Stability of neutral delay-diœerential systems with nonlinear perturbations

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1 International Journal of Systems Science, 000, volume 1, number 8, pages 961± 96 Stability of neutral delay-diœerential systems with nonlinear perturbations JU H. PARK{ SANGCHUL WON{ In this paper, the problem of the stability analysis for neutral delay-di erential systems with nonlinear perturbations is investigated. Using the L yapunov method, we present new su cient conditions for the stability, in terms of the linear matrix inequality (L MI), which can be easily solved by various convex optimization algorithms. Numerical examples are given to illustrate the application of the proposed method. 1. Introduction The stability analysis of neutral delay-di erential systems has received considerable attention over the decades. In the literature, various techniques have been utilized to develop criteria for asymptotic stability of these systems. They include the Lyapunov technique, characteristic equation approach, state trajectory approach. The developed stability criteria are often classi ed into two categories according to their dependence on the size of delays. Slemrod Infante (19), Hale et al. (1985), Kolmanovskii Myshkins (199), Hale Verdny, Lunel (199), Gopalsamy (199), Kuang et al. (1994), Li (1988), Hu Hu (1996) have presented su cient delay-independent conditions for the asymptotic stability of neutral delay-di erential systems. Brayton Willoughby (196) have also exploited the delay-dependent su cient condition. In addition Khusainov Yun kova (1988) extended the criteria to the neutral delay-di erential systems with nonlinear perturbations bounded in magnitude. However, most of the su cient conditions are expressed in terms of a matrix norm or matrix measure of the system matrices. Unfortunately, the matrix measure matrix norm operations usually make the criteria more conservative. In this paper, the problem of the stability analysis for neutral delay-di erential systems with nonlinear perturbations is investigated. The bound of the nonlinear perturbation is assumed as a function of states. Using the Lyapunov method, we present two new su - Revised 9 August Accepted 19 August { Department of Electrical Engineering, Pohang University of Science Technology, San 1, Hyoja Dong, Pohang 90-84, South Korea. won@postech.ac.kr cient conditions for asymptotic stability. One is a delayindependent criterion, the other is a delay-dependent one. Also, the derived su cient conditions are expressed in terms of LMIs to make the criteria less conservative. The solutions of the LMIs can be easily solved by various e ective optimization algorithms (Boyd et al. 1994). Two numerical examples are presented to show the applications of the proposed method. Notation R n denotes an n-dimensional Euclidean space, R n m is the set of all n m real matrices, I denotes the identity matrix of the appropriate order, k k refers to either the Euclidean vector norm or the induced matrix -norm, denotes the matrix measure of the corresponding matrix. The notation X Y (respectively, X > Y ), X Y are matrices of equal dimensions, means that the matrix X Y is positive semide nite (respectively, positive de nite).. Main results Consider a perturbed neutral delay-di erential system of the form _x t ˆ Ax t Bx t h C _x t h Q x t ;x t h ; _x t h ; 1 with the initial condition function x t ˆ t ; 8t ;0Š; x t R n is the state vector; A;B C R n n are constant matrices; h is a positive constant timedelay; t is the given continuously di erentiable function on ;0Š; Q x t ;x t h ; _x t h represents International Journal of Systems Science ISSN 000± 1 print/issn 1464± 519 online # 000 Taylor Francis Ltd

2 96 J. H. Park S. W on the nonlinear perturbations is assumed to satisfy the inequality kq x t ;x t h ; _x t h k µ 0 kx t k 1 kx t h k k _x t h k; 0 ; 1 are positive constants. For the de nition of asymptotic stability of neutral delay-di erential systems, refer to Hale Verduyn Lunel (199) Gopalsamy (199). The following lemmas are necessary to develop the new stability criteria. U 1 ˆ A T P PA R A T A 0å 1 ; U ˆ B T B R 1å 1 ; U ˆ C T C I å 1 å 1 ˆ 1 " 1 " " " 4 I: Lemma 1 (Khargonekar et al. 1990): L et D E be real matrices of appropriate dimensions. T hen, for any scalar " > 0, DE E T D T µ "DD T " 1 E T E: Lemma (Albert 1969): For the symmetric matrices Y R, the linear matrix inequality is equivalent to Y S S T R > 0 R > 0 Y SR 1 S T > 0: First we establish a delay-independent criterion for the asymptotic stability of the neutral delay-di erential system (1) using the Lyapunov method. Theorem 1: For the given scalars 0, 1, system 1 is asymptotically stable for all h > 0, if there exist n n symmetric positive de nite matrices P R, positive scalars " 1 ;" ;" " 4 satisfying the following L MI: U 1 P A T PB A T B P " 4 I 0 0 A 0 " 1 I 0 B T P B T A 0 0 U B 64 C T P C T A 0 0 C T B PC A B T B T C 0 " I U C T 0 C " I > 0; 4 5 Proof: Let the Lyapunov functional cidate be V ˆ x T t Px t _x T t s _x t s ds x T t s Rx t s ds: For simplicity, de ne W 1 ˆ W ˆ _x T t s _x t s ds x T t s Rx t s ds: 5 6 Then the time derivative of V along the solution of (1) is _ V ˆ x T A T P PA x x T PBx h x T PC _x h x T PQ _ W 1 _ W ; 8 x;x h _x h denote x t ; x t h _x t h, respectively. From (6) (), we obtain _ W 1 ˆ _x T _x _x T h _x h ˆ x T A T Ax x T h B T Bx h _x T h C T C _x h Q T Q _x T h _x h x T A T Bx h x T A T C _x h x T h B T C _x h x T A T Q x T h B T Q _x T h C T Q 9 _ W ˆ x T Rx x T h Rx h : 10 But by Lemma 1, the term Q T Q on the right-h side of (9) satis es the following inequalities

3 Q T Q ˆ kqk µ 0kxk 1kx h k k _x h k 0 1 kxkkx h k 1 kx h kk _x h k 0 kxkk _x h k µ 0kxk 1kx h k k _x h k 0kxk 1kx h k 1kx h k k _x h k 0kxk k _x h k ˆ 0x T x 1x T h x h _x T h _x h : 11 Also, by Lemma 1 inequality (11), the other terms of (8) (9) satisfy the following inequalities x T A T Q µ " 1 1 x T A T Ax " 1 Q T Q µ " 1 1 x T A T Ax " 1 0x T x 1x T h x h _x T h _x h 1 x T h B T Q µ " 1 x T h B T Bx h " 0x T x 1x T h x h _x T h _x h 1 _x T h C T Q µ " 1 _x T h C T C _x h " 0x T x 1x T h x h _x T h _x h 14 x T PQ µ " 1 4 x T PPx " 4 0x T x 1x T h x h _x T h _x h : 15 Using (9)± (15), we obtain a new bound of V_ as 0 A T 1 P PA 1 " 1 1 A T A B R " 1 C 4 A T x 0å 1 V _ 6 µ 4x h 5 _x B T P C T A h 6 64 C T P C T A Neutral delay-di erential systems with nonlinear perturbations 96 T x x ² 64 5 M 64 5 : 