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 4 FUZZY OBSERVER DESIGN In practical applications, the state of a system is often not readily available. Under such circumstances, the question arises whether it is possible to determine the state from the system response to some input over some wx period of time. For linear systems, a linear observer 1 provides an affirma- tive answer if the system is observable. Likewise, a systematic design method of fuzzy regulators and fuzzy observers plays an important role for fuzzy control systems. This chapter presents the concept of fuzzy observers and two wx design procedures for fuzzy observer-based control 2, 3 . In linear system theory, one of the most important results on observer design is the so-called separation principle, that is, the controller and observer design can be carried out separately without compromising the stability of the overall closed-loop system. In this chapter, it is shown that a similar separation principle also holds for a large class of fuzzy control systems. 4.1 FUZZY OBSERVER Up to this point we have mainly dealt with LMI-based fuzzy control designs involving state feedback. In real-world control problems, however, it is often the case that the complete information of the states of a system is not always available. In such cases, one need to resort to output feedback design methods such as observer-based designs. This chapter presents fuzzy ob- server design methodologies involving state estimation for T-S fuzzy models. Alternatively, output feedback design can be treated in the framework of dynamic feedback, which is the subject of Chapter 12. 83 FUZZY OBSERVATION DESIGN 84 wx As in all observer designs, fuzzy observers 4 are required to satisfy x t y x t ™ 0ast ™ ϱ, Ž. Ž. ˆ Ž. where x t denotes the state vector estimated by a fuzzy observer. This ˆ Ž. Ž. condition guarantees that the steady-state error between x t and x t ˆ converges to 0. As in the case of controller design, the PDC concept is employed to arrive at the following fuzzy observer structures: CFS Obser©er Rule i Ž. Ž. IF ztis M and иии and ztis M 1 i1 pip THEN ˆ x t s Ax t q Bu t q Kyt y y t , Ž. Ž. Ž. Ž. Ž. Ž. ˙ˆ ˆ iii y t s Cx t , i s 1,2, .,r.4.1 Ž. Ž. Ž . ˆˆ i DFS Obser©er Rule i Ž. Ž. IF ztis M and иии and ztis M 1 i1 pip THEN x t q 1 s Ax t q Bu t q Kyt y y t , Ž . Ž. Ž. Ž. Ž. Ž. ˆˆ ˆ iii y t s Cx t , i s 1,2, .,r.4.2 Ž. Ž. Ž . ˆˆ i The fuzzy observer has the linear state observer’s laws in its consequent Ž. Ž. parts. The steady-state error between x t and x t will be discussed in the ˆ next section. 