The application of different Lyapunov-like functionals and some aggregate norm approximations of the...

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The application of different Lyapunov-like functionalsand some aggregate norm approximations of the delayedstates for finite-time stability analysis of linear discrete

time-delay systems

Sreten B. Stojanovica,n, Dragutin Lj. Debeljkovicb, Dragan S. Anticc

aUniversity of Nis, Faculty of Technology, 16000 Leskovac, SerbiabUniversity of Belgrade, Faculty of Mechanical Engineering, 11000 Belgrade, Serbia

cUniversity of Nis, Faculty of Electronic Engineering, 18000 Nis, Serbia

Received 28 August 2013; received in revised form 17 February 2014; accepted 21 March 2014

Abstract

In this paper, the problem of finite-time stability analysis for linear discrete time-delay systems is studied. Byusing the classical Lyapunov-like functional and Lyapunov-like functionals with power or exponential functions,some sufficient conditions for finite-time stability of such systems are proposed in the form of the linear matrixinequalities. The six aggregate norm approximations of the delayed states are introduced to establish the relationsbetween the classical Lyapunov-like functional and its difference. To further reduce the conservatism of stabilitycriteria, three inequalities with delayed states for the estimation of Lyapunov-like functional are proposed. Anumerical example is included to illustrate the effectiveness and advantage of the proposed methods.& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

During last few decades, the concept of the Lyapunov asymptotic stability (LAS) has beendominating in the field of the theory of the system stability. This stability concept considers the

rg/10.1016/j.jfranklin.2014.03.0102014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

ding author.dresses: [email protected] (S.B. Stojanovic), [email protected] (D.Lj. Debeljkovic),elfak.ni.ac.rs (D.S. Antic).

this article as: S.B. Stojanovic, et al., The application of different Lyapunov-like functionals and someorm approximations of the delayed states for finite-time stability analysis of linear discrete time-s, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.03.010


dx.doi.org/10.1016/j.jfranklin.2014.03.010





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dynamic behavior of the system in the infinite time interval. However, in many practicalapplications there are some cases where the dynamic behavior of the system in the infinite timeinterval is not acceptable. For example, in order to avoid excitation of nonlinear dynamic orsaturation in the control loop during the transients at a specified finite interval of time, the statetrajectories need to be kept within a given region in the state space. Similarly, for the chemicalprocesses, rockets, airplanes and space vehicles, it is expected that their state variables will havebeen controlled within certain bounds over a finite time interval. In these cases, it is important todetermine conditions under which trajectories of a given dynamical system are bounded within aspecified region in the state space, when the system is operating over a specified finite intervalof time.In the existing literature, these conditions are served as a definition for a new stability concept

on a finite-time interval, which is called the finite-time stability (FTS). It is important toemphasize that FTS and LAS are independent concepts; neither one of them implies nor excludesthe other. In fact, a system can be FTS, i.e. a state starting within a “specified” bound α does notexceed a “specified” bound β in a specified time interval, but may become unstable after thespecified interval of time. On the other hand, the state trajectory might exceed the given boundover a certain time interval, but asymptotically go to zero. Thus, FTS is a practical stabilityconcept; it finds application whenever it is desired that the state variables do not exceed a givendomain over a finite time interval [1–6].It should be noted that in the existing literature, parallel to the previously defined FTS, there is

another stability concept denoted by FTS, but with a different meaning. The second FTS conceptis used to describe the system which states approach equilibrium point in a finite time [7–8]. Inour paper, we consider the first concept of FTS.The FTS concept has appeared in the middle of the last century [1–3] where only the class of

regular continuous systems is discussed. The derived stability criteria have been conservative andpractically useless for numerical calculation. Recently, this concept has been extended to theclass of regular discrete systems and revisited in the light of linear matrix inequalities (LMIs),which have allowed finding less conservative criteria. The consequence of this is the appearanceof many useful results; see, for instance [4–8] for continuous and [9–14] for discrete-timesystems. However, in all these works, the time delay is not considered.The phenomenon of time delay has been widely investigated in the last three decades.

Practical examples of time-delay systems include chemical engineering, communications andbiological systems. It has been ascertained that the existence of time delay may lead toinstability, oscillation and poor performance of control systems. Therefore, considerableattention has been devoted to the problem of stability of time-delays systems. Several categoriesof asymptotic and exponential stability and stabilization of time-delay systems have beenstudied: delay-independent, delay-dependent and robust [15–17].During the time, the class of time-delay systems has become interesting for studying the finite-

time stability and the existing concept of FTS has been extended to this system class. However,up to date, there are few works dealing with the FTS of time-delay systems. Some early results ofFTS for time-delay systems can be found in [18–20]. The results of these investigations areconservative because they are based on the application of simple algebraic inequalities or theknowledge of the system response. Recently, using the theory of linear matrix inequality, morepractical and precise results are obtained for continuous [21–29] and discrete-time delay systems[30–35].Despite the fact that the digital control has increased in importance for practical control applications

in last decades, there are few papers dealing with FTS of discrete systems with time-delay. Reference

Please cite this article as: S.B. Stojanovic, et al., The application of different Lyapunov-like functionals and someaggregate norm approximations of the delayed states for finite-time stability analysis of linear discrete time-delay systems, Journal of the Franklin Institute. (2014), http://dx.doi.org/10.1016/j.jfranklin.2014.03.010




S.B. Stojanovic et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] 3

[31] considers the FTS of discrete-time systems with unknown constant delay and uses classical LLFwith delay. In [32], FTS for discrete-time systems with time-varying delay are considered, while theconstant time-delay is taken as a special case.

