On the controllability of discrete linear systems with output feedback

A discrete time linear system , with output feedback , call be regarded as a nonlinear system with "control" Gt. Weak sufficient conditions are given for the existence of a finite sequence of gains for which every initial state can be driven to the origin. For a one input, one output system, the question of what terminal states can be reached from a given initial state is resolved. It is shown that an important ingredient for these problems is the semigroup of integers generated by the set (for a single input, single output system of dimension ). It is also natural to use a pair of "canonical forms," in the guise of polynomials, to represent states. One is useful for input considerations and the other for output considerations. For output feedback problems one must further distinguish between two polynomials which are equivalent in the sense that they represent the same state. This is due to the fact that some polynomials are ill-conditioned in that they would have us use a nonzero input when the output vanishes.