Event-Triggered <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 26.6531 15.7437" width="26.6531pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.012,0,0,-0.012,13.299,4.134)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"/></g></svg> Filtering for Markovian Jump Neural Networks under Random Missing Measurements and Deception Attacks

Wang, Jinxia; Gao, Jinfeng; Tan, Tian; Wang, Jiaqi; Ma, Miao

doi:https://doi.org/10.1155/2020/4151542

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Abstract Introduction Numerical Simulation Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 4151542 | https://doi.org/10.1155/2020/4151542

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

Jinxia Wang,¹Jinfeng Gao,¹Tian Tan,¹Jiaqi Wang,¹and Miao Ma¹

Academic Editor: Sergey Dashkovskiy

Received14 May 2020

Revised03 Dec 2020

Accepted11 Dec 2020

Published29 Dec 2020

Abstract

This paper concentrates on the event-triggered 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 performance. In the end, three numerical examples are given to verify the proposed theoretical results.

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 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 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 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 of 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: Section 2 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; denotes -dimensional Euclidean space; denotes the set of real matrices with rows and columns; () means that is a real symmetric positive definite matrix; denotes the space of square-integrable vector functions over ; 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:where stands for the state vector, is the measured output, and denotes the output to be estimated. The nonlinear vector-valued functions and are the neuron activation functions. represents the exogenous disturbance with . denotes the time-varying bounded delay of neural network. and represent the upper and lower bounds, respectively. , , , , , and are known real constant matrices with appropriate dimension. The parameter is constrained by a homogeneous Markovian jump process and generally takes values from the finite set . represents the transition probability matrix, which is given bywhere , () and , .

Construct the following filter:where denotes state vector of filter, represents the output of the filter, represents the actual input of filter, and , , and are appropriate matrices to be designed.

For convenience, , is denoted by , by , 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]:where are the positive definite weighting matrices that will be determined later, are presupposed scalars, represents the immediate sampled sensor output, and is the latest transmitted data. It should be noted that the immediate sampled measurement output 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 . The release period of the event trigger is given as .

Remark 3. The set of data release instants meets . The amount of is decided by the value of the variation of the measured output. Setting implies that all sampled signals are released and conveyed to the filter smoothly.
Refer to [40], suppose that is the time delay in the network communication at the moment with , where is a positive real number. Case 1. If , introduce a function: Apparently, Case 2. If , consider the two intervals: .Since , obviously there exists positive integer thatand () meets the condition (4).
LetDefine the function,From the (9), we can getBecause , the third row of (10) holds. Then, we have .
For Case 1, , and the error of measurement output is defined.
For Case 2, defineCombining the algorithm (4) and the definition (11), it can be concluded that, for ,

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 input of the filter is expressed as follows:where represents the time-discrete signal injected to the measured outputs by attackers. stands for the delay of cyber attacks. Bernoulli variable is shown as follows:

Remark 4. According to equality (14), the measurement output can be conveyed to the filter smoothly with the probability in case of . while 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

Define and , we can obtain the augmented filtering error system as follows:where

The design problems of filter can be summarized the conditions as follows:(i)The augmented system (15) with is asymptotically stable for any initial conditions.(ii)Given a scalar and , the filtering error satisfies

Next, some assumptions and lemmas which are conducive to deriving the theorem are introduced.

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

Assumption 2 (see [41]). The neural functions and in (1) meet the initial value setting and the following sector bounded condition:, where , , , and are constant real matrices satisfying and .

Lemma 1 (see [42]). For any symmetric positive-definite matrix , scalars and , vector function , such that the following inequality holds:

Lemma 2 (see [43]). For any matrix and constant , the following inequality holds:

3. Main Results

3.1. Asymptotical Stability Analysis

First, the asymptotic stability of the filtering error system (15) with is discussed.

Theorem 1. For given delay bounds , , , , trigger parameters , and matrix , system (15) is asymptotically stable with an performance index , if there exist matrices , , , and with appropriate dimensions and scalars , satisfying the following equation:where

Proof. Construct the following Lyapunov functional as follows:whereunder the condition of , , and .
We can getLettingSo, it is obvious thatwhereThen, one can get the following equation:By using Assumption 2, for scalars , , we can getwhereRecalling 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:whereBy employing Schur Complement lemma, the asymptotic stability of system (15) with is guaranteed if condition (22) holds.