16 x h _x h x h _x h Therefore, V_ is negative if the matrix M is negative de nite. By Lemma, inequality (4) is equivalent to M < 0. This completes the proof. Remark 1: In the case there is no perturbation, i.e. Q x t ;x t h ; _x t h ˆ 0, then, the su cient condition (4) is simpli ed as A T P PA R A T A PB A T B PC A T C 64 B T P B T A B T B R B T C 5 < 0: C T P C T A C T B C T C I 1 This implies that A is a Hurwitz matrix. Li (1988) Hu Hu (1996) also derived su cient conditions for this case as kbk kak kck A < 0 f or kck < kck kcak kcbk A kbk < 0 1 kck f or kck < 1; 19 respectively. These conditions may be more conservative due to the matrix measure matrix norm operations. Note that (18) (19) imply that A < 0. Next, we develop a delay-dependent stability criterion for system (1). For simplicity, we de ne the matrix functions S 1 S as S 1 P;" 1 ;" ;" ;" 4 ;" 5 ;h PB A T B PC A T C x 4 6 x h 5 1 " 1 B R 1 P B T C _x h 1 C T 1 " 1 C T C 5 B I P 1 ˆ A T 0 P PA 0 ha T A hpbb T P " " 5 PP " 1 PBB T P " 1 A T A 0 1 " 1 4 " 1 5 I h 0 1 I 4 h B T B " 1 1 C T C 0 S " ;" ;" 4 ;" 5 ;h ˆ " 1 " " 4 C T C h I " 1 4 " 1 5 I I; 1

4 964 J. H. Park S. W on P is a symmetric positive de nite matrix, " 1 ;", " ;" 4, " 5 are positive scalars, A 0 ˆ A B. Note that S 1 S are monotonic non-decreasing with respect to h (in the sense of positive semi-de niteness). The following theorem gives a delay-dependent su cient condition for asymptotic stability of the system (1). Theorem : For the given scalars 0 ; 1 ; h, - the system is asymptotically stable for any constant time-delay 0 < h µ h - if there exist a symmetric positive de nite matrix X, positive scalars " 1 ;", ", " 4 " 5, satisfying the following L MIs: N XA T XA T XA T B X X X X XB T XB T XC T AX ¼I AX 0 1 I AX 0 0 " I B ¼ I X I X " 4 I < 0 X " 5 I X ¼ I BX I BX ¼I 0 5 CX " 1 I ¼ ˆ 1= - h. " " 4 C T C I C T I I I I C " I ¼ I 0 I I I 0 0 < 0 I " I 0 I " 5 5 I N ˆ XA T 0 A 0 X " 1 BB T " " 5 I ˆ 0 1 Proof: Without loss of generality, it is assumed that x t is continuously di erentiable on ; Š. Then, system (1) can be rewritten as _x t ˆ A B x t B _x s ds C _x t h Q ˆ A 0 x t B fax s Bx s h

5 Neutral delay-di erential systems with nonlinear perturbations 965 C _x s h Q g ds C _x t h Q ˆ A 0 x t B ² 1 ² ² BC _x s h ds C _x t h Q ˆ A 0 x t B ² 1 ² ² BCx t h BCx t h C _x t h Q ; 4 ² 1 ˆ ² ˆ ² ˆ Ax s ds; Bx s h ds; Q x s ;x s h ; _x s h ds: 5 Now let the Lyapunov functional cidate for system (4) be V ˆ x T t Px t _x T t s _x t s ds x T t s R 1 x t s ds x T t s R x t s ds kax s k ds kbx s h k ds kq x s ;x s h ; _x s h k ds d ; 6 P is the matrix given in (0), R 1 R are the positive semi-de nite matrices. For simplicity, de ne Z 1 ˆ Z ˆ _x T t s _x t s ds x T t s R 1 x t s ds 8 Z ˆ x T t s R x t s ds 9 Z 4 ˆ kax s k ds kbx s h k ds kq x s ;x s h ; _x s h k ds d : 84 Then, V_ ˆ x T A T 0 P PA 0 x x T PB ² 1 ² ² x T PBCx h x T PBCx h x T PC _x h x T PQ _Z 1 _Z _Z _Z 4 ; 1 x h ˆ x t h. From ()± (0), we have _Z 1 ˆ x T A T Ax x T h B T Bx h _x T h C T C _x h Q T Q _x T h _x h x T A T Bx h x T A T C _x h x T h B T C _x h x T A T Q x T h B T Q _x T h C T Q ; _Z ˆ x T R 1 x x T h R 1 x h ; _Z ˆ x T R x x T hr x h ; _Z 4 ˆ hkax t k hkbx t h k hkq x s ;x s h ; _x s h