International Conference on Control, Automation and Systems 7 Oct. 7-,7 in COEX, Seoul, Korea Static Output Feedback Controller for Nonlinear Interconnected Systems: Fuzzy Logic Approach Geun Bum Koo l, Jin Bae Park l and Young Roon Jo I Department of Electrical and Electronic Engineering, Yonsei University, Seodaemun-gu, Seoul, -749, Korea, (E-mail: {milbam.jbparkj rs'yonsci.ac.kr) School of Electronic and Information Engineering, Kunsan National University, Kunsan, Chonbuk, 573-7, Korea, (E-mail: yhjoo@kunsan.ac.kr) Abstract: This paper discusses the robust decentralized static output feedback stabilization for nonlinear interconnected systems with parametric uncertainties. Each subsystems of interconnected systems are described by Takagi-Sugeno(T-S) fuzzy models. From T-S fuzzy models, it designs the controllers of each subsystems by parallel distributed compensation. For robust stabilization of interconnect systems, sufficient conditions are derived to obtain the values of controller gains and representedto linear matrix inequalities(lmis). An example is given to show the experimentally verification discussed throughout the paper. Keywords: interconnected systems, Takagi-Sugeno fuzzy model, parametric uncertainty, robust decentralized static output feedback control, linear matrix inequality. INTRODUCTION During the recent years, large-scale interconnected system has a serious application to many spheres which are electrical power systems, industrial processes, networks and so on. Modelling as dynamic systems composed of interconnections with subsystems provides a lot of merits to apply systems taking the interconnected with other systems and are convenient to analysis the complex system divided a few subsystems. From this reason, many people have concerned with the topic of stabilization for interconnected systems[-3]. But it is hard to design a controller, called a decentralized controller, for interconnected systems, since it needs to consider the nonlinearity and the uncertainty. Thus, it is pertinent to study the decentralized controller for the nonlinear interconnected systems with uncertainties. A large number of research results concerning interconnected systems exist in the literature[4-8]. Tseng[ 4] designed the decentralized fuzzy model reference tracking controller for nonlinear interconnected systems. Hsiao and Chen[5, 6] studied the Takagi-Sugeno(T-S) fuzzy controller for nonlinear interconnected systems with delays. Yan[7,8] studied the decentralized control for nonlinear interconnected systems by static output feedback. But, there not has study of the static output feedback controller for interconnected systems using T S fuzzy models. In this paper, the purpose is to develop a static output feedback controller and obtain some sufficient conditions in the linear matrix inequality(lmi) for nonlinear interconnected systems, which is concerned about the parametric uncertainties. Concerning the general nonlinear This work was supported by the Brain Korea Project in 7. This work was supported by MOCIE through EIRC program with Yonsei Electric Power Research Center(YEPRC) at Yonsei University, Seoul, Korea. interconnected systems, it obtains the real world application. For objective, T-S fuzzy model is employed for the nonlinear interconnected systems. Then, a robust decentralized static output feedback controller is developed about each subsystems. Obtaining control gains, it derives in terms of linear matrix inequalities(lmis) technique and proves the justness of this paper with a simulation example. This paper is organized as follows: Section describes the T-S fuzzy model and fuzzy model based controllerfor interconnected systems. The asymptotic stability conditions are proposed with the LMI form in Section 3. In Section 4, simulation examples are