International Journal of Application or Innovation in Engineering & Management (IJAI
(IJAIE
AIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
ISSN 2319 - 4847
Special Issue for International Technological Conference-2014
Optimization of PID Parameters using Dual
Population Genetic Algorithm
Dr. Shanta Sondur1, Anil Kumar R. Sudele2
1
Information Technology Dept., VES Institute of Technology, Mumbai, India
shantasondur@gmail.com
2
CRG Engineer, SMEC Automation Pvt. Ltd., Mumbai, India
anilsudele@gmail.com
ABSTRACT
The existing Dual Population Adaptive Genetic Algorithm (DPAGA) requires more number of iterations and longer time duration
for optimization. A modification has been suggested for overcoming the same. In the existing algorithm the adaptive crossover and
adaptive mutation is done in the main population and immediately the convergence condition is checked by evaluating the individuals
of the main population only whereas, in the proposed algorithm, the adaptive crossover and adaptive mutation are performed for
both the main population and the subordinate population. Then the individuals of both the populations are evaluated for checking
the convergence condition. This improves the efficiency of the proposed algorithm and it gives solution in lesser iterations. The
existing DPAGA and proposed algorithm has been applied for optimization of the PID parameters for some Plant Transfer
Functions. The performance of both the methods of optimization of PID parameters has been compared.
Keywords: DPAGA, PID parameters, Optimization, Adaptive Mutation, Adaptive Crossover etc.
1. INTRODUCTION
The Dual-Population Genetic Algorithm (DPGA) was originally proposed for stationary optimization problems [3, 4]. It
consists of two distinct populations with different evolutionary objectives. In DPGA [5], the main population plays the
same role as that of the population of an ordinary Genetic Algorithm. It evolves to find a good solution with a high fitness
value. The additional population called the reserve population provides additional diversity to the main population. In
order to allow the main population to use the diversity in the reserve population, there must be a method of exchanging
information between the populations. The migration method used for most multi-population GAs (MPGAs), however, is
not suitable for DPGA because the two populations have different evolutionary objectives. An individual of one population
can hardly survive in the other because the methods for evaluating fitness are different for both populations. Therefore,
DPGA uses crossbreeding as a means of information exchange.
In DPGA, offspring are produced by mating not only the individuals of the same population but also the individuals of
different populations. Since the crossbred offspring contain the genetic material of both populations, their fitness values
are not too low and thus they assimilate relatively easily into the new population. Another Dual Population Genetic
Algorithm with Evolutionary Density (DPGA-ED) [3] has been introduced which is the improved version of DPGA [5].
In DPGA-ED the reserve population evolves by itself to maintain the diversity whereas that in the DPGA [5] cannot evolve
on its own and depends of the genetic material imported from the main population. DPGA-ED shows performance
improvements and avoids the premature convergence. The DPGA [6] generalizes DPGAs and provides a more thorough
analysis on a wider variety of problem classes using binary, real-valued, and order-based representations. It has been shown
that the main population of DPGA can become more exploitative than the population of standard GAs when it approaches
a peak, but can also become much more explorative when it tries to escape from a local peak. This change of mode is made
possible by the existence of the reserve population that adaptively maintains appropriate distance to the main population
and thus provides controlled diversity to the main population. Latest version of Dual Population Adaptive Genetic
Algorithm (DPAGA) [7] involves adaptive adjustment of the crossover probabilities and mutation probabilities for both
main and the reserved population. The crossover probability and mutation probability get adjusted dynamically in the
course of evolution according to the actual situation of population. The DPGA [3] and DPGA-ED [6] take thousands of
iterations required for the high dimensional unimodal and multimodal benchmark test functions. Whereas in DPAGA [7],
the maximum number of iterations required for finding the solutions for the benchmark test functions was 300. Looking
at the slow convergence rates (More number of iterations) we would like to propose some changes to improve the
convergence rate and time by optimizing the number of iterations.
The objective is to optimize the number of iterations of the algorithm discussed in [7], apply that for the optimization of
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PID parameters and compare the performance of the existing method with the modified method.
