Authentic Simulation Studies of Periodic Adaptive Learning Compensation of Cogging Effect in PMSM Position Servo System Ying Luo, YangQuan Chen and Youguo Pi

Abstract— This paper presented a detailed authentic simulation model of a permanent magnet synchronous motor control system based on the SimPowerSystems toolbox in Simulink library. We then focus on the periodic adaptive learning compensation (PALC) for cogging effect on PMSM position and velocity servo tasks. The cogging force is considered as a position-dependent disturbance that is periodic and not necessarily sinusoidal. The key idea of the implemented cogging disturbance compensation method is to use past information for one position period along the state axis to update the current adaptation law. Simulation results are presented to illustrate the effectiveness of the adaptive cogging compensation scheme. Furthermore, the advantage of the PALC is demonstrated. We also reported an initial result on high-order PALC. Index Terms— Cogging force, permanent magnet synchronous motor (PMSM), adaptive control, periodic adaptive learning control, state-dependent disturbance.

I. I NTRODUCTION Permanent magnet synchronous motors (PMSM) are the most popularly used electromechanical devices for high performance industrial servo applications of accurate speed and position control because of their high efficiency, highpower density and high torque-to-inertia ratio in comparison with other machines. However, the cogging forces, which is the main disadvantage of PMSM, result in periodical torque ripples pulsates with rotor position [1]. These ripples appear as periodic oscillations in the motor speed and limit the position servo control performance of application especially in a high-precision tracking applications. So, as the increasing popularity of PMSM in industrial applications, the suppression of cogging effect has received much attention in recent years. The cogging forces are due to the interaction between the permanent magnets and the stator; thus the cogging effects minimization techniques, broadly speaking, can be divided into two major groups [2]. The first group focuses on the techniques by improvement of motor design [3], and the second group is to design additional control algorithms to minimize the cogging effect [2], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. In this paper, we first present a detailed authentic simulation model of a PMSM control system based on the SimPowerSystems toolbox in Simulink library. We then focus on the periodic adaptive learning compensation (PALC) of cogging effect on PMSM position and velocity servo tasks. The cogging force is considered as a position-dependent

disturbance that is periodic and not necessarily sinusoidal. The key idea of the implemented cogging disturbance compensation method is to use past information for one position period along the state axis to update the current adaptation law. Simulation results are presented to illustrate the effectiveness of the adaptive cogging compensation scheme. Furthermore, the advantage of the PALC is demonstrated. We also reported an initial result on high-order PALC. The major contributions of this paper include 1) An authentic PMSM position servo control system model built in Simulink using SimPowerSystems toolbox in Simulink library; 2) Simulation demonstration of periodic adaptive learning compensation of cogging effect described by a state-period sinusoidal signal of single harmonic; 3) Simulation demonstration of a general case as in 2) but with multiple harmonics; and 4) Initial simulation results of high order PALC scheme for improved cogging compensation where the information of more than one previous periods is used in the PALC updating law. The rest of this paper is organized as follows. In Sec. II, PMSM position servo system is introduced with detailed Simulink model implementation in SimPowerSystems toolbox in Simulink library. Section III presents briefly an existing periodic adaptive learning cogging compensation scheme implemented in position control system. Simulation tests are performed in Sec. IV for cogging effects described in both single harmonic and multiple harmonics using adaptive compensation schemes with and without periodic learning mechanism. Concluding remarks are given in Sec. V with an initial simulation study on high order PALC. II. PMSM P OSITION S ERVO S YSTEM A. Model of PMSM Permanent magnetic (PM) synchronous motor consists of three-phase stator windings and a PM rotor. The variables and parameters represented in the stator frame can be transformed into those in the synchronous reference frame. Therefore, the well-known mechanical equations and electrical equations of a permanent-magnet synchronous motor are given: dθ = ω, dt

Tm = Kt iqs =

Ying Luo and YangQuan Chen are with Center for Self-Organizing and Intelligent Systems (CSOIS), Dept. of Electrical and Computer Engineering, Utah State University, 4120 Old Main Hill, Logan, UT 84322-4160. Youguo Pi is with the Dept. of Automation Science and Engineering, South China University of Technology, China. Ying Luo is currently a Ph.D. candidate on leave from South China University of Technology. Corresponding author: Prof. YangQuan Chen. Email: [email protected]; Tel. 01(435)797-0148; Fax: 01(435)7973054. URL: http://www.csois.usu.edu.

