Application of linearised hierarchical models to ... - Inderscience Online

Int. J. Powertrains, Vol. 5, No. 4, 2016

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Application of linearised hierarchical models to indicated torque modelling for a turbocharged engine Mark Cary* Ford Motor Company, Dunton Technical Centre, GB 15/2A-G13-C, Laindon, Basildon, Essex, SS15 6EE, UK Email: [email protected] *Corresponding author

Byron Mason and Peter Schaal Department of Aerospace and Automotive Engineering, Loughborough University, Stewart Miller Building, University Way, Loughborough, LE11 3TU, UK Email: [email protected] Email: [email protected] Abstract: This paper shows the application of linearised hierarchical models to the estimation of indicated torque data obtained from a large scale engine mapping experiment conducted on a turbocharged spark-ignition engine. Unlike previous studies that have utilised two-stage regression techniques for analysis, the use of linearised methods provides a framework to directly address the issues of sparseness of sweep-specific data and mixed effects modelling. In addition, spline models are presented at both levels of the model hierarchy that possess the required smoothness, are capable of capturing physical behaviour and simultaneously yield models sufficiently accurate for calibration work. The paper considers, at length, the required model fitting procedures which are founded on an iterative generalised least-squares approach. Further, a model building case study is presented addressing the issue of which factors should be modelled as fixed or mixed. This utilises information criteria to identify the most parsimonious model. Finally, the fit provided by the model to the data is demonstrated, which is seen to satisfy the engineering measure of success applicable to the application. Keywords: hierarchal models; two-stage regression; torque; engine mapping and calibration; data sparseness. Reference to this paper should be made as follows: Cary, M., Mason, B. and Schaal, P. (2016) ‘Application of linearised hierarchical models to indicated torque modelling for a turbocharged engine’, Int. J. Powertrains, Vol. 5, No. 4, pp.395–411. Biographical notes: Mark Cary received his MEng degree from City University in 1986 and since then has worked at Ford Motor Company’s Dunton Technical Centre in the fields of engine calibration and engine mapping. He received his PhD degree from the University of Bradford in the application of longitudinal models to spark ignition engine modelling, Copyright © 2016 Inderscience Enterprises Ltd.

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M. Cary et al. calibration and optimisation in 2003. His areas of interest include longitudinal modelling, nonlinear repeated measurements, power train modelling, power train dynamics and controller calibration. Byron Mason is a Senior Lecturer in Advanced Propulsion. He completed his PhD on the topic of real time powertrain dynamic modelling. Since this time his work on powertrain model development, testing, control and calibration has continued. He is actively involved in a number of funded research working alongside industrial and academic partners small and large. Peter Schaal is a PhD student at Loughborough University where his research has focused on the development of a rapid engine characterisation methodology. He completed his Mechanical Engineering in Ulm, Germany June 2013. This paper is a revised and expanded version of a paper entitled ‘Application of linearised hierarchical models to indicated torque modelling for a turbocharged engine’ presented at the 2nd International Conference: Powertrain Modelling and Control, Testing, Mapping and Calibration, Bradford, UK, 10–12 September 2014.

1

Introduction

Engine mapping is the process by which the engine responses (outputs of interest) are mapped to control inputs and the engine state. This information is held in the form of a model that is ultimately used for parameterising control strategies that may contain presently in excess of 50,000 parameters. The core of many modern strategies is a torque model that is used by the controller to derive and optimally set engine actuator positions to achieve for example minimum BSFC, emissions, etc. Torque models are derived from mapping experiments that at the most fundamental level record engine torque at some fixed actuator settings as spark is varied from its maximum to minimum levels. In practice during testing, knock, exhaust temperatures and combustion stability limit the range of the spark actuation over the engine operating space. Data is then fitted to models using a two stage regression-based approach. This approach has largely superseded single stage approaches reducing the number of sweeps required and the number of model parameters used to describe the engine behaviour. As noted by Holliday (1995) spark sweep data is fundamentally hierarchical. At level 1, curves are fitted to each of the spark sweep response profiles. These describe how brake torque varies with spark advance and a set of response features are calculated from the fit coefficients. At level 2, individual response feature models are built as a function of for example of speed, inducted air mass and air-fuel ratio (Cary, 2003). Then, as is usually the case maximum likelihood estimates (MLE) are used to estimate level 2 model parameters. At the end of this process one is able to estimate response features as a function of engine speed, inducted mass and air-fuel ratio from which spark-torque curves can be reconstructed.

