a novel method for the identification of the rotor

492

https://doi.org/10.6113/JPE.2018.18.2.492

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718

JPE 18-2-16

Journal of Power Electronics, Vol. 18, No. 2, pp. 492-501, March 2018

A Novel Method for the Identification of the Rotor

Resistance and Mutual Inductance of Induction

Motors Based on MRAC and RLS Estimation

Gwon-Jae Jo* and Jong-Woo Choi†

†,*Department of Electrical Engineering, Kyungpook National University, Daegu, Korea

Abstract

In the rotor-flux oriented control used in induction motors, the electrical parameters of the motors should be identified. Among

these parameters, the mutual inductance and rotor resistance should be accurately tuned for better operations. However, they are

more difficult to identify than the stator resistance and stator transient inductance. The rotor resistance and mutual inductance can

change in operations due to flux saturation and heat generation. When detuning of these parameters occurs, the performance of

the control is degenerated. In this paper, a novel method for the concurrent identification of the two parameters is proposed based

on recursive least square estimation and model reference adaptive control.

Key words: Induction motor, Model reference adaptive control, Mutual inductance, Parameter identification, Recursive least square

estimation, Rotor resistance

NOMENCLATURE

Stator resistance

Rotor resistance

Mutual inductance

Stator self-inductance

Rotor self-inductance

Stator leakage inductance

Rotor leakage inductance

Stator transient inductance

Rotor time constant

, Superscripts: denote the stationary and rotor

reference frames

, Subscripts: denote the stator and rotor

^ Denotes estimated quantities

Derivative operator

Bold form Vector

I. INTRODUCTION

Induction motors are widely used in industrial and home

appliances. For the speed control of induction motors,

rotor-flux oriented control is usually utilized due to its

simplicity and its ability to control flux and torque components

separately. In the rotor-flux control algorithm, the precise

tuning of and is very important for efficient control of

the currents of both the flux and torque components. Moreover,

these parameters are essential in flux observers. However,

identifying them is more difficult than identifying stator

parameters such as and . This is attributed to the fact

that varies due to nonlinear characteristics in the B-H

curve. It is also due to the fact that is more affected by

thermal temperature in operation than since the rotor is

positioned inside the motor.

Several identification methods for and have been

reported. A method by which can be estimated using the

relationship between the flux and the developed torque was

proposed in [1]. A method, which helps identify by solving

simultaneous equations for the rotor voltage and the torque,

was proposed in [2]. Methods where is identified by

adopting adaptive observers were reported in [3], [4]. A

method using a particle swarm optimization algorithm was

reported in [5]. Methods for the identification of based on

© 2018 KIPE

Manuscript received Jun. 16, 2017; accepted Oct. 6, 2017

Recommended for publication by Associate Editor Kwang-Woon Lee. †Corresponding Author: [email protected]

Tel: +82-53-950-5515, Kyungpook National University *Dept. of Electrical Engineering, Kyungpook National University, Korea

A Novel Method for the Identification of the Rotor Resistance and Mutual Inductance … 493

model reference adaptive control (MRAC) were proposed in

[6], [7], where the variables observed in the MRAC are the

voltage [6] and the instantaneous reactive power [7]. A method

that compares fluxes estimated by flux observers, was reported

in [8]. The rotor time constant can be identified from a

model reference adaptive system speed estimator in sensorless

systems [9]. Using the dynamics of a synchronous reference

frame current controller, the identification of was studied

in [10].

In addition to the identification of single rotor parameters,

techniques for the identification of various other parameters

were reported in [11]-[18]. Parameters have been identified

offline based on fitting the experimental data and numerical

analysis [11]. Voltage equations of the stator and rotor have

been solved directly by integration to identify and

[12], [13]. Based on the MRAC technique, currents have been

compared for identification in [14]. Furthermore, parameters

have been identified using recursive least square (RLS)

estimation [15], [16]. Methods using sliding mode observers

and artificial neural networks have been proposed [17], [18] for

parameter identification.

