On the usefulness of persistent excitation in ARX adaptive tracking

TL;DR

This paper demonstrates that persistent excitation in ARX adaptive tracking enables parameter estimation without strong controllability assumptions, providing convergence results and asymptotic properties.

Contribution

It introduces a persistently excited adaptive tracking control that relaxes controllability requirements in multidimensional ARX models.

Findings

Almost sure convergence of estimators

Central limit theorem established

Law of iterated logarithm proved

Abstract

The usefulness of persistent excitation is well-known in the control community. Thanks to a persistently excited adaptive tracking control, we show that it is possible to avoid the strong controllability assumption recently proposed in the multidimensional ARX framework. We establish the almost sure convergence for both least squares and weighted least squares estimators of the unknown parameters. A central limit theorem and a law of iterated logarithm are also provided. All this asymptotical analysis is related to the Schur complement of a suitable limiting matrix.