The document discusses methods to improve the accuracy of reconstructing transient emissions measurements from heavy-duty vehicles. It examines using higher order derivatives and different numerical differentiation methods in the differential coefficients method. Using backward differences for numerical differentiation and including higher order derivatives improved the reconstruction accuracy by about 10% compared to just the first two derivatives. This margin of improved accuracy may be important for model accuracy or assessing emissions criteria compliance.

1. Attempts to enhance the differential coefficients
method for reconstruction of transient emissions
from heavy-duty vehicles
M R Madireddy and N N Clark*
Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia, USA
The manuscript was accepted after revision for publication on 29 October 2008.
DOI: 10.1243/14680874JER01808
Abstract: Engine tailpipe emissions as measured by a sampling system and analyser do not
clearly reflect the actual transient emissions from the engine at the tailpipe of the vehicle
exhaust. With increasing demand for accurate emissions measurements, several research
efforts are being made to compensate for the measurement distortions of the analyser system.
The differential coefficients method is one such approach which reconstructs the input signal
by approximating the analyser input as a linear combination of the output and the first two
derivatives of the output. While the results with this approach were found to be acceptable, this
paper focuses on the possibilities of improving the accuracy of the reconstruction by altering
the ways in which the numerical derivatives are computed and by considering higher order
derivatives in the linear combination. It was found that the use of backward differences in
computing the numerical derivatives proved more effective than forward differences. Using
higher order derivatives has shown improvement of about 10 per cent over using just the first
two derivatives, but this margin may be important for model accuracy or for assessing pass/fail
emissions criteria.
Keywords: differential coefficients method, time delay, time alignment, cross-correlation,
instantaneous input, dispersion, reconstructed input, dispersed output, central/ forward/
backward differences
1 INTRODUCTION
‘Instantaneous’ emissions are the actual emissions
produced by the engine (or at the tailpipe) due to
combustion of the fuel. They may be modified by
after-treatment systems before they leave the engine
stack or tailpipe. A measurement system can be used
to record these instantaneous emissions, but the
system will distort the signal that corresponds to the
instantaneous emissions and the system will produce
an output signal, which represents the ‘measured’
emissions. For an ideal analyser system, the measured
emissions will be the same as the instantaneous
emissions. However, any real analyser system will
report a distorted signal during the process of
measurement. For example, the emissions reported
by the analyser may be delayed and dispersed relative
to the instantaneous emissions [1–5]. Knowledge of
the instantaneous emissions may play an important
role in developing both inventory models and superior
engine control strategies, and the prediction of
instantaneous emissions from measured emissions.
This paper presents conclusions related to a single gas
species, nitric oxide (NO), but the results are equally
applicable to all species encountered in engine
exhaust.
2 TIME DELAY AND COMPENSATION FOR THE
TIME DELAY
When the steady state condition of the engine
changes, the corresponding engine-out emissions
*Corresponding author: Center for Alternative Fuels, Engines,
and Emissions, Department of Mechanical and Aerospace
Engineering, West Virginia University, Morgantown, West Virgi-
nia, USA. email: Nigel.Clark@mail.wvu.edu
65
JER01808 F IMechE 2009 Int. J. Engine Res. Vol. 10

2. levels also change. There is a time delay between the
point when the engine/vehicle experiences an
operating condition and the point when the corre-
sponding analyser measures the emissions related to
that operating condition. The time delay is a
combination of transport time taken by the exhaust
to reach the appropriate gas analyser and the
response time of the analyser. The sum total of
these two measurement delay times should account
for the time shift between the instantaneous (actual)
emissions at the tailpipe and measured emissions.
Time delay was addressed in detail by Messer et al.
[3, 4]. In order to compensate for the delay, the
emissions data should be shifted back in time to
align with the engine operating variables such as
power and speed. The importance of time alignment
was addressed in detail by Hawley et al. [5]. There
are several ways in which the emissions data can be
time aligned. Some researchers prefer visual time
alignment, which is usually implemented by match-
ing the crests and troughs of the continuous
emissions dataset with that of an engine or vehicle
operating parameter such as power. The most widely
followed procedure for time alignment is cross-
correlation, where two datasets are compared
against a common variable and the time shift that
best matches the two sets is calculated. It is based on
the assumption that a correlation exists between the
two datasets, but this may fail for the case of
emissions species that do not correlate well with
engine power. For example, emissions of carbon
monoxide or particulate matter from diesel engines
may depend both on power and the rate of change of
power with respect to time. Cross-correlation be-
tween the time derivatives of power and emissions
mass flowrate has also been used.