4.2 DESIGN OF AUGMENTED SYSTEMS This section presents LMI-based designs for an augmented system containing both the fuzzy controller and observer. The dependence of the premise variables on the state variables makes it necessary to consider two cases for fuzzy observer design: Ž. Ž. Case A zt, ., ztdo not depend on the state variables estimated by a 1 p fuzzy observer. DESIGN OF AUGMENTED SYSTEMS 85 Ž. Ž. Case B zt, ., ztdepend on the state variables estimated by a fuzzy 1 p observer. Obviously the stability analysis and design of the augmented system for Case A are more straightforward, whereas the stability analysis and design for Case B are complicated since the premise variables depend on the state variables, which have to estimated by a fuzzy observer. This fact leads to Ž.Ž . Ž.Ž . significant difference between z t Case A and z t Case B in the design ˆ of fuzzy observer and controller. 4.2.1 Case A The fuzzy observer for Case A is represented as follows: CFS r w z t Ax t q Bu t q Kyt y y t Ä4 Ž. Ž. Ž. Ž. Ž. Ž. Ž . ˆˆ Ý iiii i s1 ˆ x t s Ž. ˙ r w z t Ž. Ž. Ý i i s1 r s h z t Ax t q Bu t q Kyt y y t ,4.3 Ä4 Ž. Ž. Ž. Ž. Ž. Ž . Ž. Ž . ˆˆ Ý iiii i s1 r y t s h z t Cx t .4.4 Ž. Ž. Ž. Ž . Ž. ˆˆ Ý ii i s1 DFS r w z t Ax t q Bu t q Kyt y y t Ä4 Ž. Ž. Ž. Ž. Ž. Ž. Ž . ˆˆ Ý iiii i s1 x t q 1 s Ž. ˆ r w z t Ž. Ž. Ý i i s1 r s h z t Ax t q Bu t q Kyt y y t ,4.5 Ä4 Ž. Ž. Ž. Ž. Ž. Ž . Ž. Ž . ˆˆ Ý iiii i s1 r y t s h z t Cx t .4.6 Ž. Ž. Ž. Ž . Ž. ˆˆ Ý ii i s1 ŽŽ We use the same weight w z t as that of the ith rule of the fuzzy models i Ž. Ž. Ž. Ž. 2.3 and 2.4 , and 2.5 and 2.6 . The fuzzy observer design is to determine the local gains K in the consequent parts. i In the presence of the fuzzy observer for Case A, the PDC fuzzy controller FUZZY OBSERVATION DESIGN 86 Ž. takes on the following form, instead of 2.23 : r w z t Fx t Ž. Ž. Ž. ˆ Ý ii r i s1 u t sy sy h z t Fx t .4.7 Ž. Ž. Ž. Ž . Ž. ˆ Ý r ii i s1 w z t Ž. Ž. Ý i i s1 Ž. Ž.Ž. Combining the fuzzy controller 4.7 and the fuzzy observers 4.3 ᎐ 4.6 Ž. Ž. Ž. and denoting e t s x t y x t , we obtain the following system representa- ˆ tions: CFS rr x t s h z thz t A y BF x t q BFe t , Ž. Ž. Ž. Ž. Ž. Ž.Ž. Ä4 ˙ Ž. ÝÝ ij iij ij i s1 js1 rr e t s h z thz t A y KC e t . Ä4 Ž. Ž. Ž. Ž. Ž.Ž. ˙ ÝÝ ij iij i s1 js1 DFS rr x t q 1 s h z thz t A y BF x t q BFe t , Ž . Ž. Ž. Ž. Ž. Ž.Ž. Ä4 Ž. ÝÝ ij iij ij i s1 js1 rr e t q 1 s h z thz t A y KC e t . Ä4 Ž . Ž. Ž. Ž. Ž.Ž. ÝÝ ij iij i s1 js1 Therefore, the augmented systems are represented as follows: CFS rr x t s h z thz t Gx t Ž. Ž. Ž. Ž. Ž.Ž. ˙ ÝÝ aijija i s1 js1 r s h z thz t Gx t Ž. Ž. Ž. Ž.Ž. Ý ii iia i s1 r G q G ij ji q 2 h z thz t x t ,4.8 Ž. Ž. Ž. Ž . Ž.Ž. ÝÝ ij a 2 i s1 i-j DFS rr x t q 1 s h z thz t Gx t Ž . Ž. Ž. Ž. Ž.Ž. ÝÝ aijija i s1 js1 r s h z thz t Gx t Ž. Ž. Ž. Ž.Ž. Ý ii iia i s1 r G q G ij ji q 2 h z thz t x t ,4.9 Ž. Ž. Ž. Ž . Ž.Ž. ÝÝ ij a 2 i s1 i-j DESIGN OF AUGMENTED SYSTEMS 87 where x t Ž. x t s , Ž. a e t Ž. A y BF BF iij ij G s . 4.10 Ž. ij 0 A y KC iij Ž. Ž. By applying Theorems 7 and 8 to the augmented system 4.8 and 4.9 , respectively, we arrive at the following theorems. wx THEOREM 16 CFS The equilibrium of the augmented system described by Ž. 4.8 is globally asymptotically stable if there exists a common positi®e definite matrix P such that G T P q PG - 0, 4.11 Ž. ii ii T G q GGq G ij ji ij ji P q P - 0, ž/ž/ 22 i - j s.t. h l h / ␾ . 