Rotor Position Tracking Control for Low Speed Operation of Direct-Drive PMSM Servo System

In this article, a rotor position tracking control (RPTC) strategy is proposed to effectively reduce the speed fluctuation for a direct-drive permanent magnet synchronous motor servo system operating at a low speed with different torque disturbances. In this article, considering the derivative relationship between the rotor position and speed, a speed command is converted to a real-time rotor position trajectory, and then a position-current two-loop control with the RPTC controller is proposed based on the internal model method to smoothly track the rotor position. In addition, the parameter design of the RPTC controller from the perspectives of robust stability and antidisturbance capability is investigated as well. The comparative simulation and experimental results demonstrate that, at a low speed, the proposed RPTC strategy has a good speed performance for both periodic and nonperiodic torque disturbances. Moreover, it enjoys a simple implementation for not requiring the precise speed feedback and specific torque disturbance information.

response, high reliability, low noise, and so on [2]. So it is widely used in high-performance servo equipment for industry, aerospace, etc. [3], [4]. However, for low speed or even ultralow speed operations, the motor is usually subject to torque disturbances from the system itself (such as cogging torque and harmonic torque) and outside loads (such as load torque and friction torque) [5]. And because there is no transmission or reduction mechanism, these disturbances would directly act on the loads, and the load disturbances would also be directly transmitted to the motor shaft, which seriously affects the speed stability and performance, especially at the low speed [2].
A conventional speed control strategy of the permanent magnet synchronous machine (PMSM) servo system can be briefly divided into three parts: a speed outer loop, a current inner loop, and a pulsewidth modulation module. Among them, the speed outer-loop controller is usually based on a proportional-integral (PI) regulator, which is simple and reliable but in some highperformance applications, its disturbance rejection performance is not satisfactory [6]. Therefore, various methods are proposed to reject torque disturbances. For periodic disturbances, such as cogging torque, flux harmonics, current measurement errors, and phase unbalancing, Xia et al. [7]- [9], respectively, explore the proportional resonance control, repetitive control, and iterative learning control to improve the speed performance. In general, the lower the speed is, the larger the speed fluctuation caused by the periodic disturbances would arise. Unfortunately, to overcome this problem, the abovementioned methods usually need to know the specific information of disturbances [10]- [12]. For nonperiodic disturbances, such as step-change loads and random disturbances, some scholars also propose several approaches. In [13]- [16], different disturbance observers, including the reduced-order observer, extended state observer, sliding-mode disturbance observer, and so on, are studied for direct compensation to reject nonperiodic disturbances. However, the accuracy of observers is usually affected by the decrease of speed and inaccurate models, which reduces the compensation effect [17]- [19], and may increase the control complexity.
Generally, for the traditional control strategy with a speed outer loop, the accurate speed feedback is a basic requirement. For common speed or high speed, the speed information is usually easy to be acquired and its accuracy can be guaranteed. But at low speed or even ultralow speed, even if the error of position measurement is very small, it would cause a large error in speed information due to the derivative operation in speed calculation [20]- [22], which usually cannot be neglected. Thus, if such an inaccurate speed signal is used as the feedback, not only the speed stability would be affected but also the capability of the disturbance rejection would be weakened.
Based on the abovementioned analysis, in order to effectively improve the low-speed performance of the DD-PMSM servo system under different torque disturbances, this article proposes a novel strategy that realizes the speed control through the position control, namely the rotor position tracking control (RPTC) strategy. In this strategy, based on the derivative relationship between the rotor speed and position, the speed control is converted to the position control. And with the help of the internal model method, a position-current two-loop control is employed, and the RPTC controller is proposed, so as to guarantee the continuous rotor position trajectory tracked smoothly. Using this strategy, for the DD-PMSM servo system operating at low speed, the periodic and nonperiodic torque disturbances can be effectively suppressed, and a good speed performance can be obtained as well. In addition, it has the advantage of simple implementation because the precise speed feedback and specific torque disturbance information are not required.
The rest of this article is organized as follows. In Section II, the traditional PI-based speed control is analyzed. Section III presents the proposed RPTC strategy and the design of the RPTC controller is studied in Section IV. Sections V and VI give the simulation and experiments, respectively. Finally, Section VII concludes this article.

A. PMSM Model
The model of a PMSM in the synchronous d-q rotating frame is given as follows [9]: where ω is the rotor mechanical angular speed; u d and u q are the d-axis and q-axis stator voltages, respectively; i d and i q are the d-axis and q-axis stator currents, respectively; L d , L q , R s , p n , and ψ f are the d-axis inductance, q-axis inductance, resistance, pole pairs, magnet flux linkage, respectively; T e , T L , J, and B a are the electromagnetic torque, load torque, inertia, and viscous coefficient, respectively.

