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Delay-dependent robust BIBO stabilization of uncertain system via LMI approach
Chaos, Solitons and Fractals 40 (2009) 1021–1028 www.elsevier.com/locate/chaos
Delay-dependent robust BIBO stabilization of uncertain system via LMI approach Ping Li a
a,* ,
Shou-ming Zhong a, Jin-zhong Cui
b
School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China b School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China Accepted 29 August 2007
Abstract This paper addresses the problem of robust BIBO stabilization for the uncertain time-delay system. A novel delaydependent stabilizable criterion is presented by a quadratic Lyapunov function and the method of the variation of parameters to guarantee that bounded input can lead to bounded output. The proposed design condition is formulated in terms of linear matrix inequality (LMI) which can be easily solved by LMI Toolbox in Matlab. Finally, numerical examples are given to illustrate the effectiveness of our results. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Stability is one of the most important problems in the synthesis of control systems. In the context of time-delay system, the issues of stability analysis and controller design have been under investigation in the last decades. Undoubtedly, a systematic way to study the stability and control problems for the time-delay system is provided by the use of Lyapunov method; see the papers [1–5] and the references cited therein. It is worth mentioning that most of these issues are mainly in Lyapunov meaning. In this paper, we shall study the stability analysis of delayed system from an input–output point of view. By this we mean that it is possible for some bounded input to lead to unbounded output. Many valuable results have appeared in these areas which guarantee the bounded-input bounded-output (BIBO) stability of control systems, in particular, the useful but unstable bilinear systems through equations involving the zeros of the systems; see the papers [6–8]. In fact, in order to track out the reference input signal in real word, the BIBO stability analysis of delay-differential system has become an important topic of theoretical studies; see the papers [6–14]. However, to the best of our knowledge, there are very few reports on the problem of BIBO stabilization of delayed systems. Therefore, it is important and necessary to carry out the investigation of BIBO stabilization. In [9,10], the sufficient conditions to guarantee the BIBO stabilization of large-scale system are derived by applying a stabilizing local state feedback. Every subsystem is required to be uniformly completely controllable to ensure the existence of the solution to certain *
Corresponding author. E-mail address: [email protected] (P. Li).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.059
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P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
Riccati equation by virtue of Lyapunov function and Bihari inequality. However, the delayed cases are not discussed. In [11,12], BIBO stabilization criteria for time-delay system are derived by the Lyapunov functional and given in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. In [13], frequency-domain tests for the H1 and BIBO stability of large classes of delay systems of neutral type are derived by the transfer function. As we all know, over the past results on time-delay systems, LMI approach is an efficient method to solve many control problems. Then, compared with the existing results on the analysis of BIBO stabilization for timedelay system, the work of our paper has one feature, i.e., the BIBO stabilization criterion is given in terms of LMI which can be easily solved by LMI Toolbox in Matlab. At last, the issue of robust BIBO stabilization of uncertain time-delay system is considered in this paper since uncertainties frequently occurs in the various control system due to modelling errors, measurement errors, linearization approximations, and so on; see the papers [16– 18].
2. System description and preliminaries We consider the uncertain delayed system described by 8 0 > < x ðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðt sÞ þ ðC þ DCðtÞÞuðtÞ yðtÞ ¼ ðG þ DGðtÞÞxðtÞ > : xðtÞ ¼ uðtÞ t0 s 6 t 6 t0