In this paper, we apply the concept of FTS to a class of linear discrete-time systems withunknown constant time-delay. We analyze the impact of several Lyapunov-like functionals(LLFs) and some aggregate norm approximations (ANAs) of the delayed states to the FTS oflinear discrete time-delay systems. Three classes of LLFs are used as follows: the classical LLFfor time-delay systems [31], LLF with power function [32] and LLF with exponential function.Also, six aggregate norm approximations of the delayed states (Lio∑k�1

j ¼ k�dxT ðjÞQxðjÞ,

i¼ 1; 2;…; 6) are introduced to establish the relations between the classical LLF and itsdifference. In addition to this, to reduce the conservatism of stability criteria, three inequalitieswith delayed states, V xðkÞð ÞZλminðPÞxT ðkÞxðkÞ þ λminðQÞ∑k�1

j ¼ k�df iðk�1; j; γÞxT ðjÞxðjÞ, withf 1ðx; y; zÞ ¼ 1, f 2ðx; y; zÞ ¼ zx� y and f 3ðx; y; zÞ ¼ ezðx� yÞ, are proposed for the estimation of theLyapunov-like functional. These inequalities are less restrictive compared to the next inequalitywith nondelayed states VðxðkÞÞZλminðPÞxT ðkÞxðkÞ [32]. Based on theoretical and numericalconsiderations, it has been shown that the derived results are less conservative than [31,32].

1.1. Notations

The following notations will be used throughout this paper. ℜn and ℜn�m denote then-dimensional Euclidean space and the set of all real matrices of dimension n� m. Superscript“T” stands for matrix transposition. X40 means that X is real symmetric and positive definiteand X4Y means that the matrix X�Y is positive definite. λminðUÞ and λmaxðU Þ are the minimumand the maximum eigenvalues. In symmetric block matrices we use an asterisk ðnÞ to represent aterm that is induced by symmetry. If dimensions of the matrices are not explicitly stated, they areassumed to be compatible for algebraic operations.

2. Preliminaries and problem formulation

Consider the following discrete time-delay system:

xðk þ 1Þ ¼ AxðkÞ þ Adxðk�dÞxðkÞ ¼ ϕðkÞ; kA �d; �d þ 1;…; �1; 0f g ð1Þ

where xðkÞAℜn is the state vector, AAℜn�n and AdAℜn�n are the known constant matrices,and d is an unknown, constant time-delay. The initial condition, ϕðkÞ, is a vector-valued functionof kAf�d; �d þ 1;…; �1; 0g.

In the following, the concept of finite-time stability for the time-delay system (1) is introduced.This concept can be formalized through the following definition.

Definition 1. System (1) is said to be finite-time stable with respect to ðα; β;NÞ, where 0rαoβ,if 8kA 1; 2;…;Nf g

supjA �d;�dþ1;…;0f g

ϕT ðjÞϕðjÞrα ) xT ðkÞxðkÞoβ ð2Þ

Firstly, we introduce the following lemma which will be used later. Note that, as opposed toMoon's inequality [37], which is used in the proof of Lemma 1, the more restrictive inequalityaTbraTQaþ bTQb, Q40 is used in [36].






Lemma 1. For any constant symmetric matrix QAℜn�n, Q¼QT40 and any appropriatedimensional matrices, M1Aℜn�n, M2Aℜn�n,

Z ¼Z11 Z12

n Z22

" #Aℜ2n�2n;

W ¼W11 W12

n W22

" #Aℜ2n�2n;

Y ¼ M1 M2� �

Aℜn�2n, if

Q Y

n Z

� �40;

Q �Y

n W

� �40 ð3Þ

we have

∑k�1

j ¼ k�dxT ðjÞQxðjÞZ 1

2ξT ðkÞ

Λ11 Λ12

n Λ22

" #ξðkÞ ð4Þ

with ξT ðkÞ ¼ xT ðkÞ xT ðk�dÞh i

, where

Λ11 ¼ �M1�MT1 �Q�dðZ11 þW11Þ

Λ12 ¼ þMT1 �M2�dðZ12 þW12Þ

Λ22 ¼ þM2 þMT2 þ Q�dðZ22 þW22Þ ð5Þ

Proof. Using the following identity

xðkÞ�xðk�dÞ� ∑k�1

j ¼ k�dxðjþ 1Þ�xðjÞ½ � ¼ 0 ð6Þ

for any N1; N2Aℜn�n, we get

0¼ 2 xT ðkÞNT1 þ xT ðk�dÞNT

2

� �xðkÞ�xðk�dÞ� ∑

k�1

j ¼ k�dxðjþ 1Þ�xðjÞ½ �

( )

¼ 2ξT ðkÞNT I � I� �

ξðkÞ�2ξT ðkÞNT ∑k�1

j ¼ k�dxðjþ 1Þ þ 2ξT ðkÞNT ∑

k�1

j ¼ k�dxðjÞ ð7Þ

where N ¼ N1 N2� �

Aℜn�2n, and ξT ðkÞ ¼ xT ðkÞ xT ðk�dÞh i

. Based on the Moon'sinequality [37] and

Q Y

n Z

� �40;

