A less conservative criterion for stability analysis of linear ...scientiairanica.sharif.edu/article_3792_555bb8089868de0...attracted to the analysis and control of time-delay systems

Scientia Iranica D (2015) 22(6), 2415{2423

Sharif University of TechnologyScientia Iranica

Transactions D: Computer Science & Engineering and Electrical Engineeringwww.scientiairanica.com

A less conservative criterion for stability analysis oflinear systems with time-varying delay

A. Farnama, R. Mahboobi Esfanjanib;� and M. Farsib

a. SYSTeMS Research Group, Ghent University, Ghent, Belgium.b. Department of Electrical Engineering, Sahand University of Technology, Tabriz, Iran.

Received 5 April 2014; received in revised form 21 January 2015; accepted 22 August 2015

KEYWORDSStability analysis;Interval time-varyingdelay;Linear MatrixInequalities (LMIs);Maximum AllowableDelay Bound(MADB).

Abstract. This paper investigates the problem of stability analysis for linear systemswith time-varying delay. To reduce the conservativeness of su�cient stability conditions, anovel augmented Lyapunov-Krasovskii Functional (LKF) which includes quadratic termsof double-integral phrases is introduced; as well, the technique of free-weighting matriceswith new slack variables is employed; moreover, a tighter integral inequality is derived forbounding the cross-product terms in the derivative of chosen LKF. Numerical examplesare presented to illustrate the superiority of the proposed method compared to some of thepreviously developed approaches.© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Variable time-delays are encountered in many practicalsystems, especially in the processes that involve trans-portation or propagation of material and data such aschemical reactors, combustion engines, and networkedcontrol systems. The presence of time-delay in thedynamic equations of the system brings importantchallenges in its stability analysis and stabilization.Therefore, recently, considerable attention has beenattracted to the analysis and control of time-delaysystems [1-8].

Most of the time-domain stability criteria forsystems with time-varying delay are based primarilyon the Lyapunov-Krasovskii Theorem, combined withthe model transformation schemes and bounding tech-niques for the cross-terms in the derivative of theLyapunov-Krasovskii Functional (LKF) [1]. In orderto obtain less conservative delay-dependent conditions,

*. Corresponding author. Tel.: +98 41 33459356;Fax: +98 41 33444322E-mail address: [email protected] (R. MahboobiEsfanjani)

more e�cient augmented LKFs were constructed; be-sides more accurate bounding methods were developedfor the cross-terms in the derivative of the energyfunctional [9,10]. In [11], delay-dependent stabilitycriteria were derived for linear systems with a constantdiscrete delay, wherein a new simple LKF was intro-duced by uniformly dividing the delay interval into mul-tiple segments and choosing di�erent weight matricescorresponding to each segment. The idea of [11] wasgeneralized in [12] to obtain less conservative stabilityconditions for linear systems with time-varying delay inwhich the delay interval is decomposed into equidistantsubintervals; then, choosing di�erent matrix pairs foreach subinterval, a new LKF was constructed. In [13],Wirtinger inequality was used for the �rst time toreduce the conservatism in computing the derivative ofLKFs. This more accurate integral inequality dependsnot only on the state, but also on the integral of thestate over the delay interval.

The method of free-weighting matrices that in-jects additional variables to add extra-degree of free-dom in the su�cient stability condition made an im-portant progress in delay-dependent stability criteria ofsystems with time-varying delay [14,15]. The stability

2416 A. Farnam et al./Scientia Iranica, Transactions D: Computer Science & ... 22 (2015) 2415{2423

measures were further improved by involving novelslack matrices in the derivative of energy functionals.On the other hand, the technique of free-weighting ma-trices was combined with the conventional ideas, suchas descriptor transformation approach and augmentedvectors method, to further reduce the conservativenessof stability conditions [16,17]. But, in some cases, usingfunctional parameters and/or free-weighting matricesleads to complex stability criteria. Therefore, in orderto reduce computational burden, a simpli�ed criterionwas obtained in [18].

Most of the mentioned methods, such as [19]utilize LKFs, include double-integral terms. For the�rst time, triple-integral terms were added to the LKFin [20] to enhance stability criterion for the constantdelay system. This type of augmented LKF was usedto develop the stability measure for the linear systemswith time varying-delay in [21]. It is worth noting thatsome types of LKFs yield in�nite dimensional LMIs.That is why many authors have considered the simpleform of LKFs and thus derived simpler, but moreconservative, conditions [22].

In [23], delay-range-dependent stability criterionfor linear systems with time-varying delay was derived,where a new estimation method, along with the convexcombination and delay partitioning was employed toobtain a less conservative stability condition. Morerecently, the stability of a linear system with intervaltime-varying delay was investigated in [24], wherea new LKF was presented and some novel integralinequalities were established for bounding the crossterms appearing in the derivative of the chosen LKF.Moreover, the matrix-based quadratic convex approachwas used to ensure both the positive de�niteness ofLKF and negative de�niteness of its derivative.

This paper presents an improved stability condi-tion for systems with time-varying delay compared tosome of the currently available conditions in literature.To develop this novel condition, a new type of LKFcontaining quadratic phrases of double-integral terms isintroduced and the method of free-weighting matriceswith novel slack variables is utilized. Moreover, atighter integral inequality is derived to bound cross-terms in the derivative of LKF. E�ciency of thesuggested approach is demonstrated by two illustrativeexamples. Note that since less conservativeness ofthe proposed stability condition is the consequence ofincreasing the number of matrix parameters in thederived LMIs, more computation time is needed in thetesting procedure.

This paper is organized as follows: The problemis described in Section 2. Section 3 introduces thesu�cient condition for stability analysis of the linearsystem with time-varying delay. In Section 4, twonumerical examples are presented to demonstrate theadvantages of the proposed method compared to some

of the existing approaches. Section 5 concludes thepaper.

Notations: In this paper, R denotes real numbersset. The symbol * stands for symmetric block inthe symmetric matrices. I is identity matrix withappropriate dimensions. The notation P > 0 (P � 0)means that P is real symmetric and positive de�nite(positive semide�nite). The superscript T stands formatrix transposition. colfg shows column vector of theelements in the bracket.

2. System description and preliminaries

The linear system with time-varying delay is describedas follows:

_x(t) = Ax(t) +A1x(t� �(t)); t � 0;x(t) = '(t); t 2 [��2; 0); (1)

where x(t) 2 Rn is the state vector; A and A1 aresystem matrices with appropriate dimensions; and �(t)denotes time-varying delay and satis�es:

0 � �1 � �(t) � �2; (2)where, �1 < �2 are the constant upper and lowerbounds of time-varying delay.

The problem is to �nd a new delay-dependentcondition in terms of linear matrix inequalities toinvestigate the asymptotic stability of the system(Eqs. (1)) with the delay satisfying Relation (2). Beforeproceeding further, inspired by Jensen's inequalityin [23], the following lemma is extracted.

Lemma 1. Suppose 0 � �1 � �2 and !(t) 2 Rn; forany positive de�nite matrix Z 2 Rn�n, the followinginequality holds:

(�22 � �21)��1Z��2

tZt+�

!T (�)Z!(�)d�d�

� 20B@ ��1Z��2

tZt+�

!(�)d�d�

1CAT

Z

0B@ ��1Z��2

tZt+�

!(�)d�d�

1CA :Proof: Regarding the Schur complement, the follow-ing holds:�

!T (�)Z!(�) !T (�)!(�) Z�1

�� 0;

A. Farnam et al./Scientia Iranica, Transactions D: Computer Science & ... 22 (2015) 2415{2423 2417

double integrating both sides of the above relationyield:26664

��1R��2

tRt+�

!T (�)Z!(�)d�d��1R��2

tRt+�

!T (�)d�d�

��1R��2

tRt+�

!(�)d�d� (�22��21)

2 Z�1

37775�0:Applying the Schur complement completes the proof.�

3. Main results

In this section, a new delay-dependent stability condi-tion is derived to check the asymptotic stability of thelinear system (Eqs. (1)) using appropriate LKF andLemma 1.