B. Limitations of the Traditional PI-Based Speed Control
1) Contradiction Between the Tracing and Antidisturbance: Fig. 1 shows the conventional speed control block diagram with a PI regulator based on (1), where the load torque is regarded as the disturbance torque T d ; τ i is the inverse of the current loop bandwidth; K T is the torque constant, K T = 1.5p n ψ f ; k p and k i are the proportional and integral coefficients, respectively. Mendoza-Mondragón et al. [23] and Cui et al. [24] explain the contradiction between the tracing and antidisturbance for the speed control shown in Fig. 1. As k i /k p increases, the speed tracing performance would be improved but the antidisturbance performance would contrarily show a deterioration trend. In other words, the traditional PI-based speed control has a limitation: it is hard to obtain a good speed tracing and antidisturbance performance at the same time. This is an unfavorable factor for the high-performance speed control, especially at low speed.
2) Error in the Speed Acquisition at Low Speed: In general, the position error is small because of high acquisition accuracy. However, because the speed is obtained by the position derivative, it can be known from (2) that this error would be amplified by nearly 1/T c times in the speed acquisition [20], [22]. It means that the error would be up to 2000× if the speed loop control frequency is 2 kHz where ω err and θ err are the speed error and position error, respectively; T c is the speed loop control period.
In other words, due to the derivative calculation process, the abovementioned large error in speed acquisition is difficult to be avoided. At common speed or high speed, this kind of error has a little influence because it accounts for a small proportion of the speed itself. However, it cannot be ignored at low speed, thus requiring much more attention then. If such an inaccurate speed signal is used as the outer-loop feedback at low speed, the speed stability would not be guaranteed, let alone the antidisturbance performance.

A. Basic Idea and Structure of the RPTC Strategy
On the one hand, according to the derivative relationship mentioned in (2), the smooth movement of the rotor position determines the speed stability. On the other hand, the position error is much smaller than the speed error. Thus, smoothly tracking the rotor position trajectory may achieve a better speed performance at low speed with different torque disturbances. Based on the abovementioned idea, this article proposes a novel low-speed control strategy, which is called the RPTC strategy, as shown in Fig. 2.
In this RPTC strategy, apart from the traditional inner current control loop, there are other two important parts: first, the integrator for converting the speed reference to the real-time position trajectory; second, the RPTC controller for the outer position control loop. For the first part, after the speed command is converted to the position trajectory, the actual position can be directly fed back for subsequent control. Therefore, as long as the RPTC controller in the second part can smoothly track the position trajectory, the stable speed can be achieved, which is an important and key factor for the proposed RPTC strategy. Meanwhile, this strategy is introduced into i d = 0 vector control scheme to complete motor control.

B. Type Selection of the RPTC Controller 1) Important Requirements for the RPTC Controller:
From the analysis mentioned above, the command to be tracked in the RPTC strategy is no longer a constant speed signal but a real-time position trajectory. So, in order to implement the RPTC strategy, the following conditions (3) and (4) must be simultaneously satisfied. In (3), θ * p (s) is the Laplace transform of any position point on the real-time trajectory; θ(s) is the actual position; R is a constant; Φ(s) is the closed-loop transfer function of the system. In (4), θ * (s) can be regarded as smooth links between the points on this trajectory. In other words, the tracking of the real-time position trajectory needs to be theoretically free of static and dynamic errors.
Equation (3) lays emphasis on the accurate position of each point on the trajectory and (4) means the smoothness of the rotor movement between these points. However, as the torque disturbances would cause speed fluctuations, the abovementioned two conditions often cannot be strictly satisfied. Therefore, the controller should be designed to meet (3) and (4) as much as possible, so as to reject torque disturbances.
In addition, some application areas, such as aerospace, have certain limitations on control resources and require high reliability, so the RPTC controller structure should also be simple and easy to be implemented.
In summary, to meet the abovementioned requirements, the choice of the controller type is very critical.
2) Introduction of the Internal Model Method: Many methods often focus on the abovementioned condition (3), which is mainly to realize position locating [25]- [27], and the smoothness of rotor tracking is not strictly required. One reason is that their control goal is to accurately reach the position command, not to obtain a good speed performance. This is the main difference from the proposed RPTC strategy in this article. The other reason is that if the methods themselves do not possess  strong robustness, it is difficult to guarantee the smoothness of the position movement once the disturbances occur. Even if it is possible to satisfy abovementioned conditions (3) and (4) by combining many kinds of methods, the algorithm complexity would be inevitable.
The principle of the internal model control is to track the input command by feeding back the deviation between the actual plant G p (s) and the normal model of plant G m (s) [28], [29], as shown in Fig. 3, so its tracking effect does not depend on the form of the input signal r(s). What is more, the internal model control itself has strong robustness [30], [31], which is very beneficial for the satisfaction of abovementioned conditions (3) and (4). In addition, it has the advantages of no need for an accurate object model and few online adjustment parameters.
Therefore, in this article, the internal model method is selected to design the RPTC controller.

