# Dynamical analysis of the permanent-magnet synchronous motor chaotic system

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## 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 simulations## 1 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, 2, 3, 4, 5, 6, 7, 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, 10, 11, 12, 13, 14, 15, 16, 17, 18, 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, 23, 24]. However, little seems to be known about the global exponential attractive set of chaotic systems [22, 23, 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

*ω*are the state variables, which represent currents and motor angular frequency, respectively; \(u_{d}\) and \(u_{q}\) represent the direct- and quadrature-axis stator voltage components, respectively;

*J*represents the polar moment of inertia; \(T_{L}\) represents the external load torque;

*β*represents the viscous damping coefficient; \(R_{1}\) represents the stator winding resistance; \(L_{d}\) and \(L_{q}\) represent the direct- and quadrature-axis stator inductors, respectively; \(\psi_{r}\) represents the permanent-magnet flux, and \(n_{p}\) represents the number of pole-pairs, the parameters \(L_{d}\), \(L_{q}\),

*J*, \(T_{L}\), \(R_{1}\), \(\psi_{r}\),

*β*are positive.

*x*,

*y*and

*z*are the new variables of the system (3), and the parameters

*γ*and

*σ*are positive constants.

*x*,

*y*and

*z*are the new variables of the system (4), and the parameters

*γ*and

*σ*are positive constants. There exist complex nonlinear dynamical behaviors in the system (4) including chaos and periodic orbit. The butterfly chaotic attractor of the system (4) with \(\gamma= 100\) and \(\sigma= 10\) in the \(xoyz\) space is shown in Figure 1. Chaotic attractors of the system (4) with \(\gamma= 100\) and \(\sigma= 10\) on the

*x*-

*y*,

*x*-

*z*, and

*y*-

*z*planes are shown in Figure 2.

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

*For*\(\forall\lambda> \frac{1}{\vert \varepsilon \vert } > 0\), \(L_{q} > 0\), \(L_{d} > 0\), \(\sigma> 0\),

*with*

*When*\(V_{\lambda} ( X ( t ) ) > L_{\lambda}\), \(V_{\lambda} ( X ( t_{0} ) ) > L_{\lambda}\),

*we can get the exponential estimate of the system*(2),

*given by*

*That is to say*,

*the set*

*is the global exponential attractive set of the permanent*-

*magnet synchronous motor system*(2).

### Proof

*t*along the trajectory of system (2) yields

This completes the proof. □

### Theorem 2

*For*\(\forall\lambda> 0\), \(\sigma> 0\),

*with*

*When*\(V_{\lambda} ( X ( t ) ) > M_{\lambda}\), \(V_{\lambda} ( X ( t_{0} ) ) > M_{\lambda}\),

*we can get the exponential estimate of the system*(3),

*given by*

*That is to say*,

*the set*

*is the global exponential attractive set of the permanent*-

*magnet synchronous motor system*(3).

### Proof

This completes the proof. □

### Remark 1

*x*and

*z*in system (4), the PMSM model (4) can be written in the form of the Lorenz system (11) as follows:

## 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.

## Notes

### 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.

## References

- 1.Lorenz, EN: Deterministic nonperiodic flows. J. Atmos. Sci.
**20**, 130-141 (1963) CrossRefGoogle Scholar - 2.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) MathSciNetCrossRefMATHGoogle Scholar - 3.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) MathSciNetCrossRefGoogle Scholar - 4.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) CrossRefGoogle Scholar - 5.Leonov, GA: Bounds for attractors and the existence of homoclinic orbits in the Lorenz system. J. Appl. Math. Mech.
**65**, 19-32 (2001) MathSciNetCrossRefMATHGoogle Scholar - 6.Hu, J, Chen, SH, Chen, L: Adaptive control for anti-synchronization of Chua’s chaotic system. Phys. Lett. A
**339**, 455-460 (2005) CrossRefMATHGoogle Scholar - 7.Leonov, G, Bunin, A, Koksch, N: Attractor localization of the Lorenz system. Z. Angew. Math. Mech.
**67**, 649-656 (1987) MathSciNetCrossRefMATHGoogle Scholar - 8.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) MathSciNetCrossRefGoogle Scholar - 9.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) MathSciNetCrossRefMATHGoogle Scholar - 10.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) MathSciNetCrossRefMATHGoogle Scholar - 11.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) MathSciNetCrossRefMATHGoogle Scholar - 12.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) CrossRefGoogle Scholar - 13.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) CrossRefMATHGoogle Scholar - 14.Elsayed, EM: Solutions of rational difference system of order two. Math. Comput. Model.
**55**, 378-384 (2012) MathSciNetCrossRefMATHGoogle Scholar - 15.Elsayed, EM: Solution for systems of difference equations of rational form of order two. Comput. Appl. Math.
**33**(3), 751-765 (2014) MathSciNetCrossRefMATHGoogle Scholar - 16.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) CrossRefMATHGoogle Scholar - 17.Lin, D, Zhang, FC, Liu, JM: Symbolic dynamics-based error analysis on chaos synchronization via noisy channels. Phys. Rev. E
**90**, 012908 (2014) CrossRefGoogle Scholar - 18.Zhang, FC, Zhang, GY: Dynamics of a low-order atmospheric circulation chaotic model. Optik
**127**(8), 4105-4108 (2016) CrossRefGoogle Scholar - 19.Niu, YJ, Wang, XY: An anonymous key agreement protocol based on chaotic maps. Commun. Nonlinear Sci. Numer. Simul.
**16**(4), 1986-1992 (2011) MathSciNetCrossRefMATHGoogle Scholar - 20.Jing, ZJ, Yu, C, Chen, GR: Complex dynamics in a permanent-magnet synchronous motor model. Chaos Solitons Fractals
**22**, 831-844 (2004) MathSciNetCrossRefMATHGoogle Scholar - 21.Chen, Q, Ren, XM, Na, J: Robust finite-time chaos synchronization of uncertain permanent magnet synchronous motors. ISA Trans.
**58**, 262-269 (2015) CrossRefGoogle Scholar - 22.Wang, XY, Wang, MJ: A hyperchaos generated from Lorenz system. Physica A
**387**(14), 3751-3758 (2008) MathSciNetCrossRefGoogle Scholar - 23.Wang, XY, Wang, MJ: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos
**17**(3), 033106 (2007) CrossRefMATHGoogle Scholar - 24.Zhang, YQ, Wang, XY: A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice. Inf. Sci.
**273**, 329-351 (2014) CrossRefGoogle Scholar - 25.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) CrossRefMATHGoogle Scholar - 26.Wang, XY, He, YJ: Projective synchronization of fractional order chaotic system based on linear separation. Phys. Lett. A
**372**(4), 435-441 (2008) CrossRefMATHGoogle Scholar - 27.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 - 28.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) MathSciNetCrossRefMATHGoogle Scholar - 29.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) MathSciNetCrossRefMATHGoogle Scholar - 30.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) MathSciNetCrossRefMATHGoogle Scholar - 31.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

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