Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2005; 15:923–933Published online 4 November 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1039

Augmented Lyapunov functional and delay-dependentstability criteria for neutral systems

Yong He1,2, Qing-Guo Wang1,n,y, Chong Lin1 and Min Wu2

1Department of Electrical and Computer Engineering, National University of Singapore,

10 Kent Ridge Crescent, Singapore 119260, Singapore2School of Information Science and Engineering, Central South University, Changsha 410083, China

SUMMARY

In this paper, an augmented Lyapunov functional is proposed to investigate the asymptotic stability ofneutral systems. Two methods with or without decoupling the Lyapunov matrices and system matrices aredeveloped and shown to be equivalent to each other. The resulting delay-dependent stability criteria areless conservative than the existing ones owing to the augmented Lyapunov functional and the introductionof free-weighting matrices. The delay-independent criteria are obtained as an easy corollary. Numericalexamples are given to demonstrate the effectiveness and less conservativeness of the proposed methods.Copyright # 2005 John Wiley & Sons, Ltd.

KEY WORDS: neutral systems; delay-dependent stability; linear matrix inequality (LMI); Lyapunovfunctional

1. INTRODUCTION

Neutral systems are frequently encountered in various engineering systems, includingpopulation ecology, heat exchangers and repetitive control [1–3]. The existing stability criteriafor neutral systems can be classified into two types: delay-dependent ones which includeinformation on the size of delays [4–17], and delay-independent ones which are applicable todelays of arbitrary size [18–20]. Delay-dependent results are usually less conservative than delay-independent ones, especially when the size of the delay is small.

In the past two decades, the fixed model transformation and parameterized modeltransformation are usually employed to obtain the delay-dependent conditions for neutralsystem. Four basic fixed model transformations are addressed in Reference [12]. Among them,the descriptor model transformation method combined with Park’s or Moon et al.’s inequalities[21, 22] is the most efficient [11, 12, 14]. Recently, in order to reduce the conservatism, a free-weighting matrix method is proposed in References [16, 23–25], in which the boundingtechniques on some cross-product terms are not involved. Especially, Wu et al. [17] combined

Received 2 August 2005Revised 13 September 2005

Accepted 26 September 2005Copyright # 2005 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: Q.-G. Wang, Department of Electrical and Computer Engineering, National University ofSingapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore.

the parameterized model transformation into the Lyapunov functional and employed the free-weighting matrix method to obtain less conservative delay-dependent stability criteria forneutral systems. We notice that it is hard to further reduce the conservatism by using the sametypes of Lyapunovs as in the above works.

In this paper, we propose an augmented Lyapunov functional, which takes into account thedelay term. Owing to the augmented Lyapunov functional, improved delay-dependent stabilitycriteria for neutral systems are derived by using the free-weighting matrix method. We first developa delay-dependent condition by introducing some free-weighting matrices with the aid of theLeibniz–Newton formula. Then, we derive an alternative delay-dependent criterion by retainingthe derivative of the state and introducing additional free-weighting matrices with the aid of thesystem equation. The relationship between two methods is clearly established. Finally, numericalexamples are given to demonstrate the effectiveness and the merit of the proposed method.

2. PROBLEM FORMULATION

Consider the following neutral system S:(’xðtÞ � C ’xðt� tÞ ¼ AxðtÞ þ Adxðt� tÞ; t > 0

xðtÞ ¼ fðtÞ; t 2 ½�t; 0�ð1Þ

where xðtÞ 2 Rn is the state vector; t > 0 is a constant time delay; and A;Ad and C are constantmatrices with appropriate dimensions. fðtÞ denotes an initial condition which is continuousvector-valued initial function of t 2 ½�t; 0�: Define operator D: Cð½�t; 0�;RnÞ ! Rn as

Dxt ¼ xðtÞ � Cxðt� tÞ

Definition 1 (Hale and Verduyn Lund [2])Operator D is said to be stable if the zero solution of the homogeneous difference equationDxt ¼ 0; t50; x0 ¼ c 2 ff 2 Cð½�t; 0� : Df ¼ 0g is uniformly asymptotically stable.

