state-feedback controller design for discrete-time fuzzy systems using

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 2, APRIL 2003

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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, state-feedback fuzzy controller associated we propose an 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).

I. INTRODUCTION

S

INCE 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 for all the local linear models must be definite matrix 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 when 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 790-784, Korea (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TFUZZ.2003.809903

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) are normalized where are positive definite matrices and 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 where 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 , which will lead us to easily consider the case where 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

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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 disturbance attenuation (see a fuzzy control system is the [12], [14] and [15]), which has used when 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 state-feedback fuzzy controller associated with the new 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 demonfuzzy synstrates the performance of the proposed new thesis method using the relaxed controller and semidefinite programming [19]. The notation of this paper is fairly standard. In as an ellipsis for terms symmetric block matrices, we use that are induced by symmetry.

techniques used in [12], [13], we represent the discrete-time TS fuzzy system (3) such as (5) where

and denotes a fuzzy weighting function vector of in time . time-varying fuzzy weighting functions Moreover, the goal of this paper is to find a state-feedback to controller minimizing the induced norm from the input performance criterion. Based on the output , which is the 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

II. PROBLEM STATEMENT A general discrete-time TS fuzzy system can be described such as

(6) is a time-varying fuzzy weighting function vector. where The following two statements are equivalent. performance . • The system (6) is stable with the such that • There exists a matrix

(3) is the state, is the control input, is the exogenous input (such as reference signal, disis the output, is the turbance signal, sensor noise), is the premise varinumber of system rules, able 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]):

(7)

where

for for

which can be directly derived by (8) Proof: See Appendix. III.

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 and the the current-time fuzzy weighting function vector for time : one-step-past fuzzy weighting function vector (9)

(4) are measurable in current time , the system (3) Since 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 fuzzy controllers. Consequently, for applying PLMI new

is dependent on the fuzzy weighting where and . Consequently, the resulting function vectors closed-loop fuzzy system has the following form: (10) where

CHOI AND PARK:

STATE-FEEDBACK CONTROLLER DESIGN

And then, based on the new FWDLF candidate to in time such as from

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mapping

Then, with vector sidered as a quadratic condition:

, this condition can be con-

(11) we shall consider a new FWDLF-dependent Lyapunov stability. In the following theorem, we shall summarize how to find an state-feedback fuzzy controller for the closed-loop optimal performance with refuzzy system (10) to minimize the and . spect to all admissible grades Theorem 1: The closed-loop fuzzy system (10) is quadratperformance for all admissible ically stable with the and , if there exist matrices , , grades and satisfying such that

(18) where

Moreover, by Schur complements technique for (18), the -suboptimal condition becomes (12) (13) (14) where

Furthermore, the minimum performance can be obtained via the following optimization programming: Minimize

subject to (12), (13), and (14)

(19) And then, we proceed according to the congruence transformations given in [12] and [17]. The left-hand side of the inand , reequality (19) can be pre- and post-multiplied by blockdiag . spectively, where The resulting condition is

(15)

Consequently, when the above conditions are satisfied and opstate-feedback fuzzy controller (9) is timized, an optimal given by (20)

(16) Proof: The corresponding Lyapunov difference along any trajectory of the closed-loop system (10) is given by

Furthermore, for addressing the practical performance problem, we have the resulting condition from lemma 1 for all :

And, after the definitions of

(21) we obtain (12). Consequently, if the conditions (12), (13) and at any nonzero . (14) hold for all time , then And from (19), it is straightforward

(17) which yields the following

(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 performance in the sense of Lyapunov. The proof with the is completed. Note that since and appear in the closed-loop and system, the PLMIs of Theorem 1 may be nonlinear in . 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.

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IV. RELAXATION OF THE PROPOSED CONTROLLER USING LMIS

where

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) are constant matrices, that is, , are polynomially dependent on and . Furthermore, by using constraint-elimination methods for the conditions (4) of the current-time and the one-step-past fuzzy fuzzy weighting function with the -procedure, we relax weighting function 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 conperformance for all troller (9) is quadratically stable with and , if there exist constant matrices admissible grades , , , , , , , and satisfying such that , (24) and for all where

,

and

and

(25) (26)

.. .

.. .

..

.

..

.

..

.

.. .

.. . .. .

.. . .. . (24)

.. .

.. . .. .

.. . .. .

.. . .. .

.. . .. .

.. .

..

.

..

.

..

.

.. .

CHOI AND PARK:


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And since the above conditions are linear in all variables, the performance can be obtained via the following minimum semidefinite programming: Minimize

subject to (24), (25), and (26)

(27)

Consequently, when the above conditions are satisfied and state-feedback fuzzy controller (9) optimized, an optimal is given by

where

Also, based on the constraint-elimination methods [19] and [20], we consider the constraints (4) for all time such as

(30)

(28) where

(31)

(32) Proof: Based on the structural assumptions (23), we can rewrite (12) more detail and rearrange as a following form (33)

(34) 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:

where (29) , where stands for the decision variables, i.e., , and . And , , , and are real symmetric matrix-valued and linear functions of :

,

,

,

and

,

(35) Furthermore, we can sum and rewrite the five constraint conditions as the following condition:

(36)

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where and

stands for the decision variables, i.e., , , , . 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

when when when when

(38)

when when 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:

where

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 profuzzy synthesis method with the problem of balposed new ancing 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 and are respectively defined weighting functions as (39)

(37)

CHOI AND PARK:


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VI. CONCLUDING REMARKS

TABLE I COMPARISONS

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 state-feedback fuzzy controller associated with the a new 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 5.1], 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 , based on (8) by following procedure. When the difference (40) the (8) yields the following:

0

Fig. 3. Control inputs for initial condition x = [0:1; 1] .

where is the first element of the state (see Fig. 1). Consequently, by the LMI toolbox in the Matlab 5.3, we calperformance of Theorem 2 with culated the minimized , , , for . In Table I, we compared the performance of the proposed new synthesis method with the Corollary 3 of Cao and Frank [5], in state-feedback fuzzy conwhich they proposed a relaxed troller. 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 sam. pling time with Furthermore, for the initial condition , we simulated the behaviors of the controller of Theorem closed-loop systems using the each 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].

(41) Then, with vector sidered as a quadratic condition:

, this condition can be con-

(42) where

Furthermore, with taking proper Schur complements for (42), we simply obtain the final condition (7).

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Doo Jin Choi received the B.S. degree in control and instrumentation enginering from Kwangwoon University, Seoul, Korea, and the M.S. 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 Ph.D. 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 B.S. and M.S. degrees in control and instrumentation engineering from Seoul National University, Seoul, Korea, and the Ph.D. 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), where 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).