Self-Tuning DC Servo Motor Design Based on Radial Basis

Transcription

INTERNATIONAL JOURNAL OF CONTROL, AUTOMATION AND SYSTEMS
VOL.4 NO.2
ISSN 2165-8277 (Print) ISSN 2165-8285 (Online)
http://www.researchpub.org/journal/jac/jac.html
April 2015
Self-Tuning DC Servo Motor Design Based on Radial Basis Function
Neural Network
Abd-Elmeged mohamed 1, Gaber Elsaady 2, Ashraf Hemeida3, Asmaa Fawzy4*

Pal and youssef [15, 16] proposed a simple self-tuning
scheme for PI-type fuzzy logic controllers (FLCs) for a real
time water pressure control system and induction motor speed
drive improvement. Zaky [17] proposed a self-tuning PI
controller for the speed control of electrical motor drives and its
parameters will be updated online depending on the speed error.
Kota [18] used PID controller and fuzzy logic controller for
control separately excited dc motor. Fuzzy self-tuning PID has
better dynamic response curve, shorter response time, small
overshoot, and small steady state error compared to the
conventional PID controller. Fawaz [19] presented a simulation
and hardware implementation of a closed loop control of a
separately excited dc motor using a self-tuning PID controller.
It gives very acceptable results in the reduction of overshoot,
stability time and the steady-state transient response of the
controlled plant. Saad [20] proved that the proposed Neural
Network (NN) self-tuning PID controller is more efficient to
control the robot manipulator to follow the desired trajectory
compared to classical tuning method of PID controller. Alfonso
[21] introduced a new self-tuning algorithm is developed for
determining the Fourier Series Controller coefficients with the
aim of reducing the torque ripple in a Permanent Magnet
Synchronous Motor (PMSM), thus allowing for a smoother
operation. This algorithm adjusts the controller parameters
based on the component's harmonic distortion in time domain
of the compensation signal also [22] used NNPID like
controller which is tuned when the controller is operating in an
on line mode for high performance permanent magnet
synchronous motor PMSM position control.
In this paper a new technique is proposed that gives a good
control for the dc motor. An online control algorithm is
structured using the radial basis function neural network
(RBFNN). The dc motor parameters are estimated on line and
are used to update the weights of the RBFNN. The weight
update equations are derived based on the least mean squares
principle. The RBFNN virtually models the inverse of the dc
motor and thus the output tracks the reference trajectory. This
scheme is exposed to several types of disturbances for wide
range of operating conditions. The armature resistance and
rotor inertia for the dc motor are varied due to the temperature
deviation and also, take in account the changing of disturbance
torque. The rest of the paper is organized as follow: In section
2 presents the proposed structure. The simulation results are
presented in Section 3 and the conclusion is presented in
section 4.
Abstract—This paper introduces the inverse control design using
neural network based self tuning regulator (STR) for control the
DC servo motor. The controller is the radial basis function
neural network (RBFNN) and acts as inverse of the DC servo
motor. The dc motor parameters are estimated online using the
system identification method where uses the Auto-regressive
moving average (ARX) model which depends on the input and
output values of the DC servo motor. The difference between the
output of the dc motor and the reference signal is used to adjust
coefficients of ARX model. These coefficients of ARX are used to
update the weights of the RBFNN. The weight update equations
are derived based on the least mean squares principle. The speed
output tracks the reference trajectory though the self tuning
regulator (STR) structure exposed to different types of
disturbances for wide range of operating conditions. Then
compared
its
result
with
the
result
of
using
proportional-plus-integral feedback (PI) self tuning regulator.
Keywords — STR, RBFNN, ARX, LMS, DC servo motor
I. INTRODUCTION
A
daptive control schemes are used for the control of plants,
where the parameters of the plant are not known exactly or
slowly time varying. Some reasons for using Adaptive control
such as variations in process dynamics and variations in the
character of the disturbances [1, 2]. Enzeng and others [3]
present a neural network based self tuning PID controller for
autonomous underwater vehicle, the control system consists of
neural network identifier and neural network controller, and the
weights of neural networks are trained by using Davidon least
square method, also[4].
Neural network (NN) is a good structure for control the
nonlinear plants and has many types [5, 6]. Kumar [7] used
neural network for modeling the retention process and as
controller. In this paper, we used the RBFNN as a controller.
This type is faster one and uses least number of neurons at
hidden layer [8, 9]. The inverse control means that the
controller (RBFNN) acts the inverse of the plant (dc motor) so
the output tracks the reference input [10].
The DC motors have been extensively used in control
systems. The main advantages of dc motors are easy speed or
position control and wide adjustable range Therefore, DC
motors are often used in a variety of industrial applications such
as robotic manipulator, where a wide range of motions are
required to follow a predetermined speed or position trajectory
under variable load [11,12,13]. Sabahi [14] used a new adaptive
and nonlinear control based on neural network approaches, this
method has been named feedback error learning (FEL)
approaches, that classical controller is used for training of
neural network feedforward controller.
II. PROPOSED STRUCTURE
Figure (1) is the proposed structure. Autoregressive with
exogenous input (ARX) is used to identify the dc motor and
found the model. The model coefficients are updated online
depending on dc motor parameters variation. These coefficients
are fed the weight update block which trains the controller
1
Engineering faculty, Aswan university, Egypt; 2Engineering faculty, Assiut
university, Egypt; 3,4Energy engineering faculty, Aswan university, Egypt
*
Correspondence to (e-mail: [email protected]).
1
VOL.4 NO.2
April 2015
whether RBFNN or PI controller using the least mean square
LMS algorithm.
Fig. 2: A general RBF network
C. Single-phase full-converter drive
The dc motor dynamic are given by the following
equations:
di (t )
(6)
K w(t )   Ri (t )  L a
 v (t )
Fig. 1: Proposed self-tuning dc motor regulator structure
A. ARX model
The process is modeled by an ARX model [23], whose
output is given by
n
m
y (t )   ai y (t  i)   b j x(t  j )
i 1
K m ia (t )  J
(1)
j 1
Or in terms of
y (t ) 
b
q operator
(2)
B. Radial basis functions neural networks
A single input single output radial basis function neural
network (SISO RBFNN) is shown in Fig. 2. It consists of an
input node r (t ) , a hidden layer with n neurons and an output
node x(t ) . Each of the input nodes is connected to all the nodes
in the hidden layer through unity weights (direct connection).
While each of the hidden layer nodes is connected to the output
node through some weights w1 ,, wn .




