H_inf Reduced Order Control for Nanopositioning: Numerical Implementability

Abstract

In this paper we discuss how model reduction affects the stability and computational complexity of controllers for nanopositioning systems. A robust H∞ multiple-input multiple-output controller is designed and implemented for the lateral stage of an atomic force microscope. A model-based controller can often be of high order and may be difficult to run in real-time on hardware with limited computational power. The resulting controller can be considered to be stiff, which is characterized by a large spread of eigenvalues. Continuous-time systems running in real-time are often solved using explicit Runge-Kutta (ERK) methods, which easily becomes unstable for stiff systems. We show how small the time-step for a given controller needs to be for a selection of ERK methods. We also consider how model reduction affects the computational complexity of the controller, and show how the reduction can alter the placement of the eigenvalues and thus the required step-size for implementability. We demonstrate that the original 18th-order H∞ controller can be reduced to a 10th-order controller without any significant reduction in performance or stability, which results in a 46.7% reduction in execution time, partly because the order reduction enables the use of a simpler solver type.