Exact calculation of the multiloop stability margin

Abstract

A mapping theorem by L.A. Zadeh and C.A. Desoer (1963) serves as the basis for an algorithm that computes the stability margin k/sub m/ of diagonally perturbed multivariable feedback systems without conservatism. The stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs. The essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images. The technique is illustrated in an example with a plant model having three uncertainties. In general, the method is applicable to either SISO or MIMO LTI systems whose uncertainties can be modeled with noninteracting coefficients in the open-loop transfer function representations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>