Delay-Dependent Robust Asymptotically Stable for Linear Time Variant Systems

Delay-Dependent Robust Asymptotically Stable for Linear

Time Variant Systems

D. Behmardii, Y. Ordokhaniii , S. Sedaghatiii

i D. Behmardi, Maths Dept., Alzahra Univ., vanak, Theran, Iran, (e-mail: [email protected] or [email protected] ) ii Y. Ordokhani, Maths Dept., Alzahra Univ., vanak, Theran, Iran, (e-mail: [email protected] )

iii S. Sedaghat, Maths Dept., Alzahra Univ., vanak, Theran, Iran, , (e-mail: [email protected] )

ABSTRACT

In this paper, the problem of delay dependent robust asymptotically stable for uncertain linear time-variant system with multiple delays is investigated. A new delay-dependent stability sufficient condition is given by using the Lyapunov method, linear matrix inequality (LMI), parameterized first-order model transformation technique and transformation of the interval uncertainty in to the norm bounded uncertainty.

A numerical example is presented to illustrate our present stability criterion allows an upper bound which is bigger on the size of the delay in comparison with those in the literature.

KEYWORDS

Lyapunov-Krasovskii functional, Linear matrix inequality, Parameterized first-order model transformation, Time-delay systems.

1. INTRODUCTION

Time delay is frequently a source of instability and it is often encountered in various areas of control systems, such as economical systems, biology [1, 10], engineering, neural network [12], transport phenomena and population dynamics [3, 7].

A system is said to be stable independent of delay (delay-independent stable) if it is stable when delay parameter assumes all nonnegative values. The stability of a system is delay-dependent if it is stable in some domain of delays [9]. Delay independent or delay-dependent stability can be easily derived by an appropriate choice of the terms involved in the Lyapunov-Krasovskii functional. Many criteria for checking the stability of time delay systems have been given so far.

In this paper, the linear matrix inequality (LMI) method with the parameterized first order model transformation technique is employed to derive a new delay-dependent robust asymptotically stable condition for the linear time-variant systems with multiple delays.

2. NOTATIONS

The following standard notation will be used throughout the paper. Let Rn m× denotes the set of all

n×m real matrices, AT be the transpose of matrix,

0

τ > , and A <B (resp.,A B≤ ) means that the matrix B−A is positive definite (resp., positive semi-definite)

for any two symmetric matrices

alongwith

Let

[ ],

N = _nij M =[m_ij]∈ n m× and n_ij ≤m_ij, We define the set matrices

[N M, ]={ A = [a_ij ]_{n m×} : n_ij ≤a_ij ≤m_ij} . In addition 0

0 PC

t (resp., 1

0 PC

t ) denotes the space of all uniformly bounded piecewise continuous (resp., piecewise continuous differentiable) real matrix-valued functions defined on [ , ).

0

t ∞ The Banach space

([ , 0]) ([ , 0], n)

C C

n −τ −τ

of continuous vector functions mapping the delay interval into n with uniform convergence topology, where

0

τ> with standard supremum norm,

|| ||φ =Sup_{θ∈ −[ τ,0]}|| ( )||φ θ

for given φ∈C_n([−τ, 0]) and ||.|| refers to either the Euclidean vector norm or the induced matrix 2-norm.

3. STABILITY CRITERIA FOR TIME-VARIANT SYSTEMS WITH MULTIPLE DELAYS

Let us consider the linear time-variant system with multiple delays as follows,

( ) ₀( ) ( ) ( ) ( - ) 1

m

x t A t x t A_k t x t _k k

τ

= + ∑

=

with initial conditions of time instant t₀

( ) (₀ ) ( ),

0

x_t θ =x t +θ =φ θ θ∈ −[ τ,0] (2) where x t( ) (=x t₁( ),...,x_n( )) t T is state vector at time

t in usual sense. A_k( )t 0 0 PC

t

∈

, are n n× state matrixes such that their components are not known precisely but satisfying A ( )t

k ∈[Nk( ),t Mk( )],t for 0,1,...,

k= m where

( ) [ k( )],

N t n t

k = ij ( ) [ ( )]

k

M t m t

k = ij

0 0 PC

t

∈

with

( ) k n t

ij ( )

k

m t

ij

≤ for all [ , ].

