 Open Access
 Total Downloads : 14
 Authors : Ashutosh Prasad Yadav, Alok Kumar, Raj Kumar
 Paper ID : IJERTCONV4IS15005
 Volume & Issue : ACMEE – 2016 (Volume 4 – Issue 15)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
PID Controller Tuning using Genetic Algorithm for Coupled Tank System
Ashutosh Prasad Yadav Electrical and Instrumentation Department SLIET, Longowal, Sangrur, Punjab India148106
Alok Kumar
Electrical and Instrumentation Department SLIET, Longowal, Sangrur, Punjab India148106
Raj Kumar
Electrical and Instrumentation Department SLIET, Longowal, Sangrur, Punjab India148106
AbstractPerformance of Proportional Integral Derivative (PID) controllers is used in process control industries mainly suffers due to the controller tuning parameter selection. So genetic algorithm (GA) based PID controller is proposed and investigated in this work. Genetic algorithm based PID controller is designed for coupled tank system (non interacting system). The transfer function of this process is obtained. The transfer function is approximated into first order plus delay time (FOPDT) model as per equipment specification. PID parameters are obtained by multi objective genetic algorithm (MOGA) techniques. In this work genetic algorithm operators are binary tournament selection, simulated binary crossover (SBX), polynomial mutation and elitism through nondominated sorting crowding distance nearest neighbor. Simulations are performed in MATLAB/Simulink to compare the closed loop performance results of genetic algorithm PID tuning with ZeiglerNichols (ZN), CohenCoon, and TyreusLuyben tuning methods in terms of time response characteristics and performance indices like integral of absolute error (IAE), integral squared error (ISE), and integral time absolute error (ITAE). The results are compared with experimental work and confirm the validity of this technique. The results indicate that PID controller tuned by genetic algorithm provides better performance and robustness as compared to other techniques.
KeywordsPID controller, genetic algorithm, ZeiglerNichols tuning, CohenCoon tuning, TyreusLuyben tuning, coupled tank system, roubstness.

INTRODUCTION
PID controller widely used in industries due to their design simplicity and its reliable operation. A simple PID controller consists of three terms Kp, Ki, Kd referring to proportional, integral, derivative gain respectively. To implement PID controller, the gain of PID controller must be determined. The adjustment process of the value Kp, Ki, Kd is called tuning or design of PID controller. So, tuning of PID controllers has always been an area of active interest in the industry. Great effort has been devoted to develop methods to reduce the time spent on optimizing the choice of PID controller parameters. There are many tuning techniques based on several methods. These methods classified as: i) conventional tuning approaches such as manual tuning, ZeiglerNichols method, TyreusLuyben parameter rules and CohenCoon method [1] ,
ii) stochastic tuning approaches such as genetic algorithm, particle swarm optimization(PSO), ant colony optimization, bacteria foraging based optimization and simulated annealing optimization. Among these methods one of the most successful and oldest classical techniques is ZeiglerNichols method [2]. For a wide range of industry processes, ZN tuning
method works quite well. Before applying ZN method prior knowledge regarding plant model is necessary. Once tuned controller by ZN method good but not optimum system response will be reached, the transient response and robustness can be even worse if the plant dynamics are environmentally change. So, recent year optimization techniques are used.
In this paper our main motto is control the level of coupled tank system. GA based PID tuning was implemented in this paper.
This paper is organized as follows. The section II describes coupled tank system and its mathematical modeling, Section III describes Genetic Algorithm for PID tuning, Section IV describes simulation results, Section V describes experimental results and Section VI describes conclusion.

COUPLED TANK SYSTEM
The level control problem in coupled tank system is featured as a benchmark problem in the category of nonlinear and unstable control systems. Process industries play a significant role in economic growth of a nation. Control of liquid level in tanks and fluid flow between tanks is a fundamental requirement in almost all process industries such as waste water treatment, chemical, petrochemical, pharmaceutical, food, beverages, etc. Mostly, level and flow control in tanks are popular in all process control systems.

