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This article was downloaded by: [University Of Pittsburgh]On: 18 April 2013, At: 21:54Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Systems SciencePublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tsys20
Delay-derivative-dependent stability criterion forneural networks with probabilistic time-varying delayGuobao Zhang a , Ting Wang a , Tao Li b & Shumin Fei aa Key Laboratory of Measurement and Control of CSE (School of Automation, SoutheastUniversity), Ministry of Education, Nanjing 210096, Chinab School of Automation Engineering, Nanjing University of Aeronautics and Astronautics,Nanjing 210007, ChinaVersion of record first published: 08 May 2012.
To cite this article: Guobao Zhang , Ting Wang , Tao Li & Shumin Fei (2012): Delay-derivative-dependent stabilitycriterion for neural networks with probabilistic time-varying delay, International Journal of Systems Science,DOI:10.1080/00207721.2012.685198
To link to this article: http://dx.doi.org/10.1080/00207721.2012.685198
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International Journal of Systems Science2012, 1–12, iFirst
Delay-derivative-dependent stability criterion for neural networks with probabilistic
time-varying delay
Guobao Zhanga*, Ting Wanga, Tao Lib and Shumin Feia
aKey Laboratory of Measurement and Control of CSE (School of Automation, Southeast University),Ministry of Education, Nanjing 210096, China; bSchool of Automation Engineering,
Nanjing University of Aeronautics and Astronautics, Nanjing 210007, China
(Received 15 February 2011; final version received 6 November 2011)
In this article, based on Lyapunov–Krasovskii functional approach and improved delay-partitioning idea, a newsufficient condition is derived to guarantee a class of delayed neural networks to be asymptotically stable in themean-square sense, in which the probabilistic time-varying delay is addressed. Together with general convexcombination method, the criterion is presented via LMIs and its solvability heavily depends on the sizes of bothtime delay range and its derivative, which has wider application fields than those present ones. It can be shown bythe numerical examples that our method reduces the conservatism much more effectively than earlier reportedones. Especially, the conservatism can be further decreased by thinning the delay intervals.
Keywords: delayed neural networks; asymptotical stability; probabilistic time-varying delay; LMI approach
1. Introduction
Recently, neural networks (NNs) have been applied to
various signal processing problems such as optimisa-
tion, image processing, associative memory design and
other engineering fields. In those applications, the keyfeature of the designed neural network is to be stable
(Zhao, Zhang, Shen, and Gao 2011). Meanwhile, since
there inevitably exist communication delay which is the
main source of oscillation and instability in variousdynamical systems, great efforts have been made to
analyse the dynamics of delayed systems including
delayed neural networks (DNNs) and many elegant
results have been reported; see Zhang, Boukas, andHaidar (2008), Zuo and Wang (2007), Cao, Ho,
Daniel, and Huang (2007), Mou, Gao, and Lam
(2008a), Hu, Gao, and Zheng (2008), Qiu and Cao
(2009), Rakkiyappan, Balasubramaniam, and Cao(2010), Qiu, Cui, and Wu (2009), Syed Ali and
Balasubramaniam (2009), Tian, Xu, and Zu (2009),
Zuo, Yan, and Wang (2010), Yucel and Arik (2009),
Ozcan and Arik (2009), Zhang, Yue, and Tian(2009a,b), Yang, Gao, and Shi (2009), Mou, Gao,
and Qiang (2008b), Zhang, Liu, and Huang (2010),
Chen and Zheng (2008), Zhang and Han (2009), Shao
(2009), Peng, Yue, Tian, and Gu (2009), Yue, Tian,Wang, and James (2009a), Yue, Tian, Zhang, and Peng
(2008, 2009b), Yue, Zhang, Tian, and Peng (2009c),
Tang, Fang, Xia, and Yu (2009), Zhao, Gao, James,
and Chen (2009), Mahmoud, Selim, and Shi (2010),Fu, Zhang, and Ma (2009) and Bao and Cao (2011)and the references therein. In practical applications,though it is difficult to describe the form of time-delay
precisely, the bounds of delay and variant rate still can
be measured. Since Lyapunov functional approach
imposes no restriction on delay derivative and presents
simple stability criteria, Lyapunov–Krasovskii func-
tional (LKF) one has been widely utilised due to that
its analysis can fully make use of the information on
time delay of DNNs as much as possible. Thus,
recently, the delay-dependent stability has become an
important topic of primary significance, in which the
main purpose is to derive an allowable delay upper
bound guaranteeing the global stability of addressed
systems in Zuo and Wang (2007), Cao et al. (2007),
Mou et al. (2008a,b), Qiu and Cao (2009),
Rakkiyappan et al. (2010), Qiu et al. (2009), Syed Ali
and Balasubramaniam (2009), Tian et al. (2009), Zuo
et al. (2010), Zhang et al. (2010), Yucel and Arik
(2009), Ozcan and Arik (2009), Zhang et al. (2009a),
Yang et al. (2009), Chen and Zheng (2008), Zhang and
Han (2009) and Hu et al. (2008).Presently, since delay-partitioning ideas in Mou
et al. (2008a,b), Zhang et al. (2009) and Yang et al.(2009) have been proven to be more effective than theearlier ones and received some great developments(Hu et al. 2008; Zhang and Han 2009). Yet these ideas
*Corresponding author. Email: [email protected]
ISSN 0020–7721 print/ISSN 1464–5319 online
� 2012 Taylor & Francis
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in Zhang and Han (2009) and Hu et al. (2008) still needsome further improvements, since they cannot effec-tively tackle interval variable delay. Meanwhile, in areal system, time-delay often exists in a random form,i.e. some values of time-delay are very large but theprobability taking such large values is very small,which will lead to some conservatism if only theinformation of variation range of time-delay is consid-ered. Thus, recently, some researchers has consideredthe stability for various systems including NNs withprobability-distribution delays (Yue et al. 2008, 2009a–c; Fu et al. 2009; Peng et al. 2009; Tang et al. 2009;Zhang et al. 2009a; Zhao et al. 2009; Mahmoud et al.2010; Bao and Cao 2011). In Zhang et al. (2009b),Zhao et al. (2009), Tang et al. (2009) and Yue et al.(2009c), the authors have analysed the stability and itsapplications for networked control systems, uncertainlinear systems, and T-S fuzzy systems, in whichprobability delay was involved. As for discrete-timeNNs with probabilistic delay, the global stability havebeen considered and some pretty results have beenproposed in Fu et al. (2009) and Mahmoud et al.(2010). It has come to our attention that only Fu et al.(2009), Mahmoud et al. (2010) and Bao and Cao(2011) has investigated the dynamics of continuous-time NNs with probabilistic delay, in which the lowerbound of delay derivative was not considered and infact, such information could play an important rolein reducing the conservatism, which has beenillustrated in Fridman, Shaked, and Liu (2009). Tothe best of authors’ knowledge, through using upperand lower bounds on delay derivative, no researchershave employed the delay-partitioning idea toinvestigate the stability of DNNs with probabilisticdelay, which constitutes the focus of this presentedwork.
