Adaptive integral backstepping controller for linear induction motors

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 10, No. 2, June 2019, pp. 709~719
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v10.i2.pp709-719  709
Journal homepage: http://iaescore.com/journals/index.php/IJPEDS
Adaptive integral backstepping controller for linear induction
motors
Omar Mahmoudi, Abdelkrim Boucheta
Department of Electrical Engineering, University of Tahri Mohamed Bechar, Algeria
Article Info ABSTRACT
Article history:
Received Jan 21, 2019
Revised Mar 1, 2019
Accepted Mar 10, 2019
Linear induction motors offer the possibility to perform a direct linear motion
without the nead of mechanical rotary to linear motion transformers. The
main problem when controlling this kind of motors is the existence of
indesirable behaviours such as end effect and parameter variations, which
makes obtaining a precise plant model very complicated. This paper proposes
an adaptive backstepping control technique with integral action based on
lyapunov stability approach, which can guarantee the convergence of
position tracking error to zero despite of parameter uncertainties and external
load disturbance. Parameter adaptation laws are designed to estimate mover
mass, viscous friction coefficient and load disturbance, which are assumed to
be unknown constant parameters; as a result the compensation of their
negative effect on control design system. The performance of the proposed
control design was tested through simulation. The numerical validation
results have shown good performance compared to the conventional
backstepping controller and proved the robustness of the proposed controller
against parameter variations and load disturbance.
Keywords:
Adaptive backstepping
Field oriented control
Integral action
Linear induction motor
Copyright © 2019 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Omar Mahmoudi,
Department of Electrical Engineering,
University of Tahri Mohamed Bechar,
B.P 417 Bechar 08000, Algeria.
Email: omaritomah@gmail.com
1. INTRODUCTION
Linear induction motors (LIMs) are widely used in industrial processes and transportation
applications [1, 2]. The main advantage of the LIM is its ability to perform a direct linear motion without the
need of any gear box or mechanical rotary to linear motion transformation [3].
The driving principle of the LIM is similar to that of a rotary induction motor (RIM), however, the
control characteristics are more complicated and the parameters are time varying due to change of operating
conditions, such as speed of mover, temperature and configuration of rail.[4, 5]
Field oriented control (FOC) is considered as the most popular high performance control method for
induction machines. The main idea of the FOC is to decouple the dynamics of thrust force and rotor flux to
make the control as simple as in a separately excited DC machine [6-8]. Unfortunately, a good performance
of FOC can only be achieved by a precise plant model and it can be disturbed by parameter variations, which
presents a big issue, because unlike RIM the model of LIM presents more complexity. Moreover, there are
significant parameter variations in the reaction rail resistivity, the dynamics of the air gap, slip frequency,
phase unbalance, saturation of the magnetizing inductance, and end-effects phenomenon [9-12], causing a
difficulty in achieving a good control performance. Many researches have been done in modeling the LIM
performance taken into consideration time varying parameters and end effect phenomenon [3, 8-10, 13].
However, there are always some modeling uncertainties due to undesirable behaviors.

