CAREER: Model Reduction for Control of Fluids

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Model reduction for control of fluidsC. W. RowleyAbstractMany powerful techniques are available for analyzing and controlling dynamical systems, if one has a tractable mathematical model, for instance in the form of a differential equation. For fluids, and many other systems of interest, models are well known, but are too complex for traditional techniques from dynamical systems and control theory to be practical. This program develops techniques for obtaining simpler models of complex systems, in order to provide a bridge between interesting multiscale applications, and the mathematical tools useful for their analysis and control. The techniques will then be used to address problems in fluid mechanics, focusing on controlling transition to turbulence in a plane channel.Specific model reduction techniques addressed will include balanced truncation and Proper Orthogonal Decomposition (POD), and symmetry reduction methods from geometric mechanics. Each of these techniques has severe limitations when applied to fluids problems. POD often produces highly unreliable models, and often the accuracy of the models does not improve as the number of modes increases. Balanced truncation applies to stable linear systems, and has good error bounds that improve as the number of modes increases, but the method is not tractable for very large systems such as fluids. A central component of the research involves a new method for performing "balanced POD," which may be viewed as form of weighted POD that more heavily weights states of large dynamical importance. In addition, ideas from symmetry reduction, and heirarchical finite-element modeling combine to guide the development of local versions of POD and balanced truncation, which will capture traveling structures better than the global basis functions produced by standard POD and balanced truncation.The educational components of the program include developing an exchange program with Brazil, in order to promote engineering education in an international setting. In addition, a new outreach program will help local elementary school teachers introduce engineering into K-6 classrooms, by organizing undergraduates to volunteer their time to assist with the technical aspects of engineering projects.