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dω 1 = (Tm − Tl ), dt J 3P ψdm iq , 22

(1) (2)

did 1 = (Vd + ωLq iq − Rid − ωψqm ), dt Ld

(3)

diq 1 = (Vq − ωLd id − Riq − ωψdm ), dt Lq

(4)

where (1) represents the mechanical subsystem, and equations (3) and (4) represent the electrical subsystem. θ and ω are motor rotor angular position and speed, respectively; J is the moment of inertia of the rotor; Tm is the motor electromagnetic torque generated, and Tl is the load torque applied; id and iq are stator currents along the d and q axes, respectively; Vd and Vq are the voltages along d and q axes, respectively; R is the stator resistance; Ld and Lq are the stator self-inductances in the d and q axes, respectively, it has been assumed that as the surface mounted PMSM is non-salient, Ld and Lq are the same denoted by L. By using the concept of field oriented control of the PMSM, the daxis current is controlled to be zero to maximize the output torque. Under this assumption, the motor electromagnetic torque is given in (2), where Kt is the actually torque coefficient and P is the number of poles in the motor. However, in practice the motor torque can be expressed as 3P ψdm iq + Fcogging , (5) Tm = 22 where Fcogging is the periodic torque pulsation due to cogging. Cogging force is produced by the magnetic attraction between the rotor mounted permanent magnets and the stator. It is the circumferential component of attractive force that attempts to maintain the alignment between the stator teeth and the permanent magnets. Cogging force harmonics appear at frequencies that are multiple of Nlot/pp fs , where Nlot/pp is the number of slots per pole pair and fs is the electrical frequency of the rotor. Analytical modeling of the cogging force is challenging since its production mechanism involves complex field distributions around stator slots. However, one thing that can be assured is that the cogging force is a periodic function of rotor position due to the slotted nature of the primary core. In control strategies development, it has been approximately modeled as a simple sinusoidal signal [19] as follows: Fcogging = A sin(ωx + ϕ),

closed-loops for position, speed and current. The position reference is given, we can get the rotor angular information as the position feedback from the PMSM module, and the derivative of the rotor angle is the speed feedback, then the position and speed closed-loops controls can be performed. Note that the position is adjusted by a proportional controller, the speed and current (detailed current closed-loop control is introduced in II-B.2) are both adjusted by PID control strategy. The setting of the controllers’ parameters is not the focus of this paper. The following procedures are followed during the initial controller parameter setting. • Firstly, tune the current loop to make sure that Idf db and Iqf db can follow Idref and Iqref as quickly as possible. • Secondly, tune the speed loop to achieve the tradeoff between the response time and the overshoot of speed step response • Thirdly, tune the position loop to guarantee the position tracking precision and the allowable maximum controller output which serves as the speed reference. 2) Current Closed-Loop Control: In the inner current closed-loop, idref is given as 0, iqref is obtained from the speed PID controller; then the stator line currents iA and iB are measured and the following coordinate transformations between stationary a − b − c frame, stationary α − β frame and synchronously rotating d − q frame [21] are performed. • “Clark block”: Stationary a − b − c frame to stationary α − β frame.      3 0 iA iα 2 ; = √ √ iB iβ 2 2 2 •

“Park block”: Stationary α − β frame to rotating d − q frame.      cos θr iα sin θr id = ; iq − sin θr cos θr iβ



“I-Park block”: Rotating d−q frame to stationary α−β frame.      cos θr − sin θr Vd Vα = . Vβ sin θr cos θr Vq

(6)

where x is the linear motor displacement and ω = 2 ∗ pi/xp with xp the PM pitch. However, as pointed in [20], in order to get a cogging model closer to the true physical situation of PMSM, the cogging force should be described with multiple harmonics according to Fourier expansion as follows: ∞  Ai sin(ωi x + ϕi ), (7) Fcogging = i=1

where Ai is the amplitude, ωi is the state-dependent cogging force frequency, and ϕi is the phase angle. In order to compensate the cogging force of general signal shape, it is suggested to make use of the periodicity of the positiondependent cogging disturbance. Again, note that the cogging force waveform is periodic over a pole-pitch in PM motor [2]. B. Space Vector Pulse Width Modulation Control Model 1) Position and Speed Closed-Loops Control: Figure 1 shows the control structure of system model 1 with three 1 The Simulink models developed in this paper are available upon email request.