provided to demonstrate the design procedures. Finally, the conclusions are given in Section 5.. PROBLEM FORMULATION AND PRELIMINARIES Let the ith subsystems of a continuous- nonlinear interconnected system which is composed of N subsystems with parametric uncertainties be give by the following form: N x Ui + :: j=l,j#i fij(xj) + ~fij(xj), where Xi and u; are the state vector and the input vector of ith subsystem. fi (.) and gi(.) are nonlinear vector functions, ~fi (.) and ~gi (.) are uncertain vector functions of ith subsystem. fij(x) and ~fij(x) denote the vector function and uncertain vector function of the interconnection of the ith subsystem with other subsystems, and x == [Xl T XT... XNT]T. We assume that C, is the linear matrix with full-row rank. () 978-89-9538-6--9856/7/$5 @ICROS 46
This nonlinear interconnected system with parametric uncertainties can be represented by a T-S fuzzy model. In each fuzzy rule, a linear model is represented to the local dynamical system ant the overall system is obtained by fuzzy blending of all local linear models. The following form is the kth rule of the fuzzy model for the subsystem of the nonlinear interconnected system: Plant Rule k: IF z i is Γ k i,, and z ip is Γ k ip, THEN ẋ i =(A ik + A ik )x i +(B ik + B ik ) u i + (A ijk + A ijk )x j, j=,j i y i =C i x i, () where Γ k iq (q =,,...,p) is a fuzzy set for q I p = {,,...,p}, k I r = {,,...,r}, r is the number of fuzzy rules, A ik, B ik, C i and A ijk are the constant matrices of the full rank with appropriate dimensions. A ik, B ik and A ijk are unknown matrices representing the uncertainties in the ith subsystem of the interconnected system. We assume that the uncertain matrices A ik and B ik are norm-bounded and structured, A ijk is upper bounded. Assumption : [9] The parameter uncertainty matrices are represented as follow: [ ] Aik B ik = Dik F ik [ ] E ik E ik A T ijk A ijk R T ijkr ijk, F T ikf ik I where D ik, E ik, E ik and R ijk are known real constant matrices of appropriate dimensions, and F ik is an unknown matrix function with Lebesgue-measurable elements and s the identity matrix of appropriate dimension. Through the defuzzification of T S fuzzy system, the nonlinear interconnected system is represented in the following form: r ẋ i = µ k (z i )((A ik + A ik )x i +(B ik k= + B ik )u i + j=,j i (A ijk + A ijk )x j ), y i =C i x i (3) where µ k (z i ) = ω k (z i ) = ω k (z i ) r k= ω k(z i, p Γ k iq(z iq ) (4) q= in which Γ k iq (z i) is the fuzzy membership grade of z iq in Γ k iq and z i =[z i... z ip ]. For the static output feedback decentralized controller, model-based fuzzy controllers is synthesized via the technique of parallel distributed compensation(pdc)[]. The fuzzy model of a static output feedback controller for the ith fuzzy subsystem is formulated as follow: Controller Rule k: IF z i is Γ k i,, and z ip is Γ k ip, THEN u i =K ik x i (k =,,...,L) (5) where F ik is the constant control gain of kth fuzzy rule in ith subsystem. The defuzzified output of fuzzy model controller is represented as follows: r u i = µ k (z i )K ik x i (6) k= By substituting (4) into ith subsystem, the form of the ith subsystem of the interconnected system used static output feedback controller as: r r ẋ i = µ k (z i )µ m (z i ) k= m= (A ik + A ik +(B ik + B ik )K im C i )) x i + (A ijk + A ijk )x j ) (7) j=,j i We need to find the static output feedback control gain K im for global asymptotically stability of nonlinear interconnected systems (7). 3. MAIN RESULTS The purpose of this paper is to obtain static output feedback controller gain K ik for the global asymptotic stability of interconnected system. To obtain the gain, we need to represent the LMI and consider the Lyapunov function candidate for LMI as follows: V = V i (8) i= V i =x T i P i x i for i =,,...,N (9) where P i (i =,,...,N) is a symmetric and positive definite matrix. Lemma : [] For real matrices X and Y with appropriate dimensions, the following inequality is always satisfied. X T Y + Y T X σx T X + σ Y T Y where σ is a positive constant. Lemma : [] Given constant symmetric matrices N, O and L of appropriate dimensions, the following inequalities O>, N + L T OL < 47