Inspired by above discussions, in the article, withtaking lower bounds of probabilistic delay derivativeinto consideration, we make some great efforts toinvestigate the mean-squared stability for DNNs, inwhich one improved delay-partitioning idea is utilised.Through applying the general convex combinationtechnique and choosing one novel LKF, one lessconservative condition is given in terms of LMIs,which can present the pretty delay-dependenceand computational efficiency. Finally, we give fournumerical examples to illustrate that our derivedresults can be less conservative than someexistent ones.
The notations in this article are really standard. Forsymmetric matrices X,Y,X4Y (respectively, X �Y)means that X�Y4 0 (X�Y � 0) is a positive-definite(respectively, positive-semidefinite) matrix; and *denotes the symmetric term in a symmetric matrix,i.e.
�X YYT Z
�¼�X Y� Z
�.
2. Problem formulations and preliminaries
Consider the DNNs as follows:
_zðtÞ ¼ �CzðtÞ þ AgðzðtÞÞ þ Bgðzðt� �ðtÞÞÞ þ I, ð1Þ
where z¼ [z1, . . . , zn]T2Rn is a real n-vector denoting
the state variables associated with the neurons,
g(z)¼ [g1(z1), . . . , gn(zn)]T represents the neuron activa-
tion function, I¼ [I1, . . . , In]T2Rn is a constant input
vector; C¼ diag{c1, . . . , cn}4 0, and A, B are the
appropriately dimensional constant matrices.The following assumptions on the system (1) are
made throughout this article.
Assumption 1: The time-varying delay �(t) satisfies0� �1� �(t)� �3. Moreover, consider the information of
probability distribution of �(t), two sets and functions aredefined as �1¼ {t, �(t)2 [�1, �2)}, �2¼ {t, �(t)2 [�2, �3]},and
�1ðtÞ ¼�ðtÞ, for t 2 �1
��1, for t 2 �2
�and
�2ðtÞ ¼�ðtÞ, for t 2 �2
��2, for t 2 �1
� ð2Þ
�1 � _�1ðtÞ � �2, �3 � _�2ðtÞ � �4, ð3Þ
where �22 [�1, �3], ��1 2 ½�1, �2Þ, and ��2 2 ½�2, �3�. It is easyto check that t2�1 means that the event �(t)2 [�1, �2)occurs and t2�2 means that the event �(t)2 [�2, �3]occurs. Therefore, a stochastic variable �(t) can be
defined as
�ðtÞ ¼1, for t 2 �1
0, for t 2 �2:
�ð4Þ
Assumption 2: �(t) is a Bernoulli-distributed sequence
with
Probf�ðtÞ ¼ 1g ¼ Ef�ðtÞg ¼ �0,
Probf�ðtÞ ¼ 0g ¼ 1� Ef�ðtÞg ¼ 1� �0,
where 0� �0� 1 is a constant and E{�(t)} is themathematical expectation of �(t). It is easy to check
that E{�(t)� �0}¼ 0.
Assumption 3: For the constants �þj , ��j , the nonlinear
function gj(�) in (1) satisfies the following condition
��j �gjð�Þ� gjð�Þ
���� �þj , 8�,� 2R,� 6¼ �, j¼ 1,2, . . . ,n:
Here, we denote �� ¼ diagf�þ1 , . . . , �þn g, � ¼ diagf��1 , . . . , ��n g, and
�1 ¼ diag��þ1 �
�1 , . . . , �þn �
�n
�,
�2 ¼ diagn �þ1 þ ��1
2, . . . ,
�þn þ ��n
2
o: ð5Þ
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It is clear that under Assumptions 1–3, system (1) has
one equilibrium point z� ¼ ½z�1, . . . , z�n�T. For conve-
nience, we shift the equilibrium point z* to the origin
by letting x¼ z� z*, f(x)¼ g(xþ z*)� g(z*), and the
system (1) can be converted to
_xðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt� �ðtÞÞÞ:
Based on the methods in Zhang et al. (2009b), Yue
et al. (2009c), Tang et al. (2009), Zhao et al. (2009), the
system above can be equivalently converted to
_xðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ �ðtÞBf ðxðt� �1ðtÞÞÞ
þ ð1� �ðtÞÞBf ðxðt� �2ðtÞÞÞ,ð6Þ
where f(x)¼ [f1(x1), . . . , fn(xn)]T. It is easy to check that
the function fj(�) satisfies fj(0)¼ 0, and
��j �fjð�Þ
�� �þj , 8� 2 R,� 6¼ 0, j ¼ 1, 2, . . . , n: ð7Þ
Then the problem to be addressed in this article can be
formulated as developing a condition ensuring that the
system (6) is asymptotically stable.
In order to obtain the stability criterion for system
(6), the following lemmas are introduced.
Lemma 1 (Shao 2009): For any constant matrix
X2Rn�n, X¼XT� 0, a scalar functional h :¼ h(t)� 0,
and a vector function _x : ½�h, 0� ! Rn such that the
following integration is well defined, then �hR ht�h
_xTðsÞX _xðsÞds ��xðtÞ � xðt� hÞ
�TX�xðtÞ � xðt� h
��:
Lemma 2: Suppose that �, �i1, �i2(i¼ 1, 2, 3, 4) are
the constant matrices of appropriate dimensions,
�2 [0, 1], �2 [0, 1], � 2 [0, 1], �2 [0, 1], then
�þ[��11þ (1� �)�12]þ [��21þ (1� �)�22]þ [��31þ
(1� �)�32]þ [��41þ (1� �)�42]5 0 holds, if the
following matrix inequalities �þ�1iþ�2jþ�3kþ
�4l5 0(i, j, k, l¼ 1, 2) hold simultaneously.
Remark 1: As an extension of Lemma 2 in Yue et al.
(2009a), the proof of this lemma can be evidently
derived.
