H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions

Transcription

1 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL H State-Feedback Controller Design for Discrete-Time Fuzzy Systems Using Fuzzy Weighting-Dependent Lyapunov Functions Doo Jin Choi and PooGyeon Park Abstract For discrete-time Takagi Sugeno (TS) fuzzy systems, we propose an state-feedback fuzzy controller associated with a fuzzy weighting-dependent Lyapunov function The controller, which is designed via parameterized linear matrix inequalities (PLMIs), employs not only the current-time but also the one-step-past information on the time-varying fuzzy weighting functions Appropriately selecting the structures of variables in the PLMIs allows us to find an LMI formulation as a special case Index Terms control, discrete-time Takagi Sugeno fuzzy systems, fuzzy weighting-dependent Lyapunov function (FWDLF), parameterized linear matrix inequalities (PLMIs) et al [10] checked only the picecewise stability conditions to guarantee the stability of a given closed-loop system and Cao et al [7] [9] presented some step-by-step controller design algorithms based on their piecewise stability analysis In the FWDLF approach [11], which was developed only for continuous-time TS fuzzy system, a Lyapunov function is based on a mapping from fuzzy weighting functions to a Lyapunov matrix (1) I INTRODUCTION SINCE the middle of the 1980s, there have appeared a number of analysis/synthesis problems for Takagi-Sugeno (TS) fuzzy systems [1] These problems were usually handled via the common quadratic Lyapunov function (CQLF) approach [2] [6], which required that a common positive definite matrix for all the local linear models must be found to satisfy the Lyapunov stability condition Although this approach allowed one to apply convex optimization for solving the problems, it was mostly found to be conservative because of the common (or strict) structure of the Lyapunov function independent from the fuzzy weighting functions To relax this conservatism, therefore, one has considered the time-varying information on the fuzzy weighting functions constructing a Lyapunov function Consequently, there were proposed two different kinds of approaches based on a piecewise Lyapunov function (PLF) and a fuzzy weighting-dependent Lyapunov function (FWDLF) In the PLF approach [7] [10], a Lyapunov function is based on a finite combination of Lyapunov matrices, say, dependent on a dominant fuzzy weighting function at a time But, although the piecewise stability analysis results were much more powerful and flexible than their common quadratic counterpart and also obtained efficiently via convex optimization, the corresponding controller synthesis results were hardly found via nonconvex optimization For example, Johansson Manuscript received January 16, 2002; revised June 10, 2002 and July 25, 2002 This work was supported by the Ministry of Education of Korea toward the Division of Electrical and Computer Engineering at POSTECH through its BK21 program The authors are with the Division of Electrical and Computer Engineering, Pohang University of Science and Technology, Pohang , Korea ( chdj@postechackr; ppg@postechackr) Digital Object Identifier /TFUZZ are positive definite matrices and are normalized weighting functions for each rule Especially, at the cost of constraint on time-derivative of fuzzy weighting functions, Tanaka et al [11] discussed more flexible stability conditions than the previous (PLF or CQLF based) works for continuous-time TS fuzzy systems via the FWDLF approach However, in [11], they only proposed two improved stability analysis techniques and remained controller synthesis techniques as future works We remark again that there is no such approach for discrete-time TS fuzzy systems In this paper, we suggest the FWDLF approach for discretetime TS fuzzy systems Whereas, in the continuous-time FWDLF approach, the bound of time-derivatives of fuzzy weighting functions may be infinite, in the discrete-time FWDLF approach, the bound of absolute time-difference of fuzzy weighting functions is less than or equal to one Therefore, one can always employ the FWDLF approach for the discretetime TS fuzzy systems, which is main difference from the continuous-time FWDLF approach In this approach, we shall propose a new candidate of discrete-time FWDLF as follows (2) is a Lyapunov matrix and is the -step-past fuzzy weighting function vector in time, which fully contains the time-varying information on the fuzzy weighting functions For simplicity, in this paper, we shall consider the case, which will lead us to easily implement a new state-feedback fuzzy controller with only the current-time and one-step-past fuzzy weighting functions in terms of PLMIs or a finite number of LMIs Because we were motivated by the similar structures between linear parameter varying (LPV) systems and TS fuzzy systems, we shall use the PLMI techniques developed in [12] and [13] for implementing the new controller Furthermore, appropriately selecting the /03$ IEEE

