Estimation of stability region for a class of switched linear systems with multiple equilibrium points

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1 (2012) Control Theory & Applications Vol 29 No 4 Apr ( ; ) ; ; ; 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 China; 2 School of Science Shandong Polytechnic University Jinan Shandong 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 ; (G G G ); (G ); (ZR2010FM013)

2 [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 = 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 = ) {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

3 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 = (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 = 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

4 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 = 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 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)

5 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) 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) ( ) ( ) A 1 = A 2 = P ( ) A 1 A P = 4 2 ( ) P 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 = C = Ω 1 = {x R 2 : x x 79924} A 1 A 2 Lyapunov V (x) = x T x Q = P Q1 1 ( ) ( ) 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 ( ) Ω 2 ={x R 2 : x 1 x x 2 x 2 29} Ω 2 Ω 1 (2) 1 [x 1 (0) x 2 (0)] = [51 10]

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