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Smooth Indirect Solution Method for State-constrained Optimal Control Problems with Nonlinear Control-affine Systems

Kenshiro Oguri

Abstract

This paper proposes a new indirect solution method for solving state-constrained optimal control problems by revisiting the well-established optimal control theory and addressing the long-standing issue of discontinuous control and costate due to pure state inequality constraints. It is well-known that imposing pure state path constraints in optimal control problems introduces discontinuities in the control and costate, rendering the classical indirect solution methods ineffective to numerically solve state-constrained problems. This study re-examines the necessary conditions of optimality for a class of state-constrained optimal control problems, and shows the uniqueness of the optimal control input that minimizes the Hamiltonian on constrained arcs. This analysis leads to a unifying form of optimality necessary conditions and thereby address the issue of discontinuities in control and costate by modeling them via smooth functions, transforming the originally discontinuous problems to smooth two-point boundary value problems (TPBVPs), which can be readily solved by existing nonlinear root-finding algorithms. The proposed solution method is shown to have favorable properties, including its anytime algorithm-like property, which is often a desirable property for safety-critical applications, and numerically demonstrated by an optimal orbital transfer problem.

Smooth Indirect Solution Method for State-constrained Optimal Control Problems with Nonlinear Control-affine Systems

Abstract

This paper proposes a new indirect solution method for solving state-constrained optimal control problems by revisiting the well-established optimal control theory and addressing the long-standing issue of discontinuous control and costate due to pure state inequality constraints. It is well-known that imposing pure state path constraints in optimal control problems introduces discontinuities in the control and costate, rendering the classical indirect solution methods ineffective to numerically solve state-constrained problems. This study re-examines the necessary conditions of optimality for a class of state-constrained optimal control problems, and shows the uniqueness of the optimal control input that minimizes the Hamiltonian on constrained arcs. This analysis leads to a unifying form of optimality necessary conditions and thereby address the issue of discontinuities in control and costate by modeling them via smooth functions, transforming the originally discontinuous problems to smooth two-point boundary value problems (TPBVPs), which can be readily solved by existing nonlinear root-finding algorithms. The proposed solution method is shown to have favorable properties, including its anytime algorithm-like property, which is often a desirable property for safety-critical applications, and numerically demonstrated by an optimal orbital transfer problem.
Paper Structure (19 sections, 2 theorems, 41 equations, 2 figures)

This paper contains 19 sections, 2 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Problem Statement
  4. Optimality Necessary Conditions
  5. Unconstrained arc
  6. Constrained arc
  7. Discontinuity at corners
  8. Uniqueness of Optimal Control and Multiplier
  9. Optimal Control on Unconstrained Arcs
  10. Optimal Control on Constrained Arcs
  11. Unified Form of Optimality Necessary Conditions
  12. Analytical Example
  13. Solution Method
  14. Smooth Approximation of Discontinuous Multiplier
  15. Smooth Approximation of Discontinuous Control
  16. ...and 4 more sections

Key Result

Lemma 1

Under the assumption that the unique solution to eq:unconstOptimalControl is available in a closed form eq:unconstSolution, the solution to eq:constrainedOptimalControl is uniquely determined in the following form: where $h$ is continuously differentiable.

Figures (2)

  • Figure 1: Activation functions with various sharpness parameters $\rho$
  • Figure 2: Numerical example results with various $\rho$.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2