Download Intelligent Computing Methods for Indicated Torque Reconstruction

Intelligent computing methods for indicated
torque reconstruction
#
Elton Gani and Chris Manzie
Department of Mechanical and Manufacturing Engineering
The University of Melbourne, Victoria 3010, Australia
[email protected], [email protected]
Abstract
This paper proposes using a support vector machine to
reconstruct indicated torque from the crank angle signal in
an automotive engine. Support vector machines have been
shown to perform extremely well in many classification and
regression applications. The relationship between indicated
torque and crankshaft angular velocity is a current research
topic, and is also a nonlinear problem. In a typical
combustion engine, cycle-by-cycle variations of combustion
events occur, even when running at a fixed operating point.
Engine idle speed controllers capable of reducing the
variability have been proposed, and rely on indicated torque
information. Furthermore, real-time indicated torque
knowledge is important for engine diagnostics. The proposed
approach provides the potential for real-time reconstruction
of indicated torque and reduction in costs for manufacturers.
A comparison between the proposed approach and another
popular model estimation approach, K-means clustering with
RBF centres trained using a least mean squares algorithm, is
presented. Reduction in the input data resolution and its effect
on reconstruction accuracy is also investigated.
1.
INTRODUCTION
Support Vector Machines (SVMs) are a relatively new
approach of machine learning methodology. Originally
introduced by Vapnik [1], SVMs were developed for the
purpose of statistical learning theory and structural risk
minimisation. SVMs have since then evolved to solving both
classification and nonlinear function estimation problems,
with a performance often superior to other machine learning
methodology, such as neural networks and K-means clustered
radial basis function kernels [1].
SVMs work in a dimensional space called feature space,
which is effectively a transform of the input data. The applied
transformation, in the case of regression, is nonlinear.
Conceptually, regression is achieved through specifying a
hyperplane that attempts to resolve the input data to the
output values in the feature space, similar to a curve fitting
methodology. SVMs are trained and optimised through a
convex optimisation problem, typically through quadratic
programming.
0-7803-8894-1/04/$20.00  2004 IEEE
259
SVMs’ regression capabilities have shown potential to
generalise a system with sufficient accuracy, while
maintaining simplicity in its estimation process [1]. As such,
implementation in a real-time framework is often possible.
One such problem of interest in automotive research is the
real-time estimation of indicated torque in an engine.
Combustion of the air-fuel mixture creates a volumetric
expansion inside the engine cylinder, thereby creating a force
at a rod connecting the piston to the crankshaft. The rod is
connected to the crankshaft at a distance away from its centre,
creating a torque at the crankshaft, called indicated torque.
However, combustion in an engine cylinder, even at a fixed
operating point, is subject to cycle-by-cycle variations, due to
factors such as internal exhaust gas recirculation. As such,
any variability in combustion will then be evident in the
indicated torque, and crankshaft angular velocity.
During low-load idle conditions, air density and fuel mass fed
into the cylinder are at their lowest, thereby increasing the
likelihood of uneven distribution of the mixture throughout
the cylinder. Consequently, this causes variability in
combustion process, and subsequently, indicated torque. Due
to the increased variability of indicated torque at idle,
fluctuations of crankshaft speed are also increased. These
speed fluctuations may lead to unnecessary control action [2],
or requirements such as higher idle set points to counteract
engine stall, and reduce vehicle noise vibration and harshness
quality (NVH). Obviously, increasing idle set points also
increases emissions and fuel consumption, and should be
avoided.
An example of variability in the engine torque for the test
engine described in the Appendix is shown on Figure 1. The
data is obtained at an engine speed of 800 rpm and manifold
absolute pressure (MAP) of 30kPa, with a spark advance of
15° before top dead centre (BTDC).
