Elsevier

Fuzzy Sets and Systems

Volume 421, 30 September 2021, Pages 158-177
Fuzzy Sets and Systems

Fault detection in finite frequency domain for T-S fuzzy systems with partly unmeasurable premise variables

https://doi.org/10.1016/j.fss.2020.08.014Get rights and content

Abstract

This paper is concerned with the problem of finite frequency fault detection (FD) for a class of T-S fuzzy systems with partly unmeasurable premise variables. To fully use the available information of the considered T-S fuzzy system, an FD observer whose premise variables consist of the measurable premise variables and the estimations of unmeasurable ones of the fuzzy model is designed. However, the membership functions of the fuzzy model and the observer to be designed are unsynchronized, such that the existing FD methods based on parallel distribute compensation (PDC) strategy may be infeasible. To overcome this difficulty, a novel non-PDC H/H unknown input observer is designed to achieve the purpose of fault detection and provide less conservative results. Finally, two simulation examples are given to illustrate the validity and merits of the presented FD scheme.

Introduction

Due to the wide existence of nonlinearity in practical applications, considerable research effort has been devoted to the investigation of nonlinear systems [4], [5], [6]. Particularly, T-S fuzzy model [1], which is represented by locally time-invariant subsystems connected by fuzzy rules, provides a general framework to deal with nonlinear dynamics. Consequently, diverse meaningful results on the T-S fuzzy systems have been published. For example, the problem of designing controller for the fuzzy systems was studied in [7]; the stabilization and stability analysis of the fuzzy systems were investigated in [8], [9], [10]; the authors in [11] investigated the H fuzzy filter design problem. Note that, the aforementioned references mainly consider the T-S fuzzy systems in the fault-free case.

However, with the increasing complexity of practical systems, faults usually occur and may result in system performance degradation [12], [43]. Therefore, it is important to detect faults in time so as to decrease the influences of them. Among various FD methods, the model-based FD technique has attracted considerable attention, and based on which some remarkable results have been published, such as [13], [14], [15], and the references therein. It should be mentioned that the above FD schemes are all considered in entire frequency range, which cannot always meet the requirements of practical systems, because the frequency domains of some external inputs may be known beforehand [16], [17], [18]. Specially, the generalized Kalman-Yakubovich-Popov (GKYP) lemma was proposed in [19] to characterize frequency range inequalities with (semi)finite frequency domains in the formulation of linear matrix inequalities (LMIs), and based on which a great number of effective FD schemes have been developed, for instance [21], [22], [23], [24], [25].

The aforementioned full frequency and finite frequency FD methods have made significant progress in the study of FD for T-S fuzzy systems, however, only the T-S fuzzy systems with measurable premise variables were considered. Indeed, it is not the case that the premise variables of the fuzzy systems are always available. For instance, they may include unmeasurable system states [38], [39]. In this case, the so-called PDC strategy based FD methods [18], [21], [23], [24] cannot be directly applied. Then, numerous valuable results have been reported to solve this problem. For example, in [31], the Lipschitz method is used to solve the observer design problem for T-S fuzzy systems with unmeasurable premise variables, where the premise variables of the designed observer depend on the estimations of the ones of the fuzzy model. A linear filter with fixed gains was designed in [32] to detect the sensor faults. Based on a sliding mode observer, the adaptive control problem for T-S fuzzy systems with semi-Markov switching was considered in [42]. For a class of uncertain T-S fuzzy systems, the robust stability problem was studied in [44]. However, the partly available premise variables are not sufficiently used in these works, which may result in some conservative results. With that in mind, a switching-type FD filter was designed in [33], which has a promising feature that the bounds of the unknown weight functions can be used in the FD filter. Recently, the problem of sensor FD for the fuzzy systems with partly available premise variables was considered in [34], where the T-S fuzzy systems are described with the help of some concepts of the set, and the measurable premise variables are fully used for fault detection. However, the FD scheme in [34] cannot be directly applied to detect actuator or process faults due to the non-convex problem included in the H performance. Furthermore, the frequency ranges of external inputs were not considered, which might be conservative in some degree.

