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A Framework for Approximating Perturbed Optimal Control Problems

Riley Link, Ethan Ebbighausen

TL;DR

This paper tackles the challenge of adapting optimal control trajectories under parameter uncertainty when in-transit recomputation is expensive. It introduces a framework that combines global sensitivity-based parameter reduction, interpolation of post-optimal sensitivities, and a differential equation–based interpolation (Interpolated Step) to rapidly approximate updated optimal controls. The Space Shuttle 2-DOF re-entry case demonstrates that reducing to a small set of important parameters and precomputing Jacobian sensitivities yields fast, accurate trajectory adjustments with substantial computation time savings. The approach is designed to complement existing feedback control methods and is applicable to aerospace trajectory control under uncertainty.

Abstract

We consider trajectory optimal control problems in which parameter uncertainty limits the applicability of control trajectories computed prior to travel. Hence, efficient trajectory adjustment is needed to ensure successful travel. However, it is often prohibitive or impossible to recalculate the optimal control in-transit due to strict time constraints or limited onboard computing resources. Thus, we propose a framework for quick and accurate trajectory approximations by using post-optimality sensitivity information. This allows the reduction of uncertain parameter space and an instantaneous approximation of the new optimal controller while using sensitivity data computed and stored pretransit.

A Framework for Approximating Perturbed Optimal Control Problems

TL;DR

This paper tackles the challenge of adapting optimal control trajectories under parameter uncertainty when in-transit recomputation is expensive. It introduces a framework that combines global sensitivity-based parameter reduction, interpolation of post-optimal sensitivities, and a differential equation–based interpolation (Interpolated Step) to rapidly approximate updated optimal controls. The Space Shuttle 2-DOF re-entry case demonstrates that reducing to a small set of important parameters and precomputing Jacobian sensitivities yields fast, accurate trajectory adjustments with substantial computation time savings. The approach is designed to complement existing feedback control methods and is applicable to aerospace trajectory control under uncertainty.

Abstract

We consider trajectory optimal control problems in which parameter uncertainty limits the applicability of control trajectories computed prior to travel. Hence, efficient trajectory adjustment is needed to ensure successful travel. However, it is often prohibitive or impossible to recalculate the optimal control in-transit due to strict time constraints or limited onboard computing resources. Thus, we propose a framework for quick and accurate trajectory approximations by using post-optimality sensitivity information. This allows the reduction of uncertain parameter space and an instantaneous approximation of the new optimal controller while using sensitivity data computed and stored pretransit.
Paper Structure (19 sections, 1 theorem, 28 equations, 2 figures, 10 tables, 1 algorithm)

This paper contains 19 sections, 1 theorem, 28 equations, 2 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Generalized Framework for Approximation of the Optimal Control
  3. Problem Formulation
  4. Dimension Reduction
  5. Hyper-Differential Sensitivity Analysis
  6. Sensitivity Equations
  7. Approximating the Optimal Control Problem
  8. Generalizing an Approximation Method
  9. Interpolation of the Key Jacobian
  10. Numerical Results
  11. Space Shuttle Problem
  12. Problem Formulation
  13. Parameter Space Reduction
  14. Space Shuttle Results
  15. Space Shuttle Discussion
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\boldsymbol{u}$ be a vector-valued random variable and let $\Gamma$ be its covariance matrix. Assume the inputs $\theta_i$ are independent and identically distributed uniform random variables $\mathcal{U}(a,b)$. Then, where $c_{j}^{p} = \frac{(b-a)^{2}}{\pi^2}$.

Figures (2)

  • Figure 1: Controller, $\boldsymbol{u}$, and state variables, $\boldsymbol{x}$, of the optimal solution and approximated solution for parameter shift $\boldsymbol{\theta}_0\to\boldsymbol{\theta}_1$ at $t=2000$. The terminal constraints are represented by the thin dashed lines.
  • Figure 2: Cost difference (left) and error norm (right) between optimal and approximated solutions for 500 parameter perturbations. Note that the cost is on the scale of $2$ and $||u||$ is on the scale of $0.68$.

Theorems & Definitions (1)

  • Theorem 2.1