arXiv subjects

Daiki Tsuzuki

Publications and source records attributed to Daiki Tsuzuki.

4 recordsLinked to original sources

arXiv2025

Convergence Analysis of Natural Power Method and Its Applications to Control

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.

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arXiv2025

Global Convergence of Oja's Component Flow for General Square Matrices and Its Applications

In this study, the global convergence properties of the Oja flow, a continuous-time algorithm for principal component extraction, was established for general square matrices. The Oja flow is a matrix differential equation on the Stiefel manifold designed to extract a dominant subspace. Although its analysis has traditionally been restricted to symmetric positive-definite matrices, where it acts as a gradient flow, recent applications have extended its use to general matrices. In this non-symmetric case, the flow extracts the invariant subspace corresponding to the eigenvalues with the largest real parts. However, prior convergence results have been purely local, leaving the global behavior as an open problem. The findings of this study fill this gap by providing a comprehensive global convergence analysis, establishing that the flow converges exponentially for almost all initial conditions. We also propose a modification to the algorithm that enhances its numerical stability. As an application of this theory, we developed novel methods for model reduction of linear dynamical systems and the synthesis of low-rank stabilizing controllers. The study advances the theoretical understanding of the Oja flow and demonstrates its potential as a reliable and versatile tool for analyzing and controlling complex linear systems.

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arXiv2024

Low-rank approximated Kalman filter using Oja's principal component flow for discrete-time linear systems

The Kalman filter is indispensable for state estimation across diverse fields but faces computational challenges with higher dimensions. Approaches such as Riccati equation approximations aim to alleviate this complexity, yet ensuring properties like bounded errors remains challenging. Yamada and Ohki introduced low-rank Kalman-Bucy filters for continuous-time systems, ensuring bounded errors. This paper proposes a discrete-time counterpart of the low-rank filter and shows its system theoretic properties and conditions for bounded mean square error estimation. Numerical simulations show the effectiveness of the proposed method.

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arXiv2024

Low-rank approximated Kalman-Bucy filters using Oja's principal component flow for linear time-invariant systems

The Kalman-Bucy filter is extensively utilized across various applications. However, its computational complexity increases significantly in large-scale systems. To mitigate this challenge, a low-rank approximated Kalman--Bucy filter was proposed, comprising Oja's principal component flow and a low-dimensional Riccati differential equation. Previously, the estimation error was confirmed solely for linear time-invariant systems with a symmetric system matrix. This study extends the application by eliminating the constraint on the symmetricity of the system matrix and describes the equilibrium points of the Oja flow along with their stability for general matrices. In addition, the domain of attraction for a set of stable equilibrium points is estimated. Based on these findings, we demonstrate that the low-rank approximated Kalman--Bucy filter with a suitable rank maintains a bounded estimation error covariance matrix if the system is controllable and observable.

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