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Eighth International Symposium on Turbulence and Shear Flow Phenomena
August, 28-30, 2013, Poitiers, Futuroscope, France

DOI: 10.1615/TSFP8

APPLICATION OF KALMAN FILTERING AND PARTIAL LEAST SQUARE REGRESSION TO LOW ORDER MODELING OF UNSTEADY FLOWS

pages 1-6
DOI: 10.1615/TSFP8.860
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RESUMO

This study takes place in the framework of the enhancement of the stability of POD-Galerkin Low Order Dynamical System (LODS) identified from Time-Resolved particle Image Velocimetry (TR-PIV) measurements.
As proved by Noack et. al. (2005), the dynamics of incompressible flows can be efficiently represented by empirical LODS containing constant, linear and quadratic terms. However, POD-Galerkin LODS are known to diverge, or damp, rapidly in time if left uncontrolled, which led a number of authors to introduce regularization terms in order to stabilize the models via constrained optimization problems (Bergmann et al. 2005).
From a stochastic point of view, the Kalman Filter (Kalman, 1960) and its derivatives can be used to stabilize the evolution of dynamical systems, provided that state laws and observable data are available. In this formalism, LODS are modelised as discrete time hidden Markov processes in which the data of interest are represented by hidden variables that are only accessible from the state law and their relationships with observable variables. The purpose of Kalman filtering is to operate from both a state law and measured data in order to generate the best possible estimate of the hidden variables. In this study the observations are provided by hot-film measurements and linked to the Kalman filter using the partial Least Square Regression (PLSR) (Wold, 1966) as an alternative to the Linear Stochastic Estimation (LSE) (Taylor and Glauser, 2004) for over-determined data. Thus, applying a combination of PLSR and Kalman filters to POD-Galerkin LODS can be seen as a solution to reconstruct and analyse unsteady flows while preserving the physics governing their dynamics and without inserting adjustment or regularisation terms.

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