Recursive Least Squares with Fading Regularization for Finite-Time Convergence without Persistent Excitation

TL;DR

This paper introduces a novel recursive least squares algorithm with a fading regularization that guarantees finite-time convergence of parameter estimates without requiring persistent excitation, enhancing robustness and adaptability.

Contribution

It extends classical RLS to include a time-varying fading regularization, enabling finite-time convergence without persistent excitation, and proposes a computationally efficient rank-1 fading regularization method.

Findings

Fading regularization converges to zero, ensuring error convergence without persistent excitation.

Finite-time convergence of parameter estimates achieved with fading regularization.

Numerical examples validate the theoretical guarantees and robustness of the proposed method.

Abstract

This paper extends recursive least squares (RLS) to include time-varying regularization. This extension provides flexibility for updating the least squares regularization term in real time. Existing results with constant regularization imply that the parameter-estimation error dynamics of RLS are globally attractive to zero if and only the regressor is weakly persistently exciting. This work shows that, by extending classical RLS to include a time-varying (fading) regularization term that converges to zero, the parameter-estimation error dynamics are globally attractive to zero without weakly persistent excitation. Moreover, if the fading regularization term converges to zero in finite time, then the parameter estimation error also converges to zero in finite time. Finally, we propose rank-1 fading regularization (R1FR) RLS, a time-varying regularization algorithm with fading regularization that converges to zero, and which runs in the same computational complexity as classical RLS. Numerical examples are presented to validate theoretical guarantees and to show how R1FR-RLS can protect against over-regularization.