State Observer Design for a Class of Takagi-Sugeno Discrete-Time Descriptor Systems

Transcription

1 Applied Mathematical Sciences, Vol. 9, 2015, no. 118, HIKARI Ltd, State Observer Design for a Class of Takagi-Sugeno Discrete-Time Descriptor Systems Ilham Hmaiddouch ECPI, Departement GE, ENSEM University Hassan II of Casablanca B.P 8118,Oasis, Casablanca Morocco Boutayna Bentahra ECPI, Departement GE, ENSEM University Hassan II of Casablanca B.P 8118,Oasis, Casablanca Morocco Abdellatif El Assoudi ECPI, Departement GE, ENSEM University Hassan II of Casablanca B.P 8118,Oasis, Casablanca Morocco Jalal Soulami ECPI, Departement GE, ENSEM University Hassan II of Casablanca B.P 8118,Oasis, Casablanca Morocco El Hassane El Yaagoubi ECPI, Departement GE, ENSEM University Hassan II of Casablanca B.P 8118,Oasis, Casablanca Morocco Copyright c 2015 Ilham Hmaiddouch 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.

2 5872 I. Hmaiddouch, B. Bentahra, A. El Assoudi, J. Soulami and E. El Yaagoubi Abstract This paper deals with the design of explicit observers for a class of discrete-time dynamical implicit systems described by Takagi-Sugeno TS model in the two cases where the premise variable are measurable and the premise variables are unmeasurable. The idea of the proposed approach is based on the singular value decomposition. The convergence of the state estimation error is studied using the Lyapunov theory and the stability conditions are given in terms of Linear Matrix Inequalities LMIs. Finally, an example is given to illustrate the proposed approach. Keywords: Takagi-Sugeno model, discrete-time system, descriptor system, fuzzy observer, linear matrix inequality LMI, singular value decomposition 1 Introduction It is well known that descriptor systems variously called singular systems, implicit systems, or differential algebraic equations have been receiving a great deal of attention for many decades as a representation of dynamical systems [1], [2], [3]. This formulation includes both dynamic and static relations. Consequently this formalism is much more general than the usual one and can model physical constraints or impulsive behavior due to an improper part of the system. Note that many physical systems are naturally modeled as descriptor systems such as chemical, electrical and mechanical systems. The numerical simulation of such descriptor models usually combines an ODE numerical method together with an optimization algorithm. Over the two last decades, the nonlinear observer synthesis and its application for dynamical systems described by T-S fuzzy models [4], [5] has received a great deal of attention. For continuous and discrete-time nonlinear systems, many results have been reported in observer design [6], [7], [8], [9], [10]. Concerning nonlinear descriptor systems, several research work concerning the problem of observer design and applications exist in the literature see for instance [11], [12], [13], [14], [15], [16], [17], [18], [19]. Based on the singular value decomposition approach, the aim of this paper is to give a fuzzy observer design to a class of T-S discrete-time descriptor systems in the two cases where the premise variable are measurable and the premise variables are unmeasurable permitting to estimate the unknown state without the use of an optimization algorithm. The remainder of the paper is structured as follows. The class of studied systems is defined in section 2 and the main result about fuzzy observer design for T-S discrete-time descriptor systems in the two cases where the premise variable are measurable and the premise variables are unmeasurable is exposed

3 State observer design for a class of T-S discrete-time descriptor systems 5873 in section 3. Section 4 is devoted to a numerical example to demonstrate the validity of our results. 2 Problem statement In this paper, the class of T-S discrete-time descriptor systems that we consider is in the following form: Ex k+1 = µ i ξ k A i x k + B i u k 1 y k = Cx k where x k R n is the state variable, u k R m is the control input, y k R p is the measured output. E R n n is constant matrix with ranke = r. A i R n n, B i R n m, C R p n are real known constant matrices. ξ k represent the premise variable. The µ i ξ k are the weighting functions that ensure the transition between the contribution of each sub model: { Exk+1 = A i x k + B i u k y k = Cx k 2 They depend on measurable or unmeasurable premise variables state of the system, and have the following properties: 0 µ i ξ k 1 µ i ξ k = 1 Then, before giving the main results, let us make the following hypotheses for each sub-model 2, i = 1,..., q: H1 E, A i is regular, i.e. detze A i 0 z C H2 All sub-models are impulse observable, i.e. E A i rank 0 E = n + ranke 0 C 3 H3 All sub-models are detectable, i.e. rank ze Ai C = n z C

