Adaptive synchronization for uncertain chaotic neural networks with ...
Applied Mathematics and Computation 219 (2013) 5984–5995
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Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer S.C. Jeong a, D.H. Ji b, Ju H. Park c,⇑, S.C. Won a,⇑ a
Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, San 31, Hyoja-Dong, Pohang 790-784, Republic of Korea Mobile communication Division, Digital Media and Communications, Samsung Electronics, Co. Ltd., Maetan-dong, Suwon 416-2, Republic of Korea c Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University 214-1 Dae-dong, Kyongsan 712-749, Republic of Korea b
a r t i c l e
i n f o
Keywords: Adaptive synchronization Mixed time delay Chaotic neural networks Fuzzy disturbance observer
a b s t r a c t This paper proposes a robust adaptive control method for synchronization of uncertain chaotic neural networks with mixed delays. Uncertainty and disturbance in the networks are estimated by fuzzy disturbance observer without any prior information about them. The proposed control scheme with adaptive laws is derived based on Lyapunov–Krasovskii stability theory to guarantee the globally asymptotical synchronization between the networks. An example is illustrated to show the effectiveness of the proposed method. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In the past few decades, there has been considerable attention in the study of neural networks due to their potential applications in various areas, such as signal processing pattern recognition, static image processing, associative memory and combinatorial optimization [1–4]. It has been shown that artificial neural network models can exhibit some chaotic behaviors [5–8]. Since the pioneering works of Pecora and Carroll [9], Synchronization of chaotic neural networks has been intensively investigated in many fields [10–13]. In the implementation of the neural networks, time delays between neurons in the networks often arise in the processing of information storage and transmission, which may lead to instability, oscillation, and bifurcation of the neural network model [8,14]. Many studies have been developed for the synchronization problem of delayed chaotic neural networks. Some have considered the networks with time-varying delays [15–19]. However, there exist various chaotic neural networks with both time-varying delays and distributed delays in realistic network models. Therefore, it is worth taking into account the chaotic neural networks with the mixed time delays including time-varying and distributed delays [20–29]. A control method with two sufficient conditions to ensure the globally exponential stability for the error system has been proposed based on the drive-response concept [20,21]. In [22], a synchronization problem of the networks with mixed delays has been discussed by using an adaptive feedback control technique. Sufficient conditions for asymptotical or exponential synchronization are derived in terms of Linear matrix inequalities (LMIs) by constructing proper Lyapunov–Krasovskii functional [23–25]. Sliding mode control technique is proposed to synchronize nonidentical chaotic neural networks with mixed delays [26,27]. The synchronization problems of stochastic perturbed chaotic neural networks with mixed delays have been investigated in [28,29]. It is known that the uncertainty and disturbance are unavoidable factors in many practical situations and they can destroy the network stability or can make the synchronization more difficult. Some works for uncertain neural networks have been developed to overcome their effects [15–32]. They often require some prior information of the uncertain factors, such as its structure or upper bound. However, the information may not be available due to physical limitations in practical cases. ⇑ Corresponding authors. E-mail addresses: [email protected] (S.C. Jeong), [email protected] (D.H. Ji), [email protected] (J.H. Park), [email protected] (S.C. Won). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.12.017
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
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Fuzzy logic system can be a good solution to be used in the situations because it can provide an estimator for a unknown function or value. Fuzzy disturbance observer (FDO) has been proposed to estimate uncertainty and disturbance without requiring any prior information about them [33]. The estimated values have been used to compensate the uncertain factors via state feedback controller. In [34], a robust tracking control approach using a discrete-time FDO has been proposed for nonlinear sampled systems. Recently, a more precise FDO has been constructed by modifying the law used to update the parameter vector and the modified FDO showed better performances, compared with the conventional one [35]. Even though the FDO presented good performances to overcome the unknown factor, applications of the existing research are still limited. Especially, there has been still no research using the technique for uncertain chaotic neural networks with mixed time delays. In this paper, we propose a robust adaptive synchronization method for uncertain chaotic neural networks with timevarying delays and distributed delays. The uncertain factors including uncertainties and disturbances are estimated by the FDO without requiring any prior knowledge about the factors. The estimated values are used to compensate the factors in the proposed method. Based on Lyapunov–Krasovskii stability theory, the control scheme with adaptive laws is derived and guarantees the globally asymptotical synchronization between the networks. An example is illustrated to show the effectiveness of the proposed method. 2. Problem statement Consider the following chaotic neural network with time-varying delay and distributed delay:
_ xðtÞ ¼ CxðtÞ þ Af ðxðtÞÞ þ Bgðxðt sðtÞÞÞ þ D
Z
t
hðxðsÞÞds þ I;
ð1Þ
trðtÞ
where xðtÞ ¼ ½x1 ðtÞ; . . . ; xn ðtÞT 2 Rn is the neuron state vector and C ¼ diagðc1 ; c2 ; . . . ; cn Þ is a positive diagonal matrix. A ¼ ðaij Þnn ; B ¼ ðbij Þnn , and D ¼ ðdij Þnn are the connection weight matrix, the time varying delayed connection weight matrix and distributively delayed connection weight matrix, respectively. I ¼ ½I1 ; I2 ; . . . ; In T 2 Rn is an external input vector, sðtÞ is the distributed time-delay. The initial conditions are given by is the time-varying delay, the positive constant 0 6 rðtÞ 6 r xi ðtÞ ¼ wxi ðtÞ 2 Cð½r; 0; RÞ, where Cð½r; 0; RÞ denotes the set of all continuous functions from ½r; 0 to R and r ¼ maxfsðtÞ; rðtÞg. f ðxðtÞÞ; gðxðt sðtÞÞÞ, and hðxðtÞÞ are the activation functions of the neurons and described as
f ðxðtÞÞ ¼ ½f1 ðx1 ðtÞÞ; f2 ðx2 ðtÞÞ; . . . ; fn ðxn ðtÞÞT ; gðxðt sðtÞÞÞ ¼ ½g 1 ðx1 ðt sðtÞÞÞ; g 2 ðx2 ðt sðtÞÞÞ; . . . ; g n ðxn ðt sðtÞÞÞT ; hðxðtÞÞ ¼ ½h1 ðx1 ðtÞÞ; h2 ðx2 ðtÞÞ; . . . ; hn ðxn ðtÞÞT : We consider the network (1) as the drive system. The response system having the uncertainty and disturbance is established as follows:
_ yðtÞ ¼ ðC þ DCÞyðtÞ þ ðA þ DAÞf ðyðtÞÞ þ ðB þ DBÞgðyðt sðtÞÞÞ þ ðD þ DDÞ ¼ CyðtÞ þ Af ðyðtÞÞ þ Bgðyðt sðtÞÞÞ þ D
Z
Z
t
hðyðsÞÞds þ I þ dðtÞ þ uðtÞ
trðtÞ t
hðyðsÞÞ þ XðtÞ þ I þ uðtÞ;
ð2Þ
trðtÞ
where yðtÞ ¼ ½y1 ðtÞ; y2 ðtÞ; . . . ; yn ðtÞT 2 Rn is the neuron state vector of the response system. C; A; B, and D are matrices which are the same as in (1). f ðyðtÞÞ; gðyðt sðtÞÞÞ, and hðyðtÞÞ are the activation functions of the response system neurons which are defined in the same manner with the drive system (1). The initial conditions are given by yi ðtÞ ¼ wyi ðtÞ 2 Cð½r; 0; RÞ. Rt DCyðtÞ; DAf ðyðtÞÞ, and DB trðtÞ hðyðsÞÞds are the uncertainties and dðtÞ is the disturbance. The overall disturbance is defined Rt as XðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ ¼ DCyðtÞ þ DAf ðyðtÞÞ þ DB trðtÞ hðyðsÞÞds þ dðtÞ. Define the synchronization error as eðtÞ ¼ yðtÞ xðtÞ 2 Rn . Subtracting the drive system (1) from the response system (2) yields the dynamical system
_ eðtÞ ¼ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D
Z
t
hðeðsÞÞds þ XðtÞ þ uðtÞ;
ð3Þ
trðtÞ
where f ðeðtÞÞ ¼ f ðyðtÞÞ f ðxðtÞÞ; gðeðtÞÞ ¼ gðyðt sðtÞÞÞ gðxðt sðtÞÞÞ; hðeðtÞÞ ¼ hðyðtÞÞ hðxðtÞÞ. Then, our goal is to design the controller uðtÞ which makes the error dynamical system (3) stabilized, that is,
limkeðtÞk ¼ limkyðtÞ xðtÞk ¼ 0:
t!1
t!1
ð4Þ
This means that the response system (2) is synchronized with the drive system (1). Throughout this paper, the activation functions f ðÞ; gðÞ, and hðÞ and the delay sðtÞ satisfy the following assumptions. Assumption 1. The activation functions f ðÞ; gðÞ, and hðÞ satisfy the Lipschitz condition with positive constants kfi ; kgi , and an n n constant matrix L, respectively, i.e., for i ¼ 1; 2; . . . ; n
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jfi ðaÞ fi ðbÞj 6 kfi ja bj;
jg i ðaÞ g i ðbÞj 6 kgi ja bj;
6 Lja Þ hi ðbÞj bj; jhi ða
ð5Þ
2 Rn . ; b where a; b 2 R and a Assumption 2. The time delay 0 6 s_ ðtÞ 6 l < 1.
sðtÞ is a bounded and continuously differentiable function such that 0 6 sðtÞ 6 s and
The following lemmas are essential in establishing our result. Lemma 1 [36]. Given any vector x; y of appropriate dimensions and a positive definite matrix P > 0 with compatible dimensions, then the following inequality holds,
2xT y 6 xT Px þ yT P1 y:
ð6Þ
Lemma 2 [37]. For any positive definite matrix W T ¼ W 2 Rmm , scalar h > 0, and vector function x : ½0; h ! Rm , such that the integrations concerned are well defined, the following inequality holds
Z
!T
h
xðsÞds
W
Z
0
!
h
xðsÞds 6 h
Z
0
h
xT ðsÞW xðsÞds:
ð7Þ
0
3. Adaptive synchronization using fuzzy disturbance observer In this section, we propose an adaptive synchronization method for the uncertain chaotic neural networks (1) and (2). The first step for the synchronization is how well we can overcome the overall disturbance XðtÞ. We will use the fuzzy logic system (FLS) to accomplish that [38]. First, let us briefly describe the basic configuration of the FLS used in this paper. The FLS performs a mapping from a compact set X ¼ X 1 . . . X n Rn to a compact set V R. The fuzzy rule base consists of a collection of M fuzzy If–Then rules:
RðlÞ : If x1 is Al1 ; and . . . and; xn is Aln ; Then y is Gl ;
ð8Þ
T
Ali
l
where x ¼ ½x1 ; . . . ; xn 2 X is the input of the FLS and y 2 V is its output, where and G are labels of fuzzy sets in X i and R for l ¼ 1; 2; . . . ; M. By using a product inference engine, a center-average defuzzifier, and a singleton fuzzifier, the output of the fuzzy system can be expressed as
PM
l¼1 yl
Q n
yðxÞ ¼ P Q M n l¼1
i¼1
i¼1
lAli ðxi Þ
lAli ðxi Þ
¼ hT nðxÞ;
ð9Þ
where lAl ðxi Þ is membership function value of the fuzzy variable xi ; M is the number of fuzzy rules, h ¼ ½y1 ; y2 ; . . . ; yM T is an i adjustable parameter vector, and nðxÞ ¼ ½n1 ðxÞ; n2 ðxÞ; . . . ; nM ðxÞT is a regressive vector defined as
Qn nl ðxÞ ¼
i¼1
M X Qn l¼1
lAli ðxi Þ
i¼1 lAli ðxi Þ
;
ð10Þ
which are called fuzzy basis functions (FBFs). It is well known that the fuzzy system (9) can estimate unknown function with an arbitrarily small error based on ‘universal approximation theorem’ [38]. This characteristic provides that the overall disturbance XðtÞ including uncertainties and disturbances of the neural network (2) can be estimated by the FLS. Remark 1. In the existing studies [15,27,30–32], they need some information about the uncertainties or disturbances, such as the upper bound and the structure of them. However, the information may be not available in many practical cases. The approach using the FLS can be a good way to estimate the disturbance in such cases.
