Computation of the minimum destabilizing volume for interval and affine families of polynomials

We study the computation of the minimum destabilizing volume for interval and polytopic families of polynomials. Roughly speaking, this is equivalent to determining the smallest box in parameter space which contains unstable polynomials. This new concept is an alternative to the robustness margin for the case when the radii of the box are unknown but only a lower bound for each of them is given. As stated, this problem requires the solution of a nonlinear optimization problem. We show that via a proper reformulation, it can be recast as a one-dimensional optimization problem which requires, at each step, checking a vertex condition. It is interesting to observe that the vertices involved are artificially constructed and they do not correspond to the vertices of the box in parameter space. Finally, we show that in the case of interval polynomials the number of vertices required is linear in the number of uncertain parameters while in the polytopic case this number may be not polynomial in the worst case. An example concludes the paper