Approximation by Simple Poles -- Part I: Density and Geometric Convergence Rate in Hardy Space

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