Fast-Lifting Approach to the Computation of the Spectrum of Retarded Time-Delay Systems

This paper discusses a new method for computing the spectrum of retarded time-delay systems based on the discrete-time viewpoint provided by the application of the lifting technique. Stimulated by a recent study that applied (conventional) fast-sample/fast-hold (FSFH) approximation, we consider applying a somewhat alike but completely different method with modified FSFH approximation developed in the fast-lifting approach to sampled-data systems. We show that this leads to a finite-dimensional method that gives an asymptotically exact result. Compared with other existing methods that also lead to discretization approaches, the arguments in this paper are based on the bounded operator treatment of dynamical systems, and more system theoretic. The associated convergence analysis is also carried out in an operator-theoretic framework in a straightforward fashion. An advantage of the method is discussed in dealing with discrete-time controllers, and the effectiveness of the method is demonstrated with numerical examples.