Generalized system poles and zeros: the generic and global connection

For regular generalized systems, it is known that the total number of poles, finite and infinite, can exceed the total number of zeros, finite and infinite. From the outset of studies on system zeros, it has been known that this defect is related to the existence of nonzero kernels and cokernels of the system transmission function. The resolution of this relationship on the system level can be made considerably more complicated by the twin possibilities of noncausality and non-minimality. Building upon the concepts of generic system zero modules and their associated Wedderburn-Forney spaces together with module theoretic descriptions of finite and infinite system zeros and poles, this paper establishes a fundamental and intrinsic connection between all these quantities in a global sense. From an intuitive point of view, this result constitutes a extension to the generalized system level of previously known concepts for single transfer functions and transfer function matrices. From an algebraic point of view, the short exact sequence that carries the principal conclusions is yet another indication of the utility of tools from commutative algebra in settling intricate issues of system theory