Xi'an Technological University
Subject: Computer Science, Software Engineering
eISSN: 2470-8038
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Ding Dawei ^{*} / Zhang Yaqin ^{*} / Wang Nian ^{*}
Keywords : Exponential Synchronization, Memristor-based Neural Networks, Fractional-order, Linear Error Feedback Control, Time-varying Delays
Citation Information : International Journal of Advanced Network, Monitoring and Controls. Volume 3, Issue 3, Pages 1-15, DOI: https://doi.org/10.21307/ijanmc-2019-001
License : (CC-BY-NC-ND 4.0)
Published Online: 01-October-2019
Considering the fact that the exponential synchronization of neural networks has been widely used in theoretical research and practical application of many scientific fields, and there are a few researches about the exponential synchronization of fractional-order memristor-based neural networks (FMNN). This paper concentrates on the FMNN with time-varying delays and investigates its exponential synchronization. A simple linear error feedback controller is applied to compel the response system to synchronize with the drive system. Combining the theories of differential inclusions and set valued maps, a new sufficient condition concerning exponential synchronization is obtained based on comparison principle rather than the traditional Lyapunov theory. The obtained results extend exponential synchronization of integer-order system to fractional-order memristor-based neural networks with time-varying delays. Finally, some numerical examples are used to demonstrate the effectiveness and correctness of the main results.
Chua already supposed the existence of memristor in 1971 [1], however, the practical device of memristor in electronics is obtained in [2] until 2008. In addition to the existing three kinds of circuit elements, memristor is regarded as the fourth basic circuit element and is defined by a nonlinear charge-flux characteristic. As everyone knows, resistors can be used to work as connection weights so that it can emulate the synapses in artificial neural networks. However, in the neural networks of biological individual, long-term memories is essential in the synapses among neurons, but for the general resistors, it is impossible to have the function of memory. Recently, due to the memory characteristics of memristor, memristor can replace the resistor to develop a new neural networks that is memristor-based neural networks (MNN) [3-6].
In recent years, more and more attentions have been put on the dynamical analysis of memristor-based neural networks, such as the investigation of stability [7-10], periodicity [11-13], system synchronization [14-22], passivity analysis [23], dissipativity [24-25] and attractivity [26]. Particularly, the stability and synchronization of MNN has been widely studied in [27-30]. In fact, synchronization means the dynamics of nodes share the common time-spatial property. Therefore we can understand an unknown dynamical system by achieving the synchronization with the well-known dynamical systems [18]. Moreover, in the transmission of digital signals, communication will become security, reliable and secrecy by achieving synchronization between the various systems. Therefore, the synchronization of MNN is still worth further research.
Moreover, the fractional-order models can better describe the memory and genetic properties of various materials and process, so the fractional-order models have received a lot of research attentions than integer-order models. In recent years, with the improvement of fractional-order differential calculus and fractional-order differential equations, it is easy to model and analyze practical problems [31, 32]. Therefore, there have been a lot of researches about the dynamical analysis and synchronization of fractional-order memristor-based neural networks (FMNN) [34-39]. Finite-time synchronization, hybrid projective synchronization and adaptive synchronization of FMNN have all been researched [34-36]. However, there are only a very few research results on exponential synchronization of FMNN. In fact, the exponential synchronization of neural networks has been widely used in the theoretical research and practical application of many scientific fields, for example, associative memory, ecological system, combinatorial optimization, military field, artificial intelligence system and so on [40-43]. So the exponential synchronization of FMNN is still worth further studying as it is a significant academic problem.
On the other hand, the stability and synchronization of FMNN without time delay have been deeply studied such as in [33]. However, in hardware implementation of neural networks, time delay is unavoidable owing to the finite switching speeds of the amplifiers. And it will cause instability, oscillation and chaos phenomena of systems. So the investigation for stability and synchronization of FMNN cannot be independent on the time delay.
Motivated by the above discussion, this paper studies the exponential synchronization of FMNN with time-varying delays. The main contributions of this paper can be listed as follow. (1) This is the first attempt to achieve exponential synchronization of FMNN with time-varying delays by employing a simple linear error feedback controller. (2) The sufficient condition for exponential synchronization of FMNN with time delays is obtained based on comparison principle instead of the traditional Lyapunov theory. (3) Some previous research results of exponential synchronization for integer-order memristor-based system are the special cases of our results. Furthermore, some numerical examples are given to demonstrate the effectiveness and correctness of the main results.
The rest of this paper is organized as follows. Preliminaries including the introduction of Caputo fractional-order derivative, model description, assumptions, definitions and lemmas are presented in Section 2. Section 3 introduces the sufficient condition for exponential synchronization of the FMNN. In Section4, the numerical simulations are presented. Section5 gives the conclusion of this paper.
Compared to the integer-order derivatives, we know the distinct advantage of Caputo derivative is that it only requires initial conditions from the Laplace transform of fractional derivative, and it can represent well-understood features of physical situations and making it more applicable to real world problems [36]. So in the rest of this paper, we apply the Caputo fractional-order derivative for the fractional-order memristor-based neural networks (FMNN) and investigate the exponential synchronization of FMNN.
Definition1 [32] The Caputo fractional-order derivative is defined as follows:
Particularly, when 0<q<1,
In this paper, referring to some relevant works on FMNN [35,36], we consider a class of FMNN with time-varying delays described by the following equation,
In the rest of paper, we first make following assumption for system (4).
Assumption1: For j ∈ N, ∀s_{1}, s_{2} ∈ R, the neuron activation functions f_{j}, g_{j} bounded, f_{i}(0) = g_{j}(0) = 0 and satisfy
We consider system (4) as drive system and corresponding response system is given as follows:3
Where
According to the theories of differential inclusions and set valued maps [40], if x_{i}(t) and y_{i}(t) are solutions of (4) and (7) respectively, system (4) and system (7) can be written as follow:
And
Where
And
where co{u, v} denotes the closure of convex hull generated by real numbers u and v or real matrices u and v. Then the synchronization error system can be described as follows:
Definition2 [8] For ∀t ≥ 0, the exponential synchronization of system (4) and system (7) can be transformed to the exponential stability of the error system (9) (error approaches to zero). The error system (9) is said to be exponentially stable, if there exist constant Q_{i} > 0, P_{i} > 0, such that the solution e(t) = (e_{1}(t),e_{2}(t),…,e_{n}(t))^{T} of error system (9) with initial condition e(s) = ϕ(s) ∈ ([t_{0} − τ, t_{0}],R^{n}) satisfies
Lemma1 [14] Under the assumption1, the following estimation can be obtained:
(i) co[a_{ij}(y_{j}(t))]f_{j}(y_{j}(t)) − co[a_{ij}(x_{j}(t))]f_{j}(x_{j}(t)) ≤ A_{ij}F_{j}(e_{j}(t)),
(ii) co[b_{ij}(y_{j}(t − τ_{j}(t)))]g_{j}(y_{j}(t − τ_{j}(t))) − co[b_{ij}(x_{j}(t − τ_{j}(t)))]g_{j}(x_{j}(t − τ_{j}(t))) ≤ B_{ij}G_{j}(e_{j}(t − τ_{j}(t))),
where
Proof: If y_{i}(t) = 0, x_{i}(t) = 0, i ∈ N we can easily have part(i) hold. From (9) and (10), we can get
(1) For y_{i}(t) < 0, x_{i}(t) < 0, then
(2) For y_{i}(t) > 0, x_{i}(t) > 0, then
(3) For x_{i}(t) < 0 < y_{i}(t) or y_{i}(t) < 0 < x_{i}(t), then
Then complete the proof of part (i). In the similar way, part(ii) can be easily hold.
We present the exponential stability results for the synchronization error system of FMNN when the error system (9) is exponentially stable, the system (4) and system (7) will achieve the exponential synchronization.
Theorem1 If there exist positive constant ε,η_{1},η_{2},…,η_{n} such that for any
Proof: Consider W_{i}(t) = |e_{i}(t)|/η_{i},i = 1,2,…,n, according to the error system (9) or (14) and lemma1, we can get the following inequality
Evaluating the fractional order derivative of W_{i}(t) along the trajectory of error system, then
Define ${\tilde{W}}_{i}(t)={W}_{i}(t)-\overline{W}({t}_{0})\mathrm{exp}\{-\epsilon (t-{t}_{0})\},t\ge {t}_{0}>0,i=1,2,\dots ,n$, where
We will prove that ${\tilde{W}}_{i}(t)\le 0,i=1,2,\dots ,n$, for any t≥t_{0}>0. Otherwise, since ${\tilde{W}}_{i}(t)\le 0,i=1,2,\dots ,n$ for t ∈ [t_{0} − τ, t_{0}], there must exist t_{1} ≥ t_{0} and some ς such that ${D}^{q}{\tilde{W}}_{\varsigma}({t}_{1})\ge 0$ and ${\tilde{W}}_{\varsigma}({t}_{1})=0$. Then
Moreover, from inequality(15), we have
Therefore
It shows
This completes the proof.
