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Model reduction of port-Hamiltonian fluid dynamics

Project members: R. Lopezlena, J.M.A. Scherpen

Sponsored by: Instituto Mexicano del Petróleo

Fluid Mechanics is one of the cornerstones of engineering. Through the fluid dynamics equations: continuity, momentum and energy; it is possible to model with fair precision a high variety of engineering problems. Especially due to the advances of Computational Fluid Dynamics (CFD) engineers could take full advantage of this theory. Applications of CFD can be seen customarily in the aerospace, the petroleum and others industries. Nevertheless, traditionally these models could not be used for real-time control purposes due to the high dimension of the systems and the high computational demand of the corresponding algorithms. In order to deal with these high dimensional models and to adapt them for analysis and control purposes model reduction tools have to be further developed.

Over the years some techniques for model reduction of fluids have been developed, but still the necessary order reduction is mainly done by physical insight and experience or with empirical methods (e.g. Karhunen-Loève expansion) which may provide accurate models but fail to provide a reduced model interpretable in physical terms. Since physical conceptualization of results is important, there is a need for a more systematic approach which preserves the physical structure in the reduced system.

A fairly general approach to model physical phenomena with control purposes is known as the Port-Hamiltonian approach which has been useful to describe the dynamic behavior of a very diversified class of engineering systems, including mechanical and electrical, resulting in a structured representation of possibly nonlinear differential equations. Recently in the literature, the extensions of this theory to distributed parameter systems -- including fluids, -- were formalized, providing with this a uniform and structured representation of the dynamic behavior of physical systems.

The method based on symmetries can be used for reduction of Hamiltonian systems. Though very formal and general, it fails to be applicable for control engineering purposes due in part to the fact that important input-output control concepts, like controllability and observability, were not considered in their formulation.

During the course of this research we have shown that the Dissipativity Theory provides a very general framework to deal with the input-output control properties while preserving the physical structure of the system. Moreover, based on this theory several successful control algorithms can be used.

Based on the framework of Dissipativity Theory and the resulting models from the Port-Hamiltonian paradigm in fluids, the main objective of this research lies on the design of structure-preserving nonlinear balanced reduction algorithms.

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Next: Nonlinear control systems analysis Up: Modeling Previous: Modeling

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