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1.1 Problem statement Several investigators have applied nonlinear hierarchical modelling techniques to the analysis of data from engine mapping experiments (Holliday, 1995; Cary, 2003; Holliday et al., 1998; Rose et al., 2002). All of these studies are predominantly based on the two-stage regression approach. A requirement for the use of the two-stage method is the availability of sufficient data to obtain regression parameter estimates for each sweep. For naturally aspirated engines this proves not to be a significant restriction. However, for turbocharged engines sweep-specific model fitting will not be feasible for a large number of sweeps and the two-stage regression approach is no longer possible. For example, consider Figure 1(a) presents a typical light-to-moderate load spark sweep profile. Under these conditions the spark sweep is characterised by a well-defined maximum with the indicated torque response falling away on either side. The number of test points varies due to the limitations highlighted on Figure 1. The rate of decay of indicated torque with spark timing is usually steeper to the left of the maximum than to the right. Figure 1(b) presents a high load sweep, which is clearly severely limited in extent. At low speed, high load spark sweeps will be limited primarily by the onset of spark-knock. Similarly, at higher speeds and loads, limits due to knock, exhaust temperature and peak-cylinder pressure apply. The net result is to severely limit the scope of these sweeps; perhaps to as few as three of four points which may necessitate the removal of a number of spark sweeps. Figure 1

(a) Typical light load spark sweep profile. Curve is characterised by a well-defined maximum (b) Typical high load spark sweep profile. Curve is limited by exhaust temperature and knock or peak cylinder pressure constraints (see online version for colours)

Sparseness of sweep-specific data is not the only motivation for considering an analysis based on linearisation methods. In particular, model building in mixed effects models involves questions that do not have a direct analogue in both fixed effects linear and nonlinear analyses (Pinheiro et al., 1994); including which effects should have an

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associated random component and which should be regarded as purely fixed. As will be demonstrated, analyses based on linearised methods can incorporate both mixed and fixed effects in a straightforward manner. The paper is organised as follows: in Section 2 the form of the nonlinear hierarchical model employed for analysis is introduced. First order linearisation techniques applied to this model yield an approximate marginal probability density for the sweep-specific response data. Treating this approximation as exact represents the basis of the analysis methods presented.

1.2 Structure Section 2 briefly describes the experimental design and the data collection techniques. The specific form of the nonlinear hierarchical model to be trained is detailed in Section 3. Both level 1 and level 2 models are formulated using splines. At level 1 torque response to spark timing is considered and at level 2 the factors engine speed and normalised load. Section 4 and subsequent subsections detail the training algorithms employed. First, in Section 4.1 we introduce an iterative generalised least squares (IGLS) approach. Then in Section 4.2 we modify the basic algorithm by introducing the profile likelihood function. Section 4.3 addresses some miscellaneous points regarding starting values and practical aspects related to the implementation of the IGLS algorithm. Section 5 presents a case study in model building. Here the question of interest is which components of the sweep-specific parameter vector should be considered as fixed and which are mixed; i.e., possess a fixed and random component. The structure of the linearised hierarchical model chosen is demonstrated to accommodate these aspects. Information criteria are used to guide the selection of a parsimonious model, i.e., the most economical in terms of parameters whilst maintaining sufficient accuracy. Section 6 presents diagnostic data to verify the fit quality to the training data. Sweep recovery is seen to be excellent for a variety of spark sweep profiles. In addition, response surface plots presented are shown to agree with physical knowledge and exhibit the desired level of smoothness demanded by the application. Finally, Section 7 presents a summary of the paper and the appropriate conclusions.

