Convergence Analysis of Natural Power Method and Its Applications to Control

TL;DR

This paper studies the convergence of the natural power method for eigenvalue problems, applying it to model reduction and control of discrete-time systems, including time-varying cases.

Contribution

It provides a convergence analysis of the natural power method and introduces new applications to control and model reduction for discrete-time and time-varying systems.

Findings

Converges to the dominant r-dimensional subspace.

Preserves key system properties in model reduction.

Effective for tracking time-varying dominant subspaces.

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.