OVER the past one decade, Takagi Sugeno (T-S) fuzzy

2838 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006 Discrete H 2 =H Nonlinear Controller Design Based on Fuzzy Region Concept and Takagi Sugeno Fuzzy Framework Sheng-Ming Wu, Chein-Chung Sun, Hung-Yuan Chung, and Wen-Jer Chang Abstract The purpose of this paper is to develop a fuzzy controller to stabilize a discrete nonlinear model in which the controller rule is adjustable and it is developed for stabilizing Takagi Sugeno (T-S) fuzzy models involving lots of plant rules. The design idea is to partition the fuzzy model into several fuzzy regions, and regard each region as a polytopic model. The proposed fuzzy controller is called the T-S fuzzy region controller (TSFRC) where the controller rule has to stabilize all plant rules of the fuzzy region and guarantee the whole fuzzy system is asymptotically stable. The stability analysis is derived from Lyapunov stability criterion in which the robust compensation is considered and is expressed in terms of linear matrix inequalities. Comparing with parallel distributed compensation (PDC) designs, TSFRC is easy to be designed and to be implemented with simple hardware or microcontroller. Even if the controller rules are reduced, TSFRC is able to provide competent performances as well as PDC-based designs. Index Terms Fuzzy region concept, linear matrix inequality (LMI) and 2 control, Takagi Sugeno (T-S) fuzzy systems. I. INTRODUCTION OVER the past one decade, Takagi Sugeno (T-S) fuzzy control techniques have been applied to many nonlinear control problems [1] [6]. T-S fuzzy model consists of several linear subsystems, and it approximates a nonlinear model by using IF-THEN fuzzy rules. In the past, the majority of T-S fuzzy controller designs were developed by using the concept of parallel distributed compensation (PDC) [6] [9] and the Lyapunov stability criterion. This kind of problems can be converted into linear matrix inequalities (LMIs) which are solved by LMI optimization [9] [12]. Its obvious drawback is that the number of LMIs is increased rapidly when the fuzzy model involving lots of plant rules. Even if the PDC-based controller can be obtained, it is still difficult to implement with some simple hardware or a cheap microcontroller because the defuzzification procedure becomes complex. This paper attempts to solve these foregoing problems by combining fuzzy region concept and robust compensation. The concept of fuzzy region is employed to partition the original Manuscript received December 5, 2004; revised June 19, 2005 and August 23, 2005. This work was supported in part by the R.O.C. National Science Council under Grant NSC 93-2213-E-008-043. This paper was recommended by Associate Editor Y. Nishio. S. M. Wu, C. C. Sun, and H. Y. Chung are with the Department of Electrical Engineering, National Central University, Chung-li 320, Taiwan, R.O.C. (e-mail:hychung@ee.ncu.edu.tw). W. J. Chang is with the Department of Marine Engineering, National Taiwan Ocean University, Keelung 202, Taiwan, R.O.C. Digital Object Identifier 10.1109/TCSI.2006.883868 plant rules into several fuzzy regions [13] so that only one partial region is fired at the instant of each input vector being coming. This kind of fuzzy model is called the T-S fuzzy region model (TSFRM) and each fuzzy region can be regarded as a polytopic model. The TSFRC is designed to stabilize the TSFRM and to minimize the mixed norm of the closed-loop fuzzy region system. The controller rule of TSFRC corresponds to a robust controller because it has to stabilize several plant rules. For the closed-loop fuzzy system, the stability conditions with performances are derived from the Lyapunov criterion, which are expressed in terms of LMIs. It is important to emphasize that the controller rules of the TSFRC is adjustable. This feature allows us to determine the trade-off between hardware complexity and accurate performances. Finally, a numerical example is used to verify the validity and applicability of the proposed idea. II. PRELIMINARIES AND PROBLEMS DESCRIPTION A. Descriptions of T-S Fuzzy Systems A discrete nonlinear model can be represented as the following T-S fuzzy model by using modeling techniques which include both fuzzy inference rules and local analytic linear models Plant Rule is and is where and is the total number of IF-THEN rules; is the state vector, is the input vector, is the disturbance input vector, and are the controlled output vectors for and norm, respectively. is the premise variable vector, in which each element is a linear combination of states; is a standard fuzzy set, where and denotes the total number of membership functions in ; the parameter is defined as follows: where all elements of known dimensions. (1) are assumed to be of appropriate and 1057-7122/$20.00 2006 IEEE

