New Results on Synchronization of Fractional-Order Memristor‐Based Neural Networks via State Feedback Control

Li, Xiaofan; Ge, Yuan; Liu, Hongjian; Li, Huiyuan; Fang, Jian-an

doi:https://doi.org/10.1155/2020/2470972

Complexity

On this page

Abstract Introduction Preliminaries Numerical Simulation Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 2470972 | https://doi.org/10.1155/2020/2470972

New Results on Synchronization of Fractional-Order Memristor‐Based Neural Networks via State Feedback Control

Xiaofan Li,^1,2Yuan Ge ,¹Hongjian Liu,¹Huiyuan Li,³and Jian-an Fang³

Academic Editor: Xianming Zhang

Received13 May 2020

Accepted22 Jul 2020

Published09 Sept 2020

Abstract

This paper addresses the synchronization issue for the drive-response fractional-order memristor‐based neural networks (FOMNNs) via state feedback control. To achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. Then, by adopting nonsmooth analysis, fractional Lyapunov’s direct method, Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization of the drive-response FOMNNs. Lastly, for illustrating the effectiveness of the obtained theoretical results, an example is given.

1. Introduction

In recent years, fractional calculus has become a useful tool in the analysis of slow relaxation phenomena. As we all know, fractional derivative has two main advantages: infinite memory and more degrees of freedom [1, 2]. Hence, fractional derivative plays a critical part in the depiction of memory and hereditary characteristics of multifarious processes. Compared with dynamic systems described by the classical integer-order derivative, dynamic systems described by fractional derivative can accurately reflect the actual dynamic properties of real systems due to their memory and hereditary characteristics. Recently, as an extension of the classical integer-order calculus, fractional calculus has many practical applications in many interdisciplinary areas, such as fractional-order sinusoidal oscillators [3], transient wave propagation [4], fractional relaxation-oscillation and fractional diffusion-wave phenomena [5], drug release and absorption [6], and so on. Moreover, dynamic behaviors of fractional-order systems have attracted the attention of many researchers because of their practical applications. In the past decade, the dynamic analysis of the fractional-order systems has achieved many outstanding results [7–10].

In the past few years, dynamic behaviors of neural networks (NNs) have gained many attentions [11–21], since NNs have lots of applications [22, 23]. In addition, fractional derivative has been introduced to NNs, and dynamic analysis of fractional-order neural networks (FONNs) has become a focus of research and many results have been obtained [24, 25]. Among these dynamic behaviors, as a significant dynamic characteristic, synchronization was firstly introduced [26]. Since then, research on synchronization of NNs has become a hot topic because of their wide potential applications in a large number of real systems [27, 28].

On the other hand, memristor was firstly predicted by Chua [29], and a practical memristor device was successfully obtained [30]. The memristor exhibits the characteristics of pinch hysteresis, which is possessed by the human brain. It is more practical to construct artificial NNs by replacing the resistor with the memristor, that is, memristor‐based neural networks (MNNs). In recent years, dynamic behaviors of MNNs have caused widespread concern around the world [31–34]. Accordingly, it is very valuable to study the synchronization problem for fractional-order memristor‐based neural networks (FOMNNs), and many excellent works have been conducted [35, 36].

Inspired by the discussions given above, this paper studies the synchronization issue for FOMNNs via feedback control. Firstly, to achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. Then, by adopting nonsmooth analysis, fractional Lyapunov’s direct method and Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization for the drive-response FOMNNs.

2. Preliminaries and Model Description

The following preliminaries on fractional calculus are recalled.

Definition 1 (see [37]). Given an arbitrary integrable function , its Riemann–Liouville fractional integral with fractional order is defined aswhere , .

