STUDY OF THE DESIGN AND TUNING FUZZY LOGIC AND GENETIC ALGORITHM

STUDY OF THE DESIGN AND TUNING FUZZY LOGIC AND GENETIC ALGORITHM
STUDY OF THE DESIGN AND TUNING
METHODS OF PID CONTROLLER BASED ON
FUZZY LOGIC AND GENETIC ALGORITHM
A thesis submitted in partial fulfillment of the requirements for
the degree of
Bachelor in Technology
in
Electronics and Instrumentation Engineering
by
Sangram Keshari Mallick
and
Mehetab Alam Khan
Department of Electronics and Communication Engineering
National Institute of Technology, Rourkela
May, 2011
STUDY OF THE DESIGN AND TUNING
METHODS OF PID CONTROLLER BASED ON
FUZZY LOGIC AND GENETIC ALGORITHM
A thesis submitted in partial fulfillment of the requirements for the
degree of
Bachelor in Technology
in
Electronics and Instrumentation Engineering
by
Sangram Keshari Mallick
and
Mehetab Alam Khan
under the guidance of
Prof. G. S. Rath
Department of Electronics and Communication Engineering
National Institute of Technology, Rourkela
May, 2011
CERTIFICATE
This is to certify that the thesis entitled “Study of the design and tuning methods of PID
controller based on fuzzy logic and genetic algorithm” submitted by Sangram Keshari
Mallick (107EI027) and Mehetab Alam Khan (107EI028) in partial fulfillment of the
requirements for the award of Bachelor of Technology Degree in Electronics and
Instrumentation Engineering at National Institute of Technology, Rourkela is an authentic work
carried out by them under my supervision and guidance.
To the best of my knowledge, the matter embodied in thesis has not been submitted to any other
university/ Institute for the award of any degree or Diploma.
Place: Rourkela
Date:
Prof. G. S. Rath
Dept. of Electronics & Communication Engineering
National Institute of Technology
Rourkela – 769008
ABSTRACT
This project tries to explore the potential of using soft computing methodologies in controllers
and their advantages over conventional methods. PID controller, being the most widely used
controller in industrial applications, needs efficient methods to control the different parameters
of the plant. This thesis asserts that the conventional approach of PID tuning is not very efficient
due to the presence of non-linearity in the system of the plant. The output of the conventional
PID system has a quite high overshoot and settling time.
The main focus of this project is to apply two specific soft-computing techniques viz. fuzzy logic
and genetic algorithm to design and tuning of PID controller to get an output with better dynamic
and static performance. The application of fuzzy logic to the PID controller imparts it the ability
of tuning itself automatically in an on-line process while the application of genetic algorithm to
the PID controller makes it give an optimum output by searching for the best set of solutions for
the PID parameters.
The project also discusses the benefits and the short-comings of both the methods. The
simulation outputs are the MATLAB results obtained for a step input to a third-order plant.
Page | iv
ACKNOWLEDGEMENT
We would like to express our gratitude and sincere thanks to Prof. G. S. Rath whose
guidance, support and continuous encouragement helped us being motivated towards
excellence throughout the course of this work. We wish to extend our gratitude to Prof. T.
K. Dan for his timely suggestions and guidance during the project work.
Finally, we would like to thank all of them who have been helpful and were associated
with us directly and indirectly throughout the course of our work.
Sangram Keshari Mallick
107EI027
Mehetab Alam Khan
107EI028
Page | v
TABLE OF CONTENTS
CERTIFICATE
iii
ABSTRACT
iv
ACKNOWLEDGMENT
v
TABLE OF CONTENTS
vi
LIST OF FIGURES
viii
INTRODUCTION
1
1.1 Conventional PID controllers
2
1.1.1 Preliminary and background
2
1.1.2 Tuning of PID parameters
3
1.2 PID controller tuning using various soft-computing techniques
5
1.3 Thesis organization
5
FUZZY SELF-TUNING PID CONTROLLER
6
2.1 Fuzzy logic background
7
2.2 A self-tuning fuzzy PID controller
9
2.2.1 Self-tuning principle of fuzzy PID controller
9
2.2.2 Design and structure of the self-tuning fuzzy PID controller
10
2.2.3 Simulation results
13
2.3 Advantages of fuzzy self-tuning PID controller
20
2.4 Shortcomings of fuzzy self-tuning PID controller
21
GENETIC ALGORITHM BASED PID CONTROLLER
22
3.1 Genetic algorithm based PID controller
23
3.1.1 Preliminary and background of Genetic Algorithm
3.2 A GA-PID controller
23
23
3.2.1 Principles of GA-PID controller
23
3.2.2 Structure and design of GA-PID controller
24
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3.2.3 Simulation results
30
3.2.4 Advantages of GA-PID controller
32
3.2.5 Shortcomings of GA-PID controller
32
CONCLUSION
33
4.1 Conclusion
34
4.2 Future scope
34
REFERENCES
35
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LIST OF FIGURES
Fig. 1.1.
Basic block diagram of a conventional PID controller
2
Fig. 1.2.
Block diagram of the example system
4
Fig. 1.3.
MATLAB simulation result for a conventional PID controller
4
Fig. 2.1.
A pure fuzzy system
7
Fig. 2.2.
min and max Mamdani fuzzy inference system
9
Fig. 2.3.
Basic structure of a fuzzy PID controller
11
Fig. 2.4.
Fuzzy rule tables for the PID parameters
11
Fig. 2.5.
Mamdani fuzzy system
13
Fig. 2.6.
Plots for the membership functions
15
Fig. 2.7.
MATLAB simulation results for a step input for a fuzzy self-tuning PID
19
controller
Fig. 3.1.
Structure of GA-PID controller
25
Fig. 3.2.
Simluation flowchart for auto-tuning GA-PID controller
29
Fig. 3.3.
Fitness function plot
30
Fig. 3.4.
PID parameters (Optimized values)
31
Fig. 3.5.
Output response of the GA-PID controller
31
LIST OF TABLES
Table 2.1
Fuzzy rules for Kp
11
Table 2.2
Fuzzy rules for Ki
12
Table 2.3
Fuzzy rules for Kd
13
Page | viii
CHAPTER 1
INTRODUCTION
1.1 Conventional PID controllers
1.1.1 Preliminary and background
PID controllers are the most widely-used type of controller for industrial applications. They are
structurally simple and exhibit robust performance over a wide range of operating conditions. In
the absence of the complete knowledge of the process these types of controllers are the most
efficient of choices. The three main parameters involved are Proportional (P), Integral (I) and
Derivative (D). The proportional part is responsible for following the desired set-point, while the
integral and derivative part account for the accumulation of past errors and the rate of change of
error in the process respectively.
Kp e(t)
Setpoint
Error
Output
𝑡
Ki ∫ 𝑒(𝑥) 𝑑𝑥
Kd
Plant
𝑑𝑒
𝑑𝑡
Fig. 1.1 Basic block diagram of a conventional PID controller
For the PID controller presented in Fig. 1.1,
Output of the PID controller, u(t) = Kp e(t) + Ki ∫
( )
+ Kd
( )
……1.1
Where,
Error, e(t) =Setpoint- Plant output
Kp = proportional gain, Ki = integral gain, Kd = derivative gain
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1.1.2 Tuning of PID parameters
Tuning of a PID controller refers to the tuning of its various parameters (P, I and D) to achieve
an optimized value of the desired response. The basic requirements of the output will be the
stability, desired rise time, peak time and overshoot. Different processes have different
requirements of these parameters which can be achieved by meaningful tuning of the PID
parameters. If the system can be taken offline, the tuning method involves analysis of the stepinput response of the system to obtain different PID parameters. But in most of the industrial
applications, the system must be online and tuning is achieved manually which requires very
experienced personnel and there is always uncertainty due to human error. Another method of
tuning can be Ziegler-Nichols method[8]. While this method is good for online calculations, it
involves some trial-and-error which is not very desirable.
The transfer function describing the plant for our example is as follows:
G(s) =
……1.2
The above transfer function will be used for the study and comparison of the output response of
conventional PID, fuzzy PID and PID using genetic algorithm.
Following is an example of a conventional PID controller and its output to a step input response
as achieved with some particular control parameter values. The output is the simulation result
obtained with the help of MATLAB.
PID
Controller
500
𝑠 + 30𝑠 2 + 1000𝑠
Fig. 1.2 Block diagram of the example system
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Kp = 0.6
Ki=0.5
Kd = 0.001
Fig. 1.3 MATLAB simulation result for a conventional PID controller
Here, the output response of the given third order system has an overshoot of more than 40%.
