State and parameter estimation of microalgal photobioreactorcultures based on local irradiance measurement
Wei Wen Su *, Jian Li, Ning-Shou Xu
Department of Molecular Biosciences and Bioengineering, University of Hawaii at Manoa, 1955 East West Road Ag. Sci. 218, Honolulu,
Hawaii, HI 96822, USA
Received 14 January 2003; received in revised form 13 June 2003; accepted 27 June 2003
Abstract
Local photosynthetic photon flux fluence rate (PPFFR) determined by a submersible 4p quantum micro-sensor was
used in developing a versatile on-line state estimator for stirred-tank microalgal photobioreactor cultures. A marine
micro-alga Dunaliella salina was used as a model organism in this study. On-line state estimation was realized using the
extended Kalman filter (EKF), based on a state model of the photobioreactor and on-line local PPFFR measurement.
The dynamic state model for the photobioreactor was derived based on mass-balance equations of the relevant states.
The measurement equation was established based on an empirical correlation between the microalgal biomass
concentration and the local PPFFR measured at a fixed point inside the photobioreactor. An internal model approach
was used to estimate the specific growth rate without the need of state-based kinetic expression. The estimator was
proven to be capable of estimating biomass concentration and specific growth rate, as well as phosphate and dissolved
oxygen concentrations in a photobioreactor illuminated with either fixed or time-varying incident radiation. The
quantum sensor was shown to be robust and able to quickly respond to dynamic changes in local PPFFR. In addition,
the quantum sensor outputs were not affected by bubble aeration or agitation within the typical operating range. The
strong filtering capacity of EKF gives the state estimator superior performance compared to direct calculation from the
empirical biomass/local PPFFR correlation. This state estimation system makes use of inexpensive and reliable sensor
hardware to report key process dynamics of microalgal photobioreactor cultures on-line, enabling improved operation
of such a process.
# 2003 Elsevier B.V. All rights reserved.
Keywords: Kalman filter; Microalgae; Photobioreactor; Quantum sensor; State estimation
1. Introduction
Process sensing is an essential and integral
component of photobioreactor technology. Indus-
trial photobioreactor systems demand accurate
tracking of multiple culture parameters to cor-
rectly reflect the process dynamics, and from
which appropriate actions can be implemented to
enhance process performance. In addition to the
standard sensors for temperature, dissolved oxy-
gen, and pH, many sophisticated electronic sensors
for measuring a variety of culture and process
* Corresponding author. Tel.: �/1-808-956-3531; fax: �/1-
808-956-3542.
E-mail address: [email protected] (W.W. Su).
Journal of Biotechnology 105 (2003) 165�/178
www.elsevier.com/locate/jbiotec
0168-1656/03/$ - see front matter # 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0168-1656(03)00188-3
parameters (such as viability, metabolic activities,
biomass and nutrient concentrations) are avail-
able. However, most of these sensors are limited
by high cost, poor long-term stability, or both.
Furthermore, installation of the required manifold
of electronic hardware sensors would make the
photobioreactor system prohibitively expensive
not only in terms of design and construction but
also in maintenance. These considerations necessi-
tate innovative tactics to tackle process sensing
problems in photobioreactors. To this end, soft-
ware sensors or state estimators represent an
appealing solution. To achieve state estimation,
one uses appropriate process models and filtering
Nomenclature
A system dynamics matrix in the Riccati equation (Eq. (8))C observation matrix in the Riccati equation (Eq. (8))E0 parameter in the light model (Eq. (1)) (gn l�n)E1 parameter in the light model (Eq. (1)) (Ln g�n)IL local irradiance light intensity in the culture (mE m�2 s�1)I0 incident light intensity on the reactor external surface (mE m�2 s�1)K Kalman filtering gain matrix defined in Eq. (7)kla volumetric oxygen mass transfer coefficient (h�1)Km constant in the phosphate uptake model in Eq. (9) (mg l�1)M amplitude in Eq. (2)n parameter in the light model (Eq. (1))O dissolved oxygen concentration (mg l�1)O* saturated oxygen concentration in the media (mg l�1)P covariance matrix of state estimation errorQ covariance matrix of system noisesqm specific phosphate uptake rate (mg g�1 h�1)qj variance of system noiseR covariance matrix of measurement noises (mE2 m�4 s�2)Ro max maximum oxygen generation rate (mg g�1 h�1)Ro min equivalent to the specific respiration rate (mg g�1 h�1)Rum ratio between maximum oxygen generation rate and maximum specific growth rate (mg g�1)S phosphate concentration (mg l�1)T time (h)Var VarianceVI measurement noise of IL (mE m�2 s�1)
Wj system noise of state variable j
X cell density (cell dry weight) (g l�1)Y measurement of IL (mE m�2 s�1)g changing rate of m in Eq. (3) (h�2)
d tunable parameter in Eq. (10) (h�1)m specific growth rate (h�1)mmax maximum specific growth rate (h�1)v instantaneous angular frequency of m variation (rad h�1)
8 phase angle in Eq. (2) (rad)/j/ estimate of state variable j
j0 initial value of state variable j
F changing rate of m in Eq. (10)(h�2)
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178166
algorithms, together with limited and often simpleprocess measurements to estimate immeasurable
system states, reduce measurement noise, and
identify uncertain system dynamics (Stephanopou-
los and Park, 1991).
