Exact and Approximate Convex Reformulation of Linear Stochastic Optimal Control With Chance Constraints

Abstract

In this letter, we present an equivalent convex optimization formulation for discrete-time stochastic linear systems subject to linear chance constraints, alongside two tight convex relaxations for quadratic chance constraints. By lifting the state vector to explicitly encode moment information, the formulation captures linear chance constraints on states and controls across multiple time steps exactly, without conservatism, yielding strict improvements in both feasibility and optimality. For quadratic chance constraints, we derive convex approximations that are provably less conservative than existing methods. We validate the framework for minimum-snap trajectory generation in a quadrotor, demonstrating that the proposed approach achieves cost reductions of ${41}\%$ - ${67}\%$ and remains feasible at noise levels an order of magnitude beyond the operating range of prior formulations.