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

Authors: Sisi Lin; Mengzhuo Luo; Shouming Zhong; Kaibo Shi; Xingxu Tan
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## 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.


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## Body

## 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 ℋ∞ 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 andϱ-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 Section2, 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 ϱ-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.l20,∞ denotes the space of square summable sequences. · represents the Euclidean norm of a variable. A function f:ℝ0+⟶ℝ0+ is said to be a Q-function if it satisfies f0=0, continuous, and strictly increasing. Furthermore, f is said to be a Q∞-function if it is a Q-function and satisfies ft⟶∞ when t⟶∞. In addition, a function c:ℝ0+×ℝ0+⟶ℝ0+ is said to be a P-function if c·,n is a Q-function for each fixed n∈ℝ0+, and for each fixed m∈ℝ0+, cm,· is decreasing and satisfies cm,n⟶0 when n⟶∞. The maximum eigenvalue and minimum eigenvalue of a matrix is represented by λmax· and λmin·, respectively. symX=X+XT 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 rulei: if η1k is gi1 and ⋯ and ηlk is gil, then(1)xk+1=Aixk+Aτixk−τk+Biuk+Diϑk,yk=Cxk,xl=φl,l=−τM,…,−1,0,where giji∈M≜1,2,…,r,j∈≜1,2,…,l is the fuzzy set; r represents the number of IF-THEN rules; ηk=η1k,η2k,…,ηlkT denotes the vector consisting of the premise variables; xk∈ℝnx and yk∈ℝn represent, respectively, the state vector and the measurement output; uk∈ℝnu is the control input; τk denotes the discrete-time delay; and φll=−τM,…,−1,0 are the initials conditions. ϑk∈l20,∞,ℝnv denotes the disturbance input satisfying(2)ϑTkϑk≤ϑ¯,where ϑ¯ is a given scalar. Ai,Aτi,C,Di are the known constant matrices with compatible dimensions and assume the matrix Bi is of full column rank, namely, rankBi=nu.Assumption 1.
(See [12]). The discrete-time delay τk satisfies τm≤τk≤τMk∈ℕ+, where τm and τM are the two known scalars, representing the upper bound and the lower bound of τk, respectively.
By conducting the singleton fuzzifier, center-average defuzzifier, and product fuzzy inference, system (1) can be represented by the following compact form:(3)xk+1=∑i=1rhiηkAixk+Aτixk−τk+Biuk+Diϑk,yk=Cxk,where the normalized membership functions are given by(4)hiηk=∏j=1lgijηjk∑i=1r∏j=1lgijηjk,and gijηjk refers to the grade of the membership function of ηjk in gij, hiηk satisfying hiηk≥0, and ∑i=1rhiηk=1. For simplicity, we use hi to represent hiηk.
From Figure1, we know that the measurement output y(k) is grouped into yk=y1k,y2k,…,ynkT. 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 sths∈L≜1,2,…,n node is given by(5)fsk,ϕsk,ζk=ϕsTkϕsk−1σζk−εysTkysk,where ϕsk≜yskts−ysk; kts denotes the latest triggered time of the sth sensor, and yskts is the corresponding measurement output. σ and ε are the two given positive scalars. Obviously, as long as the trigger condition fsk,ϕsk,ζk>0 is satisfied, the data will be transmitted to the buffer.
We define the triggering time sequence of thesths∈L≜1,2,…,n sensor as follows: 0<k0s<k1s<⋯<kts<⋯. Then, the next transmitted instants kt+1s for the sth sensor can be described as(6)kt+1s=infk∈ℕ|k>kts,fsk,ϕsk,ζk>0.
