Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can expressed in terms of semialgebraic sets, that is as the intersection of described by means of polynomial inequalities. Important examples are for the solution set of linear matrix inequalities or the Schur/Hurwitz domains. These sets often have very complicated shapes (non-convex, even non-connected), which renders very difficult their manipulation. It is of considerable importance to find simple-enough approximations of sets, able to capture their main characteristics while maintaining a low of complexity. For these reasons, in the past years several convex, based for instance on hyperrect-angles, polytopes, or have been proposed. In this work, we move a step further, and possibly non-convex approximations, based on a small volume polynomial set of a single positive polynomial of given degree. We show how sets can be easily approximated by minimizing the L1 norm of the over the semialgebraic set, subject to positivity constraints. , this corresponds to the trace minimization heuristic commonly in minimum volume ellipsoid problems. From a computational viewpoint, design a hierarchy of linear matrix inequality problems to generate these, and we provide theoretically rigorous convergence results, in sense that the hierarchy of outer approximations converges in volume (or, , almost everywhere and almost uniformly) to the original set. Two applications of the proposed approach are considered. The first one aims reconstruction/approximation of sets from a finite number of samples. In the one, we show how the concept of polynomial superlevel set can be used to samples uniformly distributed on a given semialgebraic set. The of the proposed approach is demonstrated by different numerical.