Synopses & Reviews
Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems presents a number of techniques for robustness analysis of uncertain systems in a unified framework. The theoretical basis for their development is derived from the application of convex optimization tools to problems involving positivity of homogeneous polynomial forms. 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 book convex relaxations for several robustness problems are derived by exploiting classical and new results 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.
From the reviews: "The book describes convex optimization techniques to deal with stability and performance evaluation for linear dynamical systems subject to parametric uncertainty. ... In summary, the book is a welcome and timely addition to the technical literature, written by leading experts in the field. It is aimed at systems and control researchers willing to explore further than mainstream LMI Lyapunov analysis techniques, with the possibility of trading off between conservatism and computational burden." (Didier Henrion, Mathematical Reviews, Issue 2010 g)
This book presents a number of techniques for robustness analysis of uncertain systems. In it, convex relaxations for several robustness problems are derived by exploiting and providing new results on the theory of homogenous polynomial forms.
Table of Contents
Positive Forms.- Positivity Gap.- Robustness with Time-varying Uncertainty.- Robustness with Time-invariant Uncertainty.- Robustness with Bounded-rate Time-varying Uncertainty.- Distance Problems with Applications to Robust Control.- Appendices: Basic Tools; SMR Algorithms.