Dual Random Noise Suppression Strategy for Permanent Magnet Synchronous Motor Based on Random Pulse Width Modulation and Random High-frequency Injection Method

This study proposes an innovative dual control strategy to tackle the challenge of electromagnetic noise in the sensorless low-speed operation of Permanent Magnet Synchronous Motors (PMSM). The strategy combines random Pulse Width Modulation (PWM) technology with a stochastic high-frequency injection approach. By incorporating two distinct randomized frequency switching techniques into the high-frequency injection circuit, the method integrates two-state Markov chains into both PWM technology and the random high-frequency injection method. Subsequently, a mathematical model for signal processing is developed, and extensive simulation and experimental verification are conducted. The experimental findings affirm the effectiveness of this approach in effectively mitigating electromagnetic noise while ensuring accurate location estimation within an acceptable range.

technique effectively expands the frequency range of harmonics while maintaining the total harmonic energy at the same level.
In [3], the functioning of a permanent magnet synchronous motor is described, wherein the real-time understanding of the rotor position is crucial.Traditionally, this knowledge was obtained through the use of high-precision mechanical position sensors.However, in order to improve cost-effectiveness, reliability, and applicability of PMSM, sensorless algorithms are commonly utilized for estimating the rotor position.Specifically, during the zero-speed or low-speed range, a fixed-frequency high-frequency signal is typically introduced into the stator winding using the high-frequency signal injection method.Subsequently, the rotor position information is derived from the resultant high-frequency response.
Based on the findings in [4], when a fixed-frequency high-frequency signal is injected into the stator winding via the inverter, it causes periodic stator vibration and generates high-frequency electromagnetic noise.Therefore, it is crucial to mitigate this electromagnetic noise.One effective approach is to either decrease the harmonic amplitude or increase the harmonic frequency.However, reducing the harmonic amplitude has a negative impact on the signal-to-noise ratio of the rotor position information, while increasing the harmonic frequency places higher demands on frequency and switching device losses.Another option, as mentioned in [5], is to employ a randomized frequency injection technique similar to PWM technology, which eliminates harmonic spikes and disperses harmonic energy across different frequencies.As a result, this paper adopts the dual-frequency random switching injection method, where two high-frequency sinusoidal signals with distinct frequencies and amplitudes are randomly injected into the control system.The objective is to minimize electromagnetic noise during low-speed noninductive operations of permanent magnet synchronous motors.
In reference [6], the generation of a random number plays a crucial role in the random control algorithm.In digital systems, pseudo-random algorithms are commonly used to produce pseudo-random sequences with periodic characteristics.However, this approach often leads to consecutive random numbers deviating significantly from the expected average.Consequently, it diminishes the effectiveness of harmonic spike reduction and results in increased ripple in the input current, thereby impacting the system's stability.To address this limitation, reference [7] introduces a novel approach by incorporating a two-state Markov chain into the random PWM.This integration enhances the randomness of the generated random numbers and improves the algorithm's ability to suppress noise.
Within this study, a novel approach called the dual random low noise control method is presented by integrating the Markov chain with both random PWM technology and random high-frequency injection method.By combining these techniques, this method surpasses the individual application of random PWM or high-frequency injection methods.It achieves superior dispersion of harmonic energy and effectively reduces electromagnetic noise.The efficacy of this approach is demonstrated through simulations and experimental investigations, validating its effectiveness.

