Model order reduction for time-delay systems with initial history functions

This paper presents projection-based model order reduction methods for time-delay systems with initial history functions, addressing both time and frequency-domain formulations. In the time domain, a shifted Legendre polynomial expansion is employed to resolve the coupling between delay states and non-zero initial history functions. The projection matrix is derived from coefficients obtained via a Sylvester-type matrix equation, ensuring that the reduced model’s expansion coefficients match those of the original system. In the frequency domain, a unified MOR framework is proposed through generalized transfer function theory, integrating low-rank spatiotemporal decomposition of initial history functions. Moreover, a hybrid projection strategy combining Taylor and Laguerre series expansions is introduced to jointly capture transient and steady-state behaviors. Numerical validation underscores the accuracy and efficiency of both approaches.