algebraic stability criteria of linear neutral systems with multiple time delays

Applied Mathematics and Computation 155 (2004) 643–653

www.elsevier.com/locate/amc

Algebraic stability criteria of linearneutral systems with multiple time delays

Ping He a,*, D.Q. Cao b

a Department of Applied Mathematics, Southwest Jiaotong University,

Chengdu 610031, People�s Republic of Chinab Department of Applied Mechanics and Engineering, Southwest Jiaotong University,

Chengdu 610031, People�s Republic of China

Abstract

The asymptotic stability of linear neutral systems with multiple time delays is in-

vestigated in this paper. Based on the characteristic equation of the system, simple

delay-independent stability criteria are derived in terms of the spectral radius of mod-

ulus matrices. The conditions of the new stability criteria are easy to check via Routh–

Hurwitz and Schur–Cohn criteria. Numerical examples are given to illustrate our main

criteria.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Neutral systems; Algebraic criteria; Delay independent stability; Multiple time delays

1. Introduction

Mathematical models with time delays are often encountered in various

engineering systems due to measurement and computational delays. The

problem of stability analysis for neutral delay-differential systems has received

considerable attention and has been one of the most interesting topics in the

literature. There have been a number of stability criteria based on the char-

acteristic equation approach, involving the determination of eigenvalues,measures and norms of matrices, or matrix conditions in terms of Hurwitz

* Corresponding author.

E-mail addresses: [email protected] (P. He), [email protected] (D.Q. Cao).

0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0096-3003(03)00806-3

mail to: [email protected]

644 P. He, D.Q. Cao / Appl. Math. Comput. 155 (2004) 643–653

matrices (see, for example, Stroinski [1], Hale et al. [2], Li [3], Hale and Ver-

duyn Lunel [4], Hu and Hu [5], Bellen et al. [6] and Hu et al. [7]). In terms of theLyapunov function and matrix inequalities, some stability criteria (delay-in-

dependent and/or delay-dependent) have been presented in Khusainov and

Yunkova [8], Lien et al. [9], Fridman [10], Lien [11] and Niculescu [12].

The study of asymptotic stability has been extended to neutral systems with

multiple time delays. Based on the characteristic equation of the system, Hui

and Hu [13] proposed delay-independent stability criteria in terms of the matrix

measure and spectral norm of matrices. Further discussions for the criteria in

Hui and Hu [13] can be found in Chen [14]. The stability conditions in Hui andHu [13], however, are very conservative owing to the characteristics of matrix

measure and norm. Employing the linear matrix inequality (LMI) approach,

sufficient conditions for the stability of neutral systems with multiple delays

have been developed by Park and Won [15], Chen et al. [16] and Park [17] to

make the criteria less conservative.

Motivated by the work of Hu et al. [7] for a linear neutral system with a

single delay, we investigate the asymptotic stability of linear neutral systems

with multiple time delays in this article. Based on the characteristic equation ofthe system, simple delay-independent stability criteria are derived. The stability

conditions of our criteria are easy to check via Routh–Hurwitz and Schur–

Cohn criteria. Numerical examples are given to demonstrate the validity of the

criteria proposed and to compare them with the existing ones.

2. Notations and preliminaries

Let RnðCnÞ denote the n-dimensional real (complex) space and Rn�nðCn�nÞdenote the set of all real (complex) n by n matrices. I denotes the unit matrix of

appropriate order. kjðAÞ and qðAÞ denote the jth eigenvalue and the spectral

radius of A, respectively. jAj denote the modulus matrix of A; A6B represents

that the elements of A and B, satisfy the inequality aij 6 bij for all i and j.kAkð:¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikmaxðA�AÞ

pÞ and lðAÞð:¼ 1

2kmaxðAþ A�ÞÞ denote the spectral norm and

the matrix measure of A, respectively.Consider linear neutral delay-differential systems with multiple time delays

described by

_xðtÞ ¼ AxðtÞ þXmj¼1

Bjxðth

� sjÞ þ Cj _xðt � sjÞi; ð1Þ

where xðtÞ 2 Cn�1 is the state vector, the constant parameters sj P 0 withs ¼ maxfsj; j ¼ 1; 2; . . . ;mg represent the delay arguments, A, Bj and Cj 2 Cn�n

(j ¼ 1; 2; . . . ;m). The system matrix A is assumed to be a Hurwitz matrix, that

is all the eigenvalues of A have negative real parts.

