107-114

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Proc. Natl. Sci. Counc. ROC(A)

Vol. 25, No. 2, 2001. pp. 107-114

Speed Estimation of Induction Motor Using a Non-linear

Identification Technique

JUI-JUNG LIU*, I-CHUNG KUNG**, AND HUI-CHENG CHAO*

*Department of Electrical Engineering

Chung Cheng Institute of Technology

Taoyuan, Taiwan, R.O.C.**Chinese Naval Academy

Kaohsiung, Taiwan, R.O.C.

(Received October 4, 1999; Accepted February 18, 2000)

ABSTRACT

This paper considers the problem of estimating the speed of an induction motor using a non-linear identification

technique. A discrete-time non-linear identification approach, NARMAX (Non-linear Auto Regressive Moving Average

model with eXogenous inputs), is presented to describe a polynomial modeling between the speed and voltages of

an induction motor for estimating the motor speed. This approach is useful for identifying the non-linear relationship

between the speed and voltages of an induction motor. The feasibility and accuracy of the proposed method are verified

through laboratory tests. This approach will replace the speed sensor used in a speed control closed loop motor system.

In addition, a future robust controller design based on the NARMAX model will apply an innovative and simplified

speed control algorithm for an induction motor. The last research is now underway.

Key Words: induction motor, modeling, NARMAX, non-linear identification

107

I. Introduction

Motors play an important role in daily life, e.g., in

industrial manufacturing and in many other applications. In

their early days, DC motors had the advantage of precise speed

control when utilized for the purpose of accurate driving.

However, DC motors have the disadvantages of brush erosion,

maintenance requirements, environmental effects, complex

structures and power limits. On the other hand, induction

motors are robust, simple, small in size, low in cost, almost

maintenance-free and possess a wide range of speeds com-

pared to DC motors. The main obstacles to using induction

motor drives are the high cost of conversion equipment, thecomplexity of signal processing and poor precision. Neverthe-

less, control schemes have been developed which provide a

feasible approach of speed control to induction motors

(Blaschke, 1972). The equations of motion describing the

steady state behavior of an induction motor are highly non-

linear, time varying and coupled (Vas, 1990). Hasse and

Blaschke developed a vector control theory to simplify the

structure of speed control used to drive like DC motors by

using coordinate transformations. In recent years, the vector

control theory has become more feasible due to progress in

the development of electronics techniques and high speed

microprocessors. In most applications, speed sensors are

necessary and essential in the speed control loop. However,

sensors have several disadvantages in terms of drive cost,reliability, and noise immunity. Various approaches have been

proposed for estimating speed using some electric parameters,

such as current, voltage, frequency, and flux. They are based

on a combination of state estimation theory and vector control

theory known as speed sensorless motor control (Holtz, 1993;

Ilas et al., 1994; Hurst et al., 1994). However, the algorithm

of vector control theory requires manipulation of the electric

parameters of the motor so that the governing equations in

rectangular coordinates can be developed, prior knowledge

of the state equations is necessary when the estimation theory

is used to estimate the speed precisely. However, the values

of the electric parameters may deviate from the designatedvalues due to changes in the working environment, temperature,

speed, external load and noise. The equations of motion of

an induction motor, which are converted by means of vector

control to the type of DC motor control, may be not suitable

due to the same reasons, such as changes in the working

environment, etc. as mentioned above. Consequently, these

unpredicted factors make the actual behavior of a sensorless

control motor non-linear and hard to describe. The accuracy

will improve if this non-linearity can be governed using other

methods in practical applications.

System identification models the relation between the

input and output without knowledge of the equations of motion

a priori. Much work has focused on developing identification

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techniques, including those which employ linear filtering to

estimate linear transfer function (Schoukens, 1990), to esti-

mate the physical parameters (Moons and Moor, 1995), and

to estimate the coefficients of linear transfer function basedon measurements of the magnetic force and speed (Gahler and

Herzog, 1994; Lee et al., 1994), etc. Here, a non-linear iden-

tification technique, NARMAX (Non-linear Auto Regressive

Moving Average model with eXogenous inputs), is employed

(Leontaritis and Billings, 1985) to model the relation between

the speed and voltages of an induction motor.

Our research on developing an identification technique

for estimating the speed of an induction motor was divided

into two steps. The objectives of the first step included: (1)

choosing the proper parameters of the motor as inputs, cor-

responding to the output, i.e. speed, by analyzing the governing

equation of the motor, (2) designing and constructing an

induction motor system in order to obtain input/output data,

(3) modeling a NARMAX equation using designated input/

output data, and (4) validating the NARMAX model. The

second step involved designing a robust controller for con-

trolling the speed by adjusting the inputs. This was done by

transforming a non-linear, difference and polynomial NAR-

MAX model in the time domain into the frequency response

in the frequency domain using an FRF technique.

