H_ /H_ Fault detection observer for discrete-time Takagi-Sugeno fuzzy systems

Transcription

1 H_ /H_ Fault detection observer for discrete-time Takagi-Sugeno fuzzy systems Meng Zhou Mickael Rodrigues Zhenhua Wang Yi Shen Didier Theilliol To cite this version: Meng Zhou Mickael Rodrigues Zhenhua Wang Yi Shen Didier Theilliol. H_ /H_ Fault detection observer for discrete-time Takagi-Sugeno fuzzy systems. 56th IEEE Conference on Decision and Control CDC 217 Dec 217 Melbourne Australia. <hal > HAL Id: hal Submitted on 8 Oct 217 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents whether they are published or not. The documents may come from teaching and research institutions in France or abroad or from public or private research centers. L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés.

2 H /L Fault Detection Observer for Discrete-time Takagi-Sugeno Fuzzy Systems Meng Zhou 12 Mickael Rodrigues 2 Zhenhua Wang 1 Yi Shen 1 Didier Theilliol 3 Abstract In this paper an H /L fault detection observer is proposed for discrete-time Takagi-Sugeno (T-S) fuzzy systems. The main contribution of this paper lies as first the finite frequency H index is used to measure the worst-case fault sensitivity. Second unlike the common H /H fault detection observer this paper uses L norm to measure the worst effect of disturbance on residual instead of H norm. Besides a residual evaluation is synthesized based on the L norm to achieve an efficient fault detection despite of the disturbance occurrence. Besides sufficient conditions of the observer design are derived based on piecewise Lyapunov functions. The effectiveness of the proposed method is demonstrated by numerical simulations. I. INTRODUCTION Faults are usually inevitable for complex nonlinear systems. In order to prevent severe events fault detection has been extensively investigated over the last decades [1-3]. Takagi- Sugeno (T-S) fuzzy models are usually used to describe complex nonlinear systems which are usually subject to modelling uncertainties measurement noise and other disturbances. The disturbances may bring false alarms and missing alarms for fault detection such that robustness against disturbances has been one of the most important requirements. H technique is one of the most effective methods to attenuate disturbances and has been widely used in fault detection for T-S fuzzy systems (see e.g. [4-6] and the reference therein). Actually H norm is a measurement of energy-toenergy gain it requires that the exogenous disturbances are supposedly of energy-bounded signals. However in practice some signals are not energy-bounded but have bounded peak values. For such types of disturbances it is not easy to calculate the energy on entire time-domain accurately. When peak bounded input signals are considered the L norm appears to be more adequate as performance criterion for the observer design which aims to minimize the peak-to-peak gain [7]. In [8] peak-to-peak gain minimization problem of linear continuous-time systems has been studied based on the bounding reachable set with inescapable ellipsoids. [9] proposes a peak-to-peak filtering for a class of Markov *This work was partially supported by National Natural Science Foundation of China (Grant No ) and the Fundamental Research Funds for the Central Universities under Grant HIT.KLOF Meng Zhou Zhenhua Wang and Yi Shen are with the Department of Control Science and Engineering Harbin Institute of