4.12 Ž. ij Proof. It follows directly from Theorem 7. wx THEOREM 17 DFS The equilibrium of the augmented system described by Ž. 4.9 is globally asymptotically stable if there exists a common positi®e definite matrix P such that G T PG y P - 0, 4.13 Ž. ii ii T G q GGq G ij ji ij ji P y P - 0, ž/ž/ 22 i - j s.t. h l h / ␾ . 4.14 Ž. ij Proof. It follows directly from Theorem 8. Recall that Theorems 9 and 10 represent less conservative conditions than those of Theorems 7 and 8. Therefore, by applying Theorems 9 and 10 to Ž. Ž. 4.8 and 4.9 , respectively, we can obtain the following less conservative conditions: FUZZY OBSERVATION DESIGN 88 wx THEOREM 18 CFS The equilibrium of the augmented system described by Ž. 4.8 is globally asymptotically stable if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q such that G T P q PG q s y 1 Q - 0, 4.15 Ž. Ž. ii ii T G q GGq G ij ji ij ji P q P y Q F 0, ž/ž/ 22 i - j s.t. h l h / ␾ , 4.16 Ž. ij where s ) 1. Proof. It follows directly from Theorem 9. wx THEOREM 19 DFS The equilibrium of the augmented system described by Ž. 4.9 is globally asymptotically stable if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q such that G T PG y P q s y 1 Q - 0, 4.17 Ž. Ž. ii ii T G q GGq G ij ji ij ji P y P y Q F 0, ž/ž/ 22 i - j s.t. h l h / ␾ , 4.18 Ž. ij where s ) 1. Proof. It follows directly from Theorem 10. As a further refinement, we can incorporate the decay rate condition into the augmented systems as follows: ˙ ŽŽ ŽŽ CFS: The condition that V x t Fy2 ␣ V x t for all trajectories is aa equivalent to G T P q PG q s y 1 Q q 2 ␣ P - 0, 4.19 Ž. Ž. ii ii T G q GGq G ij ji ij ji P q P y Q q 2 ␣ P F 0, ž/ž/ 22 i - j s.t. h l h / ␾ , 4.20 Ž. ij where ␣ ) 0. Ž Ž Ž 2 .Ž Ž DFS: The condition that ⌬V x t F ␣ y 1 V x t for all trajectories is aa equivalent to G T PG y ␣ 2 P q s y 1 Q - 0, 4.21 Ž. Ž. ii ii T G q GGq G ij ji ij ji 2 P y ␣ P y Q F 0, ž/ž/ 22 i - j s.t. h l h / ␾ , 4.22 Ž. ij where ␣ - 1. DESIGN OF AUGMENTED SYSTEMS 89 Next we consider the controller and observer design problem. The ap- proach is to transform the conditions above for CFS and DFS into LMI ones so as to directly determine the feedback gains F and the observer gains K . ii The transformation procedure can be similarly applied to all theorems in this section. In the following, we present some representative results. Other cases are left as exercises for the readers. Design Procedure for Case A: CFS Assume that the number of rules that fire for all t is less than or equal to s, where 1 - s F r. The largest bound on the decay rate that we can find using a quadratic Lyapunov function can be found by solving the GEVP. maximize ␣ P , P , Y , Q , M , N 1 2 22 1 i 2 i subject to ␣ ) 0, P , P ) 0, Y G 0, Q G 0, 12 22 PA T y M T B T q APy BM q s y 1 Y