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Relationship Between Controllability Scoring and Optimal Experimental Design

Kazuhiro Sato

TL;DR

The paper establishes a structural link between finite-time controllability scoring and approximate optimal experimental design (OED) by showing that the finite-horizon controllability Gramian decomposes additively across nodes, yielding an affine information-matrix model analogous to OED. It identifies a direct correspondence: volumetric controllability score (VCS) aligns with D-optimality and average-energy controllability score (AECS) with A-optimality, while noting an invariance gap where VCS is coordinate-invariant but AECS is not. The work highlights a fundamental difference from OED: controllability scoring typically yields a unique optimizer, and introduces a long-horizon phenomenon where source-like nodes without negative self-loops can be downweighted by AECS, with negative self-loops mitigating this effect. Numerical examples illustrate the long-horizon downweighting and the qualitative divergence between VCS and AECS in node rankings. Overall, the paper provides a principled bridge between centrality-like controllability scores and classical experimental-design criteria, with implications for robust node-actuation strategies in large networks.

Abstract

Controllability scores provide control-theoretic centrality measures that quantify the relative importance of state nodes in networked dynamical systems. We establish a structural connection between finite-time controllability scoring and approximate optimal experimental design (OED): the finite-time controllability Gramian decomposes additively across nodes, yielding an affine matrix model of the same form as the information-matrix model in OED. This yields a direct correspondence between the volumetric controllability score (VCS) and D-optimality, and between the average energy controllability score (AECS) and A-optimality, implying that the classical D/A invariance gap has a direct analogue in controllability scoring. By contrast, we point out that controllability scoring typically admits a unique optimizer, unlike approximate-OED formulations. Finally, we uncover a long-horizon phenomenon with no OED counterpart: source-like state nodes without a negative self-loop can be increasingly downweighted by AECS as the horizon grows. Two numerical examples corroborate this long-horizon downweighting behavior.

Relationship Between Controllability Scoring and Optimal Experimental Design

TL;DR

The paper establishes a structural link between finite-time controllability scoring and approximate optimal experimental design (OED) by showing that the finite-horizon controllability Gramian decomposes additively across nodes, yielding an affine information-matrix model analogous to OED. It identifies a direct correspondence: volumetric controllability score (VCS) aligns with D-optimality and average-energy controllability score (AECS) with A-optimality, while noting an invariance gap where VCS is coordinate-invariant but AECS is not. The work highlights a fundamental difference from OED: controllability scoring typically yields a unique optimizer, and introduces a long-horizon phenomenon where source-like nodes without negative self-loops can be downweighted by AECS, with negative self-loops mitigating this effect. Numerical examples illustrate the long-horizon downweighting and the qualitative divergence between VCS and AECS in node rankings. Overall, the paper provides a principled bridge between centrality-like controllability scores and classical experimental-design criteria, with implications for robust node-actuation strategies in large networks.

Abstract

Controllability scores provide control-theoretic centrality measures that quantify the relative importance of state nodes in networked dynamical systems. We establish a structural connection between finite-time controllability scoring and approximate optimal experimental design (OED): the finite-time controllability Gramian decomposes additively across nodes, yielding an affine matrix model of the same form as the information-matrix model in OED. This yields a direct correspondence between the volumetric controllability score (VCS) and D-optimality, and between the average energy controllability score (AECS) and A-optimality, implying that the classical D/A invariance gap has a direct analogue in controllability scoring. By contrast, we point out that controllability scoring typically admits a unique optimizer, unlike approximate-OED formulations. Finally, we uncover a long-horizon phenomenon with no OED counterpart: source-like state nodes without a negative self-loop can be increasingly downweighted by AECS as the horizon grows. Two numerical examples corroborate this long-horizon downweighting behavior.
Paper Structure (25 sections, 6 theorems, 52 equations, 1 figure, 4 tables)

This paper contains 25 sections, 6 theorems, 52 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Controllability Scores
  3. Virtual actuation model and the corresponding controllability Gramian
  4. Finite-time controllability scoring problem
  5. Algorithm for solving Problem \ref{['prob:FTCSP']}
  6. Relation Between Problem \ref{['prob:FTCSP']} and Optimal Experimental Design
  7. Approximate OED
  8. Invariance of VCS and coordinate dependence of AECS
  9. Difference
  10. Different semantics
  11. Uniqueness
  12. Intrinsic time horizon $T$ and the limit $T\to\infty$
  13. Why AECS downweights source-like nodes on long horizons
  14. Explicit formula for the diagonal Gramian entry under a left-eigenvector condition
  15. AECS lower bound
  16. ...and 10 more sections

Key Result

Proposition 1

Let $A\in\mathbb{R}^{n\times n}$ be arbitrary and let $S\in\mathbb{C}^{n\times n}$ be any nonsingular matrix. Let $p_{\rm VCS}$ be an optimal solution to Problem prob:FTCSP with $h_T=f_T$ for system eq:lti, and let $\widetilde{p}_{\rm VCS}$ be an optimal solution to Problem prob:FTCSP2 with $\wideti Moreover, for almost all $T>0$, the optimal solution to Problem prob:FTCSP2 with $\widetilde{h}_T=\

Figures (1)

  • Figure 1: Network employed in the numerical experiments.

Theorems & Definitions (16)

  • Remark 1
  • Proposition 1: VCS invariance
  • Proof 1
  • Example 1
  • Remark 2: On transpose vs. Hermitian transpose
  • Lemma 1
  • Proof 2
  • Theorem 1
  • Proof 3
  • Remark 3
  • ...and 6 more