ð1Þ
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; u(Æ) is a continuous vector-valued initial function and kuks is defined by kuks ¼ sup kuðt0 þ hÞk. Here kxk, kAk s6h60
denote the standard Euclidean norm of vector x and spectral norm of matrix A, respectively; A, B, C and G are constant matrices where A, B are not necessary Hurwitz matrices, and DA(t), DB(t), DC(t), DG(t) are time-varying uncertain matrices subject to the following forms DAðtÞ ¼ D1 R1 ðtÞE1 ;
DBðtÞ ¼ D2 R2 ðtÞE2 ;
DCðtÞ ¼ D3 R3 ðtÞE3 ;
DGðtÞ ¼ D4 R4 ðtÞE4
ð2Þ
where Di and Ei, i = 1, 2, 3, 4 are known constant real matrices, and Ri(t), i = 1, 2, 3, 4 are unknown time-varying matrices satisfying RTi ðtÞRi ðtÞ 6 I;
i ¼ 1; 2; 3; 4
ð3Þ
Throughout this paper, if not explicitly stated, matrices and vectors are assumed to have compatible dimensions. Let u(t) be linear gain local state feedback with the reference input r(t) for Eq. (1), uðtÞ ¼ KxðtÞ þ rðtÞ
ð4Þ
so as to ensure stability of the closed-loop delayed system. Now, we introduce the following definitions and lemma for a precise formulation of our results. Definition 1 [15]. A real-valued vector rðtÞ 2 Ln1 , if krk1 ¼ sup krðtÞk < þ1. t0 6t<1
Definition 2 [15]. The control system (1) is BIBO stabilization by the local control law (4) if every solution of the system (1) y(t) satisfies kyðtÞk 6 h1 krk1 þ h2 where h1, h2 are positive constants for every reference input rðtÞ 2 Ln1 . Lemma 1. Suppose that the absolutely continuous function x(t) satisfies Z t f ðsÞxðsÞds ðt0 6 t 6 bÞ xðtÞ 6 gðtÞ þ t0
with gðtÞ : ½t0 ; b ! Rþ absolutely continuous and f ðtÞ : ½t0 ; b ! Rþ integrable then Rt Z t Rt f ðlÞdl f ðlÞdl þ g0 ðsÞe s ds ðt0 6 t 6 bÞ xðtÞ 6 gðt0 Þe t0 t0
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
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3. BIBO stabilization of time-delay system In this section, we wish to design the linear feedback control u(t) = Kx(t) + r(t), such that the resulting closed-loop system (1) with DA(t) = DB(t) = DC(t) = DG(t) = 0 is BIBO stabilizable. Substituting (4) to the system (1) with DA(t) = DB(t) = DC(t) = DG(t) = 0 yields a closed-loop system as follows: 8 0 > < x ðtÞ ¼ ðA þ CKÞxðtÞ þ Bxðt sÞ þ CrðtÞ yðtÞ ¼ GxðtÞ > : xðtÞ ¼ uðtÞ t0 s 6 t 6 t0
ð5Þ
Theorem 1. The time-delay system (5) with feedback gain matrix K = gCTR1 by the local control law (4) is BIBO stabilizable, if there exist some constants g > 0, b1 > 0, b2 > 0, b3 > 0 and matrix R > 0, such that b1 > b2 eb1 s and the linear matrix inequality 1 0 BR C AR þ RAT 2gCC T þ b1 R C B ð6Þ W¼@ RBT b2 R 0 A60 T C 0 b3 I holds. Proof. We define a Lyapunov function V(t) as V ðtÞ ¼ xT ðtÞPxðtÞ where P = R1Taking the time derivative of V(t) along the trajectory of system (5) gives that V 0 ðtÞ ¼ xT ðtÞðAT P þ K T C T P þ PA þ PCKÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ 1 1T 0 T 10 0 xðtÞ xðtÞ A P þ K T C T P þ PA þ PCK þ b1 P PB PC C C B CB B ¼ @ xðt sÞ A @ BT P b2 P 0 A@ xðt sÞ A rðtÞ rðtÞ CTP 0 b3 I b1 xT ðtÞPxðtÞ þ b2 xT ðt sÞPxðt sÞ þ b3 rT ðtÞrðtÞ 1T 0 T xðtÞ A P þ PA 2gPCC T P þ b1 P PB C B B ¼ @ xðt sÞ A @ BT P b2 P rðtÞ 0 CT P 0
1 10 xðtÞ PC C CB 0 A@ xðt sÞ A rðtÞ b3 I
b1 xT ðtÞPxðtÞ þ b2 xT ðt sÞPxðt sÞ þ b3 rT ðtÞrðtÞ Pre- and post-multiplying (6) by diag[R1, R1, I], we can obtain 1 0 T A P þ PA 2gPCC T P þ b1 P PB PC C ~ ¼B W BT P @ b2 P 0 A60 T C P 0 b3 I Then, V 0 ðtÞ 6 b1 V ðtÞ þ b2 V ðt sÞ þ b3 rT ðtÞrðtÞ Using the method of the variation of parameters, we get Z t Z t eb1 ðtsÞ V ðs sÞds þ b3 eb1 ðtsÞ krðsÞk2 ds V ðtÞ 6 V ðt0 Þeb1 ðtt0 Þ þ b2 t0
6
kP kkuk2s eb1 ðtt0 Þ
þ b2
Z
t0 ts b1 ðtssÞ
e t0 s Z t0
V ðsÞds þ b3 krk21
Z t0
t
eb1 ðtsÞ ds
Z t b eb1 ðtssÞ V ðsÞds þ b2 eb1 s eb1 ðtsÞ V ðsÞds þ 3 krk21 6 kP kkuk2s eb1 ðtt0 Þ þ b2 b1 t s t0 0 Z t b b 6 kP kkuk2s 1 þ 2 eb1 s eb1 ðtt0 Þ þ 3 krk21 þ b2 eb1 s eb1 ðtsÞ V ðsÞds b1 b1 t0
1024
So,
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
Z t b b eb1 s V ðsÞds V ðtÞeb1 t 6 kP kkuk2s 1 þ 2 eb1 s eb1 t0 þ 3 krk21 eb1 t þ b2 eb1 s b1 b1 t0
In view of Lemma 1, we can obtain Z t b eb1 s b b s b s 2 þ 3 krk21 eb1 t0 eb2 e 1 ðtt0 Þ þ b3 krk21 eb1 s eb2 e 1 ðtsÞ ds V ðtÞeb1 t 6 kP kkuks 1 þ 2 b1 b1 t0 Z t c b þ 3 krk21 eb1 t0 þcðtt0 Þ þ b3 krk21 ectþðb1 cÞs ds ¼ kP kkuk2s 1 þ b1 b1 t0 c b eðb1 cÞt eðb1 cÞt0 þ 3 krk21 eb1 t0 þcðtt0 Þ þ b3 krk21 ect ¼ kP kkuk2s 1 þ b1 b1 c b1 2 c b b krk þ 3 krk21 eb1 t0 þcðtt0 Þ þ 3 1 eb1 t 6 kP kkuk2s 1 þ b1 b1 c b1 where c ¼ b2 eb1 s By b1 > c ¼ b2 eb1 s , we have c b b krk2 V ðtÞ 6 kP kkuk2s 1 þ þ 3 krk21 eðb1 cÞðtt0 Þ þ 3 1 b1 b1 c b1 c ð2b1 cÞb3 2 2 þ 6 kP kkuks 1 þ krk1 b1 b1 ðb1 cÞ From kmin(P)kx(t)k2 6 V(t), it is obvious that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP kkuk2s b ð2b cÞkrk21 þ 3 1 kxðtÞk 6 b1 kmin ðP Þ b1 ðb1 cÞkmin ðP Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP k b3 ð2b1 cÞ 6 kuks þ krk b1 kmin ðP Þ b1 ðb1 cÞkmin ðP Þ 1 So, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP k b3 ð2b1 cÞ kuks þ kGk krk ¼ h1 krk1 þ h2 kyk 6 kGkkxðtÞk 6 kGk b1 kmin ðP Þ b1 ðb1 cÞkmin ðP Þ 1 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b3 ð2b1 cÞ ; h1 ¼ kGk b1 ðb1 cÞkmin ðP Þ This completes the proof.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP k h2 ¼ kGk kuks b1 kmin ðP Þ
h
4. Robust BIBO stabilization of uncertain time-delay system Based on Theorem 1, we can perform the robust BIBO stability analysis for system (1) with uncertainties (2) and (3). Substituting (4) to the system (1), we can rewrite it as 8 0 > < x ðtÞ ¼ ðA þ DAðtÞ þ CK þ DCðtÞK ÞxðtÞ þ ðB þ DBðtÞÞxðt sÞ þ ðC þ DCðtÞÞrðtÞ yðtÞ ¼ ðG þ DGðtÞÞxðtÞ ð7Þ > : xðtÞ ¼ uðtÞ t0 s 6 t 6 t0 Theorem 2. The time-delay system (7) with feedback gain matrix K = g 0 CTH1 by the local control law (4) is robust BIBO stabilizable, if there exist some constants g 0 > 0, e1 > 0, e2 > 0, e3 > 0, e4 > 0, e5 > 0, e6 > 0, e7 > 0 and matrix H > 0, such that x1 > x2 ex1 s and the linear matrix inequality
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
0 B B B B B B N¼B B B B B @
ð1; 1Þ BH HBT e2 H CT 0 DT1 0 DT2 DT3 DT3
0 0 0
C 0 e3 I 0
D1 0 0 e4 I
D2 0 0 0
D3 0 0 0
0 0 0
0 0 0
e5 I 0 0
0 e6 I 0
D3 0 0 0
1025
1
C C C C C C C60 C 0 C C C 0 A e7 I
ð8Þ
holds, where ð1; 1Þ ¼ HAT þ AH 2g0 CC T þ e1 H x 1 ¼ e1
e4 kE1 k2 þ e7 kE3 Kk2 >0 kmin ðP Þ
x 2 ¼ e2 þ
e5 kE2 k2 kmin ðP Þ
Proof. We define the Lyapunov function V(t) as V ðtÞ ¼ xT ðtÞPxðtÞ where P = H1Taking the time derivative of V(t) along the trajectory of system (7) gives that V 0 ðtÞ ¼ xT ðtÞððA þ DAÞT P þ K T C T P þ P ðA þ DAÞ þ PCK þ K T DC T P þ P DCKÞxðtÞ þ 2xT ðtÞP ðB þ DBÞxðt sÞ þ 2xT ðtÞP ðC þ DCÞrðtÞ ¼ vT N0 v e1 xT ðtÞPxðtÞ þ e2 xT ðt sÞPxðt sÞ þ e3 rT ðtÞrðtÞ þ e4 xT ðtÞET1 RT1 ðtÞR1 ðtÞE1 xðtÞ þ e5 xT ðt sÞET2 RT2 ðtÞR2 ðtÞE2 xðt sÞ þ e6 rT ðtÞET3 RT3 ðtÞR3 ðtÞE3 rðtÞ þ e7 xT ðtÞK T ET3 RT3 ðtÞR3 ðtÞE3 KxðtÞ where 0
ð1; 1Þ0 B T B B P B T B C P B B 0 N ¼ B DT1 P B T B D P B 2 B T @ D3 P DT3 P
PB e2 P 0 0 0
PC 0 e3 I 0 0
PD1 0 0 e4 I 0
PD2 0 0 0 e5 I
PD3 0 0 0 0
0 0
0 0
0 0
0 0
e6 I 0
PD3 0 0 0 0
1
C C C C C C C C C C C 0 A e7 I
where ð1; 1Þ0 ¼ AT P þ PA 2g0 PCC T P þ e1 P vT ¼ xT ðtÞ xT ðt sÞ rT ðtÞ xT ðtÞET1 RT1 ðtÞ
xT ðt sÞET2 RT2 ðtÞ
rT ðtÞET3 RT3 ðtÞ g0 xT ðtÞPCET3 RT3 ðtÞ
Pre- and post-multiplying (8) by diag[H1, H1, I, I, I, I, I], we can obtain V 0 ðtÞ 6 e1 xT ðtÞPxðtÞ þ e2 xT ðt sÞPxðt sÞ þ e3 rT ðtÞrðtÞ þ e4 kE1 k2 xT ðtÞxðtÞ þ e5 kE2 k2 xT ðt sÞxðt sÞ þ e6 kE3 k2 rT ðtÞrðtÞ þ e7 kE3 Kk2 xT ðtÞxðtÞ ¼ e1 V ðtÞ þ e2 V ðt sÞ þ ðe3 þ e6 kE3 k2 ÞrT ðtÞrðtÞ þ ðe4 kE1 k2 þ e7 kE3 Kk2 ÞxT ðtÞxðtÞ þ e5 kE2 k2 xT ðt sÞxðt sÞ e4 kE1 k2 þ e7 kE3 Kk2 e5 kE2 k2 6 e1 V ðtÞ þ e2 V ðt sÞ þ ðe3 þ e6 kE3 k2 ÞrT ðtÞrðtÞ þ V ðtÞ þ V ðt sÞ kmin ðP Þ kmin ðP Þ ! ! e4 kE1 k2 þ e7 kE3 Kk2 e5 kE2 k2 V ðtÞ þ e2 þ V ðt sÞ þ ðe3 þ e6 kE3 k2 ÞrT ðtÞrðtÞ ¼ e1 kmin ðP Þ kmin ðP Þ
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Let x 1 ¼ e1
e4 kE1 k2 þ e7 kE3 Kk2 ; kmin ðP Þ
x 2 ¼ e2 þ
e5 kE2 k2 ; kmin ðP Þ
x3 ¼ e3 þ e6 kE3 k2
Then, V 0 ðtÞ 6 x1 V ðtÞ þ x2 V ðt sÞ þ x3 rT ðtÞrðtÞ The remaining part of the proof is similar to that for Theorem 1, i.e., c x3 ð2x1 cÞ 2 V ðtÞ 6 kP kkuks 1 þ krk21 þ x1 x1 ðx1 cÞ where c ¼ x2 ex1 s So, kyk 6 kG þ DGðtÞkkxðtÞk 6 ðkGk þ kD4 kkE4 kÞkxðtÞk ¼ h1 krk1 þ h2 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3 ð2x1 cÞ h1 ¼ ðkGk þ kD4 kkE4 kÞ x1 ðx1 cÞkmin ðP Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðx1 þ cÞkP k h2 ¼ ðkGk þ kD4 kkE4 kÞ kuks x1 kmin ðP Þ This completes the proof.
