Abstract— In this paper, we consider the problem of finite-time stability for a class of linear singular time-delay systems. New delay dependent stability conditions have been derived using the approach based on the Lyapunov-like functions and their properties on the subspace of consistent initial conditions. The stability conditions are presented in the form of linear matrix inequalities. A numerical example has been provided to show the advantage of derived results.

I. INTRODUCTION

The class of singular systems is more appropriate to describe the behaviour of some practical systems like electrical systems [1], mechanical systems [2], and chemical systems [3-5]. In general, the singular representation consists of differential and algebraic equations, and hence it is a generalized representation of the state-space system. It is well known that study of singular systems is much more complicated than that of regular ones.

Time delay is commonly encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, etc. It has been shown that the existence of delay and uncertainty is the sources of instability and poor performance of control systems. The certain classes of singular systems are characterized by the phenomena of time delay. In general, the dynamic behaviour of continuous-time singular systems with delays is more complicated than that of system without any time-delay because the continuous time-delay system is infinite dimensional. For this reason, over the past decades, there has been increasing interest in the stability analysis for singular time-delay systems and many results have been reported in the literature [6–11].

Often Lyapunov asymptotic stability is not enough for practical applications, because there are some cases where large values of the state are not acceptable. For example, in a chemical process, the state variables (such as temperature, pressure …) are expected to be controlled within certain bounds for fixed time interval. For this purpose, the concept of the finite-time stability (FTS) is used. In the existing

*This work has been supported by The Ministry of Science and Technological Development of Serbia under the Project ON 174 001.

S. B. Stojanovic is with the University of Nis, Faculty of Technology, Department of Mathematical and Engineering Sciences, 16000 Leskovac, Serbia (phone: +381 16 247203; fax: +381 16 242859; e-mail: sstojanovic@tf.ni.ac.rs).

D. Lj. Debeljkovic is with the University of Belgrade, Faculty of Mechanical Engineering, Department of Control Engineering, 11000 Belgrade, Serbia (e-mail: ddebeljkovic@mas.bg.ac.rs).

T. Nestorovic is with the Ruhr-Universität Bochum, Dep. Mechanics of Adaptive Systems, Germany, (e-mail: tamara.nestorovic@rub.de).

D. Antic is with the University of Nis, Faculty of Electronic Engineering, Department of Control Engineering, 18000 Nis, Serbia (e-mail: dragan.antic@elfak.ni.ac.rs).

literature, there are two concepts of finite-time stability with a very different meaning. The first concept requires that the state of system does not exceed a specified bound in a given finite-time interval [12-20], while in the second concept the term FTS is used to describe system whose state approaches equilibrium point in a finite time [21-22]. In our paper, the first concept of FTS is considered.

A little work has been done for the finite-time stability and stabilization of singular time-delay systems. Some results on FTS and practical stability can be found in [12-18] (singular systems) and [19-20] (singular time-delay systems).

In this article, we consider the problem of finite-time stability for a class of linear singular time-delay systems. The concept of finite-time stability is extended to singular time-delay systems. New stability conditions have been derived using Lyapunov-like functions. The obtained conditions are presented in the form of linear matrix inequalities (LMIs). A numerical example has been provided to show the advantage of developed results.

The following notations will be used throughout the paper. Superscript “T” stands for matrix transposition.

n denotes the n-dimensional Euclidean space and n m is the set of all real matrices of dimension n m . 0X means that X is real symmetric and positive definite, and X Y means that the matrix X Y is positive definite. In

symmetric block matrices or long matrix expressions, we use an asterisk (*) to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations

II. PROBLEM FORMULATION

Consider a linear continuous singular system with state delay, described by

ˆ ˆˆ ˆ ˆ ˆdEx t Ax t A x t (1)

with a known compatible vector valued function of the initial conditions

ˆ ˆ , 0x t φ t t (2)

where ˆ( ) nx t is the state vector, is constant time

delay, ˆ n nA and ˆ n ndA are known constant

matrices. The matrix ˆ n nE may be singular, and it is

assumed that ˆrank( )E r n .

