Attainability of Prediction-Error Bounds in Passive Dynamical Model Reduction

Finite-horizon prediction accuracy in passive dynamical model reduction depends not only on the reduced order but also on the admissible model parameterization. We study linear passive resistor–capacitor networks under prescribed state aggregation and measure reduction error by the worst-case time-integrated squared output discrepancy over unit-norm initial states. A subspace on which the reduced prediction is independent of its conductance parameters yields a parameter-independent lower bound on this error. We characterize attainment of the bound through the coupling between this fixed-error subspace and its orthogonal complement, including an equivalent positive-semidefinite block condition and a quantitative lower bound when the coupling is nonzero. A five-node network aggregated into three states illustrates the distinction between structural error and parameter-dependent error. For a reduced chain constrained to equal conductances, computer-assisted calculations support a strictly positive gap above the fixed-error bound; when the two conductances vary independently, numerical solutions instead indicate attainment of the bound. The example highlights how restrictions within a passive model class can change attainability, while distinguishing analytically established criteria from numerical and computer-assisted claims requiring independent validation.