33 10 2016 10 DOI: 10.7641/CTA.2016.50818 Control Theory & Applications Vol. 33 No. 10 Oct. 2016,, (, 273165) :. Lyapunov, Lyapunov.,,. Lyapunov,, Wirtinger,.. : ; ; Lyapunov; : TP273 : A New conditions for exponential stability of sampled-data systems under aperiodic sampling ZHANG Dan, SHAO Han-yong, ZHAO Jian-rong (Institute of Automation, Qufu Normal University, Qufu Shandong 273165, China) Abstract: This article is concerned with the exponential stability problem of linear sampled-data systems under aperiodic sampling. Based on discrete-time Lyapunov theorem, a new Lyaponov-like functional is constructed with the following features. It explicitly depends on time t, contains the integral quadratic term of the states, and is not required to be positive definite between sampling instants. Based on this new Lyapunov-like functional, a new theorem is proposed in this paper to study the exponential stability of a class of nonlinear sampled-data systems firstly. And then new conditions for the linear sampled-data systems under aperiodic sampling to be exponentially and asymptotically stable are given respectively in terms of linear matrix inequalities by taking advantages of the new theorem and the improved Wirtinger integral inequality. Examples are provided to illustrate that the stability conditions have less conservatism compared with some existing ones. Key words: sampled-data systems; exponential stability; Lyapunov-like functional; linear matrix inequalities 1 (Introduction) [1 2],. [1].,,., : [3 5],,.,,., [5], Finsler (matrix-dilatian),. 2 [6],, [7]. 3 [8 9], Lyapunov. Lyapunov [10 12]. Lyapunov Lyapunov, Lyapunov. [12] : 2015 10 17; : 2016 07 06.. E-mail: hanyongshao@163.com. :. (61374090),,. Supported by National Natural Science Foundation of China (61374090), Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province and Taishan Scholarship Project of Shandong Province.
1400 33, [13], [11], Lyapunov. Lyapunov,, : Lyapunov,, Lyapunov. Lyapunov,. Lyapunov,, Wirtinger. : T; I ; ; ; NR ; R n, R m n S n nm nr n n n; P>0P ; ; V (t k )V (t k )= lim V (t), k N; He(A) t t k A + A T. 2 (Problem formulation) : ẋ(t) =Ax(t)+Bu(t), (1) : x(t) R n,u(t) R m ; A R n n,b R n m. u(t): u(t) =Kx( ),t [,+1 ),k N, (2) K R m n. { } k N 0 =t 0 <t 1 < < < 0 T min T k = +1 T max. (3) (1) (2) ẋ(t) =Ax(t)+A d x( ), t [,+1 ), k N, A d = BK. (4) 1 [5](3) (4). (3) (4), K. (3) (4), K, (3) (4) Lyapunov,. K, (3) (4). : 1 [14] M >0, ω :[a, b] R n,, : b a ωt (s)mω(s)ds 1 b a ΛT MΛ, b Λ = ω(s)ds. a 2 [15] M >0, ω :[a, b] R n, : b a ωt (s)m ω(s)ds 1 b a (ΘT MΘ +3Ω T MΩ), : Θ = ω(b) ω(a), Ω = ω(a)+ω(b) 2 b a Λ. (3) (4), : 1 : ẋ(t) =f(x(t),x( )), t [,+1 ), x( ) = lim x(t), k N, (5) t t k : x(t) R n, (3), f(0, 0) = 0, z(t) R n, f(x(t),x( )) f(z(t),z( )) L 1 x(t) z(t) + L 2 x( ) z( ), L 1, L 2., c 2 α, V a (x(t)), V b (x(t),t), t [,+1 ),k N, i) V a (x(t)) x( ) 2 V a (x( )) c 2 x( ) 2 ; ii) V b (x(t),t) V b (x( ), )=e 2αT k V b (x(t k+1 ),t k+1 ); iii) V (x(t),t)=e 2α(t ) [V a (x(t)) + V b (x(t),t)] V (x(t),t) < 0, t (,+1 ), (5)(3)α. iii) V (x(t k+1 ),t k+1 ) V (x( ), )= k+1 V (x(s),s)ds <0. V (x(t),t) e 2αT k [V a (x(t k+1 )) + V b(x(t k+1 ),t k+1 )] [V a (x( )) + V b (x( ), )] < 0.
