Dynamical analysis of the permanent-magnet synchronous motor chaotic system
- Fuchen Zhang^{1, 2}Email author,
- Xiaofeng Liao^{3} and
- Chunlai Mu^{4}
https://doi.org/10.1186/s13662-017-1118-1
© The Author(s) 2017
Received: 10 October 2016
Accepted: 15 February 2017
Published: 9 March 2017
Abstract
This paper is concerned with some dynamics of the permanent-magnet synchronous motor chaotic system based on Lyapunov stability theory and optimization theory. The innovation of the paper lies in that we derive a family of mathematical expressions of globally exponentially attractive sets for this chaotic system with respect to system parameters. Numerical simulations confirm that theoretical analysis results are correct.
Keywords
permanent-magnet synchronous motor chaotic attractors Lyapunov stability numerical simulations1 Introduction
Since Lorenz et al. were the first to investigate the Lorenz equations in 1963, chaotic systems have played an important role in a variety of industrial fields [1–8]. As is well known, the research on chaos is not limited to the fields of mathematics and physics. It is found that chaos widely exists in the fields of meteorology, medicine, computer science, economics, mechanical engineering, cryptography, and so on [9–19]. However, it was not until the 1990s that chaos has gradually attracted enough attention due to the findings in practical engineering. From the point of view of the potential application of chaos theory in practical engineering, many efforts have been made to study chaos in the past 20 years.
This paper mainly focuses on the chaotic system model from a permanent-magnet synchronous motor (PMSM) which is a nonlinear, multivariable, and strong coupling system. A permanent-magnet synchronous motor is a kind of highly efficient and high-powered motor, which has been widely used in the industry. Usually, the dynamics of a PMSM is modeled as a three-dimensional autonomous differential equation [20, 21]. Dynamical behaviors of the PMSM, such as periodic solutions, chaos phenomena, phase portraits, bifurcation diagrams, Lyapunov exponents, chaos anti-control and chaos synchronization, have been widely studied in [20, 21].
In recent years, dynamical behaviors of chaotic systems, such as stability, periodic solutions, circuit implementation, image encryption algorithm, chaos synchronization, chaos attractors, heteroclinic orbits and homoclinic orbits, have been extensively investigated [22–24]. However, little seems to be known about the global exponential attractive set of chaotic systems [22–24]. Despite the fact that many qualitative and quantitative results on the permanent-magnet synchronous motor system have been obtained [20, 21], there is a fundamental question that has not been completely answered so far: is there a global exponential attractive set for the permanent-magnet synchronous motor system? Global exponential attractive sets play an important role in dynamical systems. The global exponential attractive set is also very important for engineering applications, since it is very difficult to predict the existence of hidden attractors and they can lead to crashes [10]. Therefore, how to get the global attractive sets of a chaotic dynamical system is particularly significant both for theoretical research and practical applications. In [25, 26], one shows that Lyapunov functions can be used to study chaos synchronization. However, Lyapunov-like functions used in [16, 18, 25, 26] cannot be used to study the global attractive sets for the permanent-magnet synchronous motor system. In this paper, a new Lyapunov-like function is constructed to investigate the global attractive sets of the permanent-magnet synchronous motor system.
Motivated by the above discussion, we will investigate the global attractive sets of the permanent-magnet synchronous motor system. The meaning of the contribution of this article is that not only do we derive a family of mathematical expressions of global exponential attractive sets for permanent-magnet synchronous motor systems in [20, 21] with respect to the parameters of the system, but we also get the rate of the trajectories of the system going from the exterior of the trapping set to the interior of the trapping set.
The rest of the paper is organized as follows. The permanent-magnet synchronous motor (PMSM) model is given in Section 2. In Section 3, we prove that there exist global exponential attractive sets for the chaotic PMSM system. Some numerical simulations are also given in Section 3. Section 4 gives conclusions.
2 Permanent-magnet synchronous motor model
The periodic and chaos phenomena, phase portraits, bifurcation diagrams, Lyapunov exponents, chaos anti-control of the permanent-magnet synchronous motors (2), (3) and (4) are widely studied in [20, 21] in detail. But the global exponential attractive sets of systems (2)-(4) are still unknown. Our principal aim here is to investigate the global exponential attractive sets of (2), (3) and (4).
3 Dynamics of the PMSM
In this section, we will discuss the global exponential attractive sets of PMSM system (2), (3) and (4). We have the following results.
Theorem 1
Proof
This completes the proof. □
Theorem 2
Proof
This completes the proof. □
Remark 1
4 Conclusions
In this paper, the global attractive sets of the permanent-magnet synchronous motor have been obtained based on dynamical systems theory. This method can be applied to consider other chaotic systems. In the future we will conduct research on how to control the PMSM to avoid the chaotic behavior and protect the motors in practical applications.
Declarations
Acknowledgements
This work is supported by National Natural Science Foundation of China (Grant Nos. 11501064, 11426047), the Basic and Advanced Research Project of CQCSTC (Grant No. cstc2014jcyjA00040), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1500605), the Research Fund of Chongqing Technology and Business University (Grant No. 2014-56-11), China Postdoctoral Science Foundation (Grant No. 2016M590850) and the Program for University Innovation Team of Chongqing (Grant No. CXTDX201601026). We thank professors Min Xiao in the College of Automation, Nanjing University of Posts and Telecommunications and Gaoxiang Yang at the Department of Mathematics and Statistics of Ankang University for their help. The authors wish to thank the editors and reviewers for their conscientious reading of this paper and their numerous comments for improvement which were extremely useful and helpful in modifying the paper.