then the current closed-loop control can be performed, the current PID controller is introduced in II-B.1. Clearly, referring to Fig. 1, the PMSM will be decoupled if the control id = 0 and we can control a PMSM as easily as a DC motor. The electrical and mechanical equations of PMSM are written as (1)-(4). 3) Space Vector Pulse Width Modulation (SVPWM): SVPWM is a special switching scheme of a 3-phase power converter with six power transistors. As shown in Fig. 2, the SVPWM technique is applied to approximate the reference voltage Uo , and it combines with the eights basic space vectors [21]. Therefore, the motor voltage vector Uo will be located at one of the six sectors (S1, S2, S3, S4, S5 and S6) at any given time. Thus, for any PWM period, it can be approximated by the vector sum of two vector components lying on the two adjacent basic vectors. The modeling details of the SVPWM design are introduced as follow:

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P ³

dT dt

V

Fig. 1.

M

T

Block diagram of the PMSM position servo system

TABLE I T1 AND T2 IN A LL S ECTORS

ß B 010 u010 6

Sector T1 T2

110 u110 5

1 -Z X

5 Z Y

0 X Y

3 -X Z

2 -Y -Z

4 Y -X

Uo

u011

111

001 C

Fig. 2.



4 100

000

2

a

3 101 u101

u00 1

Step-1: Determination of the sector according to the rule shown below: ⎤   ⎡ 0 1   Vref 1 √ Vα 3 1 −2 ⎦ Vref 2 = ⎣ 2√ ; Vβ Vref 3 − 23 − 12 > 0, then a = 1, else a = 0; > 0, then b = 1, else b = 0; > 0, then c = 1, else c = 0. sector in the code is defined as sector = 4 ∗ c + 2 ∗ b + a. Step-2: Calculate X, Y and Z from the following ⎤   ⎡ 0 1   X √ Vα 1 ⎦ 3 ⎣ Y ; = 2√ 2 Vβ Z − 3 1 2



2

Step-3: Determine T1 and T2 , from Table I. If it is at

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Step-4: Determine the duty cycle Ta , Tb , Tc as follows: T − T1 − T2 , 2 Tbon = Taon + T1 , Tcon = Tcon + T2 .

Taon =

Basic space vectors and switch patterns

– If Vref 1 – If Vref 2 – If Vref 3 The variable



the saturation condition T1 + T2 > T , then T1 and T2 should be modified as:      T 0 T 1 T1 T1 +T2 = ; T T 2 T2 0 T1 +T2

u100 A

1

011



(8) (9) (10)

Step-5: Assign the duty cycles Ta , Tb and Tc from Table II, then compare with the given deltoid waveform, we can get the signals of PWM1, PWM3 and PWM5 as show in Fig. 3, and PWM2, PWM4 and PWM5 are the complementary signals of PWM1, PWM3 and PWM5 respectively. PWM1 PWM6 are used to control the power transistor switching time in the 3-Phase inverter. TABLE II A SSIGNING D UTY C YCLE IN A NY S ECTORS Sector Ta Tb Tc

1 Taon Tbon Tcon

5 Tbon Taon Tcon

0 Tcon Taon Tbon

3 Tcon Tbon Taon

2 Tbon Tcon Taon

4 Taon Tcon Tbon

C. 3-Phase Inverter The 3-phase inverter consists of three groups of IGBT power transistors. Every group is composed of upper and lower two transistors and its configuration is shown in Fig. 4, the 1 − 6 gates of the six IGBT power transistors are

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u=

1 1 Tm , T l  = Tl . J J

where x is the position and a(x) is the unknown positiondependent cogging disturbance which is repeating in every pole-pitch denoted as xp . v is the velocity and u is the control input signal. We can see that, the above equations are similar to our PMSM model in Sec. II. The control objective is to track or servo the given desired position xd (t) and the corresponding desired velocity vd (t) with tracking errors as small as possible. In practice, it is reasonable to assume that xd (t), vd (t) and v˙ d (t) are all bounded. The feedback controller is designed as: When x ≥ xp , u = v˙ d (t) + Tl + Fig. 3.

