Neural Networks 48 (2013) 195–203
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Global exponential synchronization of memristor-based recurrent neural networks with time-varying delays✩ Shiping Wen a,b,c , Gang Bao d , Zhigang Zeng a,b,∗ , Yiran Chen e , Tingwen Huang c a
School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China
b
Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China
c
Texas A & M University at Qatar, Doha 5825, Qatar
d
School of Electrical, Three Geoges University, Yichang, 443002, China
e
Department of Electronical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, United States
article
info
Article history: Received 14 May 2013 Received in revised form 2 September 2013 Accepted 6 October 2013 Keywords: Memristor Synchronization Recurrent neural networks Time-varying delays
abstract This paper deals with the problem of global exponential synchronization of a class of memristor-based recurrent neural networks with time-varying delays based on the fuzzy theory and Lyapunov method. First, a memristor-based recurrent neural network is designed. Then, considering the state-dependent properties of the memristor, a new fuzzy model employing parallel distributed compensation (PDC) gives a new way to analyze the complicated memristor-based neural networks with only two subsystems. Comparisons between results in this paper and in the previous ones have been made. They show that the results in this paper improve and generalized the results derived in the previous literature. An example is also given to illustrate the effectiveness of the results. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The sequential processing of fetch, decode, and execution of instructions through the classical von Neumann bottleneck of conventional digital computers has resulted in less efficient machines as their eco-systems have grown to be increasingly complex (Jo et al., 2010). Though the current digital computers can now possess the computing speed and complexity to emulate the brain functionality of animals like a spider, mouse, and cat (Ananthanarayanan, Eser, Simon, & Modha, 2009; Smith, 2006), the associated energy dissipation in the system grows exponentially along the hierarchy of animal intelligence. For example, to perform certain cortical simulations at the cat scale even at 83 times slower firing rate, the IBM team in Ananthanarayanan et al. (2009) has to employ Blue Gene/P (BG/P), a super computer equipped with 147456 CPUs and 144 TBs of main memory. On the other hand, the human brain contains more than 100 billion neurons and each
✩ This work was supported by the Natural Science Foundation of China under Grant 61125303, National Basic Research Program of China (973 Program) under Grant 2011CB710606, Research Fund for the Doctoral Program of Higher Education of China under Grant 20100142110021, the Excellent Youth Foundation of Hubei Province of China under Grant 2010CDA081, National Priority Research Project NPRP 4-451-2-168, funded by Qatar National Research Fund. ∗ Corresponding author at: School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China. Tel.: +86 18971124190; fax: +86 27 87543130. E-mail addresses: [email protected] (Z. Zeng), [email protected] (Y. Chen).
0893-6080/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.neunet.2013.10.001
neuron has more than 20000 synapses. Efficient circuit implementation of synapses, therefore, is very important to build a brainlike machine. However, since shrinking the current transistor size is very difficult, introducing a more efficient approach is essential for further development of neural network implementations. In 2008, the Williams group announced a successful fabrication of a very compact and non-volatile nano scale memory called the memristor (Strukov, Snider, Stewart, & Williams, 2008). It was postulated by Chua (1971) as the fourth basic circuit elements in electrical circuits. It is based on the nonlinear characteristics of charge and flux. By supplying a voltage or current to the memristor, its resistance can be altered (Sharifiy & Banadaki, 2010). In this way, the memristor remembers information. Several examples of successful multichip networks of spiking neurons have been recently proposed (Choi, Shi, & Boahen, 2004; Indiveri, 2001; Liu & Douglas, 2004); however there are still a number of practical problems that hinder the development of truly large-scale, distributed, massively parallel networks of very large scale integration (VLSI) neurons, such as how to set the weight of individual synapses in the network. It is well-known that changes in the synaptic connections between neurons are widely believed to contribute to memory storage, and the activity-dependent development of neural networks. These changes are thought to occur through correlatedbased, or Hebbian plasticity. In addition, we notice recurrent neural networks have been widely studied in recent years, for their immense application prospective (Bao, Wen, & Zeng, 2012; Cao, Huang, & Qu, 2005; Cao & Wang, 2005; Cao, Yuan, & Li, 2006; Chen, Cao, & Huang, 2002;
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S. Wen et al. / Neural Networks 48 (2013) 195–203
Fig. 1. Circuit of memristor-based recurrent network, where xi (.) is the state of the i-th subsystem, fj (.) is the amplifier, Rfij is the connection resistor between the amplifier fj (.) and state xi (.), Mi and Ci are the memristor and capacitor, Ii is the external input, ai , bi are the outputs, i, j = 1, 2, . . . , n.
Forti & Nistri, 2003; Hu & Wang, 2002; Huang & Cao, 2003; Li, Fei, & Zhu, 2009; Rakkiyappan, Balasubramaniam, & Cao, 2010; Shen & Wang, 2008; Zeng & Wang, 2006; Zeng, Wang, & Liao, 2005; Zhao & Zhu, 2010). Many applications have been developed in different areas such as combinatorial optimization, knowledge acquisition and pattern recognition. Recently, the problem of synchronization of coupled neural networks which is one of the hot research fields of complex networks has been a challenging issue due to its potential applications such as in information science, biological systems and so on (Balasubramaniam, Kalpana, & Rakkiyappan, 2012; Cao, Chen, & Li, 2008; Park, Kwon, Park, Lee, & Cha, 2012; Xing & Peng, 2012; Zhang, Wang, Li, & Fei, 2012). However, to the best of the authors’ knowledge, the research on global exponential synchronization of memristor-based recurrent neural networks is still an open problem that deserves further investigation. To shorten the gap, we investigate the problem of global exponential synchronization for a class of memristor-based recurrent neural networks with time-varying delays. The memristor-based recurrent network can be implemented by very large scale integration(VLSI) circuits as shown in Fig. 1. fj is the activation function, τj (t ) is the time-varying delay, for the i-th subsystem, xi (t ) is the voltage of the capacitor Ci , fj (xj (t )), fj (xj (t − τj (t ))) are the functions about xi (t ) with and without time-varying delays respectively, Rfij is the resistor between the feedback function fj (xj (t )) and xi (t ), Rgij is the resistor between the feedback function fj (xj (t − τj (t ))) and xi (t ), Mi is the memristor parallel to the capacitor Ci , Ii is an external input or bias, where i, j = 1, 2, . . . , n. The memductance of the memristors can be depicted as in Fig. 2 (Gergel-Hackett et al., 2009), which are bounded. Thus by Kirchoff’s current law, the equation of the i-th subsystem is written as the
following:
Ci x˙ i (t ) = −
n 1 Rfij
j =1
+
+ Wi (xi (t )) xi (t )
Rgij
n signij fj (xj (t ))
Rfij
j =1
+
+
1
n signij fj (xj (t − τj (t )))
Rgij
j =1
where signij =
1, i ̸= j; −1, i = j,
+ Ii ,
(1)
and Wi are the memductances of the
memristors Mi , and Wi (xi (t )) =
Wi′ , xi (t ) ≤ 0; Wi′′ , xi (t ) > 0.
Therefore x˙ i (t ) = −di (xi (t ))xi (t ) +
n
aij fj (xj (t ))
j=1
+
n
bij fj (xj (t − τj (t ))) + si ,
(2)
j=1
where signij aij = , Ci Rfij
bij =
di (xi (t )) =
Ci
=
n 1 j =1
signij Ci Rgij 1 Rfij
+
d1i , xi (t ) ≤ 0; d2i , xi (t ) > 0,
,
1 Rgij
+ Wi (xi (t ))
si =
Ii Ci
.
S. Wen et al. / Neural Networks 48 (2013) 195–203
197
Then we can get x˙ (t ) = −D(x(t ))x(t ) + Af (x(t )) + Bf (x(t − τ (t ))) + s,
(3)
where D(x(t )) = diag{d1 (x1 (t )), d2 (x2 (t )), . . . , dn (xn (t ))}, A = [aij ]n×n ,
B = [bij ]n×n ,
s = (s1 , s2 , . . . , sn )T , T f (x(t )) = f1 (x1 (t )), . . . , fn (xn (t )) , T f (x(t − τ (t ))) = f1 (x1 (t − τ1 (t ))), . . . , fn (t − τn (t )) .
