The Neural Network-Combined Optimal Control System of Induction Motor

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 9, No. 4, August 2019, pp. 2513~2522
ISSN: 2088-8708, DOI: 10.11591/ijece.v9i4.pp2513-2522  2513
Journal homepage: http://iaescore.com/journals/index.php/IJECE
The neural network-combined optimal control system of
induction motor
Trong-Thang Nguyen
Faculty of Energy Engineering, Thuyloi University, Vietnam
Article Info ABSTRACT
Article history:
Received Feb 13, 2019
Revised Feb 23, 2019
Accepted Mar 12, 2019
This research aims to propose the optimal control method combined with the
neuron network for an induction motor. In the proposed system, the induction
motor is a nonlinear object which is controlled at each working point. At
these working-points, the state equation of the induction motor is linear, so it
is possible to apply the linear quadratic regular algorithm for the induction
motor. Therefore, the parameters of the state feedback controller are the
functions. The output-input relationships of these functions are set through
the neural network. The numerical simulation results show that the quality of
the control system of the induction motor is very high: The response speed
always follows the desired speed with the short transition time and the small
overshoot. Furthermore, the system is robust in the case of changing the load
torque, and the parameters of the induction motor are incorrectly defined.
Keywords:
Induction motor
LQR control
Neural network
Optimal control
Copyright © 2019 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Trong-Thang Nguyen,
Faculty of Energy Engineering,
Thuyloi University,
175 Tay Son, Dong Da, Hanoi, Vietnam.
Email: nguyentrongthang@tlu.edu.vn
1. INTRODUCTION
The induction motor is applied widely in practical industrial [1] because of the low weight per
power- unit, high robustness, high reliability, and low cost [2]. There are many methods for controlling the
induction motor with the high performance, such as the scalar control [3], the indirect field orientation
control [4], the direct torque control [5], and the field orientation control [6, 7].
It is difficult to control the induction motor because of its nonlinearities, so some nonlinear methods
have been applied to control the induction motor. For example, the control method is based on a flatness
principle [8, 9], but the disadvantage of this method is the need of knowing exactly the parameters of the
induction motor. Another method is an accurate linearization method [10, 11], the purpose of this method is
to convert the input-output relationship of induction motor into a linear one by separating the nonlinear
components in the inner loop. The disadvantage of this method is that if the nonlinear components are
removed incorrectly, it will adversely affect the control results and reduce the sustainability of the system.
Other nonlinear control methods such as bakstepping [12-14], sliding mode control [15-16], the disadvantage
of these methods is that there are harmonic oscillations.
A method promises high-quality control and efficiency that is the control method based on Linear
Quadratic Regular (LQR) algorithm. The LQR method is used to control a lot of objects in practice as wind
power generator [17], converter [18], quad-rotor [19], two-wheels self-balancing mobile robot [20]. The LQR
method is built by using a mathematical algorithm to minimize the cost function with weighting factors
defined by the designer. The cost-function is often defined as a sum of the deviations of response output and
their desired values [21].
To apply the LQR method for the induction motor, the controller is designed for the induction motor
at the different working-points. At each different working point, the state feedback matrix of the controller is

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 9, No. 4, August 2019 : 2513 - 2522
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different, so each component in the state feedback matrix is a function. To establish the output-input
relationship of these functions, an artificial neural network is used. The result is an optimal control system for
the nonlinear induction motor with the high-quality.
2. THE LINEAR QUADRATIC REGULAR ALGORITHM
Considering the state equation of the object:
)(.)(.)( tuBtxAtx  (1)
where:
)](),...,(),([)( 21 txtxtxtx n is the state signal vector,
)](),...,(),([)( 21 tutututu m is the control signal vector.
The requirement of the control system is finding the control signal )(tu in order for the control-
object is operated from the initial-state )0()( 0 xtx  go to the end-state 0)( ftx and satisfy the
following condition (cost function):
 
ft
t
TT
ff
T
dttuRtutxQtxtxMtxJ
0
min)](.).()(.).([
2
1
)(.).(
2
1
(2)
where Q and M are the symmetric, positive semi-definite weight matrix. R is the symmetric, positive definite
weight matrix. To solve the problem, we establish the Hamilton function:
)](.)(.[)](.).()(.).([
2
1
tuBtxAtuRtutxQtxH TTT
 
(3)
The optimal control-signal must satisfy the following equations:
- The state equation:
)(.)(.)( tuBtxAtx  (4)
- The equilibrium equation:
)(.)(.)( tAtxQ
x
H
t T
 



(5)
- The optimal condition:
0)(.)(. 


tBtuR
u
H TT

(6)
From equation (6), we have:
)(..)( 1
tBRtu TT

 (7)
Replace )(tu into (4), we have:
)(...)(.)( 1
tBRBtxAtx TT

 (8)

Int J Elec & Comp Eng ISSN: 2088-8708 
The neural network-combined optimal control system of induction motor (Trong Thang Nguyen)
2515
Combining (8) and (5) we have:



















