A variable-order fractional proportional-integral controller and its application to a permanent magnet synchronous motor

The time-domain approximation of the Grünwald–Letnikov fractional derivative is intuitive and widely adopted in the design of fractional-order proportional-integral (FOPI) controllers. To solve the accuracy reduction caused by truncating the series, an optimized discrete FOPI is presented. The effectiveness of the new FOPI is highlighted and compared with the one exhibited by a controller implemented based on the Oustaloup method. Furthermore, to improve the performance of the FOPI, a variable-order fractional proportional-integral (VFPI) controller is proposed. The response of the VFPI is verified in the control of a permanent magnet synchronous motor. Simulation and experimental results show the superior performance of the VFPI.The authors would like to express their deep gratitude to the Editors and the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper. This work was supported by the National Natural Science Funds of China (No. 11971032), the Fundamental Research Funds for the Central Universities (No. JZ2019HGTB0090), Ministry of Education China Mobile Research Fund (MCM20180404) and Chongqing Basic Research and Frontier Exploration Project (cstc2018jcyjAX0167).