written as $\overline{y}(k)\stackrel{\mathrm{\Delta }}{=}[{\overline{y}}_1^T(k){\overline{y}}_2^T(k)\dots {\overline{y}}_{\mathrm{\Pi }}^T(k){]}^T$ . Under the RR protocol, let $\xi (k)\in \{1,2,\dots ,\mathrm{\Pi }\}$ be the selected node at time k, which satisfies the regularity $\xi (k)=\xi (k+\mathrm{\Pi })$ . That is to say, only the node $\xi (k)$ is activated and licensed to get the permission of using the network at every transmission instant. Affected by the RR protocol, the value of $\xi (k)$ is determined by the following principle

$$\xi (k)=\mathrm{mod}(k-1,\mathrm{\Pi })+1$$

(5)

where the function “ $\mathrm{mod}$ ” refers to the remainder of a division between two numeric expressions.

In addition, deception attacks may occur in the transmission process of measurement outputs because of the openness of the network. That is to say, the transmitted quantized signal $\overline{y}(k)$ via different nodes may be intentionally forged by injecting some pseudo data. This paper assumes that deception attacks occur randomly, whose behavior can be modeled in the following way

$${\overrightarrow{y}}_i(k)={\overline{y}}_i(k)+\iota (k){\zeta }_i(k)$$

(6)

where ${\zeta }_i(k)\stackrel{\mathrm{\Delta }}{=}-{\overline{y}}_i(k)+f_i(y(k))$ , it indicates the signal is affected by attacks with the nonzero $f_i(y(k))$ , the stochastic variable $\iota (k)$ satisfies a Bernoulli-distributed white sequence by taking values in {0,1} and follows the probability

$$\mathrm{Prob}\{\iota (k)=1\}=\overline{\iota },\mathrm{Prob}\{\iota (k)=0\}=1-\overline{\iota }$$

(7)

and has the following some statistical properties

$$\mathbb{E}\{\iota (k)\}=\overline{\iota },\mathbb{E}\left\{{(\iota (k)-\overline{\iota })}^2\right\}=\overline{\iota }(1-\overline{\iota })\stackrel{\mathrm{\Delta }}{=}{\hat{\iota }}^2$$

(8)

where $\overline{\iota }$ indicates the expected value of $\iota (k)$ , and ${\hat{\iota }}^2$ is used to denote the mathematical variance of $\iota (k)$ .

Analogously, the combination of received measurement output ${\tilde{y}}_i(k)$ comes into being $\tilde{y}(k)\stackrel{\mathrm{\Delta }}{=}[{\tilde{y}}_1^T(k){\tilde{y}}_2^T(k)\dots {\tilde{y}}_{\mathrm{\Pi }}^T(k){]}^T$ . Under the RR protocol, the update rule of ${\tilde{y}}_i(k)$ is as follows

$${\tilde{y}}_i(k)=\{\begin{array}{c}{\overrightarrow{y}}_i(k),i=\xi (k)\hfill \\ \multicolumn{1}{c}{{\tilde{y}}_i(k-1),\mathrm{otherwise}}\end{array}$$

(9)

Then, defining the update matrix ${\mathrm{\Phi }}_{\xi (k)}=\mathrm{diag}\{\varpi (\xi (k)-1)I,\varpi (\xi (k)-2)I,\cdots ,\varpi (\xi (k)-\mathrm{\Pi })I\}$ with $\xi (k)\in \{1,2,\dots ,\mathrm{\Pi }\}$ , where $\varpi (·)\in \{0,1\}$ is the Kronecker delta function. Without loss of generality, set $\tilde{y}(-1)=0$ . According to the above update rule (9) and in combination with formulas (4) and (6), $\tilde{y}(k)$ can be concluded as

$$\begin{array}{l}\tilde{y}(k)={\mathrm{\Phi }}_{\xi (k)}\overrightarrow{y}(k)+\left(I-{\mathrm{\Phi }}_{\xi (k)}\right)\tilde{y}\left(k-1\right)\hfill \\ =\left(\overrightarrow{\iota }-\tilde{\iota }(k)\right){\mathrm{\Phi }}_{\xi (k)}Cx(k)+{\overline{\mathrm{\Phi }}}_{\xi (k)}\tilde{y}\left(k-1\right)\hfill \\ +\left(\overrightarrow{\iota }-\tilde{\iota }(k)\right){\mathrm{\Phi }}_{\xi (k)}\nu (k)+\left(\overline{\iota }+\tilde{\iota }(k)\right){\mathrm{\Phi }}_{\xi (k)}f\left(y(k)\right)\hfill \end{array}$$

(10)

where $\tilde{\iota }(k)=\iota (k)-\overline{\iota }$ , $\overrightarrow{\iota }=1-\overline{\iota }$ , ${\overline{\mathrm{\Phi }}}_{\xi (k)}=I-{\mathrm{\Phi }}_{\xi (k)}$ .

Remark 1. Actually, the signal $f(y(k))$ is introduced into the network to represent the deception attacks signal, the Bernoulli variable $\iota (k)$ is used to indicate whether a deception attack occurs. As such, when $\iota (k)=0$ , then no deception attacks occur, then the transmitted data are $\overline{y}(k)$ ; when $\iota (k)=1$ , it means that deception attacks occur, the actual measurement output $\overline{y}(k)$ will be modified by $f(y(k))$ , then the deception attacks can manipulate the system arbitrarily.

Assumption 1. Liu et al. (2018) $f(y(k))$ represents the injected fake signal, which assumes that the following constraint is satisfied

$$\parallel f(y(k)){\parallel }_2\le \parallel \mathcal{R}y(k){\parallel }_2$$

(11)

where $\mathcal{R}$ is a known constant matrix.

Closed-loop system

The control strategy will be developed in this part, where a mode-dependent resilient controller with gain fluctuation is considered to stabilize system (1) and improve its robustness, which is shown as

$$u(k)=(K_i+\mathrm{\Delta }{K}_i)\tilde{y}(k)$$

(12)

where $K_i$ is the controller gain matrix to be determined and $\mathrm{\Delta }{K}_i={\mathcal{M}}_{1i}{\mathcal{F}}_2(k){\mathcal{N}}_{3i}$ represents the fluctuation of the gain matrix, where ${\mathcal{M}}_{1i}$ and ${\mathcal{N}}_{3i}$ are known matrices of appropriate dimensions and ${\mathcal{F}}_2(k)$ is the uncertain parameter satisfying ${\mathcal{F}}_2^T(k){\mathcal{F}}_2(k)\le I$ . By defining the augmented matrix $\lambda (k)=[x^T(k){\tilde{y}}^T(k-1){]}^T$ and combining the formulas (1, 4, 6, 10, 12), the resulting closed-loop system is obtained as follows

$$\begin{array}{l}\lambda \left(k+1\right)=\left({\overrightarrow{\mathcal{A}}}_{1i}-\tilde{\iota }(k){ℬ}_{1i}\right)\lambda (k)+{\mathcal{A}}_{2i}\lambda \left(k-d\right)+\left({ℬ}_{2i}-\tilde{\iota }(k){ℬ}_{3i}\right)\nu (k)\hfill \\ +\left({ℬ}_{4i}-\tilde{\iota }(k){ℬ}_{5i}\right)f\left(y(k)\right)+ℱw(k)\hfill \\ z(k)=\mathcal{D}\lambda (k)+Hw(k)\hfill \\ y(k)=\mathcal{C}\lambda (k)\hfill \end{array}$$

(13)

where

$$\begin{array}{c}\multicolumn{1}{c}{\begin{array}{c}{\mathcal{A}}_{1i}=\left[\begin{array}{cc}{\overline{A}}_i\hfill & B{\mathcal{K}}_i{\overline{\mathrm{\Phi }}}_i\hfill \\ \multicolumn{1}{c}{\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C}& {\overline{\mathrm{\Phi }}}_i\hfill \end{array}\right],{\mathcal{A}}_0=\left[\begin{array}{cc}{\overline{A}}_0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],{\mathcal{B}}_{1i}=\left[\begin{array}{cc}B{\mathcal{K}}_i{\mathrm{\Phi }}_{i}C\hfill & 0\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{i}C}& 0\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\mathcal{B}}_{2i}=\left[\begin{array}{c}\overrightarrow{\iota }B{\mathcal{K}}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{\overrightarrow{\iota }{\mathrm{\Phi }}_i}\end{array}\right],{\mathcal{B}}_{3i}=\left[\begin{array}{c}B{\mathcal{K}}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_i}\end{array}\right],{\mathcal{B}}_{4i}=\left[\begin{array}{c}\overline{\iota }B{\mathcal{K}}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{\overline{\iota }{\mathrm{\Phi }}_i}\end{array}\right],\mathcal{F}=\left[\begin{array}{c}F\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]}\\ \multicolumn{1}{c}{{\mathcal{B}}_{5i}=\left[\begin{array}{c}-B{\mathcal{K}}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{-{\mathrm{\Phi }}_i}\end{array}\right],{\overline{A}}_i=A+\mathrm{\Delta }A+\overrightarrow{\iota }B{\mathcal{K}}_i{\mathrm{\Phi }}_{i}C,{\overline{\mathrm{\Phi }}}_i=I-{\mathrm{\Phi }}_i}\\ \multicolumn{1}{c}{{\overline{A}}_0=A_0+\mathrm{\Delta }{A}_0,{\mathcal{K}}_i=K_i+\mathrm{\Delta }{K}_i,\mathcal{D}=[D0],\mathcal{C}=[C0]}\end{array}}\end{array}$$

To articulate the control aim of this paper, the following definitions should be provided first.

