Mover Position Control of Linear Induction Motor Drive ... Position Control of Linear Induction...

Mover Position Control of Linear Induction Motor

Drive Using Adaptive Backstepping Controller with

Integral Action

I. K. Bousserhane1*, A. Hazzab1, M. Rahli2, B. Mazari1 and M. Kamli2

1University center of Bechar, B.P 417 Bechar 08000, Algeria2University of Sciences and Technology of Oran, Algeria

Abstract

In this paper, an adaptive backstepping control system with a fuzzy integral action is proposed

to control the mover position of a linear induction motor (LIM) drive. First, the indirect field oriented

control LIM is derived. Then, an integral backstepping design for indirect field oriented control of

LIM is proposed to compensate the uncertainties which occur in the control. Finally, the adaptive

backstepping controller with fuzzy integral action is investigated where a simple fuzzy inference

mechanism is used to achieve the mover position tracking objective under the mechanical parameters

uncertainties. The performance of the proposed control scheme was demonstrated through

simulations. The numerical validation results of the proposed scheme have presented good

performances compared to the adaptive backstepping control with integral action.

Key Words: Linear Induction Motor, Field-Oriented Control, Adaptive Backstepping, Fuzzy

Integral-Backstepping

1. Introduction

Nowadays, Linear induction motors LIM’s are now

widely used, in many industrial applications including

transportation, conveyor systems, actuators, material

handling, pumping of liquid metal, and sliding door

closers, etc. with satisfactory performance [1]. The most

obvious advantage of linear motor is that it has no gears

and requires no mechanical rotary-to-linear converters.

The linear electric motors can be classified into the fol-

lowing: direct current (DC) motors, induction motors,

synchronous motors and stepping motors, etc. Among

these, the LIM has many advantages 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 [1,2]. The driving principles of the

LIM are similar to the traditional rotary induction motor

(RIM), but its control characteristics are more compli-

cated than the RIM, and the motor parameters are time

varying due to the change of operating conditions, such

as speed of mover, temperature, and configuration of rail.

Field-oriented control (FOC) or vector control [1�3]

of induction machine achieved decoupled torque and flux

dynamics leading to independent control of the torque and

flux as for a separately excited DC motor. This control

strategy can provide the same performance as achieved

from a separately excited DC machine. This technique can

be performed by two basic methods: direct field-oriented

control (DFO) and indirect field-oriented control (IFO).

Both DFO and IFO solutions have been implemented in

industrial drives demonstrating performances suitable for

a wide spectrum of technological applications. However,

the performance is sensitive to the variation of motor pa-

rameters, especially the rotor time-constant, which varies

with the temperature and the saturation of the magnetising

inductance. Recently, much attention has been given to

the possibility of identifying the changes in motor param-

Tamkang Journal of Science and Engineering, Vol. 12, No. 1, pp. 17�28 (2009) 17

*Corresponding author. E-mail: [email protected]

eters of LIM while the drive is in normal operation. This

stimulated a significant research activity to develop LIM

vector control algorithms using nonlinear control theory

in order to improve performances, achieve speed (or

torque) and flux tracking, or to give a theoretical justifica-

tion of the existing solutions [2,3].

Due to new developments in nonlinear control theory,

several nonlinear control techniques have been intro-

duced in the last two decades. One of the nonlinear control

methods that have been applied to linear induction motor

control is the backstepping design [4,5]. The backstep-

ping is a systematic and recursive design methodology for

nonlinear feedback control. This approach is based upon a

systematic procedure for the design of feedback control

strategies suitable for the design of a large class of feed-

back linearisable nonlinear systems exhibiting constant

uncertainty, and guaranteed global regulation and track-

ing for the class of nonlinear systems transformable into

the parametric-strict feedback form. The backstepping de-

sign alleviates some of these limitations [6,7]. It offers a

choice of design tools to accommodate uncertainties and

nonlinearities and can avoid wasteful cancellations. The

idea of backstepping design is to select recursively some

appropriate functions of state variables as pseudo-control

inputs for lower dimension subsystems of the overall sys-

tem. Each backstepping stage results in a new pseudo-

control design, expressed in terms of the pseudo-control

designs from the preceding design stages. When the pro-

cedure terminates, a feedback design for the true control

input results which achieves the original design objective

by virtue of a final Lyapunov function, which is formed by

summing up the Lyapunov functions associated with each

individual design stage [4�6].

