Structure of the class of linear functional time systems under output feedback transformation

In this paper, we discuss from a qualitative and unified viewpoint the change of system behaviour produced by output feedback primarily to clarify mutual relationships between systems under feedback transformation. We first define the class of all linear functional time systems that are characterized by four system-theoretic properties. Next we define feedback as an operation on a class of systems and discuss the change of system behaviour produced by feedback as a system transformation either between two classes of system or within one class of system. Finally we clarify the structure of a class under feedback transformation in terms of system-theoretic concepts. We obtain three major results which provide a theoretical basis for systems science: (1) we specified the subclass of linear systems on which the feedback transformation can be well defined; (2) we characterize a relation between the system transformation under dynamic output feedback and that under static output feedback; (3) we show that the subclass consisting of all symmetric systems is closed under dynamic output feedback transformation.