4 5874 I. Hmaiddouch, B. Bentahra, A. El Assoudi, J. Soulami and E. El Yaagoubi E H4 rank C = n. Note that under hypothesis H4, there exist a non-singular matrix such that: a b c d { ae + bc = I ce + dc = Fuzzy observer design Based on the singular value decomposition, our aim in this section is to design an explicit fuzzy observer for T-S descriptor system 1 in the two cases where the premise variable are measurable and the premise variables are unmeasurable. 3.1 Case 1: Measurable premise variables In this subsection, the design of fuzzy observer for T-S descriptor system 1 with measurable premise variables is addressed. The proposed observer is in an explicit form, and is defined by the following equations: z k+1 = µ i ξ k N i z k + L 1i y k + L 2i y k + G i u k 5 ˆx k = z k + by k + Kdy k where ˆx k denote the estimated state vector, N i, L 1i, L 2i, G i and K are unknown matrices of appropriate dimensions, which must be determined such that ˆxt will asymptotically converge to xt. Denoting the state estimation error by: e k = x k ˆx k 6 Then by substituting 1, 4 and 5 into 6 we obtain: e k = a + KcEx k z k 7 It follows from 1 and 5 that the dynamic of this observer error is: e k+1 = µ i ξ k a + KcA i x k + B i u k µ i ξ k N i z k + L 1i y k + L 2i y k + G i u k 8

5 State observer design for a class of T-S discrete-time descriptor systems 5875 Using 7, equation 8 can be written as: e k+1 = µ i ξ k a + KcA i x k + B i u k + µ i ξ k N i e k µ i ξ k N i a + KcE + L 1i C + L 2i Cx k + G i u k Provided the matrices G i, K, L 1i, L 2i and N i satisfy: 9 N i a + KcE + L 1i C + L 2i C = a + KcA i 10 G i = a + KcB i 11 Then, from 4 and 10, we have: N i = a + KcA i L 2i C + N i b + Kd L 1i C 12 Take: L 1i = N i b + Kd 13 Then: N i = a + KcA i L 2i C 14 It follows system 9 is equivalent to: e k+1 = µ i ξ k N i e k 15 Theorem 3.1 : There exists an observer 5 for 1 if the hypotheses H1, H2, H3 and H4 hold and there exists symmetric positive matrices P, Q and W i for i = 1,..., q, verifying the following LMI: P P aa i + QcA i W i C T < 0 i = 1,..., q 16 P aa i + QcA i W i C P The observer gains N i, L 1i, L 2i, G i and K are given by: N i = a + P 1 QcA i P 1 W i C L 1i = a + P 1 QcA i P 1 W i Cb + P 1 Qd L 2i = P 1 W i G i = a + P 1 QcB i K = P 1 Q 17 where a, b, c and d are such that equation 4 is satisfied.

6 5876 I. Hmaiddouch, B. Bentahra, A. El Assoudi, J. Soulami and E. El Yaagoubi Proof of theorem 3.1 : To prove the convergence of the estimation error toward zero, let us consider the following quadratic Lyapunov function: V k = e T k P e k, P = P T > 0 18 Estimation error convergence is ensured if the following condition is guaranteed: V = V k+1 V k < 0 19 The variation of V k along the trajectory of 15 is given by: By using 15, 20 can be written as: V = The negativity of V is guaranteed if: V = e T k+1p e k+1 e T k P e k 20 µ i ξ k e T k [Ni T P N i P ]e k 21 N T i P N i P < 0 i {1,..., q} 22 By using 14, 22 can be written as: aa i + KcA i L 2i C T P aa i + KcA i L 2i C P < 0 i {1,..., q} 23 Thus, the LMI conditions of Theorem 3.1 can be obtained by using the Schur complement [20] and the following change of variables: { Q = P K W i = P L 2i 24 This completes the proof of theorem Case 2: Unmeasurable premise variables In this section, our aim is to design an explicit fuzzy observer for nonlinear descriptor system 1 with unmeasurable decision variables. The proposed observer is in the following form: z k+1 = µ i ˆξ k N i z k + L 1i y k + L 2i y k + G i u k 25 ˆx k = z k + by k + Kdy k where ˆx k denote the estimated state vector, N i, L 1i, L 2i, G i and K are unknown matrices of appropriate dimensions, which must be determined such that ˆx k