Q �Y

n W

� �40 ð8Þ

we obtain

�2ξT ðkÞNT ∑k�1

j ¼ k�dxðjþ 1Þ ¼ ∑

k�1

j ¼ k�d�2ξT ðkÞNTxðjþ 1Þ

r ∑k�1

j ¼ k�d

xðjþ 1ÞξðkÞ

" #TQ Y�N

n Z

� � xðjþ 1ÞξðkÞ

" #






¼ ∑k�1

j ¼ k�dxT ðjÞQxðjÞ þ ξT ðkÞ

Q 0

n �Q

" #ξðkÞ

þ2ξT ðkÞ Y�Nð ÞT ∑k�1

j ¼ k�dxðjÞ þ 2ξT ðkÞðY�NÞT I � I

� �ξðkÞ þ dξT ðkÞZξðkÞ ð9Þ

2ξT ðkÞNT ∑k�1

j ¼ k�dxðjÞ ¼ ∑

k�1

j ¼ k�d2ξT ðkÞNTxðjÞ

r ∑k�1

j ¼ k�d

xðjÞξðkÞ

" #TQ �Y þ N

n W

� � xðjÞξðkÞ

" #

¼ ∑k�1

j ¼ k�dxT ðjÞQxðjÞ þ 2ξT ðkÞð�Y þ NÞT ∑

k�1

j ¼ k�dxðjÞ þ dξT ðkÞZξðkÞ ð10Þ

Using relations (7)–(10), we get

0r2ξT ðkÞNT I � I� �

ξðkÞ þ ∑k�1

j ¼ k�dxT ðjÞQxðjÞ þ ξT ðkÞ

Q 0

n �Q

" #ξðkÞ

þ2ξT ðkÞðY�NÞT ∑k�1

j ¼ k�dxðjÞ þ 2ξT ðkÞ Y�Nð ÞT I � I

� �ξðkÞ

þdξT ðkÞZξðkÞ þ ∑k�1

j ¼ k�dxT ðjÞQxðjÞ þ 2ξT ðkÞ �Y þ Nð ÞT ∑

k�1

j ¼ k�dxðjÞ þ dξT ðkÞZξðkÞ

r2 ∑k�1

j ¼ k�dxT ðjÞQxðjÞ

þξT ðkÞM1 þMT

1 þ Qþ dðZ11 þW11Þ �MT1 þM2 þ dðZ12 þW12Þ

n �M2�MT2 �Qþ dðZ11 þW11Þ

" #ξðkÞ

ð11Þwhich implies inequality (4).

3. Main results

3.1. The implementation of the classical LLF and some ANAs of the delayed states

In this section, using the classical LLF for time-delay systems

VðxðkÞÞ ¼ xT ðkÞPxðkÞ þ ∑k�1

j ¼ k�dxT ðjÞQxðjÞ ð12Þ

and six ANAs of the delayed states, (Lio∑k�1j ¼ k�dx

T ðjÞQxðjÞ, i¼ 1; 2;…; 6), some FTS stabilitycriteria for system (1) are proposed. The ANAs are introduced to establish the relations betweenthe LLFs and their difference.






The first stability criterion is based on the application of the following obvious inequality (thefirst ANA):

L1o ∑k�1

j ¼ k�dxT ðjÞQxðjÞ; L1 ¼ xT ðk�dÞQxðk�dÞ ð13Þ

Theorem 1. System (1) is FTS with respect to ðα; β;NÞ, 0rαoβ, if for scalar γ41, there existpositive scalars λ1, λ2, λ3, and λ4, and symmetric positive define matrices P and Q such that thefollowing conditions hold

�γPþ Q 0 ATP

n �γQ ATdP

n n �P

264

375o0 ð14Þ

λ1IoPoλ2I; λ3IoQoλ4I ð15Þ

�βγ�Nðλ1 þ dλ3Þ þ αðλ2 þ dλ4Þo0 ð16ÞProof. The forward difference of (12) is given by

ΔVðxðkÞÞ ¼ Vðxðk þ 1ÞÞ�VðxðkÞÞ ¼ ξT ðkÞΓξðkÞ ð17Þ

ξðkÞ ¼ xT ðkÞ xT ðk�dÞh iT

; Γ ¼ATPA�Pþ Q ATPAd

n ATdPAd�Q

" #ð18Þ

Define

Π ¼ Γ�ðγ�1ÞP 0

0 ðγ�1ÞQ

" #¼

ATPA�γPþ Q ATPAd

n ATdPAd�γQ

" #o0 ð19Þ

Then, based on the Schur complement, we have inequality (14). From relations (13), (18), and(19) and γ41, it follows:

ΔV xðkÞð Þ ¼ ξT ðkÞΓξðkÞ ¼ ξT ðkÞ Π þ ðγ�1ÞP 0

0 Q

" #( )ξðkÞ

r ðγ�1ÞξT ðkÞP 0

0 Q

" #ξðkÞ

oðγ�1Þ xT ðkÞPxðkÞ þ xT ðk�dÞQxðk�dÞ� �oðγ�1Þ xT ðkÞPxðkÞ þ xT ðk�dÞQxðk�dÞ þ ∑

k�1

j ¼ k�dþ1xT ðjÞQxðjÞ

" #

¼ ðγ�1ÞV xðkÞð Þ ð20ÞApplying the iterative procedure in inequality (20), we get

VðxðkÞÞoγkVðxð0ÞÞ ð21ÞThe following estimations of the LLF are obtained by applying some standard inequalities oflinear algebra to LLF of Eq. (12):