Definition 2 (see [37]). Given an arbitrary differentiable function , its Caputo fractional derivative is defined aswhere the fractional order , is a positive integer.
Consider an FOMNN as follows:where is the self-inhibition, refers to the Caputo fractional derivative, , refers to the state, refers to the activation function, is an external input or bias, and is the memristor connection weight. The initial value of (3) is and .
The memristor connection weight switches among different numbers, which can be simply modeled as follows:where the switching jump and and are constant numbers.
Let , , andThen, according to the theories of differential inclusion and set-valued map, for (3), we haveOr equivalently, there exists , such thatThe drive-response synchronization is considered. The corresponding response system of the drive system (3) iswhere denotes the state feedback controller, and the initial condition and . Similarly,Or equivalently, there exists , such that

Assumption 1. For , and , the function : is monotone nondecreasing and satisfies , , and , where , are positive constants.
Next, let the synchronization error . For (3) and (8), the following synchronization error dynamics system can be obtained aswhere . According to Assumption 1, we can know that is also monotone nondecreasing, bounded, and .

Definition 3 (see [37]). The drive FOMNN (3) is globally synchronized with the response FOMNN (8), if there are two positive constants and such thatwhere denotes the initial time, is locally Lipschitz on , , and .

Lemma 1 (see [38]). Let be a continuous function: and satisfy , where and is a constant. Then, the following inequality holds:

Lemma 2 (see [39]). If and , the following inequality holds:where is an arbitrary positive number and .

3. Main Results

In this section, two state feedback controllers are provided for achieving synchronization of FOMNNs.

The following two state feedback controllers are provided:where and denote the control parameters, .

Theorem 1. By holding Assumption 1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via the controller (15), if there are two positive constants and so that

Proof. Consider the Lyapunov function,Also, by calculating the Caputo fractional derivative along the trajectory of (11) with , we getBy utilizing Lemma 2, we getSimilarly,Substituting (20), (21) into (19),According to (17), we choose a constant so thatFrom (22) and (23), we getBy utilizing Lemma 1, we can knownamely,According to (26) and Definition 3, we can obtain that the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (15). The proof of Theorem 1 is completed.

Theorem 2. By holding Assumption 1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (16), if there are positive constants so that

Proof. Firstly, we can know that, on , is a differentiable and continuous function. Therefore, is piecewise continuous, and exits for . Then, we choose the Lyapunov function:Now, by calculating the Caputo fractional derivative along the trajectory of (11) with , we getFrom (27), it follows thatAccording to (28), we choose a constant so thatFrom (31) and (32),By utilizing Lemma 1, we can knowthat is,and then,According to (36) and Definition 3, we can obtain that the drive FOMNN (3) is globally synchronized with the response FOMNN (8) under the state feedback controller (16). The proof of Theorem 2 is completed.
It is well known that eigenvalue of the system matrix has a tight relation with dynamics. Next, we will obtain synchronization conditions according to it.

Theorem 3. By holding Assumption 1, the drive FOMNN (3) is globally synchronized with the response FOMNN (8) via controller (15), ifwhere

Proof. Firstly, according to the matrix theory, we can easily know that is also the eigenvalue of if is its eigenvalue. Therefore, the maximum eigenvalues of matrix is greater than zero, that is, .
Consider the Lyapunov function,Then, by calculating the Caputo fractional derivative along the trajectory of (11) with , we getwhere .
According to (37), we choose a constant so thatFrom (40) and (41), we can getBy utilizing Lemma 1, we can knownamely,and then,According to (45) and Definition 3, we can obtain that the drive FOMNN (3) is globally synchronized with the response FOMNN (8) under the state feedback controller (15). The proof of Theorem 3 is completed.

Remark 1. Free-weighting parameters and are introduced in synchronization criterion (17) of Theorem 1. Also, free-weighting parameters and can be used to reduce the conservativeness of the synchronization criterion.

Remark 2. The synchronization criteria obtained in this paper only depend on their system parameters, which are simpler in form than linear matrix inequalities [40]. In addition, these algebraic synchronization criteria are easy to check, quick to calculate, and help greatly reduce the computational burden.

4. Numerical Simulation

Now, we provide a numerical simulation to illustrate the effectiveness of results.

