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Krish final
1. DESIGN AND IMPLEMENTATION OF PI CONTROLLER
USING GENETIC ALGORITHM AND ANT COLONY
OPTIMIZATION FOR A SPHERICAL TANK PROCESS
By
A. KRISHNAMOORTHY
M.E. (Process Control & Instrumentation Engg.)
(2009-2011)
Mr. G. SAKTHIVEL
Lecturer (selection grade)
Department of Instrumentation Engg
Annamalai university
chidambaram.
2. OBJECTIVES OF THE PROJECT WORK
• To identify the model of the spherical tank process by black box modeling for
various operating region.
a) Low Level
b) Middle Level
c) High Level
• To tune the PI controller by Ziegler- Nichols method.
• To optimize the designed PI controller using ACO (Ant Colony Optimization)
Technique for various cost function like IAE, ITAE, ISE.
• To tune the PI controller by Genetic algorithm.
3. • To compare the results of ACO tuned PI controller with Z-N tuned PI
and GA tuned PI controller in terms of time domain specification and
performance indices like ISE, MSE, ITAE, IAE.
• To obtain the results form both simulation and real time process for
the corresponding models.
• To check to robustness of the above designed controller and test the
ACO under white noise.
4. PI CONTROLLER
• It consist of proportional and integral action
• PID can be implemented as a stand alone controller (or) part of the
controller
e.g. DDC (or) DCS
• Various actions
P-ACTION P = Kp* e
I-ACTION I = ki ∫e dt
D-ACTION D = Kd d(e)/ dt
where
Kp = proportional gain
KI = Integral gain
5. Closed loop Z-N tuned PI Controller
The transfer function of PI controller looks like following: U= Kp* e (t)+ki∫e(t)
Block diagram of PI controller
Recommended PID Value Setting
TYPE OF
Kp Ti Td
CONTROLLER
P 0.5 Ku ∞ 0
PI 0.45 Ku Pu/1.2 0
PID 0.6 Ku Pu/2 Pu/8
6. OBJECTIVE FUNCTIONS
The following objective function we are using for both
GA and ACO optimization.
1. Integral Absolute error
2. Integral square error
3. Integral time Multiplied by Absolute error
7. It is a type of machine learning technique
Mimics the biological process of evolution
Genetic algorithms
Software programs that learn in an evolutionary manner,
similar to the way biological systems evolve
An efficient, domain-independent search heuristic for a broad
spectrum of problem domains
Main theme: Survival of the fittes.
Moving towards better and better solutions by letting only
the fittest parents to create the future generations
8. Reproduction
• Multiple copies of the same string may be selected for
reproduction and the fittest string should begin to dominate
e.g. roulette wheel selection
Depiction of roulette wheel selection
9. CROSSOVER
•Once the selection process is completed, the crossover
algorithm is initiated.
•The crossover operations swaps certain port of the two
selected strings in a bid to capture the good parts of old
chromosomes and create better new ones.
Singe point
Multi point
Uniform
13. Ant Colony Optimization (ACO) is a paradigm for designing meta heuristic algo-
rithms for combinatorial optimization problems.
Ants travel from node to node until end
decision based on transition probability (called state transition)
Once all ants travel finished Solutions compared
Pheromone evaporation applied to all edges
Pheromone increased along each edge of best/each ant’s path
Original ant system: at each iteration, the pheromone values are updated by all the
ants that have build a solution in the iteration itself.
Daemon activities can be run (like local search)
Redo until termination criteria met
They have an advantage over simulated annealing and genetic algorithm approaches
when the graph may change dynamically. The ant colony algorithm can be run
continuously and can adapt to changes in real time.
