Applications of Kalman filtering to real-time trace gas concentration measurements.

Abstract

A Kalman filtering technique is applied to the simultaneous detection of NH3 and CO2 with a diode-laser-based sensor operating at 1.53 micrometers. This technique is developed for improving the sensitivity and precision of trace gas concentration levels based on direct overtone laser absorption spectroscopy in the presence of various sensor noise sources. Filter performance is demonstrated to be adaptive to real-time noise and data statistics. Additionally, filter operation is successfully performed with dynamic ranges differing by three orders of magnitude. Details of Kalman filter theory applied to the acquired spectroscopic data are discussed. The effectiveness of this technique is evaluated by performing NH3 and CO2 concentration measurements and utilizing it to monitor varying ammonia and carbon dioxide levels in a bioreactor for water reprocessing, located at the NASA-Johnson Space Center. Results indicate a sensitivity enhancement of six times, in terms of improved minimum detectable absorption by the gas sensor.

Full text

Page 1

DOI:10.1007/s003400100751
Appl. Phys. B 74, 85–93 (2002)
Lasers and Optics
AppliedPhysicsB
d.p.leleux1,✉
r.claps1
w.chen1
f.k.tittel1
t.l.harman2
Applications of Kalman filtering to real-time
trace gas concentration measurements
1Rice Quantum Institute, Rice University, Houston, TX 77251-1892, USA
2Computer Engineering Department, University of Houston Clear Lake, Houston, TX 77058-1098, USA
Received: 13 July 2001/Revisedversion: 11 October 2001
Published online: 29 November 2001• © Springer-Verlag 2001
ABSTRACT A Kalman filtering technique is applied to the sim-
ultaneous detection of NH3and CO2with a diode-laser-based
sensoroperatingat1.53 µm.Thistechniqueisdevelopedforim-
proving the sensitivity and precision of trace gas concentration
levels based on direct overtone laser absorption spectroscopy in
thepresenceof various sensor noise sources. Filter performance
is demonstrated to beadaptive to real-timenoise and data statis-
tics. Additionally, filter operation is successfully performed
with dynamic ranges differing by three orders of magnitude.
Details of Kalman filter theory applied to the acquired spectro-
scopic data are discussed. The effectiveness of this technique is
evaluated by performing NH3and CO2concentration measure-
ments and utilizing it to monitor varying ammonia and carbon
dioxide levels in a bioreactor for water reprocessing, located at
the NASA–Johnson Space Center. Results indicate a sensitiv-
ity enhancement of six times, in terms of improved minimum
detectable absorption by the gas sensor.
PACS 42.68.Ca; 42.62.Fi; 95.75.-z
1Introduction
The primary focus of this work is the development
of an adaptive filtering technique, known as Kalman filtering,
to the sensitive, selective and real-time detection of NH3and
CO2usingatunablediode-laser-based tracegassensor[1–3].
The motivation for the application of this technique is the
fact that these trace gas sensors have inherent laser, electri-
calandopticalnoiseassociatedwiththemthatmanifestsitself
as short-term variations in gas concentration measurements.
Examplesofthesereal-timenoisesourcesincludethermalex-
pansionorcontractionofopticalcomponents,laserfrequency
drift and etalons. These fluctuations in concentration occur
with every measurement, compared to true gas concentration
variations thatresultfromchanges inthesampleenvironment
which evolve over the course of a series of measurements
as reported in Sect. 4. This noise defines the minimum de-
tection limit of the gas sensor, which can be characterized
by calculating an Allan variance for the specific optical sen-
sor configuration [4]. The Allan variance defines the optimal
✉ Fax: +1-713/348-5686, E-mail: leleux@rice.edu
averagingtimeforthegasanalyzer,whichminimizestheshot-
to-shot variability of concentration measurements. Once this
averaging time has been optimized, further detection sensi-
tivity improvements can be achieved through the use of the
Kalman filtering technique[5].
