Journal of the Egyptian Mathematical Society (2015) xxx, xxx–xxx
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINIAL ARTICLE
Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach M. Kalpana a, P. Balasubramaniam a b
b,*
Department of Mathematics, National Institute of Technology – Deemed University, Tiruchirappalli 620 015, Tamil Nadu, India Department of Mathematics, Gandhigram Rural Institute – Deemed University, Gandhigram 624 302, Tamil Nadu, India
Received 26 November 2013; revised 5 July 2014; accepted 31 July 2014
KEYWORDS Global asymptotical stability; Fuzzy cellular neural networks; Leakage delay; Linear matrix inequalities; Sample-data; State estimation
Abstract In this paper, the sampled measurement is used to estimate the neuron states, instead of the continuous measurement, and a sampled-data estimator is constructed. Leakage delay is used to unstable the neuron states. It is a challenging task to develop delay dependent condition to estimate the unstable neuron states through available sampled output measurements such that the error-state system is globally asymptotically stable. By constructing Lyapunov–Krasovskii functional (LKF), a sufficient condition depending on the sampling period is obtained in terms of linear matrix inequalities (LMIs). Moreover, by using the free-weighting matrices method, simple and efficient criterion is derived in terms of LMIs for estimation. 2010 MATHEMATICS SUBJECT CLASSIFICATION:
03B52; 92B20; 93C57; 93D20
ª 2015 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction Cellular neural networks (CNNs), proposed by Chua and Yang in [1,2] have been extensively studied both in theory and in applications. Based on traditional CNN, the fuzzy cel* Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail address:
[email protected] (P. Balasubramaniam). Peer review under responsibility of Egyptian Mathematical Society.
Production and hosting by Elsevier
lular neural networks (FCNNs) have been introduced at the first time in 1996, proposed by Yang in [3,4]. The FCNN is a fuzzy neural networks which integrates fuzzy logic into the structure of traditional CNN. It is a very useful tool in image processing and pattern recognition. However, the existence of time delays may lead to the instability or bad performance of systems [5–7]. So, it is of prime importance to consider the delay effects on the dynamical behavior of systems. Recently, FCNNs with various types of delay have been widely investigated by many authors; see [8–12] and references therein. However, so far, there has been very little existing work on FCNNs with time delay in the leakage (or ‘‘forgetting’’) term [13–17]. In fact, time delay in the leakage term also has great impact on the dynamics of FCNNs. As pointed out by
http://dx.doi.org/10.1016/j.joems.2014.07.003 1110-256X ª 2015 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
2 Gopalsamy [18], time delay in the stabilizing negative feedback term has a tendency to destabilize a system. Moreover, the time delay involving in the first term of the state variable of dynamical networks known as leakage delay on FCNNs cannot be ignored. It is well known that the knowledge about the system state is necessary to solve many control theory problems; for example, stabilizing a system using state feedback. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by the way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer. Also, the state estimation problem for neural networks has attracted some attention in the recent years, see [19–21]. For the state estimation problem, normally the periodic type constant vector is used in the existing literatures for getting the unstable behavior in the system. Without having such constant vector the given system should be stable, see for example [19–21]. In this regard there is no meaningful idea behind for designing the estimator gain matrix H. Motivating this reason, in this paper, leakage delay in the leakage term is used to unstable the neuron states without constant vector. On the other hand, the sampled-data control technology has developed largely as the rapid development of computer hardware. The measurements used to estimate the neuron states are sampled by samplers. Based on the sampled measurements, in this paper a sampled-data estimator is constructed. By converting the sampling period into a time-varying but bounded delay, the error dynamics of the considered FCNN is derived in terms of a differential equation with two different timedelays [22]. To the best of authors’ knowledge, there was no results available in any of the existing literature dealing the state estimation for FCNNs with time delay in the leakage term, discrete and unbounded distributed delays based on sample-data. Motivated by the above discussion, in this paper leakage delay in the leakage term is used to unstable the neuron states. It is challenging to develop delay dependent condition to estimate the unstable neuron states through available sampled output measurements such that the error-state system is globally asymptotically stable. Based on the LKF which contains a triple-integral term, an improved delay-dependent stability criterion is derived in terms of LMIs. However using the freeweighting matrices method, simple and efficient criterion is derived in terms of LMIs for estimation. Finally, numerical examples and its simulations are provided to demonstrate the effectiveness and merits of the derived result. Notations Rn denotes the n-dimensional Euclidean Space. For any matrix A ¼ ½aij nn , let AT and A1 denote the transpose and the inverse of A, respectively. jAj ¼ ½jaij jnn . Let A > 0 ðA < 0Þ denotes the positive-definite (negative-definite) symmetric matrix, respectively. I denotes the identity matrix of appropriate dimension. K ¼ f1; 2; . . . ; ng and N ¼ f1; 2; . . . ; mg. denotes the symmetric terms in a symmetric matrix.
