A NOVEL FOR EXPONENTIAL STABILITY OF LINEAR SYSTEMS WITH MULTIPLE TIME-VARYING DELAYS

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1 International Journal of Pure and Applied Matheatics Volue 96 No , ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpa.eu A NOVEL FOR EXPONENTIAL STABILITY OF LINEAR SYSTEMS WITH MULTIPLE TIME-VARYING DELAYS Grienggrai Rajchakit Departent of Matheatics Maejo University Chiangai, 50290, THAILAND Abstract: This paper addresses exponential stability proble for a class of linear systes with ultiple tie-varying delays. The tie delay is any continuous function belonging to a given interval, in which the lower bound of delay is not restricted to zero. By constructing a suitable augented Lyapunov- Krasovskii functional cobined with Leibniz-Newton s forula, new delaydependent sufficient conditions for the exponential stability of the systes are first established in ters of LMIs. AMS Subject Classification: 15A09, 93D20, 37C75 Key Words: exponential stability, interval ultiple tie-varying delays, Lyapunov function, linear atrix inequalities 1. Introduction The proble of stability analysis and controller design for tie-delay systes have been given considerable attention over the past decades. The existing stabilization results for tie delay systes can be classified into two types, Received: June 3, 2014 c 2014 Acadeic Publications, Ltd. url:

2 530 G. Rajchakit i.e. delay independent stabilization and delay-dependent stabilization. The delay-independent stabilization provides a controller which stabilizes a syste irrespective of the extent of the delay. On the other hand, the delay-dependent stabilization is concerned with the size of the delay which usually provides an upper bound of the delay capable of ensuring the stability for any delay lower than the upper bound. As ost physical systes occur in continuous tie, consequently the theories for stability analysis and controller synthesis are ainly developed for the continuous tie. However, it is ore feasible that a discrete-tie approach is used for the purpose, as the controller is usually digitally ipleented. Stability analysis of linear systes with tie-varying delays ẋ(t) = Ax(t)+Dx(t h(t)) is fundaental to any practical probles and has received considerable attention [1 22]. Most of the known results on this proble are derived assuing only that the tie-varying delay h(t) is a continuously differentiable function, satisfying soe boundedness condition on its derivative: ḣ(t) δ < 1. In delay dependent stability criteria, the ain concerns is to enlarge the feasible region of stability criteria in given tiedelay interval. Interval tie-varying delay eans that a tie delay varies in an interval in which the lower bound is not restricted to be zero. By contracting a suitable arguent Lyapunov functional and utilizing free weight atrices, soe less conservative conditions for asyptotic stability are derived in [23 28] for systes with tie delay varying in an interval. However, the shortcoing of the ethod used in these works is that the delay function is assued to be differential and its derivative is still bounded: ḣ(t) δ. This paper gives the iproved results for the exponential stability of systes with interval ultiple tie-varying delays. The tie delay is assued to be a ultiple tie-varying delays continuous function belonging to a given interval, but not necessary to be differentiable. By constructing arguent Lyapunov functional cobined with LMI technique, we propose new criteria for the exponential stability of the syste. The paper is organized as follows: Section 2 presents definitions and soe well-known technical propositions needed for the proof of the ain results. Delay-dependent exponential stability conditions of the syste are presented in Section Preliinaries The following notations will be used in this paper. R + denotes the set of all real non-negative nubers; R n denotes the n-diensional space with the

