Asynchronous H∞ control of switched delay systems with average dwell time

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Journal of the Franklin Institute 349 (2012) 3159–3169 www.elsevier.com/locate/jfranklin

Asynchronous HN control of switched delay systems with average dwell time$ Yue-E Wanga, Xi-Ming Sunb,n, Jun Zhaoa a

Northeastern University, State Key Laboratory of Synthetical Automation for Process Industries, Shenyang 110819, China b School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China Received 3 July 2012; received in revised form 13 September 2012; accepted 7 October 2012 Available online 22 October 2012

Abstract In this paper, we study the issue of asynchronous H1 control for a class of switched delay systems. The switching signal of the switched controller involves time delay, which results in the asynchronous switching between the candidate controllers and the systems. By combining the piecewise Lyapunov– Krasovskii functional method with the merging switching signal technique, sufficient conditions of the existence of admissible H1 state-feedback controllers are developed for the switched delay system under an average dwell time scheme. These conditions imply the relationship among the upper bound of the state delay, the switching delay and the average dwell time. Finally, a numerical example is given to illustrate the effectiveness of the proposed theory. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Switched systems, which include a finite number of subsystems and a logical rule that regulates the switching between the subsystems, constitute a very active field of current scientific research [1–12]. On the other hand, time delays are inevitable in many practical control systems [13,14]. Time delays can downgrade the performance of a control system and sometimes even undermine the stability of the control system [15–20]. In fact, switched delay systems are ubiquitous in distributed networks containing lossless transmission lines, $

This work was supported by the NSFC under Grants 61174073, 61174058, and 61004020, and the Program for New Century Excellent Talents in University NCET-09-0257. n Corresponding author. E-mail addresses: [email protected] (Y.-E. Wang), [email protected] (X.-M. Sun). 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.10.003

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heat exchangers, and drilling system [21–23]. For example, in [21], a switched delay system was exploited to model the behavior of the oilwell drilling system at the bottom end. Due to the significance both in theory development and practical applications, switched delay systems have been attracting considerable attention, see the Refs. [24–32]. In [24], stabilization of arbitrary switched linear systems with unknown time-varying delays was addressed. Stability and L2-gain analysis for switched delay systems with average dwell time scheme were addressed in [25]. However, it is worthy pointing out that the majority of the results mentioned above were based on an ideal assumption that the switching between the controller and the system is synchronous, which is quite unpractical. For example, when the system and the controller communicate via a communication channel and the current subsystem is switched to next one, it is necessary to take some time to identify the active subsystem and then switch the controller from the current one to the corresponding one. For this case, the closed-loop system can experience asynchronous switching signal, see the Refs. [33–37]. The study of asynchronous switching is also motivated by increasing applications in digital control loops [22], pulse–width–modulation-driven boost converter [23], multi-agent network with switching topology [34], and so on. In [23], stability, L2-gain and asynchronous H1 control of discrete-time switched systems with average dwell time were considered. Asynchronous switched control of switched linear systems with average dwell time was the concern of the paper [35]. However, in [23,35], a common function was chosen for the running time of the subsystem with both the matched controller and the unmatched controller. Therefore, at the switching times of the controller, the function is required to be continuous. Ref. [34] addressed stability of time-delay feedback switched linear systems in which time delays appear in both the feedback state and the switching signal of the switched controller. The multiple Lyapunov function method and the merging switching signal technique were employed in [34]. Up to now, to the best of our knowledge, the problem of asynchronous H1 control for switched delay systems has not been well reported. The contribution of this paper lies in three aspects. First, we address the problem of H1 control for a more general class of systems, in which time delays not only appear in the state but also appear in the controller’s switching signal. Second, by choosing a new piecewise Lyapunov–Krasovskii functional and employing merging switching signal technique, we derive sufficient conditions of the existence of H1 state-feedback controller for a class of switching signals with average dwell time scheme. The Lyapunov–Krasovskii functional is required to decrease only during the time when the active subsystem and the corresponding controller match, while it is allowed to grow when the subsystem and the controller mismatch. Also jumps of the Lyapunov–Krasovskii functional are allowed at the switching instants of the system and the switching instants of the controller. Third, the inequalities derived in the literature are sufficiently coupled (see [33,38,39]). Therefore, they have higher dimensions and are very difficult to calculate. However, in this paper, by introducing the new Lyapunov–Krasovskii functional, the inequalities derived are decoupled, and therefore easy to compute. The organization of the paper is as follows. The problem formulation is stated in Section 2, followed by the main results in Section 6. A numerical example is given in Section 4. Section 5 draws the conclusion. Notations: Throughout this paper, Rn denotes the n-dimensional Euclidean space. In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. Matrix P40 means that P is positive definite; lmax ðPÞ and lmin ðPÞ denote the maximum