We consider the FOMNN (8) as the response system and the drive FOMNN as follows:where , , and

Then, we can obtain , and . Let ; Figure 1 depicts the chaotic behavior, error and state trajectories of the derive FOMNN (46), and the response FOMNN (8) without an external controller, which are not synchronous.

(a)

(b)

(c)

(d)

Now, according to (17) in Theorem 1, we can get

Next, we choose for convenient calculation, then we can obtain , , and choose . The condition of Theorem 1 is satisfied, that is, the drive FOMNN (46) and the response FOMNN (8) are globally synchronized via the controller . Let , and then, the chaotic behaviors, error and state trajectories of the drive FOMNN (46), and the response FOMNN (8) via the controller are shown in Figure 2.

(a)

(b)

(c)

(d)

Similarly, according to (27) in Theorem 2 and by selecting , we can get and choose . Moreover, according to (28) in Theorem 2, we can get and choose . The conditions of Theorem 2 are satisfied, that is, the drive FOMNN (46) and the response FOMNN (8) are globally synchronized via the controller . Let ; then, the chaotic behaviors, error and state trajectories of the drive FOMNN (46), and the response FOMNN (8) via the controller are shown in Figure 3.

(a)

(b)

(c)

(d)

Moreover, in Theorem 3, system parameters matrix can be easily obtained as

By easily calculating, the maximal eigenvalue of system parameters matrix is . Next, according to (37) in Theorem 3, choosing , we can obtain and and choose . The condition of Theorem 3 is satisfied, that is, the drive FOMNN (46) and the response FOMNN (8) are globally synchronized via the controller . Let ; then, the chaotic behaviors, error and state trajectories of the drive FOMNN (46), and the response FOMNN (8) via the controller are shown in Figure 4.

(a)

(b)

(c)

(d)

5. Conclusions

In this paper, the synchronization issue for FOMNNs has been investigated via state feedback control. To achieve the synchronization for the considered drive-response FOMNNs, feedback controllers are first introduced. Then, by adopting nonsmooth analysis, fractional Lyapunov’s direct method and Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization for the drive-response FOMNNs. Finally, an example is given to illustrate the effectiveness of the theoretical results. In future research, theoretical results here will be used to address the state estimation of FOMNNs.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Open Research Fund of the Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, Anhui Polytechnic University, under Grant GDSC202012.