2 272 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 structures of variables in the PLMIs will allow us to find an LMI formulation as a special case In practical fuzzy control system design, fuzzy controllers are usually required to satisfy various design specifications in addition to stabilization One of the most important requirements for a fuzzy control system is the disturbance attenuation (see [12], [14] and [15]), which has used external disturbances exist in TS fuzzy systems We shall adopt minimizing the performance as another purpose of this paper This paper is organized as follows Section II gives mathematical description of a discrete-time TS fuzzy system and performance criterion [16], [17] And Section III describes a new state-feedback fuzzy controller associated with the FWDLF Furthermore, as a special case, Section IV presents the relaxed convex solvability conditions of the proposed controller via convex optimization By the well-known problem of balancing an inverted pendulum on a cart, Section V demonstrates the performance of the proposed new fuzzy synthesis method using the relaxed controller and semidefinite programming [19] The notation of this paper is fairly standard In symmetric block matrices, we use as an ellipsis for terms that are induced by symmetry II PROBLEM STATEMENT A general discrete-time TS fuzzy system can be described such as is the state, is the control input, is the exogenous input (such as reference signal, disturbance signal, sensor noise), is the output, is the number of system rules, is the premise variable vector that may depend on states in many cases and denote normalized time-varying fuzzy weighting functions for each rule in time And, the fuzzy weighting functions generally have following conditions for all time (see [3], [5], [11], [13] and [14]): for for Since are measurable in current time, the system (3) also belongs to a special class of LPV systems whose state-space matrices are assumed to depend on a time-varying parameter vector In recent years, a number of interesting alternative approaches have been proposed in the context of LPV approach Especially, based on LMI and PLMI tools, various LPV control problems had been solved by Gahinet et al [16], Köse et al [17] and Tuan et al [12], [13], which motivated us to present new fuzzy controllers Consequently, for applying PLMI (3) (4) techniques used in [12], [13], we represent the discrete-time TS fuzzy system (3) such as and denotes a fuzzy weighting function vector of time-varying fuzzy weighting functions in time Moreover, the goal of this paper is to find a state-feedback controller minimizing the induced norm from the input to the output, which is the performance criterion Based on the following lemma 1, which is a simple extension of [16] and [17] to general discrete-time fuzzy systems, we shall develop our main results Lemma 1: With a quadratic Lyapunov function, consider a discrete-time open-loop fuzzy system described by (6) is a time-varying fuzzy weighting function vector The following two statements are equivalent The system (6) is stable with the performance There exists a matrix such that III which can be directly derived by Proof: See Appendix STATE-FEEDBACK FUZZY CONTROLLER DESIGN In this section, we shall present a new state-feedback fuzzy controller in terms of PLMIs, which is dependent on both the current-time fuzzy weighting function vector and the one-step-past fuzzy weighting function vector for time : (9) is dependent on the fuzzy weighting function vectors and Consequently, the resulting closed-loop fuzzy system has the following form: (5) (7) (8) (10)

