State-space formulas for the classes of Ω-stabilizing controllers and closed-loop transfer matrices of a singular system

For a completely general singular system (whose transfer matrix may be improper or even polynomial) and an arbitrary set Ω in the closed complex plane we give a state-space parametrization of the class of controllers that achieve closed-loop transfer matrices having all poles in Ω, and of the class of the respective closed-loop transfer matrices. The main result is expressed in terms of a special type of state-space realization - called centered - that exhibits the same attractive features and allows for formulas that bear essentially the same elegant simplicity of the standard (proper) case. In particular, the formulas can be specialized to get the parametrization of all controllers that achieve proper, polynomial, or proper and stable closed-loop transfer matrices for a singular system.