Simplification of the generalized state equations

Tanel Mullari; Ülle Kotta

Kybernetika (2006)

  • Volume: 42, Issue: 5, page 617-628
  • ISSN: 0023-5954

Abstract

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The paper studies the problem of lowering the orders of input derivatives in nonlinear generalized state equations via generalized coordinate transformation. An alternative, computation-oriented proof is presented for the theorem, originally proved by Delaleau and Respondek, giving necessary and sufficient conditions for existence of such a transformation, in terms of commutativity of certain vector fields. Moreover, the dual conditions in terms of 1-forms have been derived, allowing to calculate the new generalized state coordinates in a simpler way. The result is illustrated with an example, originally given by Delaleau and Respondek (see [2]), but solved in an alternative way.

How to cite

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Mullari, Tanel, and Kotta, Ülle. "Simplification of the generalized state equations." Kybernetika 42.5 (2006): 617-628. <http://eudml.org/doc/33828>.

@article{Mullari2006,
abstract = {The paper studies the problem of lowering the orders of input derivatives in nonlinear generalized state equations via generalized coordinate transformation. An alternative, computation-oriented proof is presented for the theorem, originally proved by Delaleau and Respondek, giving necessary and sufficient conditions for existence of such a transformation, in terms of commutativity of certain vector fields. Moreover, the dual conditions in terms of 1-forms have been derived, allowing to calculate the new generalized state coordinates in a simpler way. The result is illustrated with an example, originally given by Delaleau and Respondek (see [2]), but solved in an alternative way.},
author = {Mullari, Tanel, Kotta, Ülle},
journal = {Kybernetika},
keywords = {generalized dynamics; generalized state transformations; input derivatives; classical state; prolonged vector fields; generalized dynamics; generalized state transformation; input derivative; classical state; prolonged vector fields},
language = {eng},
number = {5},
pages = {617-628},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Simplification of the generalized state equations},
url = {http://eudml.org/doc/33828},
volume = {42},
year = {2006},
}

TY - JOUR
AU - Mullari, Tanel
AU - Kotta, Ülle
TI - Simplification of the generalized state equations
JO - Kybernetika
PY - 2006
PB - Institute of Information Theory and Automation AS CR
VL - 42
IS - 5
SP - 617
EP - 628
AB - The paper studies the problem of lowering the orders of input derivatives in nonlinear generalized state equations via generalized coordinate transformation. An alternative, computation-oriented proof is presented for the theorem, originally proved by Delaleau and Respondek, giving necessary and sufficient conditions for existence of such a transformation, in terms of commutativity of certain vector fields. Moreover, the dual conditions in terms of 1-forms have been derived, allowing to calculate the new generalized state coordinates in a simpler way. The result is illustrated with an example, originally given by Delaleau and Respondek (see [2]), but solved in an alternative way.
LA - eng
KW - generalized dynamics; generalized state transformations; input derivatives; classical state; prolonged vector fields; generalized dynamics; generalized state transformation; input derivative; classical state; prolonged vector fields
UR - http://eudml.org/doc/33828
ER -

References

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  2. Delaleau E., Respondek W., Lowering the orders of derivatives of control in gener- alized state space systems, J. Math. Systems Estimation and Control 5 (1995), 3, 1–27 (1995) MR1651823
  3. Dodson C. T. J., Poston T., Tensor Geometry, The Geometric Viewpoint and its Uses. Springer–Verlag, Berlin – Heidelberg – New York 1990 Zbl0732.53002MR1223091
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  5. Glad S. T., Nonlinear state space and input-output descriptions using differential polynomials, In: New Trands in Nonlinear Control Theory (Lecture Notes in Control and Information Sciences 122, J. Descusse, M. Fliess, A. Isidori, and P. Leborne, eds.), Springer–Verlag, New York 1989, pp. 182–189 (1989) Zbl0682.93030MR1229775
  6. Kotta Ü., Removing input derivatives in generalized state space systems: a linear algebraic approach, In: Proc. 4th Internat. Conference APEIE-98. Novosibirsk 1998, pp. 142–147 (1998) 
  7. Kotta Ü., Mullari T., 10.3166/ejc.11.185-193, European J. Control 11 (2005), 185–193 MR2194103DOI10.3166/ejc.11.185-193
  8. Moog C. H., Zheng Y.-F., Liu P., Input-output equivalence of nonlinear systems and their realizations, In: Proc. 15th IFAC World Congress, Barcelona 2002 
  9. Schaft A. J. van der, On realization of nonlinear systems described by higher-order differential equations, Math. Systems Theory 19 (1987), 239–275. Erratum: Math. Systems Theory 20 (1988), 305–306 (1987) MR0871787
  10. Schaft A. J. van der, Transformations and representations of nonlinear systems, In: Perspectives in Control Theory (B. Jakubczyk et al., eds.), Birkhäuser, Boston 1990, pp. 293–314 (1990) MR1046887

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