Finite-time control and synchronization for memristor-based chaotic system via impulsive adaptive strategy
- Wei Xiong^{1} and
- Junjian Huang^{2}Email author
https://doi.org/10.1186/s13662-016-0789-3
© Xiong and Huang 2016
Received: 8 November 2015
Accepted: 22 February 2016
Published: 8 April 2016
Abstract
This paper investigates the stabilizing and synchronization problems of a memristor-based Chua chaotic system in a finite time. A lemma concerning the finite-time stability for an impulsive system is proposed by extending the finite-time stability theory. Then some finite-time stabilizing and synchronization criterion are presented which guarantee the finite-time stabilization and synchronization for the model considered. Finally, the efficiency of the control scheme is further demonstrated by the simulation examples.
Keywords
finite time control and synchronization memristor impulsive adaptive strategy1 Introduction
In 1971, the memristor which is considered to be the missing fourth passive circuit element was postulated [1]. However, this important postulation has not caused attention in almost 40 years. Until in 2008, Hewlett-Packard Labs announced the development of a memristor based on nanotechnology [2]. As we know, the memristor takes its place along with the other three existing elements: the resistor, the capacitor, and the inductor. Increasing focus was put on the memristor for its potential applications in programmable logic, signal processing, neural networks, and so on [3].
Moreover, as the novel element, the circuit based on the memristor shares many interesting phenomenon. Recently, the research memristor chaotic circuits have become a focal topic [4–10]. In [4], the author presented a novel fourth-order memristor-based Chua oscillator by replacing Chua’s diode with an active two-terminal circuit. The stabilization problem of a memristor-based chaotic system was investigated in [10]. As the most important phenomenon, the synchronization was also discussed [9]. In [9], the adaptive synchronization problem of memristor-based Chua circuits was investigated.
As time goes on, more and more researchers began to realize the important role of the synchronization time. To attain a high convergence speed, many effective methods have been introduced and finite-time control is one of them. Finite-time synchronization means the optimality in convergence time. Much research work has been done on chaos synchronization based on finite time (see for instance [11–22] and the references therein). However, the finite-time synchronization problem has not been fully investigated in the literature, and it still remains open. Motivated by the above discussion, we investigate the finite-time synchronization problem for a memristor-based Chua circuit. Based on the finite-time stability theory, a novel lemma which guarantees the impulsive system is finite-time stable is presented. Then the impulsive [16–18] adaptive control law is proposed to realize finite-time synchronization of the model considered. Numerical simulations demonstrate the effectiveness and correctness of this results.
The paper is organized as follows. Some preliminaries are presented in the next section. Section 3 proposes the main results of this paper. In Section 4, the numerical simulations are presented, which is followed by the conclusion in Section 5.
2 Preliminaries
Assumption 1
Lemma 1
Proof
3 Main results
Theorem 1
- (i)
\(q = \lambda_{\max} [ A^{T} + A - ( 2k_{1} + 1 )I ] < 0\);
- (ii)
\(d = \lambda_{\max} ( I + B )^{T} ( I + B ) < 1\).
Proof
Theorem 2
- (i)
\(A^{T} + A - 2k_{1}I - 2abM_{1}M_{2}I < 0\);
- (ii)
\(d = \lambda_{\max} ( I + B )^{T} ( I + B ) < 1\).
Proof
4 Simulation results
5 Conclusion
In this paper, the finite-time control and synchronization problems of memristor-based chaotic systems have been investigated. Some novel impulsive adaptive control laws which guarantee the memristor-based Chua circuits is stabilized and synchronized in finite time have been proposed. Moreover, simulation results were given to verify the effectiveness and feasibility of the method. Our future research topics mainly consider the time delay effects on the finite-time stability of the memristor-based nonlinear system.
