Time delay systems exist in many engineering ?elds such as transportation communication
process engineering and more recently networked control s- tems. In recent years time
delaysystems haveattracted recurring interests from research community. Much of the research
work has been focused on stability analysis and stabilization of time delay systems using the
so-called Lyapunov- Krasovskii functionals and linear matrix inequality (LMI) approach. While
the LMI approach does provide an e?cient tool for handling systems with delays in state and or
inputs the LMI based results are mostly only su?cient and only numerical solutions are
available. For systems with knownsingle input delay there have been rather elegant- alytical
solutions to various problems such as optimal tracking linear quadratic regulation and H
control. We note that discrete-time systems with delays can ? usually be converted into delay
free systems via system augmentation however theaugmentationapproachleadsto muchhigher
computationalcosts especially for systems of higher state dimension and large delays. For
continuous-time s- tems time delayproblemscaninprinciple betreatedby thein?nite-dimensional
system theory which however leads to solutions in terms of Riccati type partial di?erential
equations or operator Riccati equations which are di?cult to und- stand and compute. Some
attempts have been made in recent years to derive explicit and e?cient solutions for systems
with input output (i o) delays. These include the study ontheH controlofsystemswith multiple
input delaysbased ? on the stable eigenspace of a Hamlitonian matrix [46].