Settling time is also close to 20 seconds. For practical applications in industry these values are
too high to be tolerated. An output response having a minimum overshoot and fastest response is
required.
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1.2 PID controller tuning using various soft-computing techniques
PID controller model structure needs to be very precise. But in practical applications, to a
different extent, most of the industrial processes exist to be nonlinear, the variability of
parameters and the uncertainty of model are very high, thus using conventional PID control the
precise control of the process cannot be achieved.The common methods known for tuning
require the process model to be of a certain type, for example as in the case of a ‘First order plus
dead time’ model. These methods require the process model to be reduced if it’s too complicated
originally. The above problems can be well addressed by the application of soft-computing
methods for tuning of the PID controller. These are specially useful for solving problems of
computationally complicated and mathematically intraceable. This is due to the convenience of
combining natural systems with intelligent machines effectively with the help of soft-computing
methods. Among all these soft-computing methods available Neural network ,fuzzy logic and
genetic algorithm are the most important ones.
Fuzzy logic mainly employ the mechanisms that are based on verbal power. Due to this fact it is
responsible for dealing with the uncertainties present in the system.
Genetic algorithm has its roots originated from the genetic science which is a biological
phenomenon. This method is useful for the process of random search from a huge pool of data.
By the implementation of the knowledge of fuzzy logic and genetic algorithm in PID controller,
the system response of the plant cant be improved.The overshoot and the rise time of the
response can be decreased and the dynamic performance of the system can also be improved.
1.3 Thesis organization
In this thesis, the first Chapter 1 discusses the characteristics of conventional controller and their
shortcomings with the help of simulation results obtained using MATLAB for a third-order
plant. Chapter 2 discusses about fuzzy self-tuning PID controller, their advantages over
conventional methods and their shortcomings. GA PID controller is studied in Chapter 3 which
also discusses its various advantages and disadvantages. Finally Chapter 4 concludes the thesis
by shedding some light on the future scope of this work.
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CHAPTER 2
FUZZY SELF-TUNING PID
CONTROLLER
2.1Fuzzy logic background
Fuzzy logic is a logic having many values. Unlike the binary logic system, here the reasoning is
not crisp ,rather it is approximate and having a vague boundary. The variables in fuzzy logic
system may have any value in between 0 and 1 and hence this type of logic system is able to
address the values of the variables those lies between completely truth and completely false. The
variables are called lingusistic variables and each linguistic variable is described by a
membership function which has a certain degree of membership at a particualr instance.
System based on fuzzy logic carries out the process of decision making by incorporation of
human knowledge into the system. Fuzzy inference system is the major unit of a fuzzy logic
system. The decision making is an important part of the entire system. The fuzzy inference
system formulates suitable rules and based on these rules the decisions are made. This whole
process of decision making is mainly the combination of concepts of fuzzy set thoery, fuzzy IFTHEN rules and fuzzy reasoning. The fuzzy inference system makes use of the IF-THEN
statements and with the help of connectors present (such as OR and AND), necessary decision
rules are constructed.
The basic Fuzzy inference system may take fuzzy inputs or crisp inputs depending upon the
process and its outputs, in most of the cases, are fuzzy sets.
Fuzzy rule base
Fuzzy sets in X
Fuzzy inference
engine
Fuzzy sets in Y
Fig. 2.1 A pure fuzzy system
Page | 7
The fuzzy inference system in Fig. 2.1 can be called as a pure fuzzy system due to the fact that it
takes fuzzy sets as input and produces output that are fuzzy sets. The fuzzy rule base is the part
responsible for storing all the rules of the system and hence it can also be called as the
knowledge base of the fuzzy system. Fuzzy inference system is responisble for necessary
decision making for producing a required output.
In most of the practical applications where the system is used as a controller, it is desired to have
crisp values of the output rather than fuzzy set values. Therefore a method of defuzzification is
required in such cases which converts the fuzzy values into corresponding crisp values.
In general there are three main types of fuzzy infernece systems such as :- Mamdani model,
Sugeno model and Tsukamoto model. Out of these three, Mamdani model is the most popular
one. There are also various defuzzification techniques such as :- Mean of maximum method,
Centroid of area method, Bisector of area methd etc.
