Convergence Properties of Adaptive Systems and the Definition of Exponential Stability

TL;DR

This paper clarifies the conditions under which adaptive systems achieve exponential stability, emphasizing the distinction between persistent excitation of different signals and introducing the concept of weak persistent excitation.

Contribution

It revisits the definition of persistent excitation, clarifies its implications for stability, and introduces the concept of weak persistent excitation in adaptive systems.

Findings

Persistent excitation of the regressor leads to exponential stability in adaptive control.

Weak persistent excitation ensures uniform asymptotic stability, but not exponential stability.

Existence of an infinite region with bounded state rate in both open-loop and closed-loop adaptive systems.

Abstract

The convergence properties of adaptive systems in terms of excitation conditions on the regressor vector are well known. With persistent excitation of the regressor vector in model reference adaptive control the state error and the adaptation error are globally exponentially stable, or equivalently, exponentially stable in the large. When the excitation condition however is imposed on the reference input or the reference model state it is often incorrectly concluded that the persistent excitation in those signals also implies exponential stability in the large. The definition of persistent excitation is revisited so as to address some possible confusion in the adaptive control literature. It is then shown that persistent excitation of the reference model only implies local persistent excitation (weak persistent excitation). Weak persistent excitation of the regressor is still sufficient for uniform asymptotic stability in the large, but not exponential stability in the large. We show that there exists an infinite region in the state-space of adaptive systems where the state rate is bounded. This infinite region with finite rate of convergence is shown to exist not only in classic open-loop reference model adaptive systems, but also in a new class of closed-loop reference model adaptive systems.