7 State observer design for a class of T-S discrete-time descriptor systems 5877 will asymptotically converge to x k. Let e k = x k ˆx k, then as in above subsection 3.1 see equations 6 to 14, the error dynamics is given by: e k+1 = Note that: µ i ˆξ k N i e k + µ i ξ k µ i ˆξ k a + KcA i x k + B i u k 26 µ i ξ k µ i ˆξ k A i = µ i ξ k µ i ˆξ k B i = Then, the equation 26 becomes: µ i ξ k µ j ˆξ k A i A j µ i ξ k µ j ˆξ k B i B j 27 e k+1 = µ i ˆξ k N i e k + µ i ξ k µ j ˆξ k a + Kc A ij x k + B ij u k 28 where A ij = A i A j and B ij = B i B j. Multiplying by µ i ξ k, equation 28 can be reduces to the equation: e k+1 = µ i ξ k µ j ˆξ k N j e k + Φ ij x k + Γ ij u k 29 where Φ ij = a + Kc A ij Γ ij = a + Kc B ij i, j {1,..., q} 30 Let us define the augmented state ē k = [e T k x T k ] T, we have: Ēē k+1 = µ i ξ k µ j ˆξ k A ij ē k + B ij u k 31 where Ē = A ij = B ij = I 0 0 E Nj Φ ij 0 A i Γij B i 32

8 5878 I. Hmaiddouch, B. Bentahra, A. El Assoudi, J. Soulami and E. El Yaagoubi Theorem 3.2 : There exists an observer 25 for 1 if the hypotheses H1, H2, H3 and H4 hold and there exists symmetric positive matrices P 1, P 2, matrices Q, R, W j, Z j and S j for j = 1,..., q, verifying the following LMI: S ij = m 11 m T 21 m T 31 m 21 m 22 m T 32 m 31 m 32 m 33 < 0 i, j {1,..., q} 33 where: m 11 = A T j a T P 1 aa j + A T j a T QcA j + A T j c T Q T aa j A T j a T W j C P 1 C T Wj T aa j + A T j c T RcA j A T j c T Sj T C C T S j ca j + C T Z j C m 22 = A T ija T P 1 a A ij + A T ija T Qc A ij + A T ijc T Q T a A ij + A T ijc T Rc A ij + A T i P 2 A i E T P 2 E m 33 = Bija T T P 1 a B ij + Bija T T Qc B ij + Bijc T T Q T a B ij + Bijc T T Rc B ij + Bi T P 2 B i m 21 = A T ija T P 1 aa j + A T ija T QcA j A T ija T W j C + A T ijc T Q T aa j + A T ijc T RcA j A T ijc T Sj T C m 31 = Bija T T P 1 aa j + Bija T T QcA j + Bijc T T Q T aa j Bija T T W j C + Bijc T T RcA j Bijc T T Sj T C m 32 = Bija T T P 1 a A ij + Bija T T Qc A ij + Bijc T T Q T a A ij + Bijc T T Rc A ij + Bi T P 2 A i 34 with a, b, c and d are such that equation 4 is satisfied. The observer gains N j, L 1j, L 2j, G j and K are given such that equations 11, 13, 14 and 42 are satisfied. Proof of theorem 3.2 : To prove the convergence of the estimation error toward zero, let us consider the following quadratic Lyapunov function: V k = ē T k ĒT P Ēē k, Ē T P Ē 0, P = P T > 0 35 with P = P1 0 0 P 2 36 Its difference V = V k+1 V k along the error dynamics 31 is given by: V = ē T k+1ēt P Ēē k+1 ē T k ĒT P Ēē k 37 By using 31, 37 can be written as: V = µ i ξ k µ j ˆξ k A ij ē k + B ij u k T P A ij ē k + B ij u k ē T k ĒT P Ēē k 38

