Algebraic aspects of linear control system stability

In this paper the problem of linear control system stability is formulated in very general terms and then resolved by (i) establishing a necessary and sufficient condition for stability, (ii) exploring in some detail the nature of this condition, and (iii) developing a parametric characterization of all controllers which stabilize a given plant. The problem formulation developed herein is based on a ring theoretic setting for linear systems, and is distinguished by the use of a novel control system configuration. Any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems. Moreover, no other configuration has this property. Benefits and extensions accrued as a result of this property are thoroughly discussed, as are some promising directions for future research.