Decentralised stabilisability condition for expanded and contracted LTI systems and quotient fixed modes

In this paper, a necessary and sufficient condition for stabilisation of linear time invariant systems with overlapping parts is presented. These systems are first expanded to decouple the overlapping parts and then a necessary and sufficient condition, which relates to their algebraic properties, is derived. The algebraic properties are characterised in terms of minimality (full controllability and observability) and the concept of quotient fixed modes of the expanded systems. To ensure that stabilisable decentralised controller designed for the expanded systems can be contracted to stabilising overlapping decentralized controller for the original system, we introduce a design algorithm to that guarantees contractibility.