System Theory pp 391-403 | Cite as

Categories of Nonlinear Dynamic Models

  • Ronald K. Pearson
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 518)

Abstract

Many industrially-promising model-based control strategies require nonlinear, discrete-time dynamic models of restricted complexity relative to that of detailed mechanistic descriptions. One possible route to these models is empirical, permitting explicit control of model complexity through the class C of approximating models considered. Conversely, if this choice is not made with sufficient care, the resulting model may fit available input/output data reasonably well but violate certain important behavioral requirements (e.g., stability, monotonicity of step responses, etc.). This paper proposes category theory as a general framework for simultaneously considering both the structural and behavioral aspects of empirical model identification. Two particular advantages of this use of category theory are first, that it forces the consideration of models and input sequences together and second, that it provides a simple framework for examining the behavioral differences between model classes.

Keywords

Acoustics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge, New York, 1989.MATHCrossRefGoogle Scholar
  2. [2]
    J. Adamek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1990.MATHGoogle Scholar
  3. [3]
    R. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.MATHGoogle Scholar
  4. [4]
    J. Astola, P. Heinonen, and Y. Neuvo, “On Root Structures of Median and Median-Type Filters,” IEEE Trans. Acoustics, Speech, Signal Proc., vol. 35, 1987, pp. 1199–1201.CrossRefGoogle Scholar
  5. [5]
    J.A. Bangham, “Properties of a Series of Nested Median Filters, Namely the Data Sieve,” IEEE Trans. Signal Proc.,vol. 41, 1993, pp. 31–42.MATHCrossRefGoogle Scholar
  6. [6]
    A. Benallou, D.E. Seborg, and D.A. Mellichamp, “Dynamic compartmental models for separation processes,” AIChE J., vol. 32, 1986, pp. 1067–1078.CrossRefGoogle Scholar
  7. [7]
    V.D. Blondel and J.N. Tsitsiklis, “Complexity of stability and controllability of elementary hybrid systems,” Automatica, vol. 35, 1999, pp. 479–489.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    M.S. Branicky, V.S. Borkar, and S.K. Mitter, “A Unifired Framework for Hybrid Control: Model and Optimal Control Theory,” IEEE Trans. Automatic Control, vol. 43, 1998, pp. 31–45.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    D.R. Brillinger, “The identification of a particular nonlinear time series system,” Biometrika, vol. 64, 1977, pp. 509–515.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods, 2nd ed., Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
  11. [11]
    E. Eskinat, S.H. Johnson, and W.L. Luyben, “Use of Hammerstein Models in Identification of Nonlinear Systems,” AIChE J., vol. 37, 1991, pp. 255–268.CrossRefGoogle Scholar
  12. [12]
    N.C. Gallagher, Jr. and G.L. Wise, “A Theoretical Analysis of the Properties of Median Filters,” IEEE Trans. Acoustics, Speech, Signal Proc., vol. 29, 1981, pp. 1136–1141.CrossRefGoogle Scholar
  13. [13]
    G.C. Goodwin and R.L. Payne, Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York, 1977.MATHGoogle Scholar
  14. [14]
    F. Gross, E. Baumann, A. Geser, D.W.T. Rippin and L. Lang, “Modelling, simulation and controllability analysis of an industrial heat-integrated distillation process,” Computers Chem. Eng., vol. 22, 1998, pp. 223–237.CrossRefGoogle Scholar
  15. [15]
    R.A. Horn, “The Hadamard Product,” Proc. Symposia Appl. Math., vol. 40, 1990, pp. 87–169.MathSciNetGoogle Scholar
  16. [16]
    R.R. Horton, B.W. Bequette, and T.F. Edgar, “Improvements in Dynamic Compartmental Modeling fo Distillation,” Computers Chem. Eng., vol. 15, 1991, pp. 197–201.CrossRefGoogle Scholar
  17. [17]
    L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, 1987.MATHGoogle Scholar
  18. [18]
    T.A. Johansen and B.A. Foss, “Constructing NARMAX models using AR-MAX models,” Int. J. Control, vol. 58, 1993, pp. 1125–1153.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    T.A. Johansen, “Identification of Non-Linear Systems Using Empirical Data and Prior Knowledge — An Optimization Approach,” Automatica, vol. 32, 1996, pp. 337–356.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    F.A. Michelsen and B.A. Foss, “A comprehensive mechanistic model of a continuous Kamyr digester,” Appl. Math. Modelling,vol. 20, 1996, pp. 523–533.MATHCrossRefGoogle Scholar
  21. [21]
    A.V. Oppenheim, R.W. Schafer, and T.C. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE, vol. 56, 1963, pp. 1264–1294.CrossRefGoogle Scholar
  22. [22]
    G. Pajunen, “Adaptive Control of Wiener Type Nonlinear Systems,” Automatica, vol. 28, 1992, pp. 781–785.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    R.K. Pearson, “Nonlinear input/output modelling,” J. Proc. Control, vol. 45, 1995, pp. 197–211.CrossRefGoogle Scholar
  24. [24]
    R.K. Pearson, Discrete-Time Dynamic Models, Oxford, New York, in press.Google Scholar
  25. [25]
    S.J. Qin and T.A. Badgwell, “An Overview of Industrial Model Predictive Control Technology,” in J.C. Kantor, C.E. Garcia, and B. Carnahan (eds.), Fifth Int’l. Conf. Chemical Process Control, AIChE and CACHE, 1997, pp. 232–256.Google Scholar
  26. [26]
    S.J. Qin and T.A. Badgwell, “An Overview of Nonlinear Model Predictive Control Applications,” Preprints Int’1. Symposium on Nonlinear Model Predictive Control: Assessment and Future Directions, Ascona, Switzerland, June, 1998, pp. 128–145.Google Scholar
  27. [27]
    S.Y. Shvartsman and I.G. Kevrekidis, “Nonlinear Model Reduction for Control of Distributed Systems: a Computer-Assisted Study,” AIChE J., vol. 44, 1998, pp. 1579–1595.CrossRefGoogle Scholar
  28. [28]
    H. Tong, Non-linear Time Series, Oxford, New York, 1990.MATHGoogle Scholar
  29. [29]
    H.J.A.F. Tulleken, “Grey-box Modelling and Identification Using Physical Knowledge and Bayesian Techniques,” Automatica, vol. 29, 1993, pp. 285–308.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Ronald K. Pearson
    • 1
  1. 1.ETH ZürichInstutut für AutomatikZürichSwitzerland

Personalised recommendations