Affine controller synthesis for stochastic reachability via difference of convex programming

We propose an affine controller synthesis technique that maximizes the probability of the state lying in a time-varying collection of safe sets for a Gaussian-perturbed linear time-varying system under bounded control authority. Specifically, we solve a chance constrained optimization problem obtained by relaxing the hard control bounds to a probabilistic constraint with a user-specified threshold. We also construct a lower bound for the true maximal reach probability, when the proposed affine controller is saturated to satisfy hard control bounds. For tractability, we formulate the optimization problem into an equivalent difference of convex program with second-order cone constraints, using risk allocation techniques and piecewise affine overapproximations. We then solve this program to a local optimum using a well-studied successive convexification procedure. We also show that the proposed chance constraint formulation has a tractable linear program-based restriction when the feedback gain matrix is known or when only open-loop controllers are considered. We compare the performance of our approach with existing approaches on a spacecraft rendezvous problem.