Technology 151 Harbin China zhenhua.wang@hit.edu.cn 2 Mickael Rodrigues is with the Laboratoire d Automatique et Génie des Procédés Université de Lyon Lyon F-693 France mickael.rodrigues@univ-lyon1.fr 3 Didier Theilliol is with the Centre de Recherche en Automatique de Nancy Université de Lorraine CNRS UMR 739 F-5456 Vandoeuvreles-Nancy France didier.theilliol@univ-lorraine.fr jump systems with uncertain parameters. In [7] the robust L filter is proposed to stochastic systems. In [1] residual evaluation and threshold computation methods based on H norm and L norm are compared and it is shown that L norm method has better performance in fault detection threshold than H norm method. In order to detect faults accurately and timely only considering the disturbance robustness in observer design is not enough high fault sensitivity to residual is important and necessary. A robust fault detection method should minimize the effect of the disturbance whilst maximize the effect of fault on residual simultaneously [11]. H /H fault detection observer design method has drawn much researchers attention in which H index is considered as a measurement of the worst-case fault sensitivity and H norm is used to measure the disturbance attenuation performance. A great deal of literatures can be found in [12-14] just to name a few. However to the best of authors knowledge very little work has been done on H /L fault detection observer design. In [15] an H /L fault detection observer is proposed for continuous-time LPV systems in which the L norm is used to measure the effect of the disturbance on residual. The major merit of this method is that the H /L fault detection observer can provide a dynamic threshold for residual evaluation in fault detection. However [15] is only suitable for continuous-time systems besides it is based on a common Lyapunov function approach which brings some conservatism. Motivated by [15] in this paper we extend the H /L fault detection observer designed method to discrete-time T-S fuzzy systems. The main contribution of this paper lies in the following aspects. First the fault sensitivity to residual in finite frequency domain is maximized by using H index. Then considering the peak bounded disturbance L norm is used to measure the effect of the disturbance to residual. Besides a dynamic threshold is generated for residual evaluation and decision making based on L norm. In order to reduce some conservatism the sufficient conditions of the proposed observer are derived based on piecewise Lyapunov function method and formulated as a set of LMIs. Furthermore some extra design freedom of parameters are added to reduce some conservatism. The rest of this paper is organized as follows. The preliminaries and system descriptions are given in Section II. An H /L fault detection observer is proposed for discretetime T-S fuzzy systems in Section III the sufficient conditions of the proposed observer are given and transformed as a set of LMIs. In Section IV numerical simulations are used