Remark 2. Takagi–Sugeno (T-S) fuzzy modeling is one of main methods using FLS. Many studies have proposed novel stability criteria for time-delayed uncertain neural networks modeled by T–S fuzzy system [39,40]. In this paper, we show that the FLS can be used as an estimator of unknown factors in the system.
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
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Now, we present the FDO design procedure to obtain the estimated value of the overall disturbance XðtÞ. Consider the following observer system
^_ ðtÞ ¼ CyðtÞ þ Af ðyðtÞÞ þ Bgðyðt sðtÞÞÞ þ D y
Z
t
^ ðtÞ þ uðtÞ þ PðyðtÞ y ^ðtÞÞ; hðyðsÞÞds þ X
ð11Þ
trðtÞ
^ ðtÞ ¼ ½x ^ 1 ðtÞ; ^ðtÞ ¼ ½y ^1 ðtÞ; y ^2 ðtÞ; . . . ; y ^n ðtÞT is the state of the observer system, X where P ¼ diagðp1 ; p2 ; . . . ; pn Þ > 0; y T T M ^ 2 ðtÞ; . . . ; x ^ n ðtÞ is the estimation for XðtÞ with x ^ i ðtÞ ¼ hi ðtÞni ðyðtÞ; yðt sðtÞÞÞ for i ¼ 1; 2; . . . ; n. hi 2 R is the fuzzy paramx eter vector. ni ðyðtÞ; yðt sðtÞÞÞ 2 RM is the fuzzy basis function vector. Then, by the universal approximation theorem [38], an ^ i ðtÞ exists such that FLS x
^ i ðtÞj < ei ; jxi ðtÞ x
ð12Þ
^ ðtÞ for XðtÞ with arbiwhere ei 2 R is the upper bound of fuzzy approximation error. Hence, we can obtain the estimator X ^ i ðtÞj 6 ei 2 R. trarily small error bounds jei j ¼ jxi ðtÞ x ^ðtÞ. Then, from (2) and (11), we have the error dynamics We define the observation error as uðtÞ ¼ yðtÞ y
^ ðtÞ PðyðtÞ y _ y ^_ ðtÞ ¼ XðtÞ X ^ðtÞÞ ¼ eðtÞ PuðtÞ; u_ ðtÞ ¼ yðtÞ
ð13Þ
^ ðtÞ ¼ u _ ðtÞ þ P uðtÞ. where eðtÞ ¼ ½e1 ðtÞ; e2 ðtÞ; . . . ; en ðtÞT ¼ XðtÞ X The disturbance reconstruction error eðtÞ can be rewritten as
^ ðtÞ ¼ XðtÞ X ^ ðtÞ þ X ^ ðtÞ X ^ ðtÞ ¼ lðtÞ þ mðtÞ; eðtÞ ¼ XðtÞ X
ð14Þ
T ^ ðtÞ; lðtÞ ¼ ½l1 ðtÞ; l2 ðtÞ; . . . ; ln ðtÞ ¼ XðtÞ X
ð15Þ
^ ðtÞ X ^ ðtÞ; mðtÞ ¼ ½m1 ðtÞ; m2 ðtÞ; . . . ; mn ðtÞT ¼ X h iT ¼ ~hT1 ðtÞn1 ðyðtÞ; yðt sðtÞÞÞ; ~hT2 ðtÞn2 ðyðtÞ; yðt sðtÞÞÞ; . . . ; ~hTn ðtÞnn ðyðtÞ; yðt sðtÞÞÞ ;
ð16Þ
~hi ðtÞ ¼ h ðtÞ hi ðtÞ; i
ð17Þ
where
T
^ ðtÞ ¼ x ^ 1 ðtÞ; x ^ 2 ðtÞ; . . . ; x ^ n ðtÞ ; X
ð18Þ
^ i ðtÞ ¼ x ^ i ðyðtÞ; yðt sðtÞjhi ðtÞÞ ¼ hT x i ðtÞni ðyðtÞ; yðt sðtÞÞ;
ð19Þ
" hi ðtÞ ¼ arg min hi ðtÞ
# sup
yðtÞ;yðtsðtÞÞ
^ i ðyðtÞ; yðt sðtÞÞjhi Þ xi ðyðtÞ; yðt sðtÞÞÞj : jx
^ i ðtÞ to estimate xi ðtÞ in the following theorem. We propose an adaptation law for hi ðtÞ of the estimator x Theorem 1. Consider the chaotic neural network (2) and the observer system (11). If the adaptation law for the parameter vector ^ i ðyðtÞ; yðt sðtÞjhi ðtÞÞ is chosen as hi ðtÞ for x
h_ i ðtÞ ¼ c1 ni ðyðtÞ; yðt sðtÞÞÞðui ðtÞ þ c0 ei ðtÞÞ;
ð20Þ
^ i ðyðtÞ; yðt sðtÞjhi ðtÞÞ where c0 and c1 are positive constants, then the unknown factors xi ðtÞ are estimated by x ¼ hTi ðtÞni ðyðtÞ; yðt sðtÞÞÞ guaranteeing the following robust performance as follows: n Z X i¼1
T 0
pi u2i ðtÞdt þ
Z 0
T
ci0 m2i ðtÞdt 6
n X
u2i ð0Þ þ
i¼1
1 ~T ~ hi ð0Þhi ð0Þ þ
ci1
Z 0
T
ci0 þ
1 2 li ðtÞdt : pi
ð21Þ
Proof. Choose the following Lyapunov function candidate:
V F ðtÞ ¼
n 1 T 1 X ~hT ðtÞ~hi ðtÞ; u ðtÞuðtÞ þ 2 2c1 i¼1 i
ð22Þ
where c1 is a pre-designed positive constant. By differentiating V F ðtÞ along (13) and using the adaptive law (20), we can obtain
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S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995 n n X 1X ^ ðtÞÞ þ 1 ~hT ðtÞ~h_ i ðtÞ; ¼ uT ðtÞðPuðtÞ þ XðtÞ X ~hT ðtÞ~h_ i ðtÞ; _ ðtÞ þ V_ F ðtÞ ¼ uT ðtÞu i i
c1
c1
i¼1
i¼1
n X ^ ðtÞ þ X ^ ðtÞ X ^ ðtÞÞ þ 1 ~hT ðtÞ~h_ i ðtÞ; ¼ uT ðtÞðP uðtÞ þ XðtÞ X i
c1
^ ðtÞ X ^ ðtÞÞ ¼ uT ðtÞPuðtÞ þ uT ðtÞlðtÞ þ uT ðtÞðX
i¼1
n X
~hT ðtÞni ðyðtÞ; yðt sðtÞÞÞðu ðtÞ þ c ei ðtÞÞ; i 0 i
i¼1
¼
n h X
i pi u2i ðtÞ þ ui ðtÞli ðtÞ þ ~hTi ðtÞni ðyðtÞ; yðt sðtÞÞÞui ðtÞ ~hTi ðtÞni ðyðtÞ; yðt sðtÞÞÞðui ðtÞ þ c0 ei ðtÞÞ ;
i¼1 n X ¼ pi u2i ðtÞ þ ui ðtÞli ðtÞ c0 m2i ðtÞ c0 mi ðtÞli ðtÞ :
ð23Þ
j¼1
By applying the following inequalities
1 2
ui ðtÞli ðtÞ 6 pi u2i ðtÞ þ
1 2 1 1 2 l ðtÞ and mi ðtÞli ðtÞ 6 m2i ðtÞ þ li ðtÞ; 2pi i 2 2
ð24Þ
we can rewrite (23) as follows:
1 1 1 1 2 2 pi u2i ðtÞ c0 m2i ðtÞ þ c0 m2i ðtÞ þ ci0 li ðtÞ þ pi u2i ðtÞ þ li ðtÞ 2 2 2 2pi i¼1 n X 1 1 1 1 ¼ pi u2i ðtÞ c0 m2i ðtÞ þ c þ l2 ðtÞ : 2 2 2 0 pi i i¼1
V_ F 6
n X
ð25Þ
Integrating both sides of (25) from 0 to T yields
Z T n Z T n Z T 1X 1X 1 pi u2i ðtÞdt þ c0 m2i ðtÞdt 6 V F ð0Þ V F ðTÞ þ c0 þ l2i ðtÞdt: 2 i¼1 0 2 i¼1 0 pi 0
ð26Þ
Inequality (26) is equivalent to (21) in Theorem 1, because V F ðTÞ > 0. This completes the proof. h R1 2 Based on Barbalat’s lemma [41], the robust performance inequality (21) can be explained. If li ðtÞ 2 L2 , i.e., 0 li ðtÞdt < 1, then ui 2 L2 and mi 2 L2 . This means limt!1 kui ðtÞk ¼ 0 and limt!1 kmi ðtÞk ¼ 0. Even though li R L2 , one can say u2i ðtÞ is 2 bounded by li ðtÞ. Hence, we can reduce the observation error ui ðtÞ to an arbitrarily small value by adjusting the pre-deter^ ðtÞ can estimate XðtÞ with arbitrarily small error. mined positive weighting matrix c0 þ p1i . Therefore, we can conclude that X Remark 3. The FLS has been widely applied in much literature since the work of [38]. They have used the approach to estimate the system nonlinear function. The estimated value may cause the singularity problem because it is utilized as denominator of the controller. By using the technique to estimate the overall disturbance, the problem can be prevented. Moreover, we can design the FDO and the controller, independently. From Theorem 1, we can see that the FDO with the adaptation law (20) for the parameter vector can estimate the overall disturbance XðtÞ in (2). This means that the value can be used to compensate the overall disturbance Xi . However, fuzzy approximation error eðtÞ still exists. To eliminate the remaining error and achieve the synchronization between (1) and (2), we propose a robust adaptive controller design method in the following theorem. Theorem 2. Consider the drive system (1) and the response system (2). The systems are globally asymptotically synchronized, if a robust adaptive controller and adaptation laws are chosen as
uðtÞ ¼ K 1 eðtÞ K 2
eðtÞ ^ ðtÞ; X keðtÞk
ð27Þ
k_ 1i ðtÞ ¼ ai e2i ðtÞ;
ð28Þ
e2 ðtÞ k_ 2i ðtÞ ¼ bi i ; kei ðtÞk
ð29Þ
^ ðtÞ is the output of FDO of which the parameter vector is where K 1 ¼ diagðk11 ; k12 ; . . . ; k1n Þ > 0; K 2 ¼ diagðk21 ; k22 ; . . . ; k2n Þ > 0, X adjusted by (20), and ai ; bi are positive constants. Proof. Choose the following Lyapunov–Krasovskii function candidate:
VðtÞ ¼
n n 1 T 1X 1 ~2 1X 1 ~2 1 k1i ðtÞ þ k ðtÞ þ e ðtÞeðtÞ þ 2 2 i¼1 ai 2 i¼1 bi 2i 2ð1 lÞ
Z
t
tsðtÞ
g T ðeðrÞÞgðeðrÞÞdr þ
