Ổn định hóa của hệ PTVP

Chơng Tính ổn định ổn định hoá hệ điều khiển tuyến tính có trễ biến thiên Chơng nµy lµ mét hai néi dung chÝnh cđa ln văn.Các kết mở rộng kết chơng II cho trờng hợp hệ có trễ biến thiên hàm trễ không khả vi.Nội dung chơng gồm hai phần: Phần thứ đa số tiêu chuẩn ổn định ổn định hóa cho hệ điều khiển tuyến tính có trễ biến thiên hệ không chắn có trễ biến thiên với điều kiện hàm trễ không khả vi Phần thứ hai đa tiêu chuẩn ổn định mũ ổn định hóa đợc dạng mũ hệ điều khiển tuyến tính có trễ biến thiên hệ không chắn có trễ biến thiên hàm trễ không khả vi 3.1 Tiêu chuẩn ổn định ổn định hoá hệ điều khiển tuyến tính có trễ biến thiên Xét hệ điều khiển tuyến tính cã trƠ biÕn thiªn sau:   x(t) ˙ = A0x(t) + A1x(t − h(t)) + Bu(t),  x(t) = φ(t), t ∈ [−h2, 0], t 0, (3.1) Với x(t) Rn véctơ trạng thái hệ, u(t) Rm hàm điều khiển A0 , A1 vµ B lµ ma trËn h»ng cã sè chiỊu thÝch hợp Còn (t)là hàm điều kiện ban đầu thỏa m·n φ(t) ∈ C([−h2, 0], Rn ) vµ h(t) lµ trễ biến thiên thỏa mÃn điều kiện h1 h(t) h2 Ta có định lý sau: Định lý 3.1 HƯ tun tÝnh cã trƠ biÕn thiªn (3.1) víi u(t) = ổn định tiệm cận tồn ma trận đối xứng xác định dơngP, Q, R, Z ma trận H1 , H2, H3 , H4 tháa m·n   T T T T T T Ω P A1 + R + H1 A1 + A0 H2 A0 H3 A0 H4 − H1     T T T T T T  H2 A1 + A1 H2 − (R + Z) A1 H3 + Z A1 H4 − H2   < 0, Π=     −(Q + Z) −H3   Γ (3.2) Trong ®ã Ω = P A0 + AT0 P + Q − R + H1A + AT0 H1T , Γ = h22 R + (h2 − h1 )2Z − (H4T + H4 ) Chøng minh XÐt hµm Lyapunov cđa hƯ (3.1) sau: t t T t T V (t, xt) = x (t)P x(t) + x˙ T (u)Rx(u)duds ˙ x (s)Qx(s)ds + h2 t−h1 t−h2 t−h1 s t x˙ T (u)Z x(u)duds ˙ +(h2 − h1 ) t−h2 s Ta thÊy λ1 x(t) V (t, xt) λ2 xt , t víi λ1 = λmin (P ), 1 λ2 = λmax (P ) + h1 λmax (Q) + h32λmax (R) + (h2 − h1 )3 λmax (Z) 2 LÊy đạo hàm V (.) theo t dọc theo nghiệm x(t) cđa hƯ (3.1) ta cã, V˙ (t, xt) = 2xT (t)P x(t) ˙ + xT (t)Qx(t) − xT (t − h1 )Qx(t − h1 ) + h22 x˙ T (t)Rx(t) ˙ t x˙ T (s)Rx(s)ds ˙ + (h2 − h1 )2 x˙ T (t)Z x(t) ˙ − h2 t−h2 t−h1 x˙ T (s)Z x(s)ds ˙ − (h2 − h1 ) t−h2 = 2xT (t)P (A0x(t) + A1x(t − h(t))) + xT (t)Qx(t) − xT (t − h1 )Qx(t − h1 ) t + x˙ T (t)[h22R x˙ T (s)Rx(s)ds ˙ + (h2 − h1) Z]x(t) ˙ − h2 t−h2 t−h1 x˙ T (s)Z x(s)ds ˙ − (h2 − h1 ) t−h2 V˙ (t, xt) = xT (t)[P A0 + AT0 P + Q]x(t) + x˙ T (t)[h22R + (h2 − h1 )2 Z]x(t) ˙ t + 2xT (t)P A1x(t − h(t)) − xT (t − h1 )Qx(t − h1 ) − h2 x˙ T (s)Rx(s)ds ˙ t−h2 t−h1 x˙ T (s)Z x(s)ds ˙ − (h2 − h1 ) t−h2 Theo bỉ ®Ị ( ),ta cã t t T −h2 x˙ (s)Rx(s)ds ˙ x˙ T (s)Rx(s)ds ˙ −h(t) t−h2 t−h(t) t − t T x(s)ds ˙ R x(s)ds ˙ t−h(t) t−h(t) T − x(t) − x(t − h(t)) R x(t) − x(t − h(t)) (3.3) Vµ t−h1 x˙ T (s)Z x(s)ds ˙ − (h2 − h1 ) t−h2 t−h1 x˙ T (s)Z x(s)ds ˙ −(h(t) − h1 ) t−h(t) t−h1 t−h1 T x(s)ds ˙ − Z t−h(t) x(s)ds ˙ t−h(t) T − x(t − h1 ) − x(t − h(t)) Z x(t − h1 ) − x(t − h(t)) (3.4) Tõ (3.3) vµ (3.4) suy ra, V˙ (t, xt) xT (t)[P A0 + AT0 P + Q]x(t) + x˙ T (t)[h22R + (h2 − h1 )2Z]x(t) ˙ + 2xT (t)P A1x(t − h(t)) − xT (t − h1 )Qx(t − h1 ) T − x(t) − x(t − h(t)) R x(t) − x(t − h(t)) T − x(t − h1 ) − x(t − h(t)) Z x(t − h1 ) − x(t − h(t)) xT (t)[P A0 + AT0 P + Q − R]x(t) + x˙ T (t)[h22R + (h2 − h1 )2 Z]x(t) ˙ + xT (t)[P A1 + R]x(t − h(t)) + xT (t − h(t))[AT1 P + R]x(t) − xT (t − h(t))[R + Z]x(t − h(t)) − xT (t − h1 )[Q + Z]x(t − h1) + xT (t − h1 )Zx(t − h(t)) + xT (t − h(t))Zx(t − h1 ) V˙ (t, xt) ξ T (t)∆ξ(t) (3.5) Trong ®ã ξ T (t) = xT (t) xT (t − h(t)) xT (t − h1 ) x˙ T (t) , vµ  P A + AT0 P + Q − R P A1 + R    −(R + Z) Z ∆=   −(Q + Z)   0 h22R + (h2 − h1)2 Z     (3.6)    H¬n nữa,nếu x(t) nghiệm hệ (3.1) với H1 ,H2 ,H3 H4 ,ta có: T (t)H[−x(t) ˙ + A0x(t) + A1x(t − h(t))] = hay T (t)(t) = đây,  H  1   H2   H=  , H3    H4     Σ=    H1 A0 + AT0 H1T H1 A1 + AT0 H2T AT0 H3T H2 A1 + AT1 H2T AT1 H3T AT0 H4T − H1    − H2     −H3  T −(H4 + H4 ) AT1 H4T (3.7) KÕt hỵp (3.5) ,(3.6) (3.7) ,ta đợc V (t, xt) T (t)Πξ(t), Víi  Π = ∆+Σ = AT0 H3T Ω P A1 + R + H1 A1 + AT0 H2T    H2 A1 + AT1 H2T − (R + Z) Z + AT1 H3T    −(Q + Z)  AT0 H4T − H1 AT1 H4T − H2 −H3 h22 R + (h2 − h1 )2 Z − (H4 + H4T )         Tõ (3.2), ta thÊy Π < 0,nªn V˙ (t, xt) −λmin (Π) x Do hệ (3.1) ổn định tiện cận Định lý đà đợc chứng minh Bây ,ta xét ổn định hệ (3.1) điều khiển ngợc u(t) = Kx(t) với u(t) = Kx(t) ,khi hệ (3.1) trở thành   x(t) ˙  x(t) = (A0 + BK)x(t) + A1x(t − h(t)), = φ(t), t 0, (3.8) t [h2, 0], Định lý 3.2 Hệ trễ biến thiên (3.8) với K ma trận điều khiển ngợc ổn định tiệm cận tồn ma trận đối xứng xác định dơngP, Q, R, Y1, Y2 , Z1 , Z2 ma trËn             S M L V W N  1  1  1  1  1  1             S2  M2  L2  V2  W2  N2                          N = N3  , S = S3  , M = M3  , L = L3  , V = V3  , W = W3  ,                         S4  M4  L4  V4  W4  N4              N5 S5 M5 L5 V5 W5 tháa m·n:  ∆= ∆11 ∆12 ∆T12 ∆22   < (3.9) Trong ®ã ∆11 = ∆1 + ∆2 + ∆T2 + ∆3 + ∆T3 ,   Q+R 0 P       −R 0     ∆1 =  , 0−Q 0       0 0   P 0 h2 Y1 + h1 Y2 + (h2 − h1 )(Z1 + Z2 ) ∆2 = N + S L + V − S M − L −(N + M + V ) , ∆3 = −W (A0 + BK) −W A1 W , ∆12 = h2 N h1 S (h2 − h1 )M (h2 − h1 )L (h2 − h1)V , ∆22 = diag −h2Y1 −h1Y2 −(h2 − h1 )Z1 −(h2 − h1 )Z1 −(h2 − h1 )Z2 Chứng minh Sử dụng công thức Newton-Leibniz,ta có phơng trình dới với ma trận N, S, M, L, V t 2ξ T (t)N x(t) − x(t − h2) − x(s)ds ˙ = 0, (3.10) x(s)ds ˙ = 0, (3.11) t−h2 t 2ξ T (t)S x(t) − x(t − h1 ) − t−h1 t−h(t) 2ξ T (t)M x(t − h(t)) − x(t − h2 ) − x(s)ds ˙ = 0, (3.12) t−h2 t−h1 T 2ξ (t)L x(t − h1 ) − x(t − h(t)) − x(s)ds ˙ = 0, (3.13) t−h(t) t−h1 2ξ T (t)V x(t − h1) − x(t − h2 ) − x(s)ds ˙ = 0, (3.14) t−h2 Víi ξ T (t) = xT (t) xT (t − h1) xT (t − h(t)) xT (t − h2 ) x˙ T (t) Ph−¬ng trình sau với ma trận W 2ξ T (t)W −(A0 + BK)x(t) + x(t) ˙ − A1x(t − h(t)) = (3.15) B©y giê,chóng ta xÐt hµm Lyapunov cđa hƯ (3.8) nh− sau: t t T T V (t, xt) = x (t)P x(t) + t−h2 t xT (s)Rx(s)ds x (s)Qx(s)ds + t t t−h1 t x˙ T (u)Y1 x(u)duds ˙ + + t−h2 s t−h1 t x˙ T (u)Y2 x(u)duds ˙ t−h1 s x˙ T (u)[Z1 + Z2 ]x(u)duds ˙ + t−h2 s Ta cã λ1 x(t) V (t, xt) λ2 xt , t = (P ), λ2 = λmax (P ) + h2λmax (Q) + h1 λmax (R) + [h22 λmax (Y1 ), 2 + h1 λmax (Y2 )] + (h2 − h1 ) λmax (Z1 + Z2 ) LÊy đạo hàm V (.) theo t dọc theo nghiệm x(t) hệ (3.8) ta đợc, V (t, xt) = 2x˙ T (t)P x(t) + xT (t)Qx(t) − xT (t − h2 )Qx(t − h2) + xT (t)Rx(t) ˙ + h1 x˙ T (t)Y2 x(t) ˙ − xT (t − h1 )Rx(t − h1 ) + h2 x˙ T (t)Y1 x(t) t T x˙ T (s)Y1 x(s)ds ˙ + (h2 − h1 )x˙ (t)[Z1 + Z2 ]x(t) ˙ − t−h2 t t−h1 x˙ T (s)Y2 x(s)ds ˙ − − t−h1 x˙ T (s)[Z1 + Z2 ]x(s)ds ˙ t−h2 (3.16) §Ĩ ý r»ng, t−h1 x˙ T (s)[Z1 + Z2 ]x(s)ds ˙ − t−h2 t−h(t) t−h1 x˙ T (s)Z1 x(s)ds ˙ − =− t−h2 t−h1 x˙ T (s)Z1 x(s)ds ˙ − t−h(t) x˙ T (s)Z2 x(s)ds ˙ t−h2 (3.17) KÕt hỵp (3.10)-(3.17) ,ta suy V˙ (t, xt) = 2x˙ T (t)P x(t) + xT (t)[Q + R]x(t) − xT (t − h2 )Qx(t − h2 ) ˙ − xT (t − h1 )Rx(t − h1 ) + x˙ T (t)[h2Y1 + h1 Y2 + (h2 − h1 )(Z2 + Z1 )]x(t) t t x˙ T (s)Y1 x(s)ds ˙ + 2ξ T (t)N x(t) − x(t − h2) − − t−h2 t x(s)ds ˙ t−h2 t x˙ T (s)Y2 x(s)ds ˙ + 2ξ T (t)S x(t) − x(t − h1 ) − − t−h1 t−h(t) x(s)ds ˙ t−h1 t−h(t) x˙ T (s)Z1 x(s)ds ˙ + 2ξ T (t)M x(t − h(t)) − x(t − h2 ) − − t−h2 t−h1 x(s)ds ˙ t−h2 t−h1 x˙ T (s)Z1 x(s)ds ˙ + 2ξ T (t)L x(t − h1 ) − x(t − h(t)) − − x(s)ds ˙ t−h(t) t−h1 t−h(t) t−h1 x˙ T (s)Z2 x(s)ds ˙ + 2ξ T (t)V x(t − h1) − x(t − h2 ) − − t−h2 x(s)ds ˙ t−h2 + 2ξ T (t)W −(A0 + BK)x(t) + x(t) ˙ − A1x(t − h(t))  T   ( x(t) x t)          x(t − h1 )   x(t − h1 )          T T ∆  x(t − h(t))  x(t − h(t)) + ξ (t)(∆3 + ∆3 )ξ(t)          x(t − h2 )   x(t − h2 )      x(t) ˙ x(t) ˙ t [2ξ T (t)N x(s) ˙ + x˙ T (s)Y1 x(s)]ds ˙ + 2ξ T (t)N x(t) − x(t − h2 ) − t−h2 t [2ξ T (t)S x(s) ˙ + x˙ T (s)Y2 x(s)]ds ˙ + 2ξ T (t)S x(t) − x(t − h1) − t−h1 t−h(t) [2ξ T (t)M x(s) ˙ + x˙ T (s)Z1 x(s)]ds ˙ + 2ξ T (t)M x(t − h(t)) − x(t − h2 ) − t−h2 t−h1 [2ξ T (t)Lx(s) ˙ + x˙ T (s)Z1 x(s)]ds ˙ + 2ξ T (t)L x(t − h1 ) − x(t − h(t)) − t−h(t) t−h1 [2ξ T (t)V x(s) ˙ + x˙ T (s)Z2 x(s)]ds ˙ + 2ξ T (t)V x(t − h1 ) − x(t − h2 ) − t−h2 10 V˙ (t, xt) ξ T (t)[∆1 + ∆3 + ∆T3 ]ξ(t) + 2ξ T (t)N I 0 −I ξ(t) + 2ξ T (t)S I −I 0 ξ(t) + 2ξ T (t)M 0 I −I ξ(t) + 2ξ T (t)L I −I 0 ξ(t) + 2ξ T (t)V I −I ξ(t) t t ξ T (t)NY1−1 N T ξ(t)ds + + t−h2 t−h(t) t−h1 t−h1 ξ + ξ T (t)SY2−1 S T ξ(t)ds T (t)MZ1−1 M T ξ(t)ds ξ T (t)LZ1−1 LT ξ(t)ds + t−h2 t−h1 t−h(t) ξ T (t)V Z2−1 V T ξ(t)ds + t−h2 t [ξ T (t)N + x˙ T (s)Y1 ]Y1−1 [N T ξ(t) + Y1 x(s)]ds ˙ − t−h2 t [ξ T (t)S + x˙ T (s)Y2 ]Y2−1 [S T ξ(t) + Y2 x(s)]ds ˙ − t−h1 t−h(t) [ξ T (t)M + x˙ T (s)Z1 ]Z1−1 [M T ξ(t) + Z1 x(s)]ds ˙ − t−h2 t−h1 [ξ T (t)L + x˙ T (s)Z1 ]Z1−1[LT ξ(t) + Z1 x(s)]ds ˙ − t−h(t) t−h1 [ξ T (t)V + x˙ T (s)Z2 ]Z2−1 [V T ξ(t) + Z2 x(s)]ds ˙ − t−h2 V˙ (t, xt) ξ T (t)[∆1 + ∆3 + ∆T3 ]ξ(t) + ξ T (t)[∆2 + ∆T2 ]ξ(t) + h2 ξ T (t)NY1−1 N T ξ(t) + h1 ξ T (t)SY2−1 S T ξ(t) + (h2 − h1 )ξ T (t)MZ1−1 M T ξ(t) + (h2 − h1 )ξ T (t)LZ1−1 LT ξ(t) t + (h2 − h1 )ξ T (t)V Z2−1 V T ξ(t) − [ξ T (t)N + x˙ T (s)Y1 ]Y1−1 [N T ξ(t) + Y1 x(s)]ds ˙ t−h2 t [ξ T (t)S + x˙ T (s)Y2 ]Y2−1 [S T ξ(t) + Y2 x(s)]ds ˙ − t−h1 t−h(t) [ξ T (t)M + x˙ T (s)Z1 ]Z1−1 [M T ξ(t) + Z1 x(s)]ds ˙ − t−h2 29  ρ1 A2P¯    ρB U         ρ2B2 U  ρ2 A2P¯          ρ3B2 U  ρ3 A2P¯          ¯ ρ4B2 U  ρ4 A2P  ,B ¯= , A¯ =      ρ5B2 U  ρ5 A2P¯          ¯ ρ6B2 U  ρ6 A2P          ρ7B2 U  ρ7 A2P¯      ¯ ρ8 A2P ρ8B2 U ¯ 22 = −diag h Y¯ h Y¯ Ω ¯1 Ω ¯2 Ω ¯ r1X ¯ r2 X ¯2 Ω ¯4 Ω ¯5 Ω ¯6 Ω ¯7 Ω ¯8 , Σ 1 2 ¯ = (h2 − h1 )Z¯1 , Ω ¯ = (h2 − h1 )Z¯1 , Ω ¯ = (h2 − h1 )Z¯2 , Ω ¯ = (r2 − r1)H ¯1, Ω ¯ = (r2 − r1 )H ¯ 1, Ω ¯ = (r2 − r1)H ¯2, Ω ¯ = e2αh2 Y¯3 , Ω ¯ = e2αr2 X ¯3 Ω Chøng minh Chúng ta đặt = P , = diag G ¯ G ¯ G ¯ G ¯ G ¯ G ¯ G ¯ G ¯ , G ¯Q ¯ i G, ¯R ¯ i G, ¯ Z¯i G, ¯H ¯ i G, ¯X ¯ j G, ¯ Ri = G ¯ Zi = G ¯ Hi = G ¯ (i = 1, 2), Xj = G ¯ Qi = G ¯ (j = 1, 2, 3), C = Λ ¯ C¯ G, ¯ D=Λ ¯D ¯ G, ¯ T =Λ ¯ T¯G, ¯ J=Λ ¯ J¯G, ¯ N =Λ ¯N ¯ G, ¯ ¯ Y¯j G, Yj = G ¯M ¯ G, ¯ L=Λ ¯L ¯ G, ¯ V =Λ ¯ V¯ G, ¯ W =Λ ¯W ¯ G, ¯ S=Λ ¯ S¯G, ¯ K = U G M = Theo công thức Newton-Leibniz,ta có phơng trình sau với ma trËn C, D, T, J, N, M, L, V, W, S, t 