Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P5 pdf

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 5 ROBUST FUZZY CONTROL wx This chapter deals with the issue of robust fuzzy control 1᎐3. In general, there exist an infinite number of stabilizing controllers if the plant is stabilizable. The selection of a particular controller among this group of available controllers is often decided by certain specifications of control performance. Fuzzy control designs which guarantee a number of control performance considerations were presented in Chapter 3. The LMI-based techniques ensure not only stabilization but also, for example, good speed of response, avoidance of actuator saturation, and output error constraint. In this and next chapters, a systematic treatment is given for two advanced and important issues of control performance, namely, robustness and optimality, in fuzzy control system designs. The robustness issue is dictated by practical control applications in which there are always uncertainties associated with, for example, the plant, actuators, and sensors in a control system. Robust control addresses these uncertainties and aims to derive the best design possible under the circumstances. This chapter presents such a robust fuzzy control methodology, whereas optimal fuzzy control based on quadratic performance functions will be treated in the next chapter. This chapter defines a class of Takagi-Sugeno fuzzy systems with uncer- tainty. Robust stability conditions for this class of systems are derived by applying the relaxed stability conditions described in Chapter 3. This chapter also gives a design method that selects the robust fuzzy controller so as to maximize the norm of the uncertain blocks out of the class of stabilizing PDC controllers. This chapter focuses on robust fuzzy control for CFS. For the wx design of robust fuzzy control for DFS, refer to 4, 5 . 97 ROBUST FUZZY CONTROL 98 5.1 FUZZY MODEL WITH UNCERTAINTY To address the robustness of fuzzy control systems, a first and necessary step is to introduce a class of fuzzy systems with uncertainty. For this purpose, we introduce uncertainty blocks to the Takagi-Sugeno fuzzy model to arrive at the following fuzzy model with uncertainty: Plant Rule i Ž. Ž. IF ztis M and иии and ztis M 1 i1 pip Ž. Ž Ž. . Ž. THEN xt s A q D ⌬ t Ext ˙ iaiaiai ŽŽ Ž. q B q D ⌬ t Eut , i s 1,2, .,r,5.1 Ž. ibibibi where the uncertain blocks satisfy 1 ⌬ t F ,5.2 Ž. Ž . ai ␥ ai ⌬ t s ⌬ T t ,5.3 Ž. Ž. Ž . ai ai 1 ⌬ t F ,5.4 Ž. Ž . bi ␥ bi ⌬ t s ⌬ T t 5.5 Ž. Ž. Ž . bi bi for all i. The fuzzy model is represented as r x t s h z t A q D ⌬ t Ext Ä Ž. Ž. Ž. Ž. Ž. Ž. ˙ Ý i i ai ai ai i s1 q B q D ⌬ t Eut .5.6 4 Ž. Ž. Ž . Ž. ibibibi Ž.w Ž.x The fuzzy model 5.1 or 5.6 contains uncertainty in the consequent parts. The robust stability for the fuzzy model with premise uncertainty was wx wx first discussed in 6 and 7 . This chapter will focus on the consequent uncertainty. 