3 to demonstrate the effectiveness of the proposed method. Section V gives the conclusions. Notations: The notations in this paper are standard. R n denotes the n-dimensional Euclidean space and I represents the identity matrix with appropriate dimension. For a matrix A A T denotes its transpose He(A) is expressed as He(A) := A+A T and A > (A < ) means that A is positive definite (negative definite). The asterisk represents the transposed block induced by symmetry in a symmetric matrix. For a vector x R n x 2 = x T x is used to denote its L 2 - norm. For a signal x(k) R n we define the L norm (the peak norm) of x(k) to be the sup over all time of the L 2 -norm at each time instant i.e. x(k) := sup x(k) 2. k II. SYSTEM DESCRIPTION AND PRELIMINARIES A nonlinear dynamic system can be described by Takagi- Sugeno (T-S) fuzzy models which is represented by fuzzy IF-THEN rules [14]. The i-th rule of T-S fuzzy model is expressed as Model Rule i: IF ξ 1 (k) is M i1 and and ξ s (k) is M is THEN x(k + 1) = Ai x(k) + B i u(k) + F i f(k) + D xi w(k) (1) y(k) = Cx(k) + D y v(k) i = 1 2 N x(k) R n x denotes the state vector u(k) R n u is the input vector y(k) R ny is the measured output vector f(k) R n f represents the actuator fault vector w(k) R n w denotes the unknown disturbance v(k) R nv is the measurement noise. A i R n x n x B i R n x n u F i R n x n f C R ny nx D xi R nx nw and D y R ny nv are known system matrices. M ij (i = 1 2 N j = 1 2 s) are the fuzzy sets and ξ i (k) i = 1 2 s are known premise variables. s is the number of premise variables and N is the number of model rules. Then the global fuzzy dynamics is inferred as follows x(k + 1) = µ i (ξ(k))[a i x(k) + B i u(k) (2) + F i f(k) + D xi w(k)] y(k) = Cx(k) + D y v(k) µ i (ξ(k)) = ξ(k) = [ ξ 1 (k) ξ 2 (k) ξ s (k) ] T ω i (ξ(k)) N ω i(ξ(k)) ω i(ξ(k)) = (3) s M ij (ξ j (k)) j=1 (4) For all k the term M ij (ξ j (k)) is the grade of membership of ξ j (k) ω i (ξ(k)) and µ i (ξ(k)) satisfy that ω i (ξ(k)) i = 1 2 N µ i (ξ(k)) i = 1 2 N ω i (ξ(k)) > µ i (ξ(k)) = 1 (5) For brevity of notation the following notations are defined as follows A(µ) = µ i (ξ(k))a i B(µ) = µ i (ξ(k))b i (6) F (µ) = µ i (ξ(k))f i D x (µ) = µ i (ξ(k))d xi Then the T-S fuzzy model (2) is rewritten such that x(k + 1) = A(µ)x(k) + B(µ)u(k) + F (µ)f(k) + D x (µ)w(k) y(k) = Cx(k) + D y v(k) The main objective of this paper is to design an observer for system (7) such that the residual is sensitive to faults and robust against disturbances simultaneously. In the sequel the following Assumptions and Lemma are given and used. Assumption 1. The actuator fault f(k) is assumed to belong to a low frequency range as (7) Θ f = θ R : θ ϑ l } (8) θ denotes the frequency of fault f(k) and ϑ l > is a known scalar. Assumption 2. The unknown disturbance and noise are assumed to be peak bounded i.e. w(k) ϵ w v(k) ϵ v (9) ϵ w > and ϵ v > are known scalars. Lemma 1. [13] Considering a T-S fuzzy system as x(k + 1) = A(µ)x(k) + B(µ)u(k) y(k) = C(µ)x(k) + D(µ)u(k) (1) Assuming that system (1) is asymptotically stable and controllable letting γ > be a given constant then under zero initial conditions system (1) has a finite frequency performance with γ i.e. y T (k)y(k) γ 2 u T (k)u(k) (11) k= k= if there exists a fuzzy-basis-dependent matrix P (µ) = P T (µ) and a matrix Q = Q T > µ + := (µ 1 (ξ(k + 1)) µ r (ξ(k + 1))) such that T A(µ) B(µ) A(µ) B(µ) Ξ + I I T C(µ) D(µ) C(µ) D(µ) Π I I holds for any µ Π = [ I ] γ 2 I (12) (13) and for a low-frequency domain θ ϑ l the matrix Ξ is given as P (µ Ξ = + ) Q (14) Q P (µ) 2 cos ϑ l Q

