Delay and Its Time Derivative Dependent Stable Criterion for Differential Algebraic Systems

http://dx.doi.org/10.4236/am.2016.710100

Delay and Its Time-Derivative Dependent

Stable Criterion for Differential-Algebraic

Systems

Hui Liu, Yucai Ding

School of Science, Southwest University of Science and Technology, Mianyang, China

Received 21 April 2016; accepted 21 June 2016; published 24 June 2016

Copyright © 2016 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

In this paper, the stable problem for differential-algebraic systems is investigated by a convex op-timization approach. Based on the Lyapunov functional method and the delay partitioning ap-proach, some delay and its time-derivative dependent stable criteria are obtained and formulated in the form of simple linear matrix inequalities (LMIs). The obtained criteria are dependent on the sizes of delay and its time-derivative and are less conservative than those produced by previous approaches.

Keywords

Differential-Algebraic Systems, Stability Analysis, Lyapunov-Krasovskii Functional, Delay Partitioning Approach, Linear Matrix Inequality (LMI)

1. Introduction

Differential-algebraic systems, also referred to as singular systems, descriptor systems or generalized state-space systems, arise in a variety of practical systems such as chemical processes, nuclear reactors, biological systems, electrical networks and economy systems. Differential-algebraic systems include not only dynamic equations but also static equations [1][2]; the study of such systems is much more complicated than that for standard state- space systems. The existence and uniqueness of a solution to a given differential-algebraic system are not always guaranteed and the system can also have undesired impulsive behavior, which can lead to the instability and poor performance [3].

differential-algebraic systems. In recent years, much attention has been focused on stability, robust stability and

H∞ control problems for differential-algebraic systems, and some results have been derived using the time domain method [4]-[11]. The existing results can be classified into two types: delay-dependent conditions, which include information on the size of delays [12][13], and delay-independent conditions, which are applica-ble to delays of arbitrary size [14][15]. Since the stability of systems depends explicitly on the time-delay, the delay-independent conditions are more conservative, especially for small delays. While the delay-dependent conditions are usually less conservative, and the conservatism will dependent on the chosen of Lyapunov func-tional, the inequality bounding technique and so on. Two approaches were used to prove the stability of the sys-tem in the existing literature. The first approach consists of decomposing the syssys-tem into fast and slow subsys-tems and the stability of the slow subsystem is proved using some Lyapunov functional. Then, the fast variables are expressed explicitly by an iterative equation in terms of the slow variables [16]. The second approach con-sists of constructing a Lyapunov-Krasovskii functional that corresponds directly to the descriptor form of the system [17]. Some other methods have also been provided to reduce the conservative, for example, convex analysis method [18] and delay partitioning approach [19][20]. To the best of our knowledge, most of the exist-ing delay-dependent asymptotically stable criteria only depend on the upper bound of the delay-derivative, and to the differential-algebraic systems, the delay and its time-derivative dependent stability criterion has not estab-lished, which motivates this paper.

This article deals with the problem of asymptotic stability for a class of linear differential-algebraic system with time-varying delay. The obtained criteria depend not only on the upper bound but also on the lower bound of the delay derivative. Based on the Lyapunov functional method and the delay partitioning approach, some delay and its time-derivative dependent stable criteria are obtained. One numerical example is provided to demonstrate the effectiveness of the proposed results. All the developed results are in the LMI framework which makes them more interesting since the solutions are easily obtained using existing powerful tools like the LMI toolbox of Matlab or any equivalent tool.

Notation: Throughout this paper, AT represents the transpose of A; The symbol “∗” in matrix inequality denotes the symmetric term of the matrix; A>0

( )

<0 means A is a symmetrical positive (negative) definite matrix; Sym

( )

M stands for M+MT; I is a unit matrix.

2. Problem Statement

Consider the following differential-algebraic system:

( )

( )

1

(

( )

)

.