 
)(
)(..
)(
)( 1
t
tx
AQ
BRBA
t
tx
T
T


(9)
Solving the above equations, we have the optimal control signal:
)().()(*
txtKtu  (10)
where )(..)( 1
tPBRtK T

)(tP is the positive semi-definite solution of the Riccati equation:
PBRBPQPAAPP TT
...... 1
 
(11)
In the case of the infinite time 𝑡𝑓 = ∞, the cost function is as follows:
 
ft
TT
dttuRtutxQtxJ
0
min)](.).()(.).([
2
1
(12)
The optimal control signal:
)(.)(*
txKtu  (13)
where PBRK T
..1

P is the positive semi-definite solution of the Riccati equation:
0...... 1
 
PBRBPQPAAP TT
(14)
3. THE SYSTEM MODEL OF THE INDUCTION MOTOR DRIVE
The induction motor drive is controlled based on the axis oriented along the stator-flux dq.
The control diagram is shown in Figure 1.
Figure 1. The control diagram of the induction motor drive

 ISSN: 2088-8708
2516
Considering the equations of cage induction motor on the dq-axis oriented along the stator-flux that
rotes with the speed  , the voltage vector of the stator and rotor are presented [1]:
s
s
sss j
dt
d
iRu 

 ..
)(
. 
(15)
r
r
rrr pj
dt
d
iRu 

 )...(
)(
. 
(16)
where rs uu , are the vectors of stator and rotor voltage on dq axis; rs ii , are the vectors of stator and rotor
current on dq axis;
rs
 , are the vectors of stator and rotor flux on dq axis; sR is the stator resistance,
rR is the rotor resistance;  is the angular speed of stator-flux.
The flux vector of stator and rotor:
mrsss
LiLi .. 
(17)
msrrr
LiLi .. 
(18)
where sL is the stator inductances, rL is the rotor inductances, mL is the mutual inductance
The motion equation:
LE TT
dt
d
J 

(19)
)..(
2
3
ssE ipT 
(20)
where  is the angular speed of the rotor, ET the electromagnetic torque, LT is the load torque, p is the
number of pole pairs. Because we're considering on the axis oriented along the stator-flux that rotes with the
speed of  , so
ssd   , 0sq . Rewriting the equations (15-20) according to d-axis and q-axis
components, we have:
dt
d
iRu sd
sdssd
)(
.


(21)
sdsqssq iRu  .. 
(22)
0)..(
)(
.  rq
rd
rdrrd p
dt
d
iRu 


(23)
0)..(
)(
.  rd
rq
rqrrq p
dt
d
iRu 


(24)
mrdssdsd LiLi ..  (25)

2517
mrqssqsq LiLi .. 
(26)
msdrrdrd LiLi ..  (27)
msqrrqrq LiLi .. 
(28)
Lsqsd Tip
dt
d
J  )..(
2
3


(29)
where sqsd uu , are the stator voltage component of d-axis and q-axis; rqrd uu , are the rotor voltage
component of d-axis and q-axis; sqsd ii , are the stator current component of d-axis and q-axis; rqrd ii , are the
rotor current component of d-axis and q-axis; sqsd  , are the stator-flux component of d-axis and q-axis;
rqrd  , are the rotor-flux component of d-axis and q-axis.
Eliminating the variables rqrdrqrd ii  ,,, in equations (21-29), we obtain the following equations:



























sq
s
sd
sd
s
sd
sdsds
sd
sd
mrs
r
sq
mrs
rs
sd
sq
sd
mrs
r
sd
mrs
r
sqsd
mrs
srrssd
u
RJ
p
RJ
p
dt
d
uiR
dt
d
LLL
Lp
i
LLL
RL
ip
dt
di
u
LLL
L
LLL
R
ipi
LLL
RLRL
dt
di
..2
..3
..2
..3
.
.
.
.
)..(
.
.
.
)..(
.
.
.
.
)..(.
.
..
22
222














(30)
Equation (30) is the state equation. Where
T
sdsqsd
T
iixxxxx ],,,[],,,[ 4321  is the state signal
vector.
T
sqsd
T
uuuuu ],[],[ 21  is the control signal vector.
4. DESIGNING THE NEURON NETWORK COMBINED-OPTIMAL CONTROLLER
Designing the controller for cage induction motor with the parameters shown in Table 1.
Table 1. The parameters of induction motor
Rs() Ls(H) Rr() Lr(H) Lm(H) J(kg.m^2) p
1.55 0.098 1.31 0.097 0.0917 0.14 3
We design the control signal )(.)( txKtu  in order for 0)(lim  setresponset xx .
Named ex is the error between response
x and setx .
),,,( _4_44_3_33_2_22_1_11 setresponsesetresponsesetresponsesetresponsee xxxxxxxxxxxxxx 
The state equation (30) is rewritten as follows:
eee uBxAx ..  (31)
where sete uuu  .
u and setu are control signals in order for the status signals reach the responsex and setx values respectively.

