An ANN-based Speed and Flux Controller of Three-Phase AC Motors with Uncertain Parameters

Abstract

This paper proposes a speed and flux control method of three-phase AC motors using an artificial neural network (ANN) to compensate for uncertain parameters in the motor’s dynamic model such as rotor resistance, moment of inertia, friction coefficients, and load changes during system operation. Global asymptotic stability of the overall system is proved by Lyapunov’s theory. Matlab simulation results are given to demonstrate the validity of the proposed control method. © 2015 Budapest Tech Polytechnical Institution. All rights reserved.

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Acta Polytechnica Hungarica Vol. 12, No. 2, 2015
– 179 –
An ANN-based Speed and Flux Controller of
Three-Phase AC Motors with Uncertain
Parameters
Hung Linh Le
Faculty of Automation Technology, University of Information and
Communication Technology, Thai Nguyen University
Quyet Thang, Thai Nguyen, Vietnam
lhlinh@ictu.edu.vn
Thuong Cat Pham
Institute of Information Technology, Vietnam Academy of Science and
Technology
18, Hoang Quoc Viet, Hanoi, Vietnam
ptcat@ioit.ac.vn
Minh Tuan Pham
Space Technology Institute, Vietnam Academy of Science and Technology
18, Hoang Quoc Viet, Hanoi, Vietnam
pmtuan@sti.vast.vn
Abstract: This paper proposes a speed and flux control method of three-phase AC motors
using an artificial neural network (ANN) to compensate for uncertain parameters in the
motor’s dynamic model such as rotor resistance, moment of inertia, friction coefficients,
and load changes during system operation. Global asymptotic stability of the overall
system is proved by Lyapunov’s theory. Matlab simulation results are given to demonstrate
the validity of the proposed control method.
Keywords: three-phase AC motor; artificial neural network; speed and flux control;
uncertain dynamics

H. L. Le et al. An ANN-based Speed and Flux Controller of Three-Phase AC Motors with Uncertain Parameters
– 180 –
1 Introduction
AC motor speed control has been a popular topic over the past decades, because of
uncertain parameters in the system model such as rotor resistance, flux, friction
coefficient and variable load [1][2][3][9]. Recently, to estimate motor speed
without using speed sensors, many researchers utilize Kalman filters or sliding
mode observers [8][9]. These would help reduce production costs. However, the
control performance would rely heavily on the estimation algorithm and the
accuracy of motor model. The classical control method cannot obtain effective
control, when the load of the system changes gradually due to uncertain
parameters in the dynamic model of the AC motor. In this case, self-adaptive
control [4][5][6][7], on-line identification methods and controllers with neural
networks were used.
The main focus of recent research has been to determine a control algorithm and
estimate the motor speed based on rotor flux orientation. In this method, the speed
of the flux vector is controlled to reach the synchronous speed. Thus, the AC
motor speed control is the same as the structure of DC motor control [11][12]. The
main objective was to decouple two currents isd, isq independently. However, these
two currents interact and depend on synchronous speed
s
w
. Therefore, this
method operates well in static mode and indicates clearly when the system
operates in the flux declining domain. This paper proposes a method of speed and
flux control for AC motors using an artificial neural network to compensate for
uncertain parameters in the dynamic model, such as rotor resistance, moment,
friction coefficient as well as a variable load during system operation.
The paper is organized as follows. The second section discusses the speed and
flux control models for AC motors. The third section shows the speed and flux
control methods with uncertain parameters. The last section presents simulations
to verify the efficiency of the proposed method.
2 Speed and Flux Control Model for AC Motors
Table 1
Nomenclature
Notion
Unit
Meaning
M, N, D, Q
Model Matrices
B
Nms/rad
Friction coefficient
J
Nms2/rad
Inertia moment of rotor
L
T
Nm
Load torque
,
s r
R R
Ω
Stator and rotor resistance