WU et al.: DISCRETE NONLINEAR CONTROLLER DESIGN 2839 Based on a standard fuzzy inference method [9], the inferred fuzzy model of (1) is described by where and is defined as follows: The PDC concept [6], [14] offers a simple procedure in designing T-S fuzzy controllers. That is, the fuzzy controller shares the same fuzzy sets with the fuzzy model in the premise parts Controllter Rule is and is (4) The output of the fuzzy controller (4) is represented by (2) (3) (5) where and are scale factors; and denote the closed-loop transfer functions from to and, respectively. B. Stability and Performance Analyses for T-S Fuzzy Systems It is well known that Lyapunov stability criterion is a popular approach for achieving above objectives. The stability synthesis for the mixed control problem has been investigated by use of LMI optimization [15], [16]. One can infer the following lemmas for T-S fuzzy systems by extending LTI control techniques. Lemma 1 [17] ( norm): The closed-loop fuzzy system (6) is asymptotically stable with, if there exists positive definite matrices and yield and (10) (11) By substituting (5) into (2), the closed-loop fuzzy system is obtained (12) (13) where and are represented as (6) where the symbol denotes the transposed element for the symmetric position. Proof: Based on the theory of [10], the closed-loop system is asymptotically stable with and there exists a symmetric matrix, such that (7) (14) (15) Remark 1: The symbol denotes the feedback gains of each controller rule affecting its adjacent plant rules when membership functions overlap with each other, i.e., the th control rule interferes with the th plant rule except the pairs with,. In this paper, this phenomenon is called the rule interference effect. In many literatures, the design goal of T-S fuzzy control problems is to find feedback gains such that i) the closed-loop system (6) is asymptotically stable; ii) the following cost function is minimized: (8) (9) According to the matrix properties, two arbitrary matrices and which satisfy if. Hence, the inequalities (15) is equivalent to (16) Utilizing the Schur complement [10], [11], (16) can be rearranged as follows: (17) (18) Similarly, (14) is equivalent to (10) (11). The proof is completed.

2840 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006 Lemma 2 [17] ( norm): The closed-loop fuzzy system (6) is asymptotically stable with, if there exists a common positive definite matrix and such that the following inequalities hold: (19) After rearranging (26), we have Clearly, (27) is achieved if (27) (28) ccording to property of Schur complement, (28) can be rearranged as follows: (20) Proof: According to the definition of the performance, implies that there exists a quadratic function, and such that for all [10], (29) (21) Because of, we get (30) (22) Noting that and since, (21) can be rewritten as (23) To satisfy (23), one situation is to assume for all k. Therefore, (23) can be replaced with the following strictly condition: (24) According to the above statements, corresponds that (24) is satisfied. After rearranging (24), we get (25) Based on the definitions of and, (25) can be expanded as follows: (31) After multiplying a negative sign, (31) can be represented as follows: (32) where. After pre- and post-multiplying, the inequality (32) can be converted into the inequality (19). Following the same procedures, one can infer the inequality (20) for the situation on. The proof is completed. All inequalities of Lemma 1 and Lemma 2 are converted into the following LMI representations such that,,,, and can be solved by LMI solver. Theorem 1: The closed-loop fuzzy system (6) is said to be asymptotically stable with a minimal mixed norm if the following constraints are satisfied: Minimize such that subject to (26) where,. (33)