While numerous filtering techniques can be applied to
post-processing of gas concentration measurements, only
afewsuccessfultechniqueshavebeenidentifiedthatallowef-
ficient on-line filtering of concentration measurements. One
direct technique, which can be applied in real time, is simple
averaging of the previous n measurements. The advantages
of the Kalman filtering technique over simple averaging are
three-fold:
– theKalmanfilterincorporatesallavailablemeasurements,
regardless of their precision, and is not limited to a nar-
row window of n measurements to estimate the actual gas
concentration;
– the Kalman filter (although the equations are more com-
plex than an averaging technique) is computationally effi-
cient because it is recursive and only requires two sets of
information instead of n to be transferred from measure-
mentto measurementforfilter calculations; and
– the Kalman filter is adaptive and can adjust to changes in
signalstatistics anddynamic rangeduringoperation.
AcomparisonbetweenaKalmanfiltersolutionandamov-
ing average technique is presented in Sect. 4.3. A Kalman fil-
tercombinesallavailablemeasurementdata,pluspriorknow-
ledge about the system and measuring devices, to produce an
estimate of theactual concentration suchthattheerroris min-
imized statistically.Theuseofthisalgorithmenablesthefilter
to solve the Gauss’ least-squares technique in real time with
little or nodelay in processingtime.
The filtering algorithm was first developed in 1960 by
Kalman [5] and applied to aerospace navigation problems.
In 1993, Werle et al. employed adaptive filtering techniques
to tunable diode-laser absorption spectroscopy (TDLAS) in
the infrared molecular “fingerprint”region [6]. Subsequently,
Kalmanfilteringappliedtotunablediode-laser-basedgassen-
sors was studied by Riris et al. in 1994 [7] and 1996 [8]. This
paper describes the effective approach of applying Kalman
filtering to real-time trace gas concentration measurements
using diode-laser overtonespectroscopy.

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86Applied Physics B – Lasers and Optics
InSect. 2,thetheoretical considerationsofaKalmanfilter
willbedescribed.Sect.3willexplainsensorconfigurationand
discussdata acquisition and processing. In Sect. 4, the results
of the experiments will be analyzed and a comparison will be
made between the Kalman filter and a moving average tech-
nique.The resultswill besummarized in Sect. 5.
2
2.1
Theoretical considerations
Kalman filter model for trace gas concentration
measurements
In this section, we report on the application of
Kalmanfilteringtoaddresstheproblemofdetermininganop-
timal estimate of the true gas concentration, xk, of a sample
gas with a near-infrared diode-laser-based sensor in the pres-
ence of two sources of uncertainty (measurement noise in-
troduced by the sensor, vk, and true concentration variability,
wk). This scalar optimal estimation problem can be modeled
bythelinear stochastic differenceequation [9,10]:
xk+1= xk+wk,
withmeasurements fromthetracegassensor, zk,modeled as
(1)
zk= xk+νk,
where the expected value, E[wk] = E[vk] = 0, and wkand vk
are uncorrelated random variables with white noise variance
of σ2
valid for trace gas concentration measurements [9], since the
noise variances are not correlated in time, i.e. knowing the
noise variance at time x does not aid in predicting what its
valuewillbeatanyothertime.Thevarianceσ2
true concentration variability while σ2
urementnoiseintroduced by thesensor.
ˆ x−
for a step k, with knowledge of the gas concentration prior to
step k,and ˆ xkto be thea posterioriconcentration estimate for
step k after the measurement value zkis incorporated. Note
that a hat above a variable such as ˆ xkindicates an estimated
or predicted quantity, and a superscript negative sign above
avariable such as ˆ x−
orianda posterioriestimate errorscan bedefinedas:
(2)
wand σ2
v, respectively. The assumption of “whiteness” is
wrepresentsthe
vrepresents the meas-
kcanbedefinedtobetheaprioriconcentration estimate
kindicates an a prioriquantity. Thea pri-
e−
k≡ xk− ˆ x−
and
k,
(3)
ek≡ xk− ˆ xk.
The a priori estimate error variance is then given by the ex-
pected value
(4)
P−
k= E?e−
and the a posteriori estimate error variance is given by the
expected value
ke−
k
?,
(5)
Pk= E [ekek] .
