Stability region estimates for SDRE controlled systems using sum of squares optimization

In this paper, we investigate the state-dependent Riccati equation (SDRE) method to control nonlinear systems. This method stabilizes the closed loop system around the origin. However, the global asymptotic stability is not ensured. Moreover, the stability analysis is complicated because the closed loop system is typically not known in a closed form. We present a theorem that turns stability region estimation into a functional search problem. Results on the sum of squares polynomials are used to turn this search into a semi-definite programming problem. A simple example demonstrating this method is given.