Successive Convexification of Non-Convex Optimal Control Problems and Its Convergence Properties

TL;DR

This paper introduces a successive convexification algorithm for non-convex optimal control problems, linearizing nonlinear dynamics iteratively, with convergence guarantees and applications to real-time autonomous vehicle path planning.

Contribution

The paper proposes a novel successive convexification method that linearizes nonlinear dynamics iteratively, with convergence analysis independent of discretization schemes.

Findings

Algorithm effectively solves non-convex control problems.

Convergence analysis demonstrates robustness of the method.

Numerical simulations validate the approach for trajectory optimization.

Abstract

This paper presents an algorithm to solve non-convex optimal control problems, where non-convexity can arise from nonlinear dynamics, and non-convex state and control constraints. This paper assumes that the state and control constraints are already convex or convexified, the proposed algorithm convexifies the nonlinear dynamics, via a linearization, in a successive manner. Thus at each succession, a convex optimal control subproblem is solved. Since the dynamics are linearized and other constraints are convex, after a discretization, the subproblem can be expressed as a finite dimensional convex programming subproblem. Since convex optimization problems can be solved very efficiently, especially with custom solvers, this subproblem can be solved in time-critical applications, such as real-time path planning for autonomous vehicles. Several safe-guarding techniques are incorporated into the algorithm, namely virtual control and trust regions, which add another layer of algorithmic robustness. A convergence analysis is presented in continuous- time setting. By doing so, our convergence results will be independent from any numerical schemes used for discretization. Numerical simulations are performed for an illustrative trajectory optimization example.