C. RPTC Controller Based on the Internal Model Method
The block diagram of the RPTC controller based on the internal model method is shown in Fig. 4. For the convenience of design, in Fig. 4, the current loop part is equivalent to 1.
In Fig. 4, G IMC (s) is defined as the internal model regulator.

It can be seen from Figs. 3 and 4 that the plant has become
In the design of the controller, it can be assumed that the internal model is accurate, i.e., G m (s) = G p (s) [32].
Because G m (s) is already the minimum phase system, [28], [32]. F(s) is a filter.
So, in Fig. 4, G RPTC (s) for the RPTC controller designed by the internal mode method is given as follows: In (5), only F(s) is unknown, which plays an important role in the control performance of G RPTC (s). The detailed design of the RPTC controller from the perspective of F(s) is given in the following section.

A. Structure Design of F(s) in the RPTC Controller
According to (5), the closed-loop transfer function in Fig. 4 can be derived as It is apparent from (6) that F(s) determines the tracking performance of the plant on the position trajectory and the antidisturbance performance.
F(s) usually takes the transfer function structure of 1/(λs + 1) r , where λ and r are the time constant and the order, respectively. In order to analyze the satisfaction of this kind of structure for (3) and (4), the following equations can be obtained by transforming (6): However, (8) shows that the commonly used low-pass filter structure tracks the real-time position trajectory with dynamic errors. In other words, it cannot satisfy (4).
Thus, this article selects another structure to satisfy (3) and (4) simultaneously, and it is given as follows: F(s) in (9) can conduct the tracking of the position trajectory without dynamic errors, which can be proved by From the abovementioned analysis, it can be seen that if the structure (9) is selected, the RPTC controller can achieve the smooth tracking of the position trajectory in theory and then stabilize the speed. However, in F(s), there are two parameters, the order r and the time constant λ, which need to be considered carefully. This is analyzed as follows.

B. Determining the Order r in F(s)
Although the structure of F(s) can satisfy the tracking of the position trajectory, the order r still needs to be optimally designed from the perspective of the antidisturbance performance. S RPTC (s) given in the following equation, which can be called a sensitivity function, is the transfer function between the disturbance and output in the article [24], [33]. The magnitudefrequency characteristic of S RPTC (s) is drawn when r = 2, 3, and 4, as shown in Fig. 5.  Equation (11) and Fig. 5 show the extent of the speed fluctuation caused by periodic disturbances at different frequencies.
The curves below 0 dB indicate that the controller has the ability to reject disturbances. And at the same frequency, the smaller the value of the ordinate is, the stronger the rejection ability will be. From Fig. 5, with the increase of r, the antidisturbance capability is gradually weakened, and the speed fluctuation may increase in a certain frequency band when r=3 and 4. So, in this article, r=2 is selected to improve the antidisturbance performance.

