A duality theory for robust stabilization and model reduction

Fuhrmann (1991, 1993) developed a duality theory in the context of Hanel norm approximation and Nehari complementation problems. The class of functions involved were the scalar, antistable transfer functions. This work was extended, using normalized coprime factorizations, by Fuhrmann and Ober (1993a) to the class of all minimal transfer functions. In this paper we extend the scope of the duality theory significantly. The paper presents a unified approach to problems of Hankel norm approximation, model reduction, and robust control of rational multivariable transfer functions. The unification is achieved by considering two classes of transfer functions and corresponding normalized coprime factorization. Using the Youla-Kucera parametrization of all stabilizing controllers, we single out a unique controller by imposing a McMillan degree minimization restriction on the doubly coprime factorizations. With this controller we construct an associated stable transfer function which we call the characteristic function. Many problems on the original system can be reduced to the study of the characteristic function.