In this paper, a novel method for the identification of

and based on MRAC is proposed. In the MRAC

algorithm, the mechanism is to compare the flux estimated

by the flux observers designed from current and voltage

models. Furthermore, the differences in fluxes estimated by

the two flux observers are analyzed. The proposed method

can operate at a sufficiently high frequency in order to obtain

the complete reference model in the MRAC system. Thus, it

is difficult to apply the proposed method at a low frequency

including standstill. However, most of the papers published

so far have reported the identification of single parameter,

whereas this paper identifies and concurrently by

separating the two parameters with respect to the equation.

Thus, this paper has advantage since there are no

dependencies on each other for and in the proposed

method. Additionally, the proposed method can guarantee

effective behavior in both the steady-state and transient-state

because any assumption for the steady state in the algorithm

is not involved. RLS estimation is utilized for the realization

of the parameters identification. The feasibility of the

proposed method has been verified by simulation and

experimental results under diverse conditions.

II. GOVERNING EQUATIONS

The flux observers utilized in this paper are based on the

governing equations for the model of an induction motor. The

governing equations on an arbitrary rotating axis, given by

in squirrel cage-type induction motors consist of the voltage

equations and linkage flux equations in the stator and rotor.

The voltage equations are expressed as follows:

s s s s sR s j v i λ λ (1)

( )r r r r r rR s j v i λ λ (2)

The linkage flux equations are expressed as follows:

s s s m rL L λ i i (3)

r r r m sL L λ i i (4)

III. FLUX OBSERVERS

A. Voltage Model

The voltage model flux observer is designed from the stator

voltage equation in [19]. The equation for the estimation of the

rotor flux in the voltage model can be expressed using (1), (3),

and (4) by defining these equations on the stationary reference

frame ( ). Thus, the rotor flux vector can be estimated

from the stator voltage and current as follows:

ˆ 1ˆ ˆ ˆˆ{ ( ) }ˆ

s s s srr s s s s s

m

LR L

sL λ v i i (5)

However, the voltage model flux observer cannot be

implemented because the pure integration in (5) can diverge

due to a dc offset voltage. Thus, the closed-loop Gopinath

style flux observer in [20] has been employed. This flux

observer model operates the same as the voltage model flux

observer in a sufficiently high frequency region with solving

the problem with the dc offset voltage.

B. Current Model

The current model is a typical flux observer in the rotor-flux

oriented control [21]. This model is based on the rotor equation.

A differential equation on the rotor reference frame ( )

can be expressed from (2) and (4) as follows:

ˆˆ

ˆ 1

r rmr s

r

L

T s

λ i (6)

Equation (6) yields the low-pass filter (LPF) form constructed

from only the rotor flux and stator current. The stator current

only can be used to estimate the rotor flux. The rotor flux vector

on the stationary reference frame can be calculated from in

(6) as follows:

ˆ ˆrjs r

r re

λ λ

where 1

r rs

(7)

IV. MRAC

In this paper, parameter identification is based on the

principle of MRAC. Fig. 1 shows a MRAC system that

consists of a reference model, an adjustable model, and an

adaptation mechanism. The reference model is independent

of the parameters, whereas the adjustable model is dependent

on the parameters. The adaptation mechanism generates the

estimated parameters by comparing the variables ( and

) from the reference and adjustable models.

494 Journal of Power Electronics, Vol. 18, No. 2, March 2018

Fig. 1. MRAC system.

The current model flux observer is dependent on and

according to (6). On the other hand, the voltage model flux

observer is independent of these parameters. Firstly, the

estimation equation of the voltage model (5) does not include

. Secondly, the ratio is nearly unity due to a small

leakage ratio. This is shown by the following equation:

ˆ ˆ ˆ1

ˆ ˆr m lr

m m

L L L

L L

(8)

Therefore, the detuning of barely affects the estimated

flux in the voltage model. From the principle of MRAC, the

voltage and current models can be chosen as reference and

adjustable models in the identification algorithm for and

. In the detuned case of and , the rotor flux

estimated by the current model has errors. On the other hand,

the rotor flux can be obtained quite accurately from the

voltage model. Therefore, correct values of and can

be obtained from the adaptation mechanism wherein the

fluxes estimated by the two flux observers can be compared.

As the parameters are calibrated by the adaptation mechanism,

the adjustable model will follow the reference model.

