Active Disturbance Rejection Controller for Smooth Speed Control of Electric Drives Using Adaptive Generalized Integrator Extended State Observer

To suppress the torque ripple for smooth speed control of electric drives, active disturbance rejection (ADR) controller based on generalized integrator (GI) extended state observer (ESO) (GIESO) is developed in this article. Deadbeat ADR controller is employed to control the current, allowing the current control algorithm to be simplified. An adaptive GIESO is developed to address the instability problem caused by the two-step delay in the current control loop. Based on the system stability analysis, the design guideline for the adaptive resonant gain is determined. The proposed adaptive GIESO allows the system to be stable across the entire speed range while suppressing speed ripple to within 1 r/min.


I. INTRODUCTION
I N RECENT years, permanent-magnet synchronous motor (PMSM) has attracted lots of attention in direct-drive applications [1] due to its simple structure and high-torque density [2]. However, the large low-order torque ripple due to the open slot and manufacturing imperfection causes speed ripple, resulting in large vibration, acoustic noise, and low control precision [3]. The dominant torque ripple sources include cogging torque, flux harmonics, current harmonics, and current measurement error [4]. To alleviate the effect of torque ripple, there are two types of methods that can be used [5]. The first group of technologies includes motor-design techniques, such as skewing [6], stator pole arc design [7], and permanent magnet magnetic design [8]. However, these procedures are expensive and only applicable to newly developed machines. Activecontrol techniques are the second category of methods, which are preferred due to their low costs and flexibility. Among many active control techniques, the speed-control techniques realize a smooth speed control by the torque compensation based on the observed disturbance, are attracting more attentions in many applications [9]. In these speed-control methods, the key technique is the disturbance observation.
Typically, there are two ways to observe the disturbance. One is to use the integrator in the feedback controller. When the reference is a direct signal, zero steady-state error can be secured by using a conventional proportional-integral (PI) controller. When the reference is an alternate signal, however, generalized integrator (GI) or resonant controller should be used [10]. The resonant controller in the stationary reference frame is equivalent to the integrator in the synchronous reference frame [11], which is another efficient technique in suppressing harmonics at the expense of additional frame transformation. When both direct signal and alternate signal exist in the reference, the so-called proportional-integral-resonant (PIR) controller should be employed [12]. In the case of multiple harmonics suppression, either multiple resonant controllers [13], or repetitive controller [14] can be employed.
Although these controllers have a good periodic disturbance rejection performance, they degrade the tracking performance at the same time because they are one-degree-of-freedom controllers. To solve this problem, a two-degree-of-freedom (TDOF) controllers can be employed. In [15], a tracking differentiator is used as a prefilter to smooth the speed reference so that the high-frequency components in the speed reference will not be amplified by the PIR controller. Alternatively, Hu et al. [16] used a partial prefilter, i.e., the prefilter is only used for the resonant controller while a normal PI controller is used for suppressing constant disturbances. Nevertheless, the parameter setting of the partial prefilter is troublesome.
The alternative way to observe the disturbance is to use disturbance observers (DOBs), by which TDOF control can be achieved naturally [17]. The goal of DOBs is to estimate the disturbance by forcing the estimated output to track the real system output. Similarly, regulators used in the feedback control can be used in observers. For example, proportional regulator in DOB, PI regulator in extended state observer (ESO). It is known from that different DOBs act as a low-pass filter when observing disturbances [18]. Therefore, the generalized PI observers [19] cannot well observe the period disturbances.
In [20], DOB (the alleged dual loop structure) based on resonant regulator is employed to suppress the sixth order current harmonics caused by the deadtime effect. However, the experiments are conducted in the low-speed range and the highest resonant frequency is only 60 Hz. A general design procedure of this kind of DOB and the stability analysis can be found in [21]. Compared with the DOB-based control method, active disturbance rejection (ADR) control (ADRC) based on ESO has a better measurement noise suppression performance because both the speed and disturbance are observed. In [22], generalized integrator ESO (GIESO) is employed to suppress the current harmonics caused by the unbalanced grid. Multiple GI modules are adopted to suppress the fundamental, the second order, and the sixth-order harmonics. So, the highest resonant frequency in the system reaches 300 Hz. However, higher frequency disturbances are not considered and thus the effect of the one-step delay caused by pulsewidth modulation (PWM) update is not taken into account. In [23], quasi-resonant ESO is developed to improve system's robustness to frequency deviations. The controller is designed in the discrete-time domain so that the one-step delay can be easily handled. The analysis shows that the system will be unstable if the sampling time is two large. Nevertheless, high-frequency disturbances are not considered either. To reject the high-order torque ripple harmonics, Hu et al. [4] employed the PIR controller for current control so that the delay caused by inner current loop can be eliminated. However, the PIR current controller complicates the algorithm.
In this article, active disturbance rejection (ADR) controller using GIESO is used to achieve smooth speed control of electric drives. Different from the system in [4], the system in this article adopts the widely used deadbeat active distubance rejection controller [24] for current control. To solve the instability problem caused by the two-step delay in the current control loop, an adaptive GIESO is developed. In such a case, the complicated PIR controller can be avoided. To obtain the design guideline for the adaptive resonant gain, the critical resonant frequency for stable operation is deduced. Various experiments are tested on the dSPACE DS1103 test bench to verify the effectiveness of the proposed method.
The rest of this article is organized as follows. In Section II, ADR controller based on GIESO is designed, followed by the system performance analysis in Section III. In Section IV, the design guideline for the adaptive resonant gain is presented. Experimental results are presented and analyzed in Section V. Finally, Section VI concludes this article.