1 2
Z
0 r
Z
t
eT ðgÞQeðgÞdgds; tþs
ð30Þ
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~ ¼ k ðtÞ k ; k ~ ¼ k ðtÞ k with constants k ; k which will be designed and Q ¼ dr LT L with a constant d > 0. where k 1i 1i 2i 2i 1i 2i 1i 2i By differentiating V 1 along the error dynamics (3), we can obtain n X 1~ 1 ~_ ðtÞ þ ~_ ðtÞ þ ~ ðtÞk k k2i ðtÞk g T ðeðtÞÞgðeðtÞÞ 1i 1i 2i b 2ð1 lÞ a i i¼1 i¼1 i Z 1 s_ ðtÞ T 1 T 1 t T e ðtÞQeðtÞ e ðsÞQeðsÞds; g ðeðt sðtÞÞÞgðeðsðtÞÞÞ þ r 2ð1 lÞ 2 2 tr " # Z
_ _ þ VðtÞ ¼ eT ðtÞeðtÞ
n X 1
t
¼ eT ðtÞ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D
hðeðsÞÞds þ XðtÞ þ uðtÞ
trðtÞ n X 1~ 1 1 s_ ðtÞ T ~_ ðtÞ þ ~_ ðtÞ þ ~ ðtÞk k k2i ðtÞk g T ðeðtÞÞgðeðtÞÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ 1i 1i 2i b 2ð1 2ð1 lÞ a l Þ i i¼1 i¼1 i Z t 1 T 1 e ðtÞQeðtÞ eT ðsÞQeðsÞds: þ r 2 2 tr
þ
n X 1
ð31Þ
By using the control laws (27)–(29) and the definition of eðtÞ (14), we have
" Z _ VðtÞ ¼ eT ðtÞ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D
t
hðeðsÞÞds þ XðtÞ K 1 eðtÞ K 2
trðtÞ
þ
n X 1
a i¼1 i
~_ 1i ðtÞ þ ~1i ðtÞk k
1 T 1 e ðtÞQeðtÞ þ r 2 2 "
n X 1
Z
i¼1
bi
~_ 2i ðtÞ þ ~2i ðtÞk k
# eðtÞ ^ ðtÞ X keðtÞk
1 1 s_ ðtÞ T g T ðeðtÞÞgðeðtÞÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ 2ð1 lÞ 2ð1 lÞ
t
eT ðsÞQeðsÞds;
tr
# eðtÞ hðeðsÞÞds þ eðtÞ ¼ e ðtÞ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D keðtÞk trðtÞ Z 1 1 s_ ðtÞ T 1 T 1 t T e ðtÞQeðtÞ þ e ðsÞQeðsÞds; g T ðeðtÞÞgðeðtÞÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ þ r 2ð1 lÞ 2ð1 lÞ 2 2 tr Z
T
t
K 1 eðtÞ
K 2
ð32Þ
where K 1 ¼ diagðk11 ; k12 ; . . . ; k1n Þ and K 2 ¼ diagðk21 ; k22 ; . . . ; k2n Þ. Using Lemma 1, we have the following two inequalities
h iT 1 1 eT ðtÞAf ðeðtÞÞ ¼ AT eðtÞ f ðeðtÞÞ 6 eT ðtÞAAT eðtÞ þ f T ðeðtÞÞf ðeðtÞÞ; 2 2 h iT 1 T 1 T T e ðtÞBgðeðt sðtÞÞÞ ¼ B eðtÞ gðeðt sðtÞÞÞ 6 e ðtÞBBT eðtÞ þ g T ðeðt sðtÞÞÞgðeðt sðtÞÞÞ: 2 2 By the inequalities (33), (34) and
1s_ ðtÞ 1l
ð33Þ ð34Þ
P 1 derived from Assumption 2, the following inequality is obtained
1 1 1 1 _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ f T ðeðtÞÞf ðeðtÞÞ þ eT ðtÞBBT eðtÞ þ g T ðeðt sðtÞÞÞgðeðt sðtÞÞÞ 2 2 2 2 Z t eT ðtÞK 2 eðtÞ 1 T T T þ e ðtÞD hðeðsÞÞds þ e ðtÞeðtÞ e ðtÞK 1 eðtÞ þ g T ðeðtÞÞgðeðtÞÞ keðtÞk 2ð1 lÞ trðtÞ Z 1 s_ ðtÞ T 1 T 1 t T e ðtÞQeðtÞ e ðsÞQeðsÞds 6 eT ðtÞCeðtÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ þ r 2ð1 lÞ 2 2 tr Z t 1 1 1 hðeðsÞÞds þ eT ðtÞeðtÞ þ eT ðtÞAAT eðtÞ þ f T ðeðtÞÞf ðeðtÞÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞD 2 2 2 trðtÞ Z eT ðtÞK 2 eðtÞ 1 1 T 1 t T e ðtÞQeðtÞ þ e ðsÞQeðsÞds: eT ðtÞK 1 eðtÞ g T ðeðtÞÞgðeðtÞÞ þ r keðtÞk 2ð1 lÞ 2 2 tr
ð35Þ
We can obtain the following inequalities from Assumption 1
jfi ðei ðtÞÞj ¼ jfi ðyi ðtÞÞ fi ðxi ðtÞÞj 6 kfi jei ðtÞj;
jg i ðei ðtÞÞj ¼ jg i ðyi ðtÞÞ g i ðxi ðtÞÞj 6 kgi jei ðtÞj;
ð36Þ
Then, they are rewritten as
f T ðeðtÞÞf ðeðtÞÞ ¼ g T ðeðtÞÞgðeðtÞÞ ¼
n n X X fi2 ðei ðtÞÞ 6 k2fi e2i ðtÞ ¼ eT ðtÞKf eðtÞ; i¼1 n X
i¼1 n X
i¼1
i¼1
g 2i ðei ðtÞÞ 6
k2gi e2i ðtÞ ¼ eT ðtÞKg eðtÞ;
where Kf ¼ diagðk2f 1 ; k2f 2 ; . . . ; k2fn Þ and Kg ¼ diagðk2g1 ; k2g2 ; . . . ; k2gn Þ. Hence, we have
ð37Þ ð38Þ
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1 1 eT ðtÞK 2 eðtÞ _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk Z Z t 1 1 1 T 1 t T e ðtÞQeðtÞ Kg K 1 eðtÞ þ eT ðtÞD hðeðsÞÞds þ r e ðsÞQeðsÞds: þ eT ðtÞ Kf þ 2 2ð1 lÞ 2 2 tr trðtÞ
ð39Þ
Let us denote a variable U with a positive scalar d 1
U ¼ d2
Z
t
1
hðeðsÞÞds d2 DT eðtÞ 2 Rn :
ð40Þ
trðtÞ
It follows from the matrix inequality UT U P 0 as follows:
"
1
UT U ¼ d2
Z
t
#"
1
T
h ðeðsÞÞds d2 eT ðtÞD
1
Z
trðtÞ
¼d
Z
t
t
#
1
hðeðsÞÞds d2 DT eðtÞ ;
d2 trðtÞ
T
h ðeðsÞÞds
Z
trðtÞ
t
hðeðsÞÞds
Z
trðtÞ
t
T
T
h ðeðsÞÞdsD eðtÞ eT ðtÞD
Z
trðtÞ
t
hðeðsÞÞds þ d1 eT ðtÞDDT eðtÞ
trðtÞ
P 0:
ð41Þ
Thus, we can obtain the following inequality
1 2
Z
t
1 T T h ðeðsÞÞdsD eðtÞ þ eT ðtÞD 2 trðtÞ
Z
t
hðeðsÞÞds 6
trðtÞ
d 2
Z
t
T
h ðeðsÞÞ
Z
trðtÞ
t
hðeðsÞÞds þ trðtÞ
d1 T e ðtÞDDT eðtÞ: 2
ð42Þ
Using the inequality (42) yields
1 1 eT ðtÞK 2 eðtÞ _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk Z t Z t 1 1 1 1 T T Kg K 1 eðtÞ þ h ðeðsÞÞdsD eðtÞ þ eT ðtÞD hðeðsÞÞds þ eT ðtÞ Kf þ 2 2ð1 lÞ 2 trðtÞ 2 trðtÞ Z t 1 T 1 e ðtÞQeðtÞ þ r eT ðsÞQeðsÞds 2 2 tr 1 1 eT ðtÞK 2 eðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk Z t Z t 1 1 d d1 T T þ eT ðtÞ Kf þ Kg K 1 eðtÞ þ h ðeðsÞÞds hðeðsÞÞds þ e ðtÞDDT eðtÞ 2 2ð1 lÞ 2 trðtÞ 2 trðtÞ Z 1 T 1 t T e ðtÞQeðtÞ þ r e ðsÞQeðsÞds: 2 2 tr
ð43Þ
From Assumption 1 and Lemma 2, we further have
d 2
Z
t
T
h ðeðsÞÞds trðtÞ
Z
t
hðeðsÞÞds 6
trðtÞ
6
d r 2 d r 2
Z
t
T
tr
Z
d r 2 Z t
h ðeðsÞÞhðeðsÞÞds 6 t
tr
eT ðsÞLT LeðsÞds ¼
1 2
Z
t
T
h ðeðsÞÞhðeðsÞÞds
tr
eT ðsÞQeðsÞds:
ð44Þ
tr
Therefore,
1 1 eT ðtÞK 2 eðtÞ _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk 1 1 1 d 1 T e ðtÞQeðtÞ Kg K 1 eðtÞ þ eT ðtÞDDT eðtÞ þ r þ eT ðtÞ Kf þ 2 2ð1 lÞ 2 2 " # eT ðtÞ eIn K 2 eðtÞ 1 T 1 T 1 1 d1 1 T T 6 e ðtÞ C þ AA þ BB þ Kf þ Kg K 1 þ DD þ rQ eðtÞ þ ; 2 2 2 2ð1 lÞ 2 2 keðtÞk
ð45Þ
where In 2 Rnn is identity matrix and, by the universal approximation theorem, e is the upper bound of kek, i.e., kek 6 e. Taking appropriate positive parameters k1i and k2i for i ¼ 1; 2; . . . ; n such that
1 2
1 2
1 2
W ¼ C þ AAT þ BBT þ Kf þ yields the following inequality:
1 d1 1 Q < 0; N ¼ eIn K 2 < 0; Kg K 1 þ DDT þ r 2ð1 lÞ 2 2
ð46Þ
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
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Fig. 1. Chaotic behavior of the drive system.
eT ðtÞNeðtÞ _ VðtÞ 6 eT ðtÞWeðtÞ þ 6 0: keðtÞk
ð47Þ
_ From (47), it is obvious that VðtÞ 6 0 for all eðtÞ. Moreover, the positive differentiable Lyapunov function VðtÞ is radially un_ bounded and the set S ¼ feðtÞ 2 Rn jVðtÞ ¼ 0g ¼ feðtÞ 2 Rn jeðtÞ ¼ 0g contains no solutions other than the trivial solution eðtÞ ¼ 0. According to Lasalle’s invariance principle [42], one can conclude that the synchronization error eðtÞ is globally asymptotically stable, i.e. limt!1 keðtÞk ¼ 0. Therefore, this means that the response system (2) having both uncertainties and disturbances is globally asymptotically synchronized with the drive system (1) by the control law (27) and adaptation laws (28), (29). This completes the proof. h
Fig. 2. Trajectories of the drive system and response system when the FDO is not applied.
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Fig. 3. Trajectories of the drive system and response system when the proposed method is applied.
Fig. 4. Actual and estimated values of the overall disturbance XðtÞ.
