Covariance-Regulated Recursive Koopman Learning for Nonlinear Systems with Uncertain Time-Varying Dynamics

arXiv:2606.15317v2 Announce Type: replace Abstract: Offline models for autonomous robots often fail under time-varying dynamics outside their training distribution. While Koopman operator theory offers a linear representation of nonlinear dynamics via lifting, its transition to online recursive estimation may suffer numerical vulnerabilities: covariance windup under low excitation when using exponential forgetting, and vanishing gain without forgetting. To prevent covariance explosion and parameter freezing, this paper introduces a unified Covariance-Regulated Recursive Koopman Learning (CR-RKL) framework that systematically incorporates and evaluates several covariance-regulated methods. Specifically, we propose a novel error dead-zone covariance skipping strategy, along with adaptations of constant-trace normalization and directional forgetting to the Koopman-specific setting. Validated on a non-holonomic differential-drive robot with wheel slip and Stribeck friction, as well as on a two-winged, tailless flapping-wing micro aerial vehicle, CR- RKL achieves numerically stable and accurate online modeling relative to its offline and vanilla recursive counterparts. When embedded in model predictive control, it maintains reliable tracking performance under uncertain, time-varying dynamics with bounded dictionaries. However, we also identify a structural limitation: while covariance-regulated methods achieve low off-policy prediction error, pairing them with unbounded dictionaries can lead to control failure.

Abstract:Offline models for autonomous robots often fail under time-varying dynamics outside their training distribution. While Koopman operator theory offers a linear representation of nonlinear dynamics via lifting, its transition to online recursive estimation may suffer numerical vulnerabilities: covariance windup under low excitation when using exponential forgetting, and vanishing gain without forgetting. To prevent covariance explosion and parameter freezing, this paper introduces a unified Covariance-Regulated Recursive Koopman Learning (CR-RKL) framework that systematically incorporates and evaluates several covariance-regulated methods. Specifically, we propose a novel error dead-zone covariance skipping strategy, along with adaptations of constant-trace normalization and directional forgetting to the Koopman-specific setting. Validated on a non-holonomic differential-drive robot with wheel slip and Stribeck friction, as well as on a two-winged, tailless flapping-wing micro aerial vehicle, CR- RKL achieves numerically stable and accurate online modeling relative to its offline and vanilla recursive counterparts. When embedded in model predictive control, it maintains reliable tracking performance under uncertain, time-varying dynamics with bounded dictionaries. However, we also identify a structural limitation: while covariance-regulated methods achieve low off-policy prediction error, pairing them with unbounded dictionaries can lead to control failure.

Robotics (cs.RO)

arXiv:2606.15317 [cs.RO]

(or arXiv:2606.15317v2 [cs.RO] for this version)

https://doi.org/10.48550/arXiv.2606.15317 Focus to learn more arXiv-issued DOI via DataCite

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Covariance-Regulated Recursive Koopman Learning for Nonlinear Systems with Uncertain Time-Varying Dynamics

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