State-feedback Abstractions for Optimal Control of Piecewise-affine Systems

TL;DR

This paper introduces state-feedback abstractions for piecewise-affine systems, enabling deterministic symbolic models that incorporate local controllers, under input constraints and noise, facilitating optimal control design.

Contribution

It proposes a novel method to create deterministic symbolic abstractions with local affine controllers, avoiding input discretization and providing semi-definite programming conditions for transitions.

Findings

Deterministic symbolic models can be constructed for unstable systems.

The approach reduces input space complexity by using a finite set of controllers.

Semi-definite programs characterize transition existence and costs.

Abstract

In this manuscript, we investigate symbolic abstractions that capture the behavior of piecewise-affine systems under input constraints and bounded external noise. This is accomplished by considering local affine feedback controllers that are jointly designed with the symbolic model, which ensures that an alternating simulation relation between the system and the abstraction holds. The resulting symbolic system is called a state-feedback abstraction and we show that it can be deterministic even when the original piecewise-affine system is unstable and non-deterministic. One benefit of this approach is the fact that the input space need not be discretized and the symbolic-input space is reduced to a finite set of controllers. When ellipsoidal cells and affine controllers are considered, we present necessary and sufficient conditions written as a semi-definite program for the existence of a transition and a robust upper bound on the transition cost. Two examples illustrate particular aspects of the theory and its applicability.