2. EXISTING ALGORITHM
Here we describe the existing DPAGA algorithm given in [7]. Genetic algorithm based on adaptive evolution in dual
population (DPAGA) [7] is a novel improved adaptive genetic algorithm. In this method, the authors have discussed about
the new concept of adaptive mutation and crossover for both the main and the reserved population. The crossover
probability and mutation probability get adjusted dynamically in the course of evolution according to the actual situation
of population. However the algorithm takes more number of iterations for convergence. Its flowchart is shown in figure 1.
Figure 1. The working flow of DPAGA
Following are the steps of the existing algorithm.
Steps of Algorithm:
Algorithm BEGIN
• Step 1: Code genetic individuals and initialize population ;
• Step 2: Calculate individual fitness;
• Step 3: Perform selecting operation according to the roulette wheel selection method, and gain the new population (main population) and the eliminated population (subordinate population);
• Step 4: For the main population , perform adaptive crossover operation with high probability and adaptive mutation
operation with low probability;
• Step 5: Judge convergence condition of . If it is congruous, perform Step 9;
• Step 6: For the subordinate population , perform adaptive crossover operation with low probability and adaptive
mutation operation with high probability;
• Step 7: Form the new population from the results of Step 4 and Step 6;
• Step 8: Repeat Step 2 ~ Step 6, till the convergence condition is congruous;
• Step 9: Stop.
Algorithm END
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Its population includes a new population formed by selection operation (the main population) and a population eliminated
by selection operation (the subordinate population). Dual crossover means the main crossover operator and the
subordinate crossover operator . Dual mutation operator means the main mutation operator and the subordinate
mutation operator . In the evolution of the main population, the individuals perform adaptive crossover operation with
high probability and adaptive mutation operation with low probability according to the main crossover operator and the
main mutation operator. And in the evolution of the subordinate population, the individuals perform adaptive crossover
operation with low probability and adaptive mutation operation with high probability according to the subordinate
crossover operator and the subordinate mutation operator.
The dual crossover operator and dual mutation operator are adjusted adaptively according to the individual fitness in the
evolution of population. Their formulae can be described as follows:
+
=
,
=
=
1 + exp +
,
+
,
=
− ,
(./0 =
( < ( *+
− 1 + exp !"
%& , ( ≥ ( *+
- !"
%& ,
- # $ - !"
1
1 + 1/2.−/0
( < ( *+
− 1 + exp ( ≥ ( *+
# $ !"
( < ( *+
− 1 + exp +
!"
%& ,
# $ !"
- !"
( ≥ ( *+
%& , ( ≥ ( *+
- # $ - !"
( < ( *+
(1)
(2)
(3)
(4)
(5)
Where, ( and ( denotes the maximal fitness of the main population and subordinate population respectively,
( *+ and ( *+ denotes the average fitness of the main population and subordinate population respectively, ( denotes
the higher fitness of the two crossing individuals, ( denotes the fitness of the mutating individuals, and denotes the lower limit and the upper limit of the crossover probability of the main population respectively, and
denotes the lower limit and the upper limit of the crossover probability of the subordinate population respectively,
and denotes the lower limit and the upper limit of the mutation probability of the main population
respectively, and denotes the lower limit and the upper limit of the mutation probability of the subordinate
population respectively. Generally, their values may be chosen as follows, = 0.6, = 0.9, = 0.001,
= 0.1, = 0.001, = 0.1, = 0.1, = 0.5.
From formula (1) ~ formula (4), we can see that the probability of crossover and the probability of mutation are adjusted
nonlinearly according to the function (./0 between the average fitness and the maximal fitness. When the fitness of the
majority of individuals is near and the average fitness is close to the maximal fitness, the probability of crossover and the
probability of mutation will be elevated. So, the individuals nearby the maximal fitness are preserved as many as possible.
3. THE PROPOSED ALGORITHM
In DPAGA [7] the workflow suggests that the convergence condition should be checked by evaluating the individuals of
the individuals of the main population only after the adaptive mutation and crossover have been done. Later the adaptive
mutation and crossover are performed in the individuals of the subordinate populations and then are evaluated and ranked.
