# Event-TriggeredH∞ Filtering for Markovian Jump Neural Networks under Random Missing Measurements and Deception Attacks

Authors: Jinxia Wang; Jinfeng Gao; Tian Tan; Jiaqi Wang; Miao Ma
Journal: Complexity
 (2020)

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## Abstract

This paper concentrates on the event-triggeredH∞ filter design for the discrete-time Markovian jump neural networks under random missing measurements and cyber attacks. Considering that the controlled system and the filtering can exchange information over a shared communication network which is vulnerable to the cyber attacks and has limited bandwidth, the event-triggered mechanism is proposed to relieve the communication burden of data transmission. A variable conforming to Bernoulli distribution is exploited to describe the stochastic phenomenon since the missing measurements occur with random probability. Furthermore, seeing that the communication networks are vulnerable to external malicious attacks, the transferred information via the shared communication network may be changed by the injected false information from the attackers. Based on the above consideration, sufficient conditions for the filtering error system to maintain asymptotically stable are provided with predefined H∞ performance. In the end, three numerical examples are given to verify the proposed theoretical results.


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## Body

## 1. Introduction

Neural networks (NNs) have been attached increasing importance by many researchers on account of the wide applications in robotization, deep learning, optimization problem, and pattern recognition in recent decades. Motivated by the extensive applications, many achievements about NNs have been delivered [1–4]. For instance, by using the universal approximation ability of NNs, an adaptive dynamic surface control method is provided for the nonlinear systems [3, 4]. Nevertheless, in practical application, NNs face many challenges, such as information interruption, random interference, and variations of the network environment. These impulsive effects can be simulated by a Markovian jump chain since the stochastic Markovian jump process can effectively reduce the conservatism. Markovian jump neural networks (MJNNs), which are firstly introduced in [5], have attracted many researchers to make great effort on this issue in recent years [6–12]. Researchers in [6, 7] both pay attention to the state estimation for delayed MJNNs concretely. The synchronization for MJNNs by employing a detector based on a hidden Markovian model (HMM) is studied in [9]. Researchers in [10–12] are concerned with the performance control of MJNNs based on the event-triggered mechanism (ETM). Among them, the issue of H∞ state estimation is investigated for a class of semi-Markovian jump NNs [11]. By using a novel distributed ETM, Vadivela et al. discuss the robust H∞ synchronization for MJNNs in detail [12].With the development and integration of communication engineering, computer technology, and control theory, the networked control systems (NCSs) exchange information through the shared communication network. Periodic sampling, namely, time-triggered mechanism (TTM), acts as a traditional method of signal processing. If the sampling period is relatively small, a large amount of redundant sampling data will be transmitted to the network channels with limited bandwidth, which will unavoidably cause network congestion. From the perspective of resource utilization, the ETM is introduced to control the signal transmission by setting a threshold value of signal variation. ETM can vastly decrease the transmission rates and alleviate the pressure of the communication channels on the premise of preserving the system performance [10–14]. Dai et al. ensure the passive synchronization of the MJNNs with gain varying in a random way [10]. In [14], Xu et al. discuss the design method of nonsynchronous H∞ filter for the singular Markovian jump systems, in which multiple redundant channels are utilized to enhance the quality of data transmission. Since the threshold value of ETM cannot adjust the sampling interval dynamically according to the changes of the system, the adaptive ETM is introduced for the nonlinear multiagent systems [15, 16]. And, the decentralized ETM is proposed in some results such as [12, 17–19] to ensure the ample utilization of network resource. A more optimized stochastic event-triggered strategy is developed in [20, 21], which ensures that only the innovational information is transmitted and the nontriggered information is also utilized. Of course, compared with single event-triggering mechanism or event-triggering mechanism, the advantages of hybrid driving mechanism are more obvious. Bernoulli distribution is used to unify the TTM and the ETM [22, 23]. On the premise of reducing the transmission rate of “invalid” signals and saving the network communication resources, it records the system state changes as much as possible and retains as much detailed information as possible. Therefore, the stochastic event-triggered based filter with better numerical stability and higher accuracy greatly meets the requirements of many practical systems.Owing to the limited communication bandwidth, the missing measurements that occur stochastically in NCSs are usually unavoidable. Frequent data packet loss will lead to a significant reduction in system performance and even lead to instability of the systems. Generally, missing measurements are described by variables meeting the laws of Bernoulli distribution [24–27] or a Markovian jump chain [28, 29]. In [25], Hu et al. focus on the time-varying nonlinear systems existing the phenomena of multiple packet dropout, in which a combination of independent variables conforming to Bernoulli distribution is exploited to reflect the uncertain probabilities of multiple missing measurements. And, a novel algorithm is put forward for the networked cascade control system with the purpose of suppressing the adverse effects of delays, finite channel, and missing measurements in [30]. For the sake of simulating a more general network environment, taking the randomly occurring missing measurement into account is indispensable.In practical communication, due to the openness, sharing, interconnection, and universality of the network, the communication network is vulnerable to external cyber-attacks. By definition, cyber attacks refer to the aggressive behaviors of destroying data transmission systems, authentic sampling data, communication infrastructure, and networked equipment. As a consequence, the system performance decreases seriously, and even the system may collapse. Cyber attacks are classified into three types: repeated attack, denial of service attack, and deception attack, in which the biggest threat to network security is deception attacks. Some interesting results associated with deception attacks have been given [31–34]. For instance, a secure filter is devised for the delayed stochastic nonlinear systems with a novel multiple-channel attack model [33]. For a nonlinear physical system subject to data injection attacks, an elastic filter is designed by employing a new ETM in reference [34]. The goal of [35] is to devise a filter which can guarantee the system security in the presence of stochastic sensor saturation and stochastic deception attacks. The stochastically occurring deception attacks are discussed for NNs, in which the attack signals are presumed to be norm bounded [23, 36]. With the assistance of an adaptive sliding mode controller, Chen et al. discuss the security of a Markovian jump system with injected spurious signals and semi-known transition probabilities in [37]. It should be noted that the MJNNs concerning with deception attacks and stochastic missing measurements have not been discussed in depth yet, which motives this article.Based on above discussions, the issue ofH∞ filtering for MJNNs under randomly occurring missing measurements and deception attacks is discussed. The contributions of this study are condensed into three major points: (1) an event-triggered communication strategy is proposed to reduce the redundant information exchange and relieve the pressure of network bandwidth. (2) In the design of filter, randomly occurring missing measurements and cyber attacks are considered to make it closer to the practical communication environment. (3) Based on the constructed mathematical model, sufficient conditions are derived to guarantee that the system is asymptotically stable.The main framework of this paper is generalized: Section2 presents the description of the system and the elaboration of all problems. The sufficient conditions to guarantee the asymptotic stability are given, and then a filter is derived for the MJNNs with the deception attacks and random missing measurements in Section 3. In Section 4, three numerical simulations are used to demonstrate the correctness and effectiveness of the analysis.Notations: in this paper, the superscripts “T” and “−1” represent the transpose and the inverse of a matrix; ℝn denotes n-dimensional Euclidean space; ℝn×m denotes the set of real matrices with m rows and n columns; P>0 (P∈ℝn×m) means that P is a real symmetric positive definite matrix; ℒ20,∞ denotes the space of square-integrable vector functions over 0,∞; I is a identity matrix; and ∗ represents the symmetric term of a symmetric matrix.


## 2. Problem Elaboration

### 2.1. System Description

Consider the MJNNs described as follows:(1)xk+1=Arkxk+Brkfxk+Erkgxk−ηk+Drkωk,yk=Crkxk,zk=Lrkxk,where xk=x1,x2,…,xnT∈ℝn stands for the state vector, yk∈ℝm is the measured output, and zk∈ℝp denotes the output to be estimated. The nonlinear vector-valued functions fxk=f1x1k,f2x2k,…,fnxnkT∈ℝn and gxk−ηk=g1x1k−ηk,g2x2k−ηk,…,gnxnk−ηkT∈ℝn are the neuron activation functions. ωk∈ℝq represents the exogenous disturbance with ωk∈ℒ20,∞. ηk∈ηm,ηM denotes the time-varying bounded delay of neural network. ηM and ηm represent the upper and lower bounds, respectively. Ark, Brk, Crk, Drk, Erk, and Lrk are known real constant matrices with appropriate dimension. The parameter rk is constrained by a homogeneous Markovian jump process and generally takes values from the finite set M=1,2,…,N. Π˜=πiji,j∈M represents the transition probability matrix, which is given by(2)Prrk+1=j|rk=i=πij,∀i,j∈M,where πij∈0,1, (∀i,j∈M) and ∑j=1Nπij=1, ∀i∈M.Construct the followingH∞ filter:(3)xfk+1=Afrkxfk+Bfrky¯k,zfk=Cfrkxfk,where xfk∈ℝn denotes state vector of filter, zfk∈ℝp represents the output of the filter, y¯k∈ℝm represents the actual input of filter, and Afrk, Bfrk, and Cfrk are appropriate matrices to be designed.For convenience,rk=i,i∈M, Ark is denoted by Ai, Brk by Bi, and so on.


