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Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach Kazuo Tanaka, Hua O. Wang Copyright ᮊ 2001 John Wiley & Sons, Inc. Ž. Ž . ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic CHAPTER 2 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION Recent years have witnessed rapidly growing popularity of fuzzy control systems in engineering applications. The numerous successful applications of fuzzy control have sparked a flurry of activities in the analysis and design of fuzzy control systems. In this book, we introduce a wide range of analysis and design tools for fuzzy control systems to assist control researchers and engineers to solve engineering problems. The toolkit developed in this book is based on the framework of the Takagi-Sugeno fuzzy model and the so-called parallel distributed compensation, a controller structure devised in accordance with the fuzzy model. This chapter introduces the basic concepts, analysis, and design procedures of this approach. This chapter starts with the introduction of the Takagi-Sugeno fuzzy Ž. model T-S fuzzy model followed by construction procedures of such models. Then a model-based fuzzy controller design utilizing the concept of ‘‘parallel distributed compensation’’ is described. The main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system. The design procedure is conceptually simple and natural. Moreover, it is shown in this chapter that the stability analysis and control design Ž. problems can be reduced to linear matrix inequality LMI problems. The design methodology is illustrated by application to the problem of balancing and swing-up of an inverted pendulum on a cart. The focus of this chapter is on the basic concept of techniques of stability wx analysis via LMIs 14, 15, 24 . The more advanced material on analysis and design involving LMIs will be given in Chapter 3. 5 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION 6 2.1 TAKAGI-SUGENO FUZZY MODEL The design procedure describing in this book begins with representing a given nonlinear plant by the so-called Takagi-Sugeno fuzzy model. The fuzzy wx model proposed by Takagi and Sugeno 7 is described by fuzzy IF-THEN rules which represent local linear input-output relations of a nonlinear system. The main feature of a Takagi-Sugeno fuzzy model is to express the Ž. local dynamics of each fuzzy implication rule by a linear system model. The overall fuzzy model of the system is achieved by fuzzy ‘‘blending’’ of the linear system models. In this book, the readers will find that many nonlinear dynamic systems can be represented by Takagi-Sugeno fuzzy models. In fact, it is proved that Takagi-Sugeno fuzzy models are universal approximators. The details will be discussed in Chapter 14. The ith rules of the T-S fuzzy models are of the following forms, where CFS and DFS denote the continuous fuzzy system and the discrete fuzzy system, respectively. Continuous Fuzzy System: CFS Model Rule i: Ž. Ž. IF ztis M and иии and ztis M , 1 i1 pip x t s Ax t q Bu t , Ž. Ž. Ž. ˙ ii THEN i s 1,2, .,r.2.1 Ž. ½ y t s Cx t , Ž. Ž. i Discrete Fuzzy System: DFS Model Rule i: Ž. Ž. IF ztis M and иии and ztis M , 1 i1 pip x t q 1 s Ax t q Bu t , Ž . Ž. Ž. ii THEN i s 1, 2, . . . , r.2.2 Ž. ½ y t s Cx t , Ž. Ž. i Ž. n Here, M is the fuzzy set and r is the number of model rules; x t g R is ij Ž. m Ž. q the state vector, u t g R is