are equivalent to the following inequality [ ] [ ] N L T O L L O < or L T <. N Lemma 3: [] Given constant matrices D and E, and a symmetric constant matrix S of appropriate dimensions, the following inequality S + DFE + E T F T D T <. is hold to next inequality where F satisfies F T F I, if and only if for ɛ> S + [ ɛ E T εd ] [ ] ɛ E εd T <. Theorem : In the nonlinear interconnected system (6), if positive definite matrices P i is the common solution of the following matrix inequalities: [ ] G T ikm P i + P i G ikm + Pi P i A ijk A T ijk P i A T ijk A <. () ijk where G ikm = A ik + A ik +(B ik + B ik )K im C i for i, j I N = {,,...,N} (j i) and k, m I r = {,,...,r}, then the whole nonlinear interconnected system is asymptotically stable at the its equilibrium point. Proof: It is omitted in this paper. Theorem : If there exist a positive definite matrices Q i, some matrices F imn, and some scalars ɛ ijkmn, such that the following LMIs and equations are satisfied, then Theorem is always satisfied. Θ ikm + I A T ijk Rijk T R ijk E ik Q i + E ik F im C i ɛ ijkm I Dik T ɛ ijkm I < () and C i Q i = M i C i () where Θ ikmn =A ik Q i + Q i A T ik + B ik F im C i + C T i F T imb T ik F im =K im M i for i, j I N = {,,...,N} and k, m I r = {,,...,r}, and denotes the transposed elements in the symmetric positions. Proof: It is omitted in this paper. Remark : In Theorem and Theorem, matrices C in need to have the full row rank. In the full row rank, C in has the right inverse of matrix and it can be obtained the controller gains in the following forms: K ik = F ik M i where M i = C i P i C i (C i C T i ). 4. SIMULATION RESULTS The equation of two flexible joint robot arms connected by a spring with ignoring damping for simplicity is represented by following equations: I θ +M gl sin(θ ) + k (θ θ ) + k 3 (θ θ 3 ) =, J θ k (θ θ ) = u, I θ3 +M gl sin(θ 3 ) + k 3 (θ 3 θ 4 ) + k 3 (θ 3 θ ) =, J θ4 k (θ 3 θ 4 ) = u, where k and k are the spring constants of each arms, k 3 is the constant of coupling spring between the arms. θ and θ 3 are the link angles and θ and θ 4 are the motor angles. I and I are the rotational inertias about the axis of rotations and J, J are the rotor inertias of the actuator shafts. M and M are the total mass of each arms, l and l are the distances to the joint from the mass centers of the axis of rotations, and g is the gravity constant. Consider interconnected system composed of two fuzzy subsystems as each flexible joint robot arm. An exact T S fuzzy models can be represented as follows: Subsystem : Rule :IF x is about M, THEN ẋ =A x +B u +A x y =C x Rule :IF x is about M, THEN ẋ =A x +B u +A x y =C x Subsystem : Rule :IF x is about M, THEN ẋ =A x +B u +A x y =C x Rule :IF x is about M, THEN ẋ =A x +B u +A x y =C x where x = [θ θ θ θ ] T, x = [θ 3 θ 3 θ 4 θ 4 ] T, and [M i,m i ] are chosen as [.5,.5] by simulation of ordinary system. A i = Migli ki k3 k i, k i ki A i = αmigli ki k3 k i, k i ki 48