The stability of operator D is necessary for the stability of neutral system S; which is alwayssatisfied when jjCjj51:

3. THE RESULTS

By introducing an augmented Lyapunov functional, we obtain a delay-dependent stabilitycriterion for system S as follows.

Theorem 1Given a scalar t50; the neutral system S is asymptotically stable if the operator D is stable andthere exist matrices

L ¼

L11 L12 L13

LT12 L22 L23

LT13 LT

23 L33

2664

377550 with L11 > 0

Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2005; 15:923–933

Y. HE ET AL.924

Q ¼Q11 Q12

QT12 Q22

" #50

Z ¼Z11 Z12

ZT12 Z22

" #50 and Mi; i ¼ 1; 2; 3

such that the following LMI holds:

G ¼

G11 G12 G13 G14 �tM1 ATS

GT12 G22 G23 G24 �tM2 AT

d S

GT13 GT

23 G33 G34 �tM3 CTS

GT14 GT

24 GT34 �tZ11 �tZ12 0

�tMT1 �tMT

2 �tMT3 �tZT

12 �tZ22 0

SA SAd SC 0 0 �S

26666666666664

3777777777777550 ð2Þ

where

G11 ¼ GAþ ATGT þ L13 þ LT13 þQ11 þ tZ11 þM1 þMT

1

G12 ¼ GBþ ATL12 þ LT23 � L13 þMT

2 �M1

G13 ¼ GC þ L12 þMT3

G14 ¼ tðL33 þ ATL13Þ

G22 ¼ ATd L12 þ LT

12Ad � L23 � LT23 �Q11 �M2 �MT

2

G23 ¼ LT12C þ L22 �Q12 �MT

3

G24 ¼ tð�L33 þ ATd L13Þ

G33 ¼ �Q22

G34 ¼ tðL23 þ CTL13Þ

S ¼ Q22 þ tZ22

G ¼ L11 þQ12 þ tZ12


AUGMENTED LYAPUNOV FUNCTIONAL AND DELAY-DEPENDENT STABILITY CRITERIA 925

ProofChoose a Lyapunov functional candidate to be

VðxtÞ :¼ zT1 ðtÞLz1ðtÞ þZ t

t�tzT2 ðsÞQz2ðsÞ dsþ

Z 0

�t

Z t

tþyzT2 ðsÞZz2ðsÞ ds dy ð3Þ

where

L ¼

L11 L12 L13

LT12 L22 L23

LT13 LT

23 L33

2664

377550 with L11 > 0

Q ¼Q11 Q12

QT12 Q22

" #50 and Z ¼

Z11 Z12

ZT12 Z22

" #50

are to be determined and

z1ðtÞ ¼

xðtÞ

xðt� tÞR tt�t xðsÞ ds

2664

3775; z2ðtÞ ¼

xðtÞ

’xðtÞ

" #

From the Leibniz–Newton formula, the following equation is true for any matrices Mi withappropriate dimensions, i ¼ 1; 2; 3:

