speedestimationandparametersidentificationsimultaneously … · 2017. 5. 2. · sliding mode...

Speed estimation and Parameters Identification simultaneouslyof PMSM based on MRAS

XIAOJUN LIU , GUANGMING ZHANG, LEI MEI and DEMING WANGCollege of electrical engineering and control science

Nanjing Tech UniversityNO.30 Puzhu Road(S), Nanjing, Jiangsu

[email protected]

Abstract: - This paper deals with the Permanent Magnet Synchronous Motor (PMSM)parameters identificationand speed estimation problems. It proposes a solution to adjust PI parameters of speed observer and identifiesstator resistance, d-axis and q-axis inductance. In fact, there are a lot of methods for estimating the speed. Forexample, high frequency injection, Kalman filter and model reference adaptive system(MRAS). However, withthe motor running, some parameters will change. Such as the stator resistance will change with the increase oftemperature. These parameters are useful for estimating the speed. So, it is necessary to identify parameters. Inthis paper, MRAS is the theoretical basis. First, the adaptive law of speed observer is constructed throughPopov stability and propose a method to adjust PI parameters for the speed observer. This method has a strongadaptability and transplantation. Then, the adaptive laws of stator resistance, d-axis and q-axis inductance areconstructed through Lyapunov stability and the Popov stability, respectively and compare their results. Thestability of the proposed observers are also discussed. Last, this paper uses field oriented control(FOC) strategyand illustrates the obtained results through simulation.

Key-Words: - Permanent magnet synchronous motor. Parameter identification. Speed estimation. Modelreference adaptive. Adjust PI parameters. Stability.

1 IntroductionIn recent years, with the development of power

electronic technology, AC servo system has beenpaid more and more attention and research. ThePMSM has the advantages of small size, highefficiency and high power density, which plays animportant role in AC servo system and has beenwidely used in high performance drive system[1-3].At present, FOC or direct torque control (DTC) isusually used in the drive of PMSM for controlstrategy. However, either for the control strategy, thespeed and rotor position angle are required. Atpresent, there are two schemes for obtaining the twoparameters, namely, the sensors and sensorless. Insensor, this can directly obtain the positioninformation through installing encoder and hallsensor. But this scheme will undoubtedly increasethe cost of the design of the system, and theadaptability is also relatively weak .In sensorless,first proposed and most simple method is backelectromotive force, but at the low speed, the backelectromotive force is small and the accuracy is nothigh. Another more mature method is high frequencysignal injection, This approach relies on externalincentives and also requires the motor itself has aconvex polarity, it doesn't apply at high speed[1].With the development of modern control theory,

sliding mode observer, MRAS, Kalman filter, andother sensorless scheme are also developed[4]. Thesemethods are more and more concerned byresearchers for their good robustness. On the otherhand, the parameter value of PMSM is veryimportant for the control of the system[3]. With theincrease of temperature, the resistance andinductance of the motor windings will change[5.6].For sensorless control, these parameters areimportant. So, it is necessary to identify theseparameters at the same time as estimating speed.

MRAS has the advantages of simple algorithm,easy to implement in the digital control system, andhas the advantages of faster adaptive speed. This hasbeen proposed and applied to the PMSM sensorlesscontrol[7]. There are two different models. One is thereference model and the other is the adjustable modelincluding the identified parameters. The deviationsignal of the output of the two models send to theadaptive mechanism, and then the output of theadaptive mechanism are the identified parameters [8].From the current research, the emphases of MRASmethod parameter identification are theestablishment of adjustable model and theconstruction of adaptive law in adaptive mechanism.They are related to the accuracy of the identificationand the stability of the system.

WSEAS TRANSACTIONS on SYSTEMS and CONTROLXiaojun Liu , Guangming Zhang,

Lei Mei, Deming Wang

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For speed estimation, there are two equations asadjustable model. One is stator flux linkageequation[9], and another is stator current equation.However, for parameters identification, there is onlystator current equation. So, this paper chooses statorcurrent equation as adjustable model. For theadaptive law, there are also two method, Lyapunovstability and the Popov stability, respectively. Thereis a PI regulator in adaptive law using Popov stability.But using Lyapunov stability, there is only aintegrator[10.11].

