Modeling of Nonlinear Dynamical Systems Using Koopman Operator Theory
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2021
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University of Hawaii at Manoa
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The dynamics of most mechanical systems tends to incorporate nonlinearfunctions and behaviors to model complex systems. Due to the complexity of some of these systems, only analytical solutions can be found to model, optimize and control them however the costs of doing so are not always feasible. To solve these nonlinear systems we usually need to take approximations through linearization which can lead to a loss of fidelity in modeling systems to accommodate the computing power needed to solve them. This work is the culmination of research and study involving the use of a data-driven spectral method to generate a linear approximation of nonlinear systems using Koopman operator theory. The Koopman operator is an infinite dimensional operator that ‘lifts’ the nonlinear behavior of a system to a higher dimensional state. In this lifted state, the evolution of these nonlinear dynamics progresses linearily. This work investigates the use of data driven methods such as dynamic mode decomposition (DMD), and one of its variants, to find the Koopman operator of some nonlinear dynamical systems. In this work we initially go through decomposing the linear dynamics of the above systems using DMD on two sample systems to verify the efficacy of this method of finding the Koopman operator before lifting the dynamics. Using a variant of DMD, extended dynamic mode decomposition, we decompose the lifted non-linear dynamics of a pendulum and cart-pendulum system to find the leading Koopman eigenfunctions to approximate a lower, finite-dimensional representation of the discovered dynamics. We used these methods of reconstruction on controlled and uncontrolled data of the sample systems to compare the observed dynamics to evaluate the reconstruction on a wider breadth of behaviors a system can produce. Using these reconstructed states analyze the limitations and discuss possible improvements to these methods of finding the Koopman operator.
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Mechanical engineering
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