h
Remark 1. In the work of Li et al. [11], the BIBO stabilization of system is analyzed by the Lyapunov functional and formulated in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. Obviously, if we let Ri(t) = 0, i = 1, 2, 3, 4, system (1) is reduced to the one considered in [11]. Moreover, our result is delaydependent and given in terms of LMI which can be easily solved by LMI Toolbox in Matlab. Then, the feasibility of our results can be checked at first hand without tuning any parameters. Remark 2. The criteria given in Theorems 1 and 2 are delay-dependent. Note also that the criterion ensuring BIBO stabilization given in [12] is delay-dependent and delay-independent with respect to distributed delay, discrete time delay, respectively. If the delayed system in [12] is reduced to one independent of distributed delay, we can see that our results are much less conservative than those in [11,12].
5. Illustrative examples Example 1. As an application of Theorem 1, we consider the system (5) with the following parameters: 4 1 2 0 2 1 A¼ ; B¼ ; C¼ 1 4 0 1 1 0 For b1 = 0.8, b2 = 0.4, b3 = 0.5, solving for g and R in LMI (6) gives us 0:2134 0:0043 g ¼ 4:2429; P ¼ R1 ¼ 0:0043 0:1157 Therefore, the stabilizing feedback gain matrix is given by 1:8291 0:5274 K¼ 0:9054 0:0182
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
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Table 1 smax for e1 = 8, e3 = e4 = e5 = e6 = e7 = 0.5 smax
e2 = 0.5
e2 = 1
e2 = 2
e2 = 4
0.2960
0.2349
0.1604
0.0492
Meanwhile, we obtain the maximum value smax = 0.8664. Example 2. As an application of Theorem 2, we consider the system (7) with the following parameters: 1 2 1 2 1 3 A¼ ; B¼ ; C¼ 0 4 0 1 1 0 and Di = I, i = 1, 2, 3, 4, E1 = E2 = 0.1I, E3 = 0.1K1. For e1 = 8, e2 = e3 = e4 = e5 = e6 = e7 = 0.5, solving for g 0 and H in LMI (8) gives us 0:0181 0:0406 g0 ¼ 36:2862; P ¼ H 1 ¼ 0:0406 0:5238 Therefore, the stabilizing feedback gain matrix is given by 0:8164 17:5335 K¼ 1:9703 4:4197 Meanwhile, we obtain the maximum value smax = 0.2960. In Table 1, the maximum value smax with certain ei is given.
6. Conclusions The main contribution of this letter is the result that ensures the BIBO stabilization for time-delay control system. By Lyapunov function and the method of the variation of parameters, the BIBO stabilization criterion is delay-dependent and expressed in terms of LMI. To this end, the robust BIBO stabilization for such uncertain system is studied. The method proposed first in this paper can be extended to the problems of BIBO stabilization for neutral system or other dynamical systems.
References [1] Kharitonov VL, Plischke E. Lyapunov matrices for time-delay systems. Syst Control Lett 2006;55:697–706. [2] Gao HJ, Lam J, Wang ZD. Discrete bilinear stochastic systems with time-varying delay: Stability analysis and control synthesis. Chaos, Solitons & Fractals 2007;34:394–404. [3] Phat VN, Niamsup P. Stability of linear time-varying delay systems and applications to control problems. J Comput Appl Math 2006;194:343–56. [4] Xu SY, Chen TW. Robust H1 control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers. Syst Control Lett 2004;51:171–83. [5] Liu XZ. Stability of impulsive control systems with time delay. Math Comput Modelling 2004;39:511–9. [6] Kotsios S. A note on BIBO stability of bilinear systems. J Franklin Inst 1995;332(6):755–60. [7] Bose T, Chen MQ. BIBO stability of the discrete bilinear system. Digital Signal Process 1995;5:160–6. [8] Fornasini E, Valcher ME. On some connections between bilinear input/output maps and 2D systems. Nonlinear Anal 1997;30:1995–2005. [9] Xu DY, Zhong SM. The BIBO stabilization of multivariable feedback systems. J UEST China 1995;24:90–6. [10] Xu DY, Zhong SM. BIBO stabilization of large-scale systems. Control Theory Appl 1995;12:758–63. [11] Li P, Zhong SM. BIBO stabilization of time-delayed system with nonlinear perturbation. Appl Math Comput 2007. doi:10.1016/ j.amc.2007.04.081. [12] Li P, Zhong SM. BIBO stabilization for system with multiple mixed delays and nonlinear perturbations. Appl Math Comput 2007. doi:10.1016/j.amc.2007.05.046. [13] Partington JR, Bonnet C. H1 and BIBO stabilization of delay systems of neutral type. Syst Control Lett 2004;52:283–8.
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[14] Shahruz SM, Sakyaman NA. How to have narrow-stripe semiconductor lasers self-pulsate. Appl Math Comput 2002;130:11–27. [15] Wu H, Mizukami K. Robust stabilization of uncertain linear dynamical systems. Int J Systems Sci 1993;24:265–76. [16] Corless M, Garofalo F, Gilelmo L. New results on composite control of singularly perturbed uncertain linear systems. Automatica 1993;29:387–400. [17] Lien CH. Delay-dependent and delay-independent guaranteed cost control for uncertain neutral systems with time-varying delays via LMI approach. Chaos, Solitons & Fractals 2007;33:1017–27. [18] Shi P, Dragan V. Asymptotic H1 control of singularly perturbed systems with parametric uncertainties. IEEE Trans Autom Control 1999;44(9):1738–42.
