DOI : 10.17577/IJERTV2IS60308
- Open Access
- Total Downloads : 233
- Authors : Afshan Ilyas And Mohammad Ayyub
- Paper ID : IJERTV2IS60308
- Volume & Issue : Volume 02, Issue 06 (June 2013)
- Published (First Online): 12-06-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Intelligent Methods for Tuning of Different Controllers
Afshan Ilyas and Mohammad Ayyub
Department of Electrical Engineering
Zakir Hussain College of Engineering & Technology Aligarh Uttar Pradesh, India
Abstract: This paper presents different methods for tuning of controllers. Ziegler-Nichols tuning formula has been used for predicting the range of gain coefficients of PID controller. Recently, it has been noticed that PID controllers are often poorly tuned and some efforts have been made to systematically resolve this matter. Thus Fuzzy logic can be used in context to vary the parameters values during the transient response, in order to improve the step response performances. This paper also discusses the Fuzzy control as applied to various industrial processes. Normally the control rules and membership functions of Fuzzy control are obtained by trial and error. Genetic Algorithm technique is supposed to provide better tuning of the parameters and is applied for tuning the parameter of the PID & PI-Like Fuzzy controllers. Simulation analysis has been carried out for the different processes and the results shows that the optimized fuzzy controller gives better performance than a conventional PID and fuzzy logic controller.
-
Introduction
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.
Many tuning formulae that have been devised such as the Ziegler-Nichols one, assures a good load- disturbance attenuation, but often fail to achieve satisfactory performances, and therefore the operator has to use their experience and might fail to attain the best performances [4].
In this context, the use of fuzzy logic seems to be particularly appropriate, since it allows us to make use of the operator's experience and therefore to add some sort of intelligence to the automatic control. Fuzzy logic controllers (FLCs) are used when the processes are too complex for analysis by conventional mathematical techniques.
But there are some difficulties that prevent the designing of fuzzy controllers from being
systematic. First, the choice of the overall control structure is a problem faced by many designers. Second, in designing of the FLC, not only structure parameters of the FLC need to be designed, but also the gains of the conventional controller need to be tuned. Because of its complicated cross-effects, analytical tuning algorithms for these parameters are really difficult [7].
Genetic Algorithms provide a way of surmounting this shortcoming. These algorithms use some of the concepts of evolutionary theory, and provide an effective way of searching a large and complex solution space to give close to optimal solutions in much faster times than random trial-and-error. They are also generally more effective. The main objective of this work is to investigate the use of genetic algorithms in tuning of PID controller. The algorithm searches for the controller gains Kp (proportional gain), Ki (integral gain) and Kd (derivative or differential
gain) so that specifications for the closed-loop step response are satisfied.
This paper is organized as follows. Section 2 reviews the ZN tuning formula in the context of tuning the PID controller. Section 3 presents the tuning procedure, based on fuzzy logic controller. Tuning of PID based on GA and GA-optimized PI- type FLC were present in section 4 and section 5 respectively. Conclusions follow in section 6.
-
TUNING OF PID CONTROLLER
-
Ziegler-Nichols Method
Proportional-Integral-Derivative (PID) controllers have been in existence for nearly two-thirds of a century. They remain a key component in industrial process control as over 90% of todays industrial processes are controlled by PID controllers. Due to its simplicity, versatility, speed, reliability, flexibility and robustness, many industries still rely on this stalwart controller for all types of control. Example includes temperature, engine speed and position control among many others. The PID controller is probably the most used feedback control design. PID controller has the general form
mainly trial and error method. But this method is very time consuming. Therefore we switch over to different tuning techniques which give more accurate results with less time. Ziegler-Nichols method is generally used for the purpose in which the parameters like ultimate gain Ku and ultimate period Tu is first calculated by Routh array criteria [10], and then Kp, Ki and Kd are calculated as shown below.
Controller
Gain (KP)
Integral time(Ti)
Derivative time(Td)
P
0.5Ku
–
–
PI
0.45Ku
0.8Tu
–
PID
0.6Ku
0.5Tu
0.125Tu
Controller
Gain (KP)
Integral time(Ti)
Derivative time(Td)
P
0.5Ku
–
–
PI
0.45Ku
0.8Tu
–
PID
0.6Ku
0.5Tu
0.125Tu
Table 1: Ziegler – Nichols tuning formulas based on ultimate gain (Ku) and ultimate period (Tu)
-
Simulation Result
The performances of the different controllers have been evaluated on different plants. Here, the following transfer functions, with different values of the parameters, are considered:
()
1
G1(s) = s(1+s) (1)
=
+ . +
0
e0.1s
where e(t) = ysp(t) – y(t) is the system error (difference between the reference input and the system output), u(t) the control variable, Kp the proportional gain, Td the derivative time constant and Ti the integral time constant.