0

t∈ t ∞ The vector function

(0)

φ is an element of Banach space C_n([−τ, 0]) and k k

τ ≤τ ≤ <∞τ , k=0,1,...,m are uncertain time-invariant delays where τ = max{τ_k: k=0,1,...,m}.

The time-invariant system associated with system (1) is of the form

( ) ₀ ( ) ( - )

1 m

x t A x t A x t _k

k k

τ

= + ∑

=

& (3)

where A

k ∈[Nk,Mk], for k=0,1,...,m. Now define B ( )t

k =(Nk( )t +Mk( )) / 2t [ ] k b_ij = ,

( ) H t

k =(Mk( )t −Nk( )) / 2t [ ] k h_ij

= where k = 0,1,...,m.

and E ( )t

k [ 1 , ..., ]

k k

E E

n

= ∈ n n× 2 such that each

, k E

l l 1, ... = n is an n × n array with entry

k k

e_ij = h_lj for l =iand e_ijk =0 for l ≠iwhere

, 1, ...

i j = n. Also define F_k( )t =[F₁k, ...,F_nk T] 2 n ×n ∈

such that each k

,

l

F

l 1, ... = n is an n × n array with entry

k k

f_ij = h_il for l = jand e_ijk =0 for l≠ j where

, 1, ...

i j = n. It is easy to verify that

( )

T

( )

k k

E

t E

t

=

( ₁ ,..., )

1 1

n n

k k

diag h _j h_nj

j j

∑ ∑

= =

(4)

and

( ,..., )

1

1 1

( )

( )

T

k k

n n

k k

diag h h

i in

i i

F

t F t

∑ ∑

= =

=

. (5)

Let ∑_k∈ −[ I_n2 ,I_n2 ], k=0,…m where 2 n

I

is

2 2

n

×

n

identity matrix. It is obvious that

k k k k

diag(₁₁,...,_1n,..., _n1,..., _nn)

k∈ ε ε ε ε

∑ , such that

|

ε

_ijk

| 1,

≤

, 1, ...,

i j = nand furthermore

2 , 0,1,... .

T T _{I k m}

n

k k

k = k≤ =

∑ ∑ ∑ ∑

Let N[N_k( ), ( )] {t M_k t =A_k( )t =

2 2

( ) ( ) ( ): [- , ]}

B_k t +E_k t ∑_kF_k t ∑_k∈ I_n I_n , then we have the following lemma, [4, 12].

Lemma 3.1. [4] For k=0,1,...,m the equalities

[N (t), M (t)]_{k k} =N[N (t), M (t)] _{k k} always hold.

This lemma shows that the linear time-variant interval system (1) is equivalent to the following linear system subject to norm bounded structured uncertainties described by the equation

( ) ( ₀( ) ₀( ) _{0 0}( )) ( )

( ( ) ( ) ( )) ( - ). 1

x t B t E t F t x t

m

B_k t E_k t _kF_k t x t _k k

τ

− − −

∑

= + +

∑ + ∑

=

& (6)

Correspondingly, associated with system (3) we have

( ) ( _{0 0}) ( ) ( ) ( - ),

0 0

1 m

x t B E F x t B_k E_{k k}F_k x t _k

k

τ

∑ ∑

= + + + ∑

= &

(7) where _{[ In}2 _,In2 _]

k∈ −

∑ , k=0,…m.

Therefore, when one is looking for stability condition which depends on delay, the standard step is to replace the original systems (1) and (3) by the systems (6) and (7), [6, 5, 9].

Also, the following lemmas are essential for the proof of the main theorem, [2].

Lemma 3.2. [2] Let

( )

( ) ( )

(

.

