Event-Triggered H ‘ Filtering for Markovian Jump Neural Networks under Random Missing Measurements and Deception Attacks

. This paper concentrates on the event-triggered H ∞ 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.


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][2][3][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.
ese 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][7][8][9][10][11][12]. Researchers in [6,7] both pay attention to the state estimation for delayed MJNNs concretely. e synchronization for MJNNs by employing a detector based on a hidden Markovian model (HMM) is studied in [9]. Researchers in [10][11][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][11][12][13][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][18][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. erefore, 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][25][26][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 cyberattacks. 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][32][33][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]. e 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. e 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 of H ∞ filtering for MJNNs under randomly occurring missing measurements and deception attacks is discussed. e contributions of this study are condensed into three major points: (1) an eventtriggered 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. e main framework of this paper is generalized: Section 2 presents the description of the system and the elaboration of all problems. e 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; R n denotes n-dimensional Euclidean space; R n×m denotes the set of real matrices with m rows and n columns; P > 0 (P ∈ R n×m ) means that P is a real symmetric positive definite matrix; L 2 [0, ∞) denotes the space of square-integrable vector functions over [0, ∞); I is a identity matrix; and * represents the symmetric term of a symmetric matrix.

System
Description. Consider the MJNNs described as follows: 2 Complexity . , x n ] T ∈ R n stands for the state vector, y(k) ∈ R m is the measured output, and z(k) ∈ R p denotes the output to be estimated. e nonlinear vector- . . , g n (x n (k − η(k)))] T ∈ R n are the neuron activation functions. ω(k) ∈ R q represents the ex- represents the transition probability matrix, which is given by where π ij ∈ [0, 1], (∀i, j ∈ M) and N j�1 π ij � 1, (∀i ∈ M). Construct the following H ∞ filter: where x f (k) ∈ R n denotes state vector of filter, z f (k) ∈ R p represents the output of the filter, y(k) ∈ R m represents the actual input of filter, and A f (r k ), B f (r k ), and C f (r k ) are appropriate matrices to be designed. For convenience, r k � i, (i ∈ M), A(r k ) is denoted by A i , B(r k ) by B i , and so on.

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. e following judgement algorithm is applied widely [11][12][13][14]27]: where Ψ i ∈ R m are the positive definite weighting matrices that will be determined later, σ i ∈ [0, 1) are presupposed scalars, y((k + j)h) represents the immediate sampled sensor output, and y(kh) is the latest transmitted data. It should be noted that the immediate sampled measurement output y((k + j)h) 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.

Remark 3.
e set of data release instants meets . . is decided by the value of the variation of the measured output. Setting σ i � 0 implies that all sampled signals are released and conveyed to the filter smoothly.
Refer to [40], suppose that ζ q is the time delay in the network communication at the moment k q with ζ q ∈ [0, ζ M ], where ζ M is a positive real number. Apparently, Since ζ q ≥ ζ M , obviously there exists positive integer m that (7) and y(κ q h), y(κ q h + jh) (j � 1, 2, . . . , m) meets the condition (4). Let Complexity Define the function, From the (9), we can get , and the error of measurement output e y (k) � 0 is defined.
For Case 2, define Combining the algorithm (4) and the definition (11), it can be concluded that,

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. en, the actual input y(k) of the filter is expressed as follows: where a(k) represents the time-discrete signal injected to the measured outputs by attackers.
stands for the delay of cyber attacks. Bernoulli variable ϱ(k) is shown as follows: Remark 4. According to equality (14), the measurement output can be conveyed to the filter smoothly with the probability ϱ in case of ϱ(k) � 1. while ϱ(k) � 0 implies that the filter fails to receive the transmitted measurement output.
Remark 5. e 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. e 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.

e Overall Model. Define
we can obtain the augmented filtering error system as follows: where e design problems of H ∞ filter can be summarized the conditions as follows: (i) e augmented system (15) with ω(k) � 0 is asymptotically stable for any initial conditions. (ii) Given a scalar c > 0 and ω(k) ∈ L 2 [0, ∞), the filtering error z(k) satisfies Next, some assumptions and lemmas which are conducive to deriving the theorem are introduced.

Complexity
Assumption 1 (see [34]). e deception attacks a(k) meets the following criterion: where W is a given constant matrix.

Asymptotical Stability Analysis.
First, the asymptotic stability of the filtering error system (15) with ω(k) � 0 is discussed.
We can get Complexity Letting So, it is obvious that where en, one can get the following equation: By using Assumption 2, for scalars α 1 > 0, α 2 > 0, we can get where Recalling the restrictive condition of deception attacks in (18), there exists an inequality as follows: According to Lemma 1 and combining conditions (12) and (26)- (38), the following inequality is obtained: where

Complexity
By employing Schur Complement lemma, the asymptotic stability of system (15) with ω(k) � 0 is guaranteed if condition (22) holds.

