Elsevier

Signal Processing

Volume 92, Issue 2, February 2012, Pages 575-586
Signal Processing

Non-fragile fuzzy H filter design for nonlinear continuous-time systems with D stability constraints

https://doi.org/10.1016/j.sigpro.2011.09.003Get rights and content

Abstract

This paper is concerned with the problem of designing non-fragile H filters for a class of nonlinear continuous-time systems. The considered nonlinear plant is represented by a Takagi–Sugeno (T–S) fuzzy model. Attention is focused on the design of a filter such that the filtering error system guarantees a prescribed H performance level with D stability constraints, where the filter to be designed is assumed to be with multiplicative gain variations. A sufficient condition for the non-fragile H filter design is proposed in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H fuzzy filter is given. A simulation example will be given to show the efficiency of the proposed design methods.

Highlights

► This paper investigates the problem of fuzzy H filter design with multiplicative gain variations. ► This paper shows that the solution of non-fragile fuzzy H filter design problem can be obtained by solving a set of LMIs. ► This paper might give least conservative results for fuzzy H filtering with D stability constraints.

Introduction

Over the past few decades, state estimation problems have attracted wide attention from scientists and engineers, essentially because the state variables in control systems are not always available [1], [2], [3]. So far, various methodologies have been developed for the filter designs. One approach to this problem is H filtering, and the advantage of this approach is that the noise signals in the H filtering setting are arbitrary signals with bounded energy, and no exact statistics are required to be known, which is more general than classical Kalman filtering [4]. Moreover, the H filter has been shown to be much more robust against unmodeled dynamics [5]. For the H filtering problems of linear systems, many important advances have been achieved through different techniques during the past decades [6], [7], [8], [9], [10], [11], [12], [13]. However, the problem of designing H filters for nonlinear systems still remains as open research subject [14].

On the other hand, an important approach to address the synthesis problems for nonlinear systems is to model the considered system as Takagi and Sugeno (T–S) fuzzy systems [15], which are locally linear time-invariant systems connected by if–then rules. In recent years, a great number of stability analysis and control synthesis results for the class of T–S fuzzy systems in both the continuous-time and discrete-time contexts have been extensively discussed in the literature [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. Due to the powerful approximation property of T–S fuzzy model, recently, there have been a number of results on H filtering for T–S fuzzy systems. Based on the fuzzy Lyapunov function approach, the H filtering problem was investigated for a class of discrete-time T–S fuzzy dynamic systems in [5], [27]. Liu and Wang [28] studied fuzzy H filtering for nonlinear stochastic systems with missing measurements. In [29], the design of fuzzy H filters for a class of nonlinear singularly perturbed systems with pole placement constraints was addressed. Chen et al. [30] presented a method of designing mixed H2/H filters for equalization/detection of nonlinear communication systems using fuzzy interpolation and LMI techniques. In [31], an H filter design for continuous-time T–S fuzzy models based on the notion of quadratic stability had been considered. An H filter design for discrete-time T–S fuzzy models with intermittent measurements proposed in [32]. The problem of H filter design for T–S fuzzy systems with time-delay were investigated in [33], [34]. The H fuzzy-filtering problem with D stability constraints was studied in [35].The problem of designing H filters for fuzzy singularly perturbed systems with the consideration of improving the bound of singular-perturbation parameter was considered in [36]. The H filters for a class of networked control systems (NCSs) with multiple state-delays via the T–S fuzzy model were designed in [37].

Noting that in the above-mentioned works on the H filtering problem of nonlinear systems via T–S fuzzy models are based on an implicit assumption that the filter will be implemented exactly. However, in fact, inaccuracies or uncertainties do occur in the implementation of a designed filter or controller. Such uncertainties can be due to, among other things, roundoff errors in numerical computation during the filter or controller implementation and the need to provide practicing engineers with safe-tuning margins. So a significant issue is how to design a filter or controller for a given plant such that the filter or controller is insensitive to some amount of errors with respect to its gain, i.e., the designed filter or controller is resilient or non-fragile [38], [39], [40], [41]. For the non-fragile filtering problem, some numerically effective design methods have been obtained [42], [43], [44]. The problem of non-fragile H filtering for a class of linear systems described by delta operator with circular region pole constraints was investigated in [42]. In [43], the non-fragile H filtering problem for linear continuous-time systems has been addressed, where the filter to be designed is assumed to be with additive gain variations of interval type. A robust non-fragile Kalman filtering problem was addressed in [44]. Nevertheless, despite there have been fruitful results of non-fragile filter designs for linear systems (i.e., [42], [43], [44]), it generally lacks common techniques in non-fragile filter designs for complex nonlinear systems [45].

