Bifurcation Control and Complex Dynamics in Field-Oriented Control of a PMSM

Abstract

The general purpose of this paper is to develop new aspects of bifurcation structures in a 3D parametric space. Identification of generic bifurcation structures in former studies was based on the arrangement of bifurcation curves in the parameter plane. So by analogy to such studies, we define the bifurcation surface in 3D parameter space as the main feature of the said generic structures. The implementation of this idea is made on the permanent magnet synchronous machine (PMSM) whose speed is regulated with a field-oriented control (FOC). Sufficient conditions are given for the existence of three main bifurcations: limit point (LP), Hopf (H) and Bogdanov–Takens (BT). Starting from bifurcation curves traced in a parameter plane and changing a third parameter, a qualitative bifurcation surface is constructed in a 3D parametric space. This led to underline the increasing complexity of the bifurcation structures when dealing with more than two parameters. This study put into evidence not only the complex behavior of PMSM, but stands as a starting point for a new formalism on the bifurcation structures in a 3D parametric space.