Parameterization of the class of deadbeat controllers

This technical note deals with the problem of parameterizing the class of deadbeat controllers for a given system through the minimum of parameters. It is shown that if
n
and
m
are dimensions of the state vector and control vector, respectively, and the μ
i
's are the controllability indexes, then almost the whole class of deadbeat controllers is explicitly parameterizable through
mn-\mu_{1} - 3\mu_{2} - ... -(2m - 1)\mu_{m}
parameters.