 ISSN: 2088-8694
Int J Pow Elec & Dri Syst, Vol. 10, No. 2, June 2019 : 709 – 719
710
The advancement in nonlinear control field gave a variety of solutions in dealing with parameter
variations and load disturbance. One of these solutions is backstepping control, this last is a nonlinear control
technique that has been widely applied in controlling LIMs [12, 14]. The basic idea of backstepping is the
decomposition of a high order nonlinear control design problem into smaller steps, in which every step
introduces a reference to stabilize the next one using the so called “virtual control” variables, until we reach
the real control input that insures the global stability based on lyapunov function [12, 14, 15]. In addition, the
adaptive version of backstepping control guarantees the global stability of the system despite of parameter
uncertainties and variations thanks to parameter update laws [15].
In this paper we propose an adaptive backstepping control design based on field oriented control to
achieve a desired position trajectory under the assumption of unknown mover mass, friction coefficient and
load disturbance. The rest of the paper is organized as follows: in section 2, we introduce the basics of
indirect field oriented control. Section 3 presents the proposed adaptive backstepping controller. In section 4,
we show the simulation results followed by discussion. Finally, a conclusion is drawn in section 5.
2. INDIRECT FIELD ORIENTED CONTROL
Figure 1 shows the construction of a LIM. The primary is simply a cut-open and rolled-flat rotary
motor primary, and the secondary usually consists of a sheet conductor using aluminum with an iron back for
the return path of the magnetic flux. For a zero relative velocity, the LIM can be considered as having an
infinite primary, in which case the end effects may be ignored [12].
Figure 1. Constriuction of a LIM
The fifth order model of the LIM in the d-q axis reference frame modified from a three phase Y-
connected rotary induction motor model is given as [12, 16]:
ds
V
s
σL
.v
qr
.
h
r
L
π
m
PL
dr
.
r
L
r
R
m
L
qs
.i
e
V
h
π
ds
.i
r
R
r
L
m
L
s
R
s
σL
dt
ds
di
s
s
σL
σL
1
2
2
1
























 (1)
ds
V
s
σL
qr
.
r
L
r
R
m
L
.v
dr
.
h
r
L
π
m
PL
qs
.i
r
R
r
L
m
L
s
R
s
σL
ds
.i
e
V
h
π
dt
qs
di
s
s σL
σL
1
2
2
1
























 (2)
  qr
.
v
P
e
V
dr
.
r
L
r
R
ds
.i
r
L
r
R
m
L
dt
dr
d
h


 
.



 (3)
  qr
r
L
r
R
dr
.
v
P
e
V
qs
.i
r
L
r
R
m
L
dt
qr
d
h







 . (4)
L
c
ds
qr
qs
dr
f
e F
v
f
dt
dv
M
i
i
K
F 



 .
)
( 
 (5)

Int J Pow Elec & Dri Syst ISSN: 2088-8694 
Adaptive integral backstepping controller for linear induction motors (Omar Mahmoudi)
711
where s
R is stator resistance per phase, r
R is rotor resistance per phase, s
L is the stator inductance per
phase, r
L is the rotor inductance per phase, m
L is the magnetizing inductance per phase, P is the number of
pole pairs, h is the pole pitch, )
/
(
1 2
r
s
m L
L
L


 is the leakage coefficient, e
V is the synchronous linear
velocity, v is the mover linear velocity, ds
i and qs
i are the d-axis and q-axis stator currents, dr
 and qr
 are
the d-axis and q-axis rotor fluxes, ds
V and qs
V are d-axis and q-axis stator voltages, )
2
/(
3 h
L
L
P
K r
m
f 
 is a
force constant, e
F is the electromagnetic force, L
F is the external load disturbance, M is the total mass of
the moving part and c
f is the viscous friction and iron-loss coefficient.
The main idea of the field oriented control is to render the behaviour of the LIM similar to that of a
separately excited DC machine, this last gives the ability to control the electromagnetic force and rotor flux
independently [7], [17]. The decoupling is realized by forcing the secondary flux linkage to align with the d-
axis. As a result we get,
0

qr
 , 0

dt
d qr

(6)
Using (6) and replacing in the rotor flux equations (3) and (4) we get,
ds
r
m
dr i
s
L




1
(7)
where s is the Laplace operator, and r
r
r R
L /

 is the secondary time constant.
Replacing (6) in electromagnetic force equation (5) gives,
qs
dr
f
e i
K
F 
 (8)
Viewing equations (7) and (8) we can see that the rotor flux and the electromagnetic force can now
be controlled separately using stator currents ds
i and qs
i .
Using (6) and (4) the slip velocity can be written as
iqs
L
hL
Pv
V
V
dr
r
r
m
e
sl





 (9)
3. ADAPTIVE BACKSTEPPING CONTROL WITH INTEGRAL ACTION
The basic idea of the backstepping design is the decomposition of a complex nonlinear system into
multiple single input-output first order subsystems, using the so-called “virtual control” variables [18], in
which every control variable is forced to stabilize the previous subsystem generating a corresponding error
which can be stabilized by convenient input selection via lyapunov stability [14, 16, 18, 19], The steps to
design such controller for LIM position tracking are as follows:
Define the position tracking error
d
d
e ref 