4 III. H /L FAULT DETECTION OBSERVER DESIGN In order to detect the actuator fault for system (7) an observer is proposed with the following structure ˆx(k + 1) = A(µ)ˆx(k) + B(µ)u(k) + L(µ)(y(k) C ˆx(k)) (15) r(k) = y(k) C ˆx(k) ˆx(k) R nx denotes the estimation of state vector x(k) and r(k) R n y stands for the residual. L(µ) R nx ny is the observer gain matrix to be determined with the following form L(µ) = µ i (ξ(k))l i (16) The state estimation error vector can be defined such that e(k) = x(k) ˆx(k) (17) By subtracting (15) from (7) the estimation error dynamics is obtained as e(k + 1) = [A(µ) L(µ)C]e(k) + F (µ)f(k) + [ µ x (h) L(µ) D y ]d(k) (18) r(k) = Ce(k) + D y d(k) d(k) = w(k) Dx (µ) = [ D v(k) x (µ) ] Dy = D y According to Assumption 2 it has d(k) d (19) d is the maximum scalar of ϵ w and ϵ v. According to [15] the observer (15) is an H /L observer if the following conditions are satisfied: (1) The estimation error dynamics (18) is asymptotically stable; (2) The state estimation error system (18) has the following peak-to-peak performance i.e. r(k) 2 γ 2 d(k) 2 + N (e() k) (2) N (e() k) is a quadratic function of initial estimation error e() and will converge to zero as k ; (3) The residual r(k) in (18) is sensitive to actuator faults f(k) in a finite frequency range in (8) i.e. r T (k)r(k) β 2 f T (k)f(k) θ ϑ l (21) k= k= To facilitate the H /L fault detection observer design the following Theorem is proposed. Theorem 1. Given positive scalars < α < 1 γ and β observer (15) is asymptotically stable and satisfies the H performance in (21) and L performance such that r(k) 2 γ 2 C 2 2 αk V () + (γ 2 C D y 2 2 ) d 2 (22) if there exist fuzzy-basis-dependent matrices P 1 (µ) = P1 T (µ) R nx nx and P 2 (µ) = P2 T (µ) > R nx nx and matrix Q = Q T > R n x n x such that Φ1 Φ 2 F (µ) T P 1 (µ + )F (µ) + β 2 < (23) I Π1 Π 2 < (24) Π 3 I γ 2 P 2 (µ) < (25) Φ 1 = [A(µ) L(µ)C] T P 1 (µ + )[A(µ) L(µ)C] + Q[A(µ) L(µ)C] + [A(µ) L(µ)C] T Q + P 1 (µ) 2 cos ϑ l Q C T C Φ 2 = [A(µ) L(µ)C] T P 1 (µ + )F (µ) + QF (µ) Π 1 =[A(µ) L(µ)C] T P 2 (µ + )[A(µ) L(µ)C] αp 1 (µ) Π 2 =[A(µ) L(µ)C] T P 2 (µ + )[ D x (µ) L(µ) D y ] Π 3 =[ D x (µ) L(µ) D y ] T P 2 (µ + )[ D x (µ) L(µ) D y ] (1 α)i Proof: The proof is divided into the following two parts. A. H fault sensitivity condition When only consider the effect of actuator fault by setting d(k) = and (18) becomes e(k + 1) = [A(µ) L(µ)C]e(k) + F (µ)f(k) (26) r(k) = Ce(k) In this paper the actuator fault is assumed to belong to a given low finite frequency domain (8) and according to Lemma 1 for a given positive scalar β > letting I Π = β 2 (27) I and substituting [A(µ) L(µ)C] A(µ) F (µ) B(µ) C C(µ) D(µ) and P 1 (µ) P (µ) in (12) then (12) is transformed in (23) and system (26) has a finite frequency H index performance as (21). B. L disturbance attenuation condition By setting f(k) = the estimation error is only affected by the disturbance and can be re-written as e(k + 1) = [A(µ) L(µ)C]e(k) + [ D x (µ) L(µ) D y ]d(k) (28) r(k) = Ce(k) + D y d(k) A Lyapunov candidate is chosen such that V (k) = e T (k)p 2 (µ)e(k) (29) Given a positive scalar α > under zero initial condition it is assumed that the following inequality is satisfied. V (k + 1) < αv (k) + (1 α)d T (k)d(k) (3) When k = we have V (1) < αv () + (1 α)d T ()d() (31)

