An Adaptive Controller for Minimization of Torque Ripple in PM Synchronous Motors

V. PetroviC t R. Ortega $ A. M. StankoviC t G. Tadmor j

tDepartment of Electrical and Computer Engineering 409 Dana Building, Northeastern University, Boston, MA 02115, U.S.A.

SLaboratoire des Signaux et Systemes Supelec, Gif-sur-Yvette, France

Abstract - This paper addresses the problem of torque ripple minimization in Permanent Magnet Synchronous Motors (PMSMs), and proposes an adaptive feedback structure as a solution. A model of PMSM that includes torque ripple phenomenon is first developed and tested. While being slightly different from the conventional one, our model is still compact and suitable for control. All parameters of the model have physical interpretation, and can ei- ther be measured directly, or estimated in a numerically reliable procedure. An adaptive control algorithm is then described, enabling speed tracking while minimizing the torque rip- ple. The algorithm is verified in simulations and implemented in a hardware setup for PMSM drive. Experimental results show significant re- duction of torque ripple.

I. INTRODUCTION

Permanent magnet synchronous motors are at- tractive candidates for high performance appli- cations such as machine tools, or direct drive robotics. The effects of torque ripple on PMSM drives in such applications are potentially sig- nificant, as the (unmodeled) high-frequency dy- namics of the actuated system may get excited.

In this paper we describe a novel adap- tive algorithm for minimization of torque rip- ple in PMSM drives. The controller is based on the same principles (energy shaping and damping injection) as the passivity-based con- troller in [l]. The topic has been addressed in a number of articles, and we list only pa- pers of immediate relevance to our approach.

A self-commissioning scheme for high-precision PMSM drives is presented in [a]. An adaptive linearizing controller for direct drive robots is analyzed and tested in [3]. A method based On parameter tuning (in a least square sense), and on-line current optimization is outlined in [4]. Reference [5] describes and quantifies relation- ships between the achievable minimum torque ripple and the required bandwidth of the power electronic converter supplying a PMSM. On- line torque observers in the dq frame are used in [6] and [7] to adaptively reduce torque pul- sations.

11. A DQ MODEL OF PMSM FOR TORQUE RIPPLE ANALYSIS

The torque produced by a PMSM can be divided into three components: 1) mutual torque, which is due to interaction of the rotor field and stator currents; 2) reluctance torque, which is due to rotor saliency; and 3) cogging torque, which is due to the geometry of sta- tor slots. Each component can contribute to higher harmonics in total torque, i.e., to torque ripple. The “electrical” phenomena responsi- ble for higher harmonics in mutual and reluc- tance torque are non-ideal (non-sinusoidal) sta- tor winding or rotor magnet distribution, and salient rotor. Ideally, mutual flux (the part of the flux through stator windings due to the ro- tor field) is purely sinusoidal, and only then mo- tor can produce constant mutual torque. This requires sinusoidal spatial distribution of either the stator windings, or of the field due to rotor magnets. As a perfect sinusoidal distribution is

0-7003-4409-0/90/$1Q.QQ 0 1990 IEEE 113

Table 1: Dependence of mutual and reluctance torque on motor construction

Sinusoidal distribution of stator phases I rotor magnet

Cylindrical rotor Mutual torque Reluctance torque

not achievable in practice, the resulting mutual flux contains higher harmonics, causing ripple in the mutual steady state torque in response to a purely sinusoidal voltage excitation. This ripple appears at multiples of 6fe (fe being elec- trical shaft speed in rps). Reluctance torque is nonzero only in the salient rotor case. Then, it has only a DC component if the stator wind- ings are sinusoidally distributed; otherwise, it has higher harmonics as well (at multiples of 2fe). However, PMSMs are usually built with cylindrical rotor, so reluctance torque is often negligible. All possible cases for the harmonic contents of the reluctance and mutual torque are summarized in Table 1. The cogging torque spectrum depends on geometry and on the num- ber of stator slots. This torque component has no DC value, and thus always contributes to torque ripple. The ripple frequencies are higher though, since cogging torque is periodic with N s l p p f e (Nslpp being the number of slots per pole pair). In PMSMs, cogging torque can usually be neglected when compared with torque rip- ple of the other two components. Analytical modeling of the cogging torque is challenging; this issue is usually addressed in machine design [8]. In this paper we assume the availability of a well-designed PMSM and concentrate exclu- sively on the mutual torque, and on ways to utilize control feedback to minimize its ripple components.

