On Sufficient Richness for Linear Time-Invariant Systems
Abstract
Persistent excitation (PE) is a necessary and sufficient condition for uniform exponential parameter convergence in several adaptive, identification, and learning schemes. In this article, we consider, in the context of multi-input linear time-invariant (LTI) systems, the problem of guaranteeing PE of commonly-used regressors by applying a sufficiently rich (SR) input signal. Exploiting the analogies between time shifts and time derivatives, we state simple necessary and sufficient PE conditions for the discrete- and continuous-time frameworks. Moreover, we characterize the shape of the set of SR input signals for both single-input and multi-input systems. Finally, we show with a numerical example that the derived conditions are tight and cannot be improved without including additional knowledge of the considered LTI system.
Cite
@article{arxiv.2502.04062,
title = {On Sufficient Richness for Linear Time-Invariant Systems},
author = {Marco Borghesi and Simone Baroncini and Guido Carnevale and Alessandro Bosso and Giuseppe Notarstefano},
journal= {arXiv preprint arXiv:2502.04062},
year = {2025}
}