The Simulation Study of Sensor Less Control for IM Drives Based on MRAS

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r ω is motor angular velocity.

Since rotor speed r ω is included in matrix 12 A and 22 A , (1)

can be selected as a reference model. If speed estimated valueˆ r ω replaces real speed r ω and motor parameters keep

invariable, the adaptive full-order observer of speedidentification system can be expressed as follows:

ˆ

ˆ s

r

i pϕ

11 12

21 22

ˆ ˆ

ˆ ˆ 0 s

s

r

i B A Au

A A ϕ = +

(3)

Where, “^” signifies the estimated value.According to the theory of MRAS, we can consider the

motor (1) as a reference model and the observer (3) as anadjustable model. Since stator currents are easy to bemeasured, the stator current is selected as the error feedback value. Therefore, the error between the states si and si can be

used the system error e . The error equation is defined bysubtracting (3) from (1) as

ˆ( ) s s p i i− =11

ˆ( ) s s A i i−12 12

ˆ ˆr r A Aϕ ϕ + − ˆ( ) s sG i i+ − (4)

Where, G is the observer gain matrix, which decides thestability of equation (4). Furthermore, actual rotor flux can’t

be directly measured. From equation (1), we can obtain the

rotor flux as follows:ˆ

r pϕ 21 22ˆ

s r A i A ϕ = + (5)

Because si can be measured directly, rotor flux can be

obtained from (5), and we usually think r ϕ is equal to ˆr ϕ . So

(4) is rewritten as follows:ˆ( ) s s p i i−

11ˆ( ) s s A i i= − ˆ( ) s sG i i+ −

12 12ˆ ˆ( ) r A A ϕ + −

11ˆ( )( ) s s A G i i= + −

12 12ˆ ˆ( ) r A A ϕ + −

11ˆ( )( ) s s A G i i= + − { /( )}m s r L L Lσ + ˆ( )r r ω ω − ˆ

r J ϕ (6)

Thus, the error between the states si and si can be used to

a speed adaptive control mechanism which gains and adjusts

estimated speed ˆr

ω . At the same time, the estimated speed

ˆ r ω is introduced in the adjustable model and the estimated

stator current si is changed consequently. While speed

adaptive mechanism should guarantee that the system error e

would approach zero if estimated speed ˆ r ω is asymptotic to

real speed r ω . Fig. 3 shows the total MRAS diagram. Where

ˆˆ ( )r s s f i iω = − is the expression of the estimated speed.

11 12

21 22 0 s s

sr r

i A A i B p u

A Aϕ ϕ

= +

ˆ A G+

C

+

B

e

ˆr ω

si

si

su

+

+ [ ]ˆ ˆT

s r i ϕ

ˆ r ϕ

ˆ( ) s s f i i−

[ ]ˆ ˆT

s r i p ϕ

Fig. 1. The block diagram of MRAS

III. A DAPTIVE SCHEME FOR SPEED ESTIMATION

As a MRAS, the stability is first to be considered. The (6)can be simplified as

e =m A e ( )d m A t x+ (7)

Where,

11m A A G= + ( ) { /( )}d m s r A t L L Lσ = ˆ( )r r ω ω − , ˆm r x J ϕ = .

A Lyapunov’s function [14] is selected as1

{ ( ) ( ) ( )}T

r d d V e Pe t A t F t A t −

= + (8)

Where, P , F are both positive symmetric matrixes. Thederivative of V to time is as follows:

1( ) 2 { ( )[ ( )]}

T T T T

m m r d m d V e A P PA e t A t Pex F A t −

= + + − (9)

It makes m A be Gourvatz matrix by configuring matrix G so

that matrix P can be gained by Lyapunov’s equationT

m m A P PA Q+ = − (10)

Where, Q is arbitrary positively definite matrix. The follow

equation is chosen to make V be negative definite1

( ) 0T

m d Pex F A t −

− = (11)

An adaptive control law [13] about ( )d A t is obtained

0( ) (0)

t T

d m d A t FPex d Aτ = + (12)

From (7) and (12), we can obtain as follows:

{ /( )}m s r L L Lσ ˆ( )r r ω ω −

0

ˆ ˆ( ) (0)T T

s s r

t

d FP i i J d Aϕ τ = − +

0ˆ{

r

t

FP α

ϕ = ˆ( ) s si i

β β − ˆˆ ( )} (0)

r s s d i i d A

β α α ϕ τ − − + (13)

Obviously, (13) can be equivalent as follows by a PIcontrol equation

ˆ ( / )r PS IS k k sω = + ˆˆ{ ( )r s s

i iα β β

ϕ − − ˆˆ ( )}r s s

i i β α α

ϕ − (14)

Where, PS k , IS k are PI parameters of speed adaptive

estimator and 1/ S is the integral operator. Therefore,according to Lyapunov’s theory we can conclude that a rightmatrix P is gained from (13) if a random positive matrix Qis given and global asymptotic stability of the system isguaranteed if adaptive gain F is positive matrix and input u s

is random parted continuous function. (14) can be used to

236 2008 Asia Simulation Conference — 7 th Intl. Conf. on Sys. Simulation and Scientific Computing

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estimate rotor speed conveniently.

