The polynomial chaos approach for reachable set propagation with application to chance-constrained nonlinear optimal control under parametric uncertainties - Frison, Lilli; Kirches, Christian (Optimal control, applications and methods, 2018)
Optimal control problems with parametric uncertainties frequently arise in control practice and can be addressed by means of a probabilistic robustification using the concept of chance constraints. In this article, we develop an efficient method based on the polynomial chaos expansion to compute nonlinear propagations of the reachable sets of all uncertain states and show how it can be used to approximate nonlinear and joint chance constraints. The strength of the obtained estimator in guaranteeing a satisfaction level is supported by providing an a priori error estimate with exponential convergence in case of sufficiently smooth solutions. The proposed approach is readily implemented in existing state-of-the-art direct methods to optimal control and is evaluated for 2 real-world nonlinear uncertain optimal control problems. The achieved level of robustness in terms of constraint satisfaction is verified by extensive Monte Carlo sampling.