4. Numerical examples In this section, a numerical example is presented to illustrate the effectiveness of our scheme proposed in the previous sections. The simulations are conducted in Simulink (MATLAB) using a fixed-step fourth order Runge–Kutta solver with sam-
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S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
Fig. 5. Synchronization error eðtÞ for two cases.
ple period T s ¼ 0:001s. We consider a two-dimensional chaotic neural network with the mixed delay as the drive system (1), which is described with
C¼
2 0
0 1
A¼
;
1:8
0:15
5:2
3:5
f ðxðtÞÞ ¼ gðxðtÞÞ ¼ hðxðtÞÞ ¼
B¼
;
tanhðx1 ðtÞÞ tanhðx2 ðtÞÞ
;
1:7
0:12
0:26
2:5
;
D¼
0:6
0:15
2
0:12
sðtÞ ¼ 0:1 sinðtÞ þ 1; rðtÞ ¼
;
I¼
0 0
;
2et : 1 þ et
The initial condition associated with the drive system is given as x1 ðsÞ ¼ 0:5; x2 ðsÞ ¼ 0:3 for all s 2 ½2; 0. Fig. 1 shows the chaotic behavior of the drive system. The response system (2) is affected by uncertainties and disturbances as follows:
DC ¼
DD ¼
0:5
0
0
0:5
;
0:3
0
0:1
0:03
DA ¼ ;
0:5
0
1
1
dðtÞ ¼
;
DB ¼
0:5
0
0
0:6
2 sinðtÞ þ 1:6y1 ðtÞ 2 cosð1:5tÞ þ 0:5y1 ðtÞy2 ðtÞ
;
:
Fig. 6. State trajectories of the drive system xðtÞ and response system yðtÞ.
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The initial condition of the response system is given as y1 ðsÞ ¼ 0:5; y2 ðsÞ ¼ 0:8 for all s 2 ½2; 0. The output of the FDO, ^ ðtÞ ¼ hT ðtÞn ðyðtÞ; yðt sðtÞÞÞ hT ðtÞn ðyðtÞ; yðt sðtÞÞÞ T , is applied to the control input (27). In order to construct the X 1 2 1 2 FDO (11), we use the input vector of the FLS as Z ¼ ½ z1 ðtÞ z2 ðtÞ z3 ðtÞ z4 ðtÞ T ¼ ½ y1 ðtÞ y2 ðtÞ y1 ðt sðtÞÞ y2 ðt sðtÞÞ T where z1 ðtÞ; z3 ðtÞ 2 ½0:8; 0:8 and z2 ðtÞ; z4 ðtÞ 2 ½5; 5. We choose three centers of the Gaussian membership function h i ll ðtÞ ¼ exp ðzl ðtÞ cml Þ2 =r2l for l ¼ 1; 2; 3; 4 where r1 ¼ r3 ¼ 0:5 and r2 ¼ r4 ¼ 3, i.e. C m ¼ ½ cm1 cm2 cm3 for ^ð0Þ ¼ 0, and m ¼ 1; 2; . . . ; 81 with uniform distance. The parameters and initial values in the FDO are c0 ¼ c1 ¼ 25; pi ¼ 5; y hð0Þ ¼ 0. Ones used in the proposed control scheme (27)–(29) are chosen as ai ¼ bi ¼ 1 and ki1 ð0Þ ¼ ki2 ð0Þ ¼ 0 for i ¼ 1; 2. We compare the simulation results to ones without the FDO to present its effectiveness. Fig. 2 shows the state trajectories of the drive and response system when the FDO is not applied. One can see that the errors between the systems still remain. On the other hand, we can remove the remaining error by using the proposed method with the FDO (Fig. 3). This is why the ^ ðtÞ estimated values for the overall disturbance XðtÞ by the FDO effectively compensate the actual one. The values XðtÞ and X are shown in Fig. 4. The synchronization error eðtÞ ¼ yðtÞ xðtÞ is presented to compare two cases in Fig. 5. Fig. 6 presents the synchronization between the chaotic neural networks. Therefore, from these results we conclude that the response system (2) is successfully synchronized with the drive system (1) by the proposed method. 5. Conclusion We have proposed a robust adaptive synchronization method for uncertain chaotic neural networks with both timevarying and distributed delays. By using the FDO, the uncertain factors including uncertainties and disturbances have been estimated without requiring any prior information about the factors. The estimated values have been used to compensate the factors. Based on Lyapunov–Krasovskii stability theory, the control scheme with adaptive laws has been derived, guaranteeing the globally asymptotical synchronization between the neural networks. An example has shown the effectiveness of the proposed method. Acknowledgements The work of J.H. Park was supported by 2012 Yeungnam University Research Grant. Park would like to thank Maureen Seo and E.K. Park for their valuable comments and supports. References [1] A. Cichoki, R. Unbehauen, Neural Networks for Optimization and Signal Processing, John Wiley and Sons, 2003. [2] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice Hall, 1998. [3] T. Kwok, K.A. Smith, A unified framework for chaotic neural-network approaches to combinatorial optimization, IEEE Trans. Neural Netw. 10 (1999) 978–981. [4] L. Wang, S. Li, F. Tian, X. Fu, A noisy chaotic neural network for solving combinatorial optimization problems: stochastic chaotic simulated annealing, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 34 (2004) 2119–2125. [5] K. Aihara, T. Takabe, M. Toyoda, Chaotic neural networks, Phys. Lett. A 144 (1990) 333–340. [6] F. Zou, J.A. Nossek, A chaotic attractor with cellular neural networks, IEEE Trans. Circuits Syst. 38 (1991) 811–812. [7] F. Zou, J.A. Nossek, Bifurcation and chaos in cellular neural networks, IEEE Trans. Circuits Syst. I: Fundam. Theor. Appl. 40 (1993) 166–173. [8] H. Lu, Chaotic attractors in delayed neural networks, Phys. Lett. A 298 (2002) 109–116. [9] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. [10] V. Milanovic, M. Zaghloul, Synchronization of chaotic neural networks and applications to communications, Int. J. Bifurcation Chaos 6 (1996) 2571– 2585. [11] Z. Tan, M. Ali, Associative memory using synchronization in a chaotic neural network, Int. J. Mod. Phys. C 12 (2001) 19–29. [12] G. Chen, J. Zhou, Z. Liu, Global synchronization of coupled delayed neural networks and applications to chaotic CNN model, Int. J. Bifurcation Chaos 14 (2004) 2229–2240. [13] M.J. Park, O.M. Kwon, Ju H. Park, S.M. Lee, E.J. Cha, Synchronization criteria for coupled neural networks with interval time-varying delays and leakage delay, Appl. Math. Comput. 218 (2012) 6762–6775. [14] S. Xu, J. Lam, D.W.C. Ho, Y. Zou, Delay-dependent exponential stability for a class of neural networks with time delays, J. Comput. Appl. Math. 183 (2005) 16–28. [15] H. Zhang, Y. Xie, Z. Wang, C. Zheng, Adaptive synchronization between two different chaotic neural networks with time delay, IEEE Trans. Neural Netw. 18 (2007) 1841–1845. [16] H. Huang, G. Feng, Synchronization of nonidentical chaotic neural networks with time delays, Neural Netw. 22 (2009) 869–874. [17] Z. Wang, L. Huang, Y. Wang, Robust decentralized adaptive control for a class of uncertain neural networks with time-varying delays, Appl. Math. Comput. 215 (2010) 4154–4163. [18] H. Xu, Y. Chen, K.L. Teo, Global exponential stability of impulsive discrete-time neural networks with time-varying delays, Appl. Math. Comput. 217 (2010) 537–544. [19] T. Ensari, S. Arik, New results for robust stability of dynamical neural networks with discrete time delays, Expert Syst. Appl. 37 (2010) 5925–5930. [20] L. Zhou, G. Hu, Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput. 195 (2008) 402–411. [21] T. Li, S.-M. Fei, Q. Zhu, S. Cong, Exponential synchronization of chaotic neural networks with mixed delays, Neurocomputing 71 (2008) 3005–3019. [22] K. Wang, Z. Teng, H. Jiang, Adaptive synchronization of neural networks with time-varying delay and distributed delay, Physica A 387 (2008) 631–642. [23] Q. Song, Design of controller on synchronization of chaotic neural networks with mixed time-varying delays, Neurocomputing 72 (2009) 3288–3295. [24] T. Li, A.-G. Song, S.-M. Fei, Y.-Q. Guo, Synchronization control of chaotic neural networks with time-varying and distributed delays, Nonlinear Anal.: Theory Methods Appl. 71 (2009) 2372–2384. [25] H. Chen, Y. Zhang, Y. Zhao, Stability analysis for uncertain neutral systems with discrete and distributed delays, Appl. Math. Comput. 218 (2012) 11351–11361.