In [12-15], the basic theoretical knowledge ofMRAS theory for estimating PMSM speed was given,and the q axis and d axis equation of stator currentusing for adjustable model was described. Theadaptive law was constructed by Popov stabilitytheory. These establish the theoretical foundation forthis paper. In [16], it identified the stator resistance,d-axis and q-axis inductance and rotor flux linkage,but speed is acquired by sensor. In[17], used MRASmade sensorless control in the case of load changes.This fully demonstrates the good robustness of theMRAS. In[18], used fuzzy controller to adjust theparameters of the PI regulator in the speed observerbased on MRAS. The whole system had a gooddynamic and steady performance in a wide speedrange. In[10.11], constructed adaptive law by Popovstability and Lyapunov stability analysis method forparameter identification, and made a contrast. Butthe speed was also acquired by sensor. In[19-21],they established different adaptive laws according todifferent stability. The performance of the system isdifferent from that of the adaptive law. In[22], thestability of MRAS system was analyzed and thetransfer function of the system was constructed withthe modern control theory, and analyzed the transferfunction. It also identified stator resistance based onthese.

In summary, the problem of using MRAS toestimate the speed or identify parameters is theconstruction of adaptive law. If using Popov stability,the problem is adjusting PI parameters of speedobserver.

First of all, this paper use q axis and d axisequations of stator current as adjustable model. Then,for speed estimation, the adaptive law is constructedby Popov stability. The stability is analyzed throughmodern control theory and the transfer function ofthe speed observer is constructed. The PI regulator inthe speed observer can be used as the adjustable link,and use root locus analysis to adjust the ratio of PI.This method is simple and effective, and avoid thecomplex algorithm of fuzzy PI in [18]. Forparameters identification, the adaptive laws areconstructed through Lyapunov stability and the

Popov stability, respectively. By comparison, it canbe seen that the former is better than the latter. Also,if you use the latter, it need to adjust the PIparameters and consider the problem of rank, this hasbrought inconvenience. Final, this paper use FOC asa control strategy and give the result verificationthrough simulation. It can be seen that the viewpointsand methods proposed in this paper are feasible andeffective.

2 Speed estimation2.1 Mathematical model of PMSMThe stator voltage equation of the surface mountingPMSM in d-q axis:

0d ds d r q

r d s qq q r r

u iR DL LL R DLu i

(1)

Wheredu , qu , di , qi are the stator voltage and current of the

motor in the d-q axis.sR , dL , qL are the stator resistance and the

inductance of the d-q axis.D is differential operator.

r and r are rotor electric angular speed and flux.According to the equation (1), the stator current stateequation can be obtained.

dsr

d d ss

q q q r rsr

s s

uRi i LL

Di i uR

L L

(2)

For surface mounting PMSM, dL = qL = sL Motoruse for reference model and the equation (2) use foradjustable model. For speed estimation, sR and

sL can be regarded as fixed values So, the adjustablemodel for speed estimation can be given.

dsr

sd ds

s q r rq qrs s

uRLi iL

DR ui iL L

(3)

Defined state error:

,d d d q q qi i i i

(4)The state error equation of equation (2)subtracting equation (3) can be given.



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sr q

d dsr rr

dq qsr s

s

Ri

LD

iRL

L

(5)

Written state space expression:

D A Bu (6)2.2 Adaptive of speed observer designMRAS basic block diagram of speed estimation isshown in Figure 1.

Figure 1: MRAS basic block diagram ofspeed estimation

Obviously, the stability and precision of the systemis related to the construction of the adaptivemechanism. From Figure 1, it can be seen that theadaptive mechanism is related to the state errorequation (5). The structural diagram of the equation(5) is shown in Figure 2:

Figure 2: Equation (5) structure diagram

For speed estimation, the adaptive law is constructedby Popov stability. So, the conditions for the stabilityof Figure 2 are about two sides. One is the zero poleof the transfer function of the forward channel in theleft half of the s domain. Another is feedbackchannel to satisfy Popov stability.For first condition, the transfer function of forwardchannel can be deduced according to modern controltheory. Its state space expression is:

A uy

(7)

transfer function:

2 22 2

s

s

s sr

s s

RsLH s

R Rs sL L

(8)

Its pole-zero loci is shown in Figure 3 for a range of

r

starting at -200 up to 200rad/s. From the graph,we can see that with the increase of the speed, thepoles also have negative real parts. So this conditionis confirmed.

