29 4 2012 4 1000 8152(2012)04 0409 06 Control Theory & Applications Vol 29 No 4 Apr 2012 12 1 (1 250061; 2 250353) ; ; ; TP273 A Estimation of stability region for a class of switched linear systems with multiple equilibrium points GUO Rong-wei 12 WANG Yu-zhen 1 (1 School of Control Science and Engineering Shandong University Jinan Shandong 250061 China; 2 School of Science Shandong Polytechnic University Jinan Shandong 250353 China) Abstract: We investigate the estimation of the stability region of a class of switched linear systems with multiple equilibrium points and put forward a number of new results in this research Using the component-wise ultimate-bound method we estimate the stability region for a switched linear system with multi-equilibrium points when the system is switching around any switching path or around switching paths with dwell time not less than the minimum dwell time In comparison with the existing results the obtained estimate of the stability region is more precise and less conservative in the estimation property Numerical simulation of an illustrative example shows the correctness and effectiveness of the proposed method Key words: multi-equilibrium switched system; regional stability analysis; stability region; component-wise ultimatebound method 1 (Introduction) 20 [1 7] Lyapunov(common Lyapunov function CLF) Lyapunov (multiple Lyapunov function MLF) Lyapunov Lyapunov [8] [9] LyapunovLyapunov 2011 03 23; 2011 06 22 (G61074068 G61034007 G60774009); (G20040422059); (ZR2010FM013)
410 29 [10] Lyapunov [11 12] (practical stability) [13] () [11] ẋ = A σ(t) x + E σ(t) w(t) (1) w(t) R p w(t) w(t) W W Lyapunov 2 (Preliminaries) ẋ = A σ(t) (x x e σ(t)) (2) x R n σ(t): [t 0 ) I = {1 2 3 N} σ(t) = i(i = 1 2 3 N) i A i i Hurwitz x e i = (x e 1i x e 2i x e ni) T i i I x e 1 xe N : x e 1i x e ji j {2 3 N} i = 1 2 n x e Ni x e ji j {1 2 N 1} i=1 2 n σ(t) = i m I (t [t m t m+1 ) m = 0 1 2 ) {t m } + m=0 t 0 < t 1 < t 2 < < t m < < + t m+1 = (2)i m [t m + ) (2)[t m + ) t m + m + t t 0 (2) 1 x e i 0 i I (2) ẋ = A σ(t) x (3) (2)(3) 1 () [10] x 0 R n σ(t) x σ(t) (t; t 0 x 0 ) (2)x 0 Ω k R n k N lim dis(x σ(t) (t; t 0 x 0 ) Ω k ) = 0 (2)Ω k Ω k (SRR) dis(x σ(t) (t; t 0 x 0 ) Ω k ) := inf{ x y : x = x σ(t) (t; t 0 x 0 ) y Ω k } Ω k k (2) 2 (SRR)Ω k Ω k x e i 0 i I (2) T (3) Ω k = {0} Ω k = {0} (2)(3) (2) (3) Ω k [10] 1 (2) A i (i I)LyapunovV 0 (x) (2) (SRR) Ω = {x R n : V 0 (x x) C} (4) k
4 411 x = 1 N x e i C = λ 1λ 2 4 N i=1 λ 2 2 (5) λ 4 = λ 3 max i( x x e i ) i I (6) λ i > 0 i = 1 2 3 (3) V 0 (x) λ 1 x 2 (7) V 0 (x) (i) = T V 0 (x) A i (x) λ 2 x 2 i I (8) V 0(x) λ 3 x (9) V 0 (x) = x T P x P (SRR) Ω = { x R n : (x x) T P (x x) C p } (10) C p = 4λ 1 µ 2 λ 1 = λ max (P ) (11) µ = max i I A i( x x e i ) (12) λ max ( ) RC; x(t) lim x(t); M M Re M ; nxy X Y X Y X Y X ij Y ij i j = 1 2 3 n (13) n X M(X) { Re X ij i = j M(X) = (14) X ij i j M R n n M ik i k M ik 0 MMetzler MMetzlere Mt 0 t 0 2 [12] ẋ = A(t)x(t) + v(t) (15) A(t) R n n v(t) R n x(0) 0 x(t) 0 t 0 B R n n Metzler B A(t) 0 t 0 ẏ = By(t) + v(t) (16) y(0) x(0) 0 y(t) x(t) t 0 3 [12] ẋ = Ax(t) + v(t) (17) A R n n Metzler v(t) R n w t 0 w ẏ = Ay(t) + w (18) y(0) x(0) y(t) x(t) t 0 3 (Main results) (2) 1 (2) Q C n n Λ i = Q 1 A i Q Λ = max i) i I (19) Z = max i I Q 1 A i ( x x e i ) (20) M( )(14) x(5) Λ i A i i I Λ Hurwitz (2) (SRR) Ω = {x R n : x x Q Λ 1 Z} (21) (2) ẋ = A σ(t) (x x) + E σ(t) w(t) (22) E σ(t) = A σ(t) w(t) = ( x x e σ(t) ) w(t) w(t) W W (22)(1) (2) x (2) (22) (2) (22) [11] (22) Ω = {x R n : x x Q Λ 1 Z 0 } Z 0 = max [ max Q 1 E i w ] i I w: w W E i = A i w i = ( x x e i ) i I Z 0 = Z (22) Ω = {x R n : x x Q Λ 1 Z} (2)Ω 3 (2) x (2)1
412 29 A i (i Λ) Lyapunov 1V 0 (x) Lyapunov(2) 2 (2) A i (i I)LyapunovV (i) 0 (x)(i I) (2)τ τ 0 Ω = N Ω i (23) j i=1 Ω i = {x R n : V (i) 0 (x x) C i } (24) x = 1 N x e i C i = λ(i) 1 ( 4 ) 2 N i=1 ( 2 ) 2 (25) 4 = 3 A i ( x x e i ) i I (26) > 0 j = 1 2 3 i I (3) V (i) 0 (x) 1 x 2 (27) V (i) 0 (x) (i) = T V (i) 0 (x) A i x 2 x 2 (28) T V (i) 0 (x) 3 x i I (29) τ 0 ẋ = A σ(t) x V i (x) = V (i) 0 (x x) C i V i (x) 0 x Ω i V i (x) > 0 x R n \Ω i V i (x) (i) = T V (i) 0 A i(x x e i ) = T V (i) 0 A i(x x) + T V (i) 0 A i( x x e i ) 2 x x 2 + 4 x x = g i (x x) g i (x x) = 2 x x 2 4 x x S i = {x R n : g i (x x) 0} g i (x x) 0 2 x x 2 4 x x 0 x x λ(i) 4 2 (27) V (i) 0 (x x) 1 ( 2 ) 2 1 x x 2 ( 2 ) 2 = C i x S i i I x Ω N S i Ω i=1 i I [8] τ τ 0 ẋ = A σ(t) x τ τ 0 V (i) 0 (x k+1 ) V (i) 0 (x k ) i I (31) V i (x k+1 ) V i (x k ) i I (32) R n \Ωτ τ 0 (30) τ τ 0 (2) Ω 21 3 (2) Q i C n n (i I) Λ i = Q 1 i A i Q i Λi = M(Λ i ) (33) Z = max i I Q 1 i A i ( x x e i ) (34) M( )(14) x(5) Λ i A i Λ i Hurwitz i I (2)τ τ 0 (SRR) Ω = N Ω i (35) i=1 Ω i = {x R n : x x Q i Λ 1 i Z} (36) τ 0 ẋ = A σ(t) x 2 τ τ 0 (2) (2) (2) (22) x = Q i z (22)i ż = Λ i (z z) + Q 1 i E i w(t) i I (37) z = Q 1 i x z(t) z(t) z = e jθ(t) ρ(t) ρ(t)z(t) z j j 2 = 1 θ(t) V i (x) (i) = g i (x x) < 0 x R n \Ω (30) θ(t) = diag{θ 1 (t) θ 2 (t) θ n (t)} (38)
4 413 θ k (t) = arg{z k (t)} k = 1 2 n (39) z(t) z = ρ(t) (37) ż = je jθ(t) θ(t)ρ(t) + e jθ(t) ρ(t) = Λ i e jθ(t) θ(t) + Q 1 E i w(t) (40) (40)e jθ(t) j θ(t)ρ(t) + ρ(t) = e jθ(t) Λ i e jθ(t) ρ(t)+e jθ(t) Q 1 i E i w(t) (41) (41) ρ(t) = N i (t)ρ(t) + γ i (t) (42) N i (t) = Re{e jθ(t) Λ i e jθ(t) } (43) γ i (t) = Re{e jθ(t) Q 1 E i w(t)} i I (44) (38) (39) { Re{[Λi ] lk } l = k [N i (t)] lk = Re{e j(θi(t) θ k(t)) [Λ i ] lk } l k N i (t) M(Λ i ) M(Λ i ) Λ i N i (t) Λ i γ i (t) Z ẏ(t) = Λ i y(t) + γ i (t) (45) Ẏ (t) = Λ i Y (t) + Z (46) (42)(45) 2 ρ(t) y(t) (45)(46) 3 y(0) ρ(0) Y (t) ρ(t) lim sup Q 1 i (x(t) x) = lim lim ρ(t) lim sup y(t) Y (t) = Λ 1 i Z (47) (22)i Ω i = {x R n : x x Q i 1 Λ i Z} τ τ 0 (22) Ω i i I τ τ 0 (2) Ω 4 V i V i I (2) Ω V i (i I) 3 1 3 τ 0 e Ait L i e αit L i > 0 α i < 0 i I [8] 4 (Example) 1 ẋ = A i (x x e i ) (48) i I = {1 2} x e 1 = (0 0) x e 2 = (1 2) ( ) ( ) 22 03 14 09 A 1 = A 2 = 08 08 24 56 P ( ) A 1 A 2 1 3 P = 4 2 ( ) P 1 1 0 A 1 P = P 1 A 2 P = ) ( 5 0 A 1 A 2 Lyapunov 1 V (x) = x T x λ 1 = 1 µ = 39962 C = 638785 Ω 1 = {x R 2 : x x 79924} A 1 A 2 Lyapunov V (x) = x T x Q = P Q1 1 ( ) ( ) 1 0 5 0 Λ 1 = Λ 2 = ) Λ = max i I M(Λ i) = ( 1 0 Z = max i I Q 1 A i ( x x e i ) = 1 ( ) 06 05 Ω 2 ={x R 2 : x 1 x 1 135 x 2 x 2 29} Ω 2 Ω 1 (2) 1 [x 1 (0) x 2 (0)] = [51 10]
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