It is known ([12]) that there exist invertible matrices M and N such that

Further Results on Stability of Singular Time Delay Systems in the Sense of Non-Lyapunov: a New Delay Dependent Conditions*

Sreten B. Stojanovic, Dragutin Lj. Debeljkovic, Tamara N. Nestorovic, Dragan S. Antic

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11 12

21 22

11 12

21 22

0 ˆˆ , ,0 0

ˆ

r

d dd d

d d

A AIE MEN A MAN

A A

A AA MA N

A A

(3)

Then, by the nonsingular transformation 1 ˆx N x (4)

the system (1) can be described by the following system:

1

( ) ( ) ( )

ˆ( ) ( ), ( ) ( ), [ ,0]

dEx t Ax t A x t

x t t t N t t

(5)

The following definition will be used in the proof of the main results. Definition 1. Matrix pair ( , )E A is said to be regular if

det( )sE A is not identically zero [6].

Definition 2. The matrix pair ( , )E A is said to be impulse-

free if det ( )deg sE A rank E [6]. The linear continuous singular time delay system (5) may

have an impulsive solution. However, the regularity and the absence of impulses of the matrix pair ( , )E A ensure the existence and uniqueness of an impulse-free solution of the system. The existence of the solutions is defined in the following Lemma.

Lemma 1. Suppose that the matrix pair ( , )E A ( ˆˆ( , )E A ) is

regular and impulsive free, then the solution to (5) ((1)) exists and is impulse-free and unique on [0, ) [6].

In view of this, we introduce the following definition for singular time-delay system (5) or (1). Definition 3. The singular continuous system with state delay (5) ((1)) is said to be regular and impulse-free, if the

matrix pair ( , )E A ( ˆˆ( , )E A ) is regular and impulse-free [6].

Lemma 2. Continuous singular time-delay systems (1) and (5) are regular and impulse-free if matrix 22A , which is

defined by (3), is invertible. Proof. Let 22A is invertible. Then

1 1

122

1 111 12 22 21

ˆˆdet( ) det( )det( )det( )

det( )det( )

det ( ) det( )

0

r

sE A M sE A N

M A

sI A A A A N

(6)

which implies

det( ) 0, deg det( ) ,

ˆ ˆˆ ˆdet( ) 0, deg det( )

sE A sE A r

sE A sE A r

(7)

Then, based on Definition 3, the singular continuous time-delay systems (1) and (5) are regular and impulse-free. Remark 1. Taking into account the structures of the matrices E , A and dA (see (3)), the system (5) can be represented in the following algebraic-differential form:

1 11 1 12 2 11 1 12 2

21 1 22 2 21 1 22 2

( ) ( ) ( ) ( ) ( )

0 ( ) ( ) ( ) ( )d d

d d

x t A x t A x t A x t A x t

A x t A x t A x t A x t

(8)

which implies

1

2 22 21 1 21 1 22 2( ) ( ) ( ) ( )

[ , ]d dx t A A x t A x t A x t

t T

(9)

The expression (9) defines the so-called subspace of consistent initial conditions on the time interval [ ,0]t .

It is obvious that the existence of this subspace is associated with the non-singularity of the matrix 22A .

Definition 4. Singular time-delay system (5) is said to be finite-time stable with respect to ( , , )T if

[ ,0]sup ( ) ( ) ( ) ( ) ,

[0, ]

T T T

tt t x t E Ex t

t T

(10)

Remark 2. Using the state transformation (4), the expression (10) can be written in the following equivalent form:

1

[ ,0]

ˆ ˆsup ( ) ( )

ˆ ˆˆ ˆ( ) ( ) , [0, ]

T T

t

T T T

t N N t

x t E M MEx t t T

(11)

which is suitable for the definition of FTS of the system (1). Thus, the connection between the FTS of the systems (1) and (5) can be simply established. Lemma 3. For any symmetric, positive definite matrix

0T the following condition is satisfied:

12 T T Tu t v t u t u t v t v t (12)

Lemma 4. For any constant matrix 0Z , scalar 0 and

differentiable vector function : ,0 nx such that the

following integration is well defined, then [23]

( ) ( )

( ) ( )1

( ) ( )*

tT T

t

T T T

T

x s E ZE x s ds

x t x tE ZE E ZE

x t x tE ZE

(13)

III. MAIN RESULTS

Theorem 1. Consider the singular time-delay system (5) with the nonsingular matrix 22A that is defined by (3). Then,

the system (5) is regular, impulse free and finite-time stable with respect to ( , , ),T for all 0T if there exist a

positive scalar , positive define symmetric matrices

,Q ,R and matrix P such that the following conditions

hold: 0T TPE E P (14)

TPE E RE (15)

0

* 0 0

* *

T TdA P PA Q PE PA

Q

(16)

1 2 3, ,I R PE I Q I (17)

2 3 1 0Te (18)

Proof. Based on Lemma 2, the condition 22 0A provides

that the system (5) is regular and impulse free.