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
5995
[26] Q. Gan, R. Xu, X. Kang, Synchronization of chaotic neural networks with mixed time delays, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 966–974. [27] Q. Gan, Y. Liang, Synchronization of non-identical unknown chaotic delayed neural networks based on adaptive sliding mode control, Neural Process. Lett. 35 (2012) 245–255. [28] X. Li, C. Ding, Q. Zhu, Synchronization of stochastic perturbed chaotic neural networks with mixed delays, J. Franklin Inst. 347 (2010) 1266–1280. [29] C.-D. Zheng, F. Zhou, Z. Wang, Stochastic exponential synchronization of jumping chaotic neural networks with mixed delays, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1273–1291. [30] K. Yuan, J. Cao, H.-X. Li, Robust stability of switched cohen-Grossberg neural networks with mixed time-varying delays, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 36 (2006) 1356–1363. [31] Y. Sun, J. Cao, Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation, Phys. Lett. A 364 (2007) 277–285. [32] H. Li, B. Chen, Q. Zhou, S. Fang, Robust exponential stability for uncertain stochastic neural networks with discrete and distributed time-varying delays, Phys. Lett. A 372 (2008) 3385–3394. [33] E. Kim, A fuzzy disturbance observer and its application to control, IEEE Trans. Fuzzy Syst. 10 (2002) 77–84. [34] E. Kim, C. Park, Fuzzy disturbance observer approach to robust tracking control of nonlinear sampled systems with the guaranteed suboptimal H1 performance, IEEE Trans. Syst. Man and Cybern. Part B: Cybern. 34 (2004) 1574–1581. [35] W. Yoo, D. Ji, S. Won, Synchronization of two different non-autonomous chaotic systems using fuzzy disturbance observer, Phys. Lett. A 374 (2010) 1354–1361. [36] J. Cao, W.C. Daniel Ho, A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach, Chaos Solitons Fract. 24 (2005) 1317–1329. [37] Z. Wang, Y. Liu, X. Liu, On global asymptotic stability of neural networks with discrete and distributed delays, Phys. Lett. A 345 (2005) 299–308. [38] L.X. Wang, A Course in Fuzzy Systems and Control, Prentice Hall PTR, 1997. [39] M. Syed Ali, Novel delay-dependent stability analysis of Takagi–Sugeno fuzzy uncertain neural networks with time varying delays, Chin. Phys. B 21 (2012) 070207. [40] M. Syed Ali, Robust stability analysis of Takagi–Sugeno uncertain stochastic fuzzy recurrent neural networks with mixed time-varying delays, Chin. Phys. B 20 (2011) 080201. [41] J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, 1991. [42] H.K. Khalil, Nonlinear Systems, Prentice Hall, 1996.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer S.C. Jeong a, D.H. Ji b, Ju H. Park c,⇑, S.C. Won a,⇑ a
Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, San 31, Hyoja-Dong, Pohang 790-784, Republic of Korea Mobile communication Division, Digital Media and Communications, Samsung Electronics, Co. Ltd., Maetan-dong, Suwon 416-2, Republic of Korea c Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University 214-1 Dae-dong, Kyongsan 712-749, Republic of Korea b
a r t i c l e
i n f o
Keywords: Adaptive synchronization Mixed time delay Chaotic neural networks Fuzzy disturbance observer
a b s t r a c t This paper proposes a robust adaptive control method for synchronization of uncertain chaotic neural networks with mixed delays. Uncertainty and disturbance in the networks are estimated by fuzzy disturbance observer without any prior information about them. The proposed control scheme with adaptive laws is derived based on Lyapunov–Krasovskii stability theory to guarantee the globally asymptotical synchronization between the networks. An example is illustrated to show the effectiveness of the proposed method. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction In the past few decades, there has been considerable attention in the study of neural networks due to their potential applications in various areas, such as signal processing pattern recognition, static image processing, associative memory and combinatorial optimization [1–4]. It has been shown that artificial neural network models can exhibit some chaotic behaviors [5–8]. Since the pioneering works of Pecora and Carroll [9], Synchronization of chaotic neural networks has been intensively investigated in many fields [10–13]. In the implementation of the neural networks, time delays between neurons in the networks often arise in the processing of information storage and transmission, which may lead to instability, oscillation, and bifurcation of the neural network model [8,14]. Many studies have been developed for the synchronization problem of delayed chaotic neural networks. Some have considered the networks with time-varying delays [15–19]. However, there exist various chaotic neural networks with both time-varying delays and distributed delays in realistic network models. Therefore, it is worth taking into account the chaotic neural networks with the mixed time delays including time-varying and distributed delays [20–29]. A control method with two sufficient conditions to ensure the globally exponential stability for the error system has been proposed based on the drive-response concept [20,21]. In [22], a synchronization problem of the networks with mixed delays has been discussed by using an adaptive feedback control technique. Sufficient conditions for asymptotical or exponential synchronization are derived in terms of Linear matrix inequalities (LMIs) by constructing proper Lyapunov–Krasovskii functional [23–25]. Sliding mode control technique is proposed to synchronize nonidentical chaotic neural networks with mixed delays [26,27]. The synchronization problems of stochastic perturbed chaotic neural networks with mixed delays have been investigated in [28,29]. It is known that the uncertainty and disturbance are unavoidable factors in many practical situations and they can destroy the network stability or can make the synchronization more difficult. Some works for uncertain neural networks have been developed to overcome their effects [15–32]. They often require some prior information of the uncertain factors, such as its structure or upper bound. However, the information may not be available due to physical limitations in practical cases. ⇑ Corresponding authors. E-mail addresses: [email protected] (S.C. Jeong), [email protected] (D.H. Ji), [email protected] (J.H. Park), [email protected] (S.C. Won). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.12.017
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
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Fuzzy logic system can be a good solution to be used in the situations because it can provide an estimator for a unknown function or value. Fuzzy disturbance observer (FDO) has been proposed to estimate uncertainty and disturbance without requiring any prior information about them [33]. The estimated values have been used to compensate the uncertain factors via state feedback controller. In [34], a robust tracking control approach using a discrete-time FDO has been proposed for nonlinear sampled systems. Recently, a more precise FDO has been constructed by modifying the law used to update the parameter vector and the modified FDO showed better performances, compared with the conventional one [35]. Even though the FDO presented good performances to overcome the unknown factor, applications of the existing research are still limited. Especially, there has been still no research using the technique for uncertain chaotic neural networks with mixed time delays. In this paper, we propose a robust adaptive synchronization method for uncertain chaotic neural networks with timevarying delays and distributed delays. The uncertain factors including uncertainties and disturbances are estimated by the FDO without requiring any prior knowledge about the factors. The estimated values are used to compensate the factors in the proposed method. Based on Lyapunov–Krasovskii stability theory, the control scheme with adaptive laws is derived and guarantees the globally asymptotical synchronization between the networks. An example is illustrated to show the effectiveness of the proposed method. 2. Problem statement Consider the following chaotic neural network with time-varying delay and distributed delay:
_ xðtÞ ¼ CxðtÞ þ Af ðxðtÞÞ þ Bgðxðt sðtÞÞÞ þ D
Z
t
hðxðsÞÞds þ I;
ð1Þ
trðtÞ
where xðtÞ ¼ ½x1 ðtÞ; . . . ; xn ðtÞT 2 Rn is the neuron state vector and C ¼ diagðc1 ; c2 ; . . . ; cn Þ is a positive diagonal matrix. A ¼ ðaij Þnn ; B ¼ ðbij Þnn , and D ¼ ðdij Þnn are the connection weight matrix, the time varying delayed connection weight matrix and distributively delayed connection weight matrix, respectively. I ¼ ½I1 ; I2 ; . . . ; In T 2 Rn is an external input vector, sðtÞ is the distributed time-delay. The initial conditions are given by is the time-varying delay, the positive constant 0 6 rðtÞ 6 r xi ðtÞ ¼ wxi ðtÞ 2 Cð½r; 0; RÞ, where Cð½r; 0; RÞ denotes the set of all continuous functions from ½r; 0 to R and r ¼ maxfsðtÞ; rðtÞg. f ðxðtÞÞ; gðxðt sðtÞÞÞ, and hðxðtÞÞ are the activation functions of the neurons and described as
f ðxðtÞÞ ¼ ½f1 ðx1 ðtÞÞ; f2 ðx2 ðtÞÞ; . . . ; fn ðxn ðtÞÞT ; gðxðt sðtÞÞÞ ¼ ½g 1 ðx1 ðt sðtÞÞÞ; g 2 ðx2 ðt sðtÞÞÞ; . . . ; g n ðxn ðt sðtÞÞÞT ; hðxðtÞÞ ¼ ½h1 ðx1 ðtÞÞ; h2 ðx2 ðtÞÞ; . . . ; hn ðxn ðtÞÞT : We consider the network (1) as the drive system. The response system having the uncertainty and disturbance is established as follows:
_ yðtÞ ¼ ðC þ DCÞyðtÞ þ ðA þ DAÞf ðyðtÞÞ þ ðB þ DBÞgðyðt sðtÞÞÞ þ ðD þ DDÞ ¼ CyðtÞ þ Af ðyðtÞÞ þ Bgðyðt sðtÞÞÞ þ D
Z
Z
t
hðyðsÞÞds þ I þ dðtÞ þ uðtÞ
trðtÞ t
hðyðsÞÞ þ XðtÞ þ I þ uðtÞ;
ð2Þ
trðtÞ
where yðtÞ ¼ ½y1 ðtÞ; y2 ðtÞ; . . . ; yn ðtÞT 2 Rn is the neuron state vector of the response system. C; A; B, and D are matrices which are the same as in (1). f ðyðtÞÞ; gðyðt sðtÞÞÞ, and hðyðtÞÞ are the activation functions of the response system neurons which are defined in the same manner with the drive system (1). The initial conditions are given by yi ðtÞ ¼ wyi ðtÞ 2 Cð½r; 0; RÞ. Rt DCyðtÞ; DAf ðyðtÞÞ, and DB trðtÞ hðyðsÞÞds are the uncertainties and dðtÞ is the disturbance. The overall disturbance is defined Rt as XðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞ ¼ DCyðtÞ þ DAf ðyðtÞÞ þ DB trðtÞ hðyðsÞÞds þ dðtÞ. Define the synchronization error as eðtÞ ¼ yðtÞ xðtÞ 2 Rn . Subtracting the drive system (1) from the response system (2) yields the dynamical system
_ eðtÞ ¼ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D
Z
t
hðeðsÞÞds þ XðtÞ þ uðtÞ;
ð3Þ
trðtÞ
where f ðeðtÞÞ ¼ f ðyðtÞÞ f ðxðtÞÞ; gðeðtÞÞ ¼ gðyðt sðtÞÞÞ gðxðt sðtÞÞÞ; hðeðtÞÞ ¼ hðyðtÞÞ hðxðtÞÞ. Then, our goal is to design the controller uðtÞ which makes the error dynamical system (3) stabilized, that is,
limkeðtÞk ¼ limkyðtÞ xðtÞk ¼ 0:
t!1
t!1
ð4Þ
This means that the response system (2) is synchronized with the drive system (1). Throughout this paper, the activation functions f ðÞ; gðÞ, and hðÞ and the delay sðtÞ satisfy the following assumptions. Assumption 1. The activation functions f ðÞ; gðÞ, and hðÞ satisfy the Lipschitz condition with positive constants kfi ; kgi , and an n n constant matrix L, respectively, i.e., for i ¼ 1; 2; . . . ; n
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jfi ðaÞ fi ðbÞj 6 kfi ja bj;
jg i ðaÞ g i ðbÞj 6 kgi ja bj;
6 Lja Þ hi ðbÞj bj; jhi ða
ð5Þ
2 Rn . ; b where a; b 2 R and a Assumption 2. The time delay 0 6 s_ ðtÞ 6 l < 1.
sðtÞ is a bounded and continuously differentiable function such that 0 6 sðtÞ 6 s and
The following lemmas are essential in establishing our result. Lemma 1 [36]. Given any vector x; y of appropriate dimensions and a positive definite matrix P > 0 with compatible dimensions, then the following inequality holds,
2xT y 6 xT Px þ yT P1 y:
ð6Þ
Lemma 2 [37]. For any positive definite matrix W T ¼ W 2 Rmm , scalar h > 0, and vector function x : ½0; h ! Rm , such that the integrations concerned are well defined, the following inequality holds
Z
!T
h
xðsÞds
W
Z
0
!
h
xðsÞds 6 h
Z
0
h
xT ðsÞW xðsÞds:
ð7Þ
0
3. Adaptive synchronization using fuzzy disturbance observer In this section, we propose an adaptive synchronization method for the uncertain chaotic neural networks (1) and (2). The first step for the synchronization is how well we can overcome the overall disturbance XðtÞ. We will use the fuzzy logic system (FLS) to accomplish that [38]. First, let us briefly describe the basic configuration of the FLS used in this paper. The FLS performs a mapping from a compact set X ¼ X 1 . . . X n Rn to a compact set V R. The fuzzy rule base consists of a collection of M fuzzy If–Then rules:
RðlÞ : If x1 is Al1 ; and . . . and; xn is Aln ; Then y is Gl ;
ð8Þ
T
Ali
l
where x ¼ ½x1 ; . . . ; xn 2 X is the input of the FLS and y 2 V is its output, where and G are labels of fuzzy sets in X i and R for l ¼ 1; 2; . . . ; M. By using a product inference engine, a center-average defuzzifier, and a singleton fuzzifier, the output of the fuzzy system can be expressed as
PM
l¼1 yl
Q n
yðxÞ ¼ P Q M n l¼1
i¼1
i¼1
lAli ðxi Þ
lAli ðxi Þ
¼ hT nðxÞ;
ð9Þ
where lAl ðxi Þ is membership function value of the fuzzy variable xi ; M is the number of fuzzy rules, h ¼ ½y1 ; y2 ; . . . ; yM T is an i adjustable parameter vector, and nðxÞ ¼ ½n1 ðxÞ; n2 ðxÞ; . . . ; nM ðxÞT is a regressive vector defined as
Qn nl ðxÞ ¼
i¼1
M X Qn l¼1
lAli ðxi Þ
i¼1 lAli ðxi Þ
;
ð10Þ
which are called fuzzy basis functions (FBFs). It is well known that the fuzzy system (9) can estimate unknown function with an arbitrarily small error based on ‘universal approximation theorem’ [38]. This characteristic provides that the overall disturbance XðtÞ including uncertainties and disturbances of the neural network (2) can be estimated by the FLS. Remark 1. In the existing studies [15,27,30–32], they need some information about the uncertainties or disturbances, such as the upper bound and the structure of them. However, the information may be not available in many practical cases. The approach using the FLS can be a good way to estimate the disturbance in such cases.