3 CHOI AND PARK: STATE-FEEDBACK CONTROLLER DESIGN 273 And then, based on the new FWDLF candidate from to in time such as mapping (11) we shall consider a new FWDLF-dependent Lyapunov stability In the following theorem, we shall summarize how to find an optimal state-feedback fuzzy controller for the closed-loop fuzzy system (10) to minimize the performance with respect to all admissible grades and Theorem 1: The closed-loop fuzzy system (10) is quadratically stable with the performance for all admissible grades and, if there exist matrices,, and satisfying such that (12), this condition can be con- Then, with vector sidered as a quadratic condition: (18) Moreover, by Schur complements technique for (18), the -suboptimal condition becomes (13) (14) Furthermore, the minimum performance can be obtained via the following optimization programming: Minimize subject to (12), (13), and (14) (15) Consequently, the above conditions are satisfied and optimized, an optimal state-feedback fuzzy controller (9) is given by (16) Proof: The corresponding Lyapunov difference along any trajectory of the closed-loop system (10) is given by (19) And then, we proceed according to the congruence transformations given in [12] and [17] The left-hand side of the inequality (19) can be pre- and post-multiplied by and, respectively, blockdiag The resulting condition is And, after the definitions of (20) Furthermore, for addressing the practical performance problem, we have the resulting condition from lemma 1 for all : which yields the following (17) (21) we obtain (12) Consequently, if the conditions (12), (13) and (14) hold for all time, then at any nonzero And from (19), it is straightforward (22) which guarantees that the proposed FWDLF candidate (11) is positive definite, decrescent and radially unbounded (see [11] and [18]) Hence, the closed-loop fuzzy system (10) is stable with the performance in the sense of Lyapunov The proof is completed Note that since and appear in the closed-loop system, the PLMIs of Theorem 1 may be nonlinear in and Hence, the PLMIs need to be checked for all values of and In the following section, we turn the PLMIs given in Theorem 1 into a minimization problem of a finite number of LMIs for discrete-time TS fuzzy systems

4 274 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 IV RELAXATION OF THE PROPOSED CONTROLLER USING LMIS Since solving the PLMIs of Theorem 1 is equivalent to solving an infinite number of LMIs and is thus an extremely difficult problem, it is important to develop a finite number of solvable conditions from the PLMIs In this section, based on the polynomial dependency of the fuzzy weighting functions for the discrete-time TS fuzzy system (3), the convex relaxation techniques of general fuzzy systems and the structural assumptions of some variables in the PLMIs, we shall develop the solvability of a finite number of LMIs, which are reduced from the solvability of an infinite number of LMIs in Theorem 1 For finding an LMI formulation of Theorem 1, as a special case, we select the structures of the variables (21) in the PLMIs such as (23), and are constant matrices, that is,, and are polynomially dependent on and Furthermore, by using constraint-elimination methods for the conditions (4) of the current-time fuzzy weighting function and the one-step-past fuzzy weighting function with the -procedure, we relax the conservatism of the proposed controller (9); see (24), shown at the bottom of the page Theorem 2: The closed-loop fuzzy system (10) with the controller (9) is quadratically stable with performance for all admissible grades and, if there exist constant matrices,,,,,,, and satisfying such that (24) and for all, (25) (26) (24)

5 CHOI AND PARK: STATE-FEEDBACK CONTROLLER DESIGN 275 And since the above conditions are linear in all variables, the minimum performance can be obtained via the following semidefinite programming: Minimize subject to (24), (25), and (26) (27) Consequently, the above conditions are satisfied and optimized, an optimal state-feedback fuzzy controller (9) is given by (28) Also, based on the constraint-elimination methods [19] and [20], we consider the constraints (4) for all time such as (30) (31) Proof: Based on the structural assumptions (23), we can rewrite (12) more detail and rearrange as a following form (32) (33) (34) (29) stands for the decision variables, ie,,, and And,,, and are real symmetric matrix-valued and linear functions of :,,, and are real matrices in satisfying,,, for all and Especially, based on the equality constraint of (4), the two conditions (33) and (34) were developed by the following equations: (35) Furthermore, we can sum and rewrite the five constraint conditions as the following condition: (36)