P. He, D.Q. Cao / Appl. Math. Comput. 155 (2004) 643–653 645

The characteristic equation of the neutral system (1) is described by

PðsÞ, det sI

"� A�

Xmj¼1

ðBj þ sCjÞ expð � ssjÞ#¼ 0: ð2Þ

Here P ðsÞ denotes the characteristic function. The following three lemmas arecited and will be used in the proof of our main results.

Lemma 2.1 [4]. If aD ¼ supfRðsÞ : PðsÞ ¼ 0g and aD < 0, then the neutralsystem (1) is asymptotically stable.

Lemma 2.2 [18]. Let R 2 Cn�n. If qðRÞ < 1, then ðI � RÞ�1 exists and

ðI � RÞ�1 ¼ I þ Rþ R2 þ � � � : ð3Þ

Lemma 2.3 [18]. Let R, T and V 2 Cn�n. If jRj6 V , then

(a) jRT j6 jRjjT j6 V jT j,(b) jRþ T j6 jRj þ jT j6 V þ jT j,(c) qðRÞ6 qðjRjÞ6 qðV Þ,(d) qðRT Þ6 qðjRjjT jÞ6 qðV jT jÞ, and(e) qðRþ T Þ6 qðjRþ T jÞ � qðjRj þ jT jÞ6 qðV þ jT jÞ:

Theorem 2.1. The neutral system (1) is asymptotically stable if A is Hurwitz and

sup qXmj¼1

ðsI��

"� AÞ�1ðBj þ sCjÞ

��#< 1; 8s 2 C such that RðsÞP 0:

ð4Þ

Proof. Denote njðsÞ ¼ expð�ssjÞ (j ¼ 1; 2; . . . ;m). Then, jnjðsÞj6 1 for

RðsÞP 0. In view of

PðsÞ ¼ det sI

"� A�

Xmj¼1

ðBj þ sCjÞ expð � ssjÞ#

¼ det½sI � A� det I

"� ðsI � AÞ�1

Xmj¼1

ðBj þ sCjÞnjðsÞ#

ð5Þ


and A is Hurwitz, we have P ðsÞ 6¼ 0 for RðsÞP 0 if

sup qXmj¼1

ðsI"

� AÞ�1ðBj þ sCjÞnjðsÞ#< 1; 8s 2 C such that RðsÞP 0:

ð6Þ

According to Lemma 2.3 for RðsÞP 0

qXmj¼1

ðsI"

� AÞ�1ðBj þ sCjÞnjðsÞ#6q

Xmj¼1

ðsI��


�� njðsÞ�� #

6qXmj¼1

ðsI��


��#: ð7Þ

Therefore, the condition (4) implies that the inequality (6) holds, i.e., RðsiÞ < 0

where si is the ith (i ¼ 1; 2; . . .) root of the characteristic Eq. (2).

Assume that there exists a sequence of roots sn of the Characteristic Eq. (2)whose real parts are not bounded away from zero, i.e., RðsnÞ ! 0 and

RðsnÞ < 0. Since the system matrix A is Hurwitz, each eigenvalue

kiXmj¼1

ðsI"

� AÞ�1ðBj þ sCjÞnjðsÞ#

ð8Þ

is an analytic function of s for RðsÞP 0. Thus, the condition (4) implies that

there exists a positive constant � such that for RðsÞP 0

sup qXmj¼1

ðsI"

� AÞ�1ðBj þ sCjÞnjðsÞ#

¼ supmaxi kiXmj¼1

ðsI"�� AÞ�1ðBj þ sCjÞnjðsÞ

#�� ¼ 1� �: ð9Þ

Hence, we have

sup qXmj¼1

ðnI"