PWM patterns and duty cycle [22]

a ˆ(t) − αS(t) − λev (t), J

(13)

and when x < xp , u = v˙ d (t) + Tl +

a ˆ(t) − ηex (t) − λev (t), J

(14)

where S(t) = ev (t) + λex (t),

(15)

and α and λ are positive gains; a ˆ(t) is an estimated cogging force from an adaptation mechanism to be specified later; v˙ d (t) is the desired acceleration; ex (t) = x(t) − xd (t) is the position tracking error and ev (t) = v(t) − vd (t) is the velocity tracking error. The periodic adaptive learning law is designed as follows: a ˆ(t − Pt ) − KS(t) if x ≥ xp a ˆ(t) = (16) z − g(v) if x < xp Fig. 4.

The 3-phase inverter configuration

controlled by the PWM1-PWM6 signals from the SVPWM module; DC+ and DC− ports are the DC voltage input terminals; the A, B and C outputs of the inverter connect to the 3-phase inputs of the motor. III. P ERIODIC A DAPTIVE L EARNING C OMPENSATION (PALC) The PMSM position servo system is a high-order dynamic system. The main objective of position control is to achieve high positioning accuracy and fast tracking speed. As mentioned before, the cogging effect must be rejected or compensated. Based on the simulation model for PMSM servo, we present simulation results on a known technique called periodic adaptive learning compensation (PALC) for cogging effect compensation. Details of the method can be found in [20]. In this section, we briefly summarize the scheme we used for simulation. Following the notations in [20], the cogging force in (7) is denoted as −a(x). The motion control system considered in [20], without loss of generality, is modeled as follows x(t) ˙

=

v(t) ˙

=

v(t), (11) a(x) a(x) 1 (Tm − − Tl ) = u − − Tl ,(12) J J J

ˆ(x − xp ) (Note that Pt is the trajectory where a ˆ(t − Pt ) = a cycle defined in [20]); K is a positive design parameter called the periodic adaptive learning gain; z will be defined in the following paragraph; and g(v) is a tuning function to be selected later based on the following guidelines: 0 < g  (v) < ∞, where g  (·) = ∂g(·) ∂· . The following tuning mechanism is used for z: ev z˙ = g  (v)[v˙ d − ηex − λev ] − . J

(17)

(18)

The stability of the above PALC scheme was performed in [20] using state-periodic argument. IV. S IMULATION I LLUSTRATIONS To verify the suggested method, an authentic PMSM position servo control system Simulink model was built using SimPowerSystems toolbox in Simulink library. Note that in the PALC scheme, the cogging force a(x) can take any shape not necessarily sinusoidal. Therefore, we presented two cases. • Case-1: Simulation demonstration of periodic adaptive learning compensation of cogging effect described by a state-period sinusoidal signal of single harmonic; • Case-2: Similar to Case-1 but with multiple harmonics.

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For our simulation test, the following reference trajectory and velocity signals are used: xd (t) vd (t)

= t(rad), = 1(rad/s) = 9.55(rpm).

performs better than the adaptive control method without utilizing the known periodicity; 3) PALC works effectively for general form of cogging forces. 40

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The control gains in (13)-(14) were selected as: α = 50, λ = 20 and η = 20; and g(v) was designed as g(v) = 3v to satisfy (17). The periodic adaptation gain K was selected as 0.2. The motor parameters are given in Table III and the Tl = 0.5[N m]

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A. Case-1: Single Harmonics In the simulation, the actual cogging force was modeled approximately as:

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Fcogging = 2 sin(6x) + 2 cos(6x).

B. Case-2: Multiple Harmonics In this simulation, the actual cogging force was modeled as Fcogging = 2 cos(6x) + cos(12x) + 0.5 cos(18x). Results similar to Case-1 are presented in Figs. 8(a)10(b). The PALC works efficiently for the multi-harmonics cogging case. In fact, the wave form of the cogging force can take an arbitrary form as long as it is bounded as indicated in [20] V. C ONCLUDING R EMARKS A. Conclusion This paper presented a detailed authentic simulation model of a permanent magnet synchronous motor control system based on the SimPowerSystems toolbox in Simulink library. We then focus on the periodic adaptive learning compensation (PALC) of cogging effect on PMSM position and velocity servo tasks. From the simulation results, we can conclude that, 1) the PMSM position servo control system Simulink model based on SimPowerSystems toolbox in Simulink library works authentically; 2) PALC method

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Fig. 6. Tracking errors with compensation using only adaptive law a ˆ(t) = z − g(v) for sinusoidal cogging force. No periodic learning.

