Parameterization of the class of deadbeat controllers

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> are dimensions of the state vector and control vector, respectively, and the μ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> 's are the controllability indexes, then almost the whole class of deadbeat controllers is explicitly parameterizable through <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mn-\mu_{1} - 3\mu_{2} - ... -(2m - 1)\mu_{m}</tex> parameters.