5 By iterating we can obtain that V (k) < α k V () k 1 + (1 α) α i d T (k 1 i)d(k 1 i) i= Due to d T (k)d(k) d(k) 2 d 2 then we have (32) V (k) α k V () + (1 α k 1 ) d 2 (33) As < α < 1 such that < α k 1 < 1 we can get Furthermore if it is satisfied that then it obtains V (k) < α k V () + d 2 (34) I < γ 2 P 2 (µ) (35) e T (k)e(k) < γ 2 e T (k)p 2 (µ)e(k) (36) According to the definition of residual r(k) in (28) it comes that r(k) 2 2 C 2 2 e(k) D y 2 2 d(k) 2 2 (37) By using (32) (36) and d(k) 2 2 d(k) 2 d 2 we obtain r(k) 2 C 2 2 γ2 P 2 (µ) e(k) D y 2 d 2 2 γ 2 C 22 αk V () + (γ 2 C 22 + D y 22 ) d (38) 2 By substituting (29) into (3) then (24) can be derived which completes the proof. Remark 1. Inequality in (23) and (24) of Theorem 1 are not jointly convex due to the coupling of the terms P 1 (µ)l(µ) and P 2 (µ)l(µ) which makes the observer difficult to be solved. In order to facilitate the observer design the following Theorem is given. Theorem 2. Given scalars < α < 1 γ > η 1 η 2 and matrix K R nx nu the observer (15) is an H /L observer if there exist matrices Q = Q T > R n x n x P 1i = P T 1i R nx nx P 2i = P T 2i > R nx nx G R n x n x and W i R n x n y such that Ψ ii < Ψ ij + Ψ ji < Ω ii < Ω ij + Ω ji < i = 1 N i = 1 N 1 i < j N 1 i < j N (39) (4) I γ 2 P 2i < (41) ψ 1 ψ 2 ψ 3 Ψ ij = ψ 4 K T G + Fi T GT P 1j G G T ω 1 ω 2 ω 3 Ω ij = (1 α)i DT xi G T D y T Wi T P 2j G G T ψ 1 = P 1i 2 cos ϑ l Q C T C + Heη 1 (GA i W i C)} ψ 2 = η 1 GF i + A T i G T K C T W T i K ψ 3 = η 1 G + Q + A T i G T C T Wi T ψ 4 = β 2 I + HeK T GF i } ω 1 = αp 2i + Heη 2 (GA i W i C)} ω 2 = η 2 (GD xi W i D y ) ω 3 = η 2 G + A T i G T C T Wi T The observer can be determined by maximizing the H index β and the optimization problem is expressed as max β s.t. (39) (41) Then the gain matrices L i is calculated by (42) L i = G 1 W i i = 1 2 N (43) Proof: First (23) is equivalent to Υ 1 + A T 1 Q T + QA 1 A T 1 P 1 (µ + )A 1 < (44) P1 (µ) 2 cos ϑ Υ 1 = l Q C T C β 2 (45) I A 1 = [ (A(µ) L(µ)C) F (µ) ] Q Q = (46) A sufficient condition of (44) is Υ1 + Y 1 A 1 + A1 T Y 1 Y 1 + Q + A1 T G T P 1 (µ + ) G G T < (47) It can be verified by pre- and post- multiplying (47) with I A1 T and its transpose respectively. In order to remove the coupling between G and Y 1 by denoting η1 G Y 1 = K T (48) G By substituting (48) and (56) into (47) we have ψ 1 ψ2 ψ3 β 2 I + HeK T GF (µ)} K T G + F (µ) T G T < P 1 (µ + ) G G T (49) ψ 1 = P 1 (µ) 2 cos ϑ l Q C T C + Heη 1 G[A(µ) L(µ)C]} ψ 2 = η 1 GF (µ) + A(µ) T G T K C T L(µ) T G T K ψ 3 = η 1 G + Q + A(µ) T G T C T L(µ) T G T Similarly (24) can be rewritten as follows Υ 2 + A T 2 P 2 (µ + )A 2 < (5) αp2 (µ) Υ 2 = (51) (1 α)i A 2 = [ (A(µ) L(µ)C) ( D x (µ) L(µ) D y ) ] (52)

6 A sufficient condition of (5) is Υ2 + Y 2 A 2 + A2 T Y2 T Y 2 + A2 T G T P 2 (µ + ) G G T < (53) It can also be easily shown by pre- and post- multiplying (5) with I A2 T and its transpose respectively. By denoting that η2 G Y 2 = (54) Substituting (54) into (53) we have ω 1 ω 2 ω 3 (1 α)i ( D x (µ) L(h) D y ) T G T < (55) P 2 (µ + ) G G T ω 1 = αp 2 (µ) + Heη 2 G(A(µ) L(µ)C)} ω 2 = η 2 G( D x (µ) L(µ) D y ) ω 3 = η 2 G + (A(µ) L(µ)C) T G T and by noting that GL(µ) = W (µ) (56) and combining the definitions in (6) and (16) according to [4] the conditions (49) and (55) are equivalent to (39) and (4) respectively which completes the proof. Remark 2. The tuned parameters η 1 and η 2 matrices K and G are added in Theorem 2 which can provide extra design degrees of freedom and further reduce some conservatism. In the implementation the values of η 1 η 2 and matrix K can be determined beforehand by trial and error method. Based on the L performance in (22) in this paper at time k the L 2 norm of the residual r(k) 2 is considered to evaluate the residual. And a dynamic threshold can be determined such as J th (k) = γ 2 C 2 2 αk V () + (γ 2 C D y 2 2 ) d 2 (57) As (22) the fault detection