New conditions for exponential stability of sampled-data systems under aperiodic sampling

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1 DOI: /CTA Control Theory & Applications Vol. 33 No. 10 Oct. 2016,, (, ) :. 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 , 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] : ; : hanyongshao@163.com. :. ( ),,. Supported by National Natural Science Foundation of China ( ), Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province and Taishan Scholarship Project of Shandong Province.

2 , [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.

3 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)

4 (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 αM 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.

5 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, X <0, X. 6 21,. 4 (Examples). 1 (3) (4), [ ] A =,A d = BK = 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 [11] ,, [11] 2, 2. 2 [11]2. 2 A = (3) (4), ,A d = BK = 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 [11] ,, [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 MATLAB LMI S =,

6 Q =, W = W , , , , W. S, Q, W >0, 2, T max S, QW,, 2., α =0, T min =0, 3 11 T max. 3 T max Table 3 Maximum upper bound T max of the aperiodic sampling for systems to be asymptotically stable [16] [6] [11] T max (Conclusions). Lyapunov,, Wirtinger,.. (,. H [J]., 2010, 27(11): ) [3] FUJIOKA H. A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices [J]. IEEE Transactions on Automatic Control, 2009, 54(10): [4] OISHI Y, FUJIOKA H. Stability and stabilization of aperiodic sampled-data control systems using robust linear matrix inequalities [J]. Automatica, 2010, 46(8): [5] LEE D, JOO Y. A note on sampled-data stabilization of LTI systems with aperiodic sampling [J]. IEEE Transactions on Automatic Control, 2015, 60(10): 1 6. [6] FRIDMAN E. A refined input delay approach to sampled-data control [J]. Automatica, 2010, 46(2): [7] SHAO H. Improved delay-dependent stability criteria for systems with a delay varying in a range [J]. Automatica, 2008, 44(12): [8] HU L, LAM J, CAO Y, et al. A LMI approach to robust H 2 sampleddata control for linear uncertain systems with time-varying delay [J]. IEEE Transactions on Systems, Man and Cybernetics, Part B, 2003, 33(1): [9] NAGHSHTABRIZI P, HESPANHA J P, TEEl A R. Exponential stability of impulsive systems with application to uncertain sampleddata system [J]. Systems & Control Letters, 2008, 57(5): [10] BRIAT C, SEURET A. A looped-functional approach for robust stability analysis of linear impulsive systems [J]. Systems & Control Letters, 2012, 61(10): [11] SEURET A. A novel stability analysis of linear systems under asynchronous samplings [J]. Automatica, 2012, 48(1): [12] SHAO H, LAM J, FENG Z. Sampling-interval-dependent stability for linear sampled-data systems with non-uniform sampling [J]. International Journal of Systems Science, 2016, 47(12): [13] SHAO H, ZHU X, ZHANG Z. Sampling-interval-dependent exponential stability for a networked control system model of sampleddata system [C] //Proceedings of the 33rd Chinese Control Conference. Nanjing: IEEE, 2014: [14] GU K. An integral inequality in the stability problem of time-delay systems [C] //Proceedings of the 39th IEEE Conference on Decision and Control. Sydney: IEEE, 2000: [15] SEURET A, GOUAISBAUT F. Wirtinger-based integral inequality: Application to time-delay systems [J]. Automatica, 2013, 49(9): [16] SEURET A. Stability analysis for sampled-data systems with a timevarying period [C] //The 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference. Shanghai: IEEE, 2009: [1] XUE Yan, LIU Ke. Variable-sampling-rate networked control systems based on prediction-value control [J]. Control Theory & Applications, 2009, 26(6): (,. [J]., 2009, 26(6): ) [2] DAI Jianguo, CUI Baotong. Robust H-infinity control of nonlinear networked systems with uncertainty [J]. Control Theory & Applications, 2010, 27(11): : (1989 ),,,, zhangdanjisuan@163.com; (1964 ),,,,, hanyong shao@163.com; (1990 ),,,, zhaojianrong6666@163.com.

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