H filtering for networked systems with hybrid-triggered communication mechanism and stochastic cyber attacks

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Abstract

This paper concentrates on investigating H filtering for networked systems with hybrid-triggered communication mechanism and stochastic cyber attacks. Random variables satisfying Bernoulli distribution are introduced to describe the hybrid-triggered scheme and stochastic cyber attacks, respectively. Firstly, a mathematical H filtering error model with hybrid-triggered communication mechanism is constructed under the stochastic cyber attacks. Secondly, by using Lyapunov stability theory and linear matrix inequality (LMI) techniques, the sufficient conditions which can guarantee the stability of augmented filtering error system are obtained and the parameters of the designed filter can be presented in an explicit form. Finally, numerical examples are given to demonstrate the feasibility of the designed filter.

Introduction

Networked control systems (NCSs) are a kind of control systems wherein feedback signals and control signals are exchanged through a network in a form of information package. It’s characterized by enabling the execution of several tasks far away when it connects cyber space to physical space [1]. Due to the advantages such as high flexibility, low cost and simple installation [2], [3], NCSs are applied in different kinds of fields referring to aircrafts, automobiles, vehicles and so on [4], [5], [6]. Therefore, NCSs are attracting more and more interest owing to the development and the advantages of the Internet [7]. In [8], the robust H controller is designed for networked control systems with uncertainties such like network-induced delay and data dropouts. The synchronization problem of feedback control is investigated in [9] with time-varying delay for complex dynamic networks. The authors in [10] study the robust fault tolerant control problem for distributed network control systems.

In the past few years, time-triggered method (periodic sampling) is widely adopted for system modeling and analysis in the NCSs [11], however, periodic sampling will generate lots of redundant signals if all the sampled data is transmitted through the network. To make full use of the limited network resource, lots of researchers propose the event-triggered schemes to overcome the problem caused by periodic sampling, for example, a novel event-triggered scheme is proposed in [12], in which a H controller is designed for NCSs. The core idea of the novel event-triggered scheme in [12] is that whether the newest sampled data is released or not is dependent on a threshold, and the adoption of event-triggered scheme can largely help alleviate the burden of the network [13]. Consequently, there are large numbers of researchers interested in the investigations about the novel event-triggered scheme proposed in [12]. An event-triggered non-parallel distribution compensation control problem in [14] is addressed for networked Takagi–Sugeno (T–S) fuzzy systems. The authors of [15] consider the event-triggered filtering problem for discrete-time linear system with package dropouts satisfying Bernoulli distribution. In [16], a discrete event-triggered scheme is proposed for fuzzy filter design in a class of nonlinear NCSs. Inspired by the aforementioned event-triggered scheme in [12], the hybrid-triggered scheme which consists of time-triggered scheme and event-triggered scheme is firstly proposed in [17], which investigates the problem of control stabilization for networked control systems under the hybrid-triggered scheme. Based on the hybrid-triggered scheme above, the authors in [18] are concerned with the hybrid-driven-based reliable control design for a class of T–S fuzzy systems with probabilistic actuator faults and nonlinear perturbations. Motivated by the proposed hybrid-triggered scheme in [17], this paper is devoted to the hybrid-triggered H filtering subject to stochastic cyber attacks on the measurement outputs.

Due to the insertion of the network in the control systems, challenges including packet dropouts, network-induced delay and randomly occurring nonlinearities [19], [20], [21], [22] are inevitable. It is nonnegligible that another phenomena named cyber attacks can be more destroyable. Cyber attacks are offensive maneuvers which target networked information systems, infrastructures and networked devices by various means of malicious acts. By hacking into a susceptible system, cyber attacks can be the biggest threat to the security of network. As the description in [23], there are three kinds of common attacks containing denial of service attacks [24], [25], relay attacks [26], [27] and deception attacks [28], [29]. With the rapid development of the network, the influence of cyber attacks can not be neglected any more. Based on the cyber attacks mentioned above, lots of researches are investigated and impressive results are yielded. The authors are concerned with extended Kalman filter design for stochastic nonlinear systems under cyber attacks in [30]. The distributed recursive filtering problem is studied in [31] with quantization and deception attacks for a class of discrete time-delayed systems. In [32], a novel state filtering approach and sensor scheduling co-design with random deception attacks are presented.

