Automatica 43 (2007) 371–376www.elsevier.com/locate/automatica

Technical communique

Delay-range-dependent stability for systems with time-varying delay�

Yong Hea,b, Qing-Guo Wangb,∗, Chong Linb,c, Min Wua

aSchool of Information Science and Engineering, Central South University, Changsha 410083, ChinabDepartment of Electrical and Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

cInstitute of Complexity Science, College of Automation Engineering, Qingdao University, Qingdao 266071, China

Received 16 September 2005; received in revised form 15 December 2005; accepted 18 August 2006

Abstract

This paper is concerned with the stability analysis for systems with time-varying delay in a range. An appropriate type of Lyapunov functionalsis proposed to investigate the delay-range-dependent stability problem. The present results may improve the existing ones due to a methodto estimate the upper bound of the derivative of Lyapunov functional without ignoring some useful terms and the introduction of additionalterms into the proposed Lyapunov functional, which take into account the range of delay. Numerical examples are given to demonstrate theeffectiveness and the benefits of the proposed method.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Delay-range-dependent; Time-varying delay; Linear matrix inequality (LMI); Stability

1. Introduction

During the last decade, considerable attention has beendevoted to the problem of delay-dependent stability analysisand controller design for time-delay systems, see e.g. Park(1999), Moon, Park, Kwon, and Lee (2001), Fridman andShaked (2002, 2003), Kharitonov and Niculescu (2003), Gu,Kharitonov, and Chen (2003), Gao, Lam, Wang, and Wang(2004), Han (2004a,b), He, Wu, She, and Liu (2004a,b),Wu, He, She, and Liu (2004), Wu, He, and She (2004), Xuand Lam (2005), Xu, Lam, and Zou (2005), Jiang and Han(2005). Recently, a free-weighting matrix method is proposedin He et al. (2004b) and Wu, He, She, and Liu (2004) tostudy the delay-dependent stability for systems with time-varying delay, in which the bounding techniques on somecross product terms are not involved. But there is room for

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Keqin Guunder the direction of Editor André Tits. This work with Y. He and M. Wuwas supported in part by the National Science Foundation of China underGrants 60574014 and 60425310 and in part by the Doctor Subject Foundationof China under Grant 20050533015.

∗ Corresponding author. Tel.: +65 6516 2282; fax: +65 6779 1103.E-mail address: elewqg@nus.edu.sg (Q.-G. Wang).

0005-1098/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2006.08.015

further investigation. First, when estimating the upper boundof the derivative of Lyapunov functional, some useful termsare ignored. For example, in Fridman and Shaked (2003),Han (2004a), He et al. (2004b) and Wu, He, She, and Liu(2004), the derivative of

∫ 0−h

∫ t

t+� xT(s)Zx(s) ds d� is often

estimated as hxT(t)Zx(t) − ∫ t

t−d(t)xT(s)Zx(s) ds and the

term − ∫ t−d(t)

t−hxT(s)Zx(s) ds is ignored, which may lead to

considerable conservativeness. On the other hand, the rangeof time-varying delay considered in these papers is from 0 toan upper bound. In practice, the range of delay may vary in arange for which the lower bound is not restricted to be 0. Inthis case, the criteria in the previous work such as Fridman andShaked (2002, 2003), Han (2004a), He et al. (2004b) and Wu,He, She, and Liu (2004) are conservative because they do nottake into account the information of the lower bound of delay.

Recently, systems with time-varying delay in a range arestudied in Han and Gu (2001), Kharitonov and Niculescu(2003), Michiels, Assche, and Niculescu (2005) and Jiang andHan (2005). However, a transformation with additional eigen-values is employed in Kharitonov and Niculescu (2003) sothat the transformed system is not equivalent to the originalone, which may lead to conservativeness (Gu & Niculescu,2000). A frequency-domain method is presented in Michielset al. (2005), where the considered time-varying delay is a

372 Y. He et al. / Automatica 43 (2007) 371–376

periodic function and so its applicability is narrow. A discretizedLyapunov functional approach is employed in Han and Gu(2001), but the criteria in it are not applicable to systems withtime-varying delay without restriction on its derivative (i.e. fasttime-varying delay). Jiang and Han (2005) employed the free-weighting matrix method to investigate the robust stability andH∞ control problem for systems with interval time-varying de-lay. But the results were obtained by neglecting some usefulterms in the derivative of Lyapunov functional. In addition, thecriteria in Jiang and Han (2005) are only available to systemswith fast time-varying delay.

