Unknown Input Observer Design for a Class of Takagi-Sugeno Descriptor Systems

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1 Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 10, HIKARI Ltd, Unknown Input Observer Design for a Class of Takagi-Sugeno Descriptor Systems Karim Bouassem 1,2, Jalal Soulami 1,2,, Abdellatif El Assoudi 1,2 and El Hassane El Yaagoubi 1,2 1 Laboratory of High Energy Physics and Condensed Matter, Faculty of Science Hassan II University of Casablanca, B.P 5366, Maarif, Casablanca, Morocco 2 ECPI, Department of Electrical Engineering, ENSEM Hassan II University of Casablanca, B.P 8118, Oasis, Casablanca Morocco Corresponding author Copyright c 2016 Karim Bouassem et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we address the problem of unknown inputs observer (UIO) design for a class of nonlinear descriptor systems described by Takagi-Sugeno (T-S) structure with measurable premise variables. The unknown inputs affect both state and output of the system. The basic idea of the proposed approach is based on the separation between dynamic and static relations in the T-S descriptor model. First, the method used for separate the differential part from the algebraic part is developed. Secondly, a fuzzy observer design permitting to estimate simultaneously the system state and the unknown inputs is proposed. The developed approach for the observer design is based on the synthesis of an augmented fuzzy model which regroups the differential variables and unknown inputs. The exponential stability of the estimation error is studied by using the Lyapunov theory and the stability conditions are given in terms of LMIs. Finally, numerical simulations using a rolling disc descriptor model are given in order to highlight the performance of the proposed UIO design. Keywords: Descriptor system, Takagi-Sugeno model, unknown input observer, linear matrix inequality

2 478 K. Bouassem, J. Soulami, A. El Assoudi, E. El Yaagoubi 1 Introduction Descriptor systems variously called implicit systems or singular systems or differential-algebraic equations (DAEs) have been widely used in the modeling of dynamic processes to describe the behavior of many chemical and physical processes see for example [1], [2], [3] and [4] and references therein. This formulation includes both dynamic and static relations. The numerical simulation of such descriptor models usually combines an ODE numerical method together with an optimization algorithm. The aim of the paper is the development of an UIO for a class of continuoustime nonlinear descriptor systems described by T-S structure with measurable premise variables. However, the basic idea of the T-S approach is to apprehend the global behavior of a process by a set of locals model [5], [6]. The advantage of such approach relies on the fact that once the T-S fuzzy models are obtained, some analysis and design tools developed in the linear case can be used, which facilitates observer or/and controller synthesis for complex nonlinear systems see for example [7], [8] and the references therein. The UIO design problem widely used in the area of fault detection and design of fault tolerant control strategy has received considerable attention and is still an active area of research. Indeed, many works on fuzzy UIO and its application to fault detection for T-S systems described by ordinary differential equations (ODEs) exist in the literature. We may cite [9], [10], [11], [12], [13], [14], [15], [16], [17]. Likewise, for T-S fuzzy descriptor systems submitted to unknown inputs, several developments exist in the literature. More precisely, various works dealing with UIO design and application to fault diagnosis in explicit form were also proposed for implicit T-S models. These results are based on the singular value decomposition approach and generalized inverse matrix and consider the output matrix without nonlinear terms see for example [18], [19], [20], [21], [22], [23], [24] and many references herein. Notice that, generally an interesting way to solve the various fuzzy UIO problems raised previously is to write the convergence conditions on the LMI form [25]. The main contribution of this paper is to give an UIO design for a class of T-S descriptor models with measurable premise variables allowing the simultaneous estimation of the unknown states and unknown inputs. A new design methodology through judicious use of the separation between the dynamic and static relations in the T-S descriptor model is proposed. Based on the Lyapunov theory, the exponential stability conditions of the UIO are given in terms of linear matrix inequalities (LMIs). Besides, for reasons of ease of the implementation, the main result of this paper consists in showing that the UIO problem for the considered class of T-S descriptor systems can be achieved by using a fuzzy observer having only an ODE structure. This paper is organized as follows. The structure of the considered class of

