On the Equivalent Relationship Between Generalized Performance, Robust Stability, and Quadratic Stability

This technical note addresses the equivalent relationship between notions of generalized performance, robust stability, and quadratic stability for the feedback connection , where is the transfer matrix of a nominal system and describes the set of uncertainty. By defining the three notions in a more general setting, the conventional equivalent relationship between robust stability and quadratic stability with respect to the norm-bound uncertainty (respectively, the positive real uncertainty) and the corresponding performance (respectively, the extended strict positive realness) has been proved only special case of the results derived in the technical note. A version of the Kalman-Yakubovich-Popov lemma, which plays a crucial role in establishing the equivalence between the generalized performance and the quadratic stability, is also presented.