  Frequency-limited model order reduction aims to approximate a high-order
model with a reduced-order model that maintains high fidelity within a specific
frequency range. Beyond this range, a decrease in accuracy is acceptable due to
the nature of the problem. The quality of the reduced-order model is typically
evaluated using absolute or relative measures of approximation error. Relative
error, which represents the percentage error, becomes particularly relevant
when reducing a plant model for the purpose of designing a reduced-order
controller. This paper derives the necessary conditions for achieving a local
optimum of the frequency-limited H2 norm for the relative error system. Based
on these optimality conditions, an oblique projection algorithm is proposed to
ensure a small relative error within the desired frequency interval. Unlike
existing algorithms, the proposed approach does not necessitate solving
large-scale Lyapunov and Ricatti equations. Instead, the proposed algorithm
relies on solving sparse-dense Sylvester equations, which typically emerge in
the majority of H2 model order reduction algorithms, but can be efficiently
solved. To evaluate the performance of the proposed algorithm, a comparison is
conducted with three existing techniques: frequency-limited balanced
truncation, frequency-limited balanced stochastic truncation, and
frequency-limited iterative Rational Krylov algorithm. The comparative analysis
focuses on designing reduced-order controllers for high-order plants. Numerical
results confirm that the reduced-order controllers obtained using the proposed
algorithm ensure superior robust closed-loop stability.
