Research articleState estimation for cyber–physical systems with limited communication resources, sensor saturation and denial-of-service attacks
Introduction
Cyber–physical systems (CPSs) are regard as a class of the multi-dimensional complex systems which integrate the physical environment, network and computational capability. Recently, the modeling and analysis of CPSs have been paid widespread attentions because of the wide spectrum of applications in areas such as transportation networks, healthcare and power generation, which leads to numbers of publications. For example, the authors in [1] investigate the transportation-cyber–physical systems which are applied in the vehicle-to-infrastructure-based communications to avert traffic disasters and traffic jams. In [2], the nonlinear CPSs with multiple stochastic incomplete measurements are discussed where a T–S fuzzy model is used for approximating the physical process. T–S fuzzy model is firstly proposed in [3], which can be used to approximate nonlinear systems by a set of local linear systems smoothly connected by fuzzy membership functions. Since T–S fuzzy model is an effective method of modeling nonlinear systems, considerable attentions have been paid by scholars and lots of achievements have been acquired [4], [5], [6], [7], [8]. For instance, a class of T–S fuzzy systems is studied with the consideration of non-fragile distributed filtering problem [4]. With the assistance of the T–S fuzzy model, the authors propose the algorithm of filtering design for networked systems in [5]. The authors in [8] address the issue of continuous-time T–S fuzzy systems by using the time derivatives. Motivated by this work, T–S fuzzy model which can be utilized to approximate nonlinear systems by a series of local linear systems is borrowed to describe CPSs in this paper.
Due to the extensive attentions paid to CPSs, the problem of state estimation for CPSs has aroused considerable interests. Lots of scholars devote themselves into investigating the issue and numerous results are available. In [9], the state estimation of the wireless CPSs subjected to DoS attacks is investigated. The authors in [10] focus on the state estimation of CPSs with DoS attacks occurring in a wireless channel. A finite time event-triggered supervised estimator is designed to estimate linear CPSs under the event-driven scheme [11]. Apart from the above statement about estimating CPSs, the issue of sensor saturation has also fascinated plenty of academics for the reason that sensor saturation is likely to undermine the stability of CPSs to some degree. It need to be pointed out that the physical process is measured by a series of sensors where there always exists sensor saturation. In fact, many efforts have been devoted into researching sensor saturation for several years and massive achievements are obtained [12], [13]. By taking the sensor saturations into account, the distributed filter is designed in [12] to estimate sensor networks. With the consideration of randomly occurring sensor saturation, the authors in [13] investigate the problem of the estimator design of complex networks. It needs to be pointed out that the problem of the state estimation for CPSs has received wide attentions in the recent years, however, the research of estimating CPSs with sensor saturation has not been fully done yet, which initially motivates this paper.
In CPSs, owing to the limitation of bandwidth and communication resources, it may bring some problems which can degrade the system performance. How to handle the problems of limited communication resources has been widely discussed. During the past years, the periodic sampling (time-triggered scheme) is a frequently-used signal processing method, where the signals are sampled and transmitted at a fixed interval. However, the periodic sampling brings the issue of redundant data transmission, which results in the great network load. For the purpose of relieving the burden of communication network, various event-triggered schemes are proposed [14], [15], [16]. One of the famous event-triggered schemes is an event-triggered scheme relying on discrete supervision of the system states, which is firstly proposed in [16]. This event-triggered scheme only supervises the difference between the states sampled in discrete instants and the data can be transmitted only when they satisfy the given condition. Based on [16], numerous researches with event-triggered scheme are carried out during the last decade [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. For instance, the networked systems with multiple sensor distortions is investigated in [22] by utilizing the event-triggered scheme mentioned [16]. Inspired by [16], the authors propose an adaptive event-triggered communication mechanism where the assumption of the prior work is relaxed in [23]. On the basis of the event-triggered scheme presented in [16], the authors in [25] propose a new discrete event-triggered transmission protocol for networked control systems (NCSs) with delays and noise. Motivated by the work in [16], the resilient event-triggering communication scheme is proposed to improve the performance of NCSs with periodic DoS attacks in [27]. Based on the event-triggered scheme proposed in [16], the hybrid-driven scheme is proposed in [28], which combines the advantages of the time-triggered scheme and the event-triggered scheme. The authors in [29] address the problem of filter design for neural networks with the hybrid-driven scheme. By adopting the hybrid-driven scheme, the issue of control for networked cascade control systems is addressed in [30]. Besides, the hybrid-driven scheme proposed in [28] is utilized to study the T–S fuzzy systems [5], [31], [32]. In addition to the event-triggered mechanism as stated above, the quantization mechanism is also adopted to relieve the influence of limited communication resources [33], [34], [35], [36], [37], [38], [39], [40]. In [33], the authors use quantization mechanism to alleviate network transmission load of NCSs. A quantization mechanism is utilized to relieve the network bandwidth burden of Markovian jump systems in [36]. By using the quantization mechanism, the authors in [40] design the robust fuzzy-model-based filter for networked systems. Motivated by the above research, the event-triggered scheme and quantization mechanism are introduced in this paper to alleviate the effects of limited communication resources.
