Performance Analysis of Multi-objective Genetic Algorithm
Tuned PID Controller for Process Control
Mohit Jain, Vijander Singh & Asha Rani
Division of Instrumentation and Control Engineering,
Netaji Subhas Institute of Technology, University of Delhi,
New Delhi-110078, India
E-mail: nsit.mohit@gmail.com, vijaydee@gmail.com, ashansit@gmail.com
Abstract
–
Proportional-Integral-Derivative
(PID)
controllers are most widely used in industrial plants due to
their ease of implementation and robust performance. This
paper presents an effective and fast tuning method using
Multi-objective Genetic Algorithm (MOGA) to find the
optimized parameters of the PID controller in order to
achieve the required performance specifications of the
processes under consideration. The improved performance
of various plants has been compared with the conventional
tuning methods, to observe the effectiveness of proposed
method.
Keywords – PID controller;
Algorithm; Tuning methods;
I.
Optimization;
overshoot is usually not desired at all. This led Tyreus
and Luyben (T-L) [3] to recommend another popular
conventional tuning approach for more conservative
process loops. In [4] Cohen–Coon proposed a different
tuning method which was based on process reaction
curve. In [5] Kitamori also suggested a new method of
PID tuning. There are several more available in
literature [6]. These conventional approaches are very
popular among control engineers since one can use them
especially when no or little information about the plant
under control is available. These methods provide
stable, robust and quite satisfactory performance but the
gains are never guaranteed for being optimal. Also,
these conventional tuning methods often fail to achieve
satisfactory performance in case of plants having
nonlinearity, higher order or time delay. Hence,
intelligent approaches have been introduced by the
researchers to make the tuning an easier one. In [7] a
new scheme of PID tuning is suggested based on Fuzzy
gain scheduling technique. A neural networks tuned PID
controller by means of fuzzy parameters is presented in
[8]. This paper presents a PID tuning method based on
Multi-objective Genetic Algorithm (MOGA) and its
performance is compared with conventional methods of
tuning. The MOGA tuned PID (MOGA-PID) controller
is tested on various complex processes often found in
control system literatures. So the next section describes
about the complex control processes used in this paper
for testing the performance of MOGA-PID controller.
Genetic
INTRODUCTION
In the generation of modern and advance
controllers, PID controllers are still most widely used
(>90%) in process industries like petrochemical, paper,
pulp, oil, gas and many more. This is due to the fact of
their ease of implementation and robust nature.
Moreover they offer very cost effective, efficient and
reliable control solution even for very complex control
problems. To implement these controllers, three
parameters which are the proportional, integral and
derivative gains should be tuned properly to provide
stable response as well as minimum error while tracking
the input. But their appropriate tuning is very difficult
since these parameters interact with each other and there
is no generalized rule defined for this purpose. However
improper PID tuning can produce the oscillatory and
unstable system behavior or sometimes causes the
system failure [1]. This problem attracted the attention
of the researchers to explore the best possible method to
tune these three parameters.
II. TEST PROBLEMS
To demonstrate the effectiveness of the presented
method three systems are considered [7]. G1(s) is a time
delayed second order system, G2(s) is a third order
system and G3(s) is a fourth order system. In process
control these systems are most commonly observed,
defined as follows:
In this scenario one of the most well known
methods of PID tuning based on quarter decay ratio was
proposed by Ziegler-Nichols (Z-N) [2]. This was too
aggressive for most of the plants where oscillations and
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
G1 ( s ) 
G2 ( s ) 
G3 ( s ) 
e 0.5s
a) Reproduction
(1)
( s  1) 2
4.288
( s  0.5)( s  1.64 s  8.456 )
2
27
( s  1)( s  3) 3
Reproduction or Selection is basically the process
of choosing two parents from the population for
crossing [11]. A part of the new population can be
created by simply copying without any change in
selected individuals from the present population. A
number of selection methods are defined in the literature
[10] and depending upon the problem at hand, one has
to decide the best suitable one. All selection methods are
based on the same principal, i.e. assigning the fittest
chromosomes a larger probability of selection. Four
most commonly used methods for selection are given as
follows:
(2)
(3)
For time delay definition in process G1(s), a second order
(t d ) 2 s 2  6t d s  12
) is
(t d ) 2 s 2  6t d s  12
used. All the three processes show a very complex
dynamic behavior and it is a good challenge for any
control engineer to design an effective controller to
control such processes with required system
specification. The next section describes about the
overview of Genetic Algorithm.
