Linear Feedback—An Algebraic Approach

The algebraic theory of linear input–output maps is reexamined with the objective of accomodating the concept of (state) feedback in this theory. The concepts of extended and restricted linear i/o maps (and linear i/s maps) are introduced and investigated. It is shown how “fraction representations” of transfer matrices arise naturally in this new theoretical framework.Conditions are given for when the change caused to a linear input-output map by an (open loop) “cascade compensator” can also be accomplished by utilization of (closed loop) state feedback. In particular, it is shown that the change caused to a linear input-output map by cascading (composing) it with an input space isomorphism, can also be effected by feedback, provided the input space isomorphism in “bicausal”, i.e. it does not change the causal structure of the input-output map. Further detailed characterizations of feedback are also given especially in connection with the newly introduced concepts of degree chain and degree list.