Estimation of non-symmetric and unbounded region of attraction using shifted shape function and R-composition

Abstract

Sum-of-squares programming is widely used for region of attraction (ROA) estimations of asymptotically stable equilibrium points of nonlinear polynomial systems. However, existing methods yield conservative results, especially for non-symmetric and unbounded regions. In this study, a cost-effective approach for ROA estimation is proposed based on the Lyapunov theory and shape functions. In contrast to existing methods, the proposed method iteratively places the center of a shifted shape function (SSF) close to the boundary of the acquired invariant subset. The set of obtained SSFs yields robust ROA subsets, and R-composition is employed to express these independent sets as a single but richer-shaped level set. Several benchmark examples show that the proposed method significantly improves ROA estimations, especially for non-symmetric or unbounded ROA without a significant computational burden.