1.3 Linearised analysis At a high level, in this investigation the focus will be nonlinear hierarchical models of the form (Rose et al., 2002; Davidian and Giltinan, 1995): Level 1 = y i = f ( βi ) + ei , ei ≈ N ( 0, σ 2 I )

(intra-sweep)

Level 2 = βi = d ( ai , β ) + Fbi , bi ≈ N (0, Γ) (inter-sweep)

(1)

where N denotes the normal distribution, y i ∈ ni is the vector of observed responses for the ith sweep, f is a nonlinear function of the sweep-specific parameter vector βi ∈ \p, ei ∈ ni a normally identically and independently distributed level 1 random error term, β ∈ \r is a vector of fixed population parameters, bi ∈ \k is a vector of

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random effects associated with the ith sweep, d(ai, β) ∈ \p ad a vector valued function that describes the systematic variation in the sweep-specific parameter vector with the level 2 covariates ai and F ∈ \p×k is a fixed design matrix. Employing the first-order linearisation scheme detailed in Sheiner et al. (1972) and Davidian and Giltinan (1995), the marginal density of yi is given by: E [ y i ] ≈ f ( d ( ai , β ) ) Var [ y i ] ≈ Zi ( β )Γ(ω)ZTi ( β ) + σ 2 I ni = W( β , ω)Zi ( β ) = J i ( β )F

where J i ( β ) ∈

ni × p

(2)

is the matrix of derivatives of f(βi) with respect to βi, evaluated at

βi = d(a, β) and ω ∈ \0.5k(k+1) a vector of level 2 covariance parameters corresponding to the unique elements of the symmetric matrix Γ. This representation affords considerable flexibility in terms of model formulation. Specifically, it permits some or all of the elements of βi to be set as purely fixed or mixed effects as desired. Clearly, if all elements of βi are considered to be mixed-effects, i.e., possess both a fixed and random component, F is the p-by-p identity matrix. Alternatively, if one or more of βi are considered to be purely fixed effects then F ∈ \p×k and the corresponding bi ∈ \k. For example, assume p = 3 and k = 2, with the third element of βi taken as a purely fixed effect. Then: ⎡1 0 ⎤ ⎡ b1 ⎤ F = ⎢⎢ 0 1 ⎥⎥ and bi = ⎢ i2 ⎥ . ⎣bi ⎦ ⎢⎣ 0 0 ⎥⎦

2

(3)

Experimental data

All data was collected on a 4-cylinder 1.6l GTDI 16-valve engine. The engine was equipped with variable valve timing; however all data presented in this study were collected at a fixed intake valve opening (IVO) and exhaust valve closing (EVC) event of 0 and 11 [oATDC] respectively. Details of the engine design are available in Appendix. Consequently, in this investigation, the level 2 factors are engine speed (N) and normalised induced air charge (L). The principle adopted for data collection was as follows. For each of the 65 fixed (N, L) combinations dictated by the experimental design detailed in Figure 2, spark timing (s) was swept from one extreme to the other. Practical limitations imposed on combustion stability, exhaust gas temperature, spark-knock and peak cylinder pressures (Heywood, 1988) bound the available range of s across sweeps. The corresponding engine response, indicated torque (T), was measured at each s. Consequently, as depicted in Figure 2, the experimental output can be viewed as a collection of data sweeps (or sweep-set) the form of which varies as a function (N, L).

