Abstract—In this paper, the class of linear discrete-time descriptor systems with time-varying exogenous disturbance are considered. Three sufficient conditions for the finite time boundedness and finite-time stability of linear discrete-time descriptor systems are presented. The stability conditions are reduced to a feasibility problem involving linear matrix inequalities. A numerical example has been provided to show the advantage of derived results.

I. INTRODUCTION

In many practical applications, it could be required that the state of a system does not exceed a certain bound during a specified time interval for given bound on the initial state. For this purpose, the concepts of the finite-time stability (FTS) and practical stability are used. While the concept of the Lyapunov stability is used for the analysis of the system behavior on infinite time interval, the concept of FTS is defined on a finite (or very short) time interval. A system is said to be finite-time stable if, once a time interval is fixed, its state does not exceed some bounds during this time interval. Some early results on FTS can be found in the 1960s [1,2]. In [3–7], the authors studied the FTS problem for continuous-time or discrete-time linear systems and proposed some criteria for the design of a controller such that the resulting closed-loop system is finite-time stable. The concept of FTS is also extended to finite-time boundedness (FTB) by introducing an exogenous input and sufficient conditions for FTB are also given in [4, 8]. Sufficient conditions for the existence of state feedback laws, which guarantee FTB of a closed-loop system, are given for linear continuous-time systems [3,4,8] and for linear discrete-time systems [4].

Singular systems, which are also referred to as descriptor systems, generalized state-space systems or semi-state systems, have been extensively studied in the last few decades because this class of systems is more appropriate to describe the behavior of some practical systems like electrical systems, mechanical systems, and chemical systems [9-10]. In general, the singular representation consists of differential

* 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).

and algebraic equations, and therefore is a generalized representation of the state-space system. Numerical solution of singular systems is more difficult compared to regular models, due to the existence of linear and non-linear algebraic equations and due to discontinuities in the algebraic variables over the independent variable space. The large number of fundamental concepts and results based on the theory of the normal state-space systems has been extended to singular systems [11-12]. In recent years, the stability problems for singular systems have been investigated by many researchers, for example [13-16].

It is noted that all the above-mentioned works about the stability of singular systems are focused on Lyapunov asymptotic stability. A small number of articles have appeared on the subject of the FTS and FTB analysis of singular systems. Some results on FTS and FTB can be found in [11, 17-27]. In [22-23], the authors introduced the concept of FTS into linear time-varying singular systems. Robust finite-time stabilization problem was studied for linear singular systems with parametric uncertainties and exogenous disturbances in [24], and a sufficient condition in terms of LMIs was obtained for robust finite-time stabilization via state feedback. It should be noticed that all the FTS and FTB-related literatures for singular systems mentioned above were developed in the context of continuous singular systems while very little attention has been paid to the discrete case.

In this paper, we extend the concepts of FTS and FTB to the discrete-time singular systems. Using LMI approach, novel sufficient conditions for the FTB are derived. A numerical example has been provided to show the advantage of developed results.

The paper is organized as follows. In the second section we define the class of the considered system and give some definitions that will be used in the paper. The third section deals with the problem of FTB and FTS. Three stability criteria are derived in the form of LMIs. In the section IV, we provide a numerical example 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.

Finite-Time Boundedness Analysis of a Class of Linear Discrete Descriptor Systems: an LMI Approach*

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

Proceedings of the 3rd International Conference onSystems and Control, Algiers, Algeria,October 29-31, 2013

ThBC.7

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

II. PROBLEM FORMULATION

Generally, we consider the following linear time invariant discrete descriptor systems with exogenous disturbances:

0

1 ,

0

E x k Ax k F z k

x x

(1)

where nx k is a state space vector, mz k is a

time-varying exogenous disturbance which satisfied

, 1, ,Tz k z k k N < (2)

and n nE is a possibly singular matrix, with rank E r n . Matrices E, A and F have appropriate dimensions and are defined over the field of real numbers.

For the linear singular discrete system (1) we present the following definition that will be used in the proof of the main results.

Definition 1. [12] The matrix pair ,E A is said to be

regular if det ( )zE A is not identically zero.

Definition 2. [12] The matrix pair ,E A is said to be

causal if deg(det ( ) ) ( )zE A rank E r .

The regularity and causality of the matrix pair ,E A

ensure the existence, uniqueness and causality of solution of the system (1) for 1, 2,3,k .

Definition 3. The linear discrete descriptor system (1) with ( ) 0z k is finite time stable (FTS) with respect to

, , N , if:

0 0T Tx E E x (3)

implies:

, 1, ,T Tx k E E x k k N < (4)

Definition 4. The linear discrete descriptor system (1), subject to exogenous disturbances, is finite time bounded (FTB) with respect to , , ,N , if :

T T0 0x E Ex (5)

implies:

T T , 1, ,x k E Ex k k N < (6)

when:

, 1, ,Tz k z k k N < (7)

III. MAIN RESULT

Now we are in position to give sufficient conditions, under which the system (1) will be regular, causal and FTB (FTS) simultaneously.