5.2 ROBUST STABILITY CONDITION To begin with, this section presents a stability condition for the uncertain Ž.w Ž.x Ž. fuzzy model 5.1 i.e., 5.6 . By substituting the PDC controller 2.23 into ROBUST STABILITY CONDITION 99 Ž. 5.6 , we have rr x t s h z thz t Ž. Ž. Ž. Ž.Ž. ˙ ÝÝ ij i s1 js1 = E ⌬ 0 ai ai DD A y BFq x t Ž. ai bi iij ½5 yEF 0 ⌬ bi j bi r ⌬ 0 E ai ai 2 DD s h z t A y BFq x t Ž. Ž. Ž. Ý ai bi iiii ½5 0 ⌬ yEF bi bi i i s1 r q h z thz t Ž. Ž. Ž.Ž. ÝÝ ij i s1 i-j = E ⌬ 0 ai ai DD A y BFq A y BFq ai bi iijjji ½ yEF 0 ⌬ bi j bi ⌬ 0 E aj aj DD q x t .5.7 Ž. Ž . aj bj 5 0 ⌬ yEF bj bj i The following theorem presents robust stability conditions for the fuzzy Ž.w Ž.x Ž. model 5.1 i.e., 5.6 with a given PDC fuzzy controller 2.23 . This theorem provides a basis for the robust stabilization problem which is considered in the next section. Ž.w Ž.x THEOREM 22 The fuzzy system 5.1 i.e., 5.6 is stabilized ®ia the PDC Ž. controller 2.23 if there exist a common positi®e definite matrix P and a common positi®e semidefinite matrix Q satisfying 0 S q s y 1 Q - 0,5.8 Ž. Ž. ii 1 T y 2Q - 0, i - j s.t. h l h / ␾ ,5.9 Ž. ij 2 ij where s ) 1, T TTT A y BF Pq PAy B F PD PD E yFE Ž.Ž. iii iii ai bi ai ibi T DP yI 00 0 ai T S s , DP 0 yI 00 ii bi 2 E 00y ␥ I 0 ai ai 2 yEF 00 0y ␥ I bi i bi ROBUST FUZZY CONTROL 100 T A y BF P Ž. iij q PAy BF Ž. iij TTTTTT PD PD PD PD E yFE E yFE ai bi aj bj ai j bi aj i bj T q A y BF P Ž. jji 0 q PAy BF Ž. jji T DP yI 000 0 0 0 0 ai T DP 0 yI 00 0 0 0 0 bi T s , ij T DP 00yI 00 0 0 0 aj T DP 000yI 0000 bj 2 E 0000y ␥ I 000 ai ai 2 y EF 0000 0y ␥ I 00 bi j bi 2 E 0000 0 0y ␥ I 0 aj aj 2 y EF 0000 0 0 0y ␥ I bj i bj Q 0000 Q s block-diag , Ž. 0 1 Q 00000000 Q s block-diag . Ž. 0 2 Ž. Proof. Consider the T-S fuzzy control system with uncertainty 5.1 , where Ž. Ž. ⌬ t and ⌬ t are the uncertain blocks satisfying ai bi 1 T ⌬ t F , ⌬ t s ⌬ t , Ž. Ž. Ž. ai ai ai ␥ ai 1 T ⌬ t F , ⌬ t s ⌬ t . Ž. Ž. Ž. bi bi bi ␥ bi T Ž. Ž. Consider a candidate of Lyapunov functions x t Px t . Then, 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. ˙˙ T r ⌬ 0 E ai ai 2 T DD s h z t x t A y BFq P Ž. Ž. Ž. Ý ai bi iiii ½ ž/ 0 ⌬ yEF bi bi i i s1 ⌬ 0 E ai ai DD qPAy BFq x t Ž. ai bi iii 5 ž/ 0 ⌬ yEF bi bi i ROBUST STABILITY CONDITION 101 r T q h z thz t x t Ž. Ž. Ž. Ž.Ž. ÝÝ ij i s1 i-j T ° E ⌬ 0 ai ai ~ DD = A y BFq P ai bi iij yEF ¢ 0 ⌬ ž/ bi j bi E ⌬ 0 ai ai DD qPAy BFq ai bi iij yEF 0 ⌬ ž/ bi j bi T ⌬ 0 E aj aj DD q A y BFq P aj bj jji 0 ⌬ yEF ž/ bj bj i ¶ ⌬ 0 E aj aj • DD qPAy BFq x t Ž. aj bj jji ß 0 ⌬ yEF ž/ bj bj i r 2 T s h z t x t Ž. Ž. Ž. Ý i i s1 = T D ai T DD A y BF Pq PAy BF q PP Ž.Ž. ai bi iii iii T ½ D bi T ⌬ 0 ⌬ 0 E ai ai ai T T q E y EF Ž. ai bi i 0 ⌬ 0 ⌬ yEF bi bi bi i T T D ⌬ 0 E ai ai ai y P y T 0 ⌬ yEF ž/ D bi bi i bi T ¶ D ⌬ 0 E ai ai ai • = P y x t Ž. T ß 0 ⌬ yEF ž/ D bi bi i bi r T q h z thz t x t Ž. Ž. Ž. Ž.Ž. ÝÝ ij i s1 i-j ROBUST FUZZY CONTROL 102 = ° T D T ai ~ DD A y BF Pq PAy BF q PP Ž.