delay decomposition approach to state estimation of neural networks with mixed time-varying delays...

Research Article

Received 26 October 2010 Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2597MOS subject classification: 34K36; 92B20

Delay decomposition approach to stateestimation of neural networks with mixedtime-varying delays and Markovian jumpingparameters‡

S. Lakshmanana,b, V. Vembarasanb and P. Balasubramaniamb*†

Communicated by Ta-Tsien Li

This paper investigates the state estimation of neural networks with mixed time-varying delays and Markovian jumpingparameters. By developing a delay decomposition approach, the information of the delayed plant states can be takeninto full consideration. On the basis of the new Lyapunov–Krasovskii functional, some inequality techniques, stochasticstability theory and delay-dependent stability criteria are obtained in terms of linear matrix inequalities. Finally, threenumerical examples are given to illustrate the less conservative and effectiveness of our theoretical results. Copyright© 2012 John Wiley & Sons, Ltd.

Keywords: linear matrix inequality; Lyapunov–Krasovskii functional; state estimation; neural networks; Markovian jumpingparameters

1. Introduction

Various kinds of neural networks (NNs) have received much attention in the past decades because of their extensive applicationsin signal processing, pattern recognition, static image processing, associative memory and combinatorial optimization [1, 2]. Timedelays are unavoidably encountered both in biological and artificial neural systems, which may lead oscillation and instability; hence,increasing attention has been focused on stability analysis of NNs with time delays in recent years [3, 4]. It is well known that delay-dependent results are generally less conservative than delay-independent results. Recently, delayed neural networks (DNNs) have beenfocused on the stability analysis, and a large amount of results have been available in the literature, see for example [5–14]. Further, mostof the research on DNNs has been restricted to simple cases of discrete delays. Because an NN usually has a spatial nature due to thepresence of an amount of parallel pathways of a variety of axon sizes and lengths, it is desired to model them by introducing distributeddelays (see [15, 16] and the references therein).

Naturally, the neuron states are not often fully available in the network outputs in many applications, the neuron state estimationproblem becomes important to utilize the estimated neuron state. Initially, Wang et al. [17] have investigated the state estimationproblem for continuous-time NNs with time-varying delay through available output measurements and derived some sufficientconditions for the existence of the desired estimators. Therefore, delay-dependent state estimation problem has been studied widelyfor continuous-time DNNs, see [18–32]. Recently, authors in [24, 26] have studied state estimation for jumping recurrent NNs withdiscrete and distributed constant time delays. In practice, sometimes an NN has finite state representations (also called modes, patterns,or clusters), and modes may switch (or jump) from one to another at different times [33–36]. It is shown that the switching (or jumping)between different NNs modes can be governed by a Markovian chain, such kind of NNs are of great significance [37–45].

aPost-Doctoral Research Associate, Department of Information and Communication Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749,South Korea

bDepartment of Mathematics, Gandhigram Rural Institute - Deemed Univesity, Gandhigram - 624 302, Tamilnadu, India*Correspondence to: P. Balasubramaniam, Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia.†E-mail: [email protected]‡The research work of S. Lakshmanan is supported by the Department of Science and Technology, New Delhi, India under the sanctioned No. SR/S4/MS:485/07.

The research work of V. Vembarasan is supported by DST INSPIRE Fellowship, Ministry of Science and Technology, Government of India under the grant no. DST/INSPIREFellowship/2011/278 dated 21.12.2011.

Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012

S. LAKSHMANAN, V. VEMBARASAN AND P. BALASUBRAMANIAM

In [10], authors have discussed delay decomposition approach and tuning parameter ˛ to derive stability results. In this paper, sametype of method is applied for state estimation of NNs with mixed time-varying delays and Markovian jumping parameters (MJPs)[24,26]. To the best of our knowledge, the state estimation of NNs with mixed time-varying delays and MJPs using delay decompositionmethod has not been investigated yet. Motivated by the previous discussions, the state estimation for NNs with mixed time-varyingdelays and MJPs are considered in this paper. By constructing a novel Lyapunov–Krasovskii functional (LKF), delay decompositionapproach and tuning parameter ˛ and some analysis techniques are employed to derive sufficient conditions for delay-dependentstability criteria for the considered NNs in terms of linear matrix inequalities (LMIs), which can be easily calculated by MATLAB LMIControl Toolbox. Numerical examples are given to illustrate the effectiveness of the proposed method.

Notations: Throughout this paper, Rn and Rn�m denote, respectively, the n-dimensional Euclidean space and the set of all n �mreal matrices. The superscript T denotes the transposition and the notation X � Y (respectively, X > Y), where X and Y are symmetricmatrices, means that X�Y is positive semi-definite (respectively, positive definite). In is the n�n identity matrix. j�j is the Euclidean normin Rn. �max.�/ and �min.�/ are the maximal and the minimal eigenvalues of a matrix. Moreover, let .�,F , P/ be a complete probabilityspace with a filtration fFtgt�0 satisfying the usual conditions. That is the filtration contains all P-null sets and is right continuous.Denote LP

F0.Œ�ı, 0�/ the family of all F0-measurable C.Œ�ı, 0�;Rn/-valued random variables & D f&.�/ : �ı � � � 0g such that

sup�ı��0Ek&.�/kP <1, where E stands for the mathematical expectation operator with respect to the given probability measure

P. The notation � always denotes the symmetric block in one symmetric matrix.

2. Problem description and preliminaries

Stability is one of the most fundamental concepts both in theoretical research and practical applications. Therefore, in this paper,we aim to study the stability of n-dimensional delay differential system with MJPs as follows:

Px.t/D h.x.t/, x.t� �.t//, t, r.t//, (1)

on t 2 Œ0, T� with initial data fx.�/ : �� 0g D & 2 L2F0.Œ�ı, 0�,Rn/. Here, �.t/ : Œ0, T�! Œ0, �� is a Borel measurable function that

stands for time lag, whereas h : Rn �Rn �RC � S!Rn. Moreover, the coefficients ‘h’ have to be sufficiently smooth. The well-knownconditions imposed for the existence and uniqueness of the solutions are the local Lipschitz condition and the linear growth condition,which are stated as follows.