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and minimum eigenvalues of P respectively; I is the identity matrix with appropriate dimensions; J  J denotes Euclidean vector norm or induced matrix norm; n denotes the symmetric part of a subblock in a block symmetric matrix; diagf  g stands for a blockdiagonal matrix; N denotes the set of nonnegative integer numbers and Nþ ¼ N=f0g. 2. Problem formulation Consider the following switched delay system: _ ¼ AsðtÞ xðtÞ þ BsðtÞ xðttd Þ þ EsðtÞ uðtÞ þ FsðtÞ wðtÞ, xðtÞ zðtÞ ¼ CxðtÞ þ DwðtÞ, xðyÞ ¼ cðyÞ, y 2 ½td ,0, n

ð1Þ

q

where xðtÞ 2 R is the state, uðtÞ 2 R is the control input, w(t) is the disturbance input which belongs to L2 ½0, þ 1Þ,; z(t) is the output to be controlled, cðyÞ is the initial vector function that is continuously differentiable on ½td ,0; sðtÞ : ½0,1Þ-M ¼ f1,2, . . . ,mg is a piecewise constant function of time t, called switching signal. Ali ,Bli ,Eli ,Fli ,C,D are known real constant matrices of appropriate dimensions; td is the state delay. Corresponding to sðtÞ, we have the switching sequence fxt0 : ðl0 ,t0 Þ, . . . ,ðli ,ti Þ, . . . ,jli 2 M,i 2 Ng, which means that the lith subsystem is active when t 2 ½ti ,tiþ1 Þ, i 2 N. We assume that the state of the system does not jump at the switching instants and that only finitely many switches can occur in any finite interval. In this paper, the control input is of the form uðtÞ ¼ Ksðtts ðtÞÞ xðtÞ,

ð2Þ

where ts ðtÞ : ½0,1Þ-½0,ts  is the switching delay, satisfying ts rtiþ1 ti ,i 2 N, and the corresponding closed-loop system is _ ¼ ðAsðtÞ þ EsðtÞ Ksðtts ðtÞÞ ÞxðtÞ þ BsðtÞ xðttd Þ þ FsðtÞ wðtÞ, xðtÞ zðtÞ ¼ CxðtÞ þ DwðtÞ, xðyÞ ¼ cðyÞ, y 2 ½td ,0:

ð3Þ

The following definitions will be used in the sequel. Definition 1 (Sun et al. [25]). The equilibrium xn ¼ 0 of system (1) is said to be exponentially stable under sðtÞ, if the solution x(t) of system (1) with wðtÞ ¼ 0 satisfies JxðtÞJrkelðtt0 Þ Jxðt0 ÞJtd , 8tZt0 , for constants kZ1 and l40, where JxðtÞJtd ¼ suptd ryr0 Jxðt þ yÞJ. Definition 2 (Liberzon [1]). For any t2 4t1 Z0, let Ns ðt1 ,t2 Þ denotes the number of switching of sðtÞ over ðt1 ,t2 Þ. If Ns ðt1 ,t2 ÞrN0 þ ðt2 t1 Þ=ta holds for ta 40, N0 Z0, then ta is called average dwell time and N0 is called a chatter bound. Denoted by Save ½ta ,N0  the class of switching signals with average dwell time ta and chattering bound N0. We employ the merging signal technique in [34] to deal with the mismatched switching signal. Similarly, we create a virtual switching signal s0 ðtÞ : ½0,1Þ-M  M as follows: s0 ¼ ðs1 ðtÞ,s2 ðtÞÞ. The merging action is denoted by  such that s0 ¼ s1  s2 . The definition implies that the set of switching times of s0 is the union of the sets of switching times of s1 and of s2 .