Vðxð0ÞÞrα λmaxðPÞ þ dλmaxðQÞ½ � ð22Þ






VðxðkÞÞZλminðPÞxT ðkÞxðkÞ þ λminðQÞ ∑k�1

j ¼ k�dxT ðjÞxðjÞ ð23Þ

By combining inequality (21) with the estimations of inequalities (22) and (23), it yields

λminðPÞxT ðkÞxðkÞ þ λminðQÞ ∑k�1

j ¼ k�dxT ðjÞxðjÞoγkVðxð0ÞÞoγkα λmaxðPÞ þ dλmaxðQÞ½ �

8kA 1; 2;…;Nf g ð24ÞIf the following condition is satisfied for αoβ:

γkα λmaxðPÞ þ dλmaxðQÞ½ �oλminðPÞβ þ λminðQÞβd; 8kA 1; 2;…;Nf g ð25Þthen

xT ðkÞxðkÞoβ ð26ÞThe condition (25) is satisfied if the following condition is true:

γNα λmaxðPÞ þ dλmaxðQÞ½ �oβ λminðPÞ þ λminðQÞd½ �; γ41 ð27ÞBy defining

0oλ1oλminðPÞ; λ24λmaxðPÞ; λ34λmaxðQÞ; λ44λmaxðQÞ ð28Þwe get

λ1IoPoλ2I; λ3IoQoλ4I ð29Þand the condition (27) becomes inequality (16). The proof is completed.

Remark 1. Let us adopt

Π ¼ Γ� ðγ�1ÞP 0

0 0

� �¼


n ATdPAd

" #o0 ð30Þ

instead of inequality (19). By using the second ANA

L2o ∑k�1

j ¼ k�dxT ðjÞQxðjÞ; L2 ¼ 0 ð31Þ

for γ41, we get

ΔVðxðkÞÞr ðγ�1ÞξT ðkÞ P 0

0 0

� �ξðkÞoðγ�1Þ xT ðkÞPxðkÞ� �

oðγ�1Þ xT ðkÞPxðkÞ þ ∑k�1


" #oðγ�1ÞV xðkÞð Þ ð32Þ

Evidently, the conditions (31) and (32) are more restrictive than inequalities (13) and (20).Accordingly, the next more conservative criteria, which is based on the second ANA inequality(31), can be obtained.

Corollary 1. System (1) is FTS with respect to ðα; β;NÞ, 0rαoβ, if for scalar γ41, there existpositive scalars λ1, λ2, λ3, and λ4, and symmetric positive define matrices P and Q such that






inequalities (15) and (16) and the following condition hold

�γPþ Q 0 ATP

n �Q ATdP

n n �P

264

375o0 ð33Þ

Remark 2. The last result is less restrictive compared to [31], [Theorem 5]. Namely, [31] usesthe following more conservative conditions with three parameters λ1;…; λ3 and nondelayedstates:

V xðkÞð ÞZλminðPÞxT ðkÞxðkÞ ð34Þ

γNα λmaxðPÞ þ dλmaxðQÞ½ �oβλminðPÞ; γ41; λ1IoPoλ2I; 0oQoλ3I; ð35Þwhile Corollary 1 is based on the condition (25) with delayed states and four parametersλ1;…; λ4.

Remark 3. The condition αoβ can be eliminated from the statements of the stability criteriathat are proposed in [31] (and existing literature which deals with this issue), because inequality(35) is derived without the use of the condition αoβ. In contrast to this, the inequality inequality(25) (Theorem 1; Corollary 1) is based on the condition αoβ since the inequality∑k�1

j ¼ k�dxT ðjÞxðjÞodβ is valid for 8k40 if αoβ. Accordingly, in [31] (and existing literature),

arbitrary relationships between the parameters α and β are permitted; βoα allows that the statenorm contracts from the initial condition, while αoβ allows contrariwise. We can reformulateTheorem 1 and Corollary 1 so that the condition αoβ is excluded from new criteria. This can beachieved if the conditions (23) and (25) are replaced with more restrictive conditions (34) and (35).Based on the previous consideration, the following corollary can be stated instead of Theorem 1.

Corollary 2. System (1) is FTS with respect to ðα; β;NÞ, if for γ41, there exist positive scalars λ1,λ2, and λ3, and symmetric positive define matrices P and Q such that (14) and the followingconditions hold

λ1IoPoλ2I; Qoλ3I ð36Þ

�βγ�Nλ1 þ αðλ2 þ dλ3Þo0 ð37ÞRemark 4. If the following inequality is adopted


0 dQ

" #¼


n ATdPAd�ð1þ dðγ�1ÞÞQ

" #ð38Þ

then, for γ41and by using the third ANA

L3o ∑k�1

j ¼ k�dxT ðjÞQxðjÞ; L3 ¼ d xT ðk�dÞQxðk�dÞ ð39Þ

we get

ΔVðxðkÞÞ ¼ ξT ðkÞΓξðkÞ ¼ ξT ðkÞ Π þ ðγ�1ÞP 0

0 dQ

" #( )ξðkÞ


0 dQ

" #ξðkÞ






oðγ�1Þ xT ðkÞPxðkÞ þ dxT ðk�dÞQxðk�dÞ� �oðγ�1Þ xT ðkÞPxðkÞ þ ∑

k�1


" #

¼ ðγ�1ÞV xðkÞð Þ ð40ÞIn a special case, inequality (39) is satisfied if the following condition is true:

xT ðk� jÞQxðk� jÞxT ðk�dÞQxðk�dÞ41; j¼ 1; 2;…; d�1 ð41Þ

Using the above conditions, the following theorem can be stated. Evidently, this theorem limitsthe ratio of the norms using the unit function f 1 ¼ 1.

