Applied Mathematics and Computation 214 (2009) 374–380 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Exponential stability criterion for time-delay systems with nonlinear uncertainties Phan T Nam Department of Mathematics, Quynhon University, 170 An Duong Vuong Road, Binhdinh, Vietnam a r t i c l e i n f o a b s t r a c t Exponential stability of time-delay systems with nonlinear uncertainties is studied in this paper Based on the Lyapunov method and the approaches of decomposing the matrix, a new exponential stability criterion is derived in terms of a matrix inequality, which allows to compute simultaneously the two bounds that characterize the exponential nature of the solution Some numerical examples are also given to show the superiority of our result to those in the literature Ó 2009 Elsevier Inc All rights reserved Keywords: Exponential stability Time-delays Nonlinear uncertainties Lyapunov function Linear matrix inequality Introduction Consider the following time-delay systems with nonlinear uncertainties: _ xtị ẳ Axtị ỵ A1 xt hị ỵ f t; xtịị ỵ f1 t; xt hịị; x0 hị ẳ /hị; 1:1ị where xtị Rn is the state, A; A1 are given matrix, and initial condition is x0 hị ẳ /hị Cẵh; 0; Rn Þ The time-varying parameter uncertainties f ; f1 are assumed to be bounded kf ðt; xðtÞÞk akxðtÞk; kf1 ðt; xðt À hÞÞk a1 kxðt À hÞk; where a; a1 are positive numbers Definition 1.1 The system (1.1) is d-stable, with d > 0, if there is a positive number N such that for each /ð:Þ, the solution xðt; /Þ of the system (1.1) satisfies kxðt; /Þk Nedt k/k 8t P 0; where k/k ẳ maxfk/tịk : t ½Àh; 0g N is called Lyapunov factor Because of data errors, environmental noises, the difficulty of measuring various parameters, unavoidable approximation, etc., most real problems are modeled by time-delay systems with nonlinear uncertainties So, the stability problem of timedelay system with nonlinear uncertainties has been an interesting problem in the recent years (see in [2–12] and reference therein) It is well known that the widely used method is the approach of Lyapunov functions with Razumikhin techniques and the stability conditions are presented in terms of the solution of either linear matrix inequalities or Riccati equations [5,8,9] By using parameterized neutral models, some less conservative criteria, which are dependent on the stability of the operator, have proposed in [2,10–12] By proposing a technique to adjust the Lyapunov functionals in [2,11,12], the E-mail address: pthnam@yahoo.com 0096-3003/$ - see front matter Ó 2009 Elsevier Inc All rights reserved doi:10.1016/j.amc.2009.04.004 P.T Nam / Applied Mathematics and Computation 214 (2009) 374–380 375 authors in [6,7] have reduced the stability of the operator and given some less conservative more criteria However, this technique has not applied to the Lyapunov functional in [10] Therefore, the first purpose of this paper is to find an improvement the criterion in [10] by using this technique Inspired by the method of decomposing the matrix in [1], we have decomposed the matrix A1 into two parts A11 ; A12 Using the operator Dxt ị ẳ xtị A11 Z t xðsÞds; ð1:2Þ tÀh we get a generalization of the result in [10] Combining an exponential translation variable and the technique to reduced the stability of the operator, we get a new d-stability criterion for the system (1.1) As a consequence of the this criterion, we also obtain an asymptotical stability criterion Compared with the results in [10], our result has the following advantages: Rt First, by decomposing the matrix A1 into two part A11 ; A12 and