Acta Polytechnica Hungarica Vol. 12, No. 2, 2015
– 181 –
,
s r
L L
H
Stator and rotor inductance
m
L
H
Inductance between stator winding and rotor winding
ref ,w w
Rad/s
Reference angular velocity, rotor angular velocity
,
r ra b
y y
Wb
Horizontal and vertical parts of rotor flux
,
r r
i i
a b
A
Horizontal and vertical components of rotor current
,
r r
u u
a b
V
Horizontal and vertical components of rotor voltage
Consider a dynamic model of a three phase motor with a squirrel-cage rotor as
follows:
1
1
s s r r
m s r r
s r r s
ssr r
m s r r
s r r s
r r r
r r m s
r r
rr r
r r m s
r r
di R R R
L i u
dt L L L L
di RR R
L i u
dt L L L L
d R R L i
dt L L
dR R L i
dt L L
a
a a b a
b
b a b b
a
a b a
b
a b b
b b y bwy
s s
b bwy b y
s s
yy wy
ywy y
 


      



 


 

  
     





 




   





  







(1)
 
L r s r s
d
J B T K i i
dt a b b a
ww y y   
(2)
where
2
1m
s r
L
L L
s 
;
m
s r
L
L L
bs

;
3
2m
r
L
P
KL

is the moment coefficient.
s
L
,
r
L
,
m
L
and
s
R
rarely change and can be measured accurately. However, rotor
resistance
r
R
often changes in accordance with motor temperature during
operation.
The principle goal of this paper is to determine control signals
,
s s
u u
a b
to regulate
the speed and flux of the motor reach these desired values
ref
w w
,
 
2 2 2 2
refr r r ra b
y y y y  
, where
, ,
r
R J B
and
L
T
are unknown.
Assuming that
,
s s
i i
a b
are known and motor speed
w
can be measured or
estimated, taking the derivative of equation (2), we obtain:
 
L r s r s r s r s
J B T K i i i i
a b a b b a b a
w w y y y y     
 
 
 
(3)
Substituting equation (1) into (3) and setting
1
xw
yields the speed equation:

H. L. Le et al. An ANN-based Speed and Flux Controller of Three-Phase AC Motors with Uncertain Parameters
– 182 –
 
 
 
 
 
1 1 1
2 2
1
1
L r s r s
srm r s r s
s r
r r r r
s
Jx Bx T Kx i i
RR
K L i i
L L K
K x u u
L
a a b b
a b b a
a b a b b a
y y
b y y
s
b y y y y
s
    
 


   



 
   

 
(4)
By setting
2 2
2r r
xa b
y y 
, the flux equation can be written as:
 
 
 
 
 
2
2 2
2 2
1
2
2
2 2
2 1
2
2 2
r r m s s
r r
s
r r
m m r r r r
r s r
rm r s r s
r
r m
rm r r
r r s
R R
x x L i i
L L
R
R R
L L i i
L L L
RL x i i
L
R L
RL x u u
L L L
a b
a a b b
a b b a
a a b b
b y y
s
y y
b y y
s
 


   



 
 
 
   
 
 
 
 


  



 
 
(5)
From equation (4) and (5), we obtain a state equation:
1
x Mx + Nx Q D u

  
 
(6)
where
 
T
1 2
x xx
;
T
u uua b
 
 
 
 
1 0
0 2
M
srm
s r
r
r
RR
BL
J L L
R
L
b
s
 
 
  
 
 
 
 
 
 
 
 
2
1 0
0 2
srm
s r
rm
r
RR
BL
J L L
RL
L
b
s
b
 
 


 

 

 


 
 
 
 
 


 


 


 
 
 
N
 
 
 
 
 
 
11 2
2
2 2
1
1
2 1
2 2
r s r s sr L L
m
s r
s
r r
m m r r r r
r s r
r r
m r s r s m s s
r r
Kx i i R
K x x R T T
L
J J L L J J
R
R R
L L i i
L L L
R R
L x i i L i i
L L
a a b b
a a b b
a b b a a b
y y bb
s
b y y
s
y y
 
 
 
 


    

 



 
 
 
 
 
 
 
   
 
 
 
 
 
 
 


 

   

 


 
 
 
Q


Acta Polytechnica Hungarica Vol. 12, No. 2, 2015
– 183 –
11
2 2
r r
r m r m
sr r
r r
K K
J J
R L R L
L
L L
b a
a b
y y
sy y

 
 

 
 
 
 
 
 