Remark 1. With the property of non-volatile, for each subsystem, only one memristor may be employed to store multiple states of the input information. Therefore, the synapses of the neural networks need not to be designed with a memristor, which will obviously reduce the cost to produce the memristor-based neural networks in future. Based on the analysis above, D(x(t )) in system (3) is changed according to the state of the system, so this network based on memristors is a state-dependent switching system. System (3) represents a class of memristor-based recurrent neural networks with time-varying delays. Taking advantage of the properties of state-dependence, a fuzzy method can be employed to investigate this system. To solve the problem about nonlinear control, fuzzy logic has attracted much attention as a powerful tool. Among various kinds of fuzzy methods, the Takagi–Sugeno fuzzy systems are widely accepted as a useful tool for design and analysis of fuzzy control system (Chiu & Chiang, 2009; Dong, Wang, & Yang, 2009; Liu & Zhong, 2007; Park & Cho, 2004; Takagi & Sugeno, 1985). Currently, some control methods for memristive systems have been proposed (Wen, Zeng, & Huang, 2012a), in which, the number of the linear subsystem is decided by how many minimum nonlinear terms should be linearized in original system. Then, the memristor-based neural network (2) can be exactly represented by the fuzzy model as follows: Rule 1: IF xi (t ) is N1i , THEN x˙ i (t ) = −d1i xi (t ) +
n
n
aij fj (xj (t ))
bij fj (xj (t − τj (t ))) + si ,
(4)
j =1
Rule 2: IF xi (t ) is N2i , THEN x˙ i (t ) = −d2i xi (t ) +
n
aij fj (xj (t ))
j=1
+
n
bij fj (xj (t − τj (t ))) + si ,
(5)
j =1
where N1i is xi (t ) ≤ 0, N2i is xi (t ) > 0. With a center-average defuzzier, the over fuzzy system is represented as x˙ i (t ) = −
2
ηli (t )dli xi (t ) +
l=1
+
n
in the T–S fuzzy system. If n is large, the number of linear subsystems in the T–S fuzzy system is huge. For this problem, Li and Ge proposed a fuzzy modeling method and applied in the synchronization problem of two totally different chaotic systems (Li & Ge, 2011). Based on this work, a new fuzzy model is proposed to linearize memristive systems, in which only two subsystems are included. Furthermore, through this model, the idea of PDC can be applied to achieve between subsystems. Therefore, system (6) can be represented by x˙ (t ) = −
2
where Ψl (t ) = diag{ηl1 (t ), . . . , ηln (t )}, and 1, . . . , n, l = 1, 2, and
n
aij fj (xj (t ))
j=1
bij fj (xj (t − τj (t ))) + si ,
(6)
1, xi (t ) ≤ 0, 0, xi (t ) > 0,
2
l =1
ηli (t ) = 1, i =
Denote u = (u1 , . . . , un )T , |u| as the absolute-value vector; i.e., |u| = (|u1 |, |u2 |, . . . , |un |)T , ∥x∥p as the p-norm of the vector x with p, 1 ≤ p < ∞. ∥x∥∞ = maxi∈{1,2,...,n} |xi | is the vector infinity norm. Denote ∥D∥p as the p-norm of the matrix D with p. Denote C as the set of continuous functions. The following assumptions will be needed throughout the paper: A1. For i ∈ {1, 2, . . . , n}, the activation function fi is Lipschitz continuous; and ∀r1 , r2 ∈ R, there exists real number ιi such that fi (r1 ) − fi (r2 ) 0≤ ≤ ιi . r1 − r2 A2. For i ∈ {1, 2, . . . , n}, τi (t ) satisfies 0 ≤ τi (t ) ≤ τ¯i ,
η2i (t ) =
0, xi (t ) ≤ 0, 1, xi (t ) > 0.
When the system becomes complicated with n memristors, there are 2n subsystems (according to 2n fuzzy rules) and 2n equations
τ˙i (t ) ≤ µi .
(8)
In this paper, we consider system (7) as the master system, and through electronic inductors, the values of memristor will be presented in the corresponding slave system, then the slave system is given as: y˙ (t ) = −
where
η1i (t ) =
(7)
Dl = diag{dl1 , dl2 , . . . , dln }.
j =1
Ψl (t )Dl x(t ) + Af (x(t )) + Bf (x(t − τ (t ))) + s,
l =1
2. Preliminaries
j=1
+
Fig. 2. Typical i–v characteristic of memristor (Gergel-Hackett et al., 2009). The pinched hysteresis loop is due to the nonlinear relationship between the memristance current and voltage. The memristor exhibits the feature of pinched hysteresis, which means that a lag occurs between the application and the removal of a field and its subsequent effect, just as with the neurons in the human brain.
2
Ψl (t )Dl y(t ) + Af (y(t ))
l =1
+ Bf (y(t − τ (t ))) + s + u(t ),
t ≥ 0,
(9)
where y(t ) = (y1 (t ), y2 (t ), . . . , yn (t )) , u(t ) = (u1 (t ), u2 (t ), . . . , un (t ))T is the control input that will be designed. The initial conT
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S. Wen et al. / Neural Networks 48 (2013) 195–203
ditions of system (7) are in the form of y(t ) = φ(t ) ∈ C ([−τ¯ , 0], Rn ), τ¯ = max1≤i≤n {τ¯i }. In order to derive sufficient conditions for the global exponential synchronization of system (7) with system (9), we will need the following lemmas. Lemma 1 (Zhao & Tan, 2007). It is given any real matrices X , Z , P of appropriate dimensions and a scalar ε0 > 0, where P > 0. Then the following inequality holds: X T Z + Z T X ≤ ε0 X T PX + ε0−1 Z T P −1 Z . In particular, if X and Z are vectors, X T Z ≤
1 2
(X T X + Z T Z ).
3. Main results As synchronization has been applied in many real applications, this paper aims to design a memoryless state-feedback controller as 2
u(t ) =
Ψl (t )Kl e(t ),
(10)
l =1
where e(t ) = (e1 (t ), e2 (t ), . . . , en (t ))T , is the synchronization error, and ei (t ) = xi (t ) − yi (t ), Kl = (klij )n×n is a constant gain matrix to be determined to synchronize the drive and response systems. With controller (10), the error system is transformed into e˙ (t ) = −
2
Ψl (t )Dˆ l e(t ) + AΩ (e(t )) + BΩ (e(t − τ (t ))),
(11)
where the scalar α > 0. Then, V˙ (t ) = α V˙ 1 (t ) + V˙ 2 (t ), where V˙ 1 (t ) = −
2
Ψl (t )eT (t )Dˆ l e(t ) + eT (t )AΩ (e(t ))
l =1 T
+ e (t )BΩ (e(t − τ (t ))), n n V˙ 2 (t ) = 2 qi ΩiT (ei (t ))˙ei (t ) + ri ΩiT (ei (t ))Ωi (ei (t )) i=1 i=1 T − (1 − τ˙ (t ))Ωi (ei (t − τ (t )))Ωi (ei (t − τ (t ))) = 2Ω T (e(t ))Q e˙ (t ) + Ω T (e(t ))RΩ (e(t )) − (1 − τ˙ (t ))Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))) 2 = −2 Ψl (t )Ω T (e(t ))Q Dˆ l e(t ) + 2Ω T (e(t ))QAΩ (e(t )) l =1
+ 2Ω T (e(t ))QBΩ (e(t − τ (t ))) + Ω T (e(t ))RΩ (e(t )) − (1 − τ˙ (t ))Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))). As V˙ 1 (t ) can be rewritten as V˙ 1 (t ) = −
2
Ψl (t )eT (t )Dˆ l e(t )
l =1
+ eT ( t )
2 l =1
Ψl (t )Dˆ l √
1/2 −1/2 2 √ Ψl (t )Dˆ l 2 l =1
2
l =1
where
Ω (e(t )) = (Ω1 (e1 (t )), Ω2 (e2 (t )), . . . , Ωn (en (t ))) = f (e(t ) + y(t )) − f (y(t )), Ω (e(t − τ (t ))) = (Ω1 (e1 (t − τ1 (t ))), Ω2 (e2 (t − τ2 (t ))), . . . , Ωn (en (t − τn (t ))))T = f (e(t − τ (t )) + y(t − τ (t ))) − f (y(t − τ (t ))).
√ ×
V˙ 1 (t ) ≤ −
×
˜ l = λI + Kl , λ = minl=1,2 {λmin {Dl }}, I is the identity mawhere D trix, z = (zij )n×n with zii = −2qi aii , zij = −(qi aij + qj aji ), for i ̸= j, then there exists a positive definite diagonal matrix R = ˆ l e(t ) where diag(r1 , r2 , . . . , rn ), such that V˙ (t )|(11) ≤ − α2 eT (t )D α 2
+
eT (t )e(t ) + 2
i =1 n i=1
t −τi (t )
i=1
Ωi2 t −τi (t )
(t )e(t ), V2 (t ) = 2
(ei (s))ri ds. Then
V (t ) = α V1 (t ) + V2 (t ),
BΩ (e(t − τ (t ))).