Definition 1. Sathishkumar et al. (2019) For given scalars $\gamma >0$ , system (13) is called stochastically stable for any initial state $\lambda (k)$ when $w(k)=0$ , the following condition holds

$$\mathbb{E}\{\sum _{k=0}^{\infty }\parallel \lambda (k){\parallel }^2|\lambda (0)\}<\infty$$

(14)

Definition 2. For given scalars $\gamma >0$ , $\alpha \in [0,1]$ and matrix X, system (13) is said to be stochastically stable with a mixed ${\mathcal{H}}_{\infty }$ and passivity performance index $\gamma$ under zero initial conditions when nonzero $w(k)\in l_2[0,\infty )$ , the following condition is met

$$\begin{array}{c}\mathbb{E}\{\sum _{k=0}^{\infty }{\gamma }^{-1}\alpha z^T(k)z(k)-2(1-\alpha )z^T(k)\mathrm{Xw}(k)\}\hfill \\ \multicolumn{1}{c}{\le \mathbb{E}\{\sum _{k=0}^{\infty }\gamma w^T(k)w(k)\}}\end{array}$$

(15)

Remark 2. Index (15) mentioned above covers two properties, the ${\mathcal{H}}_{\infty }$ norm constraint and the passivity performance, respectively, which can be stated as follows: (1) if $\alpha =0$ , then the condition in Definition 2 becomes the passivity case; (2) if $\alpha =1$ , then the index changes into the ${\mathcal{H}}_{\infty }$ attribute; and (3) if $\alpha \in (0,1)$ , then the index represents a mixed ${\mathcal{H}}_{\infty }$ and passivity index.

Then, the objective of this paper can be summarized to design a desired controller, so that resulting system (13) can guarantee the stochastic stability and a mixed ${\mathcal{H}}_{\infty }$ and passivity performance index under the unreliable communication network. Before proceeding further, some indispensable technical lemmas are introduced.

Lemma 1. Wang et al. (2007) Let ℜ be a given symmetric matrix, and the matrices ${\Re }_1,{\Re }_2$ with appropriate dimensions, then for all ${\Re }_3$ satisfying ${\Re }_3^T{\Re }_3\le I$ , the following inequality holds

$$\Re +{\Re }_1{\Re }_3{\Re }_2+{\Re }_2^T{\Re }_3^T{\Re }_1^T<0$$

(16)

the necessary and sufficient conditions for the above establishment if there exits a scalar $\varsigma >0$ , satisfying

$$\Re +\varsigma {\Re }_1{\Re }_1^T+{\varsigma }^{-1}{\Re }_2^T{\Re }_2<0$$

(17)

Lemma 2. Chang (2014) For matrices $\mathcal{T}$ , $\mathcal{W}$ , $\mathcal{Z}$ , and $\mathcal{V}$ with appropriate dimensions and scalar $\hslash$ . Inequality

$$\mathcal{T}+\mathcal{W}\mathcal{V}+{\mathcal{V}}^T{\mathcal{W}}^T<0$$

(18)

is fulfilled if the following condition holds

$$\left[\begin{array}{cc}\mathcal{T}\hfill & *\hfill \\ \multicolumn{1}{c}{\hslash {\mathcal{V}}^T+\mathcal{Z}\mathcal{W}}& -\hslash \mathcal{Z}-\hslash {\mathcal{Z}}^T\hfill \end{array}\right]<0$$

(19)

Main results

Mixed ${\mathcal{H}}_{\infty }$ and passivity performance analysis

In this section, some principal results are performed. First, a new performance analysis criteria are provided for closed-loop system (13) under the effect of system uncertainties, controller gain perturbation, time delay, dynamic quantization, deception attacks, and RR protocol. Then based on this important result, sufficient conditions are given for parameterized representations of the controller and quantizer.

Theorem 1. For the given parameters $\gamma >0$ , $\overline{\iota }$ , $M_y$ , ${\mathrm{\Delta }}_y$ , $\alpha \in [0,1]$ , and matrix X, resulting system (13) is guaranteed to be stochastically stable and preserve a mixed ${\mathcal{H}}_{\infty }$ and passivity performance index $\gamma$ , if the unknown variable $\rho$ is selected to be $\rho >M_y^{-1}$ , and there exist matrices $\mathcal{P}>0$ , $\mathcal{S}>0$ , positive constants ${ϵ}_0$ and ${ϵ}_1$ satisfying

$$\left[\begin{array}{ccc}{\overline{\mathrm{\Xi }}}_{11}\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Xi }}}_{21i}}& {\overline{\mathrm{\Xi }}}_{22}\hfill & *\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Xi }}}_{31}}& 0\hfill & {\overline{\mathrm{\Xi }}}_{33}\hfill \end{array}\right]<0$$

(20)

where

$$\begin{array}{c}\begin{array}{l}{\overline{\mathrm{\Xi }}}_{11}=\left[\begin{array}{cc}-\mathrm{diag}\left\{(\mathcal{P}-\mathcal{S}),\mathcal{S},{ϵ}_{0}I,{ϵ}_{1}I\right\}& *\\ \left[\begin{array}{cccc}-(1-\alpha )X^T\mathcal{D}& 0& 0& 0\end{array}\right]& -2(1-\alpha )He\left\{H^{T}X\right\}-\gamma I\end{array}\right]\hfill \\ {\overline{\mathrm{\Xi }}}_{21i}=\left[\begin{array}{ccccc}{\mathcal{A}}_{1i}& {\mathcal{A}}_0& {ℬ}_{2i}& {ℬ}_{4i}& ℱ\\ \widehat{\iota }{ℬ}_{1i}& 0& \widehat{\iota }{ℬ}_{3i}& \widehat{\iota }{ℬ}_{5i}& 0\end{array}\right],{\overline{\mathrm{\Xi }}}_{22}=-\mathrm{diag}\left\{{\mathcal{P}}^{-1},{\mathcal{P}}^{-1}\right\}\hfill \\ {\overline{\mathrm{\Xi }}}_{31}=\left[\begin{array}{ccccc}\sqrt{\alpha }\mathcal{D}& 0& 0& 0& \sqrt{\alpha }H\\ 2{ϵ}_0\rho {\mathrm{\Delta }}_y\mathcal{C}& 0& 0& 0& 0\\ {ϵ}_1ℛ\mathcal{C}& 0& 0& 0& 0\end{array}\right],{\overline{\mathrm{\Xi }}}_{33}=-\mathrm{diag}\left\{\gamma I,{ϵ}_{0}I,{ϵ}_{1}I\right\}\hfill \end{array}\end{array}$$

and dynamic quantizer’s parameters $\mu (k)$ meets the online adjustment strategy as

$$\rho \parallel y(k)\parallel \le \mu (k)\le 2\rho \parallel y(k)\parallel .$$

(21)

Proof. First of all, it will be confirmed that if the conditions in Theorem 1 can be guaranteed, system (13) is stochastic stable when $w(k)=0$ by Definition 1. Based on the definition of $\nu (k)$ and the homogeneity property of Euclidean norm, we obtain

$${\nu }^T(k)\nu (k)\le 4{\rho }^2{\mathrm{\Delta }}_y^2{y}^T(k)y(k)$$

(22)

Then, defining ${\overline{\phi }}^T(k)=[{\lambda }^T(k){\lambda }^T(k-d){\nu }^T(k)f^T(y(k))]$ can result in

$${\overline{\phi }}^T(k){\mathrm{\Theta }}_0\overline{\phi }(k)\ge 0$$

(23)

where ${\mathrm{\Theta }}_0=\mathrm{Ha}[2\rho {\mathrm{\Delta }}_y\mathcal{C}000]-\mathrm{diag}\{0,0,I,0\}$ .

On the other side, the restrictions given in Assumption 1 can be further described as

$${(\mathcal{R}y(k))}^T(\mathcal{R}y(k))-f^T(y(k))f(y(k))\ge 0$$

(24)

Similar to the above, we have

$${\overline{\phi }}^T(k){\mathrm{\Theta }}_1\overline{\phi }(k)\ge 0$$

(25)

where ${\mathrm{\Theta }}_1=\mathrm{Ha}[\mathcal{R}\mathcal{C}000]-\mathrm{diag}\{0,0,0,I\}$ .

