I–f closedloop starting strategy of highspeed PMSM based on current vector adaptive regulation
Abstract
Conventional I–f startup control algorithms often lead to significant speed fluctuations, extended convergence times, and sluggish dynamic responses when the motor is working at the acceleration process. To address these challenges, the article constructs a discrete kinetic model of the motor, extracts and summarizes the change law of the motor torque angle fluctuation amplitude and frequency with the motor parameters and the reference current vector during I–f starting process based on the Runge–Kutta method, reveals the fundamental mechanism of the pole slipping of the I–f control. On this basis, the article proposes a closedloop I–f starting control strategy for highspeed permanent magnet motors based on current vector adaptive regulation, The strategy notably improves speed tracking performance and current dynamic response to I–f startup control. Experimental results validate the efficacy of the proposed algorithm.
1 INTRODUCTION
As the central component of the fuel cell air system, its performance directly influences the energy conversion efficiency of the cell system and the overall performance of hydrogen fuel cell vehicles. The development trend is toward a high flowtopressure ratio, high dynamic performance, and significant dependability [13]. The ultrahighspeed permanent magnet motor has become the standard drive motor for centrifugal air compressors because of its high power density, compact design, superior transmission efficiency, prolonged durability, and rapid dynamic response [1, 4, 5]. Most centrifugal compressors operate at speeds exceeding 150,000 rpm, which complicates the use of position sensors with frequency ranges up to 2500 Hz. Therefore, sensorless control has become a necessity for highspeed permanent magnet motor drives [6]. Due to the characteristics of highspeed motors, such as the absence of a saliency effect and the small amplitude of the back EMF at low speeds [7, 8], there is a pressing need to transition quickly to an idle speed to elevate compressor pressure, especially for centrifugal air compressor applications. Hence, devising a method for a rapid and stable start of highspeed motors without sensors remains a significant research challenge [9, 10].
In recent years, research on lowspeed startup technology for permanent magnet synchronous motors (PMSM) without position sensors has deepened, and I–f startup control is increasingly adopted for ultrahighspeed PMSM with surfacemounted structures [11]. The application of the I–f startup method to PMSM was first introduced by Professor Marius Fatu of Aalborg University [12]. The fundamental control strategy supplies a rotating reference current vector within the synchronous rotating coordinate system, where the current vector amplitude remains constant, and its rotating speed aligns with the I–f control reference speed [13]. However, the conventional openloop I–f starting control strategy has a limitation: the current amplitude and frequency cannot be automatically adjusted, resulting in large fluctuations in output torque, acceleration, and speed during the dynamic process [14]. Especially for centrifugal air compressor applications, once the motor starts, it is necessary to rapidly accelerate to the idling state, and the angle fluctuations can induce pole slipping. Pole slipping denotes an instability condition when the generated torque is insufficient to maintain the rotor in synchronism with the rotating stator magnetic field. This leads to a pulsating torque on the rotor, resulting in severe acceleration and deceleration. Finally, the motor halts. Unsuitable reference current vectors and accelerations can induce pole slipping during openloop I–f startup control.
To mitigate acceleration fluctuations in openloop I–f startup, a closedloop I–f control is proposed to prevent pole slipping and reduce speed oscillations. Frede Blaabjerg provided an upper limit for the rotational speed selection of the current vector for stable I–f control based on the principle of constant average torque in a single control cycle [15]. In [16], the prerequisites that must be met by the reference current vector and motor parameters for stable operation during I–f starting process are discussed. In [17], a mathematical derivation and stability analysis are conducted for the openloop control system of PMSM, considering the constraints set by the vector and motor parameters. In addition, research indicates that realtime speed fluctuations can be reduced by making closedloop adjustments to the current vector amplitude and frequency during I–f startup control. In [18], acceleration is regulated through reactive power, using the voltagecurrent vector angle to indirectly measure torque angle changes to adjust the current amplitude. In [19], the feedback of the disturbance component of the motor activation active power is employed to increase the system damping torque. In [20], the highfrequency component of the active power is used to increase the damping torque of the system by feeding back the disturbance. It also compensates the reference rotational speed of the current vector by assessing the load variations, aiming to achieve stable operation under sudden loads in a steady state. In [21], an offline selflearning method was adopted to obtain a noload torque angle curve for online position angle compensation, which can reduce the risk of the pole slipping under shock loads.