InderivingtheKalman filterequations,theobjectiveistofind
an equation that computes an a posteriori concentration es-
timate ˆ xkas a linear combination of an a priori estimate ˆ x−
(6)
k
and a weighted difference between an actual concentration
measurement zkand a measurement prediction ˆ x−
below:
kas shown
ˆ xk= ˆ x−
The difference (zk− ˆ x−
residual. The residual reflects the discrepancy between the
predicted measurement ˆ x−
A residual of zero means that the predicted and actual meas-
urements are in complete agreement. The value of Kk in
(7) is called the Kalman gain and is chosen to minimize
the a posteriori error variance from (6). This minimiza-
tion procedure can be accomplished by substituting (7) into
(4), which defines ek. Next, this result is substituted in (6),
followed by performing the indicated expectations, taking
the derivative of the result with respect to Kk, and set-
ting this value equal to zero. Solving for Kk yields the
definition of the Kalman gain, which minimizes (6), given
by:
k+ Kk
?zk− ˆ x−
k
?.
(7)
k) in (7) is called the measurement
kand the actual measurement zk.
Kk= P−
k
?P−
k+σ2
v
?−1=
P−
k
k+σ2
?P−
v
? .
(8)
From (8), it can be seen that the gain Kkweights the residual
(zk− ˆ x−
approaches zero. In practical terms, this can be interpreted as
a sensor that is reliable and provides measurements with low
variability. Specifically, from(8),
k) more in (7) as the measurement error variance σ2
v
lim
σ2
v→0Kk= 1.
However,asthemeasurement errorvariance σ2
gain Kkweights the residual (zk− ˆ x−
tical terms, this can be interpreted as a sensor that provides
measurements with significant variability. In the limit, as σ2
approaches infinity, from(8),
vincreases,the
k) less. Again, in prac-
v
lim
σ2
v→∞Kk= 0.
Another way of interpreting the weighting by Kk is that
as the measurement error variance σ2
actual measurement zk is “trusted” more because the sen-
sor becomes more reliable and the predicted measurement
ˆ x−
ance σ2
because the sensor is considered less reliable in terms of
variability, while the predicted measurement ˆ x−
more.
By substituting (8) into (7) and using (3) through (6) to
solve for Pk, the solution provides the a posteriori estimate
error varianceas:
vapproaches zero, the
kis trusted less. However, as the measurement error vari-
vincreases, the actual measurement zkis trusted less
kis trusted
Pk= (1− Kk) P−
With (8) and (9), the complete a posteriori estimate and error
variance are defined as a function of the weighting factor Kk
that is conditioned based on both the previous uncertainty of
theestimate and theuncertainty ofthemeasuring device.
k.
(9)

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LELEUXet al. Applications of Kalman filtering to real-time trace gas concentration measurements 87
2.2
The Kalman filteralgorithm
The Kalman filter estimates the gas concentration
by using a form of feedback control. The filter estimates the
concentrationatsomegiventimeandthenobtainsfeedbackin
the form of measurements possessing a noise component. As
such,theKalmanfilterequationsfallintotwogroups:timeup-
date equations and measurement update equations. The time
update equations are responsible for projecting forward in
timecurrentconcentration and error varianceestimates to ob-
tainaprioriestimatesforthenexttimestep.Table 1showsthe
equationsused inthetime updateportion ofthealgorithm.
If the gas concentration is determined by a process that
can be modeled directly or inferred (e.g. from temperature),
this information can be used to obtain a better predictive ca-
pability instead of assuming that the next concentration will
be equal to the previous concentration plus some noise, as in
(1). If a control input can be modeled by a variable, uk(e.g.
thegasconcentrationisdeterminedbythetemperatureinare-
action chamber at a chemical processing plant), then the time
update (10) can be modified to include this information, as
shownin (12).
Themeasurementupdateequationsareresponsibleforthe
feedback, i.e. for incorporating a new measurement into the
a priori concentration estimate to obtain an improved a pos-
terioriestimate, and aregivenin Table 2.
The time update equations act then as predictor equa-
tions, while the measurement update equations act as correc-
tor equations. The final estimation algorithm is a predictor–
corrector algorithm for solving the least-squares method in
realtimeas shownin Fig. 1.
The first step in the sequence involves selecting initial
a priori values for the concentration estimate, ˆ x−
error variance, P−
critical, since the filter will converge to an appropriate value.
However, they should be chosen to be within the dynamic
range of the expected gas concentrations and errors to speed
convergence. The next step involves taking a measurement
update from the sensor by first computing the Kalman gain,
Kk, using (13). Subsequently, the sensor performs a concen-
trationmeasurementtoobtainzk.Thisisfollowedbycalculat-
ing an a posteriori concentration estimate using (14), and the
final step is to obtain an a posteriori error variance estimate
from(15).
