Skip to main content
SpringerLink
Log in

Model Reduction and Analytical Rule Extraction with Evolutionary Algorithms

  • Chapter
Book cover Intelligent Fractional Order Systems and Control

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

  • 1679 Accesses

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) PIλDμ 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 PIλDμ 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • 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 

  • Astrom, K.J., Hagglund, T.: PID controllers: theory, design and tuning. Instrument Society of America (1995)

    Google Scholar 

  • Åström, K., Hägglund, T.: Revisiting the Ziegler-Nichols step response method for PID control. Journal of Process Control 14, 635–650 (2004)

    Article  Google Scholar 

  • 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 

  • Caponetto, R., Dongola, G., Fortuna, L.: Fractional order systems: modeling and control applications. World Scientific Pub. Co. Inc. (2010)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Chen, Y.: Ubiquitous fractional order controls. In: Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications 2 (2006)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Das, S.: Functional fractional calculus. Springer (2011)

    Google Scholar 

  • 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 

  • 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 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • 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)

    Article  Google Scholar 

  • 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 

  • 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 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Ho, W., Lim, K., Xu, W.: Optimal gain and phase margin tuning for PID controllers. Automatica 34, 1009–1014 (1998)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Kaczorek, T.: Selected problems of fractional systems theory. Springer (2011)

    Google Scholar 

  • 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 

  • Koza, J.R.: Human-competitive results produced by genetic programming. Genetic Programming and Evolvable Machines 11, 251–284 (2010)

    Article  Google Scholar 

  • 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 

  • 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)

    Article  Google Scholar 

  • 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 

  • 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 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • 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)

    Article  MATH  Google Scholar 

  • Monje, C.A., Chen, Y.Q., Vinagre, B.M., et al.: Fractional-order systems and controls: fundamentals and applications. Springer (2010)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • O’Dwyer, A.: Handbook of PI and PID controller tuning rules. Imperial College Pr. (2006)

    Google Scholar 

  • 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 

  • Padula, F., Visioli, A.: Tuning rules for optimal PID and fractional-order PID controllers. Journal of Process Control 21, 69–81 (2011)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Podlubny, I.: Fractional-order systems and PIλDμ controllers. IEEE Transactions on Automatic Control 44, 208–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Skogestad, S.: Simple analytic rules for model reduction and PID controller tuning. Journal of Process Control 13, 291–309 (2003)

    Article  Google Scholar 

  • Tan, W., Liu, J., Chen, T., Marquez, H.J.: Comparison of some well-known PID tuning formulas. Computers & Chemical Engineering 30, 1416–1423 (2006)

    Article  Google Scholar 

  • 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 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • Valério, D., da Costa, J.S.: Introduction to single-input, single-output fractional control (2011)

    Google Scholar 

  • 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 

  • Valério, D., da Costa, J.S.: Tuning of fractional PID controllers with Ziegler-Nichols-type rules. Signal Processing 86, 2771–2784 (2006)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • 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 

  • Zhuang, M., Atherton, D.: Automatic tuning of optimum PID controllers. IEE Proceedings Control Theory and Applications D 140, 216–224 (1993)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Power Engineering, Jadavpur University, Kolkata, West Bengal, India

    Indranil Pan & Saptarshi Das

Authors
  1. Indranil Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saptarshi Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Indranil Pan .

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Pan, I., Das, S. (2013). Model Reduction and Analytical Rule Extraction with Evolutionary Algorithms. In: Intelligent Fractional Order Systems and Control. Studies in Computational Intelligence, vol 438. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31549-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31549-7_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31548-0

  • Online ISBN: 978-3-642-31549-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation