Reach Control on Simplices by Piecewise Affine Feedback

TL;DR

This paper addresses the reach control problem for affine systems on simplices, introducing a geometric approach and piecewise affine feedback to solve cases where continuous feedback fails, assuming open-loop solvability.

Contribution

It introduces reach control indices and a subdivision algorithm to design piecewise affine feedback, solving previously unsolvable cases by continuous feedback.

Findings

The method solves all cases where open-loop control is possible.

Reach control indices characterize structural properties causing feedback failure.

Piecewise affine feedback extends control solutions beyond continuous feedback limitations.

Abstract

We study the reach control problem for affine systems on simplices, and the focus is on cases when it is known that the problem is not solvable by continuous state feedback. We examine from a geometric viewpoint the structural properties of the system which make continuous state feedbacks fail. This structure is encoded by so-called reach control indices, which are defined and developed in the paper. Based on these indices, we propose a subdivision algorithm and associated piecewise affine feedback. The method is shown to solve the reach control problem in all remaining cases, assuming it is solvable by open-loop controls.