Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P6 pptx

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 6 OPTIMAL FUZZY CONTROL In control design, it is often of interest to synthesize a controller to satisfy, in an optimal fashion, certain performance criteria and constraints in addition to stability. The subject of optimal control addresses this aspect of control system design. For linear systems, the problem of designing optimal con- Ž. trollers reduces to solving algebraic Riccati equations AREs , which are usually easy to solve and detailed discussion of their solutions can be found wx in many textbooks 1 . However, for a general nonlinear system, the optimiza- Ž. tion problem reduces to the so-called Hamilton-Jacobi HJ equations, which Ž.wx are nonlinear partial differential equations PDEs 2 . Different from their counterparts for linear systems, HJ equations are usually hard to solve both numerically and analytically. Results have been given on the relationship between solution of the HJ equation and the invariant manifold for the Hamiltonian vector field. Progress has also been made on the numerical wx computation of the approximated solution of HJ equations 3 . But few results so far can provide an effective way of designing optimal controllers for general nonlinear systems. In this chapter, we propose an alternative approach to nonlinear optimal control based on fuzzy logic. The optimal fuzzy control methodology pre- wx sented in this chapter is based on a quadratic performance function 4᎐7 utilizing the relaxed stability conditions. The optimal fuzzy controller is designed by solving a minimization problem that minimizes the upper bound of a given quadratic performance function. In a strict sense, this approach is a suboptimal design. One of the advantages of this methodology is that the wx design conditions are represented in terms of LMIs. Refer to 8 for a more thorough treatment of optimal fuzzy control. 109 OPTIMAL FUZZY CONTROL 110 6.1 QUADRATIC PERFORMANCE FUNCTION AND STABILIZING CONTROL The control objective of optimal fuzzy control is to minimize certain perfor- mance functions. In this chapter, we present a fuzzy controller design to minimize the upper bound of the following quadratic performance function Ž. 6.1 : ϱ TT J s y t Wy t q u t Ru tdt,6.1 Ä4 Ž. Ž. Ž. Ž. Ž . H 0 where r y t s h z t Cx t . Ž. Ž. Ž. Ž. Ý ii i s1 The following theorem presents a basis to the optimal fuzzy control problem. The set of conditions given herein, however, are not in terms of LMIs. The LMI-based optimal fuzzy control design will be addressed in the next section. Ž. Ž. THEOREM 24 The fuzzy system 2.3 and 2.4 can be stabilized by the PDC Ž. fuzzy controller 2.23 if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q satisfying 0 U q s y 1 Q - 0 6.2 Ž. Ž. ii 3 V y 2Q - 0, i - j s.t. h l h / ␾ ,6.3 Ž. ij 4 ij where s ) 1, T A y BF P Ž. iii TT C yF ii ž/ qPAy BF Ž. iii U s ,6.4 Ž. ii y1 C yW 0 i y1 yF 0 yR i T A y BF P Ž. iij qPAy BF Ž. iij TTTT C yFCyF ijji T q A y BF P Ž. jji 0 qPAy BF Ž. jji V s ,6.5 Ž. ij y1 C yW 000 i y1 yF 0 yR 00 j y1 C 00yW 0 j y1 yF 000yR i QUADRATIC PERFORMANCE FUNCTION AND STABILIZING CONTROL 111 Q 00 Q s block-diag , Ž. 0 3 Q 