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

TL;DR

The paper introduces a DW-shell–based framework to unify graphical stability analysis for MIMO LTI feedback, linking 2-D representations such as SRG and numerical range to the 3-D DW geometry. It defines the rotated θ-SRG as a mixed gain-phase representation that yields the least conservative stability criterion among SRG-based tests for bi-component loops, and derives a frequencywise stability condition that aligns with the generalized Nyquist criterion. An inverse-DW shell perspective and a comprehensive set of separation theorems connect existing conditions (small/large gain, sectorial/segmental phase, and SRG-based tests) into a single cohesive picture, with explicit proof sketches and logical implications. The authors also provide a tractable SDP-based tomography algorithm to plot θ-SRGs, along with a texture-based generalization of convexity to handle non-convex SRGs, enabling robust numerical visualization. Potential extensions include nonlinear and semistable systems, supported by an SDP framework for boundary computation and visualization of DW-derived graphical sets. This fusion of geometry, algebra, and optimization equips practitioners with unified tools for assessing and visualizing the stability of complex MIMO interconnections.

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.

Figures (12)

• Figure 1: (a): A slice of the DW shell of a scaled matrix (solid red). (b): Pointwise behavior of $f_{\mathrm{inv}}^{2}$.
• Figure 2: A DW perspective on various graphical representations/ measures, separation conditions and their relationships: gray boxes depict the graphical representation of a standalone matrix. White labeled boxes (and the corresponding nodes in the graph of implications) represent separation conditions derived from the corresponding graphical representations. Exisiting conditions are labeled in black, and newly derived ones are in green. The least conservative conditions are highlighted with red circles.
• Figure 3: Optical configurations for seeing the gain interval, numerical range, SRG, SSG as shadows of the DW shell.
• Figure 4: (a)-(c): Observe squared gain interval (side view), numerical range (top view), and scaled relative graph (side and top views combined) from the DW shell. (d): SRG and $\theta$-SRG of a real matrix $A$ (example drawn from patesScaledRelativeGraph2021).
• Figure 5: Joint visualization of the DW shell, normalized numerical range (singular angle), and SRG (SRG phases) for one example matrix.

Theorems & Definitions (41)

• Definition 1: Davis-Wielandt Shell
• Remark II.1
• Lemma 1: Basic Properties of DW Shells
• Remark II.2
• Lemma 2: Nonsingularity of Unitary Orbit
• Proposition 1
• Proposition 2
• proof
• Proposition 3
• Theorem IV.1: DW Separation Condition