2η T (t)N x(t) − x(t − h1 ) − x(s)ds ˙ = 0, (3.47) x(s)ds ˙ = 0, (3.48) t−h1 t T 2η (t)M x(t) − x(t − h2 ) − t−h2 t−h(t) 2η T (t)C x(t − h(t)) − x(t − h2) − x(s)ds ˙ = 0, (3.49) x(s)ds ˙ = 0, (3.50) t−h2 t−h1 T 2η (t)D x(t − h1) − x(t − h(t)) − t−h(t) t−h1 2η T (t)L x(t − h1) − x(t − h2 ) − x(s)ds ˙ = 0, t−h2 (3.51) 30 t T 2η (t)V x(t) − x(t − r1 ) − x(s)ds ˙ = 0, (3.52) x(s)ds ˙ = 0, (3.53) t−r1 t 2η T (t)W x(t) − x(t − r2 ) − t−r2 t−r(t) 2η T (t)T x(t − r(t)) − x(t − r2 ) − x(s)ds ˙ = 0, (3.54) x(s)ds ˙ = 0, (3.55) t−r2 t−r1 2η T (t)J x(t − r1) − x(t − r(t)) − t−r(t) t−r1 2η T (t)S x(t − r1) − x(t − r2) − (3.56) x(s)ds ˙ = 0, t−r2 Víi η T (t) = xT (t) xT (t − h1 ) xT (t − h2 ) xT (t − h(t)) xT (t − r1) xT (t − r2) xT (t − r(t)) x˙ T (t) Ta còng cã phơng trình sau với G ρ3 G ¯ ρ4 G ¯ ρ5 G ¯ ρ6 G ¯ ρ7 G ¯ ρ8 G ¯ Ξ = ρ1 G T , ¯ ¯ + A1 x(t − h(t)) + B1 U Gx(t − r(t)) − x(t) ˙ 2η T (t)Ξ (A0 + B0 U G)x(t) t t ¯ x(s)ds + B2 U G + A2 t−h(t) x(s)ds = 0, t−r(t) Hay ¯ ¯ + A1 x(t − h(t)) + B1 U Gx(t − r(t)) − x(t) ˙ 2η T (t)Ξ (A0 + B0 U G)x(t) t (3.57) t ¯ x(s)ds + 2η (t)ΞB2U G T T + 2η (t)ΞA2 t−h(t) x(s)ds = t−r(t) ¸p dơng bỉ đề (.)(BĐT cauchy cho ma trận) bổ đề(.)(BĐT tích ph©n),chóng ta cã t 2η T (t)ΞA2 x(s)ds t−h(t) t t T η T (t)ΞA2e2αh2 Y3−1 AT2 ΞT η(t) + e−2αh2 x(s)ds t−h(t) t Y3 x(s)ds t−h(t) η T (t)ΞA2e2αh2 Y3−1 AT2 ΞT η(t) + e−2αh2 h(t) xT (s)Y3 x(s)ds t−h(t) t η T (t)ΞA2e2αh2 Y3−1 AT2 ΞT η(t) + e−2αh2 h2 xT (s)Y3 x(s)ds t−h2 (3.58) 31 T−¬ng tù th× t ¯ 2η T (t)ΞB2U G x(s)ds t−h(t) t t T ¯ T ΞT η(t) + e−2αr2 ¯ 2αr2 X −1 U T GB η T (t)ΞB2U Ge x(s)ds X3 t−r(t) t x(s)ds t−r(t) ¯ T ΞT η(t) + e−2αr2 r(t) ¯ 2αr2 X −1 U T GB η T (t)ΞB2U Ge x(sT X3 ds t−r(t) t ¯ 2αr2 X −1 U T GB ¯ T ΞT η(t) + e−2αr2 r2 η T (t)ΞB2U Ge xT (s)X3 x(s)ds t−r2 (3.59) KÕt hỵp tõ (3.57) ®Õn (3.59) chóng ta cã ¯ T ΞT η(t) ¯ 2αr2 X −1 U T GB η T (t)ΞA2e2αh2 Y3−1 AT2 ΞT η(t) + η T (t)ΞB2U Ge t t + e−2αh2 h2 xT (s)Y3 x(s)ds + e−2αr2 r2 t−h2 xT (s)X3 x(s)ds t−r2 ¯ ¯ + 2η (t)Ξ (A0 + B0 U G)x(t) + A1x(t − h(t)) + B1 U Gx(t − r(t)) − x(t) ˙ T (3.60) XÐt hµm Lyapunov-Krasovskii nh− sau V (t, xt) = V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8 + V9 + V10 + V11 + V12 + V13 , (3.61) ë ®©y , ¯ V1 = xT (t)Gx(t), t e2α(s−t)xT (s)Q1x(s)ds, V2 = t−h1 t e2α(s−t)xT (s)Q2x(s)ds, V3 = t−h2 t e2α(s−t)xT (s)R1x(s)ds, V4 = t−r1 t e2α(s−t)xT (s)R2x(s)ds, V5 = t−r2 t t e2α(u−t+h1 ) x˙ T (u)Y1 x(u)duds, ˙ V6 = t−h1 s 32 t t e2α(u−t+h2 ) x˙ T (u)Y2 x(u)duds, ˙ V7 = t−h2 t s t e2α(u−t+r1 ) x˙ T (u)X1 x(u)duds, ˙ V8 = t−r1 t s t e2α(u−t+r2 ) x˙ T (u)X2 x(u)duds, ˙ V9 = t−r2 s t−h1 t e2α(u−t+h2 ) x˙ T (u)[Z1 + Z2 ]x(u)duds, ˙ V10 = t−h2 t−r1 s t e2α(u−t+r2 ) x˙ T (u)[H1 + H2 ]x(u)duds, ˙ V11 = s t−r2 t t e2α(u−t) xT (u)Y3x(u)duds, V12 = h2 t−h2 t s t e2α(u−t)xT (u)X3 x(u)duds V13 = r2 tr2 s Đạo hàm hàm Vk , k = 1, , däc theo nghiƯm cđa (3.44) ®−ỵc cho bëi ¯ ¯ ¯ V˙1 = 2x˙ T (t)Gx(t) = 2x˙ T (t)Gx(t) + 2αxT (t)Gx(t) − 2αV1 , V˙2 = xT (t)Q1x(t) − e−2αh1 xT (t − h1 )Q1x(t − h1) − 2αV2 , V˙3 = xT (t)Q2x(t) − e−2αh2 xT (t − h2 )Q2x(t − h2) − 2αV3 , (3.62) V˙4 = xT (t)R1 x(t) − e−2αr1 xT (t − r1 )Q1x(t − r1) − 2αV4 , V˙5 = xT (t)R2 x(t) − e−2αr2 xT (t r2 )Q2x(t r2) 2V5 Bên cạnh ,lấy đạo hàm hàm Vj , j = 6, , 13 dọc theo nghiệm (3.44) ta