Ex t =Ax t +A x t−τ t (1) where x t

( )

∈Rn is the state vector, the matrix n n

E∈R× may be singular, and we assume that

( )

rank E = ≤r n, A and A₁ are constant matrices with appropriate dimensions.

τ

( )

t is time-varying delay,

( )

t

[

h ha, b

]

τ

∈ , ha ≥0, and it is assumed to satisfy d1≤

τ



( )

t ≤d2, where d1, d2 are constants. The initial con-

dition is given by x t

(

₀+

θ

)

=

φ θ

( )

,

θ

∈ −

[

hb, 0

]

,

φ θ

( )

∈W , where W is the space of absolutely continuous function φ:

[

−hb, 0

]

→Rn with the square integrable derivative and with the normal

( )

2 0

( )

2

( )

2 2

0 d

b

W h s s s

φ = φ + ₋ _φ +φ _

 

∫

 .

The following definition, lemmas and notation are introduced, which will be used in the proof of the main results.

Definition 1 ([6]) System (1) is said to be regular if the characteristic polynomial, det

(

sE−A

)

is not identically zero.

2) System (1) is said to be impulse-free if deg det

(

(

sE−A

)

)

=rank

( )

E .

Lemma 1 ([4]) Let H, F and G be real matrices of appropriate dimensions then, for any scalar

ε

>0 for all matrices F satisfying F FT ≤I, we have:

T T T T 1 T

.

HFG+G F H ≤εHH +ε−G G

3. Main Results

Theorem 1 System (1) is asymptotically stable for all differentiable delays

τ

( )

t ∈

[

h ha, b

]

with d1≤

τ



( )

t ≤d2,

( )

T T

0

k k

E P = P E≥

(

k=1, 2, 3

)

, and appropriately dimensioned matrices P_2j, P_3j, Y_1j, Y_2j, Z_1j, Z_2j, j

T

(

j=1, 2

)

, W₁ and W₃ such that LMIs: _{( )} 2

|τ <t h 0

Ψ < , for

τ

( )

t →h₁ and

τ

( )

t →h₂, and _{( )} 2

|τ >t h 0

Ψ < , for

τ

( )

t →h2 and

τ

( )

t →h3, where

τ



( )

t =d d1, 2. Ψ|τ <( )t h₂<0 and Ψ|τ >( )t h₂<0 are denoted in (8) and

(11), respectively.

Proof. The proof of this theorem is divided into two parts. First, we prove that the results, when

τ

( )

t <h2,

( )

t h2

τ

> and

τ

( )

t =h2, respectively, ensure that the derivative of Lyapunov functional is negative. Finally,

we prove the results given in the first part ensure that system is regular and impulse-free.

First of all, we divide the delay interval

[

h ha, b

]

into two segments:

[

h h1, 2

]

and

[

h h2, 3

]

, where we denote

1 a

h =h , h₃=h_b, and 2

2

a b

h h

h = + . Then, system (1) can be represented as

( )

( )

[h h1,2] 1

(

( )

)

1 [h h1,2] 1

(

( )

)

Ex t =Ax t +

χ

A x t−

τ

t + −_

χ

_A x t−

τ

t (2) where _{[ ]}

{ }

1,2 : 0,1

h h

χ R→ is the characteristic function of

[

h h1, 2

]

[1 2]

( )

[

]

1 2

,

1, if ,

0, otherwise.

h h

s h h

s

χ _{= } ∈

 Consider the following Lyapunov functional:

( )

t 1

( )

t 2

( )

t 3

( )

t 4

( )

t 5

( )

t

V x =V x +V x +V x +V x +V x (3) where

( )

T T

(

( )

)

( )

1 t ,

V x =x E P τ t x t

( )

_{( )}1 T

( ) ( )

2 d ,

t h

t _{t t}

V x −_τ x s Qx s s −

=

_∫

( )

( )

( )

1

T

3 0 d ,

t t _{t h}

V x =

∫

₋ x s S x s s

( )

