On the sensitivity of the subspace predictor to behavioral perturbations

TL;DR

This paper investigates the robustness of the subspace predictor in behavioral systems, establishing its invariance to basis changes and deriving explicit bounds on prediction error due to behavioral perturbations.

Contribution

It proves the predictor's invariance under basis change and introduces the first explicit error bounds based on behavioral distance, enhancing robustness analysis.

Findings

Predictor is invariant under basis change.

Prediction error bounds grow linearly with behavioral distance.

Theoretical bounds are validated by numerical experiments.

Abstract

Behavioral systems define discrete-time Linear Time-Invariant systems in terms of a set of trajectories, which forms a linear subspace. This subspace underlies the subspace predictor used in data-driven prediction and control. In practice, such subspaces are typically represented through data matrices. For robustness certification and uncertainty quantification, however, these matrix representations are coordinate-dependent and therefore do not provide a coordinate-free way to quantify uncertainty. In this work, we establish two key properties of the subspace predictor. We first show that the subspace predictor is invariant under change of basis and depends only on the underlying behavioral subspace. We then derive the first explicit prediction error bound in terms of behavioral distance between the true subspace and an estimate, showing that the predictor is locally Lipschitz with respect to behavioral perturbations. We also present a one-step prediction error bound that is relevant for receding-horizon implementations and can be computed directly. Numerical experiments show that the theoretical bound upper-bounds the prediction error, and that the average prediction error bound grows linearly with the behavioral distance.