Control of High Speed Linear Induction Motor Using

Artificial Neural Networks

Mohammadali Abbasian, Jafar Soltani, and Ali Salarvand

Engineering Department, Islamic Azad University- Khorasgan Branch- Isfahan, Iran

a_abbasian@ec.iut.ac.ir, salarvand@lh.elec.ui.ac.ir

Abstract- This paper presents a controller for speed and flux control of a high speed Linear Induction Motor (LIM) Drive considering end effect. The backstepping control and Artificial Neural Networks (ANN ) are combined in order to design a robust controller that is capable of preserving the drive system robustness subject to all parameter variations and uncertainties. The overall system stability is proved by Lyaponuv theory. The effectiveness and validity of the proposed controller is supported by computer simulation results.

I. INTRODUCTION

The Linear Induction Motor (LIM) has excellent

performance features such as high-starting thrust force,

alleviation of gear between motor and the motion devices,

reduction of mechanical losses and the size of motion devices,

high speed operation, silence, and so on [1]. Because of above

the advantages, the primary type LIM, shown in Fig. 1, has

been widely used in the field of industrial processes and

transportation applications. The driving principles of the LIM

are similar to the traditional Rotary type Induction Motor

(RIM), but its control characteristics are more complicated

than the RIM, and the motor parameters are time-varying due

to the change of operating conditions, such as speed mover,

temperature, and configuration of rail [2].

In a LIM, the primary winding corresponds to the stator

winding of a rotary induction motor (RIM), while the

secondary corresponds to the rotor. There are some

characteristics differences between the RIM and LIM. The

main difference is that the primary of the LIM has a finite

length, and therefore, there is a fringing field at both ends of

the primary. The infinitely long secondary enters the air-gap

field, carries the magnetic flux along with it, and makes, the

distribution of the electromagnetic quantities nonuniform,

resulting in considerable electric and force losses [3]. The

losses, as well as the flux-profile attenuation, become severer

as the speed increases. Such a phenomena is called end effect

of LIM.

An accurate equivalent circuit model is indispensable for

high performance control for LIM drive. Most of the existing

models of a LIM depend on field theory [4,5]. Hence, they can

not be directly applied for the nonlinear control and in the most

of the present studies, the RIM model is used in order to

control LIM [6-8]. so, because of the end effect, they can not

be valid in high speed operation of LIM. In [6], an adaptive

controller is used in order to control LIM at low speeds. This

controller is robust only against the mechanical parameter

uncertainties. In [7], an indirect field oriented control is used to

control LIM at low speed which is not robust in respect to all

machine parameter variations and uncertainties. In [8], a robust

nonlinear controller has been proposed for a primary type LIM

which is based on combination of SM control and input-output

feedback linearization. Using the RIM model, the drive system

of [8] is robust to parameters variation only at low speed

operation. Using the control method presented in [8], the

main objective of this paper is to introduce a robust nonlinear

controller for LIM which will be robust and stable subject to

the parameter uncertainties and external unknown load force at

low and high speed operation. Reference [9] has described a

secondary field-oriented control scheme for LIM with the end

effect, but the robustness is not considered.

II. LIM MODEL

The LIM fifth order model with end effect, in stationary d:q

axis reference frame with secondary fluxes and primary

currents as state variable can be expressed by[9]:

1qr mf

qs p e qr qr

r r

d Li n V

dt T h T

= + (1)

1dr mf

ds dr p e qr

r r

d Li n V

dt T T h

= (2)

1( ) ( )

1

ds s mfds dr

s r S s r r S

p mfe qr ds

s r s

di R Li

dt L T L L L T L

n LV v

L L h L

= + + +

+ +

& &

Fig. 1. Linear Induction Motor.

