Optimal Control of Hybrid Systems via Measure Relaxations

TL;DR

This paper introduces a convex relaxation method for trajectory optimization in hybrid systems using measure relaxations, enabling efficient global solutions and improved scalability over traditional mixed-integer approaches.

Contribution

It presents a novel convex relaxation framework based on occupation measures for hybrid systems, improving scalability and solution quality in trajectory planning.

Findings

Convex relaxation provides bounds close to nonconvex solutions.

Significant speedup over mixed-integer formulations.

Applicable to planning with temporal logic specifications.

Abstract

We propose an approach to trajectory optimization for piecewise polynomial systems based on the recently proposed graphs of convex sets framework. We instantiate the framework with a convex relaxation of optimal control based on occupation measures, resulting in a convex optimization problem resembling the discrete shortest-paths linear program that can be solved efficiently to global optimality. While this approach inherits the limitations of semidefinite programming, scalability to large numbers of discrete modes improves compared to the NP-hard mixed-integer formulation. We use this to plan trajectories under temporal logic specifications, comparing the computed cost lower bound to a nonconvex optimization approach with fixed mode sequence. In our numerical experiments, we find that this bound is typically in the vicinity of the nonconvex solution, while the runtime speedup is significant compared to the often intractable mixed-integer formulation. Our implementation is available at https://github.com/ebuehrle/hpoc.