3. Delay-dependent stability criterion
Firstly, we can rewrite system (6) as
_xðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ �0Bf ðxðt� �1ðtÞÞÞ
þ ð1� �0ÞBf ðxðt� �2ðtÞÞÞ þ ð�ðtÞ � �0Þ
� B�f ðxðt� �1ðtÞÞÞ � f ðxðt� �2ðtÞÞÞ
�: ð8Þ
Now letting l, m be positive integers, we, respectively,
divide the delay intervals [�1, �2] and [�2, �3] into l, m
segments averagely. Moreover, we introduce some
following denotations:
�1 ¼�2 � �1
l, �2 ¼
�3 � �2m
, �1ðtÞ ¼�1ðtÞ � �1
l,
�2ðtÞ ¼�2ðtÞ � �2
m: ð9Þ
Then based on (7) and (9), we can construct the
following LKF candidate:
VðxtÞ ¼ V1ðxtÞ þ V2ðxtÞ þ V3ðxtÞ, ð10Þ
where
V1ðxtÞ ¼ xTðtÞPxðtÞþ
Z 0
��1
Z t
tþ
�1 _xTðsÞQ _xðsÞdsd
þ2Xni¼1
ki
Z xi
0
½ fiðsÞ���i s�ds
þ2Xni¼1
li
Z xi
0
½�þi s� fiðsÞ�ds,
V2ðxtÞ ¼
Z t
t��1
xðsÞ
fðxðsÞÞ
� T P1 H1
� Q1
� xðsÞ
fðxðsÞÞ
� ds
þX2i¼1
Z t��i
t��i��i
iðsÞ
hiðiðsÞÞ
� T ~Pi~Hi
� ~Qi
" #
�iðsÞ
hiðiðsÞÞ
� dsþ
Xli¼1
Z t��1�ði�1Þ�1
t��1�ði�1Þ�1��1ðtÞ
xðsÞ
fðxðsÞÞ
� T
�X1i Y1i
� Z1i
� xðsÞ
fðxðsÞÞ
� ds
þXli¼1
Z t��1�ði�1Þ�1��1ðtÞ
t��1�i�1
xðsÞ
fðxðsÞÞ
� T
�X2i Y2i
� Z2i
� xðsÞ
fðxðsÞÞ
� ds
þXmi¼1
Z t��2�ði�1Þ�2
t��2�ði�1Þ�2��2ðtÞ
xðsÞ
fðxðsÞÞ
� T
�X3i Y3i
� Z3i
� xðsÞ
fðxðsÞÞ
� ds
þXmi¼1
Z t��2�ði�1Þ�2��2ðtÞ
t��2�i�2
xðsÞ
fðxðsÞÞ
� T
�X4i Y4i
� Z4i
� xðsÞ
fðxðsÞÞ
� ds,
V3ðxtÞ ¼Xli¼1
Z ��1�ði�1Þ�1��1�i�1
Z t
tþ
�1 _xTðsÞW1i _xðsÞdsd
þXmi¼1
Z ��2�ði�1Þ�2��2�i�2
Z t
tþ
�2 _xTðsÞW2i _xðsÞdsd
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with K¼diag{k1, . . . , kn}, L¼ diag{l1, . . . , ln}, n� n
constant matrices P,Q,P1,H1,Q1,Xji,Yji,Zji(j¼ 1, 2,
3, 4), Wji(j¼ 1, 2), ln� ln constant matrices ~P1, ~Q1, ~H1,
mn�mn constant matrices ~P2, ~Q2, ~H2, and
T1 ðsÞ ¼ xTðsÞ xTðs� �1Þ � � � xTðs�ðl�1Þ�1Þ
� �,
hT1 ð1ðsÞÞ ¼ fTðxðsÞÞ fTðxðs� �1ÞÞ � � � fTðxðs�ðl�1Þ�1ÞÞ� �
,
T2 ðsÞ ¼ xTðsÞ xTðs� �2Þ � � � xTðs�ðm�1Þ�2Þ
� �,
hT2 ð2ðsÞÞ ¼ fTðxðsÞÞ fTðxðs� �2ÞÞ � � � fTðxðs�ðm�1Þ�2ÞÞ� �
:
Denoting a parameter set
� ¼
(P,Q,K,L,W1i,W2h,
P1 H1
� Q1
� ,
~Pj~Hj
� ~Qj
" #,
Xji Yji
� Zji
� ,
Xkh Ykh
� Zkh
� ,
j ¼ 1, 2 i ¼ 1, . . . , l; k ¼ 3, 4, h ¼ 1, . . . ,m
),
then we give one proposition which is essential in the
following deduction.
Proposition 1: If the parameter set � satisfies the
following condition:
P4 0,Q4 0,K4 0,L4 0,W1i 4 0,W2h 4 0,
P1 H1
� Q1
" #4 0,
~Pj~Hj
� ~Qj
" #4 0,
Xji Yji
� Zji
" #4 0,
Xkh Ykh
� Zkh
" #4 0, j ¼ 1, 2, k ¼ 3, 4;
i ¼ 1, . . . , l; h ¼ 1, . . . ,m,
then the LKF (10) is definitely positive.
Moreover, in order to simplify the subsequent
proof, we also give some notations in the following:
��1¼�2��1, ��3¼�4��3, ~�i¼ diagf�i, . . . ,�igln�ln,��i¼ diagf�i, . . . ,�igmn�mn,
~Xi¼ diagfXi1, . . . ,Xilg, ~Yi¼diagfYi1, . . . ,Yilg,
~Zi¼ diagfZi1, . . . ,Zilg,
~Wi¼ diagfWi1, . . . ,Wilg, ~V1¼ diagfV11, . . . ,V1lg,
~R1¼ diagfR11, . . . ,R1lg, i¼ 1,2, ð11Þ
~Xj ¼ diagfXj1, . . . ,Xjmg, ~Yj ¼ diagfYj1, . . . ,Yjmg,
~Zj ¼ diagfZj1, . . . ,Zjmg,
~Wj ¼ diagfWj1, . . . ,Wjmg, ~V2 ¼ diagfV21, . . . ,V2mg,
~R2 ¼ diagfR21, . . . ,R2mg, j ¼ 3, 4: ð12Þ
Theorem 1: For two given positive integers l, m, and
