A Generalization of the Passivity Theorem and the Small Gain Theorem Based on $\rho $-Stability, with Application to a Parameter Adaptation Algorithm for Recursive Identification
Optimization and Control
2019-02-11 v1
Abstract
The usual passivity theorem considers a closed-loop, the direct chain of which consists of a strictly passive stable operator , and the feedback chain of which consists of a passive operator . Then the closed-loop is stable. Let . We show here that the closed-loop is still stable when the direct chain consists of a strictly -passive -stable operator (a weaker condition than above) and the feedback chain consists of a -passive operator (a stronger condition than above). Variations on the theme of the small gain theorem (incremental or not) can be made similarly. This approach explains the results obtained in a paper on identification which was recently published.
Cite
@article{arxiv.1902.03015,
title = {A Generalization of the Passivity Theorem and the Small Gain Theorem Based on $\rho $-Stability, with Application to a Parameter Adaptation Algorithm for Recursive Identification},
author = {Henri Bourlès},
journal= {arXiv preprint arXiv:1902.03015},
year = {2019}
}