[6] - Comparison of Torque Estimators for PMSM.pdf

Comparison of Torque Estimators for PMSM

Kheng Cher Yeo, Greg Heins, Friso De Boer IEEE Conference Publishing

School of Electrical Engineering Charles Darwin University Darwin NT 0909 Australia

Abstract- One of the disadvantages of Permanent Magnet Synchronous Motors (PMSM) is the output torque ripples which degrade the performance of the system. Actual torque feedback to the system reduces output torque ripple but increases costs through the use of a torque transducer. Thus, a torque estimator which can provide accurate instantaneous torque feedback to the system will be highly desirable. However, there is no previous work comparing the different methods in estimating torque for a PMSM. This paper will compare four torque estimators using MATLAB/Simulink to test how well the estimated torques are when compared to the actual output torque. The four criteria used for comparing the performance of the torque estimators are accuracy, robustness, complexity and cost. Depending on the type of operations, the torque estimators have different performance. From the simulations, it is found that if high accuracy and robustness is needed, Model Reference Adaptive System (MRAS) and Sliding Mode Observation (SMO) may be the best choices. However, if low cost and low complexity is desired, General Torque Equation (GTE) and Flux estimation with Compensation Scheme (FCS) should be chosen.

I. INTRODUCTION

PMSM drives have been used in many high performance servo and robotic applications. This is due to the many advantages of PMSM such as high dynamics, low maintenance, high efficiency, high torque to current ratio, high power density [1]. However, the torque ripple resulted from imperfections of the motor design, controllers or other factors leads to speed oscillations in the system. This in turn degrades the performance of the system, especially in applications where high precision is required.

There are many ways whereby torque ripples can be reduced and these have been widely discussed in papers [2] [9]. One way is to improve the design of the motor, which can be costly and complex. Another way is the active control of stator current excitations [2]. This allows more flexibility in the design of the motor system and is the preferred method to reduce torque ripples. There are many variations of this second method. This includes using preprogrammed current waveforms to cancel the harmonics, the direct control of torque to achieve the desired current waveform [3], [4], using different kinds of observers and intelligent control such as fuzzy [5], neural [6] or iterative learning schemes [7], [8].

A good way to reduce torque ripple is to feedback the actual torque into the system. However, this requires the use of a torque transducer, which is very costly. Thus, many schemes have been developed to estimate the torque so that torque control can be achieved using the estimated torque. Some

papers such as [7] and [8] uses torque estimator in the control of their system. However, the reason why that particular torque estimator was used was not explained.

Thus, an accurate torque estimator which can provide instantaneous torque feedback to the system will be highly desirable. However, there is no previous work comparing the different methods in estimating torque for a PMSM. This paper will compare four torque estimators and their performance in following the actual torque using MATLAB/Simulink TM. The four torque estimators are based on the General Torque Equation (GTE), Flux estimation with Compensation Scheme (FCS), Sliding Mode Observation (SMO) [7] and Model Reference Adaptive System (MRAS) [8], [9].

The layout of the remaining paper is as follows: section II covers the model of PMSM. Section III describes in detail the working principles of the torque estimators, the necessary inputs and the assumptions used in estimating the torque. Section IV discusses the simulation results according to the four criteria of accuracy, robustness, complexity and cost. Lastly, section V concludes the discussion.

II. MODEL OF PMSM

A PMSM can be described by a dynamic subsystem, (1) and (2) and a dynamic electrical subsystem, (3) and (4):

(1)

(2)

(3)

(4)

where vd and vq are the dq axes stator voltages, id and iq are the dq axes stator currents, Rs is the stator resistance, Ld and Lq are the dq axes stator self inductances, B is the damping factor, J is the inertia angular momentum, d and q is the dq axes flux linkage due to the permanent magnet, wr is the rotor electrical angular velocity, is the rotor electrical position, Te is the electromagnetic torque and TL is the load torque. The equation for the electromagnetic torque is given by:

(5)

rdtd

=

( )rLe BTTJdt

d

=

1

( )qrdsdd

d iRvLdt

di += 1

( )drqsqq

q iRvLdt

di= 1

( )dqqde iiPT = 23

2008 Australasian Universities Power Engineering Conference (AUPEC'08) Paper P-090 Page 1

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where P is the number of pole pairs in the motor.