The tuning problem consists of determining the values of these three parameters with the aim of satisfying different control specifications such as set-point following, load disturbance attenuation, robustness to model uncertainties and rejection of measurement noise. The value of controller parameters like KP, Kd and Ki are reached by
G2(s) = (1+10s)2 (2)
e0.4s
G3(s) = (1+10s)2 (3)
G4(s) =
G5(s) =
G6(s) =
1
(1+s)3 (4)
(10.5s)
(1+s)3 (5)
1
1+s 1+0.5s 1+0.25s (1+0.125s) (6)
1.5
1
amplitude
0.5
unit step response
ZN
G7(s) =
e0.1s
1+s 1+0.5s 1+0.25s (1+0.125s) (7)
0
0 10 20 30 40 50
time
After the tuning phase, accomplished the unit step responses have been simulated with Matlab and Simuink. The step responses for plants described by transfer function of equation (5) and (7) are shown in Fig.1 and Fig.2.
unit step response
Fig. 2: Step Response of G7(s) with Ziegler-Nichols controller
-
-
DESIGN AND TUNING OF FUZZY LOGIC CONTROLLER
Fuzzy logic controllers (FLCs) are increasingly
applied to many systems with nonlinearity and
1.5
1
amplitude
0.5
0
-0.5
ZN
uncertainty and it isbased on experience of a human operator. While controlling a plant a skilled human operator manipulates the output of the controller based on error and change in error with an aim to reduce the error with a shortest possible time.
The two types of structure of FLC have been studied so far: one is position-type fuzzy controller
0 10 20 30 40 50
time
Fig.1: Step Response of G5(s) with Ziegler-Nichols Controller
which generates control input (u) from error (e) and change in error, and the other is velocity-type fuzzy controller which generates incremental control input (u) from error and change in error. The former is called PD type FLC and the latter is called PI type FLC according to the characteristics of information that they process. In the viewpoint that the FLC is based on the knowledge of human experts, and generally FLCs applied to unknown or partially known systems, PI type FLC is known to be more practical than PD type FLC [9].
r + e
–
+
–
1/Z
Ke
Kce
Mux
Fuzzy logic controller
Kdu
+
+
1/Z
Plant
-
Simulation Results
Simulation of the example systems was being carried with the conventional controllers was being replaced by Fuzzy logic Controller (FLC). FLC was implemented with the required block available in MATLAB/SIMULINK. Mamdani type of rule-
base model is used for this FIS system. This
Fig 3: PI type fuzzy logic controller
One of the well accepted rule base is the linear rule base which appears in many research work and applications. As the rule base conveys a general control policy, it should be sustained and leaves most of design and tuning work to the scaling gains.
An input SF transforms a crisp input into a normalized input in order to keep its value within the universe. An output SF provides a
produces output in fuzzified form. It is having five input membership functions for both input variables leading to 25 rules. Input and output scaling factors are tuned manually to get the desired response. It is clear from the figures that overshoot reduces significantly with a fuzzy controller but the values of other performance parameters like IAE, rise-time and settling-time increases as compare to that of conventional controller.
transformation of the defuzzied crisp output from
1.5
unit step response
G5(s)
G5(s)
the normalized universe of the controller output G4(s)
into an actual physical output [9]. There is always an input limitation for FLC, so that conventional inputs i.e. e and ce and the controller output (du) are defined on the common normalized domain [-1,1]. The set of rules which define the relation between the input and output of fuzzy controller can be found using the available knowledge in the area of designing the system. These rules are defined using the linguistic variables. All the 25 rules governing the mechanism for each output are explained in Table 2.
