  A general numerical method using sum of squares programming is proposed to
address the problem of estimating the region of attraction (ROA) of an
asymptotically stable equilibrium point of a nonlinear polynomial system. The
method is based on Lyapunov theory, and a shape function is defined to enlarge
the provable subset of a local Lyapunov function. In contrast with existing
methods with a shape function centered at the equilibrium point, the proposed
method utilizes a shifted shape function (SSF) with its center shifted
iteratively towards the boundary of the newly obtained invariant subset to
improve ROA estimation. A set of shifting centers with corresponding SSFs is
generated to produce proven subsets of the exact ROA and then a composition
method, namely R-composition, is employed to express these independent sets in
a compact form by just a single but richer-shaped level set. The proposed
method denoted as RcomSSF brings a significant improvement for general ROA
estimation problems, especially for non-symmetric or unbounded ROA, while
keeping the computational burden at a reasonable level. Its effectiveness and
advantages are demonstrated by several benchmark examples from literature.
