Design of State Observer for a Class of Non linear Singular Systems Described by Takagi-Sugeno Model

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1 Contemporary Engineering Sciences, Vol. 6, 213, no. 3, HIKARI Ltd, Design of State Observer for a Class of Non linear Singular Systems Described by Takagi-Sugeno Model Mohamed Essabre ECPI, Departement GE, ENSEM Univ. Hassan II Casablanca B.P 8118,Oasis, Casablanca Morocco Jalal Soulami ECPI, Departement GE, ENSEM Univ. Hassan II Casablanca B.P 8118,Oasis, Casablanca Morocco jalal.soulami@gmail.com Elhassan Elyaagoubi ECPI, Departement GE, ENSEM Univ. Hassan II Casablanca B.P 8118,Oasis, Casablanca Morocco Copyright c 213 Mohamed Essabre et al. This is an open access article 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 propose a new method to design an observer for a class of non linear singular systems described by Takagi-Sugeno TS model, with measurable decision variables. 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: observer, differential-algebraic system, descriptor system, Takagi- Sugeno model, linear matrix inequality LMI

2 1 M. Essabre, J. Soulami and E. Elyaagoubi 1 Introduction All recently there has been a great deal of interest in using dynamic Takagi- Sugeno fuzzy models [11] to approximate nonlinear systems. This interest relies on the fact that once the Takagi-Sugeno fuzzy models are obtained, linear control methodology can be used to design local state feedback controllers for each linear model. Aggregation of the fuzzy rules results in a generally nonlinear model, but in a very special form, which is exactly the same as a time varying and nonlinear system described by a set of Polytopic Linear Inclusions. The Takagi-Sugeno model has been generalized to Singular systems. Many physical systems are naturally modeled as systems of differential and algebraic equations DAE such as chemical, electrical and mechanical engineering systems [8], [4], [5]. These systems are variously called descriptor systems, singular systems, implicit, or differential algebraic equations DAEs. 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. Many control issues have been extended to the descriptor case, in particular the observer design for descriptor systems has been intensively addressed see e.g. [6], [9], [1], a linear fractional transformation parameterization of linear observers is done in [12] and [3] introduces the proportional integral PI observer. This paper presents a method for state-estimation of Takagi-Sugeno descriptor systems TSDS with measurable premise variables. Under some sufficient conditions, the design of the observer is reduced to the determination of some matrices. The choice of these matrices is performed by solving strict LMIs. The proposed observer design method is based on the use of the second method of Lyapunov and a quadratic function. The paper is organized as follows: the class of studied systems is defined in section II and the main results about observers for Takagi-sugeno fuzzy descriptor systems with measurable premise variables design are detailed in section III. Section IV is devoted to a numerical example. 2 Takagi-sugeno descriptor systems In this paper, we consider the following general form of singular nonlinear systems: { Eẋt = fxt + gxtut yt = Cxt 1

3 Observer for a class of non linear singular systems 11 where x R n is the state variable, u R m is the control input, y R p is the measured output. f and g are nonlinear functions. E R n n is constant matrix with ranke = r. To obtain a Takagi-Sugeno model of the system 1 we use the procedure of fuzzy model construction given in [1]. The multiple model structure is given by: Eẋt = h i xta i xt + B i ut yt = Cxt where A i R n n, B i R n m, C R p n are real known constant matrices. The h i xt are the weighting functions that ensure the transition between the contribution of each sub model: 2 { Eẋt = Ai xt + B i ut yt = Cxt 3 The activating functions, denoted h i xt for i = 1,..., q, are normalised, and satisfy the following constraints: h i xt 1, h i xt = 1, t 4 In the sequel, we assume the following Hypotheses for each sub model 3, i = 1,..., q: H1 E, A i is regular, i.e. detse A i s C H2 All sub models are impulse observable, i.e. E A i rank E = n + ranke C H3 All sub models are R detectable, i.e. E H4 rank C = n. rank se Ai C = n s C

