The zero module and essential inverse systems

The theory of inverse dynamical systems has been and is continuing to be a keystone in the development of the theories of multivariable feedback control systems and of coding theory for reliable cornmunication. Advances in understanding of the role of plant inverses In control system design have brought about additional insights, for example, In the general areas of decoupled design and of realistic possibilities for closed-loop dynamical performance. Surprisingly enough, almost all the existing literature on inverse systems is cast in terms of matrices. Though the weD-known module theoretic approach to systems has been in place for a decade or more, this approach has not been fully exploited to bring out the foundations of a theory for inverse systems. This paper begins to lay such a foundation by developing a definition of zero module for a system. When inverse systems exist, their "pole modules" can be shown to contain the zero module in an appropriate algebraic sense. If the containment is tight, these inverses are called essential. Existence of and constructions for essential inverses are provided.