time-delay satisfying (2)–(3), the system (8) is globally
asymptotically stable in the mean square sense, if there
exist one parameter set � satisfying Proposition 1, n� n
matrices Ei(i¼ 1, 2), and n� n diagonal matrices
Ui4 0(i¼ 1, 2, 3, 4, 5), V1i4 0, R1i4 0(i¼ 1, . . . , l),
V2j4 0, R2j4 0( j¼ 1, . . . ,m) such that the following
LMIs in (13) hold
�1��T1 þ�2��T
2 �Xli¼1
(ITjiW1iIji�
��1
lIT5i
Xei Yei
� Zei
� I5i
)
�Xmi¼1
(IThiW2iIhi�
��3
mIT6i
Xfi Yfi
� Zfi
� I6i
)50,
j¼ 1,2;h¼ 3,4;e¼ 1,2;f¼ 3,4, ð13Þ
where I1i¼ [0n� (1þ i)n� In 0n� (lþm� 1)n In * ], I2i¼ [0n� inIn 0n� (lþm)n� In * ], I3i¼ [0n� (1þ lþ i)n� In 0n� (m� 1)n In *
], I4i¼ [0n� (lþ i)n In 0n� mn� In * ] and
I5i ¼0n�ðmþlþ1þiÞn � In0n�ð2lþ2mþ1Þn � In�
0n�ðmþlþ1þiÞnIn0n�ð2lþ2mþ1ÞnIn�
� I6i ¼
0n�ðmþ2lþ1þiÞn � In0n�ð2lþ2mþ1Þn � In�
0n�ðmþ2lþ1þiÞnIn0n�ð2lþ2mþ1ÞnIn�
�
� ¼
�11 Q 0 0 �15 0 0 0 �19 0 0 �0ET1B �1,13
� �22 0 0 0 �H1 0 0 0 0 0 0 0
� � �U2�1 0 0 0 U2�2 0 0 0 0 0 0
� � � �U3�1 0 0 0 U3�2 0 0 0 0 0
� � � � �55 0 0 0 �59 0 0 0 0
� � � � � �Q1 0 0 0 0 0 0 0
� � � � � � �U2 0 0 0 0 0 0
� � � � � � � �U3 0 0 0 0
� � � � � � � � �99 0 0 �0ET2B �9,13
� � � � � � � � � �U4�1 0 U4�2 0
� � � � � � � � � � �U5�1 0 U5�2
� � � � � � � � � � � �U4 0
� � � � � � � � � � � � �U5
2666666666666666666666666664
3777777777777777777777777775
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� ¼
�11 0 0 0 ~W1 0 �17 0 0 0 0 0
� �22 0 0 ~W1 0 0 �28 0 0 0 0
� � �33 0 0 ~W2 0 0 �39 0 0 0
� � � �44 0 ~W2 0 0 0 �4,10 0 0
� � � � �55 0 0 0 0 0 �5,11 0
� � � � � �66 0 0 0 0 0 �6,12
� � � � � � �77 0 0 0 0 0
� � � � � � � �88 0 0 0 0
� � � � � � � � �99 0 0 0
� � � � � � � � � �10,10 0 0
� � � � � � � � � � �11,11 0
� � � � � � � � � � � �12,12
2666666666666666666666666664
3777777777777777777777777775
�1 ¼
In �
0n�n In �
0n�ðlþ1Þn In �
0n�ðmþlþ1Þn In �
0n�2ðmþlþ1Þn In �
0n�ð2mþ2lþ3Þn In �
0n�ð2mþ3lþ3Þn In �
0n�3ðmþlþ1Þn In �
0n�4ðmþlþ1Þn In �
0n�ð4mþ4lþ5Þn In �
0n�ð4mþ4lþ6Þn In �
0n�ð4mþ4lþ7Þn In �
0n�ð4mþ4lþ8Þn In
266666666666666666666666666664
377777777777777777777777777775
�2 ¼
0ln�n Iln �
0ln�2n Iln �
0mn�ðlþ1Þn Imn �
0mn�ðlþ2Þn Imn �
0ln�ðmþlþ2Þn Iln �
0mn�ðmþ2lþ2Þn Imn �
0ln�ð2mþ2lþ3Þn Iln �
0ln�ð2mþ2lþ4Þn Iln �
0mn�ð2mþ3lþ3Þn Imn �
0mn�ð2mþ3lþ4Þn Imn �
0ln�ð3mþ3lþ4Þn Iln �
0mn�ð3mþ4lþ4Þn Imn �
266666666666666666666666664
377777777777777777777777775
with the notations * in Iji(j¼ 1, . . . , 6) and �g(g¼ 1, 2) denoting the appropriately dimensional 0 matrix guaranteeing
the relevant matrices of (4lþ 4mþ 9)n columns.
�11 ¼ �ET1C� CTE1 þ P1 �Q�U1�1,�15 ¼ ET
1AþH1 þU1�2,�19 ¼ P��Kþ ��L� ET1 � CTE2,
�1,13 ¼ ð1� �0ÞET1B,�22 ¼ �P1 �Q,�55 ¼ �U1 þQ1,�59 ¼ K� Lþ ATE2,
�99 ¼ �ET2 � E2 þ �
21Qþ
Xli¼1
�21W1i þXmi¼1
�21W2i,�9,13 ¼ ð1� �0ÞET2B,�11 ¼ ~P1 þ ~X1 � ~W1 � ~V1
~�1,
�22 ¼ � ~P1 � ~X2 � ~W1,�33 ¼ ~P2 þ ~X3 � ~W2 � ~V2��1,�44 ¼ � ~P2 � ~X4 � ~W2,
�55 ¼ ð1��2
lÞ ~X2 � ð1�
�1
lÞ ~X1 � 2 ~W1 � ~R1
~�1,�66 ¼ ð1��4
mÞ ~X4 � ð1�
�3
mÞ ~X3 � 2 ~W2 � ~R2
��1,
�77 ¼ ~Q1 þ ~Z1 � ~V1,�88 ¼ � ~Q1 � ~Z2,�99 ¼ � ~V2 þ ~Q2 þ ~Z3,�10,10 ¼ � ~Q2 � ~Z4,
�11,11 ¼ ð1��2
lÞ ~Z2 � ð1�
�1