9 State observer design for a class of T-S discrete-time descriptor systems 5879 Multiplying by V = µ i ξ k µ j ˆξ k, equation 38 can be reduces to the equation: µ i ξ k µ j ˆξ k [ ē T k u T k ] Sij [ ēk u k ] 39 where S ij = A T ij P A ij ĒT P Ē AT ijp B ij BijP T A ij BijP T B ij i, j {1,..., q} 40 The negativity of V is guaranteed if: S ij < 0 i, j {1,..., q} 41 Then, the use of the changes of variables: Q = P 1 K R = K T P 1 K W j = P 1 L 2j Z j = L T 2jP 1 L 2j S j = L T 2jP 1 K 42 and from 14, 30, 32 and 36 we establish the LMIs conditions given in 33 in the theorem 3.2. This completes the proof of theorem Numerical example In this section, we consider the following T-S discrete-time descriptor model with unmeasurable premise variables to illustrate the efficiency of the proposed fuzzy observer given by system 25: where A 1 = Ex k+1 = 4 h j x k A j x k + Bu k j=1 43 y k = Cx k , A 2 =

10 5880 I. Hmaiddouch, B. Bentahra, A. El Assoudi, J. Soulami and E. El Yaagoubi A 3 = E = , B = The membership functions are given by:, A 4 = , C = h 1 x k = 1 x2 4k 1x 2 4k h 2 x k = 1 x2 4k 11 + x 2 4k 5 16 h 3 x k = x2 4kx 2 4k h 4 x k = x2 4k1 + x 2 4k 5 16 A fuzzy observer for system 43 permitting to estimate the unknown states x 1k, x 2k, x 3k and x 4k can be designed using theorem 3.2. It takes the following form: 4 z k+1 = µ i ˆx k N i z k + L 1i y k + L 2i y k + G i u k 44 ˆx k = z k + by k + Kdy k where b and d satisfying the equation 4 are as follows: b = , d = In order to illustrate the performances of the fuzzy observer 44, we solve the LMIs given in the theorem 3.2. The observer gains N j, L 1j, L 2j, G j and K are given by: N 1 = N 2 =

11 State observer design for a class of T-S discrete-time descriptor systems 5881 L 11 = N 3 = N 4 = L 13 = L 21 = L 23 = G 1 = G 4 =, L 12 =, L 14 =, L 22 =, G 2 = , L 24 = , K = The initial conditions of the T-S model 43 are: x 0 = [ ] T , G 3 = The initial conditions of the fuzzy observer 44 are: ˆx 0 = [ ] T

12 5882 I. Hmaiddouch, B. Bentahra, A. El Assoudi, J. Soulami and E. El Yaagoubi The simulation results are given in figure 1 where the dotted lines denote the state variables estimated by the fuzzy observer 44. This simulation shows that the estimation states converge to their corresponding state variables. Figure 1 : : T-S model,... : Fuzzy observer

13 State observer design for a class of T-S discrete-time descriptor systems Conclusion Based on the singular value decomposition method and solving a system of LMIs for the determination of the observer parameters, two state observers design for a class of T-S discrete-time descriptor systems with measurable and unmeasurable premise variables are proposed in this paper. This work is an extension of the result developed in [15] and [19] which extend the method of the observer developed for linear systems in [21] to nonlinear T-S systems. To illustrate the proposed methodology with unmeasurable premise variables, a numerical example of a T-S discrete-time descriptor model is considered. The effectiveness of the proposed fuzzy observer used for the on-line estimation of unknown states in proposed model is verified by numerical simulation. References [1] S. L. Campbell, Singular Systems of Differential Equations, Pitman, London, UK, [2] F. L. Lewis, A survey of linear singular systems, Circuits, Systems and Signal Processing, , no. 1, [3] L. Dai, Singular Control Systems, Springer, Berlin, Germany, [4] T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE Trans. Syst., Man and Cyber., SCM , [5] T. Taniguchi, K. Tanaka, H. Ohtake and H. Wang, Model construction, rule reduction, and robust compensation for generalized form of Takagi- Sugeno fuzzy systems, IEEE Transactions on Fuzzy Systems, , no. 4, [6] P. Bergsten, R. Palm, and D. Driankov, Fuzzy observers, IEEE International Conference on Fuzzy Systems, Melbourne Australia, [7] K. Tanaka, and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, John Wiley & Sons, [8] D. Ichalal, B. Marx, J. Ragot, and D. Maquin, Design of observers for Takagi-Sugeno systems with immeasurable premise variables: an

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