k kax s k ds kbx s h k ds kq x s ;x s h ; _x s h k ds: 4 5 By Lemma 1, the terms on the right-h side of (1) () satisfy the following inequalities: x T PB² 1 µ hx T PBB T Px h 1 ² T 1 ² 1 x T PB² µ hx T PBB T Px h 1 ² T ² x T PB² µ hx T PBB T Px h 1 ² T ² x T PBCx h µ " 1 x T PBB T Px " 1 1 x T h C T Cx h x T PBCx h µ " 1 x T PBB T Px " 1 1 x T hc T Cx h x T PC _x h µ " x T PPx " 1 _x T h C T C _x h x T A T Bx h µ x T A T Ax x T h B T Bx h x T A T C _x h µ " 1 x T A T Ax " _x T h C T C _x h x T h B T C _x h µ x T h B T Bx h _x T h C T C _x h : 6 Also, by inequality (11), the other terms of (1), () (5) satisfy the following inequalities x T A T Q µ x T A T Ax 0x T x 1x T h x h _x T h _x h x T h B T Q µ x T h B T Bx h 0x T x 1x T h x h _x T h _x h _x T h C T Q µ " 4 _x T h C T C _x h " 1 4 0x T x 1x T h x h _x T h _x h

6 966 J. H. Park S. W on x T PQ µ " 5 x T PPx " 1 5 0x T x 1x T h x h _x T h _x h hkq x s ;x s h ; _x s h k Furthermore, we have ² T 1 ² 1 µ µ h 0x T x 1x T h x h _x T h _x h : Ax s ds µ h kax s k ds µ ² T ² µ Bx s h ds µ h kbx s h k ds ² T ² µ kax s k ds Q x s ;x s h ; _x s h ds µ h kq x s ;x s h ; _x s h k ds; the Schwartz inequality is utilized. Using (11) ()± (40), we have _ V µ x T A T 0 P PA 0 R 1 R ha T A hpbb T P " " 5 PP " 1 PBB T P " 1 A T A 0å h 0IŠx _x T h " 1 " " 4 C T C h I å IŠ _x h x T h 4 h B T B " 1 1 C T C h 1I 1å R 1 Šx h x T h " 1 1 C T C R Šx h ; 41 å ˆ " 1 4 " 5 1 I. By choosing the matrices R 1 R as 4 h B T B " 1 1 C T C h 1I 1å " 1 1 C T C, respectively, (41) is simpli ed to _ V µ x T A T 0 P PA 0 ha T A hpbb T P " " 5 PP " 1 PBB T P " 1 A T A 0 1 å h 0 1 I 4B T B hb T B " 1 1 C T CŠx _x T h " 1 " " 4 C T C h I å IŠ _x h ˆ x T S 1 P;" 1 ;" ;" ;" 4 ;" 5 ;h x _x T h S " ;" ;" 4 ;" 5 ;h _x h µ x T S 1 P;" 1 ;" ;" ;" 4 ;" 5 ; - h x _x T h S " ;" ;" 4 ;" 5 ; - h _x h ˆ x T PP 1 S 1 P;" 1 ;" ;" ;" 4 ;" 5 ; - h P 1 Px _x T h S " ;" ;" 4 ;" 5 ; - h _x h ; 4 S 1 S are de ned in (0) (1). Therefore, V_ is negative if the following two inequalities are satis ed P 1 S 1 P;" 1 ;" ;" ;" 4 ;" 5 ; - h P 1 < 0; S " ;" ;" 4 ;" 5 ; - h < 0: 4 44 Let P 1 ˆ X. Substituting (0) (1) into (4) (44), respectively, we have XA T 0 A 0 X - hxa T AX - hbb T " " 5 I " 1 BB T " 1 XA T AX 0 1 Xå X - h 0 1 XX 4XB T BX - hxb T BX " 1 1 XC T CX < 0; 45 " 1 " " 4 C T C - h I å I < 0: 46 Then, by Lemma, inequalities (45) (46) are equivalent to () (), respectively. This completes the proof. Remark : Note that LMI () implies that the matrix A 0 ˆ A B is Hurwitz. Remark : In general, the delay-dependent criteria are expected to be less conservative than the delay-independent criteria when the time-delay is relatively small. Remark 4: The LMIs given in Theorems 1 can be solved e ciently using the interior point algorithm (Boyd et al. 1994). In addition, the maximum h - that satis es the su cient condition in Theorem can be easily found by minimizing ¼ subjected to X > 0; " 1 > 0; " > 0; " > 0; " 4 > 0; " 5 > 0; ¼ > 0, 6. Note that this optimization problem has the form of an eigenvalue problem. For details, refer to Boyd et al. (1994).. Numerical examples In this section, we present two examples. One is for the delay-independent criterion, the other is for the delaydependent criterion.