This paper will focus on the attained objectives in the

first step and will discuss the research, traditional control of

speed and sensorless drive motors, and the application of the

NARMAX model to speed estimation. The procedure for

constructing and validating a NARMAX model of the speedand voltages of a motor is demonstrated through an experi-

mental case study. The results of the NARMAX model shown

in this paper show that it can replace the speed sensor, such

as tachometer or encoder in a closed loop speed control motor.

Based on the NARMAX model, an innovative speed control

algorithm for induction motors will be presented in the near

future when the robust controller design is completed. Before

this, the motor speed is still controlled by the traditional way.

II. NARMAX Method

Successful system identification requires correct model-ing. In this paper, a NARMAX modeling of identification

is proposed. For a non-linear system, representing the current

output by mapping the previous input, outputs and prediction

error can be done precisely and efficiently using a NARMAX

model.

1. NARMAX Model

A wide range of discrete time multiple variable non-

linear stochastic systems can be represented by the following

NARMAX model:

y(k) = + Fl[y(k1), ...,y(ky), u(k), ..., u(ku),

(k1), ..., (k)] + (k), (1)

wherey(k), u(k) and (k) represent the system output, input,

and prediction error, respectively. Also, l is the degree of non-linearity, is a constant dc level, Fl[.] is some vector valuednon-linear function, and u, y and represent the numberof lags in the input, output and prediction error, respectively.

The prediction error term (k), defined as (k) = y(k) (k) ,is included in the model to accommodate noise, where (k)

is the prediction output. Expanding Eq. (1) by defining the

function Fl[.] as a polynomial of degree l gives a representation

of all the possible combinations ofy(k), u(k) and (k) up todegree l. For example, the current output can be presented

as

y(k) = + 1y(k1) +

2u(k1) +

3u(k1)y(k1)

+ 4u(k1)(k1) + 5(k1) + (k),

by definingp1(k) = y(k1),p2(k) = u(k1),p3(k) = u(k1)y.(k1),p4(k) = u(k1)(k1),p5(k) = (k1),p0(k) = 1, and0 = . IfNinput and output measurements are available,and if there areMterms in the model, then the above equation

can be written in a matrix form as

Y=p+, (2)

where

YT = [y(1)y(2) ...y(N)]

T = [01 ... M]

T = [(1) (2) ... (N)]

=

p 0(1) p 1(1) pM(1)

p 0(2) p 1(2) pM(2)

. . . .

. . . .

. . . .

p 0(N) p 1(N) pM(N)

,

where p represents a term in the NARMAX model, and represents unknown parameters to be estimated. The param-

eter vector in Eq. (2) can be estimated using some well-known methods, such as a least-squares-based or prediction

error method, Choleski or U-D factorization, the Q-R algorithm,

singular value decomposition or principle component regression.

The present study employed an orthogonal estimator algorithm

to conduct parameter estimation (Korenberg et al., 1998; Bil-

lings and Leontaritis, 1981, 1982).

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2. Orthogonal Parameter Estimation

The orthogonal algorithm estimates the parameters

by transforming Eq. (2) into an equivalent auxiliary model:

(k) = g iw i(k) + (k)

i = 1

n

, k= 1, 2, ...,N, n=M, (3)

where wi(k) are constructed so as to be orthogonal over the

data record and gi represents constant coefficients. The pa-

rameters can be estimated by computing one parameter es-

timate at a time because of the orthogonal property. Conse-

quently, a family of orthogonal vectors can be constructed over

the given data records as

w1(k) = p1(k); wj(k) =pj(k) ijw i(k)i = 1

j 1

, (4)

where

ij =w i(k)pj(k)

k= 1

N

w i2(k)

k= 1

N

for j = 1, 2, ..., n; i = 1, 2, ..., j 1, j,

and in this case the orthogonality property holds, i.e.,

wi(k)wj(k) = 0, i j. (5)

The second step consists of estimating the coefficients gi and

transforming them back into the system parameters i. Theparameters gi in the auxiliary model are given by

g i =w i(k)y(k)

k= 1

N

w i2(k)

k= 1

N. (6)

Therefore, the original unknown system parameters can be

obtained from g i according to the following formulas:

n = g n

i = g i ijj

j = i + 1

n

, i = n1, n2, ..., 1. (7)

The auxiliary regressors wi(k) are orthogonal, so additional

terms can be added to the model without computing all the

previous gj ,j < i. Therefore, the orthogonal parameter es-

timation algorithm is very simple and easy to implement.

