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BIBO stabilization of time-delayed system with nonlinear perturbation
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Applied Mathematics and Computation 195 (2008) 264–269 www.elsevier.com/locate/amc
BIBO stabilization of time-delayed system with nonlinear perturbation Ping Li *, Shou-ming Zhong School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China
Abstract Time-delayed control system with the nonlinear perturbation is considered. Novel BIBO stabilization criteria are established by the Lyapunov functional and formulated in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. The robust quadratic stability for such systems is also discussed. A numerical example is given to illustrate the effectiveness of our result. 2007 Elsevier Inc. All rights reserved. Keywords: Lyapunov functional; Algebraic Riccati matrix equation; BIBO stabilization
1. Introduction As the sources of oscillations, instability and poor performance of the networks, time delays and nonlinear perturbation in many dynamical systems have attracted the interest of many researchers in recent years. A substantial part of the literature on control system has been devoted to stability analysis and stabilization; see the survey papers [1–6] and references therein. However, these results are mainly in allusion to the Lyapunov stability in general. On the other hand, it is quite useful to carry out the investigation for Bounded-Input Bounded-output (BIBO) stabilization in order to track out the reference input signal in real word, see [7–12] and some references therein. For instance, a self-pulsating laser, whose output is a periodic train of pulses, is nothing but a nonlinear system with the limit-cycle behavior. In order to apply the procedure for making lasers self-pulsate, it is necessary to establish the BIBO stability of the laser to get the rate equations representing the laser dynamics. However, the BIBO stabilization for control systems has not been further investigated and many valuable results have not been obtained. We note, in particular, the work of [7] who studied the problem of BIBO stabilization of multivariable feedback control systems by virtue of Lyapunov function and Bihari inequality.
*
Corresponding author. E-mail address: [email protected] (P. Li).
0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.081
P. Li, S.-m. Zhong / Applied Mathematics and Computation 195 (2008) 264–269
265
In fact, there is some strict limitation to the construction of Lyapunov function by this method, i.e., Lyapunov function only in the form of V(t) = xT(t)Px(t). Here, Pis a positive matrix. Additionally, the problem of BIBO stabilization for system with time-delay and nonlinear perturbations is dealt with in [8] through the use of method of the variation of parameters and Gronwall inequality. As we all know, the Lyapunov function is proposed as a powerful tool to obtain stability conditions for the stability of control systems. At last, we wish to establish sufficient BIBO stabilization condition by using the Lyapunov functional technique combined with the algebraic Riccati matrix equation in this paper. 2. System description and preliminaries Consider the control system with time-delay given by 8 0 > < x ðtÞ ¼ AxðtÞ þ Bxðt sÞ þ CuðtÞ; yðtÞ ¼ DxðtÞ; > : xðtÞ ¼ /ðtÞ 8t 2 ½t0 s; t0 : When the perturbation is introduced, it is presented by 8 0 > < x ðtÞ ¼ AxðtÞ þ Bxðt sÞ þ CuðtÞ þ f ðxðt sÞ; tÞ; yðtÞ ¼ DxðtÞ; > : xðtÞ ¼ /ðtÞ 8t 2 ½t0 s; t0 ;