6 276 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 stands for the decision variables, ie,,,, and Also,,,,, and are real symmetric matrix-valued and linear functions of : Fig 1 Fuzzy weighting functions Consider a following discrete-time inverted pendulum fuzzy system, which is based on the bilinear transformation of the numerical example in [6] with sampling time (38) Consequently, combining all the conditions provide (37), as shown at the bottom of the page Then, the (37) can be considered as a quadratic condition (24) with the following vector: Therefore if the conditions of Theorem 2 hold, the -performance condition (19) is always satisfied for all possible and And we can use (27) to minimize, which completes the proof V AN EXAMPLE In this section, we demonstrate the performance of the proposed new fuzzy synthesis method with the problem of balancing an inverted pendulum on a cart Remark 1: Since solving the PLMIs of Theorem 1 is an extremely difficult problem, we demonstrate the performance of the proposed new method by the controller of Theorem 2 using semidefinite programming Also based on the premise variable, the fuzzy weighting functions and are respectively defined as (39) (37)

7 CHOI AND PARK: STATE-FEEDBACK CONTROLLER DESIGN 277 TABLE I COMPARISONS VI CONCLUDING REMARKS In the field of discrete-time TS fuzzy analysis/synthesis problems, there have been no results using the time-varying information on the fuzzy weighting functions Therefore, we proposed anew state-feedback fuzzy controller associated with the new FWDLF for the discrete-time TS fuzzy systems Because, in the discrete-time FWDLF approach, the absolute value of time-difference of fuzzy weighting functions is bounded by one, the new proposed FWDLF can always be applied for the discrete-time TS fuzzy systems The controller employed not only the current-time but also the one-step-past information on the time-varying fuzzy weighting functions, which was designed via PLMIs Furthermore, based on the polynomial structure of the discrete-time TS fuzzy systems, appropriately selecting the structures of variables in the PLMIs allowed us to find an LMI formulation as a special case Also, since the resulting convex solvability conditions have been expressed as a finite number of LMIs, we obtained simply the gain matrices of the fuzzy controller by semidefinite programming Fig 2 State responses for initial condition x =[0:1; 01] APPENDIX PROOF OF LEMMA 1 From [16, Lemma 51], the two statements are equivalent Furthermore, the proof can be found in [21, Sec 5] or in [22, Lemma 13] And the condition (7) can be directly derived from (8) by following procedure When, based on the difference (40) the (8) yields the following: Fig 3 Control inputs for initial condition x =[0:1; 01] is the first element of the state (see Fig 1) Consequently, by the LMI toolbox in the Matlab 53, we calculated the minimized performance of Theorem 2 with,,, for In Table I, we compared the performance of the proposed new synthesis method with the Corollary 3 of Cao and Frank [5], in which they proposed a relaxed state-feedback fuzzy controller Additionally in Table I, we showed the minimized of Theorem 2 and the Corollary 3 of [5] based on the bilinearly transformed system of the numerical example in [6] with sampling time Furthermore, for the initial condition with, we simulated the behaviors of the closed-loop systems using the each controller of Theorem 2 and Cao and Frank [5] The state and control input profiles are shown by Figs 2 3 Note that the proposed new controller based on Theorem 2 shows not only better performance but also better simulation results than Corollary 3 of Cao and Frank [5], this condition can be con- Then, with vector sidered as a quadratic condition: (41) (42) Furthermore, with taking proper Schur complements for (42), we simply obtain the final condition (7)