Now, consider the Lyapunov function candidate as

$$V(\lambda (k))={\lambda }^T(k)\mathcal{P}\lambda (k)+\sum _{ı=k-d}^{k-1}{\lambda }^T(ı)\mathcal{S}\lambda (ı)$$

(26)

with $\mathcal{P}>0$ , $\mathcal{S}>0$ . From equation (7), it shows the fact that $\mathbb{E}\{{\tilde{\iota }}^2(k)\}=\overline{\iota }(1-\overline{\iota })={\hat{\iota }}^2$ , one gets

$$\begin{array}{c}\mathbb{E}\{\mathrm{\Delta }V(\lambda (k))\}\hfill \\ \multicolumn{1}{c}{=\mathbb{E}\{(({\mathcal{A}}_{1i}-\tilde{\iota }(k){\mathcal{B}}_{1i})\lambda (k)+{\mathcal{A}}_0\lambda (k-d)+({\mathcal{B}}_{2i}-\tilde{\iota }(k){\mathcal{B}}_{3i})\nu (k)}\\ \multicolumn{1}{c}{+({\mathcal{B}}_{4i}-\tilde{\iota }(k){\mathcal{B}}_{5i})f(y(k{)))}^T\mathcal{P}(({\mathcal{A}}_{1i}-\tilde{\iota }(k){\mathcal{B}}_{1i})\lambda (k)+{\mathcal{A}}_0\lambda (k-d)}\\ \multicolumn{1}{c}{+({\mathcal{B}}_{2i}-\tilde{\iota }(k){\mathcal{B}}_{3i})\nu (k)+({\mathcal{B}}_{4i}-\tilde{\iota }(k){\mathcal{B}}_{5i})f(y(k)))-{\lambda }^T(k)\mathcal{P}\lambda (k)}\\ \multicolumn{1}{c}{+{\lambda }^T(k)\mathcal{S}\lambda (k)-{\lambda }^T(k-d)\mathcal{S}\lambda (k-d)\}}\\ \multicolumn{1}{c}{={\overline{\phi }}^T(k)\mathrm{\Theta }\overline{\phi }(k)}\end{array}$$

(27)

with

$$\begin{array}{c}\begin{array}{c}\mathrm{\Theta }={\left[{\mathcal{A}}_{1i}{\mathcal{A}}_0{\mathcal{B}}_{2i}{\mathcal{B}}_{4i}\right]}^T\mathcal{P}\left[{\mathcal{A}}_{1i}{\mathcal{A}}_0{\mathcal{B}}_{2i}{\mathcal{B}}_{4i}\right]+{\hat{\iota }}^2{\left[{\mathcal{B}}_{1i}0{\mathcal{B}}_{3i}{\mathcal{B}}_{5i}\right]}^T\hfill \\ \multicolumn{1}{c}{\times \mathcal{P}\left[{\mathcal{B}}_{1i}0{\mathcal{B}}_{3i}{\mathcal{B}}_{5i}\right]-\mathrm{diag}\{(\mathcal{P}-\mathcal{S}),\mathcal{S},0,0\}}\end{array}\hfill \end{array}$$

It can be noted that the following inequality is derived from equation (20) with $w(k)=0$

$$\left[\begin{array}{ccc}{\hat{\mathrm{\Xi }}}_{11}\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\hat{\mathrm{\Xi }}}_{21i}}& {\overline{\mathrm{\Xi }}}_{22}\hfill & *\hfill \\ \multicolumn{1}{c}{{\hat{\mathrm{\Xi }}}_{31}}& 0\hfill & {\hat{\mathrm{\Xi }}}_{33}\hfill \end{array}\right]<0$$

(28)

where

$$\begin{array}{c}\begin{array}{l}{\widehat{\mathrm{\Xi }}}_{11}=-\mathrm{diag}\left\{(\mathcal{P}-\mathcal{S}),\mathcal{S},{ϵ}_{0}I,{ϵ}_{1}I\right\},{\widehat{\mathrm{\Xi }}}_{33}=-\mathrm{diag}\left\{{ϵ}_{0}I,{ϵ}_{1}I\right\}\hfill \\ {\widehat{\mathrm{\Xi }}}_{21i}=\left[\begin{array}{cccc}{\mathcal{A}}_{1i}& {\mathcal{A}}_0& {ℬ}_{2i}& {ℬ}_{4i}\\ \widehat{\iota }{ℬ}_{1i}& 0& \widehat{\iota }{ℬ}_{3i}& \widehat{\iota }{ℬ}_{5i}\end{array}\right],{\widehat{\mathrm{\Xi }}}_{31}=\left[\begin{array}{cccc}2{ϵ}_0\rho {\mathrm{\Delta }}_y\mathcal{C}& 0& 0& 0\\ {ϵ}_1ℛ\mathcal{C}& 0& 0& 0\end{array}\right]\hfill \end{array}\end{array}$$

Accordingly, in the light of Schur complement, it is equivalent to the inequality $\mathrm{\Theta }+{ϵ}_0{\mathrm{\Theta }}_0+{ϵ}_1{\mathrm{\Theta }}_1<0$ . Moreover, reviewing the S-procedure in Chang (2014) and Boyd et al. (1994) it can be concluded that $\mathrm{\Theta }<0$ , which means $\mathbb{E}=\{\mathrm{\Delta }V(\lambda (k))\}<0$ based on equation (27).

Furthermore, letting $\chi =\underset{k}{min}\{{\tau }_{\min }(-\mathrm{\Theta })\}$ , where ${\tau }_{\min }(-\mathrm{\Theta })$ represents the minimum eigenvalue of $-\mathrm{\Theta }$ , the above analysis will establish the sufficient condition as $\mathbb{E}\{\mathrm{\Delta }V(\lambda (k))\}\le -\chi \mathbb{E}\{{\lambda }^T(k)\lambda (k)\}$ . Then, for any $n\ge 1$ , we sum up the two sides of the above formula for 0 to n simultaneously, which leads to $\mathbb{E}\{V(\lambda (n+1))-V(\lambda (0))\}\le -\chi \mathbb{E}\{\sum _{k=0}^n\parallel \lambda (k){\parallel }^2\}$ . Obviously, it is easy to obtain that $\underset{n\to \infty }{lim}\mathbb{E}\{\sum _{k=0}^n\parallel \lambda (k){\parallel }^2\}\le (1/\chi )\mathbb{E}\{V(\lambda (0))\}$ . As a result, the conclusion can be drawn from Definition 1 that resulting system (13) with $w(k)=0$ is robustly stochastically stable.

In the coming part, the mixed ${\mathcal{H}}_{\infty }$ and passivity performance analysis for system (13) when $w(k)\ne 0$ will be conducted. Defining ${\tilde{\phi }}^T(k)=[{\lambda }^T(k){\lambda }^T(k-d){\nu }^T(k)f^T(y(k))w^T(k)]$ , when $w(k)\ne 0$ , inequalities (23) and (25) are able to be reformed as

$${\tilde{\phi }}^T(k){\overline{\mathrm{\Theta }}}_0\tilde{\phi }(k)\ge 0$$

(29)

$${\tilde{\phi }}^T(k){\overline{\mathrm{\Theta }}}_1\tilde{\phi }(k)\ge 0$$

(30)

where ${\overline{\mathrm{\Theta }}}_0=\mathrm{Ha}[\begin{array}{ccccc}2\rho {\mathrm{\Delta }}_y\mathcal{C}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}]-\mathrm{diag}\{0,0,I,0,0\}$ , ${\overline{\mathrm{\Theta }}}_1=\mathrm{Ha}[\begin{array}{ccccc}\mathcal{R}\mathcal{C}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}]-\mathrm{diag}\{0,0,0,I,0\}$ .

For another, the performance index $\mathcal{J}(k)$ is considered as $\mathcal{J}(k)={\gamma }^{-1}\alpha z^T(k)z(k)-2(1-\alpha )z^T(k)\mathrm{Xw}(k)-\gamma w^T(k)w(k)$ , then

$$\begin{array}{c}\mathbb{E}\{\mathrm{\Delta }V(\lambda (k))+\mathcal{J}(k)\}\hfill \\ \multicolumn{1}{c}{=\mathbb{E}\{(({\mathcal{A}}_{1i}-\tilde{\iota }(k){\mathcal{B}}_{1i})\lambda (k)+{\mathcal{A}}_0\lambda (k-d)+({\mathcal{B}}_{2i}-\tilde{\iota }(k){\mathcal{B}}_{3i})\nu (k)}\\ \multicolumn{1}{c}{+({\mathcal{B}}_{4i}-\tilde{\iota }(k){\mathcal{B}}_{5i})f(y(k))+\mathcal{F}w(k{))}^T\mathcal{P}(({\mathcal{A}}_{1i}-\tilde{\iota }(k){\mathcal{B}}_{1i})\lambda (k)}\\ \multicolumn{1}{c}{+{\mathcal{A}}_0\lambda (k-d)+({\mathcal{B}}_{2i}-\tilde{\iota }(k){\mathcal{B}}_{3i})\nu (k)+({\mathcal{B}}_{4i}-\tilde{\iota }(k){\mathcal{B}}_{5i})f(y(k))}\\ \multicolumn{1}{c}{+\mathcal{F}w(k))-{\lambda }^T(k)\mathcal{P}\lambda (k)+{\lambda }^T(k)\mathcal{S}\lambda (k)-{\lambda }^T(k-d)\mathcal{S}\lambda (k-d)}\\ \multicolumn{1}{c}{+{\gamma }^{-1}\alpha {(\mathcal{D}\lambda (k)+\mathrm{Hw}(k))}^T(\mathcal{D}\lambda (k)+\mathrm{Hw}(k))}\\ \multicolumn{1}{c}{-2(1-\alpha )(\mathcal{D}\lambda (k)+\mathrm{Hw}(k{))}^T\mathrm{Xw}(k)-\gamma w^T(k)w(k)\}}\\ \multicolumn{1}{c}{={\tilde{\phi }}^T(k)\overline{\mathrm{\Theta }}\tilde{\phi }(k)}\end{array}$$