14. •Ants choose paths depending on pheromone
•After collecting food, paths are marked
•After some time, the shortest path has the highest probability
15. When ants travel they mark their path with substance
called pheromone
Attracts other ants
When an ant reaches a fork in its path the direction it
follows is based on amount of pheromone it detects
Decision probabilistically made
This causes positive feedback situation
(i.e. Choosing a path increases the probability it will be
chosen)
16. • While ( termination not satisfied )
– create ants
– Find solutions
Quantity of
• Transition probability:
pheromone
β
1
τij (t )α Heuristic
Pij (t ) = dij distance
β
1
∑
j∈allowed nodes
τij (t )α
dij α,β constants
– Pheromone update
– Daemon activities {optional}
17. • While ( termination not satisfied )
– create ants
– Find solutions
– Pheromone update Pheromone laid by
Evaporation rate each ant that uses
edge (i,j)
Q
τij (t +1) = (1 − ρ)τij (t ) +
k∈
∑
Colony that Lk
used edge ( i , j )
– Daemon activities {optional}
18. RESULTS AND DISCUSION
• In this section the result of the implemented ACO (ant colony
optimization) tuned PI Controller was obtained.
• The ACO designed PI controller is initialized with 10 Ants and 100
iterations then response is analyzed.
• From the ACO-PI controller Reponses it is compared with GA
designed PI and ZN – tuned PI controller. The various cost functions
are plotted belowin the given figure with different tabulations.
4.5
Model 1 G(s) = e −120 s
440 s + 1
19. Initialization of Parameters
To start up with GA, certain parameters need to be defined. Initializing value
of the parameters for this project for is as follows:
Population size - 80
Bit length of considered chromosome - 6
Number of Generations - 100
Selection Method - ‘Roulette wheel selection
Crossover type - ‘Single point crossover’
Crossover probability - 0.8
Mutation type - ‘Uniform mutation’
Mutation probability - 0.05
21. S p R sp n
te e o se
1.5
ZN
AC -ita
O e
GA-itae
1
m litu e
A p d
0.5
0
5 0
0 1 0
0 0 1 0
5 0 2 0
0 0 2 0
5 0 3 0
0 0 3 0
5 0 4 0
0 0 4 0
5 0 5 0
0 0
T e (se
im c)
Step response for the closed loop system for the PI controller tuned with different methods
S p R sp n e
te e o s
1.4
A O e
C -ita
A O e
C -ia
1.2
A O e
C -is
1
m litu e
A p d
0.8
0.6
0.4
0.2
0
5 0
0 1 0
0 0 1 0
5 0 2 0 T e (se
0 0 im2 0 c)
5 0 3 0
0 0 3 0
5 0 4 0
0 0 4 0
5 0 5 0
0 0
Step response for the closed loop system for the ACO -PI controller tuned
with different cost function
22. Step Response
1.4
GA-itae
GA-iae
1.2
GA-ise
1
Amplitude
0.8
0.6
0.4
0.2
0
500 1000 1500 2000Time2500
(sec) 3000 3500 4000 4500 5000
Step response for the closed loop system for the GA- PI controller
tuned with different cost function
23. Kp distribution
1.5
Kpdistribution
1
0.5
0
0 10 20 30 40 50 60 70 80 90 100
-3 Ki distribution
x 10
6
Ki distribution
4
2
0
0 10 20 30 40 50 60 70 80 90 100
nuber of iterations
Initial distribution of Kp, Ki for AC
24. Kp setteled
0.5
0.45
0.4
0.35
0 20 40 60 80 100 120
-4 Ki settelled
x 10
10
9
gain
8
7
6
0 20 40 60 80 100 120
number of iterations
Kp, Ki settled for ACO
25. Kp Value
6
4
Gain 2
0
0 10 20 30 40 50 60 70 80 90 100
Ki Value
0.04
0.03
Gain
0.02
0.01
0
0 10 20 30 40 50 60 70 80 90 100
Generations
Kp, Ki settled for GA
28. Step Response
