Automatic Synthesis Using Genetic Programming of Improved PID Tuning Rules Matthew J. Streeter...
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Automatic Synthesis Using Genetic Programming of Improved PID Tuning Rules
Matthew J. StreeterGenetic Programming, Inc.Mountain View, California
Martin A. KeaneEconometrics, Inc.
Chicago, [email protected]
John R. KozaStanford UniversityStanford, California [email protected]
ICONS 2003, Faro Portugal, April 8-11
Outline
• Overview of Genetic Programming (GP)
• Controller Synthesis using GP
• Improved PID Tuning Rules
Overview of Genetic Programming (GP)
Overview of GP
• Breed computer programs to solve problems
• Programs represented as trees in style of LISP language
• Programs can create anything (e.g., controller, equation, controller+equations)
Pseudo-code for GP
1) Create initial random population
2) Evaluate fitness
3) Select fitter individuals to reproduce
4) Apply reproduction operations (crossover, mutation) to create new population
5) Return to 2 and repeat until solution found
Random initial population
• Function set: {+, *, /, -}
• Terminal set: {A, B, C}
+ +
*
1
2
+
*
A B
C
(1) Choose “+” (2) Choose “*” (3-5) Choose “A”, “B”, “C”
Fitness evaluation
• 4 random equations shown
• Fitness is shaded areaTarget curve
(x2+x+1)
Crossover
• Subtrees are swapped to create offspring
0.234Z + X – 0.789
X 0.789
–
0.234 Z
*
+
ZY(Y + 0.314Z)
Z Y
*
0.314 Z
*Y
+
*1 1
2 25 5
8 9
3 34 46 7 76
X 0.789
–
+
0.314 Z
*Y
+
Y + 0.314Z + X – 0.789
Z Y
*
*
0.234 Z
*
0.234Z Y2
Pickedsubtree
Parents
Offspring
Pickedsubtree
Controller Synthesis Using GP
• Program tree directly represents control block diagram
• Special functions for internal feedback / takeoff points
• Fitness measured in terms of ITAE, sensitivity, stability
Control problems solved
• Control of two and three lag plants, non-minimal phase plant, three lag plant w/ 5 second delay
• Parameterized controllers for three lag plant with variable internal gain, . . .
• Parameterized controllers for broad families of plants
Improved PID Tuning Rules
Basis for Comparison: the Åström-Hägglund controller
• Applied dominant pole design to 16 plants from 4 representative families of plants
• Used curve-fitting to obtain generalized solution
• Equations are expressed in terms of ultimate gain (Ku) and ultimate period (Tu)
The Åström-Hägglund controller
0.56 0.12+ 2
0.25*Ku KueEquation 1 (b):
Equation 2 (Kp) :
1.6 1.2+
20.72* *
Ku KuuK e
Equation 3 (Ki):
Equation 4 (Kd):
1.6 1.2+ 2
1.3 0.38+ 2
0.72* *
0.59* *
Ku Kuu
Ku Kuu
K e
T e
1.6 1.2 1.4 0.56+ +
2 20.108* * * *
K Ku uK Ku uu uK T e e
Experiment 1: Evolving tuning rules from scratch
• 4-branch program representing 4 equations (for K, Ki, Kd, and b) in terms of Ku & Tu
• Different from other GP work in that we are evolving tuning, not topology
• Fitness in terms of ITAE, sensitivity, stability
Function & terminal sets
• Function set: {+, *, -, /, EXP, LOG, POW}
• Terminal set: {KU, TU, }
Fitness measure
• ITAE penalty for setpoint & disturbance rejection
• Penalty for minimum sensor noise attenuation (sensitivity)
• Penalty for maximum sensitivity to noise (stability)
• Evaluation on 30 plants (superset of A-H’s 16 plants)
• Controllers simulated using SPICE
Reference signal
Disturbance signal
1.0 1.0
10-3 10-3
-10-6 10-6
1.0 -0.6
-1.0 0.0
0.0 1.0
Six combinations of reference and
disturbance signal heights
22
20
10
10
0
)()10()(
u
u
u
u
u
T
T
Ttu
T
T
t
CdtteTtBdttet
• Penalty is given by:
• B and C are normalizing factors
Fitness measure: ITAE penalty
Fitness measure: stability penalty
• 0 reference signal, 1 V noise signal
• Maximum sensitivity is maximum amplitude of noise signal + plant response
• Penalty is 0 if Ms < 1.5
2(Ms-1.5) if 1.5 Ms 2.0
20(Ms-1.0) is Ms > 2.0
Fitness measure: sensitivity penalty
• 0 reference signal, 1 V noise signal
• Amin is minimum attenuation of plant response
• Penalty is 0 if Amin > 40 db
(40-Amin)/10 if 20 db Amin 40 db
2+(20-Amin) if Amin < 20 db
Experimental setup
• 1000 node Beowulf cluster with 350 MHz Pentium II processors
• Island model with asynchronous subpopulations
• Population size: 100,000
• 70% crossover, 20% constant mutation, 9% cloning, 1% subtree mutation
Åström-Hägglund equations
K Ki
Kd b
Evolved equations
K Ki
Kd b
Experiment 1: Conclusions
• Evolved tuning rules are better on average than A-H, but not uniformly better
• Dominant pole design provides optimal solution for individual plants
• Maybe we can improve on A-H curve-fitting
Experiment 2: Evolving increments to A-H equations
• Same program structure, fitness measure, etc.
• Values of evolved equations are now added to A-H equations
Evolved adjustments to A-H equations
K Ki
Kd b
Results
• 91.6% of setpoint ITAE of Åström-Hägglund (89.7% out-of-sample)
• 96.2% of disturbance rejection ITAE of A-H (95.6% OOS)
• 99.5% of 1/(minimum attenuation) of A-H (99.5% OOS)
• 98.5% of maximum sensitivity of A-H (98.5% OOS)
Conclusions
• Evolved controller is slightly better than Åström-Hägglund
• Not much room for improvement (in terms of our fitness measure) with PID topology
• We have gotten better results evolving tuning+topology (also bootstrapping on A-H)