Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Best Approximation Optimal Control for Infeasible Double Integrator and Douglas--Rachford Algorithm

Regina S. Burachik, Bethany I. Caldwell, C. Yalçın Kaya, Walaa M. Moursi

TL;DR

This work addresses best-approximation control for an infeasible double integrator where the control bounds are too tight to satisfy the dynamics and boundary constraints. It provides a complete analytical treatment: the best-approximation control in the infeasible regime is bang-bang with at most one switching, with the switching time and the gap function obtained by solving two algebraic equations, and it characterizes the asymptotic switching behavior as the bound tends to zero. In addition, the paper formulates and implements a Douglas--Rachford splitting method for the infeasible problem, detailing the proximal projections onto the two constraint sets and presenting extensive numerical experiments that compare DR with direct discretization methods like AMPL--Ipopt. The results demonstrate accurate switching-time computation and competitive, often superior, performance of DR for moderately fine discretizations, while noting that near the critically feasible case $a_c$ convergence can become challenging. Overall, the work advances both analytical understanding and operator-splitting algorithms for infeasible optimal control problems in linear dynamics, with potential applicability to broader systems beyond the double integrator.

Abstract

We consider the problem of finding (in some sense) the best approximation control for an infeasible double integrator. The control function is constrained by upper and lower bounds that are too tight and thus cause infeasibility. The infeasibility is characterized by a gap function (representing the separation between two constraint sets) whose squared ${\cal L}^2$-norm is to be minimized to find the best approximation control solution. First, we review the existing results for problems involving a general linear control system. Then, for the infeasible double integrator problem, we present an analytical solution for the bang--bang control with at most one switching. The infinite-dimensional optimization problem is reduced to the problem of solving two algebraic equations in two variables, to compute the switching time and gap function. We discuss numerical approaches to solving the system of equations. Finally, we describe the (relaxed) Douglas--Rachford algorithm for the double integrator problem and carry out numerical experiments to illustrate the implementation of the algorithm and test performance.

Best Approximation Optimal Control for Infeasible Double Integrator and Douglas--Rachford Algorithm

TL;DR

This work addresses best-approximation control for an infeasible double integrator where the control bounds are too tight to satisfy the dynamics and boundary constraints. It provides a complete analytical treatment: the best-approximation control in the infeasible regime is bang-bang with at most one switching, with the switching time and the gap function obtained by solving two algebraic equations, and it characterizes the asymptotic switching behavior as the bound tends to zero. In addition, the paper formulates and implements a Douglas--Rachford splitting method for the infeasible problem, detailing the proximal projections onto the two constraint sets and presenting extensive numerical experiments that compare DR with direct discretization methods like AMPL--Ipopt. The results demonstrate accurate switching-time computation and competitive, often superior, performance of DR for moderately fine discretizations, while noting that near the critically feasible case convergence can become challenging. Overall, the work advances both analytical understanding and operator-splitting algorithms for infeasible optimal control problems in linear dynamics, with potential applicability to broader systems beyond the double integrator.

Abstract

We consider the problem of finding (in some sense) the best approximation control for an infeasible double integrator. The control function is constrained by upper and lower bounds that are too tight and thus cause infeasibility. The infeasibility is characterized by a gap function (representing the separation between two constraint sets) whose squared -norm is to be minimized to find the best approximation control solution. First, we review the existing results for problems involving a general linear control system. Then, for the infeasible double integrator problem, we present an analytical solution for the bang--bang control with at most one switching. The infinite-dimensional optimization problem is reduced to the problem of solving two algebraic equations in two variables, to compute the switching time and gap function. We discuss numerical approaches to solving the system of equations. Finally, we describe the (relaxed) Douglas--Rachford algorithm for the double integrator problem and carry out numerical experiments to illustrate the implementation of the algorithm and test performance.
Paper Structure (12 sections, 6 theorems, 51 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 12 sections, 6 theorems, 51 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Review of the Case With General Linear Dynamics
  3. The setting
  4. Problem of minimizing gap
  5. Componentwise controllability
  6. Best approximation control
  7. A Double Integrator Problem
  8. A case study
  9. Douglas--Rachford Algorithm
  10. DR Algorithm for the Infeasible Problem (PDI)
  11. Case study
  12. Conclusion

Key Result

Theorem 1

With the notation of Problem (Pf), assume that ${\cal A} \cap {\cal B} = \varnothing$. Then the optimal gap vector is given by for all $t\in[0,1]$, where the adjoint variable vector $\lambda(\cdot)$ solves $\dot{\lambda}(t) = -A^T(t)\,\lambda(t)$, with appropriate transversality conditions. Moreover, suppose that $A(\cdot)$ and $B(\cdot)$ are sufficiently smooth and that the control system $\do

Figures (4)

  • Figure 1: The best approximation control and the gap function for the double integrator for various values of $a$BurKayMou2024.
  • Figure 2: Portraits of the colour-coded number of iterations required to reach a solution of \ref{['eqn1']}--\ref{['eqn2']} via the Newton method.
  • Figure 3: Portraits of the colour-coded number of iterations required to reach a solution of \ref{['eqn1']}--\ref{['eqn2']} via the generalized Newton method with $s(y) = (y_1^3, y_2)$.
  • Figure 4: Parameter curves for (PDI) using the DR algorithm with $\epsilon=10^{-6}$.

Theorems & Definitions (8)

  • Theorem 1: Gap Vector and Best Approximation Control in ${\cal B}$ BurKayMou2024
  • Corollary 1: Best Approximation Control in ${\cal A}$ BurKayMou2024
  • Corollary 2: Projection of $u_{\cal B}$ onto ${\cal A}$
  • Remark 1: Types of Control According to the Gap Function
  • Remark 2: Setting $t_s = 1$ in Certain Cases
  • Theorem 2: Best Approximation Solution to Infeasible Problem (PDI)
  • Corollary 3: Switching Time as $a \to 0$
  • Theorem 3: Convergence of the DR Algorithm