M. Kalpana, P. Balasubramaniam 2. Model formulation and preliminaries Consider the following FCNNs with leakage delay, discrete and unbounded distributed delays 8 n X > > > x_ i ðtÞ ¼ ai xi ðt rÞ þ b0ij gj ðxj ðtÞÞ > > > j¼1 > > > > n n ^ X > Rt > < kj ðt sÞg ðxj ðsÞÞds þ b1 g ðxj ðt s1 ðtÞÞÞ þ aij j
ij
1
þ
n _ j¼1
j
j¼1
j¼1
> > > > > > > > > > > > :
bij
Rt
1
kj ðt sÞgj ðxj ðsÞÞds; i 2 K;
xi ðsÞ ¼ ui ðsÞ; s 2 ð1;0; ð1Þ
where ui ðÞ 2 Cðð1; 0; RÞ; aij and bij are the elements of fuzzy feedback MIN template, fuzzy feedback MAX template, respectively; b0ij and b1ij are the elements of feedback V W denote the fuzzy AND and fuzzy OR operatemplate; , tion, respectively; xi denotes the state of the ith neuron; ai is a diagonal matrix, ai represents the rates with which the ith neuron will reset their potential to the resting state in isolation when disconnected from the networks and external inputs; gj represents the neuron activation function; ki ðsÞ P 0 is the feedback kernel and satisfies Z 1 ki ðsÞds ¼ 1; i 2 K: ð2Þ 0
ðA1 Þ The transmission delay s1 ðtÞ is a time varying delay, and it satisfies 0 6 s1 ðtÞ 6 s1 , where s1 is a positive constant; ðA2 Þ The leakage delay satisfies r P 0. Also, it is assumed that the neuron activation function gðÞ satisfies the following Lipschitz condition jgðxÞ gðyÞj 6 jLðx yÞj;
ð3Þ
nn
where L 2 R is a known constant matrix. Our aim in this paper is to investigate an efficient estimation algorithm in order to observe the neuron states from the available network outputs. Therefore, the network measurements are assumed to satisfy yl ðtÞ ¼ clj xi ðtÞ; l 2 N; i; j 2 K, where yl 2 Rm is the measurement output of the lth neuron and clj is the element of a known constant matrix with appropriate dimension. In this paper, the measurement output is sampled before it enters the estimator. The sampled measurement is assumed to be generalized by a zero-order hold function with a sequence of hold times 0 ¼ t0 < t1 < < tk < yl ðtk Þ ¼ clj xi ðtk Þ;
tk 6 t < tkþ1 ;
ð4Þ
where tk denotes the sampling instant and satisfies 0 ¼ t0 < t1 < t2 < < tk < < limk!þ1 tk ¼ þ1. Moreover, the sampling period under consideration is assumed to be bounded by a known constant s2 , that is tkþ1 tk 6 s2 for k P 0. The full order state estimation of system (1) is given as follows
Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
Asymptotical state estimation of fuzzy cellular neural networks 8 n n X X > > > ^_ i ðtÞ ¼ ai x^i ðt rÞ þ b1ij gj ð^ b xj ðt s1 ðtÞÞÞ x x 0ij gj ð^ j ðtÞÞ þ > > > j¼1 j¼1 > > > > n > ^ Rt > > > xj ðsÞÞds aij 1 kj ðt sÞgj ð^ þ > < j¼1
> > > > > > > > > > > > > > > :
n _ Rt xj ðsÞÞds þ bij 1 kj ðt sÞgj ð^ j¼1
þhil yl ðtk Þ clj x^i ðtk Þ ; l 2 N; i;j 2 K; x^i ðsÞ ¼ vi ðsÞ; s 2 ð1; 0; ð5Þ