n0
(7)
2
D. Parameters estimation for self-tuning of RBFNN/PI
The parameters of the DC servo motor model are estimated
online and are used to update the coefficients of the controller
(weights of the RBFNN / parameters of PI). The
weight/coefficient update equations are derived based on a
recursive scheme (least mean squares principle). This previous
parameters are updated by minimizing the performance index
I given by [9]
(10)
I  12 e2 (t )
(4)
th
ci is the center of i hidden layer node where i  1,2,, n0 , 
is the norm matrix and  (.) is the nonlinear basis function.
Normally this function is taken as a Gaussian function of width
 . The output x(t ) is a weighted sum of the outputs of the
hidden layer, given by
x(t )   wi ( r (t )  ci  )
dw(t )
 bw(t )
dt
If the armature circuit of a DC servo motor is connected to the
output of a single-phase controlled rectifier, the armature
voltage can be varied by varying the delay angle of the
converter, α, as [24]:
2Vm
(9)
Va 
cos 
for 0    

0
 r (t )  cn0
 r (t )  c1 
    
d  
 1 
 2
a
Fig. 3: The block diagram of the dc motor
Each neuron finds the distance d of the input and its center
and passes the resulting scalar through nonlinearity. So the
output of the hidden neuron is given by [8, 20]
2
(3)
 (d )  exp(  12 d 2 )  exp(  12 r (t )  ci )
2
dt
Where w, va , ia , R, L, b, J , K m and K b the rotor speed,
terminal voltage, armature current, armature resistance,
armature inductance, damping constant, rotor inertia, torque
constant and back emf constant, respectively. Fig. 3 describes
the block diagram of the DC servo motor
1
B(q 1 )  d
q x(t )
A(q 1 )
a
(5)
e(t )  r (t )  w(t )
i 1
As we see the radial basis function (RBF) network utilized a
radial construction mechanism. This gives the hidden layer
parameters of RBF networks a better interpretation than for the
multilayer perceptron network MLP, and therefore allows new,
faster training methods.
(11)
Where r (t ) is the reference input signal and w(t ) is the
output speed of the DC motor model. The coefficients of the
ARX model and the weights of the RBFNN/parameters of the
PI are updated in the negative direction of the gradient as,
I
(12)
 ( K  1)   ( K )  
 ( K )
W ( K  1)  W ( K )  
I
W ( K )
PI ( K  1)  PI ( K )  
2
I
PI ( K )
(13)
(14)
VOL.4 NO.2
Where   a1  a n b1  bm  is the parameter vector,
is the weight vector for RBFNN,
W  w1 w2  wn