Recently, an engine idle speed controller capable of reducing
the effects of cyclic combustion variability has been proposed
[2]. This relies on information about indicated torque in order
to predict and compensate for cyclic variations. Further
information about torque production is useful for engine
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diagnostics like cylinder imbalances [3]. While there exist incylinder pressure sensors, and there are direct correlation
between in-cylinder pressure and indicated torque, these
sensors are generally deemed too expensive and impractical
for implementation in a production line engine. Consequently,
there is considerable research into methods that will allow the
indicated torque to be inferred from available measurements.
30
Indicated Torque (Nm)
Frequency analysis methods benefit from the use of the
frequency domain to improve accuracy by considering only a
few frequency components of the measured crankshaft speed
signals. Through using Discrete Fourier Transforms, the
speed signal is filtered and only the information that is strictly
synchronous with the engine firing frequency is preserved.
Implementation of this algorithm is experimented in [6], and
although the authors demonstrate the method is capable of
estimating the continuous engine torque accurately, the
approach is not an efficient way of obtaining the cycleaveraged indicated torque required for an idle speed control
application.
25
20
15
10
5
0
20
40 Event 60
80
this method, all complexities of the physical system are
extracted through a number of correlation functions. Having
determined the correlation models, the estimation procedure
reduces to simple evaluation of polynomial forms, allowing
real-time estimation. It is, however, evident that the method is
incapable of modelling cyclic variability in a four cylinder
engine, although the average RMS error is small [8].
10 0
Figure 1: Indicated torque variability, 800rpm at 30kPa
The proposed methods developed for torque or in-cylinder
pressure estimation can be grouped as in the following list,
and are described in the following paragraphs:
ƒ Engine dynamics model based estimation [4]
ƒ Stochastic Estimation [5] [6]
ƒ Frequency domain analysis [3] [6] [7] [8] [9]
ƒ Computational model based on thermodynamic
principles [10]
Traditional engine dynamics model based estimation
techniques typically require accurate knowledge of engine
parameters [4], and in some cases are computationally
expensive. There are two different model complexities
identified in [11]; lumped parameter with rigid crankshaft
model, and a complex model that incorporates flexible
crankshaft and dynamics of the engine. The drawback of
implementing a complex model is the computational cost and
the number of parameters involved, whereas for a lumped
parameter model, accuracy is sacrificed at higher speeds
where torsional compliance of the crankshaft affects the
fluctuations in engine speed.
The estimation process is further complicated when four
stroke engines have more than four cylinders, as the
separability of each cylinder’s combustion event reduces.
Indicated torque is generated during the expansion stroke.
Due to the overlaps in expansion strokes between engine
cylinders, there are reductions in the amount of available
independent data. Model based estimation typically requires
as much input data as possible, and may not be able to
perform effective estimation of indicated torque in the case of
overlap.
A computational model, as presented in [10], attempts to
accurately predict the in-cylinder pressure using
thermodynamic principles. Taking into account the four
strokes of the engine cycle, combined with the combustion
process, the model predicts the pressure in the cylinder fairly
accurately. Further assessment of accuracy of the indicated
torque derived from in-cylinder pressure is required.
This paper attempts to address the shortcomings of the above
approaches by the use of a SVM to perform reconstruction of
indicated torque from crankshaft speed. In other applications,
SVMs have the potential to generalise a system and have
shown better performance than other machine learning
methods. Due to its inherent optimisation and shrinkage
capabilities, the resultant matrix is capable of general
reconstruction, while maintaining a smaller size than the
initial training set. The proposed approach does not suffer
from any simplification assumptions that are tied to model
based estimations.
The remainder of the paper is organised as follows. Section 2
will discuss the fundamentals of the support vector machines.
Section 3 will discuss the basis for K-means clustering as well
as the least mean square (LMS) algorithm as an alternative
method of machine learning. Section 4 will detail the
experiment test cell and its instrumentation. Section 5 will
compare the results of the machine learning methodologies,
and explore the potential reduction of SVM dimensionality.
Conclusions about the potential of SVM to reconstruct engine
torque are then summarised.
2.