III. TORQUE ESTIMATORS FOR PMSM

The details for the four torque estimators are discussed as follow:

A. General Torque Equation (GTE) This estimator uses (5) to calculate the estimated torque and

assumes a constant value for the flux linkage of the permanent magnet. Although this may not be true after the motor runs for some time, nevertheless, GTE provides a good reference for the estimation of the torque. Another advantage is that it can be implemented easily. Two inputs are necessary: the current and position of the rotor. Parks transformation is used to convert the stator currents from the abc frame to the dq frame [10]. The flux linkages are calculated as follow:

(6) (7)

where f is the flux linkage of the permanent magnet. Besides, f, Ld and Lq are also assumed to be constant.

B. Flux estimation with Compensation Scheme (FCS) By rearranging (3) and (4), the flux linkage of the magnet

can be estimated with the following equations:

(8)

(9)

The torque can then be calculated using (5). This method allows more flexibility in the estimation of the flux linkage which may varies with time. In addition, FCS can also be implemented easily. However, due to integration, errors like drift and offset may be brought into the estimation of the torque. Three inputs are necessary for the estimation of the torque. They are the current, voltage and position of the rotor. The current and voltage are transformed from the abc frame to frame. The speed is estimated through the differentiation of the rotor position using (1). The only assumed constant variable is the stator resistance, Rs.

C. Sliding Mode Observation (SMO) The design of the SMO estimator follows [7] which uses the

gain scheduled SMO in designing the torque estimator. Using (3) and (4), the electrical dynamics of a PMSM can be re-written as follow:

(10)

The disturbance observer is designed as:

(11)

where ,

,

,

is the estimated stator vector, K1 is the decoupling matrix, K2 is the switching gain matrix and kv is the estimated flux linkage of the magnet. An assumption is made:

(12) and its derivative is norm-bounded. Details are found in [7]. As only the flux in the d-axis is needed, q = 0. Thus, k2d and kvd also equals to 0. In addition, 2 low pass filters are also necessary to generate kvq and k2q. If the observer remains in the sliding mode, the error will tend to 0. Thus, by using the correct filter parameters 1, 1, 2 and 2, the flux d can be achieved where kvq, the estimated flux linkage becomes dq(t). Equation (12) then becomes

(13)

Thus, the estimated torque can be found using

(14)

The details for proving the stability and convergence of the SMO scheme can be found in [7]. Three inputs are necessary for the torque estimation. They are current, voltage and the rotor position. Parks transformation is also used to convert the stator currents and voltages from the abc frame to the dq frame. Speed is estimated using (1). Unlike GTE and FCS, there are an additional 8 variables for the tuning of the SMO scheme. They are k1d, k1q, D0, D1, 1, 1, 2 and 2. Ld, Lq and Rs are assumed to be constant.

D. Model Reference Adaptive System (MRAS) The design of the MRAS estimator follows [8] whereby the

MRAS technique is used to estimate the torque. The reference and adjustable models for MRAS are:

(15)

(16)

where the matrices A, B, D, and are the same as that of

the SMO, . The error .

( )dtwRiv qrsddd += ( )dtwRiv drsqqq =

DBuAxx ++=

=

=

=

q

d

q

d

q

d

v

vu

ii

x

,,

=

=

=

0

0,10

01

,

q

r

d

r

q

d

qq

rd

d

rq

d

s

Lw

Lw

D

L

LB

LRs

LwL

LwL

LR

A

vxxsignKxxKBuxAx k++++=

)()( 21

=

=

= )()(

,00

,

2

22

1

1

1 tdtd

Dkk

Kk

LwL

LwL

kK

q

d

q

d

qq

d

q

dd

T

vqvdv

T

qd kkkiix

=

=

,,,

x

DBuAxx ++=

FeDBuxAx ++=

x

x

= xxe

qde iPT

= 23

fddd iL +=qqq iL= d

rq Lq

wtd =)(

^

xxe =

vqr

qd k

L

=^

=

=

2

1

2

1

00

,

GG

GFwwF

Fr

r


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F is the feedback gain matrix. The adaptation rule used in estimating is:

(17) where is the adaptation gain and G is a solution of the Lyapunov equation:

(18)

where Q is a positive definite matrix. More details can be found in [8]. As shown in fig. 2, the estimated torque can then be found using (14). Three inputs are necessary for the torque estimation. They are current, voltage and the rotor position. Parks transformation is also used to convert the stator currents and voltages from the abc frame to the dq frame. Speed is estimated using (1). There are 6 variables necessary for the tuning of the MRAS: F1, F2, G1, G2, Q and . Ld, Lq and Rs are assumed to be constant.