2 1
2 l =1 2
Ψl (t )eT Dˆ l e(t ) + Ω T (e(t ))AT
1 T T Ψl (t )Dˆ − l AΩ (e(t )) + Ω (e(t − τ (t )))B
×
2
1 Ψl (t )Dˆ − l BΩ (e(t − τ (t )))
l =1
≤−
2 1
2 l =1
Ψl (t )eT (t )Dˆ l e(t ) +
× ∥A∥22 Ω T (e(t ))Ω (e(t )) +
Ωi (s)ds
2
1 Ψl (t )∥Dˆ − l ∥
l =1 2
1 Ψl (t )∥Dˆ − l ∥
l =1
Ωi2 (ei (s))ri ds.
1 T e 2
Ψl (t )Dˆ l
0
t
Proof. Let V1 (t ) =
n t
qi
−1/2
l =1
Πl = −2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I < 0,
V (t ) =
2
2
By Lemma 1,
Lemma 2. For the synchronization error system (11), if there exist a positive number α and positive definite diagonal matrices L = diag {ι1 , ι2 , . . . , ιn }, Q = diag(q1 , q2 , . . . , qn ), and
ei (t )
1/2
l =1
In order to gain the main results, the following lemma is introduced.
2
Ψl (t )Dˆ l T l =1 × AΩ (e(t )) + e (t ) √ 2
ˆ l = Kl + D l , D
n
(12)
n
i=1 qi
ei (t ) 0
× ∥B∥22 Ω T (e(t − τ (t )))Ω (e(t − τ (t ))) 2 2 1 1 ≤− Ψl (t )eT (t )Dˆ l e(t ) + Ψl (t )∥Dˆ − l ∥ 2 l =1
Ωi (s)ds +
× ∥A∥22 Ω T (e(t ))Ω (e(t )) +
l =1 2
1 Ψl (t )∥Dˆ − l ∥
l =1
(13)
× ∥B∥22 Ω T (e(t − τ (t )))Ω (e(t − τ (t )))
S. Wen et al. / Neural Networks 48 (2013) 195–203
≤−
2 1
2 l =1
Furthermore, we have the following theorem.
Ψl (t )eT (t )D˜ l e(t ) + M Ω T (e(t ))Ω (e(t ))
+ M Ω T (e(t − τ (t )))Ω (e(t − τ (t ))), 2 2 −1 −1 2 2 ˜ ˜ where M = max Ψ ( t )∥ D ∥ ∥ A ∥ , Ψ ( t )∥ D ∥ ∥ B ∥ l l l l 2 2 l =1 l =1 ≥ 0. 1 2 As ei (t )Ωi (ei (t )) ≥ ι− i (Ωi (ei (t ))) , we have 2
−2Ω T (e(t ))Q
˜ l }}. Obviously, V (t ) which is deProof. Let ρ = min1≤l≤2 {λmin {D fined in Lemma 2 is a positive definite and radially unbounded Lyapunov functional. A positive number ϵ > 0 is chosen to satisfy
d dt
{eϵ t V (T )} = eϵ t (ϵ V (t ) + V˙ (t ))
Ψl (t )Dˆ l L−1 Ω (e(t )).
2
≤
l =1
ϵt
Ψl (t )e
Therefore V˙ 2 (t ) ≤ −2Ω T (e(t ))Q
+2
Ψl (t )Dˆ l L−1 Ω (e(t ))
+ 2Ω (e(t ))QAΩ (e(t )) + 2Ω (e(t ))QB × Ω (e(t − τ (t ))) + Ω T (e(t ))RΩ (e(t )) − (1 − µ)Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))) 2 T −1 ˜ ≤ −Ω (e(t )) 2Q Ψl (t )Dl L Ω (e(t )) T
−
− Ω (e(t ))zΩ (e(t )) + 2∥Q ∥ ∥B∥2 ∥Ω (e(t ))∥ × ∥Ω (e(t − τ (t )))∥ + Ω T (e(t ))RΩ (e(t )) T
− (1 − µ)Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))).
2
+ 4ϵ
Ψl (t )e
n
+ ϵ eϵ t
≤ ∥Q ∥∥B∥2 Ω T (e(t ))Ω (e(t )) + Ω T (e(t − τ (t )))Ω (e(t − τ (t ))) .
(15)
Since i=1 qi we can get d
From (14) and (15),
0
dt
(e V (t )) ≤
2 l =1
1−µ
t −τi (t )
Ψl (t )e
+ ϵ eϵ t
s
e 0
α=
β
,
H > 0,
2H 1, H = 0.
Then V˙ (t ) ≤ − α2
T ˜ l=1 Ψl (t )e (t )Dl e(t ).
ei (t ) 0
ιi sds ≤
1 T e 2
(t )LQe(t ),
ϵt
ϵt
ϵα eT (t )e(t ) − α eT (t )D˜ l e(t )
n
t t −τi (t )
Ωi2
(ei (s))ri ds
n
=ϵ
ϵt
n
t t −τi (t )
Ωi2 (ei (s))ri ds.
n
≤ϵ
n i=1
t t −τi (t )
i=1
i =1
2
qi
(17)
Estimating the second term on the right-hand side of (17) by changing the integrals, we can get
ϵ
By choosing
i=1
1 ≤ eϵ t ϵα − ρα + 2ϵ∥LQ ∥ eT (t )e(t ) 2
i =1
V˙ 2 (t ) ≤ −β Ω T (e(t ))Ω (t ) − β Ω T (e(t − τ (t )))Ω (e(t − τ (t ))).
Ωi2 (ei (s))ri ds.
n
i=1
∥Q ∥∥B∥2 + β I ,
which is a positive definite diagonal matrix, we can obtain
t
+ 2ϵ e (t )LQe(t ) + ϵ e
As Πl < 0, there exists β > 0 such that Πl + 2β I < 0. Define 1
Ωi (s)ds
0
n
T
× Ω (e(t − τ (t ))).
qi
˜ −1 − z + R + ∥Q ∥∥B∥2 Ω (e(t )) V˙ 2 (t ) ≤ Ω T (e(t )) −2Q DL − Ω T (e(t − τ (t ))) (1 − µ)R − ∥Q ∥∥B∥2
ϵα eT (t )e(t ) − α eT (t )D˜ l e(t ) ei (t )
Ωi (s)ds ≤
2 1
ϵt
ϵt
i =1
ei (t )
n
(ei (s))ri ds
i=1
2∥Q ∥ ∥B∥2 ∥Ω T (e(t ))∥ ∥Ω (e(t − τ (t )))∥
Ωi2
˜ l e(t ) e (t )D
Furthermore, we can obtain
t
2 l =1
(14)
Ωi (s)ds
T
2 1
≤
eT (t )e(t )
0
t −τi (t )
i =1
α
ei (t )
α 2
qi
n
+
l =1 T
n i =1
l =1 T
R=
ϵ
l =1
2
(16)
By Lemma 2, we can get
l=1
≤ −2Ω T (e(t ))Q
Theorem 1. Assume the conditions in Lemma 2 hold, then the drive system (7) is globally exponentially synchronized with the response system (9).
ϵα − ρα + 2ϵ∥LQ ∥ + 2ϵ τ¯ eϵ τ¯ ∥L2 R∥ < 0.
Ψl (t )Dˆ l e(t ) 2
199
s
eϵ t
Ωi2 (ei (ς ))ri dς dt t
t −τi (t )
0 s
−τ¯
Ωi2 (ei (ς ))ri dς dt
min{ς+τ¯ ,s} max{ς ,0}
eϵ t dt Ωi2 (ei (ς ))ri dς
200
S. Wen et al. / Neural Networks 48 (2013) 195–203
≤ϵ
n
s
≤ ϵ τ¯ eϵ τ¯ ∥L2 R∥ ≤ ϵ τ¯ eϵ τ¯ ∥L2 R∥
s
0
eϵς eT (ς )e(ς )dς
−τ¯
Proof. By Lemma 1, 1
ϵς T
e e (ς )e(ς )dς .
+
(18)
From (16)–(18), 1
ϵα − ρα + 2ϵ∥LQ ∥ + 2ϵ τ¯ eϵ τ¯ ∥L2 R∥ 2 s × eϵ t eT (t )e(t )dt + ϵ τ¯ eϵ τ¯ ∥L2 R∥
0
×
− 2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I ≤ −2Q D˜ l L−1 − z + N + ∥Q ∥2 ∥N −1 ∥ ∥B∥22 I < 0. (25)
−τ¯
This proof is completed.
eϵ t eT (t )e(t )dt
Let Q = N = I in Corollary 1, we obtain the following corollary.
