Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems

The stability and performance analysis of dynamic systems affected by structured uncertainties usually requires the solution of non-convex optimization problems. A possible way to tackle such problems is to construct a family of convex relaxations which provide upper or lower bounds to the solution of the original problem. In this talk, convex relaxations for several robustness problems are presented by exploiting classical results and providing new insights on the theory of homogeneous polynomial forms. In particular, a general framework is introduced for dealing with positivity of forms via the solution of linear matrix inequalities, which are a special class of convex problems. Lyapunov analysis of uncertain systems affected by time-invariant or time-varying uncertainty, computation of the parametric robust stability margin, robust performance analysis for polytopic systems, are examples of problems addressed within the proposed framework. This talk is based on the new book 'Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems' by G. Chesi et al. (Springer, 2009, in press