Séance Séminaire

Séminaire ACSIOM

Polynomial inner approximations of robust stability regions

Linear system stability can be formulated semi algebraically in the space of coefficients of the characteristic polynomial. The region of stability is generally nonconvex in this space, and this is a major obstacle when solving fixed-order and/or robust controller design problems. Using the Hermite stability criterion, these problems can be formulated as parametrized polynomial matrix inequalities (PMIs) where parameters account for uncertainties and the decision variables are controller coefficients. Recent results on real algebraic geometry and generalized problems of moments can be used to build up a hierarchy of convex linear matrix inequality (LMI) outer approximations of the region of stability, with asymptotic convergence to its convex hull. Whereas outer approximations of nonconvex semialgebraic sets can be readily constructed with these LMI relaxations, inner approximations are much harder to obtain. For controller design purposes, inner approximations are however essential since they correspond to sufficient conditions and hence guarantees of stability or robust stability. In the robust systems control literature, convex inner approximations of the stability region were proposed in the form of polytopes, ellipsoids, or more general LMI regions derived from polynomial positivity conditions. In this work we propose more general polynomial inner approximations of parametrized PMI feasibility sets, including stability regions. We describe a hierarchy of inner approximations with polynomial sublevel sets of increasing degrees. Each polynomial sublevel set in the hierarchy is constructed by solving an LMI problem. In addition, we can easily enforce that the inner approximations are nested and/or convex. Finally, and most importantly, we can prove uniform convergence of the hierarchy.