Theorem 1. Given 0 < �1 < �2, A and A1, linearsystem in Eqs. (1) with time-varying delay satisfyingRelation (2) is asymptotically stable if there are matri-ces Y0, Y1, Y2, M , X0, X1, X2, and symmetric matrices:

P = [Pij ]7�7 > 0 (for i; j = 1; 2; � � � ; 7);Q1 = [Q1ij ]2�2 > 0; Q2 = [Q2ij ]2�2 > 0;

T1 = [T1ij ]2�2 > 0; T2 = [T2ij ]2�2 > 0;

Z1 = [Z1ij ]2�2 > 0; Z2 = [Z2ij ]2�2 > 0;

(for i; j = 1; 2);

with appropriate dimensions satisfying the followinginequality:26666664

� �1 �Y0 �21 �Y1

� ��1�T111 T112 +X0� T122

�0

� � ��21�T211 T212 +X1� T222

��

�21 �Y200

��21�T211 T212 +X2� T222

�377775 < 0; (3)

where:�21 = �2 � �1;

and:�Y T0 =

� ~Y T0 0 0 0 0� ;�Y T1 =

� ~Y T1 0 0 0 0� ;�Y T2 =

� ~Y T2 0 0 0 0� ;

with:

~Y0 =�0 Y0

�; ~Y1 =

�0 Y1

�; ~Y2 =

�0 Y2

�;

also:X=X̂

+ + T ;

with:

T =��MA 0 0 M 0 0�MA1 0 0 0 0� ;

and:

�̂ =

2666666666666666664

�̂11 �̂12 �̂13 �̂14 P12 P13� �̂22 �̂23 �̂24 �̂25 P23� � �̂33 �̂34 PT23 �̂36� � � �̂44 0 0� � � � �̂55 0� � � � � �Q222� � � � � ��

0 �̂18 �̂19 �̂110 �̂1110 �̂28 �̂29 �̂210 �̂2110 �̂38 �̂39 �P56 �P570 �̂48 �̂49 �̂410 �̂4110 P24 P25 P26 P270 P34 P35 P36 P37

�̂77 0 0 0 0� �̂88 �̂89 �̂810 �P67� � �̂99 �PT67 �̂911� � � �Z111 0� � � � �Z211

3777777777777777775;

in which:

�̂11 =PT14 + P14 +Q111 + �1T111 + �21T211

+�414Z111 +

(�22 � �21)24

Z211 � �21Z122� �221Z222 +X0;

�̂14 =P11 + �1P16 + �21P17 +Q112 + �1T112

+ �21T212 +�414Z112 +

(�22 � �21)24

Z212 ;

�̂22 =� PT24 + PT25 � P24 + P25 �Q111 +Q211�X0 +X1;


�̂44 =Q122 + �1T122 + �21T212 +�414Z122

+��22 � �21�2

4Z222 ;

�̂12 = PT24 � P14 + P15;

�̂13 = PT34 � P15;

�̂18 = P44 � P16 + �1Z122 ;

�̂19 = P45 � P17 + �21Z222 ;

�̂110 = P46 � �1ZT112 ;

�̂111 = P47 � �21ZT212 ;

�̂23 = �PT34 + PT35 � P25;

�̂24 = �PT12 + �1P26 + �21P27;

�̂25 = �Q112 +Q212 ;

�̂28 = �P44 + PT45 � P26;

�̂29 = �P25 + P55 � P27;

�̂210 = �P46 + P56;

�̂211 = �P47 + P57;

�̂33 = �PT35 � P35 �Q211 �X2;

�̂34 = PT13 + �1P36 + �21P37;

�̂36 = P33 �Q212 ;

�̂38 = �PT45 � P36;

�̂39 = �P55 � P37;

�̂48 = P14 + �1PT46 + �21P

T47;

�̂49 = P15 + �1PT56 + �21P

T57;

�̂410 = P16 + �1P66 + �21PT67;

�̂411 = P17 + �1P67 + �21P77;

�̂55 = �Q122 +Q222 ;�̂77 = �X1 +X2;�̂88 = �PT46 � P46 � Z122 ;�̂89 = �PT56 � P47;�̂810 = �P66 + ZT112 ;�̂99 = �PT57 � P57 � Z222 ;�̂911 = �P77 + ZT212 ;

Proof: The LKF candidate is constructed as follows:

V (xt) = V1(xt) + V2(xt) + V3(xt) + V4(xt); (4)

V1(xt) = �T (t)P�(t); (5)