Table 2: Basic rules table for fuzzy inference system
1
amplitude
0.5
0
0 5 10 15 20 25 30
time
Fig 4: Step Responses of PI-type FLC of G4(s) and G5(s)
ce e
NB
NS
Z
PS
PB
NB
NVB
NB
NM
NS
Z
NS
NB
NM
NS
Z
PS
Z
NM
NS
Z
PS
PM
PS
NS
Z
PS
PM
PB
PB
Z
PS
PM
PB
PVB
1.5
1
amplitude
unit step response
G6(s)
G7(s)
The structure of the control system with GA-PID controller is shown in Fig 5. It consists of a conventional PID controller with auto-tuning its gain coefficients based on GA and a control plant.
Genetic Algorithm
Genetic Algorithm
0.5
r(t) e(t) u(t) y(t)
0
0 5 10 15 20 25 30
time
+ _ PID controller
Plant
Fig 5: Step Responses of PI-type FLC of G6(s) and G7(s)
-
-
DESIGNING OF PID CONTROLLER USING GENETIC ALGORITHM
Genetic Algorithm (GA) is a stochastic global adaptive search optimization technique based on the process of natural evolution. It is one of the methods used for optimization. John Holland formally introduced this method in the United States in the 1970 at the University of Michigan.
The genetic algorithm starts with no knowledge of the correct solution and depends entirely on responses from its environment and evolution operators such as reproduction, crossover and mutation to arrive at the best solution [9].
The basic goal of GA is to optimize functions called fitness functions. A possible solution to a specific problem is seen as an individual. A collection of a number of individuals is called a population [7]. Each solution can be represented by a binary string of ones and zeros, real number or other forms, depending on the application data. In these algorithms the fittest among a group of individuals survive and are used to form new generations of individuals with improved fitness vales. The fitness of an individual is a measure of how well the individual has performed in the problem domain [8].
Fig 5: Block diagram of GA-PID controller
The steps involved in creating and implementing a genetic algorithm are as follows:
-
Generate an initial, random population of individuals for a fixed size.
-
Evaluate their fitness.
-
Select the fittest members of the population.
-
Reproduce using a probabilistic method (e.g., roulette wheel).
-
Implement crossover operation on the reproduced chromosome.
-
Execute mutation operation with low probability.
-
Repeat step 2 until a predefined convergence criterion is met.
Flowchart of genetic algorithm process is shown in Fig 6.
Initialize Population
4.3 DESIGNING OF PID CONTROLLER USING GENETIC ALGORITHM
Evaluate Fitness
Select Fittest
Mutation
Crossover/ Reproduction
Non Optimum Solution
The main objective of this work is to investigate the use of genetic algorithms in the tuning of PID controller. The algorithm searches for the controller gains Kp (proportional gain), Ki (integral gain) and Kd (derivative or differential gain) so that specifications for the closed-loop step response are satisfied. Due to their widespread use in industry, tuning procedures for PID controllers are always a
topic of interest [11].
Optimum Solution
Optimum Solution
Fig .6: Flowchart of GA process
-
ADVANTAGES OF GENETIC ALGORITHM
-
It is a simple algorithm that is easily understood and implemented.
-
The algorithm is robust.
-
GA is a non-linear process that could be applied to most industrial processes with good results.
-
GA searches a population of points instead of a single solution. The GA is therefore not easily sidetracked to obtain a local optimal solution instead of a global optimal solution.
-
GA does not need information about the system except the fitness function [7].
-
-
SHORTCOMINGS OF GENETIC
In GA based tuning method, it cannot be guaranteed that the result obtained for the process is the most optimized values although its 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 han the global optimum of the problem. Therefore, the result may not be the perfectly optimized one [10].
The implementation of the tuning procedure through genetic algorithms starts with the definition of the chromosome representation. As illustrated in Fig.7, the chromosome is formed by three values that correspond to the three gains to be adjusted in order to achieve a satisfactory behavior. The gains Kp, Ki and Kd are binary strings numbers and characterize the individual to be evaluated. Each gain is represented by ten binary numbers each chromosome has thirty genes [8].
|
Kp |
Ki |
Kd |
Fig 7: Chromosomes definition
A GA-PID controller can be implemented as follows:
-
Initialization
Initialization of the population size, variable bounds and string-length are required. These are the initial inputs that are required for the Genetic Algorithm process to start.
-
Encoding
Encoding techniques in genetic algorithms (GAs) are problem specific, which transforms the problem solution into chromosomes. One binary string consisting of the three PID gain coefficients: Kp, Ki
and Kd. The length of string depends on defining domain of the variables and the precision of calculation. There are many chromosome encode methods in the GA. In this paper binary number encoding method was adopted as it is easy for genetic algorithms operation.