Primary(Mover)

Secondary

(3)

HSI 2008 Krakow, Poland, May 25-27, 2008

1( ) ( )

1

qs s mfqs qr

s r S s r r S

p mfe dr qs

s r s

di R Li

dt L T L L L T L

n LV v

L L h L

= + + +

+

& &

3( )

2

p ee dr qs qr ds e l

n dVF i i M DV F

h dt

= = + + (5)

where , , , , , ds qs dr qr ds qsi i v v and eV are the primary d-q currents, secondary d-q fluxes, primary d-q voltages and

mover speed, respectively. In addition, sR is the primary

resistance, rR is the secondary resistance, pn is the number

of pole pairs, h is the pole pitch, lF is the external force

disturbance, M is the total mass of mover and D is the

viscous friction coefficient, r r rT L R= is the secondary

time constant, s mf lsL L L= + is the primary inductance,

r mf lrL L L= + is the secondary inductance, lsL is the primary leakage inductance, lrL is the secondary leakage

inductance, 21 , , mf s r mf r sL L L L L L = = = . mfL is a time variant parameter as a function of eV given

by [9]:

(1 ( ))mf mL L f Q= (6)

where 1.5m moL L= and moL is the magnetizing inductance

at zero speed, and:

1

( ) , ( )

Qr

m lr e

lRef Q Q

Q L L V

= =

+ (7)

and l is the primary length.

In equations 1 to 5, & and & are functions of other system states and can be written as follows:

2

2

2( ){1 }mf r mfm

r

L L LL f Q

L = && (8)

2

( ) lrmr

LL f Q

L = && (9)

where ( )f Q& is also a function of the system states as:

( ) (1 )

3 { ( ) }

2

Q Qmo lr

r

pdr qs qr ds e l

L Lf Q e Qe

MlR

ni i DV F

h

+=

&

III. ROBUST BACKSTEPPING CONTROLER

A. ANN Basics

Define W as the controller of ANN weights, then the net

output is [10]:

( )Ty W x= (11)

Let S be a compact simply connected set of n , with map

:nf S , define ( )mC s the functional space such that

( ) ( )mf x C s , ( )x t S can be approximated by a neural

network as:

( ) ( ) ( )Tf x W x x = + (12)

With ( )x a ANN functional reconstruction error vector and ( )x is sigmoid activation function. B. Robust Backstepping Control of LIM Using ANN

Using the well known fifth order LIM model in a stationary

two axis reference frame where the secondary fluxes and

primary currents are assumed as state variables, the robust

nonlinear controller is designed in the following way:

Dividing the above LIM model into two nonlinear

subsystems, where , ds qsi i are the outputs for the first

subsystem which are simultaneously assumed the fictitious

input of the second sub-system.

Assume that:

1. The reference trajectories eV and *r are differentiable

and bounded.

2.the Load force is an unknown constant and resistances,

inductances and moment of inertia are unknown and bounded.

In the first step of the controller design, , ds qsi i are

assumed as fictions controls for the second sub-system. The

main objective is to obtain these controls so that the desired

secondary speed and amplitude signals are perfectly tracked in

spite of machine parameters and external load force

uncertainties. Considering eV and *r , the tracking error

equations are obtained as:

1

2 2 2 2 22

( )

e e

dr qr r r r

e V V

e

=

= + = (13)

Derivating (13) with respect to time (t), and using equations

(1), (2) and (5) yields:

1

2

3( )

2

2

2 2

p ldr qs qr ds e

r rdr dr mf ds

r r

r rqr qr mf qs r r

r r

nde Fi i V

dt hM M

de R RL i

dt L L

R RL i

L L

=

= + + +

&

&

(14)

(4)

(10)

416

or:

1 1 1D e F G i= +& (15)

with:

12

1

11

2

, 2 2

3 3

, 2 2

2 2

0

, , 1

0

le

r r rm r

p p

qr dr

dr qr

ds

qs

r

F MV

F

L R

n n

G h h

M

i eD i e

i e

R

=

= = = =

&

&

(16)

From (16) it is clear that 1G is known and invertible matrix.