### 2.2. Event-Triggered Mechanism

An event trigger is exploited in this work, by which the data releasing events are generated according to whether the variation of the immediate sampling data and the latest publishing data exceeds the set threshold. The following judgement algorithm is applied widely [11–14, 27]:(4)yk+jh−ykhTΨiyk+jh−ykh≤σiyTk+jhkΨiyk+jh,where Ψi∈ℝm are the positive definite weighting matrices that will be determined later, σi∈0,1 are presupposed scalars, yk+jh represents the immediate sampled sensor output, and ykh is the latest transmitted data. It should be noted that the immediate sampled measurement output yk+jh satisfying the inequality (4) will not be conveyed to the communication channels.Remark 1.
In the actual NCSs, the limited data transmission capacity usually cannot meet the transmission requirements of a large number of data. ETM can judge whether the currently sampled data is valid. Only the one that exceeds the threshold in (4) will be conveyed to the filter via the network channels; otherwise, it will be discarded.Remark 2.
Based on the similar threshold screening principle, a stochastic event-triggered strategy is proposed in [38, 39], which establishes an innovational set to judge whether the measured outputs are transferred to the network channels. Nevertheless, it is noted that this method improves the system performance by not only selecting the sampled data with more innovation but also making full use of the information contained in the innovational set. As discussed in [40], the ETM in algorithm (4) only monitors the difference between states sampled at discrete time and does not care what happens between updates. In addition, because the event trigger acts between the sensor and the filter, the expensive work of modifying the existing system is avoided.
According to the algorithm (4), suppose that the data releasing instants are κ0h,κ1h,κ2h,…,κ0=0. The release period of the event trigger is given as kqh=κq+1−κqh.Remark 3.
The set of data release instants meetsκ0,κ1,κ2,…⊆0,1,2,…. The amount of κ0,κ1,κ2,… is decided by the value of the variation of the measured output. Setting σi=0 implies that all sampled signals are released and conveyed to the filter smoothly.
Refer to [40], suppose that ζq is the time delay in the network communication at the moment kq with ζq∈0,ζM, where ζM is a positive real number.Case 1. If κqh+h+ζM≥κq+1h+ζq+1, introduce a function:(5)ζk=k−κq,k∈κqh+ζq,κq+1+ζq+1.Apparently,(6)ζq≤ζk≤κq+1−κqh+ζq+1≤h+ζM.Case 2. If κqh+h+ζM<κq+1h+ζq+1, consider the two intervals: κqh+ζq,κqh+h+ζM,κqh+jh+ζM,κqh+jh+h+ζM.
Sinceζq≥ζM, obviously there exists positive integer m that(7)κqh+mh+ζM<κq+1h+ζq+1≤κqh+mh+h+ζM,and yκqh,yκqh+jh (j=1,2,…,m) meets the condition (4).
Let(8)I0=κqh+ζq,κqh+h+ζM,Ij=∪j=0m−1κqh+jh+ζM+t,κqh+jh+h+ζM,IM=κqh+mh+ζM,κq+1h+ζq+1.
Define the function,(9)ζk=k−κqh,k∈I0,k−κqh−jh,k∈Ijj=1,…,m−1,k−κqh−mh,k∈IM.
From the (9), we can get(10)0≤ζq≤ζk≤h+ζM,k∈I0,0≤ζq≤ζM≤ζk≤h+ζM,k∈Ij,0≤ζq≤ζM≤ζk≤h+ζM,k∈IM.
Becauseζq+1h+ζq+1≤ζqh+m+1h+ζM, the third row of (10) holds. Then, we have 0≤ζq≤ζk≤h+ζM≜ζM.
For Case 1,k∈κqh+ζq,κq+1h+ζq+1, and the error of measurement output eyk=0 is defined.
For Case 2, define(11)eyk=0,k∈I0,yκqh+jh−yκqh,k∈Ij,yκqh+mh−yκqh,k∈IM.
Combining the algorithm (4) and the definition (11), it can be concluded that, for k∈κqh+ζq,κq+1h+ζq+1,(12)eyTkΨieyk≤σiyTk−ζkΨiyk−ζk.


### 2.3. Deception Attacks and Missing Measurements

In the process of data transmission, the missing measurements that occur with random probability are taken into consideration. Attackers attempt to decrease the stability and reliability of the system by injecting false signals into the measurement data in the communication network. It is assumed that the injected attack signals are not involved with the missing sensor measurements.Then, the actual inputy¯k of the filter is expressed as follows:(13)y¯k=ϱkyκq+ak−dk,where ak represents the time-discrete signal injected to the measured outputs by attackers. dt∈0,dM stands for the delay of cyber attacks. Bernoulli variable ϱk is shown as follows:(14)Probϱk=ϱ,ϱk=1,1−ϱ,ϱk=0.Remark 4.
According to equality (14), the measurement output can be conveyed to the filter smoothly with the probability ϱ in case of ϱk=1. while ϱk=0 implies that the filter fails to receive the transmitted measurement output.Remark 5.
The occurrence of missing measurements is unavoidable during the data transmission, which makes the actual input of the filter not equivalent to the measured output of the system. The data injection attacks are presumed to be conveyed to the filter via the network channels successfully, which are not in the consideration of packet dropout.


### 2.4. The Overall Model

Definex¯k+1=xTk,xfTkT and z¯k=zk−zfk, we can obtain the augmented filtering error system as follows:(15)x¯k+1=A¯ix¯k+B¯ifHx¯k+E¯igHx¯k−ηk+B¯11i+ϱk−ϱB¯12ix¯k−ζk+B¯fiak−dk+B¯21i+ϱk−ϱB¯22ieyk+D¯iωk,z¯k=L¯ix¯k,where(16)A¯i=Ai00Afi,B¯i=Bi0,B¯11i=00ϱBfiCi0,B¯12i=00BfiCi0,B¯21i=0−ϱBfi,B¯22i=0−Bfi,B¯fi=0Bfi,E¯i=Ei0,D¯i=Di0,H=I0,L¯i=Li−Cfi.The design problems ofH∞ filter can be summarized the conditions as follows:(i)
The augmented system (15) with ωk=0 is asymptotically stable for any initial conditions.(ii)
Given a scalarγ>0 and ωk∈ℒ20,∞, the filtering error z¯k satisfies(17)∑k=0+∞z¯k2≤γ2∑k=0+∞ωk2.Next, some assumptions and lemmas which are conducive to deriving the theorem are introduced.Assumption 1 (see [34]).
The deception attacksak meets the following criterion:(18)ak2≤Wxk2,where W is a given constant matrix.Assumption 2 (see [41]).
The neural functionsf· and g· in (1) meet the initial value setting f0=g0=0 and the following sector bounded condition:(19)fx−fy−U1x−yTfx−fy−U2x−y≤0,gx−gy−V1x−yTgx−gy−V2x−y≤0,∀x,y∈ℝn, where U1, U2, V1, and V2 are constant real matrices satisfying U2≥U2 and V1≥V2.Lemma 1 (see [42]).
For any symmetric positive-definite matrixG∈ℝn×n, scalars γ1 and γ2γ2>γ1, vector function χi:γ1,γ1+1,…γ2⟶ℝn, such that the following inequality holds:(20)−γ2−γ1+1∑i=γ1γ2χTiGχi≤−∑i=γ1γ2χTiG∑i=γ1γ2χi.Lemma 2 (see [43]).
For any matrixM>0 and constant ϵ, the following inequality holds:(21)−XTM−1X≤ε2X−2εX.


## 2.1. System Description

Consider the MJNNs described as follows:(1)xk+1=Arkxk+Brkfxk+Erkgxk−ηk+Drkωk,yk=Crkxk,zk=Lrkxk,where xk=x1,x2,…,xnT∈ℝn stands for the state vector, yk∈ℝm is the measured output, and zk∈ℝp denotes the output to be estimated. The nonlinear vector-valued functions fxk=f1x1k,f2x2k,…,fnxnkT∈ℝn and gxk−ηk=g1x1k−ηk,g2x2k−ηk,…,gnxnk−ηkT∈ℝn are the neuron activation functions. ωk∈ℝq represents the exogenous disturbance with ωk∈ℒ20,∞. ηk∈ηm,ηM denotes the time-varying bounded delay of neural network. ηM and ηm represent the upper and lower bounds, respectively. Ark, Brk, Crk, Drk, Erk, and Lrk are known real constant matrices with appropriate dimension. The parameter rk is constrained by a homogeneous Markovian jump process and generally takes values from the finite set M=1,2,…,N. Π˜=πiji,j∈M represents the transition probability matrix, which is given by(2)Prrk+1=j|rk=i=πij,∀i,j∈M,where πij∈0,1, (∀i,j∈M) and ∑j=1Nπij=1, ∀i∈M.Construct the followingH∞ filter:(3)xfk+1=Afrkxfk+Bfrky¯k,zfk=Cfrkxfk,where xfk∈ℝn denotes state vector of filter, zfk∈ℝp represents the output of the filter, y¯k∈ℝm represents the actual input of filter, and Afrk, Bfrk, and Cfrk are appropriate matrices to be designed.For convenience,rk=i,i∈M, Ark is denoted by Ai, Brk by Bi, and so on.