Remark 2. Takagi–Sugeno (T-S) fuzzy modeling is one of main methods using FLS. Many studies have proposed novel stability criteria for time-delayed uncertain neural networks modeled by T–S fuzzy system [39,40]. In this paper, we show that the FLS can be used as an estimator of unknown factors in the system.
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
5987
Now, we present the FDO design procedure to obtain the estimated value of the overall disturbance XðtÞ. Consider the following observer system
^_ ðtÞ ¼ CyðtÞ þ Af ðyðtÞÞ þ Bgðyðt sðtÞÞÞ þ D y
Z
t
^ ðtÞ þ uðtÞ þ PðyðtÞ y ^ðtÞÞ; hðyðsÞÞds þ X
ð11Þ
trðtÞ
^ ðtÞ ¼ ½x ^ 1 ðtÞ; ^ðtÞ ¼ ½y ^1 ðtÞ; y ^2 ðtÞ; . . . ; y ^n ðtÞT is the state of the observer system, X where P ¼ diagðp1 ; p2 ; . . . ; pn Þ > 0; y T T M ^ 2 ðtÞ; . . . ; x ^ n ðtÞ is the estimation for XðtÞ with x ^ i ðtÞ ¼ hi ðtÞni ðyðtÞ; yðt sðtÞÞÞ for i ¼ 1; 2; . . . ; n. hi 2 R is the fuzzy paramx eter vector. ni ðyðtÞ; yðt sðtÞÞÞ 2 RM is the fuzzy basis function vector. Then, by the universal approximation theorem [38], an ^ i ðtÞ exists such that FLS x
^ i ðtÞj < ei ; jxi ðtÞ x
ð12Þ
^ ðtÞ for XðtÞ with arbiwhere ei 2 R is the upper bound of fuzzy approximation error. Hence, we can obtain the estimator X ^ i ðtÞj 6 ei 2 R. trarily small error bounds jei j ¼ jxi ðtÞ x ^ðtÞ. Then, from (2) and (11), we have the error dynamics We define the observation error as uðtÞ ¼ yðtÞ y
^ ðtÞ PðyðtÞ y _ y ^_ ðtÞ ¼ XðtÞ X ^ðtÞÞ ¼ eðtÞ PuðtÞ; u_ ðtÞ ¼ yðtÞ
ð13Þ
^ ðtÞ ¼ u _ ðtÞ þ P uðtÞ. where eðtÞ ¼ ½e1 ðtÞ; e2 ðtÞ; . . . ; en ðtÞT ¼ XðtÞ X The disturbance reconstruction error eðtÞ can be rewritten as
^ ðtÞ ¼ XðtÞ X ^ ðtÞ þ X ^ ðtÞ X ^ ðtÞ ¼ lðtÞ þ mðtÞ; eðtÞ ¼ XðtÞ X
ð14Þ
T ^ ðtÞ; lðtÞ ¼ ½l1 ðtÞ; l2 ðtÞ; . . . ; ln ðtÞ ¼ XðtÞ X
ð15Þ
^ ðtÞ X ^ ðtÞ; mðtÞ ¼ ½m1 ðtÞ; m2 ðtÞ; . . . ; mn ðtÞT ¼ X h iT ¼ ~hT1 ðtÞn1 ðyðtÞ; yðt sðtÞÞÞ; ~hT2 ðtÞn2 ðyðtÞ; yðt sðtÞÞÞ; . . . ; ~hTn ðtÞnn ðyðtÞ; yðt sðtÞÞÞ ;
ð16Þ
~hi ðtÞ ¼ h ðtÞ hi ðtÞ; i
ð17Þ
where
T
^ ðtÞ ¼ x ^ 1 ðtÞ; x ^ 2 ðtÞ; . . . ; x ^ n ðtÞ ; X
ð18Þ
^ i ðtÞ ¼ x ^ i ðyðtÞ; yðt sðtÞjhi ðtÞÞ ¼ hT x i ðtÞni ðyðtÞ; yðt sðtÞÞ;
ð19Þ
" hi ðtÞ ¼ arg min hi ðtÞ
# sup
yðtÞ;yðtsðtÞÞ
^ i ðyðtÞ; yðt sðtÞÞjhi Þ xi ðyðtÞ; yðt sðtÞÞÞj : jx
^ i ðtÞ to estimate xi ðtÞ in the following theorem. We propose an adaptation law for hi ðtÞ of the estimator x Theorem 1. Consider the chaotic neural network (2) and the observer system (11). If the adaptation law for the parameter vector ^ i ðyðtÞ; yðt sðtÞjhi ðtÞÞ is chosen as hi ðtÞ for x
h_ i ðtÞ ¼ c1 ni ðyðtÞ; yðt sðtÞÞÞðui ðtÞ þ c0 ei ðtÞÞ;
ð20Þ
^ i ðyðtÞ; yðt sðtÞjhi ðtÞÞ where c0 and c1 are positive constants, then the unknown factors xi ðtÞ are estimated by x ¼ hTi ðtÞni ðyðtÞ; yðt sðtÞÞÞ guaranteeing the following robust performance as follows: n Z X i¼1
T 0
pi u2i ðtÞdt þ
Z 0
T
ci0 m2i ðtÞdt 6
n X
u2i ð0Þ þ
i¼1
1 ~T ~ hi ð0Þhi ð0Þ þ
ci1
Z 0
T
ci0 þ
1 2 li ðtÞdt : pi
ð21Þ
Proof. Choose the following Lyapunov function candidate:
V F ðtÞ ¼
n 1 T 1 X ~hT ðtÞ~hi ðtÞ; u ðtÞuðtÞ þ 2 2c1 i¼1 i
ð22Þ
where c1 is a pre-designed positive constant. By differentiating V F ðtÞ along (13) and using the adaptive law (20), we can obtain
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S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995 n n X 1X ^ ðtÞÞ þ 1 ~hT ðtÞ~h_ i ðtÞ; ¼ uT ðtÞðPuðtÞ þ XðtÞ X ~hT ðtÞ~h_ i ðtÞ; _ ðtÞ þ V_ F ðtÞ ¼ uT ðtÞu i i
c1
c1
i¼1
i¼1
n X ^ ðtÞ þ X ^ ðtÞ X ^ ðtÞÞ þ 1 ~hT ðtÞ~h_ i ðtÞ; ¼ uT ðtÞðP uðtÞ þ XðtÞ X i
c1
^ ðtÞ X ^ ðtÞÞ ¼ uT ðtÞPuðtÞ þ uT ðtÞlðtÞ þ uT ðtÞðX
i¼1
n X
~hT ðtÞni ðyðtÞ; yðt sðtÞÞÞðu ðtÞ þ c ei ðtÞÞ; i 0 i
i¼1
¼
n h X
i pi u2i ðtÞ þ ui ðtÞli ðtÞ þ ~hTi ðtÞni ðyðtÞ; yðt sðtÞÞÞui ðtÞ ~hTi ðtÞni ðyðtÞ; yðt sðtÞÞÞðui ðtÞ þ c0 ei ðtÞÞ ;
i¼1 n X ¼ pi u2i ðtÞ þ ui ðtÞli ðtÞ c0 m2i ðtÞ c0 mi ðtÞli ðtÞ :
ð23Þ
j¼1
By applying the following inequalities
1 2
ui ðtÞli ðtÞ 6 pi u2i ðtÞ þ
1 2 1 1 2 l ðtÞ and mi ðtÞli ðtÞ 6 m2i ðtÞ þ li ðtÞ; 2pi i 2 2
ð24Þ
we can rewrite (23) as follows:
1 1 1 1 2 2 pi u2i ðtÞ c0 m2i ðtÞ þ c0 m2i ðtÞ þ ci0 li ðtÞ þ pi u2i ðtÞ þ li ðtÞ 2 2 2 2pi i¼1 n X 1 1 1 1 ¼ pi u2i ðtÞ c0 m2i ðtÞ þ c þ l2 ðtÞ : 2 2 2 0 pi i i¼1
V_ F 6
n X
ð25Þ
Integrating both sides of (25) from 0 to T yields
Z T n Z T n Z T 1X 1X 1 pi u2i ðtÞdt þ c0 m2i ðtÞdt 6 V F ð0Þ V F ðTÞ þ c0 þ l2i ðtÞdt: 2 i¼1 0 2 i¼1 0 pi 0
ð26Þ
Inequality (26) is equivalent to (21) in Theorem 1, because V F ðTÞ > 0. This completes the proof. h R1 2 Based on Barbalat’s lemma [41], the robust performance inequality (21) can be explained. If li ðtÞ 2 L2 , i.e., 0 li ðtÞdt < 1, then ui 2 L2 and mi 2 L2 . This means limt!1 kui ðtÞk ¼ 0 and limt!1 kmi ðtÞk ¼ 0. Even though li R L2 , one can say u2i ðtÞ is 2 bounded by li ðtÞ. Hence, we can reduce the observation error ui ðtÞ to an arbitrarily small value by adjusting the pre-deter^ ðtÞ can estimate XðtÞ with arbitrarily small error. mined positive weighting matrix c0 þ p1i . Therefore, we can conclude that X Remark 3. The FLS has been widely applied in much literature since the work of [38]. They have used the approach to estimate the system nonlinear function. The estimated value may cause the singularity problem because it is utilized as denominator of the controller. By using the technique to estimate the overall disturbance, the problem can be prevented. Moreover, we can design the FDO and the controller, independently. From Theorem 1, we can see that the FDO with the adaptation law (20) for the parameter vector can estimate the overall disturbance XðtÞ in (2). This means that the value can be used to compensate the overall disturbance Xi . However, fuzzy approximation error eðtÞ still exists. To eliminate the remaining error and achieve the synchronization between (1) and (2), we propose a robust adaptive controller design method in the following theorem. Theorem 2. Consider the drive system (1) and the response system (2). The systems are globally asymptotically synchronized, if a robust adaptive controller and adaptation laws are chosen as
uðtÞ ¼ K 1 eðtÞ K 2
eðtÞ ^ ðtÞ; X keðtÞk
ð27Þ
k_ 1i ðtÞ ¼ ai e2i ðtÞ;
ð28Þ
e2 ðtÞ k_ 2i ðtÞ ¼ bi i ; kei ðtÞk
ð29Þ
^ ðtÞ is the output of FDO of which the parameter vector is where K 1 ¼ diagðk11 ; k12 ; . . . ; k1n Þ > 0; K 2 ¼ diagðk21 ; k22 ; . . . ; k2n Þ > 0, X adjusted by (20), and ai ; bi are positive constants. Proof. Choose the following Lyapunov–Krasovskii function candidate:
VðtÞ ¼
n n 1 T 1X 1 ~2 1X 1 ~2 1 k1i ðtÞ þ k ðtÞ þ e ðtÞeðtÞ þ 2 2 i¼1 ai 2 i¼1 bi 2i 2ð1 lÞ
