Sufficient conditions for stabilisability of decentralised LTI systems: Weighted directed graph approach

In this paper we represent a decentralised control LTI system by a weighted directed graph. Then we derive sufficient conditions for such systems with decentralised fixed modes (DFMs) to be stabilisable. The sufficient conditions relate to whether the DFMs are quotient fixed modes (QFMs) of the system and thus can or cannot be shifted by any type of controller. It is shown that a DFM is a QFM when some graphical properties, expressed in terms of newly introduced concept of complementary cut set, exist. The sufficient conditions are stated in terms of a theorem with a constructive proof. Numerical examples are given to illustrate all aspects of the paper.