Later on the populations are sorted in to main and subordinate population using roulette wheel selection. Checking the
convergence condition before performing crossover, mutation and evaluating the individuals in the subordinate population
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and considering only the individuals of the main population reduces the chances of selection of the best individual that
may have been evolved in the subordinate population. Therefore this algorithm takes many iterations for solving the
problems. The modification suggested is to perform crossover and mutation of the individuals of the main population and
form a new main population. Then perform crossover and mutation of the subordinate population and form a new
subordinate population. Later on convergence condition is checked by evaluating individuals of both the populations i.e.,
the main population and subordinate population. This can reduce the number of iterations. In terms of steps, we propose
to carry out step number 5 after step number 6 and 7.
Steps 1, 2, 3 and 4 will remain the same in the proposed algorithm. However, instead of checking the convergence condition
at step 5 we prefer it to be done after step 6 and 7. Following are the steps of the proposed algorithm. The flowchart of the
proposed DPAGA is shown in Figure 2. The selection method remains the same Roulette Wheel Selection. Also formulae
for the mutation and crossover probabilities of both the population remains same as DPAGA [7].
Figure 2. The working flow of the proposed DPAGA
Following are the steps of the proposed algorithm.
Steps of Algorithm:
Algorithm BEGIN
• Step 1: Code genetic individuals and initialize population ;
• Step 2: Calculate individual fitness;
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• Step 3: Perform selecting operation according to the roulette wheel selection method, and gain the new population (main population) and the eliminated population (subordinate population);
• Step 4: For the main population , perform adaptive crossover operation with high probability and adaptive mutation
operation with low probability;
• Step 5: For the subordinate population , perform adaptive crossover operation with low probability and adaptive
mutation operation with high probability;
• Step 6: Form the new population from the results of Step 4 and Step 5;
• Step 7: Judge convergence condition of . i.e., by considering the individuals of the main population and
subordinate population . If it is congruous, perform Step 9;
• Step 8: Repeat Step 2 ~ Step 7, till the convergence condition is congruous;
• Step 9: Stop.
Algorithm END
4. IMPLEMENTATION AND RESULTS
The DPAGA [7] and the Proposed Algorithm have been used for optimization of PID Parameters for various Plant transfer
functions. The following plant transfer functions (TFs) were used.
3 .40 =
20
324 8 + 24 + 9
(6)
342.5
+ 7.44 + 324.5
1
3> .40 =
4.4 + 20.4 + 40
38 .40 =
48
(7)
(8)
400
+ 304 8 + 2004
4.228
3@ .40 =
8
.4 + 0.50.4 + 1.644 + 8.4560
3? .40 =
3C .40 =
4>
(9)
(10)
27
.4 + 10.4 + 30>
(11)
1.6
4 8 + 2.5844 + 1
406.14 + 178.9
3E .40 = ?
>
4 + 7.14 + 121.94 8 + 71.84 + 21.5
3D .40 =
(12)
(13)
Here, F denotes the population size and 3 denotes the maximum number of Generations. Two trials (Trial 1 and Trail 2)
are performed for optimizing the PID Parameters using the DPAGA [7] based PID and Proposed DPAGA based PID. F
is taken as 100. Trail 1 was performed till the 30 generations. Trial 2 was performed till there was no significant difference
between the best individuals for 5 consecutive generations or till 10,000 generations were reached. 3 denotes the number
of iterations required for optimization while, GH , GI and GJ denotes Proportional, Integral and Derivative gains of the PID
Controller. The selected domain for the parameters, GH , GI and GJ of the Transfer Functions 3 .40 to 3E .40 for the DPAGA
[7] and Proposed Algorithm are shown in Table 1.