The Markov Chain and Generation of Random Numbers
In [7], the Markov process is stated, named after Andre Markov (1856-1922), which is a discrete-time and continuous numerical stochastic process in mathematics.It has essential applications in machinery, biology, modern physics, biological student extinction process, information processing, automatic control, and high-voltage power transmission.
, t∈ is a discrete random process, and its state space is .
On microprocessors, the linear congruence random number generation function is usually used to generate random numbers.Its mathematical definition equation is: As proposed in [8], T is generally 16 or 32 considering the microprocessor bits.Because this is a pseudorandom number with a period, randomness is the best when the period is the most prolonged.At this time, B should be coprime with 2 , A=4k+1, ∈  * , so A=29, B=37.When the system is in state 1, the probability of the system transferring to state 2 at the next time is  , and the likelihood of the system still in state 1 is 1-P .Similarly, when the system is in state 2, the probability of the system transferring to state 1 at the next time is  , and the probability of the system still in state 2 is 1-P .The transition between these states can be described as the state transition graph shown in Figure 1.Thus, the one-step transition probability matrix of the Markov chain with system state and time as variables can be obtained: The state diagram can be expressed as follows.
State transition graph of Markov chains By employing the linear congruence method, we generate a set of 100 random numbers ranging from 0 to 10,000, with a transition probability denoted as "P" _1=0.8. Figure 2 showcases two distinct random sequences: one generated using the two-state Markov chain, and the other without incorporating the Markov chain.A comparative analysis of the two figures reveals notable differences.In the random sequence derived from the Markov chain, the occurrence of consecutive random numbers gravitating towards the expected value of 5,000 is noticeably reduced.Additionally, the distribution of numbers becomes more balanced on both sides of the expected value.This strengthened randomness theoretically enhances the ability of random PWM technology and random high-frequency injection methods to suppress electromagnetic noise.

Random PWM based on the two-state Markov chain
In reference [10], an extensive exploration of random PWM technology is presented, which can be categorized into three distinct types: random switching frequency PWM, random pulse position PWM, and random zero vector action time PWM.The random switching frequency modulation technique involves introducing varying frequencies to the triangular carrier signal while maintaining a consistent duty cycle, resulting in a randomized sequence of switching frequencies.Importantly, the PWM waveform generated using this method demonstrates a balanced distribution within each carrier period, thereby minimizing disruptions to current stability.Additionally, this approach shows promising potential for enhancing harmonic content and is comparatively more straightforward to implement in practical engineering applications.As a consequence, the random switching frequency PWM control method has gained widespread acceptance as the predominant approach in the field.The mathematical representation for random switching frequency PWM is given by: where  is the randomized switching frequency,  is the expected switching frequency,  is a random number with a uniform distribution in [-1, 1], and  is the random gain, whose size determines the range of random frequency change.Experimental results in [11] indicate that the more extensive the content of random frequency changes, the better the electromagnetic noise generated by the inverter drive motor will be suppressed, and the electromagnetic compatibility performance of the whole system will be enhanced.However, when the random gain increases, the motor's efficiency decreases irregularly.If the frequency is too low, it will resonate with the mechanical frequency of the motor.If the frequency is too high, it will bring more significant switching loss.Research shows that when condition  ∈ 0.5, 1.5 is met, the balance point of low harmonic, switching loss, and noise suppression capability can be achieved.It is specified that when the frequency of switch is more than  , it is state 1; when it is less than  , it is state 2. Hence, the integration of a two-state Markov chain into random PWM technology can bolster its capacity for expanding harmonic frequencies.Figure 3 illustrates the control block diagram of the random PWM approach augmented with a two-state Markov chain.