Padé approximation ( e td s 
1.
Roulette Wheel selection
2.
Stochastic Universal sampling
3.
Normalized geometric selection
4.
Tournament selection
b) Crossover
III. PRINCIPLES OF GENETIC ALGORITHM
Crossover also known as Recombination is basically
the process of taking two parent solutions and producing
from them a child [11]. The selection process enriches
the population with better individuals. Basically selection
process makes clones of good strings but it does not
create new ones. So crossover operation is applied to the
mating pool with the belief that it creates a better
offspring. Under this operator, a selected chromosome is
splitted into two parts at a selected crossover site and
recombines with another selected chromosome which
has been splitted at the same crossover site. There are
various types of crossover operators like single point
crossover, two point crossover, arithmetic crossover and
many more [10].
Genetic Algorithm abbreviated as GA, is known as a
stochastic global adaptive search optimization technique
based on the process of evolution [9]. Evolution is itself
an optimization process. It was first proposed by John
Holland and his colleagues in 1975. GA has been
considered as an adequate and efficient technique for
solving complex optimization problems. By carefully
avoiding local minima, it converges to global minima. It
begins with an initial population having a number of
chromosomes where each one corresponds to a solution
of the given problem. Then the performance of each
individual is evaluated by using a suitable fitness
function. Basically, GA consists of three important
phases: Selection, Crossover and Mutation. These are
also called as GA operators. The application of these
three basic operators allows the creation of new children,
which may be better than their parents. This algorithm is
repeated for many generations and finally stops when
meeting offspring that represent the optimum solution to
the problem [10].
c)
Mutation
After crossover operation mutation is applied to the
binary strings. Mutation is a process that involves
flipping a bit i.e. changing it from 0 to 1 and vice-versa.
The mutation operator plays a secondary role in the
evolution. It basically prevents the algorithm from
trapping into the local minima. It also helps in
maintaining the diversity in the population by finding
new or restoring lost genetic materials by searching the
neighbourhood solution space. Although mutation plays
a very important role in a genetic algorithm, it should be
noted that it must be introduced with a small probability
rate of the entire population [10]. Now the next section
describes about the basics of PID controller and its
tuning by MOGA.
A. Genetic Operators
In each generation, the genetic operators are applied
to select the best children from the current population in
order to create a new population [9]. In most of the cases,
the three main genetic operators named as reproduction,
crossover and mutation are used. Crossover and mutation
operators must be carefully chosen, since their choice
greatly affects the performance of the whole genetic
algorithm. The three operators are described in brief as
follows:
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
IV. MOGA-PID CONTROLLER
T
ISE   e 2 (t )dt
A. Basic PID Control Algorithm
PID controller consists of three interacting
parameters: proportional, integral and derivative gains.
Appropriate setting of these parameters can provide
good dynamic response of a system with minimized
overshoot and steady state error. It can also increase
stability of the system under consideration [12]. The
transfer function of a PID controller has the following
form:
Gc ( s)  K p  Ki / s  K d s
In fitness function f1 maximum weight is assigned
to OS which means minimum overshoot is the main
requirement of the system under consideration.
Similarly other objective functions can be interpreted. In
place of ISE one can use Integral of the time-weighted
absolute error (ITAE) and Integral of absolute value of
the error (IAE) also. It is the choice of the control
engineer that which particular parameter of the control
system needs more attention. So as per the requirement
a higher weight can be assigned while considering the
other necessary specification simultaneously. However
the total sum of the weights in an objective function
must be equal to one, so that the overall performance of
the system should be justified. It shows the flexibility in
PID tuning while using MOGA. The Fig. 1 shows the
flow chart of MOGA-PID controller.
(4)
Where Kp, Ki and Kd are known as the proportional,
integral and derivative gains, respectively. Another very
useful equivalent form of the PID controller is defined
as follows:
Gc ( s)  K p (1  1 /(Ti s)  Td s)
(11)
0
(5)
Where Ti=Kp/Ki and Td=Kd/Kp. Ti and Td are called as
the integral and derivative time constants, respectively
[7]. The basic block diagram of a plant controlled by
PID controller is shown in Fig. 1.
