Conditions for Convergence of Dynamic Regressor Extension and Mixing Parameter Estimator Using LTI Filters

TL;DR

This paper analyzes the convergence conditions of the DREM parameter estimator when using LTI filters, relating them to classical PE and IE conditions, and proves finite-time convergence under IE.

Contribution

It establishes the relationship between PE and IE conditions of the original regressor and the DREM scalar regressor, extending convergence analysis to LTI filtered regressors.

Findings

PE of $ extbf{φ}(t)$ implies PE of $ riangle(t)$, ensuring exponential convergence.

IE of $ extbf{φ}(t)$ implies IE of $ riangle(t)$, guaranteeing convergence.

Finite-time convergence of DREM under IE conditions.

Abstract

In this note we study the conditions for convergence of recently introduced dynamic regressor extension and mixing (DREM) parameter estimator when the extended regressor is generated using LTI filters. In particular, we are interested in relating these conditions with the ones required for convergence of the classical gradient (or least squares), namely the well-known persistent excitation (PE) requirement on the original regressor vector, $\phi(t) \in \mathbb{R}^q$, with $q \in \mathbb{N}$ the number of unknown parameters. Moreover, we study the case when only interval excitation (IE) is available, under which DREM, concurrent and composite learning schemes ensure global convergence, being the convergence for DREM in finite time. Regarding PE we prove that, under some mild technical assumptions, if $\phi(t)$ is PE then the scalar regressor of DREM, $\Delta(t) \in \mathbb{R}$, is also PE, ensuring exponential convergence. Concerning IE we prove that if $\phi(t)$ is IE then $\Delta(t)$ is also IE. All these results are established in the almost sure sense, namely proving that the set of filter parameters for which the claims do not hold is of zero measure. The main technical tool used in our proof is inspired by a study of Luenberger observers for nonautonomous nonlinear systems recently reported in the literature.