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Bejomeh, Mohammad Yaghoubi, Ganji, Behnam and Kouzani, Abbas Z. 2012, Evaluation of
analogue PID, digital PID and fuzzy controllers for a servo system, International review of
automatic control, vol. 5, no. 2, pp. 255-261.
Available from Deakin Research Online:
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Copyright: 2012, Praise Worthy Prize S.r.L.
International Review of Automatic Control (I.RE.A.CO.), Vol. 5, N. 2
ISSN 1974-6059
March 2012
Evaluation of Analogue PID, Digital PID
and Fuzzy Controllers for a Servo System
Mohammad Yaghoubi Bejomeh, Behnam Ganji, Abbas Z. Kouzani
Abstract – This paper presents design and experimental evaluation of an analogue PID, a digital
PID and a fuzzy controller for a servo system. First, a servo system model is presented and the
stability of the system is described. Next, a PID controller is devised and then tuned by the
Ziegler-Nichols method for controlling the servo system. Then, a digital PID is formulated and
subsequently tuned with the gradient descent method for controlling the system. Afterward, the self
fuzzy tuning method is applied to the digital PID controller. Finally, to improve the servo system’s
dynamic response parameters, a fuzzy controller is proposed for controlling the system.
Performance comparisons between the controllers are carried out. The results are presented and
discussed. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.
Keywords: Servo System, Fuzzy Controller, Dynamic Response Parameters
In practice, most systems are non-linear for which PID
controllers give an inexact control. Fuzzy control is a
powerful approach that can provide a better solution for
the non-linear and imprecise models. The advantages of
the fuzzy control include its robustness due to its
tolerance to imprecise measurements and component
variations. Fuzzy rules can be easily tuned, if required,
therefore adaptation is an additional benefit of the fuzzy
control. Also, the fuzzy control has been used for high
order and time delay systems [6]-[7].
This paper presents design and experimental
evaluation of an analogue PID, a digital PID and a fuzzy
controller for a servo system. It investigates the
capability of the fuzzy approach in tuning of a digital
PID controller. In addition, it compares the performance
of a fuzzy controller against that of a conventional PID
controller on a servo system plant.
The paper is organized as follows. First, a servo
system is considered and its stability is discussed. Next, a
PID controller is formulated and tuned by the ZieglerNichols method for controlling the servo system. Then, a
digital PID is designed and tuned with the gradient
descent method for controlling the system. After that, the
self-fuzzy tuning method is used for tuning the digital
PID controller. Finally, a fuzzy controller is proposed for
controlling the system. Simulations are performed, and
the results are presented and discussed. A comparison of
the performance of the presented controller is given.
Nomenclature
a0, a1, a2
C
e
IAE
k, K
Kp
Ki
Kd
R
T
TF, H, G
τ
V
ωc
ωs
Constant parameters of digitized controller
Capacitance
Error
Integral absolute error
Constant
Proportional gain
Integral gain
Derivative gain
Resistance
Sampling period
Transfer function
Time constant
Voltage
Cut-off angular frequency
Sampling angular frequency
I.
Introduction
The most frequent controller used in industrial control
systems is the proportional-integral-derivative (PID)
controller. It has been reported that more than 95% of the
controllers used in industrial process control scenarios
are PID type [1]. The advantages of PID controllers
include their simplicity, clear functionality, applicability
and ease of use [2].
PID controllers provide robust and reliable
performance for most systems, if their PID parameters
are tuned properly. Various tuning methods are reported
in the literature [3]-[4]. The most practical tuning method
is the Ziegler-Nichols (Z-N) technique.
Also, the decent gradient method has been used for
tunning digital PID controllers [5]. However, the
application of PID controllers is limited to a precise
linear model of a plant.
II.
A Servo System and its Stability
A servo system is an example of a single-input-singleoutput system in which a DC motor converts electrical
power to mechanical movement [8]. A sensor is
employed to compare the output of the system against a
Manuscript received and revised February 2012, accepted March 2012
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
255
M. Y. Bejomeh, B. Ganji, A. Z. Kouzani
reference input. The amplitude of the error signal is used
to determine whether the output is at a desired value or
not. A block diagram description of the system [9] is
illustrated in Figs. 1.
The transfer function for time constant part circuit is:
1
+_
Fig. 2. Standard closed-loop control system
Although the system is stable, its response to the input
step shows oscillations, and long settling time, and high
overshoot (Fig. 4). To reduce the oscillations, improve
the settling time, and decrease overshoot, a PID
controller is developed.
where V1 is the input voltage, V2 is the output voltage, R
is the resistance and C is the capacitance. The ratio of the
output to the input voltage in the integrator part can be
shown as follow:
1/
1
G(s)
H(s)
(1)
1
Y
r
(2)
where τ = RC is the time constant and considered to be
0.04 s.