Table1. The initial domain selected for GH , GI and GJ
Plant TFs
3 .40
38 .40
3> .40
3? .40
3@ .40
3C .40
3D .40
3E .40
Range of KL
2 - 20
0 - 120
0 - 50
2-8
1-3
0 - 18
0 - 31
0 -16
Range of KM
0-6
0 - 25
0 - 25
0-5
0-1
0 - 17
0 - 19
0 -11
Range of KN
0-6
0 - 10
0 - 25
0-1
0-2
0 - 10
0 - 19
0-1
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The ‘Integral of Absolute Magnitude of the Error’ (IAE) has been used as the evaluation criteria for the individuals of both
the populations. Other specifications, such as overshoot, rise time and settling time may also be taken into account. In
order to overcome the large energy of the controller, the square term of control output O.P0 is added to the fitness function
[9]. The Objective function is
U
Q = R [T |1.P0| + T8 O8 .P0]YP + T> . PZ
(14)
Q = ^V [T |1.P0| + T8 O8 .P0 + T? |1[.P0|]YP + T> . PZ , 1[.P0 < 0
(15)
V
Where, w1, w2, w3 are weight coefficients, O.P0is the output of controller, PZ is rise time and 1.P0 is the system error. In
order to get satisfactory transient response and to suppress overshoot, Objective function [10] [11] was revised as follows:
U
1[.P0 = [.P0 − [.P − 10
(16)
Where T? is the weight coefficient, and T? >>T , [ is the output of the plant. Where there is overshoot, a punishment term
1[.P0 is added at once. Weight coefficients are T = 0.999, T8 = 0.001, T> = 0.2 and T? = 100. The fitness function
is chosen as
(=
1
Q
(17)
The fitness Function is the inverse of IAE. The elapsed time and optimum values of GH , GI and GJ parameters obtained
using the DPAGA [7] based PID and the Proposed DPAGA based PID for the selected transfer functions in Trial 1 is
shown in Table 2 and that for Trial 2 is shown in Table 3. Figure 3 to 6 shows response for the transfer functions 3C .40
and 3E .40 in Trial 1 and trail 2.
Table 2. The PID parameters obtained in Trial 1
Trial 1
Plant
TFs
3 .40
38 .40
3> .40
3? .40
3@ .40
3C .40
3D .40
3E .40
The Existing DPAGA[7] – PID
Time
KL
KM
KN
taken in
seconds
4.8370
2.7032
1.2465
54.144
5.2043
1.0842
0.4337
54.144
27.3441 16.3570 13.7539
40.860
6.2562
0.9195
0.2373
39.707
1.3737
0.4018
1.8577
39.894
16.2000 15.3000
9.0000
40.268
1.9758
1.2110
1.2110
46.086
14.4000
9.9000
0.9000
49.204
\]
30
30
30
30
30
30
30
30
The Proposed DPAGA[7] – PID
Time
KL
KM
KN
taken in \]
seconds
18.200 5.4000 5.4000
36.697
30
108.00 22.500 9.0000
36.796
30
45.000 22.500 22.500
27.038
30
7.4000 0.5000 0.9000
27.038
30
2.8000 0.9000 1.8000
27.491
30
1.3207 1.2474 0.7337
17.845
30
27.900 17.100 17.100
31.721
30
1.1732 0.8065 0.0733
33.604
30
Table 3. The PID parameters obtained in Trial 2
Trial 2
Plant
TFs
3 .40
38 .40
3> .40
3? .40
3@ .40
3C .40
3D .40
3E .40
KL
3.8
12.000
45.000
6.2562
1.2
16.200
3.1000
14.400
The Existing DPAGA[7] – PID
Time
KM
KN
taken in
seconds
0.6
0.6
1530.2
2.5000
1.0000
18.271
22.500
22.500
19.845
0.9195
0.2373
40.588
0.1
0.2
1398.4
15.300
9.0000
15.405
1.9000
1.9000
106.37
9.9000
0.9000
18.192
\]
1104
14
15
16
1105
9
57
9
The Proposed DPAGA[7] – PID
Time
KL
KM
KN
taken in \]
seconds
18.200 5.4000 5.4000
9.1364
9
108.00 22.500 9.0000
20.557
22
45.000 22.500 22.500
15.853
17
7.4000 4.5000 0.9000
14.296
15
2.8000 0.9000 1.8000
16.271
14
1.3207 1.2474 0.7337
299.34
334
27.900 17.100 17.100
28.379
21
1.1740 0.8071 0.0734
231.01
138
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Figure 3. Step response of the Transfer function 3C .40 with
the values of GH , GI and GJ parameters obtained by the