Mathematical modeling of coupled two tank non interacting level process[3,4]
Consider the process consisting of two noninteracting liquid tanks in the Fig.1. The objective of the process is control the level of tank. Here load changes in first tank affect the second tank but not the viceversa. Qi is the volumetric flow rate into Tank1, Q is the volumetric flow rate from tank 1 to tank 2 and Qo is the volumetric flow rate out of Tank 2. Height of liquid level in tank1 is H1 and in tank 2 is H2. Both tanks are having same crosssectional area A. Two ball valves V1 and V2 having hydraulic resistances R1 and R2 are connected at the outlet of each tanks. Vi is the control input voltage to pump.
Table 1: Parameters of coupled tank system
Parameter
Description
Value
Unit
A
Crosssectional area of tanks
138.9
cm2
R1
Hydraulic resistance of ball valve 1
0.081
sec/ cm2
R2
Hydraulic resistance of ball valve 2
0.081
sec/ cm2
Pump constant related to flow rate into tank
36.6
cm3/v.sec
Table 1: Parameters of coupled tank system
Parameter
Description
Value
Unit
A
Crosssectional area of tanks
138.9
cm2
R1
Hydraulic resistance of ball valve 1
0.081
sec/ cm2
R2
Hydraulic resistance of ball valve 2
0.081
sec/ cm2
Pump constant related to flow rate into tank
36.6
cm3/v.sec
Qi
A
Pump
From reservoir
Tank 1
Tank 2
H1
R1 V1
Q
A
H2

FOPDT model approximation
Industrial processes are of higher order so finding a real value of it is very difficult. The transfer functions of plants that can be approximately modeled by some definite transfer function. Sundaresan and Krishnaswamy [5] have proposed a simple method for fitting the dynamic response of higher order systems in terms of first order plus time delay transfer functions. The obtained second order transfer function of the coupled tank system is approximated into a FOPDT transfer function using the same method as:
The method is based on times, t1 and t2, which can be estimated from a step response curve (Fig.2), corresponding to the 35.3% and 85.3% response times, respectively.
R2 V2
Qo To reservoir
Fig.1: Two tank noninteracting process
Assuming linear resistance to flow, transfer function of the coupled tank system through mathematical modeing is
Fig.2: FOPDT approximation curve
G(s) = H2(s) = R2
(1)
The time delay and time constant are then estimated from
Qi(s) (1s+1)(2s+1)
the following equations:
Where 1 = AR1 and 2 = AR2 are the time constants of Tank 1 and Tank 2 related to operating levels in the tank
Flow rate of the pump is related as:
Qi(s) = Vi(s); is pump constant relating to control
d = 1.3t1 .29t1 (4)
= .67(t2 t1) (5)
The FOPDT Transfer function is given by:
voltage
Hence, overall transfer function of the process becomes
(+1)
(6)
() = 2() = 2
FOPDT model of Coupled Tank System is represented as:
()
(1+1)(2+1) (2)
() 2.9646
16.22+1
7.1 (7)
Here H2 is controlled variable and Vi is manipulated variable
Therefore obtained Transfer function of coupled two tank system noninteracting level process using coupled tank parameters from Table 1 is


GENETIC ALGORITHMS FOR PID TUNING

Introduction of genetic algorithm
Genetic algorithms (GAs) are computerized search and optimization algorithms based on the mechanics of natural
2
2
() = 2.9646
126.5827 +22.5018+1
(3)
genetics and natural selection [6] .GAs are very different from most of the traditional optimization methods. GAs need design space to be converted into genetic space. So, GAs work with a coding of variables. A more striking difference between GAs and most of the traditional optimization methods are that GA uses a population of points at one time in contrast to the single point approach by traditional optimization methods. This means that GA processes a number of designs at the same
time. In general, GA consists of several important parts such as initialization, objective function, fitness assignment, genetic operators like crossover and mutation, elitism and termination. Note that the three terms, solution, individual and chromosome are exchangingly used in the next sections which represent a same element. GAs use three fundamental operators: selection, crossover, mutation. Selection operator is used to select the best individuals (solutions) in a population. The crossover operator creates new individuals by mixing couple of selected individuals in a population and the mutation operator creates a new individual by randomly mutating a randomly selected part of a selected chromosome. Better convergence of GA is achieved by both exploiting the search space by selection and crossover operators and exploring the search space for new information by mutation operator.

Implementation of GA
The optimal values of the PID controller parameters Kp, Ki and Kd is found using GA. All possible sets of controller
indroduce a real coded crossover inspired by the binary coded of one point crossover to employed in the real coded GA.

Polynomial mutation: Like in the SBX operator, the polynomial mutation changes the chromosome values based on the user defined mutation index.

Elitism: The crowding distance was introduced by [7] in nondominated sorting genetic algorithm 2 (NSGAII) in improving the niche counting which used NSGA. Crowding distance calculation requires the sorting of the population according to ascending order of each objective. Consider a population of N individuals with M objective values. The smallest and largest values (boundaries) will be assigned as an infinite distance value . For other immediate individuals, the distance of each objective, is calculated based on

equation 8.
parameter values are chromosomes whose values are adjusted so as to minimize objective function, which in this case is settling time, rise time, integral time absolute error(ITAE). For
M
=
=
d
d
i m=1
f(m+1)mf(m1)m fmaxmfminm
(8)
the PID controller design, it is ensured the controller settings estimated results in a stable closed loop system. In this investigation three objective have been used. So multi objective genetic algorithm (MOGA) has been investigated for tuning of parameter. The steps of implementing MOGA are as follows:

Initialization of GAs: To start up with GA, certain parameters need to be defined. It includes the population size, number of iterations, operator types etc. the range of tuning parameter Kp(02), Ki(00.01) and Kd(03). Number of iteration=100 and other initialization shown in table 1.

