Book ChapterDOI
The Local Realization of Generating Series of Finite Lie Rank
Christophe Reutenauer
- pp 33-43
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The problem of local realization of nonlinear systems by state-space is a classical problem in control theory as discussed by the authors, which has been completely solved by Kaiman [10] in the case of linear systems.Abstract:
Realization of nonlinear systems by state-space is a classical problem in control theory This problem has been completely solved by Kaiman [10] in the case of linear systems Similarly, it was solved for bilinear systems (see Brockett [2], d’Alessandro, Isidori, Ruberti [1], Fliess [3], Sussmann [12]) In the general case, let us mention the work of Sussmann [13], Hermann, Krener [7] and Jakubczyk [9]: they assume that the solutions are regular at any time and for any inputs This restriction lead Fliess to study local realization of nonlinear systems [5]read more
Citations
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Journal ArticleDOI
Realization theory for linear and bilinear switched systems: A formal power series approach - Part I: Realization theory of linear switched systems
TL;DR: In this paper, necessary and sufficient conditions are formulated for a family of input-output maps to be realizable by a linear switched system and a characterization of minimal realizations is presented.
Journal ArticleDOI
The realization of input-output maps using bialgebras
TL;DR: In this article, the theory of bialgebras is used to prove a state space realization theorem for input/output maps of dynamical systems, allowing for the consideration of the classical results of Fliess and more recent results on realizations involving families of trees.
Journal ArticleDOI
On a conjecture by Pierre Cartier about a group of associators
TL;DR: Bourbaki et al. as mentioned in this paper proved that φ exists and that it is a free Lie exponential map over a non-commutative formal power series and gave a complete description of the kernel of ζ and drew some consequences about the arithmetical nature of the Euler constant.
Book ChapterDOI
Spaces of Observables in Nonlinear Control
TL;DR: In this article, a representation of a linear shift operator is specified by a triple of linear maps, where the linear map is defined by a linear operator and the linear operator is represented by a set of linear operators.
References
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Book
Introduction to Lie Algebras and Representation Theory
TL;DR: In this paper, Semisimple Lie Algebras and root systems are used for representation theory, isomorphism and conjugacy theorem, and existence theorem for representation.
Journal ArticleDOI
Nonlinear controllability and observability
R. Hermann,Arthur J. Krener +1 more
TL;DR: The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems as discussed by the authors.
Book
Combinatorics on words
TL;DR: Perrin and Perrin this article showed that square free words and idempotent semigroups can be expressed in terms of free monoids, and the critical factorization theorem of Van der Waerden's theorem.
Journal ArticleDOI
Mathematical description of linear dynamical systems
TL;DR: In this paper, it is shown that the input/output relations determine only one part of a system, that which is completely observable and completely controllable, and methods are given for calculating irreducible realization of a given impulse-response matrix.
Journal ArticleDOI
Fonctionnelles causales non linéaires et indéterminées non commutatives
TL;DR: In this paper, the foundations of non-lmear causal functionals are laid down using non-commutative indeterminates, and a theory of causal functions is presented.