The Phantom of Davis-Wielandt Shell: A Unified Framework for Graphical Stability Analysis of MIMO LTI Systems

Abstract

This paper presents a unified framework based on Davis-Wielandt (DW) shell for graphical stability analysis of multi-input and multi-output linear time-invariant feedback systems. Connections between DW shells and various graphical representations, as well as gain and phase measures, are established through an intuitive geometric perspective. Within this framework, we map the relationships and relative conservatism among various separation conditions. A rotated scaled relative graph ($θ$-SRG) concept is proposed as a mixed gain-phase representation, from which a closed-loop stability criterion is derived and shown to be the least conservative among the existing 2-D graphical conditions for bi-component feedback loops. We also propose a reliable and generalizable algorithm for visualizing the $θ$-SRGs and include a system example to demonstrate the reduced conservatism of the proposed condition.