Ž. ai bi iij iij T ¢ D bi T E ⌬ 0 ⌬ 0 ai T ai ai T q E y EF Ž. ai bi j yEF 0 ⌬ 0 ⌬ bi j bi bi T TT EE D ⌬ 0 D ⌬ 0 ai ai ai ai ai ai y P y P y TT yEF yEF 0 ⌬ 0 ⌬ ž/ž/ DD bi j bi j bi bi bi bi T D aj T DD q A y BF Pq PAy BF q PP Ž.Ž. aj bj jji jji T D bj T ⌬ 0 ⌬ 0 E aj aj aj T T q E y EF Ž. aj bj i 0 ⌬ 0 ⌬ yEF bj bj bj i T T D ⌬ 0 E aj aj aj y P y T 0 ⌬ yEF D 0 bj bj i bj T ¶ D ⌬ 0 E aj aj aj • = P y x t . 5.10 Ž. Ž . T 0 ⌬ yEF ß D 0 bj bj i bj If T D T ai DD A y BF Pq PAy BF q PP Ž.Ž. ai bi iij iij T D bi 1 I 0 2 E ␥ ai ai T T q E y EF Ž. ai bi j 1 yEF bi j 0 I 2 ␥ bi T D aj T DD q A y BF Pq PAy BF q PP Ž.Ž. aj bj jji jji T D bj 1 I 0 E aj 2 ␥ aj T T qy2Q - 0, 5.11 E y EF Ž. yEF Ž. 0 aj bj i bj i 1 0 I 2 ␥ bj ROBUST STABILITY CONDITION 103 then r d T 2 T x t Px t - h z t x t Ž. Ž. Ž. Ž. Ž. Ý i dt i s1 T ° D ai T ~ DD = A y BF Pq PAy BF q PP Ž.Ž. ai bi iii iii T ¢ D bi T ⌬ 0 ⌬ 0 E ai ai ai T T q E y EF Ž. ai bi i 0 ⌬ 0 ⌬ yEF bi bi bi i T T D ⌬ 0 E ai ai ai y P y T 0 ⌬ yEF ž/ D bi bi i bi T ¶ D ⌬ 0 E ai ai ai • = P y x t Ž. T ß 0 ⌬ yEF ž/ D bi bi i bi r TT q2 h z thz t x t Qx t Ž. Ž. Ž. Ž. Ž.Ž. ÝÝ ij 0 i s1 i-j r 2 T F h z t x t Ž. Ž. Ž. Ý i i s1 T ° D ai T ~ DD = A y BF Pq PAy BF q PP Ž.Ž. ai bi iii iii T ¢ D bi T ⌬ 0 ⌬ 0 E ai ai ai T T q E y EF Ž. ai bi i 0 ⌬ 0 ⌬ yEF bi bi bi i T T D ⌬ 0 E ai ai ai y P y T 0 ⌬ yEF ž/ D bi bi i bi T ¶ D ⌬ 0 E ai ai ai • = P y x t Ž. T ß 0 ⌬ yEF ž/ D bi bi i bi r 2 T q s y 1 h z t x t Qxt Ž . Ž. Ž. Ž. Ž. Ý i 0 i s1 ROBUST FUZZY CONTROL 104 r 2 T s h z t x t Ž. Ž. Ž. Ý i i s1 ° T ~ = A y BF Pq PAy BF q s y 1 Q Ž.Ž.Ž. iii iii 0 ¢ T D ai DD qPP ai bi T D bi T ⌬ 0 ⌬ 0 E ai ai ai T T q E y EF Ž. ai bi i 0 ⌬ 0 ⌬ yEF bi bi bi i T T D ⌬ 0 E ai ai ai y P y T 0 ⌬ yEF ž/ D bi bi i bi T ¶ D ⌬ 0 E ai ai ai • = P y x t . Ž. T ß 0 ⌬ yEF ž/ D bi bi i bi If T A y BF Pq PAy BF q s y 1 Q Ž.Ž.Ž. iii iii 0 T D ai DD q PP ai bi T D bi 1 I 0 2 ␥ E ai ai T T q - 0, 5.12 Ž. E y EF Ž. ai bi i 1 yEF bi i 0 I 2 ␥ bi then d T x t Px t - 0 Ž. Ž. dt ROBUST STABILIZATION 105 Ž. at x t / 0. Since 11 TT ⌬ t ⌬ t F I, ⌬ t ⌬ t F I, Ž. Ž. Ž. Ž. ai ai bi bi 22 ␥␥ ai bi T T D ⌬ 0 E ai ai ai y P y T 0 ⌬ yEF ž/ D bi bi i bi T D ⌬ 0 E ai ai ai = P yF0. T 0 ⌬ yEF ž/ D bi bi i bi Ž. Ž. Ž. Ž. By the Schur complement, 5.12 and 5.11 are rewritten as 5.8 and 5.9 , respectively. Q.E.D. When Q s 0 and Q s 0, that is, Q s 0, the relaxed robust stability 12 0 conditions are reduced to just the robust conditions: P ) 0, S - 0, T - 0, i - j s.t. h l h / ␾ . ii ij i j As a result, by utilizing the relaxed stability conditions, less conservative results can be obtained in the robust stability analysis. 5.3 ROBUST STABILIZATION We define a robust stabilization problem so as to select a PDC fuzzy Ž. controller, in the class of PDC controllers 2.23 satisfying the robust stability Ž. Ž. conditions 5.8 and 5.9 , to maximize the norm of the uncertainty blocks, or Ž. equivalently, to minimize ␥ and ␥ in 5.7 . The following theorem provides ai bi a solution to the robust stabilization problem. Ž. THEOREM 23 The feedback gains F that stabilize the fuzzy