logic is given as r(k) 2 > J th (k) Faulty r(k) 2 J th (k) Fault-free IV. SIMULATION RESULTS (58) In this section numerical simulations are given to illustrate the effectiveness of the proposed method. Borrowed from [16] a nonlinear system is expressed as x(k + 1) = A(ξ(k))x(k) + B(ξ(k))u(k) + F (ξ(k))f(k) + D x (ξ(k))d(k) (59) y(k) = Cx(k) + D y d(k) the system matrices are.3.2 ξ 2 (k) A(ξ(k)) =.6 ξ 1 (k).1 (6) 2ξ 2 (k) ξ 1 (k) B(ξ(k)) = 1 F (ξ(k)) = B(ξ(k)) (61) C = 1 D 1 x =.2 D y =.1.1. (62) In this paper the nonlinear system is represented by the following subsystems. Plant rule 1: IF ξ 1 (k) is.1 and ξ 2 (k) is.1 THEN x(k + 1) = A 1 x(k) + B 1 u(k) + F 1 f(k) + D x d(k) Plant rule 2: IF ξ 1 (k) is.1 and ξ 2 (k) is.3 THEN x(k + 1) = A 2 x(k) + B 2 u(k) + F 2 f(k) + D x d(k) Plant rule 3: IF ξ 1 (k) is.3 and ξ 2 (k) is.1 THEN x(k + 1) = A 3 x(k) + B 3 u(k) + F 3 f(k) + D x d(k) Plant rule 4: IF ξ 1 (k) is.3 and ξ 2 (k) is.3 THEN x(k + 1) = A 4 x(k) + B 4 u(k) + F 4 f(k) + D x d(k) The subsystem matrices A i B i and F i i = 1 4 can be easily obtained based on (6) and (61). In the simulations the premise variables ξ 1 (k) and ξ 2 (k) are assumed as Fig. 1. The membership functions for rule i i = 1 4 are given as µ 1 (ξ(k)) = (.3 ξ 1(k))(.3 ξ 2 (k)).4 (.3 ξ1(k))(ξ2(k).1) µ 2 (ξ(k)) =.4 µ 3(ξ(k)) = (ξ 1(k).1)(.3 ξ 2 (k)).4 µ 4 (ξ(k)) = (ξ 1(k).1)(ξ 2 (k).1)..4 In this paper the actuator fault is assumed to belong to a given frequency domain in θ [.1.1]. In the simulations the parameters are chosen as α =.9 η 1 =.2 η 2 =.2 K = 9F 1 and γ =.3. By solving the optimization problem in (42) the maximum H index is calculated as β = and the gain matrices of the proposed observer are obtained as L 1 = L 2 = L 3 = L 4 = In order to demonstrate the effectiveness of the proposed method it is compared with the L observer without considering the H fault sensitivity. With the same disturbance attenuation performance as γ =.3 the gain matrices of L observer are calculated as ˆL 1 = ˆL2 = ˆL 3 = ˆL4 =

7 Premise variables ξ 1(k) ξ 2(k) Fault detection result ˆr(k) 2 by L observer Ĵ th(k) by L observer r(k) 2 by H /L observer J th(k) by H /L observer Fault detection result Fig. 1. k Premise variables ˆr(k) 2 by L observer Ĵ th(k) by L observer r(k) 2 by H /L observer J th(k) by H /L observer Fig. 2. k Fault detection result for abrupt fault In this paper the initial condition of the nonlinear system is chosen as x() = [ ] T the disturbance is simulated by a random vector bounded by [ 1 1] such that d = 1. To demonstrate the effectiveness of the proposed method two fault scenarios are considered. First an abrupt fault with the amplitude of.2 is assumed to happen in the second actuator after time k = 5 such that f 2 (k) =.2. In the second fault scenario a time-varying fault is considered to occur in the first actuator after time k = 5 the fault is formulated as f 1 (k) =.2.5 sin(.1πk 5). Simulation results are depicted in Fig Herein the solid lines and dash-dot lines represent the residual evaluation and threshold generated by the proposed method respectively. The dashed lines and dotted lines are the residual evaluation and threshold generated by L observer respectively. From Fig. 2 and Fig. 3 we can see that the proposed H /L fault detection observer can detect the abrupt fault and timevarying fault accurately and timely while the L observer method is failed to detect the faults. The proposed method has better fault