1 (10)
Its dynamic can be written as
v
d
d
d
e ref
ref 


 


1 (11)
Consider the lyapunov function candidate
2
1
1
2
1
e
V  (12)
Deriving V and using the error dynamic equation (11) we obtain

 ISSN: 2088-8694
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)
(
1
1
1
1 v
d
e
e
e
V ref 

 

 (13)
Choosing the velocity stabilising function as
ref
ref d
e
k
v 

 1
1 (14)
where 1
k is a positive design constant to be determined later, gives
0
2
1
1
1 

 e
k
V
 (15)
witch guarantees the stabilisation of our closed-loop control.
However the velocity v is not a real input, it’s a “virtual control” input [19, 20], so we define the
following velocity tracking error:
v
v
e ref 

2 (16)
To ensure the convergence of the position tracking error to zero at the steady state despite the
presence of disturbance and modeling inaccuracy of the system we introduce an integral action to the desired
velocity as follows [15]
'
' 1
1
1
1 e
k
d
e
k
v ref
ref 

  (17)
where 
 dt
t
e
e )
(
' 1
1 is the integral action of the position tracking error and '
1
k is a positive design
constant.
The dynamics of the velocity tracking error give,
M
F
v
M
f
M
F
e
k
v
d
k
d
v
v
e
L
c
e
ref
ref
ref









1
1
1
2
'
)
( 





(18)
Then the dynamics of 1
e can be rewritten as
2
1
1
1
1
1 '
' e
e
k
e
k
e 



 (19)
Define another lyapunov function
2
1
1
2
2
2
1
2 '
2
'
2
1
2
1
e
k
e
e
V 

 (20)
Its time derivative is
1
1
1
2
2
1
1
2 '
' e
e
k
e
e
e
e
V 

 

 (21)
Using (19), (20) and (21) we get,
 














 L
v
D
M
F
d
k
k
e
e
k
k
k
k
e
e
e
k
e
k
V e
ref .
)
(
'
'
)
'
1
( 2
1
2
1
1
1
1
2
1
1
2
2
2
2
2
1
1
2


 (22)

713
where 2
k is a positive design parameter, M
f
D c /
 is the normalized friction coefficient and M
F
L L /
 is
the normalized Load disturbance.
Now we can choose our force control input ref
e
F _ to cancel the undesirable behaviours and
guarantee achieving the desired position trajectory.
If we choose
 
L
v
D
d
k
k
e
e
k
k
k
k
e
M
F ref
ref
e 







 .
)
(
'
'
)
'
1
( 2
1
2
1
1
1
1
2
1
1
_

 (23)
we get
0
2
2
2
2
1
1
2 


 e
k
e
k
V
 (24)
witch guarantees the stabilization of our control in the closed-loop.
However, the real values of parameters M , D and L are unknown and can’t be used, so we use
their estimates M̂ , D̂ and L̂ instead:
 
L
v
D
d
k
k
e
e
k
k
k
k
e
M
F ref
ref
e
ˆ
.
ˆ
)
(
'
'
)
'
1
(
ˆ
ˆ
2
1
2
1
1
1
1
2
1
1
_ 







 
 (25)
The next step is to extract the update laws of estimated parameters, for that purpose we define the
following parameter estimation errors:
M
M
M ˆ
~

 (26)
D
D
D ˆ
~

 (27)
L
L
L ˆ
~

 (28)
Using these definitions and substituting the force control in (25) to the velocity tracking error in (18)
the dynamics of 2
e become,
L
v
D
M
M
M
e
k
e
e
k
e
k
k
d
e ref 








 .
~
'
)
'
'
( 1
1
2
1
1
1
1
1
2 


 (29)
where  
L
v
D
d
k
k
e
e
k
k
k
k
e ref
ˆ
.
ˆ
)
(
'
'
)
'
1
( 2
1
2
1
1
1
1
2
1
1 