Once a posteriori estimates of the concentration and vari-
ance are obtained, a time update is processed using (10) and
(11), or (10) and (12) if a control input is modeled. This step
k, and the
k. The selection of these quantities is not
ˆ x−
P−
ˆ x−
TABLE 1
k+1= ˆ xk
k+1= Pk+σ2
k+1= ˆ xk+uk
(10)
(11)
(12)
w
Kalman filter time update equations
Kk= P−
ˆ xk= ˆ x−
Pk= (1−Kk)P−
TABLE 2
k(P−
k+Kk(zk− ˆ x−
k+σ2
v)−1
(13)
(14)
(15)
k)
k
Kalman filter measurement update equations
FIGURE 1
rent concentration estimate ahead in time. The measurement update adjusts
the projected concentration estimate by an actual measurement at that time.
Note: σ2
ware constants
Complete Kalman filter cycle. The time update projects the cur-
vand σ2
projects the concentration and variance estimates from time
step k to stepk+1.
After each time and measurement update pair, the pro-
cess is repeated with the previous a posteriori estimates used
to project or predict the new a priori estimates. This recur-
sive nature is one of the appealing features of the Kalman
filter, because it recursively conditions the current concentra-
tion estimate on all of the past measurements. Furthermore,
the selection of σ2
tuning the Kalman filter to provide the optimum sensor per-
formance, asdiscussedin Sect. 4.
wand σ2
vis one of the central features in
3
3.1
Experimental details
Diode-laser-based gas sensor
The diode-laser-based sensor shown in Fig. 2 for
gas detection and quantification using vibrational overtone
spectroscopy at 1.53µm was used to investigate the Kalman
FIGURE 2
NH3and CO2concentration measurements at 1.53 µm
Diode-laser-based trace gas sensor configuration for continuous

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88Applied Physics B – Lasers and Optics
LinestrengthFrequency
cm−1
Wavelength
nm
Ref.
cm−1/(molecule/cm2)cm−2atm−1(@ 296 K)
A
B
C
2.33×10−21
1.24×10−21
5.19×10−25
0.0578
0.0307
1.29×10−5
6528.76
6528.89
6528.895
1531.68
1531.65
1532.65
[11]
[11]
[12]
TABLE 3
sorption lines used in this work
NH3and CO2overtone ab-
filtering technique. This sensor has been described in detail
previously [1–3]. An ammonia (or any other gas) concentra-
tion measurement for narrow-linewidth radiation sources is
obtainedbyusingtheBeer–Lambertabsorptionlaw,whichre-
latestheintensityoflightenteringintoanabsorptionmedium,
I0,to thetransmitted intensity, I, asfollows:
I
I0
whereSνisthetemperature-dependentabsorptionlinestrength
for a specific transition denoted by ν[cm−2atm−1], φνis the
line shape function (cm), χjis the fractional concentration of
the species j, P is pressure (atm), and l is the optical path
length throughthemedium(cm).
Table 3 depicts the three NH3and CO2absorption lines
thatareusedinthiswork.LinesAandBarethetwoammonia
lines used: A is an isolated line useful for measuring ammo-
nia concentrations while line B overlaps with line C, which
isthe(00001)← (30011), R(36),overtonetransition ofCO2.
These three lines are accessible using a 1.53µm telecommu-
nications diode-laser.
The sensor consists of three main components: a single-
frequency, fiber pig-tailed, tunable, distributed feedback
(DFB) diode-laser (NTT-NEL Electronics), a multi-pass ab-
sorption cell and a dual-beam auto-balanced InGaAsdetector
(Nirvana 2017, New Focus, Inc). The laser diode is driven
by a compact current supply (MPL-250, Wavelength Elec-
tronics) with ripple noise < 1 µA, so that the frequency fluc-
tuations of the laser line due to current noise are negligible
(< 1 MHz). The current supply is scanned at 20Hz about an
averagecurrentof67mAwithasaw-toothrampfunction.The
laser temperature was controlled to within 0.003◦C, close to
32◦C,byacurrentmodule(HTC-1500,WavelengthElectron-
ics). The scan range of the laser under these conditions was
0.3cm−1, which allowed all the spectral lines of interest to be
accessedon everyscan.
