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Controllability scores for selecting control nodes of large-scale network systems

Kazuhiro Sato, Shun Terasaki

TL;DR

Numerical experiments demonstrate that the proposed algorithm is more efficient than an existing interior point method, and the proposed scores can correctly capture the importance of each state node on controllability, outperforming existing control centralities.

Abstract

To appropriately select control nodes of a large-scale network system, we propose two control centralities called volumetric and average energy controllability scores. The scores are the unique solutions to convex optimization problems formulated using the controllability Gramian. The uniqueness is proven for stable cases and for unstable cases that include multi-agent systems. We show that the scores can be efficiently calculated by using a proposed algorithm based on the projected gradient method onto the standard simplex. Numerical experiments demonstrate that the proposed algorithm is more efficient than an existing interior point method, and the proposed scores can correctly capture the importance of each state node on controllability, outperforming existing control centralities.

Controllability scores for selecting control nodes of large-scale network systems

TL;DR

Numerical experiments demonstrate that the proposed algorithm is more efficient than an existing interior point method, and the proposed scores can correctly capture the importance of each state node on controllability, outperforming existing control centralities.

Abstract

To appropriately select control nodes of a large-scale network system, we propose two control centralities called volumetric and average energy controllability scores. The scores are the unique solutions to convex optimization problems formulated using the controllability Gramian. The uniqueness is proven for stable cases and for unstable cases that include multi-agent systems. We show that the scores can be efficiently calculated by using a proposed algorithm based on the projected gradient method onto the standard simplex. Numerical experiments demonstrate that the proposed algorithm is more efficient than an existing interior point method, and the proposed scores can correctly capture the importance of each state node on controllability, outperforming existing control centralities.
Paper Structure (15 sections, 11 theorems, 42 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 15 sections, 11 theorems, 42 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Controllability scoring problem and controllability score
  4. Volumetric Controllability Score (VCS)
  5. Average Energy Controllability Score (AECS)
  6. Finite-time controllability scoring problem
  7. Algorithm for Controllability Scores
  8. Numerical experiments
  9. Time performance of Algorithm \ref{['alg:projgrad']}
  10. Comparison to other control centralities
  11. Concluding remarks
  12. Proof of Theorems \ref{['thm:strictlyconvex']} and \ref{['thm:strictlyconvex2']}
  13. Proof of Lemma \ref{['Lem_F0']} and Theorem \ref{['thm:unique_existence']}
  14. Proof of Theorems \ref{['thm:unstable_realeigen']} and \ref{['Thm_Laplacian']}
  15. Proof of Theorem \ref{['main_thm']}

Key Result

Lemma 1

For any $p^{(0)}\in \mathcal{F}$, and $\mathcal{F}_0$ is a closed, bounded, and convex set in ${\mathbb R}^n$.

Figures (3)

  • Figure 1: Network structure with $n=10$.
  • Figure : A projected gradient method
  • Figure : Armijo rule along the projection arc

Theorems & Definitions (26)

  • Lemma 1
  • Proof 1
  • Theorem 1
  • Proof 2
  • Theorem 2
  • Proof 3
  • Remark 1
  • Remark 2
  • Theorem 3
  • Proof 4
  • ...and 16 more