the input vector, y t g R is the output n=nn=mq=n Ž. Ž. vector, A g R , B g R , and C g R ; zt, ., ztare known ii i1 p premise variables that may be functions of the state variables, external Ž. disturbances, andror time. We will use z t to denote the vector containing Ž. Ž. all the individual elements zt, ., zt. It is assumed in this book that the 1 p Ž. premise variables are not functions of the input variables u t . This assump- tion is needed to avoid a complicated defuzzification process of fuzzy wx controllers 12 . Note that stability conditions derived in this book can be TAKAGI-SUGENO FUZZY MODEL 7 applied even to the case that the premise variables are functions of the input Ž. Ž. variables u t . Each linear consequent equation represented by Axt q i Ž. But is called a ‘‘subsystem.’’ i ŽŽ. Ž Given a pair of x t , u t , the final outputs of the fuzzy systems are inferred as follows: CFS r w z t Ax t q Bu t Ä4 Ž. Ž. Ž. Ž. Ý iii i s1 x t s Ž. ˙ r w z t Ž. Ž. Ý i i s1 r s h z t Ax t q Bu t ,2.3 Ä4 Ž. Ž. Ž. Ž . Ž. Ý iii i s1 r w z t Cx t Ž. Ž. Ž. Ý ii i s1 y t s Ž. r w z t Ž. Ž. Ý i i s1 r s h z t Cx t .2.4 Ž. Ž. Ž . Ž. Ý ii i s1 DFS r w z t Ax t q Bu t Ä4 Ž. Ž. Ž. Ž. Ý iii i s1 x t q 1 s Ž. r w z t Ž. Ž. Ý i i s1 r s h z t Ax t q Bu t ,2.5 Ä4 Ž. Ž. Ž. Ž . Ž. Ý iii i s1 r w z t Cx t Ž. Ž. Ž. Ý ii i s1 y t s Ž. r w z t Ž. Ž. Ý i i s1 r s h z t Cx t ,2.6 Ž. Ž. Ž . Ž. Ý ii i s1 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION 8 where z t s ztztиии zt, Ž. Ž. Ž. Ž. 12 p p w z t s Mzt, Ž. Ž. Ž. Ž. Ł iijj j s1 w z t Ž. Ž. i h z t s 2.7 Ž. Ž . Ž. r i w z t Ž. Ž. Ý i i s1 ŽŽ Ž. for all t. The term Mzt is the grade of membership of zt in M . ij j j ij Since r ° w z t ) 0, Ž. Ž. Ý i ~ 2.8 Ž. i s1 ¢ w z t G 0, i s 1,2, .,r , Ž. Ž. i we have r ° h z t s 1, Ž. Ž. Ý i ~ 2.9 Ž. i s1 ¢ h z t G 0, i s 1,2, .,r , Ž. Ž. i for all t. Example 1 Assume in the DFS that p s n, zts xt, zts xty 1 , ., zts xty n q 1. Ž. Ž. Ž. Ž . Ž. Ž . 12 n Then, the model rules can be represented as follows. Model Rule i: Ž. Ž . IF xt is M and иии and xty n q 1isM , i1 in x t q 1 s Ax t q Bu t , Ž . Ž. Ž. ii THEN i s 1,2, .,r, ½ y t s Cx t , Ž. Ž. i Ž. w Ž. Ž . Ž .x T where x t s xt xty 1 иии xty n q 1. Remark 1 The Takagi-Sugeno fuzzy model is sometimes referred as the Ž. Takagi-Sugeno-Kang fuzzy model TSK fuzzy model in the literature. In this Ž. Ž. book, the authors do not refer to 2.1 and 2.2 as the TSK fuzzy model. The CONSTRUCTION OF FUZZY MODEL 9 reason is that this type of fuzzy model was originally proposed by Takagi and wx w x Sugeno in 7 . Following that, Kang and Sugeno 8, 9 did excellent work on identification of the fuzzy model. From this historical background, we feel Ž. Ž. that 2.1 and 2.2 should be addressed as the Takagi-Sugeno fuzzy model. On the other hand, the excellent work on identification by Kang and Sugeno is best referred to as the Kang-Sugeno fuzzy modeling method. In this book the authors choose to distinguish between the Takagi-Sugeno fuzzy model and the Kang-Sugeno fuzzy modeling method. 