Frequency response of interval plant-controller families

In this paper a general frequency domain result for families of interval plant with a fixed linear controller is given. It is shown that the locus of the polar diagrams of frequency responses of the transfer functions of an interval plant-controller family is bounded by the polar plots relative to the 32 segments of transfer functions of the interval plant family. Easy proofs of several important results, such as the generalization of the Theorem of Kharitonov for feedback systems with interval plants or the robust version of the small gain theorem for the same class of systems, are constructed by using the general result. More importantly, an immediate consequence of the main theorem is that extremal phase/gain margins or sensitivity/complementary sensitivity peaks for systems of the family can be deduced from those of the 32 segments of the interval plant family.