Theorem 2. System (1) with Eq. (39) is FTS with respect to ðα; β;NÞ, 0rαoβ, if for scalarγ41, there exist positive scalars λ1, λ2, λ3, and λ4, and symmetric positive define matrices P andQ such that inequalities (15) and (16) and the following condition hold

�γPþ Q 0 ATP

n � 1þ dðγ�1Þð ÞQ ATdP

n n �P

264

375o0 ð42Þ

Proof. The proof is similar to that of Theorem 1, and thus is omitted.

Remark 5. Define


0 ð1þ γ þ γ2 þ⋯þ γd�1ÞQ

" #¼


n ATdPAd�γdQ

" #o0 ð43Þ

instead of inequality (19). For γ41 and by using the fourth ANA

L4o ∑k�1

j ¼ k�dxT ðjÞQxðjÞ; L4 ¼ xT ðk�dÞQxðk�dÞ ∑

d

j ¼ 1γd� j ð44Þ

we get

ΔVðxðkÞÞ ¼ ξT ðkÞΓξðkÞ ¼ ξT ðkÞ Π þ ðγ�1ÞP 0


" #( )ξðkÞ



" #ξðkÞ

oðγ�1Þ xT ðkÞPxðkÞ þ xT ðk�dÞQxðk�dÞ þ γxT ðk�dÞQxðk�dÞ�þ⋯þγd�1xT ðk�dÞQxðk�dÞ�oðγ�1Þ xT ðkÞPxðkÞ þ ∑

k�1


" #

¼ ðγ�1ÞV xðkÞð Þ ð45ÞIn a special case, inequality (44) is satisfied if the following condition is true:

xT ðk� jÞQxðk� jÞxT ðk�dÞQxðk�dÞ4γd� j; j¼ 1; 2;…; d�1 ð46Þ






Using the above conditions, the following criterion can be stated. Based on inequality (46), thiscriterion limits the ratio of the norms by the power function f 2ðd; j; γÞ ¼ γd� j.


�γPþ Q 0 ATP

n �γdQ AdTP

n n �P

264

375o0 ð47Þ

Proof. The proof is similar to that of Theorem 1, and is thus omitted.

Remark 6. If we use 1þ eγ þ e2γ þ⋯þ eðd�1Þγ instead of 1þ γ þ γ2 þ⋯þ γd�1 ininequality (43), the previous result can be expressed in a somewhat different manner, by thenext theorem. In this case, the inequality inequality (44) becomes the fifth ANA

L5o ∑j ¼ k�1

j ¼ k�dxT ðjÞQxðjÞ; L5 ¼ xT ðk�dÞQxðk�dÞ ∑

d

j ¼ 1eðd� jÞγ ð48Þ

In a special case, inequality (48) is satisfied if the following condition is true:

xT ðk� jÞQxðk� jÞxT ðk�dÞQxðk�dÞ4eγðd� jÞ; j¼ 1; 2;…; d�1 ð49Þ

Based on inequality (49), it can be concluded that the new criterion limits the ratio Eq. (49) bythe exponential function f 3ðd; j; γÞ ¼ eγðd� jÞ.


�eγPþ Q 0 ATP

n �eγdQ ATdP

n n �P

264

375o0 ð50Þ

Remark 7. Theorems 3 and 4 differ only in the functions f 2ðd; j; γÞ and f 3ðd; j; γÞ. It should beexpected that their results are mutually similar, because the following relationship between thisfunction be established:

f 3ðd; j; γÞ ¼ f 2ðd; j; eγÞ ð51ÞRemark 8. Theorems 1 and 2 can be observed as special cases of Theorems 3 and 4 for thecorresponding values of the parameter γ. The values of γ are determined by solving the nonlinearalgebraic equations that are shown in Table 1. The equations are obtained by comparing ANAinequalities (13) and (39) with ANA inequalities (44) and (48).The following theorem is based on the result of Lemma 1, which represents the sixth ANA.

Theorem 5. System (1) is FTS with respect to ðα; β;NÞ, 0rαoβ, if for scalar γ41, there existpositive scalars λ1, λ2, λ3, and λ4, symmetric positive define matrices PAℜn�n and QAℜn�n,symmetric matrices Z11Aℜn�n, Z22Aℜn�n, W11Aℜn�n, W22Aℜn�n and any appropriatedimensional matrices Z12Aℜn�n, W12Aℜn�n, M1Aℜn�n, M2Aℜn�n, such that inequalities





Table 1The relationships between Theorems 1 and 2 and Theorems 3 and 4.