0
300  t  400 , where the disturbance torque has the range

Td  1.5 for good performance. The previous mentioned
disturbances are applied synchronized. Fig. 8 presents the
output signal in this case, Fig. 8 when R  10 Ω,
at
and Td  1 N.m
J  0.05 Kg.m2
t  300
at 300  t  400 . The speed of DC servo motor has high noise
at the trajectory in the period time where all disturbances
occurred together.
The proposed PI self-tuning regulator (STR) structure
exposed to different disturbances as previous counterpart. Fig.
9 shows the efficient response of the proportional-integral (PI)
to the square input and the error signal between the reference
and the PI self tuning output without disturbance which the
tracking error of RBFNN controller is less than it.
Figure 10 shows the effect of the armature resistance
fluctuation at maximum value R=15Ω at specified period
t  300 , where the resistance has the range 2  R  15 . The
output system mimics the trajectory exactly until the armature
resistance became 15 Ω. But the error between the response of
the system and the reference signal is not smaller than radial
basis regulator.
Figure 11 shows the effect of variance for the rotor inertia at
specified period t  300 . The speed of the dc drive follows the
excitation signal but with the tracking error greater than
RBFNN tracking error; however, the ARX model parameters
value fluctuate sharply and this is not good.
Fig. 12 illustrates the efficient output while this disturbance
reaches to maximum value Td  1.7 N.m at specified period
300  t  400 , where the disturbance torque has the range
Td  1.7 give good DC servo motor speed with small square
error.
The previous mentioned disturbances are applied
synchronized. Fig. 13 presents the output signal in this case, Fig.
13 when R  3 Ω, J  0.03 Kg.m 2 at t  300 and
PI  k p ki vector for the parameters proportional-integral
(PI) and  is the learning parameter. The variable K is used to
show the iteration number of training.
Keeping the regressions of the variables in the system in a
regression vector  as  (t )   w(t  1)  w(t  n) Va (t  d )  Va (t  m  d )
and finding partial derivatives.
I 1 e 2 (t )
(15)

 2 

(16)
 e(t )
(r (t )  w(t ))

 e(t )


1
n
  r (t )   a1q   an q w(t )  
  b1q 1    bm q m q dVa (t ) 


I
 e(t ) (t )