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Lemma 1 (Vu and Kristi [34]). Given s1 ðtÞ 2 Save ½ta ,N0 , and s2 ðtÞ ¼ s1 ðtts ðtÞÞ, then, it has s2 2 Save ½ta ,N0 þ ðts =ta Þ, s0 2 Save ½t a ,N 0 , where t a ¼ ta =2, N 0 ¼ 2N0 þ ðts =ta Þ. Proof. By Lemmas 1 and 2 in [34], the results can be easily obtained.

&

Lemma 2 (Vu and Kristi [34]). Let s1 ðtÞ 2 Save ½ta ,N0 , and s2 ðtÞ ¼ s1 ðtts ðtÞÞ. Suppose that 0rts ðtÞrts for all t, and ts otkþ1 tk ,k 2 N. For an interval ðt0 ,tÞ, let mðt0 ,tÞ be the total time for which s1 ðtÞ ¼ s2 ðtÞ, and let m ðt0 ,tÞ ¼ tt0 mðt0 ,tÞ . If ts ðlm þ lm Þrðlm lÞta for some positive constants lm , lm , and l 2 ½0,lm , then lm mðt0 ,tÞ þ lm m ðt0 ,tÞ rcT lðtt0 Þ,

8tZt0 ,

ð4Þ

where cT ¼ ðlm þ lm ÞðN0 þ 1Þts . The H1 control problem for the switched delay system (1) is stated as follows: (i) The closed-loop system is exponentially stable when wðtÞ ¼ 0. (ii) Under zero initial condition, the closed-loop system has weighted-L2-gain satisfying Z 1 Z 1 as T 2 e z ðsÞzðsÞ dsrg wT ðsÞwðsÞ ds, ð5Þ 0

0

for a prescribed scalar g40, where a40.

3. Main results In this section, we derive conditions under which the asynchronous H1 control problem of system (3) under an average dwell time scheme is solvable. Theorem 1. Consider the switched delay system (1). Let td Z0, ts Z0, ls 40, lu 40, and mZ1 be given constants. Suppose that there exist matrices P li li 40, Q li li 40, Q li lj 40, and Mli li , 8li li ,li lj 2 M  M,li alj , such that 3 2 1 S11 Bli Q li li Fli þ P li li C T D P li li P li li C T 7 6 6 n els td Q li li 0 0 0 7 7 6 ð6Þ Pli li ¼ 6 n DT Dg2 I 0 0 7 7o0, 6 n 7 6 4 n n n Q li li 0 5 n

2

Pli lj

S211

6 6 n 6 6 ¼6 n 6 6 4 n n

n

n

n

I

Bli Q li lj

Fli þ P lj lj C T D

P lj lj

P lj lj C T

elu td Q li lj

0

0

0

n

DT Dg2 I

0

0

n

n

Q li lj

n

n

n

0 I

P li li rmP lj lj ,Q li lj rmQ li li ,Q lj lj rmQ li lj ,li lj ,li li ,lj lj 2 M  M,

3 7 7 7 7 7o0, 7 7 5

ð7Þ

ð8Þ

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where S111 ¼ Ali P li li þ P li li ATli þ Eli Mli li þ MlTi li ElTi þ ls P li li , Eli Mlj lj þ MlTj lj ElTi lu P lj lj :

3163

S211 ¼ Ali P lj lj þ P lj lj ATli þ

Then the H1 state-feedback controller given in Eq. (2) can guarantee that system (1) is exponential stable and has weighted L2-gain for any switching signal with average dwell time defined by ta 4tna ¼