According to (6) and the use of the buffers, for all sensors, we can derive the following event-triggered condition:(7)ϕTkϕk−nσζk−εyTkyk>0,where ϕk=ϕ1Tk,ϕ2Tk,…,ϕnTkT, nn>0 representing the number of sensors, and ζk is an internal dynamical variable satisfying(8)ζk+1=λζk+εyTkyk−ϕTkϕk,ζ0=0,where λ∈0,1 is a given constant.Figure 1
The structure of the T-S fuzzy system under DETM and deception attacks.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 ζk 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 σ 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 inputy˜k can be expressed as(9)y˜k=y¯k+αkωk,where y¯k=y1Tkt1,y2Tkt2,…,ynTktnT is the real measured data via the distributed event-triggered scheme; ωk denotes the deception attacks information launched by the attacker described by(10)ωk=−y¯k+ξk,where ξk≠0 represents an arbitrary bounded signal satisfying(11)ξTkξk≤ξ¯,where ξ¯ is a given positive scalar. αk∈0,1 is a Bernoulli distributed variable with the following probabilities:(12)Probαk=1=α¯,Probαk=0=1−α¯,where α¯∈0,1 is a known constant, and the expectation of αk is easily obtained as Eαk=α¯.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 informationξk 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 rulei: if η1k is gi1 and ⋯ and ηlk is gil, then(13)x^k+1=Aix^k+Aτix^k−τk+Biuk+∑j=1rhjLjy˜k−y^k,y^k=Cx^k,x^l=0,l=−τM,…,−1,0,where x^k∈ℝnx is the observer states of xk; Lj denotes the observer gains which will be designed later. Then, the fuzzy observer can be constructed as follows:(14)x^k+1=∑i=1r∑j=1rhihjAix^k+Aτix^k−τk+Biuk+Ljy˜k−y^k,y^k=Cx^k.
In this study, for the discrete-time T-S fuzzy system (1), we adopt the following observer-based PID control law.
Controller rulei: if η1k is gi1 and ⋯ and ηlk is gil, then(15)uk=KPix^k+KIi∑m=k−dk−1x^m+KDix^k−x^k−1,where KPi,KIi,KDi 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≥τM in this study.
The considered fuzzy PID control law can be rewritten as follows:(16)uk=∑i=1rhiKPix^k+KIi∑m=k−dk−1x^m+KDix^k−x^k−1.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 componentsKPix^k,KIi∑m=k−dk−1x^m, and KDix^k−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.
Denotingek≜xk−x^k, taking (3), (9), (10), and (14) into account, the estimation error system could be described as follows:(17)ek+1=∑i=1r∑j=1rAiek+Aτiek−τk+Diϑk−1−αk×Ljϕk−1−αkLjCxk−αkLjξk+LjCx^k.
Now, for simplicity to the following work, we introduce two augmentation vectors:(18)x˜k=xTkeTkT,Φk=ϑTkξTkT.
Then, based on estimation error system (17) and system (3), we can obtain the closed-loop system of the following form:(19)x˜k+1=∑i=1r∑j=1rhihjAij+α˜kA¯jx˜k+ℬix˜k−τk+Dij+α˜kD¯jΦk+ℒj+α˜kℒ¯jϕk+ℛijΨk,x˜l=gl,l=−d,…,−1,0,where(20)Aij=Ai+BiKPj+KDj−BiKPj+KDjα¯LjCAi−LjC,A¯j=0nx×nx0nx×nxLjC0nx×nxℬi=Aτi0nx×nx0nx×nxAτi,Dij=Di0nx×nyDi−α¯Lj,D¯j=0nx×nv0nx×ny0nx×nv−Lj,ℒj=0nx×nyα¯−1Lj,ℒ¯j=0nx×nyLj,ℛij=BiMj0nx×2dnx,Ψk=x˜Tk−1x˜Tk−2…x˜Tk−dT,gl=φlTφlTT,Mj=M1j−M2jM1jM1j…M1j︸d−1,M1j=KIj−KIj,M2j=KDj−KDj,α˜k=αk−α¯,Eα˜2k≜α^.
It follows from (2) and (11) and one has that(21)ΦTkΦk≤Φ¯,Φ¯=ϑ¯+ξ¯.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 κ∈QJ and ι∈Q∞, such that the system dynamic x˜k satisfies the following inequality:(22)Ex˜k2≤κmaxl∈−d,0Egl2,k+ιΦk∞2,k≥0,where Φk∞=supkΦk.Definition 2.