0000 Q s block-diag . Ž. 0 4 Then, the performance function satisfies J - x T 0 Px 0, Ž. Ž. T Ž. Ž. where x 0 Px 0 acts as an upper bound of J. Proof. Let us define the following new variable r C y t Ž. i y t ssh z t x t . Ž. Ž. Ž. Ž. ˆ Ý i yF u t Ž. i i s1 Ž. Equation 6.1 can be rewritten as ϱ W 0 T J s y t y tdt. Ž. Ž. ˆˆ H 0 R 0 Assume that there exists a common positive definite matrix P and a common Ž. Ž. positive semidefinite matrix Q satisfying 6.2 and 6.3 . Then, from Schur 0 complements, we have T A y BF Pq PAy BF q s y 1 Q Ž.Ž.Ž. iii iii 0 W 0 C i TT q C yF - 0 6.6 Ž. ii 0 R yF i and T A y BF Pq PAy BF Ž.Ž. iij iij T q A y BF Pq PAy BF y 2 Q Ž.Ž. jji jji 0 W 0 C i TT q C yF ij 0 R yF j W 0 C j TT q C yF - 0.6.7 Ž. ji 0 R yF i Ž. Ž. From 6.6 and 6.7 , we obtain T A y BF Pq PAy BF q s y 1 Q - 0 6.8 Ž.Ž.Ž. Ž. iii iii 0 OPTIMAL FUZZY CONTROL 112 and T A y BF Pq PAy BF Ž.Ž. iij iij T q A y BF Pq PAy BF y 2 Q - 0.6.9 Ž. Ž.Ž. jji jji 0 It is clear from Theorem 9 in Chapter 3 that the fuzzy control system is Ž. Ž. globally asymptotically stable if 6.2 and 6.3 hold. Next, it will be proved that the quadratic performance function satisfies T Ž. Ž. T Ž. Ž. J - x 0 Px 0 . Consider a Lyapunov function candidate x t Px t . Then, Ž.Ž. from 6.6 , 6.7 , and the Appendix, d T x t Px t Ž. Ž. dt s x T t Px t q x T t Px t Ž. Ž. Ž. Ž. ˙˙ rr T T s h z thz t x t A y BF Pq PAy BF x t Ž. Ž. Ž. Ž. Ž.Ž. Ž.Ž. ÝÝ ½5 ij iij iij i s1 js1 r T 2 T s h z t x t A y BF Pq PAy BF x t Ž. Ž.Ž . Ž . Ž. Ž. Ä4 Ý iiiiiii i s1 r T T q h z thz t x t A y BF Pq PAy BF x t Ž. Ž. Ž. Ž. Ž.Ž. Ž.Ž. ÝÝ ½5 ij iij iij i s1 i/j r T 2 T - h z t x t A y BF Pq PAy BF x t Ž. Ž.Ž . Ž . Ž. Ž. Ä4 Ý iiiiiii i s1 r W 0 C i TTT y x thz thz t C yF Ž. Ž. Ž. Ž.Ž. ÝÝ ij ij ½ 0 R yF j i s1 i-j r W 0 C j TT q h z thz t C yFxt Ž. Ž. Ž. Ž.Ž. ÝÝ ij ji 5 0 R yF i i s1 i-j r T q 2 h z thz t x t Qxt Ž. Ž. Ž. Ž. Ž.Ž. ÝÝ ij 0 i s1 i-j r W 0 C i T 2 TT - yx thz t C yFxt Ž. Ž. Ž. Ž. Ý iii ½5 0 R yF i i s1 r W 0 C i TTT yx thz thz t C yF Ž. Ž. Ž. Ž.Ž. ÝÝ ij ij ½ 0 R yF j i s1 i-j QUADRATIC PERFORMANCE FUNCTION AND STABILIZING CONTROL 113 r W 0 C j TT q h z thz t C yFxt Ž. Ž. Ž. Ž.Ž. ÝÝ ij ji 5 0 R yF i i s1 i-j r 2 T y s y 1 h z t x t Qxt Ž . Ž. Ž. Ž. Ž. Ý i 0 i s1 r T q2 h z thz t x t Qxt Ž. Ž. Ž. Ž. Ž.Ž. ÝÝ ij 0 i s1 i-j r W 0 C i T 2 TT Fyx thz t C yFxt Ž. Ž. Ž. Ž. Ý iii ½5 0 R yF i i s1 r W 0 C j TTT y x thz thz t C yF Ž. Ž. Ž. Ž.Ž. ÝÝ ij ij ½ 0 R yF i i s1 i-j r W 0 C i TT q h z thz t C yFxt Ž. Ž. Ž. Ž.Ž. ÝÝ ij ji 5 0 R yF j i s1 i-j r 2 T y s y 1 h z t x t Qxt Ž . Ž. Ž. Ž. Ž. Ý i 0 i s1 r T q 2 h z thz t x t Qxt Ž. Ž. Ž. Ž. Ž.Ž. ÝÝ ij 0 i s1 i-j rr C j W 0 TTT syx thz thz t C yFxt Ž. Ž. Ž. Ž. Ž.Ž. ÝÝ ij ii ½5 0 R yF j i s1 js1 rr 2 T y s y 1 h z t y 2 h z thz t x t Qxt Ž . Ž. Ž. Ž. Ž. Ž. Ž. Ž.Ž. ÝÝÝ iij0 ž/ i s1 is1 i-j rr C i W 0 TTT syx thz t C yF h z t x t Ž. Ž. Ž. Ž. Ž. Ž. ÝÝ iii i ½5 ž/ ž/ 0 R yF i i s1 is1 rr 2 T y s y 1 h z t y 2 h z thz t x t Qxt Ž . Ž. Ž. Ž. Ž. Ž. Ž. Ž.Ž. ÝÝÝ iij0 ž/ i s1 is1 i-j W 0 T syy t y t Ž. Ž. 0 R rr 2 T y s y 1 h z t y 2 h z thz t x t Qxt Ž . Ž. Ž. Ž. Ž. Ž. Ž. Ž.