The studied system is compared against a standard
closed-loop control system shown in Fig. 2. From the
closed-loop system, the parametric transfer function is as
follow:
1
·
(3)
where H(s) is 1 and G(s) is L{Gain} × L{Time Constant}
× L{Integrator}. The entire transfer function for the
system is:
1
0.04
0.04
Fig. 3. Nyquist plot
(4)
If the value of K, R, and C are 1, 100kΩ and 1µF,
respectively, then:
250
10
250
(5)
To investigate the stability of this second order
system, the Nyquist method [10] is applied, and the result
is presented in Fig. 3.
Fig. 4. Servo system response
III. Analogue PID Controller
The transfer function of an analogue PID controller
can be written as follows:
(a)
(6)
The Ziegler-Nichols (Z-N) method [10] is used to find
the parameters of the PID controller. The results are as
follows: KP = 0.4, Ki = 0.032, KD = 0.057. The step
response to the system with the PID controller in loop,
and with the listed tuned parameters is shown in Fig. 5.
(b)
Figs. 1. Structure of the closed-loop servo system and associated model
[9] (a) Position control system. (b) Time Constant and integrator
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
International Review of Automatic Control, Vol. 5, N. 2
256
M. Y. Bejomeh, B. Ganji, A. Z. Kouzani
(13)
1
Bode plot method is employed to extract sampling
period [10]. The Bode diagram of the servo system is
presented in Fig. 6.
Fig. 5. Servo system response with the PID controller in loop
Comparing Fig. 4 and Fig. 5, the rise time, settling
time and overshoot are improved by 115%, 5.8% and
72%, respectively. Moreover, the oscillation is omitted
relatively.
IV.
Frequency (rad/s)
Digital PID Controller
Fig. 6. Bode Diagram of G(s)
In modern control problems, more common PID
controllers are digital controllers. In order to investigate
the performance of a digital PID controller against the
analogue PID controller described in Section 3, the servo
system is considered in a digital domain. Therefore, the
transfer function of the system and the PID controller
from s domain need to be transferred to z domain. The
zero order hold [11] is applied in conversion from
analogue to digital. Thus, the plant transfer function can
be driven from:
Using the cut-off frequency and analogue system
frequency, sampling time can be determined via the
following relations:
2
2
22.8 rad/s
1
(15)
0.12 s
(16)
(7)
In this study, the sampling time is considered to be
0.01s.
In order to find optimal values of a0, a1 and a2, the
steepest descent gradient method (SDG) is used [5]. In
the SDG method, an error function is defined as follow:
The digital transfer function of the servo system and
the PID controller can be given as follows:
25
(14)
1
1
2.5 1
1
(8)
Integral Absolute Error (IAE) =
∑ k ek
(17)
(9)
1
where: ek = rk – Yk.
The problem parameters are a0, a1 and a2. The goal is
to find the parameters such that the error is minimised.
Respectively, the SGD method is used to solve the error
function for finding the parameters for which the error is
in its minimum value. The sketch of the minimum error
is shown in Fig. 7, and the value of the parameter in the
minimum error value is a1= 0.1.
The step response is obtained for the servo model with
the digital PID controller as shown in Fig. 8. As can be
seen from Fig. 8, the digital closed-loop system which
includes the digital PID controller tuned by the SDG
method has achieved a better overshoot of 100% and
settling time of 6.3% in the expense of a worsen rise
time.
where the constant parameters in the digitized PID can be
shown as follows:
(10)
2
2
2
(11)
(12)
where T is the sampling period in seconds. The digital
closed-loop system can be presented as follows:
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
International Review of Automatic Control, Vol. 5, N. 2
257
M. Y. Bejomeh, B. Ganji, A. Z. Kouzani
The most frequent method of fuzzy inference system,
Mamdani model, is applied for inferring. The fuzzy
inference block is shown in Fig. 10.
Fig. 7. Error sum by changing a1
in the steepest descent gradient method
Fig. 10. Fuzzy inference block
In order to achieve a feasible range for the PID
parameters, the controller is simulated with the servo
model with different parameter quantities. Respectively,
Kp, Ki and Kd are considered in the range of [Kp min, Kp
max], [Ki min, Ki max] and [Kd min, Kd max]. The
values of e(t) ,ec(t), Kp, Ki and Kd are interpreted as the
linguistic variables such as, Positive Big (PB), Positive
Middle (PM), Positive Small (PS), Zero (Z), Negative
Small (NS), Negative Middle (NM), and Negative Big
(NB). The alterations of membership functions are
defined based on the influence on the control
performance. All of the fuzzy subsets of the inputs and
the outputs of the fuzzy controller are represented by
triangular membership functions. A sample of such
membership functions is presented in Fig. 11.