2.2 CONSTRUCTION OF FUZZY MODEL Figure 2.1 illustrates the model-based fuzzy control design approach dis- cussed in this book. To design a fuzzy controller, we need a Takagi-Sugeno fuzzy model for a nonlinear system. Therefore the construction of a fuzzy model represents an important and basic procedure in this approach. In this section we discuss the issue of how to construct such a fuzzy model. In general there are two approaches for constructing fuzzy models: Ž. 1. Identification fuzzy modeling using input-output data and 2. Derivation from given nonlinear system equations. There has been an extensive literature on fuzzy modeling using input-out- wx put data following Takagi’s, Sugeno’s, and Kang’s excellent work 8, 9 . The procedure mainly consists of two parts: structure identification and parame- ter identification. The identification approach to fuzzy modeling is suitable Fig. 2.1 Model-based fuzzy control design. TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION 10 for plants that are unable or too difficult to be represented by analytical andror physical models. On the other hand, nonlinear dynamic models for mechanical systems can be readily obtained by, for example, the Lagrange method and the Newton-Euler method. In such cases, the second approach, which derives a fuzzy model from given nonlinear dynamical models, is more appropriate. This section focuses on this second approach. This approach utilizes the idea of ‘‘sector nonlinearity,’’ ‘‘local approximation,’’ or a combi- nation of them to construct fuzzy models. 2.2.1 Sector Nonlinearity The idea of using sector nonlinearity in fuzzy model construction first wx appeared in 10 . Sector nonlinearity is based on the following idea. Consider Ž. Ž Ž Ž. a simple nonlinear system xt s fxt , where f 0 s 0. The aim is to find ˙ Ž. Ž Ž wxŽ. the global sector such that xt s fxt g aaxt. Figure 2.2 illustrates ˙ 12 the sector nonlinearity approach. This approach guarantees an exact fuzzy model construction. However, it is sometimes difficult to find global sectors for general nonlinear systems. In this case, we can consider local sector nonlinearity. This is reasonable as variables of physical systems are always bounded. Figure 2.3 shows the local sector nonlinearity, where two lines Ž. become the local sectors under yd - xt - d. The fuzzy model exactly Ž. represents the nonlinear system in the ‘‘local’’ region, that is, yd - xt - d. The following two examples illustrate the concrete steps to construct fuzzy models. Fig. 2.2 Global sector nonlinearity. CONSTRUCTION OF FUZZY MODEL 11 Fig. 2.3 Local sector nonlinearity. Example 2 Consider the following nonlinear system: xt yxtq xtx 3 t Ž. Ž. Ž. Ž. ˙ 1112 s . 2.10 Ž. 3 ž/ž / xt yxtq 3 q xt xt Ž. Ž. Ž. Ž. Ž. ˙ 22 21 Ž. wx Ž. wx For simplicity, we assume that xtgy1, 1 and xtgy1, 1 . Of 12 Ž. Ž. course, we can assume any range for xt and xtto construct a fuzzy 12 model. Ž. Equation 2.10 can be written as 2 y1 xtx t Ž. Ž. 12 x t s x t , Ž. Ž. ˙ 2 3 q xt xt y1 Ž. Ž. Ž. 21 Ž. w Ž. Ž.x T Ž. 2 Ž. Ž Ž 2 Ž. where x t s xtxt and xtx tand 3 q xtxt are nonlinear 12 12 2 1 Ž. Ž. 2 Ž. Ž. Ž terms. For the nonlinear terms, define zt' xtx t and zt' 3 q 112 2 Ž 2 Ž. xtxt. Then, we have 21 y1 zt Ž. 1 x t s x t . Ž. Ž. ˙ zt y1 Ž. 2 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION 12 Ž. Ž. Next, calculate the minimum and maximum values of ztand ztunder 12 Ž. wx Ž. wx xtgy1, 1 and xtgy1, 1 . They are obtained as follows: 12 max zts 1, min ztsy1, Ž. Ž. 11 Ž. Ž. Ž. Ž. xt, xt xt, xt 12 12 max zts 4, min zts 0. Ž. Ž. 22 Ž. Ž. Ž. Ž. xt, xt xt, xt 12 12 Ž. Ž. From the maximum and minimum values, ztand ztcan be represented 12 by zts xtx 2 t s Mzt и 1 q Mzt и y1, Ž. Ž. Ž. Ž. Ž. Ž . Ž. Ž. 112 11 21 zts 3 q xt x 2 t s Nz t и 4 q Nzt и 0, Ž. Ž. Ž. Ž. Ž. Ž. Ž.