## 2.2. Event-Triggered Mechanism

An event trigger is exploited in this work, by which the data releasing events are generated according to whether the variation of the immediate sampling data and the latest publishing data exceeds the set threshold. The following judgement algorithm is applied widely [11–14, 27]:(4)yk+jh−ykhTΨiyk+jh−ykh≤σiyTk+jhkΨiyk+jh,where Ψi∈ℝm are the positive definite weighting matrices that will be determined later, σi∈0,1 are presupposed scalars, yk+jh represents the immediate sampled sensor output, and ykh is the latest transmitted data. It should be noted that the immediate sampled measurement output yk+jh satisfying the inequality (4) will not be conveyed to the communication channels.Remark 1.
In the actual NCSs, the limited data transmission capacity usually cannot meet the transmission requirements of a large number of data. ETM can judge whether the currently sampled data is valid. Only the one that exceeds the threshold in (4) will be conveyed to the filter via the network channels; otherwise, it will be discarded.Remark 2.
Based on the similar threshold screening principle, a stochastic event-triggered strategy is proposed in [38, 39], which establishes an innovational set to judge whether the measured outputs are transferred to the network channels. Nevertheless, it is noted that this method improves the system performance by not only selecting the sampled data with more innovation but also making full use of the information contained in the innovational set. As discussed in [40], the ETM in algorithm (4) only monitors the difference between states sampled at discrete time and does not care what happens between updates. In addition, because the event trigger acts between the sensor and the filter, the expensive work of modifying the existing system is avoided.
According to the algorithm (4), suppose that the data releasing instants are κ0h,κ1h,κ2h,…,κ0=0. The release period of the event trigger is given as kqh=κq+1−κqh.Remark 3.
The set of data release instants meetsκ0,κ1,κ2,…⊆0,1,2,…. The amount of κ0,κ1,κ2,… is decided by the value of the variation of the measured output. Setting σi=0 implies that all sampled signals are released and conveyed to the filter smoothly.
Refer to [40], suppose that ζq is the time delay in the network communication at the moment kq with ζq∈0,ζM, where ζM is a positive real number.Case 1. If κqh+h+ζM≥κq+1h+ζq+1, introduce a function:(5)ζk=k−κq,k∈κqh+ζq,κq+1+ζq+1.Apparently,(6)ζq≤ζk≤κq+1−κqh+ζq+1≤h+ζM.Case 2. If κqh+h+ζM<κq+1h+ζq+1, consider the two intervals: κqh+ζq,κqh+h+ζM,κqh+jh+ζM,κqh+jh+h+ζM.
Sinceζq≥ζM, obviously there exists positive integer m that(7)κqh+mh+ζM<κq+1h+ζq+1≤κqh+mh+h+ζM,and yκqh,yκqh+jh (j=1,2,…,m) meets the condition (4).
Let(8)I0=κqh+ζq,κqh+h+ζM,Ij=∪j=0m−1κqh+jh+ζM+t,κqh+jh+h+ζM,IM=κqh+mh+ζM,κq+1h+ζq+1.
Define the function,(9)ζk=k−κqh,k∈I0,k−κqh−jh,k∈Ijj=1,…,m−1,k−κqh−mh,k∈IM.
From the (9), we can get(10)0≤ζq≤ζk≤h+ζM,k∈I0,0≤ζq≤ζM≤ζk≤h+ζM,k∈Ij,0≤ζq≤ζM≤ζk≤h+ζM,k∈IM.
Becauseζq+1h+ζq+1≤ζqh+m+1h+ζM, the third row of (10) holds. Then, we have 0≤ζq≤ζk≤h+ζM≜ζM.
For Case 1,k∈κqh+ζq,κq+1h+ζq+1, and the error of measurement output eyk=0 is defined.
For Case 2, define(11)eyk=0,k∈I0,yκqh+jh−yκqh,k∈Ij,yκqh+mh−yκqh,k∈IM.
Combining the algorithm (4) and the definition (11), it can be concluded that, for k∈κqh+ζq,κq+1h+ζq+1,(12)eyTkΨieyk≤σiyTk−ζkΨiyk−ζk.


## 2.3. Deception Attacks and Missing Measurements

In the process of data transmission, the missing measurements that occur with random probability are taken into consideration. Attackers attempt to decrease the stability and reliability of the system by injecting false signals into the measurement data in the communication network. It is assumed that the injected attack signals are not involved with the missing sensor measurements.Then, the actual inputy¯k of the filter is expressed as follows:(13)y¯k=ϱkyκq+ak−dk,where ak represents the time-discrete signal injected to the measured outputs by attackers. dt∈0,dM stands for the delay of cyber attacks. Bernoulli variable ϱk is shown as follows:(14)Probϱk=ϱ,ϱk=1,1−ϱ,ϱk=0.Remark 4.
According to equality (14), the measurement output can be conveyed to the filter smoothly with the probability ϱ in case of ϱk=1. while ϱk=0 implies that the filter fails to receive the transmitted measurement output.Remark 5.
The occurrence of missing measurements is unavoidable during the data transmission, which makes the actual input of the filter not equivalent to the measured output of the system. The data injection attacks are presumed to be conveyed to the filter via the network channels successfully, which are not in the consideration of packet dropout.


## 2.4. The Overall Model

Definex¯k+1=xTk,xfTkT and z¯k=zk−zfk, we can obtain the augmented filtering error system as follows:(15)x¯k+1=A¯ix¯k+B¯ifHx¯k+E¯igHx¯k−ηk+B¯11i+ϱk−ϱB¯12ix¯k−ζk+B¯fiak−dk+B¯21i+ϱk−ϱB¯22ieyk+D¯iωk,z¯k=L¯ix¯k,where(16)A¯i=Ai00Afi,B¯i=Bi0,B¯11i=00ϱBfiCi0,B¯12i=00BfiCi0,B¯21i=0−ϱBfi,B¯22i=0−Bfi,B¯fi=0Bfi,E¯i=Ei0,D¯i=Di0,H=I0,L¯i=Li−Cfi.The design problems ofH∞ filter can be summarized the conditions as follows:(i)
The augmented system (15) with ωk=0 is asymptotically stable for any initial conditions.(ii)
Given a scalarγ>0 and ωk∈ℒ20,∞, the filtering error z¯k satisfies(17)∑k=0+∞z¯k2≤γ2∑k=0+∞ωk2.Next, some assumptions and lemmas which are conducive to deriving the theorem are introduced.Assumption 1 (see [34]).
The deception attacksak meets the following criterion:(18)ak2≤Wxk2,where W is a given constant matrix.Assumption 2 (see [41]).
The neural functionsf· and g· in (1) meet the initial value setting f0=g0=0 and the following sector bounded condition:(19)fx−fy−U1x−yTfx−fy−U2x−y≤0,gx−gy−V1x−yTgx−gy−V2x−y≤0,∀x,y∈ℝn, where U1, U2, V1, and V2 are constant real matrices satisfying U2≥U2 and V1≥V2.Lemma 1 (see [42]).
For any symmetric positive-definite matrixG∈ℝn×n, scalars γ1 and γ2γ2>γ1, vector function χi:γ1,γ1+1,…γ2⟶ℝn, such that the following inequality holds:(20)−γ2−γ1+1∑i=γ1γ2χTiGχi≤−∑i=γ1γ2χTiG∑i=γ1γ2χi.Lemma 2 (see [43]).
For any matrixM>0 and constant ϵ, the following inequality holds:(21)−XTM−1X≤ε2X−2εX.