The stochastic estimation approach procedure is based on a
signal processing method. This approach has been primarily
used for estimating conditional averages from unconditional
statistics, particularly cross correlation functions. Through
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260
SUPPORT VECTOR MACHINE
This section presents the basics of support vector machines
and its formulation for regression problems. For a more indepth discussion, interested readers may refer to [12].
Learning the training data set Z, based on input vectors X and
desired output, Y, is the goal of machine learning
methodologies. Z is defined as follows:
(1)
Z ^ X1 , Y1 , , X m , Ym `
In order to learn and generalise a system, a training goal is
needed. One possible goal often used in SVM environment is
the H-insensitive loss function. With this method, the goal of
the training is to train the SVM such that the solution yields a
function that would have at the most H deviation away from
the desired value Yi.
A nonlinear function approximating Y can be defined as:
f X W, ĭX b
(2)
Where:
The optimisation problem defined in equation (4) is
computationally expensive, as it involves evaluation of many
features from input data. An alternative representation, called
dual formulation presents a less computationally complex
solution. Dual formulation is formulated by constructing a
Lagrange function from both the objective function and the
corresponding constraints, through the introduction of a dual
set of variables. The problem then becomes, in dual form:
­ 1 m
D i D i D j D j ĭX i , ĭ X j
° 2
° i, j 1
maximise
®
m
m
°
D i D i Yi D i D i
H
°
(5)
i 1
i 1
¯
¦
¦
<W, )(X)> signifies an inner product between the weight
matrix and the mapping function, )(X), mapping input
vectors, X, to feature space.
In order to obtain the function approximating Y, as defined in
equation (2), a transformation into a constrained convex
optimisation problem is required. It can be done through
choosing the flattest function to represent Z. This is formally
defined in equation (3):
minimise
subject to
1
2
W
2
­°Yi W, ĭX i b d H
®
°̄ W, ĭX i b Yi d H
(3)
It is assumed in equation (3) that such a function exists that
can model the relationship within H precision. This is not true
in all cases, and subsequently, slack variables are added to
cope with an otherwise infeasible problem. This leads to the
formulation stated in [13], defined as follows
minimise
1
2
W
2
m
C
¦
[ i [ i
i 1
subject to
­ Yi W, ĭX i b d H [ i
°
° W, ĭX i b Yi d H [ i
®
t0
°[ i , [ i
°C
t0
¯
(4)
Where:
C = A factor determining trade off between flatness
of function and the amount of deviation larger
than H tolerated
[ i , [ i = slack variables
261
¦ ­ m
D i D i
0
°
®i 1
°D , D  >0, C @
¯ i i
¦
subject to
W
= weight matrix
)X) = nonlinear function mapping input vector, X,
to feature space
b
= bias value
Hence, the approximation function then becomes
m
¦ D
f X i
D i
ĭX i , ĭX b
(6)
i 1
Through the dual formulation, the weight matrix is no longer
needed in evaluating f(X), and the complete algorithm can be
described through terms of inner products, and hence it is
more computationally efficient.
Calculation of the bias, b, can be done by exploiting the
Karush-Kuhn-Tucker (KKT) conditions, which states that at
the optimal solution, the product between dual variables and
constraints has to vanish. For the support vectors, the KKT
conditions are:
D i H [ i Yi W, ĭX b 0
D i H [ i Yi W, ĭX b
C D i [ i
C D [
i
i
0
(7)
0
0
Another representation that was developed from dual
formulation is the kernel representation. The benefit of kernel
representation lies in the fact that the need to evaluate features
based on input data is removed. The use of the kernel
function, K(Xi, X), to represent the features also allows the
decoupling between the learning algorithm and the theorems
from the specifics of the application. Through the proper
selection of kernel functions, the problem of architecture
selection that is prevalent in neural networks is replaced by
the selection of appropriate kernel. The use of kernels can
also overcome the curse of dimensionality, where
computational and generalisation performance degrades as the
number of features increases [12].
