H∞ control in a behavioral context: the full information case

The authors formulate the H_∞-control problem in a behavioral setting. Given a mathematical model, say a set of higher order differential equations together with some static equations, the vector of manifest variables is partitioned into yet to be controlled variables, unknown exogenous variables, and interconnection variables. The interconnection variables are available for interconnection, in the sense that they can be made to obey certain differential or static equations, to be specified by the designer. Such a system of differential equations and static equations is called a controller. The design problem that we consider is to find controllers such that the size of the to be controlled variables is less than a given tolerance, for all disturbances in the unit ball, and such that the interconnection is a stable system. We find necessary and sufficient conditions for the existence of suitable controllers, under the hypothesis that we have a full information problem. These conditions involve indefinite factorizations of polynomial matrices and a test on a given Pick matrix