Graduation Year
2023
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Razvan Teodorescu, Ph.D.
Co-Major Professor
Joel A. Rosendfeld, Ph.D.
Committee Member
Catherine A. Beneteau, Ph.D.
Committee Member
Dmitry Khavinson, Ph.D.
Committee Member
Benjamin P. Russo, Ph.D.
Keywords
System Identification, Dynamical Systems, Integral Transform, Kernelized Reconstruction
Abstract
Consider a nonautonomous nonlinear evolution $\dot{x}=f(x,t,\mu)$, where the vector $x(t) \in \mathbb{R}^n$ represents the state of the dynamical system at time $t$, $\mu$ contains system parameters, and $f(\cdot)$ represents a dynamic constraint. In most practical applications, the nonlinear dynamic constraint $f$ is unknown analytically. The problem of approximating $f$ directly from data measurements generated by the system is a main goal of this manuscript. In the postulates of the Nonlinear Autoregressive (NAR) framework, we show that the problem of approximating $f$ can be studied through symbols of densely defined multiplication operators over a Reproducing Kernel Hilbert Spaces (RKHS). In this formulation, data is mapped into a RKHS by virtue of occupation kernels which are special functions that reside in a RKHS owing to an integration functional. The resulting scheme is a parameter identification algorithm where system parameters are approximated according to some induced structure on the symbols of the operator. The action of the adjoint multiplication on an occupation kernel induces a kernelized transform which is the subject of study in the second part of the dissertation.
Scholar Commons Citation
Kyei, John, "Data-Driven Learning Algorithm Via Densely-Defined Multiplication Operators and Occupation Kernels." (2023). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10440