## 3. Main Results

### 3.1. Asymptotical Stability Analysis

First, the asymptotic stability of the filtering error system (15) with ωk=0 is discussed.Theorem 1.
For given delay boundsηm, ηM, dM, ζM, trigger parameters σi, and matrix W, system (15) is asymptotically stable with an H∞ performance index γ, if there exist matrices Pi>0, Ψi>0i∈M, Qs>0, and Rs>0s=1,2,3,4 with appropriate dimensions and scalars α1>0α2>0, satisfying the following equation:(22)Σ=Ω1Ω2∗Ω3<0,(23)∑j=1NπijPj≤Pi,where(24)Ω1=Ξ1Ξ2∗Ξ3,Ω2=Φ1TPiΦ3TPiη¯Φ2TR1η¯Φ3TR1ηmΦ2TR2ηmΦ3TR2ζMΦ2TR3ζMΦ3TR3dMΦ2TR4dMΦ3TR4,Ω3=diag−Pi,−Pi,−R1,−R1,−R2,−R2,−R3,−R3,−R4,−R4,Ξ1=Θ10R20R30R4∗Θ2R1+R20000∗∗Θ3R1000∗∗∗Θ4000∗∗∗∗Θ5R30∗∗∗∗∗Θ60∗∗∗∗∗∗Θ7,Ξ2=0−α1U¯20000000000−α2V¯200000000000000000R40000,Ξ3=diag−Q4−R4,−α1I,−α2I,−Ψi,−I,(25)Θ1=−Pi+Q1+Q2+Q3+Q4−R2−R3−R4−α1U¯1,Θ2=−Q1−R1−R2,Θ3=−2R1−2R2−α2V¯1,Θ4=−Q2−R1,Θ5=−2R3+σiΨ^i,Ψ^i=CiTΨiCi000,Θ6=−Q3−R3,Θ7=−2R4+HTWH,η¯=ηM−ηm,c2=ϱ1−ϱ,Φ1=A¯i000B¯11i000B¯iE¯iB¯21iB¯fi,Φ2=A¯i−I000B¯11i000B¯iE¯iB¯21iB¯fi,Φ3=0000cB¯12i00000cB¯22i0.Proof.
Construct the following Lyapunov functional as follows:(26)Vk=V1k+V2k+V3k,where(27)V1k=x¯TkPix¯k,V2k=∑i=k−ηmk−1x¯TiQ1x¯i+∑i=k−ηMk−1x¯TiQ2x¯i+∑i=k−ζMk−1x¯TiQ3x¯i+∑i=k−dMk−1x¯TiQ4x¯i,V3k=ηM−ηm∑j=k−ηMk−ηm−1∑λ=jk−1μTλR1μλ+ηm∑j=k−ηmk−1∑λ=jk−1μTλR2μλ+ζM∑j=k−ζMk−1∑λ=jk−1μTλR3μλ+dM∑j=k−dMk−1∑λ=jk−1μTλR4μλ,μλ=x¯λ+1−x¯λ,under the condition of Pi>0, Qs>0, and Rs>0s=1,2,3,4.
We can get(28)ΔV1k=V1k+1−V1k=x¯Tk+1∑j=1NπijPjx¯k+1−x¯TkPix¯k,(29)ΔV2k=V2k+1−V2k=x¯TkQ1x¯k−x¯Tk−ηmQ1x¯k−ηm+x¯TkQ2x¯k−x¯Tk−ηMQ2x¯k−ηM+x¯TkQ3x¯k−x¯Tk−ζMQ3x¯k−ζM+x¯TkQ4x¯k−x¯Tk−dMQ4x¯k−dM,(30)ΔV3k=V3k+1−V3k=ηM−ηm2μTkR1μk−ηM−ηm∑λ=k−ηMk−ηm−1μTλR1μλ+ηm2μTkR2μk−ηm∑λ=k−ηmk−1μTλR2μλ+ζM2μTkR3μk−ζM∑λ=k−ζMk−1μTλR3μλ+dM2μTkR4μk−dM∑λ=k−dMk−1μTλR4μλ.
Letting(31)ξTk=x¯Tkx¯Tk−ηmx¯Tk−ηkx¯Tk−ηMx¯Tk−ζkx¯Tk−ζMx¯Tk−dkx¯Tk−dMfTHx¯kgTHx¯k−ηkeyTkak−dk.
So, it is obvious that(32)x¯k+1=Φ^ξk,μk=x¯k+1−x¯k=Φ^−I1ξk,where(33)Φ^=A¯i000B¯1i000B¯iE¯iB¯2iB¯fi,B¯1i=B¯11i+ϱk−ϱB¯12i,B¯2i=B¯21i+ϱk−ϱB¯22i,I1=I00000000000.
Then, one can get the following equation:(34)μTkRiμk=ξTkΦ^−ITRiΦ^−Iξk=ξTkΦ2TRiΦ2ξk+ξTkΦ3TRiΦ3ξk.
By using Assumption2, for scalars α1>0, α2>0, we can get(35)−α1x¯kfHx¯kTU¯1U¯2∗Ix¯kfHx¯k≥0,−α2x¯k−ηkgHx¯k−ηkTV¯1V¯2∗Ix¯k−ηkgHx¯k−ηk≥0,where(36)U¯1=HTU^1H,U^1=U1TU2+U2TU12,U¯2=−HTU^2,U^2=U1T+U2T2,V¯1=HTV^1H,V¯2=−HTV^2,V^1=V1TV2+V2TV12,V^2=V1T+V2T2.
Recalling the restrictive condition of deception attacks in (18), there exists an inequality as follows:(37)aTk−dkak−dk≤x¯Tk−dkHTWHx¯k−dk.
According to Lemma1 and combining conditions (12) and (26)–(38), the following inequality is obtained:(38)ΔVk≤ξTkΠξk,where(39)Π=Ω1+Φ1TPiΦ1+Φ3TPiΦ3+ηM−ηm2Φ2TR1Φ2+ηM−ηm2Φ3TR1Φ3+ηm2Φ2TR2Φ2+ηm2Φ3TR2Φ3+ζM2Φ2TR3Φ2+ζM2Φ3TR3Φ3+dM2Φ2TR4Φ2+dM2Φ3TR4Φ3.
By employing Schur Complement lemma, the asymptotic stability of system (15) with ωk=0 is guaranteed if condition (22) holds.


### 3.2.H∞Performance Analysis

TheH∞ performance of the augmented system (15) is analysed in the second theorem.Theorem 2.
For given delay boundsηm, ηM, ζM, dM, trigger parameters σi, and matrix W, the system (15) is asymptotically stable with an H∞ performance index γ, if there exist matrices Pi>0, Ψi>0i∈M, Qs>0, and Rs>0s=1,2,3,4 with appropriate dimensions and scalars α1>0α2>0, satisfying the following:(40)Σ¯=Ω¯1Ω¯2∗Ω¯3<0,(41)∑j=1NπijPj≤Pi,where(42)Ω¯1=Ω10∗−γ2I,Ω¯3=Ω30∗−I,Ω¯2=Φ¯1TPiΦ¯3TPiη¯Φ¯2TR1η¯Φ¯3TR1ηmΦ¯2TR2ηmΦ¯3TR2ζMΦ¯2TR3ζMΦ¯3TR3dMΦ¯2TR4dMΦ¯3TR4ΓT,Φ¯1=Φ1D¯i,Φ¯2=Φ2D¯i,Φ¯3=Φ30,Γ=L¯i000000000000.Proof 2.
Based on the same Lyapunov functional as Theorem1, one can derive the following inequality:(43)ΔVk−γ2ωTkωk+z¯kz¯k≤ξ¯TkΠ¯ξ¯k,where(44)ξ¯k=ξTkωTkT,Π¯=Ω¯1+Φ¯1TPiΦ¯1+Φ¯3TPiΦ¯3+ηM−ηm2Φ¯2TR1Φ¯2+ηM−ηm2Φ¯3TR1Φ¯3+ηm2Φ¯2TR2Φ¯2+ηm2Φ¯3TR2Φ¯3+ζM2Φ¯2TR3Φ¯2+ζM2Φ¯3TR3Φ¯3+dM2Φ¯2TR4Φ¯2+dM2Φ¯3TR4Φ¯3+ΓTΓ.
According to Schur Complement lemma, (40) and (41) are sufficient conditions to guarantee Π¯<0, and the following inequality can be ensured:(45)ΔVk−γ2ωTkωk+z¯kz¯k<0,under the zero initial condition, one can get(46)∑k=0∞z¯k2≤γ2∑k=0∞ωk2.
The proof is complete.


