Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fast and memory-efficient optimization for large-scale data-driven predictive control

Philipp Schmitz, Manuel Schaller, Matthias Voigt, Karl Worthmann

TL;DR

This work proposes an iterative solver for the underlying data-driven optimal control problems resulting from linear discrete-time systems and applies factorizations based on the discrete Fourier transform of the Hankel-like matrices, which enable fast and memory-efficient computations.

Abstract

Recently, data-enabled predictive control (DeePC) schemes based on Willems' fundamental lemma have attracted considerable attention. At the core are computations using Hankel-like matrices and their connection to the concept of persistency of excitation. We propose an iterative solver for the underlying data-driven optimal control problems resulting from linear discrete-time systems. To this end, we apply factorizations based on the discrete Fourier transform of the Hankel-like matrices, which enable fast and memory-efficient computations. To take advantage of this factorization in an optimal control solver and to reduce the effect of inherent bad conditioning of the Hankel-like matrices, we propose an augmented Lagrangian lBFGS-method. We illustrate the performance of our method by means of a numerical study.

Fast and memory-efficient optimization for large-scale data-driven predictive control

TL;DR

This work proposes an iterative solver for the underlying data-driven optimal control problems resulting from linear discrete-time systems and applies factorizations based on the discrete Fourier transform of the Hankel-like matrices, which enable fast and memory-efficient computations.

Abstract

Recently, data-enabled predictive control (DeePC) schemes based on Willems' fundamental lemma have attracted considerable attention. At the core are computations using Hankel-like matrices and their connection to the concept of persistency of excitation. We propose an iterative solver for the underlying data-driven optimal control problems resulting from linear discrete-time systems. To this end, we apply factorizations based on the discrete Fourier transform of the Hankel-like matrices, which enable fast and memory-efficient computations. To take advantage of this factorization in an optimal control solver and to reduce the effect of inherent bad conditioning of the Hankel-like matrices, we propose an augmented Lagrangian lBFGS-method. We illustrate the performance of our method by means of a numerical study.
Paper Structure (10 sections, 1 theorem, 23 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 10 sections, 1 theorem, 23 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation: Data-driven predictive control
  3. DFT-based factorization of Hankel-like matrices
  4. Factorization
  5. Complexity
  6. Augmented Lagrangian with lBFGS
  7. Outer loop: Augmented Lagrangian
  8. Inner loop: Limited memory BFGS
  9. Numerical simulations
  10. Conclusion

Key Result

Lemma 1

Suppose that the system eq:LTI is controllable and let $(\widetilde{\bm x}_N, \widetilde{\bm u}_N)$ be a trajectory of eq:LTI such that $\widetilde{\bm u}_N$ is persistently exciting of order $L+n$. Then $(\bm x_L,\bm u_L)$ is a length-$L$ trajectory of system eq:LTI if and only if there is $z\in\ma

Figures (5)

  • Figure 1: Residuals over iterations for the proposed method (aL lBFGS), MINRES, and gradient descent employing an augmented Lagrangian approach (aL GD) for $n=100$, $m=L = 50$.
  • Figure 2: The condition number of the matrix $\mathcal{S}$ (blue circles) and $\mathcal{B}\mathcal{S}$ (red diamonds) for increasing state dimension $n$ and fixed values $m = L = 50$.
  • Figure 3: Total number of BFGS iterations in dependence of the state dimension $n$ in \ref{['eq:docp']} for fixed $m=L=50$ (blue circles) and for $m=L=100$ (red diamonds).
  • Figure 4: Total execution time in dependence of the state dimension $n$ in \ref{['eq:docp']} for fixed $m=L=50$ (blue circles) and for $m=L=100$ (red diamonds).
  • Figure 5: Mean execution time per BFGS iteration (total execution time divided by the total number of iterations divided) for increasing $n$ and fixed $m,L$ with $m=L=50$ (blue circles) and $m=L=100$ (red diamonds).

Theorems & Definitions (2)

  • Lemma 1
  • Remark 2