V. PARAMETER IDENTIFICATION

A. Expression of the Flux Difference

The estimation equation given by (6) can be rearranged if

the current model is used as an adjustable model, as follows:

ˆˆˆ

ˆ ˆrmr

adj adj r s

r r

LRs R

L L λ λ i (9)

where denotes from the current model. Equivalently,

if the voltage model is used as a reference model, (9) is

rewritten as (10) given by:

rmrref ref r s

r r

LRs R

L L λ λ i (10)

where denotes from the voltage model. Equation

(10) consist of actual parameters because this model is robust

to the detuning of and . Fig. 2 shows the proposed

MRAC system for parameter identification. When (9) is

subtracted from (10), an equation with a flux difference

( ) is obtained as follows:

Fig. 2. MRAC system for the proposed parameter identification.

ˆˆ ˆˆ( ) ( ) ( )

ˆ ˆ ˆrm mr r r

ref r r s

r rr r r

L LR R Rs R R

L LL L L λ λ i (11)

Equation (11) can be rearranged to obtain the following

equation:

ˆ ˆ ˆˆ ˆ( ) (1 ) (1 )

ˆ ˆ ˆm m r rr r

ref ref

mr r m r r

L L L RR Rs s

LL L L L R λ λ λ (12)

The ratio in (12) is approximately unity

due to a small leakage ratio as follows:

ˆ 1 ( / )1

ˆ ˆ ˆ1 ( / )

m r lr m

m r lr m

L L L L

L L L L

(13)

Equations (12) and (13) lead to the following rearranged

equation:

ˆ ˆ ˆ1(1 ) (1 )

ˆ ˆ1 1

m r rref ref

m rr r

L R T s

L RT s T s

λ λ λ (14)

Equation (14) implies that the flux difference can be

regarded as a linear equation with the two filtered signals,

and the two coefficients given by:

1

ˆ1 m

m

La

L and

2

ˆ1 r

r

Ra

R (15)

B. RLS Estimation

For solving (14), RLS estimation, which is an algorithm

for estimating unknown coefficients in solving regression

problems, is employed. In this paper, RLS estimation with a

forgetting factor is utilized in order to enhance the rate of

convergence [22]. A regression equation is expressed by (16),

where is the output, is the regression vector, and is

the coefficient vector: Ty X A (16)

In this system, the unknown coefficients are estimated by

(17) as follows, where is the gain vector and is the

error:

ˆ ˆ[ ] [ 1] [ ] [ ]n n n e n A A K (17)

Here, is defined by (18) and is calculated from (19),

where is the covariance matrix and is the forgetting

factor.


Fig. 3. Calculation of the regression vector and output.

T ˆ[ ] [ ] [ ] [ 1]e n y n n n X A (18)

T

[ 1] [ ][ ]

[ ] [ 1] [ ]

n nn

n n n

P XK

X P X (19)

Equation (17) implies that the estimated coefficient is

compensated by the gain times the error so that the error

becomes zero. plays the role of putting more weight on

current state than the previous state, which promotes the

convergence of . Lastly, is renewed as follows:

T1[ ] ( [ 1] [ ] [ ] [ 1])n n n n n

P P K X P (20)

On application of the RLS algorithm to (14), the output

and regression vector in the equation become:

y λ (21)

Tˆ1

ˆ ˆ1 1

rref ref

r r

T s

T s T s

X λ λ (22)

The output and regression vector can be calculated from

the fluxes estimated by the flux observers and the filters

which have a known time constant . The process for the

calculation of the output and regression vector is depicted in

Fig. 3. In this way, the coefficients in (15) are estimated by

RLS. Note that the parameters ( and ) to be identified

are separate in terms of the equation. Therefore, the detuning

of does not affect the identification of and vice

versa.

C. Coefficient Controller

The compensation components of the respective

parameters are calculated from the coefficients in (15) for

tuning them. These compensation components are obtained

by regulating the estimated coefficients ( and ). When

and are reduced to zero, and are

implicated from (15). Based on MRAC, the flux difference

becomes zero according to (14). A proportional-integral (PI)

controller is employed for the regulation of and as

shown Fig. 4. The outputs of the controller ( and )

are used as compensation components of the parameters as

follows:

ˆ ˆ [0]m m mL L L (23)

ˆ ˆ [0]r r rR R R (24)

Fig. 4. Coefficient controller.