D
B,J,
r
R
and variable load TLare uncertain parameters such as:
ˆ
ˆ
r r r
B B B
J J J
R R R
 
 
 

ˆ ˆ ˆ
, , r
B J R
are known parameters.
, , r
J B R  
are unknown parameters.
From known parameters, the components of flux
ˆ ˆ
,
r ra b
y y
can be determined
following the equation:
ˆˆ ˆ
ˆ ˆ
ˆˆ ˆ
ˆ ˆ
r r r
r r m s
r r
rr r
r r m s
r r
d R R L i
dt L L
dR R L i
dt L L
a
a b a
b
a b b
yy wy
ywy y



   






  




(7)
Matrices in equation (6) can be represented as follows:
ˆ ˆ
;
ˆˆ
;
N = N +ΔN M = M + ΔM
Q = Q + ΔQ D = D + ΔD
(8)
where
ˆˆ ˆ ˆ
, , ,Q D M N
are known matrices;
, , ,Q D M N   
are unknown.
 
ˆ
ˆ1 0
ˆ
ˆˆ
0 2
srm
s r
r
r
RR
BL
L L
J
R
L
b
s
 
 
  
 
 
 
 
 
 
 
M
 
2
ˆ
ˆ1 0
ˆ
ˆˆ
0 2
srm
s r
rm
r
RR
BL
L L
J
RL
L
b
s
b
 
 


 

  
 


 
 
 
 
 
 




 



 
 
 
N

H. L. Le et al. An ANN-based Speed and Flux Controller of Three-Phase AC Motors with Uncertain Parameters
– 184 –
   
 
 
 
 
2 2
1 1
2
2 2
1
ˆ ˆ ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ ˆ
ˆ2 1
ˆ ˆ
ˆ ˆ
2 2
r s r s r r
s
r r
m m r r r r
r s r
r r
m r s r s m s s
r r
Kx i i K x
J J
R
R R
L L i i
L L L
R R
L x i i L i i
L L
a a b b a b
a a b b
a b b a a b
y y b y y
b y y
s
y y
 
  
 
 

 
 
 
 
 
 
   
 
 
 
 
 
 
 
 
 



   
 



 

 
 
Q
 
2 2
ˆˆ ˆ
2
ˆˆ
ˆˆ
ˆ ˆ
ˆ
2ˆ ˆ
2ˆ
r m r r
r
s r
r m
m r r r r r
r
R L K
L
L L J J
R L
KL R K
LJ
D
b a
a b
a b
y y
s
y y y y
 
 

 
 
 
 
 
 
 
Let us choose
 
ˆ
ˆ
u D v Q 
(9)
with
T
vv v
a b
 
 
 
being an augmented control signal.
Substituting equation (9) into (6) we obtain:
ˆ ˆ
v x Mx+ Nx f  
 
(10)
with
1 1 ˆ
f = ΔMx +ΔNx D Dv D DQ Q
 
    

being an unknown element
that can be estimated later.
In summary, the motor control problem becomes determining the control signal v
that regulates motor speed and motor flux reach their respective desired values
ref
w w
,
 
2 2 2 2 refr r r ra b
y y y y  
where
, , r
J B R
and changeable load
L
T
are
unknown.
3 Speed and Flux Control Method for AC Motors
with Uncertain Parameters
We denote:
s = e +Ce

(11)
where Cis the positive definite diagonal matrix;
ref
e x x 
is the error between
the actual value
 
T
T2
1 2 r
x xxw y
 
  
 

Acta Polytechnica Hungarica Vol. 12, No. 2, 2015
– 185 –
and the desired value
 
T
T2
ref 1ref 2ref ref refr
x xxw y
 
  
 