1.8
ZN
1.6 ACO-itae
GA -itae
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
500 1000 1500 2000Time2500
(sec) 3000 3500 4000 4500 5000
Step response for the closed loop system for the
PI controller tuned with different
29. Step Response
1.5
ACO-itae
ACO-iae
ACO-ise
1
Amplitude
0.5
0
500 1000 1500 2000 2500
Time (sec) 3000 3500 4000 4500 5000
Step response for the closed loop system for the ACO -PI controller tuned
with different cost function
30. Step Response
1.8
GA-itae
1.6 GA-iae
GA-ise
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
500 1000 1500 2000Time2500
(sec) 3000 3500 4000 4500 5000
Step response for the closed loop system for the GA- PI controller
tuned with different cost function
31. Kp distribution
2.5
2
Ki distribution 1.5
1
0.5
0 10 20 30 40 50 60 70 80 90 100
-3 Ki distribution
x 10
8
6
Kp distribution
4
2
0
0 10 20 30 40 50 60 70 80 90 100
number of iterations
Initial distribution of Kp, Ki for AC
32. Kp setteled
0.98
0.96
0.94
0.92
0.9
0 20 40 60 80 100 120
-3 Ki settelled
x 10
2.5
2
gain
1.5
1
0 20 40 60 80 100 120
number of iterations
Kp, Ki settled for ACO
33. Kp Value
5
4
Gain
3
2
1
0 10 20 30 40 50 60 70 80 90 100
Ki Value
0.2
0.15
Gain
0.1
0.05
0
0 10 20 30 40 50 60 70 80 90 100
Generations
Kp, Ki settled for GA
36. Step Response
1.8
ZN
1.6 ACO-itae
GA-itae
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
500 1000 1500 2000Time2500
(sec) 3000 3500 4000 4500 5000
Step response for the closed loop system for the PI controller tuned with different methods
37. Step Response
1.4
ACO-itae
ACO-iae
1.2
ACO-ise
1
0.8
Amplitude
0.6
0.4
0.2
0
500 1000 1500 2000Time2500
(sec) 3000 3500 4000 4500 5000
Step response for the closed loop system for the ACO -PI controller tuned
with different cost function
38. S p R sp n
te e o se
1.5
GA-itae
GA-iae
GA-ise
1
m litu e
A p d
0.5
0
5 0
0 1 0
0 0 1 0
5 0 2 0 T e (se
0 0 im2 0 c)
5 0 3 0
0 0 3 0
5 0 4 0
0 0 4 0
5 0 5 0
0 0
Step response for the closed loop system for the GA- PI controller
tuned with different cost function
39. Kp,Ki distributions
4
Ki distribution
3
2
1
0 10 20 30 40 50 60 70 80 90 100
0.015
Kp distribution
0.01
0.005
0
0 10 20 30 40 50 60 70 80 90 100
number of generation
Initial distribution of Kp, Ki for AC
40. Kp setteled
2
1.8
1.6
1.4
0 20 40 60 80 100 120
-3 Ki settelled
x 10
3
gain
2.5
2
0 20 40 60 80 100 120
number of iterations
Kp, Ki settled for ACO
41. K V lu
p a e
8
G in
a 6
4
2
0
0 10 20 30 40 50 60 70 80 90 1 0
0
K V lu
i a e
0.8
0.6
G in
a
0.4
0.2
0
0 10 20 30 40 50 60 70 80 90 1 0
0
G n ra n
e e tio s
Kp, Ki settled for GA
43. obustness of the controller is defined as its ability to tolerate a certain amount of
change in the process parameters without causing the feedback system to go
unstable
n order to investigate the robustness of the proposed method in the model parameters were
altered.
hence
ain constant K,
ime const T,
elay time Td
re deviated by ±15% of its nominal values. Therefore
44. ACTUAL MODELS ALTERED MODELS
4.5
Model 1 G(s) = e −120 s 5.7
440 s + 1 Model 1 G(s) = e −102 s
506 s + 1
6 6.9
Model2 G(s) = e −130 s Model2 G(s) = e −110 s
1200 s +1 1380 s +1
2.7 3.15
Model3 G(s) = e −150 s Model3 G(s) = e −127 s
1050 s + 1 1207 s +1
45. ALTERED MODELS
Case (i)
Gain, K value is incremented by 15%.