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Figures 5(a)-7(b) show the position/speed error without compensation, with compensation using only adaptive law a ˆ(t) = z − g(v) and with compensation using PALC, respectively. From these figures, we observe that the servo system is stable, and the position/speed errors in Figs. 6(a) and 6(b) decreased efficiently when the adaptive law a ˆ(t) = z − g(v) is used comparing with the tracking errors without compensation in Figs. 5(a) and 5(b). It is particularly interesting to observe that when PALC is used, as time increases, the position/speed errors become smaller and smaller as shown in Figs. 7(a) and 7(b), as if it is learning from pitch to pitch. The advantage of utilizing the periodicity in adaptive control is clearly demonstrated.

0

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Fig. 7. Tracking errors with compensation using PALC for sinusoidal cogging force

B. Future work Our future work will focus on the higher-order periodic adaptive learning compensation (HOPALC) of cogging effect where the information of more than one previous periods is used in the PALC updating law. For example, our adaptation law of HOPALC can be designed as follows:

ˆ(t − Pt ) + β2 a ˆ(t − Pt ) β1 a if x ≥ xp −K1 S(t) − K2 S(t − Pt ) a ˆ(t) = z − g(v) if x < xp (19) We implemented the above HOPALC in our Simulink model and the initial simulation results are compared in Fig. 11, where K is the number of periods. The red line represents the root mean squares (RMS) of the position tracking error of PALC, and the green line is for the RMS of the position error of HOPALC. It is obvious that the HOPALC performs better than PALC. The convergence proof, further simulation results and long-term stability issues on HOPALC will be reported elsewhere.

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Fig. 8. Tracking errors without compensation for multi-harmonic cogging force

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Fig. 11. RMS of position tracking errors versus the number of periods. Red line (top curve) for PALC; green line (bottom curve) for HOPALC

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Fig. 9. Tracking errors with compensation using only adaptive law a ˆ(t) = z − g(v) for multi-harmonic cogging force. No periodic learning.

ACKNOWLEDGEMENT Ying Luo would like to thank to the China Scholarship Council (CSC) for the financial support and to Dr. Huifang Dou for her expertise in PMLM (permanent magnet linear motor) cogging compensation. R EFERENCES [1] T. M. Jahns and W. L. Soong, “Pulsating torque minimization techniques for permanent magnet ac motor derives-a review,” IEEE Trans. Ind. Electron, vol. 43, pp. 321–330, 1996. [2] C. G. Kim S. K. Chung, H. S. Kim and M.-J. Youn, “A new instantaneous torque control of pm synchronous motor for highperformance direct-drive applications,” IEEE Trans. Power Electron., vol. 13, pp. 388–400, 1998. [3] A. Keyhani. T. Sebastian C. Studer and S. K. Murlhy, “Study of cogging torque in permanent magnet machines,” in IEEE 32nd Ind. Appl. Sociery (/AS) Annual Meeting, Louisiana, New Orleans, Oct. 1997, pp. 42–49. [4] T. H. Lee K. W. Lim T. S. Low, K. J. Tseng and K. S. Lock, “Strategy for the instantaneous torque control of permanent-magnet brushless dc drives,” Proc, Inst. Elec. Eng, vol. 137(6), pp. 355–363, 1990. [5] K. J. Tseng T. S. Low, T. H. Lee and K. S. Lock, “Servo performance of a bldc drive with instantaneous torque control,” IEEE Trans. Ind. Application, vol. 28(2), pp. 455–462, 1992. [6] J. Y. Hung and Z. Ding, “Design of currents to reduce torque ripple in brushless permanent magnet motors,” Proc. Inst. Elect. Eng. B, vol. 140, pp. 260–266, 1993.