(31)

where

$$\begin{array}{c}\begin{array}{c}\overline{\mathrm{\Theta }}={\left[\begin{array}{ccccc}{\mathcal{A}}_{1i}\hfill & {\mathcal{A}}_0\hfill & {\mathcal{B}}_{2i}\hfill & {\mathcal{B}}_{4i}\hfill & \mathcal{F}\hfill \end{array}\right]}^T\mathcal{P}\left[\begin{array}{ccccc}{\mathcal{A}}_{1i}\hfill & {\mathcal{A}}_0\hfill & {\mathcal{B}}_{2i}\hfill & {\mathcal{B}}_{4i}\hfill & \mathcal{F}\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{+{\hat{\iota }}^2{\left[\begin{array}{ccccc}{\mathcal{B}}_{1i}\hfill & 0\hfill & {\mathcal{B}}_{3i}\hfill & {\mathcal{B}}_{5i}\hfill & 0\hfill \end{array}\right]}^T\mathcal{P}\left[\begin{array}{ccccc}{\mathcal{B}}_{1i}\hfill & 0\hfill & {\mathcal{B}}_{3i}\hfill & {\mathcal{B}}_{5i}\hfill & 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{+{\gamma }^{-1}\alpha \mathrm{Ha}\left\{\left[\begin{array}{ccccc}\mathcal{D}\hfill & 0\hfill & 0\hfill & 0\hfill & H\hfill \end{array}\right]\right\}+\tilde{\mathrm{\Xi }}}\\ \multicolumn{1}{c}{\tilde{\mathrm{\Xi }}=\left[\begin{array}{cc}-\mathrm{diag}\{(\mathcal{P}-\mathcal{S}),\mathcal{S},0,0\}\hfill & *\hfill \\ \multicolumn{1}{c}{\left[\begin{array}{cccc}-(1-\alpha )X^T\mathcal{D}\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}& -2(1-\alpha )\mathrm{He}\{H^{T}X\}-\gamma I\hfill \end{array}\right]}\end{array}\hfill \end{array}$$

Similarly, the above-mentioned analysis process is reviewed again based on the S-procedure, if inequality (20) is established, the following condition can be obtained

$$\overline{\mathrm{\Theta }}+{ϵ}_0{\overline{\mathrm{\Theta }}}_0+{ϵ}_1{\overline{\mathrm{\Theta }}}_1<0$$

(32)

which indicates that $\overline{\mathrm{\Theta }}<0$ . Therefore, it is obvious that $\mathbb{E}\{\mathrm{\Delta }V(\lambda (k))+\mathcal{J}(k)\}<0$ , that is

$$\begin{array}{c}\mathbb{E}\{V(\lambda (k+1))-V(\lambda (k))+{\gamma }^{-1}\alpha z^T(k)z(k)\hfill \\ -2(1-\alpha )z^T(k)\mathrm{Xw}(k)-\gamma w^T(k)w(k)\}<0\hfill \end{array}$$

(33)

Then, by summing the left and right sides of equation (33) for $k=1,2,\dots ,\infty$ at the same time, we have that

$$\begin{array}{c}\mathbb{E}\{V(\lambda (\infty ))-V(\lambda (0))\}-\mathbb{E}\{\gamma \sum _{k=0}^{\infty }{w}^T(k)w(k)\}\hfill \\ \multicolumn{1}{c}{+\mathbb{E}\{\sum _{k=0}^{\infty }\{{\gamma }^{-1}\alpha z^T(k)z(k)-2(1-\alpha )z^T(k)\mathrm{Xw}(k)\}<0}\end{array}$$

(34)

Because of the zero-initial condition $V(\lambda (0))=0$ and $\mathbb{E}\{V(\lambda (\infty ))\}\ge 0$ , we can get

$$\begin{array}{c}\mathbb{E}\{\sum _{k=0}^{\infty }{\gamma }^{-1}\alpha z^T(k)z(k)-2(1-\alpha )z^T(k)\mathrm{Xw}(k)\}\hfill \\ \multicolumn{1}{c}{<\mathbb{E}\{\sum _{k=0}^{\infty }\gamma w^T(k)w(k)\}}\end{array}$$

(35)

which means that resulting system (13) is stochastically stable and satisfies the prescribed mixed ${\mathcal{H}}_{\infty }$ and passivity performance index by Definition 2. The proof is completed.

Mixed ${\mathcal{H}}_{\infty }$ and passive control design

From what has been mentioned in the past, an important result of the mixed ${\mathcal{H}}_{\infty }$ and passivity performance analysis has been established. Since Theorem 1 involves the uncertainties and coupling terms, condition (20) cannot be applied to resilient controller design immediately. In what follows, a two-step uncertainty elimination method combined with nonlinear decoupling method is adopted to solve the multiple uncertainty products and nonlinear coupling issue, so as to give the design conditions for the resilient controller and dynamic quantizer. The following theorem is given to construct the desired result.

Theorem 2. For known scalars $\eta ,\overline{\iota },M_y,{\mathrm{\Delta }}_y$ , and $\alpha \in [0,1]$ and matric X, the closed-loop system is stochastically stable with a mixed ${\mathcal{H}}_{\infty }$ and passivity performance index $\gamma$ , if there exist symmetric positive-definite matrices $P_1$ , $P_2$ , $P_3$ , $S_1$ , $S_2$ , $S_3$ , $Q_1$ , $Q_2$ , $Q_3$ , $Q_4$ and positive constants ${ϵ}_0,{ϵ}_1,\sigma ,{\epsilon }_1,{\epsilon }_2$ , so that the following conditions hold simultaneously

$${ϵ}_0-\sigma M_y<0$$

(36)

$$\begin{array}{c}\left[\begin{array}{ccc}{\mathrm{\Omega }}_{11i}\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Omega }}_{21i}}& {\mathrm{\Omega }}_{22}\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Omega }}_{31i}}& 0\hfill & {\mathrm{\Omega }}_{33i}\hfill \end{array}\right]<0\hfill \end{array}$$

(37)

