Rational Lyapunov functions for estimating and controlling the robust domain of attraction

This paper addresses the estimation and control of the robust domain of attraction (RDA) of equilibrium points through rational Lyapunov functions (LFs) and sum of squares (SOS) techniques. Specifically, continuous-time uncertain polynomial systems are considered, where the uncertainty is represented by a vector that affects polynomially 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 nonconvex 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 sufficient condition for establishing tightness of this lower bound. Lastly, the paper discusses the search for optimal rational LFs using the proposed strategy.