From 2D to 3D Bifurcation Structures in Field Oriented Control of a PMSM

Abstract

This paper presents new aspects of bifurcation structures in a Permanent Magnet Synchronous Machine (PMSM) in both motor and generator operating modes. It considers the case where the PMSM speed is regulated with a Field-oriented control (FOC). A period doubling bifurcation cascade under the variation of a proportional control parameter leading to chaotic states, was identified. The investigation of parametric singularities allows us to identify a complex bifurcation structure including three generic bifurcations. Such structure is made of the Limit Point (LP), the Hopf (H) and the Bogdanov-Takens (BT) bifurcation sets. Sufficient conditions of the existence of the main bifurcations are given analytically. An overview of the phase space singularities associated to the parametric singularities is presented. Moreover, embedding 2D bifurcation sets in a 3D parametric space, led to identify certain bifurcation surface structures. The bifurcation surfaces established correspond to limit point bifurcation for the motor and generator operating mode, and for period doubling period bifurcation. The 3D bifurcation sets play an important role to study the combined effect of three different parameters on the PMSM dynamics, and permit to control the complex PMSM dynamics to a stable equilibrium dot by an adequate tuning of either control and system parameters.