ESTONIAN ACADEMY
PUBLISHERS
eesti teaduste
akadeemia kirjastus
PUBLISHED
SINCE 1952
 
Proceeding cover
proceedings
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Model matching problem for discrete-time nonlinear systems; pp. 457–472
PDF | doi: 10.3176/proc.2015.4.01

Authors
Juri Belikov, Miroslav Halás, Ülle Kotta ORCID Icon, Claude H. Moog
Abstract

This paper addresses the model matching problem (MMP) for nonlinear single-input single-output discrete-time systems. The approach is based on the infinitesimal system description in terms of the one-forms that is converted into polynomial system representation by interpreting the polynomial indeterminate as the forward shift operator acting on the one-forms. The polynomial description is then used to derive the generalized transfer function. The problem statement of the MMP (both for the feedforward and feedback cases) is given in terms of the generalized transfer function. In general, the feedforward solution exists under restrictive conditions. Therefore, the subclass of nonlinear control systems is specified for which the solution is guaranteed to exist. The feedback solution exists always. The additional restrictions are specified for the existence of a proper compensator (in both cases). The results of the paper are illustrated by numerous examples, and the feedback solution is compared to the earlier results.

References

  1. Anderson, S. R. and Kadirkamanathan, V. Modelling and identification of nonlinear deterministic systems in delta-domain. Automatica, 2007, 43(11), 1859–1868.
http://dx.doi.org/10.1016/j.automatica.2007.03.020

  2. Aranda-Bricaire, E., Kotta, Ü., and Moog, C. H. Linearization of discrete-time systems. SIAM J. Contr. Optim., 1996, 34(6), 1999–2023.
http://dx.doi.org/10.1137/S0363012994267315

  3. Baheti, R. S., Mohler, R. R., and Spang, H. A. Second-order correlation method for bilinear system identification. IEEE Trans. Autom. Contr., 1980, 25(6), 1141–1146.
http://dx.doi.org/10.1109/TAC.1980.1102516

  4. Belikov, J., Kotta, Ü., and Leibak, A. Transformation of the transfer matrix of the nonlinear system into the Jacobson form. In The International Congress on Computer Applications and Computational Science, Singapore. 2010, 495–498.

  5. Belikov, J., Halás, M., Kotta, Ü., and Moog, C. H. Model matching problem for discrete-time nonlinear systems: transfer function approach. In The 9th International Conference on Control and Automation, Santiago, Chile. 2011, 360–365.
http://dx.doi.org/10.1109/icca.2011.6137955

  6. Belikov, J., Kotta, P., Kotta, Ü., and Zinober, A. S. I. Minimal realization of bilinear and quadratic input-output difference equations in state-space form. Int. J. Contr., 2011, 84(12), 2024–2034.
http://dx.doi.org/10.1080/00207179.2011.631587

  7. Belikov, J., Kotta, Ü., and Leibak, A. Transfer matrix and its Jacobson form for nonlinear systems on time scales: Mathematica implementation. In The 18th International Conference on Process Control, Tatranská Lomnica, Slovak Republic. 2011, 141–146.

  8. Belikov, J., Kotta, Ü., and Tõnso, M. Adjoint polynomial formulas for nonlinear state-space realization. IEEE Trans. Autom. Contr., 2014, 59(1), 256–261.
http://dx.doi.org/10.1109/TAC.2013.2270868

  9. Di Benedetto, M. D. Nonlinear strong model matching. IEEE Trans. Autom. Contr., 1990, 35(12), 1351–1355.
http://dx.doi.org/10.1109/9.61014

10. Cohn, R. M. Difference Algebra. Wiley-Interscience, New York, USA, 1965.

11. Conte, G., Moog, C. H., and Perdon, A. M. Algebraic Methods for Nonlinear Control Systems. Springer-Verlag, London, UK, 2007.
http://dx.doi.org/10.1007/978-1-84628-595-0

12. Farb, B. and Dennis, R. K. Noncommutative Algebra. Springer-Verlag, New York, USA, 1993.
http://dx.doi.org/10.1007/978-1-4612-0889-1

13. Glad, S. T. Nonlinear regulators and Ritt’s remainder algorithm. In Colloque international sur l’analyse des systems dynamiques controlés, Lyon, France. 1990, 687–692.

14. Halás, M. An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica, 2008, 44(5), 1181–1190.
http://dx.doi.org/10.1016/j.automatica.2007.09.008

15. Halás, M. and Kotta, Ü. Transfer functions of discrete-time nonlinear control systems. Proc. Estonian Acad. Sci. Phys. Math., 2007, 56(4), 322–335.

16. Halás, M. and Kotta, Ü. Extension of the transfer function approach to the realization problem of nonlinear systems to discrete-time case. In The 8th IFAC Symposium on Nonlinear Control Systems, Bologna, Italy. 2010, 179–184.

17. Halás, M. and Kotta, Ü. A transfer function approach to the realisation problem of nonlinear control systems. Int. J. Contr., 2012, 85(3), 320–331.
http://dx.doi.org/10.1080/00207179.2011.651748

18. Halás, M., Kotta, Ü., and Moog, C. H. Transfer function approach to the model matching problem of nonlinear systems. In The 17th IFAC World Congress, Seoul, Korea. 2008, 15197–15202.

