Control interpretations of products in the Hopf algebra

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

Families of differential operators, like those defining affine, generally nonlinear, control systems are known to have natural Hopf algebra structures. These provide deeper insight into relationships and properties of such objects as the Chen-Fliess series and state space realizations of systems defined by input-output operators. A starting point for this work are representations of control objects by formal powerseries in, generally noncommuting, indeterminates. Such series are elements of a free associative algebra which contains several rich substructure that are equipped with various products. This article summarizes correspondences between objects from control theory and algebraic combinatorics, with special focus on elements of the two Hopf algebra structures on this algebra of formal power series, including the antipode and two convolution products. Newly found relationships are discussed, and open questions are posed. Recent work has demonstrated that these abstract combinatorial algebra structures help obtain effective solution formulas for various control problems including questions about controllability and algorithms for path planning. They also find applications in the field of geometric numerical integration algorithms.

Original languageEnglish (US)
Title of host publicationProceedings of the IEEE Conference on Decision and Control
Pages7503-7508
Number of pages6
DOIs
StatePublished - 2009
Event48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009 - Shanghai, China
Duration: Dec 15 2009Dec 18 2009

Other

Other48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009
CountryChina
CityShanghai
Period12/15/0912/18/09

Fingerprint

Hopf Algebra
Algebra
Free Associative Algebras
Antipode
Convolution Product
Nonlinear Control Systems
Series
Formal Power Series
Path Planning
Substructure
Combinatorics
Control Theory
Controllability
Numerical integration
Differential operator
Control Problem
State Space
Correspondence
Nonlinear control systems
Motion planning

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

Kawski, M. (2009). Control interpretations of products in the Hopf algebra. In Proceedings of the IEEE Conference on Decision and Control (pp. 7503-7508). [5400441] https://doi.org/10.1109/CDC.2009.5400441

Control interpretations of products in the Hopf algebra. / Kawski, Matthias.

Proceedings of the IEEE Conference on Decision and Control. 2009. p. 7503-7508 5400441.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kawski, M 2009, Control interpretations of products in the Hopf algebra. in Proceedings of the IEEE Conference on Decision and Control., 5400441, pp. 7503-7508, 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009, Shanghai, China, 12/15/09. https://doi.org/10.1109/CDC.2009.5400441
Kawski M. Control interpretations of products in the Hopf algebra. In Proceedings of the IEEE Conference on Decision and Control. 2009. p. 7503-7508. 5400441 https://doi.org/10.1109/CDC.2009.5400441
Kawski, Matthias. / Control interpretations of products in the Hopf algebra. Proceedings of the IEEE Conference on Decision and Control. 2009. pp. 7503-7508
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