using the operator Dxt ị ẳ xtị A11 th xðsÞds, our criterion will be less restricted than the criterion in [10] Second, by reducing the stability of the operator, our criterion will be less conservative than the criterion in [10] The following lemma is needed for our main results Lemma 1.1 [6] Assume that S RnÂn is a symmetric positive-definite matrix Then for every Q RnÂn , 2hQy; xi À hSy; yi hQSÀ1 Q T x; xi 8x; y Rn : If we take S ¼ I then we have j2hQy; xij kyk2 þ kQxk2 By decomposing the matrix A1 into two parts A11 ; A12 and using the Rt operator Dðxt Þ ¼ xðtÞ À A11 tÀh xðsÞds, we get a generalization of the result in [10] This generalization also need for our main results Theorem 1.2 For given h > and a; a1 , the system (1.1) is asymptotically stable if the exist the positive-definite matrices X; Z ; Z ; M and positive scalars 0 ; 1 and < b < satisfying the following two matrix inequalities: T ÀbM hA11 M H M ! < 0; ð1:3Þ R0 < 0; ð1:4Þ where N11 I I N14 N15 hX 0 aX X H À0 I 0 0 0 H H À1 I 0 0 H H H Àh Z À1 0 ÀhZ AT11 À0 aZ AT11 B B B B B B B B R0 ¼ B B B B B B B B @ H H H H ÀZ H H H H 1 a1 Z À1 I H 0 H H H H H H ÀhZ H H H H H H H H H H H H H H À0 I H C C C C C C ÀZ AT11 C C C C C C C C C C A ÀZ with N11 ẳ A ỵ A11 ịX ỵ XA ỵ A11 ịT ; N14 ẳ A ỵ A11 ịA11 Z ; N15 ẳ A1 A11 ịZ : Proof Consider the following Lyapunov functional: V ẳ V ỵ V ỵ V 3; 1:5ị where V1 ẳ V2 ẳ Z t s t ỵ hịxT sịR1 xsịds; th Z t th xT sịR2 xsị; V ẳ DT ðxt ÞPDðxt Þ: 376 P.T Nam / Applied Mathematics and Computation 214 (2009) 374–380 By the computations, which are similarly to the proof in [10], we have V P0 and Z V_ < k kDxt ịk2 ỵ t th ! 2 < kkDxt ịk2 ; xsịds ỵ kxt À hÞk where k is a positive number Combining with (1.3), the system (1.1) is asymptotically stable h Rt Remark 1.3 Since xtị ẳ Dxt ị A11 Z kxtịk kDxt ịk ỵ A 11 t th th xsịds, we have xðsÞds : Applying the Bunhiakovski’s inequality, we have Z kxtịk2 kDxt ịk2 ỵ A 11 t tÀh This implies that ! Z 2 kDx xsịds ịk ỵ kA k t 11 t tÀh ! xðsÞds : Z t : kDxt ịk2 kxtịk2 ỵ kA11 k2 ỵ xsịds th If kA11 k < then we have, Z t : kDxt ịk2 kxtịk2 ỵ xsịds th This follows k V_ < kxtịk2 ỵ kxðt À hÞk2 : ð1:6Þ If kA11 k P then we have, ÀkDðxt Þk2 À kA11 k2 kDðxt Þk2 : This follows V_ < À k 2kA11 k kxtịk2 ỵ kxt hịk2 : ð1:7Þ Thus if (1.4) holds then there exists a positive number k0 ¼ (k ; k 2kA11 k2 ð1:8Þ ; such that V_ < Àk0 kxðtÞk2 þ kxðt À hÞk2 : ð1:9Þ Main results Combining Remark 1.3 with the following change of the state variable: ztị ẳ edt xtị; t Rỵ ; we get main results as follows: Theorem 2.1 The system (1.1) is d-stable if the exist the positive-definite matrices X; Z ; Z , and positive scalars following matrix inequality: Rd < 0; 0 ; 1 and the ð2:1Þ 377 P.T Nam / Applied Mathematics and Computation 214 (2009) 374–380 where with N11 I I N14 N15 hX 0 aX X H À0 I 0 0 0 B B B B B B B B Rd ¼ B B B B B B B B @ H H À1 I 0 0 H H H H H H Àh Z H À1 ÀZ 1 a1 edh Z ÀhZ AT11 edh À0 aZ AT11 edh H H H H H À1 I 0 H H H H H H ÀhZ H H H H H H H À0 I H H H H H H H H C C C C C T dh C ÀZ A11 e C C C; C C C C C C C A ÀZ N11 ¼ A ỵ dI ỵ A11 edh ịX ỵ XA ỵ dI ỵ A11 edh ịT ; N14 ẳ A ỵ dI ỵ A11 edh ịA11 edh Z ; N15 ¼ edh ðA1 À A11 ÞZ : Proof First, using the above change of variable then the system (1.1) is transformed to the following system: z_ tị ẳ A þ dIÞzðtÞ þ edh A1 zðt À hÞ þ f t; ztịedt ịedt ỵ edt f1 t; edthị zt hịị: Let us denote A0d ẳ A ỵ dI; A1d ¼ e A1 ; a1d ¼ a1 e dh dh ð2:2Þ and consider the following Lyapunov functional for system (2.2): V ẳ V ỵ V ỵ V ỵ V 4; 2:3ị where V1 ẳ V2 ẳ Z t s t ỵ hịzT sịR1 zsịds; th Z t zT sịR2 zsị; th T