Finite-time filtering for fuzzy nonlinear semi-Markov jump systems with deception attacks and aperiodical transmission

https://doi.org/10.1016/j.jfranklin.2021.07.055Get rights and content

Abstract

This paper investigates finite-time event-triggered filtering problems for fuzzy nonlinear semi-Markov jump systems (S-MJSs) subject to deception attacks. First, an interval type-2 (IT2) fuzzy S-MJSs is presented that can capture parameter uncertainties, in which the stochastic behavior is described by semi-Markov process while the sojourn-time obeys non-exponential distribution. Second, we design an adaptive event-triggered mechanism (AETM) for IT2 fuzzy S-MJSs that can save network resources, in which the triggered parameter can be automatically adjusted with the error information rather than a fixed value. Third, a hidden-Markov model is employed to represent the asynchronous situation between the filter and system. Then, a new IT2 fuzzy filtering error system is modeled by taking the effects of random deception attacks and AETM into consideration. Moreover, finite-time boundedness (FTB) analysis and a filter design method are proposed. Finally, an example is given to verify the effectiveness of the finite-time event-triggered filtering algorithm.

Introduction

Over the past half-century, the type-1 Takagi-Sugeno (T-S) fuzzy systems have received extensive attention in the control community because of their ability to deal with nonlinear systems well [1]. Although the type-1 fuzzy set theory can be used to handle nonlinear systems, the parameter uncertainties of nonlinear systems cannot be dealt with well. Fortunately, an interval type-2 (IT2) fuzzy model was developed to model the parameter uncertainties [2]. Subsequently, some research works based on the IT2 fuzzy sets were reported and can be seen in [3], [4], [5], [6].

On the other hand, some random factors directly affect the practical system, such as accidental changes in the environment and damage to system components [7], [8]. As we can observe, the Markov jump systems (MJSs) are an efficacious tool for dealing with these unexpected phenomena [9], [10], [11], [12]. In order to describe the model more accurately, many researchers have given the results of the semi-Markov jump model [13], [14], [15], [16], [17], [18]. The reliable filtering problem was considered for uncertain S-MJSs in [13]. In [14], the fault detection problem was solved for continuous-time IT2 fuzzy S-MJSs. A mixed estimator was designed for a class of nonlinear S-MJSs in [15]. Some other results on S-MJSs can be found in [19], [20], [21]. Moreover, most of the classical control and filtering problems only give stability conditions but do not consider the time for state convergence. Nevertheless, it should worth mentioning that there are certain requirements for the control accuracy and convergence time. Kamenkov proposed the concept of finite time stability, which guarantees the convergence in finite time [22]. Due to the superiority of finite time convergence, it has also been applied to the stability analysis of fuzzy systems, and many results have emerged [23], [24], [25], [26]. The authors in [23] addressed the filtering problem for discrete-time fuzzy systems and proposed a new finite-time stability conditions. For continuous-time MJSs, the authors in [24] provided a robust finite-time control algorithm.

However, the aforementioned work was based on a common assumption that the sensor is periodic-triggered, which implies that all the sampled signals need to be transmitted into the next node through communication networks without concerning the limited network bandwidth. In recent years, the event-triggered scheme (ETS) has drawn considerable [27], [28], [29], [30], [31], [32]. In [27], the H filtering problem was addressed via a weighted try-once-discard protocol. A dissipative asynchronous controller was developed for the IT2 MJSs with sensor saturation and actuator nonlinearity [29]. In [32], a team-triggered algorithm was proposed for double-integrator agents. The event-triggered mechanism has also been extended to the study of filtering problems S-MJSs [33], [34], [35], [36]. In [34], the guaranteed cost finite-time control problem was considered for S-MJSs by an event-triggered scheme and the reliable extended passive control problem was addressed in [35]. In [36], a finite-time sliding mode controller was designed to solve actuator-faults problems. Furthermore, a new ETS with an adaptive law was developed to further save computing resources and transmission resources [37], [38], [39], [40], [41], [42]. Herein, the event-triggered function is developed by designing the adaptive law for the threshold. Furthermore, for the wireless channel, the cyber-attacks can not be ignored when the signal is transmitted, especially considering the event-triggered control. Among the attacks, deception attacks are what we need to consider most because it leads to serious threats to systems [43], [44], [45], [46]. In [43] an event-triggered dynamic output feedback controller was designed for MJSs with deception attacks. The nonhomogeneous MJSs with asynchronous modes was considered in [44]. For the Markov jump singularly perturbed systems, the nonstationary quantized control problem was solved in [45]. In [47], the static output feedback control problem was addressed for disturbed MJSs with randomly deception attacks asynchronous mode information.

