Model order reduction for discrete time-delay systems based on Laguerre function expansion

We investigate model order reduction for discrete time-delay systems via orthogonal polynomial expansion in the frequency domain. The transfer function of systems is expanded in the framework of Laguerre function basis. We show that Laguerre coefficients of the states satisfy a linear system and reduced models generated by projection methods can preserve some Laguerre coefficients of the original systems. We prove that the subspace spanned by Laguerre coefficients is exactly a high order Krylov subspace, thereby leading to an efficient computation of projection matrices and a more accurate coefficient-matching property in the two-sided framework. Further the Laguerre expansion of systems is employed to enable an approximate but fast execution of balanced truncation for discrete time-delay systems. Specifically, Gramians are approximated based on the derived Laguerre coefficients, which circumvents the main bottleneck of balanced truncation methods. Finally, two numerical examples are simulated to demonstrate the feasibility and effectiveness of the proposed methods.