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 still open for important classes of plants. Hence, we may wonder whether or not it is possible to parametrize all stabilizing controllers of an internally stabilizable plant which does not necessarily admit doubly coprime factorizations. The aim of this paper is to give an elementary proof of the existence of such a general parametrization. This parametrization is obtained by solving the general conditions for internal stabilizability developed within the fractional representation approach to synthesis problems. We show how such ideas can be traced back to the pioneering work of G. Zames and B. Francis on H∞-control. Finally, if the transfer matrix admits a doubly coprime factorization, then we show that the Q-parametrization becomes the Youla-Kučera parametrization. Copyright c ©2005 IFAC