3 A NOVEL FOR EXPONENTIAL STABILITY OF scalar product.,. and the vector nor. ; M n r denotes the space of all atrices of (n r)-diensions; A T denotes the transpose of atrix A; A is syetric if A = A T ; I denotes the identity atrix; λ(a) denotes the set of all eigenvalues of A; λ in/ax (A) = in/ax{reλ;λ λ(a)}; x t := {x(t+s) : s [ h,0]}, x t = sup s [ h,0] x(t + s) ; C([0,t],R n ) denotes the set of all R n -valued continuous functions on [0,t]; Matrix A is called sei-positive definite (A 0) if Ax,x 0, for all x R n ;A is positive definite (A > 0) if Ax,x > 0 for all x 0;A > B eans A B > 0. denotes the syetric ter in a atrix. Consider a linear syste with interval ultiple tie-varying delays of the for ẋ(t) = Ax(t)+ D i x(t h i (t)), t R +, i = 1,2,...,, (1) x(t) = φ(t),t [ h 2i,0], i = 1,2,...,, where x(t) R n is the state; A,D i M n n, i = 1,2,...,, and φ(t) C([ h 2i,0],R n ) is the initial function with the nor φ = sup s [ h2i,0] φ(s) ; the ultiple tie-varying delays function h i (t),i = 1,2,..., satisfies the condition 0 h 0i h i (t) h 1i, i = 1,2,...,. Definition 1. Givenα > 0.Thezerosolutionofsyste(1)isα-exponentially stable if there exist a positive nuber N > 0 such that every solution x(t,φ) satisfies the following condition: x(t,φ) Ne αt φ, t R +. We end this section with the following technical well-known propositions, which will be used in the proof of the ain results. Proposition 1. (Cauchy inequality) For any syetric positive definite atrix N M n n and a,b R n we have +a T b a T Na+b T N 1 b. Proposition 2. [29] For any syetric positive definite atrix M M n n, scalar γ > 0 and vector function ω : [0,γ] R n such that the integrations concerned are well defined, the following inequality holds ( γ T ( γ ) ( γ ) ω(s)ds) M ω(s) ds γ ω T (s)mω(s)ds

4 532 G. Rajchakit Proposition 3. [29] Let E,H and F be any constant atrices of appropriate diensions and F T F I. For any ǫ > 0, we have EFH +H T F T E T ǫee T +ǫ 1 H T H. Proposition 4. (Schur copleent lea [29]). Given constant atrices X,Y,Z with appropriate diensions satisfying X = X T,Y = Y T > 0. Then X +Z T Y 1 Z < 0 if and only if ( ) X Z T < 0 or Z Y ( ) Y Z Z T < 0. X 3. Main Results Let us set λ 1 = λ in (P), λ 2 = λ ax (P)+2h 2i λ ax (Q)+2h 2 2iλ ax (R)+(h 2i h 1i )λ ax (U). Theore 1. Given α > 0. The zero solution of the syste (1) is α- exponentially stable if there exist syetric positive definite atrices P, Q, R, U, and atrices S i,i = 1,2,...,5 such that the following LMI holds M 11i M 12i M 13i M 14i M i = where S 1 S 3 A M 22i 0 M 24i S 2 M 33i M 34i S 3 M 44i S 4 S 5 D i M 55i < 0, i = 1,2,...,, M 11i = A T P +PA+2αP (e 2αh 1i +e 2αh 2i )R S 1 A+S 1 +2(Q+R), M 12i = e 2αh 1i R S 2 A,M 13i = e 2αh 2i R S 3 A,M 14i = P D i S 1 D i S 3 A (2)

5 A NOVEL FOR EXPONENTIAL STABILITY OF M 15i = S 1 S 3 A,M 22i = e 2αh 1i (Q+R),M 24i = S 2 D i +e 2αh 2i U, M 33i = e 2αh 2i (Q+R+U),M 34i = S 3 D i +e 2αh 2i U, M 44i = S 4 D i e 2αh 2i U,M 55i = S 5 +S5 T +(h2 1i +h2 2i )R+(h 2i h 1i ) 2 U. Moreover, the solution x(t, φ) of the syste satisfies x(t,φ) λ1 λ 2 e αt φ, t R +. Proof. We consider the following Lyapunov-Krasovskii functional for the syste (1) 6 V(t,x t ) = V i, where V 1 = x T (t)px(t), V 2 = V 3 = t h 1i e 2α(s t) x T (s)qx(s)ds, V 4 = h 1i 0 V 5 = h 2i 0 t h 2i e 2α(s t) x T (s)qx(s)ds, h 1i t+s h 2i V 6 = (h 2i h 1i ) It easy to check that e 2α(τ t) ẋ T (τ)rẋ(τ)dτ ds, e 2α(τ t) ẋ T (τ)rẋ(τ)dτ ds, t+s h1i t h 2i t+s e 2α(τ t) ẋ T (τ)uẋ(τ)dτ ds. λ 1 x(t) 2 V(t,x t ) λ 2 x t 2, t 0, (3) Taking the derivative of V i,i = 1,2,...,6 along the solution of syste (1) we

