The Popov criterion for multiple-loop feedback systems

Although analytic versions of the Popov criterion are available for multiple-loop feedback systems, it is difficult to get a graphical representation which is as useful as the single-loop one. This paper presents a graphical representation which provides a sufficient condition for the Popov criterion to hold in the multiple-loop case. Two versions are given: the first is suited to design methods based on diagonal dominance and the second is suited to those based on characteristic loci. When the linear part of the system is normal, the second version is necessary as well as sufficient for the criterion to hold.