Distributionally Robust Density Control with Wasserstein Ambiguity Sets

Precise control under uncertainty requires a good understanding and characterization of the noise affecting the system. This paper studies the problem of steering state distributions of dynamical systems subject to partially known uncertainties. We model the distributional uncertainty of the noise process in terms of Wasserstein ambiguity sets, which, based on recent results, have been shown to be an effective means of capturing and propagating uncertainty through stochastic LTI systems. To this end, we propagate the distributional uncertainty of the state through the dynamical system, and, using an affine feedback control law, we steer the ambiguity set of the state to a prescribed, terminal ambiguity set. We also enforce distributionally robust CVaR constraints for the transient motion of the state so as to reside within a prescribed constraint space. The resulting optimization problem is formulated as a semi-definite program, which can be solved efficiently using standard off-the-shelf solvers. We illustrate the proposed distributionally-robust framework on a path planning problem and compare with non-robust solutions.