| Title:
|
Simplification of the generalized state equations (English) |
| Author:
|
Mullari, Tanel |
| Author:
|
Kotta, Ülle |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
42 |
| Issue:
|
5 |
| Year:
|
2006 |
| Pages:
|
617-628 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
generalized dynamics |
| Keyword:
|
generalized state transformations |
| Keyword:
|
input derivatives |
| Keyword:
|
classical state |
| Keyword:
|
prolonged vector fields |
| MSC:
|
93B11 |
| MSC:
|
93B17 |
| MSC:
|
93B29 |
| MSC:
|
93C10 |
| idZBL:
|
Zbl 1249.93094 |
| idMR:
|
MR2283509 |
| . |
| Date available:
|
2009-09-24T20:19:17Z |
| Last updated:
|
2015-03-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135739 |
| . |
| Reference:
|
[1] Crouch P. E., Lamnabhi-Lagarrigue F.: State space realizations of nonlinear systems defined by input-output differential equations.In: Analysis and Optimization Systems (Lecture Notes in Control and Information Sciences 111, A. Bensousan and J. L. Lions, eds.), Springer–Verlag, Berlin – Heidelberg – New York 1988, pp. 138–149 Zbl 0675.93031, MR 0956266 |
| Reference:
|
[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 MR 1651823 |
| Reference:
|
[3] Dodson C. T. J., Poston T.: Tensor Geometry.The Geometric Viewpoint and its Uses. Springer–Verlag, Berlin – Heidelberg – New York 1990 Zbl 0732.53002, MR 1223091 |
| Reference:
|
[4] Fliess M.: Automatique et corps différentielles.Forum Math. 1 (1989), pp. 227–238 MR 1005424, 10.1515/form.1989.1.227 |
| Reference:
|
[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 Zbl 0682.93030, MR 1229775 |
| Reference:
|
[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 |
| Reference:
|
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| Reference:
|
[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 |
| Reference:
|
[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 MR 0871787 |
| Reference:
|
[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 MR 1046887 |
| . |
