Fuzzy Observer Design for a Class of Takagi-Sugeno Descriptor Systems Subject to Unknown Inputs

Transcription

1 Nonlinear Analysis and Differential Equations, Vol. 5, 217, no. 3, HIKARI Ltd, Fuzzy Observer Design for a Class of Takagi-Sugeno Descriptor Systems Subject to Unknown Inputs 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 216 Karim Bouassem, Jalal Soulami, Abdellatif El Assoudi and El Hassane El Yaagoubi. 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 [1], the authors proposed a study on the design of Unknown Inputs Observer (UIO for a class of Takagi-Sugeno descriptors systems (TSDS with measurable premise variables. The aim of this paper is to extend this study to a class of TSDS when the premise variables are not measurables. The approach is based on the separation between dynamic and static relations in the Takagi-Sugeno (T-S descriptor model. First, the method used to separate the differential part from the algebraic part is developed. Secondly, we give an observer design permitting simultaneous estimate of the system state and the unknown inputs (UI. Based on the Lyapunov theory and L 2 -gain technique, the exponential stability conditions of this observer synthesis are given in terms of Linear Matrix inequalities (LMIs. Finally, a numerical example is introduced to validate the performance of the proposed UIO design. Keywords: Takagi-Sugeno descriptor systems, unmeasurable premise variables, unknown inputs observer, linear matrix inequality

2 118 K. Bouassem, J. Soulami, A. El Assoudi and E. El Yaagoubi 1 Introduction It is well known that the monitoring and diagnosis of industrial processes have taken a major importance in the last two decades. Indeed, the constant requirement in terms of processes reliability and performance has led to the use of highly nonlinear models to describe the behavior of processes. Consequently, the resulting models are very complex and the fault diagnosis task based on the model becomes more difficult to achieve. In this work, the aim is to propose an UIO design for a class of TSDS with unmeasurable premise variables where the UI affect both state and output of the system. Notice that, the basic idea of the T-S approach is to apprehend the global behavior of a process by a set of local models [2], [3]. 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 [4], [5] and the references therein. On the other hand, descriptor systems variously called implicit systems or singular systems or differential-algebraic equations (DAEs have been widely used in the dynamic processes modeling to describe the behavior of many chemical and physical processes see for example [6], [7], [8] and [9] 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. Moreover, the UIO design problem widely used in the area of fault detection and design of fault tolerant control strategy has received considerable attention in the literature and is still an active area of research. The recourse to the use of an UIO is necessary in order to be able to estimate the unknown state of the process. The basic idea of the UIO approach consists in associating to each local model a local UIO. The fuzzy observer design is then a convex interpolation of these local observers. 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 [1], [11], [12], [13], [14], [15], [16], [17]. Likewise, for TSDS submitted to UI, 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], [2], [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 work is to propose an UIO design for a class

3 Fuzzy observer design for a class of TSDS subject to UI 119 of TSDS with unmeasurable premise variables allowing the simultaneous estimation of the unknown states and UI. The method is based on the separation between the differential and algebraic equations in the T-S descriptor model. Based on the Lyapunov theory and L 2 -gain technique, the exponential stability conditions of the proposed observer are given in terms of LMIs. Besides, for ease of implementation reasons, the main result of this paper consists in showing that the UIO problem for the considered class of TSDS can be achieved by using a fuzzy observer having only an ODE structure. The rest of this work is organized as follows. The structure of the class of TSDS with unmeasurable premise variables subject to unknown inputs is introduced in section 2. The main result about the UIO design is presented in section 3. Simulation results to show the good estimation of both state and UI are given in section 4. Finally, a conclusion is given in section 5. In this paper, some notations used are fair standard. For example, X > means the matrix X is symmetric and positive definite. X T denotes the transpose of X. The symbol I (or 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 System description In this paper, the following class of T-S descriptor systems with unmeasurable decision variables in presence of unknown inputs is considered: Eẋ = µ i (x(a i x + B i u + D i d (1 y = µ i (x(c i x + F i d where x = [X1 T X2 T ] 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 r is the unknown control input, y R p is the measured output. E R n n with rank(e = n 1, A i R n n, B i R n m, C i R p n, D i R n r, F i R p r, are real known constant matrices with: ( ( ( I A11i A E = ; A i = 12i B1i ; B i = (2 A 21i A 22i B 2i C i = ( C 1i C 2i ; Di = ( D1i D 2i (3