Delay-dependent robust BIBO stabilization of uncertain system via LMI approach Ping Li a
a,* ,
Shou-ming Zhong a, Jin-zhong Cui
b
School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China b School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China Accepted 29 August 2007
Abstract This paper addresses the problem of robust BIBO stabilization for the uncertain time-delay system. A novel delaydependent stabilizable criterion is presented by a quadratic Lyapunov function and the method of the variation of parameters to guarantee that bounded input can lead to bounded output. The proposed design condition is formulated in terms of linear matrix inequality (LMI) which can be easily solved by LMI Toolbox in Matlab. Finally, numerical examples are given to illustrate the effectiveness of our results. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Stability is one of the most important problems in the synthesis of control systems. In the context of time-delay system, the issues of stability analysis and controller design have been under investigation in the last decades. Undoubtedly, a systematic way to study the stability and control problems for the time-delay system is provided by the use of Lyapunov method; see the papers [1–5] and the references cited therein. It is worth mentioning that most of these issues are mainly in Lyapunov meaning. In this paper, we shall study the stability analysis of delayed system from an input–output point of view. By this we mean that it is possible for some bounded input to lead to unbounded output. Many valuable results have appeared in these areas which guarantee the bounded-input bounded-output (BIBO) stability of control systems, in particular, the useful but unstable bilinear systems through equations involving the zeros of the systems; see the papers [6–8]. In fact, in order to track out the reference input signal in real word, the BIBO stability analysis of delay-differential system has become an important topic of theoretical studies; see the papers [6–14]. However, to the best of our knowledge, there are very few reports on the problem of BIBO stabilization of delayed systems. Therefore, it is important and necessary to carry out the investigation of BIBO stabilization. In [9,10], the sufficient conditions to guarantee the BIBO stabilization of large-scale system are derived by applying a stabilizing local state feedback. Every subsystem is required to be uniformly completely controllable to ensure the existence of the solution to certain *
Corresponding author. E-mail address: [email protected] (P. Li).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.059
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Riccati equation by virtue of Lyapunov function and Bihari inequality. However, the delayed cases are not discussed. In [11,12], BIBO stabilization criteria for time-delay system are derived by the Lyapunov functional and given in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. In [13], frequency-domain tests for the H1 and BIBO stability of large classes of delay systems of neutral type are derived by the transfer function. As we all know, over the past results on time-delay systems, LMI approach is an efficient method to solve many control problems. Then, compared with the existing results on the analysis of BIBO stabilization for timedelay system, the work of our paper has one feature, i.e., the BIBO stabilization criterion is given in terms of LMI which can be easily solved by LMI Toolbox in Matlab. At last, the issue of robust BIBO stabilization of uncertain time-delay system is considered in this paper since uncertainties frequently occurs in the various control system due to modelling errors, measurement errors, linearization approximations, and so on; see the papers [16– 18].
2. System description and preliminaries We consider the uncertain delayed system described by 8 0 > < x ðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðB þ DBðtÞÞxðt sÞ þ ðC þ DCðtÞÞuðtÞ yðtÞ ¼ ðG þ DGðtÞÞxðtÞ > : xðtÞ ¼ uðtÞ t0 s 6 t 6 t0
ð1Þ
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; u(Æ) is a continuous vector-valued initial function and kuks is defined by kuks ¼ sup kuðt0 þ hÞk. Here kxk, kAk s6h60