V2(xt) =tZ

t��1�T (�)Q1�(�)d�

+t��1Zt��2

�T (�)Q2�(�)d�; (6)

V3(xt) =0Z

��1

tZt+�

�T (�)T1�(�)d�d�

+��1Z��2

tZt+�

�T (�)T2�(�)d�d�; (7)

V4(xt) =�212

0Z��1

0Z�

tZt+�

�T (�)Z1�(�)d�d�d�

+(�22 � �21)

2

��1Z��2

0Z�

tZt+�

�T (�)Z2�(�)d�d�d�;(8)

wherein, augmented vectors �(�) and �(t) are de�nedas:

�(�) = colfx(�); _x(�)g;


and:

�(t) = col

(x(t); x(t� �1); x(t� �2);

tZt��1

x(�)d�;t��1Zt��2

x(�)d�;

0Z��1

tZt+�

x(�)d�d�;

��1Z��2

tZt+�

x(�)d�d�

):

First, time derivative of V (xt) is computed along thetrajectories of Eqs. (1) as the following:

_V (xt) =2�T (t)P (t)� + �T (t)Q1�(t)

� �T (t� �1)Q1�(t� �1)+ �T (t� �1)Q2�(t� �1)� �T (t� �2)Q2�(t� �2)+ �T (t)(�1T1 + �21T2)�(t)

�tZ

t��1�T (�)T1�(�)d�

�t��1Z

t��(t)�T (�)T2�(�)d�

�t��(t)Zt��2

�T (�)T2�(�)d�

+ �T (t)��414Z1 +

(�22 � �21)24

Z2��(t)

� �212

0Z��1

tZt+�

�T (�)Z1�(�)d�d�

� (�22 � �21)2

��1Z��2

tZt+�

�T (t)Z2�(t)d�d�:

Note that for any of the matrices Y0, Y1, Y2, M , X0,X1, and X2 with appropriate dimensions, the following

equalities hold:

"1(t) =2�T (t)Y0

x(t)� x(t� �1)

�tZ

t��1_x(�)d�

!= 0; (9)

"2(t) =2�T (t)Y1

x(t� �1)� x(t� �(t))

�t��1Z

t��(t)_x(�)d�

!= 0; (10)

"3(t) =2�T (t)Y2

x(t� �(t))� x(t� �2)

�t��(t)Zt��2

_x(�)d�

!= 0; (11)

"4(t)=2�T (t)M ( _x(t)�Ax(t)�A1x(t��(t)))=0;(12)

"5(t) =xT (t)X0x(t)� xT (t� �1)X0x(t� �1)

� 2tZ

t��1_xT (�)X0x(�)d� = 0; (13)

"6(t) =xT (t� �1)X1x(t� �1)� xT (t� �(t))X1x(t� �(t))

� 2t��1Z

t��(t)_xT (�)X1x(�)d� = 0; (14)

"7(t) =xT (t� �(t))X2x(t� �(t))� xT (t� �2)X2x(t� �2)

� 2t��(t)Zt��2

_xT (�)X2x(�)d� = 0; (15)

in which:

�(t) =colfx(t); x(t� �1); x(t� �2); _x(t);_x(t� �1); _x(t� �2); x(t� �(t)g:

Regarding Eqs. (9)-(15) and utilizing Lemma 1, the


upper bound of _V (xt) is written as follows:

_V (xt) �2�T (t)P�(t) + �T (t)Q1�(t)� �T (t� �1)Q1�(t� �1)+ �T (t� �1)Q2�(t� �1)� �T (t� �2)Q2�(t� �2)+ �T (t)(�1T1 + (�2 � �1)T2)�(t)

�tZ

t��1�T (�)T1�(�)d�

�t��1Z

t��(t)�T (�)T2�(�)d�

�t��(t)Zt��2

�T (�)T2�(�)d�

+ �T (t)��414Z1 +

(�22 � �21)24

Z2��(t)

�

266640R��1

tRt+�

x(�)d�d�

�1x(t)�tR

t��1x(�)d�

37775T

:�Z111 Z112� Z122

�

:

266640R��1

tRt+�

x(�)d�d�

�1x(t)�tR

t��1x(�)d�

37775

�

26664��1R��2

tRt+�

x(�)d�d�

(�2 � �1)x(t)�t��1Rt��2

x(�)d�

37775T

:�Z211 Z212� Z222

�

:

26664��1R��2

tRt+�

x(�)d�d�

(�2 � �1)x(t)�t��1Rt��2

x(�)d�

37775+ i=7Xi=1

"i(t):(16)

On the other hand, the following inequalities are true:

�2�T (t) ~Y0tZ

t��1�(�)d�

� �1�T (t) ~Y0�T111 T112 +X0� T122

��1~Y T0 �(t)

+tZ

t��1�T (�)

�T111 T112 +X0� T122

��(�)d�;

~Y0 =�0 Y0

�; (17)

�2�T (t) ~Y1t��1Z

t��(t)�(�)d�

� �21�T (t) ~Y1�T211 T212 +X1� T222

��1~Y T1 �(t)

+t��1Z

t��(t)�T (�)

�T211 T212 +X1� T222

��(�)d�;

~Y1 =�0 Y1

�; (18)

�2�T (t) ~Y2t��(t)Zt��2

�(�)d�

� �21�T (t) ~Y2�T211 T212 +X2� T222

��1~Y T2 �(t)

+

t��(t)Zt��2

�T (�)�T211 T212 +X2� T222

��(�)d�;

~Y2 =�0 Y2

�: (19)

Now, the upper bound of _V (xt) is restated as followsby substituting Relations (17)-(19) in Relation (16):

_V (xt) � #T (t)(

� + �1 �Y0�T111 T112 +X0� T122

��1�Y T0

+ �21 �Y1�T211 T212 +X1� T222

��1�Y T1

+ �21 �Y2�T211 T212 +X2� T222

��1�Y T2

)#(t); (20)

where, �Y0, �Y1, and �Y2 were de�ned previously in


Relation (3) and:

#(t) = col

(x(t); x(t� �1); x(t� �2); _x(t);

_x(t� �1); _x(t� �2); x(t� �(t);tZ

t��1x(�)d�

t��1Zt��2

x(�)d�;

0Z��1

tZt+�

x(�)d�d�;

��1Z��2

tZt+�

x(�)d�d�

):

Regarding Relation (20), the Lyapunov-KrasovskiiTheorem [1] guarantees the asymptotic stability of thesystem in Eqs. (1), provided that:

� + �1 �Y0�T111 T112 +X0� T122

��1�Y T0

+ �21 �Y1�T211 T212 +X1� T222

��1�Y T1

+ �21 �Y2�T211 T212 +X2� T222

��1�Y T2 < 0;

which can be transformed easily to Relation (3) by theSchur Complement.�

Remark. The novelty in the derivation of stabilitycriterion in Relation (3) is threefold. First, new LKFwas constructed by including the double-integral terms

0R��1

tRt+�

x(�)d�d� and��1R��2

tRt+�

x(�)d�d� in the aug-

mented vectors �(t) which di�erently from [21], createinnovative quadratic terms in the energy functional.Second, novel inequality was introduced in Lemma 1to bound the cross terms appearing in the derivativeof V due to the existence of newly appended terms.Finally, inventive relations were presented in Eqs. (13)-(15) to inject new free weights in the derivative of thechosen LKF in order to create more degree of freedomin the resulting su�cient condition. Combination ofthe mentioned tricks decreases the conservativeness ofthe stability condition.

Corollary: For �1 = 0, the stability condition ofTheorem 1 is rewritten as follows where all the no-

tations were de�ned previously.26666664� �2 �Y1

� ��2�T211 T212 +X1� T222

��

�2 �Y2

0

��224T211 T212 +X2� T222

35

3777777775 < 0: (21)

Proof. The upper bound of _V in Relation (20) isrewritten as follows for �1 = 0 which is a result ofomitting some terms in the chosen LKF in Eq. (4):

_V � #T (t)"

� + �2 �Y1�T211 T212 +X1� T222

��1�Y T1

+ �2 �Y2�T211 T212 +X2� T222

��1�Y T2

##(t):

So, regarding the Schur complement, it can be easilyveri�ed that if Relation (21) holds, then _V < 0.�

4. Illustrative examples

Two numerical examples are presented to comparethe proposed method with some existing ones. TheLMI Toolbox of Matlab is utilized to solve the LMIfeasibility problems [25]. The maximum value ofdelay that retains the stability of system is called asMaximum Allowable Delay Bound (MADB). MADB,which is the common performance index to evaluatethe conservativeness of stability tests in the literature,is computed for the proposed and rival stability crite-ria.

