Krylov-Based Model Order Reduction of Time-delay Systems

• Authors: Wim Michiels;Elias Jarlebring;Karl Meerbergen
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2011

We present a model order reduction method which allows the construction of a reduced, delay-free model of a given dimension for linear time-delay systems, whose characteristic matrix is nonlinear due to the presence of exponential functions. The method builds on the equivalent representation of the time-delay system as an infinite-dimensional linear problem. It combines ideas from a finite-dimensional approximation via a spectral discretization, on the one hand, and a Krylov-Padé model reduction approach, on the other hand. The method exhibits a good spectral approximation of the original model, in the sense that the smallest characteristic roots are well approximated and the nonconverged eigenvalues of the reduced model have a favorable location, and it preserves moments at zero and at infinity. The spectral approximation is due to an underlying Arnoldi process that relies on building an appropriate Krylov space for the linear infinite-dimensional problem. The preservation of moments is guaranteed, because the chosen finite-dimensional approximation preserves moments and, in addition, the space on which one projects is constructed in such a way that the preservation of moments carries over to the reduced model. The implementation of the method is dynamic, since the number of grid points in the spectral discretization does not need to be chosen beforehand and the accuracy of the reduced model can always be improved by doing more iterations. It relies on a reformulation of the problem involving a companion-like system matrix and a highly structured input matrix, whose structure are fully exploited.