A. Mathematical Model of Mechanical System
The motion equation of the PMSM system with known moment of inertia can be given aṡ (1) where Ω, B, J, and T L denote the mechanical angular velocity, the moment of inertia, the viscous friction torque coefficient, and the load torque, respectively; b = 1/J is the control gain, T e and T * e represent the electromagnetic torque and its reference value; the viscous friction torque coefficient is very small and is usually assumed to be zero, so the viscous friction torque and the load torque are lumped as the nominal disturbance is the total disturbance considering the torque tracking error.
In many applications, conventional ESO-based ADR controller is employed to reject a constant disturbance d con . However, when both a constant disturbance d con and a sinusoidal disturbance d sin with a frequency of ω h exists in the system, the mathematical model should be modified as where d cos is the derivative of d sin .

B. Feedback Control Law
Defining the reference of mechanical angular velocity as Ω * , then the tracking error of mechanical angular velocity can be expressed as e s = Ω * − Ω and we havė The expected error convergence law can be designed aṡ where k ps is the proportional gain. Substituting (4) into (3) yields In (5), the disturbance d to is unknown. Generally, the rotor position is obtained from the position sensors such as the encoder or resolver, and the speed Ω can be calculated by the derivative of the mechanical angular position θ m . Due to the quantization error in the measurement of the position, the speed calculated by the classical frequency method is contaminated by the measurement noise [25], [26]. Denoting the measurement noise in position and speed as δ p and δ n , δ n (s) = sδ p (s), then the measured speed can be expressed as Ω m = Ω + δ n . When Ω and d to are substituted by their estimated valueΩ andd to , the torque reference is modified as In the practical system, the reference torque limit is usually applied as follows: where T * esat and T * e max are the saturated torque reference and the maximum torque reference, respectively. According to (1), the relationship between the torque reference and the total disturbance can be expressed by Without considering the saturation of torque reference, the measured speed can be obtained by substituting (8) are the speed observation error and the disturbance observation error.
From (9), it can be seen that the control performance depends not only on the feedback control law but also on the speed observation error and the disturbance observation error.

C. Speed and Disturbance Observation Using GIESO
Considering the sinusoidal disturbance, a GIESO is constructed as where variables withˆrepresents the estimated value, k r is the resonance gain of the GI. In this article, the ADR controllers based on the conventional ESO and the GIESO are called the conventional ADR controller and the modified ADR controller, the control systems using these two controllers are noted as the conventional ADRC system and the modified ADRC system, respectively. The block diagram of the modified ADRC system is shown in Fig. 1. When k r = 0, the modified ADRC system is the same as the conventional ADRC system. From Fig. 1, the speed observation error, the estimated speed and disturbance can be deduced as is the transfer function of the GI.
Generally, Δ 2 is set as (s + ω o ) 2 , thus, k 1 and k 2 can be calculated by k 1 = 2ω o and k 2 = ω 2 o , ω o is the natural frequency of the conventional ESO.