0
≤ ϵ τ¯ ∥L2 R∥
eϵ t dt ∥ψ∥2
−τ¯
Corollary 2. The synchronization error system (11) is globally exponentially stable if z = (zij )n×n is positive definite, and ∥B∥2 ≤
≡ H1 ∥ψ∥2 .
√
V (t ) ≤ V (0) + H1 ∥ψ∥2 e−ϵ t ,
ρ 2π − 1, where π = min1≤i≤n ι , and zii = −2qi aii , zij = i −(qi aij + qj aji ), for i ̸= j.
Thus,
∀t > 0,
(19) 4. Comparison and example
where
α 2
+
eT (0)e(0) + 2
n
ei (0)
qi
Ωi (s)ds
0
i =1 n i=1
0
−τi (t )
Ωi2 (ei (s))ri ds
1 2
β + 2∥QL∥ + 2τ¯ ∥L2 R∥ ∥ψ∥2
≡ H2 ∥ψ∥2 . It follows from (12) and (19) that
2
(23)
Then
0
α
N + ∥Q ∥ ∥B∥2 N −1 ≥ 2I ,
N + ∥Q ∥2 ∥N −1 ∥ ∥B∥22 I ≥ N + ∥Q ∥2 ∥B∥22 N −1 ≥ 2∥Q ∥ ∥B∥22 I . (24)
eϵ s V (s) − V (0) ≤
≤
∥Q ∥ ∥B∥2
which implies that
0
V (0) =
(22)
˜ l = λI + Kl , λ = λmin {Dl }, I is the identity matrix, z = where D (zij )n×n with zii = −2qi aii , zij = −(qi aij + qj aji ), for i ̸= j.
eϵς eT (ς )e(ς )dς
−τ¯
ˆ l = −2Q D˜ l L−1 − z + N + ∥Q ∥2 ∥N −1 ∥ ∥B∥22 I < 0, Π
s
Corollary 1. The synchronization error system (11) is globally exponentially stable if there exist positive definite diagonal matrices L = diag{ι1 , ι2 , . . . , ιn }, Q = diag{q1 , q2 , . . . , qn } and a positive definite symmetric matrix N, such that
τ¯ eϵ(ς+τ¯ ) e2i (ς )ι2i ri dς
−τ¯
i=1
By Theorem 1, we can obtain the following corollary.
eϵ t dt Ωi2 (ei (ς ))ri dς
τ¯ eϵ(ς+τ¯ ) Ωi2 (ei (ς ))ri dς
−τ¯
i=1
≤ϵ
s
n
ς
−τ¯
i=1
≤ϵ
ς +τ¯
n s
eT (t )e(t ) ≤ V (t ) ≤ (H1 + H2 )∥ψ∥2 e−ϵ t ,
∀t > 0.
(20)
Thus, we have
∥e(t )∥ ≤
2
α
ϵ
(H1 + H2 )∥ψ∥e− 2 t ,
(21)
which implies the synchronization error system (11) is globally exponentially stable. This completes the proof. Remark 2. Ensari, Arik, and Faydasicok have obtained some excellent results about the asymptotical stability of delayed neural networks such as Theorem 2 in Ensari and Arik (2010) and Theorem 4 in Faydasicok and Arik (2013). Based on these great works, this paper extends the results to the exponential synchronization of memristor-based neural networks with time-varying delays.
In this section, we will compare our results with those in Arik (2005); Cao et al. (2005); Ozcan and Arik (2006); Wen and Zeng (2012); Wen et al. (2012a); Wen, Zeng, and Huang (2012b); Wu and Cui (2008) in the case of no control input as K = 0, and show the effectiveness of our results with an example. For simplicity, the synchronization error system (11) can be seen as a class of recurrent neural networks with time delays in Arik (2005); Cao et al. (2005); Ozcan and Arik (2006); Wen and Zeng (2012); Wen et al. (2012b); Wu and Cui (2008). At first, some globally exponential stability criteria are derived for the synchronization error system (11) with bounded activation functions by the methods obtained in Wen and Zeng (2012); Wu and Cui (2008). On the other hand, in Corollary 2, the synchronization error system (11) is derived to be globally exponentially stable. However, under the same conditions and bounded activation functions, the synchronization error system (11) can only be proved to be globally asymptotically stable in Cao et al. (2005). Secondly, the globally asymptotically stability conditions for the systems which can be seen as a special case of the synchronization error system (11) in Arik (2005); Ozcan and Arik (2006) are presented as follows: Theorem 2 (Arik, 2005). The synchronization error system (11) is globally asymptotically stable if z = (zij )n×n is positive definite,
ρ and ∥B∥2 ≤ ϖ , where ϖ = min1≤i≤n ι , and zii = −2qi aii , i zij = −(qi aij + qj aji ), for i ̸= j.
S. Wen et al. / Neural Networks 48 (2013) 195–203
201
Fig. 3. Transient behavior of memristor-based system (7).
Theorem 3 (Ozcan & Arik, 2006). The synchronization error system (11) is globally asymptotically stable if there exists a positive definite diagonal matrix Q = diag{q1 , q2 , . . . , qn }, such that
˜ = −2ϖ I − z + 2∥Q ∥ ∥B∥2 I < 0, Π (26) qρ where ϖ = min1≤i≤n ιi , I is the identity matrix, z = (zij )n×n i with zii = −2qi aii , zij = −(qi aij + qj aji ), for i ̸= j. Remark 3. The following inequality holds
˜ = −2ϖ I − z + 2∥Q ∥ ∥B∥2 I Π ≥ −2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I , (27) qρ where ϖ = min1≤i≤n ιi . This implies that the conditions in i
Arik (2005); Ozcan and Arik (2006) are stronger than those in Theorem 1, and the results in Arik (2005); Ozcan and Arik (2006) can only guarantee the synchronization error system (11) to be globally asymptotically stable. In addition, in order to show the effectiveness of the obtained results, an illustrative example is given as follows: Example 1. Consider memristor-based system (7) with 1.8 10 −1.5 0.1 A= , B= , 0.1 1.8 0.1 −1.5 1 fi (xi ) = |xi + 1| − |xi − 1| , 2 τi (t ) = 0.97, si = 0, i = 1, 2.
d2 (x2 (t )) =
1.0, 1.2,
x1 (t ) ≤ 0; x1 ( t ) > 0 ,
1.2, 1.0,
x2 (t ) ≤ 0; x2 ( t ) > 0 .
The initial values of master system (7) is set to be [0.1 0.1], and the dynamical behavior of this system is shown as in Fig. 3, which is chaotic and can be used in secure communications. As
λ = min{λmin {Dl }} = 1. l=1,2
−1.8 z= −0.1
−10 , −1.8
˜Dl = 3 0
0 , 3
to make
Πl = −2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I = −Kl +
2.4 0.1
10 < 0, 2.4
therefore 2.4 Kl > 0.1
10 , 2.4
can make the synchronization error system (11) globally exponentially stable. However, ∥B∥2 = 1.6 > ϖ , which means that the results in Arik (2005) cannot be used for the synchronization error system (11). To simulate the obtained result, we choose 4.4 Kl = 0.1
10 . 4.4
Set the initial states of slave system (9) as [0.3 0.3]. The state and error trajectories of master system and slave system, are presented in Fig. 4, which illustrate the effectiveness of the obtained results. 5. Conclusions
Let d1 (x1 (t )) =
Obviously, there exists a positive definite diagonal matrix Q = diag{0.5, 0.5} such that
As the applications of memristor-based networks become more and more popular, the synchronization problem of such networks becomes necessary. Therefore, the authors investigate the problem of global exponential synchronization of a class of memristorbased recurrent neural networks with time-varying delays. A new scheme of memristor-based recurrent neural networks is designed, a corresponding dynamics equation is set up, and the PDC fuzzy strategy is taken to analyze this system. Furthermore, comparisons between obtained results and the previous ones have been made. Several numerical examples show that obtained results in this paper improve and generalize the previous ones derived in the literature. Acknowledgments The authors would like to thank the anonymous reviewers and editors. Their great suggestions are helpful to improve the quality of this paper.
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S. Wen et al. / Neural Networks 48 (2013) 195–203
Fig. 4. State and error trajectories of master system (7) and slave system (9).