In this work Mamdani fuzzification technique[1] is used. There are two types of Mamdani fuzzy
inference system such as, “min and max” and “product and max”. In our example, the “min and
max” Mamdani system is used. For this type of system, min and max operators are used for
AND and OR methods respectively. Fig. 2.2 explains the min and max Mamdani fuzzification
technique. µ is the membership value for the linguistic varibales A1, B1, A2, B2, C1, C2 and C’.
The fuzzy rules for the system are as follows:
1. If x is A1 and y belongs to B1, then z is C1.
2. If x is A2 and y belongs to B2, then z is to C2.
The defuzzification technique used is mean of maximum (MOM) method. As its name suggests
it takes the mean value of z for which µ has maximum value.
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Fig. 2.2 min and max Mamdani fuzzy inference system
2.2 A self-tuning fuzzy PID controller
2.2.1 Self-tuning principle of fuzzy PID controller
Fuzzy logic has been useful in recent years to formalize the ad-hoc approach of PID control. A
fuzzy PID controller takes the conventional PID controller as the foundation which uses the
fuzzy reasoning and variable universe of discourse to regulate the PID parameters. The
chracteristics of a fuzzy system such as robustness and adaptabilty can be successfully
incorporated into the controlling method for better tuning of PID parameters.
Page | 9
The term self-tuning refers to the characteristics of the controller to tune its controlling
parameters on-line automatically so as to have the most suitable values of those parameters
which result in optimization of the process output. Fuzzy self-tuning PID controller works on the
control rules designed on the basis of theoretical and experience analysis. Therefore,it can tune
the parameters Kp, Ki, and Kd by adjusting the other controlling parameters and factors on-line.
This, in result makes the precision of overall contol higher and hence gives a better performance
than the conventioanl PID controller or a simple fuzzy PID controller without self-tuning ability.
2.2.2 Design and structure of the self-tuning fuzzy PID controller
The Self-tuning fuzzy PID controller, which takes error "e" and rate of change-in-error "ec" as
the input to the controller makes use of the fuzzy control rules to modify PID parameters on-line.
The self-tuning of the PID controller refers to finding the fuzzy relationship between the three
parameters of PID, Kp, Ki, and Kd and "e" and "ec", and according to the principle of fuzzy
control modifying the three parameters in order to meet different requirements for control
parameters when "e" and "ec" are different and making the control object produce a good
dynamic and static performance. Selecting the language variables of “e” ,”ec”, Kp, Ki, and Kd is
choosing seven fuzzy vlaues (NB, NM, NS, ZO, PS, PM, PB). The region of these varibales, in
this case, is taken to be {-3,-2,-1,0,1,2,3}. Here (NB, NM, NS, ZO, PS, PM, PB) is the set of
linguistic values which respectively represent “negative big”, ”negative medium”, ”negative
small”, ”zero”, ”positive small”, ”positive medium” and “positive big”.
The following figure is the block diagram of a fuzzy self-tuning PID controller. As it can be seen
from the block diagram, the fuzzification takes two inputs ( e and ec) and gives three outputs
(Kp, Ki, Kd).
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Fuzzy System
𝑑𝑢
𝑑𝑡
PID Controller
Error, e =r(t)-c(t)
Plant
Rate of change of error, ec=de/dt
Fig. 2.3 Basic structure of a fuzzy PID controller
The set of linguistic rules is the essential part of a fuzzy controller. In many cases it’s easy to
translate an expert’s experience into these rules and any number of such rules can be created to
define the actions of the controller. In the designed fuzzy system, conventional fuzzy conditions
and relations such as :- “If e is A and ec is B,then Kp is C, Ki is D and Kd is E.” are used to create
the fuzzy rule table[3].