.
Therefore, when , then
e 0
.
From equation (10), fis an unknown function which includes physical motor
parameters such as flux, current, voltage and speed. However, in practice the
variation of these parameters can be considered bounded and continuous. The
motor speed and flux are bounded quantities, so fis also bounded and continuous:
max
ff
. The solution is to determine the control signal vwhich drives error e
to approach 0when
lim ( )
tt
 e 0
without knowing fexactly. This corresponds to
finding the control signal vassuring
lim ( )
tt
 s 0
. Applying the universal
approximation capacity of artificial neural networks for continuous, bounded
unknown nonlinear functions, we can use an artificial neural network with self-
adaptation to approximate the unknown parameter fof system (10) based on
known signal s(t). From [10], the artificial neural network structure is an RBF
network. We chose a RBF network as seen in Figure 1 with two inputs, two
outputs and three layers to approximate f. The input layer of the neural network
consists of the two elements of s(t) and the output layer has two linear neurons.
The hidden layer is composed of two neurons having the following Gauss
distribution function:
 
2
2
exp ; 1,2
j j
j
j
s c jql
 




  





 
where cj,

jare the expectation and variance of the Gaussian distribution function
that are freely chosen.
Figure 1
The neural network structure
The form of the neural network:
ˆ
f f ε Wθ ε
   
(12)
s 0
1
s
1

2

Σ
w11
w22
w12
w21
Σ
2
s
2
2 2
1i i
i
f w

=
=∑
2
1 1
1i i
i
f w

=
=∑

H. L. Le et al. An ANN-based Speed and Flux Controller of Three-Phase AC Motors with Uncertain Parameters
– 186 –
where
11 12
21 22
w w
w w
 
 
 
 
W
is a weighted matrix;
 
T
1 2
θq q
is an output function vector of input neuron;
ε
is a bounded approximation error
0
eε
.
Therefore, to make
s 0
and error
e 0
, we need to choose vand the learning
rule for the weighted
W
to make the system (10) asymptotically stable.
Theorem: Speed and flux of the AC motor in equation (2) approach the desired
values
ref
w w
and
 
2 2 2 2 refr r r ra b
y y y y  
while
,J
,B
r
R
and changeable
load
L
T
are unknown if the control signal vand weighted
W
are defined as
below:
ref
ˆ ˆ 1
v Hs Mx Nx x Ce v      

 
(13)
 
11s
v W
θs
m g  
(14)
w s
i i
m q

(15)
where His a positive definite diagonal matrix,
i
w
is the ith column of the
weighted matrix
W
,
0m
and
0
g e r 
with
0r
.
Proof:
Applying Lyapunov’s stability theory, we chose a positive definite function V
such as:
2
T T
1
1 1
2 2
s s w w
i i
i
V

  
(16)
Taking the derivative of both sides of the equation (16) yields:
2
T T
1
s s w w
i i
i
V

  

(17)
Substituting derivatives
,s w

into (17) yields:
 
 
T T
ref ref
s x x C x x w s
i i
i
Vm q     
   
(18)
From equation (10), (12), (13) and (18), we obtain:
 
T T 11s Hs s v W θ ε
Vm
 
     
 
 

(19)

Acta Polytechnica Hungarica Vol. 12, No. 2, 2015
– 187 –
Substituting equation (14) into (19), results in:
 
T
T T
T T 0
T T
0
. .
0
Vs s
s Hs s ε
s
s Hs s s ε s Hs s s
s Hs s s Hs s
g
g g e
g e r
   
       
       

(20)
It is clear that
0V

when
s 0
and
0V

if and only if
s 0
. Following
Lyapunov’s theory, we have
s 0
and error
e 0
. Therefore,
ref
x x
. In
other words, rotor speed and flux converge to their respective desired values with
error
e 0
.
Figure 2 shows the overall motor control system.
Figure 2
The overall motor control system
4 Simulation
Simulation was conducted using a four-pole squirrel-cage induction motor from
LEROY SOMER with the parameters shown in Table 2. The reference angular
velocity
ref
w
varies in a trapezoid shape as seen in Figure 3 with the maximum
speed
ref 100w
(rad/s) and reference flux
 
f22re 2.25 Wb
r
y
.
Motor model
Rotor flux
estimator
Speed and flux
controller
2
e
L
T
+
1
e
+
s
ub
s
ua
ws
u
sv
u
2
refr
y
2
ˆ
r
y
w
ˆra
y
ˆrb
y
-
abc

