Optimized State Space Grids for Abstractions

TL;DR

This paper introduces a method to optimize the aspect ratio of hyper-intervals in state space grids to minimize computational effort in abstraction-based controller synthesis, making the process more efficient.

Contribution

It proposes a predictive functional for computational effort and a convex optimization approach to select hyper-interval aspect ratios for more efficient abstractions.

Findings

The proposed method reduces the number of transitions in abstractions.

The optimization problem is convex and globally solvable.

Demonstrated effectiveness on a practical example.

Abstract

The practical impact of abstraction-based controller synthesis methods is currently limited by the immense computational effort for obtaining abstractions. In this note we focus on a recently proposed method to compute abstractions whose state space is a cover of the state space of the plant by congruent hyper-intervals. The problem of how to choose the size of the hyper-intervals so as to obtain computable and useful abstractions is unsolved. This note provides a twofold contribution towards a solution. Firstly, we present a functional to predict the computational effort for the abstraction to be computed. Secondly, we propose a method for choosing the aspect ratio of the hyper-intervals when their volume is fixed. More precisely, we propose to choose the aspect ratio so as to minimize a predicted number of transitions of the abstraction to be computed, in order to reduce the computational effort. To this end, we derive a functional to predict the number of transitions in dependence of the aspect ratio. The functional is to be minimized subject to suitable constraints. We characterize the unique solvability of the respective optimization problem and prove that it transforms, under appropriate assumptions, into an equivalent convex problem with strictly convex objective. The latter problem can then be globally solved using standard numerical methods. We demonstrate our approach on an example.