E. Summary The table below summarizes the important features of the

torque estimators: Table 1: Summary of important features

Torque Estimators

Number of inputs

Number of assumed fixed

variables

Number of adjustable variables

GTE 2 3 0 FCS 3 1 0 SMO 3 3 8

MRAS 3 3 6

IV. SIMULATIONS

The simulations were created in MATLAB/Simulink with a constant speed and load torque of 200 rpm, and 5 Nm respectively. A normal PI speed controller is used (P = 5 & I = 100) with a hysteresis controlled inverter (h = 0.1). Parks transformation is also used, setting Id to 0. The setup is as shown below, modified from [11]. The sampling times for the controllers are: speed controller 0.14 ms and the current controller 20 s. Standard SimPower Systems blockset has been used for the blocks shown in the figure below.

Figure 1. Comparing the accuracy of the torque estimators.

A. Parameters The constants used in the simulations and the values used in

tuning the torque estimators are shown in table 2. The values for the parameters for PMSM are taken from the Simulink blockset. For MRAS estimator, the solutions of (18) are chosen so that the resulting matrix (A + F) has stable poles and are time invariant. For SMO estimator, the parameters are chosen to satisfy the condition for convergence and stability as shown in [7].

Table 2: Parameters Variables Values

Ld 8.5 mH Lq 8.5 mH Rs 0.2 f 0.175 Wb P 4 J 0.089 kgm2

PMSM

B 0.005 Nms 2000 F1 1024 F2 1024 G1

G2

MRAS

Q

k1d 150 k1q 150 D0 1750 D1 1000 1 0.1 s 1 8000 2 0.25 s

SMO

2 100

B. Performance Criteria The four main criteria for comparing the torque estimators

are accuracy, robustness, complexity and cost. Accuracy is the most important feature of the torque estimator. It should follow as closely as possible to the actual torque. The estimator must also be robust in terms of changes to the motor parameters and to external disturbances. Five components are tested to compare the robustness of the torque estimators. They are the load torque, reference speed, flux linkage of the magnet, inductance and the stator resistance. Complexity depends on the number of parameters needed to tune the torque estimators. Cost is related to the number of inputs needed to estimate the torque.

A. Accuracy Table 3 shows how accurate the torque estimators are in

following the actual torque.

GeDT =

( ) ( ) QFAGGFA T =+++

1001

20001

20001


Table 3: Simulation results Torque Error Torque

Estimator Min (Nm)

Max (Nm)

RMS

GTE -0.790 0.675 0.4862 FCS -4.000 5.500 2.903 SMO -0.760 0.800 0.4895 MRAS -0.640 0.580 0.3871

The root mean square (RMS) of the torque error for the four estimators is calculated. It can be seen that MRAS has the best performance, followed by GTE and SMO, both have approximately similar performance. The last goes to FCS. The simulations showing the difference between the estimated torque and the actual torque are shown in fig. 2a and fig. 2b:

It can be seen from fig. 2a that FCS has the worst accuracy. Fig. 2b shows a close up of the accuracy for the other 3 estimators.

Figure 2b. Simulated results comparing the accuracy (close-up).

From the results above, the MRAS estimator is able to follow the actual torque with the least error. FCS has the worst result, indicating that it is not suitable for applications requiring high precision. However, in schemes where only an approximation is required, FCS may be useful. GTE and SMO lies in the middle for this criterion.

B. Robustness This component tests how robust the torque estimators are in

response to external disturbances such as load torque and reference speed and internal disturbances such as parameter changes of the motor. This includes changes to the flux linkage of the permanent magnet, the inductance of the coil and the stator resistance. Table 4 shows the results for the RMS of the torque error.

Table 4: RMS of the torque error Torque Error (RMS) Robustness

GTE FCS SMO MRAS TL X 2 0.496 6.726 0.529 0.336 * X 2 0.476 2.919 0.477 0.322 + 5% 0.555 3.084 0.514 0.415 L + 5% 0.461 2.887 0.464 0.376 Rs + 5% 0.494 3.008 0.497 0.391

It can be seen from table 4 that MRAS still has the best performance. However, the percentage change in the torque error is of interest since it determines whether or not the torque estimator will be affected by the disturbances. A good robust torque estimator should have minimal changes in their ability to estimate Te despite of the disturbances i.e. Te error should be minimal. This aspect disregards their accuracies but is interested in their resistance to disturbances. Thus, the smallest percentage change in absolute value will determine the estimator performance with regards to a particular disturbance. The results are shown in table 5.