� AÞ�1ðBj þ nCjÞ#6 1� � for RðnÞ ¼ 0: ð10Þ

Since RðsnÞ ! 0, for a given positive constant �1 < �, there exists an integer

n� such that jRðsnÞj is sufficiently small, RðsnÞ < 0 and

maxi

kiXmj¼1

ðsnI"��

�� AÞ�1ðBj þ snCjÞ#�� supq

Xmj¼1

ðnI"

� AÞ�1ðBj þ nCjÞ#�� < �1

for n > n� and RðnÞ ¼ 0: ð11Þ


Thus, we have

kiXmj¼1

ðsnI"�� AÞ�1ðBj þ snCjÞ

#��6 supqXmj¼1

ðnI"

� AÞ�1ðBj þ nCjÞ#þ �1

6 1� �þ �1 < 1; for n > n� and RðnÞ ¼ 0:

ð12Þ

By choosing nð> n�Þ large enough, this leads to

kiXmj¼1

ðsnI"�� AÞ�1ðBj þ snCjÞ

#njðsnÞ

�� < 1: ð13Þ

Therefore, for RðsnÞ ! 0 and RðsnÞ < 0

PðsÞ ¼ det½sI � A� det I

"� ðsI � AÞ�1

Xmj¼1

ðBj þ sCjÞnjðsÞ#6¼ 0: ð14Þ

This contradicts the assumption that there exists a sequence of roots sn such

that RðsnÞ ! 0 and RðsnÞ < 0. In view of Lemma 2.1, the proof is com-

pleted. �

Since the system matrix A is assumed to be Hurwitz, the matrix I � A is non-singular, i.e., ðI � AÞ�1

exists. Let us define N ¼ ðI � AÞ�1ðI þ AÞ and

Lj ¼ ðI � AÞ�1ðBj þ CjÞMj ¼ ðI � AÞ�1ðBj � CjÞ

�; j ¼ 1; 2; . . . ;m: ð15Þ

Using the very similar ways as the Proof of Theorem 2.2 of Hu et al. [7], we

can easily obtain the following lemma.

Lemma 2.4. If the system matrix is Hurwitz, then for RðsÞ > 0, jzj6 1 andj ¼ 1; 2; . . . ;m, the following equality

ðsI � AÞ�1ðBj þ sCjÞ ¼ ðI � zNÞ�1ðLj þ zMjÞ ð16Þ

holds, where Lj and Mj are defined by (15), and

s ¼ 1� z1þ z

for RðsÞ > 0 and jzj6 1: ð17Þ

The following theorem immediately follows from Theorem 2.1 and Lemma

2.4.


Theorem 2.2. The neutral system (1) is asymptotically stable if A is Hurwitz and

sup qXmj¼1

ðI��

"� zNÞ�1ðLj þ zMjÞ

��#< 1; 8z 2 C and jzj6 1: ð18Þ

3. Main stability criteria

Assume that qðjN jÞ < 1. Then, it follows from Lemma 2.2 that ðI � jN jÞ�1

exists. For an integer qP 1, let us define the matrices

Xjð0Þ ¼ ðI � jN jÞ�1ðjNLjj þ jNMjjÞ;XjðqÞ ¼

Pqk¼1ðjNkLjj þ jNkMjjÞ þ ðI � jN jÞ�1 jNqþ1Ljj þ jNqþ1Mjj

� �;

(

ð19Þ

where the matrices Lj and Mj (j ¼ 1; 2; . . . ;m) are defined by (15).

Lemma 3.1. For integer qP 1, Xjðqþ 1Þ6XjðqÞ (j ¼ 1; 2; . . . ;m).

Proof. It follows from the definition (19) that

Xjðqþ 1Þ �XjðqÞ ¼ jNqþ1Ljj þ jNqþ1Mjj þ ðI � jN jÞ�1 jNqþ2Ljj�

þ jNqþ2Mjj�

� ðI � jN jÞ�1 jNqþ1Ljj�

þ jNqþ1Mjj�

6 jNqþ1Ljj þ jNqþ1Mjjþ ðI � jN jÞ�1ðjN j � IÞ jNqþ1Ljj

�þ jNqþ1Mjj

�¼ I�

� ðI � jN jÞ�1ðI � jN jÞ�

jNqþ1Ljj�

þ jNqþ1Mjj�¼ 0:

ð20Þ

Thus, for integer qP 1, Xjðqþ 1Þ6XjðqÞ. The proof is completed. �

Lemma 3.2. Assume that qðjN jÞ < 1. Then, for z 2 C, jzj6 1, integer qP 1 andj ¼ 1; 2; . . . ;m, the following inequality

jðI � zNÞ�1ðLj þ zMjÞj6 jLjj þ jMjj þ XjðqÞ6 jLjj þ jMjj þ Xjð0Þ ð21Þ

holds, where Lj and Mj, Xjð0Þ and XjðqÞ are defined by (15) and (19), respectively.

