Absolute Stability Theory, Theory, and State-Space Verification of Frequency-Domain Conditions: Connections and Implications for Computation

Abstract

The main contribution of the paper is to show the equivalence between the following two approaches for obtaining sufficient conditions for the robust stability of systems with structured uncertainties: (i) apply the classical absolute stability theory with multipliers; (ii) use the modern theory, specifically, the upper bound obtained by Fan, Tits and Doyle [IEEE TAC, Vol. 36, 25-38]. 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 parameterized family of complex LMIs (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, and in particular, yields a sequence of computable ﲭixed- -norm upper bounds , defined with guaranteed convergence from above to the supremum over frequency of the aforementioned upper bound.