Torque Ripple Reduction

in Permanent Magnet Synchronous Machines

Using Angle-Based Iterative Learning Control

y. YUAN(1), F. AUGER(l), L. LORON(l), s. MOISy(2), M. HUBERT(2) (1) LUNAM Universite, IREENA, CRTT, 37 Bd de I'Universite, BP 406, 44602 Saint-Nazaire cedex, France

{yi.yuan, francois.auger, luc.loron}@univ-nantes.fr

(2) SKF France, Automotive Development Centre - Sensor Integration, 204 boulevard Charles de Gaulle, 37540 Saint-Cyr-sur-Loire, France

{ stephane.moisy, mathieu.hubert}@skf.com

Abstract-Permanent Magnet Synchronous Machines (PMSM) are used in many applications, particularly in high-performance drive systems. However, one inherent problem of PMSM is its parasitic undesired torque ripple, which reduces the PMSM per­formances. Several kinds of parasitic torque ripples are periodic functions of the rotor position. To reduce them, a method called Iterative Learning Control (ILC) seems well suited, but is efficient for only one constant speed chosen by the designer (and all the integer multiples of this speed). To design a solution for a varying speed, an ILC variant named angle-based ILC is proposed, which no longer uses time as the reference, but the angular position. This angle-based ILC can simply be implemented and does not require a heavy computational load. Simulations of a PMSM variable speed drive system and experiments have been done to assess the performances of the proposed technique.

I. INTRODUCTION

PMSMs are appealing candidates for many high perfor­mance applications, because of their attractive characteristics such as high efficiency, high torque density and high reliability. However, inherent torque ripples of PMSM are considered as a problem in many industrial applications, particularly in the low speed range. These torque ripples usually lead to a degradation of PMSM drive system performance and may bring vibrations and noise.

Torque ripple reduction is still a popular topic in today's PMSM research. Currently, two kinds of methods are used to reduce the torque ripple [I]. The first one focuses on the machine itself, optimizing the machine structure. However, special designs increase the complexity of the machine, hence it is a kind of tailored and expensive method. The other one is based on the use of advanced control methods. Since the controller is a necessary part of the PMSM drive system, this kind of solution requires no additional hardware (except RAM, if there is not enough already). Moreover, active control algorithms can easily be matched to any kind of machine.

Several sources of torque ripples such as cogging torque, harmonic torque, current scaling error, inverter defect and mechanical unbalance may occur in a shaft driven by a PMSM. Most of them depend on the rotor position only. Hence, they result in periodical oscillations of the machine speed and

torque. This paper reports on the possibility to use the ILC technique to reduce these torque ripples. Since torque ripple reduction can be considered as a kind of periodic disturbance rejection problem, the ILC technique, as an iterative control method, naturally fits this goal, as already reported in [2], [3], [4], [5]. In these publications, the torque ripple reduction is always obtained for a given speed. As a memory-based method, ILC needs to align its memory length to the period of these repetitive torque ripples. Since the torque ripple period depends on speed, a speed variation breaks the fit between the ILC memory length and the torque ripple period, which prevents the ILC based controller from reducing torque ripples. However, PMSMs are very often used as variable speed drives. To extend the ILC functionality to variable speed applications, an angle-based ILC using the angular position as the reference is proposed here. Simulations and experimental results will also be presented to compare time-based and angle-based ILC controllers.

The idea of using angle-based ILC was already mentioned in [4], [3], [6] and [7]. The first one proposes to reduce the torque ripples by an angle-based ILC, but this algorithm uses a computationally expensive Fourier series expansion of the control signal applied during the previous iteration. The second and third ones use ILC to improve the control of Switched Reluctance Machines (SRM). Since the phase torque reference in a SRM is a periodic function of the rotor position, this paper proposes to use an angle-based ILC called spatial ILC. Meanwhile, a simple interpolation technique is used to implement this angle-based ILC in a digital processor. The fourth one focuses on the reduction of the periodic disturbances produced by the inverter dead-time in an induction machine (1M). Since the time-based ILC can only compensate for fluctuations at only one frequency, an angle-based ILC is proposed to compensate for disturbances at any frequency. However, the author did not mention very clearly how to realize the angle-based ILC in a digital target.