400 Figure 2

M. Cary et al. Level 2 experimental design (see online version for colours)

Throughout each sweep, air-fuel-ratio was held at stoichiometric or enriched at the leanest value required for component protection by the test automation software. Component temperatures were continually monitored and the sweep terminated if they were exceeded. Flush mounted combustion pressure transducers permitted combustion stability, knock intensity levels (Brunt et al., 1998) and peak cylinder pressure limits to be monitored. Again the sweep was terminated if engine hardware limits were violated. Reflecting on the obvious composition of the data, it seems sensible to employ inferential procedures that can naturally accommodate the imposed structure. As first suggested by Holliday (1995) and Holliday et al. (1998), it is in this context that nonlinear models postulated for the analysis of repeated measurement data seem ideal (Davidian and Giltinan, 1995, 1993 Vonesh and Carter, 1992). The relevant sweep-set is presented in Figure 3. Figure 3

Sweep set comprised of 65 spark sweeps collected at fixed camshaft timing (IVO, EVC) = (0, 11) (see online version for colours)

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Specific form of the model

In this section, we detail the specific form of the level 1 and level 2 models utilised. Both levels of the model hierarchy utilise spline formulations. The intent of the level 1 model is to explain the relationship between spark advance and indicated torque. In particular we are interested in specifying the form of the fixed function f(βi); this is assumed to be a spline model of the form: Tij = εoi +

ε εci ε ( sij − ki )3− + qi ( sij − ki )2− + ri ( sij − ki )2+ 1, 000 100 100

(4)

where Tij represents the jth indicated torque value for the ith sweep, sij denotes the corresponding spark timing, ki the spline knot value, and εoi, εci, εq and εr are sweep-specific fit coefficients. Further, (sij – ki)– = min [0 sij – ki] and (sij – ki)+ = max [0 sij – ki]. Clearly for this model: βi = [εoi εci εqi εri ki]T. Note parameters, εci, εqi and εri are scaled accordingly to ensure that all element of βi possess similar magnitude. Similarly at level 2 in the model we are particularly interested in specifying the form of the vector valued function d(ai, β). Again use is made of spline formulations. For εoi appropriate model is: εoi = po + p1 Ni + p2 Ni2 + p3 Li + p4 Li N i + p5 ( N i − k p )+ + p6 Li ( N i − k p )+ 2

2

(5)

This choice is based on knowledge regarding the physical interpretation of this parameter. Essentially, εoi represents the peak torque (PKTQ) achievable on the ith sweep. It is well known that PKTQ varies linearly with normalised air charge. Likewise, PKTQ varies approximately in a cubic fashion with engine speed. Use of a quadratic spline in engine speed allows the model to mimic the asymmetry suggested by a cube but, unlike a cubic polynomial, rules out any upturn in the curve at high engine speeds; such behaviour is clearly non-physical. The remaining elements of βi are modelled according to equation (6). This is a two-dimensional spline basis. A conventional quadratic spline is employed in the N direction. However, for L below kL a cubic response is permitted. However, above kL ly a linear response is permitted. This permits the response in L to be potentially very steep and highly curved below the knot, while retaining a linear characteristic at high values of L. εci , εqi , εri , ki = co + c1 Ni + c1 N i2 + c3 Li + c4 Li Ni + c5 Li N i2 + c6 ( Ni − k N )+ 2

+c7 Li ( N i − k N )+ + c8 ( Li − k L )− + c9 Ni ( Li − k L )− 2

4

3

3

(6)

Model training procedures

In the following subsections, we discuss the applied model fitting algorithms, which are based on maximum likelihood methods. We begin, in Section 4.1, with a general algorithm provided by Davidian and Giltinan (1995). This details an iterative sequential process for minimisation of the appropriate log-likelihood functions. In Section 4.2, we

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introduce the use of the profile likelihood (Lindstrom and Bates, 1990, 1988) which permits a scaling or so-called nuisance variable to be eliminated from the problem. This results in a more robust algorithm from the perspective of convergence. Finally, in Section 4.3 methods for obtaining suitable starting values are discussed.

