Keywords

Carleman Linearization, graph signal processing, inverse filter, distributed algorithms, Wiener filter

Abstract

Carleman linearization is a mainstream approach for transforming finite-dimensional nonlinear dynamical systems into infinite-dimensional linear systems. It provides accurate approximations of the original nonlinear system over larger regions around the equilibrium for longer time horizons, surpassing the capabilities of conventional first-order linearization. In Chapter 1, we derive explicit error bounds for the finite-section approximation, demonstrating its exponential convergence concerning the finite-section order. We illustrate that for a specific class of nonlinear systems, exponential convergence can be achieved across the entire time horizon, extending to infinity. These results hold significant practical utility, as the proposed error bounds can be used to determine appropriate truncation lengths for applications like model predictive control and reachability analysis for safety verification.

As graph signals have gained significant attention and applications in recent years, investigating inverse filtering procedures for graph signal recovery becomes crucial. A key challenge lies in the potentially exponential computational cost associated with direct matrix inversion. Chapter 2 proposes iterative polynomial approximation algorithms that facilitate the distributed implementation of an inverse filter. These algorithms enable the distributed implementation of an inverse filter using a polynomial graph filter with commutative graph shifts. The proposed algorithms exhibit exponential convergence properties, and they can be implemented on distributed networks in which agents are equipped with a data processing subsystem for limited data storage and computation power, and with a one-hop communication subsystem for direct data exchange only with their adjacent agents. Chapter 3 presents Wiener filters for the recovery of deterministic and wide-band stationary graph signals from noisy observations. We then propose distributed algorithms to implement these Wiener filters. Furthermore, the proposed algorithms can be implemented on distributed networks in which agents are equipped with a data processing subsystem for limited data storage and computation power, and with a one-hop communication subsystem for direct data exchange only with their adjacent agents.

Completion Date

2025

Semester

Summer

Committee Chair

Qiyu Sun

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Format

PDF

Identifier

DP0029632

Language

English

Document Type

Thesis

Campus Location

Orlando (Main) Campus

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