Fault diagnosis in Takagi-Sugeno nonlinear systems

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1 Fault diagnosis in akagi-sugeno nonlinear systems Dalil Ichalal Benoit Marx José Ragot Didier Maquin Centre de Recherche en Automatique de Nancy (CRAN), Nancy-Université, CNRS, 2 avenue de la forêt de Haye F-5456 Vandoeuvre-les-Nancy. {dalil.ichalal, benoit.marx, jose.ragot, didier.maquin}@ensem.inpl-nancy.fr Abstract: his paper addresses a new scheme for fault diagnosis in nonlinear systems described by a akagi-sugeno multiple models. wo cases are considered, the first one concern the -S models with known premise variables (the input or the output of the system). he second case suppose that the weighting functions depend on unmeasurable premise variables (state of the system). he approach is based on the design of observer-based residual generator by minimization of the disturbances and maximizing the effects of the faults. he Lyapunov method is used to stability analysis and design of the residual generator. he convergence conditions are given in LMI formulation.. INRODUCION Linear models are largely studied and an important literature is devoted to this class of systems. Although they provide solutions for many problems, nonlinear behavior are often present in practical systems then reduces the domain of applicability and performances of tools developed for linear models. Indeed, linear models only represent the behavior of the system around a local operating point. It is known that nonlinear systems are complex and difficult to study, so all the works on the nonlinear systems concerned only specific classes but there is no a general framework like for linear systems. One of the more interesting classes of nonlinear systems is the akagi-sugeno (S) multiple model form which is introduced in akagi and Sugeno [985]. It is proved in anaka and Wang [2] that often nonlinear behaviors can be represented exactly or approximated by S multiple models. he most advantage of these models is the ability to extend the tools designed in the linear system framework. Indeed, many topics of control are extended to S systems, such as stability and stabilization in anaka et al. [998], Guerra et al. [26], Chadli et al. [22], observers and state estimation in Akhenak et al. [27], Bergsten et al. [22]. Due to an increasing demand for higher performances, as well as for higher safety and reliability, the modelbased approaches to fault diagnosis for dynamic systems have received more attention these last years Patton et al. [989], Chen and Zhang [99], Chen et al. [996], Ding and Frank [989], Marx et al. [23]. Concerning the S fuzzy systems few efforts have been made in fault detection and isolation. Nevertheless we can cite the method based on observers in Akhenak et al. [27]. In this paper an observer-based approach is developed for robust residual generator and diagnosis which minimizes the sensitivity to the disturbances and maximizes the sensitivity to the faults. wo cases are studied. he first case concern the -S systems with measurable premise variables and the second one deals with the systems with unmeasurable premise variables. he paper is organized as follows, section 2 gives some notations and states the problem. Robust residual generation is tackled in section 3. An LMI-based design of the residual generator is proposed. 2. PROBLEM SAEMEN Consider the following continuous-time S nonlinear system subject to faults and disturbances given by ẋ µ i (ξ) (A i x + B i u + E i d + F i f) y µ i (ξ) (C i x + D i u + G i d + R i f) () A i R n n, B i R n nu, C i R ny n, D i R ny nu, E i R n n d, F i R n n f and G i R ny n d, and R i R ny n f. he weighing functions µ i are nonlinear and depend on the decision variable ξ which can be measurable like {u,y} or not measurable like the state x of the system. he weighting functions satisfy the following properties: µ i (ξ) µ i (ξ) (2) hus the structure of the multiple model is simple and is considered as a universal approximator since it can represent any nonlinear behavior according to an adequate number r of the local models. he multiple model structure provides a mean to generalize the tools developed for linear systems to nonlinear systems due to the properties expressed in (2).

