The feedback invariant measures of distance to uncontrollability and unobservability

The selection of systems of inputs and outputs forms part of the early system design that is important since it preconditions the potential for control design. Existing methodologies for input, output structure selection rely on criteria expressing distance to uncontrollability, unobservability. Although controllability is invariant under state feedback, its corresponding degrees expressing distance to uncontrollability is not. The paper introduces new criteria for distance to uncontrollability (dually for unobservability) which is invariant under feedback transformations. The approach uses the restricted matrix pencils developed for the characterisation of invariant spaces of the geometric theory and then deploys exterior algebra to define the invariant input and output decoupling polynomials. This reduces the overall problem of distance to uncontrollability (unobservability) to two optimisation problems: the distance from the Grassmann variety and distance of a set of polynomials from non-coprimeness. Results on the distance of Sylvester Resultants from singularity provide the new measures.