The remainder of this paper is organized as follows. The PMSM model in the d-q reference is introduced firstly. Then the features of cogging torque and harmonic torque are re-

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called briefly, and the relationships between these torques and rotor angle are demonstrated. Furthermore, fundamental knowledge of ILC is presented, and its limitation is carefully discussed. The possibility of using angle-based ILC for the PMSM torque ripple reduction is analyzed. Details on the discrete-time implementation of this controller and on its loca­tion in the whole control system are provided. To investigate the etlectiveness of this method, simulations performed with the MATLAB/SIMULINK environment and experimental results obtained on a test bed controlled with a dSPACE board are presented.

II. PMSM MODEL

Under the assumptions that the PMSM is unsaturated and that eddy currents and hysteresis losses are negligible, the sta­tor d-q axis voltage equations of the PMSM in the synchronous reference frame are given by [8]

. d1j;d . . d1j;q Ud = Rs Zd+Tt -We 1j;q and uq = Rs Zq+Tt+We 1j;d (1)

whereljJd = Ld id + W andljJq = Lq iq are the d-q axis fluxes, W is the rotor flux, We is the electric angular speed, Ud and uq are the d-q axis voltages, id and iq are the d-q axis currents, Ld and Lq are the d-q axis inductances and Rs is the stator resistance. In the synchronous rotating reference frame, the rotor flux W is a periodic function of the electric angular position ee (with We = dee/ dt), the order of its harmonics being an integer multiple of 6 [9],

where Wo is the fundamental component of the rotor flux, W6 and W12 are the coefficients of 6th and 12th harmonic components. The electromagnetic torque Te resulting from the currents id and iq is [8]

3 - np(ljJdiq - ljJqid) 2 3 "2np (Wiq + (Ld - Lq)id iq)

(3)

(4)

where np is the number of pole pairs. Eq. 4 shows that the PMSM electromagnetic torque consists of two parts: the main part and the saliency torque. With a usual field oriented control of the PMSM, the stator current will be controlled to obtain a right angle between stator current and rotor flux (id = 0 A), and therefore will not contribute to magnetization, but only to torque production:

and 3 . Te = "2 np W Zq (5)

Finally, the mechanical angular speed is related to the torques through the mechanical equation [8]

dWm . .J ill = (Te - Tl - Tfric), wIth np wm = we (6)

where Wm is the mechanical angular speed, .J is the rotor inertia, T[ is the load torque and Tfric is the friction torque.

III. TORQUE RIPPLES OVERVIEW

Several kinds of parasitic torque ripples [4] such as cog­ging torque, harmonic torque, otlset torque and mechanical bias torque exist in PMSM. In many commercially available machines, cogging torque has a nominal value of 5 to 10% of the rated torque [9], therefore it is the main target of the torque ripple reduction. On the other hand, compared to other torque ripples, the harmonic torque is relatively more important. Their nature and model are briefly presented in this section.

A. Cogging Torque

Cogging torque [9] is manifested by the tendency of the rotor to align with a number of stable positions, even when the machine is unexcited. It is caused by the interaction between the magnet flux and the stator slots [10]. The cogging torque Tcog is defined as the negative partial derivative of the magnetic energy W to the mechanical angular position em, when there is no exciting current [11]:

(7)

According to the analysis presented in [12], the cogging torque can be modelized as a periodic function of em which depends on the stator slot number z and the rotor pole pair number np

where leff is the effective length of the iron core, flo is the air permeability, Ri and Ra are respectively the inner and the outer radius of the air gap (Ra being also the inner radius of the stator), Br is the magnetic flux density, Br,k are the coefficients of the Br Fourier series expansion and n is the number which makes nz/2np an integer. G is the air gap relative permeability and Gn are the coefficients its Fourier series expansion.