Ž. ÝÝÝ iij0 ž/ i s1 is1 i-j W 0 T Fyy t y t . Ž. Ž. 0 R OPTIMAL FUZZY CONTROL 114 Therefore, d W 0 TT x t Px t - yy t y t . Ž. Ž. Ž. Ž. ˆˆ 0 R dt Integrating both side from 0 to ϱ, we get ϱ ϱ W 0 TT J s y t y tdt- yx t Px t . Ž. Ž. Ž. Ž. ˆ H 0 0 R 0 Since the fuzzy control system is stable, ϱ W 0 TT J s y t y tdt- x 0 Px 0 . 6.10 Ž. Ž. Ž. Ž. Ž . ˆ H 0 R o Q.E.D. T Ž. Ž. Remark 18 The above design procedure guarantees J - x 0 Px 0 for all ŽŽ wx the values of h z t g 0, 1 . i When Q s 0 and Q s 0, that is, Q s 0, the relaxed conditions in 34 0 Theorem 24 are reduced to the following conditions: P ) 0, U - 0, V - 0, i - j s.t. h l h / ␾ . ii ij i j XX T Ž. Ž. Then, the performance function J satisfies J - x 0 Px 0. 6.2 OPTIMAL FUZZY CONTROLLER DESIGN We present a design problem to minimize the upper bound of the perfor- mance function based on the results derived in Theorem 24. As shown in the T Ž. Ž. previous section, x 0 Px 0 gives an upper bound of J under the conditions of Theorem 24. The optimal fuzzy controller to be introduced is in the strict T Ž. Ž. sense a ‘‘sub-optimal’’ controller since x 0 Px 0 will be minimized instead of J in the control design procedure. The following theorem summarizes the design conditions for such scheme. THEOREM 25 The feedback gains to minimize the upper bound of the performance function can be obtained by sol®ing the following LMIs. From the solution of the LMIs, the feedback gains are obtained as F s MX y1 ii T Ž. Ž. for all i. Then, the performance function satisfies J - x 0 Px 0 - ␭ . OPTIMAL FUZZY CONTROLLER DESIGN 115 minimize ␭ X , M , ., M , Y 1 r 0 subject to X)0, Y G 0, 0 T ␭ x 0 Ž. ) 0, 6.11 Ž. x 0 X Ž. ˆ U q s y 1 Y - 0, 6.12 Ž. Ž. ii 3 ˆ V y 2Y - 0, i - j s.t. h l h / ␾ , 6.13 Ž. ij 4 ij where s ) 1, T XA q AX ii TT XC yM ii TT ž/ yBMy MB ii ii ˆ U s , ii y1 CX yW 0 i y1 yM 0 yR i T XA q AX ii TT yBMy MB ij ji TTTT XC yMXCyM ijji T qXA q AX jj 0 TT yBMy MB ji i j ˆ V s , ij y1 CX yW 000 i y1 yM 0 yR 00 j y1 CX 00yW 0 j y1 yM 000yR i Y 00 Y s block-diag , Ž. 0 3 Y 0000 Y s block-diag . Ž. 0 4 T Ž. Ž. Proof. The main idea here is to transform the inequality J - x 0 Px 0 - ␭ and the conditions of Theorem 24 into LMIs: OPTIMAL FUZZY CONTROL 116 wx wx block-diag XII и U q s y 1 Q и block-diag XII Ä4 Ä4 Ä4 Ž. ii 3 T XA q AX ii TT XC yM ii TT ž/ yBMy MB ii ii s y1 CX yW 0 i y1 yM 0 yR i q s y 1 и block-diag XQ X 00 Ž. Ž . 0 ˆ s U q s y 1 Y , Ž. ii 3 where Y s XQ X . 00 We obtain the following condition as well: wx wx block-diag XII и V y 2Q и block-diag XII Ä4Ä4 Ä4 ij 4 T XA q AX ii TT yBMy MB ij ji TTTT XC yMXCyM ijji T qXA q AX jj 0 TT yBMy MB ji i j s y1 CX yW 000 i y1 yM 0 yR 00 j y1 CX 00yW 0 j y1 yM 000yR i y 2 и block-diag XQ X 0000 Ž. 0 ˆ s V y 2Y . ij 4 Then, the quadratic performance function satisfies J - x T 0 X y1 x 0 - ␭ . Q.E.D. Ž. Ž. Theorem 25 shows that by minimizing ␭ , we obtain the feedback gains which minimize the upper bound of J. To solve this design problem, Ž. the initial values x 0 are assumed known. If not so, Theorem 25 is not Ž. directly applicable. In this case, however, if all the vertex points x 0of k Ž. a polyhedron containing the unknown initial values x 0 are known, OPTIMAL FUZZY CONTROLLER DESIGN 117 that is, l x 0 s ␳ x 0, Ž. Ž. Ý kk k s1 ␳ G 0, k l ␳ s 1, Ý k k s1 x 0 g R n . Ž. k Theorem 25 can be modified as follows to handle