Table 2.1 Fuzzy rules for Kp
e
ec
NB
NM
NS
ZO
PS
PM
PB
NB
PB
PB
PM
PM
PS
PS
ZO
NM
PB
PB
PM
PM
PS
ZO
ZO
NS
PM
PM
PM
PS
ZO
NS
NM
ZO
PM
PS
PS
ZO
NS
NM
NM
PS
PS
PS
ZO
NS
NS
NM
NM
PM
ZO
ZO
NS
NM
NM
NM
NB
PB
ZO
NS
NS
NM
NM
NB
NB
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Table 2.2 Fuzzy rules forKi
e
ec
NB
NM
NS
ZO
PS
PM
PB
NB
NB
NB
NB
NM
NM
ZO
ZO
NM
NB
NB
NM
NM
NS
ZO
ZO
NS
NM
NM
NS
NS
ZO
PS
PS
ZO
NM
NS
NS
ZO
PS
PS
PM
PS
NS
NS
ZO
PS
PS
PM
PM
PM
ZO
ZO
PS
PM
PM
PB
PB
PB
ZO
ZO
PS
PM
PB
PB
PB
PS
NB
NM
NS
NS
ZO
PS
PS
PM
NM
NS
NS
NS
ZO
PS
PS
PB
PS
ZO
ZO
ZO
ZO
PB
PB
Table 2.3 Fuzzy rules forKd
e
ec
NB
NM
NS
ZO
PS
PM
PB
NB
PS
PS
ZO
ZO
ZO
PB
PB
NM
NS
NS
NS
NS
ZO
NS
PM
NS
NB
NB
NM
NS
ZO
PS
PM
ZO
NB
NM
NM
NS
ZO
PS
PM
Adaptive corrections can be made by the following methods,
Kp = Kp’ + Kp
Ki = Ki’ + Ki
Kd = Kd’ + Kd
Here Kp’, Ki’, and Kd’ refer to the previous value of the PID parameters whereas Kp, Ki, and Kd
refer to the new corrected values of the parameters after a particular tuning step was completed.
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2.2.3 Simulation results
The rsponse of the fuzzy self-tuning PID controller is obtained using Matlab. A two-input and
three-output fuzzy controller is created and the membership functions and fuzzy rules are
determined.
Fig. 2.4 Mamdani fuzzy system
The controller object is taken to be the third-order transfer fucntion,
G(s) =
Page | 13
Debugging and setting the three parametrs Kp, Ki and Kd of the PID controller,their values are as
follows :
Kp=0.6
Ki=0.5
Kd=0.001
The membership function of the language variables “e”, “ec”, Kp, Ki and Kd are in the given
range {-3,3} and their plots are as follows:
(a) Membership function of e
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(b) Membership function of ec
(c) Membership function of Kp
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(d) Membership function of Ki
Fig. 2.5 Membership functions plots
The output is the response of the system to a step input.The simluation results as obtained by
Matlab are given in Fig. 2.6.
Page | 16
(a) Optimized value of Kp
(b) Optimized value of Ki
Page | 17
(c) Optimized value of Ki
(d) Manipulated variable u(t)
Page | 18
(e) Error function e(t)
(f) Output response c(t)
Fig 2.6 Matlab simulation result for the system for a step input response
Page | 19
As can be seen from the simulation results, the output response of the system to a step input has a
lower rise time and lower overshoot than the output response of the conventioanl PID controller.
It has better dynamic properties and steady-state properties.
2.3 Advantages of fuzzy self-tuning PID controller
The followings are the advantages of fuzzy self-tuning PID controller over the conventional PID
controller :
i)
The traditional PID controller cannot self-tune parameters Kp, Ki and Kd while
operating.
ii)
Combining fuzzy inference with traditional PID method, self-tuning of PID
parameters can be realized.
iii)
By designing a fuzzy self-tuning PID controller based on conventional PID, the
decisions can be made through fuzzy reasoning rules according to the size, the
direction and the changing tendency of the system error together with the dynamic
changing of process characteristics.
iv)
In practical applications, to different extent, most of the industrial processes exist to
be nonlinear, the variability of parameters and the uncertainty of model are very high,
thus using conventional PID control the precise control of the process cannot be
achieved.
v)
The dependence of fuzzy control on the mathematical model is weak, so it isn't
necessary to establish the precise mathematical model of the process, and the fuzzy
control has a good robustness and adaptability.
vi)
The simulation results shows that: compared with the traditional PID controller, fuzzy
self-tuning PID controller has a better dynamic response curve, shorter response time,
small overshoot, high steady precision, good static and dynamic performance.
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2.4 Shortcomings of fuzzy self-tuning PID controller
Although fuzzy self-tuning PID controller gives a better output response than the
conventional PID controller, it also has some problems with its design and tuning methods.
These problems are mainly due to the vagueness associated with the fuzzy method.
