On a Small Phase Theorem with Necessity over Phase Bounded Nonlinear Uncertainties

In this paper, we consider the feedback interconnection of a linear time- invariant (LTI) system and phase bounded uncertainties that are possibly nonlinear and/or time-varying. We devise a small phase condition and show that it is necessary and sufficient for robust stability over such phase bounded uncertainties, in a similar spirit to the well-known small gain condition being necessary and sufficient for robust stability over norm bounded uncertainties. Moreover, we show that the small phase condition can be verified using a state space characterization via linear matrix inequalities (LMIs), which extends the sectored real lemma in previous work. Copyright (c) 2023 The Authors. This is an open access article under the CC BY-NC- ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)