As a special case of Theorem 3 As a special case of Theorem 4

Theorem 1 ∑d�1j ¼ 1γ

d� j ¼ 0 ∑d�1j ¼ 1e

γðd� jÞ ¼ 0

Theorem 2 ∑d�1j ¼ 1γ

ðd� jÞ ¼ d�1 ∑d�1j ¼ 1e

γðd� jÞ ¼ d�1


(15) and (16) and the following conditions hold

Λ¼Λ11 Λ12 ATP

n Λ22 ATdP

n n �P

264

375o0

Λ11 ¼ �γPþ γ þ 12

Qþ γ�12

ðM1 þMT1 þ dðZ11 þW11ÞÞ

Λ12 ¼γ�12

ð�MT1 þM2 þ dðZ12 þW12Þ

Λ22 ¼ � γ þ 12

Qþ γ�12

ð�M2�MT2 þ dðZ22 þW22Þ ð52Þ

Q Y

n Z

� �40;

Q �Y

n W

� �40; Z ¼

Z11 Z12

n Z22

" #; W ¼

W11 W12

n W22

" #; Y ¼ M1 M2

� �ð53Þ

Proof. The LLF (12) is used in this proof. Let us adopt

Π ¼ Γ�ðγ�1Þ P 0

n 0

� �þ 1

2

Λ11 Λ12

n Λ22

" #( )

¼ATPA�γPþ Qþ γ�1

2 Λ11 ATPAd þ γ�12 Λ12

n ATdPAd�Qþ γ�1

2 Λ22

" #o0 ð54Þ

where the elements of the block matrix Λ are defined by Eq. (5). Based on the Schurcomplement, inequality (54) is equivalent to inequality (52). Using the inequality Eq. (4) (thesixth ANA) and the expressions (18) and (54), we get

ΔVðxðkÞÞ ¼ ξðkÞTΓξðkÞ ¼ ξT ðkÞ Π þ ðγ�1Þ P 0

0 0

� �þ ðγ�1Þ

2

Λ11 Λ12

n Λ22

" #( )ξðkÞ

r ðγ�1ÞξT ðkÞ P 0

0 0

� �þ 1

2

Λ11 Λ12

n Λ22

" #( )ξðkÞ

oðγ�1Þ xT ðkÞPxðkÞ þ 12ξT ðkÞΛξðkÞ

� �

oðγ�1Þ xT ðkÞPxðkÞ þ ∑k�1


" #

¼ ðγ�1ÞV xðkÞð Þ ð55ÞThe remaining part of the proof is similar to that of Theorem 1, and is thus omitted.






Remark 9. The inequality (4) (the sixth ANA) represents the alternative to inequalities (13),(31), (39), (44) and (49) (ANAs 1–5). It should be expected that the result of Theorem 5 is moreconservative than the results of Theorems 1–4, because condition (52) is derived by usingMoon's inequality.When the time-delay d is a known constant value, then system (1) can be transformed into the

following augmented system without delay:

Xðk þ 1Þ ¼ AXðkÞ ð56Þwhere XT ðkÞ ¼ xðkÞ xðk�1Þ … xðk�dÞ� �

is augmented state vector and

A¼

A 0 ⋯ 0 Ad

I 0 ⋯ 0 0

0 I ⋯ 0 0

⋮ ⋮ ⋱ ⋮ ⋮0 0 ⋯ I 0

26666664

37777775

ð57Þ

By using LLF VðXðkÞÞ ¼ XT ðkÞPXðkÞ, the following criterion is proposed for system (56). Theproof is simple and thus omitted.

Theorem 6. System (1) is FTS with respect to ðα; β;NÞ, if for scalar γ40, there exist positivescalars λ1, and λ2 and symmetric positive define matrix P such that the following conditions hold

�γP ATP

n �P

" #o0 ð58Þ

λ1IoPoλ2I ð59Þ

�βγ�Nλþ αλ2o0 ð60Þwhere the matrix A is defined by Eq. (57).

3.2. Implementation of LLFs with power and exponential functions

With the exception of Theorem 6, the previous stability criteria are derived by selecting Eq.(12) as the LLF and using the appropriate ANAs of the delayed states (inequalities (13), (31),(39), (44), (48) and (4)). Specifically, in the proof of Theorems 2–4, the relationships f 1, f 2 andf 3 are used between the norms of the delayed state. On the other hand, the functions f 2 and f 3 canbe directly inserted into LLF without the use of additional ANAs. This leads to the LLFs with thepower function f 2ðk�1; j; γÞ ¼ γk�1� j or the exponential function f 3ðk�1; j; γÞ ¼ eγðk�1� jÞ. Thenext criteria use just these types of LLFs. Notice that the classical LLF, which is proposed in theprevious section, contains the unit function f 1 ¼ 1.