) b t t

t f s dsd

a t t

ω θ

θ

= ∫ ∫

− Then the

following is satisfied,

( ) ( ) ( ) (1 ) d

t b a f t b

dtω = − − −& ( ) t a

f s ds t b

− ∫

−

( ) ( )

t b a f s ds

t a + − ∫

−

& _& (8)

Since x t( )is continuously differentiable fort ≥0, by using the Leibnitz-Newton formula we have

0

( ) ( ) ( )

x t −τ =x t −∫−_τx t& +θ θd =

x t( )−∫−0_τAx t( + +θ) A x t_d ( + −θ τ θ)d which is used in reference [9] to transfer the system

( ) ( ) ( - ),

x t& =A x t +A x t_d θ (9) into the distributed delay system,

( ) ( ) ( ) ( )

x& = A+C x t + A_d −C x t−τ

∫−0_τAx t( + +θ) A x t_d ( + −θ τ θ)d , (10) where C is a parametric matrix which derives the stability less restrictive to some degree. Since in this process only one integration over one delay interval is used, the process is called parameterized first-order model transformation.

( ) [ ₀( ) ₀( ) _{0 0}( ) ] ( ) 1 m

x t B t E t F t C_k x t

k ∑ ∑ = + + + = & [ ( ) ( ) ( ) ] ( ) 1 m

B_k t E_k t _kF_k t C_k x t k k

τ

∑ + ∑ − − −

=

[( ₀( ) ₀( ) _{0 0}( )) ( ) 1

t m

C_k B E F x

k t _k

θ θ θ θ

τ

∑ ∫ + ∑ +

= − (11)

( ( ) ( ) ₁( )) ( _)] .

1 m

B_k E_{k k}F x _{k d}

k

θ θ θ θ τ _θ

∑ + ∑ −

=

The stability of this system implies the stability of the system (6). Therefore we focus on stability of the last one. Theorem 3.3. The interval system (6) is robust

asymptotically stable for any [0, ],

i i

τ ∈ τ i=1,...,m with

( )

A_k t ∈[N_k( ),t M_k( )],t k =0,1, ...,m,if there exist positive constant scalars λ_j,α_ij, positive definite matrices

0, T

P=P > R RT 0,

i = i > 0,

T

Q Q

ij = ij > and constant matrix W n n,

i ×

∈ for i j, =1, ...msuch that

( ) ( ) ( ) ( )

1 0 0 0 0 0

T T

PB t B t P λ F t F t

Ω = + + +

( ) ₀( )

1 m

T

W k W_k b S_{k k} t f_{k k k}S t

k τ

∑ _{+ +} +

=

1 ( )

1 1

m m _{j T}

W Q_{i ij}W_{i j ij ij}L S t Lij i j τ τ − ∑ ∑ + + = =

1 ( ) ( )

1 1

m m _{t T T}

W E_{i j} E_j W_i d t L i ij ij i j λ λ λ τ α ∑ ∑ _{∫ −} + = =

(1[( ( ) ) 1( ( ) ) 1

m

T PB_i t W_i R_j PB_i t W_i bi

i

−

∑ − − +

=

1 ( ) ( ) ] 1 )

0 j

T T

PE t E t P W Q W

i i f i i i

i i

τ

λ + − +

1 ₀ ( ) 1 ₀( ) 0

0 0

1

m _{T T}

t _{W E W d PE t P}

i _i t f i i i i λ λ τ

α +λ <

∑ _{∫ −} =

for i j, =1, ...mwhere

( )S_i0 t =BT₀ ( )t Q_{i0 0}B ( )t +α_{i0 0}FT ( )t F t₀( ) (12)

( ) T( ) ( ) T( ) ( )

S_ij t =B_j t+τ_i Q B_{ij j} t+τ_j +α_ijF_j t+τ_j F_j t+τ_j

(13)

( ) ( ) ( ),

i i

T

S t R F t F t

i = i +λi i +τ i +τ (14) , ,

0 1 2

b b b are any positive constant,

d

is any real constant and the corresponding model transformation matrices in (11) is given by C P 1W

i i

− = .