3 UIO design for a class of T-S descriptor systems 479 nonlinear descriptor systems described by T-S descriptor models with unknown input and measurable premise variables is presented in section 2. The main result concerning the design of the proposed fuzzy observer permitting to estimate unknown states and unknown inputs is established in section 3. In section 4, we illustrate the performance of the proposed UIO in simulation through a rolling disc descriptor model. Finally, some conclusions are drawn in section 5. Throughout the paper, some notations used are fair standard. For example, X > 0 means the matrix X is symmetric and positive definite. X T denotes the transpose of X. The symbol I (or 0) represents the identity matrix (or zero matrix) with appropriate ( dimension. ) ( ) X X Z T µ i µ j = µ i µ j and =. Z Y Z Y i,j=1 2 Takagi-Sugeno descriptor systems In the present work, the aim consist to consider the problem of UIO design for a class of nonlinear descriptor systems described by Takagi-Sugeno (TS) structure with measurable premise variables. For this objective, the following class of nonlinear descriptor systems with unknown inputs is considered: { Eẋ = A(X1 )x + B(X 1 )u + D(X 1 )d y = C(X 1 )x + F (X 1 )d (1) where x = [X T 1 X T 2 ] T R n is the state vector with X 1 R n 1 is the vector of differential variables, X 2 R n 2 is the vector of algebraic variables with n 1 +n 2 = n, u R m is the control input, d R σ is the unknown control input, y R p is the measured output. A(.) R n n, B(.) R n m, C(.) R p n, D(.) R n σ, F (.) R p σ are continuous functions which depend only on the vector of differential variables X 1. E R n n such that rank(e) = n 1 is a real known constant matrix with: E = ( I To design a T-S fuzzy observer, we need a T-S fuzzy model for the nonlinear descriptor systems (1). In general, there are two approaches for constructing fuzzy models: identification (fuzzy modeling) using input-output data and derivation from given nonlinear system equations. In this paper, we use the second approach which derives a fuzzy model from given nonlinear dynamical models (1). ) (2)

4 480 K. Bouassem, J. Soulami, A. El Assoudi, E. El Yaagoubi By the sector nonlinearity approach [7], the nonlinear descriptor system (1) can be exactly represented by the T-S fuzzy descriptor systems: Eẋ = y = µ i (ξ)(a i x + B i u + D i d) µ i (ξ)(c i x + F i d) where A i R n n, B i R n m, C i R p n, D i R n σ, F i R p σ, are real known constant matrices with: ( ) ( ) ( ) A11i A A i = 12i B1i D1i ; B i = ; D i = ; C i = ( ) C 1i C 2i (4) A 21i A 22i B 2i where constant matrices A 22i are supposed invertible. q is the number of submodels. The premise variable ξ which depends here only on X 1 is supposed to be real-time accessible. The µ i (ξ) (i = 1,..., q) are the weighting functions that ensure the transition between the contribution of each sub model: { Eẋ = Ai x + B i u + D i d (5) y = C i x + F i d D 2i (3) They verify the so-called convex sum properties: µ i (ξ) = 1 0 µ i (ξ) 1 i = 1,..., q (6) Before giving the main result, let us make the following assumption [1], [21]: Assumption 2.1 : Suppose that: (E, A i ) is regular, i.e. det(se A i ) 0 s C All sub-models (5) are impulse observable and detectable. In order to investigate the UIO design for T-S descriptor system (3), the approach is based on the separation between difference and algebraic equations in each sub-model (5) and the global fuzzy model is obtained by aggregation of the resulting sub-models. So, using (2) and (4), system (5) can be rewritten as follows: Ẋ 1 = A 11i X 1 + A 12i X 2 + B 1i u + D 1i d 0 = A 21i X 1 + A 22i X 2 + B 2i u + D 2i d y = C 1i X 1 + C 2i X 2 + F i d (7)

5 UIO design for a class of T-S descriptor systems 481 The form (7) for system (5) is also known as the second equivalent form [1]. From (7) and using the fact that A 1 22i exists, the algebraic equations can be solved directly for algebraic variables, to obtain: X 2 = A 1 22iA 21i X 1 A 1 22iB 2i u A 1 22iD 2i d (8) Substitution of the resulting expression for X 2 (equation (8)) in equation (7) yields the following model: Ẋ 1 = M i X 1 + N i u + P i d X 2 = J i X 1 + K i u + L i d y = R i X 1 + S i u + T i d (9) where M i = A 11i A 12i A 1 N i = B 1i A 12i A 1 P i = D 1i A 12i A 1 J i = A 1 22iA 21i K i = A 1 22iB 2i L i = A 1 22iD 2i R i = C 1i C 2i A 1 S i = C 2i A 1 22iB 2i T i = F i C 2i A 1 22iA 21i 22iB 2i 22iD 2i 22iA 21i 22iD 2i (10) In descriptor form, sub system (9) takes the following equivalent form of sub model (5): { Eẋ = Mi x + N i u + P i d y = R i x + S i u + T i d (11) where ( Mi 0 M i = J i I ) ; Ni = ( Ni K i ) ( Pi ; Pi = L i ) ; Ri = ( R i 0 ) (12) Then, fuzzy descriptor system (3) can be rewritten in the following equivalent form: Eẋ = y = µ i (ξ)( M i x + N i u + P i d) µ i (ξ)( R i x + S i u + T i d) (13)