In this section, we will give two numerical examples to demonstrate our analysis on exponential synchronization of FMNN.
Example1 Consider two-dimension fractional-order memristor-based neural networks
We consider system (21) as the drive system and corresponding response system is defined as Eq.(7). And for the controller u_{i}(t) = ω_{i}(y_{i}(t) − x_{i}(t)), the parameter ω_{i} is chosen as ω_{1} = −9.5, ω_{2} = −10.5. From Theorem1, when we take ε = 0.7, τ_{j}(t) = 1, η_{1} = η_{2} = ρ_{1} = ρ_{2} = σ_{1}= σ_{2} = 0.1, we can easily know $(-{c}_{i}+{\omega}_{i}+\epsilon ){\eta}_{i}+{\displaystyle {\sum}_{j=1}^{n}{A}_{ij}{\sigma}_{j}{\eta}_{j}}+{\displaystyle {\sum}_{j=1}^{n}{B}_{ij}{\rho}_{j}{\eta}_{j}}\mathrm{exp}\{\epsilon {\tau}_{j}(t)\}<0$ ω_{1} = −9.5, ω_{2} = −10.5, we can get is true when ω_{1} < −1.703, ω_{2} < −0.232. So when
It satisfies the condition of Theorem 1, then the exponential synchronization of drive-response system is achieved.
When the response system with this controller, we get state trajectories of variable x_{1}(t), y_{1}(t) and x_{2}(t), y_{2}(t) are depicted in Figure2a and 2b. Moreover, Figure3a and 3b depict the synchronization error curves e_{1}(t), e_{2}(t) between the drive system and response system. These numerical simulations show the state trajectories of variable x_{1}(t), y_{1}(t) and x_{2}(t), y_{2}(t) are synchronous and synchronization error e_{1}(t), e_{2}(t) are converge to zero. These prove the correctness of the Theorem1.
Example2 Consider three-dimension fractional-order memristor-based neural networks
And τ_{j}(t) = e^{t}/1 + e^{t}, I = (I_{1},I_{2},I_{3})^{T} = (0,0,0)^{T} q = 0.92 and take the activation function as f_{i}(x_{i}) = g_{i}(x_{i}) = tanh(x_{i}),i = 1,2,3. We consider system(22) as the drive system and the corresponding response system is defined in Eq.(7). And for the controller u_{i}(^{t}) = ω_{i}(y_{i}(t) − x_{i}(t)), ω_{i} is chosen as ω_{1} = −9.5, ω_{2} = −10.5, ω_{3} = −11. From Theorem1, we take ε = 0.7, τ_{j}(t) = 1 and choose η_{1} = η_{2} = 0.1 ρ_{1} = ρ_{2} = σ_{1} = σ_{2} = 0.1. According to
It suggests the condition of Theorem 1 is satisfied, then drive-response system achieves the synchronization.
When the response system with this controller, we get state trajectories of variable x_{1}(t), y_{1}(t) and x_{2}(t), y_{2}(t) and x_{3}(t), y_{3}(t) are depicted in Figure 4a,4b,4c. Moreover, Figure 5a,5b,5c depict the synchronization error curves e_{1}(t), e_{2}(t), e_{3}(t) between the drive system and response system. It’s easy to see that the state trajectories of variable x_{1}(t), y_{1}(t), x_{2}(t), y_{2}(t), and x_{3}(t), y_{3}(t) are synchronous and synchronization error e_{1}(t), e_{2}(t), e_{3}(t) are converge to zero. So the Theorem1 is proved to be correct.
In addition, we choose ω_{1} = −9.5, ω_{2} = −10.5, ω_{3} = −11, according to the Theorem1, it needs the following inequalities to hold:
So, we just need $\tau <\frac{1}{\epsilon}\mathrm{ln}\left(\frac{102}{3}-\frac{10}{3}\epsilon \right)$ holds. We have the exponential convergence rate 0 < ε < 1, figure 6 depicts the relation of time-varying delay τ and exponential convergence rate ε.
This paper achieves the exponential synchronization of a class of FMNN with time-varying delays by using linear error feedback controller. Based on comparison principle, the new theorem is derived to guarantee the exponential synchronization between the drive system and response system. The methods proposed for synchronization is effective and it is easy to achieve than other complex control methods. Moreover, it can be extended to investigate other dynamical behaviors of fractional-order memristive neural networks, such as realizing the lag synchronization or anti-synchronizaton of this system based on the suitable controller. These issues will be the topic of future research. Finally, numerical examples are given to illustrate the effectiveness of the proposed theory.