where vi ðÞ 2 Cðð1; 0; RÞ; x^i ðtÞ is the estimation of the ith neuron state; hil is the element of an estimator gain matrix to be designed. Define the error ei ðtÞ ¼ xi ðtÞ x^i ðtÞ; /j ðtÞ ¼ gj ðxj ðtÞÞ gj ð^ xj ðtÞÞ; i; j 2 K; then it follows from (1), (4), and (5) that 8 n X > > > _ e b0ij /j ðtÞ ðtÞ ¼ a e ðt rÞ h c e ðt Þ þ i i i il lj i k > > > > j¼1 > > > n n > ^ X Rt > > > þ b1ij /j ðt s1 ðtÞÞ þ aij 1 kj ðt sÞgj ðxj ðsÞÞds > > > > j¼1 j¼1 > > > n > ^ Rt > > > xj ðsÞÞds aij 1 kj ðt sÞgj ð^ > < j¼1
> > > > > > > > > > > > > > > > > > > > > > > > > > :
þ
n _
bij
j¼1
n _
bij
j¼1
Rt
kj ðt sÞgj ðxj ðsÞÞds
Rt
xj ðsÞÞds; kj ðt sÞgj ð^
1
1
i; j 2 K; l 2 N; ei ðsÞ ¼ ui ðsÞ vi ðsÞ ¼ ui ðsÞ;
s 2 ð1; 0: ð6Þ
Clearly, it is difficult to analyze the state estimation for FCNNs based on error system (6) because of the discrete term eðtk Þ. Therefore, the input delay approach [23] is applied, so that s2 ðtÞ ¼ t tk ; tk 6 t < tkþ1 . It is easily seen that 0 6 s2 ðtÞ < s2 . Therefore, the error estimator takes the following form ei ðtk Þ :¼ ei ðt s2 ðtÞÞ;
tk 6 t < tkþ1 ;
i 2 K:
Consequently, connecting (7) to system (6) yields 8 e_i ðtÞ ¼ ai ei ðt rÞ hil clj ei ðt s2 ðtÞÞ > > > n n > X X > > > b1ij /j ðt s1 ðtÞÞ þ b0ij /j ðtÞ þ > > > > j¼1 j¼1 > > > > n ^ > Rt > > aij 1 kj ðt sÞgj ðxj ðsÞÞds þ > > > > j¼1 > > > > n ^ > Rt > < xj ðsÞÞds aij 1 kj ðt sÞgj ð^ j¼1 > > > n _ > Rt > > þ bij 1 kj ðt sÞgj ðxj ðsÞÞds > > > > j¼1 > > > > n _ > Rt > > > bij 1 kj ðt sÞgj ð^ xj ðsÞÞds; > > > j¼1 > > > > > i; j 2 K; l 2 N; > > : ei ðsÞ ¼ ui ðsÞ; s 2 ð1; 0:
ð7Þ
3 Using a simple transformation, system (8) leads the following equivalent form 8 n X Rt > d > > b0ij /j ðtÞ > dt ei ðtÞ ai tr ei ðsÞds ¼ ai ei ðtÞ hil clj ei ðt s2 ðtÞÞ þ > > j¼1 > > > n > X > > > þ b1ij /j ðt s1 ðtÞÞ > > > > j¼1 > > > > > n ^ > Rt > > þ aij 1 kj ðt sÞgj ðxj ðsÞÞds > > > > j¼1 > > < n ^ Rt aij 1 kj ðt sÞgj ð^ xj ðsÞÞds > > > j¼1 > > > n > _ Rt > > > þ bij 1 kj ðt sÞgj ðxj ðsÞÞds > > > > j¼1 > > > n > _ Rt > > > bij 1 kj ðt sÞgj ð^ xj ðsÞÞds; > > > > j¼1 > > > > i;j 2 K; l 2 N; > > : ei ðsÞ ¼ ui ðsÞ; s 2 ð1;0: ð9Þ
It is clear from (3) that /T ðtÞ/ðtÞ ¼ jgðxðtÞÞ gð^ xðtÞÞj2 6 jLeðtÞj2 ¼ eT ðtÞLT LeðtÞ:
ð10Þ
3. Main results Theorem 3.1. Assume that assumptions ðA1 Þ ðA2 Þ and the Lipschitz condition (3) hold. The error dynamical system (8) is globally asymptotically stable, if there exist n n positive diagonal matrices P; Q, some n n positive definite symmetric matrices R; W; N; M1 ; M2 , three scalars l > 0; 1 > 0; T11 T12 2 > 0, and a 2n 2n matrix > 0; T22 V11 V12 > 0 such that the following LMI has feasible V22 solution " # W CT < 0; ð11Þ X¼ ln1 I where ðWÞ1313 with
ð8Þ