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The optimisation problem then becomes:
­ 1 m
D i D i D j D j K X i , X j ° 2
° i, j 1
maximise
®
m
m
° H
D i D i Yi D i D i °
i 1
i 1
¯
m
­
D i D i 0
°
subject to
® i1
°D , D  >0, C @
¯ i i
¦
¦
¦
(8)
WLS
¦
f X ¦ D
i
D i K X i , X b
G
(9)
i 1
One of the commonly used kernel functions, which is also
used for a generalisation network, is the radial basis function
(RBF). A typical RBF feature has a Gaussian shape, and
hence the closer the input value to the feature space, the more
it contributes to the general output scheme. An RBF feature is
defined as
§ XX 2 ·
C
¸
g( X) exp¨¨ (10)
2
¸¸
¨
2V
©
¹
Where
X = input vector
XC = centre of feature
V width of RBF feature.
3.
G G T
1
GT Y
>W1 Wn b@T
(11)
Where the matrix G is defined as
With the function represented as
m
mean squares (LMS) algorithm is often chosen as to minimise
the average prediction error of the RBF network. In addition,
an additional bias factor is added into the training matrix to
improve prediction capability. The weight matrix is then
obtained through LMS algorithm defined as:
K-MEANS CLUSTERING AND LMS ALGORITHM
This section will present an alternative RBF network centre
selection method and an alternative algorithm for training a
generalisation network.
A. K-means Clustering
Centre selection in an RBF network requires a separate
method, and should suitable centres be chosen, a network of
RBF features can be trained to model and generalise a system.
Subsequently, RBF networks are highly dependant on the
choice of centres.
A popular approach to centre selection is through “K-means
clustering” algorithm [14]. The algorithm selects K arbitrary
centres, and minimises the Euclidean distance from each data
point to its nearest centre. Another implementation of Kmeans clustering involves inclusion of the output value in
consideration for evaluation of the Euclidean distance. In this
paper, the output value is not taken into consideration, as
variation in output values are reflected in the variation of
input values.
>g1 X g N X 1@T
(12)
Where g(x) represents a vector of continuous transfer
function, which in this case, is the RBF feature defined in
equation (10), while Y represents the desired output vector of
the system.
4.
EXPERIMENTAL APPARATUS AND INSTRUMENTATION
The experiment was performed on a six cylinder in-line, four
stroke, naturally aspirated spark ignition engine. The test
engine specifications are listed in Table 1 in the appendix.
The engine is coupled to a magnetic dynamometer (HeenanFroude dynamatic dynamometer, Type G.V.A.L Mark II),
which acts as a speed governor and a torque-loading unit.
Pressure measurements were taken from cylinder 6, using a
piezoelectric sensor (Kistler 601B1). In-cylinder pressure
measurements are then corrected of drift errors, processed to
evaluate IMEP, and converted to indicated torque. Pressure
measurements are not used in the estimation process, other
than to evaluate indicated torque for error calculation
purposes.
Crankshaft angle is sampled using a Hall effect sensor on a
180 teeth flywheel, giving an effective maximum resolution
of 2° per sample. The timing data is then converted to
instantaneous crankshaft speed.
5.
RESULTS AND DISCUSSION
The available engine data (1278 combustion events) is
separated into training data (1218 events) and test data sets
(60 events). The SVM was then trained using Stefan Ruping’s
mySVM program [15] using the training data set.
The prediction accuracy is the result of using the SVM to
reconstruct indicated torque from the input data of the test set.
The data is then compared to the torque evaluated through incylinder pressure.
C. Comparison between SVM and LMS trained K-means
clustered RBF networks.
B. Least Mean Squares algorithm
After selection of RBF centres, a training method is required
to build the RBF network to form a generalised model. Least
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262
Comparisons between the performances of K-means
clustering, trained using least mean squares (LMS) algorithm,
and the support vector machine are shown in Figure 2 and
From Figure 2, it is evident that the LMS trained K-means
clusters consistently produce a lower mean absolute error.
SVM also performs well, with errors in the range of 0.2Nm
higher, provided that the numbers of centres selected exceed a
certain value.