IV. S TATOR R ESISTANCE ESTIMATION THEORY

From (4), if rotor speed w is invariable, it can be described:

s s( )i i− =11 s 11 s

ˆ ˆ A i A i−12 r r ˆ( ) A ϕ ϕ + −

s s( )G i i+ −

(15)From (1) and (3), we have

ˆ( )r r

p ϕ ϕ − =21 22

ˆ ˆ( ) ( ) s s r r

A i i A ϕ ϕ − + − (16)

When r ϕ is nearly equal to ˆr ϕ , it can be expressed1

22 21ˆˆ ( )r r s s A A i iϕ ϕ −

− = − − (17)

Substituting (17) into (16) gives

s s( ) p i i−11 11

ˆ( ) s A A i= − 1

22 21 12ˆ( ) s s A A A i i−

− −

s s( )G i i+ − (18)

From (1), it can be simplified as

s s( )i i− = 1

22 21 12( )G A A A−

− ˆ( ) s si i−

ˆ( / )( ) s s s si L R Rσ − − (19)

So, (6) can be rewritten by (19), whereˆ

s se i i= − 1

22 21 12m A G A A A−

= −m s x i= −

ˆ( ) ( ) /( )d s s s A t R R Lσ

= −

Therefore, according to the same Lyapunov’s theory, stator resistance s R can be estimated by a PI control equation as

rotor speed

s R ( / ) PR IRk k s= + ˆ( ( ) s s si i iα α α − ˆ( )) s s si i i β β β

+ − (20)

V. S IMULATION R ESULTS

The proposed above speed adaptive estimation is applied tothe indirect FOC of an IM drive. Fig. 2 shows an overallcontrol diagram of the sensorless induction motor drivesystem based on slip frequency FOC.

Speedcontroller

Currentcontroller

Currentcontroller

Speedestimator

Adjustablemodel

PWMinverter

M

θ

+

+

+

+

Coordinateinverse

transform

*

au

*

bu

*

cu

*

squ

*

sd u

*

sqi

sqi

*

r ω

ˆr ω

*

sd i

sd i

sai sau

sbu sbi

su α si α

su si

si α si

sϕ sα ϕ

ˆr ω

Slipfrequencycalculation

+ 1ω Park

transformClark

transform

Fig. 2. The overall diagram of sensorless vector controlled system

The simulation is performed for the verification of theabove control scheme. It is simulated by a sampling period of 25 s μ .

Table I shows the induction motor specification used insimulation system. The PI gains of the speed adaptive scheme are:

K PS = 0.02, K IS = 500.

TABLE ICONSTRUCTED MOTOR SPECIFICATION

Stator resistance 1.48 [ ]Rotor resistance 2.62 [ ]

Stator inductance 210 [mH]Rotor inductance 210 [mH]

Mutual inductance 200 [mH]Rated voltage 380 [V]

Rated frequency 50 [Hz] Number of pole 4

Rated speed 1450[rpm]

Fig. 3, 4 and 5 are simulation results of rotor speed, stator current, and electromagnetic torque at 50 rad/s. Fig. 3indicates that the estimated and real speed track each other in

both steady state and dynamic operation. Fig. 4 shows that just like speed the estimated stator current coincides with thereal one and Fig. 5 reveals the high performance of torque atstartup.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

Time (sec)

S p e e

d ( r a d / s )

realestimated

Fig. 3. Estimated and real of rotor speed waveforms at 50 rad/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20

-15

-10

-5

0

5

10

Time (s)

S t a t o r c u r r e n

t ( A )

realestimated

Fig. 4. Estimated and real stator current waveforms at 50 rad/s

2008 Asia Simulation Conference — 7 th Intl. Conf. on Sys. Simulation and Scientific Computing 237




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

0

2

4

6

8

10

12

14

16

18

Time (sec)

T e

( N . m

)