Most enhanced algorithms cater solely to constant acceleration modes, neglecting validation for alternative acceleration modes, such as centrifugal air compressor applications, whose acceleration continues rising during the starting process.

The reasons for highfrequency torque angle fluctuations and the fluctuation pattern have not been systematically investigated, resulting in unclear mechanisms of pole slipping during I–f startup control.
In this article, the motor I–f starting strategy in variable acceleration mode is investigated. Based on the motor dynamics model, the numerical Runge–Kutta solution method is used to extract the fluctuation laws of the actual speed, acceleration, torque angle, and motor parameters. The influence relationship between the fluctuation frequency and amplitude of torque angle with the reference current vector and motor parameters is analysed. This provides a theoretical basis for selecting adaptive filter parameters to extract the highfrequency component of the instantaneous power, implements adaptive adjustments to the I–f closedloop control parameters according to the reference acceleration and speed during the acceleration interval, and enhances motor speed convergence rate and system stability effectively. Experimental findings demonstrate that the control strategy has good dynamic performance in both fixed and variable acceleration mode, as well as robustness against unexpected acceleration process.
2 MATHEMATICAL MODEL OF PMSM UNDER I–F CONTROL
The paper introduces a virtual d^{v}q^{v} current vector coordinate system with rotor flux orientation to investigate current vector characteristics of the highspeed SPMSM during I–f startup control. As shown in Figure 1, the d^{v} axis is the direct axis in the virtual coordinate system, the q^{v} axis is the quadrature axis, the d axis is the direct axis in the actual coordinate system, the q axis is the quadrature axis, and the directions of q^{v} axis and current vector i_{s} remain identical. The torque angle $${\hat \theta _e}$$ defines the angle between the q^{v} axis and the d axis. Both i_{d} and i_{q} represent the projection of i_{s} from the d^{v}–q^{v} to the d–q coordinate system.
As shown in Figure 1a, the torque angle $${\hat \theta _e}$$ is 0 at the instant of I–f startup control. The actual d–q coordinate system is 90° ahead of the virtual d^{v}–q^{v} current vector coordinate system, and the reference current vector i_{s} acts on the daxis of the actual coordinate system. Therefore, no torque is generated initially. After a reference current vector command is issued, as in Figure 1b, acceleration in the d^{v}–q^{v} coordinate system aligns it with the actual dq coordinate system, and $${\hat \theta _e}$$ will increase. The reference current vector operates on the actual qaxis, leading to a proportional increase of the current i_{s}sin $${\hat \theta _e}$$ with torque angle $${\hat \theta _e}$$. The rotor commences its rotation when the electromagnetic torque generated by the actual qaxis current exceeds load torque.
3 THEORETICAL ANALYSIS OF OPENLOOP I–f CONTROL METHOD
3.1 General openloop I–f control scheme
The overall control block diagram of the openloop I–f control system is depicted in Figure 2. Combining its space vector diagram and mathematical model shows that specific offline settings for the current vector amplitude i_{s} and rotational angular velocity ω_{i} must be based on specific motor parameters for practical applications. The “torque selfbalancing” function is employed to minimize the likelihood of pole slipping during the motor starting process.
3.2 Analysis of the openloop I–f control
According to the theory of differential equation solution, it is challenging to derive a precise solution for the torque angle$${\hat \theta _e}$$. Therefore, in order to investigate the variation characteristics of each parameter during the I–f control operation, the numerical solution method is utilized to analyse the torque angle fluctuation law here.