đợc t V˙6 = h1e2αh1 x˙ T (t)Y1 x(t) ˙ − e2α(s−t+h1 ) x˙ T (s)Y1 x(s)ds ˙ − 2αV6 t−h1 t 2αh1 T h1 e (3.63) T x˙ (t)Y1 x(t) ˙ − x˙ (s)Y1 x(s)ds ˙ − 2αV6 t−h1 t ˙ − V˙7 = h2e2αh2 x˙ T (t)Y2 x(t) e2α(s−t+h2 ) x˙ T (s)Y2 x(s)ds ˙ − 2αV7 t−h2 t 2αh2 T h2 e (3.64) T x˙ (t)Y2 x(t) ˙ − x˙ (s)Y2 x(s)ds ˙ − 2αV7 t−h2 33 t V˙8 = r1 e2αr1 x˙ T (t)X1x(t) ˙ − e2α(s−t+r1 ) x˙ T (s)X1 x(s)ds ˙ − 2αV8 t−r1 t r1 e2αr1 x˙ T (t)X1 x(t) ˙ − x˙ T (s)X1 x(s)ds ˙ − 2αV8 t−r1 (3.65) t V˙9 = r2 e2αr2 x˙ T (t)X2x(t) ˙ − e2α(s−t+r2 ) x˙ T (s)X2 x(s)ds ˙ − 2αV9 t−r2 t r2 e2αr2 x˙ T (t)X2 x(t ˙ − x˙ T (s)X2 x(s)ds ˙ − 2αV9 t−r2 (3.66) V˙10 = (h2 − h1 )e2αh2 x˙ T (t)[Z1 + Z2 ]x(t) ˙ − 2αV10 t−h1 e2α(s−t+h2 ) x˙ T (s)[Z1 + Z2 ]x(s)ds ˙ − t−h2 t−h1 ˙ − (h2 − h1)e2αh2 x˙ T (t)[Z1 + Z2 ]x(t) x˙ T (s)[Z1 + Z2 ]x(s)ds ˙ − 2αV10 t−h2 t−h1 (h2 − h1)e2αh2 x˙ T (t)[Z1 + Z2 ]x(t) ˙ − x˙ T (s)Z2 x(s)ds ˙ t−h2 t−h(t) t−h1 x˙ T (s)Z1 x(s)ds ˙ − − x˙ T (s)Z1 x(s)ds ˙ − 2αV10 t−h(t) t−h2 (3.67) V˙11 = (r2 − r1)e2αr2 x˙ T (t)[H1 + H2 ]x(t) ˙ − 2αV11 t−r1 e2α(s−t+r2 )x˙ T (s)[H1 + H2 ]x(s)ds ˙ − t−r2 t−r1 (r2 − r1 )e2αr2 x˙ T (t)[H1 + H2 ]x(t) ˙ − x˙ T (s)[H1 + H2 ]x(s)ds ˙ − 2αV11 t−r2 t−r1 (r2 − r1 )e2αr2 x˙ T (t)[H1 + H2 ]x(t) ˙ − x˙ T (s)H2 x(s)ds ˙ t−r2 t−r(t) t−r1 x˙ T (s)H1 x(s)ds ˙ − − t−r2 x˙ T (s)H1 x(s)ds ˙ − 2αV11 t−r(t) (3.68) 34 t V˙12 = h22 xT (t)Y3 x(t) − h2 e2α(s−t)xT (s)Y3 x(s)ds − 2αV12 t−h2 (3.69) t h22 xT (t)Y3 x(t) −2αh2 T − h2 e x (s)Y3 x(s)ds − 2αV12 t−h2 t V˙13 = r22 xT (t)X3 x(t) − r2 e2α(s−t)xT (s)X3 x(s)ds − 2αV13 t−r2 (3.70) t r22 xT (t)X3 x(t) −2αr2 T − r2e x (s)X3 x(s)ds − 2αV13 t−r2 KÕt hợp từ (3.47) đến (3.56) từ (3.60) đến (3.70) ,chúng ta thu đợc kết sau V (t, xt) + 2αV (t, xt) ¯ ¯ 2x˙ T (t)Gx(t) + xT (t)[Q1 + Q2 + R1 + R2 + h22Y3 + r22 X3 + 2αG]x(t) − e−2αh1 xT (t − h1 )Q1x(t − h1 ) − e−2αh2 xT (t − h2)Q2 x(t − h2 ) − e−2αr1 xT (t − r1 )Q1 x(t − r1) − e−2αr2 xT (t − r2)Q2 x(t − r2) + x˙ T (t) h1e2αh1 Y1 + h2 e2αh2 Y2 + r1e2αr1 X1 + r2 e2αr2 X2 + (h2 − h1 )e2αh2 (Z1 + Z2 ) + (r2 − r1 )e2αr2 (H1 + H2 ) x(t) ˙ t t x˙ T (s)Y1 x(s)ds ˙ + 2η T (t)N x(t) − x(t − h1) − − t−h1 t x(s)ds ˙ t−h1 t x˙ T (s)Y2 x(s)ds ˙ + 2η T (t)M x(t) − x(t − h2 ) − − t−h2 t x(s)ds ˙ t−h2 t x˙ T (s)X1 x(s)ds ˙ + 2η T (t)V x(t) − x(t − r1 ) − − t−r1 t x(s)ds ˙ t−r1 t x˙ T (s)X2 x(s)ds ˙ + 2η T (t)W x(t) − x(t − r2 ) − − t−r2 t−h(t) x(s)ds ˙ t−r2 t−h(t) T T x˙ (s)Z1 x(s)ds ˙ + 2η (t)C x(t − h(t)) − x(t − h2 ) − − t−h2 t−h1 x(s)ds ˙ t−h2 t−h1 x˙ T (s)Z1 x(s)ds ˙ + 2η T (t)D x(t − h1 ) − x(t − h(t)) − − t−h(t) t−h1 x(s)ds ˙ t−h(t) t−h1 x˙ T (s)Z2 x(s)ds ˙ + 2η T (t)L x(t − h1) − x(t − h2 ) − − t−h2 t−r(t) x(s)ds ˙ t−h2 t−r(t) T T x˙ (s)H1 x(s)ds ˙ + 2η (t)T x(t − r(t)) − x(t − r2 ) − − t−r2 x(s)ds ˙ t−r2 35 t−r1 t−r1 T T x˙ (s)H1 x(s)ds ˙ + 2η (t)J x(t − r1 ) − x(t − r(t)) − − t−r(t) t−r1 x(s)ds ˙ t−r(t) t−r1 x˙ T (s)H2 x(s)ds ˙ + 2η T (t)S x(t − r1) − x(t − r2 ) − − t−r2 x(s)ds ˙ t−r2 ¯ 2αr2 X −1 U T GB ¯ T ΞT η(t) + η T (t)ΞA2e2αh2 Y3−1 AT2 ΞT η(t) + η T (t)ΞB2U Ge ¯ ¯ + 2η T (t)Ξ (A0 + B0 U G)x(t) + A1 x(t − h(t)) + B1 U Gx(t − r(t)) − x(t) ˙ ¯ + xT (t)∆x(t) + x˙ T (t)Ωx(t) 2x˙ T (t)Gx(t) − e−2αh1 xT (t − h1 )Q1 x(t − h1 ) − e−2αh2 xT (t − h2 )Q2x(t − h2 ) − e−2αr1 xT (t − r1 )Q1x(t − r1 ) − e−2αr2 xT (t − r2 )Q2x(t − r2 ) ¯ 2αr2 X −1 U T GB ¯ T ΞT η(t) + η T (t)ΞA2e2αh2 Y3−1 AT2 ΞT η(t) + η T (t)ΞB2U Ge ¯ 0 A1 0 B1U G ¯ −I η(t) + 2η T (t)Ξ (A0 + B0 U G) t ˙ + 2η T (t)N x(s) ˙ ds + 2η T (t)N x(t) − x(t − h1 ) x˙ T (s)Y1 x(s) − t−h1 t ˙ + 2η T (t)M x(s) ˙ ds + 2η T (t)M x(t) − x(t − h2 ) x˙ T (s)Y2 x(s) − t−h2 t x˙ T (s)X1 x(s) ˙ + 2η T (t)V x(s) ˙ ds + 2η T (t)V x(t) − x(t − r1 ) − t−r1 t x˙ T (s)X2 x(s)ds ˙ + 2η T (t)W x(s) ˙ ds + 2η T (t)W x(t) − x(t − r2 ) − t−r2 t−h(t) x˙ T (s)Z1 x(s)ds ˙ + 2η T (t)C x(s) ˙ ds + 2η T (t)C x(t − h(t)) − x(t − h2) − t−h2 t−h1 x˙ T (s)Z1 x(s)ds ˙ + 2η T (t)Dx(s) ˙ ds + 2η T (t)D x(t − h1) − x(t − h(t)) − t−h(t) t−h1 ˙ + 2η T (t)Lx(s) ˙ ds + 2η T (t)L x(t − h1 ) − x(t − h2 ) x˙ T (s)Z2 x(s)ds − t−h2 t−r(t) x˙ T (s)H1 x(s)ds ˙ + 2η T (t)T x(s) ˙ ds + 2η T (t)T x(t − r(t)) − x(t − r2 ) − t−r2 t−r1 ˙ + 2η T (t)J x(s) ˙ ds + 2η T (t)J x(t − r1) − x(t − r(t)) x˙ T (s)H1 x(s)ds − t−r(t) t−r1 ˙ + 2η T (t)S x(s) ˙ ds + 2η T (t)S x(t − r1) − x(t − r2 ) , x˙ T (s)H2 x(s)ds − t−r2 36 Trong ®ã, ¯ ∆ =Q1 + Q2 + R1 + R2 + h22 Y3 + r22 X3 + 2αG, Ω =h1 e2αh1 Y1 + h2 e2αh2 Y2 + r1 e2αr1 X1 + r2 e2αr2 X2 + (h2 − h1 )e2αh2 (Z1 + Z2 ) + (r2 − r1 )e2αr2 (H1 + H2 ) Suy V˙ (t, xt) + 2αV (t, xt) η T (t)Σ1 η(t) + η T (t)[Σ2 + ΣT2 ]η(t) + η T (t)[Σ3 + ΣT3 ]η(t) ¯ 2αr2 X −1 U T GB ¯ T ΞT η(t) + η T (t)ΞA2e2αh2 Y3−1 AT2 ΞT η(t) + η T (t)ΞB2U Ge t t η T (t)NY1−1 N T η(t)ds − + [η T (t)N + x˙ T (s)Y1 ]Y1−1 [N T η(t) + Y1 x(s)]ds ˙ t−h1 t t−h1 t η T (t)MY2−1 M T η(t)ds − + [η T (t)M + x˙ T (s)Y2 ]Y2−1 [M T η(t) + Y2 x(s)]ds ˙ t−h2 t t−h2 t η T (t)V X1−1 V T η(t)ds − + [η T (t)V + x˙ T (s)X1 ]X1−1 [V T η(t) + X1 x(s)]ds ˙ t−r1 t t−r1 t η T (t)W X2−1 W T η(t)ds − + t−r2 t−h(t) [η T (t)W + x˙ T (s)X2 ]X2−1 [W T η(t) + X2 x(s)]ds ˙ t−r2 t−h(t) η T (t)CZ1−1C T η(t)ds − + t−h2 t−h1 [η T (t)C + x˙ T (s)Z1 ]Z1−1[C T η(t) + Z1 x(s)]ds ˙ t−h2 t−h1 η T (t)DZ1−1 DT η(t)ds − + t−h(t) t−h1 [η T (t)D + x˙ T (s)Z1 ]Z1−1 [DT η(t) + Z1 x(s)]ds ˙ t−h(t) t−h1 η T (t)LZ2−1 LT η(t)ds − + [η T (t)L + x˙ T (s)Z2 ]Z2−1 [LT η(t) + Z2 x(s)]ds ˙ t−h2 t−r(t) t−h2 t−r(t) η T (t)T H1−1T T η(t)ds − + t−r2 t−r1 [η T (t)T + x˙ T (s)H1 ]H1−1 [T T η(t) + H1 x(s)]ds ˙ t−r2 t−r1 η T (t)JH1−1 J T η(t)ds − + t−r(t) t−r1 [η T (t)J + x˙ T (s)H1 ]H1−1 [J T η(t) + H1 x(s)]ds ˙ t−r(t) t−r1 η T (t)SH2−1 S T η(t)ds − + t−r2 [η T (t)S + x˙ T (s)H2 ]H2−1 [S T η(t) + H2 x(s)]ds ˙ t−r2 ¯ 2αr2 X −1 U T GB ¯ T ΞT ]η(t) η T (t)Σ11 η(t) + η T (t)[ΞA2e2αh2 Y3−1 AT2 ΞT + ΞB2 U Ge t + η T (t)h1NY1−1 N T η(t) − [η T (t)N + x˙ T (s)Y1 ]Y1−1 [N T η(t) + Y1 x(s)]ds ˙ t−h1 37 t +η T (t)h2MY2−1 M T η(t) [η T (t)M + x˙ T (s)Y2 ]Y2−1 [M T η(t) + Y2 x(s)]ds ˙ − t−h2 t + η T (t)r1V X1−1 V T η(t) − [η T (t)V + x˙ T (s)X1 ]X1−1 [V T η(t) + X1 x(s)]ds ˙ t−r1 t + η T (t)r2W X2−1 W T η(t) − [η T (t)W + x˙ T (s)X2 ]X2−1 [W T η(t) + X2 x(s)]ds ˙ t−r2 t−h(t) + η T (t)(h2 − h1 )CZ1−1 C T η(t) − [η T (t)C + x˙ T (s)Z1 ]Z1−1[C T η(t) + Z1 x(s)]ds ˙ t−h2 t−h1 + η T (t)(h2 − h1 )DZ1−1 DT η(t) − [η T (t)D + x˙ T (s)Z1 ]Z1−1[DT η(t) + Z1 x(s)]ds ˙ t−h(t) t−h1 + η T (t)(h2 − h1 )LZ2−1 LT η(t) − [η T (t)L + x˙ T (s)Z2 ]Z2−1 [LT η(t) + Z2 x(s)]ds ˙ t−h2 t−r(t) + η T (t)(r2 − r1 )T H1−1 T T η(t) − [η T (t)T + x˙ T (s)H1 ]H1−1 [T T η(t) + H1 x(s)]ds ˙ t−r2 t−r1 + η T (t)(r2 − r1 )JH1−1 J T η(t) − [η T (t)J + x˙ T (s)H1 ]H1−1 [J T η(t) + H1 x(s)]ds ˙ t−r(t) t−r1 + η T (t)(r2 − r1 )SH2−1 S T η(t) − [η T (t)S + x˙ T (s)H2 ]H2−1 [S T η(t) + H2 x(s)]ds, ˙ t−r2 (3.71) Trong ®ã, Σ11 = Σ1 + Σ2 + ΣT2 + Σ3 + ΣT3 ,  ∆ 0 0    −e−2αh1 Q1 0 0    −e−2αh2 Q2 0 0   0 0 0 Σ1 =   0 0 0 −e−2αr1 R1   0 0 0 −e−2αr2 R2   0 0 0  ¯ G 0 0 ¯ G 0 0 0    0   0   0 ,  0   0   0  Ω Σ2 = Π Φ −M − C − L C − D −(V + J + C) − W − T − S T − J , Π = M + N + V + W, Φ = −N + D + L, 38  ¯ + B0 U G) ¯ 0 ρ1 GB ¯ 1U G ¯ ¯ 0 ρ1 GA ¯ −ρ1G ρ1 G(A   ¯ ¯ ρ2 G(A0 + B0 U G)   ¯ ¯ ρ3 G(A0 + B0 U G)   ¯ ¯ ρ4 G(A0 + B0 U G) Σ3 =   ¯ ¯ ρ5 G(A0 + B0 U G)   ¯ ¯ ρ6 G(A0 + B0 U G)   ¯ ¯ ρ7 G(A0 + B0 U G)  ¯ + B0 U G) ¯ ρ8 G(A   ¯ −ρ2G ¯ 0 ρ2 GB ¯ 1U G ¯  0 ρ2 GA   ¯ ¯ ¯ ¯ 0 ρ3 GA1 0 ρ3 GB1U G −ρ3G   ¯ ¯ ¯ ¯ 0 ρ4 GA1 0 ρ4 GB1U G −ρ4G  ¯ −ρ5G ¯ 0 ρ5 GB ¯ 1U G ¯  0 ρ5 GA   ¯ ¯ ¯ ¯ 0 ρ6 GA1 0 ρ6 GB1U G −ρ6G  ¯ 0 ρ7 GB ¯ 1U G ¯ ¯ −ρ7G  0 ρ7 GA  ¯ ¯ ¯ ¯ 0 ρ8 GA1 0 GB1U G 8G Từ (3.71) ta có đợc kÕt qu¶ V˙ (t, xt) + 2αV (t, xt) T η T (t)[Σ11 − Σ12Σ−1 22 Σ12 ]η(t) t [η T (t)N + x˙ T (s)Y1 ]Y1−1 [N T η(t) + Y1 x(s)]ds ˙ − t−h1 t [η T (t)M + x˙ T (s)Y2 ]Y2−1 [M T η(t) + Y2 x(s)]ds ˙ − t−h2 t [η T (t)V + x˙ T (s)X1 ]X1−1 [V T η(t) + X1 x(s)]ds ˙ − t−r1 t [η T (t)W + x˙ T (s)X2 ]X2−1 [W T η(t) + X2 x(s)]ds ˙ − t−r2 t−h(t) [η T (t)C + x˙ T (s)Z1 ]Z1−1 [C T η(t) + Z1 x(s)]ds ˙ − t−h2 t−h1 [η T (t)D + x˙ T (s)Z1 ]Z1−1 [DT η(t) + Z1 x(s)]ds ˙ − t−h(t) t−h1 [η T (t)L + x˙ T (s)Z2 ]Z2−1 [LT η(t) + Z2 x(s)]ds ˙ − t−h2 t−r(t) [η T (t)T + x˙ T (s)H1 ]H1−1 [T T η(t) + H1 x(s)]ds ˙ − t−r2 t−r1 [η T (t)J + x˙ T (s)H1 ]H1−1 [J T η(t) + H1 x(s)]ds ˙ − t−r(t) t−r1 [η T (t)S + x˙ T (s)H2 ]H2−1 [S T η(t) + H2 x(s)]ds, ˙ − t−r2 (3.72) 39 Víi, r2 W ∆4 ∆5 ∆6 ∆7 ∆8 , Σ12 = h1 N h2 M ∆1 ∆2 ∆3 r1V ∆1 = (h2 − h1)C, ∆2 = (h2 − h1)D, ∆3 = (h2 − h1)L, ∆4 = (r2 − r1 )T, ¯ ∆5 = (r2 − r1 )J, ∆6 = (r2 − r1)S, ∆7 = e2αh2 ΞA2, ∆8 = e2αr2 ΞBU G, ¯ 22 = −diag h Y h Y Ω Ω Ω r X r X Ω Ω Ω Ω Ω , Σ 1 2 1 2 Ω1 = (h2 − h1 )Z1 , Ω2 = (h2 − h1 )Z1 , Ω3 = (h2 − h1 )Z2 , Ω4 = (r2 − r1)H1 , Ω5 = (r2 − r1 )H1 , Ω6 = (r2 − r1)H2 , Ω7 = e2αh2 Y3 , = e2r2 X3 Bởi ,các ma trËn Yi , Xi , Zi , Hi , i = 1, đối xứng xác định dơng nên tích phân (3.72) bé không.Do từ (3.72) ta suy V (t, xt) + 2αV (t, xt) T η T (t)[Σ11 − Σ12Σ−1 22 12 ](t) (3.73) Mặt khác,ta thấy −1 ¯ T ¯ 11 − Σ ¯ −1 Σ ¯T ¯ 12 Σ ¯ ¯ Γ[Σ 22 12 ]Γ = ΓΣ11 Γ − ΓΣ12 ΓΓ Σ22 Γ ΓΣ12 ]Γ = Σ11 − (3.74) T Σ12Σ−1 22 Σ12 , ë ®©y ¯ G ¯ G ¯ G ¯ G ¯ G ¯ G ¯ G ¯ Γ = diag G Kết hợp (3.73) (3.74) ,chúng ta có kết qu¶ V˙ (t, xt) + 2αV (t, xt) ¯ −1 Σ ¯T ¯ 11 − Σ ¯ 12 Σ η T (t)Γ[Σ 22 12 ]Γη(t) (3.75) ¯ 11 − Σ ¯ 12Σ ¯ −1 Σ ¯T ξ (t)[Σ 22 12 ]ξ(t), T ¯ 11 − víi ξ(t) = Γη(t).Ngoµi ra, từ (3.46) áp dụng bổ đề phần bù Schur ,ta ®−ỵc Σ ¯ −1 Σ ¯T ¯ 12 Σ Σ 22 12 < Bëi thÕ ,tõ (3.75) suy r»ng V˙ (t, xt) + 2αV (t, xt) ®ã V (t, xt) V (0, x0 )e−2αt, t λ1 x(t) 0, t 0, 0.V× r»ng V (t, xt) λ2 xt 2s , t ∈ R+ 40 nªn chóng ta thu đợc t e s, t x(t, ) ,1 , nh định lý.Khi ,điều khiển ngợc cho u(t) = U P x(t) Định lý đà đợc chứng minh Bây giờ,ta xét hệ phơng trình điều khiển không chắn hỗn hợp có trễ biến thiên trạng thái điều khiển sau x(t) = (A0 + ∆A)(t))x(t) + (A1 + ∆A1(t))x(t − h(t)) + (B0 + ∆B0(t))u(t)      t   +(B1 + ∆B1 (t))u(t − r(t)) + ∆A2(t)) t−h(t) x(s)ds  t   +(B2 + ∆B2 (t)) t−r(t) u(s)ds, t 0,      x(t) = φ(t), t ∈ [−h, 0], h = max{h , r } 2 (3.76) Trong dã, x(t) ∈ Rn lµ véctơ trạng thái hệ,u(t) Rm véctơ điều khiĨn cđa hƯ A0, A1, A2 , B0, B1 , B2 ma trận có số chiều thích hợp (t) hàm điều kiện ban đầu thỏa mÃn (t) C([h, 0], Rn ).Hàm h(t) trễ biến thiên trạng thái thỏa mÃn điều kiƯn h1 r1 r(t) h(t) h2 ,vµ r(t) lµ trƠ biến thiên điều khiển r2 Tính không chắn chắn hệ đợc cho A0(t) A1(t) A2(t) B0(t) B1(t) ∆B2 (t) = HF (t) E1 E2 E3 E4 E5 E6 Ta xét với điều khiển ngợc u(t) = Kx(t), t R+ Khi ,ta có định lý sau Định lý 3.9 Cho số > số không âm i , i = 1, , 8.Hệ (3.76) -ổn định hóa đợc dạng mũ bền vững tồn ma trận đối xứng xác định dơng i, R i , Zi , H ¯ i , (i = 1, 2), X ¯ j , Y¯j , (j = 1, 2, 3) ,vµ ma trận U, C, D, T¯ , J, ¯N ¯, P¯ , Q ¯ , L, ¯ V¯ , W ¯ , S¯ tháa m·n LMI sau M = + H H T Σ ET E −λI   < (3.77) 41 cho nh định lý (3.8) Trong ®ã Σ H = ρ1 H T ρ2 H T ρ3 H T ρ4 H T ρ5H T ρ6 H T ρ7 H T ρ8 H T 0 T 0 0 0 0 0 E = E1 P¯ T + E4 U 0 E2 P¯ T 0 E5U 0 0 0 0 0 e2αh2 E3 P T e2r2 E6U T Hơn nữa,với nghiệm x(t, φ) cđa hƯ ®ãng tháa m·n x(t, φ) λ2 −αt e φ s, t λ1 Trong ®ã ¯ λ1 = λ−1 max (P ) ¯ ¯ ¯ ¯ ¯ λ2 = λ−1 (P ) + h1 λmax (Q1 ) + h2 λmax (Q2 ) + r1 λmax (R1 ) + r2 λmax (R2 ) 1 + h21 e2αh1 λmax (Y¯1 ) + h22 e2αh2 λmax (Y¯2 ) + h32 λmax (Y¯3 ) 2 1 2αr1 ¯ ) + h2 e2αr2 λmax (X ¯ ) + r3 λmax (X ¯3 ) λmax (X + r1 e 2 2 1 ¯ ) + (H ¯ )] λ−2 (P¯ ) + (h2 − h1)2 e2αh2 λmax [(Z¯1 ) + (Z¯2 )] + (r2 − r1 )2 e2r2 max [(H 2 Và chúng có hàm điều khiển ngợc u(t) = U P x(t) Chứng minh Lặp lại chứng minh nh định lý (3.8) việc thay A0, A1, A2, B0 , B1 , B2, t−¬ng øng bëi A0 + HF (t)E1, A1 + HF (t)E2, A2 + HF (t)E3, B0 + HF (t)E4, B1 + HF (t)E5, B2 + HF (t)E6, ,ta thu đợc V (t, xt) + 2V (t, xt) T (t)(t), T T ¯ ¯ ¯ Σ1 + Σ2 + Σ2 + Σ3 + Σ3 Σ12 , Π= ¯ 22 ΣT12 Σ (3.78) (3.79) 42 ¯ 1, Σ ¯ + Σ3 , Σ12 = Σ ¯ 12 + Σ12 víi Σ ¯ 2, Σ ¯ 3, Σ ¯ 12, Σ ¯ 22 cho nh định lý Trong = Σ (3.8) vµ Σ3 =   ρ1 (HF (t)E1P¯ + HF (t)E4U ) 0 ρ1 HF (t)E2P¯ 0 ρ1HF (t)E5U     ρ2 (HF (t)E1P¯ + HF (t)E4U ) 0 ρ2 HF (t)E2P¯ 0 ρ2HF (t)E5U 0     ¯ ¯ ρ3 (HF (t)E1P + HF (t)E4U ) 0 ρ3 HF (t)E2P 0 ρ3HF (t)E5U 0     ρ4 (HF (t)E1P¯ + HF (t)E4U ) 0 ρ4 HF (t)E2P¯ 0 ρ4HF (t)E5U 0  ,   ρ5 (HF (t)E1P¯ + HF (t)E4U ) 0 ρ5 HF (t)E2P¯ 0 ρ5HF (t)E5U 0     ¯ ¯ ρ6 (HF (t)E1P + HF (t)E4U ) 0 ρ6 HF (t)E2P 0 ρ6HF (t)E5U 0     ρ7 (HF (t)E1P¯ + HF (t)E4U ) 0 ρ7 HF (t)E2P¯ 0 ρ7HF (t)E5U 0   ¯ ¯ ρ8 (HF (t)E1P + HF (t)E4U ) 0 ρ8 HF (t)E2P 0 ρ8HF (t)E5U   2αh2 2αr2 0 0 0 0 0 ρ1 e HF (t)E3 ρ1 e HF (t)E6     2αh 2αr 2 0 0 0 0 0 ρ1 e HF (t)E3 ρ1 e HF (t)E6     0 0 0 0 0 ρ1 e2αh2 HF (t)E3 ρ1 e2αr2 HF (t)E6     2αh 2αr 2 0 0 0 0 0 ρ1 e HF (t)E3 ρ1 e HF (t)E6  Σ3 =    0 0 0 0 0 ρ1 e2αh2 HF (t)E3 ρ1 e2αr2 HF (t)E6     2αh 2αr 2 0 0 0 0 0 ρ1 e HF (t)E3 ρ1 e HF (t)E6     2αh 2αr 2 0 0 0 0 0 ρ1 e HF (t)E3 ρ1 e HF (t)E6   2αh2 2αr2 0 0 0 0 0 ρ1 e HF (t)E3 ρ1 e HF (t)E6 Do ®ã,tõ (3.79) ta suy       T T Σ Σ12 Σ Σ + Σ3 Σ12 ¯ + ¯ + =Σ +  Π=Σ T T ¯ Σ12 Σ22 Σ12 0 ¯ + HF (t)E + (HF (t)E)T =Σ ¸p dơng bỉ ®Ị ( ) ta suy Π ¯ + λH H T + E T E (3.80) 43 Mặt khác ,áp dụng bổ đề phần bù Schur cho (3.77) ta đợc + H H T + E T E < Σ Kết hợp với (3.80) ta có < 0.Vì vËy tõ (3.78) suy V˙ (t, xt) + 2αV (t, xt) 0, t λ2 −αt e φ s, t Do đó,theo chứng minh định lý (3.9) x(t, ) Định lý đợc chứng minh