On a generalization of the Youla–Kučera parametrization. Part II: the lattice approach to MIMO systems

Abstract

Within the lattice approach to analysis and synthesis problems recently developed in Quadrat (Signal Syst, to appear), we obtain a general parametrization of all stabilizing controllers for internally stabilizable multi input multi output (MIMO) plants which do not necessarily admit doubly coprime factorizations. This parametrization is a linear fractional transformation of free parameters and the set of arbitrary parameters is characterized. This parametrization generalizes for MIMO plants the parametrization obtained in Quadrat (Syst Control Lett 50:135–148, 2003) for single input single output plants. It is named general Q-parametrization of all stabilizing controllers as we show that some ideas developed in this paper can be traced back to the pioneering work of Zames and Francis (IEEE Trans Automat control 28:585-601, 1983) on H ∞-control. Finally, if the plant admits a doubly coprime factorization, we then prove that the general Q-parametrization becomes the well-known Youla-Kucera parametrization of all stabilizing controllers (Desoer et al. IEEE Trans Automat control 25:399–412, 1980; Vidyasagar, Control system synthesis: a factorization approach MIT Press, Cambridge 1985).