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The Divide-and-Conquer Method for Modelling and Control of Nonlinear Systems: Some Important Issues Concerning Its Application

  • Juš Kocijan
Part of the Advances in Industrial Control book series (AIC)

Abstract

Divide-and-conquer methods, in the context of nonlinear dynamic systems modelling and control design, break the problem down into a number of simpler, frequently linear modelling and control design problems, each associated with a restricted operating region. The most widespread divide-and-conquer nonlinear control design method is probably gain-scheduling. The gain-scheduling method has been successfully applied in many fields, ranging from process control to aerospace engineering. The basic idea behind the method is to divide the nonlinear system to be controlled into local subsystems described by linear dynamic models. A linear control problem is then solved for each of these subsystems. The global control solution—called gain-scheduling control—is obtained by merging partial local solutions. The idea of divide-and-conquer seems very attractive, but its application is not straightforward. The following issues arising during system modelling and control design are treated in this chapter: the properties of the global nonlinear model vis à vis the properties of the local linear models; controller design from first-principles nonlinear continuous-time models; and control design based on local models identified from measured data. The problems are elaborated on simulated examples that include vehicle dynamics and a continuous stirred tank reactor. How to properly use these methods is illustrated by pressure control design on a semi-industrial gas–liquid separator unit.

Keywords

Nonlinear System Control Design Local Model Local Controller Linear Subsystem 
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.

Notes

Acknowledgements

The financial support of the Slovenian Research Agency through Programme P2-0001 is gratefully acknowledged.