The random high-frequency injection method based on a two-state Markov chain
The high-frequency signal injection technique entails the introduction of a high-frequency voltage signal into the motor winding, from which the rotor position information is derived through analysis of the ensuing high-frequency response.Presently, this approach encompasses diverse methods, including the injection of sinusoidal, square wave, or rotating voltage signals.In the case of an embedded permanent magnet synchronous motor, its high-frequency mathematical model can be simplified as follows: In the equation,  and  are the stator voltage of the  axis of the rotating coordinate system;  and  is the stator current of the dq axis of the rotating coordinate system;  and  are the stator inductance of  axis;  is the Laplace operator.The conventional approach of injecting a fixedfrequency sinusoidal signal is utilized to estimate the d-axis of the coordinate system during two-phase rotation.In this method, a sine signal with a constant frequency and amplitude is injected into the shaft: Subsequently, the excitation components in each coordinate system can be derived from the highfrequency mathematical model of the PMSM as follow: (10) where  is the estimated rotor position angle;  is the actual rotor position angle;    is the rotor position error angle;  is the injected high-frequency voltage signal;  and  is the highfrequency component of the stator voltage of the  axis of the rotating coordinate system;  and  is the high-frequency component of the stator current of the  axis;  and  is the high-frequency component of stator current in the  coordinate system.We set the injected signal as   cos   when  has enough hours and  2 0, which can be deduced as: That is, the high-frequency component of the d-axis current is related to the position angle error information.To demodulate the position information of the rotor from the high-frequency excitation, we set the demodulation coefficient as  =sin   .After the low-pass filtered  and  are modulated and linearized, the following equation can be obtained: where  and  are demodulated as  current in the coordinate system;  is a constant independent of motor parameters.The estimated rotor position after demodulation can be obtained as follows: In [12] and [13], it was observed that the application of a fixed-frequency high-frequency injection method led to the concentration of harmonic energy at the injected signal's frequency, resulting in audible noise.To overcome this limitation, the present research introduces a random high-frequency injection method.In this method, the injected signal randomly switches between two different signals, effectively dispersing the harmonic energy and reducing electromagnetic noise.Specifically, the high-frequency signal injected into the estimated d-axis of the two-phase rotation coordinate system is: where  and  are two sinusoidal high-frequency signals with different amplitude and angular frequencies, which meet  ＞ .It is specified that the signal with high frequency in the two signals is state 1, and the signal with low frequency is state 2. First, the linear congruence method generates random numbers to judge transition probability, which obeys the uniform distribution between 0 and 1.We take the transfer probability  0.8 as an example.If the current is state 1 when the random number is less than 0.8, the following switch cycle will be transferred to state 2, and when it is greater than or equal to 0.8, the following switch cycle will still be state 1.Thus, the two-state Markov chain can be introduced so that the injected high-frequency signal can be switched directly and randomly in each switching cycle as far as possible between the two signals.Because the frequency of the injected signal is different, it is necessary to switch the corresponding demodulation coefficient during signal switching, which can be described as f   and    .To overcome the limitations associated with the direct calculation of rotor position using the arc tangent method, this study proposes the utilization of a normalized orthogonal phase-locked loop structure observer for accurate estimation.The structure of this observer can be seen in Figure 4, where its closed-loop transfer function is solely dependent on the control loop parameters, decoupled from the injection signal and motor parameters.This decoupling enables more precise estimation of rotor position information.The control block diagram of the random high-frequency injection technique, which utilizes a two-state Markov chain, is depicted in Figure 5.This method combines random PWM technology with the random high-frequency injection approach to effectively suppress noise when operating a permanent magnet synchronous motor at low speeds without the requirement for supplementary hardware circuits.