 ISSN: 2088-8708
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The matrices of state equations are as follows:































0
..2
..3
.00
000
0
.
)..(
.
.
).(
0..
.
..
22
2
s
sd
s
mrs
r
mrs
rs
mrs
srrs
RJ
p
R
LLL
Lp
LLL
RL
p
pp
LLL
RLRL
A









(32)


















s
sd
mrs
r
RJ
p
LLL
L
B
..2
..3
0
01
00
0
. 2

(33)
Suppose we always control the system to satisfy the condition constsetsd  , so the matrix
constB  . The matrix A is changed because of the change of two components ).(  p and  .
Therefore, in the case of each pair of values ).(  p and  , we will find the optimal control signals
respectively.
The main missions of the control system are that setsetsd   , , so the output of system
is as follows:
setxCry . (34)
T
C ]1100[ (35)
Considering the state equation (31), at the equilibrium, we have:
setset uBAx ..1
 (36)
Combining equation (34) and equation (36), we have:
rBACuset
11
]..[ 
 (37)
The optimal control signal for the system (31) is as follows:
ee xKu . (38)
where:
PBRK T
..1
 (39)
P is the positive semi-definite solution of the Riccati equation:

2519
0...AP.A 1T
 
PBPBRQP T
(40)
According to equation (38), we have:
rKxK
rBACBAKrBACxK
xKuxK
xxKuu
rresponse
response
setsetresponse
setresponseset





.
]...[..]..[.
..
).(
11111
(41)
where:
111
]..].[..[ 
 BACIBAKK LQRr (42)
Therefore, the control diagram is shown as Figure 2.
Figure 2. The control diagram
In the above control diagram, the matrices (A, B, C) are known, we must find the matrix K
according to the formula (39), then find the matrix rK according to the formula (42). Because of the control
purposes are that setsetsd   , , we install the Q, R matrices that meet the requirements are
as follows:
)10,10.2,10,10( 2233 
 diagQ (43)
)10.2,10.2( 77 
 diagR (44)
The matrix A is changed due to the variation of two components ).(  p and  . Therefore,
in the case of each pair ).(  p and  , we will find each feedback matrix K with the size is 2x4. We
set up an off-line trained neuron network to describe the relationship between the matrix K and
]),.[(   p . A neural network includes many simple components, these components are the single-
input neuron or the multiple-input neuron, are shown as the Figure 3 [22].
Figure 3. The component of neural network: a) the case of single-input, b) the case of multiple-input

 ISSN: 2088-8708
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Usually, a neural network has several layers. For example, Figure 4 shows the neural network with
three layers, the outputs of the first layer is the input of the second layer, the outputs of second layer is the
input of the third layer. The output layer is at once with whose output is the network output, the other layers
are named hidden layer.
Figure 4. The neural network with three layers
To meet the requirements of the control system, we set up the neuron network as follows: The feed-
forward neural network includes three layers: the first layer is the input which includes two signals
).(  p and  ; The second layer is hidden; the last layer is the output, it includes 8 signals which are
the components of the matrix K(2x4). The transfer-function of the first layer and second layer are tansig. The
transfer-function of the last layer is pureli.
The diagram of neural network training is shown in Figure 5. The training results are characteristics
of the relationship between the matrix K and ]),.[(   p , which are shown in Figure 6.
Figure 5. The diagram of neural network training
Figure 6. The characteristics of the relationship between the matrix K and ]),.[(   p

2521
5. RESULTS AND ANALYSIS
Running the system with the parameters of the induction motor listed in Table 1 and the controller
parameters are set up in section 4. The simulation results are shown in Figure 7(a), it includes the following
characteristic: the desired speed, the response speed and the load torque. The results show that the response
speed always follows the set speed with very short transition time, there is not overshot. Furthermore, when
changing the disturbance (load torque TL), the response speed is almost unaffected by the change of the load
torque. To more clearly, the zoom-in simulation results are shown in Figure 7(b).
The results of controlling the stator-flux are shown in Figure 8. The results show that the response
flux always follows the set flux with the static error of zero. To investigate the sustainability of the proposed
control method, the author runs the control system in the case of the difference between the actual parameter
and the parameter used for the controller. The details are as follows: Rr.70%; Lr.130%; Rs.130%; J.130%.
The results are shown in Figure 9. The results show that the quality of the system is almost unaffected when
there are inaccuracies of induction motor parameters. Therefore, we can conclude that the proposed control
system has a high quality and high sustainability.
(a) (b)
Figure 7. The simulation results in the case of the accurate parameters:
a) the normal view, b) the zoom-in view
(a) (b)
Figure 8. The results of controlling the stator-flux: a) the normal view, b) the zoom-in view
(a) (b)
Figure 9. The simulation results in the case of the inaccuracies parameters:
a) the normal view, b) the zoom-in view

 ISSN: 2088-8708
2522
6. CONCLUSION
In this study, the author has succeeded in building the optimal controller combined with the neuron
network for the induction motor. The results show that the control quality of the system is very high:
The response speed always follows the desired speed with the short transition time and the small overshoot.
Moreover, the control system is very stable in the case of changing the load torque, and in the case of
inaccuracies of induction motor parameters. The success of the proposed method is base for applying the
control algorithm to electric motion using the induction motor in the industries.
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