An elementary proof of the general Q-parametrization of all stabilizing controllers

It is becoming to be well-known that an internally stabilizable transfer matrix does not necessarily admit doubly coprime factorizations. The equivalence between these two concepts is stillopen for important classes of plants. Hence, we may wonder whether or not it ispossible to parametrize all stabilizing controllers of an internallystabilizable plant which does not necessarily admit doubly coprime factorizations.The aim of this paper is to give an elementary proof of the existence ofsuch a general parametrization. This parametrization is obtained bysolving the general conditions for internal stabilizabilitydeveloped within the fractional representation approach to synthesisproblems. We show how such ideas can be traced back to thepioneering work of G. Zames and B. Francis on $H_{\infty}$-control. Finally,if the transfer matrix admits a doubly coprime factorization, then we showthat the Q-parametrization becomes the Youla-Kucera parametrization.