Table 5: Robustness of the Torque Estimators Torque Error (RMS) Robustness

GTE (%)

FCS (%)

SMO (%)

MRAS (%)

TL X 2 2.0 131.7 8.1 -13.3 * X 2 -2.2 0.6 -2.6 -16.9 + 5% 14.2 6.2 4.9 7.1 L + 5% -5.2 -0.6 -5.3 -3.0 Rs + 5% 1.6 3.6 1.5 1.0

For the first component: change in load torque, GTE performs the best follow by SMO, MRAS and FCS. FCS performs very badly in this component. For the second component: change in reference speed, FCS performs the best follow by GTE, SMO and MRAS. For the third component: change in flux, SMO performs the best, follow by FCS, MRAS and GTE. Since GTE assumes a constant flux linkage in the torque estimation, it naturally has the worst performance in this component. For the fourth component: change in inductance,

0 1 2 3 4 5 6 7 8 9 10-1

0

1GTE

Te(N

m)

0 1 2 3 4 5 6 7 8 9 10-1

0

1FCS

Te(N

m)

0 1 2 3 4 5 6 7 8 9 10-1

0

1SMO

Te(N

m)

0 1 2 3 4 5 6 7 8 9 10-1

0

1MRAS

time(s)

Te(N

m)

Error of the Torque Estimators

0 1 2 3 4 5 6 7 8 9 10-4-20246

GTE

Te(N

m)

0 1 2 3 4 5 6 7 8 9 10-4-20246

FCS

Te(N

m)

0 1 2 3 4 5 6 7 8 9 10-4-20246

SMO

Te(N

m)

0 1 2 3 4 5 6 7 8 9 10-4-20246

MRAS

time(s)

Te(N

m)

Error of the Torque Estimators

Figure 2a. Simulated results comparing the accuracy.


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FCS has the best performance followed by MRAS, GTE and SMO. FCS does not depend on inductance for its torque estimation while the other three do. Thus, it is also the least affected by the change in inductance. For the last component: change in stator resistance, MRAS has the best performance follow by SMO, GTE and FCS. Figures 3a to 3e shows the percentage change in torque error over a wider range of the 5 components.

Figure 3a. Plot showing % change in TL error over an increase in TL

Figure 3b. Plot showing % change in TL error over an increase in w*

Figure 3c. Plot showing % change in TL error over % increase in

Figure 3d. Plot showing % change in TL error over % increase in L

Figure 3e. Plot showing % change in TL error over % increase in Rs

To determine which estimator is the best for this criterion is a tough decision. Depending on the type of operations, the significance of the 5 components can be different. In many operations, the reference speed may not change frequently or suddenly. Thus it will have a low percentage in testing the robustness. The other four components could have almost similar importance and have about the same weightings. An example is given in table 6 where the best estimator for a particular component is awarded 4 points and the worst has 1 point (from the average of the plots from fig. 3a to fig. 3e):

Table 6: Example in determining the best robust estimator Components weightings GTE FCS SMO MRAS TL 20% 4 1 3 2 * 5% 4 2 3 1 25% 1 3 4 2 L 25% 2 4 1 3 Rs 25% 2 1 3 4

TOTAL 2.25 2.3 2.75 2.7

In this particular example, SMO has the best performance, follow by MRAS, FCS and GTE. In terms of resistance to changes to internal disturbances, MRAS is the best and GTE the worst. However, for external disturbances, GTE is the best, FCS and MRAS are the worst. Thus, for the criteria of

Change in Load Torque

-50

0

50

100

150

200

250

5 10 12.5 15

Load Torque (Nm)

% Ch

ange

in

To

rqu

e Er

ror

GTEFCSSMOMRAS

Change in Reference Speed

-40

-20

0

20

40

60

80

100

200 400 600 800

Reference Speed (rpm)

% Ch

ange

in

To

rqu

e Er

ror

GTEFCSSMOMRAS

Change in Flux Linkage

0

10

20

30

40

50

60

70

80

0 5 10 15

% Increase in Flux Linkage

% Ch

ange

in

To

rqu

e Er

ror

GTEFCSSMOMRAS

Change in Inductance

-14

-12

-10

-8

-6

-4

-2

00 5 10 15

% Increase in Inductance

% Ch

ange

in

To

rqu

e Er

ror

GTEFCSSMOMRAS

Change in Stator Resistance

-2

0

2

4

6

8

10

12

0 5 10 15

% Increase in Stator Resistance

% Ch

ange

in

To

rqu

e Er

ror

GTEFCSSMOMRAS


robustness, it has to depend on the mode of operations to decide which the best estimator is.

