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Controllability score for linear time-invariant systems on an infinite time horizon

Kota Umezu, Kazuhiro Sato

TL;DR

This work addresses the challenge of computing controllability-based centrality scores for linear time-invariant networks when the system matrix $A$ is unstable. It introduces a scaled controllability Gramian that neutralizes the divergence caused by unstable modes and reformulates the volumetric controllability score (VCS) and average energy controllability score (AECS) into numerically stable optimization problems. By taking the infinite-horizon limit, the authors prove convergence and, under suitable conditions, uniqueness of the infinite-horizon scores, clarifying why $T=\infty$ is principled even for non-Hurwitz $A$. They also show that VCS and AECS remain distinct on the infinite horizon, since VCS enforces controllability of the full system while AECS focuses only on stable modes; this distinction is illustrated with Laplacian dynamics. A numerically stable algorithm based on real Schur decomposition and Sylvester equations is provided, and experiments on a Laplacian-based network validate convergence and reveal the practical differences between the two scores, supporting the method's relevance to real-world networks.

Abstract

We introduce a scaled controllability Gramian that can be computed reliably even for unstable systems. Using this scaled Gramian, we reformulate the controllability scoring problems into equivalent but numerically stable optimization problems. Their optimal solutions define dynamics-aware network centrality measures, referred to as the volumetric controllability score (VCS) and the average energy controllability score (AECS). We then formulate controllability scoring problems on an infinite time horizon. Under suitable assumptions, we prove that the resulting VCS and AECS are unique and that the finite-horizon scores converge to them. We further show that VCS and AECS can differ markedly in this limit, because VCS enforces controllability of the full system, whereas AECS accounts only for the stable modes. Finally, using Laplacian dynamics as a representative example, we present numerical experiments that illustrate this convergence.

Controllability score for linear time-invariant systems on an infinite time horizon

TL;DR

This work addresses the challenge of computing controllability-based centrality scores for linear time-invariant networks when the system matrix is unstable. It introduces a scaled controllability Gramian that neutralizes the divergence caused by unstable modes and reformulates the volumetric controllability score (VCS) and average energy controllability score (AECS) into numerically stable optimization problems. By taking the infinite-horizon limit, the authors prove convergence and, under suitable conditions, uniqueness of the infinite-horizon scores, clarifying why is principled even for non-Hurwitz . They also show that VCS and AECS remain distinct on the infinite horizon, since VCS enforces controllability of the full system while AECS focuses only on stable modes; this distinction is illustrated with Laplacian dynamics. A numerically stable algorithm based on real Schur decomposition and Sylvester equations is provided, and experiments on a Laplacian-based network validate convergence and reveal the practical differences between the two scores, supporting the method's relevance to real-world networks.

Abstract

We introduce a scaled controllability Gramian that can be computed reliably even for unstable systems. Using this scaled Gramian, we reformulate the controllability scoring problems into equivalent but numerically stable optimization problems. Their optimal solutions define dynamics-aware network centrality measures, referred to as the volumetric controllability score (VCS) and the average energy controllability score (AECS). We then formulate controllability scoring problems on an infinite time horizon. Under suitable assumptions, we prove that the resulting VCS and AECS are unique and that the finite-horizon scores converge to them. We further show that VCS and AECS can differ markedly in this limit, because VCS enforces controllability of the full system, whereas AECS accounts only for the stable modes. Finally, using Laplacian dynamics as a representative example, we present numerical experiments that illustrate this convergence.
Paper Structure (23 sections, 25 theorems, 116 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 23 sections, 25 theorems, 116 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Controllability score
  4. Remaining challenges and our approach
  5. Scaling method of the controllability Gramian and reformulation of \ref{['prob:vcs', 'prob:aecs']}
  6. Problem settings
  7. Infinite-horizon scaled controllability Gramian
  8. Reformulation of controllability scoring problems
  9. Extension to infinite-horizon settings
  10. Problem formulation
  11. Convergence of controllability scores
  12. Uniqueness analysis
  13. Algorithms
  14. Numerical experiments
  15. Conclusion
  16. ...and 8 more sections

Key Result

Lemma 1

Let $C_1\in\mathbb{C}^{\ell_1\times\ell_1}, C_2\in\mathbb{C}^{\ell_2\times\ell_2}$, and $P\in\mathbb{C}^{\ell_1\times\ell_2}$. If $\mu_1+\mu_2\neq0$ holds for any eigenvalues $\mu_1$ of $C_1$ and $\mu_2$ of $C_2$, then has a unique solution $Y\in\mathbb{C}^{\ell_1\times\ell_2}$.

Figures (2)

  • Figure 1: Virtual system for defining controllability scores. The node $x_i$ denotes a state variable, and the node $u_i$ denotes a virtual control signal. The coefficients $p_i$ specify how a fixed amount of control resources is allocated to the nodes, and each input $u_i$ corresponds one-to-one with node $x_i$ through the weight $\sqrt{p_i}$.
  • Figure 2: Directed network employed in the numerical experiments

Theorems & Definitions (44)

  • Remark 1
  • Lemma 1: HornJohnson2012
  • Theorem 1
  • Proof 1
  • Lemma 2
  • Lemma 3
  • Proof 2: \ref{['prop:finite_nonempty_lemma', 'prop:infinite_nonempty_lemma']}
  • Remark 2
  • Theorem 2
  • Theorem 3
  • ...and 34 more