On the sensitivity of the subspace predictor to behavioral perturbations

Behavioral systems define discrete-time LTI 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 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, which becomes computable when combined with existing behavioral-distance certificates. Numerical studies show that our bound is tighter than an existing data-matrix perturbation bound when the behavioral distance is available, and remains computable, though more conservative, when combined with an existing behavioral distance certificate.