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Next, we prove the stability. Let us consider the following Lyapunov-like functional:

( ) ( )t

T T

t

V x t x t PE x t x s Qx s ds

(19)

Total derivative ,V t x t along the trajectories of the

system (5) is:

2

2

T T T T

T T

T T T Td

T T

T T T Td

T

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

x t Qx t x t Qx t

x t A P PA x t x t PA x t

x t Qx t x t Qx t

x t A P PA Q x t x t PA x t

x t Qx t

(20)

Based on Lemma 3, we get:

1

1

2

11

2

0,

0

,

T T T

T T Td d

T

T

T T T Td d

T

V x t x t A P PA Q x t

x t PA A P x t

x t Q x t

t t

A P PA Q PA A P

Q t x t x t

(21)

Using Schur complement, from (16) we get

1 00

0

PE

Q

(22)

From (16) and (22), one can have:

0

0 0T

T

PEV x t t t

x t PE x t V x t

(23)

After integrating the previous inequality we get: ( ) (0) tV x t V x e (24)

Then:

0

max

0

max

0

max max

max max

(0) (0) (0) ( ) ( )

( ) (0) (0)

( ) ( ) ( )

( ) ( )

( ) ( )

T T

T

T

V x x PEx x s Qx s ds

PE x x

Q x s x s ds

PE Q d

PE Q

(25)

On the other hand,

min

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

tT T

t

T T T T T

V x t x t PEx t x s Qx s ds

x t E REx t R x t E E x t

(26)

Combining (24)-(26) we get:

min max max( ) ( ) ( ) ( ) ( )T T T

t

R x t E E x t PE Q

e

(27) If the following condition is satisfied:

max max min( ) ( ) ( )TPE Q e R (28)

then: ( ) ( )T Tx t E Ex t , for all [0, ]t T (29)

Let: 1 min 2 max 3 max0 ( ), ( ), ( )R PE Q (30)

Then, the inequalities (17) and (18) hold [0, ]t T . This

completes the proof. Remark 3. The previous result is partially delay-dependent criteria because the delay is not included in the basic inequality (16).

Based on the above results, by using an extended Lyapunov-like functional, a new delay-dependent stability criterion is obtained. Theorem 2. Consider the singular time-delay system (5) with the nonsingular matrix 22A that is defined by (3). Then,

the system (5) is regular, impulse free and finite-time stable with respect to ( , , ),T for all 0T if there exist a

positive scalar , positive define symmetric matrices ,Q ,R

, Z and matrix P such that the following conditions hold:

0T TPE E P (31)

TPE E RE (32)

11 12

22

11

12

22

* 0 0

* *

/

/

/

d

T T

T T

T Td

T Td d

PA

A P PA Q

A ZA E ZE PE

A ZA E ZE

Q A ZA E ZE

(33)

1 2 3 4, , ,I R PE I Q I Z I (34)

22 3 4 1

0

0.5 0,

sup ( ) ( )

T

T T

s

e

x s E Ex s

(35)

Proof. Based on Lemma 2, the condition 22 0A provides

that the system (1) is regular and impulse free. Let us consider the following Lyapunov-like function:

0

( ) ( )t

T T

t

V x t V x t x s E ZEx s dsd

(36)

where V x t is defined by (19). The total derivative

V x t along the trajectories of the system (36), yields:

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0

0

0

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

tT T

t

T

T T T T

T T T

T T

dV x t V x t x s E ZEx s dsd

dt

t t

x t E ZEx t x t E ZEx t d

t t x t E ZEx t

x t E ZEx t d

(37)

Furthermore by using Lemma 4 and (37) we have: 0

( ) ( )

( ) ( )

( ) ( )1

( ) ( )*

T T

tT T

t

T T T

T

x t E ZEx t d

x s E ZEx s ds

x t x tE ZE E ZE

x t x tE ZE

(38)

i.e.