Motivated by the aforementioned observations, in this paper, we investigate the problem of non-fragile H filter design for nonlinear continuous-time systems. The nonlinear plant is represented by a T–S fuzzy model. Given a T–S fuzzy system, our objective is to design an H filter with the gain variations such that the filtering error system is quadratically D stable and guarantees a prescribed H performance level. The quadratic Lyapunov function approach is developed to design a desired H filter, and we show that the solution of non-fragile H filter design problem can be obtained by solving a set of LMIs. Finally, a simulation example is provided to illustrate the feasibility of the proposed design methods.

The remainder of this paper is organized as follows. The problem formulation and preliminaries are presented in Section 2. In Section 3, a design condition of non-fragile H filter with D stability constraint is derived by using the quadratic Lyapunov function approach. In contrast to the existing methods, an H filter design condition published recently in the literature is extended in Section 4. An illustrative example is given in Section 5, and we conclude the paper in Section 6. The proof of Theorem 2, Theorem 3 is given in Appendixes A and B, respectively.

Notations: For a matrix A, AT and A1 denote its transpose and inverse if it exists, respectively. The symbol () induces a symmetric structure in LMIs. I is the identity matrix with appropriate dimension. L2[0,) is the space of square-integrable vector functions over [0,). denotes the Kronecker product of matrices.

Section snippets

Problem formulation and preliminaries

Consider the following nonlinear continuous-time system represented by T–S fuzzy dynamic model, in which the ith rule is described as follows:Ri:ifξ1(t)isM1iandξp(t)isMpithenx˙(t)=Aix(t)+Biw(t)y(t)=Cix(t)+Diw(t)z(t)=Lix(t)where x(t)Rn is the state variable, w(t)Rv is the noise signal that is assumed to be the arbitrary signal in L2[0,), z(t)Rq is the signal to be estimated, y(t)Rf is the measurement output, ξ(t)=[ξ1(t),ξ2(t),,ξp(t)],ξd(t),d=1,2,,p are premise variables vector and

H filtering analysis with quadratic D stability

In this subsection, the filtering analysis problem is concerned. More specifically, we assume that the filter matrices in (6) are known, and we will study the condition under which the filtering error system (7) is quadratically stable in the given LMI stability region D and satisfies an H performance bounded by γ. The following theorem shows that the stability with D constraint and H performance of the filtering error system (7) can be guaranteed if there exists a matrix variable satisfying

Comparison with the existing design methods

In this section, in order to make an effective comparison with the existing design method in [50] (i.e., Proposition 2), we extend the result in [50] to non-fragile H filter design with D stability constraints for T–S fuzzy systems, and the following condition for designing the filter in (6) will be given.

Theorem 3

Consider the filtering error system (7). For a given γ>0 and any LMI stability region D, if there exist matrices P˜1,P˜2,P˜3, AFj,BFj,CFj,DFj, scalars εAj,εBij,εCj,εDij, δAj,δBij for i,j=1,2,

Simulation example

To investigate the effectiveness of the proposed non-fragile H filter design condition, a simulation example is considered to study the H performance and D stability.

Consider a tunnel diode circuit shown in Fig. 1, where x1(t)=vC(t),x2(t)=iL(t), w(t) is the disturbance noise input, y(t) is the measurement output, and z(t) is the controlled output. The nonlinear system can be approximated by the following T–S with two fuzzy rules [35]: x˙(t)=A(h)x(t)+B(h)w(t)y(t)=C(h)x(t)+D(h)w(t)z(t)=L(h)x(t)

Conclusion

The problem of designing H filter for nonlinear continuous-time systems has been addressed, which is assumed to have gain variations. The T–S fuzzy system is utilized to model the nonlinear plant. The quadratic Lyapunov function has been used to design a non-fragile H filter with D stability constraints. The sufficient condition for the existence of the filter has been given in terms of LMIs. A simulation result demonstrates the successful application of the proposed design methods.

Acknowledgment

This work Xiao-Heng Chang was supported in part by the Funds of National Science of China (Grant No. 61104071).

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