W1;1 ¼ 2PA þ P þ W þ r2 N 2M1 2M2 þ 1 LT L; W1;3 ¼ TT12 ; W1;4 ¼ RC þ VT12 ; 2 2 M1 ; W1;6 ¼ AT PA; W1;7 ¼ M1 ; W1;8 ¼ s1 s1 2 2 W1;9 ¼ M2 ; W1;10 ¼ M2 ; W1;11 ¼ PB0 ; s2 s2 W1;12 ¼ PB1 ; W2;2 ¼ W; W2;5 ¼ AT PT ; W3;3 ¼ s1 T11 2TT12 þ 2 LT L; W4;4 ¼ s2 V11 2VT12 ; W4;5 ¼ CT RT ; W4;6 ¼ CT RT A; W5;5 ¼ 2P þ s1 T22 s2 s2 þ s2 V22 þ 1 M1 þ 2 M2 ; W5;11 ¼ PB0 ; 2 2 W5;12 ¼ PB1 ; W6;6 ¼ AT PA N; W6;11 ¼ AT PB0 ; 2 W6;12 ¼ AT PB1 ; W7;7 ¼ 2 M1 ; s1 2 2 2 W7;8 ¼ 2 M1 ; W8;8 ¼ 2 M1 ; W9;9 ¼ 2 M2 ; s1 s1 s2 2 2 W9;10 ¼ 2 M2 ; W10;10 ¼ 2 M2 ; s2 s2
Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
4
M. Kalpana, P. Balasubramaniam
W11;11 ¼ Q 1 ; W12;12 ¼ 2 ; W13;13 ¼ 2nST PS þ lI Q; ( ) n n n X X X jain j ; jai2 j; . . . ; jai1 j; jajs ¼ diag i¼1
i¼1
i¼1
(
) n n n X X X jbin j ; jbi2 j; . . . ; jbi1 j; jbjs ¼ diag i¼1
i¼1
i¼1
S ¼ jajs þ jbjs ; CT ¼ 0 0 0 0
ðPSÞ
T
0 0 0
0 0 0 0
0
T
:
_ V_ 6 ðtÞ ¼ s1 e_T ðtÞT22 eðtÞ
2
ð12Þ
þ
i¼1
where V1 ðtÞ ¼ eðtÞ A
t
Z
tr
T Z eðsÞds P eðtÞ A
V2 ðtÞ ¼ V4 ðtÞ ¼
þ
Z
t 0
2
Z
t
1 0
t
"
#T "
_ eðsÞ " # eðh s1 ðhÞÞ dsdh _ eðsÞ " #T " Z h V11 eðh s2 ðhÞÞ hs1 ðhÞ
0
V7 ðtÞ ¼
Z Z
þ
0 s1 0 s1
Z
_ eðsÞ
hs2 ðhÞ
V6 ðtÞ ¼
t
t
Z
_ e_T ðsÞT22 eðsÞdsdh þ 0
h
0
s2
eT ðsÞNeðsÞdsdh;
h
Z
Z
0
h
t
T12
T22
V12
#"
V22 Z 0 Z
#
eðh s2 ðhÞÞ _ eðsÞ
#
dsdh;
0
s2
Z
t
_ e_T ðsÞM2 eðsÞdsdh;
ð19Þ
tþh
ð22Þ
Z
ts1 ðtÞ
eT ðsÞds; ts1
t
1
Z
t
eT ðsÞds; ts2 ðtÞ
Z
ts2 ðtÞ
eT ðsÞds; /T ðtÞ;
ts2
/T ðt s1 ðtÞÞ; T Kðt sÞ/T ðsÞds ;
X ¼ W þ CT l1 nC:
t
_ e_T ðsÞV22 eðsÞdsdh;
tþh
_ 6 nT ðtÞ XH nðtÞ; t > 0, By (11), it yields VðtÞ H X ¼ X > 0. Thus, it can be deduced that Z t nT ðsÞXH nðsÞds 6 Vð0Þ < 1; t P 0; VðtÞ þ
where
ð23Þ
0
T
_ e_ ðsÞM2 eðsÞdsdkdh:
where
tþk
By calculating the time derivation of Vi ðtÞ along the trajectory of system (9), one can obtain Z t n X ei ðsÞds V_ 1 ðtÞ ¼ 2 pi ei ðtÞ ai tr
Z t d ei ðtÞ ai ei ðsÞds ; dt tr
ð13Þ
V_ 2 ðtÞ ¼ eT ðtÞWeðtÞ eT ðt rÞWeðt rÞ; Z t Z t V_ 3 ðtÞ 6 r2 eT ðtÞNeðtÞ eT ðsÞds N eðsÞds; tr t
ð14Þ ð15Þ
tr
Kðt sÞ/ðsÞds V_ 4 ðtÞ ¼ /T ðtÞQ/ðtÞ 1 Z t Q Kðt sÞ/ðsÞds ; 1
_ e_T ðsÞM1 eðsÞdsdh
tþh
ts1 ðtÞ
tr
t
Z
t
_ 6 nT ðtÞ½W þ CT l1 nC nðtÞ ¼ nT ðtÞ X nðtÞ; VðtÞ
Z
_ e_T ðsÞM1 eðsÞdsdkdh
i¼1
Z
ð18Þ
nðtÞ ¼ eT ðtÞ; eT ðt rÞ; eT ðt s1 ðtÞÞ; eT ðt s2 ðtÞÞ; e_T ðtÞ; Z t Z t eT ðsÞds; eT ðsÞds;
tþk
Z
s22 T _ e_ ðtÞM2 eðtÞ 2
Z
where
T11
s2
tþh
Z
t
Z
/2j ðsÞdsdh;
eðh s1 ðhÞÞ
s1
_ e_T ðsÞV22 eðsÞds; ts2
0 6 1 eT ðtÞLT LeðtÞ /T ðtÞ/ðtÞ ; ð20Þ 0 6 2 eT ðt s1 ðtÞÞLT Leðt s1 ðtÞÞ /T ðt s1 ðtÞÞ/ðt s1 ðtÞÞ : ð21Þ
th
h
Z
Z
kj ðhÞ
0
t
Hence, from (13)–(21) we have
Z
tr
qj
Z
Z
On the other hand, it is clear from (10) that the following is true for j > 0; j ¼ 1; 2
;
eT ðsÞWeðsÞds; V3 ðtÞ ¼ r
tr n X
Z
ei ðsÞds
t
j¼1
V5 ðtÞ ¼
t
tr
i¼1
Z
eðsÞds
tr
Z n X pi ei ðtÞ ai
¼
t
_ e_T ðsÞT22 eðsÞds
ts1
s _ V_ 7 ðtÞ ¼ 1 e_T ðtÞM1 eðtÞ 2
7 X Vi ðtÞ;
t
Z
_ þ s2 e_T ðtÞV22 eðtÞ
Proof. Consider the following LKFs
ð17Þ
ts2
Moreover, the estimation gain is H ¼ P1 R.
