Fast-lifting approach to time-delay systems: Fundamental framework

This paper gives a novel approach to time-delay systems consisting of a linear time-invariant (LTI) system and a pure delay. The fast-lifting technique, introduced recently in the context of the study of sampled-data systems, is applied to the monodromy operator of a time-delay system to define the fast-lifted monodromy operator, and a stability condition is given in terms of the spectral radius of the latter operator. Then, by investigating the properties of this operator, an operator class described by two finite-dimensional matrices is introduced as candidates for a solution to the associated operator Lyapunov inequality. It is then established that the analysis restricted to such a class gives an asymptotically exact and nonconservative method for stability analysis if the integer N for fast-lifting is taken sufficiently large.