The fiber-pigtailed DFB laser diode delivers 15mW at
1531.7nm with a specified linewidth of < 10MHz. The fiber
was fusion-spliced to a 70/30% directional coupler with an
insertion loss of less than 2%. The 70% power arm of the di-
rectional coupler was sent to the multi-pass cell, using a lens
( f = 7mm, 0.5NA) mounted in a precision holder with 5
degrees of freedom (Optics For Research) to set the beam
waist at the midpoint of the multi-pass cell. The 30% power
arm was used as the reference beam for the balancing detec-
tor. Such an integrated laser and optical fiber configuration
delivers 10mW of laser light at the input of the multi-pass
cell. The output power of the beam obtained from the cell
after 182 passes was 200µW, resulting in a throughput ef-
ficiency of 2.0% (a factor of 10 better than for the cell de-
scribed in [1]). The output beam from the cell is focused on
the signal and reference photodiodes of the dual beam de-
tector by a gold-coated, parabolic mirror of 2.54cm diam-
= exp?−SνφνPχjl?
(16)
eter ( f-number= 2)and alens ( f = 7mm, 0.5NA)mounted
on a precision holder, respectively. Since the reference beam
power (Pref) was much greater than the power of the sig-
nal beam coming from the cell (Psignal), it was attenuated
using a variable fiber attenuator. For optimum performance
of the auto-balanced detector, the reference power was set as
Pref= 2.2× Psignal, at the center frequency of the laser scan.
This in turn reduces the attenuation level required at the ref-
erence input of the detector to ∼ 3dB, a range where the
operation ofthefiberattenuator isreliableandfreeoffringing
effects.
All elements, except the input and output of the multi-
pass cell, are coupled to optical fibers in order to make the
system suitable for field applications. A two-stage, micro-
mechanical diaphragm pump mounted next to the multi-pass
cell provides a continuous flow of sample gas at a rate that
is controlled by two needle valves at the input and output
of the multi-pass cell, which has a volume of 0.3liters. For
these experiments, an optimal flow rate of about 100sccm
allows for a fast response of the instrument to true con-
centration changes, while keeping the concentration rates
of the gas sample stable. The multi-pass cell was heated to
a temperature of 40◦C to minimize ammonia adsorption on
its glass walls. All the measurements presented here were
obtained with the sample gas inside the multi-pass cell at
a fixed pressure of 100Torr. For the measurements, a cylin-
der containing certified 99.99% pure CO2(Scott Specialty
Gases, Inc.) was connected in parallel with an ammonia gas-
dilution system [1]. The ammonia sample was obtained by
diluting a certified, 100ppm NH3 mixture in N2 (Mathe-
son), with a pure N2 sample. The fraction of NH3 intro-
duced in the mixture was controlled to within ±0.2ppm
by using a mass flow controller [MKS Instruments, Inc.],
whereas the CO2 concentration was regulated to within
±1000ppm by adjusting the outlet pressure of the cylinder
regulator.
Other modifications were introduced in order to improve
the operation of the instrument that was reported in [1–3]:
A pure ammonia cell (1 Torr) was used for real-time fre-
quency calibration. This cell was attached to a motorized
translational stage (New Focus, Model No. 8892) that inserts
the reference cell periodically into the optical path of the sig-
nal beam. This is accomplished using a software-generated
TTLpulsefromthedataacquisition routine.Toautomatically
account for operational fluctuations in the gas handling sys-
tem, pressure and temperature are measured in real time and
their values are updated in the data analysis routine for each
concentration measurement. By implementing these modifi-
cations, the system is now a stand-alone instrument that can
be used for on-line remote operation over extended periods
of time, monitoring the concentration of trace gases such as
NH3 and CO2. This is particularly useful for demonstrat-
ing the capabilities of the Kalman filtering technique, which

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LELEUXet al. Applications of Kalman filtering to real-time trace gas concentration measurements 89
provides an optimal estimate of the true gas concentration
continuously.
