Control structures

Robert Bryant; Robert Gardner

Banach Center Publications (1995)

  • Volume: 32, Issue: 1, page 111-121
  • ISSN: 0137-6934

Abstract

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We define an extension of the classical notion of a control system which we call a control structure. This is a geometric structure which can be defined on manifolds whose underlying topology is more complicated than that of a domain in ℝ n . Every control structure turns out to be locally representable as a classical control system, but our extension has the advantage that it has various naturality properties which the (classical) coordinate formulation does not, including the existence of so-called universal objects and classifying maps. This more general viewpoint simplifies the study of the invariants of even classical control systems. Its main technical advantage is that tools like the method of equivalence can be directly and easily applied to the study of control structures.

How to cite

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Bryant, Robert, and Gardner, Robert. "Control structures." Banach Center Publications 32.1 (1995): 111-121. <http://eudml.org/doc/262532>.

@article{Bryant1995,
abstract = {We define an extension of the classical notion of a control system which we call a control structure. This is a geometric structure which can be defined on manifolds whose underlying topology is more complicated than that of a domain in $ℝ^n$. Every control structure turns out to be locally representable as a classical control system, but our extension has the advantage that it has various naturality properties which the (classical) coordinate formulation does not, including the existence of so-called universal objects and classifying maps. This more general viewpoint simplifies the study of the invariants of even classical control systems. Its main technical advantage is that tools like the method of equivalence can be directly and easily applied to the study of control structures.},
author = {Bryant, Robert, Gardner, Robert},
journal = {Banach Center Publications},
keywords = {geometric structure; feedback equivalence; nonlinear control system},
language = {eng},
number = {1},
pages = {111-121},
title = {Control structures},
url = {http://eudml.org/doc/262532},
volume = {32},
year = {1995},
}

TY - JOUR
AU - Bryant, Robert
AU - Gardner, Robert
TI - Control structures
JO - Banach Center Publications
PY - 1995
VL - 32
IS - 1
SP - 111
EP - 121
AB - We define an extension of the classical notion of a control system which we call a control structure. This is a geometric structure which can be defined on manifolds whose underlying topology is more complicated than that of a domain in $ℝ^n$. Every control structure turns out to be locally representable as a classical control system, but our extension has the advantage that it has various naturality properties which the (classical) coordinate formulation does not, including the existence of so-called universal objects and classifying maps. This more general viewpoint simplifies the study of the invariants of even classical control systems. Its main technical advantage is that tools like the method of equivalence can be directly and easily applied to the study of control structures.
LA - eng
KW - geometric structure; feedback equivalence; nonlinear control system
UR - http://eudml.org/doc/262532
ER -

References

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  1. [1] W. Blaschke, Vorlesungen Über Differentialgeometrie II, Springer, Berlin, 1923. Zbl49.0499.01
  2. [2] R. Brockett, Feedback Control of Linear and Non-linear Systems, Lecture Notes in Control and Information Science, vol. 39, Springer, New York, 1982. Zbl0493.93011
  3. [2a] R. Brockett and X. Dai, The dynamics of the ball and plate problem, preprint, 1993. 
  4. [3] R. Bryant, On notions of equivalence of variational problems with one independent variable, Contemporary Mathematics 68 (1987), 65-76. 
  5. [4] R. Gardner, The Method of Equivalence and Applications, SIAM-CBMS Regional Conf. Ser. in Appl. Math. 58, Philadelphia, 1989. 
  6. [5] R. Gardner, W. Shadwick and G. Wilkens, A geometric isomorphism with applications to closed loop controls, SIAM J. Control and Optim. 27 (1989), 1361-1368. Zbl0697.93017
  7. [6] R. Gardner and G. Wilkens, Classical geometries arising in feedback equivalence, in: Proceedings of the 32nd IEEE-CDC, San Antonio, Texas, 1993. 

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