V ẳ D zt ịPDzt ị: V ẳ bkztịk2 ; with b ẳ k0 @ kAd k ỵ aị ỵ By Remark 1.3, we have q1 kAd k ỵ aị2 þ ðkA1d k þ a1d Þ2 A > 0: ðkA1d k ỵ a1d ị2 V_1 ỵ V_ þ V_ k0 kzðtÞk2 þ kzðt À hÞk2 : Using Lemma 1.1, we have V_ à 2bz_ tịT ztị ỵ k0 kztịk2 ỵ kzt hịk2 ẳ 2b zT tịATd ztị þ xT ðt À hÞAT1d zðtÞ þ f T ðt; ztịedt ịedt ztị ỵ f1T t; edthị zt hịịedt ztị k0 kztịk2 ỵ kzt hịk2 2bkAd kkztịk2 ỵ 2bkA1d kkzt hịkkztịk ỵ 2bakztịk2 ỵ 2ba1d kzT ðt À hÞkkzðtÞk À k0 kzðtÞk2 À k0 kzðt hịk2 k0 ỵ 2bkAd k ỵ 2baịkztịk2 ỵ 2bkA1d k ỵ a1d ịkzt hịkkztịk k0 kzt hịk2 ! kA1d k ỵ a1d ị2 kztịk2 0: k0 ỵ 2bkAd k ỵ aị ỵ b2 k0 Integrating both sides of the above inequality from to t, we have V à ðtÞ À V à 0ị 8t Rỵ : Since V ðtÞ P bkzðtÞk2 , we have bkzðtÞk2 V à 0ị ẳ bkz0ịk2 ỵ V 0ị ỵ V 0ị ỵ V 0ị: 378 P.T Nam / Applied Mathematics and Computation 214 (2009) 374–380 By simple computation, we have V ð0Þ k/k2 kR1 kh ; V ð0Þ k/k2 kR2 kh: Moreover, we have Z V 0ị ẳ z0ị ỵ h T Z A11d zsịds P z0ị ỵ h ẳ zT 0ịX z0ị ỵ zT 0ịX A11d Z Z A11d zsịds ẳ z0ị ỵ h zsịds ỵ h Since R0 h zsịds ẳ R0 Àh xðsÞeds ds k/k R0 Àh Z h T Z A11d zsịds X z0ị ỵ A11d zðsÞds Àh Z zT ðsÞds AT11d X À1 z0ị ỵ eds ds ẳ k/k edh ị, we have d 0 Àh Z zT ðsÞds AT11d X À1 A11d zðsÞds: Àh ! Àdh À eÀdh Þ ð1 À e À1 k/k2 : kX k ỵ 2kX kkA11d k ỵ kX kkA11d k d d2 À1 V ð0Þ À1 We denote N ¼ kR1 kh ; N ẳ kR2 kh, N3 ẳ kX k ỵ 2kX À1 kkA11d k N2 ¼ À eÀdh ð1 edh ị2 ; ỵ kX kkA11d k2 d d2 b ỵ N1 ỵ N2 ỵ N3 : b Then, we have bkzðtÞk2 V à ð0Þ ðb ỵ N1 ỵ N ỵ N3 ịk/k2 ẳ bN k/k2 : Hence, kzðtÞk Nk/k This implies kxðt; /ịk Nk/kedt 8t Rỵ Thus, the system (1.1) is d-stable with Lyapunov factor s b ỵ N1 þ N2 þ N3 : N¼ b The proof is completed h Remark 2.2 Assume that the matrix inequality (1.4) holds Since A0 ; A1 ; X; Z ; Z are constant matrices and 1 ; 2 ; h are constants, we can choose d0 > is small enough such that Rd0 < Hence, we have an asymptotical stability criterion Corollary 2.3 System (1.1) is asymptotically stable if LMI (1.4) holds Remark 2.4 In cases A11 ¼ A1 ; A12 ¼ and A11 ¼ 0; A12 ¼ A1 , we obtain improvements of the results in [5,10] Numerical examples Example 3.1 Consider the following linear systems, which is considered in [3]: _ xtị ẳ À2 À1 xðt À hị ỵ f t; xtịị ỵ f1 t; xt hịị; xtị ỵ 1 where kf t; xðtÞÞ 0:05kxðtÞk; kf1 ðt; xðt À hÞÞ 0:1kxðt À hÞk By decomposing the matrix A1 into following two parts: A11 ¼ À0:01 À0:01 À0:01 ; A12 ¼ À0:99 À0:99 À0:99 : The max time-delay which can ensure the system asymptotical stability is 19.80 By decomposing the matrix A1 into following two parts: A11 ¼ À0:3 À0:3 À0:3 ; A12 ¼ À0:7 À0:7 À0:7 : The max time-delay which can ensure the system 0.1-stability is 2.01 By decomposing the matrix A1 into following two parts: A11 ¼ À0:4 À0:4 À0:4 ; A12 ¼ À0:6 À0:6 À0:6 : ð3:1Þ 379 P.T Nam / Applied Mathematics and Computation 214 (2009) 374–380 Table Comparison between results of our result and recent ones Method Year d¼0 d ¼ 0:1 d ¼ 0:3 d ¼ 0:5 Park et al [10] Park et al [11] Kwon et al [3] Kwon et al [4] Corollary 2.3 2005 2005 2008 2008 2009 0.99 1.33 3.40 3.40 19.80 – – – 1.70 2.01 – – – 1.09 1.25 – – – 0.85 0.96 Table Comparison between results of our result and recent ones Method Year Convergence rate d Mondie et al [5] Theorem 2.1 2005 2009 0.47 1.153 The max time-delay which can ensure the system 0.3-stability is 1.25 By decomposing the matrix A1 into following two parts: A11 ¼ À0:47 À0:47 À0:47 ; A12 ¼ À0:53 À0:53 À0:53 : The max time-delay which can ensure the system 0.5-stability is 0.96 From Table 1, it can be seen that Corollary 2.3 gives larger delay bounds than the recent results in [10,11,3,4] Example 3.2 Consider the following linear systems, which is considered in [5]: _ xtị ẳ 0:1 xt hị ỵ f t; xtịị ỵ f1 t; xt hịị; xtị ỵ 0:1 ð3:2Þ where kf ðt; xðtÞÞ 0:2kxðtÞk; kf1 ðt; xðt À hÞÞ 0:2kxðt À hÞk By decomposing the matrix A1 into following two parts: A11 ¼ 0; A12 ¼ A1: The max convergence rate is 1.153 From Table 2, we can see Theorem 2.1 gives larger convergence rate than the result in [5] Conclusion This paper proposed a new criterion for exponential stability of time delay systems with nonlinear uncertainties By combining the method of decomposing the matrix in [1] with the technique to reduced the stability of the operator in [6], we have improved and generalized some previous results Some numerical examples are also given to show the superiority of our results Acknowledgement The author would like to thank anonymous referees for valuable comments and suggestions, which have improved the paper This work was supported by the National Foundation for Science and Technology Development, Vietnam References [1] Jiuwen Cao, Shouming Zhong, New delay-dependent condition for absolute stability of Lurie control systems with multiple time-delays and nonlinearities, Appl Math Comput 194 (2007) 250–258 [2] O.M Kwon, J.H Park, Robust stabilization criterion for uncertain systems with delay in control input, Appl Math Comput 172 (2006) 1067– 1077 [3] O.M Kwon, J.H Park, On robust stability criterion for dynamic systems with time-varying delays and nonlinear perturbations, Appl Math Comput 203 (2008) 937–942 [4] O.M Kwon, J.H Park, Exponential stability for time-delay systems with interval time-varying delays and nonlinear perturbations, J Theory Optim Appl 139 (2) (2008) 277–293 [5] S Mondie, V.L Kharitonov, Exponential estimates for retarded time-delay systems: an LMI Approach, IEEE Trans Automat Control 50 (2) (2005) 268– 273 [6] P.T Nam, V.N Phat, Robust stabilization of linear systems with delayed state and control, J Theory Optim Appl 140 (2009) 287–299 [7] P.T Nam, An improved criterion for exponential stability of linear systems with multiple time delays, Appl Math Comput 202 (2008) 870–876 380 P.T Nam / Applied Mathematics and Computation 214 (2009) 374–380 [8] P Niamsup, K Mukdasai, V.N Phat, Improved exponential stability for time-varying systems with nonlinear delayed perturbations, Appl Math Comput 204 (2008) 490–495 [9] Ju H Park, Robust stabilization for linear dynamic systems with multiple time-varying delays and nonlinear uncertainties, J Theory Optim Appl 108 (1) (2001) 155–174 [10] Ju H Park, O.M Kwon, Novel stability criterion of time delay with nonlinear uncertainties, Appl Math Lett 18 (2005) 683–888 [11] Ju H Park, O.M Kwon, Matrix inequality approach to a novel stability criterion for time-delay systems with nonlinear uncertainties, J Optim Theory Appl 126 (3) (2005) 643–655 [12] Fengli Ren, Jinde Cao, Novel a-stability of linear systems with multiple time delays, Appl Math Comput 181 (2006) 282–290 ... a new criterion for exponential stability of time delay systems with nonlinear uncertainties By combining the method of decomposing the matrix in [1] with the technique to reduced the stability. .. time- varying delays and nonlinear perturbations, Appl Math Comput 203 (2008) 937–942 [4] O.M Kwon, J.H Park, Exponential stability for time- delay systems with interval time- varying delays and nonlinear. .. novel stability criterion for time- delay systems with nonlinear uncertainties, J Optim Theory Appl 126 (3) (2005) 643–655 [12] Fengli Ren, Jinde Cao, Novel a -stability of linear systems with multiple