Observer-Based PID Security Control for Discrete-Time T-S Fuzzy Systems under Distributed Dynamic Event-Triggered Mechanism

Abstract

This study addresses the observer-based proportional-integral-derivative (PID) security control problem for nonlinear discrete-time fuzzy systems subject to deception attacks. In order to reduce the burden of communication, a distributed dynamic event-triggered mechanism (DETM) is proposed, and each sensor node is equipped with an event-trigger generator. A new discrete-time Takagi–Sugeno (T-S) fuzzy model is established, in which both DETM and deception attacks are taken into consideration. Then, based on Lyapunov stability theory, the sufficient conditions for the input-to-state stability (ISS) are proposed, and the desired fuzzy PID controllers are given. Finally, a simulation example is provided to show the usefulness of the proposed approach.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Since the Takagi–Sugeno (T-S) fuzzy model is first proposed in [1], it has received considerable research attention in the past few decades. In practical applications, systems with complex structures usually have nonlinearities, which make analysis and synthesis difficult. As one of the effective methods to deal with nonlinear systems, T-S fuzzy system technology is able to approximate nonlinear systems by a set of local linear systems and a group of if-then rules. Therefore, many stability and synchronization analysis methods of linear systems can be easily extended to the study of nonlinear systems. To cope with nonlinearity, the T-S fuzzy model has been extensively studied in the processing of complex nonlinear systems, and a lot of results have been proposed in the literature [2–5]. For instance, the T-S fuzzy technology has been utilized in [2] to construct a new system model with distributed DETM and multisensor saturations. In [3], by combining the fuzzy control with the sliding mode control, a new fuzzy sliding mode control law is designed to ensure the desired ${\mathrm{\mathscr{H}}}_{\infty }$ performance.

In the industrial processes, proportional-integral-derivative (PID) control is extensively utilized owing to its simple structure, high reliability, and convenience of parameter adjustment [6]. However, with the controlled plants in modern industry becoming more and more complex, traditional PID control mechanisms may not be able to provide satisfactory control performance for complex systems. As such, many researchers have devoted to combined traditional PID control methods with other advanced control schemes to improve PID control performance [7–10]. It should be pointed out that due to the effectiveness of T-S fuzzy technology in processing nonlinear systems, much effort has been devoted to the research of nonlinear system performance under the framework of the combination of T-S fuzzy control and PID control. For example, in [7], the T-S fuzzy PID control problem is tackled in the framework of time domain, which greatly simplifies the description of the closed-loop system. Based on the T-S fuzzy PID control method, a novel multiloop fuzzy PID-like controller has been designed in [10] to improve the fault tolerance rate of the overall system. In addition, most articles on PID are based on state feedback. In practice, however, due to the existence of economic limitations or technological constraints, the system states might be immeasurable. Therefore, the observer-based PID control scheme has important research significance, and a great number of results have been available in the literature [11–14].

Different from traditional control systems, network control systems (NCSs) have significant advantages in terms of simpler installation, lower cost, and convenient maintainability. In this way, network control systems have been widely applied in various fields such as the Internet, power grids, and transportation networks. However, the communication capacity and computing resources of the network control system are usually limited, and various network-induced phenomena have inevitably appeared, such as time-varying delays [15–17], channel fading [18, 19], and actuator saturations [20]. As is well known, if handled improperly, the network-induced phenomenon may cause the decline of the performance of the entire network control system. In this case, various communication protocols have been proposed to reduce the burden of network transmission. Some rather favorable communication protocols are event-triggered protocol (ETP) [21, 22], redundant channel transmission protocol (RCTP) [23, 24], round-robin (RR) protocol [25, 26], weighted try-once-discard (WTOD) protocol [3, 27], and stochastic communication protocol (SCP) [28, 29]. In particular, under the event-triggered mechanism (ETM), the control information will be transmitted only when the measurement error value exceeds the threshold, thereby greatly reducing the data release rate and the burden on the network. It should be pointed out that the above articles are almost focused on the ETM under the nonlinear continuous systems, while the dynamic event-triggered mechanism (DETM) under the nonlinear discrete systems has not gained sufficient research attention so far despite the great practical significance. This is one of the research motivations of this article.

Moreover, as network systems become more and more open, networks security has attracted widespread attention. In particular, network attacks have become a major threat to network security, which aimed to affect control performance by destroying/modifying some important data transmitted on the network. In general, cyber-attacks include three main types, mainly, denial of service (DoS) attacks [30–32], repeated attacks [33, 34], and deception attacks [35–37]. Under the replay attacks, the attacker forces the target to receive data repeatedly. DoS attacks primarily intend to hinder the transmission over the communication networks, while deception attacks inject malicious and falsified data into the sensor or control data transmission channels. Among them, deception attacks have been widely studied for their great destructiveness to the system stability. For example, the ISS problems of the discrete-time linear PID control systems under deception attacks have been addressed [11]. Both system state saturation and deception attacks are considered for a class of time-varying stochastic nonlinear systems [35]. Unfortunately, when it comes to the observer-based PID control issues via deception attacks, only a fraction of results have been achieved in existing articles. This is another motivation of our research.

Based on the above discussion, in this article, we are interested in investigating the observer-based PID control for discrete-time T-S fuzzy systems with multisensors under deception attacks and distributed DETM. The main contributions of this study are as follows: (1) in order to reduce the burden on the network, a distributed DETM is introduced. The event generator in the framework of the discrete-time T-S fuzzy system corresponding to each sensor can determine whether to transmit data based on local information; (2) in the presence of distributed DETM, communication delay, and stochastic cyber-attacks, the input-to-state stability and $\mathrm{\varrho }$-security issue have been considered for TCS fuzzy systems in the discrete time domain; (3) a feasible PID control strategy is proposed by considering the effects of distributed DETM and deception attacks.

The remainder of this study is arranged as follows. In Section 2, a controlled model is constructed for the discrete-time T-S fuzzy system with DETM and deception attacks. Some sufficient conditions which can guarantee the ISS and $\mathrm{\varrho }$-security of the controlled systems are obtained in Section 3. A numerical example is shown in Section 4. Finally, we conclude this study in Section 5.

Notations: the notation used in this study is fairly standard except where otherwise stated. $l_{2}0,\infty$ denotes the space of square summable sequences. $·$ represents the Euclidean norm of a variable. A function $f:{ℝ}_0^{+}⟶{ℝ}_0^{+}$ is said to be a $\mathcal{Q}$-function if it satisfies $f0=0$, continuous, and strictly increasing. Furthermore, $f$ is said to be a ${\mathcal{Q}}_{\infty }$-function if it is a $\mathcal{Q}$-function and satisfies $ft⟶\infty$ when $t⟶\infty$. In addition, a function $c:{ℝ}_0^{+}\times {ℝ}_0^{+}⟶{ℝ}_0^{+}$ is said to be a $\mathrm{P}$-function if $c·,n$ is a $\mathcal{Q}$-function for each fixed $n\in {ℝ}_0^{+}$, and for each fixed $m\in {ℝ}_0^{+}$, $cm,·$ is decreasing and satisfies $cm,n⟶0$ when $n⟶\infty$. The maximum eigenvalue and minimum eigenvalue of a matrix is represented by ${\lambda }_{\max }·$ and ${\lambda }_{\min }·$, respectively. $\text{sym}X=X+X^T$ and the symbol $\inf$ represent the infimum.

2. Problem Formulation and Preliminaries

Consider the discrete-time nonlinear systems modeled by the T-S fuzzy model:

Plant rule i: if ${\eta }_{1}k$ is $g_{i1}$ and $\cdots$ and ${\eta }_{l}k$ is $g_{il}$, then$$\begin{array}{}\text{(1)}& \begin{array}{c}xk+1=A_{i}xk+A_{{\tau }_i}xk-\tau k+B_{i}uk+D_i\vartheta k,\\ yk=Cxk,\\ xl=\phi l,l=-{\tau }_M,\dots ,-\mathrm{1,0},\end{array}\end{array}$$where $g_{ij}i\in \mathbb{M}\triangleq \mathrm{1,2},\dots ,r,j\in \triangleq \mathrm{1,2},\dots ,l$ is the fuzzy set; $r$ represents the number of IF-THEN rules; $\eta k={{\eta }_{1}k,{\eta }_{2}k,\dots ,{\eta }_{l}k}^T$ denotes the vector consisting of the premise variables; $xk\in {ℝ}^{n_x}$ and $yk\in {ℝ}^n$ represent, respectively, the state vector and the measurement output; $uk\in {ℝ}^{n_u}$ is the control input; $\tau k$ denotes the discrete-time delay; and $\phi ll=-{\tau }_M,\dots ,-\mathrm{1,0}$ are the initials conditions. $\vartheta k\in l_{2}0,\infty ,{ℝ}^{n_v}$ denotes the disturbance input satisfying$$\begin{array}{}\text{(2)}& {\vartheta }^{T}k\vartheta k\le \overline{\vartheta },\end{array}$$where $\overline{\vartheta }$ is a given scalar. $A_i,A_{{\tau }_i},C,D_i$ are the known constant matrices with compatible dimensions and assume the matrix $B_i$ is of full column rank, namely, $\text{rank}{B}_i=n_u$.

Assumption 1.

(See [12]). The discrete-time delay $\tau k$ satisfies ${\tau }_m\le \tau k\le {\tau }_{M}k\in {\mathbb{N}}^{+}$, where ${\tau }_m$ and ${\tau }_M$ are the two known scalars, representing the upper bound and the lower bound of $\tau k$, respectively.

By conducting the singleton fuzzifier, center-average defuzzifier, and product fuzzy inference, system (1) can be represented by the following compact form:$$\begin{array}{}\text{(3)}& \begin{array}{c}xk+1={\displaystyle \sum _{i=1}^r{h}_i}\eta k{A}_{i}xk+A_{{\tau }_i}xk-\tau k+B_{i}uk+D_i\vartheta k,\\ yk=Cxk,\end{array}\end{array}$$where the normalized membership functions are given by$$\begin{array}{}\text{(4)}& h_i\eta k=\frac{{\displaystyle {\prod }_{j=1}^l{g}_{ij}{\eta }_{j}k}}{{\displaystyle {\sum }_{i=1}^r{\displaystyle {\prod }_{j=1}^l{g}_{ij}{\eta }_{j}k}}},\end{array}$$and $g_{ij}{\eta }_{j}k$ refers to the grade of the membership function of ${\eta }_{j}k$ in $g_{ij}$, $h_i\eta k$ satisfying $h_i\eta k\ge 0$, and ${\displaystyle {\sum }_{i=1}^r{h}_i\eta k=1}$. For simplicity, we use $h_i$ to represent $h_i\eta k$.