where

$$\begin{array}{c}\begin{array}{l}{\mathrm{\Omega }}_{11i}=\left[\begin{array}{ccc}{\overline{\Upsilon }}_{11}& *& *\\ {\overline{\Upsilon }}_{21i}& {\overline{\Upsilon }}_{22}& *\\ {\overline{\Upsilon }}_{31}& 0& {\widehat{\mathrm{\Xi }}}_{33}\end{array}\right],{\mathrm{\Omega }}_{21i}=\left[\begin{array}{ccc}{\overrightarrow{\Upsilon }}_{11}& {\overrightarrow{\Upsilon }}_{12}& 0\\ 0& {\overrightarrow{\Upsilon }}_{22i}& 0\\ {\overrightarrow{\Upsilon }}_{31i}& 0& 0\end{array}\right]\hfill \\ {\mathrm{\Omega }}_{31i}=\left[\begin{array}{ccc}{\tilde{\Upsilon }}_{11i}& {\tilde{\Upsilon }}_{12i}& 0\end{array}\right],{\mathrm{\Omega }}_{33i}=\mathrm{diag}\left\{-\eta U_i-\eta U_i^T,-\eta U_i-\eta U_i^T\right\}\hfill \\ {\mathrm{\Omega }}_{22}=\mathrm{diag}\left\{-{\epsilon }_{1}I,-{\epsilon }_{2}I,-{\epsilon }_{2}I\right\},{\overline{\Upsilon }}_{21i}=\left[\begin{array}{cccc}{\Gamma }_{11i}& {\Gamma }_{12}& {\Gamma }_{13i}& {\Gamma }_{14}\\ {\Gamma }_{21i}& 0& {\Gamma }_{23i}& 0\\ {\Gamma }_{31}& 0& 0& {\Gamma }_{41}\end{array}\right]\hfill \\ {\overline{\Upsilon }}_{11}=\left[\begin{array}{cc}\mathrm{diag}\left\{\mathbb{P},-\mathcal{S},-{ϵ}_{0}I,-{ϵ}_{1}I\right\}& *\\ \left[\begin{array}{ccc}\mathbb{D}& 0& 0\end{array}\right]& -2(1-\alpha )He\left\{H^{T}X\right\}-\gamma I\end{array}\right]\hfill \\ \mathbb{P}=\left[\begin{array}{cc}-P_1+S_1& *\\ -P_2+S_2& -P_3+S_3\end{array}\right],{\overline{\Upsilon }}_{22}=\mathrm{diag}\left\{\overline{\mathcal{P}},\overline{\mathcal{P}},-\gamma I\right\}\hfill \\ {\overline{\Upsilon }}_{31}=\left[\begin{array}{cccc}\stackrel{·}{\Gamma }& 0& 0& 0\end{array}\right],{\overrightarrow{\Upsilon }}_{12}=\left[\begin{array}{ccc}{\overrightarrow{\Gamma }}_{11}& 0& 0\end{array}\right],{\overrightarrow{\Upsilon }}_{11}=\left[\begin{array}{cccc}{\overline{\Gamma }}_{11}& {\overline{\Gamma }}_{12}& 0& 0\end{array}\right]\hfill \\ {\overrightarrow{\Upsilon }}_{31i}=\left[\begin{array}{cccc}{\stackrel{⌣}{\Gamma }}_{11i}& 0& {\stackrel{⌣}{\Gamma }}_{13i}& 0\end{array}\right],{\Gamma }_{31}=\left[\begin{array}{cc}\sqrt{\alpha }D& 0\end{array}\right],{\overrightarrow{\Upsilon }}_{22i}=\left[\begin{array}{ccc}{\tilde{\Gamma }}_{11i}& {\tilde{\Gamma }}_{12i}& 0\end{array}\right]\hfill \\ {\Gamma }_{41}=\sqrt{\alpha }H,\mathbb{D}=\left[\begin{array}{cc}-(1-\alpha )X^{T}D& 0\end{array}\right],{\tilde{\Upsilon }}_{11i}=\left[\begin{array}{cccc}{\widehat{\Gamma }}_{11i}& 0& {\widehat{\Gamma }}_{13i}& 0\end{array}\right]\hfill \\ {\tilde{\Upsilon }}_{12i}=\left[\begin{array}{ccc}{\stackrel{\prime }{\Gamma }}_{11i}& {\stackrel{\prime }{\Gamma }}_{12i}& 0\end{array}\right],\overline{\mathcal{P}}=\left[\begin{array}{cc}-Q_1^T-Q_1+P_1& *\\ -Q_4^T-Q_2+P_2& -Q_3^T-Q_3+P_3\end{array}\right]\hfill \\ {\Gamma }_{11i}=\left[\begin{array}{cc}{Q}_{1}A+B\overrightarrow{\iota }{V}_i{\mathrm{\Phi }}_{i}C+Q_4\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C& Q_4{\overline{\mathrm{\Phi }}}_i+B{V}_i{\overline{\mathrm{\Phi }}}_i\\ Q_{2}A+s\overrightarrow{\iota }{V}_i{\mathrm{\Phi }}_{i}C+Q_3\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C& Q_3{\overline{\mathrm{\Phi }}}_i+s{V}_i{\overline{\mathrm{\Phi }}}_i\end{array}\right]\hfill \\ {\Gamma }_{12}=\left[\begin{array}{cc}{Q}_1{A}_0& 0\\ Q_2{A}_0& 0\end{array}\right],{\Gamma }_{13i}=\left[\begin{array}{cc}{Q}_4\overrightarrow{\iota }{\mathrm{\Phi }}_i+B\overrightarrow{\iota }{V}_i{\mathrm{\Phi }}_i& Q_4\overline{\iota }{\mathrm{\Phi }}_i+B\overline{\iota }{V}_i{\mathrm{\Phi }}_i\\ Q_3\overrightarrow{\iota }{\mathrm{\Phi }}_i+s\overrightarrow{\iota }{V}_i{\mathrm{\Phi }}_i& Q_3\overline{\iota }{\mathrm{\Phi }}_i+s\overline{\iota }{V}_i{\mathrm{\Phi }}_i\end{array}\right]\hfill \\ {\Gamma }_{14}=\left[\begin{array}{c}{Q}_{1}F\\ Q_{2}F\end{array}\right],{\Gamma }_{21i}=\left[\begin{array}{cc}\widehat{\iota }{Q}_4{\mathrm{\Phi }}_{i}C+B{V}_i{\mathrm{\Phi }}_{i}C& 0\\ \widehat{\iota }{Q}_3{\mathrm{\Phi }}_{i}C+s{V}_i{\mathrm{\Phi }}_{i}C& 0\end{array}\right],\stackrel{·}{\Gamma }=\left[\begin{array}{cc}2\sigma \mathrm{\Delta }C& 0\\ {ϵ}_1ℛC& 0\end{array}\right]\hfill \\ {\Gamma }_{23i}=\left[\begin{array}{cc}\widehat{\iota }{Q}_4{\mathrm{\Phi }}_i+B{V}_i{\mathrm{\Phi }}_i& -\widehat{\iota }{Q}_4{\mathrm{\Phi }}_i-B{V}_i{\mathrm{\Phi }}_i\\ \widehat{\iota }{Q}_3{\mathrm{\Phi }}_i+s{V}_i{\mathrm{\Phi }}_i& -\widehat{\iota }{Q}_3{\mathrm{\Phi }}_i-s{V}_i{\mathrm{\Phi }}_i\end{array}\right],{\overline{\Gamma }}_{11}=\left[\begin{array}{cc}0& 0\\ {\epsilon }_1{\mathcal{N}}_1& 0\end{array}\right]\hfill \\ {\overline{\Gamma }}_{12}=\left[\begin{array}{cc}0& 0\\ {\epsilon }_1{\mathcal{N}}_2& 0\end{array}\right],{\stackrel{⌣}{\Gamma }}_{11i}=\left[\begin{array}{cc}{\epsilon }_2{\mathcal{N}}_{3i}\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C& {\epsilon }_2{\mathcal{N}}_{3i}{\overline{\mathrm{\Phi }}}_i\\ {\epsilon }_2{\mathcal{N}}_{3i}{\mathrm{\Phi }}_{i}C& 0\end{array}\right]\hfill \\ {\widehat{\Gamma }}_{11i}=\left[\begin{array}{cc}{V}_i\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C& V_i{\overline{\mathrm{\Phi }}}_i\\ V_i{\mathrm{\Phi }}_{i}C& 0\end{array}\right],{\overrightarrow{\Gamma }}_{11}=\left[\begin{array}{cc}{ℳ}^T{Q}_1^T& {ℳ}^T{Q}_2^T\\ 0& 0\end{array}\right]\hfill \end{array}\end{array}$$

$$\begin{array}{c}\begin{array}{c}{\stackrel{⌣}{\Gamma }}_{13i}=\left[\begin{array}{cc}{\epsilon }_2{\mathcal{N}}_{3i}\overrightarrow{\iota }{\mathrm{\Phi }}_i\hfill & {\epsilon }_2{\mathcal{N}}_{3i}\overline{\iota }{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{{\epsilon }_2{\mathcal{N}}_{3i}{\mathrm{\Phi }}_i}& -{\epsilon }_2{\mathcal{N}}_{3i}{\mathrm{\Phi }}_i\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{{\tilde{\Gamma }}_{11i}=\left[\begin{array}{cc}{(Q_{1}B{\mathcal{M}}_{1i})}^T\hfill & {(Q_{2}B{\mathcal{M}}_{1i})}^T\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\hat{\Gamma }}_{13i}=\left[\begin{array}{cc}{V}_i\overrightarrow{\iota }{\mathrm{\Phi }}_i\hfill & V_i\overline{\iota }{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{V_i{\mathrm{\Phi }}_i}& -V_i{\mathrm{\Phi }}_i\hfill \end{array}\right],{\tilde{\Gamma }}_{12i}=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{\hat{\iota }{(Q_{1}B{\mathcal{M}}_{1i})}^T}& \hat{\iota }{(Q_{2}B{\mathcal{M}}_{1i})}^T\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{{\Gamma }^{\prime }}_{11i}=\left[\begin{array}{cc}\eta {(Q_{1}B-B{U}_i)}^T\hfill & \eta {(Q_{2}B-s{U}_i)}^T\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{{\Gamma }^{\prime }}_{12i}=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{\eta {(\hat{\iota }{Q}_{1}B-B{U}_i)}^T}& \eta {(\hat{\iota }{Q}_{2}B-s{U}_i)}^T\hfill \end{array}\right].}\end{array}\hfill \end{array}$$

Furthermore, the online adjustment strategy of quantizer parameter is given in equation (21) with $\rho =\sigma /{ϵ}_0$ . And the controller gain matrix can be correspondingly given by $K_i=U_i^{-1}{V}_i$ .

