Tài liệu Hệ thống điều khiển mờ - Thiết kế và phân tích P10 ppt

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 10 FUZZY DESCRIPTOR SYSTEMS AND CONTROL This chapter deals with a fuzzy descriptor system defined by extending the original Takagi-Sugeno fuzzy model. A number of stability conditions for the fuzzy descriptor system are derived and represented in terms of LMIs. A motivating example for using the fuzzy descriptor system instead of the original Takagi-Sugeno fuzzy model is presented. An LMI-based design approach is employed to find stabilizing feedback gains and a common Lyapunov function. The descriptor system, which differs from a state-space representation, has generated a great deal of interest in control systems design. The descriptor system describes a wider class of systems including physical models and wx nondynamic constraints 1 . It is well known that the descriptor system is much tighter than the state-space model for representing real independent parametric perturbations. There exist a large number of papers on the stability analysis of the T-S fuzzy systems based on the state-space represen- tation. In contrast, the definition of a fuzzy descriptor system and its stability wx wx analysis have not been discussed until recently 2 . In 2 we introduced the fuzzy descriptor systems and analyzed the stability of such systems. This wx chapter presents both the basic framework of 2, 3 as well as some new developments on this topic. Ž. As mentioned in Chapter 1, h l® / or denotes all the pairs i, k ik ŽŽ ŽŽ Ž. excepting h z t ® z t s 0 for all z t ; h l h l® / or denotes all ik ijk Ž. ŽŽ ŽŽ ŽŽ Ž. the pairs i, j, k excepting h z thz t ® z t s 0 for all z t ; and ijk ŽŽ ŽŽ ŽŽ i - j s.t. h l h l® / or denotes all i - j excepting h z thz t ® z t ijk i j k Ž. s 0, ᭙ z t . 195 FUZZY DESCRIPTOR SYSTEMS AND CONTROL 196 10.1 FUZZY DESCRIPTOR SYSTEM wx In 4, 5 , a fuzzy descriptor system is defined by extending the T-S fuzzy Ž. Ž. model 2.3 and 2.4 . The ordinary Takagi-Sugeno fuzzy model is a special case of the fuzzy descriptor system. We derive stability conditions for the fuzzy descriptor system, where the E matrix in the fuzzy descriptor system is assumed to be not always nonsingular. The fuzzy descriptor system is defined as r e r ® z t Ext s h z t Ax t q Bu t , Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. ˙ ÝÝ kk i i i k s1 is1 10.1 Ž. r y t s h z t Cx t , Ž. Ž. Ž. Ž. Ý ii i s1 where x t g R n , y t g R q , u t g R m , Ž. Ž. Ž. r h z t G 0, h z t s 1, Ž. Ž. Ž. Ž. Ý ii i s1 r e ® z t G 0, ® z t s 1. Ž. Ž. Ž. Ž. Ý kk k s1 Here x g R n is the descriptor vector, u g R m is the input vector, y g R q is the output vector, E g R n=n , A g R n=n , B g R n=m , and C g R