q 2 ␣ P - 0, Ž. 1 i 1 ii i1 i 1i 1 A T P y C T N T q PAy NCq s y 1 Q q 2 ␣ P - 0, Ž. i 2 i 2 i 2 i 2 ii 22 2 PA T y M T B T q APy BM y 2 Y q 4 ␣ P 1 i 1 ji i1 i 1 j 1 qPA T y M T B T q APy BM - 0, 1 j 1ij j1 j 1 i i - j s.t. h l h / ␾ , ij A T P y C T N T q PAy NCy 2Q q 4 ␣ P i 2 j 2 i 2 i 2 ij 22 2 qA T P y C T N T q PAy NC- 0, j 2 i 2 j 2 j 2 ji i - j s.t. h l h / ␾ , ij where s ) 1, M s FP, N s PK, and Y s PQ P. 1ii12i 2 i 1111 The matrices P , P , Q , M , N , and Y can be found by using convex 1 2 22 1i 2 i optimization techniques involving LMIs if they exist. The feedback gains and the observer gains can then be obtained as F s MP y1 and K s P y1 N . i 1i 1 i 22i The design conditions above address decay rate and relaxed stability condi- tions and are reduced to the stable controller design problem if we set ␣ s 0, Y s 0, and Q s 0. 22 The design problem for discrete systems can be handled similarly. Design Procedure for Case A: DFS P , P ) 0, 12 TTT PPAy MB 11i 1ii ) 0, 4.23 Ž. APy BM P i 1 i 1i 1 FUZZY OBSERVATION DESIGN 90 TTT PAPy CN 2 i 2 i 2 i ) 0, 4.24 Ž. T PAy NC P 2 i 2 ii 2 TTT PA y MB 1 i 1 ji 4 P 1 TTT ž/ qPA y MB 1 j 1ij ) 0, 4.25 Ž. APy BM i 1 i 1 j P 1 qAPy BM ž/ j 1 j 1i TT APy CN i 2 j 2 i 4 P 2 TTT ž/ qAPy CN j 2 i 2 i ) 0. 4.26 Ž. T PAy NC 2 i 2 ij P 2 T ž/ qPAy NC 2 i 2 ij Remark 15 Note that in the designs above the controller gains and the observer gains can be determined separately. This powerful result is similar to the well-known separation principle for linear systems. Unfortunately, such a separation principle only holds for Case A and does not hold for Case wx B3. Finally, we would like to point out, as in Chapter 3, that a variety of control performance specifications can be incorporated into the LMI-based observer and controller design. 4.2.2 Case B Ž. In Case B we deal with the situation when the premise variables z t are unknown since they depend on the state variables to be estimated by fuzzy Ž Ž Ž Ž observers. As a result, we must use w z t instead of w z t . In other ˆ ii ŽŽ ŽŽ Ž. Ž. words, in Case B, h z t / h z t because of z t / z t in general. ˆˆ ii Ž. The fuzzy observers for Case B are of the following forms, instead of 4.3 Ž. or 4.5 : CFS r ˆ x t s h z t Ax t q Bu t q Kyt y y t , 4.27 Ä4 Ž. Ž. Ž. Ž. Ž. Ž. Ž . Ž. Ž . ˙ˆˆ ˆ Ý iiii i s1 r y t s h z t Cx t . Ž. Ž. Ž. Ž. ˆˆˆ Ý ii i s1 DFS r x t q 1 s h z t Ax t q Bu t q Kyt y y t , 4.28 Ä4 Ž . Ž. Ž. Ž. Ž. Ž. Ž . Ž. Ž . ˆˆˆ ˆ Ý iiii i s1 r y t s h z t Cx t . Ž. Ž. Ž. Ž. ˆˆˆ Ý ii i s1 DESIGN OF AUGMENTED SYSTEMS 91 Ž. Accordingly, instead of 4.7 , the PDC fuzzy controller becomes r w z t Fx t Ž. Ž. Ž. ˆˆ Ý ii r i s1 u t sy sy h z t Fx t . 4.29 Ž. Ž. Ž. Ž . Ž. ˆˆ Ý r ii i s1 w z t Ž. Ž. ˆ Ý i i s1 Then the augmented systems are obtained as follows: CFS rr r x t s h z thz thz t Gxt Ž. Ž. Ž. Ž. Ž. Ž.Ž.Ž. ˙ˆˆ ÝÝÝ aijkijka i s1 js1 ks1 rr s h z thz thz t Gxt Ž. Ž. Ž. Ž. Ž.Ž.