(See [11]). Denote the desired safety level as ϱ>0; the closed-loop system (19) is said to be mean-square ϱ-secure if the following inequality holds:(23)Ex˜k2≤ϱ,∀k≥0.Lemma 1.
(See [38]). For positive definite matrix Z with suitable dimensions and any real-valued variables x, y, the following inequality holds:(24)xTZy+yTZx≤ϵxTZx+1ϵyTZy,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 matricesR, G such that(25)−RMT∗−G−1<0.(2)
There exist two symmetric and positive-definitive matricesR, G and constant matrix Z satisfying(26)−RZMT∗sym−Z+G<0,Lemma 3.
(See [11]). For the DETC scheme consisting of (7) and (8) with the initial value ζ0≥0, the internal dynamic variable satisfies ζk≥0 for all k≥0 if the parameters σσ>0 and λ0<λ<1 satisfy λσ≥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 ϱ-secure in mean-square sense. Then, the desired observer gains and PID controller gains are designed.Theorem 1.
Let the scalarsλ0<λ<1 and σσ>0 satisfy λσ≥1, the observer gain Lj, and the PID controller gains KPj, KIj, KDj given. For scalar ϱ (ϱ > 0), the closed-loop system (19) is ϱ-secure in mean-square sense if, for all s∈L, i,j∈M, there exist positive definite matrices P, Qr¯r¯=1,2,…,d, R, positive integer n, and positive scalars ε, θ, ϵ, γ1, γ2, γ3 satisfying(27)Πij1+Πji1<0,i≤j,(28)Πij2+Πji2<0,i≤j,(29)λ+nθ−1σ<−γ3.(30)ϖγ2Φ¯λ¯ϖ−1+λ¯+ϖγ¯−ϖ−1ρ¯1λ¯maxl∈−d,0Egl2≤ϱ,where(31)Πij1=Π111∗Πij121Π122,Πij2=−γ2Inv+ny∗Πij221Π222,in which(32)Π111=diag−P+Q¯+τ¯R+ε¯ℂTℂ+γ1I2nx×2nx,−R,−Q,−1σ+θInyΠij121=PAijPℬiPℛijPℒjϵPAijϵPℬiϵPℛijϵPℒjα^PAj¯02nx×2nx02nx×2dnxα^Pℒj¯ϵα^PAj¯02nx×2nx02nx×2dnxϵα^Pℒ¯jΠ122=−I4⊗P,Πij221=1+1/ϵPDijα^1+1/ϵPD¯j,Π222=−I2⊗P,Q¯=∑r¯=1dQr¯,ℂ=C0ny×nx,τ¯=τM−τm+1,ε¯=ε1σ+θ,Q=diagQ1,Q2…sQd,ρ1¯=maxλmaxP,1σ,P=diagP^,P^,ρ2¯=maxλmaxQd,λmaxRτM−τm+1,Proof.
Consider the Lyapunov functional as follows:(33)Vk=∑i=15Vik,where(34)V1k=x˜TkPx˜kV2k=∑r¯=1d∑q=k−r¯k−1x˜TqQr¯x˜qV3k=∑q=k−τkk−1x˜TqRx˜qV4k=∑l=k−τM+1k−τm∑q=lk−1x˜TqRx˜qV5k=1σζk.
First, calculating the difference and mathematical expectation ofVk, we have(35)EΔV1k=Ex˜Tk+1Px˜k+1−x˜kTPx˜k=EAij+nkĀjx˜k+Bix˜k−Tk+Dij+ãkD¯jΦk+Lj+ãkL¯jϕk+RijΨkTPAij+ãkĀjx˜k+Bix˜k−Tk+Dij+ãkD¯jΦk+Lj+ãkL¯jϕk+RijΨk−x˜kTPx˜k=∑i=1r∑j=1rhihjx˜TkAijTPAij+ãĀjTPĀj−Px˜k+x˜Tk−TkBiTPBix˜k−k+ΦTkDijTPDij+ãDjTPD¯j+Ψk+ϕTkLjTPLj+ãL¯jTPL¯jϕk+ΨTkRijTPRijΨk+2x˜TkAijTPBix˜Tk−Tk+2x˜TkAijTPDij+ãĀjTPDjΦk+2x˜TkAijTPLj+ãAjTPL¯jϕk+2x˜TkAijTPRijΨk+2x˜Tk−Tk×BiTPDijΦk+2x˜Tk−TkBiTPLjϕk+2ϕTkDijTPRijΨk+2TkLjTPRijΨk.(36)E△V2k=E∑r¯=1d∑q=k−r¯+1kx˜TqQr¯x˜q−∑r¯=1d∑q=k−r¯k−1x˜TqQr¯x˜q=∑r¯=1dE∑q=k−r¯+1kx˜TqQr¯x˜q−∑q=k−r¯k−1x˜TqQr¯x˜q=∑r¯=1dEx˜TkQr¯x˜k−x˜Tk−r¯Qr¯x˜k−r¯,(37)E△V3k=E∑q=k+1−τk+1kx˜TqRx˜q−∑q=k−τkk−1x˜TqRx˜q=Ex˜TkRx˜k−x˜Tk−τkRx˜k−τk+∑q=k+1−τk+1k−1x˜TqRx˜q−∑q=k+1−τkk−1x˜TqRx˜q≤Ex˜TkRx˜k−x˜Tk−τkRx˜k−τk+∑q=k−τM+1k−τmx˜TqRx˜q,(38)E△V4k=E∑l=k−τM+2k−τm+1∑q=lkx˜TqRx˜q−∑l=k−τM+1k−τm∑q=lk−1x˜TqRx˜q=E∑l=k−τM+1k−τm∑q=lkx˜TqRx˜q−∑q=k−τM+1kx˜TqRx˜q+∑q=k−τm+1kx˜TqRx˜q−∑l=k−τM+1k−τm∑q=lk−1x˜TqRx˜q=E∑l=k−τM+1k−τmx˜TqRx˜q−∑q=k−τM+1k−τmx˜TqRx˜q=EτM−τmx˜TqRx˜q−∑q=k−τM+1k−τmx˜TqRx˜q,(39)E△V5k=E1σζk+1−1σζk=E1σλζk+εyTkyk−ϕTkϕk−ζk=Eλ−1σζk+εσxTkCTCxk−1σϕTkϕk.
Then, taking (7), (33)–(39) into account, we can obtain(40)E△Vk≤∑i=1r∑j=1rhihjψTkΛ1+Λ2+Λ3ψk+2ψTkΛ4Φk+ΦTkΛ5Φk+λ+nθ−1σζk,where(41)ψk=x˜Tkx˜Tk−τkΨTkϕTkT,Λ1=diag−P+Q¯+τ¯R+ε¯ℂTℂ,−R,−Q,−1σ+θIny,Λ2=AijTPAij∗∗∗ℬiTPAijℬiTPℬi∗∗ℛijTPAijℛijTPℬiℛijTPℛij∗ℒjTPAijℒjTPℬiℒjTPℛijℒjTPℒj,Λ3=α^A¯jTPA¯jT∗∗∗02nx×2nx02nx×2nx∗∗02dnx×2nx02dnx×2nx02dnx×2dnx∗α^ℒ¯jTPA¯jT0ny×2nx0ny×2dnxα^ℒ¯jTPℒ¯jT,Λ4=Σ1TP˜Σ1,Λ5=Σ2TP˜Σ2,Σ2=Dijα^D¯j,P˜=I2⊗P,Σ1=Aijℬiℛijℒjα^Aj¯02nx×2nx02nx×2dnxα^ℒj¯.
According to Lemma1, the term 2ψTkΛ4Φk can be computed as(42)2ψTkΛ4Φk=2ψTkΣ1TP˜Σ2Φk≤ϵψTkΣ1TP˜Σ1ψk+1ϵΦTkΣ2TP˜Σ2Φk.