2½xTðtÞM1 þ xTðt� tÞM2 þ ’xTðt� tÞM3� xðtÞ �Z t

t�t’xðsÞ ds� xðt� tÞ

� �¼ 0 ð4Þ

Calculating the derivative of VðxtÞ along the solution of S yields

’VðxtÞ ¼ 2zT1 ðtÞL’z1ðtÞ þ zT2 ðtÞQz2ðtÞ � zT2 ðt� tÞQz2ðt� tÞ

þ tzT2 ðtÞZz2ðtÞ �Z t

t�tzT2 ðsÞZz2ðsÞ ds

¼ 2zT1 ðtÞL

’xðtÞ

’xðt� tÞ

xðtÞ � xðt� tÞ

2664

3775þ zT2 ðtÞQz2ðtÞ � zT2 ðt� tÞQz2ðt� tÞ

þ tzT2 ðtÞZz2ðtÞ �Z t

t�tzT2 ðsÞZz2ðsÞ ds

þ 2½xTðtÞM1 þ xTðt� tÞM2 þ ’xTðt� tÞM3�

� xðtÞ �Z t


� �

:¼1

t

Z t

t�tZT1 ðt; sÞ #GZ1ðt; sÞ ds ð5Þ


Y. HE ET AL.926

where

Z1ðt; sÞ ¼ ½xTðtÞ xTðt� tÞ ’xTðt� tÞ xTðsÞ ’xTðsÞ�T

#G ¼

G11 þ ATSA G12 þ ATSAd G13 þ ATSC G14 �tM1

GT12 þ AT

d SA G22 þ ATd SAd G23 þ AT

d SC G24 �tM2

GT13 þ CTSA GT

23 þ CTSAd G33 þ CTSC G34 �tM3

GT14 GT

24 GT34 �tZ11 �tZ12

�tMT1 �tMT

2 �tMT3 �tZT

12 �tZ22

26666666664

37777777775

S ¼Q22 þ tZ22

and Gij ; i ¼ 1; 2; 3; i4j43 are defined as in Theorem 1. By Schur complement, the inequalityG50 is equivalent to #G50; which gives ’VðxtÞ5� ejjxðtÞjj2 for a sufficiently small e > 0: Thisproves the asymptotic stability of system S: &

It is seen that Theorem 1 is established by adopting the Leibniz–Newton formula whichprovides free-weighting matrices Mi; i ¼ 1; 2; 3 as in (4). In the following, we derive analternative delay-dependent criterion by taking into account the system equation S; whichprovides another set of free-weighting matrices.

Theorem 2Given a scalar t50; the neutral system S is asymptotically stable if operator D is stable andthere exist matrices

L ¼

L11 L12 L13

LT12 L22 L23

LT13 LT

23 L33

2664

377550 with L11 > 0

Q ¼Q11 Q12

QT12 Q22

" #50

Z ¼Z11 Z12

ZT12 Z22

" #50; U; Mi and Tj ; i ¼ 1; 2; 3; j ¼ 1; . . . ; 6


F ¼

F11 F12 F13 F14 F15 F16

FT12 F22 F23 F24 F25 F26

FT13 FT

23 F33 F34 F35 F36

FT14 FT

24 FT34 F44 F45 F46

FT15 FT

25 FT35 FT

45 �tZ11 �tZ12

FT16 FT

26 FT36 FT

46 �tZT12 �tZ22

266666666664

37777777777550 ð6Þ



whereF11 ¼ L13 þ LT

13 þQ11 þ tZ11 þM1 þMT1 � T1A� ATTT

1

F12 ¼ L11 þQ12 þ tZ12 þUT þ T1 � ATTT2

F13 ¼ LT23 � L13 þMT

2 �M1 � T1Ad � ATTT3

F14 ¼ L12 þMT3 � T1C � ATTT

4

F15 ¼ tL33 � ATTT5

F16 ¼ �tM1 � ATTT6

F22 ¼ Q22 þ tZ22 þ T2 þ TT2

F23 ¼ L12 �U þ TT3 � T2Ad

F24 ¼ �T2C þ TT4

F25 ¼ tL13 þ TT5

F26 ¼ �tU þ TT6

F33 ¼ �L23 � LT23 �Q11 �M2 �MT

2 � T3Ad � ATd T

T3

F34 ¼ L22 �Q12 �MT3 � T3C � AT

d TT4

F35 ¼ �tL33 � ATd T

T5

F36 ¼ �tM2 � ATd T

T6

F44 ¼ �Q22 � T4C � CTTT4

F45 ¼ tL23 � CTTT5

F46 ¼ �tM3 � CTTT6

ProofChoose the same augmented Lyapunov functional as in (3). It is clear that the followingequation holds by system equation S:

2

Z t

t�tZT2 ðt; sÞT ½ ’xðtÞ � C ’xðt� tÞ � AxðtÞ � Adxðt� tÞ� ds ¼ 0 ð7Þ


Y. HE ET AL.928

where

T ¼ ½TT1 TT

2 TT3 TT

4 TT5 TT

6 �T

Z2ðt; sÞ ¼ ½xTðtÞ ’xTðtÞ xTðt� tÞ ’xTðt� tÞ xTðsÞ ’xTðsÞ�T

On the other hand, ’xðtÞ in ’VðxtÞ is retained (contrasting with the proof of Theorem 1, in which

’xðtÞ is replaced with the system equation S) and (4) is slightly modified to

2½xTðtÞM1 þ ’xTðtÞU þ xTðt� tÞM2 þ ’xTðt� tÞM3�

� xðtÞ �Z t


� �¼ 0 ð8Þ

Similar to the proof of Theorem 1 with (7) and (8) being added into the expression of ’VðxtÞ; theresult follows immediately. &

Remark 1Theorems 1 and 2 are based on a newly proposed augmented Lyapunov functional of form (3),which contains a structure more general than the traditional ones as those in References[8, 16, 17]. For instance, the first term of (3) involves the delay term xðt� tÞ; and thus is moregeneral than the simply case of operator D: This new type of Lyapunov functional enables us toestablish less conservative results. For theoretical comparison, let us take Reference [17] forexample. It can be verified that the result in Reference [17] is recovered by setting L11 ¼ P11;L12 ¼ �P11C; L13 ¼ P12; L22 ¼ CTP11C; L23 ¼ �CTP12; L33 ¼ P22; Q12 ¼ Z12 ¼ 0; Z11 ¼W ;Z22 ¼ Z; Q11 ¼ Q; Q22 ¼ R; M1 ¼ N1; U ¼ N2; M2 ¼ N3; M3 ¼ N4; T5 ¼ 0 and T6 ¼ 0:

In fact, Theorems 1 and 2 are equivalent to each other, which is shown as follows. Setting

J ¼

I AT 0 0 0 0

0 I 0 0 0 0

0 ATd I 0 0 0

0 CT 0 I 0 0

0 0 0 0 I 0

0 0 0 0 0 I

2666666664

3777777775

ð9Þ

then it follows that

P ¼ JFJT ¼

P11 P12 P13 P14 G14 P16

PT12 F22 P23 P24 F25 F26

PT13 PT

23 P33 P34 G24 P36

PT14 PT

24 PT34 P44 G34 P46

GT14 FT

25 GT24 GT

34 �tZ11 �tZ12

PT16 FT

26 PT36 PT

46 �tZT12 �tZ22

266666666664

377777777775

ð10Þ



where

P11 ¼ G11 þUTAþ ATU þ ATSA

P12 ¼ Gþ T1 þUT þ ATS þ ATT2

P13 ¼ G12 � ATU þUTAd þ ATSAd

P14 ¼ G13 þUTC þ ATSC

P16 ¼ �tðM1 þ ATUÞ

P23 ¼ L12 �U þ TT2 Ad þ SAd þ TT

3

P24 ¼ SC þ TT2 C þ TT

4

P33 ¼ G22 �UTAd � ATdU þ AT

d SAd

P34 ¼ G23 �UTC þ ATd SC

P36 ¼ �tðM2 þ ATdUÞ

P44 ¼ G33 þ CTSC

P46 ¼ �tðM3 þ CTUÞ

and S;G and Gij ; i ¼ 1; 2; 3; i4j43 are defined as in Theorem 1. From (10), it remains to showthat P50 is equivalent to #G50: To this end, if P50 holds, then #G50 holds since P reduces to#G by removing the second row and the second column in P and replacing variables M1;M2 andM3 in #G by M1 þ ATU; M2 þ AT

dU and M3 þ CTU; respectively. On the other hand, if #G50;P50 is feasible by setting T1 ¼ �G ¼ �ðL11 þQ12 þ tZ12Þ; T2 ¼ �S ¼ �ðQ22 þ tZ22Þ;T3 ¼ �L12; T4 ¼ 0; T5 ¼ �tL13; T6 ¼ 0 and U ¼ 0: Therefore, this shows the equivalenceof Theorems 1 and 2.

