  Optimal linear feedback control design is a valuable but challenging problem
due to nonconvexity of the underlying optimization and infinite dimensionality
of the Hardy space of stabilizing controllers. A powerful class of techniques
for solving optimal control problems involves using reparameterization to
transform the control design to a convex but infinite dimensional optimization.
To make the problem tractable, historical work focuses on Galerkin-type finite
dimensional approximations to Hardy space, especially those involving Lorentz
series approximations such as the finite impulse response appproximation.
However, Lorentz series approximations can lead to infeasibility, difficulty
incorporating prior knowledge, deadbeat control in the case of finite impulse
response, and increased suboptimality, especially for systems with large
separation of time scales. The goal of this two-part article is to introduce a
new Galerkin-type method based on approximation by transfer functions with a
selection of simple poles, and to apply this simple pole approximation for
optimal control design. In Part I, error bounds for approximating arbitrary
transfer functions in Hardy space are provided based on the geometry of the
pole selection. It is shown that the space of transfer functions with these
simple poles converges to the full Hardy space, and a uniform convergence rate
is provided based purely on the geometry of the pole selection. This is then
specialized to derive a convergence rate for a particularly interesting pole
selection based on an Archimedes spiral. In Part II, the simple pole
approximation is combined with system level synthesis, a recent
reparameterization approach, to develop a new control design method. This
technique is convex and tractable, always feasible, can include prior
knowledge, does not result in deadbeat control, and works well for systems with
large separation of tim

Systems and Control;Systems and Control;Optimization and Control