Followings are the disadvatages of a PID controller based on fuzzy logic method :
i) It uses the mode of approximate reasoning and decisions are made on vague and
incomplete information similar to that of human beings.
ii) The choice of overall control structure can also be a big problem in some cases.
iii) In designing of the fuzzy logic controller not only the structural parameters need to be
designed but also the gain of the conventional controller need to be tuned.
iv) Because of its complicated cross-effects analytical tuning algorithm for these
parameters are really difficult.
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CHAPTER 3
GENETIC ALGORITHM
BASED PID CONTROLLER
3.1 Genetic algorithm based PID controller
3.1.1 Preliminary and background of Genetic Algorithm
Genetic algorithm(GA) uses the principles of evolution, natural selection and genetics from
natural biological systems in a computer algorithm to simulate evolution.Essentially, the genetic
algorithm is an optomization technique that performs a parallel, stochastic, but directed search to
evolve the fittest population.
The idea, in all the system based on Genetic algorithm, was to evolve a population of candidate
solutions to a given problem, using operators inspired by natural genetic variation and natural
selection.
Biological evolution is an appealing source of inspiration for addressing optimization problems.
Evolution is, in effect, a method of searching among an enormous number of possibilities for
"solutions." In biology the enormous set of possibilities is the set of possible genetic sequences,
and the desired "solutions" are highly favourable organisms—organisms, which are able to
survive and reproduce in their environments. Evolution can also be seen as a method for
designing innovative solutions to complex problems. The fitness criteria continually change as
creatures evolve, so evolution is searching a constantly changing set of possibilities. Searching
for solutions in the face of changing conditions is precisely what is required for adaptive
computer programs. Furthermore, evolution is a massively parallel search method rather than a
work on one species at a time. Evolution tests and changes millions of species in parallel.
Finally, viewed from a high level, the "rules" of evolution are remarkably simple: species evolve
by means of random variation (via mutation, recombination, and other operators), followed by
natural selection in which the fittest tend to survive over others.
3.2 A GA-PID controller
3.2.1 Principles of GA-PID controller
Genatic algorithm is a robust optimization technique based on natural selection. The basic
objective of GA is to optimize fitness function. In genetic algorithms, the term chromosome
Page | 23
typically refers to a candidate solution to a problem. In this thesis GA works directly on real
parameters. Decimal type GA are equivalent to the traditionally used binary-type GA’s in
optimization[6]. Decimal-type GA’s for computer-based numerical simulation lead to high
computational efficiency, smaller computer requirements with no reduction of precision and
greater freedom in selecting genetic operator. GA has been successfully implemented in the area
of industrial electronics, for instance, parameter and system identification, control robotics,
pattern recognition,planning and scheduling.For its use in control engineering, GA can be
applied to a number of control methodologies for the improvement of the overall system
performance.
3.2.2 Structure and design of GA-PID controller
The structure of a control system with GA-PID as a controller is shown in the figure below. It
consists of a conventional PID controller with its parameter optimized by genetic algorithm. The
initial population of size N is generated randomly to start the optimization process. The next
generation can be obtained through the genetic operators. The genetic opearators are the most
important features of GA and are described below.
Genetic operator
The decision to make during implentation of genetic algorithm is the choice of genetic operators
that are to be used. The basic genetic operators are;
Reproduction :- By using the values of the performance fitness functions,select the best N/2
induviduals of the current generation to be the parents for producing the next generation.This
means that only genetically good individuals are selected to become parents.
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Genetic
algorithm
Kp
u(t)
y(t)
r(t)
1/s
Ki
s
Kd
Plant
Fig. 3.1 Structure of GA-PID controller
Crossover :-Two parents are randomly selected to exchange the genetic information with each
other and two new individuals are generated so as to keep the population size at constant value
N.
Cross over opeartion[7] can be mathematically desribed as follows :
If parents are (wn1,kf1) and (wn2,kf2), then
Child-1: wn= r *wn1 +(1-r)*wn2
kf= r* kf1 +(1-r)*kf2
Child-2: wn =(1-r) *wn1 +r*wn2
kf = (1-r) *kf1+ r*kf2
,where r ϵ (0,1) is a random number
……..3.1
Here the crossover operator works with real decimal pairs instead of any coded strings.