3 TIME DISPERSION AND ESTIMATION OF TIME
DISPERSION
The measured response from a transient (oscillating)
instantaneous emissions output experiences ampli-
tude reduction in addition to the time delay. The
amplitude of a peak or a trough in the measured
response is smaller than the one actually experi-
enced by the engine. Hence, there is a need to
compensate for the time dispersion of the data by
the analyser system. For this purpose, estimating the
transient response of the measurement system is
required [6, 7].
The current researchers examined the response
characteristics of two analysers (with the sampling
system) used to collect the data that were processed
in this study. NO was measured using a Rosemount
955 analyser and CO2 was measured using a Horiba
AIA 210 analyser. Data analysed were measured by
only these analysers connected to a 0.45 m diameter
dilution tunnel. More detail is given in reference [8].
The response of the Rosemount 955 NO analyser
(manufactured by Emerson Electric Co., St Louis,
MD, USA) to an instantaneous pulse input of NO is
shown in Fig. 1(a). The response was obtained
through the following experiment conducted in the
engine test cell at West Virginia University. A balloon
was filled with approximately 1 litre of NO with a
concentration of 1000 p/million. and inserted in the
dilution tunnel. The balloon was burst to simulate an
instantaneous pulse. The pulse travelled via the
sampling lines, and the output of the Rosemount 955
was collected. The time delay due to transport of the
pulse through the tunnel depends on the length of
the tunnel. The time delay shown in Fig. 1(a) was a
function of the length of the sampling lines and the
Fig. 1 Response of (a) the Rosemount 955 NO analyser and (b) the Horiba AIA 210 analyser to an
instantaneous input of NO at time t 5 0
66 M R Madireddy and N N Clark
Int. J. Engine Res. Vol. 10 JER01808 F IMechE 2009

3. speed of the exhaust gas travel through the lines, as
well as a contribution from the analyser. Only the
dispersion of the instantaneous pulse is of interest in
this analysis. The response was found to be
dispersed over a period of 6 s. The probability
density function (pdf) is shown in Fig. 1 as a set of
probabilities with 0.2 s increments. If the fractions
(probabilities) of the response were less than 0.05
per cent, all such fractions were considered to be
insignificant and were added as one fraction on
either side of the response. In other words, if there
was a negligible response from t 5 0 to t 5 7 as in
Fig. 1(a), the sum of all those small responses was
shown in t 5 7. In Fig. 1(a), the function is normal-
ized such that the area under the curve is unity,
which is a definitive characteristic of a pdf. A similar
response for the Horiba AIA 210 analyser (manufac-
tured by Horiba Automotive Test Systems, Ann
Arbor, MI, USA) for CO2 is shown in Fig. 1(b). A
comprehensive description of these types of analyser
and their operation is provided in reference [9].
4 DATA USED FOR ANALYSIS
The data used for testing these techniques were
collected from the Federal Test Procedure (FTP)
cycle on a 1992 Detroit Diesel Corporation (DDC)
Series 60 engine and a 2004 Cummins ISM 370
engine and the testing was conducted on a dynam-
ometer in a transient cell. Further details of the test
cell and typical operation can be obtained elsewhere
[10]. The NO data were collected using a fast NO
analyser, which was manufactured by Cambustion
Ltd, Cambridge, UK [11, 12]. Since the response
from the fast NO analyser was rapid and had low
diffusion (T10–90% 5 12 ms), the data were treated as
instantaneous data without dispersion of the type
shown in Fig. 1(a). No NO2 to NO converter was
used with the Cambustion fast NO analyser. With
early model year engines, the NO is a reasonable
approximation of NOx because the NO2 to NO ratio
is generally small [13]. This paper demonstrates the
methodology for NO, but the methodology may be
applied to NOx or to any other emission species that
is measured continuously.
The dispersion of the instantaneous emissions to
yield the measured emissions in this study includes
dispersion in the exhaust system, dispersion in the
tunnel, dispersion in the sample lines, and disper-
sion due to the nature of the analyser function. It is
true that each of these could be studied separately
using the fast NO analyser, but from a practical
perspective they are all used in concert and the
system needs to be considered as a whole for real-
world data. However, the methods used in this paper
could also be applied to each separate dispersion
system using an appropriate dispersion function if
desired. The dispersion function shown in Fig. 1(a)
was used to diffuse the instantaneous (fast NO) data
to obtain the diffused NO data, so that they
simulated the data that would be measured by the
Rosemount 955 NO analyser. This analyser was set in
NO mode without an NO2 to NO converter. In this
way the authors had companion sets of both
instantaneous and (simulated) measured emissions
to test instantaneous emissions reconstruction algo-
rithms.