ð1Þ
ð2Þ
where x(t), u(t), y(t) are the state vector, control input, control output of the system, respectively; s > 0 is the constant delay; /(Æ) is a continuous vector-valued initial function and k/ks is defined by k/ks ¼ sups6h60 k/ðt0 þ hÞk; Here kxk, kAk denote the standard Euclidean norm of vectorxand spectral norm of matrix A, respectively; A, B, C and D are constant matrices where A, B are not necessary Hurwitz matrices; f(x(t s), t) is the nonlinear vector-valued perturbation bounded in magnitude as kf ðxðt sÞ; tÞk 6 nkxðt sÞk
8t > 0;
ð3Þ
where n is a positive scalar. Throughout this paper, if not explicitly stated, matrices and vectors are assumed to have compatible dimensions. To obtain the control law described by (4) of tracking out the reference input of the system, we let uðtÞ ¼ KxðtÞ þ rðtÞ; where K is the feedback gain matrix, r(t) is the reference input. Substituting (4) to the above two systems respectively, we can rewrite them as ( x0 ðtÞ ¼ ðA þ CKÞxðtÞ þ Bxðt sÞ þ CrðtÞ yðtÞ ¼ DxðtÞ;
ð4Þ
ð5Þ
and
x0 ðtÞ ¼ ðA þ CKÞxðtÞ þ Bxðt sÞ þ CrðtÞ þ f ðxðt sÞ; tÞ yðtÞ ¼ DxðtÞ:
ð6Þ
Now, we introduce the following definitions for a precise formulation of our results. Definition 1 ([12]). A real-valued vector rðtÞ 2 Ln1 , if krk1 ¼ supt0 6t<1 krðtÞk < þ1. Definition 2 ([12]). The control system with reference input r(t) is BIBO stable if there exist some positive constants h1 and h2 satisfying kyðtÞk 6 h1 krk1 þ h2 for every reference input rðtÞ 2 Ln1 .
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3. BIBO stabilization In this section, we wish to design a linear feedback control u(t) = Kx(t) + r(t), such that the resulting closedloop systems (5) and (6) are BIBO stable. We first discuss the system with no perturbation. Theorem 1. The control system (5) with feedback gain matrix K = g1CTP is BIBO stable, if there exist some positive constants e, g1, g2 and positive matrix Q satisfying 1 g2 kmin ðQÞ ekPBk2 > 0; e where P is the symmetric positive solution of the Riccati equation AT P þ PA 2g1 PCC T P ¼ g2 Q:
ð8Þ
Proof. We define a Lyapunov functional V(t) as V ðtÞ ¼ xT ðtÞPxðtÞ þ ekPBk2
Z
t 2 kxðsÞk ds:
ts
The time derivative of V(t) (the right-hand derivative) along solutions to (5) is computed as V 0 ðtÞ ¼ xT ðtÞðAT P þ PA þ K T C T P þ PCKÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ 2
2
2
þ ekPBk kxðtÞk ekPBk kxðt sÞk
2
¼ xT ðtÞðAT P þ PA 2g1 PCC T P ÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ 2
2
2
2
þ ekPBk kxðtÞk ekPBk kxðt sÞk 1 2 2 T þ ekPBk kxðtÞk þ 2kPCkkrk1 kxðtÞk 6 g2 x ðtÞQxðtÞ þ e 1 2 2 6 ðg2 kmin ðQÞ ekPBk ÞkxðtÞk þ 2kPCkkrk1 kxðtÞk: e Let 1 2 a ¼ g2 kmin ðQÞ ekPBk ; e we have
b ¼ 2kPCkkrk1 ;
2
V 0 ðtÞ 6 akxðtÞk þ bkxðtÞk: Let 2
2
2
L1 ¼ ðkmax ðP Þ þ sekPBk Þk/ks ; L2 ¼ ðkmax ðP Þ þ sekPBk Þ
2 b : a
Now, we consider V ðtÞ 6 L ¼ L1 þ L2 : Indeed, under an assumption thatV(t) 6 V(t0) for all t P t0, we get 2
2
V ðtÞ 6 V ðt0 Þ 6 ðkmax ðP Þ þ sekPBk Þk/ks 6 L: If not, there exists t > t0, such that V(t) > V(s) for "s 2 [t0, t). Then by 2
0 6 V 0 ðtÞ 6 akxðtÞk þ bkxðtÞk; we have kxðtÞk 6 ba.