8 278 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 11, NO 2, APRIL 2003 REFERENCES [1] T Takagi and M Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans Syst, Man, Cybern, vol SMC-15, pp , Jan 1985 [2] H O Wang, K Tanaka, and M F Griffin, An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE Trans Fuzzy Syst, vol 4, pp 14 23, Feb 1996 [3] K Tanaka, T Ikeda, and H O Wang, Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs, IEEE Trans Fuzzy Syst, vol 6, pp , Feb 1998 [4] K Tanaka, T Kosaki, and H O Wang, Backing control problem of a mobile robot with multiple trailers: Fuzzy modeling and LMI-based design, IEEE Trans Syst, Man, Cybern C, vol 28, pp , 1998 [5] Y-Y Cao and P M Frank, Robust H disturbance attenuation for a class of uncertain discrete-time fuzzy systems, IEEE Trans Fuzzy Syst, vol 8, pp , Aug 2000 [6] A Jadbabaie, M Jamshidi, and A Titli, Guaranteed-cost design of continuous-time Takagi-Sugeno fuzzy controllers via linear matrix inequalities, in Proc Fuzz IEEE, 1998, pp [7] S-G Cao, N W Ress, and G Feng, Stability analysis and design for a class of continuous time fuzzy control system, Int J Control, vol 64, pp , 1996 [8], Analysis and design for a class of complex control systms, part II: Fuzzy controller design, Automatica, vol 33, pp , 1997 [9], Analysis and design of fuzzy control systems using dynamic fuzzy-state space models, IEEE Trans Fuzzy Syst, vol 7, pp , Apr 1999 [10] M Johansson, K-E Arzen, and A Rantzer, Piecewise quadratic stability of fuzzy systems, IEEE Trans Fuzzy Syst, vol 7, pp , Dec 1999 [11] K Tanaka, T Hori, and H O Wang, A fuzzy Lyapunov approach to fuzzy control system design, in Proc Amer Control Conf, 2001, pp [12] H D Tuan, P Apkarian, and M James, Parameterized linear matrix inequalities for nonlinear discrete H, in Proc Conf Decision Control, 1999, pp [13] H D Tuan, P Apkarian, T Narikiyo, and Y Yamamoto, Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Trans Fuzzy Syst, vol 9, pp , Apr 2001 [14] S-G Cao, N W Ress, G Feng, and W Liu, H control of nonlinear discrete-time systems basded on dynamical fuzzy models, Int J Syst Sci, vol 31, no 2, pp , 2000 [15] J Yoneyama, M Nishikawa, H Katayama, and A Ichikawa, H control for Takagi-Sugeno fuzzy systems, Int J Syst Sci, vol 32, no 7, pp , 2001 [16] P Gahinet and P Apkarian, A LMI approach to H control, Int J Robust Nonlinear Control, vol 4, pp , 1994 [17] I E Köse, F Jabbari, and W E Schmitendorf, A direct characterization of L -gain controllers for LPV systems, IEEE Trans Automat Contr, vol 43, pp , Sept 1998 [18] M Vidyasagar, Nonlinear Systems Analysis, 2nd ed Upper Saddle River, NJ: Prentice-Hall, 1993 [19] S Boyd, L El Ghauoi, E Feron, and V Balakrishan, Linear Matrix Inequalities in System and Control Theory Philadelphia, PA: SIAM, 1994 [20] P Park and D J Choi, LPV controller design for the nonlinear RTAC system, Int J Robust Nonlinear Control, vol 11, pp , 2001 [21] J Doyle, A Packard, and K Zhou, Review of LFTS, LMIS, and, in Proc Conf Decision Control, 1991, pp [22] K Zhou, P P Khargonekar, J Stoustrup, and H H Niemann, Robust preformance of systems with structured uncertainties in state space, Automatica, vol 31, no 2, pp , 1995 Doo Jin Choi received the BS degree in control and instrumentation enginering from Kwangwoon University, Seoul, Korea, and the MS degree in electrical and electronic engineering from Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1997 and 1999, respectively He is currently working towards the PhD degree at POSTECH Since 1999, he has been affiliated with the Division of Electrical and Computer Engineering, POSTECH His research interests are in analysis and synthesis of fuzzy systems, LPV systems, hybrid systems, intelligent transportation system (ITS), and communication networks PooGyeon Park received the BS and MS degrees in control and instrumentation engineering from Seoul National University, Seoul, Korea, and the PhD degree in electrical engineering from Stanford University, Stanford, CA, in 1988, 1990, and 1995, respectively Since 1996, he has been affiliated with the Division of Electrical and Computer Engineering at Pohang University of Science and Technology (POSTECH), he is currently an Associate Professor His current research interests include robust, LPV, RHC, intelligent and network-related control theories, signal processing, and wireless communications for personal area networks (PANs)