Proof. Consider the Lyapunov-Krasovskii functional defined as follows

( ( )) ( ) ( ) ( ) ( ) ( )

1 m

T t T

V x t x t Px t b_i x S_i x d

t _i i θ θ θ θ τ = + ∑ _{∫ −} = +

( ) ₀( ) ( )

1 m

T t t

f_{i t} x S_i x d

i i θ θ θ θ λ τ ∑ ∫_{− ∫ +} +

= (15)

( ) ( ) ( ) ,

1 1

m m _{j T}

t

L_{ij t} x S_ij x d d

i j i j τ θ θ θ θ λ λ τ τ ∑ ∑ ∫_{− − ∫ +} = =

where b f_i, _i,L_ij for i j, =1, ...,mare arbitrary constant coefficients.

The time-derivative of this functional along with the positive half trajectories of the systems (11), can be expressed as follows:

( ( )) ( )( ₀( ) ₀ ( ) ( )) ( ) 1

m

T T T

V x t x t PB t B t P W_i W_i x t i

= + +∑ + +

=

&

2xT( )t PE₀( )t ∑₀F t x t₀( ) ( )+

2 ( )( ( ) ) ( - )

1 m _T

x t PB_i t W_i x t _i i

τ

− +

∑ =

2 ( )( ( ) ( )) ( - )

1 m

T

x t PE_i t _iF t x t_{i i} i

τ

∑ ∑ −

=

2 ( ) ₀( ) ( )

1

2 ( ) ₀( ) _{0 0}( ) ( )

1 m

T

t _{x t W B x d}

i t _i

i m

T

t _{x t W E F x d}

i t _i i λ λ λ τ λ λ λ λ τ ∑ _{∫ −} − = ∑ _{∫ −} ∑ − =

2 ( ) ( ) ( )

1 1

m m _{t T}

x t W B_{i j} x _j d t _i

i j

λ λ τ λ

τ − −

∑ ∑ ∫ −_{= =}

2 ( ) ( ) ( ) ( )

1 1

m m _T

t _{x t W E F x d}

i j j j j

t _i i j λ λ λ τ λ τ − + ∑ ∑ _{∫ −} ∑ = = ( )( ( )) ( ) 1 m T

x t b S_{i i} t x t i

∑ −

= (16)

( ) ( ) ( )

1 m

T

b x_i t _i S_i t _i x t _i i

τ τ τ

∑ − − − −

=

( ) ( ) ( ) ₀( ) ( )

0

1 1

m _T m

T t

f_{i i}x t S x t f_i x S_i x d

i t _i

i i τ _τ λ λ λ λ ∑ − ∑ _{∫ −} − = = ( ) ( ) ( ) 1 1 ( ) ( ) ( ). 1 1

m m t _{j T}

L_ij x S_ij x d

t _{i j} i j

m m _T

L x t S t x t

i ij ij

i j τ λ λ λ λ τ τ τ − ∑ ∑ _{∫ − −} + = = ∑ ∑ = =

Using the following inequalities for any positive real number β>0 and any positive definite matrixD ,

where u v R, ∈ n, [8, 11]. We have

2 ( )( ( ) ) ( - )

1 m

T

x t PB_i t W_i x t _i i

τ

∑ − ≤

=

1 _{( )( ( ) )} 1_{( ( ) ) ( )}

1 1 ( - ) ( - ). 1 m T T

b_i x t PB_i t W_i R_i PB_i t W_i x t i

m T

b x_i t _i R_i x t i i τ τ − − ∑ − − + = − ∑ = (17)

2 ( ) ( ) ( ) ( - )

1 m

T

x t PE_i t _iF t x t_{i i} i

τ

∑ ∑ ≤

=

1 1 _{( ) ( ) ( ) ( )} 1

m

T T

b x t PE_i t E t Px t

i i i

i

λ− −

∑ +

= (18)

( - ) ( ) ( ) ( - ). 1

m

T T

b x t F t F t x t

i i i _i i _i

i

λ τ τ

∑

=

2 ( ) ₀( ) ( )

