The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (µ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an "approximate GCD" of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the "approximate GCD" of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations.

Implicit systems provide a general framework in which many important properties of dynamic systems can be studied. Implicit systems are especially relevant to behavioural systems theory, the analysis and synthesis of complex interconnected systems, systems identification and robust control. By incorporating algebraic constraints, implicit models provide additional versatility relative to the standard input–output framework. Problems of robust stability in implicit systems lead in a natural way to non‐standard structured singular value (μ) formulations. In this note, it is shown that for a class of uncertainty structures involving repeated scalar parameters, these problems reduce to a standard μ problem which is well studied and for the solution of which several numerical algorithms are available. Our results are based on a matrix dilation technique and the redefinition of the uncertainty structure of the transformed problem. The main results of the paper are illustrated with a numerical example.

The selection of systems of inputs and outputs (input and output structure) forms part of the early system design and this 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 which is invariant under feedback transformations. The approach uses the restricted input-state, state-output matrix pencils developed for the matrix pencil 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 to two optimisation problems: the distance from the Grassmann variety and distance of a set of polynomials from non-coprimeness that relates to the notion of approximate Greatest Common Divisor. Results on the distance of Sylvester Resultants from singularity provide the new measures. By duality, the results also apply to the problem of invariant distance to unobservability related to the selection of the output structure.

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