4.1 An IGLS for linearised models IGLS is preferred over joint maximum likelihood because it exhibits less sensitivity to non-normality of the marginal distribution of yi and misspecification of the covariance structure (Davidian and Giltinan, 1995). For the purposes of inference the approximate first two moment specifications defined by equation (2) are taken as exact. The basic IGLS algorithm is as follows: 1

Form preliminary estimates βˆ ( p ) by minimising the ordinary least squares (OLS) cost function: m

∑(y

− f ( d ( ai , β ) ) )

T

i

( y i − f ( d ( ai , β ) ) )

(7)

i =1

2

Using these preliminary βˆ ( p ) , determine the corresponding estimates ωˆ by minimising the pseudo-likelihood (LPL) cost function: m

LLP =

∑ ⎡⎣log W ( βˆ i

i =1

( p)

, ω ) + riT Wi−1 ( βˆ ( p ) , ω ) ri ⎤⎦

(8)

where ri = yi – f(d(ai, β)) 3

Using the estimates ωˆ , form the estimated weight matrices Wi−1 ( βˆ ( p ) , 0, ωˆ ), i = 1, 2,… , m.

4

Using the Wi−1 ( βˆ ( p ) , ωˆ ) from step 3, re-estimate β by minimising: m

∑ ⎡⎣( y i =1

5

i

T − fi ( d ( ai , β ) ) ) Wi −1 ( βˆ ( p ) , ωˆ ) ( y i − fi ( d ( ai , β ) ) ) ⎤⎦

(9)

Treating this estimate as a new preliminary estimate return to step 2. This scheme may be iterated a fixed number of times or to convergence, with at least one iteration recommended. Note, convergence is not guaranteed (Davidian and Giltinan, 1995).

4.2 A modified IGLS algorithm based on the profile likelihood Lindstrom and Bates (1990, 1988) suggest scaling the level 2 covariance matrix so that bi ≈ N (0, σ 2 Γ). Under these circumstances, a closed form expression can be derived for the maximum likelihood estimator of σ2. That is:

∑ ( y − f (d ( a, θ, 0) )) W ( y = ∑n T

2 σ PL

−1 i

i

i

i

− f ( d ( a, θ , 0 ) ) )

(10)

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with Wi = Zi ΓZTi + I ni . Equation (9) can be back substituted into (7) to obtain the

corresponding profile likelihood function denoted by LpPL . Hence: LpPL

=

∑

+

⎡ ni log ⎢ ⎢ ⎣

∑(y

− f ( d ( a, θ , 0 ) ) ) Wi−1 ( y i − f ( d ( a, θ , 0 ) ) ) ⎤ ⎥ ⎥ ni ⎦ T

i

∑ log W + ∑ n i

∑

(11)

i

In their article, Lindstrom and Bates (1988) recommend the use of LpPL over LPL. For the linear case considered, Lindstrom and Bates suggest that optimising the profile likelihood requires less iteration and provides for more consistent convergence. They also cite experience where the application of the Newton-Raphson algorithm to the likelihood functions failed to achieve convergence, but application to the equivalent profile likelihood succeeded in yielding an optimal solution. Use of the profile likelihood requires modification to the IGLS algorithm presented in Section 4.1. Step 2 of the algorithm described now involves minimisation of equation (10).

4.3 Choice of starting values for the modified IGLS algorithm and miscellaneous points related to model training Numerical techniques to implement the IGLS algorithm stated in the previous section often require that appropriate starting values are available. In this regard, the preliminary estimators in step 1 of the algorithm represent one possible starting value for β. In this study, a sequential optimisation strategy was adopted to determine the OLS estimates (Ross, 1970). To begin we developed a crude estimate of both boi and ki by first fitting a second order polynomial to each individual spark sweep. Sweep-specific parameters boi and ki were then determined by differentiation; εoi being set to the curve maximum and ki to the corresponding spark value at which the maximum is attained. The model specified by equation (4) was then fitted to these preliminary estimates to obtain initial values for the corresponding level 2 fixed parameter estimates. Denote these estimates by εop and kp respectively. 1

Then assuming the sets of coefficients εop and kp are fixed, next minimise the least squares cost function for the remaining unknown elements of β; that is, determine the corresponding coefficient sets εqp and εrp .

2

Now assuming εqp and εrp are fixed determine εop +1 and kp+1 by minimising the equation (4) with respect to these parameter estimates. Then treating εop +1 and kp+1 as preliminary estimates return to step 1 and establish εqp and εrp by once again minimising the OLS cost function. Repeat steps 1 and 2 several times or until convergence if preferred.