1

1 1

2 ( ) ₀ ( )

1

( ) ₀ ( ) _{0 0}( ) ( ) 1

m T

t _{x t W B x d}

i t _i

i m

T T

f x t W Q W x t

i _i i _{i i}

i

m _{t T T}

f_i x B Q_i B x d

t _i i λ λ λ τ τ λ λ λ λ λ τ ∑ − _{∫ −} ≤ = − − ∑ + = ∑ _{∫ −} = (19)

2 ( ) ₀( ) _{0 0}( ) ( )

1

1

( ) ₀( ) ₀( ) ( ) 0 0

1

( ) ( ) ( ) ( ) .

0 ₀ 0

1 m

T

t _{x t W E F x d}

i t _i

i

m _{T T T}

t _{x t W E E W x t d}

i _i t f i i i m T T t

f x F F x d

i _{i t i} i λ λ λ λ τ λ λ λ τ α α _τ λ λ λ λ λ ∑ − _{∫ −} ∑ ≤ = + ∑ _{∫ −} = ∑ _{∫ −} = (20)

2 ( ) ( ) ( )

1 1

1 _{( )} 1 _{( )}

1 1

( ) ( ) ( ) ( ) .

1 1

m m _T

t _{x t W B x d}

i j j

t _i i j

m m _{T T}

x t Q W x t

i _{ij i}

Lij i j

m m _{T T}

t

L_ij x _j B_j Q B_{ij j} x _j d

t _i i j λ λ τ λ τ τ λ τ λ λ λ τ λ τ ∑ − _{∑ ∫ −} − ≤ = = − ∑ ∑ + = = ∑ ∑ _{∫ −} − − = = (21)

2 ( ) ( ) ( ) ( )

1 1 1 ( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) . 1 1

m m _{t T}

x t W E_{i j j}F_j x d

j t _i

i j

m m _{T T T}

t _{x t W E E W x t d}

i j _{j i}

t

L i

ij ij i j

m m _{t T T}

L x F F x t d

ij ij t j j j j j

i j λ λ λ τ λ τ λ λ λ τ α α _τ λ τ λ λ τ λ ∑ − ∑ _{∫ −} ∑ − ≤ = = ∑ ∑ _{∫ −} + = = ∑ ∑ _{∫ −} − − = = (22) Substituting (17-22) into (16), we get,

( ( ) T ( ) ₁ ( ), V x t& ≤x t Ωx t where

( ) ( ) ( ) ( )

1 0 0 0 0 0

T T

PB t B t λ F t F t

Ω = + + +

( ) 0( )

1 m

T

W k W_k b S_{k k} t f_{k k k}S t

k τ

∑ _{+ +} +

=

1 ( )

1 1

m m _{j T}

W Q_i W _{j ij ij}L S t ij i Lij i j τ τ − ∑ ∑ + + = =

1 ( ) ( )

1 1

m m _{T T}

t _{W E E W d}

i j _{j i}

t L i ij ij i j λ λ λ τ α ∑ ∑ _{∫ −} + = =

(1[( ( ) ) 1( ( ) ) 1

m

T PB_i t W_i R PB_i t W_i

i bi i − ∑ − − + =

1 PE_i( )t E_iT ( ) ]t P jW Q_{i i0}1W_iT ) f

i i

τ

λ + − +

1 ₀( ) ₀ ( )

0 1 m

T T

t _{W E E W d}

i _i t f i i i j λ λ λ τ α ∑ _{∫ −} + =

1 ₀( ) ₀( ) . 0

PE t E t P λ

Since 0

1

Ω < , it is easy to show that ˙V x t&( ( ))<0 if

( ) 0

x t ≠ and ˙V x t&( ( )) 0= , if and only if x(t) = 0. Therefore by Lyapunov-Krasovskii stability theorem, the origin of the system (11) is robust asymptotically stable for

( ) A t

k ∈[Nk( ),t Mk( )],t k =0,1, ...,m,and τi ∈[0,τi], consequently the origin of the system (6) is robust asymptotically stable which completes the proof. In the above theorem, if we letm =1,W =W1,

,

0 10

S =τQ

,

1 11

S =τQ ,

1

R =R ,

1

τ τ= ,

0 10

α =α α₁=τα₁₁then the following result is immediate.