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
- Lorenz, EN: Deterministic nonperiodic flows. J. Atmos. Sci. 20, 130-141 (1963) View ArticleGoogle Scholar
- Zhang, FC, Mu, CL, Zhou, SM, Zheng, P: New results of the ultimate bound on the trajectories of the family of the Lorenz systems. Discrete Contin. Dyn. Syst., Ser. B 20(4), 1261-1276 (2015) MathSciNetView ArticleMATHGoogle Scholar
- He, P, Jing, CG, Fan, T, Chen, CZ: Robust decentralized adaptive synchronization of general complex networks with coupling delayed and uncertainties. Complexity 19, 10-26 (2013) MathSciNetView ArticleGoogle Scholar
- Leonov, GA, Kuznetsov, NV, Mokaev, TN: Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top. 224(8), 1421-1458 (2015) View ArticleGoogle Scholar
- Leonov, GA: Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. J. Appl. Math. Mech. 65, 19-32 (2001) MathSciNetView ArticleMATHGoogle Scholar
- Hu, J, Chen, SH, Chen, L: Adaptive control for anti-synchronization of Chua’s chaotic system. Phys. Lett. A 339, 455-460 (2005) View ArticleMATHGoogle Scholar
- Leonov, G, Bunin, A, Koksch, N: Attractor localization of the Lorenz system. Z. Angew. Math. Mech. 67, 649-656 (1987) MathSciNetView ArticleMATHGoogle Scholar
- Kuznetsov, NV, Mokaev, TN, Vasilyev, PA: Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor. Commun. Nonlinear Sci. Numer. Simul. 19, 1027-1034 (2014) MathSciNetView ArticleGoogle Scholar
- Leonov, GA: General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems. Phys. Lett. A 376, 3045-3050 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Bragin, V, Vagaitsev, V, Kuznetsov, N, Leonov, G: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 50, 511-543 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Leonov, GA, Kuznetsov, NV: Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23, 1330002 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Leonov, GA, Kuznetsov, NV, Kiseleva, MA, Solovyeva, EP, Zaretskiy, AM: Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn. 77, 277-288 (2014) View ArticleGoogle Scholar
- Liu, HJ, Wang, XY, Zhu, QL: Asynchronous anti-noise hyper chaotic secure communication system based on dynamic delay and state variables switching. Phys. Lett. A 375, 2828-2835 (2011) View ArticleMATHGoogle Scholar
- Elsayed, EM: Solutions of rational difference system of order two. Math. Comput. Model. 55, 378-384 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Elsayed, EM: Solution for systems of difference equations of rational form of order two. Comput. Appl. Math. 33(3), 751-765 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, FC, Mu, CL, Li, XW: On the boundedness of some solutions of the Lu system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22, 1250015 (2012) View ArticleMATHGoogle Scholar
- Lin, D, Zhang, FC, Liu, JM: Symbolic dynamics-based error analysis on chaos synchronization via noisy channels. Phys. Rev. E 90, 012908 (2014) View ArticleGoogle Scholar
- Zhang, FC, Zhang, GY: Dynamics of a low-order atmospheric circulation chaotic model. Optik 127(8), 4105-4108 (2016) View ArticleGoogle Scholar
- Niu, YJ, Wang, XY: An anonymous key agreement protocol based on chaotic maps. Commun. Nonlinear Sci. Numer. Simul. 16(4), 1986-1992 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Jing, ZJ, Yu, C, Chen, GR: Complex dynamics in a permanent-magnet synchronous motor model. Chaos Solitons Fractals 22, 831-844 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Chen, Q, Ren, XM, Na, J: Robust finite-time chaos synchronization of uncertain permanent magnet synchronous motors. ISA Trans. 58, 262-269 (2015) View ArticleGoogle Scholar
- Wang, XY, Wang, MJ: A hyperchaos generated from Lorenz system. Physica A 387(14), 3751-3758 (2008) MathSciNetView ArticleGoogle Scholar
- Wang, XY, Wang, MJ: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos 17(3), 033106 (2007) View ArticleMATHGoogle Scholar
- Zhang, YQ, Wang, XY: A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice. Inf. Sci. 273, 329-351 (2014) View ArticleGoogle Scholar
- Wang, XY, Song, JM: Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3351-3357 (2009) View ArticleMATHGoogle Scholar
- Wang, XY, He, YJ: Projective synchronization of fractional order chaotic system based on linear separation. Phys. Lett. A 372(4), 435-441 (2008) View ArticleMATHGoogle Scholar
- Leonov, GA, Kuznetsov, NV: On differences and similarities in the analysis of Lorenz, Chen, and Lu systems. Appl. Math. Comput. 256, 334-343 (2015) MathSciNetMATHGoogle Scholar
- Algaba, A, Fernandez-Sanchez, F, Merino, M, Rodríguez-Luis, AJ: Chen’s attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system. Chaos 23(3), 033108 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Chen, YM, Yang, QG: The nonequivalence and dimension formula for attractors of Lorenz-type systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23(12), 1350200 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, FC, Zhang, GY: Further results on ultimate bound on the trajectories of the Lorenz system. Qual. Theory Dyn. Syst. 15(1), 221-235 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Liao, XX: Globally exponentially attractive sets and positive invariant sets of the of the Lorenz system and its application in chaos control and synchronization. Sci. China, Ser. E, Inf. Sci. 34, 1404-1419 (2004) Google Scholar