Relationship Between Controllability Scoring and Optimal Experimental Design

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.

💡 Research Summary

The paper establishes a rigorous link between controllability scoring—a method for assigning importance to individual state nodes in a networked dynamical system—and the well‑studied field of approximate optimal experimental design (OED). By augmenting an autonomous linear time‑invariant system (\dot x = A x) with a set of “virtual” input channels, one per node, the authors define a diagonal input matrix (B(p)=\operatorname{diag}(\sqrt{p_1},\dots,\sqrt{p_n})) where the vector (p) distributes a fixed actuation budget across the nodes. The finite‑time controllability Gramian for horizon (T) then takes the form

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