Theorem 7. System (1) is FTS with respect to ðα; β;NÞ, 0rαoβ, if for scalar γ40, there existpositive scalars λ1, λ2, λ3,and λ4, and symmetric positive define matrices P and Q such thatinequalities (15) and (47) and following condition hold

�βγ�Nðλ1 þ γdλ3Þ þ αðλ2 þ γdλ4Þo0; γd ¼γd�1γ�1

ð61Þ






Proof. Let us consider the following LLF with the power function f 2ðk�1; j; γÞ ¼ γk�1� j andγ40


j ¼ k�dγk�1� jxT ðjÞQxðjÞ ð62Þ

Taking the forward difference of Eq. (62) yields

ΔVðxðkÞÞ�ðγ�1ÞVðxðkÞÞ ¼ ξT ðkÞΓξðkÞ

ξðkÞ ¼ xT ðkÞ xT ðk�dÞh iT

; Γ ¼ ATPA�γPþ Q ATPAd

n AdTPAd�γdQ

" #ð63Þ

If Γo0, then

ΔVðxðkÞÞoðγ�1ÞVðxðkÞÞ ð64ÞBy applying iterative procedure in inequality (64), we get

V xðkÞð ÞoγkV xð0Þð Þ ð65ÞIf xT ðkÞxðkÞoα; kA �d;…; 0f g, then

Vðxð0ÞÞrα λmaxðPÞ þ γd λmaxðQÞ� �

; γd ¼γd�1γ�1

ð66Þ

On the other hand

VðxðkÞÞZλminðPÞxT ðkÞxðkÞ þ λminðQÞ ∑k�1

j ¼ k�dγk�1� jxT ðjÞxðjÞ ð67Þ

By combining inequalities (65)–(67) together, it yields

λminðPÞxT ðkÞxðkÞ þ λminðQÞ ∑k�1

j ¼ k�dγk�1� jxT ðjÞxðjÞrVðxðkÞÞ

oγkV xð0Þð Þoγkα λmaxðPÞ þ γd λmaxðQÞ� � ð68Þ

Let us adopt

γkα λmaxðPÞ þ γd λmaxðQÞ� �

oλminðPÞxT ðtÞxðtÞ þ λminðQÞ ∑k�1

j ¼ k�dγk�1� jxT ðjÞxðjÞ;

8kA 1; 2;…Nf g; γ41 ð69Þthen

xT ðkÞxðkÞoβ ð70ÞIf condition (61) is satisfied, where λ1,…,λ4 are defined by inequalities (28) and (29), thencondition (69) is true. This completes the proof of Theorem 7.

Theorem 8. System (1) is FTS with respect to ðα; β;NÞ, 0rαoβ, if there exist a scalar γ,positive scalars λ1, λ2, λ3, and λ4, and positive define symmetric matrices P and Q such thatinequalities (15) and (50) and the following condition hold

�βγ�Nðλ1 þ dλ3Þ þ αðλ2 þ edλ4Þo0; ed ¼eγd�1eγ�1

ð71Þ






Proof. Let us select the following LLF with the exponential function f 3ðk�1; j; γÞ ¼ eγðk�1� jÞ


j ¼ k�deγðk�1� jÞxT ðjÞQxðjÞ ð72Þ

The rest of the proof is similar to the proof of the previous theorem.

Remark 10. By comparing the results of Theorems 3 and 4 with Theorems 7 and 8,respectively, we can see that there are minor differences between them – Theorems 3 and 4 usethe inequality (16), until Theorems 7 and 8 use inequalities (61) and (71), respectively. Thisassertion can be easily checked by numerical calculations (see Example 1).

Remark 11. It is known that selection of Lyapunov functionals (LFs) has predominant influenceon the conservatism of criteria for asymptotic stability; better results (delay-dependent stabilitycriteria) are achieved by selecting delay-dependent LFs. However, in the analysis of FTS fortime-delay systems, the application of new methods for the estimation of LLF at times k ¼ 0 (forVðxð0ÞÞ) and k¼ k (for VðxðkÞÞ), can significantly reduce the conservatism of obtained results. IfVðxð0ÞÞ and VðxðkÞÞ are poor estimated, more conservative result can be obtained, even thoughthe delay-dependent LLF is used. For example, the following restrictive inequality without time-delay:

VðxðkÞÞZλminðPÞxT ðkÞxðkÞ ð73Þis used in [32] for the estimation of LLF, while

VðxðkÞÞZλminðPÞxT ðkÞxðkÞ þ λminðQÞ∑k�1j ¼ k�df iðk�1; j; γÞxT ðjÞxðjÞ; i¼ 1; 2; 3 ð74Þ

where f 1ðk�1; j; γÞ ¼ 1, f 2ðk�1; j; γÞ ¼ γk�1� j and f 3ðk�1; j; γÞ ¼ eγðk�1� jÞ (see inequalities(23) and (67)) are used in this paper.

Remark 12. The case of the constant time-delay is considered in Theorems 2–4, 7 and 8 as wellas in [32], [Corollary 1]. The theorems give less conservative results in comparison with [32],[Corollary 1], because [32] uses restrictive inequality (73) instead of inequality (74). Weconsidered the case of the constant time-delay and delay-independent stability criteria in order tofocus attention on the influence of LLF and ANA to the conservatism of the stability criteria. Anextension of our results to the time-varying delays systems can be carried out by using somelatest results in the field of stability of time-varying delay systems [see [15–17] and referencewithin them]. This extension exceeds the aim of our work at this time and it will be the subject ofour future research.

Remark 13. It should be noted that two sets of the stability criteria may be selected with respectto the range of the parameter γ. The first set of criteria is defined by the condition γ41(Theorems 1–5 and Corollaries 1 and 2), while the condition γ40 determines the second set ofcriteria (Theorems 6–8). Stability criteria, which satisfy the condition γ41, are derived by usingthe ANAs, while the remaining criteria do not require the utilization of ANAs in their proofs.

