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

2026-06-13

AI Summary

This work addresses the challenges of offline model invalidation in nonlinear systems under time-varying uncertain dynamics and numerical instability in recursive Koopman-based learning—specifically, covariance explosion or vanishing gain. To overcome these issues, the authors propose a covariance-regulated recursive Koopman learning framework that leverages lifting maps for online linearized modeling. The approach introduces two novel mechanisms: an error-dead-zone gating strategy and constant-trace covariance normalization, which jointly prevent parameter freezing and covariance divergence while preserving the geometric structure of uncertainty. Integrated with recursive least squares estimation and model predictive control, the framework demonstrates robust online modeling accuracy, numerical stability, and effective tracking performance under time-varying uncertainties, as validated on both differential-drive robots and bio-inspired flapping-wing micro aerial vehicles.

Abstract

Offline models for autonomous robots often fail under time-varying dynamics outside their training distribution. Koopman operator theory offers a linear representation of nonlinear dynamics via lifting, but its transition to real-time recursive estimation may suffer numerical vulnerabilities: covariance windup under low excitation when using exponential forgetting, and vanishing gain without forgetting. This paper introduces a Covariance-Regulated Recursive Koopman Learning (CR-RKL) framework with two complementary strategies--error dead-zone gating and constant-trace normalization--each independently capable of preventing covariance explosion and parameter freezing, with the latter additionally preserving the geometric structure of uncertainty. Validated on a non-holonomic differential-drive robot with wheel slip and Stribeck friction and on a 26-gram butterfly-inspired flapping-wing micro aerial vehicle, CR-RKL achieves numerically stable and accurate online modeling, and when embedded in model predictive control, it maintains reliable tracking performance under uncertain, time-varying dynamics.

Problem

• time-varying dynamics
• Koopman operator
• recursive estimation
• covariance windup
• numerical stability

Innovation

• Covariance Regulation
• Recursive Koopman Learning
• Time-Varying Dynamics
• Error Dead-Zone Gating
• Constant-Trace Normalization