Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories

Hector J. Sussmann; Wensheng Liu

Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories

Hector J. Sussmann, Wensheng Liu

School of Arts and Sciences, Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

84 Scopus citations

Abstract

The authors describe sufficient conditions, extending earlier work by J. Kurzweil and J. Jarnik (Reults in Mathematics, vol. 14, pp. 125-137, 1988), for a sequence of inputs to be such that, for every m-tuple of smooth vector fields, the trajectories of the time derivative of x(t) converge to those of an extended system, where the new vector fields are Lie brackets of the original m-tuples. Using these conditions, the inverse problem is solved, wherein given a trajectory γ of the extended system, one must find trajectories of the original system that converge to γ. This is done by means of a universal construction that only involves knowledge of the coefficients of the extended system. These results can be applied to solve the problem of approximate tracking for a controllable system without drift.

Original language	English (US)
Title of host publication	Proceedings of the IEEE Conference on Decision and Control
Publisher	Publ by IEEE
Pages	437-442
Number of pages	6
ISBN (Print)	0780304500
State	Published - Jan 1 1992
Event	Proceedings of the 30th IEEE Conference on Decision and Control Part 1 (of 3) - Brighton, Engl Duration: Dec 11 1991 → Dec 13 1991

Other

Other	Proceedings of the 30th IEEE Conference on Decision and Control Part 1 (of 3)
City	Brighton, Engl
Period	12/11/91 → 12/13/91

All Science Journal Classification (ASJC) codes

Chemical Health and Safety
Control and Systems Engineering
Safety, Risk, Reliability and Quality

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title = "Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories",

abstract = "The authors describe sufficient conditions, extending earlier work by J. Kurzweil and J. Jarnik (Reults in Mathematics, vol. 14, pp. 125-137, 1988), for a sequence of inputs to be such that, for every m-tuple of smooth vector fields, the trajectories of the time derivative of x(t) converge to those of an extended system, where the new vector fields are Lie brackets of the original m-tuples. Using these conditions, the inverse problem is solved, wherein given a trajectory γ of the extended system, one must find trajectories of the original system that converge to γ. This is done by means of a universal construction that only involves knowledge of the coefficients of the extended system. These results can be applied to solve the problem of approximate tracking for a controllable system without drift.",

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N2 - The authors describe sufficient conditions, extending earlier work by J. Kurzweil and J. Jarnik (Reults in Mathematics, vol. 14, pp. 125-137, 1988), for a sequence of inputs to be such that, for every m-tuple of smooth vector fields, the trajectories of the time derivative of x(t) converge to those of an extended system, where the new vector fields are Lie brackets of the original m-tuples. Using these conditions, the inverse problem is solved, wherein given a trajectory γ of the extended system, one must find trajectories of the original system that converge to γ. This is done by means of a universal construction that only involves knowledge of the coefficients of the extended system. These results can be applied to solve the problem of approximate tracking for a controllable system without drift.

AB - The authors describe sufficient conditions, extending earlier work by J. Kurzweil and J. Jarnik (Reults in Mathematics, vol. 14, pp. 125-137, 1988), for a sequence of inputs to be such that, for every m-tuple of smooth vector fields, the trajectories of the time derivative of x(t) converge to those of an extended system, where the new vector fields are Lie brackets of the original m-tuples. Using these conditions, the inverse problem is solved, wherein given a trajectory γ of the extended system, one must find trajectories of the original system that converge to γ. This is done by means of a universal construction that only involves knowledge of the coefficients of the extended system. These results can be applied to solve the problem of approximate tracking for a controllable system without drift.

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