Simulation
In order to validate the efficacy of the dual random method in mitigating harmonic spikes, a simulation model is constructed using the MATLAB & Simulink platform, following the structure illustrated in Figure 5.The simulation focuses on an embedded permanent magnet synchronous motor, with its corresponding parameters provided in Table 1.Firstly, simulations are conducted to compare the performance of three control strategies: conventional PWM technology, random PWM technology without a Markov chain, and random PWM technology utilizing a two-state Markov chain.The expected frequency is set at 5 kHz, with the switching frequency randomly varying between 3 kHz and 7 kHz.The transition probability (P) in the two-state Markov chain is set to 0.8.The permanent magnet synchronous motor (PMSM) operates at a fixed speed of 300 r/min with a constant load torque of 1 Nꞏm. Figure 6 illustrates a partial zoom-in of the A-phase current waveform and its power spectral density under the three control strategies: fixed frequency PWM, PWM without a Markov chain, and PWM based on a two-state Markov chain.The waveform and enlarged diagram clearly demonstrate that random PWM significantly reduces current harmonics, diminishes waveform distortion, and enhances current quality.The power spectrum comparison reveals that without employing a random strategy, the harmonic energy concentrates primarily at integral multiples of the switching frequency (5 kHz).However, after applying random PWM, the harmonic energy at these integral multiples disperses significantly into other frequencies.Furthermore, with the inclusion of the two-state Markov chain treatment, the random PWM's ability to disperse harmonic frequencies is further enhanced, resulting in reduced electromagnetic noise.
Then, this paper conduct simulations to compare the performance of different control strategies: conventional PWM technology, random PWM technology without a Markov chain, and random PWM technology based on a two-state Markov chain.The expected frequency is set at 5 KHz, while the switching frequency varies randomly between 3 kHz and 7 kHz, with a transition probability of P in the two-state Markov chain set to 0.8.The PMSM operates at a constant speed of 300 r/min under a load torque of 1 Nꞏm.In Figure 6, we present a partial enlargement and power spectral density analysis of the A-phase current waveform under these three control strategies: fixed frequency PWM, PWM without a Markov chain, and PWM based on a two-state Markov chain.The waveform and partial enlargement reveal that random PWM effectively reduces current harmonics, diminishes waveform irregularities, and enhances current quality.The power spectrum comparison demonstrates that, without randomization, harmonic energy concentrates at the integral multiples of the switching frequency 5 kHz.However, after applying random PWM, the harmonic energy at these integral multiples disperses significantly into other frequencies.Furthermore, by incorporating the two-state Markov chain, the random PWM exhibits improved harmonic frequency dispersion capabilities, leading to further attenuation of electromagnetic noise.Figure 6.Simulation results of the conventional PWM technology, the random PWM technology without the Markov chain, and the random PWM technology based on a two-state Markov chain Subsequently, simulations are conducted for the fixed frequency approach, frequency injection method, and dual-frequency random injection method utilizing a two-state Markov chain and dualrandom strategy.The simulation conditions are as follows:  The target speed is 300 rpm;  The fixed injection signal frequency is 1, 000 Hz;  The dual frequency injection signal frequency is 750 Hz and 1, 250 Hz, respectively;  The constant load torque is 1 Nꞏm.The simulation results, as depicted in Figure 7, showcase the phase A current waveform, current power spectral density, and rotor position estimation under the three control strategies.When employing the fixed-frequency high-frequency injection, the application of a high-frequency signal on the stator side introduces numerous harmonics throughout the system.Consequently, a pronounced distribution of harmonic energy spans across the entire frequency band.However, with the adoption of the dualfrequency random injection method, the range of harmonic distribution improves significantly, dispersing approximately 37.64% of the harmonic energy originally brought by the high-frequency injection to other frequencies.Upon implementing the dual random control strategy, combining random PWM and random high-frequency injection methods, the strength of the harmonic spike diminishes, resulting in effective reduction of high-frequency noise.Furthermore, in terms of rotor position observation accuracy, the random strategy introduces minimal error, with deviations not exceeding 0.1 rad.Thus, it maintains excellent accuracy in estimating rotor position during low-speed sensorless operation.