Proof. Here, by pre- and post-multiplying both sides inequality (20) with congruence transforms matrix $\mathrm{diag}\{I,I,I,I,I,\mathcal{Q},\mathcal{Q},I,I,I\}$ and its transpose, and keeping sight of the fact that $-\mathcal{Q}{\mathcal{P}}^{-1}{\mathcal{Q}}^T\le -\mathcal{Q}-{\mathcal{Q}}^T+\mathcal{P}$ , one obtains the following matrix inequality

$$\left[\begin{array}{ccc}{\overline{\mathrm{\Xi }}}_{11}\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\overrightarrow{\mathrm{\Xi }}}_{21i}}& {\overrightarrow{\mathrm{\Xi }}}_{22}\hfill & *\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Xi }}}_{31}}& 0\hfill & {\overline{\mathrm{\Xi }}}_{33}\hfill \end{array}\right]<0$$

(38)

where

$$\begin{array}{c}{\overrightarrow{\mathrm{\Xi }}}_{21i}=\left[\begin{array}{ccccc}\mathcal{Q}{\mathcal{A}}_{1i}\hfill & \mathcal{Q}{\mathcal{A}}_0\hfill & \mathcal{Q}{\mathcal{B}}_{2i}\hfill & \mathcal{Q}{\mathcal{B}}_{4i}\hfill & \mathcal{Q}\mathcal{F}\hfill \\ \multicolumn{1}{c}{\hat{\iota }\mathcal{Q}{\mathcal{B}}_{1i}}& 0\hfill & \hat{\iota }\mathcal{Q}{\mathcal{B}}_{3i}\hfill & \hat{\iota }\mathcal{Q}{\mathcal{B}}_{5i}\hfill & 0\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\overrightarrow{\mathrm{\Xi }}}_{22}=\mathrm{diag}\{-\mathcal{Q}-{\mathcal{Q}}^T+\mathcal{P},-\mathcal{Q}-{\mathcal{Q}}^T+\mathcal{P}\}}\end{array}$$

(39)

The removal of the uncertain terms containing ${\mathcal{F}}_1(k)$ is considered first, whereas the uncertain terms containing ${\mathcal{F}}_2(k)$ are removed in the second step. Obviously, the condition of equation (38) can further be rewritten as

$$\begin{array}{c}{\mathrm{\Lambda }}_i+\mathrm{He}\{{\mathcal{Y}}_1{\mathcal{F}}_1(k){\mathcal{Y}}_2\}<0\hfill \end{array}$$

(40)

where

$$\begin{array}{c}\begin{array}{c}{\mathrm{\Lambda }}_i=\left[\begin{array}{ccc}{\overline{\mathrm{\Xi }}}_{11}\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\stackrel{⌣}{\mathrm{\Xi }}}_{21i}}& {\overrightarrow{\mathrm{\Xi }}}_{22}\hfill & *\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Xi }}}_{31}}& 0\hfill & {\overline{\mathrm{\Xi }}}_{33}\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{{\stackrel{⌣}{\mathrm{\Xi }}}_{21i}=\left[\begin{array}{ccccc}\mathcal{Q}{\mathbb{A}}_{1i}\hfill & \mathcal{Q}{\mathbb{A}}_0\hfill & \mathcal{Q}{\mathcal{B}}_{2i}\hfill & \mathcal{Q}{\mathcal{B}}_{4i}\hfill & \mathcal{Q}\mathcal{F}\hfill \\ \multicolumn{1}{c}{\hat{\iota }\mathcal{Q}{\mathcal{B}}_{1i}}& 0\hfill & \hat{\iota }\mathcal{Q}{\mathcal{B}}_{3i}\hfill & \hat{\iota }\mathcal{Q}{\mathcal{B}}_{5i}\hfill & 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathbb{A}}_{1i}=\left[\begin{array}{cc}A+\overrightarrow{\iota }B{\mathcal{K}}_i{\mathrm{\Phi }}_{i}C\hfill & B{\mathcal{K}}_i{\overline{\mathrm{\Phi }}}_i\hfill \\ \multicolumn{1}{c}{\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C}& {\overline{\mathrm{\Phi }}}_i\hfill \end{array}\right],}\\ \multicolumn{1}{c}{{\mathbb{A}}_0=\left[\begin{array}{cc}{A}_0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],{\mathcal{Y}}_1^T=\left[\begin{array}{ccc}0\hfill & {\mathcal{X}}_1\hfill & 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathcal{Y}}_2=\left[\begin{array}{ccc}{\mathcal{X}}_2\hfill & 0\hfill & 0\hfill \end{array}\right],{\mathcal{X}}_1=\left[\begin{array}{cc}{Y}_1^T{\mathcal{Q}}^T\hfill & 0\hfill \end{array}\right],}\\ \multicolumn{1}{c}{{\mathcal{X}}_2=\left[\begin{array}{ccccc}{Y}_2\hfill & Y_3\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}\end{array}\hfill \end{array}$$

$$\begin{array}{c}\begin{array}{c}{Y}_1^T=\left[\begin{array}{cc}{\mathcal{M}}^T\hfill & 0\hfill \end{array}\right],Y_2=\left[\begin{array}{cc}{\mathcal{N}}_1\hfill & 0\hfill \end{array}\right],Y_3=\left[\begin{array}{cc}{\mathcal{N}}_2\hfill & 0\hfill \end{array}\right].\hfill \end{array}\hfill \end{array}$$

By invoking Lemma 1 and Schur complement to inequality (40), the following inequality is satisfied

$$\left[\begin{array}{cc}{\mathrm{\Lambda }}_i\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Xi }}_{41}}& -{\epsilon }_{1}I\hfill \end{array}\right]<0$$

(41)

where ${\mathrm{\Xi }}_{41}=[{\mathcal{Y}}_1{\epsilon }_1{\mathcal{Y}}_2^T{]}^T$ . From the foregoing, all uncertain terms containing ${\mathcal{F}}_1(k)$ have been removed from equation (40). Next, similar to the above derivation process, we will separate and eliminate all uncertain terms containing ${\mathcal{F}}_2(k)$ from equation (41). Therefore, above inequality (41) can be rewritten as follows

$$\left[\begin{array}{cc}{\stackrel{⌣}{\mathrm{\Lambda }}}_i\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Xi }}_{51i}}& -{\epsilon }_{2}I\hfill \end{array}\right]<0$$

(42)