VðtÞ ¼
V_ 5 ðtÞ 6 eT ðt s1 ðtÞÞ s1 T11 2TT12 eðt s1 ðtÞÞ Z t _ e_T ðsÞT22 eðsÞds þ 2eT ðtÞTT12 eðt s1 ðtÞÞ þ ts1 þ eT ðt s2 ðtÞÞ s2 V11 2VT12 eðt s2 ðtÞÞ Z t _ þ 2eT ðtÞVT12 eðt s2 ðtÞÞ þ e_T ðsÞV22 eðsÞds;
T
ð16Þ
Vð0Þ 6 2kmax ðPÞ ð1 þ r2 maxai Þ þ rkmax ðWÞ þ r3 kmax ðNÞ i2K
Z n X þ qj kj maxl2j j¼1
j2K
1
hkj ðhÞdh þ s21 kmax ðT22 Þ þ s22 kmax ðV22 Þ
0
þs31 kmax ðM1 Þ þ s32 kmax ðM2 Þ kue k2 < 1:
From the definition of V2 ðtÞ and Jensen’s inequality lemma
R t
2 [24], we have tr eðsÞds 6 kminrðWÞ VðtÞ 6 kminrðWÞ Vð0Þ, which together with the definition of V1 ðtÞ yields
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Z t
V1 ðtÞ keðtÞk 6
eðsÞds
þ kmin ðPÞ
A tr (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) Xn pffiffiffiffiffiffiffiffiffiffi r 1 Vð0Þ: ai þ 6 i¼1 k kmin ðPÞ min ðWÞ
Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
Asymptotical state estimation of fuzzy cellular neural networks The state x and its estimation
(a)
The state x and its estimation
(b)
1
20
2
50
10
amplitude
amplitude
5
0 −10 −20 0
50
0
−50
100
0
time t
(c)
(d)
The state x and its estimation
The error states
40
20
20
amplitude
amplitude
3
40
0 −20 −40
0
50
100
time t
50
100
e1 (t) e2 (t) e3 (t)
0 −20 −40
0
time t
50
100
time t
Fig. 1 (a) the true state x1 ðtÞ and its estimate state x^1 ðtÞ when r ¼ 0:2, with estimator gain matrix (31), (b) the true state x2 ðtÞ and its estimate state x^2 ðtÞ when r ¼ 0:2, with estimator gain matrix (31), (c) the true state x3 ðtÞ and its estimate state x^3 ðtÞ when r ¼ 0:2, with estimator gain matrix (31), and (d) the error trajectories of system (8) when r ¼ 0:2 with estimator gain matrix (31).
This implies that the error system (8) is locally stable. Next, one can prove that keðtÞk ! 0 as t ! 1. First, for any constant h 2 ½0; 1, it follows from (12) and Jensen’s inequality lemma [24] that Z tþ1 1 keðt þ hÞ eðtÞk2 6 nT ðsÞXH nðsÞds ! 0 H kmin ðX Þ t as t ! 1; which implies that for any > 0; h 2 ½0; 1, there exists a T1 ¼ T1 ðÞ > 0 such that keðt þ hÞ eðtÞk < ; 2
t > T1 :
ð24Þ
On the other hand, from (12) we have
Z tþ1
2 Z tþ1
1
nT ðsÞXH nðsÞds ! 0 eðsÞds
6 k ðXH Þ t t min as t ! 1; which implies
R that for any > 0, there exists a T2 ¼ T2 ðÞ > 0
tþ1
such that t eðsÞds < 2 ; t > T2 . Note that eðsÞ is continu-
ous on ½t; t þ 1; t > 0. Applying the integral mean value theorem, there exists a vector dt ¼ ðdt1 ; dt2 ; ; dtn ÞT 2 Rn ; dtj 2 ½t; t þ 1, such that
Z tþ1
eðsÞds
ð25Þ keðdt Þk ¼
< 2 ; t > T2 : t By (24), (25) and for any > 0, there T ¼ maxfT1 ; T2 g > 0 such that t > T implies
exists
a
keðtÞk 6 keðtÞ eðdt Þk þ keðdt Þk 6 þ ¼ : 2 2 This proves that keðtÞk ! 0 as t ! 1. Therefore, one can conclude that the error dynamical system (8) is globally asymptotically stable. As a result, the full order sampled state estimation
