On Robust Stability of Polynomials with Polynomial Parameter Dependency

Abstract

We consider uncertain polynomials whose coefficients depend polynomially on the elements of the parameter vector. The size of perturbation is characterized by the weighted norm of the perturbed parameter vector. The maximal 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 smallest solution. The number of systems depend crucially on the dimension of the parameter vector, whereas the complexity of systems increases mainly with the kind of polynomial dependency and the degree of the polynomial. This method also yields the critical parameter combination and the corresponding critical frequency. For a small number of parameters, this transfomed problem can be solved by symbolic computations. For a large number of parameters, numerical methods must be used.