Declarations
Acknowledgements
The work described in this paper was partially supported by the National Natural Science Foundation of China (Grant No. 61403050) and the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1501412, KJ1501409, KJ1501301), and the Foundation of CQUE (KY201519B, KY201520B).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Chua, LO: Memristor - the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507-511 (1971) View ArticleGoogle Scholar
- Strukov, DB, Snider, GS, Stewart, DR, Williams, RS: The Missing Memristor Found. Nature 453, 80-83 (2008) View ArticleGoogle Scholar
- Bayat, FM, Shouraki, SB: Programming of memristor crossbars by using genetic algorithm. Proc. Comput. Sci. 3, 232-237 (2011) View ArticleGoogle Scholar
- Itoh, M, Chua, LO: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183-3206 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Pakkiyappan, R, Sivasamy, R, Li, XD: Synchronization of identical and nonidentical memristor-based chaotic systems via active back stepping control technique. Circuits Syst. Signal Process. 34(3), 763-778 (2015) View ArticleGoogle Scholar
- Wu, HG, Chen, SY, Bao, BC: Impulsive synchronization and initial value effect for a memristor-based chaotic system. Acta Phys. Sin. 64(3), 030501 (2015) MathSciNetGoogle Scholar
- Wang, X, Li, CD, Huang, TW, Duan, SK: Predicting chaos in memristive oscillators via harmonic balance method. Chaos 22(4), 043119 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Bao, HB, Cao, JD: Projective synchronization of fractional-order memristor-based neural networks. Neural Netw. 63, 1-9 (2015) View ArticleMATHGoogle Scholar
- Wen, SP, Zeng, ZG, Huang, TW: Adaptive synchronization of memristor-based Chua’s circuits. Phys. Lett. A 376, 2275-2780 (2012) View ArticleGoogle Scholar
- Huang, JJ, Li, CD, He, X: Stabilization of a memristor-based chaotic system by intermittent and fuzzy processing. Int. J. Control. Autom. Syst. 11(3), 643-647 (2013) View ArticleGoogle Scholar
- Amato, F, Ariola, M, Abdallah, CT: Finite-time control for uncertain linear systems with disturbance inputs. In: IEEE Proceedings of the 1999 American Control Conference, vol. 3, pp. 1776-1780 (1999) Google Scholar
- Amato, F, Arioila, M, Dorato, P: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459-1463 (2001) View ArticleMATHGoogle Scholar
- Amato, F, Arioila, M, Cosentino, C: Finite-time stability of linear time-varying systems: analysis and controller design. IEEE Trans. Autom. Control 55(4), 1003-1008 (2009) MathSciNetView ArticleGoogle Scholar
- Sanjay, PB, Dennis, SB: Finite time of continuous autonomous systems. SIAM J. Control Optim. 3(3), 751-766 (2000) MathSciNetMATHGoogle Scholar
- Liu, YG: Global finite time stabilization via time varying feedback uncertain nonlinear systems. SIAM J. Control Optim. 52(3), 1886-1913 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Li, X, Bohner, M, Wang, CK: Impulsive differential equations: periodic solutions and applications. Automatica 52, 173-178 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Li, X, Song, S: Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans. Neural Netw. 24, 868-877 (2013) View ArticleGoogle Scholar
- Wen, SP, Huang, TW, Yu, XH, Chen, MZQ, Zeng, ZG: Aperiodic sampled-data sliding-mode control of fuzzy systems with communication delays via the event-triggered method. IEEE Trans. Fuzzy Syst. (2015). doi:https://doi.org/10.1109/TFUZZ.2015.2501412 Google Scholar
- Wen, SP, Yu, XH, Zeng, ZG, Wang, JJ: Event-triggering load frequency control for multi-area power systems with communication delay. IEEE Trans. Ind. Electron. 63(2), 1308-1317 (2015) View ArticleGoogle Scholar
- Song, QK, Huang, TW: Stabilization and synchronization of chaotic systems with mixed time-varying delays via intermittent control with non-fixed both control period and control width. Neurocomputing 154, 61-69 (2015) View ArticleGoogle Scholar
- Song, QK, Zhao, ZJ: Stability criterion of complex-valued neural networks with both leakage delay and time-varying delays on time scales. Neurocomputing 171, 179-184 (2016) View ArticleGoogle Scholar
- Cao, JD, Song, QK: Stability in Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays. Nonlinearity 19(7), 1601 (2006) MathSciNetView ArticleMATHGoogle Scholar