where

$$\begin{array}{c}\begin{array}{c}{\stackrel{⌣}{\mathrm{\Lambda }}}_i=\left[\begin{array}{cc}{\tilde{\mathrm{\Lambda }}}_i\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Xi }}_{41}}& -{\epsilon }_{1}I\hfill \end{array}\right],{\tilde{\mathrm{\Lambda }}}_i=\left[\begin{array}{ccc}{\overline{\mathrm{\Xi }}}_{11}\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\tilde{\mathrm{\Xi }}}_{21i}}& {\overrightarrow{\mathrm{\Xi }}}_{22}\hfill & *\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Xi }}}_{31}}& 0\hfill & {\overline{\mathrm{\Xi }}}_{33}\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\mathrm{\Xi }}_{51i}=[{\overline{\mathcal{Y}}}_{1i}{\epsilon }_2{\overline{\mathcal{Y}}}_{2i}^T{]}^T}\\ \multicolumn{1}{c}{{\tilde{\mathrm{\Xi }}}_{21i}=\left[\begin{array}{ccccc}\mathcal{Q}{\mathbb{A}}_{1i}\hfill & \mathcal{Q}{\mathbb{A}}_0\hfill & \mathcal{Q}{\mathbb{B}}_{2i}\hfill & \mathcal{Q}{\mathbb{B}}_{4i}\hfill & \mathcal{Q}\mathcal{F}\hfill \\ \multicolumn{1}{c}{\hat{\iota }\mathcal{Q}{\mathbb{B}}_{1i}}& 0\hfill & \hat{\iota }\mathcal{Q}{\mathbb{B}}_{3i}\hfill & \hat{\iota }\mathcal{Q}{\mathbb{B}}_{5i}\hfill & 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathbb{B}}_{1i}=\left[\begin{array}{cc}B{K}_i{\mathrm{\Phi }}_{i}C\hfill & 0\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{i}C}& 0\hfill \end{array}\right],{\mathbb{A}}_{1i}=\left[\begin{array}{cc}A+\overrightarrow{\iota }B{K}_i{\mathrm{\Phi }}_{i}C\hfill & B{K}_i{\overline{\mathrm{\Phi }}}_i\hfill \\ \multicolumn{1}{c}{\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C}& {\overline{\mathrm{\Phi }}}_i\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathbb{B}}_{2i}=\left[\begin{array}{c}\overrightarrow{\iota }B{K}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{\overrightarrow{\iota }{\mathrm{\Phi }}_i}\end{array}\right],{\mathbb{B}}_{3i}=\left[\begin{array}{c}B{K}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_i}\end{array}\right],{\mathbb{B}}_{4i}=\left[\begin{array}{c}\overline{\iota }B{K}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{\overline{\iota }{\mathrm{\Phi }}_i}\end{array}\right]}\\ \multicolumn{1}{c}{{\mathbb{B}}_{5i}=\left[\begin{array}{c}-B{K}_i{\mathrm{\Phi }}_i\hfill \\ \multicolumn{1}{c}{-{\mathrm{\Phi }}_i}\end{array}\right],{\overline{\mathcal{Y}}}_{1i}^T=\left[\begin{array}{cc}{\mathcal{Z}}_{1i}\hfill & 0\hfill \\ \multicolumn{1}{c}{{\mathcal{Z}}_{2i}}& 0\hfill \end{array}\right],{\overline{\mathcal{Y}}}_{2i}=\left[\begin{array}{cc}{\mathcal{Z}}_{3i}\hfill & 0\hfill \\ \multicolumn{1}{c}{{\mathcal{Z}}_{4i}}& 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathcal{Z}}_{1i}=\left[\begin{array}{ccc}0\hfill & {\mathcal{W}}_{1i}\hfill & 0\hfill \end{array}\right],{\mathcal{Z}}_{2i}=\left[\begin{array}{ccc}0\hfill & {\mathcal{W}}_{2i}\hfill & 0\hfill \end{array}\right],{\mathcal{Z}}_{3i}=\left[\begin{array}{ccc}{\mathcal{W}}_{3i}\hfill & 0\hfill & 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathcal{Z}}_{4i}=\left[\begin{array}{ccc}{\mathcal{W}}_{4i}\hfill & 0\hfill & 0\hfill \end{array}\right],{\mathcal{W}}_{1i}=\left[\begin{array}{cc}{Y}_{4i}^T{\mathcal{Q}}^T\hfill & 0\hfill \end{array}\right],{\mathcal{W}}_{2i}=\left[\begin{array}{cc}0\hfill & \hat{\iota }{Y}_{4i}^T{\mathcal{Q}}^T\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathcal{W}}_{3i}=\left[\begin{array}{ccccc}{Y}_{5i}\hfill & 0\hfill & Y_{7i}\hfill & Y_{9i}\hfill & 0\hfill \end{array}\right],Y_{4i}^T=\left[\begin{array}{cc}{\mathcal{M}}_{1i}^T{B}^T\hfill & 0\hfill \end{array}\right],}\\ \multicolumn{1}{c}{Y_{7i}={\mathcal{N}}_{3i}\overrightarrow{\iota }{\mathrm{\Phi }}_i}\\ \multicolumn{1}{c}{{\mathcal{W}}_{4i}=\left[\begin{array}{ccccc}{Y}_{6i}\hfill & 0\hfill & Y_{8i}\hfill & Y_{10i}\hfill & 0\hfill \end{array}\right],Y_{6i}=\left[\begin{array}{cc}{\mathcal{N}}_{3i}{\mathrm{\Phi }}_{i}C\hfill & 0\hfill \end{array}\right],}\\ \multicolumn{1}{c}{Y_{8i}={\mathcal{N}}_{3i}{\mathrm{\Phi }}_i}\\ \multicolumn{1}{c}{Y_{5i}=\left[\begin{array}{cc}{\mathcal{N}}_{3i}\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C\hfill & {\mathcal{N}}_{3i}{\overline{\mathrm{\Phi }}}_i\hfill \end{array}\right],Y_{9i}={\mathcal{N}}_{3i}\overline{\iota }{\mathrm{\Phi }}_i,Y_{10i}=-{\mathcal{N}}_{3i}{\mathrm{\Phi }}_i.}\end{array}\hfill \end{array}$$

Accordingly, all uncertainties existing in the system and controller have been eliminated. In inequality (42), there are still nonlinear coupling terms, so it is not a strict condition of linear matrix inequality. Next, we will resort to a decoupling method based on Lemma 2 to deal with these coupling terms. Without losing generality, choosing

$$\mathcal{P}=\left[\begin{array}{cc}{P}_1\hfill & *\hfill \\ \multicolumn{1}{c}{P_2}& P_3\hfill \end{array}\right],\mathcal{S}=\left[\begin{array}{cc}{S}_1\hfill & *\hfill \\ \multicolumn{1}{c}{S_2}& S_3\hfill \end{array}\right],\mathcal{Q}=\left[\begin{array}{cc}{Q}_1\hfill & Q_4\hfill \\ \multicolumn{1}{c}{Q_2}& Q_3\hfill \end{array}\right]$$

and defining $K_i=U_i^{-1}{V}_i$ , inequality condition (37) can be rearranged as the following form

$$\begin{array}{c}\left[\begin{array}{cc}{\mathrm{\Omega }}_{11i}\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Omega }}_{21i}}& {\mathrm{\Omega }}_{22}\hfill \end{array}\right]+\mathrm{He}\left\{{\ell }_{1i}\left[\begin{array}{cc}{U}_i^{-1}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& U_i^{-1}\hfill \end{array}\right]\left[\begin{array}{cc}{V}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& V_i\hfill \end{array}\right]{\ell }_{2i}\right\}<0\hfill \end{array}$$

(43)

is satisfied, where

$$\begin{array}{c}\begin{array}{c}{\ell }_{1i}={\left[\begin{array}{ccccc}0\hfill & {\overline{\ell }}_i\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]}^T,{\ell }_{2i}=\left[\begin{array}{ccccc}{\overrightarrow{\ell }}_i\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\overline{\ell }}_i=\left[\begin{array}{c}\begin{array}{cccc}{(Q_{1}B-B{U}_i)}^T\hfill & {(Q_{2}B-s{U}_i)}^T\hfill & 0\hfill & 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cccc}0\hfill & 0\hfill & {(Q_{1}B-B{U}_i)}^T\hfill & {(Q_{2}B-s{U}_i)}^T\hfill \end{array}}\end{array}\right]}\\ \multicolumn{1}{c}{{\overrightarrow{\ell }}_i=\left[\begin{array}{ccccccc}\overrightarrow{\iota }{\mathrm{\Phi }}_{i}C\hfill & {\overline{\mathrm{\Phi }}}_i\hfill & 0\hfill & 0\hfill & \overrightarrow{\iota }{\mathrm{\Phi }}_i\hfill & \overline{\iota }{\mathrm{\Phi }}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{\hat{\iota }{\mathrm{\Phi }}_{i}C}& 0\hfill & 0\hfill & 0\hfill & \hat{\iota }{\mathrm{\Phi }}_i\hfill & -\hat{\iota }{\mathrm{\Phi }}_i\hfill & 0\hfill \end{array}\right]}\end{array}\hfill \end{array}$$

and s denotes the dimension adjustment matrix, which be defined as the following forms, $s=B$ or $s=I\in {\mathbb{R}}^{l\times l}$ with $n=l$ ; $s=[I\in {\mathbb{R}}^{l\times l};0\in {\mathbb{R}}^{(n-l)\times l}]$ with $n>l$ . Using Lemma 2, if the condition to equation (37) is satisfied, the above inequality equation (43) can be easily obtained, which means that condition (20) is met. Therefore, we can come to a conclusion that resulting system (13) attains stochastically stable with the mixed ${\mathcal{H}}_{\infty }$ and passivity performance index. The proof is completed.

Remark 3. Furthermore, worth noting that the control problem researched in this paper is one of the special circumstances of dissipative control, which can be freely transformed into the passive control and the pure ${\mathcal{H}}_{\infty }$ control. As we all know, dissipation analysis mainly studies systems with chaotic structures, in which the interacting particles show long-range correlated characteristics. However, this paper have not taken these constraints on the system into consideration.

Remark 4. The design of resilient controller in Theorem 2 depends on mode information, which brings much less conservatism. The strict LMIs conditions are given for the controller design, so that the control gain $K_i$ can be obtained by MATLAB LMI toolbox. Furthermore, the optimal mixed ${\mathcal{H}}_{\infty }$ and passivity performance level $\gamma$ (when $0<\alpha <1$ ) can be obtained by solving the following optimization problem

$$\min \gamma \mathrm{subject}\mathrm{to}\mathrm{LMIs}(36)-(37)$$

(44)

Simulation results

In this section, a DC motor device (Yin et al., 2014) driving an inertial load is considered as an application example to demonstrate the effectiveness and applicability of the proposed results. The terminal voltage $V_a(t)$ is directly fed into the motor as a control input, whose equation of the motor/load dynamics is

$$\begin{array}{c}{G}_a\omega (t)=-L_a{\stackrel{·}{i}}_a(t)-R_a{i}_a(t)+V_a(t)\hfill \\ \multicolumn{1}{c}{G_b{i}_a(t)=J_a\stackrel{·}{\omega }(t)+Z_a\omega (t)}\end{array}$$

(45)

Then, the following state-space model for the DC motor is generally written as

$$\begin{array}{c}\stackrel{·}{x}(t)=\mathrm{Ax}(t)+\mathrm{Bu}(t)\hfill \\ \multicolumn{1}{c}{y(t)=\mathrm{Cx}(t)}\end{array}$$

(46)

where $x(t)=[\omega (t)i_a(t){]}^T$ , $u(t)=V_a(t)$ with $A=\left[\begin{array}{cc}-(Z_a/J_a)\hfill & G_b/J_a\hfill \\ \multicolumn{1}{c}{-(G_a/L_a)}& -(R_a/L_a)\hfill \end{array}\right]$ and $B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1/L_a}\end{array}\right]$ , $\omega (t)$ and $i_a(t)$ represent the angular rate of the load and current, respectively. Table 1 gives the parameters of the DC motor, and the above system could be discretized by selecting $T=0.5s$ , one gets

$$A=\left[\begin{array}{cc}0.5956\hfill & 0.4971\hfill \\ \multicolumn{1}{c}{-0.4708}& -1.1538\hfill \end{array}\right],B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0.7692}\end{array}\right].$$

Table 1.