C. Design of the Time Constant λ in F(s)
In addition, there is another adjustable parameter, namely the time constant λ in (9), that needs to be designed. Its selection is a range where the lower bound is determined by robust stability and the upper bound is determined by the antidisturbance capability.
1) Lower Bound of λ: According to the robust stability condition, i.e., ||Δ(jω f )F (jω f )|| ∞ ≤ 1, derived from the robust stability principle [24], [28], [34], in this article, the lower bound of λ can be determined by the following equation, which means that λ in this range can still make the system stable even if uncertain factors in the actual plant are considered: Δ(jω f ) in (12) is an expression containing an uncertain factor (considered as a delay in the article) [35], [36], where ω f is the circular frequency; τ d is the delay time, which is caused by modulation, sampling, etc. This article takes τ d as 0.075T c . To clearly illustrate the satisfaction for (12) on different λ, the magnitude-frequency characteristic of F (jω f ) and 1/||Δ(jω f )|| ∞ are drawn in Fig. 6 .
Combining Fig. 6 with (12), 1/||Δ(jω f )|| ∞ is equivalent to the threshold line, which means that the curve of F (jω f ) must be below it to ensure the robust stability of the RPTC controller. Hence, it can be seen from Fig. 6 that λ = 0.2T c is the lower bound in this article and as λ gets larger, the margin of robust stability increases gradually.
2) Upper Bound of λ: The antidisturbance capability is used to determine the upper bound of λ. Therefore, in this article, the rejection capability of the periodic and nonperiodic disturbances in different λ is compared with the magnitude-frequency characteristics of (13) and (14) in Fig. 7, respectively. In (14), D RPTC (s) is the load sensitivity function, which can analyze the gain relationship between the torque disturbance and speed fluctuation generated by it for the RPTC strategy in the article In Fig. 7(a), the lower the ordinate is, the stronger the capability of rejecting periodic disturbances will be. Fig. 7(b) shows that the value of the speed fluctuation caused by the unit step-type nonperiodic disturbance and steady-state recovery time. From  Fig. 7, with the increase of λ, the antidisturbance capability is weakened. Thus, 2T c is used as the upper bound to ensure a sufficient antidisturbance performance. Moreover, even if λ is selected as 2T c , the antidisturbance capability of the RPTC strategy is still much better than that of the PI control strategy.
According to the abovementioned analysis, in this article, the range of λ can be obtained, shown as But the specific determination of λ should be further conducted according to the actual operating conditions.

V. SIMULATION RESULTS AND ANALYSIS
To evaluate the performance of the proposed RPTC strategy, a simulation model for a prototype of the DD-PMSM servo system is built. The relevant parameters are given in Table I. It should be pointed out that, in this article, a prototype with large cogging torque for aerospace application is selected and its cogging torque is about 70% (the amplitude is 35 mNm) of the rated torque (50 mNm) (this large cogging torque is used to remove the electromagnetic brake of the motor).

A. Performance Comparison of Different Parameters for the RPTC Controller
As mentioned above, in this article, for F(s) in (9), the selection of two important parameters, namely r and λ, should be further optimized. Because of this, different values are considered in the RPTC controller to compare its impact on the speed stability at the low speed with different torque disturbances. The corresponding simulation results are shown in Figs. 8 and 9.
From Figs. 8 and 9, with the decrease of r and λ, the antidisturbance capabilities of the RPTC strategy increase accordingly. However, λ is not recommended to be too small, so as to avoid approaching the critical robust stability point, otherwise, when the load is suddenly changed, it may cause the speed oscillation, as shown in Fig. 9(a). This is consistent with Fig. 6. Therefore, it is better to choose r=2 and λ = T c for the RPTC strategy used in the prototype of this article.

B. Antidisturbance Capability Comparison of Two Control Methods at Low Speed 1) Simulink Result of Two Control Methods at Low Speed:
In order to better verify the effectiveness of the RPTC strategy for the disturbance rejection, it is compared with the PI control strategy. Fig. 10 shows the simulation comparison results of speed, position, current, and torque at the low speed of 1 r/min. From Fig. 10(a), for the PI control strategy, the static speed peak shows a clear periodicity and the speed drop is very apparent at the sudden loading moment. While for the RPTC strategy in Fig. 10(b), both the static speed peak and speed drop are significantly reduced, and the actual rotor position movement is smoother. In addition, the RPTC strategy has a smaller q-axis current overshoot and electromagnetic torque overshoot compared with the PI control strategy at the moment of sudden loading.
Therefore, in the simulation, the disturbance rejection capability of the RPTC strategy is obviously better than that of the PI control strategy.