References

  1. 1.
    Åström KJ, Hägglund T (1995) PID controllers: theory, design and tuning, 2nd edn. Instrument Society of America, Research Triangle Park Google Scholar
  2. 2.
    Ažman K, Kocijan J (2009) Fixed-structure Gaussian process model. Int J Inf Syst Sci 34:31–38 Google Scholar
  3. 3.
    Čokan B, Kocijan J (2002) Some realisation issues of fuzzy gain-scheduling controllers: a robotic manipulator case study. In: Roy R (ed) Soft computing and industry: recent applications. Springer, New York, pp 191–199 Google Scholar
  4. 4.
    Gregorčič G, Lightbody G (2008) Nonlinear system identification: from multiple-model networks to Gaussian processes. Eng Appl Artif Intell 21:1035–1055 CrossRefGoogle Scholar
  5. 5.
    Johansen TA, Murray-Smith R (1997) The operating regime approach. In: Murray-Smith R, Johansen TA (eds) Multiple model approaches to modelling and control. Taylor & Francis, London Google Scholar
  6. 6.
    Johansen TA, Hunt KJ, Gawthrop PJ, Fritz H (1998) Off-equilibrium linearization and design of gain scheduled control with application to vehicle speed control. Control Eng Pract 6:167–180 CrossRefGoogle Scholar
  7. 7.
    Johansen TA, Shorten R, Murray-Smith R (2000) On the interpretation and identification of dynamic Takagi–Sugeno fuzzy models. IEEE Trans Fuzzy Syst 8:297–313 CrossRefGoogle Scholar
  8. 8.
    Johansen TA, Babuška R (2003) Multi-objective identification of Takagi–Sugeno fuzzy models. IEEE Trans Fuzzy Syst 11:847–859 CrossRefGoogle Scholar
  9. 9.
    Khalil HK (2002) Nonlinear systems, 3rd edn. Prentice-Hall, Upper Saddle River zbMATHGoogle Scholar
  10. 10.
    Kocijan J, Hvala N, Strmčnik S (2000) Multi-model control of wastewater treatment reactor. In: Mastorakis N (ed) System and control: theory and applications. World Scientific and Engineering Society, Singapore, pp 49–54 Google Scholar
  11. 11.
    Kocijan J, Žunič G, Strmčnik S, Vrančić D (2002) Fuzzy gain-scheduling control of a gas–liquid separation plant implemented on a PLC. Int J Control 75:1082–1091 zbMATHCrossRefGoogle Scholar
  12. 12.
    Kocijan J, Vrančić D, Dolanc G, Gerkšič S, Strmčnik S, Škrjanc I, Blažič S, Božiček M, Marinšek Z, Hadjinski MB, Boshnakov K, Stathaki A, King R (2003) Auto-tuning non-linear controller for industrial use. In: Proceedings of the IEEE international conference on industrial technology 2003, Piscataway, pp 906–910 CrossRefGoogle Scholar
  13. 13.
    Leith DJ, Leithead WE (1998) Appropriate realisation of MIMO gain-scheduled controllers. Int J Control 70:13–50 MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Leith DJ, Leithead WE (1998) Gain-scheduled controller design: an analytic framework directly incorporating non-equilibrium plant dynamics. Int J Control 70:249–269 MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Leith DJ, Leithead WE (1999) Analytic framework for blended multiple model systems using linear local models. Int J Control 72:605–619 MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Leith DJ, Leithead WE (2000) Survey of gain-scheduling analysis and design. Int J Control 73:1001–1025 MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Leith DJ, Leithead WE (2000) On the identification of nonlinear systems by combining identified linear models. In: Proceedings of the 3rd MathMod conference on mathematical modelling, Vienna. ARGESIM report No. 15(1), pp 407–410 Google Scholar
  18. 18.
    Leith DJ, Leithead WE (2002) Global reconstruction of nonlinear systems from families of linear systems. In: Proceedings of the 15th triennial world congress of the IFAC, Barcelona Google Scholar
  19. 19.
    Leith DJ, Leithead WE (2003) Necessary and sufficient conditions for the minimal state-space realisation of nonlinear systems from input–output information. In: Proceedings of the American control conference 2003, vol 4, pp 2937–2942 CrossRefGoogle Scholar
  20. 20.
    Leith DJ, Solak E, Leithead WE (2003) Direct identification of nonlinear structure using Gaussian process prior models. In: Proceedings of the European control conference 2003, Cambridge Google Scholar
  21. 21.
    McLoone SC, Irwin GW, McLoone SF (2001) Constructing networks of continuous-time velocity-based local models. IEE Proc, Control Theory Appl 148(5):397–405 CrossRefGoogle Scholar
  22. 22.
    Murray-Smith R, Johansen TA (1997) Multiple model approaches to modelling and control. Taylor & Francis, London Google Scholar
  23. 23.
    Murray-Smith R, Johansen TA, Shorten R (1999) On transient dynamics, off-equilibrium behaviour and identification in blended multiple model structures. In: Proceedings of 5th European control conference 1999, Karlsruhe, BA-14 Google Scholar
  24. 24.
    Nelles O (2001) Nonlinear system identification. Springer, Berlin zbMATHCrossRefGoogle Scholar
  25. 25.
    Nelles O, Fink A, Isermann R (2000) Local linear model trees (lolimot) toolbox for nonlinear system identification. In: IFAC symposium on system identification (SYSID), Santa Barbara, USA Google Scholar
  26. 26.
    Rosenvasser YeN, Polyakov EY, Lampe BP (1999) Application of Laplace transformation for digital redesign of continuous control systems. IEEE Trans Autom Control 44:883–886 MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Rugh WJ (1991) Analytic framework for gain scheduling. IEEE Control Syst Mag 11:79–84 CrossRefGoogle Scholar
  28. 28.
    Rugh WJ, Shamma JS (2000) Research on gain scheduling. Automatica 36:1401–1425 MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Slotine J-JE, Li W (1991) Applied nonlinear control. Prentice-Hall, Englewood Cliffs zbMATHGoogle Scholar
  30. 30.
    Takagi T, Sugeno M (1985) Fuzzy identification of systems and its application to modeling and control. IEEE Trans Syst Man Cybern 15:116–132 zbMATHCrossRefGoogle Scholar
  31. 31.
    Toivonen HT (2003) State-dependent parameter models of non-linear sampled-data systems: a velocity-based linearization approach. Int J Control 76:1823–1832 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Juš Kocijan
    • 1
    • 2
  1. 1.Department of Systems and ControlJožef Stefan InstituteLjubljanaSlovenia
  2. 2.University of Nova GoricaNova GoricaSlovenia

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