By treating i as a fictitious input, a controller for the ideal i is designed as:

11 1 1 1[ ] , 0i G F K e K= > (17)

where 1K a design parameter and 1F the estimate of 1F which

will be estimated in the next section with a two layer ANN.

Substituting (16) into (15) gives:

1 1 1 1 1D e F F K e G = +& , i i

= (18)

In the second step, the control v is obtained in such a way

that in Equation (17), becomes as small as possible. Differentiating with respect to time yields:

2 2 2D F G v = +& (19) where:

2 2

1 0 1 0, ,

0 1 0 1

ds

sqs

vv G D L

v

= = =

(20)

2

2 2

2

( )

( )

p mfmf rdr e qr

s s rs r

p mfmf rqr e dr

s s rs r

n LL RV

L L L hL LF D

n LL RV

L L L hL L

+

=

&

&

2 2

2

2 2

2

mf r r sds

ss r

mf r r sqs

ss r

L R L Ri

LL L

L R L Ri

LL L

+ + + +

&

&

()

1 12 1 1 1 2 1 1

1 12 1 1 1 1 1 1 1

( ) +

( )

D G F K e D G F

D G K D F F K e G

+ + +

+

&

&

To make as small as possible, the following control is chosen :

12 2 2 1[ ]Tv G F K G e= (22)

In (21), 2F is an estimate of 2F that is also estimated by a

second two layer ANN. In addition a term 1TG e is added in

(21) which is necessary to cancel the effect of 1G in (17). Combining (16) and (18), gives:

2 2 2 2 1 TD F F K G e = & (23)

C. 1 2,F F Approximation Using ANN

In this section, functions 1 2,F F are approximated by two

two-layer ANN. In. Using ANNs approximation property,

1 2,F F as outputs of two two-layer ANN with constant weights,

iW , is assumed to be as follows:

1 1 1 1 1 1

2 2 2 2 2 2

,

,

TN

TN

F W cte

F W cte

= + < =

= + < = (24)

where 1 2, provide suitable basis functions. From (23), one can find that net reconstruction error ( )i x is bounded by a

known constant iN .

Assumption 3: The ideal weighs are bounded by known

positive values so that:

1 1 2 2 ,M MF FW W W W (25)

or equivalently:

{ }1 2, ,MFZ Z Z diag W W = (26)

The actual inputs to ANN1 are , , ,r e r eV V & & and actual inputs to ANN2 are 1 2, , , , , , , , , .r e r e dr qr ds qsV V i i e e & &

(21)

417

Online ANN approximation of 1F is defined by:

1 1 1 TF W = (27)

Linking (18) and (2), gives:

1 1 1 1 1 1TD e W K e G = + +%& (28)

where 1 1 1W W W= % . Similarly, approximation of 2F is:

2 2 2 TF W = (29)

Then error dynamic (22) will be:

2 2 2 2 1 2T TD W K G e = +%& (30)

Note that there is a term 1G in (27) and a term 1TG e in (29). This means there are couplings between the error

dynamics (27) and (29).

D. Updating ANNs Weights

In this part, the stability of proposed controller, is proved

based on Lyapunov stability theory. This analysis shows that

tracking errors and updated weighs are Uniformly Ultimately

Bounded (UUB).

Theory : Let the desired trajectories * *,e rV be bounded. Take the control input (21) with weigh updates be provided by:

1 1 1 1 1

2 2 2 2 2

T

T

W e k W

W e k W

=

=

&

&

(31)

with any constant matrices 1 1 2 20, 0T T = > = > and

scalar positive constant k . Then the errors ( ), ( )t e t are UUB. ANN updated weights are bounded. The errors ( ), ( )t e t can be kept as small as desired by increasing gains 1 2,K K .

