Parametrization of Stabilizing Controllers with Applications

This contribution addresses, in a tutorial and retrospective manner, the parametrization of all controllers that stabilize a given plant with rational transfer function. An account of the classical results that paved the way for the parametrization is given. The parametrization result is then derived for several definitions of stability. The parameter, which is a qualified rational function, is shown to appear in the feedback system transfer functions in an affine manner. A two-step procedure for control system synthesis is then formulated, namely to determine all stabilizing controllers first, then meet additional performance specifications by selecting the parameter. Various applications of this procedure are given and illustrated by numerous examples. Advantages as well as limitations of this approach are discussed.