In this paper, we study the stability problem for systems withtime-varying delay in a range by choosing an appropriate Lya-punov functional. A new method is proposed to estimate theupper bound of the derivative of Lyapunov functional withoutignoring some useful terms. Some delay-range-dependent sta-bility criteria are derived based on the new Lyapunov functional

and the consideration of range for the time-delay. The resultingcriteria are applicable to both fast and slow time-varying de-lay. Finally, numerical examples are given to demonstrate theeffectiveness and the merit of the proposed method.

2. Problem formulation and main results

Consider the following linear system with time-varying de-lay:

{x(t) = Ax(t) + Adx(t − d(t)), t > 0,

x(t) = �(t), t ∈ [−h2, 0], (1)

where x(t) ∈ Rn is the state vector, A and Ad are constantmatrices with appropriate dimensions, the time delay, d(t), isa time-varying continuous function that satisfies

h1 �d(t)�h2 (2)

and

d(t)��, (3)

where 0�h1 < h2 and � are constants. Note that h1 may not beequal to 0. The initial condition, �(t), is a continuous vector-valued initial function of t ∈ [−h2, 0].

In the previous work such as Fridman and Shaked (2002),He et al. (2004b) and Wu, He, She, and Liu (2004), the well-used Lyapunov functional, in which the delay information is

from 0 to an upper bound, h2, is of the following form:

V (xt ) = xT(t)P x(t) +∫ t

t−d(t)

xT(s)Qx(s) ds

+∫ 0

−h2

∫ t

t+�xT(s)Zx(s) ds d�. (4)

In this paper, a new Lyapunov functional is constructed, whichcontains the information of the lower bound of delay, h1. Thefollowing theorem presents a delay-range-dependent result interms of LMIs.

Theorem 1. Given scalars 0�h1 < h2 and �, the linear sys-tem (1) with time-varying delay d(t) satisfying (2) and (3)is asymptotically stable if there exist matrices P = P T > 0,Qi = QT

i �0, i = 1, 2, 3, Zj = ZTj > 0, j = 1, 2, Ni , Mi and

Si , i = 1, 2 such that the following LMI holds:

� =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�11 �12 M1 −S1 h2N1 h12S1 h12M1 ATU

� �22 M2 −S2 h2N2 h12S2 h12M2 ATdU

� � −Q1 0 0 0 0 0� � � −Q2 0 0 0 0� � � � −h2Z1 0 0 0� � � � � −h12(Z1 + Z2) 0 0� � � � � � −h12Z2 0� � � � � � � −U

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (5)

where

�11 = PA + ATP +3∑

i=1

Qi + N1 + NT1 ,

�12 = PAd + NT2 − N1 + S1 − M1,

�22 = −(1 − �)Q3 + S2 + ST2 − N2 − NT

2 − M2 − MT2 ,

U = h2Z1 + h12Z2,

h12 = h2 − h1,

and � denotes the symmetric terms in a symmetric matrix.

Proof. Choose a Lyapunov functional candidate to be

V (xt ) = xT(t)P x(t) +2∑

i=1

∫ t

t−hi

xT(s)Qix(s) ds

+∫ t

t−d(t)

xT(s)Q3x(s) ds

+∫ 0

−h2

∫ t

t+�xT(s)Z1x(s) ds d�

+∫ −h1

−h2

∫ t

t+�xT(s)Z2x(s) ds d�, (6)

where P = P T > 0, Qi = QTi �0, i = 1, 2, 3 and Zj =

ZTj > 0, j = 1, 2, are to be determined. From the Leibniz–

Newton formula, the following equations are true for any

Y. He et al. / Automatica 43 (2007) 371–376 373

matrices Ni , Si and Mi, i =1, 2, with appropriate dimensions:

2[xT(t)N1 + xT(t − d(t))N2]

×[x(t) − x(t − d(t)) −

∫ t

t−d(t)

x(s) ds

]= 0, (7)

2[xT(t)S1 + xT(t − d(t))S2]

×[x(t − d(t)) − x(t − h2) −

∫ t−d(t)

t−h2

x(s) ds

]= 0, (8)

2[xT(t)M1 + xT(t − d(t))M2]

×[x(t − h1) − x(t − d(t)) −

∫ t−h1

t−d(t)

x(s) ds

]= 0. (9)

On the other hand, the following equations are also true:

−∫ t

t−h2

xT(s)Z1x(s) ds = −∫ t

t−d(t)

xT(s)Z1x(s) ds

−∫ t−d(t)

t−h2

xT(s)Z1x(s) ds, (10)