Z
t
tsðtÞ
g T ðeðrÞÞgðeðrÞÞdr þ
1 2
Z
0 r
Z
t
eT ðgÞQeðgÞdgds; tþs
ð30Þ
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~ ¼ k ðtÞ k ; k ~ ¼ k ðtÞ k with constants k ; k which will be designed and Q ¼ dr LT L with a constant d > 0. where k 1i 1i 2i 2i 1i 2i 1i 2i By differentiating V 1 along the error dynamics (3), we can obtain n X 1~ 1 ~_ ðtÞ þ ~_ ðtÞ þ ~ ðtÞk k k2i ðtÞk g T ðeðtÞÞgðeðtÞÞ 1i 1i 2i b 2ð1 lÞ a i i¼1 i¼1 i Z 1 s_ ðtÞ T 1 T 1 t T e ðtÞQeðtÞ e ðsÞQeðsÞds; g ðeðt sðtÞÞÞgðeðsðtÞÞÞ þ r 2ð1 lÞ 2 2 tr " # Z
_ _ þ VðtÞ ¼ eT ðtÞeðtÞ
n X 1
t
¼ eT ðtÞ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D
hðeðsÞÞds þ XðtÞ þ uðtÞ
trðtÞ n X 1~ 1 1 s_ ðtÞ T ~_ ðtÞ þ ~_ ðtÞ þ ~ ðtÞk k k2i ðtÞk g T ðeðtÞÞgðeðtÞÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ 1i 1i 2i b 2ð1 2ð1 lÞ a l Þ i i¼1 i¼1 i Z t 1 T 1 e ðtÞQeðtÞ eT ðsÞQeðsÞds: þ r 2 2 tr
þ
n X 1
ð31Þ
By using the control laws (27)–(29) and the definition of eðtÞ (14), we have
" Z _ VðtÞ ¼ eT ðtÞ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D
t
hðeðsÞÞds þ XðtÞ K 1 eðtÞ K 2
trðtÞ
þ
n X 1
a i¼1 i
~_ 1i ðtÞ þ ~1i ðtÞk k
1 T 1 e ðtÞQeðtÞ þ r 2 2 "
n X 1
Z
i¼1
bi
~_ 2i ðtÞ þ ~2i ðtÞk k
# eðtÞ ^ ðtÞ X keðtÞk
1 1 s_ ðtÞ T g T ðeðtÞÞgðeðtÞÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ 2ð1 lÞ 2ð1 lÞ
t
eT ðsÞQeðsÞds;
tr
# eðtÞ hðeðsÞÞds þ eðtÞ ¼ e ðtÞ CeðtÞ þ Af ðeðtÞÞ þ Bgðeðt sðtÞÞÞ þ D keðtÞk trðtÞ Z 1 1 s_ ðtÞ T 1 T 1 t T e ðtÞQeðtÞ þ e ðsÞQeðsÞds; g T ðeðtÞÞgðeðtÞÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ þ r 2ð1 lÞ 2ð1 lÞ 2 2 tr Z
T
t
K 1 eðtÞ
K 2
ð32Þ
where K 1 ¼ diagðk11 ; k12 ; . . . ; k1n Þ and K 2 ¼ diagðk21 ; k22 ; . . . ; k2n Þ. Using Lemma 1, we have the following two inequalities
h iT 1 1 eT ðtÞAf ðeðtÞÞ ¼ AT eðtÞ f ðeðtÞÞ 6 eT ðtÞAAT eðtÞ þ f T ðeðtÞÞf ðeðtÞÞ; 2 2 h iT 1 T 1 T T e ðtÞBgðeðt sðtÞÞÞ ¼ B eðtÞ gðeðt sðtÞÞÞ 6 e ðtÞBBT eðtÞ þ g T ðeðt sðtÞÞÞgðeðt sðtÞÞÞ: 2 2 By the inequalities (33), (34) and
1s_ ðtÞ 1l
ð33Þ ð34Þ
P 1 derived from Assumption 2, the following inequality is obtained
1 1 1 1 _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ f T ðeðtÞÞf ðeðtÞÞ þ eT ðtÞBBT eðtÞ þ g T ðeðt sðtÞÞÞgðeðt sðtÞÞÞ 2 2 2 2 Z t eT ðtÞK 2 eðtÞ 1 T T T þ e ðtÞD hðeðsÞÞds þ e ðtÞeðtÞ e ðtÞK 1 eðtÞ þ g T ðeðtÞÞgðeðtÞÞ keðtÞk 2ð1 lÞ trðtÞ Z 1 s_ ðtÞ T 1 T 1 t T e ðtÞQeðtÞ e ðsÞQeðsÞds 6 eT ðtÞCeðtÞ g ðeðt sðtÞÞÞgðeðsðtÞÞÞ þ r 2ð1 lÞ 2 2 tr Z t 1 1 1 hðeðsÞÞds þ eT ðtÞeðtÞ þ eT ðtÞAAT eðtÞ þ f T ðeðtÞÞf ðeðtÞÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞD 2 2 2 trðtÞ Z eT ðtÞK 2 eðtÞ 1 1 T 1 t T e ðtÞQeðtÞ þ e ðsÞQeðsÞds: eT ðtÞK 1 eðtÞ g T ðeðtÞÞgðeðtÞÞ þ r keðtÞk 2ð1 lÞ 2 2 tr
ð35Þ
We can obtain the following inequalities from Assumption 1
jfi ðei ðtÞÞj ¼ jfi ðyi ðtÞÞ fi ðxi ðtÞÞj 6 kfi jei ðtÞj;
jg i ðei ðtÞÞj ¼ jg i ðyi ðtÞÞ g i ðxi ðtÞÞj 6 kgi jei ðtÞj;
ð36Þ
Then, they are rewritten as
f T ðeðtÞÞf ðeðtÞÞ ¼ g T ðeðtÞÞgðeðtÞÞ ¼
n n X X fi2 ðei ðtÞÞ 6 k2fi e2i ðtÞ ¼ eT ðtÞKf eðtÞ; i¼1 n X
i¼1 n X
i¼1
i¼1
g 2i ðei ðtÞÞ 6
k2gi e2i ðtÞ ¼ eT ðtÞKg eðtÞ;
where Kf ¼ diagðk2f 1 ; k2f 2 ; . . . ; k2fn Þ and Kg ¼ diagðk2g1 ; k2g2 ; . . . ; k2gn Þ. Hence, we have
ð37Þ ð38Þ
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1 1 eT ðtÞK 2 eðtÞ _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk Z Z t 1 1 1 T 1 t T e ðtÞQeðtÞ Kg K 1 eðtÞ þ eT ðtÞD hðeðsÞÞds þ r e ðsÞQeðsÞds: þ eT ðtÞ Kf þ 2 2ð1 lÞ 2 2 tr trðtÞ
ð39Þ
Let us denote a variable U with a positive scalar d 1
U ¼ d2
Z
t
1
hðeðsÞÞds d2 DT eðtÞ 2 Rn :
ð40Þ
trðtÞ
It follows from the matrix inequality UT U P 0 as follows:
"
1
UT U ¼ d2
Z
t
#"
1
T
h ðeðsÞÞds d2 eT ðtÞD
1
Z
trðtÞ
¼d
Z
t
t
#
1
hðeðsÞÞds d2 DT eðtÞ ;
d2 trðtÞ
T
h ðeðsÞÞds
Z
trðtÞ
t
hðeðsÞÞds
Z
trðtÞ
t
T
T
h ðeðsÞÞdsD eðtÞ eT ðtÞD
Z
trðtÞ
t
hðeðsÞÞds þ d1 eT ðtÞDDT eðtÞ
trðtÞ
P 0:
ð41Þ
Thus, we can obtain the following inequality
1 2
Z
t
1 T T h ðeðsÞÞdsD eðtÞ þ eT ðtÞD 2 trðtÞ
Z
t
hðeðsÞÞds 6
trðtÞ
d 2
Z
t
T
h ðeðsÞÞ
Z
trðtÞ
t
hðeðsÞÞds þ trðtÞ
d1 T e ðtÞDDT eðtÞ: 2
ð42Þ
Using the inequality (42) yields
1 1 eT ðtÞK 2 eðtÞ _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk Z t Z t 1 1 1 1 T T Kg K 1 eðtÞ þ h ðeðsÞÞdsD eðtÞ þ eT ðtÞD hðeðsÞÞds þ eT ðtÞ Kf þ 2 2ð1 lÞ 2 trðtÞ 2 trðtÞ Z t 1 T 1 e ðtÞQeðtÞ þ r eT ðsÞQeðsÞds 2 2 tr 1 1 eT ðtÞK 2 eðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk Z t Z t 1 1 d d1 T T þ eT ðtÞ Kf þ Kg K 1 eðtÞ þ h ðeðsÞÞds hðeðsÞÞds þ e ðtÞDDT eðtÞ 2 2ð1 lÞ 2 trðtÞ 2 trðtÞ Z 1 T 1 t T e ðtÞQeðtÞ þ r e ðsÞQeðsÞds: 2 2 tr
ð43Þ
From Assumption 1 and Lemma 2, we further have
d 2
Z
t
T
h ðeðsÞÞds trðtÞ
Z
t
hðeðsÞÞds 6
trðtÞ
6
d r 2 d r 2
Z
t
T
tr
Z
d r 2 Z t
h ðeðsÞÞhðeðsÞÞds 6 t
tr
eT ðsÞLT LeðsÞds ¼
1 2
Z
t
T
h ðeðsÞÞhðeðsÞÞds
tr
eT ðsÞQeðsÞds:
ð44Þ
tr
Therefore,
1 1 eT ðtÞK 2 eðtÞ _ VðtÞ 6 eT ðtÞCeðtÞ þ eT ðtÞAAT eðtÞ þ eT ðtÞBBT eðtÞ þ eT ðtÞeðtÞ 2 2 keðtÞk 1 1 1 d 1 T e ðtÞQeðtÞ Kg K 1 eðtÞ þ eT ðtÞDDT eðtÞ þ r þ eT ðtÞ Kf þ 2 2ð1 lÞ 2 2 " # eT ðtÞ eIn K 2 eðtÞ 1 T 1 T 1 1 d1 1 T T 6 e ðtÞ C þ AA þ BB þ Kf þ Kg K 1 þ DD þ rQ eðtÞ þ ; 2 2 2 2ð1 lÞ 2 2 keðtÞk
ð45Þ
where In 2 Rnn is identity matrix and, by the universal approximation theorem, e is the upper bound of kek, i.e., kek 6 e. Taking appropriate positive parameters k1i and k2i for i ¼ 1; 2; . . . ; n such that
1 2
1 2
1 2
W ¼ C þ AAT þ BBT þ Kf þ yields the following inequality:
1 d1 1 Q < 0; N ¼ eIn K 2 < 0; Kg K 1 þ DDT þ r 2ð1 lÞ 2 2
ð46Þ
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
5991
Fig. 1. Chaotic behavior of the drive system.
eT ðtÞNeðtÞ _ VðtÞ 6 eT ðtÞWeðtÞ þ 6 0: keðtÞk
ð47Þ
_ From (47), it is obvious that VðtÞ 6 0 for all eðtÞ. Moreover, the positive differentiable Lyapunov function VðtÞ is radially un_ bounded and the set S ¼ feðtÞ 2 Rn jVðtÞ ¼ 0g ¼ feðtÞ 2 Rn jeðtÞ ¼ 0g contains no solutions other than the trivial solution eðtÞ ¼ 0. According to Lasalle’s invariance principle [42], one can conclude that the synchronization error eðtÞ is globally asymptotically stable, i.e. limt!1 keðtÞk ¼ 0. Therefore, this means that the response system (2) having both uncertainties and disturbances is globally asymptotically synchronized with the drive system (1) by the control law (27) and adaptation laws (28), (29). This completes the proof. h
Fig. 2. Trajectories of the drive system and response system when the FDO is not applied.
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Fig. 3. Trajectories of the drive system and response system when the proposed method is applied.