4 12 K. Bouassem, J. Soulami, A. El Assoudi and E. El Yaagoubi where constant matrices A 22i are supposed invertible. q is the number of sub-models. The µ i (x are the weighting functions that ensure the transition between the contribution of each sub model: { Eẋ = Ai x + B i u + D i d y = C i x + F i d (4 They verify the so-called convex sum properties: µ i (x = 1 µ i (x 1 i = 1,..., q (5 Before giving the main result, let us make the following assumption [6], [21]: Assumption 2.1 : Suppose that: (E, A i is regular, i.e. det(se A i s C All sub-models (4 are impulse observable and detectable. In order to investigate the UIO design for T-S descriptor system (1, the approach is based on the separation between differential and algebraic equations in each sub-model (4 and the global fuzzy model is obtained by aggregation of the resulting sub-models. So, using (2 and (3, system (4 can be rewritten as follows: Ẋ 1 = A 11i X 1 + A 12i X 2 + B 1i u + D 1i d = 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 (6 The form (6 for system (4 is also known as the second equivalent form [6]. From (6 and using the fact that A 1 22i exist, 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 (7 Substitution of the resulting expression for X 2 (equation (7 in equation (6 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 (8

5 Fuzzy observer design for a class of TSDS subject to UI 121 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 (9 The weighting functions µ i (x, i = 1,..., q can be rewritten as µ i (x = µ i (X 1, X 2 = J i X 1 + K i u + L i d = µ i (X 1, u, d = µ i (ξ (1 with ξ = [X T 1 u T d T ] T. Then, fuzzy descriptor system (1 can be rewritten in the following equivalent form: 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 (11 Assumption 2.2 : Suppose that d is considered as an unknown control input that may slow variation: with d( unknown. d = (12 Let define the augmented state vector Z 1 = [X T 1 d T ] T and Z 2 = X 2. Thus the system (11 can be represented as: Ż 1 = Z 2 = y = µ i (η( M i Z 1 + Ñiu µ i (η( J i Z 1 + K i u µ i (η( R i Z 1 + S i u (13

6 122 K. Bouassem, J. Soulami, A. El Assoudi and E. El Yaagoubi where η = [Z1 T ( u T ] T Mi P M i = = i ( Ni Ñ i = = J i = = ( J i L i R i = = ( R i T i (14 3 Main result Based on the transformation of the T-S descriptor system (1 into the equivalent form (13, the proposed UIO permitting the estimate of unmeasurable state and unknown inputs takes the following form: Ẑ 1 = Ẑ 2 = ŷ = µ i (ˆη( M i Ẑ 1 + Ñiu G i (ŷ y µ i (ˆη( J i Ẑ 1 + K i u µ i (ˆη( R i Ẑ 1 + S i u (15 where (Ẑ1, Ẑ 2, ŷ and ˆη denote the estimated augmented state vector, the output vector and the decision variable vector respectively. G i, i = 1,..., q are the observer gains witch are determined such that ( ˆX 1, ˆX2, ˆd asymptotically converges to (X 1, X 2, d. In order to establish the conditions for the asymptotic convergence of the observer (15, we define the state estimation error: e = ( e1 e 2 = ( Ẑ1 Z 1 Ẑ 2 Z 2 (16 It follows from (8 and (15 that the dynamics of state estimation error e is given by the differential and algebraic equations: ė 1 = e 2 = µ i (ˆη( M i Ẑ 1 + Ñiu G i (ŷ y µ i (η( M i Z 1 + Ñiu µ i (ˆη( J i Ẑ 1 + K i u µ i (η( J i Z 1 + K i u (17