FCNNs with time delay in the leakage term, discrete and unbounded distributed delays (5) is globally estimated with the FCNNs (1). This completes the proof. h Remark 3.1. It is evident from the Fig. 2(a)–(c), the true state xi ðtÞ is stable when r ¼ 0 and its estimated state is x^i ðtÞ; i ¼ 1; 2; 3. Further through Fig. 1(a)–(c), it is clear that the true state xi ðtÞ is unstable when r ¼ 0:2 and its estimated state is x^i ðtÞ; i ¼ 1; 2; 3. Moreover, the error trajectories of Figs. 1(d) and 2(d) converges to 0. The time delays which is called leakage delay r exists in the negative feedback term of system (1), which is different from the time-varying delays in other terms. It has been shown in [18,25] that the time delay in the leakage term has great impact on the dynamics of neural networks and often has a quick tendency to destabilize a system. This motivates to consider the leakage delay effects on the state estimation of FCNNs with discrete time-varying delays and continuously unbounded distributed delays. However, this paper deals for the constant leakage delay; to improve and extend the results for time-varying leakage delay may lead a challenging problem. In the near future, some further research on this topic will be investigated. When there is no time delay in the leakage term, that is r ¼ 0, the error dynamical FCNNs (8) becomes the following 8 n n X X > > b1ij /j ðt s1 ðtÞÞ b0ij /j ðtÞ þ > e_i ðtÞ ¼ ai ei ðtÞ hil clj ei ðt s2 ðtÞÞ þ > > > j¼1 j¼1 > > > > n n > ^ ^ Rt Rt > > > xj ðsÞÞds þ aij 1 kj ðt sÞgj ðxj ðsÞÞds aij 1 kj ðt sÞgj ð^ > > < j¼1 j¼1 > > > > > > > > > > > > > > > > :
n n _ _ Rt Rt xj ðsÞÞds; þ bij 1 kj ðt sÞgj ðxj ðsÞÞds bij 1 kj ðt sÞgj ð^ j¼1
j¼1
i;j 2 K; l 2 N;
ei ðsÞ ¼ ui ðsÞ; s 2 ð1;0: ð26Þ
Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
6
M. Kalpana, P. Balasubramaniam
(a)
The state x and its estimation 1
(b)
3
amplitude
amplitude
x1 (t) x ˆ1 (t)
2.5 2 1.5 1
The state x2 and its estimation
−0.5
0
5
x2 (t) x ˆ2 (t) −1 −1.5 −2
10
0
5
time t
(c)
The state x3 and its estimation
(d)
The error states
2 x3 (t) x ˆ3 (t)
amplitude
amplitude
4 2 0 −2
10
time t
0
5
10
0 e1 (t)
−2
e2 (t) −4 −6
e3 (t) 0
5
10
time t
time t
Fig. 2 (a) the true state x1 ðtÞ and its estimate state x^1 ðtÞ when r ¼ 0, with estimator gain matrix (34), (b) the true state x2 ðtÞ and its estimate state x^2 ðtÞ when r ¼ 0, with estimator gain matrix (34), (c) the true state x3 ðtÞ and its estimate state x^3 ðtÞ when r ¼ 0, with estimator gain matrix (34), and (d) the error trajectories of system (26) with estimator gain matrix (34).
In the following Corollary 3.1, global asymptotic stability criteria for error dynamical FCNNs (26) is discussed. Corollary 3.1. Assume that assumptions ðA1 Þ ðA2 Þ, the Lipschitz condition (3) hold. The error dynamical system (26) is globally asymptotically stable, if there exist n n positive diagonal matrices P; Q, some n n positive definite symmetric matrices R; M1 ; M2, three scalars l >0; 1 > 0; 2 > 0, and T11 T12 V11 V12 > 0, such > 0; a 2n 2n matrices V22 T22 that the following LMI has feasible solution X¼
"
W
CT ln1 I
#
W9;9 ¼ Q 1 ; W10;10 ¼ 2 ; W10;11 ¼ 0; W11;11 ¼ nST PS þ lI Q; ( ) n n n X X X jain j ; jai2 j; . . . ; jai1 j; jajs ¼ diag i¼1
i¼1
i¼1
(
) n n n X X X jbin j ; jbi2 j; . . . ; jbi1 j; jbjs ¼ diag i¼1
i¼1
i¼1
T S ¼ jajs þ jbjs ; CT ¼ 0 0 0 ðPSÞT 0 0 0 0 0 0 0 : Moreover, the estimation gain is H ¼ P1 R.