q=n . The kii i Ž. Ž. known premise variables zt; ztmay be functions of the states, external 1 p disturbances, andror time. wx wx Remark 32 A fuzzy descriptor system was first defined in 2 . In 2 , a Ž Ž Ž Ž e wx special case, that is, h z t s® z t and r s r , was presented. In 4, 5 , ik Ž. the fuzzy descriptor system was generalized as shown in 10.1 . Ž. w T Ž. T Ž.x T Ž. By defining x* t s x t x t , the fuzzy descriptor system 10.1 can ˙ be rewritten as rr e E*x* t s h z t ® z t A* x* t q B*u t , Ž. Ž. Ž. Ž. Ž. Ž.Ž. Ž. ˙ ÝÝ ik ik i i s1 ks1 10.2 Ž. r y t s h z t C*x* t , Ž. Ž. Ž. Ž. Ý ii i s1 where 0 I I 0 E* s , A* s , ik A yE 00 ik 0 C 0 B* s , C* s . i ii B i Ž. In the following the stability for the fuzzy descriptor system 10.2 is considered. STABILITY CONDITIONS 197 10.2 STABILITY CONDITIONS Ž. The open-loop systems of 10.2 is defined as follows: rr e E*x* t s h z t ® z t A* x* t .10.3 Ž. Ž. Ž. Ž. Ž . Ž.Ž. ˙ ÝÝ ik ik i s1 ks1 Ž. The fuzzy descriptor system 10.3 is quadratically stable if dV x* t Ž. Ž. Fy ␣ x* t , Ž. 2 dt where V x* t s x* T t E* T Xx* t , Ž. Ž. Ž. Ž. and the following conditions are satisfied: rr e Ž. 1. det sE* y h z t ® z t A* / 0 and the open-loop system Ž. Ž. Ž.Ž. ÝÝ ik ik i s1 ks1 is impulse free. 2. There exist a common matrix X and ␣ ) 0 such that X g R 2 n=2 n , E* T X s X T E* G 0, det X / 0. Ž. Theorem 33 gives a sufficient condition for ensuring the stability of 10.3 . Ž. THEOREM 33 The fuzzy descriptor system 10.3 is quadratically stable if there exists a common matrix X such that E* T Xs X T E* G 0,10.4 Ž. A* T X q X T A* - 0, h l® / or.10.5 Ž. ik ik i k Proof. Consider a candidate of the quadratic function V x* t s x* T t E* T Xx* t . Ž. Ž. Ž. Ž. Then, rr e U T U TT ˙ V x* t s h z t ® z t x* t AXq XA x* t . Ž. Ž. Ž. Ž. Ž. Ž. Ž.Ž. Ž. ÝÝ i k ik ik i s1 ks1 Therefore, we have the following stability conditions: A* T X q X T A* - 0, h l® / or. Q.E.D. Ž. ik ik i k FUZZY DESCRIPTOR SYSTEMS AND CONTROL 198 Ž. Remark 33 As mentioned before, h l® / or denotes ‘‘all the pairs i, k ik ŽŽ ŽŽ Ž. excepting h z t ® z t s 0 for all z t .’’ In other words, we can ignore the ik Ž . Ž . Ž Ž Ž Ž Ž . condition 10.5 for the pairs i, k such that h z t ® z t s 0 for all z t . ik Remark 34 In Theorem 33, X is not required to be positive definite. Corollary 5 is needed to discuss the stability of closed-loop systems. Ž. Ž. Ž. Ž. COROLLARY 5 The conditions 10.6 and 10.7 imply 10.4 and 10.5 , where S is a positi®e definite matrix: 1 S s S T ) 0,10.6 Ž. 11 TT ASq SA ) i 33i - 0, h l® / or,10.7 Ž. ik TT S q SAy ES yESy SE 11ik3 k 11k Ž. where the asterisk denotes the transposed elements matrices for symmetric Ž. Ž T . T positions. For example, in 10.7 , it represents S q SAy ES . 