System Degeneracy and the Output Feedback Problem: Parametrisation of the Family of Degenerate Compensators

This paper develops in depth the concept of degeneracy associated with the standard output feedback configuration of a proper multivariable system. The motivation stems from the very important property that degenerate feedback gains may be used for the linearisation of the pole assignment map and hence for frequency assignment. In general, the objective is the characterisation and parametrisation of all feedback gains that may allow the asymptotic linearisation of the pole placement map, as well as the families of systems for which such parametrisations are non trivial. The paper reviews the Global Asymptotic Linearisation method associated with the core versions of determinantal pole assignment problems and defines the conditions which characterises degenerate solutions of different types. Using the theory of ordered minimal bases, we provide a parametrisation of special families of degenerate compensators according to their degree. Finally, the special properties of degenerate solutions that allow frequency assignment are considered.