denote the standard Euclidean norm of vector x and spectral norm of matrix A, respectively; A, B, C and G are constant matrices where A, B are not necessary Hurwitz matrices, and DA(t), DB(t), DC(t), DG(t) are time-varying uncertain matrices subject to the following forms DAðtÞ ¼ D1 R1 ðtÞE1 ;
DBðtÞ ¼ D2 R2 ðtÞE2 ;
DCðtÞ ¼ D3 R3 ðtÞE3 ;
DGðtÞ ¼ D4 R4 ðtÞE4
ð2Þ
where Di and Ei, i = 1, 2, 3, 4 are known constant real matrices, and Ri(t), i = 1, 2, 3, 4 are unknown time-varying matrices satisfying RTi ðtÞRi ðtÞ 6 I;
i ¼ 1; 2; 3; 4
ð3Þ
Throughout this paper, if not explicitly stated, matrices and vectors are assumed to have compatible dimensions. Let u(t) be linear gain local state feedback with the reference input r(t) for Eq. (1), uðtÞ ¼ KxðtÞ þ rðtÞ
ð4Þ
so as to ensure stability of the closed-loop delayed system. Now, we introduce the following definitions and lemma for a precise formulation of our results. Definition 1 [15]. A real-valued vector rðtÞ 2 Ln1 , if krk1 ¼ sup krðtÞk < þ1. t0 6t<1
Definition 2 [15]. The control system (1) is BIBO stabilization by the local control law (4) if every solution of the system (1) y(t) satisfies kyðtÞk 6 h1 krk1 þ h2 where h1, h2 are positive constants for every reference input rðtÞ 2 Ln1 . Lemma 1. Suppose that the absolutely continuous function x(t) satisfies Z t f ðsÞxðsÞds ðt0 6 t 6 bÞ xðtÞ 6 gðtÞ þ t0
with gðtÞ : ½t0 ; b ! Rþ absolutely continuous and f ðtÞ : ½t0 ; b ! Rþ integrable then Rt Z t Rt f ðlÞdl f ðlÞdl þ g0 ðsÞe s ds ðt0 6 t 6 bÞ xðtÞ 6 gðt0 Þe t0 t0
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
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3. BIBO stabilization of time-delay system In this section, we wish to design the linear feedback control u(t) = Kx(t) + r(t), such that the resulting closed-loop system (1) with DA(t) = DB(t) = DC(t) = DG(t) = 0 is BIBO stabilizable. Substituting (4) to the system (1) with DA(t) = DB(t) = DC(t) = DG(t) = 0 yields a closed-loop system as follows: 8 0 > < x ðtÞ ¼ ðA þ CKÞxðtÞ þ Bxðt sÞ þ CrðtÞ yðtÞ ¼ GxðtÞ > : xðtÞ ¼ uðtÞ t0 s 6 t 6 t0
ð5Þ
Theorem 1. The time-delay system (5) with feedback gain matrix K = gCTR1 by the local control law (4) is BIBO stabilizable, if there exist some constants g > 0, b1 > 0, b2 > 0, b3 > 0 and matrix R > 0, such that b1 > b2 eb1 s and the linear matrix inequality 1 0 BR C AR þ RAT 2gCC T þ b1 R C B ð6Þ W¼@ RBT b2 R 0 A60 T C 0 b3 I holds. Proof. We define a Lyapunov function V(t) as V ðtÞ ¼ xT ðtÞPxðtÞ where P = R1Taking the time derivative of V(t) along the trajectory of system (5) gives that V 0 ðtÞ ¼ xT ðtÞðAT P þ K T C T P þ PA þ PCKÞxðtÞ þ 2xT ðtÞPBxðt sÞ þ 2xT ðtÞPCrðtÞ 1 1T 0 T 10 0 xðtÞ xðtÞ A P þ K T C T P þ PA þ PCK þ b1 P PB PC C C B CB B ¼ @ xðt sÞ A @ BT P b2 P 0 A@ xðt sÞ A rðtÞ rðtÞ CTP 0 b3 I b1 xT ðtÞPxðtÞ þ b2 xT ðt sÞPxðt sÞ þ b3 rT ðtÞrðtÞ 1T 0 T xðtÞ A P þ PA 2gPCC T P þ b1 P PB C B B ¼ @ xðt sÞ A @ BT P b2 P rðtÞ 0 CT P 0
1 10 xðtÞ PC C CB 0 A@ xðt sÞ A rðtÞ b3 I
b1 xT ðtÞPxðtÞ þ b2 xT ðt sÞPxðt sÞ þ b3 rT ðtÞrðtÞ Pre- and post-multiplying (6) by diag[R1, R1, I], we can obtain 1 0 T A P þ PA 2gPCC T P þ b1 P PB PC C ~ ¼B W BT P @ b2 P 0 A60 T C P 0 b3 I Then, V 0 ðtÞ 6 b1 V ðtÞ þ b2 V ðt sÞ þ b3 rT ðtÞrðtÞ Using the method of the variation of parameters, we get Z t Z t eb1 ðtsÞ V ðs sÞds þ b3 eb1 ðtsÞ krðsÞk2 ds V ðtÞ 6 V ðt0 Þeb1 ðtt0 Þ þ b2 t0
6
kP kkuk2s eb1 ðtt0 Þ
þ b2
Z
t0 ts b1 ðtssÞ
e t0 s Z t0
V ðsÞds þ b3 krk21
Z t0
t
eb1 ðtsÞ ds
Z t b eb1 ðtssÞ V ðsÞds þ b2 eb1 s eb1 ðtsÞ V ðsÞds þ 3 krk21 6 kP kkuk2s eb1 ðtt0 Þ þ b2 b1 t s t0 0 Z t b b 6 kP kkuk2s 1 þ 2 eb1 s eb1 ðtt0 Þ þ 3 krk21 þ b2 eb1 s eb1 ðtsÞ V ðsÞds b1 b1 t0
1024
So,
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
Z t b b eb1 s V ðsÞds V ðtÞeb1 t 6 kP kkuk2s 1 þ 2 eb1 s eb1 t0 þ 3 krk21 eb1 t þ b2 eb1 s b1 b1 t0