2 ln m þ ðls þ lu Þðts þ td Þ : ls

ð9Þ 1

Moreover, the H1 state-feedback controller gains are given by Kli ¼ Mli li P li li . Proof. Due to the switching delay, after the ljth subsystem has been switched to the lith subsystem, the controller Klj , i.e. Kli1 , is still active. Thus, we rewrite the closed-loop system as _ ¼ A s0 ðtÞ xðtÞ þ B s0 ðtÞ xðttd Þ þ F s0 ðtÞ wðtÞ, xðtÞ

ð10Þ

where A li li ¼ Ali þ Eli Kli , A li lj ¼ Ali þ Eli Klj , B li li ¼ B li lj ¼ Bli , F li li ¼ F li lj ¼ Fli . Construct a piecewise Lyapunov–Krasovskii functional candidate as follows: uðtÞ ¼ Vs0 ðtÞ ðtÞ

Z

T

t

¼ x ðtÞPs0 ðtÞ xðtÞ þ

xT ðsÞels0 ðtÞ ðstÞ Qs0 ðtÞ xðsÞ ds,

ð11Þ

ttd 1 1 where lli li ¼ ls , lli lj ¼ lu , Pli lj ¼ Plj lj , P1 li li ¼ P li li , Qli li ¼ Q li li , and Qli lj ¼ Q li lj , and define T 2 T GðsÞ ¼ z ðsÞzðsÞg w ðsÞwðsÞ. Along the trajectories of Eq. (10), we have V_ li li ðtÞ þ ls Vli li ðtÞ þ GðtÞ ¼ ZT ðtÞP li li ZðtÞ, where 2 3 1 S 11 þ C T C Pli li Bli Pli li Fli þ C T D 6 7 7, P li li ¼ 6 n els td Qli li 0 4 5

n

n

ZT ðtÞ ¼ ½xT ðtÞ xT ðttd Þ wT ðtÞ:

DT Dg2 I

ð12Þ

Multiplying both sides of P li li o0 by diagfP li li ,Q li li ,Ig, and using Lemma 1, and letting Kli P li li ¼ Mli li , we have that P li li o0 is equivalent to Pli li o0, which implies V_ li li ðtÞ þ ls Vli li ðtÞ þ GðtÞo0: ð13Þ Similarly, we have, if Pli lj o0 V_ li lj ðtÞlu Vli lj ðtÞ þ GðtÞo0:

ð14Þ

Combining (11) with (8), it holds Vli li ðti þ ts ðti ÞÞrmVli lj ððti þ ts ðti ÞÞ Þ, Vli lj ðti Þrmeðls þlu Þtd Vlj lj ðt i Þ:

ð15Þ

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Let t1 , . . . ,tNs0 ðt0 ,tÞ denote the switching times of s0 in ðt0 ,TÞ, and t0 ¼ t0 , tNs0 ðt0 ,TÞþ1 ¼ T  by convention. Because s0 is constant for t 2 ½tk ,tkþ1 Þ, from Eqs. (13) and (14), we have Z t els0 ðtk Þ ðtsÞ GðsÞ ds: ð16Þ uðtÞrels0 ðtk Þ ðttk Þ uðtk Þ tk

Integrating the inequalities (16) and using Eq. (15), we can derive N

uðTÞrmNs0 eNs1 ðls þlu Þtd ej0 s0 ðt0 Þuðt0 Þ Ns0 Z tkþ1 X N mNs0 ðs,TÞ eNs1 ðs,TÞðls þlu Þtd ejk s0 ðsÞGðsÞ ds,  k¼0

ð17Þ

tk

where Ns0 X

N ejk s0 ðsÞ ¼ exp

! ls0 ðtj Þ ðtjþ1 tj Þ expðls0 ðtk Þ ðtkþ1 sÞÞ:

ð18Þ

j ¼ kþ1

The condition (9) implies the existence of l such that 2 ln m ðls þ lu Þtd ðls þ lu Þts þ olols  : ta ta ta