2 he input signals u, f and d belong in L 2 set. he + L 2 -norm of u L 2 is given by u 2 u udt. In the field of observer design and diagnosis of nonlinear systems using multiple model approach, Patton et al. [998] proposed an observer-based method to generate residual generator and using an observer bank in order to achieve isolation, an application to DC motor is proposed. In Akhenak et al. [27], a sliding mode observer for S systems is proposed to detect and estimate actuator faults. In these works, the authors assumed that the weighting functions depend on measurable premise variables (input or output) of the system. It is clear that the choice of measurable premise variables offers a good simplicity to generalize the methods already developed for linear systems. But in the case the premise variables are not measurable, the problem becomes very hard. However, this formalism is very important in both the exact representation of the nonlinear behavior by multiple model (see the simulation example) and in diagnosis method based on observer banks to detect and isolate actuator and sensor faults. Indeed in this case, the use of measurable premise variables requires to develop two different multiple models, but using multiple models with unmeasurable premise variables allows to develop only one model of the system behavior to detect and isolate actuator and sensor faults using observer banks. In the literature, a few works are devoted to the case of unmeasurable decision variables, nevertheless, we can cite Bergsten et al. [22], Palm and Bergsten [2], the authors proposed the fuzzy hau-luenberger observer which is an extension of the classical Luenberger observer. he main contribution of this paper is to propose a method for fault diagnosis of nonlinear systems described by S models with measurable and unmeasurable premise variables using the standard H framework developed for linear systems. { e Aξ e + E ξ d + F ξ f r C ξ e + G ξ d + R ξ f A ξ E ξ F ξ C ξ µ i (ξ)µ j (ξ)(a i L i C k ) µ i (ξ)µ j (ξ)(e i L i G k ) j µ i (ξ)µ j (ξ)(f i L i R k ) j j µ i (ξ)mc i,g ξ R ξ µ i (ξ)mg i µ i (ξ)mr i (4) For convenience, the system (4) can be written as the following compact form r G rd d + G rf f (5) G rd represents the transfer from the disturbances d to r and defined by Aξ E G rd : ξ (6) MC i G ξ and G rf is the transfer from f to r which is defined by Aξ F G rf ξ (7) C ξ R ξ In standard H d framework (see figure.), the maxi- f W f 3. RESIDUAL GENERAOR DESIGN u System y r e 3. case : measurable premise variables + Let consider the S nonlinear system subject to disturbances and sensor and actuator faults modeled in () An observer-based residual generator is proposed in the following form ˆx µ i (ξ)(a iˆx + B i u + L i (y ŷ)) (3) ŷ µ i (ξ)(c iˆx + D i u) r M(y ŷ) ˆx R n is the estimated state vector and r R n f is the residual signal. he matrices L i R n ny and M R n f n y are the residual generator gains. he objective is to design the gains L i and M in order to minimize the transfer from the disturbances w and to maximize the transfer of the faults f to the residual signal r. Let define the state estimation error e x ˆx. Its dynamic is deduced from () and (3) as follows Residual Generator r Fig.. Scheme of robust residual generation mization of the effect of the faults f on the residual r can be expressed as a minimization problem. Indeed, by introducing a weighting parameter W f, the problem is reduced to a minimization of the effect of the faults on the residual error r e r W f f. As explained in Stoustrup and Niemann [2] that the FDI problem depends on the selected structure of the weight parameter W f. Indeed, he fault estimation problem is obtained when W f I and the detection problem is considered when W f R n f. In addition, W f can be chosen as a dynamic parameter. Consider the parameter W f defined Af B W f f (8) C f D f

3 W f S S is the set of stable filters which have been the following property W f inf w R (σ (W f (jw))) (9) (see Mazars et al. [28] and Mazars et al. [26] for more details). he interest in this kind of filters is that there is no attenuation of the faults but only an amplification on all frequency ranges which improves the problem of fault detection. he detection, isolation and estimation of the faults can be considered by an appropriate choice of the matrices A f, B f, C f and D f. he FDI problem is then formulated as the following multi-objective optimization problem Obtain L i and M which minimize aγ f + ( a)γ d a [ ] subject to the following constraints G rf W f < γ f () G rd < γ d () A i L i C j is stable for i,j,...,r (2) he theorem gives an LMI method to solve the optimization problem and provides the residual generator gains L i and M. heorem. Given a positive parameter a and a weighting function W f. he residual generator (3) exists if there exist matrices P P >, P 2 P 2 > and gain matrices K i and M and positive scalars γ f and γ d solution of the following optimization problem s.t. min L i,m,p,p 2,K i, γ f, γ d a γ f + ( a) γ d Xik P F i K i R k Ck M Xf 2 P 2 B f Cf Fi P Rk K i Bf P 2 γ f I Rk M Df MC k C f MR k D f I ( X ik P E i K i G