Corollary 3.4. System (1) with m = 1 is robust asymptotically stable for any A ( )t

k ∈[Nk( ),t Mk( )],t 0,1

k = if there exist constant scalersλ_i>0,α_i> 0,

symmetric and positive definite matrices P =PT >0, 0,

T

R R

i = i >

T S =S >0

i i fori =0,1and constant matrix n n

W ∈ × such that

( ) ( )

11 0 0

T T

PB t B t P W W

Ω = + + + +b R₁ +

( ) ( ) ( )

0 1 0 0 0

T

f F t F t

λ + α +

(b_{1 1}λ +L_{11 1) 1}α FT (t+τ)F t₁( +τ)+

( ) ( ) ( ) ( )

11 1 1 1 1 0 0 0

T T

L B t+τ S B t+τ +f B t S B t +

1 1

1

2 ₍ 1 0 _{) ( ) ( )}

1 1

11 1

S

S _{T T}

W W PE t E t P

L f _i

τ

λ

− −

1 1 1

( ) ( ( ) ) ( ( ) )

0 0 1 1

0 1

T T

PE t E P PB t W R PB t W

b

λ + − − − +

1 1

( _{1 0 0}( ) ₀ ( )

1 1 _{( ) ( )) 0.}

1

11 1 1

t _{W f E E}T

t

T T

L E E W d

τ _τ α λ λ

α λ λ λ

− − ₊

∫ −

− − _<

The corresponding model transformation matrix is given by C =P−1W.

4. EXAMPLE

Consider the same interval system as given in [12], ( ) ( ) ( ) ( ) ( - ),

1 0

x t& = A t x t +A t x t τ

where ( ) sin2 · , ( ) cos2 ·

0 0 0 2 1 1 1 2

A t = Λ +ε t I A t = Λ +ε t I

, such that Λ Λ₀, ₁are known 2×2matrices,ε ε₀, ₁are uncertain but bounded as |ε₀| ≤ 1, |ε₁| 1.≤

It is easy to see that, Ak t( )∈[N_k( ),t M_k( )]t , k = 0, 1 where

2 2

( ) - sin · ( ) sin ·

0 0 2 0 0 2

2 2

( ) - cos · ( ) cos · .

1 1 2 1 1 2

N t t I M t t I

N t t I M t t I

= Λ = Λ +

= Λ = Λ +

Hence, by assuming B₀( )t = Λ₀, B t₁( )= Λ₁,

2 2

( ) sin · , 1( ) cos · ,

0 2 2

H t = t I H t = t I and

2 ( ) ( ) ( ) ( ) sin ·

0 0 0 0 2

2 ( ) ( ) ( ) ( ) cos ·

1 1 1 1 2

T T

E t E t F t F t t I

T T

E t E t F t F t t I

= =

= =

We have

11 0 0 1

T T

P P W W b R

Ω = Λ + Λ + + + +

2

(λ₀+f_{1 0}α )sin t I· ₂+( ) cos2 ·

1 1 11 1 2

b λ +L α t I +

1 1 1 1

1 11 0 1

2

( )

11 1 1 1 1 0 0 0

T T T

L Λ S Λ + Λf S Λ +τ W

S L

− − +

S f

− − W +

1 _sin2 _· 1 _cos2 _·

2 2

0 1

P t I P P t I P

λ +λ +

1 1

( ) ( )

