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On the sensitivity of the subspace predictor to behavioral perturbations

Dian Jin, Jeremy Coulson

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.

On the sensitivity of the subspace predictor to behavioral perturbations

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.
Paper Structure (10 sections, 7 theorems, 35 equations, 3 figures)

This paper contains 10 sections, 7 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Prediction Error Bound
  4. On the quantitative observability condition
  5. Representation-free subspace predictor
  6. Basis alignment and chordal distance
  7. Proof of Theorem \ref{['thm:error-bound']}
  8. One-step prediction error bound
  9. Numerical Experiment
  10. Conclusion

Key Result

Lemma 1

Consider system eq:LTI-1. Let $(A, B)$ be controllable and $L \geq n$. Then $\mathcal{B}_{[0,L-1]}$ is a linear subspace of dimension $\dim \mathcal{B}_{[0,L-1]} = mL+n$. Moreover, let $T \in \mathbb{Z}_{>0}$ and $(u_{[0,T-1]}, y_{[0,T-1]}) \in \mathcal{B}_{[0,L-1]}$ with $u_{[0,T-1]}$ being persist

Figures (3)

  • Figure 1: The two planes represent the nominal restricted behavior $\mathbf{U}$ and its approximation $\widehat{\mathbf{U}}$, with discrepancy measured by $d(\widehat{\mathbf{U}}, \mathbf{U})$.
  • Figure 2: Single experiment for $T_\mathrm{ini}=4, T_\mathrm{f}=2$. The left panel shows the predicted outputs generated by $U$ and $\widehat{U}_{100}$ with $\kappa_{100}=0.8680$. The right panel compares the corresponding prediction error and the theoretical bound \ref{['eq:error-bound-one-step-known']}. The horizontal axis shows the time indices $[T_\mathrm{ini}, T_{\mathrm{end}}]$.
  • Figure 3: Average prediction error and average prediction error bound (vertical axis) as functions of the chordal distance for three $(T_\mathrm{ini}, T_\mathrm{f})$.

Theorems & Definitions (13)

  • Lemma 1: markovsky2022identifiability
  • Definition 1
  • Theorem 1: Local Lipschitz continuity of the subspace predictor
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • ...and 3 more