ð7Þ
P. Li, S.-m. Zhong / Applied Mathematics and Computation 195 (2008) 264–269
267
Thus, V ðtÞ 6 ðkmax ðP Þ þ sekPBk2 Þ
2 b 6L a
From kmin(P)jx(t)j2 6 V(t) 6 L, it is obvious that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ðkmax ðP Þ þ sekPBk Þðk/ks þ ðba Þ Þ kmax ðP Þ þ sekPBk b kxðtÞk 6 6 k/ks þ ; a kmin ðP Þ kmin ðP Þ So
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 kmax ðP Þ þ sekPBk kmax ðP Þ þ sekPBk b kyk 6 kDkkxðtÞk 6 kDk k/ks þ kDk : a kmin ðP Þ kmin ðP Þ
Let 2kDkkPCk h1 ¼ a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kmax ðP Þ þ sekPBk ; kmin ðP Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kmax ðP Þ þ sekPBk h2 ¼ kDk k/ks : kmin ðP Þ
We obtain jy(t)j 6 h1krk1 + h2 and the control system (5) with feedback gain matrix K = g1CTP is BIBO stable. Dealing with the similar method, we can establish a delay-independent criterion for the system (6). h Theorem 2. The control system described by (3) and (6) with feedback gain matrix K = g1CTP is BIBO stable, if there exist some positive constants e, #, g1, g2 and positive matrix Q satisfying 1 1 g2 kmin ðQÞ ekPBk2 #n2 kP k2 > 0 e # for every nonlinear vector-valued disturbance satisfying (3) and P is the symmetric positive solution of the Riccati equation AT P þ PA 2g1 PCC T P ¼ g2 Q: Proof. We define a Lyapunov functional V(t) as V ðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ; where V 1 ðtÞ ¼ xT ðtÞPxðtÞ Z t 7V 2 ðtÞ ¼ ðekPBk2 þ #n2 kP k2 ÞkxðsÞk2 ds: ts
The time derivative of V(t) (the right-hand derivative) along solutions to (6) is computed as V 01 ðtÞ ¼ xT ðtÞðAT P þ PA þ K T C T P þ PCKÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ þ 2xT ðtÞPf ðxðt sÞ; tÞ 1 6 xT ðtÞðAT P þ PA 2g1 PCC T P ÞxðtÞ þ kxðtÞk2 þ ekPBk2 kxðt sÞk2 e 1 2 2 2 þ 2kPCkkrk1 kxðtÞk þ kxðtÞk þ #n2 kP k kxðt sÞk # 2
2
2
2
2
2
V 02 ðtÞ ¼ ðekPBk þ #n2 kP k ÞkxðtÞk ðekPBk þ #n2 kP k Þkxðt sÞk :
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Thus, we have V 0 ðtÞ ¼ V 01 ðtÞ þ V 02 ðtÞ 6 xT ðtÞðAT P þ PA 2g1 PCC T P ÞxðtÞ þ
1 1 þ þ ekPBk2 þ #n2 kP k2 kxðtÞk2 þ 2kPCkkrk1 kxðtÞk e #
1 1 2 2 2 6 ðg2 kmin ðQÞ ekPBk #n2 kP k ÞkxðtÞk þ 2kPCkkrk1 kxðtÞk: e # Let 1 1 2 2 a ¼ g2 kmin ðQÞ ekPBk #n2 kP k e # b ¼ 2kPCkkrk1 2
2
2
L1 ¼ ðkmax ðP Þ þ sekPBk þ s#n2 kP k Þk/ks 2 b 2 2 2 L2 ¼ ðkmax ðP Þ þ sekPBk þ s#n kP k Þ : a The following proof runs as that of Theorem 1, and hence is omitted. This completes the proof.