From Figure 1, we know that the measurement output y(k) is grouped into $yk={y_{1}k,y_{2}k,\dots ,y_{n}k}^T$. In order to ease the burden of networked channels, the distributed dynamic event-triggered mechanism is utilized to save system resources. Inspired by [20], each sensor side is equipped with an event-trigger generator to judge whether the measured signal should be transmitted or not. The event generator function for the $s$^th$s\in \mathbb{L}\triangleq \mathrm{1,2},\dots ,n$ node is given by$$\begin{array}{}\text{(5)}& f_{s}k,{\varphi }_{s}k,\zeta k={\varphi }_s^{T}k{\varphi }_{s}k-\frac{1}{\sigma }\zeta k-\epsilon y_s^{T}k{y}_{s}k,\end{array}$$where ${\varphi }_{s}k\triangleq y_s{k}_t^s-y_{s}k$; $k_t^s$ denotes the latest triggered time of the $s$^th sensor, and $y_s{k}_t^s$ is the corresponding measurement output. $\sigma$ and $\epsilon$ are the two given positive scalars. Obviously, as long as the trigger condition $f_{s}k,{\varphi }_{s}k,\zeta k>0$ is satisfied, the data will be transmitted to the buffer.

We define the triggering time sequence of the $s$^th$s\in \mathbb{L}\triangleq \mathrm{1,2},\dots ,n$ sensor as follows: $0<k_0^s<k_1^s<\cdots <k_t^s<\cdots$. Then, the next transmitted instants $k_{t+1}^s$ for the $s$^th sensor can be described as$$\begin{array}{}\text{(6)}& k_{t+1}^s=\inf k\in \mathbb{N}|k>k_t^s,f_{s}k,{\varphi }_{s}k,\zeta k>0.\end{array}$$

According to (6) and the use of the buffers, for all sensors, we can derive the following event-triggered condition:$$\begin{array}{}\text{(7)}& {\varphi }^{T}k\varphi k-\frac{n}{\sigma }\zeta k-\epsilon y^{T}kyk>0,\end{array}$$where $\varphi k={{\varphi }_1^{T}k,{\varphi }_2^{T}k,\dots ,{\varphi }_n^{T}k}^T$, $nn>0$ representing the number of sensors, and $\zeta k$ is an internal dynamical variable satisfying$$\begin{array}{}\text{(8)}& \begin{array}{c}\zeta k+1=\lambda \zeta k+\epsilon y^{T}kyk-{\varphi }^{T}k\varphi k,\\ \zeta 0=0,\end{array}\end{array}$$where $\lambda \in \mathrm{0,1}$ is a given constant.

[figure omitted; refer to PDF]

Remark 1.

It should be noted that the measured data will be transmitted to the observer when the measured data satisfy the event-triggered condition (7). Moreover, a dynamical variable $\zeta k$ is introduced in the triggering condition (7) so that each event-trigger generator can determine when transmit data in a dynamic way. In fact, the distributed DETM can be regarded as a static one when $\sigma$ approaches to infinity.

Since the network communication channel may be attacked by opponents, in this article, we assume that the deception attacks occur randomly in the transmission channel between the sensors and the observer. Then, the actual observer input $\tilde{y}k$ can be expressed as$$\begin{array}{}\text{(9)}& \tilde{y}k=\overline{y}k+\alpha k\omega k,\end{array}$$where $\overline{y}k={y_1^T{k}_t^1,y_2^T{k}_t^2,\dots ,y_n^T{k}_t^n}^T$ is the real measured data via the distributed event-triggered scheme; $\omega k$ denotes the deception attacks information launched by the attacker described by$$\begin{array}{}\text{(10)}& \omega k=-\overline{y}k+\xi k,\end{array}$$where $\xi k\ne 0$ represents an arbitrary bounded signal satisfying$$\begin{array}{}\text{(11)}& {\xi }^{T}k\xi k\le \overline{\xi },\end{array}$$where $\overline{\xi }$ is a given positive scalar. $\alpha k\in \mathrm{0,1}$ is a Bernoulli distributed variable with the following probabilities:$$\begin{array}{c}\text{(12)}& Prob\alpha k=1=\overline{\alpha },Prob\alpha k=0=1-\overline{\alpha },\end{array}$$where $\overline{\alpha }\in \mathrm{0,1}$ is a known constant, and the expectation of $\alpha k$ is easily obtained as $\mathbb{E}\alpha k=\overline{\alpha }$.

Remark 2.

As is well known, cyber-attacks are inevitable in the network. Among various attacks, the deception attacks are considered to be the easiest to be sent by an attacker and the most destructive. Therefore, in this article, we consider that the transmission channel between the sensors and the observer is subject to deception attacks. Moreover, in engineering practice, since there exists security protection from the protection institution, the attacks launched by attackers might not be always successful. In this case, the attacks could occur in a random manner. Therefore, the randomly occurring deception attacks are described as a Bernoulli sequence in this study, and the attack information $\xi k$ sent by the attacker is bounded.

Considering that in industrial practice, not all systems state can be measured directly, then a fuzzy observer is given as follows.

Observer rule i: if ${\eta }_{1}k$ is $g_{i1}$ and $\cdots$ and ${\eta }_{l}k$ is $g_{il}$, then$$\begin{array}{}\text{(13)}& \begin{array}{c}\widehat{x}k+1=A_i\widehat{x}k+A_{{\tau }_i}\widehat{x}k-\tau k+B_{i}uk+{\displaystyle \sum _{j=1}^r{h}_j{L}_j}\tilde{y}k-\widehat{y}k,\\ \widehat{y}k=C\widehat{x}k,\\ \widehat{x}l=0,l=-{\tau }_M,\dots ,-\mathrm{1,0},\end{array}\end{array}$$where $\widehat{x}k\in {ℝ}^{n_x}$ is the observer states of $xk$; $L_j$ denotes the observer gains which will be designed later. Then, the fuzzy observer can be constructed as follows:$$\begin{array}{}\text{(14)}& \begin{array}{c}\widehat{x}k+1={\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^r{h}_i{h}_j}}{A}_i\widehat{x}k+A_{{\tau }_i}\widehat{x}k-\tau k+B_{i}uk+L_j\tilde{y}k-\widehat{y}k,\\ \widehat{y}k=C\widehat{x}k.\end{array}\end{array}$$

In this study, for the discrete-time T-S fuzzy system (1), we adopt the following observer-based PID control law.

Controller rule i: if ${\eta }_{1}k$ is $g_{i1}$ and $\cdots$ and ${\eta }_{l}k$ is $g_{il}$, then$$\begin{array}{}\text{(15)}& uk=K_{P_i}\widehat{x}k+K_{I_i}{\displaystyle \sum _{m=k-d}^{k-1}\widehat{x}m+K_{D_i}\widehat{x}k-\widehat{x}k-1},\end{array}$$where $K_{P_i},K_{I_i},K_{D_i}$ are the controller gains which will be designed later; $d$ is a known scalar representing the length of time-windows, and we assume that $d\ge {\tau }_M$ in this study.

The considered fuzzy PID control law can be rewritten as follows:$$\begin{array}{}\text{(16)}& uk={\displaystyle \sum _{i=1}^r{h}_i}{K}_{P_i}\widehat{x}k+K_{I_i}{\displaystyle \sum _{m=k-d}^{k-1}\widehat{x}m}+K_{D_i}\widehat{x}k-\widehat{x}k-1.\end{array}$$

Remark 3.

In this study, we consider the observer-based PID controller for discrete-time T-S fuzzy systems. The PID controller consists of three parts. Specifically, the components $K_{P_i}\widehat{x}k,K_{I_i}{\displaystyle {\sum }_{m=k-d}^{k-1}\widehat{x}m}$, and $K_{D_i}\widehat{x}k-\widehat{x}k-1$ are the proportional term, integral term, and derivative term, respectively. The gain matrices of the above three terms can be adjusted to meet different requirements in practical systems.

Denoting $ek\triangleq xk-\widehat{x}k$, taking (3), (9), (10), and (14) into account, the estimation error system could be described as follows:$$\begin{array}{}\text{(17)}& ek+1={\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^r\begin{array}{c}{A}_{i}ek+A_{{\tau }_i}ek-\tau k+D_i\vartheta k-1-\alpha k\times L_j\varphi k\\ -1-\alpha k{L}_{j}Cxk-\alpha k{L}_j\xi k+L_{j}C\widehat{x}k\end{array}}}.\end{array}$$

Now, for simplicity to the following work, we introduce two augmentation vectors:$$\begin{array}{c}\text{(18)}& \tilde{x}k={x^{T}k{e}^{T}k}^T,\Phi k={{\vartheta }^{T}k{\xi }^{T}k}^T.\end{array}$$

Then, based on estimation error system (17) and system (3), we can obtain the closed-loop system of the following form:$$\begin{array}{}\text{(19)}& \begin{array}{c}\tilde{x}k+1={\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^r{h}_i{h}_j\begin{array}{c}{\mathcal{A}}_{ij}+\tilde{\alpha }k{\overline{\mathcal{A}}}_j\tilde{x}k+{\mathrm{ℬ}}_i\tilde{x}k-\tau k\\ +{\mathcal{D}}_{ij}+\tilde{\alpha }k{\overline{\mathcal{D}}}_j\Phi k\\ +{\mathrm{ℒ}}_j+\tilde{\alpha }k{\overline{\mathrm{ℒ}}}_j\varphi k+{\mathrm{ℛ}}_{ij}\Psi k\end{array},}}\\ \tilde{x}l=gl,l=-d,\dots ,-\mathrm{1,0}\end{array},\end{array}$$where$$\begin{array}{c}\text{(20)}& {\mathcal{A}}_{ij}=\begin{array}{cc}{A}_i+B_i{K}_{P_j}+K_{D_j}& -B_i{K}_{P_j}+K_{D_j}\\ \overline{\alpha }{L}_{j}C& A_i-L_{j}C\end{array},{\overline{\mathcal{A}}}_j=\begin{array}{cc}{0}_{n_x\times n_x}& 0_{n_x\times n_x}\\ L_{j}C& 0_{n_x\times n_x}\end{array}\\ {\mathrm{ℬ}}_i=\begin{array}{cc}{A}_{{\tau }_i}& 0_{n_x\times n_x}\\ 0_{n_x\times n_x}& A_{{\tau }_i}\end{array},\\ {\mathcal{D}}_{ij}=\begin{array}{cc}{D}_i& 0_{n_x\times n_y}\\ D_i& -\overline{\alpha }{L}_j\end{array},\\ {\overline{\mathcal{D}}}_j=\begin{array}{cc}{0}_{n_x\times n_v}& 0_{n_x\times n_y}\\ 0_{n_x\times n_v}& -L_j\end{array},\\ {\mathrm{ℒ}}_j=\begin{array}{c}{0}_{n_x\times n_y}\\ \overline{\alpha }-1L_j\end{array},\\ {\overline{\mathrm{ℒ}}}_j=\begin{array}{c}{0}_{n_x\times n_y}\\ L_j\end{array},\\ {\mathrm{ℛ}}_{ij}=\begin{array}{c}{B}_i{M}_j\\ 0_{n_x\times 2d{n}_x}\end{array},\\ \Psi k={{\tilde{x}}^{T}k-1{\tilde{x}}^{T}k-2\dots {\tilde{x}}^{T}k-d}^T,\\ gl={\phi l^T\phi l^T}^T,\\ M_j=M_{1j}-M_{2j}\underset{d-1}{\underbrace{M_{1j}{M}_{1j}\dots M_{1j}}},\\ M_{1j}=K_{I_j}-K_{I_j},\\ M_{2j}=K_{D_j}-K_{D_j},\\ \tilde{\alpha }k=\alpha k-\overline{\alpha },\\ \mathbb{E}{\tilde{\alpha }}^{2}k\triangleq \widehat{\alpha }.\end{array}$$

It follows from (2) and (11) and one has that$$\begin{array}{c}\text{(21)}& {\Phi }^{T}k\Phi k\le \overline{\Phi },\overline{\Phi }=\overline{\vartheta }+\overline{\xi }.\end{array}$$

Remark 4.

