Estimating the domain of attraction based on the invariance principle

Estimating the domain of attraction (DA) of an equilibrium point is a long-standing yet still challenging issue in nonlinear system analysis. The method using the sublevel set of Lyapunov functions is proven to be efficient, but sometimes conservative compared to the estimate via invariant sets. This paper studies the estimation problem of the DA for autonomous polynomial system by using the invariance principle. The main idea is to estimate the DA via sublevel sets of a positive polynomial, which characterizes the boundary of invariant sets. This new type of invariant sets admits the condition that the derivative of Lyapunov functions is non-positive, which generalizes the sublevel set method via Lyapunov functions. An iterative algorithm is then proposed for enlarging the estimate of the DA. Finally, the effectiveness of the proposed method is illustrated by numerical examples.