10 : 1401 V a (x(t)), ii) e 2αT k V a (x(+1 )) V a (x( )) < 0. i) x( ) 2 1 V a (x( )) 1 V a (x(0))e 2α c 2 x(0) 2 e 2α, x( ) x(0) e α. (6), (5)[,t), t [,+1 ),, x(t) = x( )+ x( ) + x( ) + x( ) + x(t) =x( )+ f(x(s),x( ))ds, f(x(s),x( ))ds f(x(s),x( )) f(0, 0) ds L 1 x(s) 0 ds+ L 2 x( ) 0 ds L 1 x(s) ds + T k L 2 x( ) (1 + T max L 2 ) x( ) + L 1 x(s) ds. Grownwall-Bellman, x(t) (1 + T max L 2 ) x( ) e L 1(t ). L =1+T max L 2, x(t) Le L 1(t ) x( ). (7) t k+1 = T k T max, (6) (7), x(t) Le L 1(t ) x(0) e α = L x(0) e L1(t )+α(t ) e αt L e (L 1+α)T max x(0) e αt. (5)(3) α.. 2 V b (x(t), t), V a (x(t))v b (x(t), t) V (x(t), t) Lyapunov, Lyapunov.. V b (x(t),t)=0, 1 V (x(t),t)=e 2α(t ) V a(x(t)), 1. 3 [12]1, 1,., [13]1, 1,. 3 (New conditions for exponential stability) (3) (4). 2 0 T min T max α>0, (3) (4)α, P >0, S, X S n, Q R n n, R>0, Z>0, Y i R 3n n, i =1, 2, 3, T {T min, T max } : X<0, (8) Ψα(T 1 )=Π 1α + T (Π 2α + Π 3 ) < 0, (9) Ψ 11 Ψ 12 Ψ 13 Ψα(T 2 )= Ψ 22 0 < 0, (10) Ψ 33 Π 1α = Π 1 +2α(M T 1 PM 1 W T 1 RW 1 3W T 2 RW 2 )+ 2αT max He(Y 1 W 1 +3Y 2 W 2 ), Π 2α = Π 2 +2αW T 1 (SW 1 +2QM 2 ), Π 3 = M T 2 XM 2, Ψ 11 = Π 1α T (Π 3 Π 4α ), Ψ 12 = T (1 2αT max )Y 1, Ψ 13 =3T (1 2αT max )Y 2, Ψ 22 = T (1 2αT max )R, Ψ 33 = 3T (1 2αT max )R, Π 1 = Π 0 1 He(Y 1 W 1 + Y 3 W 1 +3Y 2 W 2 ), Π 2 = M T 0 RM 0 +He(M T 0 SW 1 +M T 0 QM 2 )+M T 1 ZM 1, Π 4α = Π 4 +2αT max M T 3 ZM 3, Π 0 1 = He(M T 1 PM 0 W T 1 QM 2 ) W T 1 SW 1, Π 4 = He(Y 3 M 4 ) M T 3 ZM 3, M 0 =[A A d 0], M 1 =[I 0 0], M 2 =[0 I 0], M 3 =[0 0 I], M 4 =[0 A d A], W 1 =[I I 0], W 2 =[I I 2I]. 1(3) (4)
1402 33. (3) (4) f(x(t),x( )) = Ax(t)+A d x( ), f(0, 0) = 0. L 1 = A, L 2 = A d, (3) (4)1f. k 0, T k [T min,t max ], : : V (x(t),t)=e 2α(t ) [V a (x(t)) + V b (x(t),t)], V a (x(t)) = x T (t)px(t), V b (x(t),t)= (+1 t)ζ T (t)[sζ(t)+2qx( )] + (+1 t) ẋ T (s)rẋ(s)ds + (+1 t)(t )x T ( )Xx( )+ (+1 t) x T (s)zx(s)ds, ζ(t) =x(t) x( ). = λ min (P ), c 2 = λ max (P ) x( ) 2 V a (x( )) c 2 x( ) 2. V b (x( ), )=0, e 2αT k V b(x(t k+1 ),t k+1 )=0, V b (x( ), )=e 2αT k V b (x(t k+1 ),t k+1 ). V a (x(t)), V b (x(t),t)1i) ii). V (x(t),t)1iii). : 1 x(s)ds, t, υ k (t) = t x(t), t =, η k (t) =[x T (t) x T ( ) υk T (t)] T. t (,+1 ), k N, V (x(t),t)(4), e 2α(t ) V (x(t),t)= η T k (t)[π 0 1 +2αM T 1 PM 1 + (+1 t)π 2α +(+1 + 2t)Π 3 + 2α(+1 t)(t )Π 3 ]η k (t)+ [2α(+1 t) 1] ẋ T (s)rẋ(s)ds+ [2α(+1 t) 1] x T (s)zx(s)ds. (11) (10), 1 2αT max > 0, t [, +1 ), 2α(+1 t) 1 < 0. [2α(+1 t) 1] ẋ T (s)rẋ(s)ds2, [2α(+1 t) 1] [2α(T max t + ) 1] ẋ T (s)rẋ(s)ds ẋ T (s)rẋ(s)ds 2α(T max t + ) 1 η T k (t)[w T 1 RW 1 + t 3W T 2 RW 2 ]η k (t) = 2αT max 1 η T k (t)[w T 1 RW 1 +3W T 2 RW 2 ]η k (t) t 2αη T k (t)[w T 1 RW 1 +3W T 2 RW 2 ]η k (t). Y i R 3n n, i =1, 2, : 1 W T i RW i (t )Y i R 1 Y T i He(Y i W i ), t [2α(+1 t) 1] ẋ T (s)rẋ(s)ds η T k (t){(1 2αT max )[ He(Y 1 W 1 +3Y 2 W 2 )+ (t )(Y 1 R 1 Y T 1 +3Y 2 R 1 Y T 2 )] 2α(W T 1 RW 1 +3W T 2 RW 2 )}η k (t). (12) [2α(+1 t) 1] x T (s)zx(s)ds1, [2α(+1 t) 1] x T (s)zx(s)ds [2α(T max t + ) 1] x T (s)zx(s)ds [2α(T max t + ) 1](t )υ T k (t)zυ k (t) (2αT max 1)(t )η T k (t)m T 3 ZM 3 η k (t). (13) t [,+1 ), (4) x(t) x( )= (Ax(s)+A d x( ))ds = (t )(Aυ k (t)+a d x( ))., Y 3 R 3n n η T k (t)[2(t )Y 3 M 4 2Y 3 W 1 ]η k (t) =0. (14) (12)(13)(11)(14), : e 2α(t ) V (x(t),t) η T k (t)υ α (t)η k (t), Υ α (t) = Π 1α +(+1 t)π 2α +(+1 + 2t)Π 3 + 2α(+1 t)(t )Π 3 +(t )Π 4α + (t )(1 2αT max )Π 5, Π 5 = Y 1 R 1 Y T 1 +3Y 2 R 1 Y T 2.
10 : 1403 (,+1 ), V (x(t),t) < 0, Υα (t). X <0, Υ α (t)t, Υ α (t)(,+1 ), Π 1α + T k (Π 2α + Π 3 ) < 0, Π 1α T k (Π 3 Π 4α )+T k (1 2αT max )Π 5 <0. T k, Schur, 2(9) (10). 1, (3) (4)α.. 4 [11]2, V (x(t), t), t, (+1 t) x T (s)zx(s)ds, t (+1 t)(t )x T ( )Xx( ). [12],, Lyapunov, Lyapunov. V (x(t),t),, 12,,. α =0, (3) (4) : 1 0 T min T max, (3) (4), P >0, S, X S n, Q R n n, R>0, Z>0, Y i R 3n n, i =1, 2, 3, T {T min, T max } Ψ 1 0 (T )=Π 1 + T (Π 2 + Π 3 ) < 0, Ψ 2 0 (T )= Π 1 T (Π 3 Π 4 ) TY 1 3TY 2 TR 0 < 0, 3TR Π i,i=1, 2, 3, 42. 5 X <0, X. 6 21,. 4 (Examples). 1 (3) (4), [ ] 0 1 0 0 A =,A d = BK =. 0 0.1 0.375 1.15 α, T min =0, [11]22 T max, 1. 1 T max Table 1 Maximum upper bound T max of the aperiodic sampling for systems to be exponentially stable α 0.1 0.2 0.3 0.4 [11] 1.447 1.259 1.119 1.006 1.623 1.539 1.473 1.250 1,, [11] 2, 2. 2 [11]2. 2 A = (3) (4), 2 0 1 0,A d = BK =. 0 0.9 1 1 α, T min =0, [11]22 T max, 2. 2 T max Table 2 Maximum upper bound T max of the aperiodic sampling for systems to be exponentially stable α 0.1 0.2 0.3 0.4 [11] 1.916 1.566 1.333 1.165 2.032 1.705 1.531 1.250 2,, [11] 2, 2., 2 [11]2. Lyapunov, Lyapunov, 2S, Q., 2 V b (x(t),t) S, Q : ζ T (t)[sζ(t)+2qx( )] T x(t) S Q S x(t) x( ) S Q Q T x( ) T x(t) x(t) W. x( ) x( ) 2S, Q, W. 2 α =0.2, T max 1.705. MATLAB LMI 83.0776 18.1933 S =, 18.1933 1.5318
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