It is well known that the introduction of the network has brought many advantages to CPSs, however, it also may lead to the common communication constraints such as packet dropouts, limited bandwidth and time delay [41]. In addition, the network may be susceptible to cyber attacks due to the openness of communication networks [42]. In general, there are three kinds of cyber attacks including injection attacks, deception attacks and DoS attacks [29]. The injection attacks aim to replace the transmitted signal with the false data when the data is transmitted via the network. The deception attacks replace normal data by pretending to be trusted parties to degrade system performance. Different from the above attacks, the purpose of the DoS attacks is to make efforts to block data transmission. Recently, the cyber attacks have been paid considerable attentions by researchers and a great many results have been available in the literature. For instance, considering the influence of false data injection attacks, the issue of estimating the NCSs is addressed in [43]. From an attacker’s point of view, the optimal attacking region is determined while a false data injection attack takes place in a power grid [44]. The problem of filter design for neural networks is investigated under the consideration of deception attacks [29]. By considering the effects of DoS attacks, the robust stabilization of NCSs with event-triggered scheme and quantization is studied in [26]. In [45], CPSs with DoS attacks are studied where the data is transmitted over the network from sensors to remote estimators. The event-triggered control model for CPSs subjected to DoS attacks is constructed in [46]. To the best of our knowledge, with the consideration of DoS attacks, the issue of state estimation for CPSs with limited communication resources and sensor saturation has been paid few attentions, which is the second motivation of this paper.
The rest of this paper is organized as follows. In Section 2, the mathematical models of considered problems are presented, respectively and system modeling is given. In Section 3, the sufficient conditions are acquired which can guarantee the exponential stability of the system and the desired estimator gains are accurately derived. In Section 4, a simulated example is supplied to illustrate the usefulness of designed estimator.
Notation , and represent the set of positive real scalars, the m-dimensional Euclidean space and the set of real matrices, respectively; the superscript denotes matrix transposition; is the identity matrix with appropriate dimension; for , the notation represents that the matrix is real symmetric positive definite. For a matrix and two symmetric matrices and , stands for a symmetric matrix, where refers to the entries implied by symmetry. denotes the space of square-integrable vector functions defined on . denotes the set of non-negative integers. For a scalar , , where is a continuous function denotes that belongs to time interval .
Section snippets
System description
The framework of event-triggered state estimation for CPSs under sensor saturations, quantization and DoS attacks is shown in Fig. 1, where the data transmission between the quantizer and estimator is realized over the network with DoS attacks. In the following, we will give the mathematical model of the physical plant, event-triggered scheme, sensor saturation, quantizer and DoS attacks, respectively.
Consider the physical process which can be described as the following continuous-time
Main results
The performance of estimator error system (30) is discussed and the main results are exhibited in the following three theorems with their corresponding proofs. In Theorem 1, the sufficient conditions guaranteeing the system (30) exponentially stable are derived. Then the system performance of CPSs subjected to the external disturbance is discussed in Theorem 2. Based on the analysis in Theorem 1, Theorem 2, the gains of state estimator are obtained in Theorem 3.
Theorem 1 Given the sequence ,
Simulation examples
In this section, a simulation example related to CPSs is presented to illustrate the feasibility of estimator design for CPSs, where the sensor saturation and DoS attacks are considered and the event-triggered scheme and quantization are introduced to relieve the influence brought by limited communication resources.
Consider the system (3) with the following system matrices:
In the following, we present
Conclusions
In this paper, a problem of estimator design has been investigated for CPSs with limited communication resources, sensor saturation and periodic DoS attacks. For the considered CPSs, a T–S fuzzy model is borrowed to described it. In order to relieve the influence of limited communication resources, the event-triggered scheme and quantization mechanism are adopted to reduce the amount of data transmission. A new mathematical model of quantized state estimation for CPSs is established, which
Acknowledgments
This work was supported in part by the Open Fund of Key Laboratory of Grain Information Processing and Control of Hennan Province of China under Grant KFJJ-2018-203, and in part by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 18 kJB120002, and in part by the Natural Science Foundation of Jiangsu Province of China under Grant BK20171481, and in part by the Key R&D Program (Industry Prospect and Common Key Technologies) of Jiangsu Province, China
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