11ik3 Proof. Define X as S 0 1 X s . SS 31 Ž. Ž. Then, 10.6 is obtained from 10.4 as follows: I 0 S 0 S 0 11 T E* X ssG0, 00 SS 00 31 TT T SS I0 S 0 13 1 T XE* ssG0. T 0 S 00 0 0 1 Ž. Equation 10.7 is obtained as follows: A* T X q X T A* ik ik TTT 0 AS0 SS 0 I i 113 sq TT I yESS 0 SAyE k 31 1 ik TT TT ASq SA Sq ASy SE i 33i 1 i 13k s - 0. Q.E.D. Ž. TT S q SAy ES yESy SE 11ik3 k 11k STABILITY CONDITIONS 199 Remark 35 It is stated in Remark 34 that X is not required to be posi- tive definite. However, in Corollary 5, X is assumed to be invertible since S 0 1 X s , where S ) 0. 1 SS 31 Next, we consider stability conditions for closed-loop systems. We propose Ž. Ž. a modified PDC 10.8 to stabilize the fuzzy descriptor system 10.2 : rr e u t sy h z t ® z t F* x* t ,10.8 Ž. Ž. Ž. Ž. Ž . Ž.Ž. ÝÝ ik ik i s1 ks1 F 0 where F* s . The fuzzy controller design problem is to determine ik ik the local feedback gains F . ik Ž. Ž. By substituting 10.8 into 10.2 , the fuzzy control system is represented as rrr e E*x* t s h z thz t ® z t A* y B*F* x* t . Ž. Ž. Ž. Ž. Ž. Ž.Ž.Ž. ˙ Ž. ÝÝÝ ijk ikijk i s1 js1 ks1 10.9 Ž. Ž. Theorem 34 gives a sufficient condition for ensuring the stability of 10.9 . Ž. THEOREM 34 The fuzzy descriptor system 10.2 can be stabilized ®ia the Ž. PDC fuzzy controller 10.8 if there exist Z , Z , and M such that 13 ik Z T s Z ) 0, 10.10 Ž. 11 T yZ y Z ) 33 Z q AZ - 0, 1 i 1 T yZE y EZ 1 kk1 ž/ yBM q EZ iik k3 h l® / or, 10.11 Ž. ik T y2 Z y 2 Z ) 33 2 Z q AZ 1 i 1 F 0, T yBM q AZ y2 ZE y 2 EZ ijk j1 1 kk1 0 yBM q 2 EZ jik k3 i - j F r s.t. h l h l® / or, 10.12 Ž. ijk Ž. where the asterisk denotes the transposed elements matrices for symmetric positions. FUZZY DESCRIPTOR SYSTEMS AND CONTROL 200 Proof. Consider a candidate of a quadratic function V x* t s x* T t E* T Xx* t , Ž. Ž. Ž. Ž. where S 0 1 X s . SS 31 Then, rrr e T ˙ V x* t s h z thz t ® z t x* t Ž. Ž. Ž. Ž. Ž. Ž . Ž.Ž.Ž. ÝÝÝ ijk i s1 js1 ks1 = T T A* y B*F* X q XA* y B*F* x* t Ž. Ž.Ž. ½5 ik i jk ik i jk rr e 2 T s h z t ® z t x* t Ž. Ž. Ž. Ž.Ž. ÝÝ ik i s1 ks1 = T T A* y B*F* X q XA* y B*F* x* t Ž.Ž.Ž. Ä4 ik i ik ik i ik rr e T q 2 h z thz t ® z t x* t Ž. Ž. Ž. Ž. Ž.Ž.Ž. ÝÝÝ ijk i s1 i-jks1 = T A* y B*F* q A* y B*F* ik i jk jk j ik X ½ ž/ 2 A* y B*F* q A* y B*F* ik i jk jk j ik T qXx* t . Ž. 5 ž/ 2 Therefore, the stability conditions are derived as follows: E* T Xs X T E* G 0, 10.13 Ž. G T X q X T G - 0, h l® / or, 10.14 Ž. iik iik i k T G q GGq G ijk jik ijk jik T X q X F 0, ž/ž/ 22 i - j F r s.t. h l h l® / or, 10.15 Ž. ijk where 0 I G s A* y B*F* s , ijk ik i jk A y BF yE iijk k F 0 F* s . ik ik STABILITY CONDITIONS 201 Ž. Equation 10.13 can be rewritten as X yT E* T s E*X y1 G 0. The above inequality is yT y1 S 0 I 0 I 0 S 0 11 sG0. SS 00 00 SS 31 31 Therefore, we obtain TT Z yZI0 13 T 0 Z 00 1 I 0 Z 0 Z 0 11 ssG0, 00 yZZ 00 31 where Z s S y1 and Z s S y1 SS y1 . 