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Neural Networks journal homepage: www.elsevier.com/locate/neunet
Global exponential synchronization of memristor-based recurrent neural networks with time-varying delays✩ Shiping Wen a,b,c , Gang Bao d , Zhigang Zeng a,b,∗ , Yiran Chen e , Tingwen Huang c a
School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China
b
Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China
c
Texas A & M University at Qatar, Doha 5825, Qatar
d
School of Electrical, Three Geoges University, Yichang, 443002, China
e
Department of Electronical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, United States
article
info
Article history: Received 14 May 2013 Received in revised form 2 September 2013 Accepted 6 October 2013 Keywords: Memristor Synchronization Recurrent neural networks Time-varying delays
abstract This paper deals with the problem of global exponential synchronization of a class of memristor-based recurrent neural networks with time-varying delays based on the fuzzy theory and Lyapunov method. First, a memristor-based recurrent neural network is designed. Then, considering the state-dependent properties of the memristor, a new fuzzy model employing parallel distributed compensation (PDC) gives a new way to analyze the complicated memristor-based neural networks with only two subsystems. Comparisons between results in this paper and in the previous ones have been made. They show that the results in this paper improve and generalized the results derived in the previous literature. An example is also given to illustrate the effectiveness of the results. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction The sequential processing of fetch, decode, and execution of instructions through the classical von Neumann bottleneck of conventional digital computers has resulted in less efficient machines as their eco-systems have grown to be increasingly complex (Jo et al., 2010). Though the current digital computers can now possess the computing speed and complexity to emulate the brain functionality of animals like a spider, mouse, and cat (Ananthanarayanan, Eser, Simon, & Modha, 2009; Smith, 2006), the associated energy dissipation in the system grows exponentially along the hierarchy of animal intelligence. For example, to perform certain cortical simulations at the cat scale even at 83 times slower firing rate, the IBM team in Ananthanarayanan et al. (2009) has to employ Blue Gene/P (BG/P), a super computer equipped with 147456 CPUs and 144 TBs of main memory. On the other hand, the human brain contains more than 100 billion neurons and each
✩ This work was supported by the Natural Science Foundation of China under Grant 61125303, National Basic Research Program of China (973 Program) under Grant 2011CB710606, Research Fund for the Doctoral Program of Higher Education of China under Grant 20100142110021, the Excellent Youth Foundation of Hubei Province of China under Grant 2010CDA081, National Priority Research Project NPRP 4-451-2-168, funded by Qatar National Research Fund. ∗ Corresponding author at: School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China. Tel.: +86 18971124190; fax: +86 27 87543130. E-mail addresses: [email protected] (Z. Zeng), [email protected] (Y. Chen).
0893-6080/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.neunet.2013.10.001
neuron has more than 20000 synapses. Efficient circuit implementation of synapses, therefore, is very important to build a brainlike machine. However, since shrinking the current transistor size is very difficult, introducing a more efficient approach is essential for further development of neural network implementations. In 2008, the Williams group announced a successful fabrication of a very compact and non-volatile nano scale memory called the memristor (Strukov, Snider, Stewart, & Williams, 2008). It was postulated by Chua (1971) as the fourth basic circuit elements in electrical circuits. It is based on the nonlinear characteristics of charge and flux. By supplying a voltage or current to the memristor, its resistance can be altered (Sharifiy & Banadaki, 2010). In this way, the memristor remembers information. Several examples of successful multichip networks of spiking neurons have been recently proposed (Choi, Shi, & Boahen, 2004; Indiveri, 2001; Liu & Douglas, 2004); however there are still a number of practical problems that hinder the development of truly large-scale, distributed, massively parallel networks of very large scale integration (VLSI) neurons, such as how to set the weight of individual synapses in the network. It is well-known that changes in the synaptic connections between neurons are widely believed to contribute to memory storage, and the activity-dependent development of neural networks. These changes are thought to occur through correlatedbased, or Hebbian plasticity. In addition, we notice recurrent neural networks have been widely studied in recent years, for their immense application prospective (Bao, Wen, & Zeng, 2012; Cao, Huang, & Qu, 2005; Cao & Wang, 2005; Cao, Yuan, & Li, 2006; Chen, Cao, & Huang, 2002;
196
S. Wen et al. / Neural Networks 48 (2013) 195–203
Fig. 1. Circuit of memristor-based recurrent network, where xi (.) is the state of the i-th subsystem, fj (.) is the amplifier, Rfij is the connection resistor between the amplifier fj (.) and state xi (.), Mi and Ci are the memristor and capacitor, Ii is the external input, ai , bi are the outputs, i, j = 1, 2, . . . , n.
Forti & Nistri, 2003; Hu & Wang, 2002; Huang & Cao, 2003; Li, Fei, & Zhu, 2009; Rakkiyappan, Balasubramaniam, & Cao, 2010; Shen & Wang, 2008; Zeng & Wang, 2006; Zeng, Wang, & Liao, 2005; Zhao & Zhu, 2010). Many applications have been developed in different areas such as combinatorial optimization, knowledge acquisition and pattern recognition. Recently, the problem of synchronization of coupled neural networks which is one of the hot research fields of complex networks has been a challenging issue due to its potential applications such as in information science, biological systems and so on (Balasubramaniam, Kalpana, & Rakkiyappan, 2012; Cao, Chen, & Li, 2008; Park, Kwon, Park, Lee, & Cha, 2012; Xing & Peng, 2012; Zhang, Wang, Li, & Fei, 2012). However, to the best of the authors’ knowledge, the research on global exponential synchronization of memristor-based recurrent neural networks is still an open problem that deserves further investigation. To shorten the gap, we investigate the problem of global exponential synchronization for a class of memristor-based recurrent neural networks with time-varying delays. The memristor-based recurrent network can be implemented by very large scale integration(VLSI) circuits as shown in Fig. 1. fj is the activation function, τj (t ) is the time-varying delay, for the i-th subsystem, xi (t ) is the voltage of the capacitor Ci , fj (xj (t )), fj (xj (t − τj (t ))) are the functions about xi (t ) with and without time-varying delays respectively, Rfij is the resistor between the feedback function fj (xj (t )) and xi (t ), Rgij is the resistor between the feedback function fj (xj (t − τj (t ))) and xi (t ), Mi is the memristor parallel to the capacitor Ci , Ii is an external input or bias, where i, j = 1, 2, . . . , n. The memductance of the memristors can be depicted as in Fig. 2 (Gergel-Hackett et al., 2009), which are bounded. Thus by Kirchoff’s current law, the equation of the i-th subsystem is written as the
following:
Ci x˙ i (t ) = −
n 1 Rfij
j =1
+
+ Wi (xi (t )) xi (t )
Rgij
n signij fj (xj (t ))
Rfij
j =1
+
+
1
n signij fj (xj (t − τj (t )))
Rgij
j =1
where signij =
1, i ̸= j; −1, i = j,
+ Ii ,
(1)
and Wi are the memductances of the
memristors Mi , and Wi (xi (t )) =
Wi′ , xi (t ) ≤ 0; Wi′′ , xi (t ) > 0.
Therefore x˙ i (t ) = −di (xi (t ))xi (t ) +
n
aij fj (xj (t ))
j=1
+
n
bij fj (xj (t − τj (t ))) + si ,
(2)
j=1
where signij aij = , Ci Rfij
bij =
di (xi (t )) =
Ci
=
n 1 j =1
signij Ci Rgij 1 Rfij
+
d1i , xi (t ) ≤ 0; d2i , xi (t ) > 0,
,
1 Rgij
+ Wi (xi (t ))
si =
Ii Ci
.
S. Wen et al. / Neural Networks 48 (2013) 195–203
197
Then we can get x˙ (t ) = −D(x(t ))x(t ) + Af (x(t )) + Bf (x(t − τ (t ))) + s,
(3)
where D(x(t )) = diag{d1 (x1 (t )), d2 (x2 (t )), . . . , dn (xn (t ))}, A = [aij ]n×n ,
B = [bij ]n×n ,
s = (s1 , s2 , . . . , sn )T , T f (x(t )) = f1 (x1 (t )), . . . , fn (xn (t )) , T f (x(t − τ (t ))) = f1 (x1 (t − τ1 (t ))), . . . , fn (t − τn (t )) .
