Robust domain of attraction: Computing and controlling estimates with non-polynomial Lyapunov functions

This paper addresses the estimation and control of the robust domain of attraction (RDA) of equilibrium points through rational Lyapunov functions (LFs). Specifically, continuous-time uncertain polynomial systems are considered. The uncertainty is represented by a vector that affects poly-nomially the system and is constrained in 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. It is shown that lower bounds of the LERDA in the estimation problem, or the maximum achievable LERDA in the control problem, can be obtained by solving either an eigenvalue problem or a generalized eigenvalue problem with smaller dimension. The conservatism of these lower bounds can be reduced by increasing the degree of some multipliers introduced in the construction of the optimization problems. Some numerical examples illustrate the use of the proposed results.