k Ck M Ei P G k K i γ d I G k M MC k MG k I ) < (3) < (4) X ik A i P + P A i K i C k C k K i (5) X 2 f A f P 2 + P 2 A f (6) i,k,...,r he gains L i are derived from L i P K i i,...,r (7) and the attenuation levels are given by γ d γ d γ f γ f (8) Proof. In faulty case without disturbances the residual generator is reduced to r G rf f. In order to maximize the effects of faults on the residual we consider the weighting stable filter W f (s) defined in (9). hen the maximization problem can be formulated as a minimization problem by solving (). G rf W f can be written in the following form G rf W f : A ξ F ξ A f B f (9) C ξ C f R ξ D f Let define a positive and symmetric bloc diagonal matrix P P (2) P 2 Using the bounded real lemma Boyd et al. [994], the condition () is formulated as follows A ξ P + P A ξ P F ξ Cξ A f P 2 + P 2 A f P 2 B f Cf Fξ P Bf P 2 γf 2 I R ξ < (2) D f C ξ C f R ξ D f I Using the definitions of the matrices A ξ, F ξ, C ξ and R ξ and the convex property of the weighing function, the following matrices are obtained Xik P F i P L i R k Ck M Xf 2 P 2 B f Cf Fi P Rk K i Bf P 2 γf 2 I R k M Df MC k C f MR k D f I < (22) X ik A i P + P A i P L i C k C k L i P (23) X 2 f A f P 2 + P 2 A f (24) i, k,..., r In order to obtain the linear matrix inequality (4), we use the change of variables K i PL i and γ f γ 2 f and γ d γ 2 d. In fault-free case with disturbances, a similar way, by using the bounded real lemma, allows to obtain the LMI (5). he LMI (5) ensure the stability of the observer (i.e. A i L i C k are stable i,k,...,r) and the robustness against disturbances. Now, in the faulty case with disturbances, the relative importance of minimizing the effects of the disturbances and maximizing the effects of the faults on the residual signal can be expressed as a minimization of the linear combination aγ f + ( a)γ d a [ ]. 3.2 case 2: unmeasurable premise variables In this section, it is assumed that the weighting functions of the S nonlinear system () depend on the unmeasurable state x of the system. he weighting function of the residual generator then depend on the estimated state ˆx as follows ˆx µ i (ˆξ)(A iˆx + B i u + L i (y ŷ)) (25) ŷ µ i (ˆξ)(C iˆx + D i u) r M(y ŷ) By adding and subtracting the term µ j (ˆξ) (A j x + B j u + E j d + F j f) j in state equation of the system () and the term µ j (ξ) (C j x + D j u + G j d + R j f) j

4 in the output equation of () and by some manipulations using convex property of the weighting function the following equivalent system is obtained ẋ µ i (ξ)µ j (ˆξ)(Ãijx + B ij u+e i d + F i f) j y µ i (ξ)µ j (ˆξ)( C ij x + D ij u + G i d + R i f) j (26) à ij A i + A ij, C ij C i + C ij B ij B i + B ij, D ij D i + D ij and X ij X i X j, X i {A i,b i,c i,d i } i,j,...,r After calculating the dynamic of the state estimation error, the following is obtained { e à ξˆξe + Ãξˆξ x + B ξˆξ d + Fξˆξf r C ξˆξe + C ξˆξx + G ξˆξ d + Rξˆξf (27) By adopting the following writing µ i (ξ)µ j (ˆξ)µ k (ξ)µ l (ˆξ) j k l he matrices of (28) are defined by à ξˆξ µ iˆµ j µ kˆµ l (A j L j C l ) B ξˆξ F ξˆξ G ξˆξ Ãξˆξ µ iˆµ j µ kˆµ l µ iˆµ j µ kˆµ l [ B ij L j D kl E i L j G k ] C ξˆξ R ξˆξ C ξˆξ µ iˆµ j µ kˆµ l (F i L j R k ) µ iˆµ j µ kˆµ l MC l, µ iˆµ j µ kˆµ l [M D kl MG k ] µ iˆµ j µ kˆµ l MR k µ iˆµ j µ kˆµ l ( A ij L j C kl ) µ iˆµ j µ kˆµ l C kl d [ u d ] Let define the augmented state vector x [e x ]. he residual vector r is then given by the equation G rd r G rd d + Grf f (28) à ξˆξ Ãξˆξ B ξˆξ A ξ Bξ C ξˆξ C ξ Gξˆξ (29) and A ξ G rf à ξˆξ Ãξˆξ F ξˆξ A ξ F ξ C ξˆξ C ξ Rξˆξ µ i (ξ)a i,f ξ B ξ (3) µ i (ξ)f i µ i (ξ)[ B i E i ] he FDI problem is the same as the problem given in ()-(3). In order to determine the gains L i and M of the residual generator (26), the theorem 2 gives an LMI solution of the problem ()-(3) for S nonlinear systems with unmeasurable premise variables. heorem 2. Given a positive parameter a and a weighting function W f. he residual generator (3) exists if there exist matrices P P >, P 2 P 2 > and gain matrices K i and M and positive scalars γ and γ 2 solution of the following optimization problem s.t. min L i,m,p,p 2,K i, γ f, γ d a γ f + ( a) γ d Xjl Ξ ijkl P F i K j R k Cl M Xi 2 P 2 F i Ckl M Xf 3 P 3 B f Cf γ f I (MR k D f ) I Xjl Ξ ijkl P B ij K j D ij P E i K j G k < (3) C l M X 2 i P 2 B i P 2 E i C kl M γ di γ di G k M I < (32) X jl A j P + P A j K j C l C l K j (33) X 2 i A i P 2 + P 2 A i (34) X 3 f A f P 3 + P 3 A f (35) Ξ ijkl P A ij K j C kl (36) i,j,k,l,...,r he gains L i are derived from L i P K i i,...,r (37) and the attenuation levels are given by γ d γ d γ f γ f (38) Proof. After calculating the augmented system with x [e x x f ] by including the filter W f which has x f as a state vector and calculating r e r r f r f is the output of the filter W f (see figure ). he proof follows exactly the steps which have been given for the proof of the theorem. Remark. Note that the theorem 2 is more general that the theorem. Indeed, if the weighting functions µ i of the system () depend on measurable premise variables, the problem given in the theorem can be deduced from the theorem 2 by taking i j and k l.