3. Structure Selection

Determining a simple polynomial model is vital. A

parsimonious representation of a non-linearity is desirable.

This means that the representation of the polynomial terms

must be as simple as possible. The selected model degree

l should not be too low; otherwise, none of the dynamics ofthe system will be described properly. On the other hand,

it should not be too high since the higher the model order,

the more parameters will have to be estimated, and this may

lead to numerical problems and over-fitting. There are several

possible ways to determine which terms are significant and

should be included in the model. An alternative and much

simpler method can be derived as a by-product of estimation

by using the error reduction ratio (ERR) to select the relevant

terms (Billings and Voon, 1983, 1986). To demonstrate

structure selection usingERR, Eq. (3) is multiplied by itself,

yielding

2(k) = g i2w i

2(k) + 2(k)i = 1

n

, (8)

where (k) is assumed to be a zero mean white noise sequencewhich is not correlated with the input and output data records.

The mean-squared prediction error will be the maximum error

when no terms are included in the model, that is n= np +

n= 0. As a result, we have [(k)2]n= 0 =y

2(k). Equation

(8) shows that the reduction in the mean squared error, achieved

by including the ith term, giwi(k), in the auxiliary model of

Eq. (3), is g i2w i

2(k). Expressing this quantity as a fraction of

the total mean squared error yields theERR for the ith term

as

ERRi =g i

2w i

2(k)

y 2(k) 100%=

g i2

w i2(k)

k= 1

N

y 2(k)k= 1

N 100% ,

for 1 i n. (9)

TheERRi value can be computed together with the parameter

estimates to indicate the significance of each term, and then

the terms can be ranked according to their contributions to

the overall mean-squared prediction error. Insignificant

terms can be discarded from the model structure by defin-ing a threshold value ofERRi. Terms are considered to

contribute negligible reduction to the mean-squared prediction

error when theirERRi values are smaller than the threshold

value. The process of selecting terms is continued until the

sum of the error reduction ratio reaches a value close to

100%.

4. Model Validation

Once the significant terms have been identified and the

estimates of associated parameters have been obtained, the

model can be constructed. It is important to know whether

the model has successfully captured all the system dynamics.

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Therefore, a strategy for evaluating the correctness and validity

of the model is necessary. If validation shows that the model

is not good, then some of the design variables should be

changed, and the identification procedure should be redone.

Two model validation strategies, one using the model predicted

output and the other using a model validity test, will be

discussed in the following.

A. Model Predicted Output (MPO)

The model predicted output is defined as

mpo(k) = F

l[y(k y) , ,y(k 1) , u(k u) , ,

u(k), 0 , 0] , (10)

where the measured input is used to generate the model output.For the model to be accepted, it is essential that the estimations

of the model predicted outputs be in good agreement with the

measured output.

B. Model Validity Tests

For a non-linear system, the residuals usually can not

be predicted from the linear and nonlinear correlation of past

inputs and outputs. This will be true if the following correlation

tests are passed (Billings and Chen, 1989):

() = (); u2'2'() = 0; u() = 0; u() = 0;

u2'() = 0, (11)

where ab() = E[a(k)b(k)], () is the Kronecker deltafunction, u() and () are the input and residual sequences,respectively, and the prime indicates the quantities with themean removed.

III. Induction Motor Modeling Using Vec-tor Control Theory

By choosing the stator current of the - axis, irs andis, the rotor flux of the axis,r, as state variables, and treating

the stator voltages, v s and v s , as inputs, a dynamic model

for an induction motor in a fixed reference axis can be obtained

as follows (Krause, 1987; Lorenz and Laeson, 1990; Miki et

al., 1991):

p

i s

i s

r

=

R s

L sR r(1 )

L s0

LmR r

L sL r2

0 R s

L s0

LmR r

L r0

R r

L r

i s

i s

r

+1

L s

v s

v s

0

, (12)

rm = (

1pJm +Bm

32P2

Lm

L r)i sr (13)

(no external load assumed), where

is, is : --axisstator currents;

Rr,Rs : rotor, stator resistance;

Lm : magnetized inductance;

rm : mechanic angle speed;

P : poles of induction motor;

r, r : --axis rotor flux;

Ls,Lr : stator, rotor inductance;

: total leakage factor;

vs, vs : --axis stator voltages;

s : derivative operator;

Jm,Bm : mechanical inertia moment, friction coefficient;

v s = vs +L si s +ML r

rr;

v s = v s L si s .

