When Small Gain Meets Small Phase
Abstract
In this paper, we investigate the feedback stability of multiple-input multiple-output linear time-invariant systems with combined gain and phase information. To begin with, we explore the stability condition for a class of so-called easily controllable systems, which have small phase at low frequency ranges and low gain at high frequency. Next, we extend the stability condition via frequency-wise gain and phase combination, based on which a mixed small gain and phase condition with necessity, called a small vase theorem, is then obtained. Furthermore, the fusion of gain and phase information is investigated by a geometric approach based on the Davis-Wielandt shell. Finally, for the purpose of efficient computation and controller synthesis, we present a bounded & sectored real lemma, which gives state-space characterization of combined gain and phase properties based on a triple of linear matrix inequalities.
Cite
@article{arxiv.2201.06041,
title = {When Small Gain Meets Small Phase},
author = {Di Zhao and Wei Chen and Li Qiu},
journal= {arXiv preprint arXiv:2201.06041},
year = {2022}
}