B-splines for Diesel Engine Emission Modeling

The equivalence between linear interpolation and B-spline functions of degree 1 is described. The equivalence is used to express interpolation based diesel engine NOX and soot emission models as B-spline functions, and to apply data fitting methods for B-spline functions to perform calibration of the models. This strategy leads to that the model calibration can be calculated directly by means of analytically solving a minimization problem. The B-spline representation also makes it possible to control the smoothness and extrapolation behavior of the interpolation maps in the models in a controlled manner. The models have been calibrated using measured steady-state data from a 5-cylinder Volvo passenger car diesel engine with a displacement volume of 2.4 liters, equipped with a turbocharger, an exhaust gas recirculation system, and a common rail injection system. The calibrated model for NOX emissions predicts the NOX mass flow with a root mean square error of 1.6 mg/s, and the calibrated model for soot emissions predicts the soot mass flow with a root mean square error of 0.65 mg/s. The described equivalence between linear interpolation and B-spline functions of degree 1 could also be used for calibration of other models of similar structure.