Automatic Synthesis Using Genetic Programming of Improved PID Tuning Rules

Matthew J. StreeterGenetic Programming, Inc.Mountain View, California

mjs@tmolp.com

Martin A. KeaneEconometrics, Inc.

Chicago, Illinoismartinkeane@ameritech.net

John R. KozaStanford UniversityStanford, California koza@stanford.edu

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)