Let r.t/, t � 0 be a right-continuous Markov process on the probability space that takes values in the finite space S D f1, 2, : : : , Ngwith generator � D .�ij/.i, j 2 S/ given by

P.r.tC4/D jjr.t/D i/D

��ijC o./, i¤ j,1C �iiC o./ iD j,

(2)

where > 0, lim�t!0

�o.�/�

�D 0 and �ij is transition rate from mode i to mode j satisfying �ij � 0 for i ¤ j with �ii D �

PNjD1,j¤i �ij ,

i, j 2 S.In addition, various classes of NNs have been active research topics in the past few years because of its important applications, such as

pattern recognition, associative memory, combinatorial optimization, and so on. Stability is one of the main properties of NNs, whichis a crucial feature in the design of NNs. Therefore, the stability analysis for NNs emerges as a research topic of primary significance(see [3–16] and references therein). As discussed in the previous section, DNNs with MJPs are more appropriate to describe a class ofNNs with finite state representation, where the network dynamics can switch from one to another with the switch law being a Markovlaw. On the basis of models (3)–(7), we now introduce the DNNs with MJPs and time-varying delays.

Consider the following DNNs of mixed time-varying delays with MJPs and the network measurements described by

Px.t/D�A.r.t//x.t/C B.r.t//g.x.t//C B1.r.t//g.x.t� �.t///C E.r.t//

Z t

t��.t/g.x.s//dsC J.r.t//

y.t/D C.r.t//x.t/CD.r.t//f .t, x.t//, (3)

where x.�/ D Œx1.�/, x2.�/, : : : , xn.�/�T 2 Rn is neuron state vector, x.t/ D &.t/, t � 0; &.t/ is the initial condition. A D diagfa1, : : : , ang is

a diagonal matrix with ai > 0, i D 1, : : : , n, the matrices B, B1, and E represent the connection weight matrices, the discretely delayedconnection weight matrix, and distributively delayed connection weight matrix, respectively. The matrices C 2 Rm�n and D 2 Rm�m

are the constant matrices. g.x.�// D Œg1.x1.�//, : : : , gn.xn.�//�T 2 Rn denotes the neuron activation function, J D ŒJ1, : : : , Jn�

T 2 Rn is aconstant input vector and f : R�Rn!Rm is the neuron-dependent nonlinear disturbances on the network outputs, which is assumedto satisfy the underlying Assumption (H). �.t/ and .t/ denote the discrete time-varying delays and the distributed time-varying delays,respectively, and satisfy 0� �.t/� � , P�.t/� � and 0� .t/� M, where � , � and M are constants.

For the DNNs (3), we construct the full-order state estimation as follows:

POx.t/D�A.r.t//Ox.t/C B.r.t//g.Ox.t//C B1.r.t//g.Ox.t� �.t///C E.r.t//

Z t

t��.t/g.Ox.s//ds

C J.r.t//C K.r.t//.y � Oy/

Oy.t/D C.r.t//Ox.t/C D.r.t//f .t, Ox.t//, (4)

where Ox.t/ is the estimation of the neuron state, and K.r.t// 2Rn�m is the estimator gain matrix to be designed.



Notice that the Markov process r.t/, t � 0 takes values in the finite space SD f1, 2, : : : , Ng. For notation simplicity, we denote

A.i/D Ai , B.i/D Bi , B1.i/D B1i , E.i/D Ei , J.i/D Ji , K.i/D Ki , C.i/D Ci , D.i/D Di .

Define the error e.t/D x.t/� Ox.t/, and �.t/D g.x.t//� g.Ox.t//, .t/D f .t, x.t//� f .t, Ox.t//. Thus, the error-state system is given by

Pe.t/D�.Ai C KiCi/e.t/C Bi�.t/C B1i�.t� �.t//C Ei

Z t

t��.t/�.s/ds� KiDi .t/. (5)

Obviously, e.s/ is bounded and continuously differential on Œ�ı, 0�, where ı Dmaxf� , Mg.Traditionally, the activation functions are assumed to be continuous, differentiable, monotonically increasing, and bounded, such

as the sigmoid-type of function. However, in many electronic circuits, the input–output functions of amplifiers may be neithermonotonically increasing nor continuously differentiable; hence, non-monotonic functions can be more appropriate to describe theneuron activation in designing and implementing an artificial NNs. In this paper, we make the following assumptions for the neuronactivation functions.Assumption (H): The neuron activation function g.�/ and the neuron-dependent nonlinear disturbances f .�/ in (3) are bounded and

there exist four constant matrices �� D diag˚��1 ,��2 , � � � ,��n

�, �C D diag

n�C1 ,�C2 , � � � ,�Cn

o, � D diag

˚ �1 , �2 , � � � , �n

�, and

C D diagn C1 , C2 , � � � , Cn

o, such that

��k01�

gk01.˛/� gk01

.ˇ/

˛ � ˇ� �C

k01, (6)

�k02�

fk02.˛, x.˛//� fk02

.ˇ, x.ˇ//

x.˛/� x.ˇ/� C

k02(7)

for all ˛,ˇ 2R, ˛ ¤ ˇ, k01 D 1, 2, � � � , n and k02 D 1, 2, � � � , m.By recalling definitions of �.t/ and .t/, from Assumption (H), we obtain the following inequality readily

h�i.t/� �

�k01

ek01.t/iT h

�k01.t/� �Ci ek01

.t/i� 0, k01 D 1, 2, � � � , n,

h k02

.t/� �k02ek02.t/iT h

k02.t/� C

k02ek02.t/i� 0, k02 D 1, 2, � � � , m.

Remark 2.1Assumption (H) was first introduced in [15], as pointed out by the authors, ��i ,�Ci , �i , and Ci are some constants, and they can bepositive, negative, or zero. Hence, the resulting activation functions may be non-monotonic and more general than the usual sigmoidfunctions as the following form discussed in [13, 19, 20, 32].

0�gk01.˛/� gk01

.ˇ/

˛ � ˇ� �k01

, jgk01.˛/� gk01

.ˇ/j � j�k01.˛ � ˇ/j, k01 D 1, 2, � � � , n.

Let e.t; &/ be the trajectory of system (5) under the initial condition e.�/ D &.�/ on �ı � � � 0 in L2F0.Œ�ı, 0�,Rn/. It is obvious that

the system (5) admits a trivial solution e.t; 0/D 0.

Remark 2.2To guarantee the existence and uniqueness of the solution of (5), under Assumption (H), it is easy to check that functions � and satisfythe linear growth condition. Therefore, for any initial data & 2 L2

F0.Œ�ı, 0�,Rn/, the system (5) has a unique solution (or equilibrium

point) denoted by e.t; &/ (for more details see [33–36]). Moreover, according to Assumption (H), it is clear that the system (5) has a trivialsolution, that is e.t/ D 0 corresponding to the initial data & D 0. Moreover, we assume � � 0 and � 0, then (5) has a trivial solutione.t; &/D 0.

