Convergence Properties of Adaptive Systems and the Definition of Exponential Stability
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 statespace 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 openloop reference model adaptive systems, but also in a new class of closedloop reference model adaptive systems.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.03222
 Bibcode:
 2015arXiv151103222J
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Systems and Control
 EPrint:
 22 pages, 5 figures