11 3131 Note that the following relation holds: S 0 Z 0 I 0 11 s . SS yZZ 0 I 31 31 Ž. Equation 10.14 can be rewritten as X yT G T XX y1 q X yT X T GX y1 iik iik TT TTT Z yZ 0 A y FB 13 iiki s TT 0 ZI yE 1 k Z 0 0 I 1 q A y BF yE yZZ iiik k 31 T yZ y Z ) 33 Z q AZ s - 0. 1 i 1 T yZE y EZ 1 kk1 ž/ yBM q EZ iik k3 Ž. Ž. Equation 10.12 is also derived in the same way as condition 10.11 . Ž. Q.E.D. FUZZY DESCRIPTOR SYSTEMS AND CONTROL 202 Ž The fuzzy controller design problem is to determine F i s 1,2, .,r; ik e . k s 1,2, . . . , r satisfying the conditions of Theorem 34. The feedback gains are obtained as F s MZ y1 ik ik 1 S 0 1 from the solution Z and M of the above LMIs. The matrix X s is 1 ik SS 31 obtained as S s Z y1 and S s Z y1 ZZ y1 . 11 3131 Ž. ŽŽ Next, we derive stability conditions for 10.9 in the case of h z t s i ŽŽ e Ž. ® z t and r s r . In this case, the fuzzy descriptor system 10.2 can be k rewritten as r E*x* t s h z t A* x* t q B*u t , 10.16 Ž. Ž. Ž. Ž. Ž . Ž. Ž. ˙ Ý iii i i s1 where I 00I E* s , A* s , ii 00 A yE ii 0 B* s . i B i Ž. Ž. In this case, the PDC controller 10.17 instead of 10.8 is used: r u t sy h z t F*x* t , 10.17 Ž. Ž. Ž. Ž . Ž. Ý iii i s1 w F 0 x where F* s . In this case, Theorem 34 can be simplified as follows. i ii ŽŽ ŽŽ e THEOREM 35 Assume that h z t s® z t and r s r . Then, the fuzzy ik Ž. Ž. descriptor system 10.16 can be stabilized ®ia the PDC fuzzy controller 10.17 if there exist Z , Z , and M such that 13 i Z T s Z ) 0, 10.18 Ž. 11 T yZ y Z ) 33 Z q AZ - 0, 1 i 1 T yZE y EZ 1 ii1 ž/ yBMq EZ ii i3 i s 1,2, .,r , 10.19 Ž. STABILITY CONDITIONS 203 T y2 Z y 2 Z ) 33 2 Z q AZ 1 i 1 - 0, T yBMq AZ y2 ZE y 2 EZ ij j1 1 ii1 0 yBMq 2 EZ ji i3 i - j F r s.t. h l h / or. 10.20 Ž. ij The feedback gains F are obtained as F s MZ y1 . iii1 Proof. Consider a candidate of quadratic function V x* t s x* T t E* T Xx* t . Ž. Ž. Ž. Ž. Then, rr T ˙ V x* t s h z thz t x* t Ž. Ž. Ž. Ž. Ž. Ž.Ž. ÝÝ ij i s1 js1 = T T A* y B*F* X q XA* y B*F* x* t Ž. Ž.Ž. ii i jj ii i jj r 2 T s h z t x* t Ž. Ž. Ž. Ý i i s1 = T T A* y B*F* X q XA* y B*F* x* t Ž.Ž.Ž. ii i ii ii i ii r T q 2 h z thz t x* t Ž. Ž. Ž. Ž.Ž. ÝÝ ij i s1 i-j = T A* y B*F* q A* y B*F* ii i jj jj j ii X ž/ 2 A* y B*F* q A* y B*F* ii i jj jj j ii T qXx* t - 0. Ž. ž/ 2 Therefore, we have the following stability conditions: E* T X s X T E* G 0, G T X q X T G - 0, i s 1,2, .,r , ii ii T G q GGq G ij ji ij ji T X q X F 0, i - j F r s.t. h l h / or, ij ž/ž/ 22 FUZZY DESCRIPTOR SYSTEMS AND CONTROL 204 where G s A* y B*F*, ij ii i jj 0 I 0 A* s , B* s , ii i A yEB ii i F 0 F* s . i ii Ž.