References

H. Wang, Y. Yu, and G. Wen, “Stability analysis of fractional-order Hopfield neural networks with time delays,” Neural Networks, vol. 55, pp. 98–109, 2014.
View at: Publisher Site | Google Scholar
B. Meng, X. Wang, Z. Zhang, and Z. Wang, “Necessary and sufficient conditions for normalization and sliding mode control of singular fractional-order systems with uncertainties,” Science China Information Sciences, vol. 63, no. 5, Article ID 152202, 2020.
View at: Publisher Site | Google Scholar
A. G. Radwan, A. S. Elwakil, and A. M. Soliman, “Fractional-order sinusoidal oscillators: design procedure and practical examples,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 55, no. 7, pp. 2051–2063, 2008.
View at: Publisher Site | Google Scholar
M. Fellah, Z. E. A. Fellah, and C. Depollier, “Transient wave propagation in inhomogeneous porous materials: application of fractional derivatives,” Signal Processing, vol. 86, no. 10, pp. 2658–2667, 2006.
View at: Publisher Site | Google Scholar
F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons & Fractals, vol. 7, no. 9, pp. 1461–1477, 1996.
View at: Publisher Site | Google Scholar
J. Kytariolos, A. Dokoumetzidis, and P. Macheras, “Power law IVIVC: an application of fractional kinetics for drug release and absorption,” European Journal of Pharmaceutical Sciences, vol. 41, no. 2, pp. 299–304, 2010.
View at: Publisher Site | Google Scholar
K. Rajagopal, L. Guessas, A. Karthikeyan, A. Srinivasan, and G. Adam, “Fractional-order and memristive nonlinear systems: advances and applications,” Complexity, vol. 2017, Article ID 1892618, 19 pages, 2017.
View at: Publisher Site | Google Scholar
Y. Yu, H. Bao, M. Shi, B. Bao, Y. Chen, and M. Chen, “Complex dynamical behaviors of a fractional-order system based on a locally active memristor,” Complexity, vol. 2019, Article ID 2051053, 13 pages, 2019.
View at: Publisher Site | Google Scholar
X. Li and T. Huang, “Adaptive synchronization for fuzzy inertial complex-valued neural networks with state-dependent coefficients and mixed delays,” Fuzzy Sets and Systems, 2020.
View at: Publisher Site | Google Scholar
W. Zhang, J. Cao, D. Chen, and A. Alsaedi, “Out lag synchronization of fractional order delayed complex networks with coupling delay via pinning control,” Complexity, vol. 2019, Article ID 5612150, 7 pages, 2019.
View at: Publisher Site | Google Scholar
H. Li, J.-a. Fang, X. Li, and T. Huang, “Exponential synchronization of multiple impulsive discrete-time memristor-based neural networks with stochastic perturbations and time-varying delays,” Neurocomputing, vol. 392, pp. 86–97, 2020.
View at: Publisher Site | Google Scholar
X. Li, J.-a. Fang, H. Li, and W. Duan, “Exponential stabilization of time-varying delayed complex-valued memristor-based neural networks via impulsive control,” Asian Journal of Control, vol. 21, no. 6, pp. 2290–2301, 2019.
View at: Publisher Site | Google Scholar
W. Zhang, Y. Tang, W. K. Wong, and Q. Miao, “Stochastic stability of delayed neural networks with local impulsive effects,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 10, pp. 2336–2345, 2015.
View at: Publisher Site | Google Scholar
X. Li, W. Zhang, J.-A. Fang, and H. Li, “Event-triggered exponential synchronization for complex-valued memristive neural networks with time-varying delays,” IEEE Transactions on Neural Networks and Learning Systems, 2020.
View at: Publisher Site | Google Scholar
Y. Fan, X. Huang, H. Shen, and J. Cao, “Switching event-triggered control for global stabilization of delayed memristive neural networks: an exponential attenuation scheme,” Neural Networks, vol. 117, pp. 216–224, 2019.
View at: Publisher Site | Google Scholar
J. Jia, X. Huang, Y. Li, J. Cao, and A. Alsaedi, “Global stabilization of fractional-order memristor-based neural networks with time delay,” IEEE Transactions on Neural Networks and Learning Systems, vol. 31, no. 3, pp. 997–1009, 2019.
View at: Google Scholar
Y. Fan, X. Huang, Y. Li, J. Xia, and G. Chen, “Aperiodically intermittent control for quasi-synchronization of delayed memristive neural networks: an interval matrix and matrix measure combined method,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 11, pp. 2254–2265, 2018.
View at: Google Scholar
X. Li, T. Huang, and J.-a. Fang, “Event-triggered stabilization for takagi-sugeno fuzzy complex-valued memristive neural networks with mixed time-varying delays,” IEEE Transactions on Fuzzy Systems, 2020.
View at: Publisher Site | Google Scholar
H.-H. Lian, S.-P. Xiao, H. Yan, F. Yang, and H.-B. Zeng, “Dissipativity analysis for neural networks with time-varying delays via a delay-product-type lyapunov functional approach,” IEEE Transactions on Neural Networks and Learning Systems, 2020.