The final parameter update equation will be,
April 2015
(17)
(18)
 ( K  1)   ( K )  e(t ) (t )
(19)
The partial derivatives for the weights are derived as
follows,
I
(20)
 e(t ) B(q 1 )q  d (t )
W
The final weight update equation will be,
(21)
W ( K  1)  W ( K )  e(t ) B(q 1 )q  d  (t )
But the final coefficients update equation of PI will be,
PI ( K  1)  PI ( K )  e(t ) B(q 1 )q  d
(22)
(va (t  1)  (t s  1)va (t ))
t s is the sample time.
III. SIMULATION RESULTS
Td  .3 N.m at 300  t  400 . The PI self tuning controller
The proposed self-tuning regulator (STR) structure exposed
to different disturbances as dc motor does in life. The dc motor
meets to changing in its parameters due to increased
temperature. This paper studies these variations where the
motor model parameters are estimated online and are used to
update the weights of the RBFNN at the same instant.
Fig. 4 shows the efficient response of the RBFNN to the
square input and the error signal between the reference and the
RBFNN output, in the case no disturbance. The changing of
temperature has an impact upon the parameters of a dc motor as
an armature resistance and rotor inertia. Fig. 5 show the effect
of variance for the armature resistance at maximum value
R=14.8Ω at specified period t  300 , where the resistance has
the range 2  R  14.8 . The output system takes the same
trajectory until the armature resistance became 14.9 Ω.
Fig. 6 show the effect of variance for the rotor inertia at
maximum value J  0.08 Kg.m2 at specified period t  300
the output system has small tracking error but the parameters
evolution turn into large values when the rotor inertia increases,
so the inertia has the range 0.02  J  0.08 .
In the life, we must take account of the disturbance torque.
Fig. 7 illustrates the efficient output while this disturbance
reach to maximum value Td  1.5 N.m at specified period
doesn't work well with this kind of disturbance. In aspect of its
small value versus the neural network gives bad result with big
magnitude disturbance.
a square wave input
2
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dc motor output
rad/sec
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the error between the input and the output
x 10
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time(sec)
parameters evolution
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-50
250
time(sec)
300
Fig.4 Tracking trajectory for radial basis self tuning dc motor and the error
signal
3
VOL.4 NO.2
April 2015
square wave input and dc motor output
2
dc motor output and input
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torque disturbance
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time(sec)
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N.m
0
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-1
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time(sec)
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-50
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time(sec)
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-5000
250
300
(a)
Fig.8 The effect of disturbances synchronized on the speed of
radial basis self tuning DC motor system
Fig. 5 the effect of variance for the armature
resistance at t  300 at using radial basis case
R  10
a square wave input
2
2
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0
-2
0
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-2
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6
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dc motor output
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-3
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rad/sec
-2
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time(sec)
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-50
Fig. 6 the effect of variance for the rotor inertia at t  300 at
using radial basis case J  0.09 Kg.m 2
250
time(sec)
300
Fig. 9 Tracking trajectory for PI self tuning dc motor and
the error signal
a square wave input and DC motor speed output
2
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-2
0
0
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-15
2
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x 10
torque disturbance
N.m
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-1
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time(sec)
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time(sec)
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0
-20
-40
Fig .7 the effect of variance for the torque disturbance at
300  t  400 at using radial basis case Td  1N.m
250
time(sec)
300
Fig. 10 The effect of variance for the armature resistance
at t  300 at using PI case R  10
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April 2015
IV. CONCLUSIONS
In this work the RBFNN based STR is implemented for
precise speed control of DC servo motor that updates itself
online. Recursive least mean squares method is used for
training the RBFNN controller or PI controller. The parameters
ARX model chooses arbitrary then updates on line using
recursive algorithm.
From simulation result, the DC servo motor speed mimics
the desired speed with smaller tracking error when using
RBFNN than using PI. The changing of temperature is affected
on the armature resistance and rotor inertia for the DC servo
motor; also we take in account the changing of disturbance
torque. The proposed STR structure exposed to all pervious
disturbances separately and simultaneously.
The adaptive RBFNN self-tuning regulator introduces a
good solution for control the DC servo motor even if the model
meets a different individual disturbances or synchronous
disturbances. The RBFNN is a fast neural network compared
with others type due to using least mean squares principle as
training algorithm. Its structure has two neurons in hidden
layer.
In future work, can be used genetic algorithm to determine
the parameters of the DC servo motor self tuning based on
neural network or on PI.
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time(sec)
0
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100
150
200
20
0
-20
-40
250
time(sec)
300
Fig. 11 The effect of variance for the rotor inertia at t  300 at
using PI case J  0.06 Kg.m2
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-2
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-3
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V. REFERENCES
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x 10
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torque disturbance
N.m
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-0.5
-1
0
50
100
150
200
250
300
time(sec)
0
50
100
150
200
50
0
-50
250
300
Fig. 12 The effect of variance for the torque disturbance at
300  t  400 at using PI case Td  1N.m
2
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time(sec)
0
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Fig. 13 the effect of disturbances synchronized on the speed of PI self
tuning DC motor system
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[22] Vikas Kumar , Prerna Gaur, A.P. Mittal ANN based self tuned PID like
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April 2015
Asmaa Fawzy received B.Sc. of Electrical
Engineering from Aswan Faculty of engineering,
Aswan University in 2002. M. Sc. of Electrical
Engineering from Assiut Faculty of Engineering,
Assiut University in 2008
Gaber El-Saady was born in Egypt. He received his
B.Sc. and M.Sc. degrees in Electrical Engineering
from Assiut University University, Egypt, in 1982 and
1988 respectively and Ph.D. from, Assiut University,
Egypt, in 1995. Associate Professor, Department of
Electrical Engineering , Faculty of Engineering, Assiut
University, 2000 Professor, Department of Electrical
Engineering , Faculty of Engineering, Assiut
University, 2006.
Ashraf Mohamed Hemeida, was born in Elmenia,
Egypt. He obtained his B.Sc., and M.Sc. in Electrical
Power Engineering From faculty of Engineering,
Elminia University, Elmenia, Egypt in 1992 and 1996
respectively. In Nov. 1992, He engaged Higher
Institute of Energy, Aswan as Teaching Assistant.
From 1998 to 1999 He was a full time Ph.D. student
with Electrical and Computer Engineering
Department, University of New Brunswick, Canada. In
Sept., 2000 He obtained his Ph.D. From Electrical Engineering Department,
Faculty of Engineering, Assiut University, Assiut Egypt. From Oct. 2000 till
August 2005 He was Assistant Professor at Electrical Engineering Department,
Higher Institute of Energy, South Valley University, Aswan, Egypt. In Nov.
2005 He obtained Associate Professor rank. Dr. Hemeida is a full professor
since July, 2011. His research interest power systems operational and control,
artificial intelligence application in power systems, FACTS applications,
voltage stability, and Advanced control Techniques in power systems.
Professor Hemeida is a member of editorial board of ARPN journal of
Engineering and Applied Sciences.
6

Self-Tuning DC Servo Motor Design Based on Radial Basis

Transcription

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