A Gauss-Newton-Induced Structure-Exploiting Algorithm for Differentiable Optimal Control

Abstract

Differentiable optimal control, particularly differentiable nonlinear model predictive control (NMPC), provides a powerful framework that enjoys the complementary benefits of machine learning and control theory. A key enabler of differentiable optimal control is the computation of derivatives of the optimal trajectory with respect to problem parameters, i.e., trajectory derivatives. Previous works compute trajectory derivatives by solving a differential Karush-Kuhn-Tucker (KKT) system, and achieve this efficiently by constructing an equivalent auxiliary system. However, we find that directly exploiting the matrix structures in the differential KKT system yields significant computation speed improvements. Motivated by this insight, we propose FastDOC, which applies a Gauss-Newton approximation of Hessian and takes advantage of the resulting block-sparsity and positive semidefinite properties of the matrices involved. These structural properties enable us to accelerate the computationally expensive matrix factorization steps, resulting in a factor-of-two speedup in theoretical computational complexity, and in a synthetic benchmark FastDOC achieves up to a 180% time reduction compared to the baseline method. Finally, we validate the method on an imitation learning task for human-like autonomous driving, where the results demonstrate the effectiveness of the proposed FastDOC in practical applications.