Piecewise-Affine Approximation-Based Stochastic Optimal Control with Gaussian Joint Chance Constraints

Abstract

This paper considers the problem of stochastic optimal control of a Gaussian-perturbed linear system subject to soft polytopic state constraints, hard polytopic input constraints, and a convex cost function. We propose two conservative approaches using risk allocation that can be implemented via existing solvers, and characterize the approximations. Unlike existing approaches, we do not decouple the risk allocation from the optimal controller synthesis. We first show that risk allocation in conjunction with optimal controller synthesis introduces reverse convex constraints into the optimization problem. Next, we use piecewise-affine approximations of the nonlinear terms in the optimization problem to propose a mixed-integer convex program. Our piecewise-affine approximation produces a solver-friendly convex program when the safety probability threshold is larger than 0.5. Using two stochastic motion planning problems, we demonstrate that the proposed approach outperforms existing approaches like iterative risk allocation and particle control approaches in computation time, without compromising on the solution quality.