Proof. In view of

jzN j6 jN j; 8z 2 C and jzj6 1 ð22Þ

we have qðzNÞ6 qðjN jÞ < 1. It follows from Lemma 2.2 that ðI � zNÞ�1exists,

and


jðI � zNÞ�1j ¼ jI þ zN þ z2N 2 þ � � � j6 I þ jzN j þ jz2N 2j þ � � �6 I þ jN j þ jN j2 þ � � � ¼ ðI � jN jÞ�1

: ð23Þ

Thus, we have

jðI � zNÞ�1ðLj þ zMjÞj ¼ jðI þ zN þ z2N 2 þ � � �ÞðLj þ zMjÞj

¼ Lj

�� þ zMj þXq

k¼1

zkNkðLj þ zMjÞ

þ zqþ1ðI þ zN þ z2N 2 þ � � �ÞNqþ1ðLj þ zMjÞ��

6 jLj þ zMjj þXq

k¼1

jNkðLj þ zMjÞj

þ jðI � zNÞ�1jjNqþ1ðLj þ zMjÞj

6 jLjj þ jMjj þXq

k¼1

ðjNkLjj þ jNkMjjÞ

þ ðI � jN jÞ�1 jNqþ1Ljj�

þ jNqþ1Mjj�

¼ jLjj þ jMjj þ XjðqÞ: ð24Þ

Moreover, based on Lemma 3.1, we can easily obtain XjðqÞ6Xjð0Þ for integerqP 1 and j ¼ 1; 2; . . . ;m. Hence, the inequality (21) holds. The proof is com-

pleted. �

By means of Theorem 2.2, Lemmas 2.3 and 3.2, we can directly obtain ourmain results as follows.

Theorem 3.1. The neutral system (1) is asymptotically stable if the system matrixA is Hurwitz, qðjN jÞ < 1 and

e0,qXmj¼1

ðjLjj"

þ jMjj þ Xjð0ÞÞ#< 1; ð25Þ

where Lj, Mj and Xjð0Þ (j ¼ 1; 2; . . . ;m) are defined by (15) and (19), respectively.

Theorem 3.2. The neutral system (1) is asymptotically stable if the system matrixA is Hurwitz, qðjN jÞ < 1 and

eq,qXmj¼1

ðjLjj"

þ jMjj þ XjðqÞÞ#< 1 for some integer qP 1; ð26Þ

where Lj,Mj and XjðqÞ ðj ¼ 1; 2; . . . ;mÞ are defined by (15) and (19), respectively.


Remark 3.1. According to Theorem 3.1, the neutral delay-differential system

(1) is asymptotically stable if the linear system

_uðtÞ ¼ AuðtÞ ð27Þ

and the two linear difference systems

ynþ1 ¼Xmj¼1

ðjLjj"

þ jMjj þ Xjð0ÞÞ#yn ð28Þ

and

znþ1 ¼ jN jzn ð29Þ

are asymptotically stable.

It is easy to check the stability of system (27) by Routh–Hurwitz criterion

[19] and the systems (28) and (29) by Schur–Cohn criterion [19], respectively.

According to Lemma 3.1, XjðqÞ6Xjð0Þ for integer q6 1. Hence, Theorem 3.2 is

sharper than Theorem 3.1.

Remark 3.2. Based on the characteristic Eq. (2), Hui and Hu [13] derived a

sufficient condition for asymptotic of system (1) as

eh,lðAÞ þXmj¼1

kBjk þ1

1� c

Xmj¼1

kCjAk"

þXmj;k¼1

kCjBkk#< 0; ð30Þ

where c ¼Pm

j¼1 kCjk < 1.