Remark 1. With the property of non-volatile, for each subsystem, only one memristor may be employed to store multiple states of the input information. Therefore, the synapses of the neural networks need not to be designed with a memristor, which will obviously reduce the cost to produce the memristor-based neural networks in future. Based on the analysis above, D(x(t )) in system (3) is changed according to the state of the system, so this network based on memristors is a state-dependent switching system. System (3) represents a class of memristor-based recurrent neural networks with time-varying delays. Taking advantage of the properties of state-dependence, a fuzzy method can be employed to investigate this system. To solve the problem about nonlinear control, fuzzy logic has attracted much attention as a powerful tool. Among various kinds of fuzzy methods, the Takagi–Sugeno fuzzy systems are widely accepted as a useful tool for design and analysis of fuzzy control system (Chiu & Chiang, 2009; Dong, Wang, & Yang, 2009; Liu & Zhong, 2007; Park & Cho, 2004; Takagi & Sugeno, 1985). Currently, some control methods for memristive systems have been proposed (Wen, Zeng, & Huang, 2012a), in which, the number of the linear subsystem is decided by how many minimum nonlinear terms should be linearized in original system. Then, the memristor-based neural network (2) can be exactly represented by the fuzzy model as follows: Rule 1: IF xi (t ) is N1i , THEN x˙ i (t ) = −d1i xi (t ) +
n
n
aij fj (xj (t ))
bij fj (xj (t − τj (t ))) + si ,
(4)
j =1
Rule 2: IF xi (t ) is N2i , THEN x˙ i (t ) = −d2i xi (t ) +
n
aij fj (xj (t ))
j=1
+
n
bij fj (xj (t − τj (t ))) + si ,
(5)
j =1
where N1i is xi (t ) ≤ 0, N2i is xi (t ) > 0. With a center-average defuzzier, the over fuzzy system is represented as x˙ i (t ) = −
2
ηli (t )dli xi (t ) +
l=1
+
n
in the T–S fuzzy system. If n is large, the number of linear subsystems in the T–S fuzzy system is huge. For this problem, Li and Ge proposed a fuzzy modeling method and applied in the synchronization problem of two totally different chaotic systems (Li & Ge, 2011). Based on this work, a new fuzzy model is proposed to linearize memristive systems, in which only two subsystems are included. Furthermore, through this model, the idea of PDC can be applied to achieve between subsystems. Therefore, system (6) can be represented by x˙ (t ) = −
2
where Ψl (t ) = diag{ηl1 (t ), . . . , ηln (t )}, and 1, . . . , n, l = 1, 2, and
n
aij fj (xj (t ))
j=1
bij fj (xj (t − τj (t ))) + si ,
(6)
1, xi (t ) ≤ 0, 0, xi (t ) > 0,
2
l =1
ηli (t ) = 1, i =
Denote u = (u1 , . . . , un )T , |u| as the absolute-value vector; i.e., |u| = (|u1 |, |u2 |, . . . , |un |)T , ∥x∥p as the p-norm of the vector x with p, 1 ≤ p < ∞. ∥x∥∞ = maxi∈{1,2,...,n} |xi | is the vector infinity norm. Denote ∥D∥p as the p-norm of the matrix D with p. Denote C as the set of continuous functions. The following assumptions will be needed throughout the paper: A1. For i ∈ {1, 2, . . . , n}, the activation function fi is Lipschitz continuous; and ∀r1 , r2 ∈ R, there exists real number ιi such that fi (r1 ) − fi (r2 ) 0≤ ≤ ιi . r1 − r2 A2. For i ∈ {1, 2, . . . , n}, τi (t ) satisfies 0 ≤ τi (t ) ≤ τ¯i ,
η2i (t ) =
0, xi (t ) ≤ 0, 1, xi (t ) > 0.
When the system becomes complicated with n memristors, there are 2n subsystems (according to 2n fuzzy rules) and 2n equations
τ˙i (t ) ≤ µi .
(8)
In this paper, we consider system (7) as the master system, and through electronic inductors, the values of memristor will be presented in the corresponding slave system, then the slave system is given as: y˙ (t ) = −
where
η1i (t ) =
(7)
Dl = diag{dl1 , dl2 , . . . , dln }.
j =1
Ψl (t )Dl x(t ) + Af (x(t )) + Bf (x(t − τ (t ))) + s,
l =1
2. Preliminaries
j=1
+
Fig. 2. Typical i–v characteristic of memristor (Gergel-Hackett et al., 2009). The pinched hysteresis loop is due to the nonlinear relationship between the memristance current and voltage. The memristor exhibits the feature of pinched hysteresis, which means that a lag occurs between the application and the removal of a field and its subsequent effect, just as with the neurons in the human brain.
2
Ψl (t )Dl y(t ) + Af (y(t ))
l =1
+ Bf (y(t − τ (t ))) + s + u(t ),
t ≥ 0,
(9)
where y(t ) = (y1 (t ), y2 (t ), . . . , yn (t )) , u(t ) = (u1 (t ), u2 (t ), . . . , un (t ))T is the control input that will be designed. The initial conT
198
S. Wen et al. / Neural Networks 48 (2013) 195–203
ditions of system (7) are in the form of y(t ) = φ(t ) ∈ C ([−τ¯ , 0], Rn ), τ¯ = max1≤i≤n {τ¯i }. In order to derive sufficient conditions for the global exponential synchronization of system (7) with system (9), we will need the following lemmas. Lemma 1 (Zhao & Tan, 2007). It is given any real matrices X , Z , P of appropriate dimensions and a scalar ε0 > 0, where P > 0. Then the following inequality holds: X T Z + Z T X ≤ ε0 X T PX + ε0−1 Z T P −1 Z . In particular, if X and Z are vectors, X T Z ≤
1 2
(X T X + Z T Z ).
3. Main results As synchronization has been applied in many real applications, this paper aims to design a memoryless state-feedback controller as 2
u(t ) =
Ψl (t )Kl e(t ),
(10)
l =1
where e(t ) = (e1 (t ), e2 (t ), . . . , en (t ))T , is the synchronization error, and ei (t ) = xi (t ) − yi (t ), Kl = (klij )n×n is a constant gain matrix to be determined to synchronize the drive and response systems. With controller (10), the error system is transformed into e˙ (t ) = −
2
Ψl (t )Dˆ l e(t ) + AΩ (e(t )) + BΩ (e(t − τ (t ))),
(11)
where the scalar α > 0. Then, V˙ (t ) = α V˙ 1 (t ) + V˙ 2 (t ), where V˙ 1 (t ) = −
2
Ψl (t )eT (t )Dˆ l e(t ) + eT (t )AΩ (e(t ))
l =1 T
+ e (t )BΩ (e(t − τ (t ))), n n V˙ 2 (t ) = 2 qi ΩiT (ei (t ))˙ei (t ) + ri ΩiT (ei (t ))Ωi (ei (t )) i=1 i=1 T − (1 − τ˙ (t ))Ωi (ei (t − τ (t )))Ωi (ei (t − τ (t ))) = 2Ω T (e(t ))Q e˙ (t ) + Ω T (e(t ))RΩ (e(t )) − (1 − τ˙ (t ))Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))) 2 = −2 Ψl (t )Ω T (e(t ))Q Dˆ l e(t ) + 2Ω T (e(t ))QAΩ (e(t )) l =1
+ 2Ω T (e(t ))QBΩ (e(t − τ (t ))) + Ω T (e(t ))RΩ (e(t )) − (1 − τ˙ (t ))Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))). As V˙ 1 (t ) can be rewritten as V˙ 1 (t ) = −
2
Ψl (t )eT (t )Dˆ l e(t )
l =1
+ eT ( t )
2 l =1
Ψl (t )Dˆ l √
1/2 −1/2 2 √ Ψl (t )Dˆ l 2 l =1
2
l =1
where
Ω (e(t )) = (Ω1 (e1 (t )), Ω2 (e2 (t )), . . . , Ωn (en (t ))) = f (e(t ) + y(t )) − f (y(t )), Ω (e(t − τ (t ))) = (Ω1 (e1 (t − τ1 (t ))), Ω2 (e2 (t − τ2 (t ))), . . . , Ωn (en (t − τn (t ))))T = f (e(t − τ (t )) + y(t − τ (t ))) − f (y(t − τ (t ))).
√ ×
V˙ 1 (t ) ≤ −
×
˜ l = λI + Kl , λ = minl=1,2 {λmin {Dl }}, I is the identity mawhere D trix, z = (zij )n×n with zii = −2qi aii , zij = −(qi aij + qj aji ), for i ̸= j, then there exists a positive definite diagonal matrix R = ˆ l e(t ) where diag(r1 , r2 , . . . , rn ), such that V˙ (t )|(11) ≤ − α2 eT (t )D α 2
+
eT (t )e(t ) + 2
i =1 n i=1
t −τi (t )
i=1
Ωi2 t −τi (t )
(t )e(t ), V2 (t ) = 2
(ei (s))ri ds. Then
V (t ) = α V1 (t ) + V2 (t ),
BΩ (e(t − τ (t ))).