Ž. We can obtain the conditions 10.18 ᎐ 10.20 in the same way as in Ž. Theorem 34. Q.E.D. Now consider the common B matrix case, that is, B s B s иии s B in 12 r Ž. 10.2 . The stability analysis for the common B matrix case is simpler and easier in comparison with that of the general case. Keep this in mind because we will refer to this when discussing the motivation behind the introduction of the fuzzy descriptor system. In the common B matrix case, the stability conditions of Theorems 34 and 35 can be simplified as Theorems 36 and 37, respectively. Theorem 37 gives Ž Ž Ž Ž e stability conditions for the case of h z t s® z t and r s r . ik Ž. THEOREM 36 The fuzzy descriptor system 10.2 with the common B matrix, that is, B s B s иии s B s B, can be stabilized ®ia the PDC fuzzy controller 12 r Ž. 10.8 if there exist Z , Z , and M such that 13 ik Z T s Z ) 0, 10.21 Ž. 11 T yZ y Z ) 33 Z q AZ - 0, h l® / or. 10.22 Ž. 1 i 1 ik T yZE y EZ 1 kk1 ž/ yBM q EZ ik k 3 The feedback gains F are obtained as F s MZ y1 . ik ik ik 1 Proof. Consider a candidate of quadratic function V x* t s x* T t E* T Xx* t . Ž. Ž. Ž. Ž. Then, rr e T ˙ V x* t s h z t ® z t x* t Ž. Ž. Ž. Ž. Ž. Ž.Ž. ÝÝ ik i s1 ks1 T T = A* y B*F* X q XA* y B*F* x* t - 0, Ž.Ž.Ž. ik ik ik ik where 0 B* s . B [...]... y 1 U x* Ž t since r Ý h2 Ž z Ž t y i is1 1 sy1 r Ý Ý 2 h i Ž z Ž t h j Ž z Ž t G 0, i - s F r is1 i-j Therefore, the closed-loop system is stable if E*T X s X T E* G 0, T Gii k X q X T Gii k q Ž s y 1 U - 0, ž Gi jk q Gji k 2 T / X qXT ž Gi jk q Gji k 2 h i l ®k / r , o / Ž 10.27 y U F 0, i - j F r s.t h i l h j l ®k / r o Ž 10.28 In the same way as in the proof of Theorem 34, we obtain... is1 i-j = ½ž Gi j q Gji 2 T / X qXT ž Gi j q Gji 2 /5 x*T Ž t r F T Ý h2 Ž z Ž t x*T Ž t Ž Gii X q X T Gi i x* Ž t i is1 r q2 Ý Ý h i Ž z Ž t h j Ž z Ž t x*T Ž t Ux* Ž t is1 i-j r F T Ý h2 Ž z Ž t x*T Ž t Ž Gii X q X T Gi i q Ž s y 1 U x* Ž t i is1 210 FUZZY DESCRIPTOR SYSTEMS AND CONTROL since r Ý h2 Ž z Ž t y i is1 1 sy1 r Ý Ý 2 h i Ž z Ž t h j Ž z Ž t G 0, i - s F r is1 i-j... Ei Z1 - 0, i s 1, 2, , r Ž 10.24 The feedback gains Fi are obtained as Fi s Mi Zy1 1 Proof Consider a candidate of a quadratic function V Ž x* Ž t s x*T Ž t E*T Xx* Ž t Then, ˙ V Ž x* Ž t s r Ý h i Ž z Ž t x*T Ž t is1 T = Ž A* y B*F* X q X T Ž A* y B*F* x* Ž t -0 ii ii ii ii Therefore, the fuzzy control system is stable if E*T Xs X T E* G 0, T Ž A* y B*F* X q X T Ž A* y B*F* - 0,... i Z1 y Bi M j 0 qA j Z1 y B j Mi q2 Ei Z 3 y 2 Y3 209 y 2 Z 1 EiT y 2 Ei Z 1 y 2 Y2 - 0, i - j F r s.t h i l h j / r o The feedback gains are obtained as Fi s Mi Zy1 1 Proof Consider a candidate of a quadratic function V Ž x* Ž t s x*T Ž t E*T Xx* Ž t Now assume that ž Gi j q Gji 2 T / X qXT Gi j q Gji ž 2 / i - j F r s.t h i l h j / r , o y U F 0, where 0 Gi j s A y B F i i j Us Q1 Q3 I yEi ,... jk qA j Z1 y B j Mi k q2 Ek Z 3 y 2 Y3 0 T y2 Z1 Ek y 2 Ek Z1 y 2 Y2 i - j F r s.t h i l h j l ®k / r o - 0, Ž 10.26 The feedback gains are obtained as Fi k s Mi k Zy1 1 Proof Consider a candidate of quadratic function V Ž x* Ž t s x*T Ž t E*T Xx* Ž t Now assume that ž Gi jk q Gji k 2 T / X qXT ž Gi jk q Gji k 2 / y U F 0, i - j F r s.t h i l h j l ®k / r , o where 0 Gi jk s A y B F i i ik Us... CONTROL since r Ý h2 Ž z Ž t y i is1 1 sy1 r Ý Ý 2 h i Ž z Ž t h j Ž z Ž t G 0, i - s F r is1 i-j Therefore, the closed-loop system is stable if E*T X s X T E* G 0, T Gii X q X T Gii q Ž s y 1 U - 0, ž Gi j q Gji 2 T X qXT / ž Gi j q Gji 2 i s 1, 2, , r , / Ž 10.29 y U F 0, i - j F r s.t h i l h j / r o Ž 10.30 In the same way as in the proof of Theorem 38, we obtain the LMI ŽQ.E.D conditions... Ž t i is1 ks1 re r Ý Ý h i Ž z Ž t h j Ž z Ž t ®k Ž z Ž t x*T Ž t q2 Ý is1 i-j ks1 = ½ž 2 T / X qXT ž Gi jk q Gji k 2 /5 x*T Ž t re r F Gi jk q Gji k T Ý Ý h2 Ž z Ž t ®k Ž z Ž t x*T Ž t Ž Gii k X q X T Gii k x* Ž t i is1 ks1 re r Ý Ý h i Ž z Ž t h j Ž z Ž t ®k Ž z Ž t x*T Ž t Ux* Ž t q2 Ý is1 i-j ks1 re r F Ý Ý h2 Ž z Ž t ®k Ž z Ž t x*T Ž t i is1 ks1 = Ž GiTi k X q X T Gii... is less than or equal to s, where 1 - s F r The fuzzy descriptor system Ž10.2 can be stabilized ®ia the PDC fuzzy controller Ž10.8 if there exist a common matrix Z1 , Z 3 , Y1 , Y2 , and Y3 such that T Z1 s Z1 ) 0, Y3T G 0, Y2 Y1 Y3 Ys ) T yZ 3 y Z 3 q Ž s y 1 Y1 ž Z 1 q A i Z 1 y B i Mi k / qEk Z 3 q Ž s y 1 Y3 T yZ1 Ek y Ek Z1 q Ž s y 1 Y2 h i l ®k / r , o - 0, Ž 10.25 ) T y2 Z 3 y 2 Z 3 y 2... or equal to s, where 1 - s F r Moreo®er, assume that h i Ž z Ž t s ®k Ž z Ž t and r s r e Then, the fuzzy descriptor system Ž10.16 can be stabilized ®ia the PDC fuzzy controller Ž10.17 if there exist Z1 , Z 3 , Y1 , Y2 , and Y3 such that T Z1 s Z1 ) 0, Ys Y1 Y3 Y3T G 0, Y2 T yZ 3 y Z 3 q Ž s y 1 Y1 ž Z 1 q A i Z 1 y B i Mi qEi Z 3 q Ž s y 1 Y3 / ) yZ1 EiT y Ei Z1 q Ž s y 1 Y2 - 0, i s 1, 2, , r... descriptor system instead of the ordinary fuzzy model Consider a simple nonlinear system, ¨ ˙ Ž 1 q a cos ␪ Ž t ␪ Ž t s yb␪ 3 Ž t q c␪ Ž t q du Ž t , Ž 10.31 ˙ ˙ where a - 1 and assume the range of ␪ Ž t as < ␪ Ž t < - ␾ First, we replace the nonlinear dynamics Ž10.31 with the ordinary TakagiSugeno fuzzy model From Ž10.31., we have ¨ ␪ Ž t s y b 1 q a cos ␪ Ž t c q 1 q a cos ␪ Ž t ˙ ␪ 3Ž . 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 10 FUZZY DESCRIPTOR SYSTEMS AND CONTROL. 0, i - s F r. Ž. Ž. Ž. Ž. Ž.Ž. ÝÝÝ iij s y 1 i s1 is1 i-j Therefore, the closed-loop system is stable if E* T X s X T E* G 0, G T X q X T G q s y 1 U - 0,