Proof: Consider the following Lyapunov candidate:

1 12

01 1( )

02 2

T TDV tr Z ZD

= + % % (32)

where:

{ } { }1 2 1 2, , ,Z diag W W diag= = % % % (33) { }1 2[ , ] , ,T T Te K diag K K = = (34)

Differentiate (32):

{ }( )T T TV K k e tr Z Z Z = + +& % % (35)

Now, using Schwarz inequality [10]:

{ } 2( )T FF Ftr Z Z Z Z Z Z % % % % (36)

and having in mind some basics inequalities, from (35) the

following can be obtained:

2

min

min

( )

( )

M NF F

M NF F

V k Z Z Z

k Z Z Z

+ +

= +

& % %

% %

(37)

where min is the minimum singular value of K .The right-hand side of (37) is negative as long the term inside braces is

positive. Completing the square yields:

min

2 2min

( )

( / 2) / 4

M NF F

M M NF

k Z Z Z

k Z Z k Z

+ =

+

% %

%

(38)

which is guaranteed positive if

2 min[ / 4 ] /M Nk Z > + (39) 2/ 2 / 4 /M M N

FZ Z Z k> + +% (40)

Thus, it can be concluded that and are UUB [10] and the

system is stable.

Note 1: Small tracking error bounds may be achieved by

selecting large control gain K . The parameter k offers a

design tradeoff between the relative eventual magnitudes of

and F

Z% , a smaller k yields a smaller and a larger

FZ% , and vice versa.

Note 2 : If (0)iW are taken as zeroes the linear proportional

control term K stabilizes the system on an interim basis.

IV. SYSTEM SIMULATION

Based on the proposed control strategy in this paper, the

block diagram of LIM drive control is shown in Fig. 2.

A C++ computer program was developed for system

simulation. In this program, the nonlinear equations are solved

based on static forth order Range-Kutta method. The proposed

control method, is tested by simulation for a three-phase LIM

with parameters shown in Table (1). In this simulation, the controller gains are obtained by trial

and error and are given as

1 2{1000,1000} , {2000,1000}

1 , 10i

K diag K diag

k I

= =

= = (41)

418

Table 1 : LIM PARAMETERS

5.36sR = Primary resistance 3.53rR = Secondary resistance

36.08 / secD Kg= Viscous friction coefficient

1pn = Number of pole pairs

0.0029sL H= Primary leakage inductance

0.0029rL H= Secondary leakage inductance

0 0.0681mL H= Magnetizing Inductance at zero speed

0.027h m= Pole pitch

2.78M kg= Total Mass of moving element

Simulation results shown in Fig. 3 (3a to 3g) , are obtained

in the case of an exponential reference flux rising up from zero

to 0.1 Wb t at 0st = with a time constant of 0.1sec = , an exponential reference speed from zero to 1m/sec at 0sect = ,

a step load disturbance from zero to 5Nm at 0.5sect = , step

down to 3Nm at 1sect = and motor electromechanical

parameters assumed to be twice their nominal value.

Fig. 4 (4a to 4g) shows the simulation results obtained for an

exponential reference flux rising up from zero to 0.1 Wb t at

0sect = and a sinusoidal reference speed. In flux and speed

control performance, motor electromechanical parameters

assumed to be twice their nominal values.

Referring to Fig. 3f and 4.f, it is clear that the maximum of

( )f Q function is 0.16. This means that the motor magnetizing

inductance reduces to 84 percent its original value, when motor

moves at the spped of 1m/s.

Fig. 2. The overall block diagram of LIM drive system control

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 3a. Secondary flux

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

Fig. 3b. Mover speed

0 0.5 1 1.5 2-40

-30

-20

-10

0

10

20

30

40

Fig. 3c. d-axis current

0 0.5 1 1.5 2-40

-30

-20

-10

0

10

20

30

40

Fig. 3d. q-axis current

,r eV

(sec)t

(sec)t

( )dsi A

(sec)t

(sec)t

LIM

CONTROLLER

CONTROLLER

ANN1

ANN2

PWM

INVERTER1

TG

1F

2F

eV

eV

( / )eV m s

rr

( )r Wb

( )qsi A

419

0 0.5 1 1.5 2-800

-600

-400

-200

0

200

400

600

800

Figure 3e. Inverter's output voltage

0 0.5 1 1.5 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Figure 3f. f(Q) function

