Reliable exponential H filtering for a class of switched reaction-diffusion neural networks

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Highlights

  • The reliable exponential H filtering of switched reaction-diffusion neural networks is studied.

  • An analysis result on the exponential H performance is presented.

  • A linear matrix inequalities-based design scheme for the desired Luenberger observer is proposed.

Abstract

In this paper, the reliable exponential H filtering issue is studied for switched reaction-diffusion neural networks subject to exterior interference. The purpose is to design a Luenberger observer to make sure that the filtering error system possesses a pre-defined exponential H interference-rejection level against possible sensor failures. An analysis result on the exponential H performance is presented by the use of a Lyapunov functional together with a few inequalities. On its basis, a linear matrix inequalities-based design scheme for the Luenberger observer is proposed by getting rid of the nonlinear terms composed of the Lyapunov matrix, the gain matrix, and an uncertainty matrix caused by the sensor failures. In the case when the factors of sensor failures and reaction-diffusion are not concerned, the design scheme is shown to be an improvement over an existing design scheme. Finally, two examples are given to demonstrate the applicability and reduced conservatism of the obtained results, respectively.

Introduction

So far, study efforts on dynamic neural networks have spanned several decades. These efforts have not only led to a deeper understanding of the human brain, but also promoted the use of neural networks in various areas, such as object recognition and classification, system identification and control, image encryption, and combinatorial optimization. It is sometimes the case that a neural network shows a complex behavior of mode switching owing to internal evolution or extrinsic stimulation. Under this circumstance, the whole neural network can be viewed as a switched system composed of finite sub-neural networks, which switch from one to another in an arbitrary or deterministic manner at some moments [1], [2]. There has been a growing interest in switched neural networks in recent years, along with the increasing recognition of the importance of hybrid systems in mathematics, physics, and engineering. And correspondingly, several different control mechanisms for such networks including, but not restricted to, the sampled-data control [3], [4], [5], [6], the switching event-triggered control [7], [8], the quantized control [9], [10], the switching control [11], [12], the quasi-time-dependent control [13], and the finite-time control [14], [15], [16], have been reported to date in the literature.

It is noteworthy that most of the existing control mechanisms utilize the state of the networks directly, while for a real-world controlled system, it is often expensive, or even unachievable to access the information of all its states, which is especially the case for the system with multiple state variables [17], [18]. For this reason, some investigations have been done on the estimation of switched neural networks from the control community. For instance, utilizing the average dwell time method, Chen et al explored the aperiodic sampled-data switched delayed neural networks in the case of actuator saturation and asynchronous switching and achieved the dynamic estimation of the attraction domain [19]. In the literature [20], the authors presented a resilient asynchronous filter by adopting a hierarchical structure method, which realized the estimation of the non-fragile asynchronous state of the Markov jumping neural networks. In [21], Li et al examined the issues of the state estimate of the discrete nonlinear neural network with Markov jumps and sensor failures, and gave a design scheme of a non-fragile state estimator. In [22], Ahn introduced the exponential H filtering framework for estimating the state of interference-affected switched neural networks and proposed a design scheme for the Luenberger observer via making use of linear matrix inequalities (LMIs).

As noted by Liao et al. [23], the electromagnetic environment in which a neural network is located is not uniform. The reaction-diffusion effect may occur when electrons move in such an environment; in this regard, the dynamics of the neural networks not only rely on the time variable but also is dependent on the space variable. Recently, despite the estimation of switched neural networks with reaction-diffusion has attracted a little attention [24], [25], while to our knowledge, how to deal with the H filtering issue for such networks is still not clear. On the other hand, a usual hypothesis in the estimator design of switched neural networks is that one can obtain ideal measurements of the sensors, while in real-world systems, faults of sensors sometimes occur. These sensor faults can greatly affect the estimator’s performance. Recently, there have been exploratory research activities on reliable estimation or filtering of various dynamic systems without reaction-diffusion terms. In [26], a proportional-integral observer was designed for the estimation of actuator and sensor faults of a Takagi–Sugeno fuzzy model subject to unknown premise variables. In [27], the authors designed two fuzzy observers with adaptive function for a type of switched nonlinear system subject to estimate actuator failures and sensor failures respectively. The reconstruction of actuators and sensors affected by faults and disturbances of uncertain T-S systems was examined in Brahim et al. [28] and a design method of sliding mode observers with discontinuous projects was proposed. For complex distributed parameter system models — the reaction-diffusion switched neural networks, can one develop a feasible Luenberger observer for the exponential H filtering against possible sensor faults? As far as the author knows, this question has not been theoretically addressed in the literature, which motivates the present study.

