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Fitting Turbocharger Maps with Multidimensional Rational Functions
ISSN: 0148-7191, e-ISSN: 2688-3627
Published April 14, 2015 by SAE International in United States
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Turbocharger maps measured on a gas stand test bench are commonly used to represent turbine and compressor performance. The maps are useful source of information for mean value modeling, engine calibration optimization, virtual sensing and feedback control design. For some tasks, representing the maps by fitted functional forms can be more convenient than using interpolation of the map data directly. The functional representation usually allows for wider extrapolation ranges and more reliable application of numerical optimization methods.
In literature most successful functional forms chosen to represent the compressor flow characteristics are based on rational polynomials of dimensionless head and flow parameters. Turbine flow characteristics, on the other hand, are commonly modeled as orifices or orifices with variable cross-section in case of variable geometry turbines (VGT).
In this work the approach of modeling the compressor flow with rational functions is extended and also applied to compressor efficiency characteristics as well as turbine flow and efficiency. Particularly for the case of VGT turbine flows this can lead to higher dimensional rational functions of multivariate polynomials. Using standard nonlinear optimization methods to fit the coefficients of such models can lead to issues due to complex singularities found in such multidimensional rational representations.
This work presents a robust numerical optimization approach that fits parameters of the multivariate rational functions while avoiding issues with singularities and eliminates the requirement of providing consistent initial parameter values necessary for standard optimization methods. The accuracy of the proposed modeling approach is compared to commonly applied methods for fitting turbo maps found in literature. The capability of the proposed approach for the estimation of the turbo speed in form of an online virtual sensor is briefly discussed.
While the presented method has been evaluated for the case of turbo map fitting it is also suitable for other model identification problems especially cases containing singularities.
CitationPachner, D., Lansky, L., Germann, D., and Eigenmann, M., "Fitting Turbocharger Maps with Multidimensional Rational Functions," SAE Technical Paper 2015-01-1719, 2015, https://doi.org/10.4271/2015-01-1719.
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