Cascade analysis and synthesis of transfer functions of infinite dimensional linear systems

Problems of cascade connections (synthesis) and decomposition (analysis) are analyzed for two classes of linear systems with infinite dimensional state spaces, namely, 1) admissible systems in the sense of Bart, Gohberg and Kaashoek and 2) regular systems as recently introduced by Weiss. For the class of BGK-admissible systems, it is shown that the product of two admissible systems is again admissible and that a Wiener-Hopf factorization problem can be solved just as in the finite-dimensional case. For the class of regular systems, it is shown that the cascade connection of a rational stable and antistable system has an additive stable-antistable decomposition; this involves giving a distribution interpretation to the solution of a linear Sylvester equation involving unbounded operator coefficients. As an application, some preliminary work is presented toward obtaining a state space solution of the sensitivity minimization problem for a pure delay plant.