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

The paper provides a new characterisation of constant and dynamic degenerate compensators for proper multivariable systems. The motivation stems from the very important property that degenerate feedback gains may be used for the linearisation of the pole assignment map and enable frequency assignment. The objective is the characterisation and parametrisation of all feedback gains that may allow the asymptotic linearisation of the pole placement map. Such a parametrisation introduces new degrees of freedom for the linearisation of the related frequency assignment map and plays an important role to the solvability of the output feedback pole assignment problem. 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.