Exponential stability criterion for time delay systems with nonlinear uncertainties

Compared with the results in[10], our result has the following advantages: First, by decomposing the matrix A1into two part A11;A12and using the operator DðxtÞ ¼ xðtÞ A11 Rt thxðsÞds,

Exponential stability criterion for time-delay systems

with nonlinear uncertainties

Phan T Nam

Department of Mathematics, Quynhon University, 170 An Duong Vuong Road, Binhdinh, Vietnam

a r t i c l e i n f o

Keywords:

Exponential stability

Time-delays

Nonlinear uncertainties

Lyapunov function

Linear matrix inequality

a b s t r a c t

Exponential stability of time-delay systems with nonlinear uncertainties is studied in this paper Based on the Lyapunov method and the approaches of decomposing the matrix, a new exponential stability criterion is derived in terms of a matrix inequality, which allows

to compute simultaneously the two bounds that characterize the exponential nature of the solution Some numerical examples are also given to show the superiority of our result to those in the literature

Ó2009 Elsevier Inc All rights reserved

1 Introduction

Consider the following time-delay systems with nonlinear uncertainties:

_xđtỡ Ử Axđtỡ ợ A1xđt  hỡ ợ f đt; xđtỡỡ ợ f1đt; xđt  hỡỡ;

x0đhỡ Ử /đhỡ;



đ1:1ỡ

where xđtỡ 2 Rnis the state, A; A1 are given matrix, and initial condition is x0đhỡ Ử /đhỡ 2 Cđơh; 0; Rnỡ The time-varying parameter uncertainties f ; f1are assumed to be bounded

kf đt; xđtỡỡk 6akxđtỡk; kf1đt; xđt  hỡỡk 6a1kxđt  hỡk;

wherea;a1are positive numbers

Definition 1.1 The system(1.1)is d-stable, with d > 0, if there is a positive number N such that for each /đ:ỡ, the solution xđt; /ỡ of the system(1.1)satisfies

kxđt; /ỡk 6 Nedtk/k 8t P 0;

where k/k Ử maxfk/đtỡk : t 2 ơh; 0g N is called Lyapunov factor

Because of data errors, environmental noises, the difficulty of measuring various parameters, unavoidable approximation, etc., most real problems are modeled by delay systems with nonlinear uncertainties So, the stability problem of time-delay system with nonlinear uncertainties has been an interesting problem in the recent years (see in[2Ờ12]and reference therein) It is well known that the widely used method is the approach of Lyapunov functions with Razumikhin techniques and the stability conditions are presented in terms of the solution of either linear matrix inequalities or Riccati equations

[5,8,9] By using parameterized neutral models, some less conservative criteria, which are dependent on the stability of the operator, have proposed in[2,10Ờ12] By proposing a technique to adjust the Lyapunov functionals in[2,11,12], the

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E-mail address: pthnam@yahoo.com

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Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c

authors in[6,7]have reduced the stability of the operator and given some less conservative more criteria However, this tech-nique has not applied to the Lyapunov functional in[10] Therefore, the first purpose of this paper is to find an improvement the criterion in[10]by using this technique

Inspired by the method of decomposing the matrix in[1], we have decomposed the matrix A1into two parts A11;A12 Using the operator

DðxtÞ ¼ xðtÞ  A11

Z t th

we get a generalization of the result in[10] Combining an exponential translation variable and the technique to reduced the stability of the operator, we get a new d-stability criterion for the system(1.1) As a consequence of the this criterion, we also obtain an asymptotical stability criterion Compared with the results in[10], our result has the following advantages:

 First, by decomposing the matrix A1into two part A11;A12and using the operator DðxtÞ ¼ xðtÞ  A11

Rt thxðsÞds, our criterion will be less restricted than the criterion in[10]

 Second, by reducing the stability of the operator, our criterion will be less conservative than the criterion in[10] The following lemma is needed for our main results

Lemma 1.1 [6] Assume that S 2 Rnn

is a symmetric positive-definite matrix Then for every Q 2 Rnn,

2hQy; xi  hSy; yi 6 hQS1QTx; xi 8x; y 2 Rn

:

If we take S ¼ I then we have j2hQy; xij 6 kyk2

þ kQxk2 By decomposing the matrix A1into two parts A11;A12and using the operator DðxtÞ ¼ xðtÞ  A11

Rt thxðsÞds, we get a generalization of the result in[10] This generalization also need for our main results

Theorem 1.2 For given h > 0 and a;a1, the system (1.1) is asymptotically stable if the exist the positive-definite matrices X; Z1;Z2;M and positive scalars0;1and 0 < b < 1 satisfying the following two matrix inequalities:

bM hAT11M

!

where

R0¼

H H H h1Z1 0 0 hZ1AT11 0aZ1AT11 Z1AT11

H H H H Z2 1a1Z2 0 0 0

0

B

B

B

B

B

B

B

B

@

1 C C C C C C C C A

with

N11¼ ðA þ A11ÞX þ XðA þ A11ÞT;

N14¼ ðA þ A11ÞA11Z1;

N15¼ ðA1 A11ÞZ2:

Proof Consider the following Lyapunov functional:

where

V1¼

Z t

th

ðs  t þ hÞxTðsÞR1xðsÞds;

V2¼

Z t

xTðsÞR2xðsÞ; V3¼ DTðxtÞPDðxtÞ:

By the computations, which are similarly to the proof in[10], we have

V P 0

and

_V < k kDðxtÞk2þ

Zt th

xðsÞds









2

þ kxðt  hÞk2

!

