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Douglas--Rachford algorithm for control-constrained minimum-energy control problems

Regina S. Burachik, Bethany I. Caldwell, C. Yalçın Kaya

TL;DR

This paper utilizes four popular projection algorithms, namely the Method of Alternating Projections, Dykstra, Douglas–Rachford and Arag´on Artacho–Campoy algorithms, to solve control-constrained linear-quadratic optimal control problems.

Abstract

Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize the Douglas--Rachford (DR) algorithm to solve control-constrained minimum-energy optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the DR algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.

Douglas--Rachford algorithm for control-constrained minimum-energy control problems

TL;DR

This paper utilizes four popular projection algorithms, namely the Method of Alternating Projections, Dykstra, Douglas–Rachford and Arag´on Artacho–Campoy algorithms, to solve control-constrained linear-quadratic optimal control problems.

Abstract

Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize the Douglas--Rachford (DR) algorithm to solve control-constrained minimum-energy optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the DR algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.
Paper Structure (18 sections, 16 theorems, 114 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 18 sections, 16 theorems, 114 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Minimum-energy Control Problem
  3. Optimality conditions
  4. Constraint splitting
  5. Projectors
  6. Projectors for general minimum-energy control
  7. Projectors for special cases
  8. Double integrator
  9. Pure harmonic oscillator
  10. Damped harmonic oscillator
  11. Machine tool manipulator
  12. Douglas--Rachford Algorithm
  13. Numerical Approach
  14. Background and algorithm for projector onto ${\bf\cal A}$
  15. Experiments
  16. ...and 3 more sections

Key Result

Lemma 1

Given the $n\times n$ matrix $A(t)$, consider the $n^2\times n^2$ matrix $\widetilde{A}(t)$, defined as where $\mathbf{0}$ is a zero matrix of appropriate size, and the matrix $A(t)$ appears repeatedly ($n$ times) in diagonal blocks. The state transition matrix of $\widetilde{A}(t)$ is the $n^2\times n^2$ matrix defined as where $\Phi_A(t,t_0)$ (the state transition matrix for $A(t)$), appears r

Figures (3)

  • Figure 1: Top plots for $\omega_0=5,\,\zeta=0$ where $|u(t)|\leq 0.259$. Bottom plots for $\omega_0=5,\,\zeta=0.5$ where $|u(t)|\leq 9.34\times10^{-7}$.
  • Figure 2: Control solution plot for the machine tool manipulator where $|u(t)|\leq 2000$.
  • Figure 3: Parameter curves for the harmonic oscillator with various values of $(\omega_0,\zeta)$ and for the machine tool manipulator with tolerance values.

Theorems & Definitions (36)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • ...and 26 more