Start
Generate initial
population for Kp, Ti, Td
Fig. 1 : A plant controlled by PID controller
(7)
f 3  0.10 ISE  0.20tr  0.70ts
(8)
f 4  0.10 ISE  0.80OS  0.10ts
(9)
f 5  0.10 ISE  0.60tr  0.10OS  0.20ts
(10)
Check fitness
Next
generation
No
Yes
Best PID values
The fitness function is a very important factor while
tuning the PID controller by MOGA. Various multiobjective fitness functions considered in this study are
defined as follows:
f 2  0.10 ISE  0.70tr  0.20ts
Crossover
Converged?
B. Tuning of PID by MOGA
(6)
Simulate the Process
& evaluate f
Mutation
Where r(t) is set point, y(t) is the plant output, e(t)=r(t)y(t) is called as error signal and u(t) is the controller
output for this particular error signal.
f1  0.05 ISE  0.05ts  0.90OS
selection
Stop
Fig.1 : Flow chart of MOGA-PID Controller
Performance of the MOGA-PID controller also
depends on its convergence rate. The parameters like
population size, population type and many more also
affect the convergence rate. So in order to avoid
premature convergence [13] of GA, it is very important
to select the appropriate operators and parameters for it.
Table I shows the parameters and operators of GA
which are achieved by rigorous experimentation in this
work.
Where ts is the settling time within 5 percent, tr is rise
time, OS is percentage overshoot and ISE is Integral of
the square error which can be defined as follows:
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
TABLE I. GENETIC ALGORITHM OPERATORS AND
PARAMETERS
1.4
1.2
Value/Method
Double vector
20
Feasible population
Tournament
Mutation
Crossover
Generations
Adaptive feasible
Arithmetic
65
1
Output y
Parameter
Population Type
Population Size
Creation function
Selection method
0.6
0.4
0.2
0
0
10
15
20
Time (Sec)
25
30
35
40
Table II shows the overall performance analysis of
the MOGA-PID controller and various conventional
PID controllers. In this comparative study the results
obtained by Ziegler-Nichols and Kitamori’s PID
controller have been referred from [7]. The parameters
of Tyreus and Luyben PID controller have been
determined by Kp=Ku/2.2, Ti=2.2Pu and Td=Pu/6.3
where Ku is the ultimate gain and Pu is the ultimate
period of sustained oscillation [9]. While tuning of PID
by MOGA, f1 fitness function is considered which is
already explained in section IV.
V. SIMULATION RESULTS
MOGA based PID tuning have been tested on
variety of plants which are commonly found in process
control. All these simulations have been performed on a
PC having Intel(R) Core(TM) 2Duo processor operating
at 2.00 GHz and installed with MATLAB® 7.9.0.529
(R2009b). Fig. 2, Fig. 3 and Fig. 4 show the step
response of process G1(s), G2(s) and G3(s) respectively
controlled with various conventional and MOGA PID
controller.
TABLE II. SUMMARY OF COMPARATIVE STUDY
Process
1.4
Output y
5
Fig. 4 : Step response of process G3(s)
Apart from above two stopping criterion have been used
in this paper, the first one is number of generations and
the second one is function tolerance which is 1×10 -6.
Index
Z-N
Kitamori
T-L
MOGA
Kp
2.808
2.212
2.1328
2.1245
1.2
Ti
1.64
2.039
7.194
2.3329
1
Td
0.41
0.519
0.519
0.5556
%OS
ts
(±5%)
Kp
32
6.8
0
0
4.16
2.37
16.03
1.57
2.19
-
1.659
2.3505
Ti
1.03
-
4.532
2.6756
Td
0.258
-
0.3269
0.4470
%OS
ts
(±5%)
Kp
17
-
0
0
5.45
-
12.02
4.22
3.072
2.357
2.3272
1.9827
Ti
1.352
1.649
5.9488
1.8419
Td
0.338
0.414
0.4292
%OS
32.8
10.9
0
0.4533
8.8×10-
ts
(±5%)
3.722
2.3
13.09
G1
MOGA-PID
Z-N
Kitamori
T-L
0.8
0.6
0.4
0.2
0
0
5
10
15
20
Time (Sec)
25
30
35
G2
40
Fig. 2 : Step response of process G1(s)
1.4
1.2
1
Output y
MOGA-PID
Z-N
Kitamori
T-L
0.8
G3
0.8
MOGA-PID
Z-N
T-L
0.6
0.4
0.2
14
1.52
The sign ‘-’ indicates that these values are not provided in the literature
0
0
5
10
15
20
Time (Sec)
25
30
35
It is obvious from the above results that whether it
is transient domain specification like percentage
overshoot (%OS) or steady state specification like
40
Fig. 3 : Step response of process G2(s)
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
settling time (ts), the performance of MOGA-PID
controller is far better than the conventionally tuned PID
controller. Fig. 5 shows the comparison of step response
of system G1(s) controlled with MOGA-PID controller
in which different fitness functions have been used for
the purpose of tuning.