Fig. 4. Actual and estimated values of the overall disturbance XðtÞ.
4. Numerical examples In this section, a numerical example is presented to illustrate the effectiveness of our scheme proposed in the previous sections. The simulations are conducted in Simulink (MATLAB) using a fixed-step fourth order Runge–Kutta solver with sam-
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Fig. 5. Synchronization error eðtÞ for two cases.
ple period T s ¼ 0:001s. We consider a two-dimensional chaotic neural network with the mixed delay as the drive system (1), which is described with
C¼
2 0
0 1
A¼
;
1:8
0:15
5:2
3:5
f ðxðtÞÞ ¼ gðxðtÞÞ ¼ hðxðtÞÞ ¼
B¼
;
tanhðx1 ðtÞÞ tanhðx2 ðtÞÞ
;
1:7
0:12
0:26
2:5
;
D¼
0:6
0:15
2
0:12
sðtÞ ¼ 0:1 sinðtÞ þ 1; rðtÞ ¼
;
I¼
0 0
;
2et : 1 þ et
The initial condition associated with the drive system is given as x1 ðsÞ ¼ 0:5; x2 ðsÞ ¼ 0:3 for all s 2 ½2; 0. Fig. 1 shows the chaotic behavior of the drive system. The response system (2) is affected by uncertainties and disturbances as follows:
DC ¼
DD ¼
0:5
0
0
0:5
;
0:3
0
0:1
0:03
DA ¼ ;
0:5
0
1
1
dðtÞ ¼
;
DB ¼
0:5
0
0
0:6
2 sinðtÞ þ 1:6y1 ðtÞ 2 cosð1:5tÞ þ 0:5y1 ðtÞy2 ðtÞ
;
:
Fig. 6. State trajectories of the drive system xðtÞ and response system yðtÞ.
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The initial condition of the response system is given as y1 ðsÞ ¼ 0:5; y2 ðsÞ ¼ 0:8 for all s 2 ½2; 0. The output of the FDO, ^ ðtÞ ¼ hT ðtÞn ðyðtÞ; yðt sðtÞÞÞ hT ðtÞn ðyðtÞ; yðt sðtÞÞÞ T , is applied to the control input (27). In order to construct the X 1 2 1 2 FDO (11), we use the input vector of the FLS as Z ¼ ½ z1 ðtÞ z2 ðtÞ z3 ðtÞ z4 ðtÞ T ¼ ½ y1 ðtÞ y2 ðtÞ y1 ðt sðtÞÞ y2 ðt sðtÞÞ T where z1 ðtÞ; z3 ðtÞ 2 ½0:8; 0:8 and z2 ðtÞ; z4 ðtÞ 2 ½5; 5. We choose three centers of the Gaussian membership function h i ll ðtÞ ¼ exp ðzl ðtÞ cml Þ2 =r2l for l ¼ 1; 2; 3; 4 where r1 ¼ r3 ¼ 0:5 and r2 ¼ r4 ¼ 3, i.e. C m ¼ ½ cm1 cm2 cm3 for ^ð0Þ ¼ 0, and m ¼ 1; 2; . . . ; 81 with uniform distance. The parameters and initial values in the FDO are c0 ¼ c1 ¼ 25; pi ¼ 5; y hð0Þ ¼ 0. Ones used in the proposed control scheme (27)–(29) are chosen as ai ¼ bi ¼ 1 and ki1 ð0Þ ¼ ki2 ð0Þ ¼ 0 for i ¼ 1; 2. We compare the simulation results to ones without the FDO to present its effectiveness. Fig. 2 shows the state trajectories of the drive and response system when the FDO is not applied. One can see that the errors between the systems still remain. On the other hand, we can remove the remaining error by using the proposed method with the FDO (Fig. 3). This is why the ^ ðtÞ estimated values for the overall disturbance XðtÞ by the FDO effectively compensate the actual one. The values XðtÞ and X are shown in Fig. 4. The synchronization error eðtÞ ¼ yðtÞ xðtÞ is presented to compare two cases in Fig. 5. Fig. 6 presents the synchronization between the chaotic neural networks. Therefore, from these results we conclude that the response system (2) is successfully synchronized with the drive system (1) by the proposed method. 5. Conclusion We have proposed a robust adaptive synchronization method for uncertain chaotic neural networks with both timevarying and distributed delays. By using the FDO, the uncertain factors including uncertainties and disturbances have been estimated without requiring any prior information about the factors. The estimated values have been used to compensate the factors. Based on Lyapunov–Krasovskii stability theory, the control scheme with adaptive laws has been derived, guaranteeing the globally asymptotical synchronization between the neural networks. An example has shown the effectiveness of the proposed method. Acknowledgements The work of J.H. Park was supported by 2012 Yeungnam University Research Grant. Park would like to thank Maureen Seo and E.K. Park for their valuable comments and supports. References [1] A. Cichoki, R. Unbehauen, Neural Networks for Optimization and Signal Processing, John Wiley and Sons, 2003. [2] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice Hall, 1998. [3] T. Kwok, K.A. Smith, A unified framework for chaotic neural-network approaches to combinatorial optimization, IEEE Trans. Neural Netw. 10 (1999) 978–981. [4] L. Wang, S. Li, F. Tian, X. Fu, A noisy chaotic neural network for solving combinatorial optimization problems: stochastic chaotic simulated annealing, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 34 (2004) 2119–2125. [5] K. Aihara, T. Takabe, M. Toyoda, Chaotic neural networks, Phys. Lett. A 144 (1990) 333–340. [6] F. Zou, J.A. Nossek, A chaotic attractor with cellular neural networks, IEEE Trans. Circuits Syst. 38 (1991) 811–812. [7] F. Zou, J.A. Nossek, Bifurcation and chaos in cellular neural networks, IEEE Trans. Circuits Syst. I: Fundam. Theor. Appl. 40 (1993) 166–173. [8] H. Lu, Chaotic attractors in delayed neural networks, Phys. Lett. A 298 (2002) 109–116. [9] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. [10] V. Milanovic, M. Zaghloul, Synchronization of chaotic neural networks and applications to communications, Int. J. Bifurcation Chaos 6 (1996) 2571– 2585. [11] Z. Tan, M. Ali, Associative memory using synchronization in a chaotic neural network, Int. J. Mod. Phys. C 12 (2001) 19–29. [12] G. Chen, J. Zhou, Z. Liu, Global synchronization of coupled delayed neural networks and applications to chaotic CNN model, Int. J. Bifurcation Chaos 14 (2004) 2229–2240. [13] M.J. Park, O.M. Kwon, Ju H. Park, S.M. Lee, E.J. Cha, Synchronization criteria for coupled neural networks with interval time-varying delays and leakage delay, Appl. Math. Comput. 218 (2012) 6762–6775. [14] S. Xu, J. Lam, D.W.C. Ho, Y. Zou, Delay-dependent exponential stability for a class of neural networks with time delays, J. Comput. Appl. Math. 183 (2005) 16–28. [15] H. Zhang, Y. Xie, Z. Wang, C. Zheng, Adaptive synchronization between two different chaotic neural networks with time delay, IEEE Trans. Neural Netw. 18 (2007) 1841–1845. [16] H. Huang, G. Feng, Synchronization of nonidentical chaotic neural networks with time delays, Neural Netw. 22 (2009) 869–874. [17] Z. Wang, L. Huang, Y. Wang, Robust decentralized adaptive control for a class of uncertain neural networks with time-varying delays, Appl. Math. Comput. 215 (2010) 4154–4163. [18] H. Xu, Y. Chen, K.L. Teo, Global exponential stability of impulsive discrete-time neural networks with time-varying delays, Appl. Math. Comput. 217 (2010) 537–544. [19] T. Ensari, S. Arik, New results for robust stability of dynamical neural networks with discrete time delays, Expert Syst. Appl. 37 (2010) 5925–5930. [20] L. Zhou, G. Hu, Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput. 195 (2008) 402–411. [21] T. Li, S.-M. Fei, Q. Zhu, S. Cong, Exponential synchronization of chaotic neural networks with mixed delays, Neurocomputing 71 (2008) 3005–3019. [22] K. Wang, Z. Teng, H. Jiang, Adaptive synchronization of neural networks with time-varying delay and distributed delay, Physica A 387 (2008) 631–642. [23] Q. Song, Design of controller on synchronization of chaotic neural networks with mixed time-varying delays, Neurocomputing 72 (2009) 3288–3295. [24] T. Li, A.-G. Song, S.-M. Fei, Y.-Q. Guo, Synchronization control of chaotic neural networks with time-varying and distributed delays, Nonlinear Anal.: Theory Methods Appl. 71 (2009) 2372–2384. [25] H. Chen, Y. Zhang, Y. Zhao, Stability analysis for uncertain neutral systems with discrete and distributed delays, Appl. Math. Comput. 218 (2012) 11351–11361.
S.C. Jeong et al. / Applied Mathematics and Computation 219 (2013) 5984–5995
5995
[26] Q. Gan, R. Xu, X. Kang, Synchronization of chaotic neural networks with mixed time delays, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 966–974. [27] Q. Gan, Y. Liang, Synchronization of non-identical unknown chaotic delayed neural networks based on adaptive sliding mode control, Neural Process. Lett. 35 (2012) 245–255. [28] X. Li, C. Ding, Q. Zhu, Synchronization of stochastic perturbed chaotic neural networks with mixed delays, J. Franklin Inst. 347 (2010) 1266–1280. [29] C.-D. Zheng, F. Zhou, Z. Wang, Stochastic exponential synchronization of jumping chaotic neural networks with mixed delays, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 1273–1291. [30] K. Yuan, J. Cao, H.-X. Li, Robust stability of switched cohen-Grossberg neural networks with mixed time-varying delays, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 36 (2006) 1356–1363. [31] Y. Sun, J. Cao, Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation, Phys. Lett. A 364 (2007) 277–285. [32] H. Li, B. Chen, Q. Zhou, S. Fang, Robust exponential stability for uncertain stochastic neural networks with discrete and distributed time-varying delays, Phys. Lett. A 372 (2008) 3385–3394. [33] E. Kim, A fuzzy disturbance observer and its application to control, IEEE Trans. Fuzzy Syst. 10 (2002) 77–84. [34] E. Kim, C. Park, Fuzzy disturbance observer approach to robust tracking control of nonlinear sampled systems with the guaranteed suboptimal H1 performance, IEEE Trans. Syst. Man and Cybern. Part B: Cybern. 34 (2004) 1574–1581. [35] W. Yoo, D. Ji, S. Won, Synchronization of two different non-autonomous chaotic systems using fuzzy disturbance observer, Phys. Lett. A 374 (2010) 1354–1361. [36] J. Cao, W.C. Daniel Ho, A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach, Chaos Solitons Fract. 24 (2005) 1317–1329. [37] Z. Wang, Y. Liu, X. Liu, On global asymptotic stability of neural networks with discrete and distributed delays, Phys. Lett. A 345 (2005) 299–308. [38] L.X. Wang, A Course in Fuzzy Systems and Control, Prentice Hall PTR, 1997. [39] M. Syed Ali, Novel delay-dependent stability analysis of Takagi–Sugeno fuzzy uncertain neural networks with time varying delays, Chin. Phys. B 21 (2012) 070207. [40] M. Syed Ali, Robust stability analysis of Takagi–Sugeno uncertain stochastic fuzzy recurrent neural networks with mixed time-varying delays, Chin. Phys. B 20 (2011) 080201. [41] J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice Hall, 1991. [42] H.K. Khalil, Nonlinear Systems, Prentice Hall, 1996.
