Realization Theory for Discrete-Time Semi-algebraic Hybrid Systems

  • Mihály Petreczky
  • René Vidal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4981)


We present realization theory for a class of autonomous discrete-time hybrid systems called semi-algebraic hybrid systems. These are systems in which the state and output equations associated with each discrete state are defined by polynomial equalities and inequalities. We first show that these systems generate the same output as semi-algebraic systems and implicit polynomial systems. We then derive necessary and almost sufficient conditions for existence of an implicit polynomial system realizing a given time-series data. We also provide a characterization of the dimension of a minimal realization as well as an algorithm for computing a realization from a given time-series data.


Hybrid System Formal Power Series Realization Theory Switching Signal Discrete Mode 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Isidori, A.: Nonlinear Control Systems. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  2. 2.
    Sontag, E.D.: Polynomial Response Maps. Lecture Notes in Control and Information Sciences, vol. 13. Springer, Heidelberg (1979)zbMATHGoogle Scholar
  3. 3.
    Sontag, E.D.: Realization theory of discrete-time nonlinear systems: Part I – the bounded case. IEEE Transaction on Circuits and Systems CAS-26(4) (1979)Google Scholar
  4. 4.
    Fliess, M.: Matrices de Hankel. J. Math. Pures Appl. (23), 197–224 (1973)Google Scholar
  5. 5.
    Sussmann, H.: Existence and uniqueness of minimal realizations of nonlinear systems. Mathematical Systems Theory 10, 263–284 (1977)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Jakubczyk, B.: Realization theory for nonlinear systems, three approaches. In: Fliess, M., Hazewinkel, M., (eds.) Algebraic and Geometric Methods in Nonlinear Control Theory, pp. 3–32. D. Reidel Publishing Company (1986)Google Scholar
  7. 7.
    Wang, Y., Sontag, E.: Generating series and nonlinear systems: analytic aspects, local realizability and I/O representations. Forum Mathematicum (4), 299–322 (1992)Google Scholar
  8. 8.
    Bartosiewicz, Z.: Realizations of polynomial systems. In: Algebraic and geometric methods in nonlinear control theory., Math. Appl., vol. 29, pp. 45–54. Dordrecht, Reidel (1986)Google Scholar
  9. 9.
    Wang, Y., Sontag, E.: Algebraic differential equations and rational control systems. SIAM Journal on Control and Optimization (30), 1126–1149 (1992)Google Scholar
  10. 10.
    Grossman, R., Larson, R.: An algebraic approach to hybrid systems. Theoretical Computer Science 138, 101–112 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Petreczky, M.: Realization theory for linear switched systems: Formal power series approach. Systems and Control Letters 56(9-10), 588–595 (2007)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Petreczky, M.: Realization theory for bilinear switched systems: A formal power series approach. In: Proc. of 44th IEEE Conference on Decision and Control, pp. 690–695 (2005)Google Scholar
  13. 13.
    Petreczky, M.: Realization Theory of Hybrid Systems. PhD thesis, Vrije Universiteit, Amsterdam (2006),
  14. 14.
    Petreczky, M.: Hybrid formal power series and their application to realization theory of hybrid systems. In: Proc. 17th International Symposium on Mathematical Theory of Networks and Systems (2006)Google Scholar
  15. 15.
    Petreczky, M., Pomet, J.B.: Realization theory of nonlinear hybrid systems. In: Proceedings of CTS-HYCON Workshop on Hybrid and Nonlinear Control Systems (2006)Google Scholar
  16. 16.
    Petreczky, M.: Realization theory for discrete-time piecewise-affine hybrid systems. In: Proc 17th Internation Symposium on Mathematical Theory of Networks and Systems (2006)Google Scholar
  17. 17.
    Petreczky, M., Vidal, R.: Metrics and topology for nonlinear and hybrid systems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 459–472. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Petreczky, M., Vidal, R.: Realization theory of stochastic jump-Markov linear systems. In: Proceedings 46th IEEE Conference on Decision and Control (2007)Google Scholar
  19. 19.
    Ma, Y., Vidal, R.: Identification of deterministic switched ARX systems via identification of algebraic varieties. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 449–465. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Vidal, R.S.S., Sastry, S.: An algebraic geometric approach to the identification of linear hybrid systems. In: IEEE Conference on Decision and Control, pp. 167–172 (2003)Google Scholar
  21. 21.
    Vidal, R.: Identification of PWARX hybrid models with unknown and possibly different orders. In: Proceedings of the IEEE American Conference on Control, pp. 547–552 (2004)Google Scholar
  22. 22.
    Kunz, E.: Introduction to commutative algebra and algebraic geometry. Birkhaeuser, Stuttgard (1985)Google Scholar
  23. 23.
    Cox, D., Little, J., O’Shea, D.: Ideal, varieties, and algorithms. Springer, New York (1997)Google Scholar
  24. 24.
    Brumfiel, G.W.: Partialy Ordered Rings and Semi-Algebraic Geometry. Cambridge University Press, Cambridge (1979)Google Scholar
  25. 25.
    Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  26. 26.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  27. 27.
    Collins, P., van Schuppen, J.H.: Observability of hybrid systems and Turing machines. In: Proceedings of the 43rd IEEE Conference on Decision and Control, pp. 7–12 (2004)Google Scholar
  28. 28.
    Ma, Y., Yang, A., Derksen, H., Fossum, R.: Estimation of subspace arrangements with applications in modeling and segmenting mixed data. SIAM Review (to appear, 2007)Google Scholar
  29. 29.
    Ho, B.L., Kalman, R.E.: Effective construction of linear state-variable models from input/output data. In: Proc. 3rd Allerton Conf. on Circuit and System Theory, pp. 449–459 (1965)Google Scholar
  30. 30.
    Caines, P.: Linear Stochastic Systems. John Wiley and Sons, New-York (1988)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mihály Petreczky
    • 1
    • 2
  • René Vidal
    • 2
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Center for Imaging ScienceJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations