© 2005 prs s.a. two degree of freedom pid controller design using genetic algorithms daniel...
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© 2005 PRS S.A.
TWO DEGREE OF FREEDOM PID CONTROLLER DESIGN USING GENETIC
ALGORITHMS Daniel Czarkowski
Polish Register of Shipping, Gdańsk, POLAND
[email protected] Tom O’Mahony
Cork Institute of Technology, Department of Electronic Engineering,
Cork, IRELAND
2© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Overview
2- DOF PID controller Design strategy Genetic Algorithms
– A solution of reduction computation time
Models Results Summary of work Conclusions
3© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
2-DOF PID controller
Controller structure
Control law
6 variables to tune
G(s)F(s)Y(s)
H(s)
U(s)R(s)
D(s)
( ) ( )( ) ( ) ( ) ( ) ( )
1
dip
d
p
sK c R s Y sKU s K b R s Y s R s Y s
sKsK N
, , , , ,p i dK K K b c N
4© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Design strategy
Performance & robustness
Performance– IAE servo + regulator
Robustness1) Gain and phase margin
2) Gain and phase margin
3) Modulus margin
4) Maximum value of the input sensitivity function
1 2
1
1
0
( ) ( )t t
k k t
IAE e k e k
6 45mmA dB j= = °
14 45mmA dB j= = °
0.6MM =varuM =
5© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Why Genetic Algorithms?
Avoid your local minimum!
00.1
0.20.3
0.4
0
0.2
0.420
40
60
80
100
Kp
Ki
Obj
f
local min.
global min.
152 3
1( )
( 1)sG s e
s
,min . . 0.6, 1.24, 1, 1, 29
p iMM dK K
J IAE s t MM K b c N
0
0.2
0.4
0
0.2
0.40
0.5
1
Kp
Ki
MM
6© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Direct the GA
GA optimisation problem
Penalty factors on gain and phase margins
min . . 6 , 45Am m m mJ IAE s t A dB
0 2 4 6 80
2
4
6
8
10
Am (dB)
Am
0 10 20 30 40 50 600
2
4
6
8
10
m (deg)
m
7© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Flow diagram of optimisation environmentMATLAB
GA1) Initial population
2) Ranking3) Selection4) Crossover5) Mutation6) Evaluation of the fitness function7) Reinsertion of offspring in the population8) Are the termination criteria satisfied?9) If conditions satisfied pass the controller variables outside the function
Objective function
1) Change chromosomes into controller parameters
2) Calculate robustness factors3) Calculate objective function
Controller parameters
IAE
Simulink
Calculate IAE
8© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
GA with look up table
Gray coding Population of 100 Single Point Crossover Stochastic Universal Sampling Matrix: 2342 rows and 7 columns MATLAB find function
0 20 40 60 80 1000
5
10
15
20
25
30
Generations
tim
e (
se
c)
Reduced the execution time up to 60%!
152 3
1( )
( 1)sG s e
s
9© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Models
Benchmark test– Inverse unstable system– Integrating systems– Underdamped system– Conditionally stable system– 3 models with time delay
11 models were evaluated
(K. J. Åström 1998, 2000)
10© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Results, I
1 31
( )( 1)
G ss
=+
1 min . . 6 , 45Am m m mJ IAE s t A dB
2 min . . 14 , 45Am m m mJ IAE s t A dB
3 min . . 0.6MMJ IAE s t MM
4 min . . 20Mu uJ IAE s t M
The fourth design gives sluggish response?
0 5 10 15 200
0.5
1
1.5
y(t)
0 5 10 15 20-5
0
5
10
u(t)
time (sec)
11© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Results, II
0 5 10 15 20 25 300
0.5
1
1.5
y(t)
0 5 10 15 20 25 30-5
0
5
10
u(t)
time (sec)
4 21
( )( 1)
G ss s
=+
1 min . . 6 , 45Am m m mJ IAE s t A dB
2 min . . 14 , 45Am m m mJ IAE s t A dB
3 min . . 0.6MMJ IAE s t MM
4 min . . 20Mu uJ IAE s t M
12© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Summary of work
Four robust designs have been proposed– Direct the GA
Eleven models evaluated A solution to speed up the GA optimisation The two degree of freedom controller well
performs for systems such as: stable, inverse unstable, non-minimum phase, integrating long time-delay.
Model uncertainty has not been discussed, however from my experience the fourth design performs well in this case as well as implemented to real-time systems.
13© 2005 PRS S.A. 11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Conclusions
None of the proposed methods performs significantly better. However, even though the responses from the third design are not as fast as from the first design, it can be summarised that this design gives slightly better results than the other counterparts.
The fourth design directly penalises the impact of the high frequency measurement noise on the closed-loop system.
The GA with look up table significantly reduced the computation time.
