On Regular Regressors in Adaptive Control

TL;DR

This paper introduces a new concept of regular regressors in adaptive control, addressing issues with the traditional persistent excitation property and proposing a geometric framework to improve adaptive control strategies.

Contribution

It proposes a broad class of regular regressors with confined excitation, backed by a PE decomposition, and demonstrates new adaptive control problem formulations based on this framework.

Findings

Regular regressors have excitation confined to a subspace.

PE decomposition provides a foundational geometric characterization.

New adaptive control problems are formulated using the regularity concept.

Abstract

This paper addresses a shortcoming in adaptive control, that the property of a regressor being persistently exciting (PE) is not well-behaved. One can construct regressors that upend the commonsense notion that excitation should not be created out of nothing. To amend the situation, a notion of regularity of regressors is needed. We are naturally led to a broad class of regular regressors that enjoy the property that their excitation is always confined to a subspace, a foundational result called the PE decomposition. A geometric characterization of regressor excitation opens up new avenues for adaptive control, as we demonstrate by formulating a number of new adaptive control problems.