 


After few calculations we get,

M
M
L
v
D
e
k
e
e
ˆ
ˆ
.
ˆ
2
2
1
2 





 (30)
Finally, we construct the final lyapunov function that includes all the system’s error signals in order
to derive the estimates update laws,
2
3
2
2
2
1
2
1
1
2
2
2
1
3
~
2
1
~
2
1
~
2
1
'
2
'
2
1
2
1
L
D
M
M
e
k
e
e
V









(31)
where 1
 , 2
 and 3
 are positive design constants.
Assuming that M , D and L are constant parameters, we can write,
M
M

 ˆ
~

 (32)
D
D

 ˆ
~

 (33)

 ISSN: 2088-8694
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L
L

 ˆ
~

 (34)
Using (32), (33), (34) and the error dynamics in (19) and (30) we calculate the time derivative of
(31)
)
'
'
(
ˆ
~
1
ˆ
~
1
ˆ
~
1
ˆ
.
ˆ
~
'
'
ˆ
~
1
ˆ
~
1
ˆ
~
1
'
'
2
1
1
1
3
2
1
1
2
1
1
1
2
2
2
2
1
1
3
2
1
1
1
1
2
2
1
1
3
e
e
k
e
L
L
D
D
M
M
M
L
v
D
M
M
e
e
e
e
k
e
k
e
k
L
L
D
D
M
M
M
e
e
k
e
e
e
e
V













































(35)
We finally get
































 L
e
L
D
v
e
D
M
e
M
M
e
k
e
k
V ˆ
1
~
ˆ
1
~
ˆ
1
~
3
2
2
2
1
2
2
2
2
2
1
1
3






 (36)
In order to render the global lyapunov function negative we choose the update laws of the parameter
estimates as:

 2
1
ˆ e
M 

(37)
v
e
D 2
2
ˆ 


(38)
2
3
ˆ e
L 


(39)
The resulting lyapunov function dynamics give
0
2
2
2
2
1
1
3 


 e
k
e
k
V

which guarantees the stability of the global system and the achievement of the desired position
trajectory.despite of load disturbance and undesirable behaviours.
The previeus controller designing steps are summed up in the diagram in Figure 2:
Figure 2. Adaptive backstepping controller scheme
4. SIMULATION AND DISCUSSION
To investigate the performance of the proposed adaptive backstepping controller with integral action
for the LIM position tracking, we run numerical simulation in which the effectiveness of the proposed
controller under different operating conditions is tested (external load disturbance, friction and mover mass
variation).
The obtained results are compared to those of conventional backstepping control technique. The
LIM parameters used for simulation are shown in Table. 1.

715
Table 1. LIM parameters
Parameter Value
Rs 3.4 (Ω)
Rr 1.95 (Ω)
Ls 0.1078 (H)
Lr 0.1078 (H)
Lm 0.1042 (H)
P 2
M 5.47 (Kg)
Fc 26.36 (Nm.s/rd)
4.1. Case 1 (known parameters)
First, we present the simulated results for a periodic step reference with known parameters and no
load disturbance. The controller design parameters and update laws are chosen as follows: 10
1 
k , 1
.
0
'
1 
k ,
80
2 
k 001
.
0
1 
 , 8
.
0
2 
 , 500
3 
 . The position tracking response is shown in Figure 3a. The applied
electromagnetic force is shown in Figure 3b. The stator current Iqs is shown in Figure 3c. And Figure 3d.
shows the rotor fluxes.
The obtained results show good performance for both conventional and adaptive backstepping
controllers when the parameters are known and invariant. The position tracking error converges to zero with
fast response time about 0.5s, the decoupling between electromagnetic force and rotor flux is realized, and
we can see that the rotor flux is kept constant in the steady state (Figure 3d) while the electromagnetic force
is being controlled by the variation of quadratic stator current Iqs (Figure 3b. Figure 3c.).
Next, we test the robustness of the proposed control scheme under the existence of parameter
uncertainties. For that purpose we devide the next simulation into three steps, each step presents a possible
parameter variation as Case 2, 3, 4.
(a) (b)
(c) (d)
Figure 3. Adaptive backstepping position control of LIM with known parameters,
(a) position tracking response, (b) electromagnetic force, (c) stator current Iqs, (d) rotor fluxes
4.2. Case 2 (load disturbance)
Case 2: a constant force disturbance of 10N is injected at t=5s and removed at t=7s. The obtained
results are presented in Figure 4.