3.2. Performance Analysis

The performance of the augmented system (15) is analysed in the second theorem.

Theorem 2. For given delay bounds , , , , trigger parameters , and matrix , the system (15) is asymptotically stable with an performance index , if there exist matrices , , , and with appropriate dimensions and scalars , satisfying the following:where

Proof 2. Based on the same Lyapunov functional as Theorem 1, one can derive the following inequality:whereAccording to Schur Complement lemma, (40) and (41) are sufficient conditions to guarantee , and the following inequality can be ensured:under the zero initial condition, one can getThe proof is complete.

3.3. Filter Design

Based on the derivation of Theorems 1 and 2, the filter is designed.

Theorem 3. For given delay bounds , , , , trigger parameters , and matrix , the filtering error system (15) is asymptotically stable with an performance index , if there exist real matrices , , , , , , , and with appropriate dimension, and scalars satisfying , such thatwhere

The parameter matrices of the filter are given by

Proof. Firstly, pre- and postmultiplying (40) by and , in which , , , and , one can havewithPartition the matrix as and define , .
Since , by Schur Complement, it follows that and .
DefineMultiply and on both sides of (55), respectively, (47) is obtained by definingOn the basis of the linear transformation above and according to Lemma 2, can be replaced as 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:Mode 1Mode 2Referring to [27], the activation functions of neural network are set as follows:, and , conforming to Assumption 1 are acquired easily as follows:Assuming the sampling period , the event-triggered parameters and . Set the maximum delay in communication and deception attack and , respectively, the upper and lower bounds of neural network delay , , and performance level is referring to Table 1.
We set some uncertain parameters to simulate with the initial states are assumed as , , the delay of neural network , and the communication delay and deception attacks delay are considered as and . The exogenous disturbance isIt can be proved that the occurring probability of the missing measurements is bound to influence the performance of the system. Table 1 lists the obtained minimal allowable performance level when changing the Bernoulli distribution probability . One can see that the smaller , the bigger is the performance index .
The function of deception attacks . According to Assumption 1, it satisfies , where .
Next, the influence of different missing measurements probabilities on the designed filter is discussed. Case 1. Set the Bernoulli distribution parameter , which implies that the measured output is conveyed to the filter smoothly. The matrices of filter parameters and event-trigger parameters are acquired as follows: Figure 1 presents the responses of output to be estimated and the output of the filter . Corresponding filtering error 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, of all the sampled signals are triggered in this period. Case 2. Set the Bernoulli distribution parameter ; it is easily obtained the following matrices.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 and the actual output of the filter , while the filtering error 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 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 1Mode 2Set and , which means that the missing measurements and deception attacks are not considered temporarily.
The neuron activation functions and external disturbances are given:For given parameters , , , , and , we can obtain the corresponding matrices of the filter as follows:Figures 8 and 9 depict the responses of and under the initial conditions and . 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 performance index is smaller than in [11], which indicates that, by comparison with [11], the method in this paper guarantees the system stability with better 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. The biological network is described aswhere and denote the concentrations of mRNA and protein, represents the feedback regulation of the protein, stand for the decay rates of mRNA, decay rates of protein, and the translation rate, respectively. are the coupling matrices of the genetic network. DefineAccording to the form of (1), the following matrices can be obtained:The parameters cited from [45] are listed in Table 3.As discussed in [45], the regulation function is set as , , , , and , and the transition probability matrix is given asSet , and the constraint matrix of deception attacks is . The desired filter parameters are obtained:In the simulation, the initial conditions are assumed as and , and the disturbance is . Based on these parameters, the responses of the filter and the filtering error are shown in 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]. 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-triggered 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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 62073296 and 61374083 and Zhejiang Province Natural Science Foundation of China under Grants LY20F030015 and LQ19F030014.

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Copyright © 2020 Jinxia Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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