model 5.1 and i Ž. maximize the norms of the uncertain blocks i.e., minimize ␥ and ␥ can be ai bi obtained by sol®ing the following LMIs, where ␣ , ␤ ) 0 are design parameters: ii r 22 Ä4 minimize ␣␥ q ␤␥ Ý iai i bi 22 ␥ , ␥ , X , M , .,M , Y is1 ai bi 1 r 0 subject to ˆ X ) 0, Y G 0, S q s y 1 Y - 0, 5.13 Ž. Ž. 0 ii 1 ˆ T y 2Y - 0, i - j s.t. h l h / ␾ , 5.14 Ž. ij 2 ij ROBUST FUZZY CONTROL 106 where s ) 1, T XA q AX ii )) ) ) TT ž/ yBMy MB ii ii T D yI 00 0 ai ˆ S s , ii T D 0 yI 00 bi 2 EX 00y ␥ I 0 ai ai 2 yEM 00 0 y ␥ I bi i bi T XA q AX ii TT yBMy M B ij ji TTTTTT DDDD XE yME XE yME ai bi a j bj ai j bi a j i bj T qXA q AX jj 0 TT yBMy MB ji i j T D yI 000 0 0 0 0 ai T D 0 yI 00 0 0 0 0 bi ˆ T s , ij T D 00yI 00 0 0 0 aj T D 000yI 0000 bj 2 EX 0000y ␥ I 000 ai ai 2 yEM 000 0 0 y ␥ I 00 bi j bi 2 EX 0000 0 0 y ␥ I 0 aj ai 2 yEM 0000 0 0 0 y ␥ I bj i bi Y 0000 Y s block-diag , Ž. 0 1 Y 00000000 Y s block-diag , Ž. 0 2 where Y s XQ X 00 Ž. and the asterisk denotes the transposed elements matrices for symmetric positions. Proof. The main idea is to transform the conditions of Theorem 22 into [...]...107 ROBUST STABILIZATION LMIs: Ä block-diag X I = Ä block-diag ž I I I X I I XAT q A i X i 4 Ä S i i q Ž s y 1 Q1 4 I I 4 / ) ) ) ) T Dai yI 0 0 0 D Ti b 0 yI 0 0 0 2 y␥ b i I yBi Mi y MiT BiT s Eai X 0 0 2 y␥ai I yEb i Mi 0 0 0 q Ž s y 1 и block-diag Ž XQ 0 X 0 0 0 0 ˆ s Sii q Ž s y 1 Y1 , Ä block-diag X I Ž 5.15 I I I I I I I = Ä block-diag X I I I I I 4 и Ä Ti j y 2 Q 2 4 I I I 4 XAT... Automatic Control ŽIFAC World Congress, Beijing, July 1999, pp 213᎐218 2 K Tanaka, M Nishimura, and H O Wang, ‘‘Multi-Objective Fuzzy Control of High RiserHigh Speed Elevators Using LMIs,’’ 1998 American Control Conference, 1998, pp 3450᎐3454 3 K Tanaka, T Taniguchi, and H Wang, ‘‘Model-Based Fuzzy Control of TORA System: Fuzzy Regulator and Fuzzy Observer Design via LMIs that Represent Decay Rate, Disturbance... DTj b 0 0 0 yI 0 0 0 0 qXAT q A j X j yB j Mi y MiT B T j s Eai X 0 0 0 0 2 y␥ai I 0 0 0 yEb i M j 0 0 0 0 0 2 y␥ b i I 0 0 Ea j X 0 0 0 0 0 0 y␥a2j I 0 yEb j Mi 0 0 0 0 0 0 0 2 y␥ b j I 0 0 y 2 и block-diag Ž XQ 0 X 0 0 0 0 ˆ s Ti j y 2 Y2 , 0 0 Ž 5.16 where X s Py1 , for all i Mi s Fi Py1 Q.E.D 108 ROBUST FUZZY CONTROL The feedback gains can be obtained as Fi s Mi Xy1 from the solutions X and Mi... International Conference on Fuzzy System, Vol 1, 1993, pp 29᎐34 7 K Tanaka and M Sano, ‘‘A Robust Stabilization Problem of Fuzzy Controller Systems and Its Applications to Backing up Control of a Truck-Trailer,’’ IEEE Trans on Fuzzy Syst Vol 2, No 2, pp 119᎐134, Ž1994 8 K Tanaka, T Ikeda, and H O Wang, ‘‘Robust Stabilization of a Class of Uncertain Nonlinear System via Fuzzy Control: Quadratic Stabilizability, . 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 5 ROBUST FUZZY CONTROL wx This chapter. common positi®e semidefinite matrix Q satisfying 0 S q s y 1 Q - 0,5.8 Ž. Ž. ii 1 T y 2Q - 0, i - j s.t. h l h / ␾ ,5.9 Ž. ij 2 ij where s ) 1, T TTT A y BF