Reachability Analysis for Linear Systems with Uncertain Parameters using Polynomial Zonotopes

TL;DR

This paper introduces a novel reachability analysis method for linear systems with uncertain parameters using polynomial zonotopes, providing tighter set approximations and extending to time-varying and nonlinear systems.

Contribution

The authors develop a new reachability algorithm with improved tightness for uncertain linear systems and extend it to time-varying, nonlinear, and hybrid systems, outperforming existing methods.

Findings

Superior tightness on benchmark systems

Effective handling of non-convex reachable sets

Scalable optimization for multi-affine zonotopes

Abstract

In real world applications, uncertain parameters are the rule rather than the exception. We present a reachability algorithm for linear systems with uncertain parameters and inputs using set propagation of polynomial zonotopes. In contrast to previous methods, our approach is able to tightly capture the non-convexity of the reachable set. Building up on our main result, we show how our reachability algorithm can be extended to handle linear time-varying systems as well as linear systems with time-varying parameters. Moreover, our approach opens up new possibilities for reachability analysis of linear time-invariant systems, nonlinear systems, and hybrid systems. We compare our approach to other state of the art methods, with superior tightness on two benchmarks including a 9-dimensional vehicle platooning system. Moreover, as part of the journal extension, we investigate through a polynomial zonotope with special structure named multi-affine zonotopes and its optimization problem. We provide the corresponding optimization algorithm and experiment over the examples obatined from two benchmark systems, showing the efficiency and scalability comparing to the state of the art method for handling such type of set representation.