To derive our main results, the following essential definition and Lemmas are introduced to prove the main results.

Definition 2.3 ([24])For the error-state system (5) and every & 2 L2

F0.Œ�ı, 0�,Rn/, the trivial solution is asymptotically stable in the mean square if, for every

network mode,

limt�!1

Eje.t; &/j2 D 0.

Lemma 2.4 ([46])(Schur Complement) Given constant matrices�1,�2, and�3 with appropriate dimensions, where�T

1 D�1 and�T2 D�2 > 0, then

�1C�T3��12 �3 < 0



if and only if "�1 �T

3

� ��2

#< 0, or

"��2 �3

� �1

#< 0.

Lemma 2.5 ([47])(Jensen’s inequality) For any constant matrices M 2 Rm�m, scalar � > 0, vector function ! : Œ0, �� ! Rm such that the integrationsconcerned are well defined, then

�

Z �

0!T .s/M!.s/ds�

�Z �

0!.s/ds

�T

M

�Z �

0!.s/ds

�.

3. Main results

In this section, we shall establish the main criterion on the basis of the LMI approach. For presentation convenience, we introduce thefollowing notations:

†1 D diagn��1 �

C1 ,��2 �

C2 , � � � ,��n �

Cn ,o

, †2 D diag

(��1 C �

C1

2,��2 C �

C2

2, � � � ,

��n C �Cn

2

),

†3 D diagn �1

C1 , �2

C2 , � � � , �n

Cn ,o

, †4 D diag

( �1 C

C1

2, �2 C

C2

2, � � � ,

�n C Cn

2

).

Theorem 3.1For given positive scalars � , M, ˛ .0 < ˛ < 1/, and �, the equilibrium solution of DNNs with MJPs (5) is globally asymptotically stablein the mean square if there exist positive definite matrices Pi D PT

i > 0, Rl D RTl > 0, Ql D QT

l > 0, l D 1, 2, 3, 4, diagonal matricesU > 0, V > 0, and W > 0 such that the following LMIs hold2

6664…1i

p˛�‡

p�.1� ˛/‡

p�‡

� �2Pi C R2 0 0

� � �2Pi C R3 0

� � � �2Pi C R4

37775< 0, (8)

26664…2i

p˛�‡

p�.1� ˛/‡

p�‡

� �2Pi C R2 0 0

� � �2Pi C R3 0

� � � �2Pi C R4

37775< 0. (9)

Moreover, the state estimator gain matrix is given by

Ki D P�1i Gi ,

where

…1i D

266666666666664

…11,i …12,i 0 0 …15,i …16,i …17,i …18,i

� …22,i …23,i 0 0 †2V 0 0

� � …33,i …34,i 0 0 0 0

� � � �Q2 �1

.1�˛/� R3 0 0 0 0

� � � � Q4C MR1 � U 0 0 0

� � � � � �.1��/Q4 � V 0 0

� � � � � � � 1�M

R1 0

� � � � � � � �W

377777777777775

,

…2i D

2666666666666664

…11,i 0 ….1/13,i 0 …15,i …16,i …17,i …18,i

� ….1/22,i …

.1/23,i …

.1/24,i 0 †2V 0 0

� � ….1/33,i 0 0 0 0 0

� � � �Q2 �1

.1�˛/� R3 0 0 0 0

� � � � Q4C MR1 � U 0 0 0

� � � � � �.1��/Q4 � V 0 0

� � � � � � � 1�M

R1

� � � � � � � �W

3777777777777775

,



with

…11,i D Q1CQ3 � PiAi � ATi Pi � GiCi � CT

i GTi �

1

˛�.R2C R4/C

NXiD1

�ijPj �†1U�†3W ,

…12,i D1

˛�.R2C R4/, …15,i D PiBi C†2U, …16,i D PiB1i , …17,i D PiEi …18,i D�GiDi C†4W ,

…22,i D�.1��/Q3 �1

˛�.R2C R4/�

1

˛�R2 �†1V , …23,i D

1

˛�R2,

…33,i D�Q1CQ2 �1

˛�R2 �

1

.1� ˛/�R3, …34,i D

1

.1� ˛/�R3, ….1/13,i D

1

˛�.R2C R4/,

….1/22,i D�.1��/Q3 �

1

.1� ˛/�.R3C R4/�

1

.1� ˛/�R3 �†1V , ….1/23,i D

1

.1� ˛/�.R3C R4/,

….1/24,i D

1

.1� ˛/�R3, ….1/33,i D�Q1CQ2 �

1

˛�.R2C R4/�

1

.1� ˛/�.R3C R4/,

‡ Dh�AT

i Pi � CTi GT

i 0 0 0 BTi Pi BT

1iPi ETi Pi � GT

i DTi

iT.

ProofChoose the LKF for error-state system (5) as

V.et , t, i/D V1.et , t, i/C V2.et , t, i/C V3.et , t, i/, (10)

where

V1.et , t, i/D eT .t/Pie.t/,

V2.et , t, i/D

Z t

t�˛�eT .s/Q1e.s/dsC

Z t�˛�

t��eT .s/Q2e.s/dsC

Z t

t��.t/eT .s/Q3x.s/dsC

Z t

t��.t/�T .s/Q4�.s/ds,

V3.et , t, i/D

Z 0

��M

Z t

tC��T .s/R1�.s/dsd� C

Z 0

�˛�

Z t

tC�PeT .s/R2 Pe.s/dsd�

C

Z �˛��

Z t

tC�PeT .s/R3 Pe.s/dsd� C

Z 0

��

Z t

tC�PeT .s/R4 Pe.s/dsd� .

Let L be the weak infinitesimal generator of the random process fet , t � 0g. Then, for each r.t/D i, i 2 S, it can be shown that:

LV1.et , t, i/D 2eT .t/Pi

��.Ai C KiCi/e.t/C Bi�.t/C B1i�.t� �.t//C Ei

Z t

t��.t/�..s//ds� KiDi .t/

C

NXjD1

�ijeT .t/Pje.t/, (11)

LV2.et , t, i/� eT .t/Q1e.t/� eT .t� ˛�/Q1e.t� ˛�/C eT .t� ˛�/Q2e.t� ˛�/

� eT .t� �/Q2e.t � �/C eT .t/Q3e.t/� .1��/eT .t� �.t//Q3e.t� �.t//

C �T .t/Q4�.t/� .1��/�.t� �.t//Q4� .t � �.t/ , (12)