5 REVISITING THE DIFFERENTIAL
COEFFICIENTS METHOD (DCM)
Ajtay and Weilenmann [1] and Madireddy and Clark
[2] have discussed a mathematical approach to
reconstruct the true emission signals from the
measured output of the analyser. The DCM ap-
proach of Ajtay and Weilenmann [1] defines real
input as a linear combination of the output and its
time derivatives. Let U(t) be the value measured by
the fast NO analyser, which was considered to yield
instantaneous data. Now consider that U(t) is the
input to the conventional analyser, and let Y(t) be
the output (measured) value from the conventional
analyser. The output was simulated by mathemati-
cally dispersing or convoluting the input U(t) with
the pdf (i.e. dispersion function), which was shown
in Fig. 1. Y9(t) and Y0(t) are the first and second
derivatives of the simulated measured emissions
(simulating the conventional analyser output). The
method assumes that the input can be expressed as
the sum of the output and some linear combinations
of the first and second derivatives of the output. The
input U(t) and output Y(t) and its derivatives are
related by
U tð Þ~Y tð Þza1Y ’ tð Þza2Y ’’ tð Þ ð1Þ
In order to obtain the values of the coefficients, the
known response of the analyser, which was the
response to an instantaneous pulse, was used in
equation (1). The dispersion function, which was
experimentally estimated, was considered as the
output Y(t) to a unit input U(t) and the output Y(t)
was differentiated numerically to obtain Y9(t) and
Y0(t) over a period of the dispersion. The numerical
derivatives for this study were computed using
central, forward, and backward differences in 1 s
Attempts to enhance the ‘differential coefficients method’ 67
JER01808 F IMechE 2009 Int. J. Engine Res. Vol. 10

4. time intervals. Then the derivatives were mapped
with the unit impulse input and the time sequence
was fitted over the dispersion period and the error
was then computed at each second as the absolute
value of [U(t) 2 Y(t) 2 a1Y9(t) 2 a2Y0(t)]. The least
squares error was computed as the sum of the
squares of the computed errors at all points and this
was minimized for the best fit that generated the
values of a1 and a2, the coefficients of the derivatives
of the output. Values of a1 and a2 depended on the
actual analyser response. The values of a1 and a2
were used to obtain the input U of the analyser from
any measured output Y of the analyser. The inherent
assumption in the method was that the analyser was
consistent, and linear in its dispersion behaviour
with respect to time and rate of change of input.
6 EFFECT OF FORWARD, CENTRAL, AND
BACKWARD DIFFERENCES
The DCM suggested by Ajtay and Weilenmann [1]
involved the first two differentials numerically
computed using backward differences. It was of
interest to compare the results when different
methods of computing the derivatives were used.
Continuous data from a heavy-duty engine FTP cycle
were measured by the fast NO analyser from the
engine tests conducted on two different engines
(DDC Series 60 and Cummins ISM 370). Each of
these cases was examined to understand the effect of
the method by which numerical derivatives were
computed. The forward, central, and backward
differences were used independently while comput-
ing the differentials. All the differences were com-
puted numerically with 1 Hz frequency for 1200 data
points. A forward difference at time t 5 n was given
by Dx(t 5 n) 5 x(t 5 n + 1) 2 x(t 5 n). Similarly, a
backward difference at time t 5 n was given by
Dx(t 5 n) 5 x(t 5 n) 2 x(t 5 n 2 1), and a central dif-
ference at time t 5 n was given by Dx(t 5 n) 5 1/
2[x(t 5 n + 1) 2 x(t 5 n 2 1)].
The DCM was used to reconstruct the original data
from the dispersed data and in each of the cases, the
percentage error was computed as follows. If U(t) is
the original input and Y(t) is dispersed output, and if
U*(t) represents the reconstructed input, then the
percentage error is computed according to
Percentage error~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
U tð Þ{UÃ tð Þ½ Š2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP
U tð Þ½ Š
p 2
ð2Þ
In each of the cases, percentage error (equation (2))
was computed and the results when more deriva-
tives were used are shown in Table 1. Cases 1 to 3
represent three separate runs for the DDC engine
and cases 4 to 6 represent three separate runs for the
Cummins engine. The backward differences showed
better results than the forward differences. This
could be attributed to the fact that the reconstruc-
tion at any point in time depends more on the data
points that precede than the data points that follow
as a result of the asymmetry of the diffusion
function. In addition, typical engine transients may
contribute to different behaviour under increasing
and decreasing emissions scenarios, but this is still
speculative. In all cases the backward difference
approach yielded fewer errors than the forward
difference approach. For the DDC engine, the
central difference approach was slightly superior to
the backward difference approach, but overall for
the dataset, the backward difference approach is
recommended.