Ž. 2211222 where Mzt q Mzt s 1, Ž. Ž. Ž. Ž. 11 21 Nzt q Nzt s 1. Ž. Ž. Ž. Ž. 12 2 2 Therefore the membership functions can be calculated as ztq 11y zt Ž. Ž. 11 Mzt s , Mzt s , Ž. Ž. Ž. Ž. 11 21 22 zt 4 y zt Ž. Ž. 22 Nzt s , Nzt s . Ž. Ž. Ž. Ž. 12 22 44 We name the membership functions ‘‘Positive,’’ ‘‘Negative,’’ ‘‘Big,’’ and Ž. ‘‘Small,’’ respectively. Then, the nonlinear system 2.10 is represented by the following fuzzy model. Model Rule 1: Ž. Ž. IF ztis ‘‘Positive’’ and ztis ‘‘Big,’’ 12 Ž. Ž. THEN x t s Axt. ˙ 1 Model Rule 2: Ž. Ž. IF ztis ‘‘Positive’’ and ztis ‘‘Small,’’ 12 Ž. Ž. THEN x t s Axt . ˙ 2 Model Rule 3: Ž. Ž. IF ztis ‘‘Negative’’ and ztis ‘‘Big,’’ 12 CONSTRUCTION OF FUZZY MODEL 13 Ž Ž Ž Ž Fig. 2.4 Membership functions Mzt and Mzt. 11 21 ŽŽ ŽŽ Fig. 2.5 Membership functions Nzt and Nzt. 12 22 Ž. Ž. THEN x t s Axt . ˙ 3 Model Rule 4: Ž. Ž. IF ztis ‘‘Negative’’ and ztis ‘‘Small,’’ 12 Ž. Ž. THEN x t s Axt . ˙ 4 Here, y11 y11 A s , A s , 12 4 y10y1 y1 y1 y1 y1 A s , A s . 34 4 y10y1 Figures 2.4 and 2.5 show the membership functions. The defuzzification is carried out as 4 x t s h z t Ax t , Ž. Ž. Ž. Ž. ˙ Ý ii i s1 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION 14 where h z t s Mzt = Nz t , Ž. Ž. Ž. Ž. Ž. Ž. 11112 h z t s Mzt = Nzt , Ž. Ž. Ž. Ž. Ž. Ž. 21122 h z t s Mzt = Nz t , Ž. Ž. Ž. Ž. Ž. Ž. 32112 h z t s Mzt = Nzt. Ž. Ž. Ž. Ž. Ž. Ž. 42122 This fuzzy model exactly represents the nonlinear system in the region wxwx y1, 1 = y1, 1 on the x -x space. 12 wx Example 3 The equations of motion for the inverted pendulum 21 are xts xt, Ž. Ž. ˙ 12 g sin xt y amlx 2 t sin 2 xtr2 y a cos xt ut Ž. Ž. Ž. Ž. Ž. Ž. Ž . Ž. 121 1 xts , Ž. ˙ 2 2 4lr3 y aml cos xt Ž. Ž. 1 2.11 Ž. Ž. Ž . where xtdenotes the angle in radians of the pendulum from the vertical 1 Ž. 2 and xt is the angular velocity; g s 9.8 mrs is the gravity constant, 2 m is the mass of the pendulum, M is the mass of the cart, 2 l is the length Ž. of the pendulum, and u is the force applied to the cart in newtons ; Ž. a s 1r m q M . Ž. Equation 2.11 is rewritten as 1 xts Ž. ˙ 2 2 4lr3 y aml cos xt Ž. Ž. 1 = amlx t sin 2 xt Ž. Ž. Ž. 21 g sin xt y xty a cos xt ut . Ž. Ž. Ž. Ž. Ž. Ž. 121 ž/ 2 2.12 Ž. Define 1 zt' , Ž. 1 2 4lr3 y aml cos xt Ž. Ž. 1 zt' sin xt , Ž. Ž. Ž. 21 zt' xtsin 2 xt , Ž. Ž. Ž. Ž. 32 1 zt' cos xt , Ž. Ž. Ž. 41 Ž. Ž . Ž. wx where xtgy ␲ r2, ␲ r2 and xtgy ␣ , ␣ . Note that the system is 12 Ž. uncontrollable when xts " ␲ r2. To maintain controllability of the fuzzy 1 [...]... hold for continuous-time systems as well Instead of using the Lyapunov inequality for discrete-time systems, we should use the Lyapunov inequality for continuous-time systems, AT P q PA - 0 In the next section, we apply the PDC approach to a continuous-time system 2.6 APPLICATION: INVERTED PENDULUM ON A CART To illustrate the PDC approach, consider the problem of balancing and swing-up of an inverted... is, there is only one IF-THEN rule; Ž2.5 becomes a linear time-invariant system, x Ž t q 1 s Ax Ž t q Bu Ž t Ž 2.36 ORIGIN OF THE LMI-BASED DESIGN APPROACH 37 For a given control gain F, using standard stability theory for linear time-invariant systems or Theorem 2, the system Ž2.36 is Žquadratically stable if there exists P ) 0 such that Ä A y BF 4 T P Ä A y BF 4 y P - 0 Ž 2.37 The control... easily extend the LMI-based control design approach to multiplerule Ž r ) 1 cases of the Takagi-Sugeno fuzzy models For instance, the