B. Harmonic Torque

Harmonic torque is caused by the interaction between the stator current and the rotor magnetic field [13]. Substituting Eq. 2 into Eq. 5 leads to:

3 . Te = "2npl.qWO + Tharm 3 . Tharm ="2nplq(W6 cos(6ee) + W12 cos(12ee) + ... )

(9)

This equation shows that the electromagnetic torque can be separated into two parts: one is the product of the current and the fundamental flux component, and the second one, which is called harmonic torque, is the product of the current and the flux harmonics and depends on the electrical angle. Since ee = npem, the harmonic torque can also be considered as a function of the mechanical angle.

According to this, the cogging torque, the harmonic torque and their sum are all functions of the mechanical angle.

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IV. PRINCIPLE A ND LIMITATIONS OF ILC

A. Iterative Learning Control

Iterative Learning Control (lLC) is based on the notion

that the performance of a system that executes the same

task several times can be improved by learning from the

previous executions (trials, iterations, passes) [14]. Compared

to other kinds of active control methods, it has the following

advantages: it changes the control signal, which means that it

does not need to change the structure of the previous control

system, and it is insensitive to plant parameter variations. In

fact, an iterative learning controller can even be successfully

implemented with a very weak knowledge of the controlled

process model [15]. In this section, we will consider the process as a single

input-single outpout discrete-time, linear time-invariant (LTI)

dynamic system, defined as

ydk] = P(z)uHk] + d[k] (l0)

where 'i is the current iteration index, k = 0, . . . N - 1 is the

time index relative to the beginning of the current iteration

and N is the length of an iteration. ydk] is the system

output obtained at the absolute time index n = N i + k, P represents the system transfer function, z is the forward

time-shift operator, ui is the control signal, and d is an

exogenous signal that repeats at each iteration. A widely used

ILC learning algorithm [14] is

udk] = Q(Z)(Ui-l[k] + L(z) ei-l[k + 1]) (11)

where the LTI dynamics Q(z) and L(z) are called the Q-filter

and learning function, respectively, Ui is the output signal of

the ILC algorithm and ei = yref - y7n is the control error, yref

being the reference output and y7n the measured output.

As an iterative method, ILC has a good performance when

tracking a periodic reference or rejecting a periodic distur­

bance. A PMSM with periodic torque ripple can be considered

as a process with a periodic disturbance. Therefore, it is

reasonable to choose ILC to achieve torque ripple reduction.

But for the ILC to be effective, the following basic postulates

are required [16]:

1) Every iteration (pass, cycle, batch, repetition) ends in a

fixed time of duration.

2) Invariance of the system dynamics is ensured throughout

the repetition.

3) The output ydk] is measured in a deterministic way.

The stability of this controller can be assured by the

following criterion [14], [17]:

p(Q(1 - LP)) < 1 (12)

where p(X) is the spectral radius of the matrix X and L, P, Q are the (N, N) matrices of the lifted representation as defined

in [14]. However, several papers [14], [18] argue that even

when the ILC system satisfies this stability requirement, the

performance of ILC still cannot be accepted, because of its bad

transient behavior. In practice, to avoid this kind of problem,

the ILC should satisfy the following monotonic convergence

norm

(eoo - ei+l) � ,(eoo - ei) (13)

with, = O"(PQ(I - LP)P-1) < 1, where O"(X) denotes the

maximum singular value of the matrix X. On the other hand, the steady state performance of ILC is

derived from the amplitude of the asymptotic error. The ILC

asymptotic error eoo is given by [14]

eoo = [I - P [I - Q(I - LP)-lQL]] (yref - d) (14)

Another very practical way to assess the stability and perfor­

mance of such controllers is presented in [19].

B. Working Principle of ILC

Though ILC can be used for repetitive disturbance rejection,

its effectiveness is still constrained by the requirements men­

tioned above. However, to successfully implement the ILC,

satisfying only these conditions is not enough. A controller

with a fixed memory length which can accurately record

an integer number of disturbance iterations should be also

used. This means that the ILC iteration should have the

same duration as one full rotation of the rotor (or should

be an integer multiple of it). The matching and unmatching

Angle-­

JGO .

Angl ::)(jO ..

One H.otation One H.otr .. tion One Hotcotion

Time (Iteration) One lteration One iteration One ltewtion One lter1'tion

Fig. 1. Relation between iteration and repetitive rotation.

OTl" Rot.ation One Rota!.io" One Rota!.io"

Time (Iteration)

Fig. 2. The relation between iteration and variable rotation.

durations are separately shown in Figs. 1 and 2. In both figures,

the x-axis is time and the y-axis is the mechanical angle in

degrees, and the memory size is deliberately chosen to have

the same length as the first rotation. In Fig. 1, because of

the invariability of the system and because of the constant

speed, the following rotations still have the same length as the

ILC memory. In this figure, points like h, t2, t3, t4 located at

different iteration indexes but at the same relative position

are called "counterpart points". In Eq. 11, all these points

have the same k-index but different i-indexes. In this case,

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these counterpart points can capture the same angle A, and

since torque ripples are function of the angular position, these

counterpart points record the same torque ripple information.

This is the reason why ILC can effectively reduce the torque

ripple when the machine speed is constant.

In Fig. 2, because of the speed variation, all rotations

except the first one are shorter than the ILC iteration length.

Consequently, the counterpart points correspond to different

rotation angles and the ILC functionality is lost.

V. ANGLE-BASED ILC

A. Basic idea of angle-based ILC

The former analysis showed that ILC can successfully

reduce the torque ripple only when the machine speed and the

memory length N are related to each other, while in practice

most of PMSMs are used for variable speed drives. Hence,

this limitation strongly reduces the interest of ILC for torque

ripple reduction.

The presentation of ILC above implied that the controller

iterations are expressed in the time domain. However, the

periodicity and the repetitiveness treated in ILC could more

generally be time-, state-, angle- or trajectory-dependent. In

fact, the torque ripples of PMSM are only indirectly related

to time but directly related to mechanical position. Therefore,

as index variable in Eq. 11, one may use the angular position

ern instead of the time variable t. Eq. 11 now becomes [4],

[7]

where Ts is the sampling period of the controller. Since torque

ripples are periodic functions of rotor angular position, no

matter how speed changes, the relationship between torque

ripple and mechanical angle does not change. Therefore, it is

possible to use this new algorithm to solve the torque ripple

reduction problem at a varying speed.

To make the difference between the angle-dependent ILC

and the time-dependent ILC, the former one, using Eq. 15, is

named angle-based ILC whereas the latter one, using Eq. 11, is named time-based ILC.

B. Implementation details

To provide an iterative algorithm with a low computational

cost, the angular position is quantized in N allowed values

ern,k = 27rk/N (with k = O ... N - 1), called represen­

tative positions. A rounding rule is used, which means that

a measured angular position ern is associated to an index

k = round (Nern/27r). At each iteration, the arrays Ui(em) and ei (em) are filled, in anticipation of their use to the next

iteration, and the arrays Ui-l(ern) and ei-l(ern) are used in

Eq. 15. If the rotor velocity V (in rpm) is lower than 60Fs/N, where Fs = 1/Ts is the sampling frequency of the controller,

then there are no index skipped in the tables. However, if one

or more indices are skipped between the current and previous

positions, linear interpolation is used to fill the missing table

elements.

C. Overall proposed control architecture

The proposed association of a classical PMSM vector con­

trol and an angle-based ILC controller is shown in Fig. 3. The

standard PI speed controller and the proposed ILC controller

are used in parallel. This means that a third term deduced

from the current error signal ei [k 1 is added to the control

signal defined in Eq. 11. The difference between the speed

reference and the measured speed, Werr, is used as the control

error e, and the measured angle ern is used as index for the

angle-based ILC algorithm. Several publications [4], [3], [l9]

have confirmed that this topology is particularly well suited for

an effective digital implementation: while the PI action gives

the most relevant contribution to the current reference, the

angle-based ILC is only aimed at reducing the high-frequency

periodic speed ripples that the PI controller is not able to

compensate, because of its limited bandwidth.

V I. SIMULATION AND EXPERIMENTAL RESULTS

A PMSM drive simulated with MATLAB/SIMULINK and an

experimental test bed controlled with a dSPACE board have

been used to confirm the effectiveness of the proposed method.

A. Simulation Results

Simulations have been performed using the machine pa­

rameters listed in Table. I and with a constant load torque

equal to 10 N.m and no friction torque. The cogging torque

and the harmonic torque have been both simulated by using

Eq. 8 and Eq. 9. We compared an angle-based ILC (sampled

at Fs = 3 kHz) to a time-based ILC designed for a speed

of 60 rpm (sampled at Fs = 1 kHz). In both cases, the

memory length is equal to N = 1000, Q = 0.93 and L = 5. To assess the speed and torque ripple reduction provided by

the proposed approach, we defined two performance criteria:

the speed ripple reduction ratio (SRR) and the torque ripple

reduction ratio (TRR). They are defined as the ratio of the

peak-to-peak speed (or torque) variation when an ILC is used

divided by the peak-to-peak speed (or torque) variation with

a standard vector control without ILC.

SRR =

6.wwith ILC 6.wwithout ILC

and 6.Twith ILC

TRR = (16)

6.Twithout ILC

where 6.wwith ILC is the peak-to-peak speed variation obtained

when a time- or angle-based ILC is used and 6.wwithout ILC is

the peak-to-peak speed variation obtained with a standard PI

controller without an ILC in parallel (the same for 6.Twith ILC and 6.Twithout ILc)'

Fig. 4.a shows the SRR of both angle-based and time-based

ILC as a function of the speed reference. At the speed for

which it was designed (60 rpm), the time-based ILC has a

good performance. Its SRR value is 0.12 (which means a

speed ripple reduction rate of 88 %) and the TRR value is

0.167 (which means a torque ripple reduction rate of 83.3 %). However, when the speed changes, its ripple reduction capacity

decreases quickly. Particularly, when the speed is lower than

60 rpm, the speed ripple can be almost twice larger than the

speed ripple obtained with a standard PI speed controller. By

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70 68 66 64

!62

Fig. 3. Block diagram of the proposed field oriented vector controller associated to an angle-based ILC.

70-68-66

E62 " \ .

70 68 66

160 1

60 OJ 58 (f') 58-

56 56 54 54-52 52

50 V \/. "

50 1DO 150 50 1DO 150 50 100 Time (s) Time (8) Time (s)

Angle-based ILC Time-based ILC Standard PI controller only

Fig. 5. Simulation results of the three controllers for a piece-wise linear speed reference.

80--------------

M��

� 55� " I

50

75-70 65-60-

45-

80 80 75 70

E65 "

l :� w 45 f�i 1 I

150

10 15 4000-------:-----7:10--------;',5 Time (s) Time (s) Time (s)

Angle-based ILC Time-based ILC Standard PI controller only

Fig. 6. Experimental measurements of the speed ripple at 50 rpm, 60 rpm and 70 rpm.