Condition (30) may be relatively conservative due to the matrix measure and

the matrix norm operations. Furthermore, the condition lðAÞ < 0 is necessary

to satisfy sufficient condition (30), and is a strict restriction for application,

compared with our requirement that the system matrix A is Hurwitz. The

numerical examples in next section shows the effectiveness and advantages ofour new criteria.

4. Illustrative examples

Example 4.1. Consider system (1) with

A ¼�4 1

�2 �3

� ; B1 ¼

0:2 �0:03

�0:2 0:12

� ; B2 ¼ a

0:4 0:25

�0:12 0:3

� ;

C1 ¼0:2 0:0

�0:3 0:1

� ; C2 ¼ a

�0:3 0:0

�0:1 0:2

� ;

where a is a non-zero parameter.

0 5 10 15 200.7

0.8

0.9

1

1.1

Integer q

Sta

bilit

y bo

unds

| α

|

Fig. 1. Stability bounds jaj in Example 4.1.


With simple computation, we have qðjN jÞ ¼ 0:7273 < 1. Using Theorem

3.1, the stability bound can be easily obtained as jaj < 0:7696. The criterion of

Hui and Hu [13] gives the stability bound jaj < 0:0234. This example shows

that our stability criterion is less conservative than that given in [13]. Taking

different integer q, allowable upper bounds on a can be obtained according

to Theorem 3.2. Fig. 1 shows the stability bounds jaj against the integer q inthe condition (26) of Theorem 3.2. The allowable bound on jaj is 1.0732 for

q ¼ 20.


A ¼ �8 1

�4 �0:1

� ; B1 ¼

0:2 0:0�0:2 0:1

� ; B2 ¼

0:2 0:1�0:1 0:2

� ;

C1 ¼0:2 0:00:0 0:2

� ; C2 ¼

�0:2 �0:10:1 0:1

� :

With simple computation, we have qðjN jÞ ¼ 0:9378 < 1, e0 ¼ 5:2601. Since

e0 > 1, the conditions of Theorem 3.1 are not satisfied. However, further

computation yields eq ¼ 0:9757 for q ¼ 20. According to Theorem 3.2, the

system (1) is asymptotically stable. This shows Theorem 3.2 is shaper than

Theorem 3.1. It is evident that �h is positive since lðAÞ ¼ 0:1752 > 0. Thus, thestability criterion in Hui and Hu [13] cannot be satisfied.



A ¼ �29 1

�2 �30

� ; B1 ¼

0:08 �0:03�0:20 0:12

� ; B2 ¼

0:04 0:40�0:12 0:05

� ;

C1 ¼0:24 0:00�0:30 0:20

� ; C2 ¼

�0:30 0:40�0:25 0:20

� :

It is evident that the system matrix A is Hurwitz. With simple computation, we

have qðjN jÞ ¼ 0:9378 < 1, e0 ¼ 0:9936 < 1, and eq ¼ 0:9648 for q ¼ 40. Ac-

cording to Theorem 3.1 or Theorem 3.2, the system (1) is asymptotically stable.

Since c ¼ kC1k þ kC2k ¼ 1:0074 > 1, the stability criterion of Hui and Hu [13]

are not applicable in this example. Furthermore, it is interesting to note that

for c ¼ kC1k þ kC2k ¼ 1:0074 > 1 ðkC1k ¼ 0:4176; kC2k ¼ 0:5898Þ the stabilitycriteria proposed by Park and Won [15], Chen et al. [16] and Park [17], using

the Lyapunov method and LMI approach, also cannot be applied to guaranteethe stability of the system.

5. Conclusion

In this paper, we have investigated sufficient conditions for the asymptotic

stability of neutral systems with multiple time delays. Based on the charac-

teristic equation, delay independent stability criteria have been derived in termsof scalar inequalities involving the spectral radius of modulus matrices. The

checking of stability by our criteria can be carried out rather simply. Numerical

examples have demonstrated that the new stability criteria are less conservative

and more powerful comparing to those in the literature.

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algebraic stability criteria of linear neutral systems with multiple time delays

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