Given the preliminary βˆ ( p ) , we now consider starting values for ωˆ . Throughout we initially assume Γ = Ik. This aligns with previous work presented by Rose et al. (2002). In practice Γ is factorised using the Cholesky factorisation (Gill et al., 1981). Initially, this

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implies that Γ 2 = I. Further, to address the issue of lack of uniqueness of the Cholesky factorisation (Pinheiro and Bates, 1996), in practice the log Cholesky factorisation was employed. The log Cholesky factorisation implements a constraint by taking logs of the 1

parameters on the leading diagonal of Γ 2 . The result is a unique factorisation for Γ. This approach also ensures that step 2 of the IGLS algorithm is an unconstrained optimisation. The MATLAB function fminunc from the optimisation toolbox was employed to determine ωˆ . The physical interpretation of the level 1 spline knot parameter ki is it represents the spark timing corresponding to the location of the maximum indicated torque value for the ith sweep. It is well known that this optimal spark timing parameter must decrease with increasing L. To ensure this, the minimisation of equation (8) was subject to the nonlinear constraint: ∂ki ( N , L) ≤0 ∂L

(11)

Further, for ki to correspond to the maximum value parameters εqi, εri ≤ 0. Rather than implement these constraints directly, we introduce the new variables: γqi = log10 ( εqi ) γri = log10 ( εri )

(12)

And then rewrite equation (3) in the form: Tij = εoi +

εci 10γqi 10γri ( sij − ki )3− − ( sij − ki )2− − ( sij − ki )2+ 1, 000 100 100

(13)

This action is equivalent to incorporating the constraints εqi, εri ≤ 0 the model, substantially reducing the number of constraints to be subsequently handled by the optimisation routine – considerably improving speed. Finally, minimisation of equation (8) was carried out in MATLAB using the optimisation toolbox function fmincon.

5

Model building

In this study, the model building question of interest is which parameters should be mixed effects and which should be considered purely fixed? Different prospective models can be fitted and nested models compared using a performance criterion such as Aikaike’s information criterion (AIC) or Schwarz’s Bayesian information criterion (BIC) (Burnham and Anderson, 2001). Pinheiro, Bates and Lindstrom employ this approach in their paper (Pinheiro et al., 1994). A model with a larger number of parameters is always associated with a larger (smaller) likelihood (log-likelihood) evaluated at the corresponding parameter estimates. Consequently, AIC and BIC apply a penalty related to the dimension of the unknown parameter vector to offset this advantage, permitting comparison on a more equal footing – with models exhibiting smaller AIC or BIC values being preferred.