−∫ t−h1

t−h2

xT(s)Z2x(s) ds = −∫ t−d(t)

t−h2

xT(s)Z2x(s) ds

−∫ t−h1

t−d(t)

xT(s)Z2x(s) ds. (11)

Calculating the derivative of V (xt ) along the solutions ofsystem (1) and adding the left side of (7)–(9) into it and using(10)–(11) yield

V (xt ) = 2xT(t)P x(t) +2∑

i=1

{xT(t)Qix(t) − xT(t − hi)

× Qix(t − hi)} + xT(t)Q3x(t)

− (1 − d(t))xT(t − d(t))Q3x(t − d(t))

+ h2xT(t)Z1x(t) −

∫ t

t−h2

xT(s)Z1x(s) ds

+ (h2 − h1)xT(t)Z2x(t) −

∫ t−h1

t−h2

xT(s)Z2x(s) ds

�2xT(t)P x(t) + xT(t)

3∑i=1

Qix(t)

−2∑

i=1

xT(t − hi)Qix(t − hi) − (1 − �)xT(t − d(t))

× Q3x(t − d(t)) + xT(t)[h2Z1 + (h2 − h1)Z2]x(t)

−∫ t

t−d(t)

xT(s)Z1x(s) ds

−∫ t−d(t)

t−h2

xT(s)(Z1 + Z2)x(s) ds

−∫ t−h1

t−d(t)

xT(s)Z2x(s) ds

+ 2[xT(t)N1 + xT(t − d(t))N2]

×[x(t) − x(t − d(t)) −

∫ t

t−d(t)

x(s) ds

]

+ 2[xT(t)S1 + xT(t − d(t))S2]

×[x(t − d(t)) − x(t − h2) −

∫ t−d(t)

t−h2

x(s) ds

]

+ 2[xT(t)M1 + xT(t − d(t))M2]

×[x(t − h1) − x(t − d(t)) −

∫ t−h1

t−d(t)

x(s) ds

]

��T(t)[� + AT(h2Z1 + h12Z2)A + h2NZ−11 NT

+ h12S(Z1 + Z2)−1ST + h12MZ−1

2 MT]�(t)

−∫ t

t−d(t)

[�T(t)N + xT(s)Z1]

× Z−11 [NT�(t) + Z1x(s)] ds

−∫ t−d(t)

t−h2

[�T(t)S + xT(s)(Z1 + Z2)](Z1 + Z2)−1

× [ST�(t) + (Z1 + Z2)x(s)] ds

−∫ t−h1

t−d(t)

[�T(t)M + xT(s)Z2]Z−12

× [MT�(t) + Z2x(s)] ds, (12)

where

�(t) =⎡⎢⎣

x(t)

x(t − d(t))

x(t − h1)

x(t − h2)

⎤⎥⎦ , � =

⎡⎢⎣

�11 �12 M1 −S1� �22 M2 −S2� � −Q1 0� � � −Q2

⎤⎥⎦ ,

N =⎡⎢⎣

N1N200

⎤⎥⎦ , S =

⎡⎢⎣

S1S200

⎤⎥⎦ , M=

⎡⎢⎣

M1M200

⎤⎥⎦ , A =

⎡⎢⎣

AT

ATd

00

⎤⎥⎦

T

.

Since Zi > 0, i = 1, 2, then the last three parts in (12) are allless than 0. So, if � + AT(h2Z1 + h12Z2)A + h2NZ−1

1 NT +h12S(Z1 + Z2)

−1ST + h12MZ−12 MT < 0, which is equivalent

to (5) by Schur complements, then V (xt ) < − �‖x(t)‖2 for asufficiently small � > 0, which ensures the asymptotic stabilityof system (1), see e.g. Hale and Verduyn Lunel (1993). �

Remark 2. Theorem 1 solves the stability problem for time-varying delay in a range and presents a delay-range-dependentcriterion. The previous results in Fridman and Shaked (2002,2003), Han (2004a), He et al. (2004b), Wu, He, She, and Liu

374 Y. He et al. / Automatica 43 (2007) 371–376

(2004) only consider the case that the range of the time-varyingdelay is from 0 to an upper bound. In practice, the time-varyingdelay often lies in a range, in which the lower bound is not 0.In this case, the results in Fridman and Shaked (2002, 2003),Han (2004a), He et al. (2004b), Wu, He, She, and Liu (2004)may produce conservative results. Even for h1 = 0, the resultin Theorem 1 may be less conservative than the existing delay-dependent stability criteria. In fact, if Q1 = �1I , Z2 = �2I ,with �i > 0, i = 1, 2 being sufficient small scalars, M1 =M2 =0, Theorem 1 yields the following delay-dependent stabilitycriterion.

