Phase Robustness Analysis for Structured Perturbations in MIMO LTI Systems

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 have 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 an 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 to provide less conservative robustness metrics than can be obtained via conventional$\mu$-analysis methods.