WU et al.: DISCRETE NONLINEAR CONTROLLER DESIGN 2841 (34) where the mutual system is defined below (35) (36) (37) (38) (39) (40) Fig. 1. Relationship between membership functions and fuzzy regions. end, three issues will be discussed in the following: (i) Conversion of general T-S fuzzy model into T-S Region-based Fuzzy Model (TSFRM) with fuzzy region concept [20], (ii) Definition of T-S Region-based Fuzzy Controller (TSFRC) with robust compensation [21] and (iii) Synthesis for the closed-loop fuzzy region systems. Definition 1: The membership functions of the fuzzy model (1) are defined in Fig. 1. It shows that the can be partitioned into several fuzzy regions by cutting the membership functions at the operating points of. The fuzzy region Region denotes the th region of involving the right-hand side of and the left-hand side of. It can be regarded as the crisp membership function defined as follows: (41) Region else (43) (42) Proof: Extending the proofs of above-mentioned lemmas, we let and the feedback gains of each rule is obtained as. Hence, following this sense the results (33) (39) can be obtained. Even if PDC-based design concept is very popular, the following shortcomings become much more conservative when the fuzzy system involving lots of rules: (i) The total number of LMIs will be increased rapidly so that the infeasible probability of LMI solver will be increased. (ii) The PDC-based fuzzy controller involving many IF-THEN rules is difficult to perform the defuzzification and the hardware realization. Recently, some scholars have proposed several approaches [18], [19] to overcome these shortcomings, which regard some model nonlinearities as specific model uncertainties. This kind of fuzzy models is called the fuzzy uncertainty model. Because its local models involve uncertainties, the rule interference effect becomes more complex and difficult to handle than general T-S fuzzy designs because these approaches have to add some extra constraints or redefine LMI representations. Therefore, the design procedure of these methods needs to be improved even if the total number of controller rules is reduced. III. MAIN RESULT A new fuzzy control structure is proposed in this paper that focuses on reducing the controller rules and the complexity in syntheses. The design concept is to partition the general T-S fuzzy model into several fuzzy regions, and then to design the feedback gains for each region with robust compensation. To this where. According to Definition 1, the general T-S fuzzy models can be converted into a TSFRM. A. T-S Fuzzy Region Model Region Plant Rule (44) where and is the amount of the fuzzy regions;, and is the serial fired number in each region; stands for the total number of the fired plant rules in. The parameter matrix, ] denotes the th subsystem in the th region plant rule where all matrices or vectors of are assumed to be of appropriate and known dimensions. After performing the defuzzification, the final outputs of the TSFRM are shown below (45)

2842 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006 Fig. 2. Example of illustrated fuzzy region concept (4-rule). Fig. 3. Example of illustrated fuzzy region concept (2-rule). TABLE I RELATIONSHIP BETWEEN 4-RULE TSFRM (H GENERAL T-S FUZZY MODEL (T ) ) AND TABLE II RELATIONSHIP BETWEEN 2-RULE TSFRM (H GENERAL T-S FUZZY MODEL (T ) ) AND compensated by a region controller rule. where Region Controller Rule is and is Region The final output of TSFRC is computed by (48) (49) (46) By substituting (49) into (45), the closed-loop fuzzy region system is shown as Region (47) Region is the grade of membership of in Region. Example: Assume a nonlinear model can be converted into a T-S fuzzy model where the nonlinear states are and. The model framework and membership functions are shown in Fig. 2, where means the th plant rule. The premise variable can be partitioned into four regions. (4-Rule TSFRM): Based on the definitions of TSFRM, it is known that involves,,, and. Similarly, involves,,, and. Therefore, the relationship between and is arranged in Table I. To further reduce the region plant rules, we can further combine the adjacent fuzzy regions into a new one. (2-rule TSFRM): If the membership functions of can be partitioned into two regions, the structure of TSFRM can be represented in Fig. 3 and its sub-models of each region plant rule are arranged in Table II. The control purpose is to design a fuzzy controller such that the TSFRM can be stabilized and the cost function (9) can be minimized. The fuzzy controller derived from the TSFRM is called the TSFRC, which means that each region plant rule is Remark 2: Because Region it has the following property: (50) is crisp membership function, while (51) while According to (51), one can find that (50) does not have the rule interference effect. Therefore, (50) can be rewritten as (52)

WU et al.: DISCRETE NONLINEAR CONTROLLER DESIGN 2843 where,,. Remark 3: Because the only one fuzzy region is fired at any instant, the property leads that the corresponds to the. The difference between and is that TSFRM only identifies the fired fuzzy region rather than acquires the exact value of. Because the controller rules of TSFRC have to stabilize all plant rules of the fired fuzzy region of the TSFRM, the difference does not cause any problem when replacing the PDC-based fuzzy controller with TSFRC. Now we use the following example to explain the relationship between and. Example: Suppose the magnitude of membership functions of Fig. 2 gives as,, and. It is easy to infer that only,,, and are fired and the grade of weight of each plant rule is,,,. According to the definitions of 4-rule TSFRM, one can find that only is fired, i.e., and. If we assume,,, and for the, one can find that the final output of general T-S fuzzy model and TSFRM are equivalent. B. Stability and Performance Analyses for T-S Fuzzy Region Systems Now we illustrate how to use Lyapunov stability criterion to synthesize the closed-loop fuzzy region system (52). Theorem 2: The closed-loop fuzzy region system (52) is said to be asymptotically stable and yields minimal mixed norm if the constraints shown in (53) (56), at the bottom of the page, are satisfied, where, and. Proof: Using Bounded Real Lemma [10], [17] and derivation of Lemma 2, one can infer that if is strictly Hurwitz and there exists a symmetric with (57) The left-hand side of inequalities (57) can be pre- and post- TABLE III TOTAL NUMBER OF LMIS FOR PDC AND TSFRC multipled by and, respectively, where. It leads to (58) Extending the result of Theorem 1, it is readily verified that if is stable and there exist and such that (59) (60) Based on the property of matrix, (60) is equal to the following inequalities: (61) (62) Furthermore, with the change of variables, and taking proper Schur complement for (58), (59), (61), and (62), one can obtain (53) (56). It should be noted that the proposed idea, combination of region switching concept and robust compensation, has the following advantages: 1) the total number of controller rules is adjustable; and 2) the rule interference effect of PDC-based designs can be cancelled. It gives that the decision variables and the total number of LMIs is greatly decreased. It proves that the proposed idea is able to improve the feasibility of LMI optimization. The comparison between the PDC-based design [7] [9] and the proposed one is arranged in Tables III and IV. Minimize subject to (53) (54) (55) (56)

