# Random PWM Technique Based Two-State Markov Chain for Permanent Magnet Synchronous Motor Control
## 1 Introduction  
With the deterioration of the global ecological environment and the aggravation of the energy crisis, Electric Vehicles (EVs), as an important carrier of green transportation, have shown explosive growth in market demand and technological innovation. As the core of EVs, the performance of the motor drive system directly determines the energy efficiency, reliability, and user experience of the vehicle. At present, the mainstream inverter control technology--SVPWM--is widely used for its high efficiency, low harmonics, and other advantages. However, the traditional SVPWM adopts the modulation strategy of fixed switching frequency, which leads to the generation of high-amplitude discrete harmonics at the switching frequency and its octave frequency of the inverter power devices, triggering the problems of Electromagnetic Interference (EMI), Noise Vibration Harshness (NVH), etc., and severely restricting the comprehensive control of the electric drive system. This seriously restricts the comprehensive performance of the electric drive system [1]. In recent years, in order to reduce the high-amplitude harmonics generated by conventional SVPWM, researchers have continuously optimized the conventional SVPWM andproposed various modulation techniques [2]. For example, Periodic PWM (PPWM) varies the switching frequency periodically (e.g., with the laws of triangle wave, sine wave, and sawtooth wave, etc. ), which disperses the harmonic energy that was originally concentrated at a fixed frequency into an extended frequency band centered on the switching frequency, but it introduces side effects such as the increase in current ripple and the decrease in the control accuracy [3; 4]; Selective Harmonic Elimination PWM (SHEPWM), on the other hand, directly eliminates the harmonic components in the target frequency band by optimizing the combination of switching angles. Taking the three-phase two-level inverter as an example, specific harmonic elimination can be realized without relying on filters by solving a system of nonlinear equations constructed using Fourier series. However, its algorithmic complexity is high and needs to be redesigned for different topologies, which limits its generalization [5; 6]. Passive suppression means mainly relying on external devices, such as common-mode chokes, RC absorber circuits, or active electromagnetic shields, which can mitigate the harmonic effects but increase the system size and cost and do not solve the problem at the root cause [7; 8]. Relatively speaking, the RPWM technique, by randomizing the switching frequency or pulse position, uniformly scatters the discrete harmonic energy within a broad frequency domain range, significantly reduces the amplitude of the harmonic energy, and has thereby become a hot research topic at present [9]. Reference [10] proposed a randomized switching control method to improve the high-voltage spectrum, but only three switching frequencies were used, and the spectrum distribution was not uniform. Reference [11] proposed a random switching frequency method for eliminating specific frequency harmonics so that the harmonic spectrum can be uniformly distributed, but the system oscillates at low frequencies and the output waveform fluctuates too much. Reference [12] proposed a spectral zero method based on the random switching cycle technique, which realizes spectral shaping at the resonance frequency, but it only works at resonance frequencies greater than 20 kHz. Reference [13] proposed a double random pulse width modulation technique that improves the harmonic expansion factor by about 30% compared to SVPWM, but it causes the low-frequency spreading problem of PWM harmonics, which in turn causes low-frequency noise in the motor.