Efficient Algorithms for Optimal Control of Time-Periodic and Nonlinear Systems

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Fluids often exhibit cyclical motion, either due to external periodic forces (e.g., lunar tides) or due to internal forces that naturally arise through interaction with the environment (e.g., air buffeting through a slightly opened car window or the eddies that develop from a river current interacting with a bridge pillar). The control of such flows is of great interest since in many cases small changes in a flow profile can produce either dramatic benefits or catastrophic costs. Stabilizing and tuning cyclic flow patterns to a given frequency can be useful in the design of wind farms and wave energy converters, for example, while in other circumstances eliminating oscillatory motion altogether may help reduce fatigue loads placed on critical support structures. This project specifically addresses the modeling and control of oscillatory phenomena through the development of new mathematical algorithms for simulation and control that integrate underlying periodic system behavior into the core modeling framework, thus promising both improved accuracy and reduced computational costs. This research will result in improved understanding of mathematical models of systems with periodic behavior as well as the development of new simulation and modeling tools, which will have an immediate bearing on a wide range of applications found in biology (e.g., circulatory and respiratory systems) and energy (e.g., wind turbines and power grid dynamics).Simulation and control of periodic flow structures require methods specialized to the task. The Floquet transformation has long been a theoretical tool for such problems, allowing for a change in system representation that effectively shifts the periodic time dependence out of the internal dynamics into the input/output ports. Only recently has it been practical to use this approach for small to medium scale problems. The initial focus of this research will be on the development of scalable, numerically effective algorithms for large-scale Floquet transformations, enabling the high-fidelity reduced models for time-varying periodic flow dynamics. By combining operator splitting approaches with optimal model reduction methods for quadratic systems, new capabilities for input-independent model reduction for nonlinear dynamics will be developed and analyzed. The model reduction framework that is proposed here offers improved numerical efficiency and greater accuracy at modest cost. Fundamental to this approach will be the integration of an accurate representation of dynamics into reduced-order models, better respecting the properties of the underlying optimal control problem. Robust computational tools aiding the simulation and modeling of large-scale oscillatory dynamics will be developed and provided to the science and engineering community.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.