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

Antoine P. Leeman, Johannes Köhler, Florian Messerer, Amon Lahr, Moritz Diehl, Melanie N. Zeilinger

TL;DR

A tailored algorithm for solving the corresponding disturbance feedback optimization problem for linear time-varying systems by iterating between optimizing the controller and the nominal trajectory while converging q-linearly to an optimal solution.

Abstract

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.

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

TL;DR

A tailored algorithm for solving the corresponding disturbance feedback optimization problem for linear time-varying systems by iterating between optimizing the controller and the nominal trajectory while converging q-linearly to an optimal solution.

Abstract

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 for each iterate. On a numerical example, the proposed algorithm exhibits computational speedups by a factor of up to compared to general-purpose commercial solvers.
Paper Structure (14 sections, 36 equations, 3 figures, 2 algorithms)

This paper contains 14 sections, 36 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup & System Level Synthesis
  3. Efficient algorithm for SLS
  4. SLS and its KKT conditions
  5. Iterative trajectory and controller optimization
  6. Algorithm
  7. Controller optimization via efficient Riccati recursions
  8. Scalability
  9. Numerical example
  10. Scalability with state dimension
  11. Scalability with horizon length
  12. Convergence
  13. Conclusion
  14. Nonlinear Extension

Figures (3)

  • Figure 1: Illustration how the robust MPC problem \ref{['eq:sls']} is solved by alternating between a nominal trajectory optimization problem with tightened constraints and the computation of optimal feedback gains $K_{k,j}$ using Riccati recursions.
  • Figure 2: Scalability of different solvers with horizon length for $L=10$ masses (left). Scalability of different solvers with the number of states for horizon $N=10$ (right). The proposed algorithm's (fast-SLS) computation times, and the Riccati recursions contributions \ref{['eq:riccati']} (fast-SLS: Riccati) are shown with standard deviations.
  • Figure 3: Number of iterations required until convergence for $10^3$ randomly sampled initial conditions, for horizon $N=25$, and $L=25$ masses.

Theorems & Definitions (10)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 4.1
  • Remark A.1