By combining (40) and (41), the inequality (40) can be described as(43)E△Vk≤∑i=1r∑j=1rEψTkΘ1ψk+ΦTkΘ2Φk+λ+nθ−1σζk,where(44)Θ1=Λ1+Λ2+Λ3+ϵΣ1TP˜Σ1,Θ2=1+1ϵΛ5.
Therefore, by using Schur complement and conditions (27)–(29), we know that(45)E△Vk≤∑i=1r∑j=1rEψTkΘ1ψk+ΦTkΘ2Φk+λ+nθ−1σζk≤E−γ1x˜k2+γ2Φk∞2−γ3ζk≤−γ¯EΩk2+γ2Φk∞2,where(46)γ¯=minγ1,γ3,Ωk=x˜Tkζ1/2kTT.
Next, we will discuss the ISS problem of the closed-loop system (19). According to (33), we obtain(47)EVk≤ρ¯1EΩk2+ρ¯2∑q=k−dk−1EΩq2,where(48)ρ¯1=maxλmaxP,1σ,ρ¯2=maxλmaxQd,λmaxRτ¯.
For anyρ>1, considering (45) and (47), it can be derived that(49)Eρk+1Vk+1−EρkVk=ρk+1E△Vk+ρkρ−1EVk≤ρk+1−γ¯EΩk2+γ2Φk∞2+ρkρ−1×ρ¯1EΩk2+ρ¯2∑q=k−dk−1EΩq2=σ1ρρkEΩk2+σ2ρ∑q=k−dk−1ρkEΩρ2+ρk+1γ2Φk∞2,where σ1ρ=−ργ¯+ρ¯1ρ−1,σ2ρ=ρ¯2ρ−1. In the following, for any integer N≥d+1, summing up both sides of (49) from 0 to N−1 with respect to k results in(50)EρNVN−EV0≤σ1ρ∑k=0N−1ρkEΩk2+σ2ρ∑k=0N−1∑q=k−dk−1ρkEΩρ2+ρ1−ρN1−ργ2Φk∞2.
It can be checked that(51)∑k=0N−1∑q=k−dk−1ρkEΩq2≤∑q=−d−1∑k=0q+d+∑q=0N−d−1∑k=q+1q+d+∑q=N−dN−1∑k=q+1N−1×ρkEΩq2≤d∑q=−d−1ρq+dEΩq2+d∑q=0N−d−1ρq+dEΩq2+d∑q=N−dN−1ρq+dEΩq2≤dρdmax−d≤q≤0EΩq2+dρd∑k=0N−1ρkEΩk2.
Thus, it implies from (50) and (51) that(52)EρNVN−EV0≤π1ρ∑k=0N−1ρkEΩk2+π2ρmax−d≤q≤0EΩq2+ρ1−ρN1−ργ2Φk∞2,where π1ρ=σ1ρ+dρdσ2ρ,π2ρ=dρdσ2ρ.
Sinceπ11=−γ¯<0 and limρ⟶∞π1ρ=+∞, there exists a scalar ϖ>1, such that π1ϖ=0, which yields(53)EϖNVN−EV0≤ϖ1−ϖN1−ϖγ2Φk∞2+π2ϖmax−d≤q≤0EΩq2.
Then, it is obvious that(54)EV0≤λ¯maxl∈−d,0Egl2.(55)EVN≥λ¯EΩN2,where(56)λ¯=maxλmaxP,dλmaxQ,dλmaxR,λ¯=minλminp,1σ.
Taking (53)–(55) into consideration, we have(57)EΩN2≤ϖ1−ϖNλ¯ϖN1−ϖγ2Φk∞2+π2ϖλ¯ϖNmax−d≤q≤0EΩq2+λ¯λ¯ϖNmaxl∈−d,0Egl2≤ϖλ¯ϖ−1γ2Φk∞2+λ¯+ϖγ¯−ρ¯1ϖ−1λ¯ϖNmaxl∈−d,0Egl2=ιΦk∞2+κmaxl∈−d,0Egl2,N,where(58)ιp≜ϖλ¯ϖ−1γ2p,κp,N≜λ¯+ϖγ¯−ρ¯1ϖ−1λ¯ϖNp.
Finally, considering the inequality (30), we can obtain(59)Ex˜k2≤EΩk2≤κmaxl∈−d,0Egl2,k+ιΦk2≤ϱ,based on Definition 2, and we can know that the considered system is ϱ-secure in mean-square sense. The proof is now complete.
In Theorem1, sufficient conditions which can guarantee the ISS and ϱ-secure in mean-square sense of the closed-loop system (19) have been obtained. However, due to the presence of nonlinear terms such as P^BiKPj 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 scalarsλ0<λ<1 and σσ>0 satisfy λσ≥1. For scalar ϱ (ϱ > 0), the closed-loop system (19) is ϱ-secure in mean-square sense if, for all s∈L, i,j∈M, there exist positive integer n, positive scalars ε, θ, ϵ, γ1, γ2, γ3, positive definite matrices P, Qr¯r¯=1,2,…,d, R, Y, X, and matrices K^Pj,K^Ij,K^Dj, and L^j satisfying(60)Π˜ij1+Π˜ji1<0i≤j,(61)Π¯ij2+Π¯ji2<0,i≤j,λ+nθ−1σ<−γ3.(62)ϖγ2Φ¯λ¯ϖ−1+λ¯+ϖγ¯−ϖ−1ρ¯1λ¯maxl∈−d,0Egl2≤ϱwhere(63)Π˜ij1=∏¯ij1+E1XE1T∗∗BiTBiK^PjE2sym−BiTBiY∗0nx×2nx+2dnx+nyP^Bi−BiY−X,Π˜ij2=−γ2Inu×ny∗Π˜ij221Π˜222,Π˜ij1=Π˜111∗Π˜ij121Π˜122,Π˜ij121=A˜ijB˜iR˜ijL˜j∈A˜ij∈B˜i∈R˜ij∈L˜jα˜Aj02nx×2nx02nx×2dnxα˜L˜j∈α˜Aj02nx×2nx02nx×2dnx∈α˜L˜j,Π˜ij221=1+1∈D˜ija⌢1+1∈D˜j,A˜j=0nx×nx0nx×nxL˜j0nx×nx,A˜ij=P˜Ai+BiK˜Pj+K˜Dj−BiK˜Pj+K˜Dja¯L˜jCP˜A−L˜jC,Bi=P˜ATi0nx×nx0nx×nxP˜ATi,D˜ij=P˜Di0nx×nyP˜Di−a¯L˜j,D˜j=0nx×nu0nx×ny0nx×nu−L˜j,L˜j=0nx×nya¯−1L˜j,L˜j=0nx×nyL˜j,R˜ij=BiMj0nx×2dnx,Mj=M1j−M2jM1jM1j…M1j︸d−1,Mj=K˜IjK˜Ij,Mj=K˜DjK˜Dj.
Furthermore, the observer gainsLj and the PID controller gains KPj, KIj, and KDj are given by(64)Lj=P^−1L^j,KPj=Y−1K^Pj,KIj=Y−1K^Ij,KDj=Y−1K^Dj.Proof.
The inequality (27) is equivalent to(65)Π¯ij1+Π¯ji1+symE1P^Bi−BiYY−1K^PjE2+symE1P^Bj−BjYY−1K^PiE2<0,i≤j,where(66)E1T=Inx0nx×11nx+2dnx+nyE2=0nx×2nxInx0nx×9nx+2dnx+ny.
According to the Young inequality, there exists a positive definite matrixX∈ℝnx×nx satisfying(67)Π¯ij1+E1XE1T+E2TY−1K^PjTP^Bi−BiYTX−1P^Bi−BiY×Y−1K^Pj+Π¯ji1+E1XE1T+E2TY−1K^PiTP^Bj−BjYT×X−1P^Bj−BjYY−1K^Pi<0,i≤j.