1

( )

( )

2 11 12 T 4 13 d , t h

t _{t h}

S S

V x s s s

S ξ ξ − −   = _{ } ∗  

∫

( )

(

)

(

( )

)

(

( )

)

1 2 _T 5 1 =0

d d , i

i

h t

t i i _{h t} i

i

V x h h _θ Ex s R Ex s s θ

+ −

+ _{− +}

=

∑

−

_{∫ ∫}

 

( )

_T

( )

_T

(

(

)

)

T _{11 12}

0 2 1 0

13

0, , , 0, _i 0, 0, S S 0.

h s x s x s h h Q R S

S

ξ    

= =_ − − _ > > > _{ }> ∗

 

In addition, we define the continuous function P

(

τ

( )

t

)

as the following form:

( )

(

)

[ ]

(

( )

)

( )

( )

[ ]

(

( )

)

( )

( )

1 2

1 2

1 1 2 2

,

2 1 2 1

2 3 3 1

,

3 2 3 2

1 ,

h h

h h

t h h t

P t t P P

h h h h

t h h t

t P P

h h h h

τ τ

τ χ τ

τ τ χ τ − −   = _ + _ − −   − −     + −_{ } + _ − −   (4)

When

τ

( )

t ≠h₂, one can obtain

( )

(

)

( )

(

1 2

)

(

)

(

3 1

)

2 1 3 2

1

,

P P P P

P t t

h h h h

χ χ

τ =τ  − + − − 

− −      _

( )

( )

(

( )

)

( )

( )

( )

( )

(

) ( )

( )

( )

1 2

T T T 1 2

1

2 1 2 1

2 3 3 1

3 2 3 2

2

1 ,

t

t h h t

V x x t E P t x t x t P P

h h h h

t h h t

P P Ex t

h h h h

( )

(

( )

)

(

)

(

( )

)

(

)

(

( )

)

(

( )

)

1

2 2

T 2 T

5 1 1

0 =0

d i

i t h

t i i i i i _{t h} i

i i

V x Ex t h h R Ex t h h Ex s R Ex s s

+ − + + ₋ =   = _ − _ − − 

∑



∑

∫

 _{   }

To obtain the main results, we consider the following three cases:

τ

( )

t ∈

[

h h₁, ₂

)

,

τ

( ) (

t ∈ h h₂, ₃

]

and

( )

t h2

τ

= .

Case 1:

τ

( )

t ∈

[

h h₁, ₂

)

Using the fact that

( )

( )

( )

( )

( )

1 1

d d d

j j

j j

t h t t t h

j j j

t h f s s t h f s s t t f s s τ

τ

+ +

− − −

− = − + −

∫

∫

∫

, where f_j

( )

s =

(

Ex s

( )

)

TR Ex s_i

(



( )

)

, we easily obtain the following inequalities by Jensen’s inequality:

(

) ( )

(

( )

)

(

( )

)

1 1 1

T

1 d d d ,

i i i

i i i

t h t h t h

i i i i

t h+ h h f s s t h+ Ex s sR t h+ Ex s s

− − − + − − ≥ − −

∫

∫



∫

 ( )

(

)

( )

(

( )

)(

)

T

1 d 1 1 1,

j t h

j j j j j j j j j

tτt h h f s s τ t h h h υ Rυ

− + + − − ≥ − −

∫

( )

₍

₎

( )

(

( )

)(

)

1 T

1 d 1 1 2 2,

j t t

j j j j j j j j j

t h h h f s s h t h h R

τ

τ υ υ

+ − + + + − − ≥ − −

∫

where

( )

( )

(

( )

)

1 1 d j t h

j _{t t}

j

Ex s s

t h τ

υ τ − − = −

∫

 ,

( )

( )

( )

(

)

1 2 1 1 d j t t

j _{t h}

j

Ex s s

h t τ υ τ + − − + =

−

∫

 , i=0, 2, j=1. Then, the time- derivative of V x

( )

_t along the solution of (2) is given by

( )

_{( )}

( )

_{( )}

(

( )

)

(

)

(

( )

)

(

) (

)

(

( )

)

(

( )

)

(

( )

)

( )

( )

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

(

)

)

2 2 2 T 2 1 1 | | 0 T T 1 1 T T

0 1 0 1

T 11 12 1 1 13 2 2 T

2 11 12 2

3 13 3

1

t t h t t h i i i

i

V x V x Ex t h h R Ex t

x t h Qx t h t x t t Qx t t

x t S x t x t h S x t h

S S

x t h x t h

S

x t h x t h

x t h S S x t h

x t h S x t h

τ τ

τ τ τ

+ < < =   = + _ − _   + − − − − − − + − − −  −    −  +  ₋  _∗  ₋         −    − −  ₋  _∗  ₋    

∑

  _{ } 

( ) (

)

( ) (

)

(

) (

)

(

) (

)

( )

(

)(

)

( )

(

)(

)

T _T

1 0 1

T _T

2 3 2 2 3

T

1 1 1

T

1 1 2 2,

j j j j j j

j j j j j j

x t x t h E R E x t x t h

x t h x t h E R E x t h x t h

t h h h R

h t h h R

τ υ υ

τ υ υ

+ + +       −_ − − _ _ − − _     −_ − − − _{ } − − − _ − − − − − − (5)

Letting

η

₁

( )

t =_xT

( )

t ,ExT

( ) (

t ,xT t−h₁

) (

,xT t−h₂

)

,

υ υ

₁₁T, ₁₂T,xT

(

t−

τ

( )

t

)

,xT

(

t−h₃

)

_T , and adding the following terms

( )

( )

(

( )

)

(

)

(

( )

)

(

( )

)

T T T T T T T

11 21 1 1 1 11

0=2_x t Y +x t E Y +x t−τ t T _{ } −Ex t−h +Ex t−τ t + τ t −h υ _,

( )

( )

(

)

(

)

(

( )

)

(

( )

)

T T T T T T T

11 21 3 3 2 2 12

0=2_x t Z +x t E Z +x t−h W _{ }Ex t−h −Ex t−τ t + h −τ t υ _,

( )

( )

( )

(

( )

)

( )

T T T T T

21 31 1

0=2_x t P +x t E P _{ }Ax t +A x t−τ t −Ex t _ (6) to (5) gives

( )

_{( )}

( )

_{( ) 2}

( )

( )

2

2 T

1 | 1 1

|

t t h t h

( ) 2

T

11 12 13 11 15 16 17

T T

22 21 21 25 26 27

T

33 12 1

44 48

|

55 57

66 68

T

77 3

88

0 0

0 0 0

0 0 0

0

0 0

0

t h

Z E

Y E Z E

S E T

E W

τ <

Π Π Π Π Π Π 

 

∗ Π − Π Π Π

 

 _{∗ ∗ Π −} 

 

∗ ∗ ∗ Π Π

 

Ψ =_{ }<

∗ ∗ ∗ ∗ Π Π

 

∗ ∗ ∗ ∗ ∗ Π Π

 

 _{∗ ∗ ∗ ∗ ∗ ∗ Π −} 

 

 ∗ ∗ ∗ ∗ ∗ ∗ ∗ Π 

 

(8)

where

( )

₍

₎

( )

₍

_{( )}

₎

( )

₍

_{( )}

₎

T 1

T 1 2 1

11

2 1 2 1

T

2 2 T T T

0 0 21 21

2 1

Sym

Sym ,

t t h

E P P P A

h h h h

h t

P A S E R E A P P A

h h

τ τ

τ

−

Π = − +

− −

−

+ + − + +

−



T T T

12 P21 A P31, 13 E R E0 Y E11 ,

Π = − + Π = −

( )

(

)

T

(

( )

)

