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

Abstract

In this article, we study the conditions for convergence of the recently introduced dynamic regressor extension and mixing (DREM) parameter estimator when the extended regressor is generated using linear time-invariant 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">persistent excitation</i> (PE) requirement on the original regressor vector, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi (t) \in \mathbb {R}^q$</tex-math></inline-formula> , with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q \in \mathbb {N}$</tex-math></inline-formula> the number of unknown parameters. Moreover, we study the case when only <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interval excitation</i> (IE) is available, under which DREM, concurrent, and composite learning schemes ensure global convergence, being the convergence for DREM in a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite time</i> . Regarding PE, we prove, under some mild technical assumptions, that if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> is PE, then the scalar regressor of DREM, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta _N \in \mathbb {R}$</tex-math></inline-formula> , is also PE ensuring exponential convergence. Concerning IE, we prove that if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> is IE, then <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta _N$</tex-math></inline-formula> is also IE. All these results are established in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">almost sure</i> 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.