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Optimal Control using Composite Bernstein Approximants

Gage MacLin, Venanzio Cichella, Andrew Patterson, Michael Acheson, Irene Gregory

TL;DR

The paper addresses the challenge of solving Bolza-type optimal control problems with discontinuous solutions such as bang-bang controls. It introduces composite Bernstein polynomials as a direct collocation method, proving convergence properties and demonstrating quadratic convergence with respect to the number of knots, while enabling exact constraint satisfaction and avoiding Gibbs phenomena. The authors formulate a discrete Problem $P_M$ using composite Bernstein approximants and a knotting method, and prove feasibility and convergence of the discrete solutions to the continuous optimum under Lipschitz conditions. Numerical examples, including bang-bang control and multi-vehicle motion planning, show the method’s ability to accurately capture discontinuities and replan trajectories, highlighting practical advantages over traditional pseudospectral methods.

Abstract

In this work, we present composite Bernstein polynomials as a direct collocation method for approximating optimal control problems. An analysis of the convergence properties of composite Bernstein polynomials is provided, and beneficial properties of composite Bernstein polynomials for the solution of optimal control problems are discussed. The efficacy of the proposed approximation method is demonstrated through a bang-bang example. Lastly, we apply this method to a motion planning problem, offering a practical solution that emphasizes the ability of this method to solve complex optimal control problems.

Optimal Control using Composite Bernstein Approximants

TL;DR

The paper addresses the challenge of solving Bolza-type optimal control problems with discontinuous solutions such as bang-bang controls. It introduces composite Bernstein polynomials as a direct collocation method, proving convergence properties and demonstrating quadratic convergence with respect to the number of knots, while enabling exact constraint satisfaction and avoiding Gibbs phenomena. The authors formulate a discrete Problem using composite Bernstein approximants and a knotting method, and prove feasibility and convergence of the discrete solutions to the continuous optimum under Lipschitz conditions. Numerical examples, including bang-bang control and multi-vehicle motion planning, show the method’s ability to accurately capture discontinuities and replan trajectories, highlighting practical advantages over traditional pseudospectral methods.

Abstract

In this work, we present composite Bernstein polynomials as a direct collocation method for approximating optimal control problems. An analysis of the convergence properties of composite Bernstein polynomials is provided, and beneficial properties of composite Bernstein polynomials for the solution of optimal control problems are discussed. The efficacy of the proposed approximation method is demonstrated through a bang-bang example. Lastly, we apply this method to a motion planning problem, offering a practical solution that emphasizes the ability of this method to solve complex optimal control problems.
Paper Structure (10 sections, 5 theorems, 74 equations, 2 figures, 1 table)

This paper contains 10 sections, 5 theorems, 74 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Composite Bernstein Approximation
  4. Proposed Direct Method
  5. Knotting Method
  6. Numerical Results
  7. Conclusion
  8. Proof of Lemma \ref{['lem:function']}
  9. Proof of Lemma \ref{['lem:derivative']}
  10. Proof of Lemma \ref{['lem:integral']}

Key Result

Lemma 1

Let $x(t) \in \mathcal{C}^2([0,t_f])$. Let $x_M(t)$ be the composite Bernstein polynomial approximation of $x(t)$. The following bound holds: for all $t \in [0,t_f]$, where

Figures (2)

  • Figure 1: Solution to Example 1: (a) Approximation of a bang-bang input using a single Bernstein polynomial for orders $N=10,15,30,55$. (b) Derivative of the control input for the single Bernstein polynomial solution for $N=10$. (c) Approximation of a bang-bang input using a composite Bernstein polynomial consisting of two polynomials of order $N=10$.
  • Figure 2: (a) Aerial map of runway with aircraft following 45$\degree$ entry until an intruding aircraft is detected and the initial trajectory is replanned to avoid collision. (b) Distance between the initial and replanned trajectory of the aircraft, and the trajectory of the intruder for the motion planning example, with a separation variable of 500 ft.

Theorems & Definitions (8)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2