Fig. 8. System step response for the discrete PID controller
with the tuned parameters by SGD
V.
Fuzzy Self-Tuning PID Controller
The SDG method is a recursive and time consuming
approach. It also needs an accurate estimation of the
plant transfer function. To reduce the SDG calculation
burden and the reliance on the plant model, a fuzzy selftuning method can be implemented with the PID
controller.
In the fuzzy self-tuning method, there exist two inputs
and three outputs. The inputs are the error, e(t), and the
error gradient, ec=de(t)/dt. The outputs are PID
coefficients, Kp, Ki and Kd. A conceptual presentation of
the PID self-tuning in a closed-loop system is shown in
Fig. 9. In the fuzzy self-tuning of the input signals, the
error and derivation of error are converted to fuzzy
parameters. Then, fuzzy inference provides a nonlinear
mapping from the inputs to the PID coefficients.
de/dt
KD
e/ec
NB
NM
NS
ZO
PS
PM
PB
KI
Y(t)
+
PID Controller
-
To figure out the fuzzy rules, the practical experience
approach is implemented. Accordingly, the fuzzy rules
are formulated as they are shown in Tables I, II and III.
They are based on the offered rules in previous studies
[12]-[13]. For instance, the ith rule is presented as
follow: “Rule i: If e(t) is A1i and de(t) A2i then Kp = Bi
and Ki = Ci and Kd = Di.”
To be able to achieve the accuracy for each parameter,
49 fuzzy rules are formed.
Fuzzy Controller
KP
R(t)
Fig. 11. Membership functions of the linguistic variables
for e, ec, Kp, Ki and Kd
Plant
e
NB
PB
PB
PM
PM
PS
PS
ZO
NM
PB
PB
PM
PM
PS
ZO
ZO
TABLE I
FUZZY RULES OF KP
NS
ZO
PS
PM
PM
PS
PM
PS
PS
PM
PS
ZO
PS
ZO
NS
ZO
NS
NS
NS
NM NM
NM NM NM
PM
ZO
ZO
NS
NM
NM
NM
NB
PB
ZO
NS
NS
NM
NM
NB
NB
Fig. 9. Structure of the fuzzy self-tuning PID controller
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
International Review of Automatic Control, Vol. 5, N. 2
258
M. Y. Bejomeh, B. Ganji, A. Z. Kouzani
e/ec
NB
NM
NS
ZO
PS
PM
PB
TABLE II
FUZZY RULES OF KI
NM
NS
ZO
PS
NB NM NM NS
NB NM
NS
NS
NM
NS
NS
ZO
NM
NS
ZO
PS
NS
ZO
PS
PS
ZO
PS
PS
PM
ZO
PS
PM PM
NB
NB
NB
NB
NM
NM
ZO
ZO
PM
ZO
ZO
PS
PM
PM
PB
PB
controller, whose rules are based on empirical knowledge
of experts, can give a proper solution to a nonlinear plant
[14]-[15]. The advantage of this system is its robustness,
because it is tolerant to imprecise measurements and
component variations. The associated fuzzy rules can be
tuned, if required; therefore, adaptation is an additional
benefit of this controller. The structure of the fuzzy
controller is shown in Fig. 14.
PB
ZO
ZO
PS
PM
PB
PB
PB
R(t)
e/ec
NB
NM
NS
ZO
PS
PM
PB
NB
PS
PS
ZO
ZO
ZO
PB
PB
TABLE III
FUZZY RULES OF KD
NM
NS
ZO
PS
NS
NB
NB
NB
NS
NB NM NM
NS
NM NM
NS
NS
NS
NS
NS
ZO
ZO
ZO
ZO
NS
PS
PS
PS
PM
PM
PM
PS
+
PM
NM
NS
NS
NS
ZO
PS
PS
PB
PS
ZO
ZO
ZO
ZO
PB
PB
e
de/
dt
Fuzzy
Controller
Y(t)
Plant
Fig. 14. Structure of the fuzzy controller
The configuration of the implemented fuzzy controller
in this study is presented in Fig. 15. The inputs are the
classical error, e(t) and the rate of the change of error
(de(t)/dt). The fuzzy rules based on empirical knowledge
of expert are shown in Tables IV [16]-[17]. In order to
have a fair comparison between the performance of the
proposed fuzzy controller and the conventional PID
controller, the controller with the servo system is
modelled and simulated in Simulink as shown in Fig. 16.
The proposed fuzzy self-tuning PID controller with
the servo system plant is modeled and simulated in
Simulink as shown in Fig. 12. The step response for this
model is obtained as it is shown in Fig. 13.