This paper addresses the issue about H filtering for networked systems under stochastic cyber attacks with hybrid-triggered communication mechanism. The main contributions of this paper are as follows. (1) In order to make full use of networked bandwidth and guarantee the desired system performance, the hybrid-triggered scheme which consists of time-triggered scheme and event-triggered scheme is introduced. (2) Due to the insertion of network, the stochastic cyber attacks are considered, and the launching probability of cyber attacks is governed by Bernoulli random variable. (3) By taking the hybrid-triggered communication mechanism and stochastic cyber attacks into consideration, an H filter is designed for networked systems. Although there are several researches concerned with filter design, to the best of our knowledge, there is no research investigating the H filter design for networked systems by considering both hybrid-triggered scheme and cyber attacks.

The rest of this paper is organized as follows. In the Section 2, a filtering error system is constructed by introducing the hybrid-triggered communication mechanism and taking the stochastic cyber attacks into account. Section 3 gives the sufficient conditions which can guarantee the augmented filtering system stable by using Lyapunov functional approach and LMI techniques. Moreover, the design algorithm of H filter is presented and the filtering parameters are obtained in an explicit form. Section 4 gives illustrative examples to demonstrate the usefulness of desired H filter.

Notation: Rn and Rn×m denote the n-dimensional Euclidean space, and the set of n × m real matrices; the su-perscript T stands for matrix transposition; I is the identity matrix of appropriate dimension; the notation X > 0 (respectively, X ≥ 0), for X ∈ Rn × n means that the matrix X is real symmetric positive definite (respectively, positive semi-definite). For a matrix B and two symmetric matrices A, C and [A*BC] denotes a symmetric matrix, where * denotes the entries implied by symmetry.

Section snippets

Problem description and preliminaries

In this paper, the problem of H filtering for networked system with hybrid-triggered communication mechanism and stochastic cyber attacks is investigated. The framework of hybrid-triggered H filtering for networked systems under stochastic cyber attacks is shown as Fig. 1. From Fig. 1, one can see that the framework consists of the sensor, the hybrid-triggered scheme, the filter, a zero-order-hold (ZOH), a network channel.

Consider the following continuous-time linear system. {x˙(t)=Ax(t)+Bw(t)

Main results

In this section, by using Lyapunov functional approach, the main results will be summarized and the sufficient conditions which can guarantee the stability of networked system will be obtained.

Theorem 1

For given positive parameters θ¯, α¯, τM, ηM, dM, σ, ρi (i=1,2) and matrix G, with hybrid-triggered communication mechanism and cyber attacks, system (11) is asymptotically stable with an H disturbance attenuation level γ, if there exist matrices P > 0, Qk > 0, Rk > 0 (k=1,2,3), Ω > 0 and M, N, U, S, W,

Simulation examples

In this section, three examples are given to demonstrate the effectiveness of the designed filter.

Example 1

Consider the system (1) with the following matrix parameters. A=[2.1002],B=[10.2],C=[0.80.6],L=[0.20.3],ω(t)={1,5t101,15t200,else

The function of cyber attacks f(x(t)) which is shown in Fig. 2 is supposed as f(x(t))=[tanh(0.1x2(t))tanh(0.01x1(t))]. According to Assumption 1, the upper bound G=diag{0.01,0.1} can be derived, it can be verified easily that the upper bound confirm to the

Conclusion

This paper is devoted to the investigation of the hybrid-triggered H filtering problem for networked systems under stochastic cyber attacks. In order to alleviate the burden of the network, a hybrid-triggered scheme is introduced, in which the switching rule between the time-triggered scheme and the event-triggered scheme is described by Bernoulli variable. By taking the hybrid-triggered scheme and the effects of stochastic cyber attacks into consideration, a mathematical H filtering model

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (Nos. 61403185, 61473156), the Natural Science Foundation of Jiangsu Province of China (No. BK20171481), Six Talent Peaks Project in Jiangsu Province (No. 2015-DZXX-21), major project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJA120001), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and

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