Motivated by the above discussions, the problem of finite frequency FD for T-S fuzzy systems with partly unmeasurable premise variables is studied in this paper. First, a description method based on some concepts of the set [26] is employed to describe the considered fuzzy systems. Within this framework, the measured premise variables are explicitly separated from the unmeasurable ones, then the available information is fully used in the designed observer to decrease the conservatism of the existing results [27], [28], [29], [30], [31]. Furthermore, inspired by [35], the asynchronization of the membership functions is coped with the differential mean value theorem, and some slack variables are introduced to further reduce the conservatism of the proposed methods.

This paper is organized as follows. The preliminaries and system description are given in Section 2. The main design conditions of the FD observer are shown in Section 3. Some simulation results are provided in Section 4 to illustrate the validity of the theoretical results. Finally, the conclusions are derived in Section 5.

Notation

Rn is the n-dimensional Euclidean space. “⁎” represents the symmetric term in a symmetric matrix and diag=[] is a block-diagonal matrix. AT and A respectively denote the transpose and the orthogonal complement of the matrix A, and He(A)A+AT. I represents the identity matrix with suitable dimension. For a symmetric matrix P, P>()0 and P<()0 denote positive (semi)definiteness and negative (semi)definiteness, respectively. “∧” denotes a classic logical operator “conjunction”. Some notations about the sets and the subscripts used throughout this paper are summarized in Table 1.

Section snippets

Preliminaries and problem statement

Consider a nonlinear system described by the following T-S fuzzy model:

Plant Rule i:

IF ν1(t) is M1i1, ν2(t) is M2i2, …, and νn(t) is MninTHENx˙(t)=Aix(t)+Biu(t)+Eid(t)+Fif(t)y(t)=Cx(t) where i=i1i2inS, x(t)Rnx denotes the system state, u(t)Rnu represents the input signal, y(t)Rny indicates the system output, d(t)Rnd represents the external disturbance which is assumed to belong to L2[0,), and f(t)Rnf denotes the fault signal which includes actuator fault and process fault. Ai, Bi

Fault detection observer design

Because the premise variables νj (j=1,,p) are measurable and νg (g=p+1,,n) are unmeasurable, therefore, the following unknown input observer with the measurable premise variables and the estimations of the unmeasurable ones is designed in this paper:

Plant Rule i:

IF ν1 is M1i1,,νp is Mpip and νˆp+1 is Mp+1ip+1,, νˆn is MninTHENz˙=Niz+Giu+Liyxˆ=z+Ryyˆ=Cxˆ where xˆRnx is the estimation of x(t) and νˆg represent the estimations of νg (g=p+1,,n). Ni, Gi, Li and R are the observer gain

Simulation examples

In this section, some simulation results are provided to illustrate the advantages and effectiveness of the proposed FD scheme.

Example 1

Consider the nonlinear mass-spring-damper system [8] described byMx¨(t)+c3x(t)+c4x3(t)+c1x˙(t)+c2x˙2(t)u(t)=0 where x(t) denotes the measurable displacement from a reference position, u(t) denotes the input force, and M=1, c1=3, c2=0.15, c3=5, c4=1, a=3, b=8. It is assumed that x(t)[aa] and x˙(t)[bb]. Define x1(t)=x(t), x2(t)=x˙(t) and assume that the nonlinear

Conclusion

The problem of finite frequency FD for T-S fuzzy systems with partly unmeasurable premise variables has been investigated in this paper. By employing the description scheme based on some concepts of the set, the premise variables of the considered fuzzy system are divided into the measurable and unmeasurable two parts. Within this framework, combining H/H performance, a new FD observer with measurable premise variables and the estimations of the unmeasurable ones of the fuzzy system has been

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported in part by the Funds of the National Natural Science Foundation of China (Grant Nos. 61621004 and U1908213), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2018ZCX03).

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