Model Order Reduction For Time-delay Systems And Delay Eigenvalue Problems

In this thesis, we focus on the subject of model order reduction and eigenvalue problem for time-delay systems, which can be applied in various fields, such as electrodynamics, population modeling and electronic engineering.First, we consider the model order reduction for time-delay systems. We review the Krylov-Pade method given by Michiels et al, which uses spectral discretization to get an approximate large linear dynamical system, and then reduces the dimension of the linear system via Krylov subspace method, where the transfer function of the reduced system is a Pade approximation of that of the original time-delay system. By analyzing the special structure of the linear system matrix and the Arnoldi process to compute the basis matrix of the Krylov subspace, we found that if we choose an initial vector with special structure in the Arnoldi process, then the basis matrix of the Krylov subspace is a block upper triangular matrix. Based on this observation, we present a compact expression of the orthonormal basis matrix, and then present a new efficient process, a delay two-level orthogonalization Arnoldi (TOAR) process, to calculate this compact expression. Moreover, we show the numerical stability by an error analysis and it is showed that the delay TOAR process requires less memory consumption than the Arnoldi process. The resulting memory requirement is reduced from 1/2n(k2+k) to nk+1/2k3+O(k2), where n is the dimension of the original time-delay system and k is the reduced order. Based on the result of the Arnoldi process and the delay TOAR process, we present three methods to reduce the original delay system-one uses the classical form of the Krylov subspace to reduce the approximate linear dynamical system (Michiels et al, [60]); one uses the compact expression to reduce the approximate linear dynamic system; and one uses the compact expression to directly get a reduced-order time-delay system. Furthermore, moments matching properties are proved to guarantee the accuracy of the reduced systems. And numerical experiments are given to illustrate the theoretic results and the feasibility and effectiveness of these methods.Then, we generalize the method of model reduction to solve the delay eigen-value problem. After using a large linear eigenvalue problem (LEP) to approximate the delay eigenvalue problem, we use the Arnoldi method to solve the LEP (see Jarlebring et al., [44]). During the process of solving the eigenvalues via Arnoldi method, we can use delay TOAR process instead of Arnoldi process, since the LEP has the similar structure with the system matrix of the approximate linear dynamic system used in the model order reduction. And then we compute the approximate eigenvalues of delay eigenvalue problem. We call this process as TOAR method. Numerical examples illustrate that the TOAR method is not only effective but also capable to compute more approximate eigenvalues owning to less memory cost. At last, we further extend the TOAR method to general nonlinear eigenvalue problems.