quadratic stabilizability of the Takagi-Sugeno fuzzy models via a linear state feedback can be cast as the following LMI problem in X and M: X ) 0, X Ž A i X y Bi M Ž A i X y Bi M X T ) 0, i s 1, 2, , r , with the state feedback gain F s MXy1 The LMI-based control design... fuzzy controller from a given T-S fuzzy model To realize the PDC, a controlled object Žnonlinear system is first represented by a T-S fuzzy model We emphasize that many real systems, for example, mechanical systems and chaotic systems, can be and have been represented by T-S fuzzy models In the PDC design, each control rule is designed from the corresponding rule of a T-S fuzzy model The designed fuzzy... , THEN uŽ t s yFi x Ž t , i s 1, 2, , r 2.4 A MOTIVATING EXAMPLE In this chapter, for brevity only results for discrete-time systems are presented The results, however, also hold for continuous-time systems subject to some minor modifications A MOTIVATING EXAMPLE 27 The open-loop system of Ž2.5 is r x Ž t q 1 s Ý hi Ž z Ž t A i x Ž t Ž 2.24 is1 A sufficient stability condition, derived by... fuzzy system Ž2.24., the lack of systematic procedures to find a common positive definite matrix P has long been recognized Most of the time a trial-and-error type of procedure has been used w2, 23x In w13x a procedure to construct a common P is given for second-order fuzzy systems, that is, the dimension of state n s 2 We first pointed out in w14, 15, 24x that the common P problem can be solved efficiently... design problems via parallel distributed compensations 2.5 ORIGIN OF THE LMI-BASED DESIGN APPROACH This section gives the origin of the control design approach, which forms the core subject of this book, that is, the LMI-based design approach The objective here is to illustrate the basic ideas w24x of stability analysis and 30 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED COMPENSATION Fig 2.14 Basin... , Ž 2.29 is1 i-j where Gi j s Ä A i y Bi Fj 4 q Ä A j y B j Fi 4 2 , i - j s.t hi l h j / ␾ Therefore we have the following sufficient condition THEOREM 3 The equilibrium of a fuzzy control system Ž2.27 is globally asymptotically stable if there exists a common positi®e definite matrix P such that the following two conditions are satisfied: Ä A i y Bi Fi 4 T P Ä A i y Bi Fi 4 y P - 0, i s 1, 2, ... a common positi®e definite matrix P such that the following two conditions are satisfied: Ä A i y Bi Fi 4 T P Ä A i y Bi Fi 4 y P - 0, i s 1, 2, , r GiTj PGi j y P - 0, i - j F r s.t hi l h j / ␾ For the meaning of the notation i - j F r s.t Chapter 1 Ž 2.30 Ž 2.31 h i l h j / ␾ , see Remark 7 The conditions of Theorem 3 are more relaxed than those of Theorem 2 The control design problem is to... we may simply define triangular-type membership functions On the other hand, in the fuzzy model in Example 3, the membership functions are obtained so as to exactly represent the nonlinear dynamics The following remark addresses the important issue of approximating nonlinear systems via T-S models PARALLEL DISTRIBUTED COMPENSATION 25 Fig 2.11 Membership functions of two-rule model Remark 5 Section 2.2 . John Wiley & Sons, Inc. Ž. Ž . ISBNs: 0-4 7 1-3 232 4-1 Hardback ; 0-4 7 1-2 245 9-6 Electronic CHAPTER 2 TAKAGI-SUGENO FUZZY MODEL AND PARALLEL DISTRIBUTED. local sectors under yd - xt - d. The fuzzy model exactly Ž. represents the nonlinear system in the ‘‘local’’ region, that is, yd - xt - d. The following two