  In time-limited model order reduction, a reduced-order approximation of the
original high-order model is obtained that accurately approximates the original
model within the desired limited time interval. Accuracy outside that time
interval is not that important. The error incurred when a reduced-order model
is used as a surrogate for the original model can be quantified in absolute or
relative terms to access the performance of the model reduction algorithm. The
relative error is generally more meaningful than an absolute error because if
the original and reduced systems' responses are of small magnitude, the
absolute error is small in magnitude as well. However, this does not
necessarily mean that the reduced model is accurate. The relative error in such
scenarios is useful and meaningful as it quantifies percentage error
irrespective of the magnitude of the system's response. In this paper, the
necessary conditions for a local optimum of the time-limited H2 norm of the
relative error system are derived. Inspired by these conditions, an oblique
projection algorithm is proposed that ensures small H2-norm relative error
within the desired time interval. Unlike the existing relative error-based
model reduction algorithms, the proposed algorithm does not require solutions
of large-scale Lyapunov and Riccati equations. The proposed algorithm is
compared with time-limited balanced truncation, time-limited balanced
stochastic truncation, and time-limited iterative Rational Krylov algorithm.
Numerical results confirm the superiority of the proposed algorithm over these
existing algorithms.
