On Robust Stability of Polynomials with Polynomial Parameter Dependency: Two/Three Parameter Cases

Abstract

We consider real polynomials whose coefficients depend polynomially on the elements of an uncertain parameter vector. The size of perturbation is characterized by the weighted norm of the parameter vector. The smallest destabilizing perturbation defines the stability radius of the set of uncertain polynomials. It is shown that determining this radius is equivalent to solving a finite set of systems of algebraic equations and picking out the real solution with the smallest norm. The number of systems of equations increases mainly with the kind of polynomial dependency and the degree of the polynomial. This method also yields the smallest destabilizing parameter combination and the corresponding critical frequency. For two or three parameters this transformed problem can be solved using symbolic and numeric computations.