Model Reduction and Analytical Rule Extraction with Evolutionary Algorithms

Part of the Studies in Computational Intelligence book series (SCI, volume 438)

Abstract

Genetic Algorithm (GA) has been used in this chapter for a new approach of sub-optimal model reduction in the Nyquist plane and optimal time domain tuning of PID and fractional order (FO) PI λ D μ controllers. Simulation studies show that the Nyquist based new model reduction technique outperforms the conventional H2 norm based reduced parameter modeling technique. With the tuned controller parameters and reduced order model parameter data-set, optimum tuning rules have been developed with a test-bench of higher order processes via Genetic Programming (GP). The GP performs a symbolic regression on the reduced process parameters to evolve a tuning rule which provides the best analytical expression to map the data. The tuning rules are developed for a minimum time domain integral performance index described by weighted sum of error index and controller effort. From the reported Pareto optimal front of GP based optimal rule extraction technique a trade-off can be made between the complexity of the tuning formulae and the control performance. The efficacy of the single-gene and multi-gene GP based tuning rules has been compared with original GA based control performance for the PID and PI λ D μ controllers, handling four different class of representative higher order processes. These rules are very useful for process control engineers as they inherit the power of the GA based tuning methodology, but can be easily calculated without the requirement for running the computationally intensive GA every time. Three dimensional plots of the required variation in PID/FOPID controller parameters with reduced process parameters have been shown as a guideline for the operator. Parametric robustness of the reported GP based tuning rules has also been shown with credible simulation examples.

Keywords

Fractional Order Model Reduction High Order Process Symbolic Regression Model Reduction Technique 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. de Almeida, G., Rocha e Silva, V., Nepomuceno, E., Yokoyama, R.: Application of genetic programming for fine tuning PID controller parameters designed through Ziegler-Nichols technique. In: Advances in Natural Computation, pp. 434–434 (2005)Google Scholar
  2. Astrom, K.J., Hagglund, T.: PID controllers: theory, design and tuning. Instrument Society of America (1995)Google Scholar
  3. Åström, K., Hägglund, T.: Revisiting the Ziegler-Nichols step response method for PID control. Journal of Process Control 14, 635–650 (2004)CrossRefGoogle Scholar
  4. Cao, J.Y., Liang, J., Cao, B.G.: Optimization of fractional order PID controllers based on genetic algorithms. In: Proceedings of 2005 International Conference on Machine Learning and Cybernetics, vol. 9, pp. 5686–5689 (2005)Google Scholar
  5. Caponetto, R., Dongola, G., Fortuna, L.: Fractional order systems: modeling and control applications. World Scientific Pub. Co. Inc. (2010)Google Scholar
  6. Chen, B.S., Cheng, Y.M., Lee, C.H.: A genetic approach to mixed H2/H∞ optimal PID control. IEEE Control Systems Magazine 15, 51–60 (1995)CrossRefGoogle Scholar
  7. Chen, Y.: Ubiquitous fractional order controls. In: Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications 2 (2006)Google Scholar
  8. Chen, Y.Q., Bhaskaran, T., Xue, D.: Practical tuning rule development for fractional order proportional and integral controllers. Journal of Computational and Nonlinear Dynamics 3, 21403 (2008)CrossRefGoogle Scholar
  9. Das, S.: Functional fractional calculus. Springer (2011)Google Scholar
  10. Das, S., Pan, I., Das, S., Gupta, A.: Genetic Algorithm Based Improved Sub-Optimal Model Reduction in Nyquist Plane for Optimal Tuning Rule Extraction of PID and PIλDi Controllers via Genetic Programming. In: Programming. 2011 International Conference on Process Automation, Control and Computing, PACC, pp. 1–6 (2011a)Google Scholar
  11. Das, S., Saha, S., Das, S., Gupta, A.: On the selection of tuning methodology of FOPID controllers for the control of higher order processes. ISA Transactions (2011b)Google Scholar
  12. Feliu-Batlle, V., Perez, R.R., Rodriguez, L.S.: Fractional robust control of main irrigation canals with variable dynamic parameters. Control Engineering Practice 15, 673–686 (2007)CrossRefGoogle Scholar
  13. Feliu-Batlle, V., Rivas Pérez, R., Castillo García, F., Sanchez Rodriguez, L.: Smith predictor based robust fractional order control: Application to water distribution in a main irrigation canal pool. Journal of Process Control 19, 506–519 (2009a)CrossRefGoogle Scholar
  14. Feliu-Batlle, V., Rivas-Perez, R., Castillo-Garcia, F., et al.: Robust fractional order controller for irrigation main canal pools with time-varying dynamical parameters. Computers and Electronics in Agriculture (2011)Google Scholar
  15. Feliu-Batlle, V., Rivas-Perez, R., Castillo-Garcia, F.: Fractional order controller robust to time delay variations for water distribution in an irrigation main canal pool. Computers and Electronics in Agriculture 69, 185–197 (2009b)CrossRefGoogle Scholar
  16. Gude, J.J., Kahoraho, E.: Modified Ziegler-Nichols method for fractional PI controllers. In: 2010 IEEE Conference on Emerging Technologies and Factory Automation (ETFA), pp. 1–5 (2010)Google Scholar
  17. Gude, J.J., Kahoraho, E.: Simple tuning rules for fractional PI controllers. In: IEEE Conference on Emerging Technologies & Factory Automation, ETFA 2009, pp. 1–8 (2009)Google Scholar
  18. Ho, W., Gan, O., Tay, E., Ang, E.: Performance and gain and phase margins of well-known PID tuning formulas. IEEE Transactions on Control Systems Technology 4, 473–477 (1996)CrossRefGoogle Scholar
  19. Ho, W., Hang, C., Zhou, J.: Performance and gain and phase margins of well-known PI tuning formulas. IEEE Transactions on Control Systems Technology 3, 245–248 (1995)CrossRefGoogle Scholar
  20. Ho, W., Lim, K., Xu, W.: Optimal gain and phase margin tuning for PID controllers. Automatica 34, 1009–1014 (1998)MATHCrossRefGoogle Scholar
  21. Jin, Y., Chen, Y.Q., Xue, D.: Time-constant robust analysis of a fractional order [proportional derivative] controller. Control Theory & Applications, IET 5, 164–172 (2011)CrossRefGoogle Scholar
  22. Kaczorek, T.: Selected problems of fractional systems theory. Springer (2011)Google Scholar
  23. Keane, M.A., Koza, J.R., Streeter, M.J.: Automatic synthesis using genetic programming of an improved general-purpose controller for industrially representative plants. In: 2002 Proceedings NASA/DoD Conference on Evolvable Hardware, pp. 113–122 (2002)Google Scholar
  24. Koza, J.R.: Human-competitive results produced by genetic programming. Genetic Programming and Evolvable Machines 11, 251–284 (2010)CrossRefGoogle Scholar
  25. Koza, J.R., Al-Sakran, S.H., Jones, L.W.: Cross-domain features of runs of genetic programming used to evolve designs for analog circuits, optical lens systems, controllers, antennas, mechanical systems, and quantum computing circuits. In: Proceedings 2005 NASA/DoD Conference on Evolvable Hardware, pp. 205–212 (2005)Google Scholar
  26. Koza, J.R., Keane, M.A., Streeter, M.J.: What’s AI done for me lately? Genetic programming’s human-competitive results. IEEE Intelligent Systems 18, 25–31 (2003)CrossRefGoogle Scholar
  27. Koza, J.R., Keane, M.A., Yu, J., et al.: Automatic synthesis of both the control law and parameters for a controller for a three-lag plant with five-second delay using genetic programming and simulation techniques. In: Proceedings of the 2000 American Control Conference, vol. 1, pp. 453–459 (2000)Google Scholar
