A new concept of strong controllability via the Schur complement in adaptive tracking

TL;DR

This paper introduces a novel concept of strong controllability using the Schur complement, extending results for multidimensional ARX models and analyzing convergence, CLT, and LIL for least squares algorithms.

Contribution

It develops a new strong controllability framework via the Schur complement, enhancing analysis of adaptive tracking in multidimensional ARX models.

Findings

Almost sure convergence of least squares algorithms

Central limit theorem for weighted least squares

Law of iterated logarithm for stochastic algorithms

Abstract

We propose a new concept of strong controllability associated with the Schur complement of a suitable limiting matrix. This concept allows us to extend the previous results associated with multidimensional ARX models. On the one hand, we carry out a sharp analysis of the almost sure convergence for both least squares and weighted least squares algorithms. On the other hand, we also provide a central limit theorem and a law of iterated logarithm for these two stochastic algorithms. Our asymptotic results are illustrated by numerical simulations.