Up to now, by considering the influences of time-varying delays and deception attacks, a novel observer-based PID security control model is established for the discrete-time T-S fuzzy system under distributed DETM.

Definition 1.

(See [11]). System (19) is said to be input-to-state stable (ISS) in mean-square sense if there exist functions $\kappa \in \mathcal{QJ}$ and $\iota \in {\mathcal{Q}}_{\infty }$, such that the system dynamic $\tilde{x}k$ satisfies the following inequality:$$\begin{array}{}\text{(22)}& \mathbb{E}{\tilde{x}k}^2\le \kappa \underset{l\in -d,0}{\max }\mathbb{E}{gl}^2,k+\iota {\Phi k}_{\infty }^2,\hspace{1em}k\ge 0,\end{array}$$where ${\mathrm{\Phi }k}_{\infty }={\text{sup}}_k\mathrm{\Phi }k$.

Definition 2.

(See [11]). Denote the desired safety level as $\mathrm{\varrho }>0$; the closed-loop system (19) is said to be mean-square $\mathrm{\varrho }$-secure if the following inequality holds:$$\begin{array}{}\text{(23)}& \mathbb{E}{\tilde{x}k}^2\le \mathrm{\varrho },\hspace{1em}\forall k\ge 0.\end{array}$$

Lemma 1.

(See [38]). For positive definite matrix Z with suitable dimensions and any real-valued variables $x$, $y$, the following inequality holds:$$\begin{array}{}\text{(24)}& x^{T}Zy+y^{T}Zx\le \mathrm{ϵ}{x}^{T}Zx+\frac{1}{\mathrm{ϵ}}{y}^{T}Zy,\end{array}$$where $ϵ>0$ is a given constant.

Lemma 2.

(See [39]). Suppose the matrix M with appropriate dimensions; the following two items are equivalent:

(1) There exist two symmetric and positive-definitive matrices $R$, $G$ such that$$\begin{array}{}\text{(25)}& \begin{array}{cc}-R& M^T\\ \ast & -G^{-1}\end{array}<0.\end{array}$$

(2) There exist two symmetric and positive-definitive matrices $R$, $G$ and constant matrix $Z$ satisfying

$$\begin{array}{}\text{(26)}& \begin{array}{cc}-R& {ZM}^T\\ \ast & sym-Z+G\end{array}<0,\end{array}$$

Lemma 3.

(See [11]). For the DETC scheme consisting of (7) and (8) with the initial value $\zeta 0\ge 0$, the internal dynamic variable satisfies $\zeta k\ge 0$ for all $k\ge 0$ if the parameters $\sigma \sigma >0$ and $\lambda 0<\lambda <1$ satisfy $\lambda \sigma \ge 1$.

3. Main Results

We mainly consider the security performance of the controlled plant (1) in this section. First, the sufficient condition is given to guarantee that the closed-loop system (19) is ISS and $\mathrm{\varrho }$-secure in mean-square sense. Then, the desired observer gains and PID controller gains are designed.

Theorem 1.

Let the scalars $\lambda 0<\lambda <1$ and $\sigma \sigma >0$ satisfy $\lambda \sigma \ge 1$, the observer gain $L_j$, and the PID controller gains $K_{P_j}$, $K_{I_j}$, $K_{D_j}$ given. For scalar $\mathrm{\varrho }$ ($\mathrm{\varrho }$ > 0), the closed-loop system (19) is $\mathrm{\varrho }$-secure in mean-square sense if, for all $s\in \mathbb{L}$, $i,j\in \mathbb{M}$, there exist positive definite matrices $P$, $Q_{\overline{r}}\overline{r}=\mathrm{1,2},\dots ,d$, $R$, positive integer $n$, and positive scalars $\epsilon$, $\theta$, $ϵ$, ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ satisfying$$\begin{array}{}\text{(27)}& {\Pi }_{ij}^1+{\Pi }_{ji}^1<0,i\le j,\text{(28)}& {\Pi }_{ij}^2+{\Pi }_{ji}^2<0,i\le j,\text{(29)}& \frac{\lambda +n\theta -1}{\sigma }<-{\gamma }_3.\text{(30)}& \frac{\varpi {\gamma }_2\overline{\Phi }}{\underset{¯}{\lambda }\varpi -1}+\frac{\overline{\lambda }+\varpi \overline{\gamma }-\varpi -1{\overline{\rho }}_1}{\underset{¯}{\lambda }}\underset{l\in -d,0}{\text{max}}\mathbb{E}{gl}^2\le \mathrm{\varrho },\end{array}$$where$$\begin{array}{c}\text{(31)}& {\Pi }_{ij}^1=\begin{array}{cc}{\Pi }^{111}& \ast \\ {\Pi }_{ij}^{121}& {\Pi }^{122}\end{array},{\Pi }_{ij}^2=\begin{array}{cc}-{\gamma }_2{I}_{n_v+n_y}& \ast \\ {\Pi }_{ij}^{221}& {\Pi }^{222}\end{array},\end{array}$$in which$$\begin{array}{c}\text{(32)}& {\Pi }^{111}=diag-P+\overline{Q}+\overline{\tau }R+\overline{\epsilon }{ℂ}^Tℂ+{\gamma }_1{I}_{2n_x\times 2n_x},-R,-Q,-\frac{1}{\sigma }+\theta I_{n_y}{\Pi }_{ij}^{121}=\begin{array}{cccc}P{\mathcal{A}}_{ij}& P{\mathrm{ℬ}}_i& P{\mathrm{ℛ}}_{ij}& P{\mathrm{ℒ}}_j\\ \sqrt{\mathrm{ϵ}}P{\mathcal{A}}_{ij}& \sqrt{\mathrm{ϵ}}P{\mathrm{ℬ}}_i& \sqrt{\mathrm{ϵ}}P{\mathrm{ℛ}}_{ij}& \sqrt{\mathrm{ϵ}}P{\mathrm{ℒ}}_j\\ \sqrt{\widehat{\alpha }}P\overline{{\mathcal{A}}_j}& 0_{2n_x\times 2n_x}& 0_{2n_x\times 2d{n}_x}& \sqrt{\widehat{\alpha }}P\overline{{\mathrm{ℒ}}_j}\\ \sqrt{\mathrm{ϵ}\widehat{\alpha }}P\overline{{\mathcal{A}}_j}& 0_{2n_x\times 2n_x}& 0_{2n_x\times 2d{n}_x}& \sqrt{\mathrm{ϵ}\widehat{\alpha }}P{\overline{\mathrm{ℒ}}}_j\end{array}\\ {\Pi }^{122}=-I_4\otimes P,\\ {\Pi }_{ij}^{221}=\begin{array}{c}\sqrt{1+1/\mathrm{ϵ}}P{\mathcal{D}}_{ij}\\ \sqrt{\widehat{\alpha }1+1/\mathrm{ϵ}}P{\overline{\mathcal{D}}}_j\end{array},\\ {\Pi }^{222}=-I_2\otimes P,\\ \overline{Q}={\displaystyle \sum _{\overline{r}=1}^d{Q}_{\overline{r}}},\\ ℂ=C{0}_{n_y\times n_x},\\ \overline{\tau }={\tau }_M-{\tau }_m+1,\\ \overline{\epsilon }=\epsilon \frac{1}{\sigma }+\theta ,\\ Q=diag{Q}_1,Q_2\dots s{Q}_d,\\ \overline{{\rho }_1}=max{\lambda }_{\max }P,\frac{1}{\sigma },\\ P=diag\widehat{P},\widehat{P},\\ \overline{{\rho }_2}=max{\lambda }_{\max }Qd,{\lambda }_{\max }R{\tau }_M-{\tau }_m+1,\end{array}$$

Proof.

