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Cholesky factorisation, and intrinsically sparse linear quadratic regulation

Julia Adlercreutz, Richard Pates

TL;DR

This work develops a sparsity-preserving Cholesky factorisation for matrices of shift operators, enabling exact, back-substitution-based solutions to infinite-horizon LS problems arising in distributed LQR formulations. Central to the approach is a noncommutative factorisation $\mathbf{L}\mathbf{L}^*=(\mathbf{MP})^*(\mathbf{MP})$ whose existence is characterized by the underlying graph being a tree, and which preserves sparsity so that control laws admit local implementations. The authors provide a full computational framework, including an operator-to-coefficient mapping, sequence representations, and backward-substitution procedures, along with theoretical comparisons to spectral factorisation. Through transportation-network examples, they demonstrate that optimal control laws can be highly sparse and distributed, and they discuss cases where sparse spectral factors do not exist, underscoring the value of their noncommutative approach for scalable control design. The work also offers practical software tools in Matlab/Julia and highlights implications for large-scale, locally communicative control systems in engineering networks.

Abstract

We classify a family of matrices of shift operators that can be factorised in a computationally tractable manner with the Cholesky algorithm. Such matrices arise in the linear quadratic regulator problem, and related areas. We use the factorisation to uncover intrinsic sparsity properties in the control laws for transportation problems with an underlying tree structure. This reveals that the optimal control can be applied in a distributed manner that is obscured by standard solution methods.

Cholesky factorisation, and intrinsically sparse linear quadratic regulation

TL;DR

This work develops a sparsity-preserving Cholesky factorisation for matrices of shift operators, enabling exact, back-substitution-based solutions to infinite-horizon LS problems arising in distributed LQR formulations. Central to the approach is a noncommutative factorisation whose existence is characterized by the underlying graph being a tree, and which preserves sparsity so that control laws admit local implementations. The authors provide a full computational framework, including an operator-to-coefficient mapping, sequence representations, and backward-substitution procedures, along with theoretical comparisons to spectral factorisation. Through transportation-network examples, they demonstrate that optimal control laws can be highly sparse and distributed, and they discuss cases where sparse spectral factors do not exist, underscoring the value of their noncommutative approach for scalable control design. The work also offers practical software tools in Matlab/Julia and highlights implications for large-scale, locally communicative control systems in engineering networks.

Abstract

We classify a family of matrices of shift operators that can be factorised in a computationally tractable manner with the Cholesky algorithm. Such matrices arise in the linear quadratic regulator problem, and related areas. We use the factorisation to uncover intrinsic sparsity properties in the control laws for transportation problems with an underlying tree structure. This reveals that the optimal control can be applied in a distributed manner that is obscured by standard solution methods.
Paper Structure (25 sections, 7 theorems, 111 equations, 7 figures, 1 algorithm)

This paper contains 25 sections, 7 theorems, 111 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basic notation
  3. Shift operators
  4. Results
  5. A factorisation for matrices of shift operators
  6. Computing with shift operators
  7. Shift operator algebra
  8. Shift operator computations
  9. Pseudoinverses and square roots
  10. The factorisation algorithm
  11. Back substitution computations
  12. Sparse factorisations
  13. Comparison with spectral factorisation
  14. Application in control
  15. A simple LQR example
  16. ...and 10 more sections

Key Result

Theorem 1

Given any undirected graph $\mathcal{G}$, the following two statements are equivalent:

Figures (7)

  • Figure 1: Transportation network with a line graph topology.
  • Figure 2: Graph specifying the sparsity pattern of $\mathbf{M}$ in \ref{['eq:graph']}.
  • Figure 3: Illustration of the graphs from \ref{['def:edge']}. a.) shows a graph $\mathcal{G}$, while b.) shows $\mathcal{G}$ in grey dashed lines and $\mathop{\mathrm{edge}}\nolimits\left(\mathcal{G}\right)$ in black.
  • Figure 4: Model of the transportation network studied in \ref{['sec:simple-lqr']}.
  • Figure 5: Graph representing the transportation network in \ref{['fig:state-graph']}. There is one vertex per storage facility, and one edge per transportation link. The edges have been oriented to reflect the transportation direction. The undirected version of this graph (i.e. \ref{['fig:example-graph1']}) describes the sparsity pattern of the $\mathbf{M}$ appearing in \ref{['sec:lqrleastsquares']} according to \ref{['def:mon']}.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Remark 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Remark 2
  • ...and 11 more