Automatic Synthesis Using Genetic Programming of Improved PID Tuning Rules Matthew J. Streeter Genetic Programming, Inc. Mountain View, California Martin A. Keane Econometrics, Inc. Chicago, Illinois John R. Koza Stanford University Stanford, California 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) Choose “+”(2) Choose “*”(3-5) Choose “A”, “B”, “C”

Fitness evaluation 4 random equations shown Fitness is shaded area Target curve ( x 2 +x+1 )

Crossover Subtrees are swapped to create offspring Picked subtree Parents Offspring Picked subtree

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 (K u ) and ultimate period (T u )

The Åström-Hägglund controller Equation 1 (b): Equation 2 (K p ) : Equation 3 (K i ): Equation 4 (K d ):

Experiment 1: Evolving tuning rules from scratch 4-branch program representing 4 equations (for K, K i, K d, and b) in terms of K u & T u 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 Six combinations of reference and disturbance signal heights 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 M s < 1.5 2(M s -1.5) if 1.5  M s  (M s -1.0) is M s > 2.0

Fitness measure: sensitivity penalty 0 reference signal, 1 V noise signal A min is minimum attenuation of plant response Penalty is 0 if A min > 40 db (40-A min )/10 if 20 db  A min  40 db 2+(20-A min ) if A min < 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 KKiKi KdKd b

Evolved equations KKiKi KdKd 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 KKiKi KdKd 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)