A parametrization of all one parameter stabilizing controllers and a mixed sensitivity problem, for square systems

Explicit formulas of the Parametrization of All one parameter Stabilizing Controllers (PASC) for square systems, are presented. Multi Input Multi Output (MIMO), strictly proper, lumped and Linear Time Invariant (LTI) systems with stabilizable and detectable realizations are considered. It is assumed that the state dimension is even, the input dimension is half the state dimension, and the plant is strongly stabilizable and detectable. The separation principle is applied to design a dynamic output control in a controller-observer feedback configuration. The results for the observer are gotten by duality. For both controller and observer, right and left coprime factorizations of the transfer function in terms of the state space realization are proposed, right and left Diophantine equations are solved, and the controller and observer belong to the PASC. Conditions to get strong stability are given and the free parameters of the stabilizing controllers are fixed solving a MIxed Sensitivity Problem (MISP). The results are illustrated through a simulation example of a two-cart system.