Although there are some results about filtering for S-MJSs, some issues worthy to discuss further. The first one is that the designed filter is mode-independent in [48], which means that all the system modes only correspond to one filter gain. The second one is that the threshold parameter of the designed ETS in [33] is set to be a fixed value, which cannot truly reflect the dynamics of the system, thereby leading to a certain conservatism. The third one is that the proposed method in [33], [48], [49] cannot be applied to S-MJSs with uncertainties, but the uncertainties are inevitable in practical systems.

Motivated by these observations, this paper investigates the problem of finite-time filter design for nonlinear S-MJSs with deception attacks and aperiodical transmission via IT2 T-S fuzzy models. The contributions are listed as follows:

(1) Different from Wang et al. [33], [48], [49], we consider a more practical IT2 fuzzy S-MJSs that covers more complex situations: complex stochastic process, delay, deception attacks, asynchronous modes, and the limited network resources.

(2) Compared with the existing works [33], [49], in this paper, the designed filter can guarantee the FTB of the IT2 fuzzy S-MJSs, rather than infinite-time interval. Moreover, the hidden-Markov model is introduced to describe the relationships between system mode and filter mode.

(3) The adaptive event-triggered mechanism (AETM) is designed to guarantee that the IT2 T-S fuzzy S-MJSs are finite-time boundedness (FTB). Different from the studies [48], [49], the proposed method can not only guarantee the performance of the filtering error system but also reduce unnecessary waste of communication resources.

The rest parts are organized as follows: In Section 2, some basic results are presented. The main results are given in Section 3. Section 4 provides an example. The conclusions are summarized in Section 5.

Notation: Let Rn denote n-dimensions Euclidean space; Let N denote the positive integer; M>0 (M0) and M<0 (M0) stand for positive definite (semi-definite) and negative definite (semi-definite) matrix, respectively; He{W} represents W+WT and Ξ{·} stands for mathematical expectation operator; Let ’1’ denote the inverse of the matrix and ’T’ stand for the transpose of the matrix; λmin(W) (λmax(W)) denotes the minimum (maximum) eigenvalue of W. Besides, I and 0 represent the identity matrices and zero matrix with suitable dimensions. Moreover, * is the symmetric term.

Section snippets

Fuzzy S-MJSs model

Consider the following S-MJSs modelled by IT2 fuzzy sets:

Plant Rule i: IF F1(x(t)) is Ni1, F2(x(t)) is Ni2,..., and FK(x(t)) is NiK, THEN{x˙(t)=Aϑtix(t)+Bϑtiν(t)y(t)=Cϑtix(t)+Dϑtiν(t)z(t)=Eϑtix(t) where Fi(x(t)) and Nik (iO1={1,2,,r};kO2={1,2,,K}), respectively, denote the premise variable and IT2 fuzzy set, in which r is the number of fuzzy rules; x(t)Rnx and y(t)Rny denote state vector and measured output, respectively; ν(t)Rnν represents external disturbance input that belongs to L2[0,

Main results

This section firstly presents a sufficient condition that the filtering error system (15) remains FTB via the AETM and has H performance. Then, the existence condition of the filter is provided for the filtering error system (15).

Numerical example

In this section, we will list the following numerical simulations to demonstrate the effectiveness of the proposed method.

As shown in Fig. 2, an electric circuit model is considered [53], where it contains parasitic capacitors and a nonlinear resistor. The characteristics of the resistor can be presented as IR(t)=0.2vc(t)(vc2(t)1). It follows from Kirchoff voltage that{LmI˙Lm(t)=σ1UILm(t)RvcmCv˙cm(t)=ILm(t)+σ2U0.2vcm(t)(vcm2(t)1)where Lm(t) and vcm(t) (m=1,2) represent the inductor current

Conclusion

The finite-time event-triggered filtering problems were addressed for fuzzy nonlinear S-MJSs with deception attacks. A hidden-Markov model was employed to represent the asynchronous situation between the filter and the system. Moreover, a new IT2 fuzzy filtering error system was modeled by taking the effects of random deception attacks and AETM into consideration. Furthermore, FTB analysis and filter design method were proposed. Finally, an example was given to verify the effectiveness of the

Declaration of Competing Interest

None.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61876050, the China Scholarship Council under Grant 202006120100, the Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 2020Z081, the Fundamental Research Funds for the Central Universities under Grant 2242021R20002, the China Postdoctoral Science Foundation funded project under Grant 2021M690032, the Natural Science Foundation of Jiangsu Province of China under Grant BK20210214,

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      To mention a few, event-triggered synchronization for SMJSs was probed in [31]. When taken the deception attacks into account, the event-triggered filter for SMJSs was obtained in [32]. By employing slack matrix approach and Lyapunov theory, the event-triggered controller was designed in [33] for SMJSs with time-varying delay.

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