Direct transcription of optimal control problems with finite elements on Bernstein basis

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Abstract

The paper introduces the use of Bernstein polynomials as a basis for the direct transcription of optimal control problems with Finite Elements in Time. Two key properties of this new transcription approach are demonstrated in this paper: Bernstein bases return smooth control profiles with no oscillations near discontinuities or abrupt changes of the control law, and for convex feasible sets, the polynomial representation of both states and controls remains within the feasible set for all times. The latter property is demonstrated theoretically and experimentally. A simple but representative example is used to illustrate these properties and compare the new scheme against a more common way to transcribe optimal control problems with Finite Elements in Time.
Original languageEnglish
Pages (from-to)229-243
Number of pages15
JournalJournal of Guidance, Control and Dynamics
Volume42
Issue number2
Early online date22 Oct 2018
DOIs
Publication statusPublished - 28 Feb 2019

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Bernstein Basis
optimal control
Transcription
Optimal Control Problem
Finite Element
polynomials
Bernstein Polynomials
Polynomials
Discontinuity
discontinuity
oscillation
Oscillation
oscillations
Polynomial
profiles

Keywords

  • optimal control
  • finite elements in time (FET)
  • Bernstein polynomials
  • guaranteed feasibility

Cite this

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title = "Direct transcription of optimal control problems with finite elements on Bernstein basis",
abstract = "The paper introduces the use of Bernstein polynomials as a basis for the direct transcription of optimal control problems with Finite Elements in Time. Two key properties of this new transcription approach are demonstrated in this paper: Bernstein bases return smooth control profiles with no oscillations near discontinuities or abrupt changes of the control law, and for convex feasible sets, the polynomial representation of both states and controls remains within the feasible set for all times. The latter property is demonstrated theoretically and experimentally. A simple but representative example is used to illustrate these properties and compare the new scheme against a more common way to transcribe optimal control problems with Finite Elements in Time.",
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author = "Ricciardi, {Lorenzo A.} and Massimiliano Vasile",
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AU - Vasile, Massimiliano

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AB - The paper introduces the use of Bernstein polynomials as a basis for the direct transcription of optimal control problems with Finite Elements in Time. Two key properties of this new transcription approach are demonstrated in this paper: Bernstein bases return smooth control profiles with no oscillations near discontinuities or abrupt changes of the control law, and for convex feasible sets, the polynomial representation of both states and controls remains within the feasible set for all times. The latter property is demonstrated theoretically and experimentally. A simple but representative example is used to illustrate these properties and compare the new scheme against a more common way to transcribe optimal control problems with Finite Elements in Time.

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KW - finite elements in time (FET)

KW - Bernstein polynomials

KW - guaranteed feasibility

UR - https://arc.aiaa.org/loi/jgcd

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