### 3.3.H∞Filter Design

Based on the derivation of Theorems1 and 2, the H∞ filter is designed.Theorem 3.
For given delay boundsηm, ηM, ζM, dM, trigger parameters σi, and matrix W, the filtering error system (15) is asymptotically stable with an H∞ performance index γ, if there exist real matrices P1i>0, Xi>0, Ψi>0i∈M, Q˜s>0, R˜s>0s=1,2,3,4, A^fi, B^fi, and C^fi with appropriate dimension, and scalars α1>0,α1>0 satisfying P1i−Xi>0, such that(47)Σ˜=Ω˜1Ω˜2∗Ω˜3,(48)∑j=1NπijP1j−Xj≤P1i−Xi,where(49)Ω˜1=Ξ˜1Ξ˜2∗Ξ˜3,Ω˜2=Ξ˜41Ξ˜42,Ω˜3=diag−P˜i,−P˜i,−2P˜i+R˜1,−2P˜i+R˜1,−2P˜i+R˜2,−2P˜i+R˜2,−2P˜i+R˜3,−2P˜i+R˜3,−2P˜i+R˜4,−2P˜i+R˜4,−I%,Ξ˜1=Θ˜10R˜20R˜30R˜4∗Θ˜2R˜1+R˜20000∗∗Θ˜3R˜1000∗∗∗Θ˜4000∗∗∗∗Θ˜5R˜30∗∗∗∗∗Θ˜60∗∗∗∗∗∗Θ˜7,Ξ˜2=0Θ˜8000000000000Θ˜9000000000000000000000R˜400000,Ξ˜3=diag−Q˜4−R˜4,−α1I,−α2I,−Ψi,−I,−γ2I,Θ˜1=−P˜i+Q˜1+Q˜2+Q˜3+Q˜4−R˜2−R˜3−R˜4−α1U¯1,Θ˜2=−Q˜1−R˜1−R˜2,Θ˜3=−2R˜1−2R˜2−α2V¯1,Θ˜4=−Q˜2−R˜1,Θ˜5=−2R˜3+σiΨ^i,Θ˜6=−Q˜3−R˜3,Θ˜7=−2R˜4+W¯,Θ˜8=α1U^20,Θ˜9=α2V^20,Ψ^i=CiTΨiCi000,W¯=W000,Ξ˜41=Θ˜41Θ˜11Θ˜21Θ˜31Θ˜51Θ˜61Θ˜71Θ˜81Θ˜91Θ˜01Θ˜111,Ξ˜42=Θ˜12Θ˜22Θ˜32Θ˜42Θ˜52Θ˜62Θ˜72Θ˜82Θ˜92Θ˜02O,(50)Θ˜11=ϒ11T000ϒ12T00T,ϒ11=AiTP1iAiTXiA^fiTA^fiT,ϒ12=ϱCiTB^fiTϱCiTB^fiT00,Θ˜12=0ϒ13Tϒ14Tϒ15Tϒ16Tϒ17TT,ϒ13=BiTP1iBiTXi,ϒ14=EiTP1iEiTXi,ϒ15=−ϱB^fiT−ϱB^fiT,ϒ16=B^fiTB^fiT,ϒ17=DiTP1iDiTXi,Θ˜21=0000ϒ21T00T,ϒ21=cCiTB^fiTcCiTB^fiT00,Θ˜22=000ϒ22T00T,ϒ22=−cB^fiT−cB^fiT,(51)Θ˜31=ϒ31T000ϒ32T00T,ϒ31=η¯AiTP1i−P1iη¯AiTXi−Xiη¯A^fiT−Xiη¯A^fiT−Xi,ϒ32=η¯ϱCiTB^fiTη¯ϱCiTB^fiT00,Θ˜32=0ϒ33Tϒ34Tϒ35Tϒ36Tϒ37TT,ϒ33=η¯BiTP1iη¯BiTXi,ϒ34=η¯EiTP1iη¯EiTXi,ϒ35=−η¯ϱB^fiT−η¯ϱB^fiT,ϒ36=η¯B^fiTη¯B^fiT,ϒ37=η¯DiTP1iη¯DiTXi,Θ˜41=0000ϒ41T00T,ϒ41=η¯cCiTB^fiTη¯cCiTB^fiT00,Θ˜42=000ϒ42T00T,ϒ42=−η¯cB^fiT−η¯cB^fiT,Θ˜51=ϒ51T000ϒ52T00T,ϒ51=ηmAiTP1i−P1iηmAiTXi−XiηmA^fiT−XiηmA^fiT−Xi,(52)ϒ52=ηmϱCiTB^fiTηmϱCiTB^fiT00,Θ˜52=0ϒ53Tϒ54Tϒ55Tϒ56Tϒ57TT,ϒ53=ηmBiTP1iηmBiTXi,ϒ54=ηmEiTP1iηmEiTXi,ϒ55=−ηmϱB^fiT−ηmϱB^fiT,ϒ56=ηmB^fiTηmB^fiT,ϒ57=ηmDiTP1iηmDiTXi,Θ˜61=0000ϒ61T00T,ϒ61=ηmcCiTB^fiTηmcCiTB^fiT00,Θ˜62=000ϒ62T00T,ϒ62=−ηmcB^fiT−ηmcB^fiT,Θ˜71=ϒ71T000ϒ72T00T,ϒ71=ζMAiTP1i−P1iζMAiTXi−XiζMA^fiT−XiζMA^fiT−Xi,ϒ72=ζMϱCiTB^fiTζMϱCiTB^fiT00,Θ˜72=0ϒ73Tϒ74Tϒ75Tϒ76Tϒ77TT,ϒ73=ζMBiTP1iζMBiTXi,ϒ74=ζMEiTP1iζMEiTXi,ϒ75=−ζMϱB^fiT−ζMϱB^fiT,ϒ76=ζMB^fiTζMB^fiT,ϒ77=ζMDiTP1iζMDiTXi,(53)Θ˜81=0000ϒ81T00T,Υ81=ζMcCiTB^fiTζMcCiTB^fiT00,Θ˜82=000ϒ82T00T,ϒ82=−ζMcB^fiT−ζMcB^fiT,Θ˜91=ϒ91T000ϒ92T00T,ϒ91=dMAiTP1i−P1idMAiTXi−XidMA^fiT−XidMA^fiT−Xi,ϒ92=dMϱCiTB^fiTdMϱCiTB^fiT00,Θ˜92=0ϒ93Tϒ94Tϒ95Tϒ96Tϒ97TT,ϒ93=dMBiTP1idMBiTXi,ϒ94=dMEiTP1idMEiTXi,ϒ95=−dMϱB^fiT−dMϱB^fiT,ϒ96=dMB^fiTdMB^fiT,ϒ97=dMDiTP1idMDiTXi,Θ˜101=0000ϒ101T00T,ϒ101=dMcCiTB^fiTdMcCiTB^fiT00,Θ˜102=000ϒ102T00T,ϒ102=−dMcB^fiT−dMcB^fiT,Θ˜111=ϒ111T000000Tϒ111=LiT−CfiT.The parameter matrices of the filter are given by(54)Afi=A^fiXi−1,Bfi=B^fi,Cfi=C^fiXi−1.Proof.
Firstly, pre- and postmultiplying (40) by K=diagI,…,I︷15,K1,K2,K3,K4,I and KT, in which K1=diagPiR1−1,PiR1−1, K2=PiR2−1,PiR2−1, K3=PiR3−1,PiR3−1, and K4=PiR4−1,PiR4−1,, one can have(55)Σ^=KTΣ¯K=Ω¯1Ω^2∗Ω^3<0,with(56)Ω^2=Φ¯1TPiΦ¯3TPiη¯Φ¯2TPiη¯Φ¯3TPiηmΦ¯2TPiηmΦ¯3TPiζMΦ¯2TPiζMΦ¯3TPidMΦ¯2TPidMΦ¯3TPiΓT,Ω^3=diag−Pi,−Pi,−PiR1−1Pi,−PiR1−1Pi,−PiR2−1Pi,−PiR2−1Pi,−PiR3−1Pi,−PiR3−1Pi,−PiR4−1Pi,−PiR4−1Pi,−I.
Partition the matrixPi as Pi=P1iP2iP2iTP3i and define Xi=P2iP3i−1P2iT, Ji=I00P2iTP3i−1.
SincePi>0, by Schur Complement, it follows that P1i>0 and P1i−Xi>0.
Define(57)J=diagJi,…,Ji︷8,I,…,I︷5,Ji,…,Ji︷10,I.
MultiplyJ and JT on both sides of (55), respectively, (47) is obtained by defining(58)P˜i=JiPiJiT=P1iXiXiXi,Q˜s=JiQsJi,R˜s=JiRsJis=1,2,3,4,A^fi=P2iAfiP3i−1P2iT,B^fi=P2iTBfi,C^fi=CfiP3i−1P2iT.
On the basis of the linear transformation above and according to Lemma2, −PiRs−1Pi can be replaced as −2P˜i+R˜ss=1,2,3,4 for simplicity. It can be seen that (40) is equal to (47). This proof is complete.