Various numerical techniques exist for solving secondorder differential equations have their strengths and limitations. Considering the nonlinear trigonometric functions characteristics in the openloop I–f control differential equation for torque angle $${\hat \theta _e}$$, the fourth order RungeKutta method is chosen. This method operates on its fundamental principle: the selection of distinct points proximate to various time instances, the linear combination of function values at these points, and the utilization of the resultant combined value as a substitute for the derivative in the Taylor series expansion. The Runge–Kutta approach provides a numerical solution to the differential equations in (5). Matlab is used to chart the behaviour of torque angle $${\hat \theta _e}$$, actual rotational mechanical angular velocity ω_{r}, actual rotational mechanical angular acceleration K_{a}_{r}, reference current vector i_{s}, and motor parameters.
For simplifying the numerical analysis, the load torque is set to 0 under openloop I–f control, the reference current vector is constant, the reference acceleration by the current vector is constant, and the initial torque angle $${\hat \theta _0}$$ is 0°. As illustrated in Figure 3, the numerical solution derived from the initial value of the reference current vector reveals fluctuating patterns with a certain regularity for torque angle $${\hat \theta _e}$$, actual rotor angular velocity ω_{r}, and actual rotor angular acceleration K_{a}_{r}.
Then, by selecting a fluctuation period (t_{1}–t_{4}) in Figure 3, we can analyse the fluctuation law of torque angle, actual acceleration, and motor actual speed in conjunction with the I–f control vector diagram in dual coordinates.
 Phase 1: During t_{1}t_{2}, In Figure 4a, the torque angle shows a rising trend as the virtual coordinate system edges closer to the actual coordinate system. Figure 4b shows that the actual rotor acceleration gradually increases from 0 to the constant acceleration provided by openloop I–f control at t_{2}. In Figure 4c, the actual motor speed is always less than the openloop reference speed, with the error speed peaking at t_{2}.
 Phase 2: During t_{2} t_{3}, the torque angle continues increasing in Figure 4a. Figure 4b demonstrates that the actual rotor acceleration continues to rise with both the torque angle and the actual rotor acceleration reaching respective maxima at t_{3}. In Figure 4c, the actual speed remains below reference speed but with the error decreasing to 0 at t_{3}.
 Phase 3: During t_{3}t_{4}, the torque angle in Figure 4a is decreasing, as is the actual rotor acceleration in Figure 4b, and the torque angle at t_{4} is the minimal value close to 0. The actual speed is equal to the openloop reference speed, and the actual speed error is currently 0. At this point, the actual speed acceleration reaches a value near to 0, while the acceleration error reaches a positive maximum. In Figure 4c, the actual speed exceeds the openloop reference speed, and the error speed is always more than 0.
Assuming current vector is constant, both the numerical solution results and the vector diagram of openloop I–f control indicate that the torque angle and actual speed always fluctuate periodically during motor operation at the same frequency. When the fluctuation amplitude of the torque angle surpasses 90° between the virtual and actual coordinate systems, it heightens the risk of pole slipping. Therefore, theoretical research into the high frequency fluctuation mechanism and influence factors is required.
4 I–f CURRENT VECTOR CLOSEDLOOP ADAPTIVE REGULATION ALGORITHM
4.1 Mechanism analysis of high frequency oscillations under I–f control
According to Equation (8), the actual torque angle of the motor governed by openloop I–f control is influenced not only by the motor mechanical parameters B、J、T_{L} and the reference current vector i_{s}, but also by the actual speed of the rotor ω_{r}. Unsuitable control parameters, load disturbances etc. can easily cause $${\hat \theta _e}$$ to deviate from the stable range, resulting in pole slipping. The high frequency fluctuation of ω_{r} is the fundamental cause of pole slipping during the I–f starting process. Therefore, this paper will analyse the law of high frequency fluctuation and find suitable methods to reduce the speed fluctuation.
4.2 Analysis of numerical solutions for the change law of fluctuating frequency under I–f control