lÞ ~Z1 � ~R1,�12,12 ¼ ð1�
�4
mÞ ~Z4 � ð1�
�3
mÞ ~Z3 � ~R2,
�17 ¼ ~H1 þ ~Y1 þ ~V1~�2,�28 ¼ � ~H1 � ~Y2,�39 ¼ ~H2 þ ~Y3 þ ~V2
��2,�4,10 ¼ � ~H2 � ~Y4,
�5,11 ¼ ð1��2
lÞ ~Y2 � ð1�
�1
lÞ ~Y1 þ ~R1
~�1,�6,12 ¼ ð1��4
lÞ ~Y4 � ð1�
�3
lÞ ~Y3 þ ~R2
��2:
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Proof: Firstly, through directly calculating and using the denotations in (11)–(12), the stochastic differential ofVi(xt)(i¼ 1, 2) in (10) along the trajectories of system (8) yield
LV1ðxtÞ ¼ 2xTðtÞP _xðtÞ þ _xTðtÞ�21Q _xðtÞ �
Z t
t��1
�1 _xTðsÞQ _xðsÞdsþ 2�fTðxðtÞÞðK� LÞ þ xTðtÞð ��L��KÞ
�_xðtÞ
� 2xTðtÞP _xðtÞ þ _xTðtÞ�21Q _xðtÞ ��xðtÞ � xðt� �1Þ
�TQ�xðtÞ � xðt� �1Þ
�þ 2
�fTðxðtÞÞðK� LÞ
þ xTðtÞð ��L��KÞ�
_xðtÞ, ð14Þ
LV2ðxtÞ ¼�xTðtÞP1xðtÞ þ 2xTðtÞH1f ðxðtÞÞ þ fTðxðtÞÞQ1f ðxðtÞÞ
���xTðt� �1ÞP1xðt� �1Þ
þ 2xTðt� �1ÞH1f ðxðt� �1ÞÞ þ fTðxðt� �1ÞÞQ1f ðxðt� �1ÞÞ�þ�T1 ðt� �1Þ
~P11ðt� �1Þ
þ 2T1 ðt� �1Þ~H1h1ð1ðt� �1ÞÞ þ hT1 ð1ðt� �1ÞÞ
~Q1h1ð1ðt� �1ÞÞ���T1 ðt� �1 � �1Þ
~P1
� ðt� �1 � �1Þ þ 2T1 ðt� �1 � �1Þ~H1h1ððt� �1 � �1ÞÞ þ hT1 ððt� �1 � �1ÞÞ
~Q1
� h1ððt� �1 � �1ÞÞ�þ�T2 ðt� �2Þ
~P22ðt� �2Þ þ 2T2 ðt� �2Þ~H2h2ð2ðt� �2ÞÞ
þ hT2 ð2ðt� �2ÞÞ~Q2h2ð2ðt� �2ÞÞ
���T2 ðt� �2 � �2Þ
~P2ðt� �2 � �2Þ
þ 2T2 ðt� �2 � �2Þ~H2h2ððt� �2 � �2ÞÞ þ hT2 ððt� �2 � �2ÞÞ
~Q2h2ððt� �2 � �2ÞÞ�
þ
hT1 ðt� �1Þ
~X11ðt� �1Þ þ T1
t� �1 � �1ðtÞ
�ð1�
_�1ðtÞ
lÞ
~X2 � ~X1
�1t� �1 � �1ðtÞ
�� T1 ðt� �1 � �1Þ
~X21ðt� �1 � �1Þiþ
h2T1 ðt� �1Þ
~Y1h1ð1ðt� �1ÞÞ þ 2T1t� �1 � �1ðtÞ
�� ð1�
_�1ðtÞ
lÞ
~Y2 � ~Y1
�h11ðt� �1 � �1ðtÞÞ
�� 2T1 ðt� �1 � �1Þ
~Y2h1ð1ðt� �1 � �1ÞÞi
þ
hhT1 ð1ðt� �1ÞÞ
~Z1h1ð1ðt� �1ÞÞ þ hT11ðt� �1 � �1ðtÞÞ
�ð1�
_�1ðtÞ
lÞ
�
~Z2 � ~Z1
�h11ðt� �1 � �1ðtÞÞ
�� hT1
1ðt� �1 � �1Þ
�~Z2h
T1
1ðt� �1 � �1Þ
�iþ
hT2 ðt� �2Þ
~X32ðt� �2Þ þ T2
t� �2 � �2ðtÞ
�ð1�
_�2ðtÞ
mÞ
~X4 � ~X3
�2t� �2 � �2ðtÞ
�� T2 ðt� �2 � �2Þ
~X42ðt� �2 � �2Þiþ
h2T2 ðt� �2Þ
~Y3h2ð2ðt� �2ÞÞ þ 2T2t� �2 � �2ðtÞ
�� ð1�
_�2ðtÞ
mÞ
~Y4 � ~Y3
�h22ðt� �2 � �2ðtÞÞ
�� 2T2 ðt� �2 � �2Þ
~Y4h2ð2ðt� �2 � �2ÞÞi
þ
hhT2 ð2ðt� �2ÞÞ
~Z3h2ð2ðt� �2ÞÞ þ hT22ðt� �2 � �2ðtÞÞ
�ð1�
_�2ðtÞ
mÞ
�
~Z4 � ~Z3
�h22ðt� �2 � �2ðtÞÞ
�� hT2
2ðt� �2 � �2Þ
�~Z4h
T2ðt� �2 � �2Þ
�i: ð15Þ
Moreover, together with Lemma 1 and the methods in Shao (2009), we can estimate LV3(xt) as follows:
LV3ðxtÞ ¼Xli¼1
_xTðtÞð�21W1i
�_xðtÞ �
Xli¼1
Z t��1�ði�1Þ�1
t��1�i�1
�1 _xTðsÞW1i _xðsÞdsþXmi¼1
_xTðtÞð�22W2iÞ _xðtÞ
�Xmi¼1
" Z t��2�ði�1Þ�2��2ðtÞ
t��2�i�2
zþ
Z t��2�ði�1Þ�2
t��2�ði�1Þ�2��2ðtÞ
#_xTðsÞð�2W2iÞ _xðsÞds
�Xli¼1
h_xTðtÞð�2W1i
�_xðtÞ �
�1ðtÞ � �1�2 � �1
�xt� �1 � ði� 1Þ�1 � �1ðtÞ
�� xðt� �1 � i�1Þ
�TW1i
��xt� �1 � ði� 1Þ�1 � �1ðtÞ
�� xðt� �1 � i�1Þ
���2 � �1ðtÞ
�2 � �1
�xt� �1 � ði� 1Þ�1
�� x
t� �1 � ði� 1Þ�1 � �1ðtÞ
��TW1i
�xt� �1 � ði� 1Þ�1
�� x
t� �1 � ði� 1Þ�1 � �1ðtÞ
��i�
hT1t� �1 � �1ðtÞ
�~W1
t� �1 � �1ðtÞ
�� 2T1
t� �1 � �1ðtÞ
�~W11ðt� �1 � �1Þ
þ T1 ðt� �1 � �1Þ~W11ðt� �1 � �1Þ
i�
hT1t� �1 � �1ðtÞ
�~W11
t� �1 � �1ðtÞ
�
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� 2T1t� �1 � �1ðtÞ
�~W11ðt� �1Þ þ
T1 ðt� �1Þ
~W11ðt� �1ÞiþXli¼1
h_xTðtÞð�2W2i
�_xðtÞ
��2ðtÞ � �2�3 � �2
�xt� �2 � ði� 1Þ�2 � �2ðtÞ
�� xðt� �2 � i�2Þ
�TW2i
