Scalable Zonotopic Under-approximation of Backward Reachable Sets for Uncertain Linear Systems

TL;DR

This paper introduces a scalable method using zonotopes to under-approximate backward reachable sets of uncertain linear systems, enabling improved control design despite the challenges of Minkowski difference operations.

Contribution

It presents a novel approach to approximate backward reachable sets with zonotopes by solving linear optimization problems, addressing the closure issues under Minkowski difference.

Findings

The method effectively under-approximates backward reachable sets.

The approach is computationally efficient and scalable.

It outperforms existing methods on benchmark instances.

Abstract

Zonotopes are widely used for over-approximating forward reachable sets of uncertain linear systems for verification purposes. In this paper, we use zonotopes to achieve more scalable algorithms that under-approximate backward reachable sets of uncertain linear systems for control design. The main difference is that the backward reachability analysis is a two-player game and involves Minkowski difference operations, but zonotopes are not closed under such operations. We under-approximate this Minkowski difference with a zonotope, which can be obtained by solving a linear optimization problem. We further develop an efficient zonotope order reduction technique to bound the complexity of the obtained zonotopic under-approximations. The proposed approach is evaluated against existing approaches using randomly generated instances and illustrated with several examples.