0 0.5 1 1.5 20

1

2

3

4

5

6

7

Figure 3g. External force disturbance

0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

0.12

Figure 4a. Secondary flux

0 0.5 1 1.5

-1

-0.5

0

0.5

1

Fig. 4b. Mover speed

0 0.5 1 1.5-40

-30

-20

-10

0

10

20

30

40

Fig. 4c. d-axis current

0 0.5 1 1.5-40

-30

-20

-10

0

10

20

30

40

Fig. 4d. q-axis current

0 0.5 1 1.5-600

-400

-200

0

200

400

600

Fig. 4e. Inverter's output voltage

( )f Q

( )r Wb t

(sec)t

( )ds Ai

(sec)t

( )qsi A

(sec)t

(sec)t

( . )lF N m

(sec)t

( )asv V

*r

r

eV

eV

( / )eV m s

( )asv V

(sec)t

(sec)t

(sec)t

420

0 0.5 1 1.50

0.05

0.1

0.15

0.2

Fig. 4f. f(Q) function

0 0.5 1 1.5 20

1

2

3

4

5

6

7

Fig. 4g. External force disturbance

CONCLUSIONS

In this paper, a composite nonlinear controller has been

proposed for the LIM secondary flux and speed tracking

control. The nonlinear controller is designed based on the LIM

fifth order model in a fixed two axis reference frame,

combining the backstepping control and ANN. The overall

stability of this controller is proved by Lyapunov theory.

Computer simulation results obtained, confirm the

effectiveness and validity of the proposed controller. These

results also confirm that the drive system control is robust and

stable against the parameters uncertainties and unknown load

force disturbance.

REFERENCES

[1] I. Takahashi, and Y. Ide, Decoupling control of thrust and attractive

force of LIM using a space vector control inverter, IEEE Trans. Ind.

Appl., vol.29, No. 1, pp.161-167, 1993.

[2] G. H. Abdou and S.A. Sherif, Theoretical and experimental design of LIM in automated manufacturing systems, IEEE Trans. Ind. Applicant.., Vol. 27, pp. 286-293, Mar/Apr 1991.

[3] Jan Jamali, End Effect in Linear Induction and Rotating Electrical Machines, IEEE Transaction on Energy Conversion, Vol. 18, No. 3, September 2003.

[4] T.A. Nondahl and D.W. Novotny, Three-phase pole-by-pole model of a linear induction machine, Proc. IEE, Vol. 127, Pt. B, No. 2, pp. 68-82, 1980.

[5] J.F. Gieras, G.E. Dawson and A. R. Eastham, A New Longitudinal end effect factor for Linear Induction Motors, IEEE. Trans. On Magnetics, Vol. EC-2, No. 1, pp. 152-159, 1987.

[6] F.-J. Lin and C.-C. Lee, Adaptive backstepping sliding mode control for linear induction motor drive to track periodic refrences, IEE Proc.-Electr. Power Appl., Vol. 147, No. 6, November 2000.

[7] F.-J. Lin and P.-H. Shen, Adaptive backstepping sliding mode control for linear induction motor, IEE Proc.-Electr. Power Appl., Vol. 149, No. 3, May 2002.

[8] R.-J. Wai and W.-K. Liu, Nonlinear decoupled control for linear induction motor servo-drive using the sliding-mode technique, IEE Proc.-Control Theory Appl., Vol. 148, No. 3, May 2001.

[9] G. Kang and K. Nam, Field-oriented control scheme for linear induction with the end effect, IEE Proc.-Electr. Power Appl., Vol. 152, No. 6, November 2005.

[10] C. M. Kwan, F. L. Lewis, Robust Backstepping Control of Nonlinear Systems Using Neural Networks, IEEE Trans. Systems, Man and Cybernetics, vol. 30, No. 6, Nov. 2000.

( . )lF N m

(sec)t

( )f Q

(sec)t

421

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