Theorem 1. The linear discrete descriptor system (1) is regular, causal and FTB with respect to , , ,N

if there exist a scalar 1 , two symmetric positive definite

matrices n nP , m mQ and matrix ( )r n rS such that:

* 0

* *

T T T T T T

T

E PE A RS SR A SR F A P

Q F P

P

(8)

max max min1

1

NN P Q P

(9)

where ( )n n rR is any full-column rank matrix satisfying

0TE R (10)

Proof. First, we prove the regularity and causality of the system. By Schur complement, (8) is equivalent to

0*

T T T T TE PE A RS SR A SR F

Q

(11)

i.e.

0T T T TE PE A RS SR A (12)

Whereas 1 , we have

0T T T TE PE A RS SR A (13) Now, we choose two nonsingular matrices M and N

such that:

11 12

21 22

ˆ ˆ0 ˆˆ ,ˆ ˆ0 0

r A AIE MEN A MAN

A A

(14)

11 121

12 22

ˆ ˆˆ

ˆ ˆT

T

P PP M PM

P P

(15)

1

2

ˆˆ

ˆT S

S N SS

(16)

( ) ( )2

2

0ˆ ˆ, , 0

ˆT n r n rR M R R R

R

(17)

Then:

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ 0T T T T TN N E PE A RS SR A (18)

From (18) we get

11 12

12 22

11 12

11

ˆ ˆ0 0ˆ ˆ ˆˆ ˆ0 0 0 0

ˆ ˆ 0

0 00 0

ˆ 0

0 0

r rT

T

r

P PI IE PE

P P

IP P

P

(19)

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978-1-4799-0275-0/13/$31.00 ©2013 IEEE

1 11 122

2 21 22

1 2 11 12

2 2 21 22

1 2 21 1 22 22

2 2 21 2 2 22

ˆ ˆ ˆˆ ˆˆ ˆ0

ˆ ˆ ˆ

ˆ ˆ ˆˆ0

ˆ ˆ ˆˆ0

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

T T

T

T

T T

T T

S A ASR A R

S A A

S R A A

S R A A

S R A S R A

S R A S R A

(20)

1 2 21 1 22 22

2 2 21 2 2 22

1 2 21 2 2 21

1 22 22 2 2 22

ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

TT T

TT T T

T T

T TT T

T TT T

S R A S R AA RS SR A

S R A S R A

S R A S R A

S R A S R A

(21)

11 1 22 2 2 21

21 2 2 22 2 2 22

11 11 1 2 21 1 2 21

21 2 2 21 1 22 22

ˆ ˆ ˆ ˆˆ ˆˆˆ 0

ˆ ˆ ˆ ˆˆ ˆˆ

ˆ ˆ ˆ ˆˆ ˆ ˆˆ

ˆ ˆ ˆ ˆˆ ˆˆ

TT T

TT T

TT T

TT T

S R A S R A

S R A S R A

P S R A S R A

S R A S R A

(22)

i.e.

2 2 22 22 2 2ˆ ˆ ˆ ˆˆ ˆ 0T T TS R A A R S (23)

From (23), we obtain that 22A is nonsingular and from

1 1

11 121 1

21 22

11 12

21 221 1

122 21

1

ˆˆdet(z ) det( )det( )det( )

ˆ ˆdet( )det det( )

ˆ ˆ

ˆ ˆ

ˆ ˆdet( )det det( )

0

ˆ ˆ

det( )

r

r

r

n r

E A M zE A N

zI A AM N

A A

zI A A

A AM N

I

A A I

M

1

111 12 22 21 12

22

1 122

111 12 22 21

det( )

ˆ ˆ ˆ ˆ ˆdet

ˆ0

ˆdet( )det( )det( )

ˆ ˆ ˆ ˆdet ( )

r

r

N

zI A A A A A

A

M N A

zI A A A A

(24)

we get

det( ) 0, deg det( ) zE A zE A r (25)

which implies that ( , )E A is regular and causal.

Next, we will show that the linear discrete descriptor system (1) with exogenous disturbances is FTB. Choose a Lyapunov-like function as follows:

T TV x k x k E PE x k (26)

A difference of V x k along the trajectories of the system

(1) is:

1

( 1) ( 1) ( ) ( )

( ) ( ) 2 ( 1) ( )

ˆ

T T T T

T

T T T T T

T

V x k V x k V x k

x k E PEx k x k E PEx k

Ax k F z k P Ax k F z k

x k E PEx k x k E RS x k

ξ k ξ k

(27)

where

( ) ,

ˆ

*

T TT T

T T T

T

x kk

z k

A PA E PEA PF SR F

A RS SR A

F PF

(28)

From (11) and (27) we have:

ˆ

( 1) 0

0

T

TT

V x k ξ k ξ k

E PEξ k ξ k

Q

(29)

If (8) (i.e. (11)) is satisfied, then (29) reduces to:

( 1)

( 1)

T T

T

T

V x k x k E PE x k

z k Qz k

V x k z k Qz k

(30)

so, we obtain:

1 TV x k V x k z k Qz k (31)

Applying iterative procedure on (31), we obtain:

1

1

0

0k

kk j T

j

V x k V x

z j Qz j

(32)

Furthermore:

max

max

max

max max

max max

10 0

1

0 0

1

1

1

1

1

1

kk T T

k T T

k

kk

NN

V x k x E PE x Q

P x E E x

Q

P Q

P Q

(33)

On the other hand:

Proceedings of the 3rd International Conference on ThBC.7

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

minT TP x k E E x k V x k (34)

Combining (33) and (34), yields:

max max

min

1

1

T T

NN

x k E E x k

P Q

P

(35)

Condition (9) and above inequality imply:

, 1, ,T Tx k E E x k k N (36)

Then, from Definition 4, we conclude that the system is FTB with respect to , , ,N and this completes

the proof.

Remark 1. It should be pointed out that condition (9) is not linear matrix inequality condition. In following corollary, we eliminate this problem.

Corollary 1. The linear discrete descriptor system (1) is regular, causal and FTB with respect to , , ,N ,

if there exist a scalar 1 , positive scalars 1 , 2 and 3 ,

two symmetric positive definite matrices n nP , m mQ and matrix ( )r n rS such that:

* 0

* *

T T T T T T

T

E PE A RS SR A SR F A P

Q F P

P

(37)

1 2 3, 0I P I Q I (38)

1 2 3

2

3

* 0 0

* *

NN

(39)

where:

1

1

N

N

(40)

and ( )n n rR is any full-column rank matrix satisfying

0TE R . Proof. The proof is similar to the previous theorem. Let

1 min 2 max 3 max0 ( ), ( ), ( )P P Q

Then

1 2 1 3

1 2 3

, 0, 0

0NN

I P I Q I

(41)

By Schur complement, (41) is equivalent to (39). The proof is completed.

Remark 2. For system (1) with ( ) 0z k , the following FTS criterion can be formulated and easily proved.

Corollary 2. The linear discrete descriptor system (1) is regular, causal and FTS with respect to , , ,N

if there exist a scalar 1 and symmetric positive definite

matrix n nP and matrix ( )r n rS such that:

0*

T T T T TE PE A RS SR A A P

P

(42)

1 2I P I (43)

1 2

2

0*

N

(44)

where ( )n n rR is any full-column rank matrix satisfying

0TE R .

IV. NUMERICAL EXAMPLE AND SIMULATION

Example 1. Consider the following system:

( 1) ( ) ( )

1 0 0 0.7 0.1 0.2 0.1

0 1 0 , 0.5 1.1 0.4 , 0.1

0 0 0 0.2 0 0.8 0.2

(0) 2 1 0.5 ,

( ) 5, 0, (0) 0

Ex k Ax k Fz k

E A F

x

z k k z

(45)

Fig. 1 shows time histories of the state trajectories of the system (45) with the initial condition (0) [2 1 0.5]Tx

and (0) 0z . Time dependent of norm of the state

trajectories is illustrated on Fig. 2. Obviously, the observed system is not asymptotically stable. However, in the rest of the paper we will show that this system is FTB, for certain values of the parameters , and N . In this sense, it is

necessary to investigate FTB with respect to 469(6, ,10) .

For given matrix E we choose [0 0 1]TR such that (10)

is satisfied. Solving LMIs (37)-(39) for fixed 1.42 , we

obtain the following feasible solution: 226.12 20.43 8.65

20.43 168.43 2.76

8.65 2.76 163.18

P

10.43Q ,

157.59

112.31

185.66

S

,

1 2 3161.91, 233.82, 10.439

Thus, the system (45) is causal, regular and FTB with respect to 469(6, ,10) .

Based on Corollary 1, for given numerical values 6 and 10N , the maximum allowed value for parameter

is 469m (see Fig. 2). From Fig 2, the actual values

of parameter , a , is estimated from the norm of the state

vector and its value is 224a . Using numerical values of

m and a , the percentage of estimation of the parameter is calculated:

Proceedings of the 3rd International Conference on ThBC.7

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

469 224100 100 109,38%

224m a

a

Thus, based on Corollary 1, one can get the estimate of the parameter , 469m , that is higher than its actual value,

224a , for 109.9%, which provides the satisfactory

accuracy.

0 5 10 15-10

0

10

20

30

40

x(k)

k

x1(k)

x2(k)

x3(k)

Fig. 1. The initial response of the system (45) with the initial

condition (0) [2 1 0.5]Tx .

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

V. CONCLUSION

In this paper, the finite-time boundedness and finite-time stability for singular discrete-time systems are discussed. New stability criteria are derived using Lyapunov-like function and expressed in terms of LMIs.

ACKNOWLEDGMENT

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

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0 2 4 6 8 10 12 140

100

200

300

400

500

600

700

x(k)

T ET E

x(k)

k

469m

224a

10T

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978-1-4799-0275-0/13/$31.00 ©2013 IEEE