## 3.1. Asymptotical Stability Analysis

First, the asymptotic stability of the filtering error system (15) with ωk=0 is discussed.Theorem 1.
For given delay boundsηm, ηM, dM, ζM, trigger parameters σi, and matrix W, system (15) is asymptotically stable with an H∞ performance index γ, if there exist matrices Pi>0, Ψi>0i∈M, Qs>0, and Rs>0s=1,2,3,4 with appropriate dimensions and scalars α1>0α2>0, satisfying the following equation:(22)Σ=Ω1Ω2∗Ω3<0,(23)∑j=1NπijPj≤Pi,where(24)Ω1=Ξ1Ξ2∗Ξ3,Ω2=Φ1TPiΦ3TPiη¯Φ2TR1η¯Φ3TR1ηmΦ2TR2ηmΦ3TR2ζMΦ2TR3ζMΦ3TR3dMΦ2TR4dMΦ3TR4,Ω3=diag−Pi,−Pi,−R1,−R1,−R2,−R2,−R3,−R3,−R4,−R4,Ξ1=Θ10R20R30R4∗Θ2R1+R20000∗∗Θ3R1000∗∗∗Θ4000∗∗∗∗Θ5R30∗∗∗∗∗Θ60∗∗∗∗∗∗Θ7,Ξ2=0−α1U¯20000000000−α2V¯200000000000000000R40000,Ξ3=diag−Q4−R4,−α1I,−α2I,−Ψi,−I,(25)Θ1=−Pi+Q1+Q2+Q3+Q4−R2−R3−R4−α1U¯1,Θ2=−Q1−R1−R2,Θ3=−2R1−2R2−α2V¯1,Θ4=−Q2−R1,Θ5=−2R3+σiΨ^i,Ψ^i=CiTΨiCi000,Θ6=−Q3−R3,Θ7=−2R4+HTWH,η¯=ηM−ηm,c2=ϱ1−ϱ,Φ1=A¯i000B¯11i000B¯iE¯iB¯21iB¯fi,Φ2=A¯i−I000B¯11i000B¯iE¯iB¯21iB¯fi,Φ3=0000cB¯12i00000cB¯22i0.Proof.
Construct the following Lyapunov functional as follows:(26)Vk=V1k+V2k+V3k,where(27)V1k=x¯TkPix¯k,V2k=∑i=k−ηmk−1x¯TiQ1x¯i+∑i=k−ηMk−1x¯TiQ2x¯i+∑i=k−ζMk−1x¯TiQ3x¯i+∑i=k−dMk−1x¯TiQ4x¯i,V3k=ηM−ηm∑j=k−ηMk−ηm−1∑λ=jk−1μTλR1μλ+ηm∑j=k−ηmk−1∑λ=jk−1μTλR2μλ+ζM∑j=k−ζMk−1∑λ=jk−1μTλR3μλ+dM∑j=k−dMk−1∑λ=jk−1μTλR4μλ,μλ=x¯λ+1−x¯λ,under the condition of Pi>0, Qs>0, and Rs>0s=1,2,3,4.
We can get(28)ΔV1k=V1k+1−V1k=x¯Tk+1∑j=1NπijPjx¯k+1−x¯TkPix¯k,(29)ΔV2k=V2k+1−V2k=x¯TkQ1x¯k−x¯Tk−ηmQ1x¯k−ηm+x¯TkQ2x¯k−x¯Tk−ηMQ2x¯k−ηM+x¯TkQ3x¯k−x¯Tk−ζMQ3x¯k−ζM+x¯TkQ4x¯k−x¯Tk−dMQ4x¯k−dM,(30)ΔV3k=V3k+1−V3k=ηM−ηm2μTkR1μk−ηM−ηm∑λ=k−ηMk−ηm−1μTλR1μλ+ηm2μTkR2μk−ηm∑λ=k−ηmk−1μTλR2μλ+ζM2μTkR3μk−ζM∑λ=k−ζMk−1μTλR3μλ+dM2μTkR4μk−dM∑λ=k−dMk−1μTλR4μλ.
Letting(31)ξTk=x¯Tkx¯Tk−ηmx¯Tk−ηkx¯Tk−ηMx¯Tk−ζkx¯Tk−ζMx¯Tk−dkx¯Tk−dMfTHx¯kgTHx¯k−ηkeyTkak−dk.
So, it is obvious that(32)x¯k+1=Φ^ξk,μk=x¯k+1−x¯k=Φ^−I1ξk,where(33)Φ^=A¯i000B¯1i000B¯iE¯iB¯2iB¯fi,B¯1i=B¯11i+ϱk−ϱB¯12i,B¯2i=B¯21i+ϱk−ϱB¯22i,I1=I00000000000.
Then, one can get the following equation:(34)μTkRiμk=ξTkΦ^−ITRiΦ^−Iξk=ξTkΦ2TRiΦ2ξk+ξTkΦ3TRiΦ3ξk.
By using Assumption2, for scalars α1>0, α2>0, we can get(35)−α1x¯kfHx¯kTU¯1U¯2∗Ix¯kfHx¯k≥0,−α2x¯k−ηkgHx¯k−ηkTV¯1V¯2∗Ix¯k−ηkgHx¯k−ηk≥0,where(36)U¯1=HTU^1H,U^1=U1TU2+U2TU12,U¯2=−HTU^2,U^2=U1T+U2T2,V¯1=HTV^1H,V¯2=−HTV^2,V^1=V1TV2+V2TV12,V^2=V1T+V2T2.
Recalling the restrictive condition of deception attacks in (18), there exists an inequality as follows:(37)aTk−dkak−dk≤x¯Tk−dkHTWHx¯k−dk.
According to Lemma1 and combining conditions (12) and (26)–(38), the following inequality is obtained:(38)ΔVk≤ξTkΠξk,where(39)Π=Ω1+Φ1TPiΦ1+Φ3TPiΦ3+ηM−ηm2Φ2TR1Φ2+ηM−ηm2Φ3TR1Φ3+ηm2Φ2TR2Φ2+ηm2Φ3TR2Φ3+ζM2Φ2TR3Φ2+ζM2Φ3TR3Φ3+dM2Φ2TR4Φ2+dM2Φ3TR4Φ3.
By employing Schur Complement lemma, the asymptotic stability of system (15) with ωk=0 is guaranteed if condition (22) holds.


## 3.2.H∞Performance Analysis

TheH∞ performance of the augmented system (15) is analysed in the second theorem.Theorem 2.
For given delay boundsηm, ηM, ζM, dM, trigger parameters σi, and matrix W, the system (15) is asymptotically stable with an H∞ performance index γ, if there exist matrices Pi>0, Ψi>0i∈M, Qs>0, and Rs>0s=1,2,3,4 with appropriate dimensions and scalars α1>0α2>0, satisfying the following:(40)Σ¯=Ω¯1Ω¯2∗Ω¯3<0,(41)∑j=1NπijPj≤Pi,where(42)Ω¯1=Ω10∗−γ2I,Ω¯3=Ω30∗−I,Ω¯2=Φ¯1TPiΦ¯3TPiη¯Φ¯2TR1η¯Φ¯3TR1ηmΦ¯2TR2ηmΦ¯3TR2ζMΦ¯2TR3ζMΦ¯3TR3dMΦ¯2TR4dMΦ¯3TR4ΓT,Φ¯1=Φ1D¯i,Φ¯2=Φ2D¯i,Φ¯3=Φ30,Γ=L¯i000000000000.Proof 2.
Based on the same Lyapunov functional as Theorem1, one can derive the following inequality:(43)ΔVk−γ2ωTkωk+z¯kz¯k≤ξ¯TkΠ¯ξ¯k,where(44)ξ¯k=ξTkωTkT,Π¯=Ω¯1+Φ¯1TPiΦ¯1+Φ¯3TPiΦ¯3+ηM−ηm2Φ¯2TR1Φ¯2+ηM−ηm2Φ¯3TR1Φ¯3+ηm2Φ¯2TR2Φ¯2+ηm2Φ¯3TR2Φ¯3+ζM2Φ¯2TR3Φ¯2+ζM2Φ¯3TR3Φ¯3+dM2Φ¯2TR4Φ¯2+dM2Φ¯3TR4Φ¯3+ΓTΓ.
According to Schur Complement lemma, (40) and (41) are sufficient conditions to guarantee Π¯<0, and the following inequality can be ensured:(45)ΔVk−γ2ωTkωk+z¯kz¯k<0,under the zero initial condition, one can get(46)∑k=0∞z¯k2≤γ2∑k=0∞ωk2.
The proof is complete.


