Zero-Order Robust Nonlinear Model Predictive Control with Ellipsoidal Uncertainty Sets

In this paper, we propose an efficient zero-order algorithm that can be used to compute an approximate solution to robust optimal control problems (OCP) and robustified nonconvex programs in general. In particular, we focus on robustified OCPs that make use of ellipsoidal uncertainty sets and show that, with the proposed zero-order method, we can efficiently obtain suboptimal, but robustly feasible solutions. The main idea lies in leveraging an inexact sequential quadratic programming (SQP) algorithm in which an advantageous sparsity structure is enforced. The obtained sparsity allows one to eliminate the variables associated with the propagation of the ellipsoidal uncertainty sets and to solve a reduced problem with the same dimensionality and sparsity structure of a nominal OCP. The inexact algorithm can drastically reduce the computational complexity of the SQP iterations (e.g., in the case where a structure exploiting interior-point method is used to solve the underlying quadratic programs (QPs), from O(N ⋅ (nx6 + nu3)) to (nx6 + nu3)). Moreover, standard embedded QP solvers for nominal problems can be leveraged to solve the reduced QP.