Overview of Engine Response Modelling

Once engine test data is collected from an engine test bed, empirical models of engine responses can be built by using the response surface methodology based on the test data. A variety of types of models, both parametric and non-parametric, are commonly used for engine responses (Morton, 2002 , Ward et al., 2002). Polynomial models have been commonly used to develop “local” models for engine response features; they have been found to provide an adequate representation of engineering trends and a reasonably accurate approximation over relatively narrow operating spaces (Saunders, 2004). Non-parametric estimators can be useful when the responses of interest are relatively complex or there is little prior knowledge regarding the expected behaviour of the system. The Matlab® MBC toolbox provides a non-parametric modelling capability for Radial Basis Functions (RBF) models, which is a type of neural network (Morton, 2002). While the RBF models offer the attractive feature of local adaptability, they suffer from the curse of ‘over fitting’. Research (Saunders, 2004) has shown that the choice of good measures for the quality of the model in a statistical sense (such as cross validation, Akaike Information Criteria or Bayesian Information Criteria) is very important. Such features are available in MBC and have been used in fitting models for the engine responses in this case study.

Since Holliday (Holliday, 1995) first suggested the use of a two-stage modelling method for Spark Ignition (SI) statistical engine modelling processes, this has

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become a widely used methodology for Gasoline engines.. The two-stage modelling matches the natural process of conducting gasoline engine experiments. The first stage concerns the modelling of engine responses (such as torque) over the spark sweeps, while the second stage is to fit a model of these engine responses or features as a function of other engine variables and parameters (such as engine load, speed and air/fuel ratio) (Davis and LAWRANCE, 2000). Table 3.1 shows engine input and output variables from one of the early two stage modelling applications (Davis and LAWRANCE, 2000).

Table 3. 1 Engine Mapping Inputs and Outputs

Inputs Outputs

Air/Fuel Ratio Torque

Exhaust Gas Recirculation NOx

Spark Timing CO

Speed HC

Load

The principle of two-stage modelling can be applied more generally for ‘global’

engine response models.

The introduction of the two-stage engine modelling approach (Holliday et al., 1998) for SI engine calibration required the introduction of more complex models capable of representing engine behaviour over the whole operating space. Similarly, the broad requirements for a ‘global’ model are to adequately represent the engineering trends and also to have flexibility or local adaptability to capture ‘local’ changes.

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Polynomial models are generally not adequate for global models or when more variables are involved; higher order polynomials can model the general trends, but still lack the local flexibility. Alternative modelling strategies have been introduced, such as semi-parametric models based on cubic splines (Grove et al., 2004) and non-parametric models including artificial neural networks (Han et al., 2004), radial basis functions (Morton, 2002), and stochastic process models or Kriging (Jeong et al., 2006). Of these, radial basis functions and Kriging models are the most commonly used in engine modelling practice.

As one type of Artificial Neural Networks (ANN), Radial Basis Functions (RBF) are a powerful tool, which has proved useful for fitting data points with a large number of inputs (Morton, 2002). RBF models take a general form which is similar to the polynomial models (Equation 3.2):

(‖ ‖) Equation 3.2

is the vector of inputs and is the centre of the th

radial basis function term.

‖ ‖ is give by the Euclidean distance between input and centre , shown in

Equation 3.3:

‖ ‖ √ Equation 3.3

is the width of the radial basis function, and is a strictly positive symmetric

function called the ‘kernel’ with a unique maximum at its centre . Examples of

commonly used kernels include Gaussian, multi-quadric, inverse multi-quadric, thin- plate spline, and logistic functions.

ANN methods have been used to obtain a global engine model and are statistical tools of artificial intelligence, which are efficient in solving a wide range of problems

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including many engine calibration applications (Han et al., 2004 , Howlett et al., 1999). This approach has the ability of predicting the engine response models of factors and interactions between factors over the load-speed range of the engine drive cycle (Alonso et al., 2007 , Atkinson et al., 1998) .

Kriging can be regarded as a generalisation of RBFs, and it has been introduced as an interpolation technique based on a stochastic process model (Forrester et al., 2008). The predicted response at a point x0 is calculated as the weighted average of the neighbouring points, i.e.

̂ ∑

Equation 3.4

are the interpolation weights, and is the realisation of a Gaussian stochastic

process with mean zero, variance v, and non-zero covariance (Seabrook, 2000). In simple kriging, the weights are derived based on the assumption that the mean and the covariance of y(x) are known, thus the kriging predictor minimises the variance of the prediction error.

One of the problems associated with using non-parametric models is that they suffer from the curse of over-fitting (Cary, 2003). Model selection and validation must be carefully conducted such that the model balances the error of approximation with the error due to random fluctuations. Model validation is usually based on the statistical analysis of residuals, and relies on measures such as Root Mean Squared Error (RMSE) of residuals and Predicted Residual Error Sum of Squares (PRESS). More advanced model selection and validation criteria include generalised cross-validation, Akaike information criteria (AIC) and Bayesian information criteria (BIC) (Saunders,

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2004). The information based methods (AIC and BIC) introduce a penalty for the effective number of terms in the model, so they have the effect of reducing the modelling sequence (i.e. the number of terms in the model), thus effectively addressing over-fitting.

Within the engineering arena, non-parametric RBF and kriging models have been widely used in recent years, in particular in conjunction with experiments involving computer based models where a surrogate model is sought in order to address computational expense. Within this context, studies comparing RBF and kriging models have been reported (Chandrashekarappa and Duvigneau, 2007 , Costa et al., 1999 , Peter et al., 2007) pointing to the fact that in general kriging models tend to be more accurate.