As shown in Figure 3, the torque angle fluctuation frequency aligns with the actual motor speed fluctuation frequency, Thus, this study correlates the characteristics of torque angle fluctuations with actual speed oscillations. To quantify the high frequency fluctuation of the torque angle, we define two parameters: fluctuation frequency and fluctuation amplitude. where fluctuation frequency reflects the torque angle fluctuation cycle and fluctuation amplitude reflects the range within a single cycle. The study employs four parameters to investigate the fluctuation law of torque angle: the initial value of torque angle $${\hat \theta _0}$$, the reference current vector acceleration K_{a}_{i}, the moment of inertia J, and viscous damping coefficient B of the motor.
Figure 5 displays numerical solution results for the torque angle with varying B, J, K_{a}_{i} and ω_{r}, where J_{pu} and B_{pu} are the original moment of inertia and viscosity coefficient of the motor. The reference value of Ka_{pu} is 600 r/min/s, and speed reference value is 10,000 r/min.
The variation curves for torque angle $${\hat \theta _e}$$ fluctuation frequency and amplitude under distinct B, J, K_{a}_{i} and ω_{r} are obtained in Figure 6 based on the numerical solution results in Figure 5.
Figure 6a depicts that the fluctuation frequency of the torque angle under various B, J, K_{a}_{i} and ω_{r} conditions. The frequency exhibits an approximately linear as reference acceleration K_{a}_{i} and viscosity coefficient B increase from the reference value 1, and decreases further as speed ω_{r} rises, with the slope continuously increasing.
Equation (9) shows the torque angle fluctuation frequency varies continuously with speed and reference acceleration, and the torque angle fluctuation amplitude is consistently influenced by the reference acceleration. To suppress the highfrequency fluctuations of the rotational speed effectively, we can devise a closedloop algorithm based on the fluctuation frequency variation rule of the torque angle and the influencing factors of fluctuation amplitude.
4.3 I–f closedloop current vector adaptive regulation algorithm overall scheme
An I–f closedloop control strategy is proposed here to adapt to the abrupt acceleration interval and reduce speed oscillations during I–f starting process, based on instantaneous active power feedback and adaptive current vector adjustment, the block diagram of the overall control structure is depicted in Figure 7.
The main idea of the control algorithm: Calculating the instantaneous active power P of the SPMSM in real time based on the voltage and current in the d^{v}–q^{v} virtual coordinate system. Combining the fluctuation law of Equation (9) with the realtime calculation of the adaptive filter cutoff frequency based on the reference current vector and estimated rotational speed, a feedback adjustment algorithm is constructed based on the high frequency fluctuation, and the current vector is adjusted in real time by adding the virtual damping torque component to the reference rotational speed ω_{i}. It enables the motor to converge to the synchronized state rapidly between the virtual and actual coordinate system during the I–f starting process. It can also automatically adjust control parameters to accelerate the convergence of torque angle and improve the stability and adaptability of system when faced with abrupt changes in reference acceleration.
4.4 Algorithm for current vector adaptive regulation based on instantaneous power
In Equation (14), ∆ω_{r} represents the fluctuation of rotor speed at the steadystate operating point, and k_{ω} is a positive quantity that can be adjusted. In Equation (15), ∆$$T_{\mathrm{e}}^{\mathrm{*}}$$ is proportional to ∆ω_{r}, exhibiting virtual damping torque characteristics. When the speed fluctuates, realtime compensation of the reference speed is achieved by determining the fluctuation amount. Consequently, the electromagnetic torque acquires a virtual damping component, enabling adaptive damping for rapid actual speed tracking of the reference current vector speed.
4.5 Analysis of the stability of the vector adaptive regulation algorithm
Based on Equation (31), it is inferred that k_{1} > 0, which implies that the real portion of the characteristic root is negative. According to the principle of automatic control theory, a sufficient necessary condition for the stability of a linear system is that the characteristic equation closedloop system roots have negative real components. Therefore, the closedloop system is stable.