T

15 τ t h Y1 11, 16 h2 τ t Z11,

Π = − Π = −

( )

1 1 2

( )

2 T T T

17 1 1 11 11 21 1

2 1 2 1

,

t h h t

P A P A Y E Z E P A

h h h h

τ − −τ

Π = + + − +

− −

(

)

(

( )

)

2

2

T T

22 31 31 1 25 1 21

0

, ,

i i i

i

P P h₊ h R τ t h Y

=

Π = − − +

∑

− Π = −

( )

(

)

T T T T

26 h2 τ t Z21, 27 Y E21 Z E21 P A31 1,

Π = − Π = − +

T T

33 Q S0 S11 E R E0 , 44 S11 S13 E R E2 ,

Π = − + − Π = − + −

( )

(

)

(

)

T T

48 S12 E R E2 E W3, 55 τ t h1 h2 h R1 1,

Π = − + + Π = − − −

( )

(

)

(

( )

)

(

)

57 τ t h T1 1, 66 h2 τ t h2 h R1 1,

Π = − Π = − − −

( )

(

)

(

( )

)

T T

68 h2 τ t W3, 77 1 τ t Q T E1 E T1,

Π = − Π = − − + +

T 88 S13 E R E2 .

Π = − −

Inequality (8) contains two variables which make it difficult to solve by LMI tool. In order to overcome this difficulty, we seek the sufficient conditions for inequality (8). When

τ

( )

t →h1 and

τ

( )

t →h2, the inequality

(8) leads to the following LMIs:

(

)

(

)

(

)

(

)

T T

11 12 13 11 2 1 11 17

T T T

22 21 21 2 1 21 27

T

33 12 1

44 48

1

2

2 1 1 2 1 3

T

77 3

88

0 0

0 0

0 0 0

0

Z E h h Z

Y E Z E h h Z

S E T

h h R h h W

E W

Π Π Π − Π 

 _{∗ Π − − Π} 

 

 _{∗ ∗ Π −} 

 

∗ ∗ ∗ Π Π

Ψ = <

 

∗ ∗ ∗ ∗ − − −

 

 _{∗ ∗ ∗ ∗ ∗ Π −} 

 

∗ ∗ ∗ ∗ ∗ ∗ Π

 

 

(9)

(

)

(

)

(

)

(

)

T T

11 12 13 11 2 1 11 17

T T T

22 21 21 2 1 21 27

T

33 12 1

44 48

2

2

2 1 1 2 1 1

T

77 3

88

ˆ ˆ ₀

0

0 0

0 0 0

0

Z E h h Y

Y E Z E h h Y

S E T

h h R h h T

E W

Π Π Π − Π 

 

∗ Π − − Π

 

 _{∗ ∗ Π −} 

 

 _{∗ ∗ ∗ Π Π} 

Ψ =_{ }<

 _{∗ ∗ ∗ ∗ − − −} 

 

 _{∗ ∗ ∗ ∗ ∗ Π −} 

 

 ∗ ∗ ∗ ∗ ∗ ∗ Π 

 

(10)

where

( ) 1 ( ) 1 ( ) 2 ( ) 2

11 11|τt=h, 17 17|τt=h, ˆ11 11|τt=h, ˆ17 17|τt=h.

Π = Π Π = Π Π = Π Π = Π

Note that we have omitted the zero row and zero column in Ψ₁ and Ψ₂. Letting

( )

(

( )

)

_T

(

) (

)

(

( )

)

(

)

T

T T T T T T

1i x t , Ex t ,x t h1 ,x t h2 , ji,x t t ,x t h3

η

=_  − −

υ

−

τ

− _ , the latter two LMIs imply (8), which is because

( )

_{( )}

_{( )}

_{( ) ( )}

_{( )}

( )

( ) 2

( )

( )

1 T 2 T

11 2 11 12 1 12

2 1 2 1

2 T

1 | t h 1 1 .

t h h t

t t t t

h h h h

t _τ t x t

τ τ

η η η η

η _< η α

− −

Ψ + Ψ

− −

= Ψ ≤ −

Thus, _{( )} 2

|τ <t h

Ψ is convex in

τ

( )

t ∈

[

h h₁, ₂

)

. When

τ



( )

t =di, it follows from (9) that

( )

1i 1|τ =t di 0, i 1, 2.

