I. Introduction
A permanent-magnet synchronous motor (PMSM) is of great interest, particularly for industrial applications in the low- to medium-power range, since it has superior features such as compact size, high torque/weight ratio, high torque/inertia ratio, and absence of rotor losses. Over the past few years, the secure and stable operation of the PMSM, which is an essential requirement of industrial automation manufacturing, has received considerable attention. For instance, Li et al. [1] studied the dynamic characteristics of the PMSM based on bifurcation and chaos theory. It is found that, with the operating parameters changing, the PMSM exhibits complex behaviors such as limit cycles, chaos oscillation, etc. Jing et al. [2] fully investigated the complex dynamics in the PMSM with a nonsmooth air gap, extending the work on the smooth case. Their theoretical analysis and numerical simulations show some more new results in the PMSM, including period-doubling bifurcation, cyclic-fold bifurcation, single- and double-scroll chaotic attractors, ribbon chaotic attractors, as well as intermittent chaos. Reference [3] used a linear transformation to develop a more explicit form of the PMSM model for which the stability analysis and chaotic behavior are analyzed. Zaher [4] analyzed the stability of the PMSM in the uncontrolled and controlled cases. However, in those previous studies, the effect of the time-delayed feedback current on the dynamics of the PMSM is not taken into consideration. In fact, time-delayed feedback is a ubiquitous feature of electromechanical systems. Thus, it is necessary to pay great attention to the effects of the time-delayed feedback current on the dynamic behavior of the PMSM. Since the pioneer work of Pyragas [5], [6], the time-delayed feedback on the dynamics in nonlinear systems has widely been studied. For example, Abdallah et al. [7], [8] studied the effects of time delay on the static output feedback. They have generalized the earlier research on the presence of stability “switches” using a matrix pencil approach and have provided numerous examples to illustrate the applicability of their results. Ramana Reddy et al. [9] studied time-delay-induced amplitude death in two coupled nonlinear electronic circuits. They found that the amplitude of the time delay coupled limit-cycle oscillators can shrink to zero at the proper coupling strength and delay time. Ryu et al. [10] studied the effects of time-delayed feedback on coherent and incoherent chaotic oscillators with both fixed and varying delayed time. It is found that, for the fixed delayed time, the stability islands are found when the delay time is about , where is an integer and is the average period of the chaotic oscillator. In [11] and [12], the authors investigated the effects of three different types of time-delay modulations on the stabilization of the Lorenz system and found that the time-varying delay control scheme can considerably improve the efficiency of control of unstable steady states. All those works show that the existence of time-delayed feedback plays a vital role in the systeM's dynamics. The main aim of this brief is to investigate how the dynamic behavior of the PMSM depends on current time-delayed feedback, where the delay time is both fixed (static case) and varying (dynamic case) in time. We choose model parameters for which the PMSM displays, in the absence of feedback, chaotic oscillations. The stable operation islands of the PMSM are first investigated in the parameter space of feedback gain and delay time. Then, detailed regimes of motion can be explored by bifurcation diagrams for a fixed feedback gain as the delay time varies. It is found that the dynamic delay time feedback obtained stabilization of unstable steady states over a much larger domain of parameters in comparison with the static delay time feedback. The mechanism behind the action of current time-delayed feedback is also addressed last.
Equivalent circuit for the PMSM.
PMSM ParametersParameters | Nomenclature |
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Stator inductance in the axis | |
Stator inductance in the axis | |
Stator phase resistance | |
Torque constant | |
Moment of inertia | |
Load torque | |
-axis flux | |
-axis flux | |
Viscous damping coefficient |