Koopman Operator Approach to Uncertainty Quantification and Decision-Making
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Meyers, Joseph
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Abstract
In an increasingly complex world, the control of autonomous systems subject to unexpected events or uncertainty is paramount to the success of the systems of the future. Uncertainty quantification techniques developed over the years seek efficient methods to quantify and propagate uncertainty in dynamical systems to make engineering control and design decisions. By computing expectations of the propagated state density, control inputs can incorporate system uncertainty into the decision-making process and be incorporated into optimization procedures involving expected losses and constraints on the system in question. This work demonstrates the computational advantages of a Koopman operator- based approach to uncertainty quantification and decision-making for deterministic models under parameter and/or state uncertainty in three key areas: initial state density modeling, optimal control parameter generation, and the computation of statistical quantities from propagated state densities.
The first section develops a computationally efficient approach to solving a probabilistic inverse problem using the Koopman operator for systems subject to expected value targets or probabilistic constraints on the model. A constrained quadratic programming approach is developed that allows a set of integral equations to be solved simultaneously to generate an initial probability density function that satisfies the model constraints. The second section demonstrates the computational advantages of a Koopman operator approach for generating a small number of discrete control decisions for systems under uncertainty compared with Monte Carlo simulation and direct uncertainty propagation using a direct density propagation method using the Frobenius-Perron operator. The next section com- pares a Koopman operator approach for computing statistical quantities from propagated system state densities to the Polynomial Chaos framework and Monte Carlo simulation. Several moment computation methodologies are compared in terms of computational performance in relation to the Koopman operator approach. Finally, the moment computation work is extended to marginal density propagation and quantile estimation in comparison to Monte Carlo methods. The proposed method utilizes the maximum entropy distribution as a marginal representation and accuracy is computed through quantiles and the Wasserstein distance.
Together, these novel applications of the Koopman operator to uncertainty quantification of models with state and/or parameter uncertainty provide an computationally efficient alternative method to existing methods of uncertainty quantification.
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Date
2022-07-30
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Dissertation