ð19Þ

The preceding inequalities can be rewritten as ðls þ lu Þts oðls lÞta ,

ð20Þ

ln m ðls þ lu Þtd þ : ta ta

ð21Þ

l4

Besides, from Eqs. (18) and (20), by using Lemma 2, it holds ! Ns 0 X ls0 ðtj Þ ðtjþ1 tj Þ Ns 0 ej0 ðt0 Þ ¼ exp e ¼ els mðt0 ,TÞ þlu m ðt0 ,TÞ recT lðTt0 Þ

ð22Þ

j¼0

and Ns0 X

N ejk s0 ðsÞ ¼ exp

! e

ls0 ðtj Þ ðtjþ1 tj Þ

expðls0 ðtk Þ ðtkþ1 sÞÞ

j ¼ kþ1

¼ els mðs,TÞ þlu m ðs,TÞ recT lðTsÞ : Considering Eq. (21) and the foregoing analysis, we have N

uðTÞrmNs0 eNs1 ðls þlu Þtd ej0 s0 ðt0 Þuðt0 Þ Ns0 Z tkþ1 X mNs0 ðs,TÞ eNs1 ðs,TÞðls þlu Þtd els mðs,TÞ þlu m ðs,TÞ GðsÞ ds  k¼0

tk

ð23Þ

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rg0 el0ðTt0 Þ uðt0 Þ

Z

3165

T

mNs0 ðs,TÞ eNs1 ðs,TÞðls þlu Þtd els mðs,TÞ þlu m ðs,TÞ GðsÞ ds,

ð24Þ

t0

where g0 ¼ mN 0 eðls þlu Þðtd þts ÞN0 þðls þlu Þts , ln m ls þ lu  td 40: l0 ¼ l ta ta Under zero initial condition, Eq. (24) gives Z T mNs0 ðs,TÞ eNs1 ðs,TÞðls þlu Þtd els mðs,TÞ þlu m ðs,TÞ GðsÞ dsr0,

ð25Þ

t0

which implies Z T mNs0 ðs,TÞ eNs1 ðs,TÞðls þlu Þtd els mðs,TÞ þlu m ðs,TÞ zT ðsÞzðsÞ ds t0

r

Z

T

mNs0 ðs,TÞ eNs1 ðs,TÞðls þlu Þtd els mðs,TÞ þlu m ðs,TÞ g2 wT ðsÞwðsÞ ds:

ð26Þ

t0

Multiplying both sides of Eq. (26) by mNs0 ðt0 ,TÞ eNs1 ðt0 ,TÞðls þlu Þtd yields Z T mNs0 ðt0 ,sÞ eNs1 ðt0 ,sÞðls þlu Þtd els mðs,TÞ þlu m ðs,TÞ zT ðsÞzðsÞ ds t0

r

Z

T

mNs0 ðt0 ,sÞ eNs1 ðt0 ,sÞðls þlu Þtd els mðs,TÞ þlu m ðs,TÞ g2 wT ðsÞwðsÞ ds:

ð27Þ

t0

Thus, combining Eq. (26) with Eq. (27), we have Z T mNs0 ðt0 ,sÞ eNs1 ðt0 ,sÞðls þlu Þtd els ðTsÞ zT ðsÞzðsÞ ds t0

r

Z

T

mNs0 ðt0 ,sÞ eNs1 ðt0 ,sÞðls þlu Þtd ecT lðTsÞ g2 wT ðsÞwðsÞ ds:

ð28Þ

t0

Here, suppose that t0 ¼ 0. Noticing that Ns0 ð0,sÞrN 0 þ ðs=t a Þ, Ns1 ð0,sÞrN0 þ ðs=ta Þ, and Eq. (9), it follows from Eq. (28) that Z T Z T c1 eas els ðTsÞ zT ðsÞzðsÞ dsr ecT lðTsÞ g2 wT ðsÞwðsÞ ds, ð29Þ 0

0 ððN 0 ls ta ðls þlu Þðts þtdÞ Þ=2ÞN0 ðls þlu Þtd

where c1 ¼ e and a ¼ ðls ta ðls þ lu Þðts þ td ÞÞ=2t a þ ððls þ lu Þtd Þ=ta . Integrating both sides of Eq. (29) from T ¼ 0 to 1 leads to Z 1 Z ls e c T 2 1 T as T g e z ðsÞzðsÞ dsr w ðsÞwðsÞ ds: ð30Þ c1 l 0 0 The proof is completed.