abc

ref
w
su
i
sv
i
s
ia
s
ib
su
u
-
-

H. L. Le et al. An ANN-based Speed and Flux Controller of Three-Phase AC Motors with Uncertain Parameters
– 188 –
Table 2
Motor parameters
Rated power
1.5 KW
Rated stator voltage
220/380 V
Rated stator current
6.1/3.4 A
Stator resistance (Rs)
4.58 Ω
Rotor resistance (Rr)
4.468 Ω
Stator inductance (Ls)
0.253 H
Rotor inductance (Lr)
0.253 H
Mutual inductance (Lm)
0.113 H
Motor inertia (J)
0.023 Nms2/rad
Viscous coefficient friction (B)
0.0026 Nms/rad
Figure 3
Desired rotor speed
ref
w
The motor speed control system was simulated with these assumed uncertain
parameters:
 
 
2
ˆ ˆ
; 0.85 ; 0.15 Nms/r
ˆ ˆ
; 0.85 ; 0
ad
Nms /rad.15
B B B B B B B
J J J J J J J
     
     
When the unknown changeable load was formulated as
ˆ; 1.5sin(2 ) 0.5sin(50 )
L L L L
T T T T t t     
(Nm)
L
T
had an amplitude change over time as seen in Figure 4a) and b).
0 5 10 15 20 25 30 35 40 45 50
0
20
40
60
80
100
Time (s)
Rad/s

Acta Polytechnica Hungarica Vol. 12, No. 2, 2015
– 189 –
Figure 4
Simulation with control signal using neural networks and direct rotor speed feedback signal:
a) Load changes suddenly; b) Load
L
T
changes
The coefficients of the neural network were calculated as follows:
20, 0.5, 0.001, 300
j j
cm l g   
200 0 ,
0 200
H 
 
 
 
200 0
0 200
C 
 
 
 
The simulation results are shown in Figure 5 to Figure 9.
Based on the simulation results using the neural network shown in Figure 5 to
Figure 7, rotor speed and rotor flux were close to the desired values. When the
load changed suddenly while the motor was operating normally, speed and rotor
flux had a transient period with an error of about 1.6% to rotor angular velocity
and 0.1% to rotor flux.Then, they converged rapidly to the desired speed and flux.
The results without using the neural network (v1= 0) are seen in Figure 8 and
Figure 9 which show that rotor speed and flux could not be maintained close to
desired values at times when load changed suddenly. Error of rotor angular
velocity was about 1.6% and that of the rotor flux about 0.5%.
This proves that the self-adaptive capacity of the system and the efficiency of the
proposed control method using ANN with an online learning algorithm
compensated for the impact of uncertain parameters and load changes.
a)
b)
0 5 10 15 20 25 30 35 40 45 50
-2
0
2
4
6
8
10
Time(s)
Nm

H. L. Le et al. An ANN-based Speed and Flux Controller of Three-Phase AC Motors with Uncertain Parameters
– 190 –
Figure 5
Real rotor speed
w
Figure 6
Error e1between desired rotor speed
ref
w
and real rotor speed
w
Figure 7
Error e2between desired flux
2refr
y
and estimated flux
2
r
y
Figure 8
Error e1between desired rotor speed
ref
w
and real rotor speed
w
when
1v 0
0 5 10 15 20 25 30 35 40 45 50
0
20
40
60
80
100
Time (s)
Rad/s
0 5 10 15 20 25 30 35 40 45 50
-1
-0.5
0
0.5
1
Time (s)
Rad/s
0 5 10 15 20 25 30 35 40 45 50
-5
0
5x 10-3
Time (s)
Wb2
0 5 10 15 20 25 30 35 40 45 50
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
Rad/s

Acta Polytechnica Hungarica Vol. 12, No. 2, 2015
– 191 –
Figure 9
Error e2between desired flux
2refr
y
and estimated flux
2
r
y
when
1v 0
Conclusion
This paper proposes an adaptive, non-decoupling control method based on an
ANN, for speed and flux control of AC motors, with uncertain parameters. Global
asymptotic stability of the overall control system is proven by Lyapunov’s direct
method. The proposed speed and flux control method performs well while friction,
moment of inertia, unknown rotor resistance and load change significantly in the
AC motor dynamic model. The simulation results clearly show the efficiency of
the proposed method.
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– 192 –
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