TWO DEGREE OF FREEDOM PID CONTROLLER
DESIGN USING GENETIC ALGORITHMS
Daniel Czarkowski
Polish Register of Shipping,
Gdańsk, POLAND
d.czarkowski@prs.pl
Tom O’Mahony
Cork Institute of Technology,
Department of Electronic Engineering,
Cork, IRELAND
© 2005 PRS S.A.
Overview
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2- DOF PID controller
Design strategy
Genetic Algorithms
– A solution of reduction computation time
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Models
Results
Summary of work
Conclusions
2
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
2-DOF PID controller

Controller structure
R(s)
D(s)
U(s)
F(s)
G(s)
Y(s)
H(s)

Control law
sK d  c  R(s)  Y (s) 
Ki
U ( s )  K p  b  R( s )  Y ( s )    R( s )  Y ( s )  
sK
s
1 d
KpN

6 variables to tune
K p , Ki , Kd , b, c, N
3
© 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
t1 1
t2
k 0
k t1
IAE   e(k )   e(k )
Robustness
1) Gain and phase margin Am = 6dB j m = 45°
2) Gain and phase margin Am = 14dB j m = 45°
MM = 0.6
3) Modulus margin
4) Maximum value of the input sensitivity function M u = var
4
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Why Genetic Algorithms?
1
100
0.5
Objf
MM
80
60
local min.
40
global min.
0
0.4
20
0.4
0.4
0.2
Ki
0.2
0
0
0.4
0.3
0.2
Ki
Kp
0.2
0
0.1
0
Kp
Avoid your local minimum!
1
e15 s
3
( s  1)
J  min IAE  MM  s.t. MM  0.6, K d  1.24, b  1, c  1, N  29
G2 ( s) 
K p , Ki
5
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Direct the GA
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
GA optimisation problem
J  min  IAE  Am   m 
Penalty factors on gain and phase margins
10
8
8
6
6
 Am
 m
10
4
4
2
2
0
0
s.t. Am  6dB, m  45
2
4
Am (dB)
6
8
0
0
10
20
30
 m (deg)
40
50
60
6
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Flow diagram of optimisation environment
MATLAB
GA
1) Initial population
Objective function
1) Change chromosomes into controller parameters
Controller parameters
IAE
Simulink
Calculate IAE
2) Calculate robustness factors
3) Calculate objective function
2) Ranking
3) Selection
4) Crossover
5) Mutation
6) Evaluation of the fitness function
7) Reinsertion of offspring in the population
8) Are the termination criteria satisfied?
9) If conditions satisfied pass the controller variables outside the function
7
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
GA with look up table
30
25
G2 ( s) 
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1
15 s
e
( s  1)3
time (sec)
20
15
10
Gray coding
5
Population of 100
Single Point Crossover
0
0
Stochastic Universal Sampling
Matrix: 2342 rows and 7 columns
MATLAB find function
20
40
60
80
100
Generations
Reduced the execution time up to 60%!
8
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Models
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Benchmark test
–
–
–
–
–

(K. J. Åström 1998, 2000)
Inverse unstable system
Integrating systems
Underdamped system
Conditionally stable system
3 models with time delay
11 models were evaluated
9
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Results, I
1.5
1
G1(s ) =
(s + 1)3
J1  min IAE  Am   m 
J 2  min IAE  Am   m 
s.t. Am  6dB, m  45
y(t)
1
0.5
s.t. Am  14dB, m  45
J3  min IAE  MM 
s.t. MM  0.6
J 4  min IAE  Mu 
s.t. M u  20
0
0
5
10
15
20
0
5
10
time (sec)
15
20
10
u(t)
5
0
The fourth design gives sluggish response?
-5
10
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Results, II
1.5
1
s(s + 1)2
1
y(t)
G 4 (s ) =
J1  min IAE  Am   m 
J 2  min IAE  Am   m 
J3  min IAE  MM 
J 4  min IAE  Mu 
0.5
s.t. Am  6dB, m  45
s.t. Am  14dB, m  45
0
0
5
10
15
20
25
30
0
5
10
15
time (sec)
20
25
30
s.t. MM  0.6
10
s.t. M u  20
u(t)
5
0
-5
11
© 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
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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.
12
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Conclusions
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
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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.
13
© 2005 PRS S.A.
11th IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2005
Questions?
Daniel Czarkowski
my e-mail: d.czarkowski@prs.pl
© 2005 PRS S.A.