Ψ = Ψ _ < = (11) The LMI (11) implies (9), since

( )

( )

2 2

11 12 1

2 1 2 1

0.

d t t d

d d d d

τ τ

− −

Ψ + Ψ = Ψ <

− −

 

Thus, Ψ₁ is convex in

τ



( )

t ∈

[

d d₁, ₂

]

. Similarly, we can obtain that Ψ₂ is also convex in

τ



( )

t ∈

[

d d₁, ₂

]

. Case 2:

τ

( ) (

t ∈ h h₂, ₃

]

By the definition of the characteristic function

χ

=0, we apply the above arguments and representations with i=0,1 and j=2. In addition, we replace (6) with the following equations:

( )

( )

(

( )

)

(

)

(

)

(

( )

)

(

( )

)

T T T T T T T T T

12 22 2 1 1

2 2 11

0 2

,

x t Y x t E Y x t t T x t h W

Ex t h Ex t t t h

τ

τ τ υ

 

= _ + + − + − _

 

⋅ −_ − + − + − _



( )

( )

(

)

(

( )

)

(

( )

)

T T T T T

12 22 3 3 22

0=2_x t Z +x t E Z _{ }Ex t−h −Ex t−τ t + h −τ t υ _,

( )

( )

( )

(

( )

)

( )

T T T T T

22 32 1

0=2_x t P +x t E P _{ }Ax t +A x t−τ t −Ex t _,

and letting

( )

( )

(

( )

)

(

) (

)

(

( )

)

(

)

T T

T T T T T T T

2 t x t , Ex t ,x t h1 ,x t h2 , 21, 22,x t t ,x t h3

η

=_  − −

υ υ

−

τ

− _ . It is easy to

obtain that

( )

_{( )}

( )

_{( ) 2}

( )

( )

2

2 T

2 | 2 2

| ,

t t h t h

( ) 2

T T T

11 12 0 12 15 16 17 12

T T

22 22 25 26 27 22

T

33 34 35 1

T

44 2 12

|

55 57

66

77

13

0

0 0

0 0

0

0 0

0 0

0

t h

E R E Y E Z E

Y E Z E

W E

E T S

S

τ >

Π Π − Π Π Π 

 

∗ Π − Π Π Π

 

 _{∗ ∗ Π Π Π} 

 

∗ ∗ ∗ Π − −

 

Ψ =_{ }<

∗ ∗ ∗ ∗ Π Π

 

∗ ∗ ∗ ∗ ∗ Π

 

 _{∗ ∗ ∗ ∗ ∗ ∗ Π} 

 

 ∗ ∗ ∗ ∗ ∗ ∗ ∗ − 

 

    

   

  



 

 

(12)

where

( )

₍

₎

( )

₍

_{( )}

₎

( )

₍

_{( )}

₎

T 2

T 3 1 3

11

3 2 3 2

T

3 1 T T T

0 0 22 22

3 2

Sym

Sym ,

t t h

E P P P A

h h h h

h t

P A S E R E A P P A

h h

τ τ

τ

−

Π = − +

− −

−

+ + − + +

−

 

( )

(

)

(

( )

)

T T T T

12 P22 A P32, 15 τ t h Y2 12, 16 h3 τ t Z12,

Π = − + Π = − Π = −

( )

₂

_{( )}

₃ T ₃

( )

_{( )}

₁ T _{T T T}

17 1 1 12 12 22 1

3 2 3 2

,

t h h t

P A P A Y E Z E P A

h h h h

τ − −τ

Π = + + − +

− −



(

)