## 3.3.H∞Filter Design

Based on the derivation of Theorems1 and 2, the H∞ filter is designed.Theorem 3.
For given delay boundsηm, ηM, ζM, dM, trigger parameters σi, and matrix W, the filtering error system (15) is asymptotically stable with an H∞ performance index γ, if there exist real matrices P1i>0, Xi>0, Ψi>0i∈M, Q˜s>0, R˜s>0s=1,2,3,4, A^fi, B^fi, and C^fi with appropriate dimension, and scalars α1>0,α1>0 satisfying P1i−Xi>0, such that(47)Σ˜=Ω˜1Ω˜2∗Ω˜3,(48)∑j=1NπijP1j−Xj≤P1i−Xi,where(49)Ω˜1=Ξ˜1Ξ˜2∗Ξ˜3,Ω˜2=Ξ˜41Ξ˜42,Ω˜3=diag−P˜i,−P˜i,−2P˜i+R˜1,−2P˜i+R˜1,−2P˜i+R˜2,−2P˜i+R˜2,−2P˜i+R˜3,−2P˜i+R˜3,−2P˜i+R˜4,−2P˜i+R˜4,−I%,Ξ˜1=Θ˜10R˜20R˜30R˜4∗Θ˜2R˜1+R˜20000∗∗Θ˜3R˜1000∗∗∗Θ˜4000∗∗∗∗Θ˜5R˜30∗∗∗∗∗Θ˜60∗∗∗∗∗∗Θ˜7,Ξ˜2=0Θ˜8000000000000Θ˜9000000000000000000000R˜400000,Ξ˜3=diag−Q˜4−R˜4,−α1I,−α2I,−Ψi,−I,−γ2I,Θ˜1=−P˜i+Q˜1+Q˜2+Q˜3+Q˜4−R˜2−R˜3−R˜4−α1U¯1,Θ˜2=−Q˜1−R˜1−R˜2,Θ˜3=−2R˜1−2R˜2−α2V¯1,Θ˜4=−Q˜2−R˜1,Θ˜5=−2R˜3+σiΨ^i,Θ˜6=−Q˜3−R˜3,Θ˜7=−2R˜4+W¯,Θ˜8=α1U^20,Θ˜9=α2V^20,Ψ^i=CiTΨiCi000,W¯=W000,Ξ˜41=Θ˜41Θ˜11Θ˜21Θ˜31Θ˜51Θ˜61Θ˜71Θ˜81Θ˜91Θ˜01Θ˜111,Ξ˜42=Θ˜12Θ˜22Θ˜32Θ˜42Θ˜52Θ˜62Θ˜72Θ˜82Θ˜92Θ˜02O,(50)Θ˜11=ϒ11T000ϒ12T00T,ϒ11=AiTP1iAiTXiA^fiTA^fiT,ϒ12=ϱCiTB^fiTϱCiTB^fiT00,Θ˜12=0ϒ13Tϒ14Tϒ15Tϒ16Tϒ17TT,ϒ13=BiTP1iBiTXi,ϒ14=EiTP1iEiTXi,ϒ15=−ϱB^fiT−ϱB^fiT,ϒ16=B^fiTB^fiT,ϒ17=DiTP1iDiTXi,Θ˜21=0000ϒ21T00T,ϒ21=cCiTB^fiTcCiTB^fiT00,Θ˜22=000ϒ22T00T,ϒ22=−cB^fiT−cB^fiT,(51)Θ˜31=ϒ31T000ϒ32T00T,ϒ31=η¯AiTP1i−P1iη¯AiTXi−Xiη¯A^fiT−Xiη¯A^fiT−Xi,ϒ32=η¯ϱCiTB^fiTη¯ϱCiTB^fiT00,Θ˜32=0ϒ33Tϒ34Tϒ35Tϒ36Tϒ37TT,ϒ33=η¯BiTP1iη¯BiTXi,ϒ34=η¯EiTP1iη¯EiTXi,ϒ35=−η¯ϱB^fiT−η¯ϱB^fiT,ϒ36=η¯B^fiTη¯B^fiT,ϒ37=η¯DiTP1iη¯DiTXi,Θ˜41=0000ϒ41T00T,ϒ41=η¯cCiTB^fiTη¯cCiTB^fiT00,Θ˜42=000ϒ42T00T,ϒ42=−η¯cB^fiT−η¯cB^fiT,Θ˜51=ϒ51T000ϒ52T00T,ϒ51=ηmAiTP1i−P1iηmAiTXi−XiηmA^fiT−XiηmA^fiT−Xi,(52)ϒ52=ηmϱCiTB^fiTηmϱCiTB^fiT00,Θ˜52=0ϒ53Tϒ54Tϒ55Tϒ56Tϒ57TT,ϒ53=ηmBiTP1iηmBiTXi,ϒ54=ηmEiTP1iηmEiTXi,ϒ55=−ηmϱB^fiT−ηmϱB^fiT,ϒ56=ηmB^fiTηmB^fiT,ϒ57=ηmDiTP1iηmDiTXi,Θ˜61=0000ϒ61T00T,ϒ61=ηmcCiTB^fiTηmcCiTB^fiT00,Θ˜62=000ϒ62T00T,ϒ62=−ηmcB^fiT−ηmcB^fiT,Θ˜71=ϒ71T000ϒ72T00T,ϒ71=ζMAiTP1i−P1iζMAiTXi−XiζMA^fiT−XiζMA^fiT−Xi,ϒ72=ζMϱCiTB^fiTζMϱCiTB^fiT00,Θ˜72=0ϒ73Tϒ74Tϒ75Tϒ76Tϒ77TT,ϒ73=ζMBiTP1iζMBiTXi,ϒ74=ζMEiTP1iζMEiTXi,ϒ75=−ζMϱB^fiT−ζMϱB^fiT,ϒ76=ζMB^fiTζMB^fiT,ϒ77=ζMDiTP1iζMDiTXi,(53)Θ˜81=0000ϒ81T00T,Υ81=ζMcCiTB^fiTζMcCiTB^fiT00,Θ˜82=000ϒ82T00T,ϒ82=−ζMcB^fiT−ζMcB^fiT,Θ˜91=ϒ91T000ϒ92T00T,ϒ91=dMAiTP1i−P1idMAiTXi−XidMA^fiT−XidMA^fiT−Xi,ϒ92=dMϱCiTB^fiTdMϱCiTB^fiT00,Θ˜92=0ϒ93Tϒ94Tϒ95Tϒ96Tϒ97TT,ϒ93=dMBiTP1idMBiTXi,ϒ94=dMEiTP1idMEiTXi,ϒ95=−dMϱB^fiT−dMϱB^fiT,ϒ96=dMB^fiTdMB^fiT,ϒ97=dMDiTP1idMDiTXi,Θ˜101=0000ϒ101T00T,ϒ101=dMcCiTB^fiTdMcCiTB^fiT00,Θ˜102=000ϒ102T00T,ϒ102=−dMcB^fiT−dMcB^fiT,Θ˜111=ϒ111T000000Tϒ111=LiT−CfiT.The parameter matrices of the filter are given by(54)Afi=A^fiXi−1,Bfi=B^fi,Cfi=C^fiXi−1.Proof.
Firstly, pre- and postmultiplying (40) by K=diagI,…,I︷15,K1,K2,K3,K4,I and KT, in which K1=diagPiR1−1,PiR1−1, K2=PiR2−1,PiR2−1, K3=PiR3−1,PiR3−1, and K4=PiR4−1,PiR4−1,, one can have(55)Σ^=KTΣ¯K=Ω¯1Ω^2∗Ω^3<0,with(56)Ω^2=Φ¯1TPiΦ¯3TPiη¯Φ¯2TPiη¯Φ¯3TPiηmΦ¯2TPiηmΦ¯3TPiζMΦ¯2TPiζMΦ¯3TPidMΦ¯2TPidMΦ¯3TPiΓT,Ω^3=diag−Pi,−Pi,−PiR1−1Pi,−PiR1−1Pi,−PiR2−1Pi,−PiR2−1Pi,−PiR3−1Pi,−PiR3−1Pi,−PiR4−1Pi,−PiR4−1Pi,−I.
Partition the matrixPi as Pi=P1iP2iP2iTP3i and define Xi=P2iP3i−1P2iT, Ji=I00P2iTP3i−1.
SincePi>0, by Schur Complement, it follows that P1i>0 and P1i−Xi>0.
Define(57)J=diagJi,…,Ji︷8,I,…,I︷5,Ji,…,Ji︷10,I.
MultiplyJ and JT on both sides of (55), respectively, (47) is obtained by defining(58)P˜i=JiPiJiT=P1iXiXiXi,Q˜s=JiQsJi,R˜s=JiRsJis=1,2,3,4,A^fi=P2iAfiP3i−1P2iT,B^fi=P2iTBfi,C^fi=CfiP3i−1P2iT.
On the basis of the linear transformation above and according to Lemma2, −PiRs−1Pi can be replaced as −2P˜i+R˜ss=1,2,3,4 for simplicity. It can be seen that (40) is equal to (47). This proof is complete.


