Control by coin flips: Mixed strategy for finite-horizon stochastic optimal control

It may sound counterintuitive that choosing control inputs randomly lowers cost in an optimal control problem. It can be the case in a nonconvex chance-constrained optimal control problem, including stochastic model predictive control (SMPC). This is because allowing mixed strategy convexifies a nonconvex problem; the expected cost and the probability of constraint violation of a mixed strategy control is a convex combination of pure strategy controls. Therefore the improvement in cost that mixed strategy control provides over pure strategy is equal to the duality gap. This paper presents an efficient method to compute an optimal mixed strategy solution through dual optimization. The focus of this paper is given to the solution method to finite-horizon, constrained stochastic optimal control problems, which are solved at each iteration of SMPC. We demonstrate the method on a chance-constrained trajectory planning problem with obstacles.