Consider the Lyapunov functional as follows:$$\begin{array}{}\text{(33)}& Vk={\displaystyle \sum _{i=1}^5{V}_i}k,\end{array}$$where$$\begin{array}{}\text{(34)}& \begin{array}{c}{V}_{1}k={\tilde{x}}^{T}kP\tilde{x}k\\ V_{2}k={\displaystyle \sum _{\overline{r}=1}^d{\displaystyle \sum _{q=k-\overline{r}}^{k-1}{\tilde{x}}^{T}q{Q}_{\overline{r}}\tilde{x}q}}\\ V_{3}k={\displaystyle \sum _{q=k-\tau k}^{k-1}{\tilde{x}}^{T}qR\tilde{x}q}\\ V_{4}k={\displaystyle \sum _{l=k-{\tau }_M+1}^{k-{\tau }_m}{\displaystyle \sum _{q=l}^{k-1}{\tilde{x}}^{T}qR\tilde{x}q}}\\ V_{5}k=\frac{1}{\sigma }\zeta k.\end{array}\end{array}$$

First, calculating the difference and mathematical expectation of $Vk$, we have$$\begin{array}{c}\text{(35)}& E\Delta V1k=E{\tilde{x}}^{T}k+1P\tilde{x}k+1-\tilde{x}{k}^{T}P\tilde{x}k=E\begin{array}{c}{\begin{array}{c}{A}_{ij}+nk{\text{Ā}}_j\tilde{x}k+B_i\tilde{x}k-Tk+D_{ij}+\text{ã\hspace{0.17em}}k{\overline{D}}_j\Phi k\\ +L_j+\text{ã}k{\overline{L}}_j\varphi k+R_{ij}\Psi k\end{array}}^T\\ P\begin{array}{c}{A}_{ij}+\text{ã\hspace{0.17em}}k{\text{Ā}}_j\tilde{x}k+B_i\tilde{x}k-Tk+D_{ij}+\text{ã\hspace{0.17em}}k{\overline{D}}_j\Phi k\\ +L_j+\text{ã\hspace{0.17em}}k{\overline{L}}_j\varphi k+R_{ij}\Psi k-\tilde{x}{k}^{T}P\tilde{x}k\end{array}\end{array}\\ ={\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^r{h}_i{h}_j\begin{array}{c}{\tilde{x}}^{T}k{A}_{ij}^{T}P{A}_{ij}+\text{ã}{\text{Ā}}_j^{T}P\text{Ā}j-P\tilde{x}k\\ +{\tilde{x}}^{T}k-Tk{B}_i^{T}P{B}_i\tilde{x}k-k+{\Phi }^{T}k{D}_{ij}^{T}P{D}_{ij}+\text{ã}{D}_j^{T}P{\overline{D}}_j\\ +\Psi k+{\varphi }^{T}k{L}_j^{T}P{L}_j+\text{ã}{\overline{L}}_j^{T}P{\overline{L}}_j\varphi k\\ +{\Psi }^{T}k{R}_{ij}^{T}P{R}_{ij}\Psi k+2{\tilde{x}}^{T}k{A}_{ij}^{T}P{B}_i{\tilde{x}}^{T}k-Tk\\ +2{\tilde{x}}^{T}k{A}_{ij}^{T}P{D}_{ij}+\text{ã}{\text{Ā}}_j^{T}P{D}_j\Phi k\\ +2{\tilde{x}}^{T}k{A}_{ij}^{T}P{L}_j+\text{ã}{A}_j^{T}P{\overline{L}}_j\varphi k\\ +2{\tilde{x}}^{T}k{A}_{ij}^{T}P{R}_{ij}\Psi k+2{\tilde{x}}^{T}k-Tk\\ \times B_i^{T}P{D}_{ij}\Phi k+2{\tilde{x}}^{T}k-Tk{B}_i^{T}PLj\varphi k\\ +2{\varphi }^{T}k{D}_{ij}^{T}P{R}_{ij}\Psi k+2^{T}k{L}_j^{T}P{R}_{ij}\Psi k\end{array}.}}\\ \text{(36)}& \mathbb{E}△V_{2}k=\mathbb{E}{\displaystyle \sum _{\overline{r}=1}^d{\displaystyle \sum _{q=k-\overline{r}+1}^k{\tilde{x}}^{T}q{Q}_{\overline{r}}\tilde{x}q}}-{\displaystyle \sum _{\overline{r}=1}^d{\displaystyle \sum _{q=k-\overline{r}}^{k-1}{\tilde{x}}^{T}q{Q}_{\overline{r}}\tilde{x}q}}={\displaystyle \sum _{\overline{r}=1}^d\mathbb{E}{\displaystyle \sum _{q=k-\overline{r}+1}^k{\tilde{x}}^{T}q{Q}_{\overline{r}}\tilde{x}q}-{\displaystyle \sum _{q=k-\overline{r}}^{k-1}{\tilde{x}}^{T}q{Q}_{\overline{r}}\tilde{x}q}}\\ ={\displaystyle \sum _{\overline{r}=1}^d\mathbb{E}{\tilde{x}}^{T}k{Q}_{\overline{r}}\tilde{x}k-{\tilde{x}}^{T}k-\overline{r}{Q}_{\overline{r}}\tilde{x}k-\overline{r},}\\ \text{(37)}& \mathbb{E}△V_{3}k=\mathbb{E}{\displaystyle \sum _{q=k+1-\tau k+1}^k{\tilde{x}}^{T}qR\tilde{x}q}-{\displaystyle \sum _{q=k-\tau k}^{k-1}{\tilde{x}}^{T}qR\tilde{x}q}=\mathbb{E}\begin{array}{c}{\tilde{x}}^{T}kR\tilde{x}k-{\tilde{x}}^{T}k-\tau kR\tilde{x}k-\tau k\\ +{\displaystyle \sum _{q=k+1-\tau k+1}^{k-1}{\tilde{x}}^{T}qR\tilde{x}q}-{\displaystyle \sum _{q=k+1-\tau k}^{k-1}{\tilde{x}}^{T}qR\tilde{x}q}\end{array}\\ \le \mathbb{E}\begin{array}{c}{\tilde{x}}^{T}kR\tilde{x}k-{\tilde{x}}^{T}k-\tau kR\tilde{x}k-\tau k\\ +{\displaystyle \sum _{q=k-{\tau }_M+1}^{k-{\tau }_m}{\tilde{x}}^{T}qR\tilde{x}q}\end{array},\\ \text{(38)}& \mathbb{E}△V_{4}k=\mathbb{E}{\displaystyle \sum _{l=k-{\tau }_M+2}^{k-{\tau }_m+1}{\displaystyle \sum _{q=l}^k{\tilde{x}}^{T}qR\tilde{x}q}}-{\displaystyle \sum _{l=k-{\tau }_M+1}^{k-{\tau }_m}{\displaystyle \sum _{q=l}^{k-1}{\tilde{x}}^{T}qR\tilde{x}q}}=\mathbb{E}\begin{array}{c}{\displaystyle \sum _{l=k-{\tau }_M+1}^{k-{\tau }_m}{\displaystyle \sum _{q=l}^k{\tilde{x}}^{T}qR\tilde{x}q}}-{\displaystyle \sum _{q=k-{\tau }_M+1}^k{\tilde{x}}^{T}qR\tilde{x}q}\\ +{\displaystyle \sum _{q=k-{\tau }_m+1}^k{\tilde{x}}^{T}qR\tilde{x}q}-{\displaystyle \sum _{l=k-{\tau }_M+1}^{k-{\tau }_m}{\displaystyle \sum _{q=l}^{k-1}{\tilde{x}}^{T}qR\tilde{x}q}}\end{array}\\ =\mathbb{E}{\displaystyle \sum _{l=k-{\tau }_M+1}^{k-{\tau }_m}{\tilde{x}}^{T}qR\tilde{x}q}-{\displaystyle \sum _{q=k-{\tau }_M+1}^{k-{\tau }_m}{\tilde{x}}^{T}qR\tilde{x}q}\\ =\mathbb{E}{\tau }_M-{\tau }_m{\tilde{x}}^{T}qR\tilde{x}q-{\displaystyle \sum _{q=k-{\tau }_M+1}^{k-{\tau }_m}{\tilde{x}}^{T}qR\tilde{x}q},\\ \text{(39)}& \mathbb{E}△V_{5}k=\mathbb{E}\frac{1}{\sigma }\zeta k+1-\frac{1}{\sigma }\zeta k=\mathbb{E}\frac{1}{\sigma }\lambda \zeta k+\epsilon y^{T}kyk-{\varphi }^{T}k\varphi k-\zeta k\\ =\mathbb{E}\frac{\lambda -1}{\sigma }\zeta k+\frac{\epsilon }{\sigma }{x}^{T}k{C}^{T}Cxk-\frac{1}{\sigma }{\varphi }^{T}k\varphi k.\end{array}$$

Then, taking (7), (33)–(39) into account, we can obtain$$\begin{array}{}\text{(40)}& \mathbb{E}△Vk\le {\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^r{h}_i{h}_j}}{\psi }^{T}k{\Lambda }_1+{\Lambda }_2+{\Lambda }_3\psi k+2{\psi }^{T}k{\Lambda }_4\Phi k+{\Phi }^{T}k{\Lambda }_5\Phi k+\frac{\lambda +n\theta -1}{\sigma }\zeta k,\end{array}$$where$$\begin{array}{c}\text{(41)}& \psi k={\tilde{x}}^{T}k{\tilde{x}}^{T}k-\tau k{\Psi }^{T}k{\varphi }^T{k}^T\text{,}{\Lambda }_1=\text{diag}-P+\overline{Q}+\overline{\tau }R+\overline{\epsilon }{ℂ}^Tℂ,-R,-Q,-\frac{1}{\sigma }+\theta I_{n_y},\\ {\Lambda }_2=\begin{array}{cccc}{\mathcal{A}}_{ij}^{T}P{\mathcal{A}}_{ij}& \ast & \ast & \ast \\ {\mathrm{ℬ}}_i^{T}P{\mathcal{A}}_{ij}& {\mathrm{ℬ}}_i^{T}P{\mathrm{ℬ}}_i& \ast & \ast \\ {\mathrm{ℛ}}_{ij}^{T}P{\mathcal{A}}_{ij}& {\mathrm{ℛ}}_{ij}^{T}P{\mathrm{ℬ}}_i& {\mathrm{ℛ}}_{ij}^{T}P{\mathrm{ℛ}}_{ij}& \ast \\ {\mathrm{ℒ}}_j^{T}P{\mathcal{A}}_{ij}& {\mathrm{ℒ}}_j^{T}P{\mathrm{ℬ}}_i& {\mathrm{ℒ}}_j^{T}P{\mathrm{ℛ}}_{ij}& {\mathrm{ℒ}}_j^{T}P{\mathrm{ℒ}}_j\end{array},\\ {\Lambda }_3=\begin{array}{cccc}\widehat{\alpha }{\overline{\mathcal{A}}}_j^{T}P{\overline{\mathcal{A}}}_j^T& \ast & \ast & \ast \\ 0_{2n_x\times 2n_x}& 0_{2n_x\times 2n_x}& \ast & \ast \\ 0_{2d{n}_x\times 2n_x}& 0_{2d{n}_x\times 2n_x}& 0_{2d{n}_x\times 2d{n}_x}& \ast \\ \widehat{\alpha }{\overline{\mathrm{ℒ}}}_j^{T}P{\overline{\mathcal{A}}}_j^T& 0_{n_y\times 2n_x}& 0_{n_y\times 2d{n}_x}& \widehat{\alpha }{\overline{\mathrm{ℒ}}}_j^{T}P{\overline{\mathrm{ℒ}}}_j^T\end{array},\\ {\Lambda }_4={\Sigma }_1^T\tilde{P}{\Sigma }_1,\\ {\Lambda }_5={\Sigma }_2^T\tilde{P}{\Sigma }_2,\\ {\Sigma }_2=\begin{array}{c}{\mathcal{D}}_{ij}\\ \sqrt{\widehat{\alpha }}{\overline{\mathcal{D}}}_j\end{array},\\ \tilde{P}=I_2\otimes P,\\ {\Sigma }_1=\begin{array}{cccc}{\mathcal{A}}_{ij}& {\mathrm{ℬ}}_i& {\mathrm{ℛ}}_{ij}& {\mathrm{ℒ}}_j\\ \sqrt{\widehat{\alpha }}\overline{{\mathcal{A}}_j}& 0_{2n_x\times 2n_x}& 0_{2n_x\times 2d{n}_x}& \sqrt{\widehat{\alpha }}\overline{{\mathrm{ℒ}}_j}\end{array}.\end{array}$$

