Convex Optimization of Nonlinear State Feedback Controllers for Discrete-time Polynomial Systems via Occupation Measures

In this paper, we design nonlinear state feedback controllers for discrete-time polynomial dynamical systems via the occupation measure approach. We propose the discretetime controlled Liouville equation, and use it to formulate the controller synthesis problem as an infinite-dimensional linear programming (LP) problem on measures. The LP is then approximated by a family of finite-dimensional semidefinite programming (SDP) problems on moments of measures and their duals on sums-of-squares polynomials. By solving one or more of the SDP’s, we can extract the nonlinear controllers. The advantage of the occupation measure approach is that we solve convex problems instead of generally non-convex problems, and hence the approach is more reliable and scalable. We illustrate our approach on three dynamical systems.