1

Based on the integral Ampere’s and flux con- servation laws, expressions for mutual flux can be derived for various stator winding and ro-

tor magnet configurations. In all cases these expressions are Fourier expansions with rapidly decaying coefficients, allowing good approxima- tions based only on the first few terms. Af- ter the Park transformation, the following dq model can be obtained:

d i d t

L , P -

dw d t

J - = T , - B W - T ~ (4)

Here P is the number of pole pairs, Ld and L, are stator inductances in the dq frame (which are equal in the case of a cylindrical rotor - the case we concentrate on), R, is stator winding resistance, T, is torque produced by the motor and TE is a load torque. J and B are the mo- ment of inertia and the friction constant (both normalized with P). The position 6 and an- gular velocity w are measured in electrical radi- ans and rad/sec respectively. This model differs from the standard PMSM dq model in position dependence of the back emf terms wad(@) and w@,( 8) , where:

@ d ( 6 ) = sin(66) f @d12 sin(128)

@,(e) = @,o + @,6 cos(66) + @,12 cos(128)

Off-line identification of the coefficients @dk, @,I, based on back emf measurements was per- formed on a fractional horse power PMSM in our laboratory (Pacific Scientific model R43H).

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These are the parameter settings that we use as “true” values in our simulations. In practice, however, these coefficients may (slowly) change with shifting operating conditions (e.g., due to magnetic saturation). Thus, we have explored an adaptive scheme that tracks the coefficients of @(8) on-line.

For the purpose of control design, the model (1)-(4) is compressed into two equations in vec- tor notation:

d i d t

L- = - R i - w Y L i - w @ ( 8 ) + ~ 1 ( 5 )

Parameter P Ld L , Rs J B where:

Value [Unit] 2

6.65 [ m H ] 6.65 [ m H ]

1.25 [RI 0.01 [kg - m2]

0.001 [+I

111. AN ADAPTIVE CONTROL ALGORITHM

The algorithm presented here focuses on adap- tation on the “ripple” parameters @dk and @ q l c .

The performance objective that we consider first is torque tracking, with a reference 7,.

(Speed tracking will be discussed later.) The @(e) function can be linearly parameter-

ized as follows:

were x(8) is a 2 x 5 periodic matrix function and q* is the (unknown) vector of actual coefficients of @(e), to be dynamically estimated by 6:

A desired current vector i, is selected to pro- duce the reference torque r,, using the estimate 7i

Table 2: R43H motor parameters

Parameter 11 Value (in v s)

0.00 177097004957 0.00110629463160 0.19935735831226 0.009 1 15 18948474

The voltage input is then selected to balance

(5):

d i, d t U = L - 4- Ri, twYLi,twX( e)$+ p( i,-i) (8)

where we have added a damping injection to the electrical subsystem specified by a design parameter p > 0 to enforce asymptotic stability.

Let + = 6 - 7, and i = i - i, represent devi- ations from the desired values. Subtracting (8) from (5), we have

I

da dt

L- + (R + PI)? + U Y L ; - wx(B)+ = 0 (9)

Based on analysis of the Lyapunov function candidate

an adaptation law in terms of a design param- eter a > 0 is selected as follows:

When 7, is constant, or slowly varying, we can assume that 4 FZ 6. Now, the derivative of H along trajectories is

(7)

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whereby the state = 0 is asymptotically sta- ble. Furthermore, reflecting the uniqueness of

Estimated parameters solid- Q , dashdoted- a6 0.41 0.35 t

-0.05 t 1 I

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Time [SI

-0.1 ' Figure 1: Convergence of Qq0 and ters

parame-

40, , Motor speedtransieyt , ,

0' I

Figure 2: Speed transient for a step change in

2 5 3 3.5 4 4.5 5 Time [SI

load torque at t = 3s

Fourier expansions, as long as w # 0, the sub- space { i = 0, f j = const.} is not invariant under the joint evolution of (9)-(10). Thus 7 i q*, as well. An increase in p will result with a more aggressive control and faster convergence, and an increase in a will result in more aggressive adaptation.

The proposed control law (7) and (8) with (10) enables tracking of a torque reference, and adaptation to parameters that possibly shift with operating conditions. A common way to achieve speed servo operation is to design a con- troller that produces the torque reference (and its time derivative) from the difference between

-

Motor torque spectrum with (solid) and withaut adaptation (dash-dotted) -20, I I I

1 :: I

.. .