sensitivity than the L observer method. Besides the generated dynamic threshold is able to detect the actuator faults in a timely manner. V. CONCLUSIONS This paper proposes an H /L fault detection observer design method for discrete-time T-S fuzzy systems. In this paper the fault sensitivity to residual is maximized by using the finite frequency H index and the disturbance robustness to residual is measured by L norm. The proposed method exhibits a peak-to-peak disturbance attenuation performance and a dynamic threshold is provided for residual evaluation Fig. 3. k Fault detection result for time-varying fault based on L norm it is suitable for disturbances with peak bounded. Simulations results demonstrate the effectiveness of the proposed method. REFERENCES [1] J. Chen R. J. Patton Robust Model-Based Fault Diagnosis for Dynamic Systems Kluwer Academic Publishers Boston [2] S. X. Ding Model-based diagnosis techniques: design schemes algorithms and tools Berlin Heidelberg: Springer publishing company Inc. 28. [3] M. Rodrigues H. Hamdi D. Theilliol C. Mechmeche N. BenhadjBraiek Actuator fault estimation based adaptive polytopic observer for a Class of LPV descriptor systems International Journal of Robust and Nonlinear Control. Vol. 25 no. 5 pp [4] K. Tanaka H. O. Wang Fuzzy control systems design and analysis: A linear matrix inequality approach John Wiley & Sons Inc. 21. [5] K. Zhang B. Jiang P. Shi Fault estimation observer design for discrete-time Takagi-Sugeno fuzzy Systems based on piecewise lyapunov functions IEEE Transactions on Fuzzy Systems vol. 2 no. 1 pp [6] M. Zhou Z. H. Wang Y. Shen Simultaneous fault estimation and fault-tolerant tracking control for uncertain nonlinear discrete-time systems International Journal of Systems Science vol. 48 no. 7 pp [7] M. Tabarraie Robust L -induced deconvolution filtering for linear stochastic systems and its application to fault reconstruction Signal Processing vol. 93 no. 5 pp [8] J. Abedor K. Nagpal K. Poola A linear matrix inequality approach to peak-to-peak gain minimization International Journal of Robust and Nonlinear Control vol. 6 no. 9-1 pp [9] S. P. He F. Liu Robust peak-to-peak filtering for Markov jump systems Signal Processing vol. 9 no. 2 pp [1] A. Q. Khan S. X. Ding Threshold computation for fault detection in a class of discrete-time nonlinear systems International Journal of Adaptive Control and Signal Processing vol. 25 no. 5 pp [11] M. Hou R. J. Patton An LMI approach to H /H fault detection observers Proceedings of the UKACC International Conference on Control Exeter UK pp [12] J. Liu J. L. Wang G. H. Yang An LMI approach to minimum sensitivity analysis with application to fault detection Automatica vol. 41 no.11 pp [13] D. W. Ding G. H. Yang Fuzzy filter design for nonlinear systems in finite-frequency domain IEEE Transactions on Fuzzy Systems vol. 18 no. 5 pp [14] M. Chadli A. Abdo S. X. Ding H /H fault detection filter design for discrete-time Takagi-Sugeno fuzzy system Automatica vol. 49 no. 7 pp [15] Z. H. Wang C. C. Lim P. Shi Y. Shen H /L fault detection observer design for linear parameter-varying systems 2th World Congress of the International Federation of Automatic Control accepted 217. [16] D. Rotondo F. R. L. Estrada and F. Nejjari and J. C. Ponsart and D. Theilliol and V. Puig Actuator multiplicative fault estimation in discrete-time LPV systems using switched observers Journal of the Franklin Institute vol. 353 no. 13 pp