��xt� �2 � ði� 1Þ�2 � �2ðtÞ
�� xðt� �2 � i�2Þ
���3 � �2ðtÞ
�3 � �2
�xt� �2 � ði� 1Þ�2
�� x
t� �2 � ði� 1Þ�2 � �2ðtÞ
��TW2i
�xt� �2 � ði� 1Þ�2
�� x
t� �2 � ði� 1Þ�2 � �2ðtÞ
��i�
hT2t� �2 � �2ðtÞ
�~W2
t� �2 � �2ðtÞ
�� 2T2
t� �2 � �2ðtÞ
�~W22ðt� �2 � �2Þ
þ T2 ðt� �2 � �2Þ~W22ðt� �2 � �2Þ
i�
hT2t� �2 � �2ðtÞ
�~W2
� 2t� �2 � �2ðtÞ
�� 2T2
t� �2 � �2ðtÞ
�~W22ðt� �2Þ þ
T2 ðt� �2Þ
~W22ðt� �2Þi: ð16Þ
From (7), for any n� n diagonal matrices Ui4 0(i¼ 1, 2, 3, 4, 5), V1i4 0, R1i4 0(i¼ 1, . . . , l), V2j4 0,
R2j4 0(j¼ 1, . . . ,m) and setting ~V1, ~R1 in (11), ~V2, ~R2 in (12), the following inequality holds
0 �h� xTðtÞU1�1xðtÞ þ 2xTðtÞU1�2f ðxðtÞÞ � fTðxðtÞÞU1f ðxðtÞÞ
iþ
h� xTðt� �2ÞU2�1xðt� �2Þ
þ 2xTðt� �2ÞU2�2f ðxðt� �2ÞÞ � fTðxðt� �2ÞÞU2f ðxðt� �2ÞÞiþ
h� xTðt� �3ÞU3�1xðt� �3Þ
þ 2xTðt� �3ÞU3�2f ðxðt� �3ÞÞ � fTðxðt� �3ÞÞU3f ðxðt� �3ÞÞiþ
h� xTðt� �1ðtÞÞ
�U4�1xðt� �1ðtÞÞ þ 2xTðt� �1ðtÞÞU4�2f ðxðt� �1ðtÞÞÞ � fTðxðt� �1ðtÞÞÞU4f ðxðt� �1ðtÞÞÞi
þ
h� xTðt� �2ðtÞÞU5�1xðt� �2ðtÞÞ þ 2xTðt� �2ðtÞÞU5�2f ðxðt� �2ðtÞÞÞ
� fTðxðt� �2ðtÞÞÞU5f ðxðt� �2ðtÞÞÞiþ
h� T1 ðt� �1Þ
~V1~�11ðt� �1Þ þ 2T1 ðt� �1Þ
~V1~�2h1ð1ðt� �1ÞÞ
� hT1 ð1ðt� �1ÞÞ~V1h1ð1ðt� �1ÞÞ
iþ
h� T1
t� �1 � �1ðtÞ
�~R1
~�11t� �1 � �1ðtÞ
�þ 2T1
t� �1 � �1ðtÞ
�~R1
~�2h11ðt� �1 � �1ðtÞÞ
�� hT1
1ðt� �1 � �1ðtÞÞ
�~R1h1
1ðt� �1 � �1ðtÞÞ
�iþ
h� T2 ðt� �2Þ
~V2��12ðt� �2Þ þ 2T2 ðt� �2Þ
~V2��2h2ð2ðt� �2ÞÞ � hT2 ð2ðt� �2ÞÞ
� ~V2h2ð2ðt� �2ÞÞiþ
h� T2
t� �2 � �2ðtÞ
�~R2
��12t� �2 � �2ðtÞ
�þ 2T2
t� �2 � �2ðtÞ
�� ~R2
��2h22ðt� �2 � �2ðtÞÞ
�� hT2
2ðt� �2 � �2ðtÞÞ
�~R2h2
2ðt� �2 � �2ðtÞÞ
�i: ð17Þ
Moreover, together with (8) and any constant matrices E1, E2 with compatible dimensions, one can deduce
0 ¼�xTðtÞET
1 þ _xTðtÞET2
�h� _xðtÞ � CxðtÞ þ Af ðxðtÞÞ þ �0Bf ðxðt� �1ðtÞÞÞ þ ð1� �0ÞBf ðxðt� �2ðtÞÞÞ
þ ð�ðtÞ � �0ÞB�f ðxðt� �1ðtÞÞÞ � f ðxðt� �2ðtÞÞÞ
�i: ð18Þ
Now, adding the right terms (14)–(18) to LV(xt), and taking the mathematical expectation on its both sides, we can
deduce
EfLVðxtÞg � �TðtÞ
(�1��T
1 þ�2��T2 �
�1ðtÞ � �1�2 � �1
Xli¼1
IT1iW1iI1i ��2 � �1ðtÞ
�2 � �1
Xli¼1
IT2iW1iI2i
��2ðtÞ � �2�3 � �2
Xmi¼1
IT3iW2iI3i ��3 � �2ðtÞ
�3 � �2
Xmi¼1
IT4iW2iI4i
þ_�1ðtÞ � �1
l
Xli¼1
IT5iX1i Y1i
� Z1i
� I5i þ
�2 � _�1ðtÞ
l
Xli¼1
IT5iX2i Y2i
� Z2i
� I5i
þ_�2ðtÞ � �3
m
Xmi¼1
IT6iX3i Y3i
� Z3i
� I6i þ
�4 � _�2ðtÞ
m
Xmi¼1
IT6iX4i Y4i
� Z2i
� I6i
)�ðtÞ
:¼ �TðtÞðtÞ�ðtÞ,
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where �, �, �i (i¼ 1, 2), Iki (k¼ 1, 2, 3, 4, 5, 6) arepresented in (13), and
�TðtÞ ¼
"xTðtÞ T1 ðt� �1Þ
T2 ðt� �2Þ x
Tðt� �3Þ
Tt� �1 � �1ðtÞ
�Tt� �2 � �2ðtÞ
�fTðxðtÞÞ
hT1 ð1ðt� �1ÞÞ hT2 ð2ðt� �2ÞÞ f
Tðxðt� �3ÞÞ
hT11ðt� �1 � �1ðtÞÞ
�hT22ðt� �2 � �2ðtÞÞ
�_xTðtÞ xTðt� �1ðtÞÞx
Tðt� �2ðtÞÞ
fTðxðt� �1ðtÞÞÞ fTðxðt� �2ðtÞÞÞ
#:
Then, utilising Lemma 2, the LMIs described by (13)can guarantee (t)5 0, which indicates that theremust exist a positive scalar �4 0 such thatE{LV(xt)}� �
T(t)(t)�(t)���kx(t)k25 0 for anyx(t) 6¼ 0. Then it follows from Lyapunov–Krasovskiistability theorem that the system (8) is asymptoticallystable in the mean-square sense, and it completes theproof.