Objective function: The objective functions considered are based on performance criteria. A number of such criteria are available in this paper controllers performance is evaluated in terms of integral time absolute error (ITAE), rise time and settling time. In this paper we consider the limit for equation from time t=0 to t=Ts, where Ts is settling of the system to reach steady state condition for a unit step input.

Global ranking fitness assignment: The purpose of global ranking is to rank the individuals

Binary tournament selection: Among selection techniques in GA, MOGA uses binary tournament because it is easier to modify the procedure [7] in order to handle constrained in the case of constrained optimization problems. Because of controller optimization problems always deal with the cnstraints that needs to satisfied (e.g. closed loop stability), the design of MOGA should includes the constraint handling technique in the algorithm. The binary tournament selection takes two random individuals then it compares the fitness between the two and the fitter one is selected to be reproduced.

Simulated binary crossover: Binary crossovers like one point crossover or two point crossover have a successful history in binary coded GA. Motivated from this successe, [8]
Where M is the number of objective, fmax and fmin are the values of maximum and minimum objective values respectively. The larger the value of the crowding distance, the smaller (better) its crowdedness property.
When the number of nondominated individuals is more than N, the dominated individuals are automatically rejected. At this stage, the KNN values will take the crowding distances place to descending sort the nondominated individuals. The Fig. 3 shows our proposed elitism mechanism.
Fig.3: The elitism mechanism in MOGA

Termination: Termination of optimization algorithm can take place either when maximum number of iterations gets over or with the attainment of satisfactory fitness value. In this paper termination criteria is considered to be the attainment of satisfactory fitness value which occurs with the maximum number of iterations as 100.

Complete loop: Here the complete flow chart for mechanism of MOGA is shown in fig.4. In MOGA, the objective values of every chromosome are converted into global ranking values and the binary tournament selects the potential parents to be bred.
Fig.4: Complete flowchart of MOGA
After the parents undergo genetic operations (SBX and polynomial mutation), the current population and the newly generated population are combined in the elitism mechanism. As described before, the survivors the combined populations are decided by the nondominated sorting, the crowding distance and kNN techniques.


SIMULATION RESULTS

Initialization of GA and its PID Parameters
Initialization of GA and its parameters found by MOGA technique are shown in Table 2.
Table 2: Parameters of the PID controller optimization by GA
Parameters
Values
No. of generations
100
Population size
80
Probability of Crossover
0.6
Probability of Mutation
0.1
20
Distribution in polynomial mutation
20
Kp Ki Kd
1.1529
0.0434
2.7484

Simulation Results for Performance
The controller performance is measured by calculating performance indices like ISE, IAE and ITAE and determining the time response characteristics like rise time(tr),settling time(ts) and peak overshoot(Mp) through closedloop simulation in MATLAB/Simulink. Performance results for GAPID tuning are compared with ZieglerNichols, Cohen Coon and TyreusLuyben tuning methods to see its effectiveness. The responses to setpoint of magnitude 15 cm for t=180sec has been taken. Results in Table 3 and simulation responses in Fig.57 indicates that GA tuned PID Controller provides optimum settling time, reduced overshoot and
minimized performance indices in comparison with other tuning methods. The responses to step changes in setpoint and in the disturbance at t=100sec for different tuning methods. Simulation responses in Fig.8 and Fig.9 shows set point tracking and disturbance rejection capability of GAPID tuning in comparison with other tuning methods used.
Table 3: Performance results for different tuning methods
Specifications
GAPID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise Time(sec)
5.8068
5.8267
6.2146
11.9207
Settling Time(sec)
28.7841
37.2368
23.7051
85.6635
Peak Overshoot (%)
0
37.3916
17.0065
0
IAE
116.3
164.4
117.3
232.7
ISE
1665
2015
1704
1769
ITAE
1395
1637
806.4
7205
Gain margin
1.9871
2.0882
2.0540
2.1709
Phase margin
50.9381
32.3386
45.4909
71.7253
Fig.5: Simulation response for step input for different tuning methods
Fig.6: Simulation response of integral of absolute value of error (IAE) for different tuning methods
Fig.7: Simulation response of integral square error (ISE) for different tuning methods
Fig.8: Simulation response of different tuning methods for step change in setpoint
Fig.9: Simulation response of different tuning methods for step change in disturbance