Fig. 12. Subsystem model of the fuzzy self-tuning PID controller
Fig. 15. Fuzzy inference block
From the step response, it can be seen that the
overshoot, rise time and settling time, comparing to those
of the SDG method, are improved by 27% in rise time,
23% in settling time and the same in overshoot.
e/ec
PL
PM
S
ZR
NS
NM
NL
TABLE IV
FUZZY CONTROLLER RULE-BASED [17]
NL
NM
NS
ZR
PS
PM
ZR
PS
PM
PL
PL
PL
NS
ZR
PS
PM
PL
PL
NM
NS
ZR
PS
PM
PL
NL
NM
NS
ZR
PS
PM
NL
NL
NM
NS
ZR
PS
NL
NL
NL
NM
NS
ZR
NL
NL
NL
NL
NM NS
PL
PL
PL
PL
PL
PM
PS
ZR
Fig. 16. Subsystem model of the fuzzy controller
The response of the system to the step input is
obtained and the result is shown in Fig. 17. As can be
seen from the response, the system response parameters,
overshoot, rise time and settling time, in comparison with
the results of the digital PID tuned by the SDG method
are enhanced by 34% in rise time, 40% in settling time
and nearly same in overshoot (0).
Fig. 13. Simulated fuzzy self-tuning PID controller system response
VI.
Fuzzy Controller
The conventional PID controller is not an effective
method for controlling nonlinear systems. A fuzzy
Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved
International Review of Automatic Control, Vol. 5, N. 2
259
M. Y. Bejomeh, B. Ganji, A. Z. Kouzani
applied with the plant. The controller is tuned by the
Zigler-Nicols method. Then, the plant with a digital PID
controller is modelled in the discrete domain. The
discrete controller is tuned by the steepest descent
gradient method.
Also, the PID is tuned by a fuzzy self-tuning
approach. Finally, a fuzzy controller is applied in the
closed-loop system and the model is simulated. The
output response for the step function input is obtained.
The rise time and settling time for the fuzzy controller
proves enhancement of 34% in rise time, 40% in settling
time in comparison with the digital PID.
References
Fig. 17. The simulated fuzzy controller system response
VII.
[1]
Discussions
[2]
In order to compare the results of this study, the
response time to the step signal for the controlled servo
system with the PID controller by different tuning
methods, and the fuzzy controller are shown in Fig. 18.
The dynamic parameter of the response such as rise time,
settling time and overshoot are shown for different
scenarios in Table V. From the table, it can be seen that
the application of the fuzzy controller on the system
enhances the settling time and rise time of step response.
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
Fig. 18. The system response of all controllers
[12]
TABLE V
COMPARISON OF THE RESPONSES OF THE CONTROLLERS
Settling
Rise Time
Overshoot
Controller
Time
Fuzzy Controller
0.4415
0.6944
0.009
Fuzzy Self-Tuning
0.4868
0.8851
0
PID
PID (SDG)
0.6655
1.1575
0
[13]
[14]
[15]
VIII. Conclusion
In this paper, the transfer function of a servo system is
extracted. Then, its stability is investigated. In order to
improve the response parameters such as oscillation, rise
time, overshoot and settling time, a PID controller is
[16]
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Authors’’ information
1,2,3
School of Enngineering, Deakkin University, Geelong, Victoria 3217,
Australia.
bgaanj@deakin.edu.aau
E-mails:
kouuzani@deakin.eduu.au
kouuzani@deakin.eduu.au
Mohamm
mad Yaghoubi Bejomeh
B
receiveed his
B.S. deggree in Electronics Engineering from
Inistute of
o Sadjad, Iran, 2008, and his M.Eng.
M
degree inn electrical and electronics
e
engineeering
from Deaakin University, Australia,
A
2011.
Behnam Ganji receivedd his B.S. degrree in
Engineerinng
from
Issfahan
Electroniics
Universitty of Technologyy, 1993, and his M.S.
degree inn Industrial Engiineering with hoonours
from Am
mirkabir Universitty of Technology, Iran,
2003. Hee is currently a PhD student witth the
School of Engineering,, Deakin Univeersity,
Australiaa.
Abbas Kouzani
K
receivedd his B.Sc. degrree in
computerr engineering from
m Sharif Universsity of
Technoloogy, Iran, 19900, M.Eng. degreee in
electricall and electronics engineering from
m the
Universitty of Adelaide, Australia, 19955, and
Ph.D. degree in electrrical and electrronics
engineeriing from Flinderss University, Ausstralia,
1999.
He was a lecturrer with the Schoool of Engineerinng, Deakin Univeersity,
and then a Seniior Lecturer withh the School of Electrical Engineeering
and Computer Science,
S
University of Newcastlee, Australia. Currrently,
he is an Associiate Professor wiith the School off Engineering, Deakin
D
University. He has been invoolved in several ARC, industryy, and
m
than 150 pubblications. His ressearch
university researrch grants, and more
interests includee intelligent microo electro mechaniical systems.
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