  Appropriate estimation of Region of Attraction for a nonlinear dynamical
system plays a key role in system analysis and control design. Sum-of-Squares
optimization is a powerful tool enabling Region of Attraction estimation for
polynomial dynamical systems. Employment of a positive definite function called
shape function within the Sum-of-Squares procedure helps to find a richer
representation of the Lyapunov function and a larger corresponding Region of
Attraction estimation. However, existing Sum-of-Squares optimization techniques
demonstrate very conservative results. The main novelty of this paper is the
Union theorem which enables the use of multiple shape functions to create a
polynomial Lyapunov function encompassing all the areas generated by the shape
functions. The main contribution of this paper is a novel
computationally-efficient numerical method for Region of Attraction estimation,
which remarkably improves estimation performance and overcomes limitations of
existing methods, while maintaining the resultant Lyapunov function polynomial,
thus facilitating control system design and construction of control Lyapunov
function with enhanced Region of Attraction using conventional Sum-of-Squares
tools. A mathematical proof of the Union theorem along with its application to
the numerical algorithm of Region of Attraction estimation is provided. The
method yields significantly enlarged Region of Attraction estimations even for
systems with non-symmetric or unbounded Region of Attraction, which is
demonstrated via simulations of several benchmark examples.