Remark 2: As we know, time-delays are alwaysexistent in various NNs due to the finite switchingspeed of amplifiers in electronic NNs or the finitesignal propagation time in biological networks.Furthermore, with the development of network delaytomography, the probability of time-delay distributioncan be estimated. Thus, when the probability of time-delay is available, it will be helpful to utilise suchinformation to reduce the conservatism (Fu et al. 2009;Tang et al. 2009; Yue et al. 2009c; Zhang et al. 2009b;Zhao et al. 2009; Mahmoud et al. 2010; Bao and Cao2011). Yet, up to now, few authors have utilised thedelay-partitioning idea to investigate the stability ofNNs with probabilistic time-varying delay and in thiswork, we applied one improved idea, which can fullytake the information of every delay subinterval intoconsideration. Moreover, though the stability criterionin (13) is not presented in the forms of standard LMIs,it is still convenient and straightforward to check thefeasibility without tuning any parameters by resortingto LMI in Matlab Toolbox.
Remark 3: Presently, convex combination techniquehas been widely employed to tackle time-varying delaysowing to that, it could help reduce the conservatismmore efficiently than these previous ones. Yet, thetechniques described in Mou et al. (2008b), Zhang etal. (2010), and Shao (2009) have always ignored theinformation of the lower bound on delayderivative except for few works (Zhang et al. 2008;Fridman et al. 2009), which can be fully taken intoconsideration in this work based on improved convexcombination technique over the one in
Yue et al. (2009a). Thus our methods can be much
more applicable when the lower bounds of delay
derivatives can be measured.
Remark 4: As for V2(xt) in (10), if we denote�X2i Y2i
� Z2i
�¼�X4i Y4i
� Z4i
�¼ 0 (respectively,
�X1i Y1i
� Z1i
�¼�
X3i Y3i
� Z3i
�¼ 0), our results can be true as only �2, �4
(respectively, �1, �3) are available. If we set�Xji Yji
� Zji
�¼ 0 ( j¼ 1, 2, 3, 4) in (10) simultaneously,
Theorem 1 still holds as that �i (i¼ 1, 2, 3, 4) are
unknown, or �i(t) (i¼ 1, 2) are not differentiable.
Moreover, the number of free-weighting matrices in
Theorem 1 is much smaller than the ones of these
present results (Tang et al. 2009; Yue et al. 2009b),
which can induce the computational simplicity in a
mathematical point of view.
Remark 5: In view of the delay-partitioning idea
employed in this work, with integers l, m increasing,
the dimension of the derived LMIs will become higher
and it will take more computing time to check them.
Yet, if the lower bound of �1(t) is set and l� 5, the
maximum allowable delay upper bound will become
unapparent larger and approach to an approximate
upper limitation (Mou et al. 2008a,b; Yang et al. 2009;
Zhang and Han 2009). Thus, if we want to employ the
idea to real cases, we do not necessarily partition two
delay intervals into too many segments.
Remark 6: In order to give more general results, the
delay intervals [�1, �2] and [�2, �3] are, respectively,
divided into l,m subintervals in this work, which makes
the condition of Theorem 1 seem to be very compli-
cated. Yet if we set l¼m� 5, our results will avoid the
complexity in some degree. Moreover, if we choose the
simple LKF in (10) with H1¼Y1i¼Y2i¼Y3i¼Y4i¼ 0
and ~Hi ¼ 0, the condition of Theorem 1 also will
become much less complicated.
Remark 7: It is possible to extend our main results to
more complex DNNs, such as delayed Cohen–
Grossberg NN model, uncertain parameters (Fu et
al. 2009), stochastic perturbations (Bao and Cao 2011)
or Markovian jumping parameters (Cong and Zou
2010). The corresponding results will appear in our
future works.
4. Numerical examples
In this section, four numerical examples will be
presented to illustrate the derived results. Firstly, we
will utilise a numerical example to illustrate the
significance of studying the lower bound of delay
derivative.
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Example 1: As a special case of �0¼ 1, we revisit the
DNNs considered in Zhang et al. (2009a) and Chen
and Zheng (2008) with
C ¼2 0
0 2
� A ¼
1 1
�1 �1
� B ¼
0:88 1
1 1
�
� ¼0 0
0 0
� �� ¼
0:4 0
0 0:8
�
and �1¼ 0 is set. If we do not consider the existence of
�1, then by utilising Theorem 1 and Remark 4, the
corresponding maximum allowable upper bounds
(MAUBs) �max for different �2 derived by the results
in Zhang et al. (2009a) and in this article can be
summarised in Table 1, which demonstrates that
Theorem 1 of l¼ 1 is somewhat more conservative
than the one in Zhang et al. (2009a). Yet, if we set
�1¼ 0.5, it is easy to verify that our results can yield
much less conservative results than the one in Zhang
et al. (2009a), which can be shown in Table 2.Based on Tables 1 and 2, it indicates that the
conservatism of stability criterion can be greatly
deduced if we take the available �1 into consideration.
Moreover, though the delay-partitioning idea has been
used in Chen and Zheng (2008), the corresponding
MAUBs �max derived by Chen and Zheng (2008) and
Theorem 1 are summarised in Table 3, which shows
that our idea can be more efficient than the one in
Chen and Zheng (2008) even for l¼ 1, 2.
Example 2: Considering the special case of �0¼ 1, we
consider the DNNs (1) with
C ¼ diag�1:2769, 0:6231, 0:9230, 0:4480
�� ¼ 03�3 �� ¼ diag
�0:1137, 0:1279, 0:7994, 0:2368
�
A ¼
�0:0373 0:4852 �0:3351 0:2336
�1:6033 0:5988 �0:3224 1:2352
0:3394 �0:0860 �0:3824 �0:5785
�0:1311 0:3253 �0:9534 �0:5015
26664
37775
B ¼
0:8674 �1:2405 �0:5325 0:0220
0:0474 �0:9164 0:0360 0:9816
1:8495 2:6117 �0:3788 0:8428
�2:0413 0:5179 1:1734 �0:2775
26664
37775
which has been addressed extensively, see Zhang and
Han (2009), Hu et al. (2008) and the references therein.
Together with the delay-partitioning idea and for
different �2, the work (Hu et al. 2008) has calculated
the MAUBs �max such that the origin of the system is
globally asymptotically stable for �1(t) satisfying
3¼ �1� �1(t)� �2� �max. By resorting to Theorem 1
and Remark 4, the corresponding results can be given
in Table 4, which indicates that our delay-partitioning
idea can be more effective than the relevant ones in
Hu et al. (2008) even for l¼ 1, 2 and �1¼ 0.