  28. Koza, J.R., Keane, M.A., Yu, J., et al.: Automatic synthesis of both the topology and parameters for a robust controller for a nonminimal phase plant and a three-lag plant by means of genetic programming. In: 1999 Proceedings of the 38th IEEE Conference on Decision and Control, vol. 5, pp. 5292–5300 (1999)Google Scholar
  29. Koza, J.R., Streeter, M.J., Keane, M.A.: Routine high-return human-competitive automated problem-solving by means of genetic programming. Information Sciences 178, 4434–4452 (2008)CrossRefGoogle Scholar
  30. Lin, M., Lakshminarayanan, S., Rangaiah, G.: A comparative study of re-cent/popular PID tuning rules for stable, first-order plus dead time, single-input single-output processes. Industrial & Engineering Chemistry Research 47, 344–368 (2008)CrossRefGoogle Scholar
  31. Luo, Y., Chen, Y.Q., Wang, C.Y., Pi, Y.G.: Tuning fractional order proportional integral controllers for fractional order systems. Journal of Process Control 20, 823–831 (2010)CrossRefGoogle Scholar
  32. Mann, G., Hu, B.G., Gosine, R.: Time-domain based design and analysis of new PID tuning rules. IEE Proceedings-Control Theory and Applications 148, 251–261 (2001)CrossRefGoogle Scholar
  33. Merrikh-Bayat, F.: Optimal tuning rules of the fractional-order PID controllers with application to first-order plus time delay processes. In: 2011 International Symposium on Advanced Control of Industrial Processes (ADCONIP), pp. 403–408 (2011)Google Scholar
  34. Monje, C.A., Calderon, A.J., Vinagre, B.M., et al.: On fractional PI λ controllers: some tuning rules for robustness to plant uncertainties. Nonlinear Dynamics 38, 369–381 (2004)MATHCrossRefGoogle Scholar
  35. Monje, C.A., Chen, Y.Q., Vinagre, B.M., et al.: Fractional-order systems and controls: fundamentals and applications. Springer (2010)Google Scholar
  36. Monje, C.A., Vinagre, B.M., Feliu, V., Chen, Y.Q.: Tuning and auto-tuning of fractional order controllers for industry applications. Control Engineering Practice 16, 798–812 (2008)CrossRefGoogle Scholar
  37. O’Dwyer, A.: Handbook of PI and PID controller tuning rules. Imperial College Pr. (2006)Google Scholar
  38. Ou, B., Song, L., Chang, C.: Tuning of fractional PID controllers by using radial basis function neural networks. In: 2010 8th IEEE International Conference on Control and Automation (ICCA), pp. 1239–1244 (2010)Google Scholar
  39. Padula, F., Visioli, A.: Tuning rules for optimal PID and fractional-order PID controllers. Journal of Process Control 21, 69–81 (2011)CrossRefGoogle Scholar
  40. Pan, I., Das, S., Gupta, A.: Handling packet dropouts and random delays for unstable delayed processes in NCS by optimal tuning of PIλDμ controllers with evolutionary algorithms. ISA Transactions 50, 557–572 (2011b)CrossRefGoogle Scholar
  41. Pan, I., Das, S., Gupta, A.: Tuning of an optimal fuzzy PID controller with stochastic algorithms for networked control systems with random time delay. ISA Transactions 50, 28–36 (2011a)CrossRefGoogle Scholar
  42. Podlubny, I.: Fractional-order systems and PIλDμ controllers. IEEE Transactions on Automatic Control 44, 208–214 (1999)MathSciNetMATHCrossRefGoogle Scholar
  43. Searson, D.P., Leahy, D.E., Willis, M.J.: GPTIPS: an open source genetic programming toolbox for multigene symbolic regression. In: International Multi Conference of Engineers and Computer Scientists (2010)Google Scholar
  44. Skogestad, S.: Simple analytic rules for model reduction and PID controller tuning. Journal of Process Control 13, 291–309 (2003)CrossRefGoogle Scholar
  45. Tan, W., Liu, J., Chen, T., Marquez, H.J.: Comparison of some well-known PID tuning formulas. Computers & Chemical Engineering 30, 1416–1423 (2006)CrossRefGoogle Scholar
  46. Tavakoli-Kakhki, M., Haeri, M.: Fractional order model reduction approach based on retention of the dominant dynamics: Application in IMC based tuning of FOPI and FOPID controllers. ISA Transactions (2011)Google Scholar
  47. Tavakoli-Kakhki, M., Haeri, M.: The minimal state space realization for a class of fractional order transfer functions. SIAM Journal on Control and Optimization 48, 4317 (2010)MathSciNetMATHCrossRefGoogle Scholar
  48. Tavakoli-Kakhki, M., Haeri, M.: Model reduction in commensurate fractional-order linear systems. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 223, 493–505 (2009)CrossRefGoogle Scholar
  49. Tenoutit, M., Maamri, N., Trigeassou, J.: A time moments approach to the design of robust fractional PID controllers. In: 2011 8th International Multi-Conference on Systems, Signals and Devices (SSD), pp. 1–7 (2011)Google Scholar
  50. Valério, D., da Costa, J.S.: Introduction to single-input, single-output fractional control (2011)Google Scholar
  51. Valério, D., da Costa, J.S.: A review of tuning methods for fractional PIDs. In: 4th IFAC Workshop on Fractional Differentiation and Its Applications, FDA 2010 (2010)Google Scholar
  52. Valério, D., da Costa, J.S.: Tuning of fractional PID controllers with Ziegler-Nichols-type rules. Signal Processing 86, 2771–2784 (2006)MATHCrossRefGoogle Scholar
  53. Wang, Q.G., Lee, T.H., Fung, H.W., et al.: PID tuning for improved performance. IEEE Transactions on Control Systems Technology 7, 457–465 (1999)CrossRefGoogle Scholar
  54. Xue, D., Chen, Y.Q.: Suboptimum H2 Pseudo-rational Approximations to Fractional-order Linear Time Invariant Systems. Advances in Fractional Calculus, 61–75 (2007)Google Scholar
  55. Yu, J., Keane, M.A., Koza, J.R.: Automatic design of both topology and tuning of a common parameterized controller for two families of plants using genetic programming. In: IEEE International Symposium on Computer-Aided Control System Design, CACSD 2000, pp. 234–242 (2000)Google Scholar
  56. Zhuang, M., Atherton, D.: Automatic tuning of optimum PID controllers. IEE Proceedings Control Theory and Applications D 140, 216–224 (1993)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Power EngineeringJadavpur UniversityKolkataIndia

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