From Figure 3, it is evident that SVM performs better in
reducing the maximum error. The absolute maximum error is
lower than 5 Nm, when the number of centres selected is
optimal. This presents an improvement in comparison to the
results noted in [8] through their signal processing procedure.
Support Vectors
LMS K-means Cluster
3.5
3
30
Measured Torque
20
15
10
5
Events
2.5
Figure 4: Estimated and measured torque, 800 rpm, 30 kPa
2
1.5
1
D. Effect of reducing resolution of input data
Number of Centres/Support Vectors
There is a potential for reduction in SVM dimensionality to
improve computation times. While there may be some
accuracy lost due to reduction of available information, the
reduction in dimensionality will reduce the number of
computations during each iteration and support real time
operation. The current effective resolution of the crankshaft is
2°, averaged for three points in the previous section to reduce
noise effects, which is much higher than the crankshaft
resolution used in production line engines. It is desirable to
use as low resolution as the engine control design permits to
reduce cost. In the production line engine, the crankshaft has
36 teeth, thereby having an effective resolution of 10°. Figure
5 and Figure 6 shows the comparison of two different crank
angle resolutions at the same SVM training parameters.
Figure 2:Mean absolute prediction error for test data set
Maximum Absolute Error
(Nm)
10
Estimated Torque
25
Support Vectors
LM S K-M eans
9
8
7
6
5
4
Num ber of Centres/Support Vectors
Figure 3:Maximum absolute prediction error for test data set
The results shown clearly identify the differences in training
procedures between the SVM and K-means clustering
approaches. The SVM approach aims to ensure all errors stay
within H deviation from the measured data, whereas the LMS
algorithm aims to minimise mean error for the training data.
SVM is shown to perform better, on the basis of a reduction
in maximum absolute prediction error, hence a better worstcase scenario, with a comparable mean error. This is desirable
for the application of idle speed control.
One of the main issues in generalising a system is overfitting,
evident in both Figure 2 and Figure 3 above. As the number
of centres increases, both maximum and mean error increases
as the system tries to fit specific results, hence the finalised
prediction matrix is less equipped of generalising a system
effectively.
It was observed that the reduction in input dimension resulted
in an increase in training time for the SVM. It is not a
significant issue due to SVM not requiring further training
during on-line estimation.
Mean Absolute Error (Nm)
Mean Absolute Error (Nm)
4
From Figure 4, it is evident that the SVM can predict the
torque variability effectively, yielding very small errors in the
majority of the test cases. There are some events where SVM
errors are quite significant, in the range greater than 2 Nm (or
10% error in comparison to the mean torque at this operating
point). These errors are most likely caused by the lack of
training data covering those events. Should more training data
be used to train the SVM, it is very likely that these errors
could be reduced, at the cost of a small increase in the number
of centres required.
Indicated Torque (Nm)
Figure 3. Both figures have the same X-axis scale. The
crankshaft speed data is averaged over 3 points, resulting in
an effective resolution of 6°. Both methods implement RBF
as the main basis or kernel function.
6
6° C/A
10° C/A
5
4
3
2
1
0
Number of Support Vectors
Figure 5:Comparison of Mean Error for two resolutions
263
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Maximum Absolute Error (Nm)
16
14
12
10
8
6
4
6 ° C /A
2
1 0 ° C /A
N u m b e r o f S u p p o rt V e c to rs
Figure 6:Comparison of Maximum Error for two Resolutions
6.
CONCLUSION
The SVM approach is clearly capable of reconstructing
engine torque to within acceptable limits of accuracy. The
benefit of the support vector approach is the potential for realtime performance while maintaining high accuracy. Another
benefit of this approach is that it is not dependent on any
other engine parameters other than the engine speed.
A comparison with another popular model estimation
approach is also presented. SVM is shown to perform better
with equivalent number of centres, on the basis of lower
maximum error with comparable mean error.
Reduction in SVM dimensionality was also explored in order
to support real-time operation. It was shown that SVM could
estimate unseen crank angle speed to indicated torque
relationships, suggesting good generalisation capability. The
estimation capability is also noted with a reduction of input
data dimension subject to appropriate selection of the
complexity of SVMs.