C. Complexity To determine the complexity of the torque estimator, the

parameters that have to be tuned, in order for the estimator to work properly, has to be known. As shown earlier, SMO has the most number of tuning parameters (the most complex), followed by MRAS. GTE and FCS do not require any tuning.

D. Cost The cost in implementing the torque estimators is related to

the number of inputs necessary for torque estimation. GTE only require a current transducer and position encoder to generate the necessary inputs. The other 3 estimators require an additional voltage transducer. Thus, GTE has the least cost and the other 3 estimators have roughly the same cost.

E. Summary Table 7 shows the summary of the performance of the four

estimators against the criteria. To determine the robustness of the estimators, the example given earlier is used whereby SMO is the best follow by MARS, FCS and GTE.

Table 7: Summary of the performance for the four torque estimators Torque Estimators Criteria

GTE FCS SMO MRAS Accuracy Medium Low Medium High

Robustness Low Low High High Complexity Low Low High Medium

Cost Low Medium Medium Medium

From the above results, one has to determine the type of operations, to decide on which is the best estimator to use. As a guideline, if high accuracy and robustness is needed, MRAS and SMO may be the best choices. However, if low cost and low complexity is desired, GTE and FCS should be chosen.

V. CONCLUSION

This paper compares four different torque estimators to test how well the estimated torque is in following the actual torque. As this sort of comparison has not been done previously, this paper will provide a general guideline of the suitability of different torque estimators for different applications with different cost restraints. This in turn determines their capability to reduce output torque ripples.

One difficulty of this work lies in the tuning of the parameters. The SMO and MRAS estimator may not be tuned properly and this may results in poorer performance in some aspects. Further work includes verifying the simulated results with a real PMSM and improving the method for tuning the parameters of the SMO and MRAS torque estimator.

REFERENCES [1] Dote, Y. and Kinoshita, S., Brushless Servomotors Fundamentals and

Applications, Oxford Science Publication, 1990. [2] T.M. Jahns and W.L. Soong, Pulsating torque minimization techniques

for permanent magnet AC motor drives - A Review, IEEE Transactions on Industrial Electronics, 43(2), 1996, pp. 321330.

[3] L. Zhong, M. F. Rahman, W. Y. Hu and K. W. Lim, Analysis of direct torque control in permanent magnet synchronous motor drives, IEEE Transactions on Power Electronics, vol. 12, no. 3, 1997, pp. 528 536.

[4] L. Zhong, M. F. Rahman, Md. E. Haque and M. A. Rahman, A direct torque controlled interior permanent magnet synchronous motor drive without a speed sensor, IEEE Transactions on Energy Conversion, vol. 18, no. 1, 2003, pp. 17 22.

[5] M. Nasir Uddin, An adaptive filter based torque ripple minimization of a fuzzy logic controller for speed control of a PM Synchronous Motor, Industry Applications Conference, vol. 2, 2005, pp. 1300 1306

[6] Tong Liu, Iqbal Husain, Malik Elbuluk, Torque ripple minimisation with on-line parameter estimation using neural networks in Permanent Magnet Synchronous Motors, Industry Applications Conference, vol. 1, 1998, pp. 35 40

[7] Weizhe Qian, S.K. Panda, and Jian-Xin Xu, Improved PMSM pulsating torque minimization with iterative learning and sliding mode observer, Industrial Electronics Society, 2000. IECON 2000. 26th Annual Conference of the IEEE Publication, vol. 3, 2000, pp. 1931-1936.

[8] B. H. Lam, S. K. Panda, J. x. Xu and K.W. Lim Torque ripple minimisation in PM synchronous motor using iterative learning control, Proceedings of 3rd IEEE International Conference on Power Electrics and Drives, 1999, l. 1, pp. 141-149.

[9] S. -K. Chung, H. -S. Kim, C. -G. Kim and M. -J. Youn, A new instantaneous torque control of PM synchronous motor for high-performance direct drive applications, IEEE Trans. Power Elect., vol. 13, no. 3, May 1998, pp. 388-400.

[10] B. K. Bose, Modern power electronics and AC drives: Prentice Hall, 2002.

[11] Hoang Le-Huy, Modeling and simulation of electrical drives using MATLAB/Simulink and Power system blockset, 27th Annual Conference of IEEE, vol. 3, pp. 160.-1611, Nov/Dec 2001


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[6] - Comparison of Torque Estimators for PMSM.pdf

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