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Mutation :- Mutation plays a secondary role in genetic algorithms. It is needed because,
occasionally, chromosomes may lose some potentially useful genetic material. Mutation takes
place with a certain probability; thus genetic content of a particular individual gets changed and a
new generation is produced. Mutation is important in nature as it brings a change in genetic
content of the invdividuals in order to enable them to adapt to a different environment. In the
same way, in artificial systems the mutation will direct the search algorithm to a new search
space so that a global minima can be found. In our simulations, mutation rate is set to be 0.1.
After mutation,we get a modified mating pool M(k). To form the next generation for the
population,we let
P(k+1)=M(k)
………3.2
Where M(k) is the one that was formed by selection and modified by crossover and mutation.
Then the above steps repeat, successive generations are produced, and the evolution is modelled.
Search space and fitness landscape
The idea of searching among a collection of candidate solutions for a desired solution is so
common that it has been given its own name: searching in a "search space." Here the term
"search space" refers to some collection of candidate solutions to a problem and some notion of
"distance" between candidate solutions.
Fitness function
A fitness function takes a chromosome as an input and returns a number that is a measure of the
chromosome’s performance on the problem to be solved. Fitness function plays the same role in
GA as the environment plays in natural evolution. The interaction of an individual with its
environment provides a measure of fitness to reproduce. Similarly the interaction of a
chromosome with a fitness function provides a measure of fitness that the GA uses while
carrying out reproduction. Genetic algorithm is a maximization routine; the fitness function must
be a non-negative figure of merit.
In this particular situation our main aim is to minimise error and reduce the rise time and
overshoot. Hence the fitness function, in this case, is a function of error and rise time.
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J=∫
(w1|e(t)| + w2u2(t)) dt+ w3tr
……3.3
Where w1,w2,w3,w4 are the weight cofficients.U(t) is the output of the controller.e (t) is the error.
The square term of control output is added to overcome the large energy of the controller.
In this paper fitness function is chosen as,
f=
.…..3.4
The term 10-8 is added in the denominator of fitness function to avoid it from becoming zero.
Fitness proportionate selection with "Roulette Wheel"
The original GA used fitness proportionate selection, in which the "expected value" of an
individual (i.e., the expected number of times an individual will be selected to reproduce) is that
individual's fitness divided by the average fitness of the population. The most common method
for implementing this, is "roulette wheel" sampling[9]. Each individual is assigned a slice of a
circular "roulette wheel", the size of the slice being proportional to the individual's fitness. The
wheel is spun N number of times, where N is the number of individuals in the population. On
each spin, the individual under the wheel's marker is selected to be in the pool of parents for the
next generation.
This method can be implemented as follows:
1. The total expected value of individuals in the population are summed and the sum is
called as T.
2. The above step is repeated N number of times.
A random integer r between 0 and T is choosen. Looping is carried out through the individuals in
the population, summing the expected values, until the sum is greater than or equal to r. The
individual whose expected value puts the sum over this limit is the one selected.
This stochastic method statistically results in the expected number of offspring for each
individual. However, with the relatively small populations typically used in GAs, the actual
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number of offspring allocated to an individual is often far from its expected value (an extremely
unlikely series of spins of the roulette wheel could even allocate all offspring to the worst
individual in the population).
Formalization of Genetic algorithm (GA)
We started with a random population of binary strings of length L.
1. Fitness f(x) of each string x in the population was calculated.
2. We chose (with replacement) two parents from the current population with probability
proportional to each string's relative fitness in the population.
3. Cross over was carried out between the two parents (at a single randomly chosen point)
with probability Pc to form two offspring. (If no crossover occurs, the offspring are exact
copies of the parents.) One of the offspring was selected at random and the other was
discarded.
4. We mutated the selected offspring with probability pm and place it in the new population.
5. Step 2 is repeated until a new population is complete.
6. The above process was repeated again from step 1.
Terminal conditions
While biological evolutionary process continues, perhaps indefinitely we would like to terminate
our artificial one and find the following:
1.The popualtion string, say ø(k), best maximizes the fitness function. To determine this ,we
also need to know the generation number k where the fittest individual existed (it is not
necessarily in the last generation). A computer code, implementing genetic algorithm, keeps
track of the highest J value, and the generation number and the individual that achieved this
value of J.
2.The value of the fitness function J(ø).
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The flowchart of simulation for GA-PID controller[4] is as follows:
Start
Parameter
Initialization
Generation 1
Calculate fitness
function value
Yes
Generation 1
Satisfy the
requirement?
