  We analyze and design H-infinity controllers for general time-delay systems
with time-delays in systems' state, inputs and outputs. We allow the designer
to choose the order of the controller and to introduce constant time-delays in
the controller. The closed-loop system of the plant and the controller is
modeled by a system of delay differential algebraic equations (DDAEs). The
advantage of the DDAE modeling framework is that any interconnection of systems
and controllers prone to various types of delays can be dealt with in a
systematic way, without using any elimination technique. We present a
predictor-correct algorithm for the H-infinity norm computation of systems
described by DDAEs. Instrumental to this we analyze the properties of the
H-infinity norm. In particular, we illustrate that it may be sensitive with
respect to arbitrarily small delay perturbations. Due to this sensitivity, we
introduce the strong H-infinity norm which explicitly takes into account small
delay perturbations, inevitable in any practical control application. We
present a numerical algorithm to compute the strong H-infinity norm for DDAEs.
Using this algorithm and the computation of the gradient of the strong
H-infinity norm with respect to the controller parameters, we minimize the
strong H-infinity norm of the closed-loop system based on non-smooth,
non-convex optimization methods. By this approach, we tune the controller
parameters and design H-infinity controllers with a prescribed order or
structure.
