Estimating the region of attraction of uncertain systems with invariant sets

In this article the problem of estimating the Region of Attraction (ROA) for polynomial nonlinear systems subject to modeling uncertainties is studied. Based on recent theoretical studies on the calculation of positively invariant sets, this article proposes an optimization problem which allows robust inner Estimates of the Region of Attraction (rERA) to be evaluated. The uncertainties, which can generically be time-invariant or time-varying, are described as semialgebraic sets, and the problem is solved numerically by means of Sum Of Squares relaxations, which allow set containment conditions to be enforced. The ensuing optimization entails non-convex constraints, and an iterative algorithm to enlarge the provable invariant level set is discussed. The proposed algorithm is applied to two study cases of increasing complexity. Further, in order to benchmark the proposed rERA algorithm, comparisons are shown with a class of well established algorithms based on Lyapunov functions level sets. The results showcase the prowess of the proposed approach and its advantages in terms of accuracy and computational time, particularly as the size of the system increases.