Proof. Consider the following LKFs ð27Þ
< 0;
where ðWÞ1111 with W1;3 ¼ RC þ VT12 ; W1;4 ¼ AT P; 2 2 2 W1;5 ¼ M1 ; W1;6 ¼ M1 ; W1;7 ¼ M2 ; s1 s1 s2 2 W1;8 ¼ M2 ; W1;9 ¼ PB0 ; W1;10 ¼ PB1 ; s2 W2;2 ¼ s1 T11 2TT12 þ 2 LT L; W3;3 ¼ s2 V11 2VT12 ; W3;4 ¼ CT RT ; s21 s2 M1 þ 2 M2 ; W4;9 ¼ PB0 ; 2 2
2 M1 ; s21 2 2 2 W5;6 ¼ 2 M1 ; W6;6 ¼ 2 M1 ; W7;7 ¼ 2 M2 ; s1 s1 s2 2 2 W7;8 ¼ 2 M2 ; W8;8 ¼ 2 M2 ; s2 s2 W4;10 ¼ PB1 ; W5;5 ¼
ð28Þ
i¼1
W1;1 ¼ 2PA þ P 2M1 2M2 þ 1 LT L; W1;2 ¼ TT12 ;
W4;4 ¼ 2P þ s1 T22 þ s2 V22 þ
5 X Vi ðtÞ;
VðtÞ ¼ where
V1 ðtÞ ¼ eT ðtÞPeðtÞ ¼
n X pi e2i ðtÞ; i¼1
Z n X qj V2 ðtÞ ¼
1
kj ðhÞ
0
j¼1
Z
t
/2j ðsÞdsdh; th
T T11 T12 eðh s1 ðhÞÞ eðh s1 ðhÞÞ dsdh _ _ T22 eðsÞ eðsÞ 0 hs1 ðhÞ T Z tZ h V11 V12 eðh s2 ðhÞÞ eðh s2 ðhÞÞ þ dsdh; _ _ V22 eðsÞ eðsÞ 0 hs2 ðhÞ Z 0 Z t Z 0 Z t _ _ e_T ðsÞT22 eðsÞdsdh þ e_T ðsÞV22 eðsÞdsdh; V4 ðtÞ ¼ s1 tþh s2 tþh Z 0 Z 0Z t _ V5 ðtÞ ¼ e_T ðsÞM1 eðsÞdsdkdh V3 ðtÞ ¼
Z tZ
h
s1
h
þ
Z
0
s2
Z
tþk
0
h
Z
t
_ e_T ðsÞM2 eðsÞdsdkdh:
tþk
Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
Asymptotical state estimation of fuzzy cellular neural networks The proof of this Corollary 3.1 is immediately follows from Theorem 3.1. h Remark 3.2. In this paper, delay rate independent stability conditions have been derived without involving the time-varying delay s1 ðtÞ in the LKFs. Moreover, the conditions that the time-varying delay is differentiable and the derivative is bounded or smaller than one are not required. 4. Numerical examples
Example 4.1. Consider the following simple three-dimensional FCNNs with leakage delay, discrete and unbounded distributed delays 8 n n X X > > > x_ i ðtÞ ¼ ai xi ðt rÞ þ b0ij gj ðxj ðtÞÞ þ b1ij gj ðxj ðt s1 ðtÞÞÞ > > > j¼1 j¼1 > > > > n ^ > Rt > < kj ðt sÞg ðxj ðsÞÞds þ aij j
1
j¼1
> > > > > > > > > > > > :
n _ Rt þ bij 1 kj ðt sÞgj ðxj ðsÞÞds; i 2 K; j¼1
xi ðsÞ ¼ ui ðsÞ; s 2 ð1; 0;
ð29Þ
with parameters defined as r ¼ 0:2; s1 ðtÞ ¼ 0:1 j sinðtÞj, uðsÞ ¼ ð2; 1; 1:7ÞT ; s 2 ð1; 0, and gj ðxj Þ ¼ 12 ðjxj þ 1j jxj 1jÞ; j ¼ 1; 2; 3, which satisfy the Lipschitz condition (3), we get L ¼ I, 3 3 2 2 5 1:2 1 6 0 0 7 7 6 6 A ¼ 4 0 6 0 5; B0 ¼ 4 3 1:1 4 5; 0:32 1:7 0:95 0 0 6 3 2 1:5 0:7 2:6 7 6 B1 ¼ 4 3:3 1:2 0:5 5; 0:9
2
1:5
2:3
3
1=31 1=31 1=31 7 6 a ¼ 4 1=31 1=31 1=31 5; 1=31 1=31 1=31 3 2 1=31 1=31 1=31 7 6 b ¼ 4 1=31 1=31 1=31 5; 1=31 1=31 1=31
2
5
6 C ¼ 40 0
0 0
3
7 5 0 5: 0 5
7 the LMI is feasible. Consequently, the estimator gain matrix H is designed as follows 3 2 0:0865 0:0235 0:0010 7 6 ð31Þ H ¼ P1 R ¼ 4 0:0264 0:0611 0:0165 5: 0:0007 0:0095 0:0925
By Theorem 3.1, systems (29) and (30) are asymptotically estimated. The simulation results are depicted in Fig. 1(a)–(d) by applying the estimator designed in (31) for this Example 4.1 by choosing the time step size h ¼ 0:1, and time segment T ¼ 100. Example 4.2. Consider the following simple three-dimensional FCNNs without time delay in the leakage term, discrete and unbounded distributed delays 8 n n X X > > > x_ i ðtÞ ¼ ai xi ðtÞ þ b0ij gj ðxj ðtÞÞ þ b1ij gj ðxj ðt s1 ðtÞÞÞ > > > j¼1 j¼1 > > > > n ^ > Rt > < kj ðt sÞg ðxj ðsÞÞds þ aij 1
j
j¼1
> > > > > > > > > > > > :
þ
n _ Rt bij 1 kj ðt sÞgj ðxj ðsÞÞds;
i 2 K;
j¼1
xi ðsÞ ¼ ui ðsÞ;
s 2 ð1; 0;
ð32Þ
with parameters defined as in Example 4.1 and r ¼ 0. The corresponding full order sampled state estimation of system (32) is defined as follows 8 n n X X > > > ^_ i ðtÞ ¼ ai x^i ðtÞ þ x xj ðt s1 ðtÞÞÞ x b1ij gj ð^ b 0ij gj ð^ j ðtÞÞ þ > > > j¼1 j¼1 > > > > n > ^ Rt > > > xj ðsÞÞds aij 1 kj ðt sÞgj ð^ þ > < j¼1
> > > > > > > > > > > > > > > :
n _ Rt xj ðsÞÞds þ bij 1 kj ðt sÞgj ð^ j¼1
þhil yl ðtk Þ clj x^i ðtk Þ ; x^i ðsÞ ¼ vi ðsÞ; s 2 ð1; 0;
l 2 N; i 2 K; ð33Þ
By using the Matlab LMI toolbox to solve the LMI (27) in Corollary 3.1, it can be found that the LMI is feasible. Consequently, the estimator gain matrix H is designed as follows 2