Experiment
To validate the efficacy of the dual random strategy, this study constructed a hardware platform employing TMS320F28335 as the control MCU and IR2136 drive module, as depicted in Figure 8. Experimental verification was conducted on a 500 W embedded permanent magnet synchronous motor, comparing the fixed frequency high-frequency injection method with the dual random strategy control technology.The electrical parameters of the motor were set to align with the simulation conditions.The random PWM was designed with an expected switching frequency of 5 kHz, ranging from 3 kHz to 7 kHz.In contrast, the fixed injection signal maintained a frequency of 1 kHz, while the dual-frequency injection signal alternated between 750 Hz and 1,250 Hz.An eddy current dynamometer provided a constant load torque of 1 Nꞏm.To collect noise parameters, a noise decibel tester was employed, while the upper computer recorded the current and speed outputs.Additionally, a 2,500 mesh photoelectric encoder was installed on the motor solely for comparing actual speed measurements with estimated values, without engaging in closed-loop control.
To evaluate the dynamic performance, two tests were conducted using the fixed-frequency highfrequency injection method and the dual random strategy.Figure 9 illustrates the motor's speed profile and speed error while running at 300 rpm, decelerating to 200 rpm after 3 seconds, and subsequently accelerating to 400 rpm after 3 seconds.A comparison of the images reveals that the speed error in the low-speed range is reduced by 13.7% when employing the dual random strategy, indicating favorable dynamic performance.Additionally, Figure 10 presents the A-phase current waveform of the motor, demonstrating a significant reduction in current fluctuations and irregularities.11 indicates that the implementation of the dual random strategy leads to an increase in the system's temperature rise as the switching frequency rises.This temperature variation significantly impacts the motor parameters.Consequently, there is a slight elevation in the error of rotor position estimation, resulting in a 15.3% reduction in accuracy.Nonetheless, the error remains within the acceptable range of 0.06 rad, satisfying the observation requirements of the sensorless algorithm for rotor position.Figure 12 showcases a comparison of the phase current spectrum under the two control strategies.When the dual random strategy is employed, the presence of harmonic spikes concentrated at specific frequencies disperses throughout other frequencies.The electromagnetic noise levels of the two control strategies during the motor's operation at 500 rpm were measured using a noise decibel meter.Measurements were taken every 5 seconds, and the average value of 20 noise decibel readings was recorded.The test results, as presented in Table 2, demonstrate a reduction in the average noise value by 10.52% following the implementation of the dual random strategy.This indicates a notable improvement in noise reduction efficacy.

Conclusion
This study introduces a novel dual random strategy that combines the random PWM technology based on a two-state Markov chain with the dual-frequency random injection method.By contrast to the fixedfrequency high-frequency injection method, this innovative approach offers significant advantages in distributing harmonic energy generated by the high-frequency injection method and PWM technology across various frequencies.As a result, it effectively eliminates harmonic spikes and considerably reduces electromagnetic noise during the low-speed sensorless operation of permanent magnet synchronous motors.To validate the proposed control strategy, a mathematical signal processing model is established, and a simulation model is implemented using the MATLAB Simulink platform.Moreover, experimental verification is conducted on a dedicated experimental platform.The simulation and experimental findings demonstrate a 13.7% decrease in speed estimation error, while the rotor position error increases by 15.3%, but remains within an acceptable range of 0.06 rad, meeting the operational requirements.Furthermore, the average steady-state running noise of the motor is reduced by 10.52%, effectively mitigating electromagnetic noise.

Figure 2 .
Figure 2. Comparison of the linear congruence method for random number generation before and after introducing two-state Markov chains

Figure 3 .
Figure 3.The control block diagram of the random PWM technology based on a two-state Markov chain

Figure 4 .Figure 5 .
Figure 4.The structure of an orthogonal phase-locked loop

Figure 7 .
Figure 7. Simulation results of fixed frequency high-frequency injection method, dual frequency random injection method based on the two-state Markov chain, and dual random strategy

Figure 11 .
Figure 11.Comparison between the fixed frequency high-frequency injection method and dual random control strategy for angle estimation

Figure 12 .
Figure 12.Comparison diagram of phase current frequency spectrum under two control strategies unrelated to the previous state of affairs.That is, there is no aftereffect.When applying the Markov chain to practical engineering, it is necessary to calculate the transition of the system state, so the transition probability is introduced to represent the probability of the random process from the current state to the next state.All one-step transfer probabilities of the Markov chain are formed into a matrix, which is called a one-step transfer probability matrix, namely: where   , t∈ is called a discrete Markov chain.According to the definition, the state x of the next time  in the discrete Markov chain is only related to the state  of the current time  and the current situation is

Table 1 .
Parameters of permanent magnet synchronous motor