The objective of vector control is to decouple the and -axis terms in order to simplify control of the induction motor;

otherwise, the coupled terms will spoil the simplicity and

Fig. 1. Vector control diagram of the induction motor.

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controllability by causing disturbance. Equation (13) shows

that the speed is linear to the stator current, is, if the rotor

flux coincides with the axis and stays at a fixed rating value.This means that is can dominate the dynamics of the motor

linearly, just as in DC motors, as shown in Fig. 1. The stator

currents, is

and is

, and the rotor flux, r

, are controlled by

three designed controllers, as shown in Fig. 2. The rotor flux

command keeps a fixed value to ensure linearity between rmandis, as shown in the lower half of in Fig. 2. The measurement

Eq. (13) will obviously be non-linear ifr does not stayproperly fixed.

IV. Identification of an Induction MotorUsing the NARMAX Approach

Figure 2 shows that three controllers are needed to

accomplish speed control. Figure 3 shows the identification

approach using the NARMAX model. The feasibility of the

proposed approach has been verified through an experimentalstudy, and the performance will be verified as discussed in

a previous section. The experimental setup and schematic

diagram are shown in Fig. 4(a) and (b), respectively. The

experimental apparatus consists of a personal computer con-

nected to an induction motor with the physical parameters

listed in Table 1, to some input/output and power unit interfaces.

Software written in the turbo-C language activates the system.

The commands are input using a keyboard, and the data is

read from a monitor.

The voltages vs and v s are treated as inputs and the

speed rm as output in Eqs. (12) and (13). The data of 100sets ofvs , v s and rm are available to model and verify.The parameters l, y, u and in Eq. (1) of the NARMAXmodel need to be determined first. Ideally, l equals 1 for a

linear model and equals an integer larger than 1 for a non-

Fig. 2. Vector control diagram of the induction motor with controllers.

Table 1. Physical Parameters of the Motor

Parameter Rs Rr Ls M P Bm Jm

Value 1.0977 0.6667 0.0487H 0.0463H 4 0.009 0.009 Kg-m2

Fig. 3. Diagram of NARMAX identification.

(a)

(b)

Fig. 4. (a) The experimental setup for the induction motor. (b) Schematic

diagram of the experimental induction motor.

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linear one. The other parameters have to be positive integers

and must be as small as possible. Actually, all the parameters

for the NARMAX model must represent the systematic behavior

very well. After trying several times (a detailed discussion will

be provided in the next section), the result of the NARMAX

model for the data set employed using the orthogonal esti-

mation andERR ranking ofl = 2, y = 9, u1 = 8, u2 = 14,= 9, is

y(k) = 0.21u(1,k1)u(2,k1)+0.09u(1,k2)u(2,k1)

+0.038u(1,k3)u(2,k1)+0.016u(1,k4)u(2,k1)

+0.007u(1,k5)u(2,k1)+0.003u(1,k6)u(2,k1)

+0.001u(1,k7)u(2,k1)+0.014u(2,k1)0.14

(k9)+0.12(k2),

where u(1, ka) is the input v s of lag a, and u(2, kb) is the

input v s of lag b. The result was obtained through the fol-

lowing steps:

(1) Assume the prediction errors are zeros and estimate all

the parameters that do not include .(2) Estimate the prediction error, =y .(3) Estimate all the parameters including the prediction

error, .(4) Go to (2) and continue to be converged. This depends

on the parameter change after iteration.

(5) Determine the final parameters, .

A comparison of the predicted outputs obtained using

the theoretical system and model is shown in Fig. 5(a). They

are almost identical because of the large scale of the y-axis

in relation to the wide speed range. The prediction error be-

tween the system output and model predicted output, shown

in Fig. 5(b), is almost negligible. This means that the output

predicted by the model agrees with the output predicted by

the system. The correlation tests are implemented and illus-

trated in the system very well. The correlation tests are imple-

mented and illustrated in Fig. 6. It shows the correlation be-

tween the error and two inputs (time step versus magnitude)

Fig. 5. (a) System output and model predicted output. (b) Difference between

the outputs predicted using the system and model.

Fig. 6. Model validity test results obtained in the experimental study. (the

upper left figure is first then right, and the lower figure is next,

according to the order , u1, u2,

2

u1,

2

u2,

u1

2'

, (u1u2)',

u22', (u1u2)'2, (u1u1)'2, (u2u2)'2)

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and the auto-correlation between errors. The dash lines for

the upper and lower bounds imply a deviation band of 5%.