7 Neutral delay-di erential systems with nonlinear perturbations 96 Example 1: Consider the following system _x t ˆ Ax t Bx t h C _x t h Q x;x h ; _x h A ˆ 0:5 0 1 ; B ˆ 0: :4 ; C ˆ 0: :15 the nonlinear perturbation Q is bounded as kq x;x h ; _x h k µ 0:kx t k 0:1kx t h k 0:05k _x t h k; that is, 0 ˆ 0:, 1 ˆ 0:1 ˆ 0:05. Then, the solutions of the LMI given in (4) are found to be P ˆ R ˆ 1:1190 6:6901 6:6901 9:104 ; :4 :416 :416 5:90 ; " 1 ˆ :46; " ˆ 0:116; " ˆ 0:51; " 4 ˆ 16:4516: Therefore, by Theorem 1, the system is asymptotically stable. Now suppose that the perturbation is Q ˆ 0, then Hu s criterion Theorem 1 can be applied for stability analysis. Since kcak kcbk A kbk ˆ 0:19 > 0 1 kck One cannot determine the stability of the system using Hu s criterion. However, LMI (41) gives the solutions as P ˆ 1:941 0:04 0:04 :08 ; R ˆ 1:995 0:089 0:089 1:8081 : So, a system without perturbation is also asymptotically stable. Example : Consider the following system: " _x t ˆ 0 # " x t 0 0:4 # x t h 0 0:4 0 " 0:1 0 # _x t h 0 0:1 Q x;x h ; _x h ; Q is bounded as kq x;x h ; _x h k µ 0:1kx t k 0:05kx t h k 0:05k _x t h k: Then, in light of Remark 4, the solutions of the two LMIs (60) (6) are found to be " 1 ˆ 0:044; " ˆ 0:0188; " ˆ 10:9; " 4 ˆ 1:19; " 5 ˆ 0:049; ¼ ˆ 4:5914 " # 0:1 0:098 X ˆ ; 0:098 0:1 the maximum - h is - h ˆ 1=¼ ˆ 0:18: 4. Conclusions Based on the Lyapunov method, two stability criteria for linear neutral delay-di erential systems with nonlinear perturbations have been presented. They are expressed in terms of LMIs in order to make the criteria less conservative. The numerical examples show the e ectiveness of the proposed criteria. References ALBERT, A., 1969, Conditions for positive nonnegative de niteness in terms of pseudo inverses. SIAM Journal of Applied Mathematics, 1, 44± 440. BOYD, S., G HAOUI, L. EL., F ERON, E., BALAKRISHNAN, V., 1994, L inear Matrix Inequalities in Systems Control Theory (Philadelphia: SIAM). BRAYTON, R. K., WILLOUGHBY, R. A., 196, On the numerical integration of a symmetric system of di erence-di erential equations of neutral type. Journal of Mathematical Analysis Applications, 18, 18± 189. G OPALSAMY, K., 199, Stability Oscillations in Delay Di erential Equations of Population Dynamics (Boston: Kluwer Academic Publishers). HALE, J. K., INFANTE, E. F., TSEN, F.-S. P., 1985, Stability in linear delay equations. Journal of Mathematical Analysis Applications, 105, 5± 555. H ALE, J., VERDUYN LUNEL, S. M., 199, Introduction to Functional Di erential Equations (New York: Springer-Verlag). HU, G.-DI., H U, G.-DA, 1996, Some simple stability criteria of neutral delay-di erential systems. Applied Mathematics Computation, 80, 5± 1. K HARGONEKAR, P. P., PETERSEN, I. R., ZHOU, K., 1990, Robust stabilization of uncertain linear systems: Quadratic stability H 1 control theory. IEEE Transactions on Automatic Control, 5, 56± 61. K HUSAINOV, D. YA., YUN KOVA, E. V., 1988, Investigation of the stability of linear systems of neutral type by the Lyapunov function method. Di. Uravn, 4, 61± 61. K OLMANOVSKII, V., M YSHKIS, A., 199, Applied Theory of Functional Di erential Equations (Dordrecht: Kluwer Academic Publishers). K UANG, J. X., XIANG, J. X., TIAN, H. J., 1994, The asymptotic stability of one-parameter methods for neutral di erential equations. BIT, 4, 400± 408. LI, L. M., 1988, Stability of linear neutral delay-di erential systems. Bulletin of the Australian Mathematical Society, 8, 9± 44. SLEMROD, M., INFANTE, E. F., 19, Asymptotic stability criteria for linear systems of di erence-di erential equations of neutral type their discrete analogues. Journal of Mathematical Analysis Application, 8, 99± 415.