Design of observers for Takagi–Sugeno descriptor systems with unknown inputs and application to fault diagnosis

B. Marx, J. Ragot, D. Koenig
2007 IET Control Theory & Applications  
This paper presents a method for state-estimation of Takagi-Sugeno descriptor systems (TSDS) affected by unknown inputs (UI). For ease of implementation's sake, the proposed observers are not in descriptor form but in usual form. Sufficient existence conditions of the unknown input observers are given and strict linear matrix inequalities (LMI) are solved to determine the gain of the observers. If the perfect unknown input decoupling is not possible, the UI observer is designed in order to
more » ... ed in order to minimise the L2-gain from the UI to the state estimation error. The two previous objectives can be mixed in order to decouple the estimation to a subset of the UI, while attenuating the L2 gain from the other UI to the estimation. The proposed UI observers are used for robust fault diagnosis. Fault diagnosis for TSDS is performed by designing a bank of observers. A simple decision logic and thresholds setting allow to determine the occurring fault. The results are established for both the continuous and the discrete time cases. The proposed method is illustrated by a numerical example. Index Terms Takagi-Sugeno systems, singular systems, state estimation, unknown input observers, fault diagnosis. I. INTRODUCTION The Takagi-Sugeno (TS) model proposed by [14] is a well-known structure to represent nonlinear systems into several linear fuzzy models. In the last two decades, the control and the observation of TS systems have become challenging problems that received a considerable amount of attention. In [19] , stability analysis and controller design are addressed, solutions are derived in the linear matrix inequality (LMI) formalism. Relaxed sufficient conditions for fuzzy controllers and fuzzy observers are proposed in [15] , and in [16] via a multiple Lyapunov function approach. The descriptor formalism is very attractive for system modelling, as pointed out in [4] , since it describes a wider class of systems including physical systems with non dynamic constraints (e.g. algebraic relations induced in interconnected systems such as power transfer networks or water distribution networks) or jump behaviour. The enhancement of the modelling ability is due to the structure of the dynamic equation which encompasses not only dynamic equations, but also algebraic relations. Since both TS and descriptor formalisms are attractive in the field of modelling, the TS representation has been generalised to descriptor systems. The stability and the design of state-feedback controllers for TS descriptor systems (TSDS) are characterised via LMI in [17],[18], in particular, the problem of nonlinear model following is treated in [18]. Robust output feedback, and H ∞ -control are considered for TSDS in [12] and [21] respectively. The study of TSDS is envisaged with interval methods in [20], in order to take into account the different operating points. Unfortunately, the problem of observer design, and especially the design of unknown input observers, has resulted in very few works. The design of unknown input observer (UIO) is a crucial problem since, in many practical cases, all input signals cannot be known. Moreover, this class of observers is widely used in the area of fault diagnosis, even if all the inputs are known (see chap. 3 in [13]). The design of UIO has received considerable attention in the case of usual (in opposition to descriptor) linear systems [5], descriptor systems [7], [8], [11] or TS systems [1]. Unfortunately, to the authors' knowledge, the design of UIO has not been treated in the generic case of TSDS. The aim of this paper is not only to generalise the existing works on UIO design to TSDS, but also to apply this new observer in the field of fault diagnosis of TSDS which has not been treated so far. This paper gives a simple extension to TSDS of the design of observers for the state estimation in the presence of unknown inputs (UI). Under some sufficient conditions, the design of the observer is reduced to the determination of a matrix. The choice of this parameter is performed by solving strict LMIs. If the estimation error cannot be decoupled from the UI, an L 2 observer is proposed to minimise the influence of the UI on the state estimation. The two design objectives can be mixed by decoupling the state estimation from a subset of the UI, and minimising the L 2 -gain between the other UI and the state estimation error. The designed observers are used for fault diagnosis, since the UI can encompass the faults and the disturbances affecting the system. Designing several observers attenuating the disturbance effect, and decoupling the estimation from all faults but one lead to the well-known Generalised Observer Scheme (GOS) for fault diagnosis [13] . The design of observers is detailed both in the continuous time case, and in the discrete time case. The paper is organised as follows: the class of studied systems is defined in section II and the main results about UIO design are detailed in section III. Firstly, the definition of the UIO and the sufficient existence condition are established. Secondly, the computation of the gains of the observer is established. The design of L 2 observers is treated in section IV. Section V deals with the design of observers for both UI decoupling and disturbance attenuation. The application to fault diagnosis is studied in section VI. Section VII is devoted to a numerical example.
doi:10.1049/iet-cta:20060412 fatcat:ryuzp32puvhyni2goimjvt37hi