Convergence Analysis of Natural Power Method and Its Applications to Control

Abstract

This paper analyzes the discrete-time natural power method, demonstrating its convergence to the dominant $r$-dimensional subspace corresponding to the $r$ eigenvalues with the largest absolute values. This contrasts with the Oja flow, which targets eigenvalues with the largest real parts. We leverage this property to develop methods for model order reduction and low-rank controller synthesis for discrete-time LTI systems, proving preservation of key system properties. We also extend the low-rank control framework to slowly-varying LTV systems, showing its utility for tracking time-varying dominant subspaces.