DOI: 10.5281/zenodo.13920145
title: Datasets of Paper "Data-Driven Parametric Optimization for Pre-Calibration of Internal Combustion Engine Controls"

description: This dataset consists of three parts.

The first part contains the code for Rational Bézier Regression (RBR), which is an integrated function for Rational Bézier Curve Regression (RBCR) and Rational Bézier Surface Regression (RBSR), written in MATLAB. It includes a subfunction for the Bernstein Polynomial.

The second part contains the datasets for pre-calibration measured from a real vehicle, including Measurement A and Target Calibration. Measurement A consists of 8 ECU signals in total, specifically 7 ECU inputs:

• S_n_Engine_min: engine idle state (possible values: 0 or 1)• S_Coasting: vehicle coasting state (possible values: 0 or 1)• T_Ambient: ambient temperature (unit: ◦𝐶)• n_Engine: engine speed (unit: 𝑚𝑖𝑛−1)• m_Air_Flow: air mass flow (unit: 𝑘𝑔∕ℎ)• v_Vehicle: vehicle speed (unit: 𝑘𝑚∕ℎ)• T_Engine: engine temperature (unit: ◦𝐶)

And 1 ECU output:

• T_Oil: engine oil temperature (unit: ◦𝐶)

The Target Calibration includes 3 Curves and 1 Map used in the real vehicle:• Curve_1: influence factor of ambient air temperature f_T_Ambient• Curve_2: influence factor of engine speed f_n_Engine• Curve_3: influence factor of engine temperature f_T_Engine• Map_1: influence factor of load and vehicle speed f_m_Air Flow/v_Vehicle

The third part contains the optimized parameter vectors obtained after applying different optimization procedures mentioned in the paper, with a total of 12 parameter vectors.

 
description: Implementation Idea of the RBR-AlgorithmThe Rational Bézier Regression (RBR) algorithm is a parametric regression method based on Rational Bézier Curves (RBCR) and Rational Bézier Surfaces (RBSR). It extends classical Bézier curve fitting by introducing weight parameters that allow for more flexible and precise approximations. The implementation consists of two main cases: RBCR (1D) for curves and RBSR (2D) for surfaces.

Step 1: Identifying the Regression Type



If only one independent variable 𝑥 is given, the function performs RBCR.

If two independent variables  (𝑥,𝑦) are provided, the function performs RBSR.


Step 2: RBCR (For the case where only 𝑥 is provided):

1. Initialize Parameters:



The parameter vector 𝑝 contains control weights that determine the shape of the fitted curve.

The total number of parameters is 2𝑛, where 𝑛 is the degree of the Bézier curve.


2. Construct the Rational Bézier Curve:



The function iterates through all input values of 𝑥.

The Bernstein polynomials are used as basis functions to construct both the numerator and denominator of the rational Bézier function.

The numerator is computed as a weighted sum of Bernstein polynomials, where the weights come from the parameter vector 𝑝.

The denominator is constructed in a similar way to normalize the function.

The final output is obtained by dividing the numerator by the denominator and scaling it with an additional parameter.


Step 3: RBSR (For the case where both 𝑥 and 𝑦 are provided):

1. Initialize Parameters:



The number of parameters is determined based on a Bézier surface of degree 𝑛.

The parameter vector 𝑝 is split into control weights and an additional rotation angle 𝜃.


2. Apply Park Transformation:



The input coordinates (𝑥,𝑦) are rotated using a transformation matrix based on 𝜃.

This transformation enhances the flexibility of the model by adapting to rotated data patterns.


3. Construct the Rational Bézier Surface:



The function iterates over all input values of 𝑥 and 𝑦.

The Bernstein polynomials are computed separately for each dimension and combined to form a tensor-product basis.

Similar to RBCR, the numerator and denominator are constructed using weighted Bernstein polynomials.

The final output is computed by normalizing the numerator with the denominator and applying a scaling factor.