H ∞ Performance Analysis.
e H ∞ performance of the augmented system (15) , Q s > 0, and R s > 0 (s � 1, 2, 3, 4) with appropriate dimensions and scalars α 1 > 0 α 2 > 0, satisfying the following: where Proof 2. Based on the same Lyapunov functional as eorem 1, one can derive the following inequality: According to Schur Complement lemma, (40) and (41) are sufficient conditions to guarantee Π < 0, and the following inequality can be ensured: under the zero initial condition, one can get e proof is complete.

H ∞ Filter Design.
Based on the derivation of eorems 1 and 2, the H ∞ filter is designed.  1, 2, 3, 4), A fi , B fi , and C fi with appropriate dimension, and scalars α 1 > 0, where 8 Complexity Complexity Complexity 11 e parameter matrices of the filter are given by Proof. Firstly, pre-and postmultiplying (40) by with Partition the matrix P i as P i � P 1i P 2i P T 2i P 3i and define Since P i > 0, by Schur Complement, it follows that P 1i > 0 and P 1i − X i > 0. Define Multiply J and J T on both sides of (55), respectively, (47) is obtained by defining 1, 2, 3, 4), On the basis of the linear transformation above and according to Lemma 2, −P i R −1 s P i can be replaced as −2P i + R s (s � 1, 2, 3, 4) for simplicity. It can be seen that (40) is equal to (47). is proof is complete.

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: Mode 1 Mode 2 Referring to [27], the activation functions of neural network are set as follows: U 1 , U 2 and V 1 , V 2 conforming to Assumption 1 are acquired easily as follows: Assuming the sampling period h � 0.1s, the eventtriggered parameters σ 1 � 0.05 and σ 2 � 0.1. Set the maximum delay in communication and deception attack ζ M � 1 and d M � 1, respectively, the upper and lower bounds of neural network delay η M � 4, η m � 1, and H ∞ performance level is c � 5 referring to Table 1.
We set some uncertain parameters to simulate with the initial states are assumed as x(0) � [0, 0.8] T , x(0) � [−0.1, 0.2] T , the delay of neural network η(k) � 4|sin(2πk)|, and the communication delay and deception attacks delay are considered as ζ(k) � |sin(4πk)| and d(k) � |sin(2πk)|. e exogenous disturbance is It can be proved that the occurring probability of the missing measurements is bound to influence the H ∞ performance of the system. Table 1 lists the obtained minimal allowable H ∞ performance level c when changing the Bernoulli distribution probability ϱ. One can see that the smaller ϱ, the bigger is the H ∞ performance index c. e function of deception attacks a(k) � tanh (0.5x 1 (k) + 0.2x 2 (k)). According to Assumption 1, it sat- Next, the influence of different missing measurements probabilities on the designed H ∞ filter is discussed. Case 1. Set the Bernoulli distribution parameter ϱ � 1, which implies that the measured output is conveyed to the filter smoothly. e matrices of filter parameters and event-trigger parameters Ψ i are acquired as follows: (65) Figure 1 presents the responses of output to be estimated z(k) and the output of the filter z f (k). 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. e releasing instants and its interval of ETM are displayed in Figure 3. As shown in Figure 4, the jump of Bernoulli distribution reflects the state of randomly occurring packet dropout. Figure 5 depicts the responses of output to be estimated z(k) and the actual output of the filter z f (k), while the filtering error z(k) is shown in Figure 6. is 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.
Example 2. Consider the system (1) with following parameters given in [11]: Mode 1 Set ϱ � 1 and W � 0 0 0 0 , which means that the missing measurements and deception attacks are not considered temporarily. e neuron activation functions and external disturbances are given:     14 Complexity e 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 c min � 0.8482 is smaller than c min � 1.2247 in [11], which indicates that, by comparison with [11], the method in this paper guarantees the system stability with better H ∞ performance.
Example 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. e biological network is described as where m(k) and p(k) denote the concentrations of mRNA and protein, f(k) represents the feedback regulation of the protein, A 1 (r k ), A 2 (r k ), and A 3 (r k ) stand for the decay rates of mRNA, decay rates of protein, and the translation rate, respectively. B 1 (r k ) and E 1 (r k ) are the coupling matrices of the genetic network. Define According to the form of (1), the following matrices can be obtained: e parameters cited from [45] are listed in Table 3.
As discussed in [45], the regulation function is set as f(x) � x 2 /1 + x 2 , η M � 5, η m � 1, σ 1 � σ 2 � 0.01, and c � 1, and the transition probability matrix is given as Set ϱ � 1, and the constraint matrix of deception attacks is W � diag 1, 0.5, 0.6, 0.5, 1, 0.5 { }. e desired H ∞ filter parameters are obtained:      In the simulation, the initial conditions are assumed as  Figures 11 and 12 , respectively. 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]. e 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.

Conclusion
e issue of event-triggered H ∞ filtering for the discretetime 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. e eventtriggered 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.