5 4. ROBUS FAUL DIAGNOSIS Due to the presence of exogenous disturbances, the residual signals are different from zero even in the fault-free case. In the framework of fault detection, a threshold based on the obtained attenuation levels γ f and γ d is generated. An alarm is generated by comparison between the residual signals r and the threshold. A fixed threshold is determined as follows J th γ d ρ (39) ρ is the bound of d in the measurable premise variables case and it represent the bound the bound of d in the unmeasurable premise variables. he decision logic is given by { ri < J th no fault (4) r i > J th fault 5. DISCUSSION AND REMARKS Remark 2. In order to improve the fault detection, residual generator is constructed for each fault separately. Each residual generator is designed to minimize the transfer from f i to r ei r i W fi f i, i,...,n f. Remark 3. In the unmeasurable premise variables case, the system is seen as an uncertain system. he input u then appear in the dynamic of state estimation error. he method proposed in this paper considers the input u as a perturbation as d and by considering the new perturbation vector d [u d ] the problem is solved. It is clear that considering the input u as a perturbation penalizes the fault detection because the computed threshold depends on the upper bound of d. Using the method proposed in Casavola et al. [28] for linear systems with polytopic uncertainties u is considered as a perturbation to minimize separately from d. Indeed, instead of minimizing the index (a γ f + ( a) γ d) under the LMI constraints, the index which has been used in Casavola et al. [28] described by (a γ f + b γ d +c γ u ) can be used. An adaptive threshold can be then generated using a time-windowed rms-norm (see Casavola et al. [28], Frank and Ding [997]). Remark 4. It is often considered that the fault vector f has two components, the first one noted f a represent the vector of the faults affecting only the actuator, thus, they appear in the state equation. he second component noted f s are the vector of the faults affecting only the sensors. he output of the system is given by y µ i (ξ)(c i x + D i u + G i d + R i f) (4) In the case the faults f a do not affect the output of the system, the matrices R i are not full rank. As pointed out in (Stoustrup and Niemann [2]), in this case, W f I the attenuation level γ f > or the problems in theorem and 2 have not solutions. In order to avoid this problem, a perturbation term is added in the output equation as follows ( [ ] [ ]) y µ i (ξ) C i x + D i u + G i d + ε i Ri f a (42) f s ε i are the matrices of distribution of the actuator faults f a in the output equation and are chosen as small as possible. In the context of fault isolation, this approach may generate a false alarms. o improve the isolation results, we propose to add and subtract the perturbation term and make the added term to ensure the full rank of R i and consider the subtract term as a perturbation to minimize ( [ ]) y µ i (ξ) C i x + D i u + Ḡi d + R fa i (43) f s Ḡ i [ G i bε i ], R i [ [ ] ] d ε i Ri, d f a b b is a positive real parameter. Using this second approach, the threshold J th is calculated by using the bound of the new perturbation vector d, thus the fault isolation is improved. 6. NUMERICAL EXAMPLE he proposed algorithm of robust diagnosis is illustrated by an academic example. Let consider the nonlinear system () defined by [ ] [ ] A 3, A 2 5 3, and B [ ] 5, B 2.5 C [ ] [ ] 3 7, E 5, F 7 2 [ ] [ ] 6 F 2, E 2 3, [ ], G [ ] 5, R [ ] [ ], he unknown inputs vector d is made up of d which affects only the outputs of the system and d 2 affecting only the dynamic of the system (see the matrices E, E 2 and G). he first component of the vector f is a sensor fault and the second component is an actuator fault. W f is chosen to be a diagonal of first order low-pass filters. he minimization of γ results in γ.392, the obtained residuals are displays on figure 3. A second simulation is performed in order to estimate the faults. W f is then chosen an identity matrix. he original and estimated faults are depicted in figure CONCLUSION Considering nonlinear systems represented by S systems, two methods for observer-based residual generator (RG) design are proposed. One is devoted to the systems the premise variables depend on the measured variables such as the input or the output of the system which are available, the other one concerns the systems which the premise variables depend on the unmeasured state variables. Sufficient conditions for the existence of RG were established in the LMI formalism in order to ease RG design. REFERENCES A. Akhenak, M. Chadli, J. Ragot, and D. Maquin. Design of sliding mode unknown input observer for uncertain

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