Phase Robustness Analysis for Structured Perturbations in MIMO LTI Systems

TL;DR

This work extends phase-based robustness concepts to structured perturbations in MIMO LTI systems. By defining a structured phase index $\psi_\chi(A)$ and computable upper/lower bounds via LMIs and nonconvex optimization, the authors derive a frequency-dependent phase gain framework that, when combined with the structured singular value $\mu_\chi$, yields less conservative stability guarantees for perturbed interconnections. The main results provide explicit IQC-based and FDIs-based criteria ensuring stability under phase-bounded and gain-bounded perturbations, with a concrete rotating-body example demonstrating improved conservatism over gain-only methods. The approach integrates numerical range-based phase notions, $D$-scaling multipliers, and IQCs to offer a practical, theoretically grounded method for robust stability analysis of structured uncertainties in MIMO LTI systems.

Abstract

The stability of interconnected linear time-invariant systems using singular values and the small gain theorem has been studied for many decades. The methods of mu-analysis and synthesis has been extensively developed to provide robustness guarantees for a plant subject to structured perturbations, with components in the structured perturbation satisfying a bound on their largest singular value. Recent results on phase-based stability measures have led to a counterpart of the small gain theorem, known as the small phase theorem. To date these phase-based methods have only been used to provide stability robustness measures for unstructured perturbations. In this paper, we define a phase robustness metric for multivariable linear time-invariant systems in the presence of a structured perturbation. We demonstrate its relationship to a certain class of multiplier functions for integral quadratic constraints, and show that a upper bound can be calculated via a linear matrix inequality problem. When combined with robustness measures from the small gain theorem, the new methods are able provide less conservative robustness metrics than can be obtained via conventional mu-analysis methods.

Figures (9)

• Figure 1: Block diagram of feedback interconnection $(G,\Delta)$.
• Figure 2: $A_1$ is sectorial, $A_2$ is quasi-sectorial but not sectorial, $A_3$ is semi-sectorial but not quasi-sectorial, and $A_4$ is non-semi-sectorial.
• Figure 3: Relationship between $\underline{\phi}(A)$, $\overline{\phi}(A)$ and $W(A)$.
• Figure 4: A sectorial matrix $A_1$ with phase index $\alpha$.
• Figure 5: Block diagram of perturbed rotating body system.

Theorems & Definitions (32)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5: Matrix phaseChen2024
• Lemma 1
• Definition 6: Structured singular value zhou1996
• Definition 7
• Lemma 2