LV3.et , t, i/� M�T .t/R1�.t/�

Z t

t��.t/�T .s/R1�.s/dsC PeT .t/..˛�/R2C .1� ˛/�R3C �R4/Pe.t/

�

Z t

t�˛�PeT .s/R2 Pe.s/ds�

Z t�˛�

t��PeT .s/R3 Pe.s/ds�

Z t

t��.t/PeT .s/R4 Pe.s/ds. (13)

For positive diagonal matrices U > 0, V > 0, and W > 0, we can obtain from Assumption (H) that

…1 D

"e.t/

�.t/

#T "†1U �†2U

�†2U U

#"e.t/

�.t/

#� 0, (14)

…2 D

"e.t � �.t//

�.t� �.t//

#T "†1V �†2V

�†2V V

#"e.t � �.t//

�.t� �.t//

#� 0, (15)

…3 D

�e.t/ .t/

T"

†3W �†4W

�†4W W

#"e.t/

.t/

#� 0. (16)

Now, we estimate the upper bound of the last three terms in the inequality (13) as follows:



Case 1If 0� �.t/� ˛� , we obtain

�

Z t


Z t�˛�


Z t

t��.t/PeT .s/R4 Pe.s/ds

D�

Z t

t��.t/PeT .s/.R2C R4/Pe.s/ds�

Z t��.t/


Z t�˛�

t��PeT .s/R3 Pe.s/ds. (17)

Using Jensen’s inequality Lemma 2.5, it follows

�

Z t

t��.t/PeT .s/.R2C R4/Pe.s/ds� �

1

�.t/Œe.t/� e.t� �.t//�T .R2C R4/Œe.t/� e.t� �.t//�

� �1

˛�Œe.t/� e.t � �.t//�T .R2C R4/Œe.t/� e.t � �.t//�, (18)

�

Z t��.t/

t�˛�PeT .s/R2 Pe.s/ds� �

1

˛� � �.t/Œe.t� �.t//� e.t� ˛�/�T R2Œe.t � �.t//� e.t� ˛�/�

� �1

˛�Œe.t � �.t//� e.t� ˛�/�T R2Œe.t� �/� e.t � ˛�/�, (19)

�

Z t�˛�

t��PeT .s/R3 Pe.s/ds� �

1

� � ˛�Œe.t � ˛�/� e.t� �/�T R3Œe.t � ˛�/� e.t� �/�

� �1

.1� ˛/�Œe.t � ˛�/� e.t� �/�T R3Œe.t� ˛�/� e.t� �/�. (20)

From Equations (11)–(16) and (18)–(20), we have

LV.et , t, i/� T .t/hM…1i C ˛�ˆ

T R2ˆC �.1� ˛/ˆT R3ˆC �ˆ

T R4ˆi .t/,

using Lemma 2.4, we have

LV.et , t, i/� T .t/�1i .t/. (21)

Taking mathematical expectation on both sides of Equation (21) and considering the LMI (8), applying definition 2.3, one has thefollowing inequality:

EfLV.et , t, i/g � En T .t/‰1i .t/

o��min.�‰1i/E

nje.t; &/j2

o,

where

‰1i D

2664M…1i

p˛�ˆR2

p�.1� ˛/ˆR3

p�ˆR4

� �R2 0 0� � �R3 0� � � �R4

3775 ,

M…1i D

2666666666664

„11,i …12,i 0 0 …15,i …16,i …17,i „18,i

� …22,i …23,i 0 0 †2V 0 0� � …33,i …34,i 0 0 0 0� � � �Q2 �

1.1�˛/� R3 0 0 0 0

� � � � Q4C MR1 � U 0 0 0� � � � � �.1��/Q4 � V 0 0� � � � � � � 1

�MR1 0

� � � � � � � �W

3777777777775

,

„11,i D Q1 C Q3 � Pi.Ai C KiCi/ � .Ai C KiCi/T Pi �

1˛� .R2 C R4/ C

PNjD1 �ijPj � †1U � †3W ,„18,i D �PiKiDi C †4W , ˆ D

�.Ai C KiCi/T 0 0 0 BT

i BT1i ET

i � KTi DT

i

�T, and

T .t/D

�eT .t/ eT .t� �.t// eT .t� ˛�/ eT .t� �/ �T .t/ �T .t� �.t//

Z t

t��.t/�T .s/ds T .t/

.

Pre-multiplying and post-multiplying Equation (21) by diag˚

I8, PiR�12 , PiR�1

3 , PiR�14

�and diag

˚I8, R�1

2 Pi , R�13 Pi , R�1

4 Pi�

respectively,applying the change of variable such that Ki D P�1

i Gi and using the following inequality [19],

PiR�1l Pi � 2Pi � Rl , lD 2, 3, 4.



One can deduce that the ‰1i < 0 is equivalent to Equation (8). This means that the error-state system (5) is globally asymptoticallystable in the mean square, and this completes the proof of Case 1.

Case 2If ˛� � �.t/� � , we obtain

�

Z t


Z t�˛�


Z t

t��.t/PeT .s/R4 Pe.s/ds

D�

Z t

t�˛�PeT .s/.R2C R4/Pe.s/ds�

Z t�˛�

t��.t/PeT .s/.R3C R4/Pe.s/ds�

Z t��.t/


Consider the Jensen’s inequality Lemma 2.5, we obtain

�

Z t

t�˛�PeT .s/.R2C R4/Pe.s/ds��

1

˛�Œe.t/� e.t� ˛�/�T .R2C R4/Œe.t/� e.t� ˛�/�, (23)

�

Z t�˛�

t��.t/PeT .s/.R3C R4/Pe.s/ds��

1

.1� ˛/�Œe.t� ˛�/� e.t � �.t//�T .R3C R4/

� Œe.t � ˛�/� e.t� �.t//� , (24)

�

Z t��.t/

t��PeT .s/R3 Pe.s/ds��

1

.1� ˛/�Œe.t� �.t//� e.t� �/�T R3

� Œe.t � �.t//� e.t� �/� . (25)

From Equations (11)–(16) and (23)–(25), we obtain

LV.et , t, i/� T .t/hM…2i C ˛�ˆ

T R2ˆC �.1� ˛/ˆT R3ˆC �ˆ

T R4ˆi .t/,


LV.et , t, i/� T .t/‰2i .t/. (26)

Taking mathematical expectation on both sides of Equation (26) and considering the LMI (9), applying definition 2.3, one has thefollowing inequality

EfLV.et , t, i/g � En T .t/‰2i .t/

o��min.�‰2i/E

nje.t; &/j2

o,

where

‰2i D

2664M…2i

p˛�ˆR2

p�.1� ˛/ˆR3

p�ˆR4

� �R2 0 0� � �R3 0� � � �R4

3775 ,

M…2i D

2666666666666664

„11,i 0 ….1/13 0 …15,i …16,i …17,i „18,i

� ….1/22,i …

.1/23,i …

.1/24,i 0 †2V 0 0

� � ….1/33,i 0 0 0 0 0

� � � �Q2 �1

.1�˛/� R3 0 0 0 0

� � � � Q4C MR1 � U 0 0 0

� � � � � �.1��/Q4 � V 0 0

� � � � � � � 1�M

R1 0

� � � � � � � �W

3777777777777775

,


I8, PiR�12 , PiR�1

3 , PiR�14

�and diag

˚I8, R�1

2 Pi , R�13 Pi , R�1

4 Pi�

, respectively,applying the change of variable such that Ki D P�1


PiR�1l Pi � 2Pi � Rl , lD 2, 3, 4,

one can deduce that ‰2i < 0 which is equivalent to Equation (9). This means that the error-state system (5) is globally asymptoticallystable in the mean square, and this completes the proof of Case 2. �



Remark 3.2Recently authors in [24, 26] have studied state estimation for recurrent NNs with MJPs and mixed time delays. In those papers,time-delays are assumed to be constants. However, in most of the cases in engineering problems, unknown time-varying delaysexist. Motivated by this, in this paper, we discussed state estimation for DNNs with MJPs and mixed time-varying delays viadelay decomposition approach [10]. Accordingly, numerical results are given for time-varying delay cases without considering forconstant delays.

Remark 3.3In the proof of Theorem 3.1, the interval Œt�� , t� is divided into two subintervals Œt�� , t�˛�� and Œt�˛� , t�. Because a tuning parameter ˛is introduced, the information about x.t�˛�/ can be taken into account. The parameter ˛ plays a vital role in deriving less conservativeresults.

In the following Theorem 3.4, we derive the stability condition for the error-system (5) without decomposing the time-varying delay.

Theorem 3.4For given positive scalars � , M, and�, the equilibrium solution of NNs with MJPs (5) is globally asymptotically stable in the mean squareif there exist positive definite matrices Pi D PT

i > 0, Rl D RTl > 0, l D 1, 2, Qk D QT

k > 0, k D 1, 2, 3, diagonal matrices U > 0, V > 0,and W > 0 such that the following LMIs hold

2666666666666664

‚11,i1� R2 0 PiBi C†2U PiB1i PiEi �GiDi C†4W

p��AT

i Pi � CTi GT

i

� ‚22,i

1� R2 0 †2V 0 0 0

� � �Q1 �1� R2 0 0 0 0 0

� � � Q3C MR1 � U 0 0 0p�BT

i Pi

� � � � �.1��/Q3 � V 0 0p�BT

1iPi

� � � � � � 1�M

R1 0p�ET

i Pi

� � � � � � �W �p�GT

i DTi

� � � � � � � �2Pi C R2

3777777777777775

< 0. (27)

Moreover, the state estimator gain matrix is given by Ki D P�1i Gi , where‚11,i D Q1CQ2�PiAi�AT

i Pi�GiCi�CTi GT

i �1� R2C

PNjD1 �ijPj�

†1U�†3W ,‚22,i D�.1��/Q2 �2� R2 �†1V .

ProofWe choose the LKF for error-state system (5) as

V.et , t, i/D V1.et , t, i/C V2.et , t, i/C V3.et , t, i/, (28)

where

V1.et , t, i/D eT .t/Pie.t/,

V2.et , t, i/D

Z t

t��eT .s/Q1e.s/dsC

Z t

t��.t/eT .s/Q2e.s/dsC

Z t

t��.t/�T .s/Q3�.s/ds,

V3.et , t, i/D

Z 0

��M

Z t

tC��.s/T R1�.s/dsd� C

Z 0

��

Z t

tC�PeT .s/R2 Pe.s/dsd� .

Let L be the weak infinitesimal generator of the random process fet , t � 0g. Then, for each r.t/D i, i 2 S, it can be shown that:

LV1.et , t, i/D 2eT .t/Pi

��.Ai C KiCi/e.t/C Bi�.t/C B1i�.t� �.t//C Ei

Z t

t��.t/�.e.s//ds� KiDi .t/

C

NXjD1

�ijeT .t/Pje.t/, (29)

LV2.et , t, i/� eT .t/Q1e.t/� eT .t� �/Q1e.t � �/C eT .t/Q2e.t/� .1��/eT .t� �.t//

�Q2e.t� �.t//C �T .t/Q3�.t/� .1��/�.t� �.t//Q3�.t� �.t//, (30)

LV3.x.t/, t, i/D M�T .t/R1�.t/�

Z t

t��M

�T .s/R1�.s/dsC � PeT .t/R2 Pe.t/�

Z t

t��PeT .s/R2 Pe.s/ds

� M�T .t/R1�.t/�

Z t

t��.t/�T .s/R1�.s/dsC � PeT .t/R2 Pe.t/

�

Z t

t��.t/PeT .s/R2 Pe.s/ds�

Z t��.t/




Using Jensen’s inequality, it follows

�

Z t

t��.t/PeT .s/R2 Pe.s/ds��

1

�.t/Œe.t/� e.t � �.t//�T R2Œe.t/� e.t� �.t//�

��1

�Œe.t/� e.t � �.t//�T R2Œe.t/� e.t� �.t//�, (32)

�

Z t��.t/

t��PeT .s/R2 Pe.s/ds��

1

� � �.t/Œe.t� �.t//� e.t� �/�T R2Œe.t � �.t//� e.t� �/�

��1

�Œe.t � �.t//� e.t� �/�T R2Œe.t� �/� e.t � �/�. (33)

For positive diagonal matrices U > 0, V > 0, and W > 0, we can obtain from Assumption (H) that

…1 D

"e.t/

�.t/

#T "†1U �†2U

�†2U U

#"e.t/

�.t/

#� 0, (34)

…2 D

"e.t � �.t//

�.t� �.t//

#T "†1V �†2V

�†2V V

#"e.t � �.t//

�.t� �.t//

#� 0, (35)

…3 D

"e.t/

.t/

#T "†3W �†4W

�†4W W

#"e.t/

.t/

#� 0. (36)

From Equation (29)–(36), we obtain

LV.et , t, i/� �T .t/h�i C � N̂

T R2 N̂i�.t/,


LV.et , t, i/� �T .t/ N�i�T .t/. (37)

Taking mathematical expectation on both sides of Equation (37) and considering the LMI (27), applying definition 2.3, one has thefollowing inequality:

EfLV.et , t, i/g � Ef�T .t/ N�i�.t/g � ��min.� N�i/Efje.t; &/j2g,

where

N�i D

��ip� N̂ R2

� �R2

,

�i D

266666666664

N‚11,i1� R2 0 PiBi C†2U PiB1i PiEi N‚18,i

� ‚22,i1� R2 0 †2V 0 0

� � �Q1 �1� R2 0 0 0 0

� � � Q3C MR1 � U 0 0 0

� � � � �.1��/Q3 � V 0 0

� � � � � � 1�M

R1 0

� � � � � � �W

377777777775

,

N‚11,i D Q1 C Q2 � Pi.Ai C KiCi/ � .Ai C KiCi/T Pi �

1� R2 C

PNjD1 �ijPj � †1U � †3W , N‚18,i D �PiKiDi C †4W , and N̂ D

�.Ai C KiCi/T 0 0 BT

i BT1i ET

i � KTi DT

i

�with

�T .t/D

�eT .t/ eT .t� �.t// eT .t � �/ �T .t/ �T .t� �.t//

Z t

t��.t/�.s/ds .t/

.


I7, PiR�12

�and diag

˚I7, R�1

2 Pi�

, respectively, applying the change ofvariable such that Ki D P�1


PiR�12 Pi � 2Pi � R2.

One can deduce that N�i < 0 which is equivalent to Equation (27). This means that the error-state system (5) is globally asymptoticallystable in the mean square, and this completes the proof. �



In the following, we will discuss the stability for the following error-state system (5) without MJPs

Pe.t/D�.AC KC/e.t/C B�.t/C B1�.t� �.t//C E

Z t

t��.t/�.s/ds� KD .t/. (38)

Corollary 3.5For given positive scalars � , M, ˛ .0 < ˛ < 1/ and �, the equilibrium solution of DNNs (38) is globally asymptotically stable if thereexist positive definite matrices P D PT > 0, Rl D RT

l > 0, Ql D QTl > 0, l D 1, 2, 3, 4, diagonal matrices U > 0, V > 0 and W > 0 such that

the following LMIs hold 2664…1

p˛� O‡

p�.1� ˛/ O‡

p� O‡

� �2PC R2 0 0� � �2PC R3 0� � � �2PC R4

3775< 0, (39)

2664…2

p˛� O‡

p�.1� ˛/ O‡

p� O‡

� �2PC R2 0 0� � �2PC R3 0� � � �2PC R4

3775< 0. (40)


K D P�1G,

where

…1 D

2666666666664

…11 …12 0 0 …15 …16 …17 …18

� …22 …23 0 0 †2V 0 0� � …33 …34 0 0 0 0� � � �Q2 �

1.1�˛/� R3 0 0 0 0

� � � � Q4C MR1 � U 0 0 0� � � � � �.1��/Q4 � V 0 0� � � � � � � 1

�MR1 0

� � � � � � � �W

3777777777775

,

…2 D

266666666666664

…11 0 ….1/13 0 …15 …16 …17 …18

� ….1/22 …

.1/23 …

.1/24 0 †2V 0 0

� � ….1/33 0 0 0 0 0

� � � �Q2 �1

.1�˛/� R3 0 0 0 0

� � � � Q4C MR1 � U 0 0 0� � � � � �.1��/Q4 � V 0 0� � � � � � � 1

�MR1 0

� � � � � � � �W

377777777777775

,

with

…11 D Q1CQ3 � PA� AT P� GC � CT GT �1

˛�.R2C R4/�†1U�†3W ,

…12 D1

˛�.R1C R3/, …15 D PBC†2U, …16 D PB1, …17 D PE, …18 D�GDC†4W ,

…22 D�.1��/Q3 �1

˛�.R2C R4/�

1

˛�R2 �†1V , …23 D

1

˛�R2,

…33 D�Q1CQ2 �1

˛�R2 �

1

.1� ˛/�R3, …34 D

1

.1� ˛/�R3, ….1/13 D

1

˛�.R2C R4/,

….1/22 D�.1��/Q3 �

1

.1� ˛/�.R3C R4/�

1

.1� ˛/�R3 �†1V , ….1/23 D

1

.1� ˛/�i.R3C R4/,

….1/24 D

1

.1� ˛/�R3, ….1/33 D�Q1CQ2 �

1

˛�.R2C Z4/�

1

.1� ˛/�.R3C R4/,

O‡ Dh�AT P� CT GT 0 0 0 BT P BT

1 P ET P � GT DTiT

.



ProofThe proof follows immediately from the similar way of proof of Theorem 3.1. Hence, it is omitted. �

In the following Corollary 3.6, we derive the stability condition for the error-state system (38) without decomposing the time-varying delay.

Corollary 3.6For given positive scalars � , M, and �, the equilibrium solution of DNNs (38) is globally asymptotically stable if there exist positivedefinite matrices P D PT > 0, Rl D RT

l > 0, l D 1, 2, Qk D QTk > 0, k D 1, 2, 3, diagonal matrices U > 0, V > 0 and W > 0 such that the

following LMIs hold

266666666666666664

O‚111� R2 0 PBC†2U PB1 PE �GDC†4V

p��AT P� CT GT

� O‚22

1� R2 0 †2V 0 0 0

� � �Q1 �1� R2 0 0 0 0 0

� � � Q3C MR1 � U 0 0 0p�BT P

� � � � �.1��/Q3 � V 0 0p�BT

1 P

� � � � � � 1�M

R1 0p�ET P

� � � � � � �W �p�GT DT

� � � � � � � �2PC R2

377777777777777775

< 0, (41)


K D P�1G,

where O‚11 D Q1CQ2 � PA� AT P� GC � CT GT � 1� R2 �†1U�†3W , O‚22 D�.1��/Q2 �

2� R2 �†1V .

ProofThe proof follows immediately from the similar way of proof of Theorem 3.4. Hence, it is omitted. �

Remark 3.7The traditional assumption that derivatives of the time-varying delays are less than 1 is no longer required in our analysis. Therefore,theorems and corollaries in this paper are applicable to use more generally and practically than those used in the investigated papers[17, 18]. In many cases, the information of the derivative of time delays is unknown [19, 32] because it is a difficult problem to obtain theprecise values (even their boundedness or the boundedness of their derivatives) of time-delay systems. Regarding this circumstance,rate-independent criteria for delay �.t/ satisfying the conditions 0 � �.t/ � � are derived by choosing Q3, Q4 to be zero in theoremsand corollaries of this paper.