>
Generation 2
No
Reproduce
Crossover
Genetic
algorithm
Mutation
Generation 2
End
Fig. 3.2 Simluation flowchart for auto-tuning GA-PID controller
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3.2.3 Simulation results
Simulation results are obtained using MATLAB for the same transfer function which was used in
the cases of conventional PID and fuzzy self-tuning PID controller.
G(s) =
………3.5
The size of population of GA is often chosen between [20,100]. For our simulation, we chose the
size of population as 40. The number of generation is often chosen between [100,500]. For our
case, we chose number of generatons equal to 300. The mutation rate is chosen to be 0.05. The
weight co-efficients w1, w2 and w3 are 0.988, 0.001 and 3.0 respectively.The parameter ranges of
GA-PID controller are Kp ϵ [0,20], Ki ϵ [0,1] and Kd ϵ [0,5].
Fig. 3.3 Fitness function plot
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Fig. 3.4 PID parameters (Optimized values)
Fig. 3.5 Output response of the GA-PID controller
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3.2.4 Advantages of GA-PID controller
i) It is a simple algorithm that is easily understood and implemented.
ii) The algorithm is robust.
iii) GA is a non-linear process that could be applied to most industrial processes with good
results.
iv) GA searches a population of points instead of a single solution.
v) GA does not need information about the system except for the fitness function.
3.2.5 Shortcomings of GA-PID controller
For a GA-PID controller, it cannot be guaranteed that the result obtained through the process is
the most optimized values although it’s near optimum. As GA can different result for each new
search for the same system under same conditions. In many problems, GAs may have a tendency
to converge towards local optima or even arbitrary points rather than the global optimum of the
problem.Therefore, the result may not be the perfectly optmized one.
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CHAPTER 4
CONCLUSION
4.1 Conclusion
In this project, we studied the design and tuning methods for PID controller using fuzzy logic
and Genetic algorithm. A third-order plant was taken as the control object. Simulation was
carried out using MATLAB to get the output response of the system to a step input.The
simulation results and the characteristics of both the methods were observed and compared with
that of coventional PID controller.
According to the profiling results, the use of above soft-computing techniques resulted in an
outputs better dynamic and static characteristics.The response of the system was also faster than
in the case of conventional PID controller.The amount of overshoot for the output response was
successfully decreased using the above techniques. The application of fuzzy logic to the PID
controller imparted it’s the ability to tune itself while operating on-line. Similarly, Genetic
algorithm enabled the PID controller to get an output which is robust and has faster response.
4.2 Future scope
Future scope of this project involves combination of fuzzy logic and Genetic algorithm for
tuning of PID controller. In this method, GA and fuzzy logic account for estimation of gain
parameters and ranking basement of GA respectively.
Neuro-fuzzy PID controller can be designed by the implementation of neural network to fuzzy
PID controller.
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References
[1] J.-S.R. Jang, C.-T. Sun, E. Mizutani. Neuro-fuzzy and soft computing.
[2] S. N. Sivanandam, S. Sumathi , S. N. Deepa. Introduction to fuzzy logic using Matlab.
[3] Wang-Xiao Kan, Sun Zhong-Liang, Wnglei, Feng Dong-qing. “Design and research
based on fuzzy self-tuning PID using Matlab”. Proceedings of International Conference
on Advanced Computing theory and Engineering (2008).
[4] Liu Fan, Er Meng Joo. “Design for Auto-tuning PID Controller Based on Genetic
Algorithms”. IEEE Conference on Industrial Electronics and Applications (ICIEA 2009)
[5]
B. Nagaraj, S. Subha, B.Rampriya. “Tuning Algorithms for PID Controller Using Soft
Computing Techniques”.
[6] Melanie Mitchell.”An introduction to genetic algorithms”. MIT Press.
[7] S.S. Ge,member,IEEE, T.H. Lee,Member,IEEE and G.Zhu . “Genetic algorithm tuning of
Lyapunov-based controllers:An Application to a Single-Link flexible Robot System”.
IEEE transaction on industrial electronics, VOL. 43,no.5, october 1996.
[8] J. G. Ziegler, N. B. Nichols, “Optimum setting forautomatic controllers”, Trans. ASME,
Vol. 64, pp. 759-768, 1942.
[9] Gopal M. “Digital control and state variable methods”.
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