6 482 K. Bouassem, J. Soulami, A. El Assoudi, E. El Yaagoubi So, using (2), (6) and (12), system (13) can be rewritten in the following equivalent system: Ẋ 1 = X 2 = y = µ i (ξ)(m i X 1 + N i u + P i d) µ i (ξ)(j i X 1 + K i u + L i d) µ i (ξ)(r i X 1 + S i u + T i d) (14) Assumption 2.2 : Suppose that d is considered as an unknown control input that may slow variation: with d(0) unknown. d = 0 (15) Let us define the augmented state vector Z 1 = [X T 1 d T ] T and Z 2 = X 2. Thus, the system (14) can be represented as: Ż 1 = Z 2 = y = µ i (ξ)( M i Z 1 + Ñiu) µ i (ξ)( J i Z 1 + K i u) µ i (ξ)( R i Z 1 + S i u) (16) where ( ) Mi P M i = = i 0 0 ) ( Ni Ñ i = = 0 J i = = ( ) J i L i R i = = ( R i T i ) (17) 3 Main result Based on the transformation of the T-S descriptor system (3) into the equivalent form (16), the proposed UIO permitting to estimate simultaneously the

7 UIO design for a class of T-S descriptor systems 483 unmeasurable states and unknown inputs takes the following form: Ẑ 1 = µ i (ξ)( M i Ẑ 1 + Ñiu G i (ŷ y)) Ẑ 2 = ŷ = µ i (ξ)( J i Ẑ 1 + K i u) µ i (ξ)( R i Ẑ 1 + S i u) (18) where (Ẑ1, Ẑ 2 ) and ŷ denote the estimated augmented state vector and the output vector respectively. The activation functions µ i (ξ) are the same than those used in the T-S model (16). G i, i = 1,..., q are the gains of UIO which are determined such that (Ẑ1, Ẑ 2 ) asymptotically converges to (Z 1, Z 2 ). In order to establish the conditions for the asymptotic convergence of the observer (15), we define the state estimation error: ( ) ( ) e1 Ẑ1 Z e = = 1 (19) e 2 Ẑ 2 Z 2 It follows from (16) and (18) that the observer error dynamic is given by the differential-algebraic equation: ė 1 = µ i (ξ)µ j (ξ)γ ij e 1 i,j=1 (20) e 2 = µ i (ξ)q i e 1 where Γ ij = M i G i Rj (21) To prove the convergence of the estimation error e toward zero, it suffices to prove from (20), that e 1 converges toward zero. Then, the following result can be stated. Theorem 3.1 : There exists an UIO (18) for T-S descriptor (3) if given α > 0 there exist matrices Q > 0, W i, i = 1,..., q verifying the following LMIs: { Λ ii + 2αQ < 0 i = 1,..., q (22) Λ ij + Λ ji + 4αQ < 0 i < j s.t. µ i µ j where Λ ij = M T i Q + Q M i R T i W T j W i Rj (23) The fuzzy local observer gains G i, i = 1,..., q are given by: G i = Q 1 W i (24)

8 484 K. Bouassem, J. Soulami, A. El Assoudi, E. El Yaagoubi Proof of Theorem 3.1 : Let us consider the following quadratic Lyapunov function as follows: V (e 1 ) = e T 1 Qe 1, Q = Q T > 0 (25) Estimation error convergence is exponentialy ensured if the following condition is guaranteed: V (e 1 ) = ė T 1 Qe 1 + e 1 Qė 1 < 2αV (e 1 ) α > 0 (26) By using (20), the condition (26) can be written as: V (e 1 ) = i,j=1 µ i (ξ)µ j (ξ)e T 1 (Γ T ijq + QΓ ij )e 1 < 2αV (e 1 ) (27) which is equivalent to the following stability conditions: Γ T iiq + QΓ ii + 2αQ < 0 i = 1,..., q ( Γ ij + Γ ji 2 ) T Q + Q( Γ ij + Γ ji ) + 2αQ < 0 i < j s.t. µ i µ j 2 (28) Letting W i = QG i, from (21) it follows that (28) is equivalent to (22). From the Lypunov stability theory, if the LMI conditions (22) is satisfied, the error dynamic equation (20) is exponentially asymptotically stable. 4 Application to a rolling disc process In order to demonstrate the effectiveness and applicability of the proposed approach of the UIO synthesis (18), let us to consider a rolling disc process described by the descriptor model given in [26] which is supposed to be affected by an unknown input variable as follows: { Eẋ = A(x)x + Bu + Dd (29) y = Cx where x = (x 1, x 2, x 3, x 4 ) T is the state vector with x 1 is the position of the center of the disc, x 2 is the translational velocity of the same point, x 3 is the angular velocity of the disc, x 4 is the contact force between the disc and the surface, u is the applied input force to the disc, y = x 1 is the variable of output measurement and d is the unknown input variable. A(x) = k 1 k 2 x 2 1 b m m 0 1 m 0 1 r 0 k 1 k 2 x 2 1 b m m 0 r 2 J + 1 m, B = r J