( ) ( )

1

*

( ) ( ) ( ) ( )

1

*

T T T

T TT

T

T

T

d d

T TT

T

T

V x t t t x t E ZEx t

E ZE E ZEt t

E ZE

t t

Ax t A x t Z Ax t A x t

E ZE E ZEt t

E ZE

t t

(39)

where

11 12

22

111

12 12 22 22

*

1

,

T T T Td d

T T

A P PA Q PA A P

A ZA E ZE

(40)

Using Schur complement from (33) we get

1

11 12

22

ˆ 0*

T Td dPA A P

(41)

From (33) and (41), one can get:

0ˆ0 0

T

T

T

V x t t t

PEt t

x t PE x t V x t

(42)

After integrating the previous inequality we get: ( ) (0) tV x t V x e (43)

Then:

0

0

2max max max

(0) (0) (0) ( ) ( )

( ) ( )

( ) ( ) 0.5 ( )

T T

tT T

t

V x x PEx x s Qx s ds

x s E ZEx s dsd

PE Q Z

(44)

min( ) ( ) ( ) ( ) ( ) ( )T T T T TV x t x t E REx t R x t E E x t (45)

Combining (43), (44) and (45), for [0, ]t T , we get:

min

2max max max

( ) ( ) ( )

( ) ( ) 0.5 ( )

T T T

t

R x t E E x t

PE Q Z e

(46)

If the following condition is satisfied:

2

max max max

min

( ) ( ) 0.5 ( )

( )

TPE Q Z e

R

(47)

then: ( ) ( )T Tx t E Ex t , for all [0, ]t T (48)

Let:

1 min 2 max

3 max 4 max

0 ( ), ( ),

( ), (Z)

R PE

Q

(49)

Then, the inequalities (34) and (35) hold [0, ]t T . This

completes the proof. Remark 4. Note that the FTS of the system (5) guarantees the FTS of the system (1) as a result of application of nonsingular transformation (4).

IV. NUMERICAL EXAMPLE AND SIMULATION

Example 1. Consider the following system:

(t) ( ) ( )

2 1 0 0.5 1 0

0 2 0 , 1 0.5 1 , 1

1 0 2 0.7 0.8 0

1 0 0

0 1 0 , ( ) 1 1 0.25 , ,0

0 0 0

d

d

T

Ex Ax t A x t

A A

E x

(50)

Fig. 1 shows time initial response of the system (50) with the initial condition ( ) [1 1 0.25]Tx , ,0 . It is obvious

that 0 . The time-dependent of the norm ( ) ( )T Tx t E Ex t

of the state trajectories is illustrated on Fig. 2. Obviously, the observed system is not asymptotically stable. It is necessary to investigate the FTS of the system (50). Solving LMIs (14)-(18) for 3 , 30 , 5T and fixed 0.077 the

following feasible solution is obtained: 36.9 10.6 18.4 77.5 12.4 0

10.6 64.6 25.9 12.4 88.5 0

62.0 15

,

0 0 0 7 40 .

P Q

36.9 10.6 0 58.8 6.3 11.3

10.6 64.6 0 6.3 63.2 8.5

88.6 11.3 8.5 90.8

,

0 0

R

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1 2 333.302, 68.559, 157.73

Thus, based on Theorem 1, it can be concluded that the system (50) is regular, impulse free and finite-time stable with respect to (3,30,5) .

A similar result is obtained by applying Theorem 2: a feasible solution of (31)-(35) for 0.089 and

( , , ) (3,30,5)T is:

,

0 0

8.2 2.2 2.7 16.9 2.3 0

2.2 13.9 3.1 2.3 18.9 0

16.9 32.40 0

P Q

8.2 2.2 0 12.6 1.9 2.3

2.2 13.9 0 1.9 13.1 2,

0

.1

17.3 2.3 2.1 18.70

R

1 2

3 4

0.31 0.17 0.047.43, 14.81

0.18 1.59 1.5932.62, 13.73

0.04 1.59 2

,

.01

Z

Thus, the obtained stability criteria represent the simple checking methods for the FTS.

0 2 4 6 8 100.8

0.9

1

1.1

1.2

1.3

t

x 1(t),

x2(t

)

x1(t)

x2(t)

Fig. 1. The initial response of the system (50) with the initial condition ( ) [1 1 0.25]Tx , , 0 .

0 2 4 6 8 102

2.1

2.2

2.3

2.4

2.5

t

x(t)

T ET E

x(t)

Fig. 2. The norm ( ) ( )T Tx t E Ex t of the state trajectories.

V. CONCLUSION

This paper extends some of the basic results in the area of the non-Lyapunov stability (finite-time stability) to the class of linear continuous singular time-delay systems. New sufficient delay-dependent criteria for the finite stability have been presented.

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