LMI-based estimation of the domain of attraction of equilibrium points for nonlinear non-polynomial dynamical systems

Estimating the domain of attraction of equilibrium points of nonlinear systems is an important problem in numerous fields. Recently, some methods based on LMIs have been developed for addressing this problem in the case of polynomial systems. This paper proposes an approach for extending the use of LMIs to the case of non-polynomial systems. In particular, it is shown how the estimate achievable with a chosen Lyapunov function can be obtained either via a sequence of LMI feasibility tests or via a generalized eigenvalue problem by constructing a suitable rectangle of polynomials and evaluating the positivity of its vertices. These results are also important for the case of a variable Lyapunov function because the computation of the estimate guaranteed by a chosen Lyapunov function is clearly a necessary step in the search for optimal Lyapunov functions.