Advanced-Step Real-time Iterations with Four Levels -- New Error Bounds and Fast Implementation in acados

The Real-Time Iteration (RTI) is an online nonlinear model predictive control algorithm that performs a single Sequential Quadratic Programming (SQP) per sampling time. The algorithm is split into a preparation and a feedback phase, where the latter one performs as little computations as possible solving a single prepared quadratic program. To further improve the accuracy of this method, the Advanced-Step RTI (AS-RTI) performs additional Multi-Level Iterations (MLI) in the preparation phase, such as inexact or zero-order SQP iterations on a problem with a predicted state estimate. This paper extends and streamlines the existing local convergence analysis of AS-RTI, such as analyzing MLI of level A and B for the first time, and significantly simplifying the proofs for levels C and D. Moreover, this paper provides an efficient open-source implementation in acados, making it widely accessible to practitioners.