This thesis presents a collection of methods on the topic of evidence based change detection from discrete data. Some of the methods deal with detecting changes in the data and in its derivatives. Others are focused on modeling systems with the purpose of condition monitoring. The goal in all of them is to monitor and analyze the changes in behavior of mechanical systems governed by physical principles, which themselves are described by differential equations. The methods are demonstrated through a series of peer reviewed papers. The thesis is divided in the following chapters: polynomial methods, optimal control, the variable projection method, and industrial applications.
Matrix-algebraic formulations of polynomial problems allow for the development of new approaches in analyzing data. Using double-sided constrained Taylor approximations of discrete data, a measure of discontinuity is developed similar in concept to Lipschitz continuity. Additionally, using discrete orthogonal polynomials, a generalized framework for solving constrained inverse problems emanating from cyber-physical systems is presented.
With respect to optimal control, new algebraic formulations for discretizing the Euler-Lagrange equations are presented. These provide good approximations to solutions for numerically and physically stiff systems.
Further, it was shown how to use the variable projection method to model periodic functions with, or without a background signal. In this manner, spectral leakage and the Gibbs error were avoided, which led to highly stable and numerically efficient solutions.
Finally, through a consequent use of mathematical methods, the desired information was extracted from large volumes of industrial data, which can be corrupted by both statistical and systematic noise. With this, the interdisciplinary nature of data analysis in industrial applications was demonstrated.