1 1

1

T

P W R P W

b

−

Λ − Λ − +

1 1

2 _{( sin}2 _{. cos}2 _{. ) .}

2 2

1 0 11 1

T

W I I W

f L

τ λ λ

α + α

Also if we assume

b=b ,f=f ,L=L ,2 = = ,2 = = ,

1 1 11 α α α λ λ λ0 1 0 1 then we

have

11 0 0

T T

P P W W bR

Ω = Λ + Λ + + + + (2λ+2fα)I₂+

(2bλ+2Lα)I₂+LΛ_{1 1 1}TS Λ + Λf T_{0 0 0}S Λ +

1 1 1 1

1 0

2

( ) T

W

S L

S f

W

τ − − + − − + 1 P2

λ +

1 1

(P ₁ W R) (P ₁ W)T b

−

Λ − Λ − + 2 (1 1) . 2

T W W

f L

τ

α +

For example let b =1/ 6 , f =L =1/ 2 and assume

2 0 0.5 0

,

0 _{0 1.9} 1 _{0.1 0.5}

− −

⎡ ⎤ ⎡ ⎤

Λ =_{⎢ ⎥} Λ =_{⎢ ⎥}

− −

⎣ ⎦ ⎣ ⎦

Then one feasible solution for associated linear matrix inequality (LMI) is

(172.2344, 159.9684) (37.0500, 0.1602)

(8.8125, 8.8647) (34.9834, 30.8433)

0 1

( 83.7071, 79.8313) 114.6665, 56.8986

p diag R diag

S diag S diag

W diag λ α

= =

= =

= − − = =

Therefore, by substituting into the right hand side of inequality , Ω₁₁ we get

2

-187.027570 2052.041992 7.014642528

0 2 7.014642528 -100.822578 1907.101854

. 11

τ

τ

⎡ ⎤

+

⎢ ⎥

≤_{⎢ ⎥}<

+

⎢ ⎥

⎣ ⎦

Ω

Therefore τ<0.301897627 and τ<0.229928049.

Consequently the system is robust asymptotically stable for τ∈[0, 0.229928049] which is a larger domain for delay with respect to example 1 of reference [12].

5. CONCLUSION

In this paper, we have investigated the robust asymptotical stability issue of linear interval time variant systems with uncertain delays. A new delay dependent stability condition is derived by using the Lyapunov method, (LMI), parameterized first-order model transformation technique and introducing ingeniously real constants. Based on a present criterion, a new upper bound on the size of delays is presented. A numerical example is also provided to demonstrate the effectiveness of the new result.

6. REFERENCES

[1] Campbell,S. A. and Belair, J., 1992, “Multiple-delayed differential equations as model for biological control systems.” In Proceeding World Congress of Nonlinear Analysts’ 92, 3110-3117 ,Tampa.

[2] Kim, J.-H., 2001, “Delay and Time-Derivative Dependent Robust Stability of Time-Delay Linear Systems with Uncertainty.” IEEE Trans. Autom. contr., 46,(5), 789-792.

[3] Kuang, Y., 1993, Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston. 9

[4] Li, C. D. and Liao, X. f., 2006, “A global exponential robust stability criterion for NN with variable delays.” Neurocomputating 69, 80-89.

[5] Li, X. and de Souza, C. E., 1995 “LMI approach to delay -dependent robust stability of uncertain linear systems.” in Proc. of the 34th CDC, New Orleans, 3614-3619.

[6] Li, X. and de Souza, C. E., 1997, “Delay dependent robust stability and stabilization of uncertain linear delay system: A linear

[7] Macdoonald, N., 1989, Biological Delay Systems: Linear Stability Theory, Cambridge University Press, Cambridge.

[8] Niculescu, S.-I., Doin, J.-M., Dugard, L., and Li, H., 1997, “Stability of linear systems with several delays: An L.M.I. approach.” JESA, special issue on ‘Analysis and control of time-delay systems’ 31, 955-970.

[9] Niculescu, S.-I., 2001, Delay effects on stability: A robust approach. Springer, Berlin.

[10] Stepan, G., 1998, “Retarded dynamical system stability and characteristic function.” Research Notes in Mathematics Series, John Wiley, New York, P:210.

[11] Su, J .H., 1994, “Further results on the robust stability of linear systems with a single delay.” Systems and Control Letters, 23, 375-379.