Abstract Due to the structural defects of the permanent magnet synchronous motor and various nonlinear factors in the driving circuit, the three-phase current in the motor contains current harmonics of various orders, which are mainly (6k±1) (k is a positive integer) orders. These current harmonics of (6k±1) orders in the natural coordinate system will be displayed as 6k orders after the coordinate is changed to the two-phase rotating coordinate system. At the same time, the transfer function of the repetitive controller has multiple resonance peaks, and its resonance frequency is a multiple relationship. Thus, the repetitive controller is widely used for suppressing the current harmonics of permanent magnet synchronous motors. However, the sampling period of the motor controller is fixed in the actual control system, and the current harmonic frequency changes with the motor speed, causing a fractional delay link in the repetitive controller. The most commonly used method to solve this problem is Lagrange interpolation to approximate the fractional delay link. This paper first analyzes the shortcomings of this method, and then discusses the method of Lagrange interpolation to approximate fractional order with numerical analysis. There are unnecessary specific constraints between Lagrange interpolation coefficients, and a better harmonic suppression effect can be achieved by reasonably configuring the interpolation coefficients. Then, the coefficient in the interpolation approximation method is mapped to compound circular motion, combined with the amplitude-frequency characteristic surface of the proposed repetitive controller. The effects of the interpolation coefficient on the resonant frequency and harmonic peak value of the repetitive controller are analyzed, and a fractional-order repetitive controller based on geometric constraint optimization is proposed. The resonance frequency and current harmonic frequency of the motor can be guaranteed to coincide in a broader range of speeds. Moreover, the resonance peak value is consistent with the harmonic composition of the motor, thus achieving a better harmonic suppression effect. The repetitive controller with a large resonance peak in the high-frequency part will cause system instability. This paper also introduces the combined filter, and discusses the influence of the feedback loop position of the filter on the resonance frequency and peak. This paper introduces the sensitivity function and combines the Nyquist curve to analyze the stability of the current control system, including PI, repetitive controller, and discrete motor model. First, the influence of PI controller gain K on the system’s stability without a repetitive controller is analyzed. After determining the value of K, the influence of repetitive controller gain Krc and internal model coefficient Q on the stability of the system is discussed, and the values of Krc and Q are determined. After all parameters are determined, the stability of the current control system at different motor speeds is also discussed. Finally, the experiments of no repetitive controller, rounding, Lagrange interpolation, and the improved repetitive controller proposed in this paper are carried out on the motor bench at different speeds. The current waveform and FFT analysis verify that the harmonic suppression performance of the proposed method is good.
Zhu Yuan,Zhu Liting,Xiao Mingkang等. Repetitive Controller Based on Geometric Constraint Optimization and Its Application to Current Harmonic Suppression of PMSM[J]. Transactions of China Electrotechnical Society, 2024, 39(4): 1059-1073.
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2024, Vol. 39 