This shows that the model validity test results are all inside

the 95% confidence band and indicates that the fitted modelis almost unbiased and has correctly captured the system

dynamics. The excellent output predicted by the model and

validity test results reveal that the multi-input single output

(MISO) second degree NARMAX model can sufficiently

represent the speed dynamics of an induction motor.

V. Discussion and Conclusions

In Section IV, we presented the parameters of the

NARMAX model for the induction motor assigned to the

system. How can reasonable parameters for the NARMAX

model, l, y

, u

and , be assigned in physical applications?

The non-linear order l depends on the non-linearity of the

strong or weak intensity, and the values ofy,u anddependon the available input, output and error. The other requirement

for model parameter assignment is numerical accuracy. Simply

increasing the number of parameters in the polynomial ex-

pansion to achieve the desired prediction accuracy will, in

general, result in an excessively complex model and possibly

in numerical ill-conditioning. Simulation has shown that,

usually, less than ten key terms in the NARMAX model are

dominant, and that the remainder can be deleted with little

distortion subsequent in the prediction accuracy of the model

(Billings and Fadzil, 1985). Here, in this motor system, if

Eq. (12) is linear, and the measurement Eq. (13) is non-linearwith two states multiplied by each other, then the non-linear

order assigned to l = 2 is appropriate. If the available input

and output measurements are sufficient for a NARMAX

model, then the key terms can be determined through numeri-

cal constraint withERR ranking.

The popular sensorless drive motor has been realized

by applying the estimation theory to the governing equation

in the state space. The estimation theory includes least-squares,

maximum likelihood, Kalman filtering etc. The governing

equation is corrupted by noise, as mentioned in Section I, and

the speed estimation obtained using the estimation theory also

deteriorates. The proposed method using the NARMAXmodel to estimate the motor speed constructs the governing

equation based on experimental data, including the effect of

noise. The prediction error in the NARMAX model is defined

as the difference between the real and model output, and this

term takes into consideration noise in the sense of its physical

meaning. The mathematical NARMAX model can represent

a physical motor accurately to estimate the speed.

A robust controller design is the final goal of this research.

Glass and Franchek (1999) gives a promise of success of the

robust control using a describing function representation and

loop shaping approach of a non-linear model. In other words,

once the NARMAX model has been developed, the model

is mapped into a describing function so that it can be used

to design controllers in frequency domain, and the design

satisfies a pre-specified output tolerance in time domain.

There are three major disadvantages in applying the

vector control theory to induction motors to control speed:(1) For the purpose of speed control, three controllers are

employed to keep the rotor flux and to control the stator

current according to the speed. As a result, a higher

drive cost and more complex structure are added.

(2) Rotor flux is hard to be kept at a steady value by a

controller due to noise. Therefore, the linearity of the

vector control destroys.

(3) At low speed, the current and flux have low values and

are easily disturbed. The behavior of an induction motor

at low speed deviates greatly from expectations.

Therefore, the designs of the controllers, the predictions of

the electric parameters and the responses of the whore system

are inaccurate. Consequently, the performance of the motor

deteriorates.

For the sake of improving the disadvantages of the

sensorless control motor, an identification strategy has been

presented here in which voltages and speed are used as inputs

and outputs to create a model. Modeling by means of data

collection over a wide range of speed improves the accuracy,

reduces the complexity, increases the immunity to noise, and

fewer controllers are needed. The main contribution of this

paper has been to extend the NARMAX methodology to the

application of speed estimation for induction motors. The

results obtained here have demonstrated that the estimated

model can sufficiently capture all the system dynamics overthe desired operating range. In practical applications, the

number of times that the speed sensor is measured is the lag

assigned by the parameter in the NARMAX model, i.e., y= 9 in this experimental study, so the sensor is discarded. In

other words, the sensor is employed only at the beginning or

before the beginning, and the NARMAX model with a

measurement of motor voltages can be used to estimate the

remaining speeds. After all, voltage measurement is easier

than speed measurement. Furthermore, for a black-box motor

system or a more complicated one, not much a priori infor-

mation is available, but a NARMAX model can represent the

system by simply starting with the needed inputs and outputs.A design of a closed loop speed controller based on the

NARMAX model is being studied now, and a PC-based

controller can possibly be used to integrate the whole operating

process. This paper has discussed the value of the model based

on validation test results. The NARMAX model approach has

been shown capable of accurately estimating the speed through

non-linear identification. Eventually, a robust controller will

be designed and will be presented in a forthcoming paper.

Acknowledgment

The authors would like to thank the National Science Council,

R.O.C., for financially supporting this research under contract NSC 87-2218-E-014-005.

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