Remark 3.8For presentation simplicity, the modeling of uncertainty is not taken into account in the error-state system presented in this paper.However, it is not difficult to further extend the results to uncertain NNs with MJPs, where the error-state system matrices containparameter uncertainties either in norm-bounded or polytopic uncertain forms.

4. Numerical examples

Example 1Consider the error-state system (5) with the following parameters

A1 D

�4.2 00 5.2

, A2 D

�4 00 5

, B1 D

��1.5 �0.7

0.5 0.8

, B2 D

��0.8 �0.6

0.2 �0.7

,

B11 D

�0.5 0.4�0.8 �0.5

, B12 D

�0.2 0.1�0.6 �0.5

, E1 D

�0.7 0.60.8 0.5

, E2 D

�0.4 0.70.5 0.7

,

C1 D

�0.2 00 0.2

, C2 D

�0.3 00 0.3

, D1 D D2 D

�0.2 00 0.2

, � D

��5 5

4 �4

,

J1 D J2 D

�2 sin tC 0.03t2

2 cos.1.5t/� 0.03t2

.



The activation function satisfies Assumption (H) with��D�0.5I,�C D 0.4I, and the nonlinear disturbance also satisfies Assumption (H)with � D 0.2I, C D 0.4I. Thus, we can obtain the following parameters

†1 D

��0.2 0

0 �0.2

, †2 D

��0.05 0

0 �0.05

, †3 D

�0.08 00 0.08

, †4 D

�0.3 00 0.3

.

The activation function g.x.t//D 15 Œjx.t/C 1j � jx.t/� 1j�, and the nonlinear disturbance f .t, x.t//D 0.1 cos.x.t//C 0.3 are taken into

account. The time varying delays are chosen as �.t/ D 0.5C 0.0599 sin.t/, which means � D 0.5599, derivative of time-varying delayssatisfy P�.t/ � � D 1, M D 0.5, ˛ D 0.3. Using the MATLAB LMI Control Toolbox to solve the LMIs in Theorem 3.1, we obtain thefollowing corresponding gain matrices.

K1 D P�11 G1 D

��9.5687 0.0097�0.0569 �13.6688

, K2 D P�1

2 G2 D

��6.4055 �0.0058

0.0003 �9.0884

.

We obtain the maximum allowable upper bound (MAUB) � with different � and ˛ for fixed M D 0.5, which are listed in Table I.Therefore, it follows from Theorem 3.1, that the DNNs with MJPs (5) is globally asymptotically stable in the mean square. The responseof the error dynamics for the DNNs with MJPs (5) that converges to zero asymptotically in the mean square are given in Figure 1.Simulation of system modes of Example 1 is displayed in Figure 2.

Table I. The MAUB � with different � and ˛ for fixed M D 0.5.

� 0 0.5 0.75 0.9 1� P�.t/ <1 unknown �

˛ D 0.3 0.6325 0.5670 0.5599 0.5599 0.5599 0.5599˛ D 0.5 0.6325 0.5670 0.5622 0.5622 0.5622 0.5622˛ D 0.7 0.6325 0.5670 0.5601 0.5601 0.5601 0.5601˛ D 0.9 0.6325 0.5670 0.5530 0.5530 0.5530 0.5530

MAUB, maximum allowable upper bound.

0 1 2 3 4 5 6 7 8 9 100

10

20

30The state x1(t) and its estimation

The state x2(t) and its estimation

Time (sec)

ampl

itude

0 1 2 3 4 5 6 7 8 9 100

10

20

30

Time (sec)

ampl

itude

0 1 2 3 4 5 6 7 8 9 10

10

0

20

30

40

Time (sec)

ampl

itude

True StateEstimation


The error states

e1(t)

e2(t)

Figure 1. The response of the state, estimation, and error trajectories in Example 1.



0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3Mode Graph

Time (sec)

r(t)

r(t)

Figure 2. The simulation of system mode r.t/D f1, 2g in Example 1.

Example 2Consider the error-state system (38) with parameters as follows:

AD

�3 00 5

, BD

�0.2 �0.10.1 0.3

, B1 D

�0.1 0.20.2 �0.1

, E D

�0.3 0.10.2 0.3

, (42)

C D DD

�1 00 1

, JD

�2 sin tC 0.03t2

2 cos.1.5t/� 0.03t2

. (43)

The activation function satisfies Assumption (H) with �� D�0.5I,�C D 0.4I, and the nonlinear disturbance also satisfies Assumption(H) with � D 0.2I, C D 0.4I. Thus, we can obtain the following parameters readily.

†1 D

��0.2 0

0 �0.2

, †2 D

��0.05 0

0 �0.05

, †3 D

�0.08 00 0.08

, †4 D

�0.3 00 0.3

.

Authors in [32], have been studied the state estimation for NNs with mixed interval time-varying delays by assuming that the time-varying delays are not differentiable. However, in this paper, we have discussed both the cases when the time-varying delay isdifferentiable and non-differentiable. Further in this paper, we have removed the restrictive condition � < 1. We obtain the MAUB �with different � and ˛ for fixed M D 0.5, which are listed in the following Table II.

The activation function g.x.t// D 15 Œjx.t/C 1j � jx.t/� 1j� and nonlinear disturbance is of the form f .t, x.t// D 0.1 cos.x.t//C 0.3.

The time-varying delays are chosen as �.t/D 1.5294j sin.t/j, which means � D 1.5294, when ˛ D 0.3 and M D 0.5. However, by usingmethods in [32], allowable upper bound of � is equal to 1. Hence, the delay-dependent stability result in this paper is better than thatin [32]. Using the MATLAB LMI Control Toolbox to solve the LMIs in Corollary 3.5, we obtain the corresponding gain matrix as

K D P�1GD

��1.8602 �0.0092�0.0135 �3.3663

.

Therefore, it follows from Corollary 3.5 that the DNNs (38) is globally asymptotically stable. Therefore, for this example, the resultsgiven in this paper are less conservative than those results discussed in [32]. The response of the error dynamics for the DNNs (38) thatconverges to zero asymptotically are given in Figure 3.

Table II. The MAUB � with different � and ˛ for fixed M D 0.5

� 0 0.5 0.75 0.9 1� P�.t/ <1 unknown �

˛ D 0.3 1.6575 1.5781 1.5423 1.5312 1.5294 1.5294˛ D 0.5 1.6575 1.5792 1.5450 1.5344 1.5327 1.5327˛ D 0.7 1.6575 1.5787 1.5436 1.5328 1.5310 1.5310˛ D 0.9 1.6575 1.5778 1.5374 1.5254 1.5235 1.5235




0 1 2 3 4 5 6 7 8 9 100

20

40



Time (sec)am

plitu

de

0 1 2 3 4 5 6 7 8 9 100

10

20

30

Time (sec)

ampl

itude

0 1 2 3 4 5 6 7 8 9 100

10

30

20

40

Time (sec)

ampl

itude


True State

Estimation

The error states

e1(t)

e2(t)

Figure 3. The response of the state, estimation, and error trajectories in Example 2.

