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Genetic Programming for Enhanced General-Purpose Controllers in Industrial Applications

This paper presents a novel approach to developing improved general-purpose controllers for industrial plants using genetic programming. It outlines the limitations of traditional control methods, such as P, PI, and PID controllers, and introduces the Astrom-Hagglund controller as a basis for evolution. The methodology employs LISP expression trees for controller representation, with evolved controllers assessed based on rise time, overshoot, stability, and sensitivity. Simulation results validate the effectiveness of the genetic programming approach, demonstrating its potential for optimizing control systems in diverse industrial applications.

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Genetic Programming for Enhanced General-Purpose Controllers in Industrial Applications

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  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

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