Automatic Synthesis Using Genetic Programming of an Improved General-Purpose Controller for Industrially Representative

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

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

3. 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

4. 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

5. P, PI, and PID Controllers

6. Proportional (P) Control • Leads to oscillation Figure from http://www.engin.umich.edu/group/ctm/PID/PID.html

7. Proportional-Integrative (PI) Control • Eliminates oscillation • Doesn’t anticipate future values of plant response Figure from http://www.engin.umich.edu/group/ctm/PID/PID.html

8. Proportional-Integrative-Derivative (PID) Control • With appropriate tuning, outperforms both P and PI controllers • Over 90% of modern controllers are PID Figure from http://www.engin.umich.edu/group/ctm/PID/PID.html

9. 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

10. 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 (Ku), ultimate period (Tu), time constant (Tr) and dead time (L), all readily obtainable in the field • Broadly recognized and accepted in the control world

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

12. 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

13. 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

14. Representation of Controller as LISP Expression

15. 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 20-24 plants, always including 16 Astrom-Hagglund plants

16. ITAE Penalty Six combinations of reference and disturbance signal heights • Penalty is given by: • B and C are normalizing factors

17. 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

18. 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

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

20. 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

21. 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

22. Evolved Controllers

23. Block Diagram for First Evolved Controller

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

25. 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

26. Block Diagram for Second Evolved Controller

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

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

29. Block Diagram for Third Evolved Controller

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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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