4 12 M. Essabre, J. Soulami and E. Elyaagoubi 3 Fuzzy observer design In this section, our aim is to design a fuzzy observer for system described 2. The proposed observer witch is not in descriptor form is given by the following equations: ż = h i xn i z + L 1i y + L 2i y + G i u ˆx = z + by + Kdy where N i, L 1i, L 2i, G i and K are unknown matrices of appropriate dimensions, which must be determined such that ˆx will asymptotically converge to x. : The estimation error of the observer 5 for 2 tends to zero if there exists a symmetric positive matrices P,Q and W verifying the following LMI for i = 1,..., q A T i a T P + P aa i + ca i T Q T + QcA i C T W T i W i C2αP < 6 5 E Proof: Since rank C such that: = n, there exist a nonsingular matrix a b c d ae + bc = I n 7 ce + dc = 8 The estimation error is given by: et = xt ˆxt 9 Then by substituting 2 and 4 into 7 we obtain: et = I n bc Kdcx z 1 From 7 and 8, we have: et = a + KcEx z 11 Then, the dynamic of this observer error is: ėt = a + KcEẋ ż 12 By substituting 2 into 12 we obtain: ėt = h i x[a + KcA i x + B i u N i z L 1i y L 2i y G i u] 13

5 Observer for a class of non linear singular systems 13 By substituting 11, equation 13 can be written as: ėt = h i x[a + KcA i x + a + KcB i u + N i e N i a + KcEx L 1i Cx L 2i CxG i u] 14 Which is equivalent to: ėt = h i x[a + KcA i N i a + KcE L 1i C L 2i Cx +a + KcB i G i u + N i e] 15 Equation 15 reduces to the equation: ėt = Provided the matrices G i, K, L 1i, L 2i and N i satisfy: and h i xn i e 16 G i = a + KcB i 17 a + KcA i N i a + KcE L 1i C L 2i C = 18 From 7,8 and 18, we have: Take: Then: N i = a + KcA i L 2i C N i b + Kd L 1i C 19 L 1i = N i b + Kd 2 N i = aa i + KcA i L 2i C 21 To prove the convergence of the estimation error toward zero, let us consider the following quadratic Lyapunov function: V t = e T tp et, P = P T > 22 Estimation error convergence is ensured if the following condition is guaranteed: V t < 23

6 14 M. Essabre, J. Soulami and E. Elyaagoubi Its derivative with regard to time is given by: V t = ė T tp et + e T tp ėt 24 By using the dynamic of the state estimation error 16, 24 can be written as: V t = V t = h i xe T tni T P et + e T tp N i et h i xe T tni T P + P N i et 25 According to the convex sum property of the weighting functions h i x : r h i x = 1 and h i x > the following inequalities are obtained: V t < N T i P + P N i < 26 By substituting 21, equation 26 can be written as: aa i + KcA i L 2i C T P + P aa i + KcA i L 2i C T < 27 Which be rewritten: A T i a T P + ca i T K T P C T L T 2iP + P aa i + P KcA i P L 2i C < 28 The matrices inequalities 28 are not linear with regard to the variables P, K, and L 2i. In order to solve these matrices inequalities, it is necessary to linearize them to obtain LMIs. To do this, we will use some variable changes: which allows to obtain : { Q = P K W i = P L 2i 29 A T i a T P + P aa i + ca i T Q T + QcA i C T W T i W i C < 3 To assure a speed of convergence of the error of estimation, we can define a surface S to the left of the complex plan bounded by a right of abscissa α where α R +. The LMIs 3 are then replaced by the following LMIs: A T i a T P + P aa i + ca i T Q T + QcA i C T W T i W i C2αP < 31 which completes the proof.