-350 -300 -250 -200 -150 -100 -50 0-200

-150

-100

-50

0

50

100

150

2000.140.30.440.580.720.84

0.92

0.98

0.140.30.440.580.720.840.92

0.98

50100150200250300350

Pole-Zero Map

Real Axis

Imag

inar

y Ax

is

Figure 3: The pole-zero loci of H(s) about range of

r

starting at -200 up to 200rad/sFor second condition, the derivation of Popov stabilityis introduced in above literature, and it’s notdescribed in this article. The adaptive law can beconstructed by Popov stability.

( )

0

ri d q q d q

s

rrp d q q d q

s

K i i dtL

K i iL

(9)

2.3 PI parameter adjustment of speedobserverThe adaptive law of the observer is constructed in 2.2,equation (9). So, the MRAS structure diagram can beobtained in Figure 4.



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Figure 4: MRAS structure diagram

The dotted line in Figure 4 is represented by the statespace expression.

T

A Buy B

(10)

So, its transfer function is:

22

2 22 2

s rq d

s s

s sr

s s

Rs i iL L

G sR Rs sL L

(11)

It can be find that this has the same zero poledistribution with equation (8).Further simplify about figure 4:

Figure 5: MRAS simplified structure diagram

So, this can be used to adjust the parameters of the PIregulator by using the root locus analysis method forthe simplified structure diagram of Figure 5, and makethe system to achieve the desired results. Because thevalue of di and qi are relatively small. so they can beneglected. This can get the open-loop transferfunction of system.

*

2 23 22

s

sc

s sr

s s

Rk s s zL

G sR Rs s sL L

(12)

Where,2

*2 ,

irp

s p

kk k zL k

.

So, this can determine the value of *k and z tomake the system to achieve the desired results, thusdetermining the value of ik and pk .

Firstly, it need to confirm z. With the increase of z,the root locus bend to S domain left half plane, asshown in Figure 6.

Root Locus

Real Axis

Imag

inar

y Ax

is-3000 -2500 -2000 -1500 -1000 -500 0 500

-800

-600

-400

-200

0

200

400

600

8000.350.620.780.880.930.965

0.986

0.997

0.350.620.780.880.930.965

0.986

0.997

5001e+0031.5e+0032e+0032.5e+0033e+003

z increase

Figure 6: the root locus of different z

In fact, the value of in general systems is 0.707by automatic control principle. So, this candetermine the range of z in Figure 6 to make it hasclosed loop poles of equal to 0.707. Through thetest, it can obtain that z≥670.Then, find two closed loop poles of equal to0.707(Imaginary part symmetry). This can determinethe value of *k and find another negative real poleaccording to the value of *k . Through the test, thetable 1 of the data can be earned.

Table 1: Closed loop poles of different z values

No. z Imaginarypart poles

Negativereal pole

*k(1) 670 358±358i -300 340(2) 750 292±292i -282 190(3) 800 279±278i -277 159(4) 1000 255±256i -273 105(5) 2000 227±228i -263 40.2

Finally, according to the above data can get thesystem step response, as shown in Figure 7. So, itcan be find the data of meeting the systemperformance to determine the parameters of the PIregulator.



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0 0.01 0.02 0.03 0.04 0.050

0.2

0.4

0.6

0.8

1

1.2

1.4

Time/s

Am

plitu

de

Step Response

5 10 15

x 10-3

0.8

0.85

0.9

0.95

1

1.05

1.1

(1)(2)(3)(4)(5)

Figure 7: the system step responseaccording to Table1

It can be seen from the figure that the performance ofNo.(1)is better. This means that the imaginary partpoles are in the left side of the negative real pole andthe lateral distance is the largest. This can determinethe parameters of the PI regulator.

