Triangulations, Subdivisions, and Covers for Control of Affine Hypersurface Systems on Polytopes

TL;DR

This paper provides a comprehensive method for controlling affine hypersurface systems within polytopes, using subdivision, triangulation, and covers to synthesize feedback controls that guarantee reachability of a target set.

Contribution

It introduces a systematic approach to partitioning polytopes for feedback control synthesis, ensuring correctness when open-loop control is feasible.

Findings

Systematic subdivision and triangulation methods guarantee correct feedback synthesis.

Characterization of states reachable by open-loop control.

Explicit procedures for feedback control synthesis.

Abstract

This paper studies the problem for an affine hypersurface system to reach a polytopic target set starting from inside a polytope in the state space. We present an exhaustive solution which begins with a characterization of states which can reach the target by open-loop control and concludes with a systematic procedure to synthesize a feedback control. Our emphasis is on methods of subdivision, triangulation, and covers which explicitly account for the capabilities of the control system. In contrast with previous literature, the partition methods are guaranteed to yield a correct feedback synthesis, assuming the problem is solvable by open-loop control.