Figure 3: Torque spectrum with (solid line) and without (dashed line) adaptation

the desired and actual speed. This is achieved with the following second order controller:

s + 2, Is-, . 4 s + Pc>

Notice that the control law (8) requires deriva- tive of the desired current vector i,, which in turn requires derivative of the torque reference r,. The proposed form of the speed loop con- troller is then suitable since i* can be generated from the speed error (6) using the same con- troller without integral action (i.e., without the pole at the origin). The parameter settings of the controller are chosen to achieve the desired performance in speed tracking. This method is equivalent to generating r, using a PI controller and a filtered I;, signal, and i* using the same PI controller and approximate derivative of the speed error [9].

IV. SIMULATIONS A N D

EXPERIMENTAL RESULTS

First, the controller was implemented and tested in simulations, using equations (1) through (4) to model the motor behavior. Pa- rameter values that were used are given in Table 2; those values were either measured or taken from the nameplate of our R43H motor.

Convergence of the parameters is shown in Fig. 1. Note that our adaptive controller starts

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T o m e meclrum with (solid) and without adaDtation (dash-dotted) Z w m on the 6' and 12' harmonic

-1001 I , I , I I I , , 0 20 40 60 80 100 120 140 160 180 200

Frequency [Hz]

. . -30- I

I

'

I I

g - 3 5 - . . . I ; I 4 I

C I $ 4 0 - . . . . ..

- I I I

I

I ( !

I 1 l'l

I I 1 : i

I I '

-45- 1

. I I . ! I

-50 !I; -

I I " . . I

I I I '

-55 I I I I I

25 30 35 40 45 50 55 60 65 Frequency [Hz]

Figure 4: Experimental steady state torque spectrum (fe = 5H.z and ~1 = 1.76") with adapta- tion (solid line), and without adaptation (dashed line); in the right panel expanded region around the 6th and 12th harmonic with initial estimates that are 50% or more away from the "true" values. Nevertheless, we can see that parameters converge quickly to their true values. Control design parameters ( a and p) are chosen based on a trade-off be- tween speed and oscillations in parameter con- vergence. The larger the a is, the convergence is faster, but excessive increase in this parameter leads to oscillations in the parameter response. The increase in p dampens the oscillations, but requires higher controller gain and slows down the parameter convergence if a is kept small. Thus, chosen values in simulations are cy = 10 and p = 0.

Initial estimates can be taken to be 0 for all parameters, except for aq0 because of the divi- sion in (7). cPqo is closely related to the back emf constant which is usually given in motor data sheets with some tolerance. Thus, an ini- tial estimate for this parameter close to its true value is often available.

A speed transient for a step change in torque is given in Fig. 2. It is determined by the choice of the speed controller parameters; in our sim- ulations Kc = 5.3, z, 1 4 and p , = 42. The speed reference is taken to be w, = 1 0 ~ and the torque changes from 0 to 40% of its nominal

value (1.765 N m ) at t=3s. The speed returns to its steady state value after approximately one second.

Next, the simulated steady state motor torque spectrum with and without adaptation is shown in Fig. 3, suggesting that our adaptive controller is suitably designed for torque ripple minimization.

Finally, the controller was implemented in a general purpose hardware setup for our PMSM drive. The torque spectrum in Fig. 4 shows considerable reduction in 6th (25dB) and 12th (5dB) torque harmonics. In the hardware im- plementation more harmonics appear due to mismatches with modeling assumptions (e.g., an unbalance of the three phases). After min- imization of torque ripple components at mul- tiples of 6fe, those additional harmonics may actually become dominant and require further a t tention.

V. CONCLUSIONS

A versatile PMSM model for torque ripple anal- ysis has been developed and tested in simula- tions and experiments. An adaptive controller was then designed to minimize torque pulsa-

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tions. Simulations show almost total elimina- on Power Electronics and Variable Speed tion of torque ripple indicating proper controller Drives, 1994, pp. 508-513. design. Hardware implementation is more sen- sitive to practical limitations (e.g., tolerance of electronic components, noise and measurements errors). Nevertheless, the practical results in torque ripple reduction are quite promising.

[8] C. Studer, A. Keyhani, T. Sebastian, S. K. Murthy, “Study of Cogging Torque in Per- manent Magnet Machines”, IEEE IAS An- nual Meeting, New Orleans, LA, October 1997, pp. 42-49.

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