<kkDðxtÞk2;

where k is a positive number

Combining with(1.3), the system(1.1)is asymptotically stable h

Remark 1.3 Since xðtÞ ¼ DðxtÞ  A11Rtht xðsÞds, we have

kxðtÞk 6 kDðxtÞk þ A11

Z t th

xðsÞds







:

Applying the Bunhiakovski’s inequality, we have

kxðtÞk262 kDðxtÞk2þ A11

Z t th

xðsÞds









2!

62 kDðxtÞk2þ kA11k2

Z t th

xðsÞds









2! :

This implies that

kDðxtÞk261

2kxðtÞk

2

þ kA11k2þ

Zt th

xðsÞds









2

:

If kA11k < 1 then we have,

kDðxtÞk261

2kxðtÞk

2

þ

Zt th

xðsÞds









2

:

This follows

_V < k

2 kxðtÞk

2

þ kxðt  hÞk2

If kA11k P 1 then we have,

kDðxtÞk26 1

kA11k2kDðxtÞk

2

:

This follows

_V <  k

2kA11k2 kxðtÞk

2

þ kxðt  hÞk2

Thus if(1.4)holds then there exists a positive number

k0¼

k

2;

k

2kA 11 k2;

(

ð1:8Þ

such that

_V < k0kxðtÞk2þ kxðt  hÞk2: ð1:9Þ

2 Main results

CombiningRemark 1.3with the following change of the state variable:

zðtÞ ¼ edtxðtÞ; t 2 Rþ

;

we get main results as follows:

Theorem 2.1 The system(1.1)is d-stable if the exist the positive-definite matrices X; Z1;Z2, and positive scalars0;1and the following matrix inequality:

Rd¼

H H H h1Z1 0 0 hZ1AT11edh 0aZ1AT11edh Z1AT11edh

H H H H Z2 1a1edhZ2 0 0 0

0

B

B

B

B

B

B

B

B

@

1 C C C C C C C C A

;

with

N11¼ ðA þ dI þ A11edhÞX þ XðA þ dI þ A11edhÞT;

N14¼ ðA þ dI þ A11edhÞA11edhZ1;

N15¼ edhðA1 A11ÞZ2:

Proof First, using the above change of variable then the system(1.1)is transformed to the following system:

_zðtÞ ¼ ðA þ dIÞzðtÞ þ edhA1zðt  hÞ þ f ðt; zðtÞedtÞedtþ edtf1ðt; edðthÞzðt  hÞÞ: ð2:2Þ

Let us denote A0d¼ A þ dI; A1d¼ edhA1;a1d¼a1edhand consider the following Lyapunov functional for system(2.2):

where

V1¼

Z t

th

ðs  t þ hÞzTðsÞR1zðsÞds;

V2¼

Z t

th

zTðsÞR2zðsÞ;

V3¼ DTðztÞPDðztÞ:

V4¼ bkzðtÞk2;

with

b¼ k0

ðkAdk þaÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkAdk þaÞ2þ ðkA1dk þa1dÞ2 q

ðkA1dk þa1dÞ2

0

@

1

A>0:

ByRemark 1.3, we have

_

V1þ _V2þ _V36k0kzðtÞk2þ kzðt  hÞk2:

UsingLemma 1.1, we have

_V

62b_zðtÞT

zðtÞ þ k0kzðtÞk2þ kzðt  hÞk2

¼ 2b zTðtÞATdzðtÞ þ xTðt  hÞAT1dzðtÞ þ fTðt; zðtÞedtÞedtzðtÞ þ fT

1ðt; edðthÞzðt  hÞÞedtzðtÞ

 k0kzðtÞk2þ kzðt  hÞk2

62bkAdkkzðtÞk2þ 2bkA1dkkzðt  hÞkkzðtÞk þ 2bakzðtÞk2þ 2ba1dkzTðt  hÞkkzðtÞk  k0kzðtÞk2 k0kzðt  hÞk2

6ðk0þ 2bkAdk þ 2baÞkzðtÞk2þ 2bðkA1dk þa1dÞkzðt  hÞkkzðtÞk  k0kzðt  hÞk2

6 k0þ 2bðkAdk þaÞ þ b2ðkA1dk þa1dÞ2

k0

! kzðtÞk260:

Integrating both sides of the above inequality from 0 to t, we have

VðtÞ  Vð0Þ 6 0 8t 2 Rþ

:

Since V

ðtÞ P bkzðtÞk2, we have

bkzðtÞk26Vð0Þ ¼ bkzð0Þk2þ V ð0Þ þ V ð0Þ þ V ð0Þ:

By simple computation, we have

V1ð0Þ 6 k/k2kR1kh2;

V2ð0Þ 6 k/k2kR2kh:

Moreover, we have

V3ð0Þ ¼ zð0Þ þ

Z 0

h

A11dzðsÞds

P zð0Þ þ

Z 0

h

A11dzðsÞds

¼ zð0Þ þ

Z 0

h

A11dzðsÞds

X1 zð0Þ þ

Z 0

h

A11dzðsÞds

¼ zTð0ÞX1zð0Þ þ zTð0ÞX1A11d

Z 0

h

zðsÞds þ

Z 0

h

zTðsÞds

AT11dX1zð0Þ þ

Z 0

h

zTðsÞds

AT11dX1A11d

Z 0

h

zðsÞds:

SinceR0

hzðsÞds ¼R0

hxðsÞedsds 6 k/kR0

hedsds ¼k/k

d ð1  edhÞ, we have

V3ð0Þ 6 kX1k þ 2kX1kkA11dk1  e

dh

d þ kX1kkA11dk2ð1  e

dhÞ2

d2

! k/k2:

We denote N1¼ kR1kh2;N2¼ kR2kh,

N3¼ kX1k þ 2kX1kkA11dk1  e

dh

d þ kX1kkA11dk2ð1  e

dhÞ2

d2 ;

N2¼bþ N1þ N2þ N3

Then, we have

bkzðtÞk26Vð0Þ 6 b þ Nð 1þ N2þ N3Þk/k2¼ bN2k/k2:

Hence, kzðtÞk 6 Nk/k This implies kxðt; /Þk 6 Nk/kedt8t 2 Rþ Thus, the system(1.1)is d-stable with Lyapunov factor

N ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

bþ N1þ N2þ N3

b

s

:

The proof is completed h

Remark 2.2 Assume that the matrix inequality(1.4)holds Since A0;A1;X; Z1;Z2are constant matrices and1;2;h are con-stants, we can choose d0>0 is small enough such thatRd 0<0 Hence, we have an asymptotical stability criterion

Corollary 2.3 System(1.1)is asymptotically stable if LMI(1.4)holds

Remark 2.4 In cases A11¼ A1;A12¼ 0 and A11¼ 0; A12¼ A1, we obtain improvements of the results in[5,10]

3 Numerical examples

Example 3.1 Consider the following linear systems, which is considered in[3]:

_xðtÞ ¼ 2 0

0 1

xðtÞ þ 1 0

1 1

xðt  hÞ þ f ðt; xðtÞÞ þ f1ðt; xðt  hÞÞ; ð3:1Þ

where kf ðt; xðtÞÞ 6 0:05kxðtÞk; kf1ðt; xðt  hÞÞ 6 0:1kxðt  hÞk

By decomposing the matrix A1into following two parts:

A11¼ 0:01 0

0:01 0:01

; A12¼ 0:99 0

0:99 0:99

:

The max time-delay which can ensure the system asymptotical stability is 19.80

By decomposing the matrix A1into following two parts:

A11¼ 0:3 0

0:3 0:3

; A12¼ 0:7 0

0:7 0:7

:

The max time-delay which can ensure the system 0.1-stability is 2.01

By decomposing the matrix A1into following two parts:

A11¼ 0:4 0

0:4 0:4

; A12¼ 0:6 0

0:6 0:6

:

The max time-delay which can ensure the system 0.3-stability is 1.25.

By decomposing the matrix A1into following two parts:

A11¼ 0:47 0

0:47 0:47

; A12¼ 0:53 0

0:53 0:53

:

The max time-delay which can ensure the system 0.5-stability is 0.96 FromTable 1, it can be seen that Corollary 2.3 gives larger delay bounds than the recent results in[10,11,3,4]

Example 3.2 Consider the following linear systems, which is considered in[5]:

_xðtÞ ¼ 4 1

0 4

xðtÞ þ 0:1 0

4 0:1

xðt  hÞ þ f ðt; xðtÞÞ þ f1ðt; xðt  hÞÞ; ð3:2Þ

where kf ðt; xðtÞÞ 6 0:2kxðtÞk; kf1ðt; xðt  hÞÞ 6 0:2kxðt  hÞk

By decomposing the matrix A1into following two parts:

A11¼ 0; A12¼ A1:

The max convergence rate is 1.153 FromTable 2, we can see Theorem 2.1 gives larger convergence rate than the result in[5]

4 Conclusion

This paper proposed a new criterion for exponential stability of time delay systems with nonlinear uncertainties By com-bining the method of decomposing the matrix in[1]with the technique to reduced the stability of the operator in[6], we have improved and generalized some previous results Some numerical examples are also given to show the superiority

of our results

Acknowledgement

The author would like to thank anonymous referees for valuable comments and suggestions, which have improved the paper This work was supported by the National Foundation for Science and Technology Development, Vietnam

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Table 1

Comparison between results of our result and recent ones.

Table 2

Comparison between results of our result and recent ones.

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