ACKNOWLEDGEMENT
The authors would like to thank the RABiT centre and the
Thermodynamics group in the Department of Mechanical
Engineering at the University of Melbourne for their support.
REFERENCES
[1] Suykens, J.A.K., "Support Vector Machines: A
Nonlinear Modelling and Control Perspective".
European Journal of Control, 2001. 7: p. 311-327.
[2] Manzie, C. and H.C. Watson, "A novel approach to
disturbance rejection in idle speed control towards
reduced idle fuel consumption". Proceedings of the I
MECH E Part D Journal of Automobile Engineering,
2003. 217(8): p. 677-690.
[3] Taraza, D. "Statistical Model and Simulation of
Engine Torque and Speed Correlation". in
International Fall Fuels and Lubricants Meeting and
Exposition. 2001. San Antonio, TX, USA: Society of
Automotive Engineers. SAE 2001-01-3686.
[4] Taraza, D., N.A. Henein, and W. Bryzik.
"Determination of the Gas-Pressure Torque of a
ISSNIP 2004
264
Multicylinder Engine from Measurements of the
Crankshaft's Speed Variation". in International
Congress & Exposition. 1998. Detroit, MI, USA:
Society of Automotive Engineers. SAE 980164.
[5] Chen, X.D. and M. Roskilly, "A Crank Angular
Velocity Based Method for Engine IMEP
Measurement for Idle Quality Investigation and
Adaptive Ignition Time Trimming to Improve Idle
Quality". SAE Technical Paper Series, 1999. SAE
1999-01-0855
[6] Lee, B., et al. "Engine control using torque
estimation". in SAE 2001 World Congress. 2001.
Detroit, MI, USA: SAE. SAE 2001-01-0995.
[7] Rizzoni, G. and F.T. Connolly, "Estimage of IC
engine torque from measurement of crankshaft
angular position". SAE Technical Paper Series, 1993.
SAE 932410
[8] Cavina, N., F. Ponti, and G. Rizzoni. "Fast algorithm
for on-board torque estimation". in International
Congress & Exposition. 1999. Detroit, MI, USA.
SAE 1999-01-0541.
[9] Azzoni, P.M., et al., "Indicated and Load Torque
Estimation using Crankshaft Angular Velocity
Measurement". SAE Technical Paper Series, 1999.
SAE 1999-01-0543
[10] Kuo, P.S., "Cylinder Pressure in a Spark-Ignition
Engine: A Computational Model". Journal of
Undergraduate Sciences, 1996. 3: p. 141-145.
[11] Ball, J., K., et al., "Torque Estimation and Misfire
Detection using Block Angular Acceleration". SAE
Technical Paper Series, 2000. 4. SAE 2000-01-0560
[12] Cristianini, N. and J. Shawe-Taylor, An introduction
to support vector machines and other kernel-based
learning methods. United Kingdoms: Cambridge
University Press. 2000
[13] Vapnik, V., The Nature of Statistical Learning
Theory. N.Y.: Springer. 1995
[14] Zhang, Y., et al. "A new clustering and training
method for radial basis function networks". in IEEE
International Conference on Neural Networks. 1996.
Washington, DC USA: IEEE.
[15] Rüping, S., "mySVM-Manual". 2000, University of
Dortmund, Lehrstuchl Informatik 8.
APPENDIX
TABLE 1: TEST ENGINE SPECIFICATIONS
Manufacturer
FORD
Type
Falcon AU, Naturally Aspirated
Cylinders, Type and Displacement
6, in-line, 3.9835 L
Fuel type
LPG, gaseous phase
Bore x stroke
92.26 mm x 99.31 mm
Compression Ratio
10.9
Intake Opens: 12° BTDC
Intake Closes: 72° ABDC
Valve Timing
Exhaust Opens: 58° BBDC
Exhaust Closes: 24° ATDC