## 4. Numerical Simulation

In this part, three examples are presented to verify the effectiveness of the proposed method.Example 1.
Consider the discrete-time system (1) with the relevant parameters as follows:(59)Π˜=0.50.20.30.6.
Mode 1(60)A1=0.700−0.6,B1=0.2−0.3−0.10.2,E1=0.10.150.20.1,C1=0.50.3,D1=0.2000.1,L1=0.150.3.
Mode 2(61)A2=−0.5000.8,B2=0.2−0.10.20.2,E2=0.1−0.30.30.2,C2=0.3−0.2,D2=0.2000.5,L2=−0.10.2.
Referring to [27], the activation functions of neural network are set as follows:(62)fx=0.4x1−tanh0.1x1+0.2x20.8x2−tanh0.6x2,gx=0.5x1−tanh0.2x1+0.1x20.7x2−tanh0.1x1+0.2x1.
U1, U2 and V1, V2 conforming to Assumption 1 are acquired easily as follows:(63)U1=0.30.200.2,U2=0.40.200.8,V1=0.30.10.10.7,V2=0.50.10.20.7.
Assuming the sampling periodh=0.1s, the event-triggered parameters σ1=0.05 and σ2=0.1. Set the maximum delay in communication and deception attack ζM=1 and dM=1, respectively, the upper and lower bounds of neural network delay ηM=4, ηm=1, and H∞ performance level is γ=5 referring to Table 1.
We set some uncertain parameters to simulate with the initial states are assumed asx0=0,0.8T, x^0=−0.1,0.2T, the delay of neural network ηk=4sin2πk, and the communication delay and deception attacks delay are considered as ζk=sin4πk and dk=sin2πk. The exogenous disturbance is(64)ωk=0.4e−0.2kcos0.6k0.4e−0.15kcos0.4k.
It can be proved that the occurring probability of the missing measurements is bound to influence theH∞ performance of the system. Table 1 lists the obtained minimal allowable H∞ performance level γ when changing the Bernoulli distribution probability ϱ. One can see that the smaller ϱ, the bigger is the H∞ performance index γ.
The function of deception attacksak=tanh0.5x1k+0.2x2k. According to Assumption 1, it satisfies ak2≤Wxk2, where W=0.5001.
Next, the influence of different missing measurements probabilities on the designedH∞ filter is discussed.Case 1. Set the Bernoulli distribution parameter ϱ=1, which implies that the measured output is conveyed to the filter smoothly. The matrices of filter parameters and event-trigger parameters Ψi are acquired as follows:(65)Af1=0.6799−0.2231−0.10390.1291,Bf1=−0.13560.5154,Cf1=−0.3346−0.5468,Ψ1=9.2731,Af2=−0.15720.14100.7812−1.0439,Bf2=0.34820.7943,Cf2=0.1640−0.4777,Ψ2=15.5416.Figure1 presents the responses of output to be estimated zk and the output of the filter zfk. Corresponding filtering error z¯k is shown in Figure 2, from which we can get the conclusion that the filter performs well even when the system suffers from the exogenous deception attacks. The releasing instants and its interval of ETM are displayed in Figure 3. In the simulation time, 49.01% of all the sampled signals are triggered in this period.Case 2. Set the Bernoulli distribution parameter ϱ=0.6; it is easily obtained the following matrices.(66)Af1=0.3799−0.91410.40390.3291,Bf1=−0.42080.3154,Cf1=−0.3255−0.4962,Ψ1=7.5949,Af2=−0.15720.54631.05270.6634,Bf2=0.23250.6743,Cf2=−0.2532−0.6764,Ψ2=13.0506.
As shown in Figure4, the jump of Bernoulli distribution reflects the state of randomly occurring packet dropout. Figure 5 depicts the responses of output to be estimated zk and the actual output of the filter zfk, while the filtering error z¯k is shown in Figure 6. This proposed method can still work well when the system suffers from deception attacks and partial measurements missing according to the simulations. Figure 7 displays the data releasing instants and intervals, and 42.15% of all the sampled signals are triggered in the simulated period. It reveals the fact that the method proposed in this paper reduces the frequency of data transmission and degrades the waste of network communication resources effectively.Table 1
Theγmin for different ϱ (ηm=1, ηM=4,ζM=1, dM=1,σ1=0.05,σ2=0.1).
ϱ10.90.80.70.5γmin1.48821.61391.89312.17704.0975Figure 1
zk and zfk in Case 1.Figure 2
The filtering error in Case 1.Figure 3
The release instants and their interval in Case 1.Figure 4
The jump of Bernoulli distribution in Case 2.Figure 5
zk and zfk in Case 2.Figure 6
The filtering error in Case 2.Figure 7
The release instants and their interval in Case 2.Example 2.
Consider the system (1) with following parameters given in [11]:
Mode 1(67)A1=5.1004.7,B1=1.1−0.70.91.2,E1=1.20.60.81,C1=0.90.8,D1=0.10.2,L1=0.20.
Mode 2(68)A2=1.1001.2,B2=−0.80.90.90.8,E2=0.60.60.650.6,C2=0.90.8,D2=0.10.2,L2=0.20.
Setϱ=1 and W=0000, which means that the missing measurements and deception attacks are not considered temporarily.
The neuron activation functions and external disturbances are given:(69)fx=gx=0.5x1−tanh0.5x1+0.2x20.95x2−tanh0.75x2,ωk=0.5,k∈5,10,−0.5,k∈15,20,0,else.
For given parametersγ=5, ζM=0.1, ηM=0.2, ηm=0.1, and σ1=σ2=0.1, we can obtain the corresponding matrices of the filter as follows:(70)Af1=0.2948−0.0564−0.03240.3008,Bf1=−0.1508−0.0854,Cf1=−0.09640.1449,Ψ1=4.6552,Af2=0.1475−0.03030.00140.1880,Bf2=−0.2204−0.1870,Cf2=−0.0971−0.0590,Ψ2=4.1070.
Figures8 and 9 depict the responses of zk and zfk under the initial conditions x0=0.2,−0.1T and x^0=0.5,0T. The data releasing instants and their intervals are displayed in Figure 10, and the corresponding triggered times and transmission rates obtained by the method proposed in literature [11] and this paper are listed in Table 2. Obviously, the rate of data transmission in this paper is lower than that in [11], so more communication resources are saved. Besides, the minimum allowable H∞ performance index γmin=0.8482 is smaller than γmin=1.2247 in [11], which indicates that, by comparison with [11], the method in this paper guarantees the system stability with better H∞ performance.Figure 8
The response ofzk and zfk in Example 2.Figure 9
The filtering error in Example2.Figure 10
The release instants and their interval in Example2.Table 2
The transmission rates andγmin.
This paperLiterature [11]Triggered times (v)3647Transmission rate (v/N)35.3%47%γmin0.84921.2247Example 3.
In this example, the numerical verification will be strengthened by clarifying the physical meaning of the system under the test. Referring to [44, 45], a synthetic genetic regulatory network is provided to demonstrate the application of the proposed approach. The biological network is described as(71)mk+1=A1rkmk+B1rkfpk+E1rkgpk−ηk+D1rkωk,pk+1=A2rkpk+A3rkmk+D2rkωk,where mk and pk denote the concentrations of mRNA and protein, fk represents the feedback regulation of the protein, A1rk,A2rk,andA3rk stand for the decay rates of mRNA, decay rates of protein, and the translation rate, respectively. B1rkandE1rk are the coupling matrices of the genetic network. Define(72)xk+1=mk+1pk+1,fxk=fmkfpk.
According to the form of (1), the following matrices can be obtained:(73)Ai=A1i0A2iA3i,Bi=0B2i00,Ei=0E2i00,Di=D1iD2i.
The parameters cited from [45] are listed in Table 3.(74)X=001100010,D1=−0.700.500.6000.20.3000.1T,C1=10−1001,L1=110010,D2=0.30−0.50.50000.300.50.20T,C2=0.30−0.50.500,L2=100100.
As discussed in [45], the regulation function is set as fx=x2/1+x2, ηM=5, ηm=1, σ1=σ2=0.01, and γ=1, and the transition probability matrix is given as(75)Π˜=0.80.20.30.7.
Setϱ=1, and the constraint matrix of deception attacks is W=diag1,0.5,0.6,0.5,1,0.5. The desired H∞ filter parameters are obtained:(76)Af1=0.0745−0.0404−0.0503−0.0080−0.0213−0.01020.00790.0207−0.0171−0.0023−0.0046−0.0011−0.03030.01950.05160.00020.00180.0015−0.04690.04780.03270.05100.01970.01090.1070−0.0860−0.0917−0.01580.0013−0.02300.0346−0.03600.0185−0.0163−0.01720.0318,Af2=0.05610.04740.00260.01220.02270.00400.00930.0513−0.0073−0.0025−0.00240.00070.0048−0.03020.0412−0.0090−0.0238−0.0045−0.0317−0.07400.01550.0037−0.0493−0.01910.01500.0383−0.00650.02910.07930.0097−0.0359−0.08770.0193−0.0170−0.03310.0411,(77)Bf1=0.5516−0.6524−0.4486−0.62600.72890.6551,Bf2=−0.5119−0.41670.46770.6712−0.6903−0.5975,Cf1=−0.8350−0.84610.8040−0.8107−0.8275−0.7850,Cf2=−0.93121.00650.8115−0.9155−1.0017−1.0012,Ψ1=96.5800,Ψ2=395.2375.
In the simulation, the initial conditions are assumed asx0=−0.2,0.1,−0.3,0.2,−0.1,0.2T and x^0=−0.1,−0.2,0.1,−0.1,0,−0.1T, and the disturbance is ωk=0.4e−0.2kcos0.6k,0.4e−0.15kcos0.4kT. Based on these parameters, the responses of the H∞ filter and the filtering error are shown in Figures 11 and 12 , respectively.Table 3
Parameters with different modes.
ParametersA1iA2iA3iB2iE2ii = 10.2I0.09I0.1I−0.5X−0.2Xi = 20.1I0.08I0.09I−0.8X−0.1XFigure 11
The response ofzk and zfk in Example 3.Figure 12
The filtering error in Example3.Remark 6.
By reviewing the existing research achievements on the filter design of MJNN, the dissipative filtering of MJNN under deception attacks and incomplete measurements is discussed in paper [46]. The incompleteness of actual measurements considered in [46] is dominated by randomly occurring deception attacks. In contrast, this paper considers a more complex and practical network environment including the stochastic packet dropouts due to the limited communication bandwidth and the malicious attack signals injected by attackers through the communication network. Furthermore, this paper discusses in detail the influence of changing the probability of packet dropouts on system performance and further demonstrates the effectiveness of the proposed method.


## 5. Conclusion

The issue of event-triggeredH∞ filtering for the discrete-time MJNNs under deception attacks and randomly occurring missing measurements has been investigated in this work. In order to be closer to the real network communication environment, this paper introduces deception attacks and randomly occurring missing measurements. The event-triggered communication strategy is proposed to deal with the problems caused by frequent information exchange. According to the judgement algorithm, the effective sampling data is transmitted, and the purpose of maintaining system performance and saving network resources is achieved. Based on the mathematical model, sufficient conditions are given to guarantee that the filtering augmented system is asymptotically stable. Finally, three numerical simulations are provided to clarify the influence of changing the probability of missing measurements on the system performance, the advantage of event-triggered strategy, and the physical meaning of system under test, respectively. In the next research, we will focus on the detection of cyber-attacks based on the existing research.



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