The above analysis is aimed at r

equal to 120rad/s.So in the case of the No.(1), the system step response

under different r

.

0 0.01 0.02 0.03 0.04 0.050

0.2

0.4

0.6

0.8

1

1.2

1.4

Time/s

Am

plitu

de

Step Response

4.8 5 5.2 5.4 5.6 5.8

x 10-3

0.98

0.99

1

1.01

1.02

120rad/s110rad/s100rad/s90rad/s150rad/s

Figure 8: the system step response of different r

according to No.(1)From figure, it can be seen that the step responses ofthe system are similar to 120 when under 120, andthe response time is better than 120. But when higherthan 120, the system performance becomes bad. So,

in the PMSM speed control system, the value of r

should be appropriate. And this can make theobserver have a good response in a wide range ofspeed.

3 Parameters Identification

3.1 Adaptive law parameters identificationbased on Lyapunov stability

Because speed estimation is based on Popov stability,parameters identification does not need toconsider the problem of rank based on Lyapunovstability .For parameters identification, r can be regarded asfixed values. So, the adjustable model for adjustableparameters can be given.

dd dr

q r rq qr

u bi iaD

u bi ia

(13)

Where, s

s

RaL

, sb L

.

The state error equation of equation (2) subtractingequation (13) can be given.

d d d dr

q q r rq qr

i uaD a a b b

i ua

(14)

Written state space expression:

1 1 1 2D A Bu Cu (15)So, MRAS basic block diagram of parametersidentification is shown in Figure 9.

Figure 9: MRAS basic block diagram ofparameters identification

There are two methods to construct adaptive law.One is Lyapunov stability and another is the Popovstability.Definition 1 2

T u u , 1Ts B C

Equation (15) becomes:

1TD A s (16)

The design of the Lyapunov equation:



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12

T TV X P (17)

Where,1 00 1

P

,1 00 1

.

According to Lyapunov stability second method. Ifthe system is stable, two conditions should be met.One is V X positive definite, another is

V X

negative definite. Because the P and

are positive definite, V X is positive definite.

12

12

TT

TT

V X P P

(18)

Where

1

TT T TP A P s P

1T T T TP P A P s

1 1 2 22T

T u u u u

So, Equation(18) becomes

1 1

1 1 2 2

12

12

T T

T T T

V X PA A P

s P P s u u u u

(19)

Because 1 1

00

T aPA A P

a

is negative

definite, 1 112

T TPA A P is negative

definite.

To meet the condition about V X

is negativedefinite, we can let

1 1 2 21 02

T T Ts P P s u u u u

By calculation, we can get:

1 1

2 2 0

d d q q

d d q q r r

u u i i

u u u u

1 d d q qu i i

, 2 d d q q r ru u u

So, the adaptive law can get.

0 d d q qa a i i dt

0 d d q q r rb b u u dt

3.2 Adaptive law parameters identificationbased on Popov stability

Because speed estimation is based on Popov stability,parameters identification need to consider theproblem of rank based on Popov stability .For parameters identification of stator resistance,

r and d-axis and q-axis inductance can be regardedas fixed values. So, the adjustable model foradjustable parameters can be given.

dsr

d d ss

q r rq qs

r ss

uRi i LL

Dui iR

LL

(20)

The state error equation of equation (2)subtracting equation (20) can be given.

dsr

d d sss s

q q qs

r ss

iRLL

D R RiRLL

(21)

According to 2.2, the adaptive law stator resistancecan be derived according to Popov stability.

0

d d q qs Ri

s

d d q qRp s

s

i iR K dt

L

i iK R

L

(22)

Also, for parameters identification of d-axis andq-axis inductance r and stator resistance can beregarded as fixed values. In the same way, theadaptive law stator resistance can be derived.

1

1

0

d d s d

Liq q s q r rs

d d s d

Lpq q s q r r s

u R iK dt

u R iL

u R iK

u R i L

(23)

For the adjustment of RiK , RpK , LiK , LpK , wecan use the method of 2.3.



