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

TL;DR

This paper introduces an exact convex reformulation for stochastic linear control with chance constraints, improving feasibility and optimality, and providing less conservative approximations for quadratic constraints, validated on quadrotor trajectory planning.

Contribution

It presents a novel convex optimization framework that exactly encodes linear chance constraints and offers tighter convex relaxations for quadratic constraints in stochastic control.

Findings

Exact convex reformulation improves feasibility and optimality.

Convex relaxations for quadratic constraints are less conservative.

Framework remains feasible under higher noise levels in quadrotor experiments.

Abstract

In this paper, we present an equivalent convex optimization formulation for discrete-time stochastic linear systems subject to linear chance constraints, alongside a tight convex relaxation for quadratic chance constraints. By lifting the state vector to encode moment information explicitly, 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 on minimum-snap trajectory generation for a quadrotor, demonstrating that the proposed approach remains feasible at noise levels an order of magnitude beyond the operating range of prior formulations.