h
Corollary 1. The systems (1) and (2) are also robust quadratic stable when all the conditions in Theorems 1 and 2 are satisfied, if r(t) = 0 in (4). 4. Illustrative example As an application of Theorem 1, we consider the system (5) with the following parameters 2 0 1 0 1 0 A¼ ; B¼ ; C¼ 1 1 1 1 s0 1 18 6 For g1 ¼ g2 ¼ 1 and Q ¼ , solving forP in the Riccati matrix Eq. (8) gives us 6 21 2:1117 0:2532 P¼ : 0:2532 2:7689 Therefore, the stabilizing feedback gain matrix is given by 2:1117 0:2532 K¼P ¼ : 0:2532 2:7689 Meanwhile, when 0.0830 < e < 0.7854, the inequality (8) is satisfied. So, the system (5) is BIBO stable. 5. Conclusions In this paper, sufficient condition of BIBO stabilization for time-delayed control system with the nonlinear perturbation is proposed. The key idea of our approach is to establish results in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. In particular, the system is robust quadratic stable when the reference input r(t) = 0 in (4). References [1] F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186 (1) (2007) 23–29. [2] M. Corless, F. Garofalo, L. Gilelmo, New results on composite control of singularly perturbed uncertain linear systems, Automatica 29 (2) (1993) 387–400. [3] S. Sen, K.B. Datta, Stability bounds of singularly perturbed systems, IEEE Trans. Automat. Control 38 (2) (1993) 302–304.
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[4] G. Garcia, J. Bernussou, H2 guaranteed cost control for singularly perturbed uncertain systems, IEEE Trans. Automat. Control 43 (9) (1998) 1323–1329. [5] P. Shi, V. Dragan, Asymptotic H1 control of singularly perturbed systems with parametric uncertainties, IEEE Trans. Automat. Control 44 (9) (1999) 1738–1742. [6] H. Singh, R.H. Brown, D.S. Naidu, J.A. Heinen, Robust stability of singularly perturbed state feedback systems using unified approach, IEE Proc. Control Theory Appl. 148 (5) (2001) 391–396. [7] D.Y. Xu, S.M. Zhong, The BIBO stabilization of multivariable feedback systems, J. UEST China 24 (1) (1995) 90–96. [8] K.C. Cao, S.M. Zhong, B.S. Liu, BIBO and robust stabilization for system with time-delay and nonlinear perturbations, J. UEST China 32 (6) (2003) 787–789. [9] Z.H. Guan, X.C. Wen, Y.Q. Liu, Variation of the parameters formula and the problem of BIBO for singular measure differential systems with impulse effect, Appl. Math. Comput. 60 (2–3) (1994) 153–169. [10] R.J. Partington, C. Bonnet, H1 and BIBO stabilization of delay systems of neutral type, Syst. Control Lett. 52 (3–4) (2004) 283–288. [11] S.M. Shahruz, N.A. Sakyaman, How to have narrow-stripe semiconductor lasers self-pulsate, Appl. Math. Comput. 130 (1) (2002) 11–27. [12] H. Wu, K. Mizukami, Robust stabilization of uncertain linear dynamical systems, Int. J. Syst. Sci. 24 (2) (1993) 265–276.
Applied Mathematics and Computation 195 (2008) 264–269 www.elsevier.com/locate/amc
BIBO stabilization of time-delayed system with nonlinear perturbation Ping Li *, Shou-ming Zhong School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China
Abstract Time-delayed control system with the nonlinear perturbation is considered. Novel BIBO stabilization criteria are established by the Lyapunov functional and formulated in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. The robust quadratic stability for such systems is also discussed. A numerical example is given to illustrate the effectiveness of our result. 2007 Elsevier Inc. All rights reserved. Keywords: Lyapunov functional; Algebraic Riccati matrix equation; BIBO stabilization
1. Introduction As the sources of oscillations, instability and poor performance of the networks, time delays and nonlinear perturbation in many dynamical systems have attracted the interest of many researchers in recent years. A substantial part of the literature on control system has been devoted to stability analysis and stabilization; see the survey papers [1–6] and references therein. However, these results are mainly in allusion to the Lyapunov stability in general. On the other hand, it is quite useful to carry out the investigation for Bounded-Input Bounded-output (BIBO) stabilization in order to track out the reference input signal in real word, see [7–12] and some references therein. For instance, a self-pulsating laser, whose output is a periodic train of pulses, is nothing but a nonlinear system with the limit-cycle behavior. In order to apply the procedure for making lasers self-pulsate, it is necessary to establish the BIBO stability of the laser to get the rate equations representing the laser dynamics. However, the BIBO stabilization for control systems has not been further investigated and many valuable results have not been obtained. We note, in particular, the work of [7] who studied the problem of BIBO stabilization of multivariable feedback control systems by virtue of Lyapunov function and Bihari inequality.