In the past years, fuzzy control technique has also been

successfully applied to the control of motor drives [8,9].

The mathematical tool for the FLC is the fuzzy set theory

introduced by Zadeh. It can be regarded as nonmathe-

matical control algorithms in contrast to a conventional

feedback control algorithm. In the control by fuzzy tech-

nique, the linguistic description of human expertise in con-

trolling a process is represented as fuzzy rules or relations.

This knowledge base is used by an inference mechanism, in

conjunction with some knowledge of the states of the pro-

cess (say, of measured response variables), in order to de-

termine control actions. However, the huge amount of fuz-

zy rules for a high-order system makes the analysis com-

plex. Nowadays, much attention has focused on the combi-

nation of fuzzy logic and the backstepping [10,11]. Fuzzy

inference mechanism can arbitrarily approximate nonlinear

function and has the superiority of using experiential lan-

guage information. So, the adaptive fuzzy backstepping

controller compensates the uncertainties of the system [11].

In this paper, adaptive backstepping with fuzzy integral

action is designed to control the position of mover of the

linear induction motor. The output of the proposed control-

ler is the current (iqs) required to maintain the motor speed

close to the reference speed. The current (iqs) is forced to

follow the control current by using current regulators. The

reminder of this paper is organized as follows. Section II re-

views the principle of the indirect field-oriented control

(IFOC) of linear induction motor. Section III shows the de-

velopment of the adaptive backstepping technique with in-

tegral action for mover position control of LIM. Then, the

backstepping design with fuzzy integral action for LIM

control is derived, where a simple fuzzy logic system is

used to compensate the uncertainties which occur in the

control. Section V gives some simulation results. Finally,

some conclusions are drawn in section VI.

2. Indirect Field-Oriented Control of the LIM

The dynamic model of the LIM is modified from tra-

ditional model of a three-phase, Y-connected induction

motor and can be expressed in the d-q synchronously ro-

tating frame as [2,3]:

(1)

18 I. K. Bousserhane et al.

where

� � �1 2( / )L L Lm s r

�r = Lr / Rr

Kf 3P�Lm / (2hLr)

Rs winding resistance per phase;

Rr secondary resistance per phase referred primary;

Ls primary inductance per phase;

Lr secondary inductance per phase;

Lm magnetizing inductance per phase;

h pole pitch;

P number of pole pairs;

�r Rotor time constant (Lr / Rr)

Vds, Vqs Direct and quadrature stator voltages

ids, iqs Direct and quadrature stator currents.

�dr, �qr Direct and quadrature rotor fluxes.

ve synchronous linear velocity;

vr mover linear velocity;

d mover position

J Inertia moment of the moving element

D viscous friction and iron-loss coefficient

FL external force disturbance.

The main objective of the vector control of linear in-

duction motors is, as in DC machines, to independently

control the electromagnetic force and the flux; this is

done by using a d-q rotating reference frame synchro-

nously with the rotor flux space vector [3]. In ideally

field-oriented control, the secondary flux linkage axis is

forced to align with the d-axis, and it follows that [2�4]:

(2)

�rd = �r = constant (3)

Applying the result of (2) and (3), namely field-ori-

ented control, the electromagnetic force equation be-

come analogous to the DC machine and can be desc-

ribed as follows:

Fe = K f � �r � i qs (4)

And the slip frequency can be given as follow:

(5)

3. Integral-Backstepping Control of Lim

The difficulties in designing a high-performance

controller for the aforementioned ac motor servo system

derived from the fact that several unknown parameters

and disturbances are related to the mechanical uncer-

tainties. The proposed control system is designed to

achieve position tracking objective and described step

by step as follows.