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4 Simulation results

In this paper, the field oriented control (FOC) modelof PMSM is built firstly in Matlab/Simulink. Thenbuild the adjustable model and adaptive law, asshown in Figure 10. The parameters involved in thispaper are shown in Table 2. To verify the PIparameter adjustment in the speed observer, sR and

sL can be regarded as fixed values. The PI regulatorparameters as determined by the value of the Table 1are shown in Table 3. And then the parameteridentification of sR and sL is performed on the

basis.Table 2: Simulation parameters

dcU PWMperiod sL sR r

310V 5KHZ 8.5mH 2.8758ohm 0.175V.s

Table 3: PI regulator parametersNo. pk ik(1) 0.8 536(2) 0.45 337.5(3) 0.37 296(4) 0.25 250(5) 0.1 200

Figure 10: The Control block diagram of speed estimation and parameters identification of PMSM

0.00 0.05 0.10 0.15 0.20 0.250

20

40

60

80

100

120

140

160

0.005 0.010100

120

140

160

0.200 0.202 0.204 0.206 0.208 0.210

100

120

Spee

d/(r

ad/s

)

Time/s

Sensor value

Given value

No.(1)

No.(2)

No.(3)

No.(4)No.(5)

No.(5)No.(4)

No.(3)

No.(2)

No.(1)

Sensor valueGiven value

Figure 11: Simulation results of speed without parameters identificationAs shown in Figure 11. In the case of the No.1, thespeed response of the observer has good dynamic

performance. Then, the given rotational speedmutation 100rad/s, the observer is still in good



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response.Then the parameters identification is performed. wecan use the method of 2.3 to determine the values of

RiK , RpK , LiK , LpK , as shown in Table 4.

Table 4: the values of RiK , RpK , LiK , LpK

RiK RpK LiK LpK911.2 1.36 67 0.1

0 0.05 0.1 0.15 0.2

0.00844

0.00847

0.0085

0.00853

Time/s

Ls/H

0 0.05 0.1 0.15 0.20.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

Time/s

Ls/H

Figure 12: the parameter identification of sL basedon Lyapunov stability(up) and Popov stability(down)

As can be seen from the Figure 12, the up responsespeed and accuracy are higher than the down.

0 0.05 0.1 0.15 0.22.85

2.865

2.88

2.895

Time/s

Rs/o

hm

0 0.05 0.1 0.15 0.2-4

-2

0

2

4

6

8

Time/s

Rs/

ohm

Figure 13: the parameter identification of sR basedon Lyapunov stability(up) and Popov stability(down)

As can be seen from the Figure 13, the up responsespeed and accuracy are higher than the down.

0.00 0.04 0.08 0.12 0.16 0.200

40

80

120

0.010 0.015 0.020

120

speed response withparameter identification(Popov stability)

Spe

ed re

spon

se/(r

ad/s

)

Time/s

speed response withoutparameter identification

speed response withparameter identification( Lyapunov stability)

Figure 14: the speed response of different cases

As shown in Figure 14, the speed response withparameter identification based on Lyapunov stabilityis better than Popov stability. We can use thismethod to estimate speed and identify parameterssimultaneously.



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5 ConclusionIn this paper, the model is built according to thestator current d and q axis equations, and theadaptive law of speed estimation is constructedaccording to the Popov stability. A new method forthe PI parameters of the speed observer is proposed.This method can be based on the parameters of themotor to quickly locate the parameters of the PIregulator, which provides a theoretical basis for thePMSM sensorless control.Then, for the problems of speed estimation andparameters identification simultaneously, theadaptive laws of stator resistance, d-axis and q-axisinductance are constructed through Lyapunovstability and the Popov stability, respectively andcompare their results. By comparison can be seenthat the former is better than the latter. And theformer do not need to adjust the PI parameters. Thismethod can estimate speed and identify parameterssimultaneously.So, this paper solve the problems about PIparameters adjustment of speed observer andparameter identification and speed estimationsimultaneously. The feasibility and effectiveness ofthis method are proved by simulation results.

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