Fig. 5. Real torque waveform at 50 rad/sFig. 6, 7 and 8 are simulation results of rotor speed, stator

current, and electromagnetic torque at 10 rad/s. From thesefigures, estimated and real speeds track each other by almostzero error at very low speeds.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

Time (sec)

S p e e

d ( r a d / s ) real

estimated

Fig. 6. Estimated and real of rotor speed waveforms at 10 rad/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4

-3

-2

-1

0

1

2

3

4

Time (sec)


t ( A )

realestimated

Fig. 7. Estimated and real stator current waveforms at 10 rad/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.4

0

0.4

0.8

1.2

1.6

2

Time (sec)

T e

( N . m

)

Fig. 8. Real torque waveform at 10 rad/s

In the condition of same motor parameters, simulation iscarried out for stator resistance estimation. The PI gains of thestator resistance estimation scheme are: K PR = 0.06, K I R= 50.When the rotor speed is 50 rad/s, the stator resistance isincreased by 50% above the nominal value. Fig. 9 and 10show the rotor speed and stator current waveforms in the caseof without stator resistance estimation. Obviously, theestimated speed and stator current fluctuate because of stator

resistance change. However, Fig. 11 is the simulation resultsof rotor speed after adding stator resistance estimationaccording to (20). Compared to Fig. 9, Fig. 11 indicates thatestimated speed can trace the real speed by adding stator resistance estimation.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10

0

10

20

30

40

50

60

Time (sec)

S p e e

d ( r a d / s )

estimatedreal

Fig. 9. Rotor speed waveforms without stator resistance estimationat 50 rad/s

238 2008 Asia Simulation Conference — 7 th Intl. Conf. on Sys. Simulation and Scientific Computing




0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-12

-10

-8

-6

-4

-2

0

2

4

6

Time (sec)


t ( A )

estimatedreal

Fig. 10. Stator current waveforms without stator resistance estimationat 50 rad/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

Time (sec)

S p e e

d ( r a d / s ) real

estimated

Fig. 11. Rotor speed waveforms adding stator resistance estimationat 50 rad/s

VI. C ONCLUSION

This paper has proposed a method of speed estimation for sensorless induction motor drives based on MRAS. The

proposed speed and stator resistance identification schemesare educed from and proved by the Lyapunov’s criterion and

applied to an indirect oriented induction motor controlwithout speed sensors. The performance of the proposedscheme is verified by simulation results particularly in verylow speeds.

R EFERENCES

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[3] K. Ohyama, G. M. Asher and M. Sumner, “Comparative experimentalassessment for high performance sensorless induction motor drives,”

IEEE Proceedings of ISIE’99 , pp. 386-391, 1999.[4] C. schauder, “Adaptive speed identification for vector control of

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[5] S. D. Huang, Y. N. Wang, J. Gao, et al, “The vector control based onMRAS speed sensorless induction motor drive,” Conference of the 5 th

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[7] R. Cardenas, R. Pena, G. Asher, et al, “MRAS observer for doubly fedinduction machines,” IEEE Transactions on Industrial EnergyConversion , vol. 19, no. 2, pp. 467-468, 2004.

[8] M. N. Marwali and A. Keyhani, “A comparative study of rotor flux based MRAS and back EMF base MRAS speed estimators for speedsensorless vector control of induction machines,” IEEE Proceedings of

IAS Annual Meeting , pp. 160-166, 1997.[9] X. H. Nian, T. Wang, J. Wang, et al, “Adaptive Stator Resistance

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[10] H. Tajima, G. Guidi and H. Umida, “Consideration about problems andsolutions of speed estimation method and parameter tuning for speed-sensorless vector control of induction motor drives,” IEEE Transactions on Industry Applications , vol. 38, no. 5, pp. 1282-1289,2002.

[11] M. Rashed, F. Stronach and P. Vas, “A new stable MRAS-based speedand stator resistance estimators for sensorless vector control inductionmotor drive at low speeds,” IEEE Proceedings of IAS Annual Meeting ,

pp. 1181-1188, 2003.[12] L. Zhen and L. Y. Xu, “Sensorless field orientation control of induction

machines based on a mutual MRAS scheme,” IEEE Transactions on Industrial Electronics, vol. 45, no. 5, pp. 824-831, 1998.

[13] G. Guidi and H. Vmida, “A novel stator resistance estimation methodfor speed-sensorless induction motor dirivers,” IEEE Transactions on

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University Press (in Chinese), 2003.

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The Simulation Study of Sensor Less Control for IM Drives Based on MRAS

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