According to Lemma 1, the term $2{\psi }^{T}k{\mathrm{\Lambda }}_4\mathrm{\Phi }k$ can be computed as$$\begin{array}{}\text{(42)}& 2{\psi }^{T}k{\Lambda }_4\Phi k=2{\psi }^{T}k{\Sigma }_1^T\tilde{P}{\Sigma }_2\Phi k\le \mathrm{ϵ}{\psi }^{T}k{\Sigma }_1^T\tilde{P}{\Sigma }_1\psi k+\frac{1}{\mathrm{ϵ}}{\Phi }^{T}k{\Sigma }_2^T\tilde{P}{\Sigma }_2\Phi k.\end{array}$$

By combining (40) and (41), the inequality (40) can be described as$$\begin{array}{}\text{(43)}& \mathbb{E}△Vk\le {\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^r\mathbb{E}\begin{array}{c}{\psi }^{T}k{\Theta }_1\psi k\\ +{\Phi }^{T}k{\Theta }_2\Phi k+\frac{\lambda +n\theta -1}{\sigma }\zeta k\end{array}}},\end{array}$$where$$\begin{array}{c}\text{(44)}& {\Theta }_1={\Lambda }_1+{\Lambda }_2+{\Lambda }_3+\mathrm{ϵ}{\Sigma }_1^T\tilde{P}{\Sigma }_1,{\Theta }_2=1+\frac{1}{\mathrm{ϵ}}{\Lambda }_5.\end{array}$$

Therefore, by using Schur complement and conditions (27)–(29), we know that$$\begin{array}{c}\text{(45)}& \mathbb{E}△Vk\le {\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^r\mathbb{E}{\psi }^{T}k{\Theta }_1\psi k+{\Phi }^{T}k{\Theta }_2\Phi k+\frac{\lambda +n\theta -1}{\sigma }\zeta k}}\le \mathbb{E}-{\gamma }_1{\tilde{x}k}^2+{\gamma }_2{\Phi k}_{\infty }^2-{\gamma }_3\zeta k\\ \le -\overline{\gamma }\mathbb{E}{\Omega k}^2+{\gamma }_2{\Phi k}_{\infty }^2,\end{array}$$where$$\begin{array}{c}\text{(46)}& \overline{\gamma }=\min {\gamma }_1,{\gamma }_3,\Omega k={{\tilde{x}}^{T}k{{\zeta }^{1/2}k}^T}^T.\end{array}$$

Next, we will discuss the ISS problem of the closed-loop system (19). According to (33), we obtain$$\begin{array}{}\text{(47)}& \mathbb{E}Vk\le {\overline{\rho }}_1\mathbb{E}{\Omega k}^2+{\overline{\rho }}_2{\displaystyle \sum _{q=k-d}^{k-1}\mathbb{E}{\Omega q}^2},\end{array}$$where$$\begin{array}{c}\text{(48)}& {\overline{\rho }}_1=\max {\lambda }_{\max }P,\frac{1}{\sigma },{\overline{\rho }}_2=\max {\lambda }_{\max }Qd,{\lambda }_{\max }R\overline{\tau }.\end{array}$$

For any $\rho >1$, considering (45) and (47), it can be derived that$$\begin{array}{c}\text{(49)}& \mathbb{E}{\rho }^{k+1}Vk+1-\mathbb{E}{\rho }^{k}Vk={\rho }^{k+1}\mathbb{E}△Vk+{\rho }^k\rho -1\mathbb{E}Vk\le {\rho }^{k+1}-\overline{\gamma }\mathbb{E}{\Omega k}^2+{\gamma }_2{\Phi k}_{\infty }^2\\ +{\rho }^k\rho -1\times {\overline{\rho }}_1\mathbb{E}{\Omega k}^2+{\overline{\rho }}_2{\displaystyle \sum _{q=k-d}^{k-1}\mathbb{E}}{\Omega q}^2\\ ={\sigma }_1\rho {\rho }^k\mathbb{E}{\Omega k}^2+{\sigma }_2\rho {\displaystyle \sum _{q=k-d}^{k-1}{\rho }^k\mathbb{E}{\Omega \rho }^2+{\rho }^{k+1}{\gamma }_2{\Phi k}_{\infty }^2,}\end{array}$$where ${\sigma }_1\rho =-\rho \overline{\gamma }+{\overline{\rho }}_1\rho -1,{\sigma }_2\rho ={\overline{\rho }}_2\rho -1$. In the following, for any integer $N\ge d+1$, summing up both sides of (49) from 0 to $N-1$ with respect to $k$ results in$$\begin{array}{c}\text{(50)}& \mathbb{E}{\rho }^{N}VN-\mathbb{E}V0\le {\sigma }_1\rho {\displaystyle \sum _{k=0}^{N-1}{\rho }^k\mathbb{E}{\Omega k}^2}+{\sigma }_2\rho {\displaystyle \sum _{k=0}^{N-1}{\displaystyle \sum _{q=k-d}^{k-1}{\rho }^k}}\mathbb{E}{\Omega \rho }^2+\frac{\rho 1-{\rho }^N}{1-\rho }{\gamma }_2{\Phi k}_{\infty }^2.\end{array}$$

It can be checked that$$\begin{array}{c}\text{(51)}& {\displaystyle \sum _{k=0}^{N-1}{\displaystyle \sum _{q=k-d}^{k-1}{\rho }^k\mathbb{E}{\Omega q}^2}}\le {\displaystyle \sum _{q=-d}^{-1}{\displaystyle \sum _{k=0}^{q+d}+}}{\displaystyle \sum _{q=0}^{N-d-1}{\displaystyle \sum _{k=q+1}^{q+d}+}}{\displaystyle \sum _{q=N-d}^{N-1}{\displaystyle \sum _{k=q+1}^{N-1}\times {\rho }^k\mathbb{E}{\Omega q}^2}}\le d{\displaystyle \sum _{q=-d}^{-1}{\rho }^{q+d}}\mathbb{E}{\Omega q}^2+d{\displaystyle \sum _{q=0}^{N-d-1}{\rho }^{q+d}}\mathbb{E}{\Omega q}^2+d{\displaystyle \sum _{q=N-d}^{N-1}{\rho }^{q+d}}\mathbb{E}{\Omega q}^2\\ \le d{\rho }^d\underset{-d\le q\le 0}{\text{max}}\mathbb{E}{\Omega q}^2+d{\rho }^d{\displaystyle \sum _{k=0}^{N-1}{\rho }^k}\mathbb{E}{\Omega k}^2.\end{array}$$

Thus, it implies from (50) and (51) that$$\begin{array}{c}\text{(52)}& \mathbb{E}{\rho }^{N}VN-\mathbb{E}V0\le {\pi }_1\rho {\displaystyle \sum _{k=0}^{N-1}{\rho }^k}\mathbb{E}{\Omega k}^2+{\pi }_2\rho \underset{-d\le q\le 0}{\text{max}}\mathbb{E}{\Omega q}^2+\frac{\rho 1-{\rho }^N}{1-\rho }{\gamma }_2{\Phi k}_{\infty }^2,\end{array}$$where ${\pi }_1\rho ={\sigma }_1\rho +d{\rho }^d{\sigma }_2\rho ,{\pi }_2\rho =d{\rho }^d{\sigma }_2\rho$.

Since ${\pi }_{1}1=-\overline{\gamma }<0$ and ${\text{lim}}_{\rho ⟶\infty }{\pi }_1\rho =+\infty$, there exists a scalar $\varpi >1$, such that ${\pi }_1\varpi =0$, which yields$$\begin{array}{c}\text{(53)}& \mathbb{E}{\varpi }^{N}VN-\mathbb{E}V0\le \frac{\varpi 1-{\varpi }^N}{1-\varpi }{\gamma }_2{\Phi k}_{\infty }^2+{\pi }_2\varpi \underset{-d\le q\le 0}{\text{max}}\mathbb{E}{\Omega q}^2.\end{array}$$

Then, it is obvious that$$\begin{array}{}\text{(54)}& \mathbb{E}V0\le \overline{\lambda }\underset{l\in -d,0}{\text{max}}\mathbb{E}{gl}^2.\text{(55)}& \mathbb{E}VN\ge \underset{¯}{\lambda }\mathbb{E}{\Omega N}^2,\end{array}$$where$$\begin{array}{c}\text{(56)}& \overline{\lambda }=\max {\lambda }_{\max }P,d{\lambda }_{\max }Q,d{\lambda }_{\max }R,\underset{¯}{\lambda }=\min {\lambda }_{\min }p,\frac{1}{\sigma }.\end{array}$$

Taking (53)–(55) into consideration, we have$$\begin{array}{c}\text{(57)}& \mathbb{E}{\Omega N}^2\le \frac{\varpi 1-{\varpi }^N}{\underset{¯}{\lambda }{\varpi }^{N}1-\varpi }{\gamma }_2{\Phi k}_{\infty }^2+\frac{{\pi }_2\varpi }{\underset{¯}{\lambda }{\varpi }^N}\underset{-d\le q\le 0}{\text{max}}\mathbb{E}{\Omega q}^2+\frac{\overline{\lambda }}{\underset{¯}{\lambda }{\varpi }^N}\underset{l\in -d,0}{\text{max}}\mathbb{E}{gl}^2\le \frac{\varpi }{\underset{¯}{\lambda }\varpi -1}{\gamma }_2{\Phi k}_{\infty }^2\\ +\frac{\overline{\lambda }+\varpi \overline{\gamma }-{\overline{\rho }}_1\varpi -1}{\underset{¯}{\lambda }{\varpi }^N}\underset{l\in -d,0}{\text{max}}\mathbb{E}{gl}^2\\ =\iota {\Phi k}_{\infty }^2+\kappa \underset{l\in -d,0}{\text{max}}\mathbb{E}{gl}^2,N,\end{array}$$where$$\begin{array}{}\text{(58)}& \iota p\triangleq \frac{\varpi }{\underset{¯}{\lambda }\varpi -1}{\gamma }_{2}p,\kappa p,N\triangleq \frac{\overline{\lambda }+\varpi \overline{\gamma }-{\overline{\rho }}_1\varpi -1}{\underset{¯}{\lambda }{\varpi }^N}p.\end{array}$$

Finally, considering the inequality (30), we can obtain$$\begin{array}{}\text{(59)}& \mathbb{E}{\tilde{x}k}^2\le \mathbb{E}{\Omega k}^2\le \kappa \underset{l\in -d,0}{\text{max}}\mathbb{E}{gl}^2,k+\iota {\Phi k}^2\le \mathrm{\varrho },\end{array}$$based on Definition 2, and we can know that the considered system is $\mathrm{\varrho }$-secure in mean-square sense. The proof is now complete.