In view of Lemma 1, we can obtain Z t b eb1 s b b s b s 2 þ 3 krk21 eb1 t0 eb2 e 1 ðtt0 Þ þ b3 krk21 eb1 s eb2 e 1 ðtsÞ ds V ðtÞeb1 t 6 kP kkuks 1 þ 2 b1 b1 t0 Z t c b þ 3 krk21 eb1 t0 þcðtt0 Þ þ b3 krk21 ectþðb1 cÞs ds ¼ kP kkuk2s 1 þ b1 b1 t0 c b eðb1 cÞt eðb1 cÞt0 þ 3 krk21 eb1 t0 þcðtt0 Þ þ b3 krk21 ect ¼ kP kkuk2s 1 þ b1 b1 c b1 2 c b b krk þ 3 krk21 eb1 t0 þcðtt0 Þ þ 3 1 eb1 t 6 kP kkuk2s 1 þ b1 b1 c b1 where c ¼ b2 eb1 s By b1 > c ¼ b2 eb1 s , we have c b b krk2 V ðtÞ 6 kP kkuk2s 1 þ þ 3 krk21 eðb1 cÞðtt0 Þ þ 3 1 b1 b1 c b1 c ð2b1 cÞb3 2 2 þ 6 kP kkuks 1 þ krk1 b1 b1 ðb1 cÞ From kmin(P)kx(t)k2 6 V(t), it is obvious that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP kkuk2s b ð2b cÞkrk21 þ 3 1 kxðtÞk 6 b1 kmin ðP Þ b1 ðb1 cÞkmin ðP Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP k b3 ð2b1 cÞ 6 kuks þ krk b1 kmin ðP Þ b1 ðb1 cÞkmin ðP Þ 1 So, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP k b3 ð2b1 cÞ kuks þ kGk krk ¼ h1 krk1 þ h2 kyk 6 kGkkxðtÞk 6 kGk b1 kmin ðP Þ b1 ðb1 cÞkmin ðP Þ 1 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b3 ð2b1 cÞ ; h1 ¼ kGk b1 ðb1 cÞkmin ðP Þ This completes the proof.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb1 þ cÞkP k h2 ¼ kGk kuks b1 kmin ðP Þ
h
4. Robust BIBO stabilization of uncertain time-delay system Based on Theorem 1, we can perform the robust BIBO stability analysis for system (1) with uncertainties (2) and (3). Substituting (4) to the system (1), we can rewrite it as 8 0 > < x ðtÞ ¼ ðA þ DAðtÞ þ CK þ DCðtÞK ÞxðtÞ þ ðB þ DBðtÞÞxðt sÞ þ ðC þ DCðtÞÞrðtÞ yðtÞ ¼ ðG þ DGðtÞÞxðtÞ ð7Þ > : xðtÞ ¼ uðtÞ t0 s 6 t 6 t0 Theorem 2. The time-delay system (7) with feedback gain matrix K = g 0 CTH1 by the local control law (4) is robust BIBO stabilizable, if there exist some constants g 0 > 0, e1 > 0, e2 > 0, e3 > 0, e4 > 0, e5 > 0, e6 > 0, e7 > 0 and matrix H > 0, such that x1 > x2 ex1 s and the linear matrix inequality
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
0 B B B B B B N¼B B B B B @
ð1; 1Þ BH HBT e2 H CT 0 DT1 0 DT2 DT3 DT3
0 0 0
C 0 e3 I 0
D1 0 0 e4 I
D2 0 0 0
D3 0 0 0
0 0 0
0 0 0
e5 I 0 0
0 e6 I 0
D3 0 0 0
1025
1
C C C C C C C60 C 0 C C C 0 A e7 I
ð8Þ
holds, where ð1; 1Þ ¼ HAT þ AH 2g0 CC T þ e1 H x 1 ¼ e1
e4 kE1 k2 þ e7 kE3 Kk2 >0 kmin ðP Þ
x 2 ¼ e2 þ
e5 kE2 k2 kmin ðP Þ
Proof. We define the Lyapunov function V(t) as V ðtÞ ¼ xT ðtÞPxðtÞ where P = H1Taking the time derivative of V(t) along the trajectory of system (7) gives that V 0 ðtÞ ¼ xT ðtÞððA þ DAÞT P þ K T C T P þ P ðA þ DAÞ þ PCK þ K T DC T P þ P DCKÞxðtÞ þ 2xT ðtÞP ðB þ DBÞxðt sÞ þ 2xT ðtÞP ðC þ DCÞrðtÞ ¼ vT N0 v e1 xT ðtÞPxðtÞ þ e2 xT ðt sÞPxðt sÞ þ e3 rT ðtÞrðtÞ þ e4 xT ðtÞET1 RT1 ðtÞR1 ðtÞE1 xðtÞ þ e5 xT ðt sÞET2 RT2 ðtÞR2 ðtÞE2 xðt sÞ þ e6 rT ðtÞET3 RT3 ðtÞR3 ðtÞE3 rðtÞ þ e7 xT ðtÞK T ET3 RT3 ðtÞR3 ðtÞE3 KxðtÞ where 0
ð1; 1Þ0 B T B B P B T B C P B B 0 N ¼ B DT1 P B T B D P B 2 B T @ D3 P DT3 P
PB e2 P 0 0 0
PC 0 e3 I 0 0
PD1 0 0 e4 I 0
PD2 0 0 0 e5 I
PD3 0 0 0 0
0 0
0 0
0 0
0 0
e6 I 0
PD3 0 0 0 0
1
C C C C C C C C C C C 0 A e7 I
where ð1; 1Þ0 ¼ AT P þ PA 2g0 PCC T P þ e1 P vT ¼ xT ðtÞ xT ðt sÞ rT ðtÞ xT ðtÞET1 RT1 ðtÞ
xT ðt sÞET2 RT2 ðtÞ
rT ðtÞET3 RT3 ðtÞ g0 xT ðtÞPCET3 RT3 ðtÞ
Pre- and post-multiplying (8) by diag[H1, H1, I, I, I, I, I], we can obtain V 0 ðtÞ 6 e1 xT ðtÞPxðtÞ þ e2 xT ðt sÞPxðt sÞ þ e3 rT ðtÞrðtÞ þ e4 kE1 k2 xT ðtÞxðtÞ þ e5 kE2 k2 xT ðt sÞxðt sÞ þ e6 kE3 k2 rT ðtÞrðtÞ þ e7 kE3 Kk2 xT ðtÞxðtÞ ¼ e1 V ðtÞ þ e2 V ðt sÞ þ ðe3 þ e6 kE3 k2 ÞrT ðtÞrðtÞ þ ðe4 kE1 k2 þ e7 kE3 Kk2 ÞxT ðtÞxðtÞ þ e5 kE2 k2 xT ðt sÞxðt sÞ e4 kE1 k2 þ e7 kE3 Kk2 e5 kE2 k2 6 e1 V ðtÞ þ e2 V ðt sÞ þ ðe3 þ e6 kE3 k2 ÞrT ðtÞrðtÞ þ V ðtÞ þ V ðt sÞ kmin ðP Þ kmin ðP Þ ! ! e4 kE1 k2 þ e7 kE3 Kk2 e5 kE2 k2 V ðtÞ þ e2 þ V ðt sÞ þ ðe3 þ e6 kE3 k2 ÞrT ðtÞrðtÞ ¼ e1 kmin ðP Þ kmin ðP Þ
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Let x 1 ¼ e1
e4 kE1 k2 þ e7 kE3 Kk2 ; kmin ðP Þ
x 2 ¼ e2 þ
e5 kE2 k2 ; kmin ðP Þ
x3 ¼ e3 þ e6 kE3 k2
Then, V 0 ðtÞ 6 x1 V ðtÞ þ x2 V ðt sÞ þ x3 rT ðtÞrðtÞ The remaining part of the proof is similar to that for Theorem 1, i.e., c x3 ð2x1 cÞ 2 V ðtÞ 6 kP kkuks 1 þ krk21 þ x1 x1 ðx1 cÞ where c ¼ x2 ex1 s So, kyk 6 kG þ DGðtÞkkxðtÞk 6 ðkGk þ kD4 kkE4 kÞkxðtÞk ¼ h1 krk1 þ h2 where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x3 ð2x1 cÞ h1 ¼ ðkGk þ kD4 kkE4 kÞ x1 ðx1 cÞkmin ðP Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ðx1 þ cÞkP k h2 ¼ ðkGk þ kD4 kkE4 kÞ kuks x1 kmin ðP Þ This completes the proof.