For ∆B > 0 (B is overestimated): As the characteristic roots are closer to the imaginary axis, the system responds more quickly, and the oscillation frequency may increase.

For ∆B < 0 (B is underestimated): As the characteristic roots are further from the imaginary axis, the response speed may decrease, and the oscillation frequency may decrease.

For ∆J > 0 (J is overestimated): As the characteristic roots are further from the imaginary axis, the response speed may decrease, and the oscillation frequency may reduce.

For ∆J < 0 (J is underestimated): As the characteristic roots are closer to the imaginary axis, the system responds more quickly, and the oscillation frequency may increase.
5 ANALYSIS OF EXPERIMENTAL RESULTS
In order to experimentally validate the proposed control strategy, the experimental platform in Figure 10 was built, and the motor parameters are shown in Table. 1. The motor controller consists of a chip of type TMS320F28377S and an inverter with SiC components. The switching frequency and control period are both 50 kHz, the experimental phase current waveform is measured in Figure 10b, and other experimental results were transmitted in real time through the CAN communication of the motor controller and the humanmachine interface based on Labview software development, data transmission period is 10 ms.
Parameters  Numerical values 

Stator phase inductance/H  0.000827 
Stator phase resistance/Ω  0.047 
Permanent magnet flux / Wb  0.14 
Inertia/(kg⋅m)^{2}  0.063 
Viscous friction factor/(N⋅m⋅s)  0.0054 
Number of pole pairs p  1 
Rated speed n_{r} /(r/min)  20,000 
Rated voltage U_{dc} /(V)  540 
Rated current I/(Arms)  54 
Rated torque T_{e} /(N⋅m)  11 
It should be emphasized that the aim of the research in this paper is to improve the stability of I–f starting process of highspeed PMSM and the robustness under different accelerations, so the next experiments were conducted to verify the I–f closedloop algorithm only at the idle speed (20% of rated speed).
5.1 Experimental verification and analysis under fixed acceleration of I–f startup conditions
Figure 11 shows experimental acceleration phase waveforms under openloop I–f control and closedloop adaptive regulation of the current vector at 720 r/min/s constant acceleration.
Figure 11a illustrates the experimental waveforms of the reference current vector tracking, speed tracking, and torque angle under openloop I–f control. It can be seen that maximum speed fluctuation tracking error exceeds 100 r/min, the maximum torque angle fluctuation is 90°, the reference current vector tracking error is about 10A, and the tracking error increases larger and larger during the whole I–f starting process. Figure 11b presents the supplied current, speed, and torque angle tracking waveforms based on closedloop I–f startup control. The experimental results indicate that the torque angle fluctuation only occurs during the early stage of starting, peaking at 20° and stabilizing within 0.3 s. Throughout the whole process, the speed fluctuation tracking error peaks at approximately 5 r/min, and the reference current vector always tracks around the specified current vector amplitude of 15 A.
Figure 12 shows experimental torque angle fluctuations during I–f startup acceleration, comparing the algorithm proposed in this study with conventional I–f closedloop control methods. Under the adaptive filter closedloop control algorithm proposed in this paper, the torque angle converges more rapidly, and the torque angle fluctuations have the least amplitude. If the filter frequency is chosen incorrectly, the convergence speed of torque angle is clearly affected. In comparison to the conventional closedloop methods with fixed filter cutoff frequency, the vector adaptive regulation method suggested in this study has a shorter convergence time of at least 0.5 s and more flexibility for variable acceleration of starting conditions.
5.2 Experimental verification and analysis under variable acceleration of I–f startup control conditions
In order to verify the robustness of the current vector closedloop adaptive control algorithm, experiments were conducted under specific acceleration conditions during I–f starting process. The experimental results are compared and analysed from four angles: the characteristics of torque angle convergence, the tracking performance of the reference current vector, speed tracking features, and the rate of phase current convergence.
Figures 13 and 14 are the experimental waveforms of the conventional I–f closedloop control and the current vector adaptive algorithm under the reference acceleration mutation condition. Reference acceleration is fixed at 500 r/min/s during the 0–3 s period, increased to 1000 r/min/s at t = 3 s.
In Figure 13, which depicts the conventional current vector closedloop control, the torque angle fluctuates up to 40° and takes approximately 1 s to converge, the reference current vector has a tracking error of around 2 A, and the tracking error between the reference speed and the actual speed is about 20 r/min. On the contrary, under the closedloop I–f startup control strategy based on current vector adaptive regulation in Figure 14, the maximum fluctuation of torque angle is 25° and converges within 0.4 s, current tracking error is around 1 A and speed tracking error remains under 5 r/min.
When motor speed reaches 1500 r/min, the acceleration is raised from 500 to 1000 r/min/s. Figure 13a shows that the torque angle undergoes an abrupt change before it converges slowly to a stable value with conventional closedloop I–f control algorithm. The torque angle fluctuates ±35° after a sudden change in acceleration, requiring 1.2 s to converge. The tracking error of the reference current amplitude is about 5 A, which is 30% of the 15 A fundamental value. Within the grey highlighted region, the motor speed experiences a continuous nontracking phenomenon for 0.5 s, leading to a tracking error of up to 50 r/min. From the motor phase current waveform presented in Figure 14b, it is evident that the current of motor phase A oscillates between 10 and 25 A, reaching relative stability after 1.2 s during the abrupt acceleration.
Under the adaptive closedloop control of the current vector in this paper, Figure 14a demonstrates that after the sudden change of acceleration, the torque angle fluctuates just ±10°, the torque angle quickly converges to the new stable value within 0.3 s, and the actual speed tracks the reference speed of current vector consistently throughout the entire process, the speed tracking error is only about 10 r/min even during changes in acceleration, the maximum reference current tracking error is about 1 A. Figure 14b shows the motor phase A current stabilizes within 0.4 s after acceleration changes, 0.8 s faster than the conventional I–f closedloop control.
The experimental results in Table. 2 and 3 indicate that the current vector adaptive regulation closedloop algorithm has better robustness during I–f startup under fixed and abrupt accelerations. It provides smaller torque angle fluctuations, faster motor convergence, and improved current tracking compared to conventional I–f control.
Fixed acceleration  Conventional control  Adaptive control 