In Theorem 1, sufficient conditions which can guarantee the ISS and $\mathrm{\varrho }$-secure in mean-square sense of the closed-loop system (19) have been obtained. However, due to the presence of nonlinear terms such as $\widehat{P}{B}_i{K}_{P_j}$ in Theorem 1, it cannot be solved by the method of LMIs. Therefore, in the next part, we will deal with the nonlinear terms and obtain the observer gains and the controller gains.

Theorem 2.

Let the scalars $\lambda 0<\lambda <1$ and $\sigma \sigma >0$ satisfy $\lambda \sigma \ge 1$. For scalar $\mathrm{\varrho }$ ($\mathrm{\varrho }$ > 0), the closed-loop system (19) is $\mathrm{\varrho }$-secure in mean-square sense if, for all $s\in \mathbb{L}$, $i,j\in \mathbb{M}$, there exist positive integer $n$, positive scalars $\epsilon$, $\theta$, $ϵ$, ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$, positive definite matrices $P$, $Q_{\overline{r}}\overline{r}=\mathrm{1,2},\dots ,d$, $R$, $Y$, $\mathbb{X}$, and matrices ${\widehat{K}}_{P_j},{\widehat{K}}_{I_j},{\widehat{K}}_{D_j}$, and ${\widehat{L}}_j$ satisfying$$\begin{array}{c}\text{(60)}& {\tilde{\Pi }}_{ij}^1+{\tilde{\Pi }}_{ji}^1<0i\le j,\text{(61)}& {\overline{\Pi }}_{ij}^2+{\overline{\Pi }}_{ji}^2<0,\hspace{1em}i\le j,\frac{\lambda +n\theta -1}{\sigma }<-{\gamma }_3.\\ \text{(62)}& \frac{\varpi {\gamma }_2\overline{\Phi }}{\underset{¯}{\lambda }\varpi -1}+\frac{\overline{\lambda }+\varpi \overline{\gamma }-\varpi -1{\overline{\rho }}_1}{\underset{¯}{\lambda }}\underset{l\in -d,0}{\text{max}}\mathbb{E}{gl}^2\le \mathrm{\varrho }\end{array}$$where$$\begin{array}{c}\text{(63)}& {\tilde{\Pi }}_{ij}^1=\begin{array}{ccc}{\overline{\prod }}_{ij}^1+E_1\mathbb{X}{E}_1^T& \ast & \ast \\ B_i^T{B}_i\widehat{K}{P}_j{E}_2& \text{sym}-B_i^T{B}_{i}Y& \ast \\ 0_{n_x\times 2n_x+2d{n}_x+n_y}& \widehat{P}{B}_i-B_{i}Y& -\mathbb{X}\end{array},{\tilde{\Pi }}_{ij}^2=\begin{array}{cc}\begin{array}{c}-{\gamma }_2{I}_{n_u\times n_y}\end{array}& \ast \\ {\tilde{\Pi }}_{ij}^{221}& {\tilde{\Pi }}^{222}\end{array},\\ {\tilde{\Pi }}_{ij}^1=\begin{array}{cc}{\tilde{\Pi }}^{111}& \ast \\ {\tilde{\Pi }}_{ij}^{121}& {\tilde{\Pi }}^{122}\end{array},\\ {\tilde{\Pi }}_{ij}^{121}=\begin{array}{cccc}{\tilde{\mathcal{A}}}_{ij}& {\tilde{\mathcal{B}}}_i& {\tilde{\mathcal{R}}}_{ij}& {\tilde{\mathcal{L}}}_j\\ \sqrt{\in }{\tilde{\mathcal{A}}}_{ij}& \sqrt{\in }{\tilde{\mathcal{B}}}_i& \sqrt{\in }{\tilde{\mathcal{R}}}_{ij}& \sqrt{\in }{\tilde{\mathcal{L}}}_j\\ \tilde{\alpha }{A}_j& 0_{2n_x\times 2n_x}& 0_{2n_x\times 2d{n}_x}& \tilde{\alpha }{\tilde{\mathcal{L}}}_j\\ \sqrt{\in \tilde{\alpha }{\mathcal{A}}_j}& 0_{2n_x\times 2n_x}& 0_{2n_x\times 2d{n}_x}& \sqrt{\in \tilde{\alpha }{\tilde{\mathcal{L}}}_j}\end{array},\\ {\tilde{\Pi }}_{ij}^{221}=\begin{array}{c}\sqrt{1+\frac{1}{\in }}{\tilde{\mathcal{D}}}_{ij}\\ \sqrt{\stackrel{⌢}{a}1+\frac{1}{\in }}{\tilde{\mathcal{D}}}_j\end{array},\\ {\tilde{\mathcal{A}}}_j=\begin{array}{cc}{0}_{n_x\times n_x}& 0_{n_x\times n_x}\\ {\tilde{L}}_j& 0_{n_x\times n_x}\end{array},\\ {\tilde{\mathcal{A}}}_{ij}=\begin{array}{cc}\tilde{P}{A}_i+B_i{\tilde{K}}_{P_j}+{\tilde{K}}_{D_j}& -B_i{\tilde{K}}_{P_j}+{\tilde{K}}_{D_j}\\ \overline{a}{\tilde{L}}_{j}C& \tilde{P}A-{\tilde{L}}_{j}C\end{array},\\ {\mathcal{B}}_i=\begin{array}{cc}\tilde{P}{A}_{T_i}& 0_{n_x\times n_x}\\ 0_{n_x\times n_x}& \tilde{P}{A}_{T_i}\end{array},\\ {\tilde{\mathcal{D}}}_{ij}=\begin{array}{cc}\tilde{P}{D}_i& 0_{n_x\times n_y}\\ \tilde{P}{D}_i& -\overline{a}{\tilde{L}}_j\end{array},\\ {\tilde{\mathcal{D}}}_j=\begin{array}{cc}{0}_{n_x\times n_u}& 0_{n_x\times n_y}\\ 0_{n_x\times n_u}& -{\tilde{L}}_j\end{array},\\ {\tilde{\mathcal{L}}}_j=\begin{array}{c}{0}_{n_x\times n_y}\\ \overline{a}-1{\tilde{L}}_j\end{array},\\ {\tilde{\mathcal{L}}}_j=\begin{array}{c}{0}_{n_x\times n_y}\\ {\tilde{L}}_j\end{array},\\ {\tilde{\mathcal{R}}}_{ij}=\begin{array}{c}{B}_i{M}_j\\ 0_{n_x\times 2d{n}_x}\end{array},\\ M_j=\begin{array}{cc}{M}_{1j}-M_{2j}& \underset{d-1}{\underbrace{\begin{array}{cc}{M}_{1j}& M_{1j}\dots M_{1j}\end{array}}}\end{array},\\ M_j=\begin{array}{cc}{\tilde{K}}_{I_j}& {\tilde{K}}_{I_j}\end{array},\\ M_j=\begin{array}{cc}{\tilde{K}}_{D_j}& {\tilde{K}}_{D_j}\end{array}.\end{array}$$

Furthermore, the observer gains $L_j$ and the PID controller gains $K_{P_j}$, $K_{I_j}$, and $K_{D_j}$ are given by$$\begin{array}{c}\text{(64)}& L_j={\widehat{P}}^{-1}{\widehat{L}}_j,K_{P_j}=Y^{-1}{\widehat{K}}_{P_j},\\ K_{I_j}=Y^{-1}{\widehat{K}}_{I_j},\\ K_{D_j}=Y^{-1}{\widehat{K}}_{D_j}.\end{array}$$

Proof.

The inequality (27) is equivalent to$$\begin{array}{c}\text{(65)}& {\overline{\Pi }}_{ij}^1+{\overline{\Pi }}_{ji}^1+\text{sym}{E}_1\widehat{P}{B}_i-B_{i}Y{Y}^{-1}{\widehat{K}}_{P_j}{E}_2+\text{sym}{E}_1\widehat{P}{B}_j-B_{j}Y{Y}^{-1}{\widehat{K}}_{P_i}{E}_2<0,\hspace{1em}i\le j,\end{array}$$where$$\begin{array}{}\text{(66)}& \begin{array}{c}{E}_1^T=I_{n_x}{0}_{n_x\times 11n_x+2d{n}_x+n_y}\\ E_2=0_{n_x\times 2n_x}{I}_{n_x}{0}_{n_x\times 9n_x+2d{n}_x+n_y}\end{array}.\end{array}$$

According to the Young inequality, there exists a positive definite matrix $\mathbb{X}\in {ℝ}^{n_x\times n_x}$ satisfying$$\begin{array}{c}\text{(67)}& {\overline{\Pi }}_{ij}^1+E_1\mathbb{X}{E}_1^T+E_2^T{Y^{-1}{\widehat{K}}_{P_j}}^T{\widehat{P}{B}_i-B_{i}Y}^T{\mathbb{X}}^{-1}\widehat{P}{B}_i-B_{i}Y\times Y^{-1}{\widehat{K}}_{P_j}+{\overline{\Pi }}_{ji}^1+E_1\mathbb{X}{E}_1^T+E_2^T{Y^{-1}{\widehat{K}}_{P_i}}^T{\widehat{P}{B}_j-B_{j}Y}^T\\ \times {\mathbb{X}}^{-1}\widehat{P}{B}_j-B_{j}Y{Y}^{-1}{\widehat{K}}_{P_i}<0,\hspace{1em}i\le j.\end{array}$$

Define ${\aleph }_i={\widehat{P}{B}_i-B_{i}Y}^T{\mathbb{X}}^{-1}\widehat{P}{B}_i-B_{i}Y$; applying to the Schur complement, (67) is equivalent to$$\begin{array}{}\text{(68)}& \begin{array}{cc}{\overline{\Pi }}_{ij}^1+E_1\mathbb{X}{E}_1^T& \ast \\ Y^{-1}{\widehat{K}}_{P_j}{E}_2& -{\aleph }_i\end{array}+\begin{array}{cc}{\overline{\Pi }}_{ji}^1+E_1\mathbb{X}{E}_1^T& \ast \\ Y^{-1}{\widehat{K}}_{P_i}{E}_2& -{\aleph }_j\end{array}<0,\hspace{1em}i\le j.\end{array}$$

Based on Lemma 2, we can obtain$$\begin{array}{c}\text{(69)}& \begin{array}{cc}{\overline{\Pi }}_{ij}^1+E_1\mathbb{X}{E}_1^T& \ast \\ B_i^T{B}_i{\widehat{K}}_{P_j}{E}_2& \text{sym}-B_i^T{B}_{i}Y+{\aleph }_i\end{array}+\begin{array}{cc}{\overline{\Pi }}_{ji}^1+E_1\mathbb{X}{E}_1^T& \ast \\ B_j^T{B}_j{\widehat{K}}_{P_i}{E}_2& \text{sym}-B_j^T{B}_{j}Y+{\aleph }_j\end{array}<0i\le j.\end{array}$$

Then, by utilizing the Schur complement lemma, we can obtain that (60) can be ensured by (69). So far, the proof is completed.