Defineℵi=P^Bi−BiYTX−1P^Bi−BiY; applying to the Schur complement, (67) is equivalent to(68)Π¯ij1+E1XE1T∗Y−1K^PjE2−ℵi+Π¯ji1+E1XE1T∗Y−1K^PiE2−ℵj<0,i≤j.
Based on Lemma2, we can obtain(69)Π¯ij1+E1XE1T∗BiTBiK^PjE2sym−BiTBiY+ℵi+Π¯ji1+E1XE1T∗BjTBjK^PiE2sym−BjTBjY+ℵj<0i≤j.
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 variableY 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 Theorems1-2 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:(70)xk+1=∑i=12hiηkAixk+Aτixk−τk+Biuk+Divk,yk=Cxk.The model parameters are provided as follows:(71)A1=−0.110.090.09−0.11,A2=0.110.0190.0410.31,Aτ1=−0.0210.0090.009−0.021,Aτ2=0.0110.0290.0290.011,B1=0.0110.12,B2=−0.0510.19,C=−0.010.110.210.02−0.010.12,D1=0.210.2,D2=−0.50.3,h1ηk=1−sin2x1k2,h2ηk=1−h1ηk.Assume the upper and lower bounds on the time delay asτM=3 and τm=1, respectively. The success probabilities of the deception attacks are set to be α¯=0.7. Moreover, the event-triggered parameters in (6) and (7) are chosen as σ=5, ε=0.2, and λ=0.2. The security level of this study is assumed to be ϱ=0.6, and the length of time-windows is taken as d = 3. The solutions to LMIs (60)–(62) are obtained as follows:(72)Q1=0.33600.0663−0.0057−0.00480.06630.3990−0.0041−0.0332−0.0057−0.00410.39290.0633−0.0048−0.03320.06330.3797,Q2=0.33570.0666−0.0064−0.00580.06660.3993−0.0048−0.0385−0.0064−0.00480.39260.0636−0.0058−0.03850.06360.3801,Q3=0.33570.0666−0.0055−0.00490.06660.3988−0.0043−0.0331−0.0055−0.00430.39260.0635−0.0049−0.03310.06350.3796,R=0.15620.03570.0002−0.00250.03570.1853−0.0008−0.01560.0002−0.00080.18680.0363−0.0025−0.01560.03630.1750,K˜P1=−0.38730.3367,K˜P2=−0.1988−1.4069,K˜I1=−0.12620.0955,K˜I2=0.0046−0.0039,K˜D1=−0.37920.2885,K˜D2=0.0139−0.0114,L˜i=0.0259−0.11970.0277−0.06880.1242−0.0743,L˜i=0.07140.08410.07820.17230.07610.1881,P˜=2.14180.48120.48122.7100,X=0.31670.04470.04470.3359,Y=2.7854,γ1=0.01,γ2=1,γ3=0.01,θ=0.2,∈=2.Therefore, according to Theorem2, the desired PID controller gains and observer gains can be derived as follows:(73)KP1=−0.13900.1209,KP2=−0.0714−0.5051,KI1=−0.04530.0343,KI2=0.0017−0.0014,KD1=−0.13610.1036,KD2=0.0050−0.0041,L1=0.0185−0.06890.0199−0.02860.0581−0.0309,L2=0.01990.03430.02180.06000.02200.0655.Set the initial states asx0=−0.60.6T, x−3=x−2=x−1=00T. The disturbance input is assumed to be vk=0.5sink, and the deception attacks information launched by adversary is selected as ξk=0.5cosk0.5cosk0.5cosk.The simulation results are shown in Figures2–4. 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 2
State trajectoriesxk of the system with PID control.Figure 3
The trajectory of controller inputuk.Figure 4
The time instants of successful deception attacks withα¯=0.7.Table 1
The minimum security level under differentα¯.
(0.5 mm)α¯Security levelϱ0.30.28280.40.31860.50.35520.60.42690.70.46620.80.47190.90.5058For 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 Figures5-6. 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 5
Dynamic triggering instants.Figure 6
Static triggering instants.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.



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