TABLE I

SPECIFICATIONS AND PARAMETERS OF AN INDUCTION MOTOR

Specifications and Parameters Quantities

Rated power 1.5 kW

Pole 4 pole

Rated speed 1,455 rpm

Rated torque 9.2 N·m

sR 1.67 Ω

rR 0.73 Ω

mL 137 mH

lsL , lrL 6.5 mH

where and are the compensation components of

and , and and are their initial setting

values. In addition, the final outputs and are

filtered by a LPF to prevent noise.

The process for the proposed parameter identification is

shown in Fig. 5. A block diagram of a rotor-flux oriented

control system with the proposed parameter identification

algorithm in an induction motor is shown in Fig. 6.

VI. SIMULATION RESULTS

The specifications and parameters of the utilized induction

motor are specified in Table I. All of the controllers in the

simulation are executed digitally, where the control periods

of the current controller and speed controller are 100 μs and

400 μs, respectively. The control period of the parameter

identification is synchronized with the speed controller. The

test speeds are 300 (low-speed), 600 (mid-speed), and 1200

rpm (high-speed). Furthermore, various loads (30%, 50% and

80% of the rating) are applied to the induction motor. The

forgetting factor in the RLS is set to 0.99. The initial value of

is 1.5 times (206 mH) the actual value, and is

detuned to be 0.5 times (0.37 Ω) the actual value. First, a

situation with perfectly tuned stator parameters ( and )

is tested. Second, the detuned cases for the stator parameters

are simulated, where the voltage model flux observer

generated an erroneous flux due to the detuning. The

parameter identification algorithm starts in the steady state

when the speed becomes consistent with the applied load

torque. Furthermore, and , estimated by the algorithm,

are updated to all of the controllers and the algorithm in real

time, as shown in Fig. 6.

Fig. 5. Proposed parameter identification


Fig. 5. Process of the proposed parameter identification.

Fig. 6. Rotor-flux oriented control with the proposed parameter identification in an induction motor.

(0.1

/div

.)

(0.1

/div

.)

Time (1 s/div.) Time (1 s/div.)

(a) (b)

Fig. 7. Responses of coefficients. (a) On the d-axis. (b) Upon averaging.

(Ω, m

H, A

)

(0.1

Wb

/div

.)

Time (1 s/div.) Time (1 s/div.)

(a)

(b)

Fig. 8. Well-tuned case for all of the stator parameters (600 rpm, 50 % load). (a) , , and currents. (b) |∆ |.

In fact, the vector form (14), which involves both the

d-axis and q-axis, can be analyzed on one of the two axes.

First, the identification with respect to only the d-axis is

tested. Fig. 7(a) shows the responses of the actual coefficients

( ) and the estimated coefficients ( ) for the d-axis.

followed from its initial value since the start of

identification. Although the coefficient converges to zero

asymptotically in the steady state, has oscillation in the

transient state due to the fact that is sinusoidal signal

with a slip frequency. For generating the parameter

compensation components, it is appropriate that and

to be regulated by the PI controller should follow


(Ω, m

H, A

)

Time (1 s/div.)

(a)

(Ω, m

H, A

)

Time (1 s/div.)

(b)

Fig. 9. Detuned case of the stator parameters. (a)

case (300 rpm, 80 % load). (b) case (1200 rpm,

30 % load).

and smoothly without oscillations in the transient state.

To achieve this, the average of two coefficients on the

d-axis and q-axis is considered. It is confirmed from Fig.

7(b) that the oscillation of the coefficient in the transient

state is removed after averaging the outputs on the two

axes.

Fig. 8(a) shows the result of identification for and

under given conditions. Upon the implementation of the

proposed algorithm, both of the parameters converge to their

actual values within 5 s. Although the approximation in (13)

is applied, is accurately estimated since is

simultaneously calibrated to yield the actual value. When the

parameters are calibrated, both of d-axis and q-axis currents

change slowly. Fig. 8(b) is the flux vector difference between

the reference and adjustable models. The flux vector

difference is defined as the distance between the flux

positions estimated by the voltage model and current model,

which is given by:

2 2( ) ( )d ref d adj q ref q adj λ (25)

The flux vector difference converges to zero when the

parameters are well tuned. This implies that the adjustable

model follows the reference model both on the d-axis and the

q-axis.