(

( )

)

2

2

T T

22 32 32 1 25 2 22

0

, ,

i i i

i

P P h+ h R τ t h Y

=

Π = − − +

∑

− Π = −

( )

(

)

T T T T

26 h3 τ t Z22, 27 Y E22 Z E22 P A32 1,

Π = − Π = − +

(

)

T T T

33 Q S0 S11 E R0 R E1 , 34 E R E W E1 1 S12,

Π = − + − + Π = − +

( )

(

)

T T

35 τ t h W2 1 , 44 S11 S13 E R E1 ,

Π = − Π = − + −

( )

(

)

(

)

(

( )

)

55 τ t h2 h3 h2 R2, 57 τ t h T2 2,

Π = − − − Π = −

( )

(

)

(

)

(

( )

)

T T

66 h3 τ t h3 h2 R2, 77 1 τ t Q T E2 E T2.

Π = − − − Π = − −  + +

Similar to the case I, we obtain the results when

τ

( )

t →h₂ and

τ

( )

t →h₃, which can be marked as

1 0

Ψ < and Ψ <₂ 0, respectively. Further, we can verify _{( )} 2

|τ >t h

Ψ is convex in

τ

( ) (

t ∈ h h₂, ₃

]

, Ψ₁ and Ψ₂ are also convex in

τ



( )

t ∈

[

d d₁, ₂

]

.

From Case 1 and Case 2, we have

( ) 2 [1 2]

( )

( )

( )

( ) 2

( )

(

[1 2]

( )

( )

)

( )

( ) 2

( )

( )

2

T T

1 1 2 2

| t h h h, | t h 1 h h, | t h ,

V_{τ ≠} ≤

χ

τ

t

η

t Ψ_{τ <}

η

t + −

χ

τ

t

η

t Ψ_{τ >}

η

t ≤ −

α

x t (13) for some scalar

α

>0.

Case 3:

τ

( )

t =h₂

( ) 2

{

( )

( ) 2

( )

( )

( ) 2

( )

}

( )

2

T T

1 1 2 2

| t h max | t h , | t h .

V_{τ =} ≤

η

t Ψ_{τ <}

η

t

η

t Ψ_{τ >}

η

t ≤ −

α

x t (14) Therefore, V x

( )

_t ≤ −

α

x t

( )

2 when

τ

( )

t ∈

[

h ha, b

]

,

τ



( )

t ∈

[

d d1, 2

]

.

(

)

0

11 12 ₂

2

22 1

0

0

0.

i i i

i S

h₊ h R =

 

Π Π

 _{  }

− <

 _{∗ Π  ∗ − }

 _{  }



∑



(15)

Pre- and post-multiplying (15) by _I A, T_ and _I A, T_T, respectively, we obtain

( )

₍

₎

( )

₍

_{( )}

T

₎

( )

₍

_{( )}

T

₎

1 2

T 1 2 1 2 T

0

2 1 2 1 2 1

Sym Sym 0

t t h h t

E P P P A P A E R E

h h h h h h

τ τ − −τ

− + + − <

− − −



(16) Since rankE= ≤r n, there must exist two invertible matrices G and H∈n n× such that

0

ˆ _.

0 0 r I E=GEH=  _

  (17)

Similar to (17), we define

11 12 T 1 2

21 22 3 4

ˆ _, ˆ _{, 1, 2.}

i i

i i

i i

A A P P

A GAH P G P H i

A A P P

−  

 

= =_{ } = =_{ } =

   

Then, using T k

( )

k T 0

E P = P E≥ , it can be shown that ₂i 0

P = . Pre- and post-multiplying (16) by HT and H, we can formulate the following inequality easily

(

)

(

T 1 2

)

22 4 4

0 Sym A P P

 

  < +

 

 

 

 (18)

where  will be irrelevant to the following discussion. Form (18) we get Sym

(

A22T

(

P41+P42

)

)

<0, and thus 22

A is nonsingular, which implies that det

(

sE−A

)

is not identically zero and

(

)

(

)

deg det sE−A = =r rankE. Hence, by Definition 1, the above-mentioned results guarantee system (1) is regular and impulse-free. This completes the proof.