9 UIO design for a class of T-S descriptor systems 485 E = , D = , C = ( ) To express the model of the rolling disc process as a T-S model with the measurable variable y = x 1 as decision variable, we consider the sector of nonlinearities of the term ξ = k 1 k 2 x 2 1 [ξ min, ξ max ] of the matrix A(x). m Then, we can transform the nonlinear term under the following shape: where ξ = M 1 ξ max + M 2 ξ min (30) M 1 = M 2 = Hence, the global T-S fuzzy model is inferred as: with A 1 = ξ ξ min ξ max ξ min ξ max ξ (31) ξ max ξ min Eẋ = 2 µ i (ξ)(a i x + Bu + Dd) y = Cx b m 0 1 m, A 0 1 r 0 2 = b m 0 r 2 J + 1 m ξ max ξ max (32) b m 0 1 m 0 1 r 0 b m 0 r 2 J + 1 m ξ min ξ min The weighting functions are given by: { µ1 (ξ) = M 1 µ 2 (ξ) = M 2 (33) Note that, the application of the proposed observer (18) for rolling disc process requires that the above model (32) takes the form (13). To do so, considering the following: X 1 = [x 1 x 2 ] T, X 2 = [x 3 x 4 ] T ( ) (. For i = 1, ) 2: A11i A A i = 12i Ai (1 : 2, 1 : 2) A = i (1 : 2, 3 : 4) B i = B = A 21i A 22i ( B(1 : 2) B(3 : 4) ) A i (3 : 4, 1 : 2) A i (3 : 4, 3 : 4) ( ) D(1 : 2) ; D i = D = ; D(3 : 4) C i = C = ( C(1 : 2, 1 : 2) C(1 : 2, 3 : 4) ).

10 486 K. Bouassem, J. Soulami, A. El Assoudi, E. El Yaagoubi ( ) I 0 E = with rank(e) = This shows that the model (32) is a particular case of the system (13). Hence, fuzzy descriptor system (32) can be rewritten in the following equivalent system: Ẋ 1 = µ i (ξ)(m i X 1 + N i u + P i d) X 2 = µ i (ξ)(j i X 1 + K i u + L i d) y = C 1 X 1 (34) where M i, N i, P i, J i, K i and L i are given in the above equation (10). Let define the augmented state vector Z 1 = [X1 T d] T and Z 2 = X 2. Thus the system (34) can be represented as: Ż 1 = µ i (X 1 )( M i Z 1 + Ñiu) Z 2 = µ i (X 1 )( J i Z 1 + K i u) y = C 1 Z 1 (35) where M i, Ñ i, Ji can be calculated using equation (17) and C 1 = ( C 1 0 ). The expression of control variable and values of physical parameters are from [27]. By Theorem 3.1, considering α = 2 the following observer gains G 1, G 2 are obtained: G 1 = , G 2 = The expression of unknown input signal is defined as (see Figure 3): d(t) = { 1 10 t 20 0 otherwise (36) Simulation results with initial conditions: Z 1 (0) = [ ] T, Z 2 (0) = [ ] T Ẑ 1 (0) = [ ] T, Ẑ 2 (0) = [ ] T are given in Figures 1, 2 and 3. These simulation results show the performances of the proposed UIO (18) with the gains G 1, G 2 where the dashed lines denote the state variables and unknown input estimated by the fuzzy observer. They show that the observer

11 UIO design for a class of T-S descriptor systems 487 gives a good estimation of unknown states and unknown input of the rolling disc process. Figure 1: State variables x 1, x 2 and their estimates

12 488 K. Bouassem, J. Soulami, A. El Assoudi, E. El Yaagoubi Figure 2: State variables x 3, x 4 and their estimates

13 UIO design for a class of T-S descriptor systems 489 Figure 3: Unknown input d and its estimate 5 Conclusion A new fuzzy observer design approach for a class of nonlinear descriptor systems described by T-S descriptor models with unknown input and measurable premise variables is proposed in this paper. The main idea of the present work is based on the separation between dynamic and static relations in the T-S descriptor model. The exponential convergence of the state estimation error is studied by using the Lyapunov theory and the stability conditions are given in terms of LMIs. Simulation results are given and demonstrate the good performance of the proposed UIO design. References [1] L. Dai, Singular Control Systems, Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, [2] K. E. Brenan, S. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, SIAM, New York,

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