Regularized Online Dynamic Mode Decomposition with Control for Non-Persistently Excited Systems
Online dynamic mode decomposition with control (DMDc) commonly relies on least-squares-based online updates whose numerical behavior can become fragile when the data stream is not persistently excited. In such cases, the regressor covariance may become rank-deficient or poorly conditioned, resulting in numerically unstable estimates. This paper proposes a regularized online DMDc (RO-DMDc) algorithm based on an exponentially weighted least-squares formulation with ridge regularization around a nominal model. The resulting identification law admits a closed-form recursion with a forgetting factor and fixed memory requirements, enabling real-time updates without storing past snapshots. A Lyapunov-based analysis is developed to establish deterministic boundedness of the estimation error under arbitrary excitation levels, including rank-deficient regimes. Simulations on a lane-keeping bicycle model with intermittent excitation and abrupt cornering-stiffness changes demonstrate that RO-DMDc mitigates drift and large spikes observed in a conventional online DMDc during weakly excited operation. Thus, RO-DMDc enables reliable real-time identification for adaptive control and monitoring under limited excitation.