2844 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006 TABLE IV COMPARISON OF CONTROLLER RULE REDUCTION METHODS Comparing the PDC-based approach with the proposed one, the total number of LMIs in some general cases is listed in Table III in which n denotes the total number of premise variables; and are the total number of LMIs for Theorem 1 and Theorem 2, respectively. IV. NUMERICAL EXAMPLE Consider a discrete nonlinear model as follows: (63a) (63b) is around and is around (64d) where the membership functions and the structure of fuzzy model (64) are shown in Fig. 2. The matrices of for are given below (63c) where and. Based on T-S fuzzy modeling techniques, the nonlinear model (63) can be represented as the following general T-S fuzzy model: is around and is around (64a) is around and is around (64b). is around and is around (64c)

WU et al.: DISCRETE NONLINEAR CONTROLLER DESIGN 2845 and (65d) where the relationship between and is as shown in Table II. In this case, each region controller rule has to stabilize four plant rules. After solving Theorem 2 with LMI solver, the feedback gains of TSFRC are where and Case 1: PDC-based Design (10-Rule): First, we design the fuzzy controller by using the PDC-based approach. One can obtain the following design results after solving all conditions of Theorem 1 with MATLAB s LMI control toolbox. Case 3: Region-Based Design (2-Region): By combining the adjacent regions, the original fuzzy model can be converted into the 2-region TSFRM (66a) (66b) The relationship between and is exhibited in Table III. Following the design procedures of Case 2, the TSFRC with 2 region controller rules are shown as Case 2: Region-based Design (4-Region): Based on the definitions of TSFRM, the general T-S fuzzy model (64) can be represented as the 4-rule TSFRM (65a) Case 4: Region-Based Design (1-Region): In this case, the TSFRM regards the general T-S fuzzy model as a polytopic model so that the TSFRC corresponds to a robust controller (67) (65b) (65c) where is equal to, i.e.. The 1-rule TSFRC is obtained as. The simulation results for the initial condition are shown in Figs. 4 7. The performance comparison for these four cases is arranged in Table V. From the controller complexity and synthesis point of view, the 2-rule TSFRC is highly recommended in this example, i.e., it only needs two controller rules to approximate the system performances of the PDC-based approach. From the above statements, there is two important thing should be noted that: 1) the system

2846 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006 Fig. 4. State and control responses (Case 1). Fig. 5. State and control responses (Case 2). performances are not absolutely related with the total number of controller rules because the computing efficiency of LMI solver is greatly degraded when it deals with many decision variables and LMIs simultaneously; 2) the design results of PDCbased approaches would be encumbered with the rule interference effect. V. CONCLUSION A new type of T-S fuzzy controller design approach has been proposed in this paper. According to the definition of the TSFRM, any general T-S fuzzy models can be represented as TSFRM, and each fuzzy region can be regarded as a polytopic model. Therefore, the robust compensation must be considered when designing the controller rule for each fuzzy region. Lyapunov stability criterion is employed to synthesize the TSFRC such that the closed-loop fuzzy region system is asymptotically stable and its mixed norm can be minimized. Comparing Theorem 2 with Theorem 1, one can find that the total number of IF-THEN rules of TSFRC is adjustable so that the total number of LMIs in TSFRC design is much more fewer than that of PDC-based designs, especially for complex fuzzy