Remark 5.

It is worth emphasizing that the linearization method adopted in this study can directly calculate the variable Y matrix in the LMI, thereby, quickly obtaining the PID controller gains, which greatly reduces the computational complexity.

Remark 6.

Until now, we have addressed the observer-based PID security control issues for the discrete-time T-S fuzzy system with DETM and deception attacks and proposed the desired PID controllers under the designed closed-loop system. Compared with existing articles on the performance of T-S fuzzy systems, this article has two distinctions: (1) the problem solved is new as the security problem for the discrete-time T-S fuzzy system is first constructed under multisensor and deception attacks; (2) the PID control scheme is new in the sense that the influence of discrete-time delay, distributed DETM, and deception attacks on the PID controller are considered at the same time. It is worth mentioning that the main results derived in Theorems 1-2 can be utilized to the fuzzy PID control problem for other systems, such as large-scale systems, multiagent systems, and neural networks.

4. Numerical Examples

For the purpose of illustrating the effectiveness of the designed fuzzy PID controller, we cite a simulation example in this section. Consider system (1) as follows:$$\begin{array}{}\text{(70)}& \begin{array}{c}xk+1={\displaystyle \sum _{i=1}^2{h}_i\eta k{A}_{i}xk+A_{{\tau }_i}xk-\tau k+B_{i}uk+D_{i}vk,}\\ yk=Cxk.\end{array}\end{array}$$

The model parameters are provided as follows:$$\begin{array}{c}\text{(71)}& A_1=\begin{array}{cc}-0.11& 0.09\\ 0.09& -0.11\end{array},A_2=\begin{array}{cc}0.11& 0.019\\ 0.041& 0.31\end{array},\\ A_{{\tau }_1}=\begin{array}{cc}-0.021& 0.009\\ 0.009& -0.021\end{array},\\ A_{{\tau }_2}=\begin{array}{cc}0.011& 0.029\\ 0.029& 0.011\end{array},\\ B_1=\begin{array}{c}0.011\\ 0.12\end{array},\\ B_2=\begin{array}{c}-0.051\\ 0.19\end{array},\\ C=\begin{array}{cc}-0.01& 0.11\\ 0.21& 0.02\\ -0.01& 0.12\end{array},\\ D_1=\begin{array}{c}0.21\\ 0.2\end{array},\\ D_2=\begin{array}{c}-0.5\\ 0.3\end{array},\\ h_1\eta k=\frac{1-{\sin }^2{x}_{1}k}{2},\\ h_2\eta k=1-h_1\eta k.\end{array}$$

Assume the upper and lower bounds on the time delay as ${\tau }_M=3$ and ${\tau }_m=1$, respectively. The success probabilities of the deception attacks are set to be $\overline{\alpha }=0.7$. Moreover, the event-triggered parameters in (6) and (7) are chosen as $\sigma =5$, $\epsilon =0.2$, and $\lambda =0.2$. The security level of this study is assumed to be $\mathrm{\varrho }=0.6$, and the length of time-windows is taken as d = 3. The solutions to LMIs (60)–(62) are obtained as follows:$$\begin{array}{c}\text{(72)}& Q1=\begin{array}{cccc}0.3360& 0.0663& -0.0057& -0.0048\\ 0.0663& 0.3990& -0.0041& -0.0332\\ -0.0057& -0.0041& 0.3929& 0.0633\\ -0.0048& -0.0332& 0.0633& 0.3797\end{array},Q2=\begin{array}{cccc}0.3357& 0.0666& -0.0064& -0.0058\\ 0.0666& 0.3993& -0.0048& -0.0385\\ -0.0064& -0.0048& 0.3926& 0.0636\\ -0.0058& -0.0385& 0.0636& 0.3801\end{array},\\ Q3=\begin{array}{cccc}0.3357& 0.0666& -0.0055& -0.0049\\ 0.0666& 0.3988& -0.0043& -0.0331\\ -0.0055& -0.0043& 0.3926& 0.0635\\ -0.0049& -0.0331& 0.0635& 0.3796\end{array},\\ R=\begin{array}{cccc}0.1562& 0.0357& 0.0002& -0.0025\\ 0.0357& 0.1853& -0.0008& -0.0156\\ 0.0002& -0.0008& 0.1868& 0.0363\\ -0.0025& -0.0156& 0.0363& 0.1750\end{array},\\ {\tilde{K}}_{P_1}=\begin{array}{cc}-0.3873& 0.3367\end{array},\\ {\tilde{K}}_{P_2}=\begin{array}{cc}-0.1988& -1.4069\end{array},\\ {\tilde{K}}_{I_1}=\begin{array}{cc}-0.1262& 0.0955\end{array},\\ {\tilde{K}}_{I_2}=\begin{array}{cc}0.0046& -0.0039\end{array},\\ {\tilde{K}}_{D_1}=\begin{array}{cc}-0.3792& 0.2885\end{array},\\ {\tilde{K}}_{D2}=\begin{array}{cc}0.0139& -0.0114\end{array},\\ {\tilde{L}}_i=\begin{array}{ccc}0.0259& -0.1197& 0.0277\\ -0.0688& 0.1242& -0.0743\end{array},\\ {\tilde{L}}_i=\begin{array}{ccc}0.0714& 0.0841& 0.0782\\ 0.1723& 0.0761& 0.1881\end{array},\\ \tilde{P}=\begin{array}{cc}2.1418& 0.4812\\ 0.4812& 2.7100\end{array},\\ X=\begin{array}{cc}0.3167& 0.0447\\ 0.0447& 0.3359\end{array},\\ Y=2.7854,\\ {\gamma }_1=0.01,\\ {\gamma }_2=1,\\ {\gamma }_3=0.01,\\ \theta =0.2,\\ \in =2.\end{array}$$

Therefore, according to Theorem 2, the desired PID controller gains and observer gains can be derived as follows:$$\begin{array}{c}\text{(73)}& K_{P_1}=\begin{array}{cc}-0.1390& 0.1209\end{array},K_{P_2}=\begin{array}{cc}-0.0714& -0.5051\end{array},K_{I_1}=\begin{array}{cc}-0.0453& 0.0343\end{array},\\ K_{I_2}=\begin{array}{cc}0.0017& -0.0014\end{array},\\ K_{D_1}=\begin{array}{cc}-0.1361& 0.1036\end{array},\\ K_{D_2}=\begin{array}{cc}0.0050& -0.0041\end{array},\\ L_1=\begin{array}{ccc}0.0185& -0.0689& 0.0199\\ -0.0286& 0.0581& -0.0309\end{array},\\ L_2=\begin{array}{ccc}0.0199& 0.0343& 0.0218\\ 0.0600& 0.0220& 0.0655\end{array}.\end{array}$$

Set the initial states as $x0={-\mathrm{0.60.6}}^T$, $x-3=x-2=x-1={00}^T$. The disturbance input is assumed to be $vk=0.5\text{sin}k$, and the deception attacks information launched by adversary is selected as $\xi k=0.5\text{cos}k0.5\text{cos}k0.5\text{cos}k$.

The simulation results are shown in Figures 2–4. To be specific, Figure 2 shows the state trajectories of x(k) for discrete-time fuzzy system (1) under the distributed DETM and the deception attacks with PID control. Obviously, the proposed system (19) with observer-based PID control scheme is ISS and valid. Figures 3-4 and Table 1 show the impact of deception attacks on PID control performance. Specifically, Figures 3 and 4 show the trajectories of control input $uk$ and the time instants of successful deception attacks, respectively. Table 1 provides the relationship between the probability of deception attacks and the minimum security level. It is easy to see that as the probability of attack increases, the minimum security level also increases. Therefore, we can draw the following conclusion: the security performance of PID control deteriorates when the probability of deception attacks increases.

[figure omitted; refer to PDF]

Table 1

The minimum security level under different $\overline{\alpha }$.

(0.5 mm) $\overline{\alpha }$	Security level $\mathrm{\varrho }$
0.3	0.2828
0.4	0.3186
0.5	0.3552
0.6	0.4269
0.7	0.4662
0.8	0.4719
0.9	0.5058

For the purpose to show the advantages of the DETM adopted in this study compared to the static event-triggered mechanism (SETM), we derive the comparison results shown in Figures 5-6. The dynamic triggering instants of three distributed DETM are shown in Figure 5 and the static triggering instants are shown in Figure 6, from which we can know that the number of triggering time of the DETM are 24, 12, and 27 and the number of triggering time of the DETM are 37, 39, and 37. At this point, we can conclude that the triggering time of the DETM is much less than the STEM. Furthermore, we can draw that compared with the SETM, the DETM can save networked resources more effectively.

[figure omitted; refer to PDF][figure omitted; refer to PDF]

Remark 7.

Based on the above discussion, it can be summarized that (1) under the observer-based PID control law proposed in this study, the closed-loop system (19) can achieve the expected stability; (2) system performance may be destroyed by deception attacks, and the security performance of the system decreases as the probability of the attack increases; (3) compared with the SETM, the DETM has more stringent trigger conditions, so the dynamic event-triggered mechanism can more effectively reduce pressure on the network bandwidth.

5. Conclusion

In this study, by considering DETM and deception attacks, the observer-based PID security control problem has been investigated for the discrete-time T-S fuzzy system. The distributed DETM has been employed to reduce the pressure of network bandwidth. Based on the designed T-S fuzzy model, sufficient conditions for the ISS of the closed-loop system have been derived. Moreover, the desired fuzzy controller gains have been obtained. Finally, the effectiveness of the proposed methods has been demonstrated by a numerical example. In future work, we will continue to pay attention to the observer-based PID control problem for discrete T-S fuzzy systems and combine this framework with various communication protocols, such as redundant channel transmission protocol, round-robin protocol, and stochastic communication protocol.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (11661028) and the Natural Science Foundation of Guangxi, PR China (2020GXNSFAA159141).