When the matrix E is nonsingular, the result in this case can be obtained by setting E equal to I with the appropriate transformations. The corresponding result is given by the following corollary:

Corollary 1 System (1) with E=I is asymptotically stable for all differentiable delays

τ

( )

t ∈

[

h h_a, _b

]

with

( )

1 2

d ≤

τ

 t ≤d , if there exist symmetric positive-definite matrices Q, R i_i

(

=0,1, 2

)

, S₀, S₁₁, S₁₂, S₁₃, Pk

(

k=1, 2, 3

)

, and appropriately dimensioned matrices P_2j, P_3j, Y_1j, Y_2j, Z_1j, Z_2j, T _j

(

j=1, 2

)

, W₁ and W₃ such that LMIs: _{( )}

2

|τ <t h 0

Ψ < , for

τ

( )

t →h1 and

τ

( )

t →h2, and Ψ|τ >( )t h₂ <0, for

τ

( )

t →h2 and

( )

t h3

τ

→ , where

τ



( )

t =d d1, 2. Ψ|τ <( )t h₂ <0 and Ψ|τ >( )t h₂ <0 are denoted in (8) and (11), respectively.

When d₁ is unknown, let k

P=P

(

k=1, 2, 3

)

and

τ



( )

t =d₂, we have the following corollary.

Corollary 2 System (1) is asymptotically stable for all differentiable delays

τ

( )

t ∈

[

h ha, b

]

with

τ



( )

t ≤d2,

if there exist symmetric positive-definite matrices Q, R_i

(

i=0,1, 2

)

, S₀, S₁₁, S₁₂, S₁₃, P satisfying

T T

0

E P=P E≥ , and appropriately dimensioned matrices P_2j, P_3j, Y_1j, Y_2j, Z_1j, Z_2j, T _j

(

j=1, 2

)

, W₁ and W₃ such that LMIs: _{( )}

2

|τ <t h 0

Ψ < , for

τ

( )

t →h₁ and

τ

( )

t →h₂, and _{( )} 2

|τ >t h 0

Ψ < , for

τ

( )

t →h₂ and

( )

t h3

τ

→ , where

τ



( )

t =d₂. _{( )} 2

|τ <t h 0

Ψ < and _{( )} 2

|τ >t h 0

Ψ < are denoted in (8) and (11), respectively.

Remark 1. It should be pointed out that if h_a is big enough, delay partitioning of

[

0,h_a

]

may improve the results. In addition, the better results may be obtained if we divide the delay interval

[

h ha, b

]

into N

(

N>2

)

segments.

4. A Numerical Example

In this section, a numerical example will be presented to show the validity of the main results derived above. Example Consider the following linear differential-algebraic system described by systems (1) with

1

1 0 0.3 0.5 0.5 0

, , .

0 0 0.3 1 0 0.1

E=_{ } A=_− _ A =_− _

− − −

For ha =0, choosing d1= −0.3 and d2 =0.9 and applying Theorem 1, the maximum values of hb is

5.27, which guarantees that the system is asymptotically stable.

5. Conclusion

In this paper, the asymptotic stability of differential-algebraic system with time-varying delay has been investi- gated. Some delay and its time-derivative dependent asymptotically stable criteria have been obtained by de- composing time-varying delay in a convex set. The obtained criteria depend not only on the upper but also on the lower bound of the delay derivative. One numerical example has been given to illustrate the effectiveness of the proposed main results.

Acknowledgements

We thank the Editor and the referee for their comments. This work was supported by A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department (16ZA0146) and the Doctoral Research Foundation of Southwest University of Science and Technology (13zx7141).

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