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

Published in arXiv preprint arXiv:2606.15317, 2026

Preprint Spotlight

This work develops CR-RKL, a recursive Koopman learning method for nonlinear systems whose dynamics vary over time and fall outside the training distribution. The method updates a lifted linear predictor online while regulating covariance growth and avoiding parameter freezing.

arXiv PDF

CR-RKL covariance regulation and recursive Koopman learning framework
Original paper figure illustrating covariance windup, vanishing gain, trace-normalized uncertainty, and the CR-RKL recursive update pipeline.

Why This Matters

Robotic systems and autonomous platforms often operate under changing payloads, environments, contact conditions, and aerodynamic effects. Batch-identified models can quickly become stale. This paper studies how Koopman-based predictors can be updated recursively while controlling the uncertainty introduced by new measurements.

Main Contributions

  • Proposes a covariance-regulated recursive Koopman learning framework for nonlinear systems with uncertain, time-varying dynamics.
  • Introduces two complementary covariance-regulation strategies: error dead-zone gating and constant-trace normalization.
  • Addresses two recursive-estimation failure modes: covariance windup under low excitation and vanishing gain without forgetting.
  • Validates online modeling on a differential-drive robot with wheel slip and Stribeck friction and on a 26-gram butterfly-inspired flapping-wing robot.
  • Embeds the learned model in model predictive control to evaluate closed-loop tracking under uncertain dynamics.

Key Findings

  • CR-RKL maintains numerically stable online learning in settings where conventional recursive updates can become ill-conditioned.
  • Constant-trace normalization preserves the geometric structure of uncertainty while preventing covariance explosion.
  • The approach improves online modeling and supports reliable MPC tracking under uncertain, time-varying dynamics.
CR-RKL online modeling benchmark for the flapping-wing robot
Original paper figure comparing online Koopman modeling performance for the flapping-wing robot across same-regime and cross-regime flight data.

Citation

Gu, Weibin, Chen Yang, Lu Shi, and Chao Gao. “Covariance-Regulated Recursive Koopman Learning for Nonlinear Systems with Uncertain Time-Varying Dynamics.” arXiv preprint arXiv:2606.15317 (2026).