7 EFFECT OF MULTIPLE DERIVATIVES ON DCM
Instead of using just two derivatives as in equation
(1) for the DCM, multiple derivatives were also used,
as follows
U tð Þ~Y tð Þza1Y 0
tð Þza2Y 00
tð Þza3Y 3
tð Þ
za4Y 4
tð Þza5Y 5
tð Þza6Y 6
tð Þ ð3Þ
Table 1 Errors in DCM with different numerical methods for computing the derivatives for the
DDC Series 60 and Cummins ISM engines
Numerical method
Backward differences method
of Ajtay and Weilenmann [1] Forward differences Central differences
Case 1 2.03% 2.45% 1.91%
Case 2 2.04% 2.65% 1.93%
Case 3 2.07% 2.42% 1.96%
Case 4 3.55% 4.03% 4.38%
Case 5 3.58% 4.16% 4.32%
Case 6 3.95% 4.39% 4.25%
Cases 1 to 3 are the results for the DCC engine and cases 4 to 6 are for the Cummins engine.
68 M R Madireddy and N N Clark
Int. J. Engine Res. Vol. 10 JER01808 F IMechE 2009

5. Clearly more fitting parameters were needed.
Several cases were examined for reconstruction
using DCM with more derivatives. Equation (3)
holds good for the case in which six derivatives
were used for computation. The same equation can
be used for different cases. For example, in the case
that uses only the first three derivatives, only the
first four terms exist on the right-hand side. Back-
ward differences (which were comparatively better
than forward differences from the results shown in
Table 1) were used for each of these cases and the
percentage error was again computed according to
equation (2), the results are shown in Table 2.
Employing higher order derivatives improved the
accuracy of reconstruction, but the computation
time and the complexity of the procedure increased
with the addition of higher order derivatives. Noise
levels during data measurement could affect
the accuracy of reconstruction. This effect was
discussed in detail in reference [2]. In the case
of heavy-duty-engine NO emissions reconstruc-
tion, it can be concluded that the use of more
derivatives was advantageous offering in the order
of a 10 per cent improvement. For researchers
seeking simplicity, only the first two derivatives
were recommended for reconstruction, as origin-
ally suggested by Ajtay and Weilenmann [1], but for
more accurate modelling, additional derivatives
proved desirable.
8 CONCLUSION
The original DCM method [1] was shown again [2]
by the authors to recreate instantaneous data from
dispersed data successfully. The original method was
extended to use more than two derivatives and the
improvement over the use of two differentials
offered at best a 10 per cent advantage in reducing
the root mean square error. For further application
of this procedure in emissions reconstruction, the
use of higher order derivatives can improve the
accuracy of results, given a way in which computa-
tional (rounding) errors can be minimized. This 10
per cent improvement may be important for model
accuracy or for assessing pass/fail emissions criteria.
ACKNOWLEDGEMENTS
The authors appreciate the research efforts of the West
Virginia University engine test cell for providing the
data analysed in this study. The authors also thank
Bradley Ralston and David McKain for assisting in
obtaining the transient response shown in Fig. 1.
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Table 2 Percentage errors in DCM with multiple derivatives for the DDC Series 60 and Cummins
ISM engines
Derivatives used
First two derivatives
‘suggested by Ajtay and
Weilenmann [1]’
First three
derivatives
First four
derivatives
First five
derivatives
First six
derivatives
Case 1 2.03 1.83 1.83 1.78 1.78
Case 2 2.04 1.81 1.81 1.75 1.75
Case 3 2.07 1.83 1.82 1.77 1.77
Case 4 3.55 3.48 3.29 3.29 3.28
Case 5 3.58 3.55 3.42 3.42 3.41
Case 6 3.95 3.82 3.29 3.60 3.57
Cases 1 to 3 are the results for the DDC engine and cases 4 to 6 are for the Cummins engine
Attempts to enhance the ‘differential coefficients method’ 69
JER01808 F IMechE 2009 Int. J. Engine Res. Vol. 10

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APPENDIX
Notation
U(t) original input to the analyser
U*
(t) reconstructed input
Y(t) dispersed analyser input
70 M R Madireddy and N N Clark
Int. J. Engine Res. Vol. 10 JER01808 F IMechE 2009