Algebraic aspects of linear control system stability

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Abstract
After formulating the problem of linear control system stability in very general terms, the problem is resolved by 1) establishing a necessary and sufficient condition for stability, 2) exploring in some detail the nature of this condition, and 3) developing a parametric characterization of all controllers which stabilize a given plant. The problem formulation 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. Furthermore, no other configuration has this property. Several benefits and extensions are accrued as a result of using this configuration. First, a completely general, yet physically meaningful problem formulation results. Consequently, state-space and operator theoretic approaches to linear control system stability are thoroughly reconciled. This reconciliation embraces continuous-time as well as discrete-time systems, lumped as well as distributed systems, and 1-D as well as n -D systems. Second, the controller parametrization affords four degrees-of-freedom, two more than any existing parametrization. The additional degrees-of-freedom make explicit the design opportunity associated with controller implementation, and thus determine several important control system characteristics. This result is expected to provide a theoretical basis for the development of comprehensive control system design methodologies, wherein design considerations related to controller implementation are addressed.

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