Mixed Gain/Phase Robustness Criterion for Structured Perturbations With an Application to Power System Stability

A novel conception of phase for linear time-invariant multivariable systems was recently introduced. It enables robustness of such systems to be determined in terms of a phase-bounded set of perturbations via a so-called small phase theorem, in analogy to the well-known small gain theorem. However, it requires the system’s frequency response to satisfy the relatively strong condition known as “sectoriality”, which not all practical systems have. This letter aims to show that if the perturbation is assumed to have a block diagonal structure, a matrix-valued multiplier function can be calculated that can enable phase-based robustness margins to be defined in some cases when the original system is not sectorial. A real-world power systems example is presented to show how the small phase criterion using a multiplier can significantly reduce the conservatism of the small gain theorem, providing computationally straightforward methods to inform further nonlinear stability analysis of power systems.