Graph theoretic approach to characterisation of quotient fixed modes in LTI systems

In this paper we represent a decentralised LTI control system with decentralised fixed modes (DFMs) by a weighted directed graph and derive sufficient conditions for it to be stabilisable. The sufficient conditions relate to whether the DFMs are decentralised quotient fixed modes (QFMs); decentralised QFMs cannot be shifted by any type of decentralised controller. We use a graph theoretic approach, using existing concepts, and introducing some new ones, to identify the decentralised QFMs for the class of LTI systems that have distinct, repeated, cyclic, or non-cyclic DFMs. It is shown in this paper that a DFM is a QFM, and thus cannot be moved, when some identifiable graphical properties exist. The main results are presented in terms of a Theorem with a constructive proof. Numerical examples are given to illustrate all aspects of the paper.