*
Corresponding author. E-mail address: [email protected] (P. Li).
0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.081
P. Li, S.-m. Zhong / Applied Mathematics and Computation 195 (2008) 264–269
265
In fact, there is some strict limitation to the construction of Lyapunov function by this method, i.e., Lyapunov function only in the form of V(t) = xT(t)Px(t). Here, Pis a positive matrix. Additionally, the problem of BIBO stabilization for system with time-delay and nonlinear perturbations is dealt with in [8] through the use of method of the variation of parameters and Gronwall inequality. As we all know, the Lyapunov function is proposed as a powerful tool to obtain stability conditions for the stability of control systems. At last, we wish to establish sufficient BIBO stabilization condition by using the Lyapunov functional technique combined with the algebraic Riccati matrix equation in this paper. 2. System description and preliminaries Consider the control system with time-delay given by 8 0 > < x ðtÞ ¼ AxðtÞ þ Bxðt sÞ þ CuðtÞ; yðtÞ ¼ DxðtÞ; > : xðtÞ ¼ /ðtÞ 8t 2 ½t0 s; t0 : When the perturbation is introduced, it is presented by 8 0 > < x ðtÞ ¼ AxðtÞ þ Bxðt sÞ þ CuðtÞ þ f ðxðt sÞ; tÞ; yðtÞ ¼ DxðtÞ; > : xðtÞ ¼ /ðtÞ 8t 2 ½t0 s; t0 ;
ð1Þ
ð2Þ
where x(t), u(t), y(t) are the state vector, control input, control output of the system, respectively; s > 0 is the constant delay; /(Æ) is a continuous vector-valued initial function and k/ks is defined by k/ks ¼ sups6h60 k/ðt0 þ hÞk; Here kxk, kAk denote the standard Euclidean norm of vectorxand spectral norm of matrix A, respectively; A, B, C and D are constant matrices where A, B are not necessary Hurwitz matrices; f(x(t s), t) is the nonlinear vector-valued perturbation bounded in magnitude as kf ðxðt sÞ; tÞk 6 nkxðt sÞk
8t > 0;
ð3Þ
where n is a positive scalar. Throughout this paper, if not explicitly stated, matrices and vectors are assumed to have compatible dimensions. To obtain the control law described by (4) of tracking out the reference input of the system, we let uðtÞ ¼ KxðtÞ þ rðtÞ; where K is the feedback gain matrix, r(t) is the reference input. Substituting (4) to the above two systems respectively, we can rewrite them as ( x0 ðtÞ ¼ ðA þ CKÞxðtÞ þ Bxðt sÞ þ CrðtÞ yðtÞ ¼ DxðtÞ;
ð4Þ
ð5Þ
and
x0 ðtÞ ¼ ðA þ CKÞxðtÞ þ Bxðt sÞ þ CrðtÞ þ f ðxðt sÞ; tÞ yðtÞ ¼ DxðtÞ:
ð6Þ
Now, we introduce the following definitions for a precise formulation of our results. Definition 1 ([12]). A real-valued vector rðtÞ 2 Ln1 , if krk1 ¼ supt0 6t<1 krðtÞk < þ1. Definition 2 ([12]). The control system with reference input r(t) is BIBO stable if there exist some positive constants h1 and h2 satisfying kyðtÞk 6 h1 krk1 þ h2 for every reference input rðtÞ 2 Ln1 .