Within this paper, the reliable exponential H filtering issue is studied for switched reaction-diffusion neural networks subject to exterior interference. The purpose is to design a Luenberger observer to make sure that the filtering error (FE) system possesses a pre-defined exponential H interference-rejection level against possible sensor failures. The reminder of the paper is structured as follows: Section 2 formulates the problem and gives some lemmas. In Section 3, the exponential H disturbance-attenuation analysis is carried out firstly by employing Lyapunov functional (LF) and some inequalities, and then an LMIs-based design scheme is proposed by getting rid of the nonlinear terms; moreover, it is shown in this section that, when the factors of sensor failures and reaction-diffusion are not concerned, the design scheme is an improvement over an existing design scheme. Section 4 gives two different numerical examples, and compares one of them with the theoretical results of [22] to illustrate the applicability and reduced conservatism of the Luenberger observer design method in this paper. Finally, in Section 5, we summarize the research work of this paper.

Section snippets

Problem formulation

Consider the following neural network with reaction-diffusion:ϱ(s,t)t=k=1lsk(Bkϱ(s,t)sk)+Uϱ(s,t)+Wϕ(ϱ(s,td))+J(s,t)+Gw(s,t),σ(s,t)=C1ϱ(s,t)+C2ϱ(s,td)+Fw(s,t).

The network model consists of a set of complex semi-linear partial differential equations [29], [30], where s=[s1,,sl]T Υ with Υ={s=[s1,,sl]T | φkskψk, φk, ψkRl; k=1,,l} be a bounded set in Rl with boundary Υ; ϱ(s,t)=[ϱ1(s,t),,ϱn(s,t)]TRn represents the state vector; σ(s,t)=[σ1(s,t),,σm(s,t)]TRm represents the output

Main results

This section is devoted to designing the Luenberger observer for switched reaction-diffusion neural network (5). Firstly, we are concerned with the exponential H interference-attenuation analysis of system (12). And we are capable of proposing the result in the following theorem:

Theorem 1

For given γ>0, S=ST>0, and a sufficiently small positive constant δ, assume that there exist matrices Pζ=diag{Pζ1,,Pζn}>0,Rζ=RζT>0, and scalars ε1ζ>0, such thatΦζ=[Δ11ζPζC¯2ζPζG¯ζ*ε1ζLϕ2IeδdRζ0**γ2I]<0for ζ=1,,N,

Numerical example

This section provides two examples to show the applicability and reduced conservatism of the Luenberger observer design method, respectively.

Example 1

Consider the switched reaction-diffusion neural network (5) withϱ(s,t)=[ϱ1(s,t)ϱ2(s,t)ϱ3(s,t)],ϕ(ϱ(s,t))=[tanh(ϱ1(s,t))tanh(ϱ2(s,t))tanh(ϱ3(s,t))],U1=U2=[100010002],W1=[2101.71.711.12.52.90.56],W2=[2101.71.711.12.52.90.76],C11=C12=[100010001],C21=C22=[0.10000.10000.2],G1=[0.100.1],G2=[0.20.20.1],F1=F2=[000],E^1=[0.80000.80000.8],Eˇ1=[0.20000.2000

Conclusion

The reliable exponential H filtering issue for switched neural networks concerning both reaction-diffusion term and exterior interference term has been studied in this paper. A Luenberger observer has been devised to ensure that the FE system has the pre-defined exponential H interference-rejection level for possible sensor failures. By the use of LF and some inequalities, the analysis result of exponential H performance has been obtained. Based on this, the nonlinear terms of the Lyapunov

Acknowledgment

This work was supported by High School Outstanding Young Talents Abroad Visiting Research Project of the Anhui Higher Education Institutions (Grant no. gxgwfx2018020) and the Natural Science Foundation of the Anhui Higher Education Institutions (Grant no. KJ2020A0248).

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