19. Huijberts, H. J. C. A nonregular solution of the nonlinear dynamic disturbance decoupling problem with an application to a complete solution of the nonlinear model matching problem. SIAM J. Contr. Optim., 1992, 30(2), 350–366.
http://dx.doi.org/10.1137/0330022

20. Ito, N., Schmale, W., and Wimmer, H. K. Computation of minimal state space realizations in Jacobson normal form. In Fast Algorithms for Structured Matrices, Theory and Applications (Olshevsky, V., ed.), Contemp. Math. American Mathematical Society Boston, MA, USA, 2003, 323, 221–232.
http://dx.doi.org/10.1090/conm/323/05706

21. Kailath, T. Linear Systems. Englewood Cliffs, New Jersey: Prentice Hall, USA, 1980.

22. Kimura, G., Matsumoto, T., and Takahashi, S. A direct method for exact model matching. Syst. & Contr. Lett., 1982, 2(1), 53–56.
http://dx.doi.org/10.1016/S0167-6911(82)80043-1

23. Kotsios, S. A model matching algorithm for a class of nonlinear discrete systems. A symbolic computational approach. In The 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain. 2005, 6603–6607.

24. Kotta, Ü. Inversion method in the discrete-time nonlinear control systems synthesis problems. Lecture Notes Control Inform. Sci., 1995, 205, 1–152.
http://dx.doi.org/10.1007/3-540-19966-7_1

25. Kotta, Ü. Model matching problem for nonlinear recursive systems. Proc. Estonian Acad. Sci. Phys. Math., 1997, 46(4), 251–261.

26. Kotta, Ü. and Nurges, Ü. Identification of input-output bilinear systems. In The 9th World Congress of IFAC: a Bridge Between Control Science and Technology, Budapest, Hungary. 1984, 113–117.

27. Kotta, Ü. and Tõnso, M. Realization of discrete-time nonlinear input-output equations: polynomial approach. Automatica, 2012, 48(2), 255–262.
http://dx.doi.org/10.1016/j.automatica.2011.07.010

28. Kotta, Ü., Zinober, A. S. I., and Liu, P. Transfer equivalence and realization of nonlinear higher order input-output difference equations. Automatica, 2001, 37(11), 1771–1778.
http://dx.doi.org/10.1016/S0005-1098(01)00144-3

29. Kučera, V. Analysis and Design of Discrete Linear Control Systems. Prentice Hall, New York, USA, 1991.

30. Kučera, V. and Toledo, E. C. A review of stable exact model matching by state feedback. In The 22nd Mediterranean Conference on Control and Automation, Palermo, Italy. 2014, 85–90.

31. Marinescu, B. and Bourlès, H. The exact model-matching problem for linear time-varying systems: an algebraic approach. IEEE Trans. Autom. Contr., 2003, 48(1), 166–169.
http://dx.doi.org/10.1109/TAC.2002.805654

32. McConnell, J. C. and Robson, J. C. Noncommutative Noetherian Rings. John Wiley & Sons, New York, USA, 1987.

33. Moog, C. H., Perdon, A., and Conte, G. Model matching and factorization for nonlinear systems: a structural approach. SIAM J. Contr. Optim., 1991, 29(4), 769–785.
http://dx.doi.org/10.1137/0329042

34. Moore, B. C. and Silverman, L. M. Model matching by state feedback and dynamic compensation. IEEE Trans. Autom. Contr., 1972, 17(4), 491–497.
http://dx.doi.org/10.1109/TAC.1972.1100032

35. Morse, A. S. Structure and design of linear model following systems. IEEE Trans. Autom. Contr., 1973, 18(4), 346–354.
http://dx.doi.org/10.1109/TAC.1973.1100342

36. Ore, O. Theory of non-commutative polynomials. Annals Math., 1933, 34(3), 480–508.
http://dx.doi.org/10.2307/1968173

37. Sadegh, N. Minimal realization of nonlinear systems described by input-output difference equations. IEEE Trans. Autom. Contr., 2001, 46(5), 698–710.
http://dx.doi.org/10.1109/9.920788

38. Tzafestas, S. G. Model matching in time-delay control systems. IEEE Trans. Autom. Contr., 1976, 21(3), 426–428.
http://dx.doi.org/10.1109/TAC.1976.1101216

39. Tzafestas, S. G. and Paraskevopoulos, P. N. On the exact model matching controller design. IEEE Trans. Autom. Contr., 1976, 21(2), 242–246.
http://dx.doi.org/10.1109/TAC.1976.1101169

40. van der Schaft, A. J. On realization of nonlinear systems described by higher-order differential equations. Math. Syst. Theory, 1987, 19(1), 239–275.
http://dx.doi.org/10.1007/BF01704916

41. Wang, S. H. and Desoer, C. A. The exact model matching of linear multivariable systems. IEEE Trans. Autom. Contr., 1972, 17(3), 347–349.
http://dx.doi.org/10.1109/TAC.1972.1099984

42. Wolovich, W. A. The use of state feedback for exact model matching. SIAM J. Contr. Optim., 1972, 10(3), 512–523.
http://dx.doi.org/10.1137/0310039

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