7 Observer for a class of non linear singular systems 15 Theorem 3.1 : There exists an observer 5 for 2 if the Hypotheses H1, H2, H3 and H4 holds and there exists a symmetric positive matrices P,Q and W verifying 6 for i = 1,..., q Finally, the design of the proposed observer is reduced to the following procedure. Step 1. Verify that Hypotheses H1, H2, H3 and H4 holds. Step 2. Compute a,b,c and d using 7and 8 Step 3. Solve the LMI 6 in P,Q and W i. Step 4. Compute K and L 2i using 29. the matrices N i,l 1i and G i are derived from 21, 2 and 17 respectively. 4 An illustrative example In this section, the proposed method is illustrated through an example. Consider a singular system represented by the circuit displayed in figure 1 [2]. Where a dc source with voltage E is connected in series to a linear resistor, a linear inductor and a nonlinear capacitor with q v characteristic q = zv = v v 1/3 + q Figure 1: A nonlinear circuit This system can be modeled as descriptor system by the following charge-flux description: q = φ L ẋ 2 = R L φ v + E = v v q q 3 32

8 16 M. Essabre, J. Soulami and E. Elyaagoubi where φ is the magnetic flux in the inductor. R = 1, L =.5, E = 2, v = 1, q = 1. By choosing the state and input vector as following: x = x 1 x 2 x 3 = q q φ v v, u = E v. We assume that x 1 and x 3 can be measured. The system can be described by the following implicit model: { Eẋ = fx + gxu 33 y = Cx with: E = 1 1, fx = x 2 L R L x 2 x 3 x 3 x 3 1, gx = The fuzzy discriptor system obtain from 33 is given by: Where A 1 = A 2 = C = Eẋt = 2 h i xta i xt + B i ut yt = Cxt., B 1 =, B 2 = 1 1 The weighting functions are defined by: h 1 xt = x2 1 4 h 2 xt = 4 x , C =

9 Observer for a class of non linear singular systems 17 The fuzzy observer for system 34 is given by: with ż = 2 h i xn i z + L 1i y + L 2i y + G i u ˆx = z + by + Kdy 35 d = G 1 = a = P = N 1 = L 11 = L 21 =, b =.5 1,c = Q = 1 8 W 1 = , G 2 = , L 12 =, L 22 = , K =,N 2 =., , W 2 = ,x = 2 T,xe = 2.5 T.

10 18 M. Essabre, J. Soulami and E. Elyaagoubi The simulation results are given in figure 2. Figure 2: States and the estimation performance TS: Takagi-Sugeno descriptor system. NL: non linear Descriptor system. Figure 2 display the comparison of the state variable and their estimates supplied by the proposed observer. 5 Conclusion In this paper a simple method is proposed to design observers for Takagi- Sugeno descriptor systems. This work is an extension of the observers developed for linear systems to nonlinear Takagi-Sugeno systems. The determination of the observer parameters is based on the singular decomposition and solving a system of LMI. The design example has illustrated the proposed method.

11 Observer for a class of non linear singular systems 19 References [1] C.Lin, Q.Wang and T.Lee, Stability and Stabilization of a Class of Fuzzy Time-Delay Descriptor Systems, IEEE Transactions on Fuzzy Systems, vol. 14, No.4, pp , August 26. [2] Chunyu Yang, Qingling Zhang and Tianyou Chai, Observer design for a class of nonlinear descriptor systems, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 29. [3] D Koenig and S. Mammar, Design of a proportional integral observer for unknown input descriptor systems, IEEE Transactions on Automatic Control, vol. 47, pp , 22. [4] E. L. Lewis. A survey of linear singular systems. Circuits,Systems and Signal Processing, vol. 5, no. 1, pp. 3-36, [5] L. Dai. Singular Control Systems, Berlin, Germany:Springer, [6] M. Darouach and M. Boutayeb, Design of observers for descriptor systems, IEEE Transactions on Automatic Control, vol. 4, pp , [7] S. Johan and G. Torkel, Computing the Controllability Function for Nonlinear Descriptor systems, Proceedings of the 26 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 26. [8] S. L. Campbell. Singular Systems of Differential Equations,London, UK: Pitman, 198. [9] T.Taniguchi, K.Tanaka,and H.Wang, Fuzzy Descriptor Systems and Nonlinear Model Following Control, IEEE Transactions on Fuzzy Systems, vol. 8, No.4, pp , August 2. [1] 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, vol. 9, No.4, pp , August 21. [11] T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modelling and control. IEEE Trans. Syst., Man and Cyber, Vol.1115, pp , [12] Z. Gao, Parametrization of linear observer for descriptor systems and its application, Proc. of the IFAC world congress, CD-ROM, Received: June, 212