 ISSN: 2088-8694
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(a) (b)
(c) (d)
(e) (f)
Figure 4. Adaptive backstepping position control of LIM with applied load force of 10N,
(a) position tracking response, (b) zoom in position, (c) electromagnetic force, (d) zoom in electromagnetic
force, (e) stator current Iqs, (f) rotor fluxes.
4.3. Case 3 (1.5×D)
Case 3: friction coefficient variation equals to 1.5×D. The obtained results are presented in Figure 5.
(a) (b)

717
(c) (d)
(e)
Figure 5. Adaptive backstepping position control of LIM with 1.5×D, (a) position tracking response,
(b) zoom in position, (c) electromagnetic force, (d) stator current Iqs, (e) rotor fluxes.
4.4. Case 4 (2×M)
Case 4: mover mass variation equals to 2×M. The results are presented in Figure 6.
(a) (b)
(c) (d)

 ISSN: 2088-8694
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(e)
Figure 6. Adaptive backstepping position control of LIM with 2×M, (a) position tracking response,
(b) zoom in position, (c) electromagnetic force, (d) stator current Iqs, (e) rotor fluxes.
The simulated results prove that adaptive backstepping controller is insensitive against load
disturbance and can reject it with nearly no overshoot, in contrary of conventional backstepping where the
control is sensitive causing a failure in position tracking with a non-null static error (Figure 4a. Figure4b.).
The electromagnetique force generated by the adaptive controller has risen by 10N after applying the load
force in order to cancel the disturbance (Figure 4c. Figure 4d.).
In case 3, where the friction coefficient has risen by 50%, we can see that the response time for
conventional backstepping has risen slightly to 0.7s, while for the adaptive controller it hasn’t been affected
(Figure 5a. Figure 5b.), that shows that the proposed technique is insensitive against friction force
variation too.
In the case of mover mass change to double value, we can observe that the adaptive backstepping
controller is always insensitive and can achieve the desired position trajectory with fast response time and no
overshoot, while the conventional backstepping controller response has overshot before it could achieve the
desired trajectory causing a delay in response time as well (Figure 6a. Figure 6b.).
The previous numerical validations show good performance and prove the robustness of the
adaptive backstepping controller against sudden changes thanks to parameter update lows based on
Lyapunov stability, which guarantees the stability of the global system despite of load disturbance and
parameter variations. The estimated parameters are shown in Figure 7. We can observe from the figure that
the estimated parameters converge to the real values with acceptable response time and error range allowing
the controller to track the desired trajectory despite of parameter changes.
(a) (b)
(c)
Figure 7. Estimated parameters, (a) load force disturbance, (b) viscous friction coefficient, (c) mover mass

719
5. CONCLUSION
This paper has demonstrated the application of an adaptive backstepping controller with integral
action to track a position reference for LIM. First, the indirect field oriented control was used to decouple the
control of electromagnetic force and rotor flux. Then, an adaptive backstepping controller was designed and
parameter update laws were extracted, to achieve a desired position reference under the assumption of
parameter uncertainties, and the existence of external load force disturbance. The effectiveness of the
proposed controller was tested through numerical simulation under different operating conditions. The
obtained results show good tracking performance compared to the conventional backstepping controller, and
prove the ability of the adaptive backstepping design to reject external load disturbance without overshoot,
and its insensitivity against sudden parameter variations thanks to parameter update laws. Finally, the
robustness of the adaptive backstepping controller was successfully confirmed.
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