-
Fitness Function
A fitness function could be created to find a PID controller that gives the smallest overshoot, fastest rise time or quickest settling time. However in order to combine all of these objectives it was decided to design a fitness function that will minimize the error of the controlled system. The fitness of a chromosome is calculated from the integral of the absolute error (IAE).
= .
0
Where e(t) the error between the reference signal and the system output.
In this paper the fitness function is chosen as
f = 1/ J
F. Selection
E. Mutation
Mutation changes the structure of the string by changing the value of a bit chosen at random. This operator can prevent individuals falling into a local optimum. A random string with the length L is generated. If the value in a position of this random string is less than or equal to the mutation rate pm, the gene of the child in the same position will be inverse of the original [7]. In this paper, the mutation rate is chosen as pm = 0.032.
-
Simulation Results
-
Population size: 30
-
Generations: 20
-
Mutation: 0.032
-
Reproduction method: roulette wheel
-
Crossover: Single Point Crossover [7].
-
The parameter range of GA-PID Controller for the second order and second order delay systems Kp [0,10], Ki [0,1], Kd [0,1], for the third order system Kp [0,20], Ki [0,1], Kd [0,5] and for the fourth order and fourth order delay systems Kp [0,3], Ki [0,1], Kd [0,1].
The standard roulette wheel selection is applied for selecting the chromosomes. In this method parents are selected according to the fitness values. Chromosomes with higher fitness have higher chances to be selected.
-
Crossover
1.5
1
amplitude
0.5
unit step response
G4(s)
G5(s)
It is the process in which genes are selected from the parent chromosomes and new offspring is created. But not all individuals are necessarily used for crossover. In this paper uniform crossover is chosen.
0
0 5 10 15 20 25 30
time
Fig 8: Responses of GA-PID controller of G4(s) and G5(s)
1.4
1.2
1
0.8
amplitude
0.6
0.4
0.2
0
0
unit step response
G6(s)
G7(s)
G6(s)
G7(s)
5 10 15 20 25 30
time
linguistic rules obtained from experienced human operators [12].
5.1 Simulation Results
The GA-optimization algorithm was run for 20 generations with each generation having a population size of 30. MATLAB M-files where utilized for the encoding, testing and decoding of the tuned FLC parameters. This includes the scaling gains of the controller. The output scaling gain of the controller was adjusted over a range of
Fig 9: Responses of GA-PID controller of G6(s) and G7(s)
-
OPTIMIZATION OF PI TYPE FLC USING GENETIC ALGORITHM
The combination of genetic algorithm and fuzzy logic controllers is normally shortened as GA- FLC and this intelligent hybrid controller has found application in many scenarios like motor speed control, temperature control, robotics and
1-9 while the input scaling gain and the derivative input scaling gains were adjusted over a range of 0.0003 – 0.0009 and 0.01 – 1 respectively. The suitability of the ranges of the scaling gains was determined from the prior hand tuning of the controller.
The step responses of the optimized fuzzy PI- controller are shown below for the various systems.
unit step response
in many other control systems.
This method employs the fuzzy logic technique to design a Fuzzy Proportional-Integral (PI) Controller and optimizes scaling gains of this controller using Genetic Algorithm (GA).
The inference system has three linguistic variables which are the two inputs (error signal and change in error signal) and the output (control signal), each input having five membership function sets. This results in 25-rule fuzzy inference system with inputs as the error and the rate of change in error. The output of the fuzzy logic inference system is the control action of the controller and the universe of discourse of all the variables are set within the range (-1, 1).
The Mamdani fuzzy inference method was considered to develop the model. Such method attempts to solve control problems by a set of
1.5
1
amplitude
0.5
0
G4(s)
G5(s)
0 5 10 15 20 25 30
time
Fig 10: Responses of GA tuned PI-type FLC of G4(s) and G5(s)
unit step response
G6(s)
G7(s)
G6(s)
G7(s)
1.4
1.4
unit step response
1.2
1
0.8
amplitude
0.6
0.4
1.2
1
0.8
amplitude
0.6
0.4 ZN
FLC
0.2
0
0 5 10 15 20 25 30
0.2
0
-0.2
GA-PID GA-FLC
time
Fig 11: Responses of GA tuned PI-type FLC of G6(s) and G7(s)
RESULTS AND DISCUSSION
The performance metrics of the conventional PID controller, PI-type fuzzy controller, GA-PID and GA-optimized fuzzy logic PI controller obtained from the simulation are tabulated below and the comparison of step responses of the different controllers are also shown in Fig. 12 and Fig. 13.