D. System Output Considering the Torque Control Loop
Suppose the torque control system can be modeled as a firstorder low-pass filter with a time constant of T ci . The relationship between the torque reference and the total disturbance can be expressed by (14) Substituting (13) into (12) yields where Δ cl = T ci s 2 G 3 (s) + (s + k ps )G 0 (s) is the characteristic polynomial of the closed-loop control system.

A. System Dynamic Performance
When T ci = 0, from (14) and (15), it is known that the transfer functions from the actual disturbance to the observed disturbance, from the disturbance to the output, and from the measurement noise to the output can be expressed as  It shows that the disturbance rejection ability and the measurement noise suppression performance are decided by k ps and the transfer function G 0 (s), G 3 (s).
Setting k ps = 300 rad/s and ω o = 500 rad/s, when ω h is set as 20 Hz, dynamic performances of the modified ADRC system and the conventional ADRC system are shown in Fig. 2. From Fig. 2(a), it can be seen that the disturbance of the 20 Hz resonant frequency can be well suppressed by the modified ADRC system, but cannot be suppressed by the conventional ADRC system. Using larger k r can reject a wider range of disturbances and reduce the sensitivity to the resonant frequency, but it also leads to poor measurement noise suppression performance, as shown in Fig. 2(b).
From Fig. 2(a), it is also observed that the modified ADRC system has poorer rejection property for disturbances near 8.94 Hz and 14.14 Hz than the conventional ADRC system, this is because the GIESO cannot observe the disturbance of a specific frequency that lower than the resonant frequency, as shown in Fig. 3(a). According to the first equation in (16), the specific frequency ω spe1 can be deduced as When λ is 1 and 4, the specific frequency is 0.707ω h and 0.447ω h , respectively, this explains why the frequency is 8.94 Hz and 14.14 Hz when ω h is 20 Hz. The conclusion is also true when ω h = 500 Hz, as proved by Fig. 3(b). Since the disturbance  with the frequency of ω spe1 cannot be observed by GIESO, it is suppressed by the proportional feedback control. It should be pointed out that in the conventional ADRC system, the disturbance in the low-frequency range is mainly rejected by the disturbance feedforward control, while the disturbance in the high-frequency range is mainly suppressed by the feedback control [19]. Therefore, the modified ADRC system is inferior to conventional ADRC system in rejecting the specific frequency disturbance in the low-frequency range, but its suppression performance of the specific frequency disturbance in the highfrequency range is almost the same as that of the conventional ADRC system, as shown in Fig. 2(a) and 4(a). From Fig. 3(b), it can also be found that the disturbance of another specific frequency ω spe2 that higher than the resonant frequency is obviously amplified by GIESO in the highfrequency range. This property has little effect on the disturbance rejection ability, but it has large effect on the measurement noise suppression performance, as shown in Fig. 4   decrease in the measurement noise suppression performance for noises with frequencies around ω spe2 . Using smaller λ mitigates the performance degradation, but increases the sensitivity to the resonant frequency. In the experiments, λ should be carefully tuned to achieve the desired performance.

B. System Stability of the Conventional ADRC System
When k r = 0, the modified ADRC system is the same as the conventional ADRC system and there is G 0 (s) = Δ 2 . The characteristic polynomial of the conventional ADRC system can be deduced as To evaluate the effect of T ci , root locus when T ci varies can be plotted. The open-loop transfer function in a unit negative feedback control system with the same characteristic polynomial can be expressed as Setting k ps = 300 rad/s, the plots of root locus with the variation of T ci when ω o is 500 rad/s and 1000 rad/s are shown in Fig. 8. It can be seen that the condition for stable operation is T ci < 4.52 ms and T ci < 3.47 ms when ω o is 500 and 1000 rad/s, respectively.  According to the Routh-Hurwitz stability criterion, the critical value of T ci for stable operation T ci_crit under different k ps and ω o can be calculated and the plot is shown in Fig. 9. It can be seen that T ci_crit decreases as k ps or ω o increases. From (18), it is easy to know that the system is stable when T ci = 0. As a results, it is learned that the delay in torque control loop makes the conventional ADRC system unstable.
To improve the system stability, torque control method with high bandwidth should be adopted. For example, PIR controller is employed for current regulation in [4]. However, the measures to deal with the deteriorated tracking performance caused by the PIR controller complicates the algorithm [16].
In this article, deadbeat current vector control is employed to control the torque. By using the deadbeat control, the actual torque lags the reference torque with two control steps when the voltage is not saturated [27], the minimum value of the time constant of the torque control loop is twice the sampling time, i.e., 2T s [24]. In this article, the sampling time is set as 0.1 ms, so the minimum value of T ci is 0.2 ms.