2 1
2 l =1 2
Ψl (t )eT Dˆ l e(t ) + Ω T (e(t ))AT
1 T T Ψl (t )Dˆ − l AΩ (e(t )) + Ω (e(t − τ (t )))B
×
2
1 Ψl (t )Dˆ − l BΩ (e(t − τ (t )))
l =1
≤−
2 1
2 l =1
Ψl (t )eT (t )Dˆ l e(t ) +
× ∥A∥22 Ω T (e(t ))Ω (e(t )) +
Ωi (s)ds
2
1 Ψl (t )∥Dˆ − l ∥
l =1 2
1 Ψl (t )∥Dˆ − l ∥
l =1
Ωi2 (ei (s))ri ds.
1 T e 2
Ψl (t )Dˆ l
0
t
Proof. Let V1 (t ) =
n t
qi
−1/2
l =1
Πl = −2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I < 0,
V (t ) =
2
2
By Lemma 1,
Lemma 2. For the synchronization error system (11), if there exist a positive number α and positive definite diagonal matrices L = diag {ι1 , ι2 , . . . , ιn }, Q = diag(q1 , q2 , . . . , qn ), and
ei (t )
1/2
l =1
In order to gain the main results, the following lemma is introduced.
2
Ψl (t )Dˆ l T l =1 × AΩ (e(t )) + e (t ) √ 2
ˆ l = Kl + D l , D
n
(12)
n
i=1 qi
ei (t ) 0
× ∥B∥22 Ω T (e(t − τ (t )))Ω (e(t − τ (t ))) 2 2 1 1 ≤− Ψl (t )eT (t )Dˆ l e(t ) + Ψl (t )∥Dˆ − l ∥ 2 l =1
Ωi (s)ds +
× ∥A∥22 Ω T (e(t ))Ω (e(t )) +
l =1 2
1 Ψl (t )∥Dˆ − l ∥
l =1
(13)
× ∥B∥22 Ω T (e(t − τ (t )))Ω (e(t − τ (t )))
S. Wen et al. / Neural Networks 48 (2013) 195–203
≤−
2 1
2 l =1
Furthermore, we have the following theorem.
Ψl (t )eT (t )D˜ l e(t ) + M Ω T (e(t ))Ω (e(t ))
+ M Ω T (e(t − τ (t )))Ω (e(t − τ (t ))), 2 2 −1 −1 2 2 ˜ ˜ where M = max Ψ ( t )∥ D ∥ ∥ A ∥ , Ψ ( t )∥ D ∥ ∥ B ∥ l l l l 2 2 l =1 l =1 ≥ 0. 1 2 As ei (t )Ωi (ei (t )) ≥ ι− i (Ωi (ei (t ))) , we have 2
−2Ω T (e(t ))Q
˜ l }}. Obviously, V (t ) which is deProof. Let ρ = min1≤l≤2 {λmin {D fined in Lemma 2 is a positive definite and radially unbounded Lyapunov functional. A positive number ϵ > 0 is chosen to satisfy
d dt
{eϵ t V (T )} = eϵ t (ϵ V (t ) + V˙ (t ))
Ψl (t )Dˆ l L−1 Ω (e(t )).
2
≤
l =1
ϵt
Ψl (t )e
Therefore V˙ 2 (t ) ≤ −2Ω T (e(t ))Q
+2
Ψl (t )Dˆ l L−1 Ω (e(t ))
+ 2Ω (e(t ))QAΩ (e(t )) + 2Ω (e(t ))QB × Ω (e(t − τ (t ))) + Ω T (e(t ))RΩ (e(t )) − (1 − µ)Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))) 2 T −1 ˜ ≤ −Ω (e(t )) 2Q Ψl (t )Dl L Ω (e(t )) T
−
− Ω (e(t ))zΩ (e(t )) + 2∥Q ∥ ∥B∥2 ∥Ω (e(t ))∥ × ∥Ω (e(t − τ (t )))∥ + Ω T (e(t ))RΩ (e(t )) T
− (1 − µ)Ω T (e(t − τ (t )))RΩ (e(t − τ (t ))).
2
+ 4ϵ
Ψl (t )e
n
+ ϵ eϵ t
≤ ∥Q ∥∥B∥2 Ω T (e(t ))Ω (e(t )) + Ω T (e(t − τ (t )))Ω (e(t − τ (t ))) .
(15)
Since i=1 qi we can get d
From (14) and (15),
0
dt
(e V (t )) ≤
2 l =1
1−µ
t −τi (t )
Ψl (t )e
+ ϵ eϵ t
s
e 0
α=
β
,
H > 0,
2H 1, H = 0.
Then V˙ (t ) ≤ − α2
T ˜ l=1 Ψl (t )e (t )Dl e(t ).
ei (t ) 0
ιi sds ≤
1 T e 2
(t )LQe(t ),
ϵt
ϵt
ϵα eT (t )e(t ) − α eT (t )D˜ l e(t )
n
t t −τi (t )
Ωi2
(ei (s))ri ds
n
=ϵ
ϵt
n
t t −τi (t )
Ωi2 (ei (s))ri ds.
n
≤ϵ
n i=1
t t −τi (t )
i=1
i =1
2
qi
(17)
Estimating the second term on the right-hand side of (17) by changing the integrals, we can get
ϵ
By choosing
i=1
1 ≤ eϵ t ϵα − ρα + 2ϵ∥LQ ∥ eT (t )e(t ) 2
i =1
V˙ 2 (t ) ≤ −β Ω T (e(t ))Ω (t ) − β Ω T (e(t − τ (t )))Ω (e(t − τ (t ))).
Ωi2 (ei (s))ri ds.
n
i=1
∥Q ∥∥B∥2 + β I ,
which is a positive definite diagonal matrix, we can obtain
t
+ 2ϵ e (t )LQe(t ) + ϵ e
As Πl < 0, there exists β > 0 such that Πl + 2β I < 0. Define 1
Ωi (s)ds
0
n
T
× Ω (e(t − τ (t ))).
qi
˜ −1 − z + R + ∥Q ∥∥B∥2 Ω (e(t )) V˙ 2 (t ) ≤ Ω T (e(t )) −2Q DL − Ω T (e(t − τ (t ))) (1 − µ)R − ∥Q ∥∥B∥2
ϵα eT (t )e(t ) − α eT (t )D˜ l e(t ) ei (t )
Ωi (s)ds ≤
2 1
ϵt
ϵt
i =1
ei (t )
n
(ei (s))ri ds
i=1
2∥Q ∥ ∥B∥2 ∥Ω T (e(t ))∥ ∥Ω (e(t − τ (t )))∥
Ωi2
˜ l e(t ) e (t )D
Furthermore, we can obtain
t
2 l =1
(14)
Ωi (s)ds
T
2 1
≤
eT (t )e(t )
0
t −τi (t )
i =1
α
ei (t )
α 2
qi
n
+
l =1 T
n i =1
l =1 T
R=
ϵ
l =1
2
(16)
By Lemma 2, we can get
l=1
≤ −2Ω T (e(t ))Q
Theorem 1. Assume the conditions in Lemma 2 hold, then the drive system (7) is globally exponentially synchronized with the response system (9).
ϵα − ρα + 2ϵ∥LQ ∥ + 2ϵ τ¯ eϵ τ¯ ∥L2 R∥ < 0.
Ψl (t )Dˆ l e(t ) 2
199
s
eϵ t
Ωi2 (ei (ς ))ri dς dt t
t −τi (t )
0 s
−τ¯
Ωi2 (ei (ς ))ri dς dt
min{ς+τ¯ ,s} max{ς ,0}
eϵ t dt Ωi2 (ei (ς ))ri dς
200
S. Wen et al. / Neural Networks 48 (2013) 195–203
≤ϵ
n
s
≤ ϵ τ¯ eϵ τ¯ ∥L2 R∥ ≤ ϵ τ¯ eϵ τ¯ ∥L2 R∥
s
0
eϵς eT (ς )e(ς )dς
−τ¯
Proof. By Lemma 1, 1
ϵς T
e e (ς )e(ς )dς .
+
(18)
From (16)–(18), 1
ϵα − ρα + 2ϵ∥LQ ∥ + 2ϵ τ¯ eϵ τ¯ ∥L2 R∥ 2 s × eϵ t eT (t )e(t )dt + ϵ τ¯ eϵ τ¯ ∥L2 R∥
0
×
− 2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I ≤ −2Q D˜ l L−1 − z + N + ∥Q ∥2 ∥N −1 ∥ ∥B∥22 I < 0. (25)
−τ¯
This proof is completed.
eϵ t eT (t )e(t )dt
Let Q = N = I in Corollary 1, we obtain the following corollary.