References

[1] T. Takagi, M. Sugeno, "Fuzzy identification of systems and its applications to modeling and control," IEEE Transactions on Systems, Man, and Cybernetics, vol. SMC-15 no. 1, pp. 116-132, DOI: 10.1109/tsmc.1985.6313399, 1985.

[2] L. Zha, J. Liu, J. Cao, "Security control for T-S fuzzy systems with multi-sensor saturations and distributed event-triggered mechanism," Journal of the Franklin Institute, vol. 357 no. 5, pp. 2851-2867, DOI: 10.1016/j.jfranklin.2020.02.013, 2020.

[3] J. Wang, C. Yang, J. Xia, Z.-G. Wu, H. Shen, "Observer-based sliding mode control for networked fuzzy singularly perturbed systems under weighted try-once-discard protocol," IEEE Transactions on Fuzzy Systems, vol. 1,DOI: 10.1109/TFUZZ.2021.3070125, 2021.

[4] G. Wu, G.-H. Yang, H. Wang, "ISS control synthesis of T-S fuzzy systems with multiple transmission channels under denial of service," Journal of the Franklin Institute, vol. 358 no. 6, pp. 3010-3032, DOI: 10.1016/j.jfranklin.2021.02.014, 2021.

[5] M. Gao, L. Zhang, W. Qi, J. Cao, J. Cheng, Y. Kao, Y. Wei, X. Yan, "SMC for semi-Markov jump T-S fuzzy systems with time delay," Applied Mathematics and Computation, vol. 374,DOI: 10.1016/j.amc.2019.125001, 2020.

[6] K. H. Ang, G. Chong, Y. Li, "PID control system analysis, design, and technology," IEEE Transactions on Control Systems Technology, vol. 13 no. 4,DOI: 10.1109/tcst.2005.847331, 2005.

[7] Y. Wang, L. Zou, Z. Zhao, X. Bai, "H ∞ fuzzy PID control for discrete time-delayed T-S fuzzy systems," Neurocomputing, vol. 332 no. MAR.7, pp. 91-99, DOI: 10.1016/j.neucom.2018.12.002, 2019.

[8] S. Du, Q. Yan, J. Qiao, "Event-triggered PID control for wastewater treatment plants," Journal of Water Process Engineering, vol. 38,DOI: 10.1016/j.jwpe.2020.101659, 2020.

[9] F. Wallam, C. P. Tan, "Output feedback cross-coupled nonlinear PID based MIMO control scheme for pressurized heavy water reactor," Journal of the Franklin Institute, vol. 356 no. 15, pp. 8012-8048, DOI: 10.1016/j.jfranklin.2019.06.029, 2019.

[10] Y. Wang, Z. Wang, L. Zou, H. Dong, "Multiloop decentralized H ∞ fuzzy PID-like control for discrete time-delayed fuzzy systems under dynamical event-triggered schemes," IEEE Transactions on Cybernetics,DOI: 10.1109/TCYB.2020.3025251, 2020.

[11] D. Zhao, Z. Wang, G. Wei, Q.-L. Han, "A dynamic event-triggered approach to observer-based PID security control subject to deception attacks," Automatica, vol. 120 no. 4,DOI: 10.1016/j.automatica.2020.109128, 2020.

[12] D. Zhao, Z. Wang, D. W. C. Ho, G. Wei, "Observer-based PID security control for discrete time-delay systems under cyber-attacks," IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 51 no. 6, pp. 3926-3938, DOI: 10.1109/tsmc.2019.2952539, 2021.

[13] X. Wen, L. Guo, "Attenuation and rejection for multiple disturbances of nonlinear robotic systems using nonlinear observer and PID controller," Proceedings of the 2010 8th World Congress on Intelligent Control and Automation, pp. 2512-2517, .

[14] M. Hong, W. Yao, Z. Zhu, Y. Guo, "A hybrid PID controller for flexible joint manipulator based on state observer and singular perturbation approach," Proceedings of the 2020 39th Chinese Control Conference (CCC), pp. 3599-3603, DOI: 10.23919/ccc50068.2020.9189446, .

[15] Z.-H. Pang, G.-P. Liu, D. Zhou, D. Sun, "Data-driven control with input design-based data dropout compensation for networked nonlinear systems," IEEE Transactions on Control Systems Technology, vol. 25 no. 2, pp. 628-636, DOI: 10.1109/tcst.2016.2557278, 2017.

[16] D. Mao, Y. Ma, "Dissipativity analysis for Takagi-Sugeno fuzzy system with time-varying delays and stochastic packet dropouts," Information Sciences, vol. 587, pp. 535-555, DOI: 10.1016/j.ins.2021.12.038, 2022.

[17] Z. Xu, X. Li, P. Duan, "Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control," Neural Networks, vol. 125, pp. 224-232, DOI: 10.1016/j.neunet.2020.02.003, 2020.

[18] Y. Yang, Y. Niu, Z. Zhang, "Dynamic event-triggered sliding mode control for interval Type-2 fuzzy systems with fading channels," ISA Transactions, vol. 110, pp. 53-62, DOI: 10.1016/j.isatra.2020.10.035, 2021.

[19] M. Hedayati, M. Rahmani, "H ∞ filtering for nonlinearly coupled complex networks subjected to unknown varying delays and multiple fading measurements," ISA Transactions, vol. 120, pp. 43-54, DOI: 10.1016/j.isatra.2021.03.008, 2022.

[20] J. Liu, M. Yang, L. Zha, X. Xie, E. Tian, "Multi-sensors-based security control for T-S fuzzy systems over resource-constrained networks," Journal of the Franklin Institute, vol. 357 no. 7, pp. 4286-4315, DOI: 10.1016/j.jfranklin.2020.01.017, 2020.

[21] Q. Lei, Y. Luo, "Event-triggered-based external consensus protocol of RBF-ARX-model-based networked multiagent systems with nonlinear dynamics and communication delays," Discrete Dynamics in Nature and Society, vol. 2021,DOI: 10.1155/2021/5530878, 2021.

[22] C. Nowzari, E. Garcia, J. Cortés, "Event-triggered communication and control of networked systems for multi-agent consensus," Automatica, vol. 105,DOI: 10.1016/j.automatica.2019.03.009, 2019.

[23] J. Song, Y. Niu, H.-K. Lam, "Reliable sliding mode control of fast sampling singularly perturbed systems: a redundant channel transmission Protocol Approach," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 66 no. 11, pp. 4490-4501, DOI: 10.1109/tcsi.2019.2929554, 2019.

[24] L. Shuai, S. Yan, G. Wei, D. Derui, L. Yurong, "Event-triggered dynamic output feedback RMPC for polytopic systems with redundant channels: the input-to-state stability," Journal of the Franklin Institute, vol. 354 no. 7, pp. 2871-2892, 2017.

[25] D. Li, J. Liang, F. Wang, "H ∞ state estimation for two-dimensional systems with randomly occurring uncertainties and round-robin protocol," Neurocomputing, vol. 349 no. JUL.15, pp. 2019248-2019260, 2019.

[26] J. Zhang, J. Sun, H. Lin, "Optimal DoS attack schedules on remote state estimation under multi-sensor round-robin protocol," Automatica, vol. 127,DOI: 10.1016/j.automatica.2021.109517, 2021.

[27] Y. Dong, Y. Song, "Model predictive control for interval type-2 T-S fuzzy systems under round-robin protocol," IFAC-PapersOnLine, vol. 53 no. 5, pp. 368-373, DOI: 10.1016/j.ifacol.2021.04.114, 2020.

[28] H. Fu, J. Li, F. Han, N. Hou, H. Dong, "Outlier-resistant observer-based H∞ PID control under stochastic communication protocol," Applied Mathematics and Computation, vol. 411,DOI: 10.1016/j.amc.2021.126535, 2021.

[29] S. Chen, L. Ma, Y. Ma, "Distributed set-membership filtering for nonlinear systems subject to Round-Robin protocol and stochastic communication protocol over sensor networks," Neurocomputing, vol. 385, pp. 13-21, DOI: 10.1016/j.neucom.2019.11.056, 2020.

[30] A. Lu, G. Yang, "Input-to-state stabilizing control for cyberphysical systems with multiple transmission channels under denial of service," IEEE Transactions on Automatic Control, vol. 63 no. 6,DOI: 10.1109/tac.2017.2751999, 2018.

[31] X. Shao, D. Ye, "Neural-network-based adaptive secure control for nonstrict-feedback nonlinear interconnected systems under DoS attacks," Neurocomputing, vol. 448, pp. 263-275, DOI: 10.1016/j.neucom.2021.03.087, 2021.

[32] D. Liu, D. Ye, "Pinning-observer-based secure synchronization control for complex dynamical networks subject to DoS attacks," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 67 no. 12, pp. 5394-5404, DOI: 10.1109/tcsi.2020.3016994, 2020.

[33] S. Yoon, H. Yu, "Multiple points input for convolutional neural networks in replay attack detection," Proceedings of the 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6444-6448, DOI: 10.1109/icassp40776.2020.9053303, .

[34] M. Zhou, Z. Zhang, L. Xie, "Permutation entropy based detection scheme of replay attacks in industrial cyber-physical systems," Journal of the Franklin Institute, vol. 358 no. 7, pp. 4058-4076, DOI: 10.1016/j.jfranklin.2021.02.024, 2021.

[35] L. Li, H. Yang, Y. Xia, C. Zhu, "Attack detection and distributed filtering for state-saturated systems under deception attack," IEEE Transactions on Control of Network Systems, vol. 8 no. 4, pp. 1918-1929, DOI: 10.1109/TCNS.2021.3089146, 2021.

[36] J. Wu, C. Peng, J. Zhang, B.-L. Zhang, "Event-triggered finite-time H∞ filtering for networked systems under deception attacks," Journal of the Franklin Institute, vol. 357 no. 6, pp. 3792-3808, DOI: 10.1016/j.jfranklin.2019.09.002, 2020.

[37] K. Wang, E. Tian, J. Liu, L. Wei, D. Yue, "Resilient control of networked control systems under deception attacks: a memory‐event‐triggered communication scheme," International Journal of Robust and Nonlinear Control, vol. 30 no. 4, pp. 1534-1548, DOI: 10.1002/rnc.4837, 2020.

[38] Y. Wang, L. Xie, C. E. D. Souza, "Robust control of a class of uncertain nonlinear systems," Systems & Control Letters, vol. 19 no. 2,DOI: 10.1016/0167-6911(92)90097-c, 1992.

[39] M. Shen, J. H. Park, D. Ye, "A separated approach to control of Markov jump nonlinear systems with general transition probabilities," IEEE Transactions on Cybernetics, vol. 46 no. 9, pp. 2010-2018, DOI: 10.1109/tcyb.2015.2459717, 2016.