Simulation-based reachability analysis for nonlinear systems using componentwise contraction properties

TL;DR

This paper introduces a scalable, simulation-based reachability analysis method for nonlinear systems that leverages componentwise contraction properties and adaptive weighted norms to efficiently estimate system behavior.

Contribution

It presents a novel approach combining contraction theory with componentwise analysis and adaptive norms to improve scalability and accuracy in reachability computations for nonlinear systems.

Findings

The method effectively handles systems with uncertain parameters.

Componentwise analysis reduces conservatism in reachability estimates.

Adaptive weighted norms improve the tightness of reachable set bounds.

Abstract

A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of trajectories of the system and next establish bounds on the convergence or divergence between the samples and neighboring trajectories. We compute these bounds using contraction theory and reduce the conservatism by partitioning the state vector into several components and analyzing contraction properties separately in each direction. Among other benefits this allows us to analyze the effect of constant but uncertain parameters by treating them as state variables and partitioning them into a separate direction. We next present a numerical procedure to search for weighted norms that yield a prescribed contraction rate, which can be incorporated in the reachability algorithm to adjust the weights to minimize the growth of the reachable set.