0885-8993 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPEL.2018.2876607, IEEE Transactions on Power Electronics Abstract— Torque ripple has been a critical issue for high- performance applications using permanent magnet synchronous machines (PMSMs). An efficient approach to minimize torque ripple is to control the stator current to follow an optimized current reference, which will produce an extra torque ripple to cancel the existing one. This paper proposes a multiple reference frame (MRF)-based controller for torque ripple minimization (TRM), in which the measured speed ripple is explored as the feedback control signal. The proposed MRF-based controller consists of a TRM controller whose task is to find the optimal current reference and a current controller whose task is to control the actual current to follow the optimized current reference. In TRM controller, the control of reference current magnitude and phase angle is decoupled, and PI controller is able to achieve TRM control. In current controller, MRF is adopted to convert harmonic current control into DC current control, thus PI controller is able to achieve harmonic current control with the use of MRF. Compared with existing approaches, the proposed controller is capable of TRM under both steady-state and transient conditions. The proposed controller is experimentally evaluated on a laboratory PMSM drive system. Index Terms— Harmonic current control, multiple reference frame, PMSM, speed harmonic, torque ripple minimization. I. INTRODUCTION In permanent magnet synchronous machines (PMSMs), torque ripple has been a critical issue that could prevent them from high-performance applications [1-5]. Indeed, even a PMSM is well designed, the torque ripple is still inevitable due to imperfections in the design and manufacture [6]. Hence, torque ripple modeling and minimization have received considerable research interests. In PMSMs, torque ripple is induced from many sources such as the harmonics in magnet flux linkage, cogging torque, and stator current, and the current and position measurement errors [7-11]. Here, the torque ripple due to the measurement errors can be eliminated through a compensation of these errors. However, the torque ripple due to magnet flux linkage harmonics and cogging torque, which is the dominant torque ripple in a PMSM, can only be cancelled through an injection of the harmonic current that will produce an extra torque ripple with the same magnitude but opposite in phase of the existing one [12-15]. Corresponding author: Guodong Feng G. Feng, J. Tian and N. C. Kar are with the Department of Electrical and Computer Engineering, University of Windsor, ON, Canada (emails: guodong.feng@uwindsor.ca, tian11f@uwindsor.ca, nkar@uwindsor.ca) C. Lai was with the University of Windsor and is now with the Department of Electrical and Computer Engineering, Concordia University, QC, Canada (email: chunyan.lai@concordia.ca) Therefore, optimal harmonic current design/control for torque ripple minimization (TRM) has been widely investigated in the literature. Existing approaches can be mainly divided into the feedforward-based and feedback-based approaches. In the feedforward-based approaches, a torque ripple model is firstly developed and employed to predict the torque ripple of a PMSM, and the harmonic current is then optimized to minimize the predicted torque ripple. Various optimization algorithms such as genetic algorithm, neural network and LaGrange multiplier are employed in [16-18] to optimize the harmonic current for TRM. However, these approaches are computationally complex due to iterative optimizations. In order to ensure computation efficiency, an analytical solution is proposed in [19], in which the optimal harmonic current is calculated from analytical equations. It should be emphasized that the feedforward-based approaches require comprehensive torque models as well as accurate machine parameters to provide accurate torque ripple prediction for optimal harmonic current design. However, machine parameters can vary significantly during machine operation [20-22]. Thus, the performance of these approaches is dependent on the accuracy of the torque ripple prediction for harmonic current design. In the feedback-based approaches, a feedback signal that can be used as a measure of the torque ripple is required. In the machine drive, the torque ripple can produce the speed ripple as well as vibrations [23-25], which can be explored as the feedback signal for TRM control. In [26], vibration signal measured from a piezoelectric sensor is employed to control the harmonic current to minimize the torque ripple. Compared with the vibration signal, the speed ripple can be directly obtained from the speed sensor/encoder that is available in a machine drive system. Hence, the speed ripple has been extensively explored for feedback TRM control in [27-32]. The existing studies reveal that a torque harmonic can produce a speed harmonic of the same order [22,29], and thus minimizing the torque harmonic is equivalent to minimizing the speed harmonic of the same order. In the literature, the iterative learning control, fuzzy logic, and proportional- integral-resonant based algorithms have been explored as the control strategy for TRM in [23,28,31,32]. Compared with feedforward-based approaches, the feedback-based approaches have advantages in terms of robustness to machine parameter variation and computation efficiency. Existing studies in [15, 29, 33] demonstrate that the optimal harmonic current for TRM is dependent on the operating conditions, which can also be observed from the torque ripple model in [16]. For instance, as the load torque changes, the magnitude and phase angle of the optimal harmonic current Multiple Reference Frame based Torque Ripple Minimization for PMSM Drive Under Both Steady-State and Transient Conditions Guodong Feng, Member, IEEE, Chunyan Lai, Member, IEEE, Jiangbo Tian, Student Member, IEEE, and Narayan C. Kar, Senior Member, IEEE