7 Fuzzy observer design for a class of TSDS subject to UI 123 which are equivalent to the following equations: ė 1 = e 2 = µ i (ˆη( M i e 1 G i (ŷ y (µ i (η µ i (ˆη( M i Z 1 + Ñiu µ i (ˆη J i e 1 (µ i (η µ i (ˆη( J i Z 1 + K i u (18 Let Λ i = M i, Ñ i, Ji, K i, and using the fact that: (µ i (η µ i (ˆηΛ i = i,j=1 µ i (ηµ j (ˆη Λ ij (19 where Λ ij = Λ i Λ j. Hence, the equation (18 becomes: ė 1 = e 2 = µ i (ˆη( M i e 1 G i (ŷ y µ i (ηµ j (ˆη( M ij Z 1 + Ñiju i,j=1 µ i (ˆη J i e 1 µ i (ηµ j (ˆη( J ij Z 1 + K ij u i,j=1 (2 Since ė 1 = e 2 = µ i (η = 1, equality (2 can be written as follows: µ i (ηµ j (ˆη( M j e 1 G j (ŷ y µ i (ηµ j (ˆη( M ij Z 1 + Ñiju i,j=1 i,j=1 µ i (ηµ j (ˆη( J j e 1 J ij Z 1 K ij u i,j=1 (21 Similarly, y and ŷ can be written as follows: y = µ i (ηµ k (ˆη[( R k + R ik Z 1 + (S k + S ik u] ŷ = i,k=1 i,k=1 µ i (ηµ k (ˆη( R k Ẑ 1 + S k u where R ik = R i R k and S ik = S i S k. By substituting (22 in (21, we obtain: ė 1 = e 2 = i,j,k=1 i,j=1 µ i (ηµ j (ˆηµ k (ˆη(Γ jk e 1 + Φ ijk Z 1 + Ω ijk u µ i (ηµ j (ˆη( J j e 1 J ij Z 1 K ij u (22 (23

8 124 K. Bouassem, J. Soulami, A. El Assoudi and E. El Yaagoubi where Γ jk = M j G j Rk Φ ijk = G j R ik M ij Ω ijk = G j S ik Ñij i, j, k {1,..., q} (24 Note that to prove the convergence of the estimation error e toward zero, it suffice to prove that e 1 converges to zero. Thus, let Z 1 = ( T e T 1 Z1 T, we have: Z 1 = µ i (ηµ j (ˆηµ k (ˆη(A ijk Z1 + B ijk u i,j,k=1 (25 e 1 = C Z 1 where A ijk = B ijk = ( Γjk Φ ijk Mi ( Ωijk Ñ i C = (I (26 Thus the aim is to determine the observer gains G i (i = 1,..., q to ensure the stability of (25 while attenuating the effect of the input u on e 1. Therefore, the convergence condition of (15 can be formulated by the following theorem. Theorem 3.1 : Under above Assumptions 1 and 2, the state estimation error between the T-S descriptor model (1 and its UIO (15 converges asymptotically towards zero, if given λ > there exist matrices Q 1 > and Q 2 >, W i, i = 1,..., q and a positive scalar β, such that the following LMIs hold: Π ijk = Π 11 Π 21 Π 22 Π 31 Π 32 Π 33 < (i, j, k {1,..., q} 3 (27 where Π 11 = M T j Q 1 + Q 1 Mj W j Rk R T k W T j + I + 2λQ 1 Π 21 = R T ikw T j M T ij Q 1 Π 22 = M T i Q 2 + Q 2 Mi + 2λQ 2 Π 31 = S T ikw T j Ñ T ij Q 1 Π 32 = Ñ T i Q 2 Π 33 = βi (28