View at: Publisher Site | Google Scholar
X.-M. Zhang, Q.-L. Han, X. Ge, and D. Ding, “An overview of recent developments in lyapunov-krasovskii functionals and stability criteria for recurrent neural networks with time-varying delays,” Neurocomputing, vol. 313, pp. 392–401, 2018.
View at: Publisher Site | Google Scholar
X.-M. Zhang, Q.-L. Han, Z. Wang, and B.-L. Zhang, “Neuronal state estimation for neural networks with two additive time-varying delay components,” IEEE Transactions on Cybernetics, vol. 47, no. 10, pp. 3184–3194, 2017.
View at: Publisher Site | Google Scholar
M. Egmont-Petersen, D. de Ridder, and H. Handels, “Image processing with neural networks-a review,” Pattern Recognition, vol. 35, no. 10, pp. 2279–2301, 2002.
View at: Publisher Site | Google Scholar
Y. V. Pershin and M. Di Ventra, “Experimental demonstration of associative memory with memristive neural networks,” Neural Networks, vol. 23, no. 7, pp. 881–886, 2010.
View at: Publisher Site | Google Scholar
Y. Tang, Z. Wang, and J. A. Fang, “Pinning control of fractional-order weighted complex networks,” Chaos, vol. 19, no. 1, 2009.
View at: Publisher Site | Google Scholar
C. Song and J. Cao, “Dynamics in fractional-order neural networks,” Neurocomputing, vol. 142, pp. 494–498, 2014.
View at: Publisher Site | Google Scholar
L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990.
View at: Publisher Site | Google Scholar
S. Wen, Z. Zeng, T. Huang, and Y. Zhang, “Exponential adaptive lag synchronization of memristive neural networks via fuzzy method and applications in pseudorandom number generators,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 6, pp. 1704–1713, 2014.
View at: Publisher Site | Google Scholar
X. Li, J.-a. Fang, and H. Li, “Exponential stabilisation of stochastic memristive neural networks under intermittent adaptive control,” IET Control Theory & Applications, vol. 11, no. 15, pp. 2432–2439, 2017.
View at: Publisher Site | Google Scholar
L. Chua, “Memristor-The missing circuit element,” IEEE Transactions on Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971.
View at: Publisher Site | Google Scholar
D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 80–83, 2008.
View at: Publisher Site | Google Scholar
X. Li, J.-a. Fang, and H. Li, “Exponential stabilisation of memristive neural networks under intermittent output feedback control,” International Journal of Control, vol. 91, no. 8, pp. 1848–1860, 2018.
View at: Publisher Site | Google Scholar
X. Li, J.-a. Fang, and H. Li, “Master-slave exponential synchronization of delayed complex-valued memristor-based neural networks via impulsive control,” Neural Networks, vol. 93, pp. 165–175, 2017.
View at: Publisher Site | Google Scholar
X. Yang, Y. Feng, K. F. C. Yiu, Q. Song, and F. E. Alsaadi, “Synchronization of coupled neural networks with infinite-time distributed delays via quantized intermittent pinning control,” Nonlinear Dynamics, vol. 94, no. 3, pp. 2289–2303, 2018.
View at: Publisher Site | Google Scholar
X. Li, J.-a. Fang, and H. Li, “Exponential synchronization of stochastic memristive recurrent neural networks under alternate state feedback control,” International Journal of Control, Automation and Systems, vol. 16, no. 6, pp. 2859–2869, 2018.
View at: Publisher Site | Google Scholar
S. Yang, C. Li, and T. Huang, “Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control,” Neural Networks, vol. 75, pp. 162–172, 2016.
View at: Publisher Site | Google Scholar
X. Li, J.-a. Fang, W. Zhang, and H. Li, “Finite-time synchronization of fractional-order memristive recurrent neural networks with discontinuous activation functions,” Neurocomputing, vol. 316, pp. 284–293, 2018.
View at: Publisher Site | Google Scholar
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
J. Chen, Z. Zeng, and P. Jiang, “Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks,” Neural Networks, vol. 51, pp. 1–8, 2014.
View at: Publisher Site | Google Scholar
H. Deng and M. Krstić, “Stochastic nonlinear stabilization-I: a backstepping design,” Systems & Control Letters, vol. 32, no. 3, pp. 143–150, 1997.
View at: Publisher Site | Google Scholar
W. Cui, S. Sun, J.-a. Fang, Y. Xu, and L. Zhao, “Finite-time synchronization of Markovian jump complex networks with partially unknown transition rates,” Journal of the Franklin Institute, vol. 351, no. 5, pp. 2543–2561, 2014.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Xiaofan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

237

Downloads

860

Citations