Torque angle fluctuation amplitude/(°)  40  25 
Rotational speed tracking error/(r/min)  20  5 
Torque angle convergence time/(s)  1  0.4 
Abrupt acceleration  Conventional control  Adaptive control 

Torque angle fluctuation amplitude/(°)  35  10 
Rotational speed tracking error/(r/min)  50  10 
Tracking error for a reference current (A)  ±5  ±1 
Convergence time after mutation/(s)  1.2  0.4 
6 CONCLUSION

Motor parameters, reference current vector, and actual speed determine the frequency and magnitude of speed and torque angle variations under I–f control. The highfrequency fluctuation of rotational speed is the underlying cause of the pole slipping under I–f control. The variation laws of torque angle fluctuation frequency and amplitude provide the theoretical basis for devising the I–f closedloop control algorithms.

Adjusting the current vector reference speed based on motor instantaneous active power feedback can minimize torque angle fluctuation magnitude, reduce torque fluctuation, prevent pole slipping and reduce speed oscillations.

Experimental results demonstrate this proposed control method provides good performance for fixed and varying I–f acceleration, increasing convergence speed 50% and reducing torque angle fluctuation 30% under acceleration changes versus conventional regulation. Which provides an important algorithmic basis to enhance startup stability for ultrahigh speed centrifugal air compressor applications.
Finally, it is worth noting that while significant identification errors in motor parameters can affect the performance of the closedloop algorithm, they do not compromise the stability of system. Specifically, the response, oscillation frequency, and stabilization time of system can vary based on the overestimation or underestimation of B and J.
AUTHOR CONTRIBUTIONS
Guxuan Xu: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Software; Writing—original draft; Writing—review and editing. Feng Zhao: Funding acquisition; Validation. Tao Liu: Supervision; Validation; Writing—review and editing
ACKNOWLEDGEMENTS
This work was supported in part by the National Natural Science Foundation of China under grant number 51807139.
FUNDING INFORMATION
This work was supported in part by the National Natural Science Foundation of China under grant number 51807139.
CONFLICT OF INTEREST STATEMENT
The authors declare no conflicts of interest.
Open Research
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.