3.2
Data processing: acquisition and analysis
A laptop PC running LabView 5.0 software was
usedfordataacquisitionandprocessing.Withthephotodetec-
tor operating in auto-balance mode, 500 spectral scans were
averaged for each single concentration measurement. The
total data collection time, averaging and processing to obtain
a single concentration measurement was less than 30s. For
convenience,thesensorwasconfiguredtotakemeasurements
every minute in order to correlate ammonia measurements to
external daily events. The capability to remotely monitor the
sensor performance and concentration data was implemented
byusing“PCAnywhere”software.
Thereal-time fitting routine implemented in LabView has
been reported elsewhere [13], and only minor software mod-
ifications were required for this work. Initially a direct trans-
mission spectrum from the logarithmic output of the Nirvana
detector,operatinginauto-balancemode,isobtainedanda3rd
order polynomial is fitted to the baseline of the absorption
scan. If a voltage, V∗(ν), is assigned to the baseline values,
and the logarithmic signal values are assigned a voltage V(ν),
then the corresponding optical transmission for the given fre-
quencyvalue, T(ν),in percent, is givenby [14]:
T(ν) = 100e−A
TV∗(ν)+1
e−A
TV(ν)+1
(17)
where T is the temperature of the photodetector (taken to be
300K), and A is a constant determined by the gain in the
amplifier circuit of the detector (A = 273K/V). This proced-
ure provides a balanced transmission spectrum and simultan-
eously corrects for the baseline of the absorption signal. To
thisspectrum(afterdividingby100andtakingthenaturallog-
arithm), an absorption line-shape function is fitted to obtain
thegas concentration as described below. Theconvenience of
this procedure is that only one detection channel is needed in
this configuration to accountfor both thesignal and reference
outputsin thephoto-detector.
The fit employs a non-linear, least-squares Levenberg–
Marquardtalgorithm and uses a given absorption profileto fit
thedata.Sincethepressureregimeofoperation(100 Torr)lies
between the predominantly Doppler and pressure-broadened
regimes, a triple Voigt profile with fixed pressure broadening
and Doppler width was used. Typical absorption spectra ob-
tained were fitted to a third-order polynomial and three Voigt
profiles [15,16], with a linewidth of 0.027cm−1and a pres-
sure broadening contribution of 0.020cm−1. Small pressure
fluctuations in thesystemof±1Torrareaccounted forin real
time by the data collection software. Therefore, the linewidth
of the ammonia and carbon dioxide absorption peaks was ex-
pected to fluctuate by less than ±0.001cm−1[17–19]. This
introduces an error for the ammonia and carbon dioxide con-
centration measurements in the fitting routine of 2%. The
residual of the Voigt fit in typical 2.8ppm NH3spectra (peak
absorption of 0.14%) yields an uncertainty level for a single
measurement (with 500 averages) of ±0.01% – absorption
– (note that 100×ln(I/I0) is equivalent to % absorption for
small absorptions). The rms uncertainty level is then 0.014%
absorption and a peak absorption and corresponds to 2.8ppm
of NH3. This is the minimum concentration that can be meas-
ured reliably with the reported gassensor design, withoutany
further datafiltering. Opticalfringesfrominterference effects
introduced by the multi-pass cell limit the sensitivity and are
the primary source of uncertainty in the ammonia and carbon
dioxide concentration values reported.
4
4.1
Experimental results and discussion
Kalman filter parameters and tuning
The development of a Kalman filter for real-time
measurements that may vary by an order of magnitude re-
quires the selection of appropriate values for σ2
centration variability) and σ2
by the sensor). Both of these variances represent the concen-
tration variability from one measurement to the next. For in-
stance, σ2
be expected to vary from one measurement to the next, and
σ2
introduceinto themeasured concentration fromonemeasure-
ment tothenext.
The σ2
sor at a constant gas concentration to determine the variance
of the measured concentrations. This value may then be used
as a constant in a real-time filter operation. However, such
a calculation is not valid over all dynamic ranges, since the
concentration variability changes in direct proportion to the
concentration range. This is shown in Fig. 3, where the vari-
w(true con-
v(measurement noise introduced
wrepresents how much the true concentration would
vrepresents how much noise the sensor may be expected to
v, value may be measured while operating the sen-
a
b
FIGURE 3
ments for a16-hperiod. Kalman filter results are indicated by athick bold line
while raw data measurements appear as thin lines depicting system noise
Simultaneous NH3 (a) and CO2 (b) concentration measure-