Remark 14. By using Theorems 1–8 and Corollaries 1 and 2, we can determine the parameter γ,such that the considered system is FTS with respect to ðα; β;NÞ and the parameter β reaches itslower bound βlb. Based on the value of βlb, the stability criteria can be compared according to thefollowing key: if βlb is greater, then conservativeness is greater. For the given criterion, the valueof βlb is always greater than its theoretical value βtlb which represent the minimal values of β,






such that the consider system is FTS. The system simulation can be used for the estimation of theparameter βtlb (see Example 1).

4. Illustrative numerical example and simulations

In this section, a numerical example is provided to verify the effectiveness of the theoreticalresults.

Example 1. Consider the following discrete time-delay system

xðk þ 1Þ ¼ AxðkÞ þ Adxðk�dÞ; A0 ¼0:3 0:2 0

0:3 0:5 0:4

0:2 1 0:4

264

375; A1 ¼

0:2 0:1 0:1

0:3 0:1 0:3

0:2 0:1 0:1

264

375 ð75Þ

The aim is to investigate the finite-time stability of system (75) with respect to ðα; β;NÞ ¼ ð3; β;NÞ,NA 5; 10; 20; 50; 100f g, and dA 2; 5f g and to find the lower bound of the parameter β (βlb), by usingTheorems 1–8 and Corollary 1. Table 2 shows the values of βlb, which were obtained by applyingTheorems 1–8, Corollary 1, [31], [Theorem 5] and [32], [Corollary 1], for different values of theparameters N and d. The theoretical stability limit βtlb is computed by system simulation and is shownin Table 2, for different values of the parameters N and d. By using this data, the numerical values ofβlb=β

tlb are computed and showed in Figs. 1 and 2.

From Table 1 and Figs. 1 and 2, it can be concluded that Theorems 2–4, 7 and 8 give preciseresults compared to the theoretical stability limits, βtlb, while the results of Theorems 5 and 6 andCorollary 1 are conservative. Also, from Table 1, we can see that our criteria (Theorems 2–4, 7and 8) are less conservative than the criteria which are proposed in [31,32]. For explanations seeRemarks 2 and 11.

Note that the results from the first group of criteria (Theorems 2–4, 6–8 and Corollary 1 in[32]) provide approximately the same accuracy in the whole range of changes of the parameter

Table 2The lower bound βlb, which is obtained by applying different criteria to system (75) for dA 2; 5f g, NA 5; 10; 20; 50; 100f g,and theoretical stability limit βtlb, which is obtained by system simulations.

N 5 10 20 50 100

d 2 5 2 5 2 5 2 5 2 5

Theorem 1 9 8 14 14 28 30 219 238 6730 7342Theorem 2 8 7 12 10 24 18 142 71 2786 651Theorem 3 8 7 12 10 23 17 139 65 2667 560Theorem 4 8 7 12 10 23 17 139 65 2647 558Theorem 5 10 9 16 16 36 39 410 450 23085 25464Theorem 6 18 18 27 25 51 41 306 157 5900 1349Theorem 7 8 6 12 9 23 17 139 66 2674 564Theorem 8 8 6 12 9 23 17 139 66 2654 563Corollary 1 10 9 16 16 36 39 411 452 23179 25583Theorem 5 in [31] 12 19 19 26 43 60 487 673 27440 37944Corollary 1 in [32] 11 13 15 17 28 28 164 102 3165 875System simulation 4.9 4.9 6.5 5.9 11.7 9.0 68.2 31.9 1287.1 264.0





0 20 40 60 80 1001.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

N

β lb/

βt lb

d = 2

Theorem 2

Theorems 3, 4, 7, 8

Corollary 1 [32]

Fig. 1. The ratio βlb=βtlb, which is obtained by applying Theorems 2–4, 7 and 8 and Corollary 1 in [32] to system (1) with

Eq. (75), d¼ 2 and NA 5; 10; 20; 50; 100f g.

0 20 40 60 80 1001

1.5

2

2.5

3

3.5

N

d = 5

Theorem 2

Theorems 3, 4

Theorems 7, 8

Corollary 1 [32]

β lb/

βt lb

Fig. 2. The ratio βlb=βtlb, which is obtained by applying Theorems 2–4, 7 and 8 and Corollary 1 in [32] to system (1) with

Eq. (75), d¼ 5 and NA 5; 10; 20; 50; 100f g.


N. Contrariwise, the results that are based on the second group of criteria (Theorems 1 and 5,Corollary 1 and Theorem 5 in [31]) give a greater accuracy at lower values of the parameter N.The main reason is different qualities of approximation of the ratio xT ðk� jÞQxðk� jÞ=xT ðk�dÞQxðk�dÞ, j¼ 1; 2;…; d�1, k ¼ 1; 2;…;N for different values of N, by using variousstability criteria.






5. Conclusion

In this paper, new sufficient conditions in the form of the linear matrix inequalities have beenestablished to ensure the finite-time stability of linear discrete time-delay systems. Three classesof Lyapunov-like functionals are applied: the classical, with power function and with exponentialfunction. For the classical functional, six aggregate norm approximations of the delayed statesare considered. Also, the new inequalities with delayed states for the estimation of Lyapunov-like functional are proposed. It has been shown that the proposed results for the consideredsystem are less conservative than the existing ones in the literature.

Acknowledgment

This work was supported in part by the Ministry of Science and Technological Developmentof Republic of Serbia under Grant ON174001.

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