6 534 G. Rajchakit have V 1 =2x T (t)pẋ(t) = 2x T (t)[a T P +AP]x(t)+2x T (t)p D i x(t h i (t)); V 2 =x T (t)qx(t) e 2αh 1i x T (t h 1i )Qx(t h 1i ) 2αV 2 ; V 3 =x T (t)qx(t) e 2αh 2i x T (t h 2i )Qx(t h 2i ) 2αV 3 ; V 4 =h 2 1iẋ T (t)rẋ(t) h 1i e 2αh 1i V 5 =h 2 2iẋ T (t)rẋ(t) h 2i e 2αh 2i t h 1i ẋ T (s)rẋ(s)ds 2αV 4 ; t h 2i ẋ T (s)rẋ(s)ds 2αV 5 ; V 6 =(h 2i h 1i ) 2 ẋ T (t)uẋ(t) (h 2i h 1i )e 2αh 2i Therefore, we have V(.)+2αV(.) h1i x T (t)m 11i x(t)+2x T (t)[e 2αh 1i R S 2 ]x(t h 1i ) +2x T (t)[e 2αh 2i R S 3 A]x(t h 2i )+2x T (t)[p S 1 D i S 3 A]x(t h(t)) t h 2i ẋ T (s)uẋ(s)ds 2αV 6. +2x T (t)[s 1 S 3 A]ẋ(t) x T (t h 1i )[e 2αh 1i Q+e 2αh 1i R]x(t h 1i ) +2x T (t h 1i )[S 2 D i +e 2αh 1i U]x(t h i (t))+2x T (t h 1i )S 2 ẋ(t) x T (t h 2i )[e 2αh 2i Q+e 2αh 2i R+e 2αh 2i U]x(t h 2i ) +2x T (t h 2i )S 3 ẋ(t)+x T (t h i (t))[s 4 D i e 2αh 2i U]x(t h i (t)) +2x T (t h i (t))(s 4 S 5 D i )ẋ(t) +ẋ T (t)[s 5 +S T 5 +h2 1i R+h2 2i R+(h 2i h 1i ) 2 U]ẋ(t) = ζ T i (t)m iζ i (t), where ζ i (t) = [x(t),x(t h 1i ),x(t h 2i ),x(t h i (t)),ẋ(t)]. By condition (2), we D i

7 A NOVEL FOR EXPONENTIAL STABILITY OF obtain V(t,x t ) 2αV(t,x t ), t R +. (4) Integrating both sides of (4) fro 0 to t, we obtain V(t,x t ) V(φ)e 2αt, t R +. Furtherore, taking condition (2) into account, we have then λ 1 x(t,φ) 2 V(x t ) V(φ)e 2αt λ 2 e 2αt φ 2, x(t,φ) λ2 λ 1 e αt φ, t R +, which concludes the proof by the Lyapunov stability theore [29]. 4. Conclusion In this paper, we have proposed new delay-dependent conditions for the exponential stability of linear systes with non-differentiable interval tie-varying delay. Based on the iproved Lyapunov-Krasovskii functional and linear atrix inequality technique, the conditions for the exponential stability of the systes have been established in ters of LMIs. Acknowledgeents This work was supported by the Thailand Research Fund Grant, the Coission for Higher Education and Faculty of Science, Maejo University, Thailand. The authors thank anonyous reviewers for valuable coents and suggestions, which allowed us to iprove the paper. References [1] S.G. Wang, H.S. Yao, Pinning synchronization of the tie-varying delay coupled coplex networks with tie-varying delayed dynaical nodes, Chin.Phys. B 21 (2012)

