Automatic Synthesis Using Genetic Programming of an Improved General-Purpose Controller for Industrially Representative Plants Martin A. Keane Econometrics, Inc. Chicago, Illinois John R. Koza Stanford University Stanford, California Matthew J. Streeter Genetic Programming, Inc. Mountain View, California Evolvable Hardware 2002, Washington D.C., July 15-18

Overview The problem of industrial control P, PI, and PID controllers The Astrom-Hagglund controller Genetic programming and control Evolved controllers Cross-validation Conclusions

The problem of industrial control Example: cruise control Desired speed is reference signal Flow of fuel to engine is control signal Engine/car is plant; car’s speed is plant response

Evaluating Controllers Low rise time: the plant response must rise to the desired value quickly Minimal overshoot: the plant response must not rise too far above the desired value Stability: controller should be stable with respect to noise in the feedback signals Sensitivity: controller should not be overly sensitive to small changes in reference signal or plant response Disturbance rejection: the controller must work even if its own output is offset by external forces

P, PI, and PID Controllers

Proportional (P) Control Leads to oscillation Figure from

Proportional-Integrative (PI) Control Eliminates oscillation Doesn’t anticipate future values of plant response Figure from

Proportional-Integrative-Derivative (PID) Control With appropriate tuning, outperforms both P and PI controllers Over 90% of modern controllers are PID Figure from

Tuning rules for PID controllers Original PID controllers were tuned manually Ziegler-Nichols (1942) provided generalized tuning equations Astrom-Hagglund (1995) Applied curve-fitting to values obtained by well-known “dominant pole design” to obtain improved generalized tuning rules

The Astrom-Hagglund 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 ), ultimate period (T u ), time constant (T r ) and dead time (L), all readily obtainable in the field Broadly recognized and accepted in the control world

The Astrom-Hagglund Controller Equation 1: Equation 2: Equation 3: Equation 4:

Genetic Programming and Control Controllers are represented as LISP expression trees Crossover is performed by swapping subtrees Evolution of topology, identity of each block, and equations giving parameter values of blocks Fitness incorporates rise time, overshoot, and disturbance rejection (ITAE), stability, and sensitivity

Representation of Controller as LISP Expression Direct encoding of block diagram as LISP expression tree Global variables used to create loops Special TAKEOFF function for internal feedback (takeoff points) Problem-specific: Astrom-Hagglund controller made available as primitive

Representation of Controller as LISP Expression

Fitness Measure ITAE penalty (Integral of time-weighted absolute error) for setpoint and disturbance rejection Penalty for minimum sensor noise attenuation (sensitivity) Penalty for maximum sensitivity to noise (stability) Evaluation on plants, always including 16 Astrom- Hagglund plants

ITAE Penalty Reference signalDisturbance signal Six combinations of reference and disturbance signal heights Penalty is given by: B and C are normalizing factors

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

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

Simulation Evolved controllers simulated with SPICE circuit simulator using user-defined control blocks 160 or 192 simulations per individual

Previous Work Controllers for two and three lag plants Discovery of PID and PID2 controllers Controller for highly non-linear plant Generalized controller for three lag plant with variable time constant Generalized controller for two families of plants

Control Parameters 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

Evolved Controllers

Block Diagram for First Evolved Controller

Equations for First Evolved Controller Equation 11: Equation 12: Equation 13: Equation 14: Equation 15: Equation 16: Equation 17: Equation 18:

Performance of First Evolved Controller 66.4% of setpoint ITAE of A-H controller 85.7% of disturbance rejection ITAE of A-H controller 94.6% of 1/(minimum attenuation) of A-H controller 92.9% of maximum sensitivity of A-H controller

Block Diagram for Second Evolved Controller

Equations for Second Evolved Controller Equation 21: Equation 22: Equation 23: Equation 24: Equation 25: Equation 26: Equation 27: Equation 28:

Performance of Second Evolved Controller 85.5% of setpoint ITAE of A-H controller 91.8% of disturbance rejection ITAE of A-H controller 98.9% of 1/(minimum attenuation) of A-H controller 97.5% of maximum sensitivity of A-H controller

Block Diagram for Third Evolved Controller

Equations for Third Evolved Controller Equation 31: Equation 32: Equation 33: Equation 34: NLM(x) =10 0 if x (-100/19-x/19) if -100  x < (100/19-x/19) if 5 < x  x if -5  x  5

Performance of Third Evolved Controller 81.8% of setpoint ITAE of A-H controller 93.8% of disturbance rejection ITAE of A-H controller 98.8% of 1/(minimum attenuation) of A-H controller 93.4% of maximum sensitivity of A-H controller

Comparison of Response of Evolved Controller and Astrom-Hagglund Controller for a Typical Plant Evolved controller has shorter rise time and less overshoot Comparison is similar for other plants

Cross-Validation 18 new plants selected with plant parameters in range specified by Astrom and Hagglund All evolved controllers do better than Astrom-Hagglund controller over 18 additional plants Evolved controllers outperform Astrom-Hagglund controller on out-of-sample fitness cases about 99% of the time

Cross-Validation of First Evolved Controller 64.1% of setpoint ITAE of A-H controller 84.9% of disturbance rejection ITAE of A-H controller 95.8% of 1/(minimum attenuation) of A-H controller 93.5% of maximum sensitivity of A-H controller

Cross-Validation of Second Evolved Controller 84% of setpoint ITAE of A-H controller 90.6% of disturbance rejection ITAE of A-H controller 98.9% of 1/(minimum attenuation) of A-H controller 97.5% of maximum sensitivity of A-H controller

Cross-Validation of Third Evolved Controller 81.8% of setpoint ITAE of A-H controller 94.2% of disturbance rejection ITAE of A-H controller 99.7% of 1/(minimum attenuation) of A-H controller 92.5% of maximum sensitivity of A-H controller

Conclusions Genetic programming can provide a generalized controller for a wide variety of industrially representative plants Significant improvement over Astrom-Hagglund controller as measured by ITAE for setpoint and disturbance rejection, minimum attenuation, and maximum sensitivity Evolved controller performs well on out-of-sample plants