2) Detailed Analysis of Two Control Methods for Target PMSM:
a) PI control strategy: The reason why the speed fluctuations of the PI control strategy for target PMSM is larger in the article is that the speed with errors caused by the derivative as a PI controller input will have an impact on the entire speed control link. One of the main effects is that the gain of the controller is limited to avoid greater speed fluctuations even oscillations. Therefore, the PI control strategy is not effective enough to reject the torque disturbance for PMSM with the cogging torque of 70% rated torque in the article, as shown in Fig. 11. Fig. 11 shows the magnitude-frequency characteristic of D PI (s) [the export method is the same as (14)] for the target PMSM. When the target PMSM operates at the low speed of 1 r/min, the change frequency of the cogging torque (70% of the rated torque) is about 0.6 Hz. Under this condition, it can be seen from Fig. 11 that the corresponding magnitude is about 34.4 dB, which means the corresponding speed fluctuation is about 18 r/min. Consequently, for the target PMSM with a large cogging torque, the PI-based speed loop is difficult to mitigate the effect of the low-frequency cogging torque. Especially at low speeds, it would cause a larger speed ripple, even creeping problem, as shown in Fig. 10.  b) Proposed RPTC strategy: Based on Fig. 12, the reason why the RPTC strategy has better torque disturbance rejection performance for the target PMSM can be explained from two aspects.
First of all, Part II in Fig. 12 has a coefficient λ compared with Part Ⅰ in Fig. 12, which is selected as T c in the article, and T c is the speed loop control period, 5 × 10 −4 s, which is a relatively small number. Therefore, different from the PI control strategy, the errors caused by the derivative have a little effect on the entire controller, as shown in Fig. 13(a).
However, it is indispensable and can play the role of weak damping to prevent speed oscillations in Fig. 14.
Then, according to the similarity between i * q1 in Fig. 13(a) and T d /K T in Fig. 13(b), Part I in Fig. 12 can be regarded as a good observation for the disturbance torque. Hence, from the perspective of implementation, it can be considered as adding a good current compensation link (see Part Ⅰ in Fig. 12) at the q-axis current reference end of Part II in Fig. 12.

C. Robustness Analysis of Parameter Changes for RPTC
In order to further verify, the simulation is carried out by changing the main system parameter related to the controller, i.e., inertia, as shown in Fig. 15. According to the inertia change range in [37] and [38], the total inertia is chosen here to be four times the nominal inertia.

VI. EXPERIMENTAL RESULTS
To further prove the validity of the proposed RPTC strategy, an experimental prototype is developed, as shown in Fig. 16. The proposed control method and the current loop control are implemented in digital signal processor (DSP) and fieldprogrammable gate array (FPGA), respectively. The relevant parameters are the same as those in the simulation.
Based on this prototype, some comparative experiments are carried out. Figs. 17 and 18, respectively, show the experimental results for the proposed RPTC strategy and PI control strategy at the low speed of 1 r/min with the periodic and nonperiodic torque disturbances.
From Fig. 17(a), for the PI control strategy, large speed peaks (about 30 r/min) appear periodically at the low speed of 1 r/min. Due to the large cogging torque, such speed peaks cause the creeping problem of the target motor. While for the RPTC strategy in Fig. 17(b), speed peaks (about 5 r/min) are significantly reduced and there is almost no creeping. Moreover, during the process of speed's reaching its peak, the decline of the q-axis current for the RPTC strategy is gentler, which is consistent with Fig. 10.
From Fig. 18(a) and (b), the speed drop of the PI control and RPTC strategies at the sudden loading moment is about 33 and 4 r/min, respectively. At this point, the q-axis current overshoot of the PI control strategy reaches nearly 0.82 A and drops slowly, while the q-axis current of the RPTC strategy is relatively stable.
In addition, the actual position in the RPTC strategy not only can closely track the position trajectory but also its static and transient fluctuations are much smaller than those of the PI control strategy.  The following can be seen from the abovementioned results. 1) The experimental results, simulation results, and theoretical analyses for the RPTC strategy are basically consistent. 2) At the low speed of 1 r/min, the RPTC strategy has better static and dynamic speed performance than the PI control strategy under different torque disturbances (cogging torque: 70% of the rated torque; sudden load: rated torque). 3) From the correspondence between the position and speed fluctuations, the RPTC strategy can stabilize the speed by controlling the rotor track the position trajectory smoothly at low speed. 4) From the correspondence between the actual position and q-axis current, the RPTC strategy can make effective compensation through the q-axis current under the condition of unknown torque disturbance information.

VII. CONCLUSION
This article proposes a novel RPTC strategy to stabilize the speed of a DD-PMSM servo system at the low speed with different torque disturbances. The main idea of this control strategy is that the speed control can be converted to the position control as long as the continuous position trajectory can be smoothly tracked. To achieve this goal, the position-current two-loop with the RPTC controller is proposed. In this article, the RPTC controller is based on the internal-mode method, and its parameters should be designed and optimized from the perspectives of robust stability and antidisturbance capability. Compared with the PI control strategy, the proposed RPTC strategy has a significant reduction of speed fluctuation and a better performance in the presence of periodic and nonperiodic torque disturbances at the low speed, especially for the target motor in this article with the large cogging torque. What is more, it is worth mentioning that this control strategy does not require the precise speed feedback, is insensitive to the inertia variation, and in particular, the specific information of torque disturbances is also not needed, so it is simple to be implemented.
Although the proposed RPTC strategy is verified based on the DSP+FPGA experimental platform built for aerospace applications, for civil applications, it still has certain applicability under DSP only.