Process
G1
0.8
Output y
f2
0.6
f3
0.2
f5
0
0
5
10
15
20
Time (Sec)
25
30
35
40
Best: 1.139
Best fitness
Fitness value
3
2.5
2
1.5
1
10
15
20
25
30
Generation
35
40
45
VII. REFERENCES
50
Fig. 6 : Optimization of fitness function f5 by MOGA
Fig. 6 shows the minimization process of fitness
function f5 by MOGA. Now table III presents a
comparative study of MOGA-PID controller used to
control process G1(s) by considering various objective
functions which are already explained in section IV.
[1]
K. J. Astrom and T. Hangglund, “Automatic
tuning of PID Controllers,” Instrument Society of
America, 1988.
[2]
J. G. Ziegler and N. B. Nichols, “Optimum
setting for Automatic Controllers,” ASME
Trans., vol. 64, pp. 759-768, 1942.
[3]
Tyreus, B.D. and W.L. Luyben, “Tuning PI
controllers for integrator/dead time processes,”
Ind. Eng. Chem. Res. pp. 2628–2631, 1992.
[4]
G. H. Cohen and G. A. Coon, “Theoretical
investigation of retarded control,” ASME Trans.,
vol. 75, pp. 827-834, 1953.
[5]
T. Kitamori, “A method of control system design
based upon partial knowledge about controlled
processes,” Trans. SICE Japan, vol. 15, pp. 549555, 1979.
TABLE III. COMPARATIVE STUDY OF MOGA-PID FOR
VARIOUS OBJECTIVE FUNCTIONS
Process
G1
Index
Kp
Ki
Kd
%OS
ts
(±5%)
f5
2.1068
0.59
1.1390
In this paper a comparative study has been
performed between MOGA-PID controller and
conventionally tuned PID controller. Results show that
the performance of MOGA based tuning of PID
controller is much better in comparison to the
conventional tuning approaches. So this can be
considered as a powerful tuning scheme for PID
controllers. The main advantage of using MOGA is that
it is completely independent from the complexity of
objective function under consideration. So while tuning a
PID controller, the control engineer can consider various
objectives rather than restricted to single objective as in
the conventional tuning methods.
3.5
5
f4
1.6375
1.38
0.3210
VI. CONCLUSION
Fig. 5 : Step response of process G1(s) with MOGA-PID for
various objective functions
0
f3
2.2167
0.62
1.0514
A careful observation of the table III clearly shows that
MOGA-PID controller gives the freedom to the control
engineer to tune it as per the desired requirement. For
this purpose, one just requires to create a suitable fitness
function in which a suitable weight is assigned to a
particular specification. For example in f4 fitness
function maximum weight 0.80 is assigned to the
percentage overshoot, which is a time domain
specification. In this case the target is to reduce the
overshoot which is also clear from the table III.
1
f4
f2
2.3348
0.68
0.8159
* Obj is the values of the objective function
1.2
0.4
Index
Kp
tr
Obj*
f2
2.3348
1.1161
1.4532
5.23
f3
2.2167
1.3539
1.6870
5.26
f4
1.6375
0.7778
0.7455
0
f5
2.1068
0.8930
1.8782
0.4273
1.30
1.22
2.25
3.32
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International Journal on Advanced Computer Theory and Engineering (IJACTE)
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S. N. Sivanandam and S. N. Deepa, Introduction
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G. Stephanopoulos, Chemical process control: an
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
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