Unstructured quotient fixed modes and decentralised stabilisability

In this paper we derive a sufficient condition for which unstructured decentralised fixed modes (UDFMs) of a LTI system cannot be stabilised by any form of decentralised output feedback controllers. The sufficient condition establishes the link between special UDFMs (SUDFMs) and quotient fixed modes (QFMs). The link between UDFMs and QFMs provides an unequivocal answer, which has up until now eluded researchers, as to why some UDFMs cannot be eliminated by sampled-data controllers. In addition, we use this link to highlight the role system parameter uncertainties can play in shifting stabilisable DFMs to UDMFs or QFMs and vice versa. Numerical examples are presented to illustrate all aspects of the paper.