不确定离散模糊时滞系统的时滞相关H∞控制

Ý ÌÉÖ
5
´ » À© ¨ Æ Ï Ö H ¿ Í ¤ Ë ¶Û¢­ Li Guan ¾ ËÉ Ö ÕÄÅÑ ¶ ½ ℄Î¥ ° ¢± © Í Õ Û ¹ ¶Û Ð ¨ ¥ à ± Ì ¶Ç ¢ ­ § Å T-S Í ¨
∞
[5] [6]
835
Í
Þ
q ξ (k ) = ξ1 (k ) · · · ξr (k ) Mij (j = 1, · · · , r) x(k ) ∈ Rn u(k ) ∈ Rm w (k ) ∈ Rl z (k ) ∈ Rp d>0 x(k ) = 0, ∀d k 0. ∆Ai (k ) ∆Adi (k ) ∆Bi (k ) = DF (k ) Ei Edi Ebi , D, Ei , Edi Ebi F (k ) F T (k )F (k ) I, ∀k . (PDC) Controller rule i: IF ξ1 (k ) is Mi1 and · · · and ξr (k ) is Mir THEN u(k ) = Ki x(k ),
410083)
Abstract Problems of stability and H∞ control for discrete-time T-S fuzzy systems with state delay and uncertainties are discussed. A new delay-dependent stability criterion is first proposed by defining an appropriate Lyapunov function. Then, the sufficient conditions for the existence of H∞ fuzzy state-feedback controller is investigated in terms of linear matrix inequality(LMI). Finally, an iterative algorithm is provided for the optimal design of the H∞ controller. Key words T-S fuzzy system, delay-dependent, robust control, linear matrix inequality
k −1 q q
(P + dT )y (k ) −
l= k − d
y T (l)T y(l) + 2ηT (k )M x(k + 1) −
i=1 j =1
hi (ξ (k ))hj (ξ (k ))·
k −1
Aij x(k ) + Adi x(k − d) + B1i w(k )
+ 2η T (k )N x(k ) − x(k − d) −
y (l)
l= k − d
Ð ¿ ¥ À 1,
k −1 k −1
(6)
−2ηT (k )N (7)
y (l)
l= k − d
dη T (k )N T −1 N T η(k ) +
l= k − d
y T (l)T y(l)
(7)
Æ
∆V (k )
(6),
¸´À
q q
hi (ξ (k ))hj (ξ (k ))η T (k )Ξij η (k )
∞
Ò
Lyapunov H∞
Delay-Dependent H ∞ Control for Uncertain Discrete-Time Delay Fuzzy Systems
CHEN Zhi-Sheng1
1
PENG Ke2
LI Yong-Gang3
ZHANG Tai-Shan3
(School of Energy and Power Engineering, Changsha University of Science and Technology, Changsha 410076) 2 (College of Engineering, Hunan Normal University, Changsha 410081) 3 (School of Information Science and Engineering, Central South University, Changsha (E-mail: czs9988@)
T T M3 , N T = N1 [8,9] T N2
(5)
T N3 ,
Þ
T ηT (k ) = xT (k ) xT (k − d) y T (k ) , M T = M1 Ml , Nl (l = 1, 2, 3)
±Á¸
¯ ¥ ½ ² ¥ ¹Ã
É ℄΢
∆V (k ) = V (k + 1) − V (k ) = xT (k )S x(k ) + 2xT (k )P y (k ) − xT (k − d)S x(k − d) + y T (k )·
32
Þ
°Â ∗
T T Λ11 = N1 + N1 − M1 (Aij − I ) − (Aij − I )T M1 + S; T T T T T T Λ12 = −N1 + N2 − M1 Adi − (Aij − I ) M2 ; Λ13 = P + N3 + M1 − (Aij − I ) M3 ; Λ22 = T T T T T T T −S − N2 − N2 − M2 Adi − Adi M2 ; Λ23 = −N3 + M2 − Adi M3 ; Λ33 = P + dT + M3 + M3 .