existing DPAGA and the proposed DPAGA in Trial 1
Figure 4. Step response of the Transfer function 3C .40 with
the values of GH , GI and GJ parameters obtained by the
existing DPAGA and the proposed DPAGA in Trial 2
Figure 5. Step response of the Transfer function 3E .40 with
the values of GH , GI and GJ parameters obtained by the
existing DPAGA and the proposed DPAGA in Trial 1
Figure 6. Step response of the Transfer function 3E .40 with
the values of GH , GI and GJ parameters obtained by the
existing DPAGA and the proposed DPAGA in Trial 2
The performance parameters obtained from the DPAGA [7] based PID and the Proposed DPAGA based PID in Trial 1 are
shown in Table 4 and that for Trial 2 is shown in Table 5. The comparison of performance parameters of DPAGA [7]
based PID and the Proposed DPAGA based PID in terms of settling time, time taken for convergence and number of
generations required for convergence in Trial 1 and Trial 2 is shown in Table 6. While the comparison in terms of
percentage improvements in settling time, time taken for convergence and number of generations required for convergence
in Trial 1 and Trial 2 is shown in Table 7.
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Table 4. The performance parameters obtained in Trial 1
Trial 1
Plant TFs
3 .40
38 .40
3> .40
3? .40
3@ .40
3C .40
3D .40
3E .40
_L
0.63
0
0.33
1.27
0
The Existing DPAGA[7] – PID
`a
`b
\]
0.764
17.9
30
0.0435
6.4
30
0.644
2.36
30
0.33
0.702
30
7.333
9.83
30
Unstable
30
0.05
4
2.03
30
Unstable
30
The Proposed DPAGA[7] – PID
_L
`a
`b
\]
0.3
0.363
1.27
30
0
0.0418
0.05
30
0.29
0.457
2.3
30
0.1
0.207
0.278
30
0
0.599
5.21
30
0.1
3.6
4.93
30
0.01
2.25
0.139
30
0.09
1.26
1.84
30
Table 5. The performance parameters obtained in Trial 2
Trial 2
Plant TFs
3 .40
38 .40
3> .40
3? .40
3@ .40
3C .40
3D .40
3E .40
_L
0.6
0
0.29
0.27
0
The Existing DPAGA[7] – PID
`a
`b
\]
0.96
17.1
1104
0.1048
2.06
14
0.457
2.31
15
0.144
0.702
16
44.605
45.4
1105
Unstable
9
0.04
3.33
1.5
57
Unstable
9
The Proposed DPAGA[7] – PID
_L
`a
`b
\]
0.3
0.363
1.27
9
0
0.0418
0.05
22
0.29
0.457
2.31
17
0.1
0.207
0.278
15
0
0.599
5.21
14
0.1
3.6
4.93
334
0.01
2.25
0.139
21
0.13
0.895
1.84
138
Table 6. The comparison of the improvement in the performance parameters obtained in Trial 1 and Trial 2
The Existing DPAGA[7] – PID
Plant
TFs
3 .40
38 .40
3> .40
3? .40
3@ .40
3C .40
3D .40
3E .40
Number of
Generations
Trial 1
Trial 2
30
1104
30
14
30
15
30
16
30
1105
30
9
30
57
30
9
Settling time
Trial 1
17.9
6.4
2.36
0.702
9.83
unstable
2.03
unstable
Trial 2
17.1
2.06
2.31
0.702
45.4
unstable
1.5
unstable
The Proposed DPAGA[7] – PID
Number of
Generations
Trial 1
Trial 2
30
9
30
22
30
17
30
15
30
14
30
334
30
21
30
138
Settling time
Trial 1
1.27
0.05
2.31
0.278
5.21
4.93
0.139
1.84
Trial 2
1.27
0.05
2.31
0.278
5.21
4.93
0.139
1.84
The Table 7 indicates that the Proposed DPAGA gives very good improvements in the settling time. In case of 38 .40 and
3C .40 the Proposed DPAGA takes more convergence time and more number of generations compared to the existing
DPAGA but, the performance of the Proposed DPAGA is better than the existing DAPGA. Also, in case of 3C .40 and
3E .40 the Proposed DPAGA takes more convergence time and more number of generations compared to the existing
DPAGA but, the performance of the Proposed DPAGA is acceptable while the existing DAPGA gives unstable response
in Trial 1 and Trial 2.