k 3 A ij = A ij =, B i = B i =, C = C = [ ] for i =, and α in the matrix A i is the arbitrary value excepted and. The membership functions for x and x are shown in Figs...9.8.7.6.5.4.3.. 8 6 4 4 6 8 Fig. The membership function of x and x. The parameter values of each subsystems are determined as following: I = I =.3(kgm ), J = J =.4(kgm ), M = M =.687(kg), g = 9.8(m/ sec ), l = l = (m), k = k = k 3 = 3.(Nm/rad). We assume that the parametric uncertainties is bounded within.3 of parameter values. Thus, we can define the uncertain matrices D ik, E ik and E ik for i, k =, using Assumption and the uncertain matrices of interconnection matrices R ijk are defined to.a ijk. k i M igl i k 3 D i =, k i k i αm igl i k 3 D i =, k i J i E i = E i = T 7, E i = E i = T 7, k 3 R ij = R ij =.. We can obtain the controller gain matrices K im using Theorem and solving the corresponding LMIs: K = [ 56.54.946 ], K = [ 57.5.5 ], K = [.88.43 ], K = [.54.35 ]. When the initial state conditions of subsystem are [ π π 6 π π 6 ] and initial conditions of subsystem are same with subsystem, the output results of two flexible joint robot arms with decentralized static output feedback controllers are shown in Fig. and 3. x x.5.5.5 3 5 5.5.5.5 3 x 3 x 4 x x x 4 x 3.5.5.5 3 5 5.5.5.5 3 Fig. The state responses of subsystem..5.5.5 3 5 5.5.5.5 3 4.5.5.5 3.5.5.5 3 Fig. 3 The state responses of subsystem. 5. CONCLUSION In this paper, a fuzzy decentralized output feedback controller has been proposed for nonlinear interconnected system with parametric uncertainties. It has been shown 49
that the interconnected system is possible to be controlled by static output feedback. Using T S fuzzy model, decentralized controller has been designed and stability condition has been developed to LMI form with some qualification. Solving LMIs, we can have obtained the controller gains for an asymptotically stability of interconnected system and a simulation example has shown that the results of this paper are effective and valuable. of uncertain linear systems, Systems and Control Letters, Vol. 8, pp. 35-357, 987. [] L. Xie, Output feedback H control of systems with parameter uncertainties, Int. J. Contr, Vol. 63, No. 4, pp. 74-75, 996. REFERENCES [] R. J. Wang, Nonlinear decentralized state feedback controller for uncertain fuzzy -delay interconnected systems, Fuzzy Sets and Systems, Vol. 5, pp. 9-4, 5. [] M. Benyakhlef, Decentralised nonlinear adaptive fuzzy control for a class of large-scale interconnected systems, International Journal of Computational Cognition, Vol. 4, No., pp. 4-9, 6. [3] S. S. Stanković, D. M. Stipanović, and D. D. Šiljak, Decentralized dynamic output feedback for robust stabilization of a class of nonlinear interconnected systems, Automatica, Vol. 43, pp. 86-867, 7. [4] C. S. Tseng, and B. S. Chen, H decentralized fuzzy model reference tracking control design for nonlinear interconnected systems, IEEE Transactions on Fuzzy Systems, Vol. 9, No. 6, pp. 795-89,. [5] F. H. Hsiao, C. W. Chen, Y, W. Liang, S. D. Xu, and W. L. Chiang, T S fuzzy controllers for nonlinear interconnected systems with multiple delays, IEEE Transactions on Circuits and Systems, Vol. 5, No. 9, pp. 883-893, 5. [6] C. W. Chen, W. L. Chiang, and F. H. Hsiao, Stability analysis of T S fuzzy models for nonlinear multiple delay interconnected systems, Mathematics and Computers in Simulation, Vol. 66, pp. 53-537, 4. [7] X. G. Yan, C. Edwards, and S. K. Spurgeon, Decentralised robust sliding mode control for a class of nonlinear interconnected systems by static output feedback, Automatica, Vol. 4, pp. 63-6, 4. [8] X. G. Yan, S. K. Spurgeon, and C. Edwards, Decentralized output feedback sliding mode control of nonlinear large-scale systems with uncertainties, Journal of Optimization Theory and Applications, Vol. 9, No. 3, pp. 597-64, 3. [9] H. J. Lee, J. B. Park, and G. Chen, Robust fuzzy control of nonlinear systems with parametric uncertainties, IEEE Transactions on Fuzzy Systems, Vol. 9, No., pp. 369-379,. [] K. Tanaka, T. Ikeda, and H. O. Wang, Robust stabilization of a class of uncertain nonliear systems via fuzzy control: quadratic stabilizability, H control theorym and linear matrix inequalities, IEEE Transactions on Fuzzy Systems, Vol. 4, No., pp. -3, 996. [] I. R. Petersen, A stabilization algorithm for a class 5