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[7] T. Makino N. Matsui and H. Satoh, “Auto-compensation of torque ripple of dd motor by torque observer,” IEEE Tran. Ind. Application, vol. 29(1), pp. 187–194, 1993. [8] D. C. Hanselman, “Minimum torque ripple, maximum efficiency excitation of brushless permanent magnet motors,” IEEE Trans. Ind. Electron, vol. 41, pp. 292–300, 1994. [9] S. K. Chung K. Y. Cho, J. D. Bae and M. J. Youn, “Torque harmonics minimization in permanent magnet synchronous motor with back emf estimation,” Proc. Inst. Elect. Eng., vol. 141(6), pp. 323–330, 1994. [10] J. J. Carroll D. M. Dawson and M. Schneider, “Integrator back stepping control of a brush dc motor turning a robotic load,” IEEE Trans. Contr. Syst. Technol., vol. 2, pp. 233–244, 1994. [11] Q. W. Jia J. X. Xu, T. H. Lee and M. Wang, “On adaptive robust back stepping control schemes suitable for pm synchronous motors,” Int. J. control, vol. 70(6), pp. 893–920, 1998. [12] S. K. Panda B. H. Lam and J. X. Xu, “Torque ripple minimization in pm synchronous motors - an iterative learning control approach,” in Intl. Conf. on Power Elect. And Drives PEDS’99, Hong Kong, 1999, IEEE, pp. 141–149. [13] J. X. Xu B. H. Lam, S. K. Panda and K. W. Lim, “Torque ripple minimization in pm synchronous motors using iterative learning control,” in IECON ’99 Proc. The 25th Annual Conference of the IEEE, San. Jose, California USA, 1999, pp. 1458 – 1463. [14] A. M. Stankovic G. Tadmor V. Petrovi’c, R. Ortega, “Design and implementation of an adaptive controller for torque ripple minimization in pm synchronous motors,” IEEE Trans. Power Electron, vol. 15, pp. 871–880, 2000. [15] S.K.Panda W. Qian, J. X. Xu, “Periodic torque ripples minimization in pmsm using learning variable structure control based on a torque observer,” in IECON ’03. The 29th Annual Conference of the IEEE, Roanoke, Virginia USA, 2003, pp. 2983 – 2988. [16] S. K. panda W. Qian and J.-X.Xu, “Torque ripple minimization in pm synchronous motors using iterative learning control,” IEEE Trans. Power Electron, vol. 19, pp. 272–279, 2004. [17] Y.-J. Pan J.-X. Xu, S. K. Pands and T. H. Lee, “A modular control scheme for pmsm speed control with pulsating torque minimization,” IEEE Trans. Ind. Electron., vol. 51, pp. 526–536, 2004. [18] Jong Pil Yun; ChangWoo Lee; SungHoo Choi; Sang Woo Kim, “Torque ripples minimization in pmsm using variable step-size normalized iterative learning control,” in IEEE Conference on Robotics, Automation and Mechatronics, Dec. 2006, pp. 1 – 6. [19] S. N. Huang K. K. Tan and T. H. Lee, “Robust adaptive numerical compensation for friction and force ripple in permanent-magnet linear motors,” IEEE Trans. on Magnetics, vol. 38, no. 1, pp. 221– 228, 2002. [20] Hyo-Sung Ahn; YangQuan Chen; Huifang Dou, “State-periodic adaptive compensation of cogging and coulomb friction in permanent-magnet linear motors,” IEEE Transactions on Magnetics, vol. 41, no. 1, pp. 90 – 98, 2005. [21] B.K. Bose, “Power electronics and variable frequency drives technology and application,” in IEEE Press, 1997. [22] Ying-Shieh Kung; Pin-Ging Huang, “High performance position controller for pmsm drives based on tms320f2812 dsp,” in Proceedings of the IEEE International Conference, Sept. 2004, pp. 290 – 295.

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Fig. 10. Tracking errors with compensation using PALC for multiharmonic cogging force.

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Authentic Simulation Studies of Periodic Adaptive ... - IEEE Xplore

Ying Luo, YangQuan Chen and Youguo Pi. Abstract— This paper presented a detailed authentic simula- tion model of a permanent magnet synchronous motor control system based on the SimPowerSystems toolbox in Simulink library. We then focus on the periodic adaptive learning compensation (PALC) for cogging effect ...

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