On robust stability with nonlinear parameter dependence: some benchmark problems illustrating the dilation integral method

We consider the problem of robust stability analysis with uncertain parameters entering nonlinearly into the coefficients of the polynomial of interest. In this nonlinear setting, the few results which are available in the literature apply to special cases, or, due to computational complexity, are tractable for only a few uncertain parameters. The objective of this paper is to demonstrate the efficacy of the so-called "dilation integral" method of [Barmish, BR and Shcherbakov, PS (2003)] and [Shcherbakov, PS and Barmish, BR (2003)] in a robust stability context. This rather general method involves a "softening" of the robustness formulation to allow for an "acceptably small" volume of performance violation in parameter space.