Example 1. Consider the following system from [21]:

_x(t) =�0 10 �0:1

�x(t) +

�0

0:1

�u(t); (22)

which is controlled through a communication networkwith:

u(t) =��3:75 �11:5�x(t� �(t));

wherein �(t) denotes the time-varying transmissiondelay in the communication link. So, the closed-loop


Table 1. MADBs computed by di�erent methods for Example 1 (with �1 = 0).

Method[27] [7] [16] [28] [26] [21] Proposed method

MADB 0.9412 1.0081 1.0081 1.0081 1.0432 1.0629 1.0762

Table 2. Admissible upper bound �2 for di�erent values of �1.

Method

[31] [30] [29]N = 2

[29]N = 4

[23]N = 2

[23]N = 4

Proposedmethod

�1 = 1 1.9008 1.8737 2.004 2.0273 2.0089 2.0448 2.0852�1 = 2 2.5663 2.5048 2.5650 2.5915 2.5829 2.6051 2.6767�1 = 3 3.3408 3.2591 3.2866 3.3010 3.2983 3.3098 3.3778�1 = 4 4.169 4.0744 4.0818 4.0855 4.0848 4.0877 4.123

system can be described by the following time-delaysystem as in [21]:

_x(t)=�0 10 �0:1

�x(t)+

�0 0

�0:375 �1:15�x(t��(t)):

(23)

In Table 1, the maximum allowable value of �2 obtainedfrom the proposed method along with the results fromrival methods is listed for comparison. Table 1 clearlyshows that the proposed approach outperforms themethods presented in [7,16,21,26-28].

Example 2. Considering the system in Eqs. (1) withthe following matrices [23]:

A =��2 0

0 �0:9�; A1 =

��1 0�1 �1

�:

Table 2 shows the maximum allowable delay upperbound for di�erent values of delay lower bound �1,obtained from the proposed approach and methodsof [23,29-31]. As seen, the maximum bound of delayobtained by the proposed method is larger than theexisting ones which veri�es the superiority of thesuggested delay-dependent condition.

Results of several recent papers are reported inTables 1 and 2 to reveal that the improvement ofstability conditions with regard to the nearby rivalmethods is gradual.

5. Conclusion

In this paper, a new approach has been proposed toanalyze the asymptotic stability of the linear timeinvariant systems with time-varying delay. By con-structing new augmented Lyapunov-Krasovskii func-tional and using free-weighting matrices, a novel delay-dependent stability condition has been derived in termsof LMIs. Numerical examples have been given todemonstrate that the proposed criterion is less conser-vative compared to some of the existing approaches inthe literature.

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Biographies

Arash Farnam received the BSc degree in ElectricalEngineering from Tabriz University, Iran, in 2010, andthe MSc degree in Electrical Engineering from SahandUniversity of Technology, Tabriz, Iran, in 2012. Heis currently working toward the PhD degree in theSYSTeMS Research Group, Ghent University, Ghent,Belgium. His research interests include time-delaysystems and networked control systems.

Reza Mahboobi Esfanjani received his BSc degree(with honours) in Electrical Engineering from the De-partment of Electrical Engineering, Sahand Universityof Technology, Tabriz, Iran in 2002. He received theMSc and PhD degrees from Amirkabir University ofTechnology in 2004 and 2009, respectively. He has beenholding a faculty position at the Electrical EngineeringDepartment, Sahand University of Technology, since2010. His research interests are analysis and control oftime-delay and networked systems.

Milad Farsi received his BSc degree in ElectricalEngineering from Tabriz University, Tabriz, Iran, andMSc degree from Sahand University of Technology,Tabriz, Iran, in 2010 and 2013, respectively. Hiscurrent research interests include time-delay systemsand switched systems theory with applications intoanalysis and design of networked control systems andpower electronic circuits.

Documents

A less conservative criterion for stability analysis of linear ...scientiairanica.sharif.edu/article_3792_555bb8089868de0...attracted to the analysis and control of time-delay systems