0
≤ ϵ τ¯ ∥L2 R∥
eϵ t dt ∥ψ∥2
−τ¯
Corollary 2. The synchronization error system (11) is globally exponentially stable if z = (zij )n×n is positive definite, and ∥B∥2 ≤
≡ H1 ∥ψ∥2 .
√
V (t ) ≤ V (0) + H1 ∥ψ∥2 e−ϵ t ,
ρ 2π − 1, where π = min1≤i≤n ι , and zii = −2qi aii , zij = i −(qi aij + qj aji ), for i ̸= j.
Thus,
∀t > 0,
(19) 4. Comparison and example
where
α 2
+
eT (0)e(0) + 2
n
ei (0)
qi
Ωi (s)ds
0
i =1 n i=1
0
−τi (t )
Ωi2 (ei (s))ri ds
1 2
β + 2∥QL∥ + 2τ¯ ∥L2 R∥ ∥ψ∥2
≡ H2 ∥ψ∥2 . It follows from (12) and (19) that
2
(23)
Then
0
α
N + ∥Q ∥ ∥B∥2 N −1 ≥ 2I ,
N + ∥Q ∥2 ∥N −1 ∥ ∥B∥22 I ≥ N + ∥Q ∥2 ∥B∥22 N −1 ≥ 2∥Q ∥ ∥B∥22 I . (24)
eϵ s V (s) − V (0) ≤
≤
∥Q ∥ ∥B∥2
which implies that
0
V (0) =
(22)
˜ l = λI + Kl , λ = λmin {Dl }, I is the identity matrix, z = where D (zij )n×n with zii = −2qi aii , zij = −(qi aij + qj aji ), for i ̸= j.
eϵς eT (ς )e(ς )dς
−τ¯
ˆ l = −2Q D˜ l L−1 − z + N + ∥Q ∥2 ∥N −1 ∥ ∥B∥22 I < 0, Π
s
Corollary 1. The synchronization error system (11) is globally exponentially stable if there exist positive definite diagonal matrices L = diag{ι1 , ι2 , . . . , ιn }, Q = diag{q1 , q2 , . . . , qn } and a positive definite symmetric matrix N, such that
τ¯ eϵ(ς+τ¯ ) e2i (ς )ι2i ri dς
−τ¯
i=1
By Theorem 1, we can obtain the following corollary.
eϵ t dt Ωi2 (ei (ς ))ri dς
τ¯ eϵ(ς+τ¯ ) Ωi2 (ei (ς ))ri dς
−τ¯
i=1
≤ϵ
s
n
ς
−τ¯
i=1
≤ϵ
ς +τ¯
n s
eT (t )e(t ) ≤ V (t ) ≤ (H1 + H2 )∥ψ∥2 e−ϵ t ,
∀t > 0.
(20)
Thus, we have
∥e(t )∥ ≤
2
α
ϵ
(H1 + H2 )∥ψ∥e− 2 t ,
(21)
which implies the synchronization error system (11) is globally exponentially stable. This completes the proof. Remark 2. Ensari, Arik, and Faydasicok have obtained some excellent results about the asymptotical stability of delayed neural networks such as Theorem 2 in Ensari and Arik (2010) and Theorem 4 in Faydasicok and Arik (2013). Based on these great works, this paper extends the results to the exponential synchronization of memristor-based neural networks with time-varying delays.
In this section, we will compare our results with those in Arik (2005); Cao et al. (2005); Ozcan and Arik (2006); Wen and Zeng (2012); Wen et al. (2012a); Wen, Zeng, and Huang (2012b); Wu and Cui (2008) in the case of no control input as K = 0, and show the effectiveness of our results with an example. For simplicity, the synchronization error system (11) can be seen as a class of recurrent neural networks with time delays in Arik (2005); Cao et al. (2005); Ozcan and Arik (2006); Wen and Zeng (2012); Wen et al. (2012b); Wu and Cui (2008). At first, some globally exponential stability criteria are derived for the synchronization error system (11) with bounded activation functions by the methods obtained in Wen and Zeng (2012); Wu and Cui (2008). On the other hand, in Corollary 2, the synchronization error system (11) is derived to be globally exponentially stable. However, under the same conditions and bounded activation functions, the synchronization error system (11) can only be proved to be globally asymptotically stable in Cao et al. (2005). Secondly, the globally asymptotically stability conditions for the systems which can be seen as a special case of the synchronization error system (11) in Arik (2005); Ozcan and Arik (2006) are presented as follows: Theorem 2 (Arik, 2005). The synchronization error system (11) is globally asymptotically stable if z = (zij )n×n is positive definite,
ρ and ∥B∥2 ≤ ϖ , where ϖ = min1≤i≤n ι , and zii = −2qi aii , i zij = −(qi aij + qj aji ), for i ̸= j.
S. Wen et al. / Neural Networks 48 (2013) 195–203
201
Fig. 3. Transient behavior of memristor-based system (7).
Theorem 3 (Ozcan & Arik, 2006). The synchronization error system (11) is globally asymptotically stable if there exists a positive definite diagonal matrix Q = diag{q1 , q2 , . . . , qn }, such that
˜ = −2ϖ I − z + 2∥Q ∥ ∥B∥2 I < 0, Π (26) qρ where ϖ = min1≤i≤n ιi , I is the identity matrix, z = (zij )n×n i with zii = −2qi aii , zij = −(qi aij + qj aji ), for i ̸= j. Remark 3. The following inequality holds
˜ = −2ϖ I − z + 2∥Q ∥ ∥B∥2 I Π ≥ −2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I , (27) qρ where ϖ = min1≤i≤n ιi . This implies that the conditions in i
Arik (2005); Ozcan and Arik (2006) are stronger than those in Theorem 1, and the results in Arik (2005); Ozcan and Arik (2006) can only guarantee the synchronization error system (11) to be globally asymptotically stable. In addition, in order to show the effectiveness of the obtained results, an illustrative example is given as follows: Example 1. Consider memristor-based system (7) with 1.8 10 −1.5 0.1 A= , B= , 0.1 1.8 0.1 −1.5 1 fi (xi ) = |xi + 1| − |xi − 1| , 2 τi (t ) = 0.97, si = 0, i = 1, 2.
d2 (x2 (t )) =
1.0, 1.2,
x1 (t ) ≤ 0; x1 ( t ) > 0 ,
1.2, 1.0,
x2 (t ) ≤ 0; x2 ( t ) > 0 .
The initial values of master system (7) is set to be [0.1 0.1], and the dynamical behavior of this system is shown as in Fig. 3, which is chaotic and can be used in secure communications. As
λ = min{λmin {Dl }} = 1. l=1,2
−1.8 z= −0.1
−10 , −1.8
˜Dl = 3 0
0 , 3
to make
Πl = −2Q D˜ l L−1 − z + 2∥Q ∥ ∥B∥2 I = −Kl +
2.4 0.1
10 < 0, 2.4
therefore 2.4 Kl > 0.1
10 , 2.4
can make the synchronization error system (11) globally exponentially stable. However, ∥B∥2 = 1.6 > ϖ , which means that the results in Arik (2005) cannot be used for the synchronization error system (11). To simulate the obtained result, we choose 4.4 Kl = 0.1
10 . 4.4
Set the initial states of slave system (9) as [0.3 0.3]. The state and error trajectories of master system and slave system, are presented in Fig. 4, which illustrate the effectiveness of the obtained results. 5. Conclusions
Let d1 (x1 (t )) =
Obviously, there exists a positive definite diagonal matrix Q = diag{0.5, 0.5} such that
As the applications of memristor-based networks become more and more popular, the synchronization problem of such networks becomes necessary. Therefore, the authors investigate the problem of global exponential synchronization of a class of memristorbased recurrent neural networks with time-varying delays. A new scheme of memristor-based recurrent neural networks is designed, a corresponding dynamics equation is set up, and the PDC fuzzy strategy is taken to analyze this system. Furthermore, comparisons between obtained results and the previous ones have been made. Several numerical examples show that obtained results in this paper improve and generalize the previous ones derived in the literature. Acknowledgments The authors would like to thank the anonymous reviewers and editors. Their great suggestions are helpful to improve the quality of this paper.
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S. Wen et al. / Neural Networks 48 (2013) 195–203
Fig. 4. State and error trajectories of master system (7) and slave system (9).
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