Example 3: We still consider the DNNs with the
following parameters (Zhang et al. 2010) by setting
�0¼ 1,
C ¼
4:1889 0 0
0 0:7160 0
0 0 1:9985
264
375
B ¼
�0:1052 �0:5069 �0:1121
�0:0257 �0:2808 0:0212
0:1205 �0:2153 0:1315
264
375
�� ¼
0:4129 0 0
0 3:8993 0
0 0 1:016
264
375
A¼�¼ 03�3, and time-varying delay �(t) satisfy
0 � �ðtÞ � �2, 0 � _�ðtÞ � �2. Then, the MAUBs for
this example with time-varying delay �(t) is given in
Table 5 by using the delay-partitioning idea in Zhang
et al. (2010) and the one in this article, which can
illustrate that our delay-partitioning idea is superior
over the one by Zhang et al. (2010).
Table 3. Calculated MAUBs �max for various l and �1¼ 0.5in Example 1.
Methodsn�2 0.8 0.9 unknown �2
Chen and Zheng (2009) l¼ 1 1.8496 1.1650 1.0904l¼ 2 1.9149 1.1786 1.0931Theorem 1 l¼ 1 2.8813 1.9655 1.2052l¼ 2 3.1486 2.1967 1.4073
Table 1. Calculated MAUBs �max for various l, unavailable�1 in Example 1.
Methodsn�2 0.6 0.8 0.9 1.2
Zhang et al. (2009a) 3.5209 2.8654 1.9508 –Theorem 1 l¼ 1 3.4877 2.8456 1.9149 1.1168Theorem 1 l¼ 2 3.7449 3.1141 2.1151 1.3182
Table 2. Calculated MAUBs �max for various l and �1¼ 0.5in Example 1.
Methodsn�2 0.6 0.8 0.9 1.2
Zhang et al. (2009a) 3.5209 2.8654 1.9508 –Theorem 1 l¼ 1 3.5871 2.8813 1.9652 1.2052
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Example 4: Consider DNNs (3) of the following
parameters, which has been considered in Fu et al.
(2009),
C ¼7 0
0 6
� A ¼
0:2 �4
0:1 0:3
� B ¼
0:4 0:2
0:1 0:7
�
f ðxÞ ¼tanhð�0:2x1Þ
tanhðx2Þ
�
If we set �1¼ 0, �2¼ 0.4, �1¼�3¼ 0.1 and �2¼�4, the
relevant MAUBS �max are computed and listed in
Tables 6 and 7 for various l, m, �2, and the delay
probability distribution �0¼ 0.2, which can guarantee
the addressed system to be asymptotically stable in the
mean-squared sense.Based on Tables 6 and 7, even for m¼ l¼ 1, our
results still can be less conservative than the one in
Fu et al. (2009). As for m¼ 1, l¼ 2, it is easy to verify
that our results can reduce the conservatism much
more evidently.
5. Conclusions
This article has investigated the asymptotical stabilityfor NNs with probability-distribution variable delay.Through employing the improved idea of delay parti-tioning and constructing one novel LKF, one stabilitycriterion with significantly reduced conservatism hasbeen established in terms of LMIs. The proposedstability condition benefits from the upper and lowerbounds of both time delay and its derivative. Fournumerical examples have been given to demonstratethe effectiveness of the presented criteria and theirimprovements over the existent methods.
Acknowledgements
This work is supported by the National Natural ScienceFoundation of China Grant Nos. 60875035, 60904020,61004064, 61004032 and the Special Foundation of ChinaPostdoctoral Science Foundation Project No. 201003546.
Notes on contributors
Guobao Zhang received his bachelor’sdegree from Zhejiang University in1986. Later, he received the master’sdegree and doctor’s degree fromSoutheast University in 1991 and1998, respectively, China. During1999 and 2001, he has been a seniorvisiting scholar in Hong KongPolytechnic University. Presently, he
is a professor at School of Automation, Southeast Universityin China. His research interests include complex networks,digital processing, intelligent control and so on.
Ting Wang received her BS and MSdegrees in Science from ShandongUniversity, China in 2002 and 2005,respectively. During 2005 and 2009,she was a lecturer in Anqing NormalUniversity, China. Since 2010, she wasa PhD candidate in SoutheastUniversity in China. Her currentresearch includes complex networks,
smart grid, etc.
Table 4. Calculated MAUBs �max for various l, �2 inExample 2.
Methodsn�2 0.1 0.5 0.9 Unknown �2
Hu et al. (2008) l¼ 1 3.33 3.16 3.10 3.09l¼ 2 3.65 3.32 3.26 3.24Theorem 1 l¼ 1 3.35 3.21 3.15 3.13l¼ 2 3.77 3.41 3.34 3.31
Table 6. Calculated MAUBs �max for various l, m, �2, and�0¼ 0.2 in Example 3.
Methodsn�2 0.2 0.6 1 1.5 2.5
Fu et al. (2009) 1.1281 1.1279 1.1278 1.1277 1.1276Theorem 1 l¼ 1,m¼ 1
1.3149 1.3086 1.3031 1.3026 1.3025
l¼ 1, m¼ 2 1.5242 1.5189 1.5134 1.5131 1.5130
Table 7. Calculated MAUBs �max for various l, m, �2, and�0¼ 0.8 in Example 3.
Methodsn�2 0.2 0.6 1 1.5 2.5
Fu et al. (2009) 1.7177 1.6869 1.5758 1.5757 1.5757Theorem 1 l¼ 1,m¼ 1
1.9778 1.8863 1.7742 1.7741 1.7740
l¼ 1, m¼ 2 2.2815 2.1657 2.0098 2.0089 2.0087
Table 5. Calculated MAUBs �max for various l, �2 inExample 2.
Methodsn�2 0.1 0.5 0.9 1
Zhang et al. (2010) �1 ¼12 1.9897 1.3246 1.0575 1.0571
�1 ¼13 , �2 ¼
23 2.1573 1.4276 1.1870 1.1790
Theorem 1 l¼ 2 2.0011 1.3450 1.0687 1.0685l¼ 3 2.1687 1.4465 1.1922 1.1826
10 G. Zhang et al.
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Tao Li received his PhD degree inEngineering from SoutheastUniversity in 2008 and was a post-doctoral fellowship at School ofInstrument Science and Engineeringin Southeast University during 2008and 2011. He is now with School ofAutomation Engineering, NanjingUniversity of Aeronautics and
Astronautics in China. His current research interests includeNNs, time-delay systems, networked control systems, etc.
Shumin Fei received his PhD degreefrom Beijing University ofAeronautics and Astronautics, Chinain 1995. Form 1995 to 1997, he wasdoing postdoctoral research atResearch Institute of Automation,Southeast University, China.Presently, he is a professor anddoctor adviser at Southeast
University in China. He has published more than 70 journalpapers and his research interests include nonlinear systems,time-delay system, complex systems and so on.
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