Stability multipliers and μ upper bounds: connections and implications for numerical verification of frequency domain conditions

The paper shows the equivalence between the following two approaches for obtaining sufficient conditions for the robust stability of systems with structured uncertainties: 1) apply the classical absolute stability theory with multipliers, and 2) use modern μ theory, specifically, the μ upper bound obtained by Fan et al. (1991). In particular, the relationship between the stability multipliers used in absolute stability theory and the scaling matrices used in the cited reference is explicitly characterized. The development hinges on the derivation of certain properties of a parametrized family of complex linear matrix inequalities, a result of independent interest. The derivation also suggests a general computational framework for checking the feasibility of a broad class of frequency-dependent conditions, based on which bisection schemes can be devised to reliably compute several quantities of interest for robust control