Parameters of DC motor device.

Symbol	Description	Value
$J_a$	Inertial load	0.68 $\mathrm{Kg}·{\mathrm{m}}^2$
$R_a$	Resistance	2.80 $\mathrm{\Omega }$
$Z_a$	Damping constant	0.55 $\mathrm{N}·\mathrm{m}·\mathrm{s}$
$G_a$	EMF constant	0.676 $\mathrm{V}·\mathrm{s}/\mathrm{rad}$
$G_b$	torque constant	0.612 $\mathrm{N}·\mathrm{m}/\mathrm{A}$
$L_a$	Inductance	0.65 $\mathrm{H}$

EMF: electromotive force.

The eigenvalue of A is 0.4496, −1.0078. Obviously, the system is unstable without a controller. For the model, the inevitable time delay and uncertainties caused by armature reaction are considered simultaneously, so the other parameters are given as follows

$$\begin{array}{l}\begin{array}{l}{A}_0=\left[\begin{array}{cc}-0.1014& -0.0298\\ -0.0159& -0.2198\end{array}\right],C=\left[\begin{array}{cc}0.0614& -0.0298\\ -0.1255& 0.0441\end{array}\right]\hfill \\ D=\left[\begin{array}{cc}0.1118& 0.0312\end{array}\right],F=\left[\begin{array}{c}-0.2144\\ -0.4954\end{array}\right],ℳ=\left[\begin{array}{c}-0.0244\\ -0.0143\end{array}\right]\hfill \\ X=H=1,{ℳ}_{11}=-0.0268,{\mathcal{N}}_1=\left[\begin{array}{cc}-0.0017& -0.0702\end{array}\right]\hfill \\ {\mathcal{N}}_2=\left[\begin{array}{cc}-0.0146& -0.0178\end{array}\right],{\mathcal{N}}_{31}=\left[\begin{array}{cc}-0.0046& -0.0042\end{array}\right]\hfill \\ {ℳ}_{12}=0.0068,{\mathcal{N}}_{32}=\left[\begin{array}{cc}0.0046& 0.0042\end{array}\right]\hfill \end{array}\hfill \end{array}$$

In this simulation, given some parameters $\alpha =0.5$ , $\eta =1.0392$ , $\overline{\iota }=0.3$ , $\gamma =2$ , the quantization range and quantization error bound can be selected as $M_y=200$ and ${\mathrm{\Delta }}_y=0.01$ , respectively. When RR protocol is used in the system under consideration, we take $\xi (k)\in \{1,2\}$ , then ${\mathrm{\Phi }}_{\xi (k)}\in \{\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]\}$ . Suppose the deception attack $f(y(k))=[\tan {\mathrm{h}}^T(0.2y_1(k))\tan {\mathrm{h}}^T(0.2y_2(k)){]}^T$ , there is the restriction matrix $\mathcal{R}=\mathrm{diag}\{0.2,0.2\}$ satisfying limitation condition (11). Making use of the MATLAB LMI toolbox, we can easily solve the LMIs in Theorem 2 to obtain the gain matrices of the mode-dependent controller (12) shown below

$$\begin{array}{c}{K}_1=\left[\begin{array}{cc}-11.0956\hfill & -0.7398\hfill \end{array}\right],K_2=\left[\begin{array}{cc}-0.5156\hfill & 7.1395\hfill \end{array}\right]\hfill \end{array}$$

and the designed scalar $\rho =0.0053$ . Furthermore, the results of minimum $\gamma$ value and control parameters corresponding to different $\alpha$ are also displayed in Table 2, which is available from Remark 4.

Table 2.

Optimized ${\gamma }_{\min }$ and control parameters for different $\alpha$ .

$\alpha$	$K_1$ and $K_2$	${\gamma }_{\min }$
0	$K_1=[-11.1533-0.7509]$ and $K_2=[-0.53917.1911]$	0.9225
0.3	$K_1=[-10.6742-0.6343]$ and $K_2=[-0.39356.4948]$	0.6046
0.5	$K_1=[-10.6681-0.6321]$ and $K_2=[-0.39086.4698]$	0.3963
0.8	$K_1=[-10.6588-0.6286]$ and $K_2=[-0.38656.4299]$	0.7751
1	$K_1=[-10.6537-0.6266]$ and $K_2=[-0.38416.4073]$	1.0812

Assume that the initial condition is $x(0)=[1.5-1.5]$ and the disturbance input and the uncertainties are $w(k)=0.2e^{-0.3k}\cos (0.5k)$ and $F_1(k)=0.5\sin (2k),F_2(k)=0.3$ cos(2k), respectively. Affected by external conditions, the delay is assumed to satisfy $1\le d\le 2$ . Based on the above parameters, the simulation responses are given in Figures 2–9. Figures 2 and 3 depict the state trajectories of open-loop and closed-loop system, respectively. From the state trajectory of the open-loop system, it can be seen that the equipment under consideration is an unstable system. However, using the designed controller and the quantizer, the state trajectory will eventually be in the stable region. Figures 4 and 5 describe the respective trajectory of $y(k)$ , $\overline{y}(k)$ , $\tilde{y}(k)$ , and the Bernoulli distributed variable for the deception attacks, respectively, which graphically reflect the influence of deception attacks. Moreover, Figure 6 depicts the dynamic parameter $\mu (k)$ of the quantizer, which is adaptively adjusted according to the measurement output $y(k)$ , which is consistent with the proposed adjustment rule (Chang et al., 2019). Figure 7 shows the quantization error $\nu (k)$ of the measurement output. The control responses of the system with and without quantization and the controlled output responses with and without uncertainties are depicted in Figures 8 and 9, respectively. From the simulation results, it comes to the conclusion that the designed controller is appropriate and effective since it is comprehensive and can strongly tolerate uncertainties, quantization, deception attacks, and the RR protocol scheduling.

Figure 2.

State responses of the open-loop system.

Figure 3.

State responses of the closed-loop system.

Figure 4.

Response of $y(k)$ , $\overline{y}(k)$ , and $\tilde{y}(k)$ .

Figure 5.

Occurrence instants of deception attacks.

Figure 6.

Dynamic quantizer’s parameter $\mu (k)$ .

Figure 7.

Quantization error $\nu (k)$ for measurement output $y(k)$ .

Figure 8.

Control response with and without quantization.

Figure 9.

Controlled output response with and without uncertainties.

Comparative explanations: In contrast to the existing results on mixed ${\mathcal{H}}_{\infty }$ and passive control design strategy, the controller design strategy mentioned in this paper has some advantages: (1) Compared with existing results (Li et al., 2018; Sathishkumar et al., 2019; Zheng et al., 2018), the RR protocol is considered to schedule sensors with the effect of deception attacks on measurement output, so as to avoid data conflicts in practical applications, which is necessary in networks with limited bandwidth. (2) Compared with the static quantization (Fu and Xie, 2005; Gao et al., 2008), the dynamic quantization strategy is chosen to quantize the measurement output signal, and the S-procedure is used to deal with quantization errors and deception attacks. (3) Moreover, a new performance index is developed to solve restriction of the same dimension of control output and the exogenous disturbance, which is more general than those adopted in the literature (Li et al., 2018; Sathishkumar et al., 2019). The parameters of the controller are obtained using the gradual elimination method and decoupling method, which is also applicable to the design of robust controller for other systems coupled with multiple uncertain parameters.

Conclusion

In this paper, the problem of mixed ${\mathcal{H}}_{\infty }$ and passive resilient control for uncertain discrete time-delay systems has been researched. Considering the effects induced by dynamic quantization, deception attacks and RR protocol in the network before the measurement output are transmitted to controller via communication network may reflect the reality more closely. To improve the robustness of controller, the gain perturbation is also considered. Then by adopting the S-procedure to deal with quantization error and deception attacks, a new analysis criteria are given for judging the stochastic stability and the mixed ${\mathcal{H}}_{\infty }$ and passivity performance of the closed-loop system. Furthermore, sufficient conditions for the resilient controller and dynamic quantizer are provided based on the uncertainty elimination method and nonlinearity decoupling method. Finally, an example is employed to verify the validity of the main results.