The L2 gain of a nonlinear time-dependent system measures the maximal gain in the transfer of energy from admissible input signals to the output signals, in which both the input and output signals are measured with the L2 norm. For general nonlinear systems, obtaining a sharp estimate of the L2 gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of L2 gains for systems with quadratic nonlinearity. The approach utilizes a recently developed method that solves a class of Hamilton-Jacobi-Bellman equations via a Taylor series-based approximation, which is scalable to high-dimensional problems given the utilization of linear tensor systems.
The ideas are demonstrated through a few concrete examples that include a one-dimensional problem with an explicit energy function and several Galerkin approximations of the viscous Burgers equation. / Master of Science / With nonlinear systems that are of the form of input-output models, questions often arise as to how to measure the energy that passes through such systems and determine strategies to look for specific signals that allow the designer freedom to explore certain system behaviors. The energy comes in the form of a signal. For general nonlinear systems, obtaining a sharp estimate of such energy gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of these gains for systems with quadratic nonlinearity. The approach combines fundamental theoretical understandings established in the literature with scalable software recently developed in approximating the solution of the underlying partial differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation. In this approach, the energy gain is linked to a single scalar parameter in the HJB equation. Roughly speaking, the energy gain is the lower bound of this scalar parameter above which the HJB equation always admits a non-negative solution. Thus, it boils down to approximating the HJB solution using the software while changing this scalar parameter. We will present the theoretical foundation of the approach and illustrate the foundation through several academic examples ranging from low to relatively high dimensions.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/116067 |
| Date | 21 August 2023 |
| Creators | Lang, Sydney |
| Contributors | Mathematics, Borggaard, Jeffrey T., Liu, Honghu, Ciupe, Stanca M. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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