Fast System Level Synthesis: Robust Model Predictive Control using Riccati Recursions

System level synthesis enables improved robust MPC formulations by allowing for joint optimization of the nominal trajectory and controller. This paper introduces a tailored algorithm for solving the corresponding disturbance feedback optimization problem for linear time-varying systems. The proposed algorithm iterates between optimizing the controller and the nominal trajectory while converging q-linearly to an optimal solution. We show that the controller optimization can be solved through Riccati recursions leading to a horizon-length, state, and input scalability of $\mathcal{O}(N^2 ( n_x^3 +n_u^3))$ for each iterate. On a numerical example, the proposed algorithm exhibits computational speedups by a factor of up to $10^3$ compared to general-purpose commercial solvers.