Estimate of the domain of attraction for polynomial systems using bisectional sum-of-squares optimization

This paper proposes a new approach to estimate the domain of attraction (DA) of equilibrium points for polynomial autonomous systems. The largest estimate of the DA (LEDA) for a given Lyapunov function (LF) is computed via finding the largest sublevel set of the LF, which leads to the nonlinear optimization problem and possibly becomes non-convex when the LF is allowed to vary. We simplified the complete square metrical representation (CSMR) based LMI conditions to much more simple sum-of-squares (SOS) conditions, and avoided the set containment relationship between the LF and its derivative. As a result, simpler bilinear SOS conditions are formulated to compute the LEDAs for given LFs and search for better LFs which may provide larger LEDAs than a quadratic one. Two bisectional SOS programming (BiSOSP) algorithms are given for solving those bilinear SOS conditions. The method is applied to the well known Van der pol system and the computational cost is also discussed.