C. System Stability of the Modified ADRC System
The characteristic polynomial of the modified ADRC system can be obtained as Traditionally, the delay caused by the torque control loop is not considered, i.e., T ci = 0. In this case, the characteristic  polynomial of the closed-loop control system is simplified as According to the Routh-Hurwitz stability criterion, the system is always stable regardless of the value of k r . When T ci = 0, it is known from (20) that the system stability varies with the resonant frequency ω h . To evaluate the effect of ω h , root locus when ω h varies can be plotted. The open-loop transfer function in a unit negative feedback control system with the same characteristic polynomial can be expressed as Defining the ratio k r to k 2 as λ = k r /k 2 . When k ps = 300 rad/s and ω o = 500 rad/s, the plots of root locus when ω h varies under different T ci and λ are shown in Fig. 10. It can be seen from Fig. 10(b) that ω h_crit is 338 Hz when T ci = 0.2 ms, λ = 1.0. If GI is used to suppress the 12th harmonic, then the speed limit for stable operation is 1690 r/min. However, if GI is used to suppress the 60th harmonic, then the speed limit for stable operation is only 338 r/min.
According to the Routh-Hurwitz stability criterion, the critical ω h for stable operation ω h_crit can be calculated. When k ps = 300 rad/s, ω o is 500 rad/s, and 1000 rad/s, ω h_crit under different λ and T ci is shown in Fig. 11. It shows that ω h_crit can be increased by either decreasing λ or decreasing T ci , with the latter having a greater effect.
When T ci = 0.2 ms and λ = 1, the plots of root locus when ω h varies under different k ps and ω o are shown in Fig. 12. Compared with Fig. 10(b), it can be concluded that increasing k ps from 300  to 600 rad/s decreases ω h_crit from 338 to 326 Hz, but increasing ω o from 500 to 1000 rad/s increases it from 338 to 467 Hz.
When λ = 1, T ci is 0.1 ms and 0.2 ms, the plot of ω h_crit varies with k ps and ω o is shown in Fig. 13. It indicates that increasing ω h_crit can be accomplished by either decreasing k ps or increasing ω o . It should be pointed out that decreasing k ps has little effect on increasing ω h_crit .
In summary, the modified ADRC system will be unstable as the resonant frequency increases due to the limited bandwidth of ESO and the delay caused by the torque control loop. Although the frequency limit can be increased by decreasing T ci or increasing ω o , large ω o results in poor measurement noise suppression performance, whereas small T ci has a high requirement for hardware. To this end, adaptive GIESO is developed in this article.

IV. ADAPTIVE GIESO
Since high-frequency torque ripple has little effect on the speed but leads the system to be unstable, the high-order harmonics suppression can be removed in the high-speed range. As a result, an adaptive GIESO with variable resonance gain is proposed in this article. The gain of the GI for the suppression of 60th harmonic is decreased as speed increases so that the GI only acts in the low-frequency range. The adaptive resonance gain is expressed by where k is a positive gain, p n is the polePairs. According to the motion equation, the speed ripple caused by the torque ripple depends on the amplitude of torque ripple as well as the system inertia. Under the same torque ripple, the  speed ripple in the system with larger inertia will be smaller. Therefore, the adaptive gain should be designed according to the actual system. From Fig. 10(b), it is known that the speed limit is 338 r/min when k ps = 300 rad/s, ω o = 500 rad/s, and T ci = 0.2 ms. In addition, in the experiments, we found that the 60th speed ripple at the speed over 200 r/min is smaller than 1.5 r/min. Therefore, to make the system stable in the whole speed range, the gain k can be set by making k r1 equals 0 at the speed ranges from 200 to 338 r/min. Consequently, k should be set between 0.0028 and 0.0048. In this article, the two k for the module is set as 0.0025 in the system so that k r1 equals 0 at the speed of 238.7 r/min. The saturation function can be used to keep k r1 nonnegative. The block diagram of the adaptive GIESO is shown in Fig. 14.