Remark 2Theorem 2 is suitable for dealing with uncertain systems with polytopic-type uncertainties byusing parameter-dependent Lyapunov functionals as in Reference [23]. This is because the LMIcondition of Theorem 2 does not involve the product of system matrices and Lyapunovmatrices. As for norm-bounded uncertain case, Theorems 1 and 2 are applicable for the robuststability analysis following a similar line to References [17, 24].


Y. HE ET AL.930

Before concluding this section, we would like to point out that our method can be used toestablish delay-independent criteria. To see this, setting Z ¼ eI ; where e > 0 is a sufficiently smallpositive scalar, L13 ¼ L23 ¼ L33 ¼ 0; T5 ¼ T6 ¼ 0 and U ¼M1 ¼M2 ¼M3 ¼ 0 in conditionsof Theorem 2, the following delay-independent result is straightforward.

Corollary 1For any scalar t50; the neutral system S is asymptotically stable if operator D is stable andthere exist matrices

L ¼L11 L12

LT12 L22

" #50 with L11 > 0; Q ¼

Q11 Q12

QT12 Q22

" #50 and Tj ; j ¼ 1; . . . ; 4


C11 C12 C13 C14

CT12 C22 C23 C24

CT13 CT

23 C33 C34

CT14 CT

24 CT34 C44

26666664

3777777550 ð11Þ

where

C11 ¼ Q11 � T1A� ATTT1

C12 ¼ L11 þQ12 þ T1 � ATTT2

C13 ¼ �T1Ad � ATTT3

C14 ¼ L12 � T1C � ATTT4

C22 ¼ Q22 þ T2 þ TT2

C23 ¼ L12 � T2Ad

C24 ¼ �T2C þ TT4

C33 ¼ �Q11 � T3Ad � ATd T

T3

C34 ¼ L22 �Q12 � T3C � ATd T

T4

C44 ¼ �Q22 � T4C � CTTT4



With a similar treatment, an alternative delay-independent result, which is equivalent toCorollary 1, can be obtained as a direct corollary of Theorem 1. For brevity, we omit the details.

4. NUMERICAL EXAMPLES

In this section, we use two numerical examples to show the less conservativeness of our results.

Example 1Consider the neutral system S with

A ¼�2 0

0 �0:9

" #; Ad ¼

�1 0

�1 �1

" #; C ¼

c 0

0 c

" #; 04c51

Table I lists the maximum upper bound of t for this system in case of different c’s. It is clear thatthe method in this paper produces better results than those in References [8, 12, 17].

Example 2Consider the neutral system S with

A ¼�1:7073 0:6856

0:2279 �0:6368

" #; Ad ¼

�2:5026 �1:0540

�0:1856 �1:5715

" #; C ¼

0:0558 0:0360

0:2747 �0:1084

" #

The upper bound of delay obtained in References [16, 17, 25] are 0.5735, 0.6054 and 0.5937,respectively. By Theorems 1 and 2, the computed value is 0.6189, which is larger than those inReferences [16, 25]. In addition, setting C ¼ 0; this system reduces to a linear nominal systemwith time-invariant delay. The upper bound of delay obtained in References [11, 12, 14, 23, 24]are all 0.6903, and 0.7163 in Reference [17], while the computed value is 0.7918 by Theorems 1and 2, which is larger than those in References [11, 12, 14, 17, 23, 24].

5. CONCLUSIONS

In this paper, an augmented Lyapunov functional is constructed to study the delay-dependentstability problems for neutral systems. The resulting criteria are shown less conservative thansome existing ones. Numerical examples given have convinced the less conservativeness.

Table I. Maximum upper bound of t:

c 0 0.1 0.3 0.5 0.7 0.9

Fridman et al. [12] 4.47 3.49 2.06 1.14 0.54 0.13Han [8] 4.35 4.33 4.10 3.62 2.73 0.99Wu et al. [17] 4.47 4.35 4.13 3.67 2.87 1.41Theorems 1 and 2 4.47 4.42 4.17 3.69 2.87 1.41


Y. HE ET AL.932

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Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems