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Abstract
In recent years, data-driven methods for the analysis of nonlinear systems have flourished. Derived from machine learning techniques, these methods allow one to analyze, predict, and control the behavior of a nonlinear system without prior model knowledge. The only requirement are data taken from the nonlinear system of interest; moreover, data-driven models are particularly useful for complex nonlinear systems and have seen success in many branches of engineering, from modal decomposition of fluid flows to designing stabilizing controllers for nonlinear systems. In particular, this thesis will focus on the extension of two of these data-driven methods: dynamic mode decomposition (DMD) and kernelized principal component analysis (KPCA).
DMD, which relies on representing a nonlinear system as an infinite-dimensional linear operator, has seen success in predicting the behavior of both continuous-time and discrete-time nonlinear systems without prior model knowledge; however, the extension of DMD methods to discrete-time, controlled nonlinear systems is nontrivial. In this thesis, we develop a novel operator representation of discrete-time, control-affine nonlinear dynamical systems. The representation is learned using recorded snapshots of the system state resulting from arbitrary, potentially open-loop control inputs. We thereby extend the predictive capabilities of dynamic mode decomposition to discrete-time nonlinear systems that are affine in control.
KPCA is typically a data-driven dimensionality reduction technique that allows one to study a nonlinear system via a reduced-order model in a higher-dimensional space; however, KPCA can be used for fault detection in nonlinear systems without prior model knowledge. Reliable operation of automatic systems is heavily dependent on the ability to detect faults in the underlying dynamics. While traditional model-based methods have been widely used for fault detection, data-driven approaches have garnered increasing attention due to their ease of deployment and minimal need for expert knowledge. In the latter portion of this thesis, we develop a novel fault detection method using KPCA with the occupation kernel as the feature map. Occupation kernels result in feature maps that are tailored to the measured data, have inherent noise-robustness due to the use of integration, and can utilize irregularly sampled system trajectories of variable lengths for PCA.