266
P. Li, S.-m. Zhong / Applied Mathematics and Computation 195 (2008) 264–269
3. BIBO stabilization In this section, we wish to design a linear feedback control u(t) = Kx(t) + r(t), such that the resulting closedloop systems (5) and (6) are BIBO stable. We first discuss the system with no perturbation. Theorem 1. The control system (5) with feedback gain matrix K = g1CTP is BIBO stable, if there exist some positive constants e, g1, g2 and positive matrix Q satisfying 1 g2 kmin ðQÞ ekPBk2 > 0; e where P is the symmetric positive solution of the Riccati equation AT P þ PA 2g1 PCC T P ¼ g2 Q:
ð8Þ
Proof. We define a Lyapunov functional V(t) as V ðtÞ ¼ xT ðtÞPxðtÞ þ ekPBk2
Z
t 2 kxðsÞk ds:
ts
The time derivative of V(t) (the right-hand derivative) along solutions to (5) is computed as V 0 ðtÞ ¼ xT ðtÞðAT P þ PA þ K T C T P þ PCKÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ 2
2
2
þ ekPBk kxðtÞk ekPBk kxðt sÞk
2
¼ xT ðtÞðAT P þ PA 2g1 PCC T P ÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ 2
2
2
2
þ ekPBk kxðtÞk ekPBk kxðt sÞk 1 2 2 T þ ekPBk kxðtÞk þ 2kPCkkrk1 kxðtÞk 6 g2 x ðtÞQxðtÞ þ e 1 2 2 6 ðg2 kmin ðQÞ ekPBk ÞkxðtÞk þ 2kPCkkrk1 kxðtÞk: e Let 1 2 a ¼ g2 kmin ðQÞ ekPBk ; e we have
b ¼ 2kPCkkrk1 ;
2
V 0 ðtÞ 6 akxðtÞk þ bkxðtÞk: Let 2
2
2
L1 ¼ ðkmax ðP Þ þ sekPBk Þk/ks ; L2 ¼ ðkmax ðP Þ þ sekPBk Þ
2 b : a
Now, we consider V ðtÞ 6 L ¼ L1 þ L2 : Indeed, under an assumption thatV(t) 6 V(t0) for all t P t0, we get 2
2
V ðtÞ 6 V ðt0 Þ 6 ðkmax ðP Þ þ sekPBk Þk/ks 6 L: If not, there exists t > t0, such that V(t) > V(s) for "s 2 [t0, t). Then by 2
0 6 V 0 ðtÞ 6 akxðtÞk þ bkxðtÞk; we have kxðtÞk 6 ba.
ð7Þ
P. Li, S.-m. Zhong / Applied Mathematics and Computation 195 (2008) 264–269
267
Thus, V ðtÞ 6 ðkmax ðP Þ þ sekPBk2 Þ
2 b 6L a
From kmin(P)jx(t)j2 6 V(t) 6 L, it is obvious that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ðkmax ðP Þ þ sekPBk Þðk/ks þ ðba Þ Þ kmax ðP Þ þ sekPBk b kxðtÞk 6 6 k/ks þ ; a kmin ðP Þ kmin ðP Þ So
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 kmax ðP Þ þ sekPBk kmax ðP Þ þ sekPBk b kyk 6 kDkkxðtÞk 6 kDk k/ks þ kDk : a kmin ðP Þ kmin ðP Þ
Let 2kDkkPCk h1 ¼ a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kmax ðP Þ þ sekPBk ; kmin ðP Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kmax ðP Þ þ sekPBk h2 ¼ kDk k/ks : kmin ðP Þ
We obtain jy(t)j 6 h1krk1 + h2 and the control system (5) with feedback gain matrix K = g1CTP is BIBO stable. Dealing with the similar method, we can establish a delay-independent criterion for the system (6). h Theorem 2. The control system described by (3) and (6) with feedback gain matrix K = g1CTP is BIBO stable, if there exist some positive constants e, #, g1, g2 and positive matrix Q satisfying 1 1 g2 kmin ðQÞ ekPBk2 #n2 kP k2 > 0 e # for every nonlinear vector-valued disturbance satisfying (3) and P is the symmetric positive solution of the Riccati equation AT P þ PA 2g1 PCC T P ¼ g2 Q: Proof. We define a Lyapunov functional V(t) as V ðtÞ ¼ V 1 ðtÞ þ V 2 ðtÞ; where V 1 ðtÞ ¼ xT ðtÞPxðtÞ Z t 7V 2 ðtÞ ¼ ðekPBk2 þ #n2 kP k2 ÞkxðsÞk2 ds: ts
