Data-driven invariant subspace identification for black-box switched linear systems

TL;DR

This paper introduces a data-driven method to identify invariant subspaces in black-box switched linear systems using observed trajectories, quadratic Lyapunov analysis, and scenario optimization, with applications in network consensus and opinion dynamics.

Contribution

It proposes a novel algorithmic framework for approximating invariant subspaces in switched systems from data, linking block-triangularization to invariant subspace existence under certain conditions.

Findings

Successfully identifies disconnected components in switching networks.

Determines stationary opinion vectors in switching gossip models.

Provides a confidence-based approach for subspace approximation.

Abstract

We present an algorithmic framework for the identification of candidate invariant subspaces for switched linear systems. Namely, the framework allows to compute an orthonormal basis in which the matrices of the system are close to block-triangular matrices, based on a finite set of observed one-step trajectories and with a priori confidence level. The link between the existence of an invariant subspace and a common block-triangularization of the system matrices is well known. Under some assumptions on the system, one can also infer the existence of an invariant subspace when the matrices are close to be block-triangular. Our approach relies on quadratic Lyapunov analysis and recent tools in scenario optimization. We present two applications of our results for problems of consensus and opinion dynamics; the first one allows to identify the disconnected components in a switching hidden network, while the second one identifies the stationary opinion vector of a switching gossip process with antagonistic interactions.