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5.1 Identifying mixed effects In this section, the issue of which effects should be considered mixed rather than purely fixed is addressed. A conservative strategy is to initially assume that all components should be regarded as mixed effects (Pinheiro et al., 1994). Thus, at the outset all components of βi were considered to possess random components at level 2 in the model. Thus, with reference to equation (1), at the outset F is the 5-by-5 identity matrix. This model is referred to as model ‘A’ throughout. The modified IGLS algorithm of Section 2.2 converged after only two passes of the separate variance and fixed effect parameter processes. Corresponding AIC and BIC values for model A are presented in Table 1. These values are calculated using equation (10) for –2log(L), where L denotes the likelihood function. Similarly, the matching parameter count is assumed to be 15 variance parameters and 60 coefficients for the fixed effects (spline models). Note use of the profile likelihood eliminates the necessity to include σ2 in the count for the variance parameters. An analysis of Γˆ A reveals the corresponding smallest and largest eigenvalues are 5.54 × 10–9 and 1.248 respectively, indicating that the model is over parameterised. Note, as the scales of measurement for the elements of βi are similar, there is no necessity to scale Γ prior to undertaking the Eigen-analysis as suggested by Pinheiro et al. (1994). The eigenvector associated with the smallest eigenvalue is [0.06–0.506 –0.848 –0.144 –0.004]T. Assuming the first and fifth elements of this vector are approximately zero, this indicates a clear linear dependency involving the mixed effects associated with εc, εq and εr. The mitigating strategy initially employed is to assign εc and as a fixed effect, since it possesses the smallest variance of the three, and refit the model. This second model in which εc is regarded as a fixed effect is referred to as model ‘B’ in Table 1. In this case, the smallest and largest eigenvalues of Γˆ B are 2.284 × 10–6 and 1.38. Again this suggests an over parameterised model. The eigenvector corresponding to the smallest eigenvalue is [–0.005 –0.9973 0.0714 –0.0180]T. Treating the first, third and fourth terms as zero this implies that the variance of εq is very small. In fact the estimated variance of εq is 0.0004, which we take as approximately zero. Consequently, εq is now considered to be a fixed effect. A third model, denoted by ‘C’ in Table 1, is subsequently fitted in which εc and εq are regarded as fixed. Again analysis of the eigenvalues of Γˆ C suggests the model is over parameterised. The eigenvector associated with the smallest eigenvalue is [0.1675 –0.9822 0.0846], suggesting once again that the variance of one component, εr in this case, is small and potentially negligible compared to the remainder. As a result, εr is now treated as fixed and the model retrained. Model D has only εo and k as mixed effects. The eigenvalues for Γˆ D are 0.3874 and 1.8651 respectively. The correlation between εo and k is 0.528; which is not severe. Despite this we treat k as being fixed, as it has smallest variance, and retrain the model.

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Table 1

Model selection criteria

Model

Description

Parameter count

AIC

BIC

A

All mixed effects

75

1,096.8

1,153.3

B

εc fixed effect

70

1,083.8

1,136.5

C

εc and εq fixed effects

66

1,098.2

1,147.9

D

εr, εq and εc fixed effects

63

1,095.7

1,143.2

E

εr, εq, εc and k fixed effects

61

1,213.9

1,259.9

Inspection of the relevant model selection criteria presented in Table 1 confirms that model B is preferred over the remainder. As a result this model is tentatively adopted as best, even though Γˆ B contains one parameter exhibiting relatively low variance. ⎡1.1042 ⎢ 0.019 Γˆ B = ⎢ ⎢ 0.386 ⎢ ⎣ 0.128

6

0.019 0.386 0.0212 0.005 0.005 0.3552 −0.008 0.094

0.128 ⎤ ⎥ −0.008⎥ 0.094 ⎥ ⎥ 0.8902 ⎦

(14)

Model training results

The model specified in Section 3.1 affords considerable flexibility in recovering a wide variety of sweep profiles including those with a reduced number of data points. This is demonstrated in Figure 4. Here data from a selection of training sweeps have been plotted in blue and the corresponding model predictions have been overlaid in red. Regardless of the nature of the sweep data profile the model appears to fit the data well, there is no need to eliminate spark uncharacteristic spark sweeps as would be the case with nonlinear methods. Figure 4

Example fits to a variety of spark sweeps at various (N, L) operating conditions (see online version for colours)

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Figure 5 presents a residual diagnostic plot for the entire training dataset. Superimposed in red is the desired engineering measure of success (MOS). This may be stated as ±5 [Nm] or ±5%, whichever is the largest. Note the vast majority of the data are seen to meet the MOS. Figure 5

Residual diagnostics plot for the entire training set (see online version for colours)

Note: Superimposed on the diagram is the engineering accuracy requirement shown in red. Figure 6

PKTQ response surface as a function of engine speed (N) and load (L) (see online version for colours)

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In previous sections, we provided a physical interpretation for the sweep specific parameter εoi, which represents the maximum indicated torque attained on a spark sweep. To this end, consider Figure 6. This presents the response surface for εoi(PKTQ) as a function of engine speed (N) and normalised air charge or load (L). This is reassuringly physical in appearance. For example, it is well known that εoi should vary linearly with L. This characteristic is clearly demonstrated in the plot. Similarly, the asymmetric nature of the curvature of εoi with N is also apparent. These comments are true despite fact that model has been extrapolated in some regions of the plot. A glance at the experimental design of Figure 2 confirms that the response profile has been considerably extrapolated at high loads towards the extremes of engine speed. Also, as demanded by the application, the response surface is smooth. The ability to yield physical and smooth response feature profiles appears to justify the use of spline models to construct the elements of d(ai, β). Figure 7 presents the corresponding response surface for sweep-specific parameter ki. Like εoi this also has a straightforward physical interpretation. Parameter ki represents the spark-timing corresponding to the location of the sweep maximum. This surface is quite physical in appearance in the sense that the profile corresponds to engineering knowledge. For example, note how ki is seen to initially decrease very rapidly with L and then vary linearly at higher loads. Also note the cubic nature of the response profile with engine speed. In the author’s experience such characteristics with engine speeds are typical of high tumble port designs. Figure 7

ki response surface (see online version for colours)

Note: Optimal spark timing as a function of (N, L).

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Again, as required by the engineering application, the response surface depicted in Figure 7 is also sufficiently smooth in appearance. This justifies the use of equation (5) as a suitable choice for one element of d(ai, β).

7

Summary and conclusions

A linearised hierarchical statistical model has been proposed for modelling data from large scale engine mapping experiments conducted on turbocharged spark-ignition engines. Specifically, the objective was to characterise the behaviour of a large number of spark-sweeps collected at various operating conditions across the entire (N, L) operating region. The application of linearised techniques is seen to address issues with respect to sparseness of sweep-specific data and also questions arising as to which elements of βi can be considered to possess random as well as fixed components. At the core of the model hierarchy are spline models that explain both the behaviour of indicated torque with spark advance, at level 1 in the hierarchy, and the change in character of the individual spark sweeps across (N, L) at level 2. The level 1 model is seen to be sufficiently flexible to characterise a variety of sweep behaviours. Similarly, the level 2 spline formulations are seen to yield behaviour consistent with physical knowledge and at the same time possess the desired level of smoothness demanded by the application at hand. The model is demonstrated to satisfy the engineering MOS with regard to the training data. Model training procedures predicated on IGLS techniques are seen to provide a convenient and relatively robust means of estimating the model parameters. Minimisation of the appropriate likelihood component functions was carried out using standard functions from the MATLAB optimisation toolbox. A case study in model building, addressing the issue of identifying purely fixed versus mixed elements of βi was presented. Since the elements of βi exhibited similar magnitudes there was no necessity to scale Γ prior to conducting the analysis. Large variation in the characteristic roots or eigenvalues of Γ was used to indicate possible over-specification of the model. At each stage, analysis of the eigenvector corresponding to the smallest characteristic root of Γ revealed which elements of βi to treat as fixed. Model selection was guided by information criteria (AIC and BIC) which were easily computed from knowledge of the log-likelihood function and the corresponding parameter count.

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Nomenclature AIC

Aikaike’s information criterion.

ATDC

After top-dead centre.

BIC

Bayesian information criterion.

EVC

Exhaust valve closing.

GTDI

Gasoline turbocharged direct injection engine.

IGLS

Interative generalised least squares.

IVO

Inlet valve opening.

L

Engine normalised air charge.

MLE

Maximum likelihood estimate.

MOS

Measure of success.

Application of linearised hierarchical models to indicated torque modelling N

Engine speed.

OLS

Ordinary least squares.

PKTQ

Peak torque.

s

Spark angle.

Appendix Test engine details Maximum engine speed Maximum rated power Maximum torque Idle speed Bore

6,150 RPM 130 kW @ 6,000 RPM 240 Nm @ 2,000–4,500 RPM 750 RPM 79 mm

Stroke

81.4 mm

Engine capacity

1,596 cm3

Conrod length Nominal compression ratio Inlet cam timing range of operation Maximum inlet cam lift Exhaust cam timing range of operation Maximum exhaust cam lift

133 mm 10:1 –20 to 29° ATDC 7.8 mm –8 to 40° ATDC 6.5 mm

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