WU et al.: DISCRETE NONLINEAR CONTROLLER DESIGN 2847 Fig. 6. State and control responses (Case 3). Fig. 7. State and control responses (Case 4). TABLE V COMPARISON BETWEEN PDC-BASED DESIGN AND THE PROPOSED APPROACH systems. Finally, it is important to emphasize that the proposed approach provides not only an easy design procedure but also a simple hardware implementation. ACKNOWLEDGMENT The authors wish to express their sincere gratitude to the anonymous reviewers and to the associate editor who gave so

2848 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 53, NO. 12, DECEMBER 2006 many constructive comments, criticisms and suggestions, which led to made substantial improvements to this manuscript. REFERENCES [1] Z. Li, J. B. Park, Y. H. Joo, Y. H. Choi, and G. R. Chen, Anticontrol of chaos for discrete Ts fuzzy-systems, IEEE Trans. Circuit Syst. I, Fundam. Theory Appl., vol. 49, no. 2, pp. 249 253, Feb. 2002. [2] Y. Park, M. J. Tahk, and J. Park, Optimal stabilization of Takagi sugeno fuzzy-systems with application to spacecraft control, J. Guid. Contr. Dynam., vol. 24, no. 4, pp. 767 777, 2001. [3] C. S. Tseng, B. S. Chen, and H. J. Uang, Fuzzy tracking control design for nonlinear dynamic-systems via T-S fuzzy model, IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 381 392, Mar. 2001. [4] X. J. Ma and Z. Q. Sun, Output tracking and regulation of nonlinearsystem based on Takagi sugeno fuzzy model, IEEE Trans. Syst. Man. Cybern. B, vol. 30, no. 1, pp. 47 59, Jan. 2000. 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Liu, Parameter-dependent lyapunov functional for stability of time-delay systems with polytopic-type uncertainties, IEEE Trans. Autom. Contr., vol. 49, no. 5, pp. 828 832, May 2004. Sheng-Ming Wu received the B.S. degree in mechanical engineering from Yuan-Ze University, Taoyuan, Taiwan, R.O.C.,, and the M.S. degree in the marine engineering from the National Taiwan Ocean University, Taiwan, R.O.C., in 2001, and 2003, respectively. He is currently working toward the Ph.D. degree in electrical engineering at the National Central University, Chung-li, Taiwan. R.O.C. His research interests focus on fuzzy control and control system theory. Chein-Chung Sun was born in Kaohsiung, Taiwan, R.O.C. He received the Ph. D. degree in electrical engineering from the National Central University, Tainan, Taiwan, R.O.C., in 2005. He is currently a Research Assistant in the Division of Energy Storage Materials and Technologies, which is subordinate to Material Research Laboratories (MRL) of Industrial Technology Research Institute (ITRI). His research interests focus on nonlinear control, robust control, fuzzy control, intelligent control, battery manager systems, and battery charging/discharging strategies. Hung-Yuan Chung was born in Ping-Tung, Taiwan, R.O.C. He received the Ph.D. degree in electrical engineering from the National Cheng Kung University (NCKU), Tainan, Taiwan, R.O.C., in 1987. In 1977, he was affiliated with the Chung-Shan Institute of Science and Technology as a Research Assistant. In 1982, he became an Assistant Scientist. In 1984, he was a Lecturer in the Department of Mechanical Engineering, NCKU while pursuing his doctoral degree. In August 1987, he joined the Department of Electrical Engineering at the National Central University, Chung-li, Taiwan, R.O.C., as an Associate Professor. In August 1992, he was promoted to Professor. In addition, he is a registered professional Engineer in R.O.C. His research and teaching interests include system theory and control, adaptive control, fuzzy control, neural network applications, and microcomputer-based control applications. Dr. Chung is a life member of the CIEE and the CIE. He received the outstanding Electrical Engineer award of the Chinese Institute of Electrical Engineering in October 2003. Wen-Jer Chang received the B.S. degree in marine engineering (major course) and electronic engineering (minor course) from National Taiwan Ocean University, Keelung, Taiwan, R.O.C., in 1986, and the M.S. degree in the computer science and electronic engineering and the Ph.D. degree in electrical engineering from the National Central University, Chung-li, Taiwan, R.O.C., in 1990 and 1995, respectively. He has over 110 publications including 55 journal papers. His recent research interests are fuzzy control, robust control, performance constrained control. Since 1995, he has been with National Taiwan Ocean University, where he is currently a Full Professor in the Department of Marine Engineering. Dr, Chang is currently a member of the CIEE, CACS, CSFAT, and SNAME. Since 2003, he was listed in the Marquis Who s Who in Science and Engineering. In 2005, he was selected as an excellent teacher of the National Taiwan Ocean University.