this case. THEOREM 26 The feedback gains to minimize the upper bound of the performance function can be obtained by sol®ing the following LMIs. From the solution of the LMIs, we obtain F s MX y1 ii T Ž. Ž. for all i. Then, the performance function satisfies J - x 0 Px0 - ␭ : minimize ␭ X , M , ., M , Y 1 r 0 subject to X ) 0, Y G 0, 0 T ␭ x 0 Ž. k ) 0, k s 1,2, .,l, x 0 X Ž. k ˆ U q s y 1 Y - 0, Ž. ii 3 ˆ V y 2Y - 0, i - j s.t. h l h / ␾ . ij 4 ij Proof. It directly follows from Theorem 25. Q.E.D. Remark 19 An alternative approach to handle the uncertainty in initial w condition is to employ the initial condition independent design see Chapter Ž.x 3, equation 3.56 . An interesting and important theorem is given below. THEOREM 27 The following statements are equi®alent. Ž. 1 There exist a common positi®e definite X and a common positi®e semidefi- Ž. Ž. nite Y satisfying 3.23 and 3.24 . OPTIMAL FUZZY CONTROL 118 Ž. X 2 There exist a common positi®e definite X s Xr ␧ and a common positi®e Ž. Ž. ᭚ semidefinite Y satisfying 6.12 and 6.13 , where ␧ ) 0. 0 Ž. Ž. Ž . Proof. 1 ´ 2 Assume that 3.23 is satisfied. Since W 0 CX i TT XC yM G 0, ii 0 R yM i there exists a very small ␧ ) 0 satisfying ␧ XA T q AXy BMy M T B T q s y 1 Y Ž. ii iiii 0 ž W 0 CX i TT q ␧ XC yM - 0 ii / 0 R yM i for i s 1,2, .,r. The above condition is equivalent to T XA q AX ii TT ␧␧ XC y ␧ M ii TT ž/ yBMy MB ii ii y1 ␧ CX yW 0 i y1 y ␧ M 0 yR i q ␧ s y 1 Y - 0, i s 1,2, .,r. Ž. 3 Since X X s ␧ X, M X s ␧ M , and Y X s ␧ Y can be regarded as new X, M , ii 33 i Ž. and Y , respectively, we obtain the condition 6.12 . 3 Ž. Ž. We can obtain the condition 6.13 from 3.24 as well. Ž. Ž. 2 ´ 1 . It is obvious. Q.E.D. X Ž. The theorem above says that there exists a common X satisfying 6.12 Ž. Ž. Ž. and 6.13 for any W and R if conditions 3.23 and 3.24 hold. The optimal fuzzy controller design in Theorem 25 is feasible if the stability conditions Ž. Ž. 3.23 and 3.24 hold. A design example for optimal fuzzy control will be discussed in detail in Chapter 7. APPENDIX TO CHAPTER 6 COROLLARY A.1 yC T WC y C T WC FyC T WC y C T WC , ii jj ij ji where W ) 0. [...]... An Introduction, Prentice-Hall, Englewood Cliffs, NJ, 1970 2 A J van der Schaft, ‘‘On a State Space Approach to Nonlinear Hϱ Control,’’ Syst Control Lett., Vol 16, pp.1᎐8 Ž1991 3 W M Lu and J C Doyle, ‘‘ Hϱ Control of Nonlinear Systems: A Convex Characterization,’’ IEEE Trans Automatic Control, Vol 40, No 9, pp 1668᎐1675 Ž1995 4 K Tanaka, M Nishimura, and H O Wang, ‘‘Multi-Objective Fuzzy Control of... Nishimura, and H O Wang, ‘‘Multi-Objective Fuzzy Control of High RiserHigh Speed Elevators Using LMIs,’’ 1998 American Control Conference, 1998, pp 3450᎐3454 5 K Tanaka, T Taniguchi, and H O Wang, ‘‘Model-Based Fuzzy Control of TORA System: Fuzzy Regulator and Fuzzy Observer Design via LMIs That Represent Decay Rate, Disturbance Rejection, Robustness, Optimality,’’ Seventh IEEE International Conference . 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 6 OPTIMAL FUZZY CONTROL In control design,. following conditions: P ) 0, U - 0, V - 0, i - j s.t. h l h / ␾ . ii ij i j XX T Ž. Ž. Then, the performance function J satisfies J - x 0 Px 0. 6.2 OPTIMAL FUZZY