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Table 7. The comparison of the percentage improvements in the performance parameters obtained in Trial 1 and Trial 2
Settling time
Plant TFs
3 .40
38 .40
3> .40
3? .40
3@ .40
3C .40
3D .40
3E .40
Trial 1
92.905%
99.218%
2.542%
70.085%
46.998%
∞%
93.152%
∞%
Trial 2
92.573%
97.572%
0%
60.398%
88.722%
∞%
90.733%
∞%
Convergence time
Trial 1
32.223%
32.040%
33.8277%
31.906%
31.089%
55.684%
31.169%
31.704%
Trial 2
99.403%
-12.511%
20.115%
64.777%
98.836%
-24.922%
73.320%
1169.84%
Number of Generations
Trial 1
NA
NA
NA
NA
NA
NA
NA
NA
Trial 2
99.1364%
-57.142%
-5.686%
64.777%
98.9376%
-1611.11%
63.157%
-1433.33%
5. CONCLUSION
In our work we tried to improve the performance of the existing DPAGA in terms of number of iterations and settling time.
We suggest to check the convergence condition after evaluating the individuals of both the populations i.e. the main
population and the subordinate population. This improves the performance of the existing algorithm. We also applied these
two algorithms to optimize the PID parameters. The performance of the proposed DPAGA based PID controllers gave
acceptable results for all the selected transfer functions. Moreover, in almost all the cases the Proposed DPAGA based PID
performed better than the existing DPAGA based PID in terms of settling time, number of iterations and time taken for
convergence. We observed that in few of the cases the tuning parameters obtained using existing DPAGA gave unstable
response whereas for the same transfer functions the proposed algorithm gave stable response.
6. FUTURE SCOPE
The scheme of DPAGA may be adopted for optimization of the PID Parameters in industrial controllers.
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Organized By: Vivekanand Education Society's Institute Of Technology
International Journal of Application or Innovation in Engineering & Management (IJAI
(IJAIE
AIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
ISSN 2319 - 4847
Special Issue for International Technological Conference-2014
[11] Gouhab Lin, Guofan Liu, “Tuning PID Controller using Adaptive Genetic Algorithms” The 5th International
Conference on Computer Science & Education Hefei, China, August 24-27, 2010.
AUTHORS
Shanta Sondur received the B.E. and M.E. degrees in Instrumentation and Control Engineering from
S.G.G.S.C. of Engineering and Technology, Nanded and Electronics and Telecommunications from Govt. C.O.E
Pune, in 1989 and 1997, respectively. She has received PhD degree from Systems and Control Engineering, IIT
Bombay in 2008. During 1991-1998, she worked as faculty in Dept. of Instrumentation at GCOE, Pune. Now
she is working as a Professor in Dept. of IT, at VES Institute of Technology, Mumbai, India.
Anil Kumar R. Sudele received the B.E. degree in Electrical Engineering from Yadavrao Tasgaonkar Institute
of Engineering and Technology, Karjat and M.E. degree in Instrumentation & Control Engineering from VES
Institute of Technology, Mumbai in 2010 and 2013, respectively. From 2013 he is working as Core Resource
Group Engineer at SMEC Automation Pvt. Ltd., Mumbai, India. His areas of interest are Process control and
Automation.
Organized By: Vivekanand Education Society's Institute Of Technology