For the control objective, the tracking of a given

reference mover position signal dref, we regard the ve-

locity vr as the “control” variable (called virtual control

in [12,13]). Define the position tracking error signal

e1 = dref � d (6)

Then its error dynamics is

(7)

If vr were our control, it then can be chosen as

(8)

And we will have

(9)

Let us define a virtual control as [14,15]

(10)

An integral may be added

vref = k2� + k1e1 + dref, k2 > 0

(11)

However, vr is not real control. Let us define another er-

ror as

e2 = vref � vr (12)

Then we can obtain

Mover Position Control of Linear Induction Motor Drive Using Adaptive Backstepping Controller with Integral Action 19

(13)

(14)

We can rewrite (13)

(15)

Equation (7) can be rewritten as follows

(16)

with �� e1

Let us define the following estimations errors

(17)

(18)

Then

(19)

(20)

Let us construct a Lyapunov function [14,15]

(21)

(22)

(23)

Also it is true that

(24)

(25)

As a result

(26)

Let

Fq = Fq1 + Fq2 (27)

(28)

Then, we can obtain

(29)

Next let

(30)

(31)

Then

(32)

Choose the external load estimate update law as

(33)

Hence

(34)


Any term without 1/J may be rearranged as

(35)

As a result

(36)

Choose the inertia estimate update law as

(37)

Also we can put

(38)

Then

(39)

Suppose there exists an auxiliary function f such that

[14]

(40)

This auxiliary function, f is to be determined later.

(41)

Choose

(42)

Then

(43)

The inertia estimate update law becomes

(44)

Let us put all the components of Fq together

(45)

We choose

(46)

Then we get the following

(47)

�V/�e2 can be chosen to be linear such as

(48)

We can choose

(49)

We will have then

(50)


(51)

The very last term in V is a part that we do not need

to construct explicitly. This explains the name implicit

construction of Lyapunov function. Finally, the proposed

control law can be shown in Figure 1 [14,15].

4. Fuzzy Integral-Backstepping

Control of Lim

In this section, our objective is to find a robust con-

trol law based on the controller of the previous section

for LIM position tracking. The adaptive backstepping

controller with fuzzy integral action is proposed. The

proposed backstepping controller scheme is shown in

Figure 4. In this control scheme, a PI-like fuzzy logic

controller can replace the first part of the previous con-

troller (vref = k2� + k1e1 + dref). The FLC included in the

design has two inputs error (e1) and its first time deriva-

tive (e1), where e1 is defined by e1 = dref � d. The output

of the FLC is the incremental change in speed reference

(control signal) vref the control (speed reference vref) is

obtained by integrating vref. The problem of selecting

the suitable fuzzy controller rules remain relying on ex-

pert knowledge and try and error tuning methods. The

i-th fuzzy rule, which incorporates the knowledge and

experience of the operator in position control, is ex-

pressed as

Rule i: if e is Aj and �e is Bk then vref is Cl (52)

Where Ai, Bi, Ci and Di are fuzzy sets on correspond-

ing supporting sets. Because the data manipulated in the

fuzzy inference mechanism is based on the fuzzy set

theory, the associated fuzzy sets involved in the fuzzy

control rules are defined as follows:

NB: Negative big NM: Negative medium

NS: Negative small ZE: Zero

PS: Positive small PM: Positive medium

PB: Positive big

All membership functions of the FLC inputs, e and

e, and the output, vref, are defined on the common nor-

malized domain [-1,1]. The membership functions of the

error ‘e’ and ‘e’, corresponding to the fuzzy sets, NB,

NM, NS, ZE, PS, PM and PB, are the same as shown in

Figure 2. The membership functions of vref, corre-

sponding to the same fuzzy sets as e and e are shown in

Figure 3. In this work, triangular membership functions

(MFs) are chosen for NM, NS, ZE, PS, PM fuzzy sets and

trapezoidal MFs are chosen for fuzzy sets NB and PB.

The rule-base for computing the output vref is


Figure 1. Block diagram of the adaptive backstepping control.

Figure 2. Membership functions of the error e and its derivative e.

shown in Table 1; this is a very often used rule base de-

signed with a two dimensional phase plane [11]. The

control rules in Table 1 are built based on the characteris-

tics of the step response. For example, if the output is

falling far away from the set-point, a large control signal

that pulls the output toward the set-point is expected,

whereas a small control signal is required when the out-

put is near and approaching the set-point.

