The Woodruff School

SUMMARY

Many engineering systems are challenging to model in simulation because of the difficulty of measuring elements of the environment or inherent randomness. One framework for modeling system uncertainty is through augmenting the state space to include state and parameter uncertainty in the model. This method of uncertainty quantification allows for robust system modeling and design in applications ranging from canonical spring-mass-damper models to entry vehicle dynamics. This work demonstrates the computational efficiency of a Koopman operator approach to uncertainty quantification and decision-making for models under parameter and/or state uncertainty in three key areas: input uncertainty modeling, optimal control generation, and the computation of statistical quantities from propagated uncertainty. The first section develops a computationally efficient approach to solving a probabilistic inverse problem subject to expected value targets or probabilistic constraints on the model. The constrained quadratic programming approach allows a set of integral equations to be solved simultaneously and generate an input probability distribution that satisfies the model constraints. The second section demonstrates the computational advantages of a Koopman operator approach to generating optimal control decisions for systems under uncertainty compared with Monte Carlo simulation and direct uncertainty propagation using the stochastic Liouville equation. The final section compares a Koopman operator approach for computing statistical quantities from propagated model uncertainty to the Polynomial Chaos framework and Monte Carlo simulation. Statistical quantities such as the moments, quantiles, and marginal distributions are explored and compared with equivalent, state-of-the-art methodologies. 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 propagation.

SUBJECT:	Ph.D. Dissertation Defense

BY:	Joseph Meyers

TIME:	Wednesday, July 20, 2022, 9:00 a.m.

PLACE:	https://bit.ly/3z60U5U, Virtual

TITLE:	A Koopman Operator Approach to Uncertainty Quantification and Decision-Making

COMMITTEE:	Dr. Jonathan Rogers, Chair (ME/AE) Dr. Aldo Ferri (ME) Dr. Adam Gerlach (AFRL) Dr. Panagiotis Tsiotras (AE) Dr. Ye Zhao (ME)

SUMMARY Many engineering systems are challenging to model in simulation because of the difficulty of measuring elements of the environment or inherent randomness. One framework for modeling system uncertainty is through augmenting the state space to include state and parameter uncertainty in the model. This method of uncertainty quantification allows for robust system modeling and design in applications ranging from canonical spring-mass-damper models to entry vehicle dynamics. This work demonstrates the computational efficiency of a Koopman operator approach to uncertainty quantification and decision-making for models under parameter and/or state uncertainty in three key areas: input uncertainty modeling, optimal control generation, and the computation of statistical quantities from propagated uncertainty. The first section develops a computationally efficient approach to solving a probabilistic inverse problem subject to expected value targets or probabilistic constraints on the model. The constrained quadratic programming approach allows a set of integral equations to be solved simultaneously and generate an input probability distribution that satisfies the model constraints. The second section demonstrates the computational advantages of a Koopman operator approach to generating optimal control decisions for systems under uncertainty compared with Monte Carlo simulation and direct uncertainty propagation using the stochastic Liouville equation. The final section compares a Koopman operator approach for computing statistical quantities from propagated model uncertainty to the Polynomial Chaos framework and Monte Carlo simulation. Statistical quantities such as the moments, quantiles, and marginal distributions are explored and compared with equivalent, state-of-the-art methodologies. 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 propagation.