Fig. 9(a) shows results of the detuned case of . is

set to 1.15 times the actual value. In this situation, the relative

error of is -6.7 % and that of is -0.7 % in the steady

(Ω, m

H, A

)

Time (1 s/div.)

(a)

(V,

rpm

)

Time (1 s/div.)

(b)

(0.1

Wb

/div

.)

Time (1 s/div.)

(c)

Fig. 10. Well-tuned case for all of the stator parameters (600 rpm,

50 % load). (a) , and currents. (b) Voltages and speed. (c)

|∆ |.

state. Finally, the detuned case of is shown in Fig. 9(b).

It is usually harder to tune than . Thus, is set to

-25 %, to have a higher error than the case. In this case,

the relative errors of estimation are 0.6 % and 2.5 % for

and , respectively. In the detuned cases of the stator

parameters, the errors of the estimated and at

various speeds and loads are enumerated in Table II and

Table III. In terms of their speed and load, their errors are

different. However, these errors are not as obvious as these

are low. For the detuned case of , equal errors appear as

different speeds. In conclusion, the numerical results in Table

II and Table III demonstrate the good robustness of the

proposed algorithm for detuning the stator parameters.

VII. EXPERIMENTAL RESULTS

The information on the induction motor used in the

experiment has been specified in Table I. The utilized DSP in

the experiment is a TMS320F28335. The switching frequency

of the PWM inverter is 5 kHz, and the sampling frequency of

Ro

tor

resi

stan

ce a

nd

Mu

tual

In

du

ctan

ce

Fig. 3. Calculation of the output and the input


(Ω, m

H, A

)

Time (1 s/div.)

(a)

(V,

rpm

)

Time (1 s/div.)

(b)

Fig. 11. case (300 rpm, 80 % load). (a) ,

and currents. (b) Voltages and speed.

TABLE II

ERRORS IN THE ESTIMATED VALUES OF AND IN IN VARIOUS SITUATIONS (UNIT: %)

Speeds

(rpm)

30 % load 50 % load 80 % load

300 5.0 -2.7 1.7 -4.4 -0.7 -6.7

600 2.4 -1.5 0.8 -2.4 -0.5 -3.7

1200 1.2 -0.8 0.4 -1.3 -0.3 -2.0

TABLE III

ERRORS IN THE ESTIMATED VALUES OF AND IN IN VARIOUS SITUATIONS (UNIT: %)

Speeds

(rpm)

30 % load 50 % load 80 % load

300

2.5 0.6 2.6 -0.6 2.8 -3.2 600

1200

the line currents is 10 kHz. All of the settings for the control

period are same as those for the aforementioned simulations.

The utilized load motor is a 2.26 kW synchronous motor and it

is set to run in the torque control mode to generate a specific

torque. Before the drive implementation, in order to obtain

accurate output voltages, the methods for dead time

compensation in [23] were applied, where the switch

compensation time is calibrated considering the dead time, the

voltage drops of the power devices, etc. was identified

before the speed drive. was 1.67 Ω at the beginning of the

experiment. The initial settings of and are 202.5 mH

(Ω, m

H, A

)

Time (1 s/div.)

(a)

(V,

rpm

)

Time (1 s/div.)

(b)

Fig. 12. case (1200 rpm, 30 % load). (a) ,

and currents. (b) Voltages and speed.

and 0.35 Ω which have errors of approximate 50 %.

In the well-tuned case of and , results of parameter

identification are shown in Fig. 10(a). The two parameters are

estimated correctly. These results correspond to Fig. 8(a)

from the simulation results. The currents change a bit with

the identification. The voltages and speed are plotted in Fig.

10(b). The speed remains constantly at 600 rpm during both

the steady-state and transient-state under the identification

algorithm. The voltages decrease as the parameters are being

tuned. Fig. 10(c) shows the flux vector difference between

the reference and the adjustable models. |∆ | converges to

zero asymptotically as and are being tuned.