The value of is incremented by 15%.
The value of td is decremented by 15%.
Case (ii)
Gain, K value is incremented by 10%.
The value of is incremented by 10%.
The value of td is decremented by 10%.
46. Case (iii)
Gain, K value is incremented by 25%.
and , td values no changes.
Case (iv)
Time constant is incremented by 25%.
and k, td values no changes.
Case (v)
Time delay td is incremented by 25%.
and k, values no changes.
47. Robustness check with various cost functions for various model
CASE 1
Model 1 Model 2 Model 3
Kp Ki %Mp tp Ts tr Kp Ki %Mp tp Ts tr Kp Ki %Mp tp Ts tr
ZN 0.7333 0.0018 33.5 347 1000 105 1.3846 0.0032 53.8 390 1230 105 2.29 0.0046 49.6 448 1390 123
ACO 0.4479 0.00087 1.96 484 380 201 0.9678 0.0020 30.8 504 1225 157 1.4852 0.0020 28.6 618 1310 195
GA 0.5235 0.0012 11.7 439 729 162 1.1528 0.0025 39.5 437 1230 128 1.8586 0.0024 33.9 509 1350 155
55. The following results shows different PI-tuned methods
are implemented from real time process for above said models.
%Mp ts ISE
ZN 0.2 3.5 8.5274 x108
GA 0.16 3 5.3635x 106
ACO 0.12 2.5 7.3456x 105
Comparison of Performance index and
time domain specification
56. 70
ACO
GA
ZN
60
50
40
am plitude
30
20
10
0
0 1 2 3 4 5 6 7 8
tim e 5
x 10
Step response for the closed loop system for the PI
controller tuned with deferent methods
57. In order to test the PI tuning with ant algorithm in the presence of
noise, ACO- ITAE is used
The above system is tested for three different variances σ2=0.0025
σ2=0.025, σ2=0.25
Ant algorithm was run 5 times with 10 ants and 100 iterations due to
the probabilistic nature of AA and noise.
1.2
1
0.8
Varience-0.0025
amplitude
0.6
0.4
0.2
0
-0.2
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
time
White noise for variance-0.0025
58. 1.2
1
varience-0.00025
0.8
amplitude
0.6
0.4
0.2
0
-0.2
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
time
White noise for variance-0.00025
2
V r n e0 2
aie c - .0 5
1.5
m litu e
d
1
a p
0.5
0
- .5
0
0 50
0 10
00 10
50 20
00 2050 30
00 30
50 40
00 40
50 50
00
time
White noise for variance-0.025
59. In phase 2 of this project work, the conventional PI controller was tuned by Z-N
tuning method and compared with proposed GA and ACO methods.
Then it is implemented to the first order with dead time process. Then simulation
studies are carried out to analyze the performance of the spherical tank process and
Robustness of above mentioned controller for the different set points.
It is also implemented in real time for the real time results of GA, ACO, ZN same
set points. The result of both simulation and real time process were compared.
From the output response obtained using ACO tuned PI controller gives less over
shoot, fastest settling time, fastest rise time then the other techniques.
Time domain specification and performance indices were tabulated for the above
said models.
60. Ying- Tung Haiao, (2004) Ant colony optimization for Designing of PID
controller, IEEE, internation symposium on computer Aided control system
aided control systems design Taipei, Taiwan, September 2-4, 2004.
s S. Nithya, Abhay Singh Gour, N. Sivakumaran, T.K. Radhakrishnan and N.
Anantharaman, Model Based controller Design for shell and Tube heat
exchanger, sensors and Transducers Journal, Vol.84, Issue 10, October 2007,
pp.1677-1686.
, Sigurd Skogestasd, Simple analytic rules for model reduction ad PID
controller tuning, Journal of process control, 13,2003, pp.291-309.
g P. Wang and D.P Kwok, “Optimal design of PID process controllers based on
genetic algorithms” control Engineer practices Vol.2,no.4, pp.641-648, 1994.