Ž. ˆˆ ÝÝ ijj ijja i s1 js1 rr G q G ijk ikj q 2 h z thz thz t x t . 4.30 Ž. Ž. Ž. Ž. Ž . Ž.Ž.Ž. ˆˆ ÝÝ ijk a 2 i s1 j-k DFS rr r x t q 1 s h z thz thz t Gxt Ž . Ž. Ž. Ž. Ž. Ž.Ž.Ž. ˆˆ ÝÝÝ aijkijka i s1 js1 ks1 rr s h z thz thz t Gxt Ž. Ž. Ž. Ž. Ž.Ž.Ž. ˆˆ ÝÝ ijj ijja i s1 js1 rr G q G ijk ikj q 2 h z thz thz t x t , Ž. Ž. Ž. Ž. Ž.Ž.Ž. ˆˆ ÝÝ ijk a 2 i s1 j-k 4.31 Ž. where x t Ž. x t s , Ž. a e t Ž. e t s x t y x t , Ž. Ž. Ž. ˆ A y BF BF iikik G s , ijk 12 SS ijk ijk S 1 s A y A y B y BFq KCy C , Ž. Ž.Ž. ijk i j i j k j k i S 2 s A y KC q B y BF. 4.32 Ž. Ž. ijk j j k i j k FUZZY OBSERVATION DESIGN 92 Ž. The following stability theorem for the augmented system 4.30 can be derived from Theorem 7. wx THEOREM 20 CFS The equilibrium of the augmented system described by Ž. 4.30 is globally asymptotically stable if there exists a common positi®e definite matrix P such that G T P q PG - 0, 4.33 Ž. ijj ijj T G q GGq G ijk ikj ijk ikj P q P - 0, ž/ž/ 22 ᭙i, j - k s.t. h l h l h / ␾ . 4.34 Ž. ijk Proof. It follows directly from Theorem 7. Ž. The following stability theorem for the augmented system 4.31 can be derived from Theorem 8. wx THEOREM 21 DFS The equilibrium of the augmented system described by Ž. 4.31 is globally asymptotically stable if there exists a common positi®e definite matrix P such that G T PG y P - 0, 4.35 Ž. ijj ijj T G q GGq G ijk ikj ijk ikj P y P - 0, ž/ž/ 22 ᭙i, j - k s.t. h l h l h / ␾ . 4.36 Ž. ijk Proof. It follows directly from Theorem 8. Remark 16 Consider the common C matrix case, that is, C s C s иии s 12 C s C. In this case, r S 1 s A y A y B y BF, Ž.Ž. ijk i j i j k S 2 s A y KCq B y BF. Ž. ijk j j i j k The conditions of Theorems 20 and 21 imply those of Theorems 18 and 19, respectively. Ž Remark 17 We can no longer apply the relaxed conditions Theorems 9 and .ŽŽ ŽŽ 10 to Case B because of h z t / h z t in general. ˆ ii [...]... 3 Ž t g w yb, b x 1 1 The terms M1 , M2 , M12 , and M22 can be interpreted as membership functions of fuzzy sets By using these fuzzy sets, the nonlinear system can be represented by the following T-S fuzzy model: Model Rule 1 1 1 IF x 1Ž t is M1 and x 3 Ž t is M2 , THEN ½ x ˙Ž t s A1 x Ž t q B1 u Ž t , y Ž t s C1 x Ž t Ž 4.37 Model Rule 2 1 IF x 1Ž t is M1 and x 3 Ž t is M22 , THEN ½... A using a convex optimization technique involving LMIs The designed fuzzy controller stabilizes the overall control system The fuzzy observer estimates the states of the nonlinear system without steady- 96 FUZZY OBSERVATION DESIGN Fig 4.1 Simulation result state errors for the range x 1 Ž t g w y0.8, 0.8 x , x 3 Ž t g w y0.6, 0.6 x REFERENCES 1 R E Kalman, ‘‘On the General Theory of Control Systems,’’ . Copyright ᮊ 2001 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 4 FUZZY OBSERVER DESIGN In practical. methods such as observer-based designs. This chapter presents fuzzy ob- server design methodologies involving state estimation for T-S fuzzy models. Alternatively,