Fig. 11 and Fig. 12 describe the results of the detuned case

of and , respectively. The two detuned cases show the

similar results to Fig. 10, although errors of the stator

parameters exist. Since the two parameters are tuned

simultaneously, the currents and voltages change. Meanwhile,

the speeds stay at the command speed. Data collected from 27

situations (speeds: 300/600/1200 rpm, loads: 30/50/80 %, and

the tuned-states of or ) are shown in Fig. 13(a). This

data shows consistent results under various conditions. As a

result, these experimental results proves that the proposed

algorithm is robust to the detuning of the stator parameters.

Two hours after the speed drive operation, was

identified again. increased a bit than the beginning, which

was 1.78 Ω. In this situation, experiments in the selected

condition were carried out repeatedly. Fig. 13(b) shows these

results. The results for have 6~7 % higher values than

those for the previous experiments. has values that are


(m

H)

(m

H)

(Ω) (Ω)

(a)

(b)

Fig. 13. Sets of estimated parameters under various conditions. (a) Beginning of experimental activity. (b) Two hours after experimental

activity.

(Ω, m

H, A

)

Time (1 s/div.)

(a)

(Ω, m

H, A

)

Time (1 s/div.)

(b)

(Ω, m

H, A

)

Time (1 s/div.)

(c)

Fig. 14. In cases of speed operation with fluctuating load. (a) At

300 rpm. (b) At 600 rpm. (c) At 1200 rpm.

equal to those of the previous experiment.

In practical speed drive systems, coupled loads fluctuate

for any reason. Thus, it is necessary to prove that the

proposed method operates properly under such conditions.

It is expected that the proposed algorithm can operate

smoothly because it does not have any assumptions for the

steady state, as can be seen in chapter V. In other words,

proper operation of the algorithm can be guaranteed in

states where the speed or current is not constant. In

making these situations, a fluctuating load is generated,

which has a constant component 50 % plus a sinusoidal

component 20 % for the rated torque of the induction

motor. Here, the frequency of the sinusoidal component is

synchronized with the speed. Fig. 14 shows the

experimental results at 300, 600 and 1200 rpm. The Q-axis

current fluctuates to compensate the fluctuating load. The

speed also fluctuates since load torque is not perfectly

compensated. Despite this condition, the parameters are

well identified without fluctuations. Thus, the proposed

identification method may be used in practical speed drive

systems with fluctuating loads.

VIII. CONCLUSIONS

In this paper, a novel method for the identification of

and has been proposed. In order to realize the proposed

method, the MRAC principle is applied, and RLS estimation

is utilized for estimating these parameters. Because a closed-

loop Gopinath style flux observer is employed as the

reference model in the adaptation mechanism, the proposed

algorithm can operate at a sufficiently high frequency to get

the complete reference model. However, the proposed

algorithm has the following advantages:

― Simple structure and well-known algorithms: flux

estimations, filtering calculations, and RLS estimation.

― The two parameters can be identified simultaneously.


― Independence of the two parameters: the detuning of

does not affect the estimation accuracy of and

vice versa.

― It guarantees effective operation in both the steady-

state and the transient-state.

The proposed method was verified by simulation and

experimental results where the parameters were estimated

accurately through a speed control test with a load.

Additionally, in detuned cases of the stator parameters and

cases with a fluctuating load, experimental results

demonstrated the excellence of the proposed algorithm in

diverse situations.

ACKNOWLEDGMENT

This work was supported by the Korea Institute of Energy

Technology Evaluation and Planning (KETEP) and the

Ministry of Trade, Industry & Energy (MOTIE) of the

Republic of Korea (No. 20174030201490).

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Gwon-Jae Jo was born in Daegu, Korea. He

received his B.S. and M.S. degrees in

Electrical Engineering from Kyungpook

National University, Daegu, Korea, in 2013

and 2016, respectively, where he is presently

working towards his Ph.D. degree in

Electrical Engineering. His current research

interests include AC motor drives and

custom power devices.

Jong-Woo Choi was born in Daegu, Korea.

He received his B.S., M.S. and Ph.D. degrees

in Electrical Engineering from Seoul

National University, Seoul, Korea, in 1991,

1993 and 1996, respectively. From 1996 to

2000, he worked as a Research Engineer at

LG Industrial Systems Co., Korea. Since

2001, he has been a Professor in the

Department of Electrical Engineering at Kyungpook National

University, Daegu, Korea. His current research interests include

static power conversion and electric machine drives.

a novel method for the identification of the rotor

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