Nearest common root of a set of polynomials: A structured singular value approach

Abstract

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.

Structured singular value of implicit systems

“…Part II: Constant Matrix Problems and Application to Robust H 2 Analysis. In Limantseva et al, 16 we analyse an approximate GCD problem for multiple polynomials with real coefficients and perturbations via a structured Sylvester resultant approach which has a very similar structure to the implicit systems problem considered here.…”

“…The structured singular value ( μ ) is a fundamental analysis and synthesis tool of robust control theory of input–output models with several applications in model validation and the characterisation of robust stability and performance of dynamic systems; see previous studies 4,7,9‐13 and references therein. The analysis of structured perturbations introduced in Doyle 14 has been developed and extended to various problems including general structured singular value problems, 9 mixed μ ‐value problems, 10,11 lower and upper bound estimation, 11 minimal magnitude norm perturbations in polynomials, 15,16 stability radii analysis, 12,17 pseudo spectrum 4 and many others.…”

“…Section 2 contains some background results and highlights the importance of the implicit framework. Section 3 adapts a technique established in a previous paper by the authors 16 and introduces an alternative approach for solving the generalised μ ‐value problem arising in the analysis and synthesis of uncertain implicit systems. It is shown that in the case of real repeated uncertain parameters, the problem can be expressed in the standard μ setting using a matrix dilation technique and by redefining the uncertainty structure.…”

The feedback invariant measures of distance to uncontrollability and unobservability

“…An alternative approach for the calculation of the "approximate" GCD, recently evaluated in (Halikias, Galanis, Karcanias, & Milonidis, 2012), (Limantseva, Halikias, & Karcanias, 2019), defines the distance problem as the standard structured singular value problem, considering the minimal norm perturbation in the coefficients of the nominal polynomials to the set of the polynomials with a common root. Such an approach uses powerfully notions of the µ-value in order to tackle highly structured nature of the Sylvester resultant.…”

“…Since, the problem of "approximate" GCD is closely related to the coefficients of polynomials and the Sylvester matrix is a Toeplitz structure matrix, thus stronger criteria for the distance to singularity should be considered. Motivated by the recent results of the authors in (Halikias et al, 2012), (Limantseva et al, 2019) one can define the distance to non-coprimeness as an optimisation problem, where we seek to find the minimal magnitude norm perturbation in the coefficients. Definition 7.7.…”