Interior permanent magnet synchronous motors (IPMSMs) are extensively used in several high-performance industrial applications (e.g., electric vehicle drives, robotics, traction drives, and home appliances) owing to their significant properties, such as fast dynamics, high precision, high torque and power density, dynamic performance, low maintenance cost, and high reliability. Precise rotor shaft speed and position information are needed to achieve high-performance vector control [1
]. However, at low speed, because of the presence of external disturbances, such as parasitic torque ripples or parameter uncertainties, caused by imperfect machine design, the uncertain data measured from the sensor caused by noise and the effect of mechanical load or power electronic switches [3
] cause the speed trajectory tracking performance to deteriorate and degrade the robustness. Therefore, it is necessary to reduce the periodic torque ripples and overcome the effects of parametric variation to ensure high-precision trajectory tracking performance of IPMSM drives.
Various control techniques for solving the aforementioned problems have been presented in the literature. Several control strategies based on conventional controllers, such as proportional–integral (PI) control [5
] have attracted attention owing to their simplicity and ease of implementation on hardware; however, they are highly dependent on actual drive parameters and require exact parameter values to tune the PI gain to achieve efficient closed-loop performance. To eliminate the parameter dependency on the control design, nonlinear controllers have been developed [7
]. In previous research [7
], deadbeat control showed excellent speed tracking performance, but it depends highly on the IPMSM parameters to achieve efficient tracking performance by setting the closed-loop poles to zero. Fuzzy logic controllers [10
] can effectively deal with drive nonlinearities and model unknown parameter uncertainties. However, this control scheme depends highly on gains and requires extensive knowledge to choose appropriate fuzzy interference rules to achieve excellent speed tracking. Model predictive control (MPC) [12
] is easy to implement and straightforward, but highly depends on the exact model of the IPMSM to predict the future control output variables that ensure adequate closed-loop control effects.
Variable structure control (VSC) was first studied in the 1950s for high-order differential systems, but, because of excessive chattering in the VSC system and the lack of an appropriate design approach, it has become less popular among researchers [15
]. After the 1970s, the importance of VSC was further explored because of its robustness, quick response, and easy implementation. Moreover, system parametric variation and external disturbance are not issues, and VSC shows excellent dynamic performance [16
]. In previous studies [16
], the sliding mode control (SMC) scheme was applied to various applications owing to its invariance to external disturbances and unknown model parameter variations, which ensures excellent speed tracking performance regardless of parameter uncertainties or disturbances. The robustness of SMC highly depends on large control gains, whereas a large gain value causes a chattering issue, which excites higher-frequency dynamics. To suppress this issue, a reaching law is designed based on an exponential term that adopts the change of system states and the sliding surface. Therefore, in this study, an exponential reaching law SMC was applied to eliminate the issue of high chattering and slow reaching time.
Parasitic electromagnetic torque ripples are caused by several factors, such as the cogging torque, periodic flux harmonic, sensor measurement uncertainties, and current offset [20
], which cause periodic speed ripples and degrade the control performance of the drive, especially under low-speed operating conditions. The fluctuation in speed can even cause instability in the system [21
]. Torque pulsation can be divided into two high- and low-frequency components. By increasing the bandwidth of the closed-loop control, the high-frequency torque pulsating component can be effectively controlled, whereas, for the low-frequency component, which occurs within the closed-loop control bandwidth, further attention is required [22
Several control schemes have been proposed in recent years to reduce the periodic torque ripples. The schemes are classified into two groups. The first focuses on the machine design and improvement of the machine structure (e.g., improving the winding distribution, skewing the slots or magnet, or guaranteeing the fractional number of slots per pole) [23
]. Machine design optimization is the most effective way to minimize periodic torque ripples; however, once the machine structure is designed, the performance cannot be modified. In addition, optimizing the structure design results in a higher cost and more complex realization. The other group tends to minimize periodic torque ripples by employing advanced controllers that improve the performance of the motor drive by correcting and compensating for periodic torque ripples [25
]. In an earlier study [25
], the current compensation signal was added across the q
-axis reference current to suppress the disturbance factor that helps to reduce the periodic speed oscillation caused by periodic torque ripples. MPC is designed in such a way that it reduces speed ripples because of the embedding disturbance frequency caused by the current sensor offset error [12
]. In other work [26
], a hybrid control scheme employing a finite control set MPC and lookup table was implemented to compensate for torque ripples caused by cogging.
To improve the dynamic and standstill response of an IPMSM drive at low speed, an effective compensation scheme along with a sliding-mode speed controller was implemented in this study. The proposed control schemes guarantee speed ripple reduction under low-speed working conditions. The speed control loop was designed based on exponential reaching law sliding-mode control (ERL-SMC), which is independent of the motor model, and the compensation signal was injected across the reference current and the output of the speed control loop. This was done to utilize low-frequency current and torque disturbance, which helps to modify actual reference variables in such a way that the controller rejects unwanted disturbing signals. The proposed control scheme was compared with conventional field-oriented control. The results show that the proposed scheme can effectively compensate for torque ripples that significantly increase the drive performance in terms of rotor shaft speed ripple minimization and transient/steady-state performance. Moreover, the proposed control scheme is simple and does not require a high computational cost.