0:0523
6 H ¼ P1 R ¼ 4 0:0228 0:0078
0:0187 0:0471 0:0138
0:0127
3
7 0:0275 5: 0:0603
ð34Þ
The corresponding full order sampled state estimation of system (29) is defined as follows 8 n n X X > > b0ij gj ð^ b1ij gj ð^ xj ðtÞÞ þ xj ðt s1 ðtÞÞÞ > x^_ i ðtÞ ¼ ai x^i ðt rÞ þ > > > j¼1 j¼1 > > > < n n _ ^ Rt Rt þ aij 1 kj ðtsÞgj ð^ xj ðsÞÞds xj ðsÞÞdsþ bij 1 kj ðtsÞgj ð^ > > j¼1 j¼1 > > > > > þhil ½yl ðtk Þ clj x^i ðtk Þ; l 2 N; i 2 K; > > : x^i ðsÞ ¼ vi ðsÞ; s 2 ð1;0; ð30Þ
By Corollary 3.1, systems (32) and (33) are asymptotically estimated. The simulation results are depicted in Fig. 2(a)–(d) by using the above estimator (34) for this Example 4.2 by choosing the time step size h ¼ 0:1, and time segment T ¼ 10.
where yl ðtk Þ is given by (4) and the initial condition is vðsÞ ¼ ð3; 2; 4ÞT ; s 2 ð1; 0. Moreover, the sampling period is taken as s2 ¼ 0:05. By using the Matlab LMI toolbox to solve the LMI (11) in Theorem 3.1, it can be found that
In this paper, state estimation for FCNNs is considered with time delay in the leakage term, discrete and unbounded distributed delays based on sampled-data. The sampled measurements have been used to estimate the neuron states. Also
5. Conclusion
Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
8 simple and efficient estimation criterion is derived in terms of LMIs by constructing the LKF which contains a triple-integral term and the free-weighting matrices method. Further, the differentiability of the time-varying delay s1 ðtÞ is not required in this paper. Finally, the effectiveness of the proposed sampleddata estimation approach has been verified by demonstrating numerical simulations of the derived results. Acknowledgements Authors would like to thank, anonymous referee for the constructive comments and fruitful suggestions to improve the quality of the manuscript. Moreover, the work done by Dr. M. Kalpana was supported by the Government of India, Ministry of Science and Technology, Department of Science and Technology, under Grant No. DST/INSPIRE Fellowship/2010/[293]/dt. 05/02/2013. References [1] L.O. Chua, L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst. 35 (1988) 1257–1272. [2] L.O. Chua, L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits Syst. 35 (1988) 1273–1290. [3] T. Yang, L.B. Yang, C.W. Wu, L.O. Chua, Fuzzy cellular neural networks: theory, in Proceedings of the IEEE International Workshop on Cellular Neural Networks and Applications, 1996, pp. 181–186. [4] T. Yang, L.B. Yang, C.W. Wu, L.O. Chua, Fuzzy cellular neural networks: applications, in: Proceedings of the IEEE International Workshop on Cellular Neural Networks and Applications, 1996, pp. 225–230. [5] S. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer, New York, 2001. [6] Q. Zhang, L. Yang, D. Liao, Global exponential stability of fuzzy BAM neural networks with distributed delays, Arab. J. Sci. Eng. 38 (2013) 691–697. [7] Q. Zhang, Y. Shao, J. Liu, Analysis of stability for impulsive fuzzy Cohen–Grossberg BAM neural networks with delays, Math. Meth. Appl. Sci. 36 (2013) 773–779. [8] Q. Zhang, L. Yang, D. Liao, Existence and exponential stability of a periodic solution for fuzzy cellular neural networks with time-varying delays, Int. J. Appl. Math. Comput. Sci. 21 (2011) 649–658. [9] T. Huang, Exponential stability of delayed fuzzy cellular neural networks with diffusion, Chaos Solit. Fract. 31 (2007) 658–664. [10] T. Huang, Exponential stability of fuzzy cellular neural networks with distributed delay, Phys. Lett. A 351 (2006) 48–52.
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Please cite this article in press as: M. Kalpana, P. Balasubramaniam, Asymptotical state estimation of fuzzy cellular neural networks with time delay in the leakage term and mixed delays: Sample-data approach, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.07.003