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Remark 1. It is worth pointing out that the Lyapunov–Krasovskii functional is required to decrease only during the time when the active subsystem and the corresponding controller match. If state delays are deleted and the switching delays are only constant ones, Theorem 1 will reduce to Theorem 1 of [35]. Remark 2. The condition (9), applied to guarantee system stability, implies the relationship among the state delay td , the upper bound ts of the switching delay, and average dwell time ta . Remark 3. The inequalities derived in the literature are sufficiently coupled (see [33,38,39]). Thus, they have higher dimensions and are very difficult to calculate. However, by introducing the new Lyapunov–Krasovskii functional, in this paper, the conditions are decoupled, and therefore easy to compute. In addition, in [39], the stabilization problem of switched delay systems under asynchronous switching was studied, but all the subsystems are required to be exponentially stable with both matched and mismatched controllers. In this paper, there is no restriction on subsystems with the mismatched controllers. In the absence of the switching delay, i.e. ts ðtÞ ¼ 0 in Theorem 1, the following corollary can be easily obtained. Corollary 1. Consider the switched delay system (1) with ts ðtÞ ¼ 0. Let td Z0, ls 40, lu 40, and mZ1 be given constants. Suppose that there exist matrices P li li 40, Q li li 40, and Mli li , 8li li 2 M  M,li alj , such that Eq. (6) holds. Then the state-feedback controller given in Eq. (2) can guarantee that system (1) is exponentially stable and has weighted L2-gain for any switching signal with average dwell time ta 4tna ¼ 2 ln m=ls , where m satisfies P li li rmP lj lj , Q li li rmQ lj lj , li li , lj lj 2 M  M. Moreover, the H1 state-feedback controller gains are given 1 by Kli ¼ Mli li P li li . 4. Numerical example In this section, we present an example to illustrate the effectiveness of the proposed method. Consider the switched delay system (1) with two subsystems: Subsystem 1:       9 0 0:3 0 1 0 A1 ¼ , B1 ¼ , E1 ¼ : 0 2 0:1 0:3 1 1 Subsystem 2:  3 A2 ¼ 1

2:5 1:5



 ,

B2 ¼

0:3

0

0

0:1



 ,

E2 ¼

1:5

2:5

0:5

1:5

 :

For td ¼ 0:3, ts ¼ 0:23, solving the conditions in Theorem 1 with ls ¼ 1:07, lu ¼ 5, and m ¼ 1:01, we can obtain the minimum average dwell time tna ¼ 3:0252 and the corresponding controller is     22:4847 7:5873 13:1310 0:4969 K1 ¼ , K2 ¼ : ð31Þ 24:7048 4:6564 10:9942 0:8572

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200 150 100 50 0 −50 −100 −150 −200

0

10

20

30

40

50

60

70

Fig. 1. The state responses of the closed-loop system. 150

100

50

0

−50

−100

0

10

20

30

40

50

60

70

Fig. 2. The control trajectories.

Constructing a possible switching sequence satisfying ta ¼ 3:034tna , one gets the steady state response of the closed-loop system with xt0 ¼ ½5 6T . Figs. 1 and 2 show the state responses and the control trajectories, respectively. Then, for the same parameters, turning to using the Corollary 2 in [33], it can be checked that the linear matrix inequalities have no feasible solutions. Therefore, the results in this paper may be less conservative. 5. Conclusion The asynchronous H1 control problem for a class of switched delay systems under an average dwell time scheme has been investigated in this paper. By further allowing the

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Lyapunov–Krasovskii functional to increase during the running time of the active subsystem with the mismatched controller, we have provided the existence conditions of the asynchronous H1 state-feedback controller. Then, the corresponding solvability condition for the controller has been established. Based on the method provided in this paper, fault detection, fault estimation for switched systems with time delays under asynchronous switching will be considered in the future work.

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