Corollary 3. Given scalars h2 > 0, h1 = 0 and �, the linearsystem (1) with time-varying delay d(t) satisfying (2) and (3)is asymptotically stable if there exist matrices P = P T > 0,Qi =QT

i �0, i=2, 3, Z1 =ZT1 > 0, j =1, 2,Ni and Si , i=1, 2

such that the following LMI holds:

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

11 12 −S1 h2N1 h2S1 h2ATZ1

� 22 −S2 h2N2 h2S2 h2ATdZ1

� � −Q2 0 0 0

� � � −h2Z1 0 0

� � � � −h2Z1 0

� � � � � −h2Z1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(13)

where

11 = PA + ATP +3∑

i=2

Qi + N1 + NT1 ,

12 = PAd + NT2 − N1 + S1,

22 = −(1 − �)Q3 + S2 + ST2 − N2 − NT

2 ,

and � denotes the symmetric terms in a symmetric matrix.

Remark 4. For the delay-dependent stability criteria for sys-tems with h1 =0, if Q2 = �I with � > 0 being a sufficient smallscalar, S1 = S2 = 0, Corollary 3 yields an equivalent form ofTheorem 2 in Wu, He, She, and Liu (2004) according to theproof procedure in Xu et al. (2005). That is to say, Q2, S1, S2may provide some extra freedom in the selection of them inCorollary 3.

Remark 5. Jiang and Han (2005) proposed some delay-dependent stability criteria and bounded real lemma for sys-tems with fast time-varying interval delay. However, the term− ∫ t−d(t)

t−h2xT(s)Z1x(s) ds is ignored. This may bring conserva-

tiveness. In Theorem 1, all these terms are reserved, which canreduce conservativeness in the sense that producing a largerrange of the time-varying delay to keep the stability of systems.Moreover, the results in Jiang and Han (2005) are only appli-cable to systems with fast time-varying delay. In fact, in manycases, the derivative of time-varying delay is known and may besmall. Thus, the results in Jiang and Han (2005) may have lim-ited use. In our Theorem 1, � can be any value or unknown dueto �22=−(1−�)Q3+S2+ST

2 −N2−NT2 −M2−MT

2 .Therefore,Theorem 1 is applicable to both cases of fast and slow time-varying delay.

In the following, we will show the other characteristic ofTheorem 1. When the distance between h1 and h2 is suffi-ciently small, the upper bound h2 of delay for unknown � willbe very close to the upper bound for � = 0. This character-istic is not included in previous Lyapunov functional basedwork where the upper bound of delay for � �= 0 is alwaysless than that for � = 0. In a more detail, let us add the thirdand fourth row and the third and fourth column into, respec-tively, the second row and the second column of �. Then wehave

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�11 12 M1 −S1 h2N1 h12S1 h12M1 ATU

� 22 M2 − Q1 −S2 − Q2 h2N2 h12S2 h12M2 ATdU

� � −Q1 0 0 0 0 0� � � −Q2 0 0 0 0� � � � −h2Z1 0 0 0� � � � � −h12(Z1 + Z2) 0 0� � � � � � −h12Z2 0� � � � � � � −U

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (14)

where

12 = PAd + NT2 − N1,

22 = −Q1 − Q2 − (1 − �)Q3 − N2 − NT2 ,

and �11, U and h12 are defined as in Theorem 1. Whenh12 → 0, if LMI (3) in Xu and Lam (2005) is feasible forgiven h = h2, setting Q1 = Q/2, Q2 = Q/2, Q3 = 0, N1 = Y ,N2 = W , M1 = 0, M2 = Q1, S1 = 0, S2 = −Q2, Z1 = Z andZ2 = Z, LMI (14) is also feasible. That is to say, if system(1) is stable with d(t) = h2 for � = 0, then it is stable whenh2 − ��d(t)�h2 with sufficient small scalar � > 0 no matterhow large the value of � is (Since Q3 can be set as 0.).

In fact, Theorem 1 gives a criterion for system (1) with d(t)

satisfying (2) and (3). In many cases, the information of thederivative of delay is unknown. Regarding this circumstance,a rate-independent criterion for a delay only satisfying (2) isderived as follows by choosing Q3 = 0 in Theorem 1.