The time derivative of V(t) (the right-hand derivative) along solutions to (6) is computed as V 01 ðtÞ ¼ xT ðtÞðAT P þ PA þ K T C T P þ PCKÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ þ 2xT ðtÞPf ðxðt sÞ; tÞ 1 6 xT ðtÞðAT P þ PA 2g1 PCC T P ÞxðtÞ þ kxðtÞk2 þ ekPBk2 kxðt sÞk2 e 1 2 2 2 þ 2kPCkkrk1 kxðtÞk þ kxðtÞk þ #n2 kP k kxðt sÞk # 2
2
2
2
2
2
V 02 ðtÞ ¼ ðekPBk þ #n2 kP k ÞkxðtÞk ðekPBk þ #n2 kP k Þkxðt sÞk :
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Thus, we have V 0 ðtÞ ¼ V 01 ðtÞ þ V 02 ðtÞ 6 xT ðtÞðAT P þ PA 2g1 PCC T P ÞxðtÞ þ
1 1 þ þ ekPBk2 þ #n2 kP k2 kxðtÞk2 þ 2kPCkkrk1 kxðtÞk e #
1 1 2 2 2 6 ðg2 kmin ðQÞ ekPBk #n2 kP k ÞkxðtÞk þ 2kPCkkrk1 kxðtÞk: e # Let 1 1 2 2 a ¼ g2 kmin ðQÞ ekPBk #n2 kP k e # b ¼ 2kPCkkrk1 2
2
2
L1 ¼ ðkmax ðP Þ þ sekPBk þ s#n2 kP k Þk/ks 2 b 2 2 2 L2 ¼ ðkmax ðP Þ þ sekPBk þ s#n kP k Þ : a The following proof runs as that of Theorem 1, and hence is omitted. This completes the proof.
h
Corollary 1. The systems (1) and (2) are also robust quadratic stable when all the conditions in Theorems 1 and 2 are satisfied, if r(t) = 0 in (4). 4. Illustrative example As an application of Theorem 1, we consider the system (5) with the following parameters 2 0 1 0 1 0 A¼ ; B¼ ; C¼ 1 1 1 1 s0 1 18 6 For g1 ¼ g2 ¼ 1 and Q ¼ , solving forP in the Riccati matrix Eq. (8) gives us 6 21 2:1117 0:2532 P¼ : 0:2532 2:7689 Therefore, the stabilizing feedback gain matrix is given by 2:1117 0:2532 K¼P ¼ : 0:2532 2:7689 Meanwhile, when 0.0830 < e < 0.7854, the inequality (8) is satisfied. So, the system (5) is BIBO stable. 5. Conclusions In this paper, sufficient condition of BIBO stabilization for time-delayed control system with the nonlinear perturbation is proposed. The key idea of our approach is to establish results in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. In particular, the system is robust quadratic stable when the reference input r(t) = 0 in (4). References [1] F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186 (1) (2007) 23–29. [2] M. Corless, F. Garofalo, L. Gilelmo, New results on composite control of singularly perturbed uncertain linear systems, Automatica 29 (2) (1993) 387–400. [3] S. Sen, K.B. Datta, Stability bounds of singularly perturbed systems, IEEE Trans. Automat. Control 38 (2) (1993) 302–304.
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