A. System Configuration
The PMSM studied in this article has 10 pole pairs and 12 slots [28]. The motor specifications are shown in Table I. The configuration of the test bench is shown in Fig. 15. A programmable dc power supply is used to provide 150 V dc-bus voltage. The PMSM is driven by a self-made inverter, consisting of Mitsubishi intelligent power module, current and voltage hall sensors, etc. The dSPACE DS1103-based field-oriented control box is employed for driving the PMSM. The control strategy is based on space vector PWM control with i * d = 0. Deadbeat current controllers are employed in the current-loop to control i d and i q , respectively, and different ADRC controllers are employed in the speed-loop. The current limit is set as 9 A. The sampling frequency, current control frequency, and the speed control frequency are all set to 10 kHz.
The torque ripple of the tested motor is estimated by GIESO, which is employed to suppress the speed ripple. When the speed   ripple is fully suppressed, the estimated disturbance torque is taken as the torque ripple. When k ps = 300 rad/s and ω o = 500 rad/s, the speed ripple when using ESO and GIESO at the speed of 60 r/min is shown in Fig. 16(a), and the spectrum analysis is shown in Fig. 16(b). The comparison of the speed ripple in different ADRC systems are shown in Table II. It can be seen that all the speed ripple harmonics in the GIESO-ADRC system are greatly suppressed. In such case, the estimated torque ripple and its spectrum analysis are shown in Fig. 16(c) and (d), respectively. From Fig. 16(d), it is known that there are  abundant harmonics in the torque ripple, such as 12th harmonic, 24th harmonic, 36th harmonic, 60th harmonic, and 120th harmonic. According to the analysis results in [3], these low-order harmonics are caused by manufacturing uncertainties, specifically, the imperfection of stator. The frequencies of different order harmonics are speed dependent, i.e., ω h = k h ω r , k h is the harmonic order, ω r is the rotor angular speed. Among these harmonics, the 12th harmonic with the amplitude of 0.4 N·m and the 60th harmonic with the amplitude of 0.3 N·m are the dominant harmonics. Consequently, only these two harmonics are suppressed in experiments.

B. Performance When Using the Conventional ADR Controller
Since the cogging torque contains different harmonics, the effect of torque ripple on speed feedback will be different under different speed references. Therefore, the speed references in the experiments are stepped from 0 r/min to 100 r/min and 500 r/min in a step of 20 r/min and 100 r/min, respectively.
The speed response of the conventional ADRC system is shown in Figs. 17(a) and 18(a), where the fast Fourier transform (FFT) is implemented on the speed tracking error in the steady state. It can be seen that the speed ripple decreases with the increase of the cogging torque frequency. The 12th order harmonic of the cogging torque causes the largest effect on the speed ripple when the speed is higher than 100 r/min. When the reference speed is 100 r/min, the frequency of the 12th order harmonic is 20 Hz. As shown in Fig. 2(a), the conventional ADRC system has the worst rejection ability for the disturbances at frequencies around 50 Hz, which explains why the 12th speed ripple at the speed of 300 r/min is the largest while the 60th speed ripple at the speed of 40 r/min and 60 r/min are the largest. It is noted that the 60th speed ripple is negligible when the speed is over 200 r/min.
Since the 60th speed ripple is dominant in the low-speed range while the 12th speed ripple is dominant in the high-speed range, only two GI modules are employed in the system, as shown in Fig. 14. The two resonance gain ratio λ for the module GI 12 and GI 60 are set as 1.0 and 0.1, respectively.