9 Fuzzy observer design for a class of TSDS subject to UI 125 The observer gains in (15 are derived from: and the attenuation level is: G i = Q 1 1 W i (29 α = β (3 Proof of Theorem 3.1 : Considering the following quadratic Lyapunov function: with V = Z T 1 Q Z 1, Q > (31 Q = ( Q1 Q 2 The time derivative of V along the trajectory of (25 is given by: V = µ i (ηµ j (ˆηµ k (ˆη( Z 1 T (A T ijkq + QA ijk Z 1 + Z 1 T QB ijk u + u T BijkQ T Z 1 (33 i,j,k=1 (32 In order to ensure the stability of (25 and the boundedness of the transfer from u to e 1 : e 1 2 u 2 < α, u 2 (34 we consider the following criterion: V + e T 1 e 1 α 2 u T u < (35 Estimation error convergence is exponentially ensured if the following condition is guaranteed: V + e T 1 e 1 α 2 u T u < 2λV λ > (36 From (25 and (33, inequality (36 becomes: µ i (ηµ j (ˆηµ k (ˆη ( ZT 1 u ( Z1 T Π ijk u where i,j,k=1 Π ijk = The inequality (37 is satisfied if: ( A T ijk P + QA ijk + C T C + 2λQ QB ijk BijkQ T α 2 I < (37 (38 Π ijk < i, j, k {1,..., q} (39 Then, from (24, (26, (32 and the use of the changes of variables: { Wi = Q 1 G i β = α 2 (4 we establish the LMI conditions (27. The proof is completed.

10 126 K. Bouassem, J. Soulami, A. El Assoudi and E. El Yaagoubi 4 Numerical illustration To demonstrate the effectiveness and applicability of the proposed approach of the unknown input observer synthesis, we consider the following T-S descriptor model subject to unknown input and unmeasurable premise variable: Eẋ = 2 µ i (x(a i x + Bu + Dd y = Cx (41 where x R 6, u R, d R and y R 3 are the state vector, known input, unknown input and output, respectively. The matrices numerical values are: A 1 = A 2 = E = , C =, B =, D = The membership functions are: µ 1 (x = 1 sin(x 2 and µ 2 (x = sin(x 2. x 2 x 2 Notice that, to apply the proposed UIO (21 for (41, it suffices to rewrite the model (41 in its equivalent form (2 as mentioned in section 2. Thus, by Theorem 3.1 with η =.3 the following observer gain H is obtained: G 1 = , G 2 =

11 Fuzzy observer design for a class of TSDS subject to UI 127 Simulation results with initial conditions: Z 1 ( = [.31 ] T, Z 2 ( = [ ] T Ẑ 1 ( = [.35 1] T, Ẑ 2 ( = [ ] T are given in Figures 1, 2, 3 and 4 where the input u = sin(t and the expression of unknown input signal is defined as in Figure 4. These simulation results show the performances of the proposed UIO (21 with the gain H where the dashed lines denote the state variables and unknown input estimated by the fuzzy observer. They show that the observer gives a good estimation of unknown states and unknown input of the considered fuzzy T-S descriptor model.

12 128 K. Bouassem, J. Soulami, A. El Assoudi and E. El Yaagoubi Figure 1: State variables x 1, x 2 and their estimates

13 Fuzzy observer design for a class of TSDS subject to UI 129 Figure 2: State variables x 3, x 4 and their estimates

14 13 K. Bouassem, J. Soulami, A. El Assoudi and E. El Yaagoubi Figure 3: State variables x 5, x 6 and their estimates

15 Fuzzy observer design for a class of TSDS subject to UI 131 Figure 4: Unknown input d and its estimate 5 Conclusion In this work, we propose a new fuzzy UIO design method based on explicit structure in order to estimate the state and unknown inputs for a class of TSDS with unmeasurable premise variables. The main result is based on the separation between dynamic and static equations in the TSDS. The developed approach for the UIO design is based on the synthesis of the augmented fuzzy models which regroups the differential variables and unknown inputs. The exponential convergence of the state estimation error is studied using the Lyapunov theory and L 2 -gain technique and the existence of conditions ensuring this convergence is expressed in term of LMIs. To show the good performance of the proposed fuzzy observer, a T-S descriptor model subject to unknown input variable is considered. The effectiveness of the proposed method for the on-line simultaneous estimation of unknown states and unknown input of the used fuzzy descriptor model is verified by numerical simulation.

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