Data-driven stabilization of nonlinear polynomial systems with noisy data

In a recent paper, we have shown how to learn controllers for unknown linear systems using finite length noisy data by solving linear matrix inequalities. In this note, we extend this approach to deal with unknown nonlinear polynomial systems by formulating stability certificates in the form of data-dependent sum of squares programs, whose solution directly provides a stabilizing controller and a Lyapunov function. We then derive variations of this result that lead to more advantageous controller designs. The results also reveal connections to the problem of designing a controller starting from a least-square estimate of the polynomial system.