Simulation Results for Robustness Testing
The robustness testing of GA tuned PID Controller was evaluated by incorporating uncertainty in the actual process by a factor of 15% and 30% in gain, delay time and time constant. The simulation results in Table 5.35.8 and simulation responses Fig.5.105.15 were presented show the
robustness of GA tuned PID Controller in comparison with other tuning techniques.
Table 4: Performance results with 15% change in gain()
Specifications
15% change in gain()
GA PID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
4.9963
5.3721
5.6982
9.3708
Settling time(sec)
24.7013
35.3501
21.5613
78.8279
Peak Overshoot (%)
2.3289
45.2073
22.3314
0
IAE
121.8
182.3
125.9
233.2
ISE
2070
2573
2113
1993
ITAE
1393
1747
822.7
7115
Fig.10: Simulation response for step input for different tuning methods for 15% change in gain (K)
Table 5: Performance results with 30 % change in gain()
Specifications
30% change in gain()
GAPID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
4.4680
4.8868
5.0942
7.4988
Settling time(sec)
20.5377
33.0165
19.3554
70.8924
Peak Overshoot (%)
9.8704
56.1187
30.0887
0
IAE
134.7
204.8
137.8
233.3
ISE
2729
3430
2742
2302
ITAE
1437
1850
845.6
7007
Fig.11: Simulation response for step input for different tuning methods for 30% change in gain


EXPERIMENTAL RESULTS
Experimental results of coupled tank system were taken by help of coupled tank experimental setup and MATLAB/Simulink. Fig. 16 shows the real time response for
GA based PID tuning for 15 cm level, Fig. 17 shows the ZN tuning and Fig. 18 step change in set point using GA tuning.
Table 6: Performance results with 15% change in delay time ()
Specifications
15% change in delay time()
GAPID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
6.0837
5.9492
6.1445
10.4794
Settling time(sec)
29.4148
38.3656
22.8990
84.4127
Peak Overshoot (%)
9.9513
56.2489
30.0851
0
IAE
133.6
223.1
143
223.3
ISE
2039
2793
2149
1994
ITAE
1472
2748
997.5
6986
Fig.12: Simulation response for step input for different tuning methods for 15% change in delay time ()
Table 7: Performance results with 30% change in delay time ()
Specifications
30% change in delay time()
GAPID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
6.3017
5.9522
6.0980
9.1784
Settling time(sec)
31.4866
67.5595
35.2432
84.7112
Peak Overshoot (%)
23.0506
82.9161
48.2936
0
IAE
165.8
347.5
187.6
233.4
ISE
2476
4186
/td>
2727
2227
ITAE
1790
6600
1646
6764
Fig.13: Simulation response for step input for different tuning methods for 30% change in delay time ()
Table 8: Performance results with 15% change in time constant()
Specifications
15% change in Time Constant()
GAPID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
7.2731
6.7508
7.2371
15.2679
Settling time(sec)
10.7056
43.8052
29.5353
86.3788
Peak Overshoot (%)
0.0576
40.3179
18.3729
0
IAE
116.6
190.5
134.7
233.6
ISE
1629
2064
1704
1795
ITAE
1122
2395
1125
6759
Fig.14: Simulation response for step input for different tuning methods for 15% change in time constant ()
Table 9: Performance results with 30% change in time constant ()
Specifications
30% change in Time Constant()
GAPID
Ziegler Nichols
Cohen Coon
Tyreus Luyben
Rise time(sec)
8.6682
7.5292
8.1310
17.3539
Settling time(sec)
12.5727
48.8882
34.3593
84.9611
Peak Overshoot (%)
0.7193
42.7334
19.8406
0
IAE
117.1
216.5
151.1
234.1
ISE
1636
2150
1740
1842
ITAE
850.3
3291
1452
6291
Fig.15: Simulation response for step input for different tuning methods for 30% change in time constant ()

CONCLUSION
In the application in tuning PID parameters, the MOGA has successfully provides the reliable and optimized PID parameters in the both simulation and real time results. The algorithm was derived and programmed in the MATLAB environment. GA is viable alternative to classical methods of design and parameter optimization for most of the control applications. Elitism selection strategy reduces convergence and computation time and allows fine tuning. Simulation
results were presented to illustrate the GA based PID tuning and to demonstrate its effectiveness. Coupled tank system was considered for liquid level control. The four tuning methods GAPID, ZieglerNichols, CohenCoon and TyreusLuyben considered for PID controller and are comparatively analyzed based on performance and robustness. It is evident from the simulation and results that PID controller tuned with MOGA gives better performance and robustness as compared to other tuning methods.
Fig.16: Real time response for GA based PID tuning
Fig.17: Real time response for ZieglerNichols tuning
Fig.18: Real time response for step changes in setpoint
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