Model accuracy

The general model structures are introduced in Chap. 2. In the following the particular realisations for the local models are shortly described and the model qualities are stated. The applied criteria of fit to compare the models are the coefficient of determination R2 _{and the root mean square error}

Look-up tables

The local models Pmnox;j.u/ depend on four input variables, see eq. (6.2), whereas the regarded

look-up tables are limited to two inputs. To map this higher dimensional relation, a combination of several two-dimensional look-up tables is applied. The utilised look-up table structure is derived from trial and error and depicted in Fig. 6.1.

Figure 6.1:Multiplicative local look-up table structure to model Pmnox;j.u/. Inputs of the look-

up tables and the multiplicative composition of the two-dimensional look-up tables are derived from trial and error. This look-up table structure is applied for each of the M D 21 engine operation points. The global model output is determined by eq. (6.1).

To identify this look-up table structure, at first the base look-up table is determined by a regression. The applied inputs mairand 'Q50have the highest contribution for such a regression, why these are

selected for the base look-up table. Thus, the four dimensional relation is approximated by a two- dimensional look-up table. Then the residuals are computed and the best performing correction look-up table is selected by trial and error. A multiplicative correction look-up table shows the best improvement, applying the inputs mairand p2i. Again, a regression with regard to these inputs

and the regarded residuals is performed to identify the correction look-up table. Then, a further correction look-up table is identified analogously with the selected inputs m_airand T_2i. Hence, the local model can be written as

P

mnox;j.u/D fbase.mair; 'Q50/
fcorr 1.mair; p2i/
fcorr 2.mair; T2i/: (6.4)

The look-up table inputs are each discretised in ten equidistant steps such that the discretisation is given by

uT D Œ68:8 mg

cyc; 0:11bar; 1:36

ı_{CA; 13:6}ı_{C: (6.5)}

The distribution of grid points is the same for all M D 21 local models, each defined for a fixed operation point. Since the drivability space varies with the engine operation point, there are re- gions in each look-up table that are not covered by measurements. Therefore, the regularisation as introduced in Sect. 2.1.2 is applied.

The attained model qualities are R2 _{D 0:991and RMSE D 2:08 mg=s for the training data and}

LOLIMOT

The LOLIMOT model approximates each local model Pmnox;j.u/by a superposition of affine func-

tions. The partitions on which the local affine models are valid are determined automatically by the tree construction algorithm. This algorithm divides the local drivability space by axis orthogonal splits in the input u. The maximum number of local model partitions is limited to ten and the best number of model partitions is selected by a heuristic approximation of Akaikes Information Criteria (AIC) implemented in the algorithm, see [122]. The local LOLIMOT model with p inputs and ML

partitions is then given by

P

mnox;j.u/D ML

X

iD1

ˆLOLIMOT;i.u/
w0;i C w1;iu1C : : : C wp;iup

(6.6)

For the 21 local operation points, there are in average ML D 8:38model partitions.

The attained model qualities are R2 _{D 0:992and RMSE D 1:93 mg=s for the training data R}2 _D

0:985and RMSE D 2:38 mg=s for the validation data.

LOPOMOT

Since LOPOMOT utilises local polynomial models of order three, there are no further partitions of the input space necessary. Regarding the local models, the LOPOMOT model simplifies to an adaptive polynomial model, as described in Sect. 2.2,

P

mnox;j.u/D w1x1C w2x2C : : : C wnxn (6.7)

with xi 2 A D˚1; u1; u2; : : : ; u4; u21; u1u2; : : : ; u34 : (6.8)

This model is the same as the one presented in Chap. 3 for Pmnox.

The attained model qualities are R2 _{D 0:996and RMSE D 1:45 mg=s for training data and R}2 _D

0:988and RMSE D 2:09 mg=s for validation data.

Kernel model

The Kernel model is a data driven approach, described in detail in Sect. 2.5. It is a class of model structures from which the applied type utilises one Kernel function for each measurement point .u.i /; y.i //of the training data. Therefore, the formulation of a local model is given by

P

mnox;j.u/ D

PML

iD1Kh.u; u.i //y.i /

PML

iD1Kh.u; u.i //

: (6.9)

with MLbeing the number of local measurements. The parameters to be identified are the elements

in eq. (2.67), and there is one bandwidth parameter for each dimension of u. To identify the band- widths, the ten-fold cross-validation error on the training data is minimised by applying a numerical optimisation algorithm, see Sect. 4.2.1. The bandwidths are individual identified for each of the 21 local models. The average bandwidth parameters over the 21 operation points are

Nh D  17:4 mg cyc; 0:05bar; 1:92 ı_{CA; 28:2}ı_C  : (6.10)

There are various alternative Kernel models to the one presented in Sect. 2.5, which differ in type and number of applied kernel function. The various approaches posses different properties, why the regarded Kernel model shall be regarded as an example for the class of Kernel models.

The attained model qualities are R2 _{D 0:997and RMSE D 1:24 mg=s for training data and R}2 _D

0:982and RMSE D 2:58 mg=s for validation data.

Comparison

The qualities of the various model structures are summarised in Tab. 6.1. The results show that all models attain an R2 _{above 0.99 on training data and are therefore all able to model the measured}

NOx emissions with minor differences in accuracy. The validation error on the other hand shows

some drawbacks for the Kernel model and the look-up table structure.

Table 6.1: Model qualities of the four regarded model structures for models of Pmnox.z; u/.

Model qualities are stated in coefficient of determination R2 _{and root mean squared error}

RMSE for the training and validation data as presented in Sect. 3.3.

Model Training Validation

R2 RMSE R2 RMSE

Look-up tables 0.991 2.08 mg/s 0.982 2.56 mg/s LOLIMOT 0.992 1.93 mg/s 0.985 2.38 mg/s LOPOMOT 0.996 1.45 mg/s 0.988 2.09 mg/s Kernel model 0.997 1.24 mg/s 0.982 2.58 mg/s

The LOPOMOT structure shows the best performance, which is due to its possibility to adapt to the non-linearity of the process by selecting the significant regressors for a local model. The LOLIMOT structure also adapts to the process by partitions of the input space, why this structure results in comparable results. The look-up table structure suffers from its equidistant distribution of grid points. Since for each local model the same discretisation is applied, some regions of the look- up tables are not covered with measurements. The quality of the Kernel model mainly depends on the coverage of the input space. Since the training values are part of the model structure, it shows the best performance on training data. The quality on the validation data depends on the distances to the training values and is the worst for the regarded model structures.