8 536 G. Rajchakit [2] S. Wang, H. Yao, S. Zheng, Y. Xie, A novel criterion for cluster synchronization of coplex dynaical networks with coupling tie-varying delays, Coun. Nonlinear Sci. Nuer. Siul. 17 (2012) [3] Manlika Ratchagit, ON STABILITY OF SWITCHED LINEAR SYS- TEMS, International Journal of Pure and Applied Matheatics, Vol. 78 No. 6, 2012, [4] K. Ratchagit, Asyptotic stability of delay-difference syste of Hopfield neural networks via atrix inequalities and application, International Journal of Neural Systes, 17(2007), DOI: /S [5] Kreangkri Ratchagit, STABILITY CRITERIA OF LPD SYSTEM WITH TIME-VARYING DELAY, International Journal of Pure and Applied Matheatics, Vol. 78 No. 6, 2012, [6] Kreangkri Ratchagit, STABILITY ANALYSIS OF LINEAR SYSTEMS WITH TIME DELAYS, International Journal of Pure and Applied Matheatics, Vol. 76 No. 1, 2012, [7] Kreangkri Ratchagit, STABILITY OF LINEAR TIME-VARYING SYS- TEMS, International Journal of Pure and Applied Matheatics, Vol. 63 No. 4, 2010, [8] Kreangkri Ratchagit, EXPONENTIAL STABILITY OF SWITCHED LINEAR SYSTEMS, International Journal of Pure and Applied Matheatics, Vol. 58 No. 3, 2010, [9] K. Ratchagit, THE SUFFICIENT CONDITIONS FOR STABILITY OF LINEAR TIME-VARYING SYSTEMS WITH STATE DELAYS, International Journal of Pure and Applied Matheatics, Vol. 65 No. 1, 2010, [10] G. Rajchakit, Stabilization of switched discrete-tie systes with convex polytopic uncertainties, Journal of Coputational Analysis and Applications 16 (2014) [11] K. Ratchagit and V.N. Phat, Stability criterion for discrete-tie systes, J. Ineq. Appl., 2010(2010), 1 6. doi: /2010/ [12] G. Rajchakit, Switching design for the robust stability of nonlinear uncertain stochastic switched discrete-tie systes with interval tie-varying delay. Journal of Coputational Analysis & Applications 16(2014),

9 A NOVEL FOR EXPONENTIAL STABILITY OF [13] G. Rajchakit, Robust stability and stabilization of nonlinear uncertain stochastic switched discrete-tie systes with interval tie-varying delays. APPLIED MATHEMATICS and INFORMATION SCIENCES 6(2012), [14] G. Rajchakit, Delay-dependent optial guaranteed cost control of stochastic neural networks with interval nondifferentiable tie-varying delays, ADVANCES IN DIFFERENCE EQUATIONS, 2013(2013), DOI: / [15] K. Ratchagit, A switching rule for the asyptotic stability of discrete-tie systes with convex polytopic uncertainties, Asian-European J. Math., 5(2012), (12 pages). DOI: /S [16] G. Rajchakit, Stabilization of switched discrete-tie systes with convex polytopic uncertainties, Journal of Coputational Analysis & Applications 16(2014), [17] Manlika Rajchakit, Piyapong Niasup, Grienggrai Rajchakit, A constructive way to design a switching rule and switching regions to ean square exponential stability of switched stochastic systes with non-differentiable and interval tie-varying delay, Journal of Inequalities and Applications 2013, 2013:499 doi: / x [18] Grienggrai Rajchakit, Delay-Dependent Asyptotical Stabilization Criterion of Recurrent Neural Networks, Applied Mechanics and Materials. 330(2013) doi: / [19] K. Ratchagit, Asyptotic stability of nonlinear delay-difference syste via atrix inequalities and application, International Journal of Coputational Methods, pp , DOI: /S [20] M. de la Sen, Global Stability of Polytopic Linear Tie-Varying Dynaic Systes under Tie-Varying Point Delays and Ipulsive Controls, Matheatical Probles in Engineering, vol. 2010, Article ID , 33 pages, doi: /2010/ [21] G. Rajchakit, Exponential stability of switched linear systes with interval tie-varying delays, Proceedings of the 2012 IEEE International Conference on Robotics and Bioietics Deceber 11-14, 2012, Guangzhou, China, doi: /ROBIO

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