Control-Channel Informativity for Koopman EDMDc under Behavior-Policy Data

TL;DR

This paper investigates the limitations of EDMDc in identifying control effects from behavior-policy data, introducing a certificate for control-channel identifiability and analyzing its implications through theory and experiments.

Contribution

It introduces a conditional intervention certificate for EDMDc, providing necessary and sufficient conditions for control-channel identifiability and analyzing its scaling behavior.

Findings

The intervention certificate's positivity is necessary and sufficient for identifiability.

Residual intervention information scales quadratically with dither amplitude.

Experiments confirm the theoretical scalings in linear and nonlinear systems.

Abstract

Extended dynamic mode decomposition with control (EDMDc) is often trained from trajectories generated by a behavior policy or a pre-existing feedback controller. Such data can predict the observed behavior accurately while failing to identify how new input commands change the lifted state. This paper studies that failure as a control-channel informativity problem. We introduce a conditional intervention certificate, defined as the residual input covariance after projecting the input data away from the active lifted-state feature span. The certificate is the Schur complement of the lifted-state block in the EDMDc information matrix. We prove that its strict positivity is necessary and sufficient for finite-sample sample- identifiability of the lifted control-channel block. If the certificate vanishes, distinct lifted models agree on every collected transition but disagree under counterfactual inputs. We then give a closed-loop statistical bound using predictable regressors, conditionally sub-Gaussian transition noise, and a regularized Schur complement. A scalar feedback example shows the unavoidable scaling: under dithered feedback, residual intervention information grows quadratically with the dither amplitude and the control-channel error decreases with the inverse intervention signal-to-noise scale. New experiments verify these scalings exactly in a linear system and diagnostically in controlled Duffing and Van der Pol benchmarks. A larger EDMDc acquisition grid further shows that state coverage, joint regression conditioning, and intervention excitation are complementary diagnostics rather than interchangeable performance score.