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Semi-Infinite Programs for Robust Control and Optimization: Efficient Solutions and Extensions to Existence Constraints

Jad Wehbeh, Eric C. Kerrigan

TL;DR

The paper develops a local-reduction approach to solve semi-infinite programs arising in discrete-time robust control with uncountable uncertainty, including extensions to existence constraints and non-unique trajectory propagation. By casting the min-max-min problems into SIP form and generating optimal uncertainty scenarios, the method produces violation-free, locally optimal open-loop trajectories for obstacle avoidance, input saturation, and parameter estimation. Although implemented with a local solver, random-restarts and scenario augmentation yield practically robust solutions without the computational burden of global SIP solvers. The framework can potentially extend to model predictive control, enabling later-time information to improve performance while maintaining feasibility over the horizon. Overall, the approach broadens the class of robust control problems solvable via SIPs with efficient, scenario-based reductions.

Abstract

Discrete-time robust optimal control problems generally take a min-max structure over continuous variable spaces, which can be difficult to solve in practice. In this paper, we extend the class of such problems that can be solved through a previously proposed local reduction method to consider those with existence constraints on the uncountable variables. We also consider the possibility of non-unique trajectories that satisfy equality and inequality constraints. Crucially, we show that the problems of interest can be cast into a standard semi-infinite program and demonstrate how to generate optimal uncertainty scenario sets in order to obtain numerical solutions. We also include examples on model predictive control for obstacle avoidance with logical conditions, control with input saturation affected by uncertainty, and optimal parameter estimation to highlight the need for the proposed extension. Our method solves each of the examples considered, producing violation-free and locally optimal solutions.

Semi-Infinite Programs for Robust Control and Optimization: Efficient Solutions and Extensions to Existence Constraints

TL;DR

The paper develops a local-reduction approach to solve semi-infinite programs arising in discrete-time robust control with uncountable uncertainty, including extensions to existence constraints and non-unique trajectory propagation. By casting the min-max-min problems into SIP form and generating optimal uncertainty scenarios, the method produces violation-free, locally optimal open-loop trajectories for obstacle avoidance, input saturation, and parameter estimation. Although implemented with a local solver, random-restarts and scenario augmentation yield practically robust solutions without the computational burden of global SIP solvers. The framework can potentially extend to model predictive control, enabling later-time information to improve performance while maintaining feasibility over the horizon. Overall, the approach broadens the class of robust control problems solvable via SIPs with efficient, scenario-based reductions.

Abstract

Discrete-time robust optimal control problems generally take a min-max structure over continuous variable spaces, which can be difficult to solve in practice. In this paper, we extend the class of such problems that can be solved through a previously proposed local reduction method to consider those with existence constraints on the uncountable variables. We also consider the possibility of non-unique trajectories that satisfy equality and inequality constraints. Crucially, we show that the problems of interest can be cast into a standard semi-infinite program and demonstrate how to generate optimal uncertainty scenario sets in order to obtain numerical solutions. We also include examples on model predictive control for obstacle avoidance with logical conditions, control with input saturation affected by uncertainty, and optimal parameter estimation to highlight the need for the proposed extension. Our method solves each of the examples considered, producing violation-free and locally optimal solutions.
Paper Structure (16 sections, 51 equations, 1 figure)

This paper contains 16 sections, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. State of the Art
  3. Contributions
  4. Motivational Examples
  5. Obstacle Avoidance
  6. Input Saturation
  7. Parameter Estimation
  8. Problem Formulation
  9. System Definition
  10. Robust Optimal Control Problem
  11. Local Reduction Algorithm
  12. Results
  13. Obstacle Avoidance
  14. Input Saturation
  15. Parameter Estimation
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Distribution of 1000 randomly sampled trajectories under the open loop control inputs obtained