Kybernetika 50 no. 4, 473-490, 2014

State elimination for nonlinear neutral state-space systems

Miroslav Halás and Pavol BistákDOI: 10.14736/kyb-2014-4-0473

Abstract:

The problem of finding an input-output representation of a nonlinear state space system, usually referred to as the state elimination, plays an important role in certain control problems. Though, it has been shown that such a representation, at least locally, always exists for both the systems with and without delays, it might be a neutral input-output differential equation in the former case, even when one starts with a retarded system. In this paper the state elimination is therefore extended further to nonlinear neutral state-space systems, and it is shown that also in such a case an input-output representation, at least locally, always exists. In general, it represents a neutral system again. Computational aspects related to the state elimination problem are discussed as well.

Keywords:

neutral systems, nonlinear time-delay systems, input-output representation, linear algebraic methods, Gröbner bases

Classification:

93C10, 34K40, 93B25

paper.pdf

References:

  1. M. Anguelova and B. Wennberg: State elimination and identifiability of the delay parameter for nonlinear time-delay systems. Automatica 44 (2008), 1373-1378.   CrossRef
  2. T. Becker and V. Weispfenning: Gr{ö}bner Bases. Springer-Verlag, New York 1993.   CrossRef
  3. B. Buchberger and F. Winkler: Gr{ö}bner Bases and Applications. Cambridge University Press, Cambridge 1998.   CrossRef
  4. P. M. Cohn: Free Rings and Their Relations. Academic Press, London 1985.   CrossRef
  5. G. Conte, C. H. Moog and A. M. Perdon: Algebraic Methods for Nonlinear Control Systems. Theory and Applications. Second edition. Communications and Control Engineering. Springer-Verlag, London 2007.   CrossRef
  6. D. Cox, J. Little and D. O'Shea: Ideals, Varieties, and Algorithms. Springer-Verlag, New York 2007.   CrossRef
  7. S. Diop: Elimination in control theory. Math. Contr. Signals Syst. 4 (1991), 72-86.   CrossRef
  8. S. T. Glad: Nonlinear regulators and {R}itt's remainder algorithm. In: Analysis of Controlled Dynamical Systems (B. Bournard, B. Bride, J. P. Gauthier, and I. Kupka, eds.), Progress in systems and control theory 8, Birkh{ä}user, Boston 1991, pp. 224-232   CrossRef
  9. A. Glumineau, C. H. Moog and F. Plestan: New algebro-geometric conditions for the linearization by input-output injection. IEEE Trans. Automat. Control 41 (1996), 598-603.   CrossRef
  10. M. Halás: An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica 44 (2008), 1181-1190.   CrossRef
  11. M. Halás: Nonlinear time-delay systems: a polynomial approach using {O}re algebras. In: Topics in Time-Delay Systems: Analysis, Algorithms and Control (J. J. Loiseau, W. Michiels, S. Niculescu, and R. Sipahi, eds.), Lecture Notes in Control and Information Sciences, Springer, 2009.   CrossRef
  12. M. Halás: Computing an input-output representation of a neutral state-space system. In: IFAC Workshop on Time Delay Systems, Grenoble 2013.   CrossRef
  13. M. Halás and M. Anguelova: When retarded nonlinear time-delay systems admit an input-output representation of neutral type. Automatica 49 (2013) 561-567.   CrossRef
  14. M. Halás and {Ü}. Kotta: A transfer function approach to the realisation problem of nonlinear systems. Internat. J. Control 85 (2012), 320-331.   CrossRef
  15. M. Halás, {Ü}. Kotta and C. H. Moog: Transfer function approach to the model matching problem of nonlinear systems. In: 17th IFAC World Congress, Seoul 2008.   CrossRef
  16. M. Halás and C. H. Moog: A polynomial solution to the model matching problem of nonlinear time-delay systems. In: European Control Conference, Budapest 2009.   CrossRef
  17. M. Huba: Comparing 2DOF PI and predictive disturbance observer based filtered {PI} control. J. Process Control 23 (2013), 1379-1400.   CrossRef
  18. {Ü}. Kotta, Z. Bartosiewicz, E. Pawluszewicz and M. Wyrwas: Irreducibility, reduction and transfer equivalence of nonlinear input-output equations on homogeneous time scales. Syst. Control Lett. 58 (2009), 646-651.   CrossRef
  19. {Ü}. Kotta, P. Kotta and M. Halás: Reduction and transfer equivalence of nonlinear control systems: unification and extension via pseudo-linear algebra. Kybernetika 46 (2010), 831-849.   CrossRef
  20. L. A. Márquez-Martínez, C. H. Moog and M. Velasco-Villa: The structure of nonlinear time-delay systems. Kybernetika 36 (2000), 53-62.   CrossRef
  21. L. A. Márquez-Martínez, C. H. Moog and M. Velasco-Villa: Observability and observers for nonlinear systems with time delays. Kybernetika 38 (2002), 445-456.   CrossRef
  22. T. Ohtsuka: Model structure simplification of nonlinear systems via immersion. IEEE Trans. Automat. Control 50 (2005), 607-618.   CrossRef
  23. O. Ore: Linear equations in non-commutative fields. Ann. Math. 32 (1931), 463-477.   CrossRef
  24. O. Ore: Theory of non-commutative polynomials. Ann. Math. 34(1933), 480-508.   CrossRef
  25. P. Picard, J. F. Lafay and V. Kučera: Model matching for linear systems with delays and 2D systems. Automatica 34 (1998), 183-191.   CrossRef
  26. J. Rudolph: Viewing input-output system equivalence from differential algebra. J. Math. Systems Estim. Control 4 (1994), 353-383.   CrossRef
  27. U. Walther, T. T. Georgiou and A. Tannenbaum: On the computation of switching surfaces in optimal control: a Gr{ö}bner basis approach. IEEE Trans. Automat. Control 46 (2001), 534-540.   CrossRef
  28. X. Xia, L. A. Márquez-Martínez, P. Zagalak and C. H. Moog: Analysis of nonlinear time-delay systems using modules over non-commutative rings. Automatica 38 (2002), 1549-1555.   CrossRef
  29. J. Zhang, X. Xia and C. H. Moog: Parameter identifiability of nonlinear systems with time-delay. IEEE Trans. Automat. Control 51 (2006), 371-375.   CrossRef
  30. Y. Zheng, J. Willems and C. Zhang: A polynomial approach to nonlinear system controllability. IEEE Trans. Automat. Control 46 (2001), 1782-1788.   CrossRef