Motion planning, equivalence, infinite dimensional systems

Pierre Rouchon

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 1, page 165-188
  • ISSN: 1641-876X

Abstract

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Motion planning, i.e., steering a system from one state to another, is a basic question in automatic control. For a certain class of systems described by ordinary differential equations and called flat systems (Fliess et al. 1995; 1999a), motion planning admits simple and explicit solutions. This stems from an explicit description of the trajectories by an arbitrary time function, the flat output, and a finite number of its time derivatives. Such explicit descriptions are related to old problems on Monge equations and equivalence investigated by Hilbert and Cartan. The study of several examples (the car with -trailers and the non-holonomic snake, pendulums in series and the heavy chain, the heat equation and the Euler-Bernoulli flexible beam) indicates that the notion of flatness and its underlying explicit description can be extended to infinite-dimensional systems. As in the finite-dimensional case, this property yields simple motion planning algorithms via operators of compact support. For the non-holonomic snake, such operators involve non-linear delays. For the heavy chain, they are defined via distributed delays. For heat and Euler-Bernoulli systems, their supports are reduced to a point and their definition domain coincides with the set of Gevrey functions of order 2.

How to cite

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Rouchon, Pierre. "Motion planning, equivalence, infinite dimensional systems." International Journal of Applied Mathematics and Computer Science 11.1 (2001): 165-188. <http://eudml.org/doc/207498>.

@article{Rouchon2001,
abstract = {Motion planning, i.e., steering a system from one state to another, is a basic question in automatic control. For a certain class of systems described by ordinary differential equations and called flat systems (Fliess et al. 1995; 1999a), motion planning admits simple and explicit solutions. This stems from an explicit description of the trajectories by an arbitrary time function, the flat output, and a finite number of its time derivatives. Such explicit descriptions are related to old problems on Monge equations and equivalence investigated by Hilbert and Cartan. The study of several examples (the car with -trailers and the non-holonomic snake, pendulums in series and the heavy chain, the heat equation and the Euler-Bernoulli flexible beam) indicates that the notion of flatness and its underlying explicit description can be extended to infinite-dimensional systems. As in the finite-dimensional case, this property yields simple motion planning algorithms via operators of compact support. For the non-holonomic snake, such operators involve non-linear delays. For the heavy chain, they are defined via distributed delays. For heat and Euler-Bernoulli systems, their supports are reduced to a point and their definition domain coincides with the set of Gevrey functions of order 2.},
author = {Rouchon, Pierre},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Gevrey functions; motion planning; absolute equivalence; infinite dimensional control systems; delay systems; flatness; Pfaffian systems; infinite dimensional system},
language = {eng},
number = {1},
pages = {165-188},
title = {Motion planning, equivalence, infinite dimensional systems},
url = {http://eudml.org/doc/207498},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Rouchon, Pierre
TI - Motion planning, equivalence, infinite dimensional systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 1
SP - 165
EP - 188
AB - Motion planning, i.e., steering a system from one state to another, is a basic question in automatic control. For a certain class of systems described by ordinary differential equations and called flat systems (Fliess et al. 1995; 1999a), motion planning admits simple and explicit solutions. This stems from an explicit description of the trajectories by an arbitrary time function, the flat output, and a finite number of its time derivatives. Such explicit descriptions are related to old problems on Monge equations and equivalence investigated by Hilbert and Cartan. The study of several examples (the car with -trailers and the non-holonomic snake, pendulums in series and the heavy chain, the heat equation and the Euler-Bernoulli flexible beam) indicates that the notion of flatness and its underlying explicit description can be extended to infinite-dimensional systems. As in the finite-dimensional case, this property yields simple motion planning algorithms via operators of compact support. For the non-holonomic snake, such operators involve non-linear delays. For the heavy chain, they are defined via distributed delays. For heat and Euler-Bernoulli systems, their supports are reduced to a point and their definition domain coincides with the set of Gevrey functions of order 2.
LA - eng
KW - Gevrey functions; motion planning; absolute equivalence; infinite dimensional control systems; delay systems; flatness; Pfaffian systems; infinite dimensional system
UR - http://eudml.org/doc/207498
ER -

References

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