Elsevier

Neurocomputing

Volume 270, 27 December 2017, Pages 145-151
Neurocomputing

Event-based distributed state estimation under deception attack

https://doi.org/10.1016/j.neucom.2016.12.109Get rights and content

Abstract

Recently, security issues in wireless sensor networks (WSNs) have attracted lots of attention. It is necessary to design effective measures against the attacker. In this paper, a novel distributed state estimator with an event-triggered scheme is proposed to defend against false data injection attack in WSNs. Here, each sensor detects the rightness of its estimate according to a condition at each time step. When the condition is satisfied, the sensor regards it is not attacked, then it is triggered to send the estimate to its neighboring sensors; Otherwise, the sensor is not triggered. Based on the event-triggered scheme, an optimal estimator gain is designed by minimizing the mean-squared estimation error covariance, and then a sufficient condition is provided to guarantee the stability of the proposed distributed estimator. Finally, a numerical example is provided to illustrate the effectiveness of the proposed estimator with event-based strategy.

Introduction

Recently, wireless sensor networks(WSNs), motivated by the development of wireless communication, microelectronics, sensing and embedded systems technologies, are widely used in the military, industrial and consumer applications such as battlefield surveillance, industrial process and so on [1], [2], [3]. Over the last decade, many works on consensus-based distributed state estimation emerge due primarily to its high estimation accuracy, low utilization of communication resources and high robustness when compared to centralized and decentralized state estimations [4], [5], [6], [7], [8].

In practical applications, the structure of distributed estimation is a double-edged sword. It brings better estimation performances while causes the network low robustness, specifically under some attacks such as false data injection or replay. Once one sensor is attacked successfully, then the tampered estimates will be broadcasted to the whole network like virus in an epidemic model. Thus, security issues have been a natural concern in WSNs, due to its vulnerable distributed network structure. In recent years, some researchers have studied different kinds of attacking strategies. For example, Zhang et al. [1], [9] design an optimal DoS attacking schedule which degrades the performance of the considered system. Shi et al. [10] study sensors with limited energy who need to design its data scheduling over a packet-dropping. Mo et al. [11] study false data injection attack against state estimation in WSNs, where the system is equipped with a failure detector. When the behaviors of attackers are exposed to the network, it is easy to design effective measures to detect the hostile attack. As most of the existing works focus on centralized estimation, the study of defending against attack on consensus-based distributed estimation becomes an urgently practical problem. It is known that each sensor estimates the system state using the data from its neighboring sensors in the distributed state estimation. With such a scheme, distributed filtering issues have attracted much attention [12], [13], [14], [15], [16]. In [16], an event-based recursive distributed filtering problem is investigated with a predetermined Send-on-Delta data transmission condition. The proposed mechanism can decrease the unnecessary executions of the systems and extract the information about the state vectors of the system against external disturbances.

Recently, the results on event-based mechanism have been increasing rapidly. In the earlier work [17], an send-on-delta data collecting strategy to capture information from the surroundings has been addressed. The proposed send-on-delta concept means a signal-dependent temporal sampling scheme, where the executions are triggered if an event generator condition is satisfied. The existing works cover many applications of the event-triggered schedule to kinds of practical system such as: (1) Networked control system[18], [19], [20]. In [18], the H filtering problem for a class of event-triggered time-delay stochastic systems is investigated to reduce the communication load in the network. In [19], the distributed H state estimation problem is studied and an event-triggered communication protocol is designed by utilising the available innovations from both itself and its neighbors. To eliminate the shortcomings of the deterministic event-triggering mechanism, Han et al. [20] investigate an open-loop and a closed-loop stochastic event-triggered sensor schedule for remote state estimation. (2) Multi-agent systems [21], [22], [23]. Seybotha et al. [21] propose a novel event-based control strategy for distributed multi-agent system, where each agent only needs to extract information from local agents. Nowzari et al. [22] propose an event-triggered broadcasting and controller update strategy, which does not require the sensor to obtain the priori knowledge of the global. In [23], the proposed nonlinear distributed event-triggered control protocol can achieve finite-time consensus and it only needs the information between the agent and its local neighbors. More related works have been proposed, such as in [24], an event-triggered transmission scheme appears to reduce communication load for complex networks with limited communication bandwidth. Selivanov et al. [25] consider both quantization of transmitted measurements and network-induced delays to design distributed event triggered controller of parabolic systems under point or spatially averaged discrete time measurements. In [27], a general event-trigger framework in order to reduce data communication frequency and network bandwidth usages is adopted to deal with the finite-horizon H filtering problem. Dong et al. [28] build a comprehensive model which covers event-triggered measurement transmissions, ROUs and state-dependent noises of a class of certain stochastic systems to study robust distributed state estimation problem. To the best of our knowledge, there are few studies directed to event-based schedule on distributed state estimation for defending against false data injection attack. One reason lies in the difficulty of gathering the information of sensor network topology and the attacker, the other one lies in the complicated analysis of distributed state estimator with event-triggered scheme.

Motivated by the above problem, we aim to design an event-triggered mechanism to defend against false data injection (deception) attack on distributed network. Due to the distributed network structure, lots of potential attacks exist in the applications in WSNs, such as the false data injection attacks in electricity market which brings the potential economic losses[26]. Thus, it is necessary to design effective defense strategy to guarantee the security of the networks. We focus on designing an effective event-based strategy for distributed estimator to detect the suspected data transmission on the communication link under false data injection attack. The event-based strategy consider the deviation between the innovation of each sensor and a positive scalar. An event indicator, representing whether the triggering condition is success or not, is added into the estimator. With the proposed estimator, each sensor executes the information transmission to its neighboring sensors only when its innovation exceeds a fixed threshold defined as an important discriminant value. Our main contributions can be highlighted as follows:

  • (1)

    We define an event generator function and the discriminant value for each sensor i;

  • (2)

    Under the false data injection attack, we propose a distributed state estimator equipped with an event trigger strategy for the network system;

  • (3)

    We design an optimal estimator gain by minimizing the mean-squared estimate error under a given attacking strategy and known event-triggered instants;

  • (4)

    We derive an upper bound of the expected state estimation error covariance to simplify the difficulties in analysis of the system stability and provide a sufficient condition to guarantee the system stability.

The remainder of the paper is organized as follows: We build the network topology and the attacking schedule problems, introduce an event-triggered mechanism in Section 2. In Section 3, we design an optimal distributed state estimator gain given event-triggered instants under known attack. An upper bound of the estimator error covariance is obtained to analyze the stability of the expected estimation error covariance. In Section 4, numerical examples are provided to illustrate the performance of the proposed estimator. Finally Section 5 concludes the paper.

Notation. XT is the transposition of X. vec(X) means the vector formed by stacking the columns of X in the natural order. diag(Xi) represents a block diagonal matrix with main diagonal block equal to Xi. E[X] is the mean of random variable X. For matrices X and Y, XY is their Kronecker product. We write X ≥ 0 if X is positive semi-definite, and XY if XY0. Moreover, X > 0 if X is positive-definite, and X > Y if XY>0. We use 1 to denote a vector of arbitrary dimension with each component equal to one. I denotes the identity matrix.

Section snippets

Problem formulation

Consider the following discrete linear system: x(k+1)=Ax(k)+w(k),where x(k)Rm is the system state vector, w(k)Rm is the process noise. Assume that w(k) and x(0) are irrelevant zero mean Gaussian random variables with covariance Q and Π00, respectively.

For the ith sensor node, the measurement equation is given by yi(k)=Hix(k)+vi(k),where yi(k)Rm is the measurement data of the ith sensor at time step k, vi(k)Rmi is the measurement noise. vi(k) is the zero mean white Gaussian noise with

Convergence analysis of the proposed estimator

In this section, we first give an optimal estimator gain Kpi(k) of the ith sensor by minimizing the state estimation error Pi(k) for a given sequence of γij(k) and ζj(k),jϵNi,i=1,2,,n.

To obtain the main results, we first introduce the following three assumptions.

Assumption 1

The graph G is strongly connected.

Assumption 2

{A, Hi} is observable.

Assumption 3

{A, Q1/2} is stabilizable.

For later analysis, we first define the estimation error as the following: ei(k)=x(k)x^i(k),and the mean-squared estimation error Pi(k)=E[(x(k)x^i(k)

Simulation results

To verify the main results obtained in Section 3, a WSN composed of n=5 sensors is considered. We set the system parameters as follows: A=[0.90.0050.350.9],Q=[2002],ϕi=[i00i],Hi=[hi00hi],Ri=[vi00vi],where hi,vi,iϵ(0,1] for all i.

We choose a directed network topology G=(V,E,D)(see Fig. 1), and the adjacency matrix obtained as follows: D=(0100010100010000100010000).We further obtain the Laplacian matrix L of the network whose second eigenvalue λ2(L)=1 : L=(1100012100011000101010001).The

Conclusion

In this paper, a novel event based distributed estimator has been proposed to defend against the false data injection attack. According to an event-trigger condition, each sensor judges whether its estimate is attacked or not, and then decides when to send the estimate to its neighboring sensors. By minimizing the estimation error covariance, an optimal estimator gain for each sensor i has been obtained. An upper bound of the expected error covariance has been derived for the stability

Wen Yang Assistant Professor at ECUST. She received the M.Sc. degree in control theory and control engineering from Central South University, Hunan, China, in 2005 and the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2009. She was a Visiting Student with the University of California, Los Angeles, from 2007 to 2008. She is currently with the School of Information Science and Engineering, East China University of Science and

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    Wen Yang Assistant Professor at ECUST. She received the M.Sc. degree in control theory and control engineering from Central South University, Hunan, China, in 2005 and the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2009. She was a Visiting Student with the University of California, Los Angeles, from 2007 to 2008. She is currently with the School of Information Science and Engineering, East China University of Science and Technology (ECUST), Shanghai. Her research interests include coordinated and cooperative control, consensus problems, multi-agent systems, and complex networks.

    Li Lei Master student at East China University of Science and Technology (ECUST). She majored in control theory and control engineering at ECUST. Her research interests include coordinated and cooperative control, consensus problems, multi-agent systems, and complex networks.

    Chao Yang received the B.S. degree in theoretical and applied mechanics from the Department of Mechanics and Engineering Science, Peking University, Beijing, China, in 2009 and the Ph.D. degree in electronic and computer engineering from the Hong Kong University of Science and Technology, Hong Kong, in 2013. Between September 2013 and August 2014, she was a Research Associate in the Department of Electronic and Computer Engineering, the Hong Kong University of Science and Technology. She is currently an assistant professor in the Department of Automation, East China University of Science and Technology, Shanghai, China. Her research interests include networked control systems, optimal filtering, sensor scheduling, security in cyber-physical systems, and social networks.

    This work was supported in part by the National Natural Science Foundation of China under Grant(61573143), the Innovation Program of Shanghai Municipal Education Commission under Grant no. 14zz55

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