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Optimal linear feedback control design is a valuable but challenging problemdue to nonconvexity of the underlying optimization and infinite dimensionalityof the Hardy space of stabilizing controllers. A powerful class of techniquesfor solving optimal control problems involves using reparameterization totransform the control design to a convex but infinite dimensional optimization.To make the problem tractable, historical work focuses on Galerkin-type finitedimensional approximations to Hardy space, especially those involving Lorentzseries approximations such as the finite impulse response appproximation.However, Lorentz series approximations can lead to infeasibility, difficultyincorporating prior knowledge, deadbeat control in the case of finite impulseresponse, and increased suboptimality, especially for systems with largeseparation of time scales. The goal of this two-part article is to introduce anew Galerkin-type method based on approximation by transfer functions with aselection of simple poles, and to apply this simple pole approximation foroptimal control design. In Part I, error bounds for approximating arbitrarytransfer functions in Hardy space are provided based on the geometry of thepole selection. It is shown that the space of transfer functions with thesesimple poles converges to the full Hardy space, and a uniform convergence rateis provided based purely on the geometry of the pole selection. This is thenspecialized to derive a convergence rate for a particularly interesting poleselection based on an Archimedes spiral. In Part II, the simple poleapproximation is combined with system level synthesis, a recentreparameterization approach, to develop a new control design method. Thistechnique is convex and tractable, always feasible, can include priorknowledge, does not result in deadbeat control, and works well for systems withlarge separation of tim

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