Corollary 6. Given scalars 0�h1 < h2, the linear system (1)with time-varying delay d(t) satisfying (2) is asymptotically

Y. He et al. / Automatica 43 (2007) 371–376 375

Table 1Allowable upper bound of h2 with given h1 for different �

h1 Methods � = 0.5 � = 0.9 Unknown �

0 Fridman and Shaked (2002, 2003), Han (2004a), He et al. (2004b), Wu, He, She, and Liu (2004) 2.00 1.18 0.99Jiang and Han (2005) — — 1.01Our results 2.04 1.37 1.34

1 Jiang and Han (2005) — — 1.64Our results 2.07 1.74 1.74

2 Jiang and Han (2005) — — 2.39Our results 2.43 2.43 2.43

3 Jiang and Han (2005) — — 3.20Our results 3.22 3.22 3.22

4 Jiang and Han (2005) — — 4.06Our results 4.07 4.07 4.07

4.4697 Our results 4.47 4.47 4.47

Table 2Allowable upper bound of h2 with given h1

� Method h1 0 0.3 0.5 0.8 1 2

Unknown � Jiang and Han (2005) h2 0.67 0.91 1.07 1.33 1.50 2.39Corollary 6 h2 0.77 0.94 1.09 1.34 1.51 2.40

0.3 Theorem 1 h2 2.19 2.19 2.20 2.20 2.21 2.40

stable if there exist matrices P =P T > 0, Qi =QTi �0, i=1, 2,

Zj = ZTj > 0, j = 1, 2, Ni , Mi and Si , i = 1, 2 such that the

following LMI holds:

� =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�11 �12 M1 −S1 h2N1 h12S1 h12M1 ATU

� �22 M2 −S2 h2N2 h12S2 h12M2 ATdU

� � −Q1 0 0 0 0 0� � � −Q2 0 0 0 0� � � � −h2Z1 0 0 0� � � � � −h12(Z1 + Z2) 0 0� � � � � � −h12Z2 0� � � � � � � −U

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (15)

where

�11 = PA + ATP +2∑

i=1

Qi + N1 + NT1 ,

�22 = S2 + ST2 − N2 − NT

2 − M2 − MT2 ,

and �12, U, h12 are defined in Theorem 1.

3. Numerical examples

In this section, we use two numerical examples to show thebenefits of our results.

Example 7. Consider system (1) with

A =[−2 0

0 −0.9

], Ad =

[−1 0−1 −1

].

For various �, the computed upper bounds, h2, which guar-antee the stability of system (1) for given lower bounds, h1, are

listed in Table 1. When h1 = 0, it is clear that our results areimprovement over those in Fridman and Shaked (2002, 2003),Han (2004a), He et al. (2004b) and Wu, He, She, and Liu(2004). On the other hand, for systems with time-varying delayin a range, Table 1 also lists the comparison between our resultsand those in Jiang and Han (2005) for the cases of � = 0.5,0.9 and unknown �, and shows the merits of Theorem 1 andCorollary 6.

Example 8. Consider system (1) with

A =[

0 1−1 −2

], Ad =

[0 0

−1 1

].

Let �0 ∈ (0, 140 ). It is reported in Kharitonov and Niculescu

(2003) that system (1) remains stable for h1 = (1 − 1640�0),

376 Y. He et al. / Automatica 43 (2007) 371–376

h2 = (1 + 1640�0) and � < 1 − 8�0. For example, when �0 = 1

40 ,system (1) is stable for h1 = 1 − 3.9063 × 10−5, h2 = 1 +3.9063 × 10−5 and � = 0.8. However, for � = 0.8, Theorem 1yields h1 =0.72 and h2 =1.28, which are better than those ob-tained in Kharitonov and Niculescu (2003). In addition, Table 2lists the upper bounds of h2, which guarantee the stability ofsystem (1) by using Corollary 6, for given lower bounds of h1.On the other hand, the lower bounds of h1, which guaranteethe stability of system (1) by using Theorem 1 and Corollary6 for given upper bounds of h2 can be derived similarly. Forexample, the corresponding h1 is obtained as 1.55 for h2 = 2and unknown � by using Corollary 6.

4. Conclusion

In this paper, a new class of Lyapunov functionals is con-structed to study the stability problems for systems with time-varying delay in a range. The numerical results seem to suggestthat the proposed methods may improve the results in someexisting papers.

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