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The aim of this study is to make a general exploration of the dynamic characteristics of the permanent magnet synchronous motor (PMSM) with parametric or external perturbation. The pitchfork, fold, and Hopf bifurcations are derived by using bifurcation theory. Simulation results not only confirm the theoretical analysis results but also show the Bogdanov–Takens bifurcation of the equilibrium. Dynamic behaviors, such as period three and chaotic motion of PMSM, are analyzed by using bifurcation diagram and phase portraits. The symmetric fold/fold bursting oscillation as well as two kinds of delayed pitchfork bursting oscillations is obtained, and different mechanisms are presented.

Since the permanent magnet synchronous motor (PMSM) has low inertia, low noise, high power density, and high efficiency, it is one of most important development trends in a variety of driving motors [

Scholars have done a lot of research on the dynamic characteristics of PMSM. Li et al. [

Bursting oscillation is a kind of fast-slow dynamic behavior when the dynamic system has two or more time scales [

This paper gives a general exploration of the dynamic characteristics of PMSM with parametric excitation or load excitation. PMSM parameters are not limited to a special case. The stability conditions and the bifurcation conditions of the equilibrium have been derived. The local dynamic characteristics of the equilibrium points and the global dynamic characteristics of PMSM are discussed by bifurcation set and bifurcation diagram, respectively. The dynamic behaviors at some specific parameters are studied by phase portraits and Poincaré sections, and three kinds of multiple time scale dynamic behaviors are obtained.

The dynamic model of PMSM can be described as a three-dimensional nonlinear system [

By applying an affine transformation and a time-scaling transformation, system (

System (

There are only two combined parameters,

By the analysis of the equilibrium points,

Solving the equilibrium points equations, we obtain the following:

If

If

The local stability of the equilibrium point is determined by the roots of the characteristic equation, and the equilibrium point is stable if the all the roots of the characteristic equation have negative real parts. The characteristic equation can be expressed as det (

The Jacobian matrix at the equilibrium points of system (

Substituting the Jacobian matrix and equilibrium,

According to Routh–Hurwitz stability criterion, the sufficient condition of the local stability of the equilibrium points is

By the bifurcation theory [

Equating real and imaginary parts of the equation,

Then, if Hopf bifurcation occurs at the equilibrium point of (

In addition, if

To verify the theoretical analysis results, the double-parameter bifurcation curves of system (

Bifurcation set in

Parameters of the PMSM [

Parameter | Value | Unit |
---|---|---|

_{d}/_{q} | 1.01 | mH |

Ψ_{R} | 0.06784 | Nm/A |

_{1} | 0.24 | Ω |

4.8 × 10^{−5} | kg m^{2} | |

_{n} | 4 | |

0.01619 | N/rad/s |

Multiequilibrium coexistence phenomenon.

This is a special case that the control inputs of the system are removed after the motor runs for a period of operation. Let

By analyzing the equilibrium,

According to the relationship among

According to the Routh–Hurwitz criteria, the local stability condition of the equilibrium can be written as

If fold bifurcation occurs at the equilibrium of (

If Hopf bifurcation occurs at the equilibrium of (

The bifurcation set of equilibrium of (

Bifurcation set in the

The dynamics of the PMSM model with parametric or external excitation will be discussed in this section. Based on the different order of magnitude of the excitation frequency, the dynamic behaviors are divided into two parts. The bifurcation and chaos are discussed in

In order to discuss the dynamic characteristics of the PMSM with parametric and external excitation, let

The Poincaré bifurcation diagram.

Period one motion,

Period two motion,

Period four motion,

Period three motion,

An interesting two time scale fast-slow dynamic [

Symmetric fold/fold bursting. (a) Phase portrait. (b) Time history. (c) Equilibrium curve. (d) Overlap of TPP and equilibrium curve.

Suppose the trajectory starts from “A” of the up piece of equilibrium curve “E_{+}” in Figure _{−}” as the stable equilibrium curve “E_{+}” disappears. Similar dynamics take place at the fold bifurcation point “LP2.” A cycle is completed when the trajectory is back to the initial point, and this dynamic behavior is caused by the fold bifurcation of the equilibrium curve and can be named as “symmetric fold/fold bursting.”

When considering about the parametric excitation, i.e., _{0} + Asin (

Delayed pitchfork bursting I;

Delayed pitchfork bursting II; _{0} = 1.5. (a) Phase portrait. (b) Overlap of TPP and equilibrium curve.

The nonlinear dynamic behaviors of the PMSM with parametric or load perturbation are generally studied. The primary conclusions of this paper are as follows:

Different kinds of bifurcations, such as pitchfork bifurcation, fold bifurcation, and Hopf bifurcation, are deduced. The bifurcation curves are also simulated.

The nonlinear dynamics of the PMSM with both parametric excitation and load excitation are simulated. Different kinds of motions, such as period one, period two, and period four, are obtained. The period three motion is also obtained which can prove the existence of chaotic motion.

When the frequency of the excitation is superlow, the fast-slow dynamics are obtained. Simulation results show three kinds of fast-slow dynamics: symmetric fold/fold bursting oscillation of the PMSM with external perturbation, two kinds of delayed pitchfork bursting oscillations of the PMSM with parametric perturbation. The mechanisms of those three kinds of fast-slow dynamics are also analyzed by using bifurcation curve of the equilibrium.

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

The authors declare that they have no conflicts of interest.

The authors thank Changzhou Institute of Mechatronic Technology and Jiangsu University for funding this study. The authors thank Jiangsu University Natural Science Foundation Project (16KJB580012) and Overseas Research and Training Program for Young and Middle-Aged Backbone Teachers in Jiangsu Province ((2016)13).