C. Performance When Using the Modified ADR Controller
When only 12th harmonic suppression is enabled, the speed response of the modified ADRC system is shown in Figs. 17(b) and 18(b). It can be seen that the 12th harmonics are greatly reduced in the whole speed range. Although only the module GI 12 is employed, the high-order harmonics in the low-speed range (n < 80 r/min) is also decreased, which can also be explained by the bode plots shown in Fig. 2(a). Without the 12th speed ripple, the 60th speed ripple becomes the dominant harmonic component.
When both 12th and 60th harmonic suppression is enabled and an adaptive gain k r1 is utilized, the speed response of the modified ADRC system is shown in Figs. 17(c) and 18(c). It can be found that both the 12th and the 60th speed ripple are also greatly reduced, which verifies the effectiveness of the proposed method. Fig. 19 shows the step responses of the modified ADRC system under different combinations of k and λ. It can be seen that the system becomes unstable in the high-speed range when using a constant resonance gain k r1 . It should be pointed out that the waveform is captured by using a trigger signal, which acts at the time of 0 s. This explains why there are waveforms before 0 s. In Fig. 19(a), it seems that the system can be stable at the speed of 500 r/min. This is because the system takes longer time to reach the unstable state when using a smaller λ. When λ is increased from 0.1 to 1.0, the experimental results are shown in Fig. 19(c). It is noted that the system becomes unstable at the speed of 300 r/min, which is consistent to the theoretical results. As a comparison, when using the proposed adaptive resonance gain, the system can be stable in the whole speed range, as shown in Fig. 19(b) and (d).
To show the effectiveness of the modified ADRC system in the whole speed range, the tracking performance for two kinds of sinusoidal speed references expressed by 300 + 300 sin 4πt and 600 sin 4πt are tested, the speed responses are shown in Fig. 20. It can be seen that when using the conventional ADRC controller, the speed tracking error when tracking the single direction speed and the double direction speed is ±0.4 rad/s and ±0.5 rad/s, while they can be reduced by 50% to ±0.2 rad/s and ±0.25 rad/s when using the modified ADRC controller.

VI. CONCLUSION
In this article, ADR controller based on adaptive GIESO is developed to suppress the torque ripple for smooth speed control of electric drives in the whole speed range. The GIESO with fixed resonance gain is effective for the low-order harmonics in the whole speed range, while it leads the system to be unstable in the high-speed range when rejecting high-order harmonics. The reason behind this is the limited bandwidth of the torque control loop and the limited bandwidth of ESO. High bandwidth of the torque control loop can be realized by using deadbeat control, while the increase of ESO bandwidth is restricted by the speed measurement noise. To make a stable operation in the whole speed range, an adaptive resonance gain can be adopted. By using the proposed adaptive GIESO, both the 12th and the 60th speed ripple can be suppressed within 1 r/min in the whole speed range when tracking constant speed references. Compared with the conventional ADR controller, the proposed ADR controller can reduce the speed tracking error by 50% when tracking the sinusoidal speed references. Since delay also exists in the PWM undate process, the method proposed in this article can also be applied in torque/current control system to suppress harmonics.
Wen-Hua Chen (Fellow, IEEE) received the M.Sc. and Ph.D. degrees in automatic control from Northeastern University, Shenyang, China, in 1989 and 1991, respectively.
From 1991 to 1996, he was a Lecturer with the Department of Automatic Control, Nanjing University of Aeronautics and Astronautics, Nanjing, China. He held a research position and then a lectureship in control engineering with the Centre for Systems and Control, University of Glasgow, Glasgow, U.K., from 1997 to 2000. He is currently a Professor in Autonomous Vehicles with the Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, U.K. He has authored or coauthored three books and 250 papers in journals and conferences. His research interests include the development of advanced control strategies and their applications in aerospace engineering, particularly in unmanned aircraft systems.
Dr. Chen is a Fellow of the Institution of Engineering and Technology and the Institution of Mechanical Engineers. He was a recipient of the EPSRC Established Career Fellowship Award. He is currently an Assistant Professor with Nanyang Technological University, Singapore, and an Honorary Assistant Professor with The University of Hong Kong, Hong Kong. He was a Postdoctoral Fellow and then a Visiting Assistant Professor with the Massachusetts Institute of Technology, Cambridge, MA, USA. He is a Chartered Engineer in Hong Kong. In these areas, he has authored or coauthored one book, three books chapters, and more than 100 referred papers. His research interests include electric machines and drives, renewable energies, and electromechanical propulsion technologies.
Dr. Lee was the recipient of many awards, including NRF Fellowship, Nanyang Assistant Professorship, Li Ka Shing Prize (the best Ph.D. thesis prize), and Croucher Foundation Fellowship. He is currently an Associate Editor for IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, IEEE TRANSACTIONS ON ENERGY CONVERSION, IEEE ACCESS, and IET Renewable Power Generation.