0 5 10 15 20 25 30 35 40 45 50
time
Fig .12 Step responses of G5(s) with different Controllers
unit step response
2
1.5
amplitude
1
ZN
From the tables 3, 4, 5 & 6, it can be seen that the fuzzy logic controller can produce the better performance with the use of only the proportional and Integral Component (PI). When compared to
the conventional PID controller, the PI-type fuzzy
0.5
0
FLC GA-PID
GA-FLC
0 5 10 15 20 25 30
time
logic controller shows a better performance in terms of overshoot while it exhibits a slightly lesser performance in terms of rise time and settling time. Further Genetic Algorithm technique is used to optimize the conventional PID and PI-type fuzzy logic controller. By comparing the results, it can be noted that the GA-optimized PI-type fuzzy controller produces better performance in terms of achieving the desired value with for the IAE, percentage overshoot, rise time and settling time, as compared to the PI-type FLC and GA-PID controllers.
Fig .13 Step responses of G6(s) with different controllers
Table 3: Value of IAE achieved by the examined controllers tuned by different methods
ZN
FLC
GA-PID
GA-FLC
G1(s)
1.5185
3.7559
3.4404
3.4192
G2(s)
3.7827
9.0040
8.4988
8.0983
G3(s)
4.8676
9.0212
8.6487
7.9154
G4(s)
1.9767
6.4774
6.1500
4.9485
G5(s)
2.4464
7.1983
5.5370
4.5543
G6(s)
1.3811
6.1054
4.5794
3.5108
G7(s)
2.1240
5.0163
3.8429
3.7472
Table 4: Value of Percentage Overshoot OS (%) achieved by the examined controllers tuned by different methods
ZN
FLC
GA-PID
GA-FLC
G1(s)
49.2961
0
0
0
G2(s)
45.8958
0
0
0
G3(s)
62.5152
0
0
0
G4(s)
48.8019
0
0
0
G5(s)
20.2102
0
0
0
G6(s)
53.5438
0
0
0
G7(s)
68.5653
0
0
0
Table 5: Value of Rise-time (tr) achieved by the examined controllers tuned by different methods
ZN
FLC
GA-PID
GA-FLC
G1(s)
0.7076
6.6089
6.4750
5.1251
G2(s)
1.1530
5.3924
6.7234
6.5440
G3(s)
2.1775
5.2174
6.4293
6.3521
G4(s)
0.9534
6.0942
6.4617
6.0029
G5(s)
1.4142
5.7092
6.5493
5.2133
G6(s)
0.5626
6.5305
6.6646
4.9769
G7(s)
0.6069
5.9776
5.9292
5.3606
Table 6: Value of Settling-time (ts) achieved by the examined controllers tuned by different methods
ZN
FLC
GA-PID
GA-FLC
G1(s)
8.7372
9.2893
9.1558
8.2792
G2(s)
9.8992
9.9106
9.8623
9.8597
G3(s)
9.9100
9.9131
9.8677
9.8583
G4(s)
8.2922
9.7648
9.7287
9.5387
G5(s)
7.6717
9.8411
9.5328
9.0687
G6(s)
6.2269
9.6567
9.6083
8.2824
G7(s)
9.4797
9.4678
9.4032
8.8841
-
CONCLUSION
In this paper, the designing and tuning of different controller using Ziegler-Nichols, fuzzy logic and Genetic algorithm has been presented. The performance metrics taken into consideration are the IAE, overshoot, rise time and settling time. The simulation is carried out on various systems and the results shows that the GA-optimized PI-type fuzzy controller gives better performance than PI-type
fuzzy and GA-PID controller in terms of all the metrics. According to the profiling results, the use of soft-computing technique resulted in a better outputs. The amount of overshoot for the output response was successfully decreased using the GA tuned PI-type FLC but the values of other performance parameters like IAE, rise-time and settling-time are slightly increases as compared to conventional controller. Thus a compromise has to be made between the percentage overshoot and the other performance parameters.
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