Linear networks and systems depending polynomially on parameters: Stability for large values subject to tolerance errors

摘要

AbstractThis paper studies the problem of the stability of linear networks or systems depending polynomially on parameters when considering large values of the parameters. It has been taken into account that only the specification of nominal values and tolerances—and not of actual values—is physically meaningful.Algorithms to test the existence of arbitrarily large nominal values of the parameters which ensure stability either with zero tolerance, with non‐zero tolerances depending on the particular nominal values or with a non‐zero tolerance independent of the particular nominal values are proved to exist.Depending on the type of response of the above‐mentioned algorithms, analytic, polynomial and monomial functions—in a single suitable variable—that describe arbitrarily large nominal values which ensure stability in the case of zero tolerance, polynomially decreasing non‐zero tolerance and constant non‐zero tolerance respectively are proved to exist.When a monomial description with constant non‐zero tolerance is assured, algorithms which give the degrees and—in terms of rational numbers—the coefficients of the monomials and the constant non‐zero tolerance are proved to exist.Similar algorithms when only an analytic description with zero tolerance or a polynomial description with polynomially decreasing non‐zero tolera