Estimating the Region of Attraction for Uncertain Polynomial Systems Using Polynomial Chaos Functions and Sum of Squares Method

We present a general formulation for estimation of the region of attraction (ROA) for nonlinear systems with parametric uncertainties using a combination of the polynomial chaos expansion (PCE) theorem and the sum of squares (SOS) method. The uncertain parameters in the nonlinear system are treated as random variables with a probability distribution. First, the decomposition of the uncertain nonlinear system under consideration is performed using polynomial chaos functions. This yields to a deterministic subsystem whose state variables correspond to the deterministic coefficient components of the random basis polynomials in PCE. This decomposed deterministic subsystem contains no uncertainty. Then, the ROA of the deterministic subsystem is derived using sum of squares method. Finally, the ROA of the original uncertain nonlinear system is derived by transforming the ROA spanned in the decomposed deterministic subsystem back to the original spatial-temporal space using PCE. This proposed framework on estimation of the robust ROA (RROA) is based on a combination of PCE and SOS and is specially useful, with appealing computation efficiency, for uncertain nonlinear systems when the uncertainties are non-affine or when they are associated with a specific probability distribution.