Example 3Consider the error-state system (5) with parameters as follows

A1 D

24 1.5 0 0

0 2.5 00 0 3.5

35 , B1 D

24 0.2 �0.4 0.4�0.4 0.2 0.2

0.2 0.4 �0.4

35 , B11 D

24 0.1 1 0.2�0.1 0.2 0.1

0.2 �0.1 0.4

35 ,

E1 D

24 1.5 0.6 �0.9

0.7 1.2 1.2�0.5 �0.6 1.3

35 , C1 D D1 D C2 D D2 D

24 1 0 0

0 1 00 0 0

35 , � D

��1 1

2 �2

,

A2 D

24 2.5 0 0

0 3.0 00 0 4.5

35 , B2 D

24 0.3 �0.2 0.3�0.4 0.3 0.1

0.3 0.4 �0.5

35 , B12 D

24 0.2 0.5 0.3�0.4 0.2 0.3

0.2 �0.5 0.4

35 ,

E2 D

24 1.0 0.4 �0.5

0.8 1.3 1.2�0.2 �0.4 1.5

35 , J1 D J2 D

24 2 cos.t/C 0.03t2

2 sin.t/� 0.03t2

2 cos.t/C 0.03t2

35 .

The activation function satisfies Assumption (H) with �� D�0.5I,�C D 0.4I, and the nonlinear disturbance also satisfies Assumption(H) with � D 0.2I, C D 0.4I. Thus, we can obtain the following parameters readily.

†1 D

24 �0.2 0 0

0 �0.2 00 0 �0.2

35 , †2 D

24 �0.05 0 0

0 �0.05 00 0 �0.05

35 ,

†3 D

24 0.08 0 0

0 0.08 00 0 0.08

35 , †4 D

24 0.3 0 0

0 0.3 00 0 0.3

35 .

The activation function g.x.t// D 15 Œjx.t/C 1j � jx.t/� 1j� and nonlinear disturbance is of the form f .t, x.t// D 0.1 cos.x.t//C 0.3.

The time-varying delays are chosen as �.t/ D 0.1921C 0.25 sin.t/, which means � D 0.4421, when ˛ D 0.3, � D 0.25, and M D 0.5.The response of the state and estimation for the DNNs are given in Figure 4. In Figure 5, DNNs with MJPs (5) are given, which convergesto zero asymptotically. Simulation of system modes of Example 3 is displayed in Figure 6. We obtain the MAUB � with different � and



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

50




Time (sec)am

plitu

de

Time (sec)

ampl

itude

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

Time (sec)

ampl

itude

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

True State

Estimation

True State

Estimation

True State

Estimation

Figure 4. The response of the state, estimation in Example 3.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16The error states

Time (sec)

ampl

itude

e (t)e (t)e (t)

Figure 5. The error trajectories in Example 3.

˛ for fixed M D 0.5, which are listed in the following Table III. Therefore, it follows from Theorem 3.1 that the DNNs with MJPs (5) isglobally asymptotically stable in the mean square. Using the MATLAB LMI Control Toolbox to solve the LMIs in Theorem 3.1, we obtainthe corresponding gain matrices as follows:

K1 D P�11 G1 D

24 0.4670 �0.0950 0.0475

0.0668 0.0134 0.04400.2727 1.1819 0.0306

35 , K2 D P�1

2 G2 D

24 0.7842 �1.8328 �0.2766�1.4756 1.9075 0.5338�3.4600 6.0237 0.6855

35 .



0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3Mode Graph

Time (sec)

r(t)

r(t)

Figure 6. The simulation of system mode r.t/D f1, 2g in Example 3.

Table III. The MAUB � with different � and ˛ for fixed M D 0.5.

� 0 0.25 0.5 0.75 1� P�.t/ <1 unknown �

˛ D 0.3 0.4448 0.4421 0.4376 0.4308 0.4279 0.4279˛ D 0.5 0.4448 0.4421 0.4376 0.4310 0.4283 0.4283˛ D 0.7 0.4448 0.4421 0.4376 0.4308 0.4280 0.4280˛ D 0.9 0.4448 0.4421 0.4376 0.4302 0.4269 0.4269


In [29, 30], authors have been studied the state estimation for NNs with discrete time-varying delays. The state estimation for NNswith mixed constant delays is studied in [27]. Further in [21], authors have been studied the state estimation for NNs with mixed time-varying delays. In the sequel, the state estimation problem for recurrent NNs with discrete time delays and MJPs are considered in [25].Recently authors in [24, 26] have studied state estimation for recurrent NNs with mixed time delays and MJPs. In papers [24] and [26],time-delays are assumed to be constants. However, in this paper we have considered NNs with mixed time delays and MJPs via delaydecomposition approach. Therefore, for this example, the results given in this paper are more general than those results discussed in[21, 24–27, 29, 30].

Remark 4.1In Examples 1 and 2, to reduce computational complexity, we have considered two-dimensional matrices to illustrate the theoreticalresults. Actually, for real world applications, the determination of our stability criteria can also perform well for any dimensions by usingLMI Control Toolbox in MATLAB. As pointed in [46], the powerful and efficient interior-point methods have been developed to solvethe LMIs that arise in system and control theory. Although there remains much to be performed in this area, several interior-pointalgorithms for LMI problems have been implemented and tested on specific families of LMIs that arise in control theory, and found tobe extremely efficient. If the DNNs possess a large number of neurons, the computation time of calculating MAUBs listed in Tables I, II,and III will also automatically increase. In future, we will do some further research on reducing time complexity when the DNNs possessa large number of neurons.

5. Conclusion

In this paper, we have dealt with the problem of state estimation of NNs with mixed time-varying delays and MJPs. By developing adelay decomposition approach and defining LKF, new LMI-based stability conditions are derived. The state estimation of NNs with MJPsand time-varying delays using delay decomposition approach is new in the literature. Numerical examples are demonstrated to showless conservative results by comparison of numerical results in the existing literature.

Acknowledgements

The authors would like to express their sincere gratitude to the editor-in-chief and the anonymous reviewer for their valuable commentsand suggestions to improve the quality of the manuscript.



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