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Automatic Synthesis Using Genetic Programming of Improved PID Tuning Rules Martin A. Keane Econometrics, Inc. Chicago, Illinois martinkeane@ameritech.net Matthew J. Streeter Genetic Programming, Inc. Mountain View, California mjs@tmolp.com John R. Koza Stanford University

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automatic synthesis using genetic programming of improved pid tuning rules

Automatic Synthesis Using Genetic Programming of Improved PID Tuning Rules

Martin A. Keane

Econometrics, Inc.

Chicago, Illinois

martinkeane@ameritech.net

Matthew J. Streeter

Genetic Programming, Inc.

Mountain View, California

mjs@tmolp.com

John R. Koza

Stanford University

Stanford, California

koza@stanford.edu

ICONS 2003, Faro Portugal, April 8-11

outline
Outline
  • Overview of Genetic Programming (GP)
  • Controller Synthesis using GP
  • Improved PID Tuning Rules
overview of 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
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
Random initial population
  • Function set: {+, *, /, -}
  • Terminal set: {A, B, C}
  • (1) Choose “+”
  • (2) Choose “*”
  • (3-5) Choose “A”, “B”, “C”
fitness evaluation
Fitness evaluation
  • 4 random equations shown
  • Fitness is shaded area

Target curve (x2+x+1)

crossover
Crossover

Picked

subtree

  • Subtrees are swapped to create offspring

Picked

subtree

Parents

Offspring

controller synthesis using gp
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 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
basis for comparison the str m h gglund controller
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 str m h gglund controller
The Åström-Hägglund controller

Equation 3 (Ki):

Equation 4 (Kd):

Equation 1 (b):

Equation 2 (Kp) :

experiment 1 evolving tuning rules from scratch
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 & terminal sets
  • Function set: {+, *, -, /, EXP, LOG, POW}
  • Terminal set: {KU, TU, }
fitness measure
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
fitness measure itae penalty
Fitness measure: ITAE penalty

Six combinations of reference and disturbance signal heights

  • Penalty is given by:
  • B and C are normalizing factors
fitness measure stability 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
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
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
experiment 1 conclusions
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
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
results
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
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)