5
Ý
∞
837
T
2
3
T
> 0,
T
1
2
3
i
µopt = min µ Subject to LMI(10)
µ>0
(9)
Θii < 0, i = 1, 2, · · · , q Θij + Θji < 0, 1 i<j q Hij 0 0 0 0 0 0 0 0 −I
Ø ¸¢ Ê¢ ¯ ¢ Í ¾¢ ½ µÆ ¢ ª È µÆ ¢ ½ µ ¾¢ · ű ¢ Í §Å ¸Å¡ ¨ °Â · ±Á ½²¢ ª È ½ ² ¢¡ À Ò Ð ³ ¾ ½ ¢Þ Ø ¨
T
Plant rule i: IF ξ1 (k ) is Mi1 and · · · and ξr (k ) is Mir THEN x(k + 1) = (Ai + ∆Ai (k ))x(k ) + (Adi + ∆Adi (k ))x(k − d) + (Bi + ∆Bi (k ))u(k ) + B1i w (k ) z (k ) = Ci x(k ), i = 1, 2, · · · , q
(1)
Þ
K i ∈ R m× n
Ú ¢Ë £ ℄΢
x(k + 1) =
q q
½ ¢½²¥ §Í
q
i = 1, 2, · · · , q
(2)
¦¼
(1)
¨
(2),
Ü Â
£ ¸ Æ À
i=1 j =1 q
hi (ξ (k ))hj (ξ (k )) Aij x(k ) + Adi x(k − d) + B1i w(k ) (3)

Λ12 Λ22 ∗
∆V (k ) < 0,
Λ13 Λ23 + dN T −1 N T Λ33
§Í
பைடு நூலகம்
(3)
· Ä Å¥
H∞
½ ¢ ­
Û¨
´ » À© ¨ Æ Ï Ö H ¿ Ç 2. Ì Í (1) Å ± ¸ ρ ,ρ , ¯ ±Á ½ ² P = P > 0, S = S T = T > 0, N , N , N , Y (i = 1, 2, · · · , q ) £½² X, ­ ¬£Ç ±
z (k ) =
hi (ξ (k ))Ci x(k )
i=1
°Â A
ij
= Ai + Bi Kj + DF (k )Ei + DF (k )Ebi Kj ; Adi = Adi + DF (k )Edi ; hi (ξ (k )) =
r
ωi (ξ (k ))/
i=1
ωi (ξ (k )); ωi (ξ (k )) =
∞ 2 2
[7]
T
T
T
T
−1
[8]
T
T
T
−1
T
T
T
T
T
T
T
T
1
2
3
1
2
3
i
Ψii < 0, i = 1, 2, · · · , q i<j q
Ψij + Ψji < 0, 1
(4)
836
Ë
Λ11 ∗ Ψij = ∗ ∗ Λ12 Λ22 ∗ ∗
¦
Λ13 Λ23 Λ33 ∗
Ú
dN1 dN2 , dN3 −dT i, j = 1, 2, · · · , q
j =1
Mij (ξj (k )), 0
ωi (ξ (k ))
1.
Ã Û ¹ Ç ° ± Ì Í (1), PDC ½ (2), § Í (3) Î ¬ Ä Å¢¡ À H ª È ½ Ñ Õ z < γ w . 2 ×ÆÇ Ð 1 . Ì ±Á ½ ² X ,Y µ Å ½ ² R, X Y + Y X X RX + Y R Y . Ð 2 . Å ±Á ½ ² Q = Q ,D,E R = R > 0, Ì ½ À F F R F , ¡´ ǫ>0 Q + ǫ DD + ǫE RE < 0 ¶ Å ¢ Q + DF E + E F D < 0. ² ª È µÆ w = 0 ¢ É § Í (3) Ä Å Ñ ¥ Ç 1. Ì § Å Í (1), § Í (3) · Ä Å » ±Á ½ ² P = P > 0, S = S > 0, T = T > 0, M , M , M , N , N , N , K (i = 1, 2, · · · , q), À