h
Remark 1. In the work of Li et al. [11], the BIBO stabilization of system is analyzed by the Lyapunov functional and formulated in terms of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation. Obviously, if we let Ri(t) = 0, i = 1, 2, 3, 4, system (1) is reduced to the one considered in [11]. Moreover, our result is delaydependent and given in terms of LMI which can be easily solved by LMI Toolbox in Matlab. Then, the feasibility of our results can be checked at first hand without tuning any parameters. Remark 2. The criteria given in Theorems 1 and 2 are delay-dependent. Note also that the criterion ensuring BIBO stabilization given in [12] is delay-dependent and delay-independent with respect to distributed delay, discrete time delay, respectively. If the delayed system in [12] is reduced to one independent of distributed delay, we can see that our results are much less conservative than those in [11,12].
5. Illustrative examples Example 1. As an application of Theorem 1, we consider the system (5) with the following parameters: 4 1 2 0 2 1 A¼ ; B¼ ; C¼ 1 4 0 1 1 0 For b1 = 0.8, b2 = 0.4, b3 = 0.5, solving for g and R in LMI (6) gives us 0:2134 0:0043 g ¼ 4:2429; P ¼ R1 ¼ 0:0043 0:1157 Therefore, the stabilizing feedback gain matrix is given by 1:8291 0:5274 K¼ 0:9054 0:0182
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
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Table 1 smax for e1 = 8, e3 = e4 = e5 = e6 = e7 = 0.5 smax
e2 = 0.5
e2 = 1
e2 = 2
e2 = 4
0.2960
0.2349
0.1604
0.0492
Meanwhile, we obtain the maximum value smax = 0.8664. Example 2. As an application of Theorem 2, we consider the system (7) with the following parameters: 1 2 1 2 1 3 A¼ ; B¼ ; C¼ 0 4 0 1 1 0 and Di = I, i = 1, 2, 3, 4, E1 = E2 = 0.1I, E3 = 0.1K1. For e1 = 8, e2 = e3 = e4 = e5 = e6 = e7 = 0.5, solving for g 0 and H in LMI (8) gives us 0:0181 0:0406 g0 ¼ 36:2862; P ¼ H 1 ¼ 0:0406 0:5238 Therefore, the stabilizing feedback gain matrix is given by 0:8164 17:5335 K¼ 1:9703 4:4197 Meanwhile, we obtain the maximum value smax = 0.2960. In Table 1, the maximum value smax with certain ei is given.
6. Conclusions The main contribution of this letter is the result that ensures the BIBO stabilization for time-delay control system. By Lyapunov function and the method of the variation of parameters, the BIBO stabilization criterion is delay-dependent and expressed in terms of LMI. To this end, the robust BIBO stabilization for such uncertain system is studied. The method proposed first in this paper can be extended to the problems of BIBO stabilization for neutral system or other dynamical systems.
References [1] Kharitonov VL, Plischke E. Lyapunov matrices for time-delay systems. Syst Control Lett 2006;55:697–706. [2] Gao HJ, Lam J, Wang ZD. Discrete bilinear stochastic systems with time-varying delay: Stability analysis and control synthesis. Chaos, Solitons & Fractals 2007;34:394–404. [3] Phat VN, Niamsup P. Stability of linear time-varying delay systems and applications to control problems. J Comput Appl Math 2006;194:343–56. [4] Xu SY, Chen TW. Robust H1 control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers. Syst Control Lett 2004;51:171–83. [5] Liu XZ. Stability of impulsive control systems with time delay. Math Comput Modelling 2004;39:511–9. [6] Kotsios S. A note on BIBO stability of bilinear systems. J Franklin Inst 1995;332(6):755–60. [7] Bose T, Chen MQ. BIBO stability of the discrete bilinear system. Digital Signal Process 1995;5:160–6. [8] Fornasini E, Valcher ME. On some connections between bilinear input/output maps and 2D systems. Nonlinear Anal 1997;30:1995–2005. [9] Xu DY, Zhong SM. The BIBO stabilization of multivariable feedback systems. J UEST China 1995;24:90–6. [10] Xu DY, Zhong SM. BIBO stabilization of large-scale systems. Control Theory Appl 1995;12:758–63. [11] Li P, Zhong SM. BIBO stabilization of time-delayed system with nonlinear perturbation. Appl Math Comput 2007. doi:10.1016/ j.amc.2007.04.081. [12] Li P, Zhong SM. BIBO stabilization for system with multiple mixed delays and nonlinear perturbations. Appl Math Comput 2007. doi:10.1016/j.amc.2007.05.046. [13] Partington JR, Bonnet C. H1 and BIBO stabilization of delay systems of neutral type. Syst Control Lett 2004;52:283–8.
1028
P. Li et al. / Chaos, Solitons and Fractals 40 (2009) 1021–1028
[14] Shahruz SM, Sakyaman NA. How to have narrow-stripe semiconductor lasers self-pulsate. Appl Math Comput 2002;130:11–27. [15] Wu H, Mizukami K. Robust stabilization of uncertain linear dynamical systems. Int J Systems Sci 1993;24:265–76. [16] Corless M, Garofalo F, Gilelmo L. New results on composite control of singularly perturbed uncertain linear systems. Automatica 1993;29:387–400. [17] Lien CH. Delay-dependent and delay-independent guaranteed cost control for uncertain neutral systems with time-varying delays via LMI approach. Chaos, Solitons & Fractals 2007;33:1017–27. [18] Shi P, Dragan V. Asymptotic H1 control of singularly perturbed systems with parametric uncertainties. IEEE Trans Autom Control 1999;44(9):1738–42.