TABLE I PMSM MODEL PARAMETERS

Rated power 1.4 kW Rated speed 2000 rpm

Stator resistance 0.2 n Stator inductance 4 mH

Magnetic flux linkage 0.17 Wb Number of poles 8 Inertia 0.0467 kg.lll" Number of slots 16

cons, the SRR of the angle-based ILC is nearly the same at any

speed value between 50 and 70 rpm, and with the same level

as the time-based ILC at 60 rpm. Fig. 4.b shows the TRR of

both angle-based ILC and time-based ILC as a function of the

speed reference. These results are very similar to Fig. 4, since

Eq. 6 shows that the torque and the speed are related by a first­

order transfer function. Fig. 4.a and Fig. 4.b also show that

increasing the iteration length N from 512 to N = 4096 does

not significantly improve the performance of the angle-based

ILC, whereas it increases the TRR of the time-based ILC.

Fig. 5 shows the rotor speed signals obtained for a sawtooth­

shaped speed reference when using an angle-based ILC, a

time-based ILC designed to reduce torque ripples at 60 rpm

and a standard vector control. This figure clearly indicates the

ability of the angle-based ILC controller to significantly reduce

the speed ripples when the speed is variable. Therefore, all the

simulation results presented here confirm that the proposed

angle-based ILC has better capability than a time-based ILC

to reduce torque ripple and speed ripple at any speed.

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TABLE 11 EXPERIMENTAL RESULTS

a: a: <f)

1.6

0.'

0.4

(a) Speed _ angle-based I LC with N=4000 -angle-based I LC with N=512 -. -time-based ILC with N=4000 -. -time-based ILC with N=512

��0--�52��5�4--�56��5�'--�60��6=2--�64�-6=6--�6�'�70 Speed (rpm)

1.8-

0 2

�O 52 54 56

(b) Torque -angle based ILC with N_4000 -angle-based ILC with N=512 -. -time-based ILC with N=4000 -. -time-based ILC with N=512

58 60 62 Speed (rpm)

64 66 68 70

Fig. 4. Assessment derived from simultations: speed ripple reduction Cal and the torque ripple reduction (bl provided by an angle-based ILC or a time-based ILC designed to reduce the torque ripple at 60 rpm. as a function of the rotor speed. Two iteration lengths. N = 512 and N = 4096 were used.

B. Experimental results

To check the etlectiveness of the angle-based ILC, several

experiments have been carried out on a test bed controlled by

a dSPACE system (DSlO05 and DS5202 boards). The time­

based and angle-based ILC controllers both use a memory of

length N = 1000 with Q = 0.8 and L = 0.3. The sampling

frequency of the speed controllers is Fs = 1 kHz for the time­

based ILC and 3 kHz for the angle-based ILC.

We used a piece-wise constant speed reference which is

successfully equal to 50, 60 and 70 rpm during steps of

5 s each. The experimental results are shown in Fig. 6

and analysed in Table. II. Unfortunately, the test bed still

sutlers from mechanical unbalance and the main speed ripple

component is at the fundamental frequency. Although iterative

learning controllers can also handle this kind of speed ripples,

we will try to fix this mechanical problem for our forthcoming

publications. Table. II indicates that the SRR of the time-based

ILC is around 0.45 to the speed for which it was designed.

However, the SRR is nearly equal to 1 at 70 rpm and higher

that 1 at 50 rpm. This confirms again that the controller

produces no torque ripple reduction even at speeds close to

60 rpm. On the other side, the SRR of the angle-based ILC

is approximately constant and very close to the SRR obtained

by the time-based ILC at 60 rpm. This shows that the angle­

based ILC provides the same efficiency as the time-based ILC

at any frequency.

VII. CONCLUSION

In this paper, the basic principle of ILC was first briefly

presented and the limitations of time-based ILC have been

clearly identified. To deal with PMSM torque ripple reduction

under variable speed, a simple angle-based ILC has been

proposed. Several simulation and experimental results confirm

that angle-based ILC, compared to a time-based ILC, can suc­

cessfully handle the PMSM torque ripple reduction problem

under variable speed. Further research include the assessment

of the stability and robustness of such iterative controllers.

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