By using the membership functions shown in Figure

4, we have the following conditions:

(53)

Where, i represents the membership functions grade

for ith rule.

Then, the fuzzy output vref can be calculated by

the centre of area defuzzification as:

(54)

Where � = [c1…c7] is the vector containing the output

fuzzy centers of the membership functions of vref, W =

[ ]w w

wi

i

1 7

1

2

�

�

�is the firing strength vector of the i-th

rule.

5. Simulation Results

To investigate the effectiveness of the proposed con-

trol scheme, we apply the designed controllers to the

mover position control of the linear induction motor. The

control objective is to control the mover to move 0.1

meter and 0.05 meter.

First, the simulated results of the adaptive back-

stepping controller with fuzzy integral action for peri-


Figure 4. Block diagram of the fuzzy-integral backstepping control.

Table 1. fuzzy rules

e / ë NB NM NS ZE PS PM PB

NB NB NB NB NM NS NS ZE

NM NB NM NM NM NS ZE PS

NS NB NM NS NS ZE PS PM

ZE NB NM NS ZE PS PM PB

PS NM NS ZE PS PS PM PB

PM NS ZE PS PM PM PM PB

PB ZE PS PS PM PM PB PB

Figure 3. Membership functions of vref.

odic sinusoidal and triangular inputs are proposed. The

position responses of the mover and the control effort are

shown in Figure 5. From the simulated results, the pro-

posed fuzzy-integral backstepping controller can track

periodic step, sinusoidal and triangular inputs precisely.

Next, the simulated results of the fuzzy-integral back-

stepping control system for periodic step, sinusoidal and

triangular inputs with parameters variation are shown in

Figures 6, 7 and 8. From the simulated results, the track-

ing responses of the fuzzy-integral backstepping control

system are insensitive to parameter variations. More-

over, the control efforts (ias) are larger than for Case 1

due to the existence of parameter variations. Figure 9

shows the simulated results of the fuzzy-integral back-

stepping control system for periodic step, sinusoidal

and triangular inputs with force loads disturbance ap-

plication (constant and sinusoidal force load). From

simulated results, the tracking responses of the pro-

posed controller are insensitive to force load applica-

tion. A comparison between the proposed controller

(fuzzy-integral backstepping) and the adaptive back-

stepping with integral action is shown in Figure 10 for

step and triangular references signal. In Figure 10, it

can be observed that the position response of the fuzz-

integral backstepping controller presented best track-

ing responses and very robust characteristics and bet-

ter than the conventional controller.

6. Conclusion

This paper has demonstrated the applications of a

hybrid control system to the periodic motion control of a

LIM. First, an adaptive backstepping controller with in-

tegral action for LIM control was designed. Moreover, a

PI-like type fuzzy logic controller was introduced to

construct a robust control law based on the conventional

controller for LIM position tracking. The control dy-

namics of the proposed hierarchical structure has been

investigated by numerical simulation. Simulation results

have shown that the proposed fuzzy-integral backstep-

ping controller is robust with regard to parameter varia-

tions and external load disturbance. Finally, the pro-

posed controller provides drive robustness improvement

and assures global stability.


Figure 5. Simulated results of fuzzy-integral backsteppingcontroller for LIM position control.


Figure 6. Simulated results of the fuzzy-integral backsteppingcontrol with rotor resistance variation (3 Rr n).

Figure 7. Simulated results of the fuzzy-integral backsteppingcontrol with stator resistance variation (3 Rs n).


Figure 8. Simulated results of the fuzzy-integral backsteppingcontrol with stator resistance variation (1,7 J n).

Figure 9. Simulated results of the fuzzy-integral backstep-ping control with force load variation for cases: (a)Constant force load 20N occurring at 5 sec. (b) Si-nusoidal force load 20.sin(t)N occurring at 5 sec. (c)Constant force load 20N occurring at 5 sec.

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Manuscript Received: Mar. 21, 2006

Accepted: Jan. 11, 2007


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