Rational Lyapunov Functions for Estimating and Controlling the Robust Domain of Attraction Graziano Chesi Department of Electrical and Electronic Engineering The University of Hong Kong Contact: http://www.eee.hku.hk/~chesi Abstract This paper addresses the estimation and control of the robust do- main of attraction (RDA) of equilibrium points through rational Lya- punov functions (LFs) and sum of squares (SOS) techniques. Specif- ically, continuous-time uncertain polynomial systems are considered, where the uncertainty is represented by a vector that affects polyno- mially the system and is constrained into a semialgebraic set. The estimation problem consists of computing the largest estimate of the RDA (LERDA) provided by a given rational LF. The control problem consists of computing a polynomial static output controller of given degree for maximizing such a LERDA. In particular, the paper shows that the computation of the best lower bound of the LERDA for chosen degrees of the SOS polynomials, which requires the solution of a non- convex optimization problem with bilinear matrix inequalities (BMIs), can be reformulated as a quasi-convex optimization problem under some conditions. Moreover, the paper provides a necessary and suffi- cient condition for establishing tightness of this lower bound. Lastly, the paper discusses the search for optimal rational LFs using the pro- posed strategy. 1 Introduction Studying the RDA of equilibrium points is a key problem in uncertain non- linear systems. In fact, the RDA is the set of initial conditions for which the state of the system asymptotically converges to the equilibrium point under consideration for all admissible uncertainties. Hence, when dealing with uncertain nonlinear systems, it is not sufficient to establish that the 1