Graph-theoretic characterization of fixed modes in centralized and decentralized control

A closed-loop system is represented by a weighted directed graph which exhibits all couplings of the system in a very clear and simple manner. The coefficients of the characteristic polynomial of the closed-loop system may be obtained with the aid of cycle families in this graph. Based on this result, a criterion for the existence of fixed modes is derived. The multiplicities of fixed modes may also be determined graph-theoretically. The consequences of structural constraints on the output feedback pattern become evident. Decentralized feedback is discussed as a special case.