The remainder of this article is organized as follows. A mathematical model and the maximum torque per ampere (MTPA) scheme for an IPMSM are introduced in Section 2
. In Section 3
, the source of periodic torque ripples is explained. The design process of sliding-mode speed control combined with a compensation scheme is described in Section 4
. The results obtained to verify the effectiveness of the proposed control scheme are presented and discussed in Section 5
. Finally, the contribution of this study is summarized in Section 6
5. Results and Discussion
A simulation model was developed to demonstrate the effectiveness of the proposed control design. An IPMSM drive with parameters given in Table 1
, was utilized for performance evaluation under different test conditions.
To demonstrate its effectiveness, a computer-based model of the proposed control design was implemented in MATLAB/Simulink, and a comparative study using a simulation tool was conducted to show the robustness and effectiveness of attaining excellent standstill performance under low-speed operating working conditions. The compensation gains for the electromagnetic torque and
-axis current loop are set to
, whereas cutoff frequency is set to 50 rad/s. The optimal compensator gains should be chosen in such a way that the system’s stability and robustness are guaranteed while also obtaining the highest disturbance rejection capabilities. Figure 5
shows the rate of the ripple factor for the speed and torque of the IPMSM drive. The torque ripple factor (TRF) and speed ripple factor (SRF) are expressed as the ratio of the peak–peak torque and speed ripple to the rated torque and speed of the IPMSM drive, and they are derived as follows:
a,b show the evaluation of the efficacy of the designed control algorithm with conventional control designs. For Figure 5
a the speed is varied (10, 20, 30, 40, 50, 60) rpm under constant load torque of 7 Nm, whereas in Figure 5
b the torque is varied (2, 3, 4, 5, 6, 7) Nm under constant speed of 50 rpm. The result reveals that employing the proposed control design ripple across the speed and torque effectively results in a reduction when the reference value increases. Compared with the conventional design, the ripples are greatly minimized with the proposed control design with increasing robustness, especially at low speed.
shows the speed response under various load conditions. A sliding-mode speed controller is applied to eliminate linear PI controllers that are sensitive to external disturbances. Initially, a reference speed of 50 rpm is applied under a load torque of 3 rpm that is shown in Figure 6
a. Figure 6
b,c shows the settling time and steady-state error of the drive, respectively. The settling time of the proposed design is fast, and the steady-state error is reduced. At 0.5 s, a load torque of 7 Nm is applied with a speed change from 50 to 30 rpm at 0.6 s. Figure 6
d–f shows that, with load variation, the speed response is good, and the ripples across the speed at standstill are smaller than those in the conventional control design.
a shows the a
-phase current response of the IPMSM drive. The waveform of the a
-phase current has a sinusoidal shape in the case of the proposed model, which indicates the regular operation of the IPMSM drive. A comparison of the a
-phase current shows that the designed algorithm has a smaller current ripple, and the harmonic distortion is suppressed, ensuring the efficacy of the design model.
b shows the response of the reference torque and the output electromagnetic torque across the drive. By employing the MTPA control scheme, the torque was obtained by utilizing the minimum current value. It can be seen that
has a higher ripple in the conventional method, whereas ripples are reduced in the proposed control structure design.
shows the drive phase current along the d
-axis. Based on conventional methods, the d
-axis currents show higher current ripples, whereas the proposed control design shows minimum ripples across the d
To verify the robustness of the proposed control design, the speed convergence characteristic of the IPMSM was simulated based on a sinusoidal reference speed input with an amplitude of 10 rpm and a frequency of 30 Hz. The sinusoidal speed response and error curves based on different control designs are shown in Figure 9
. The convergence of the actual to the reference speed with minimal error is observed in the proposed design. This implies that the steady-state error of the control system is minimized when the proposed control method is employed. Moreover, the controller has excellent dynamic performance and can effectively increase the robustness of the system by resisting the effect of load disturbance, which increases the overall performance of IPMSM speed control under uncertainties and load variations. The qualitative performance analyzed between the conventional and the proposed control schemes is summarized in Table 2
An effective compensation method with a sliding-mode speed controller which reduces speed and torque ripples at low-speed operation is proposed. The SMC scheme makes the speed controller parameter independent. The main sources of torque ripple, the structure of the speed controller, and the compensation technique for minimizing the ripples in the speed control were discussed in detail. The proposed method is simple and does not require additional computational cost and can be applied to drive applications where low-order harmonics are undesired. The chattering of SMC is reduced by introducing an exponential reaching law that has the advantage of fast convergence and could adapt the variation of switching function, thus, increasing the robustness of the speed controller. The feasibility of the control structure was verified through simulation results under different conditions, e.g., variable low-speed, varied load, and sinusoidal speed reference input. A comparative study with the conventional control method was conducted, and the results show that the proposed controller effectively enhances the steady-state and dynamic performance of SPMSM especially in the low-speed regions, e.g., smaller speed ripples, smaller steady-state error, and faster transient response.