The present study is an extension from our
earlier work (Li et al., 2002) in which process
sensing in microalgal photobioreactors was
achieved via state estimation in connection withdissolved oxygen measurement. Here we report the
application of a submersible quantum sensor
coupled with process modeling and extended Kal-
man filter (EKF) in establishing a versatile state
estimator for on-line estimation of multiple state
variables in microalgal photobioreactors. Quan-
tum sensors are more affordable and robust than
most electrochemical sensors, and they are espe-cially suited for photobioreactors. A submersible
spherical quantum micro-sensor was utilized in
this study to determine the local photosynthetic
photon flux fluence rate (PPFFR), which served as
the measurement signal in our EKF-based state
estimator. In this case, the photon flux fluence rate
represents the total photon flux radiance (over all
directions) intercepted by the spherical sensorsurface, divided by the cross-sectional area of the
sphere (Anon, 1990). When the wavelength range
of the photons measured is limited to the 400�/700
nm range (photosynthetically active radiations;
PAR), the term ‘PPFFR’ is used (Anon, 1990).
Herein, estimation of biomass concentration and
specific growth rate, as well as phosphate and
dissolved oxygen concentrations was demon-strated in a series of growth experiments in a
photobioreactor illuminated with constant inci-
dent radiation at a wide range of intensity levels or
with time-varying incident radiation.
2. Materials and methods
The green alga Dunaliella salina (Teod.) UTEX1644 was used throughout this study. The organ-
ism was cultured in a chemically defined hypersa-
line liquid medium as previously described (Li et
al., 2002). Bioreactor cultures of D. salina were
conducted in an instrumented bench-top photo-
bioreactor. This reactor was modified from a 3-l
stirred-tank fermenter (BiofloIII, New BrunswickScientific, Edison, NJ). The agitation and aeration
rates were fixed at 150 rpm and 0.5 vvm, respec-
tively, in all reactor experiments. Culture pH was
controlled at 7.49/0.05 by adjusting CO2 supple-
mentation to the air feed stream using a PID
controller as described in Li et al. (2002). The basic
EKF algorithm, as well as the determinations of
biomass and dissolved oxygen concentrations,were also described in Li et al. (2002). The incident
light intensity on the reactor surface was measured
using a cosine (2p) quantum sensor (LI-190SA, LI-
COR, Lincoln, NE) connected to a micrologger
(21X, Campbell Scientific, Logan, UT), and ex-
pressed as photosynthetic photon flux density
(PPFD). The local irradiance level at a fixed point
within the photobioreactor was measured using asubmersible spherical (4p collector) quantum mi-
cro-sensor (US-SQS/LI, Heinz Walz GmbH, Ef-
feltrich, Germany) connected to a micrologger
(21X, Campbell Scientific, Logan, UT), and ex-
pressed as PPFFR. This fiber-optic micro sensor
has a spherical sensor tip (with a diameter of 3
mm) that can sense light from all directions
underwater, and the sensor reading was loggedinto a supervisory computer through a data
acquisition board (AT-MIO-16DE-10, National
Instruments, Austin, TX). To protect the sensor
from the highly corrosive hypersaline medium, the
sensor was inserted into a glass optical well that
was fixed to the reactor head plate and submerged
in the culture broth. The sensor tip was placed 2
cm inward from the reactor periphery and 6 cmbelow the culture surface. Sensor placement within
the photobioreactor is shown schematically in Fig.
1. Phosphate concentration in the culture medium
was determined by the molybdate method (Tara,
1991).
3. Results and discussion
3.1. Light intensity measurement
Light availability plays a key role in photobior-
eactor performance. Therefore, it is necessary to
accurately assess this process parameter to ensure
adequate photobioreactor operation. Quantum
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178 167
sensors have been widely used to measure the level
of irradiance either on the external photobioreac-
tor surface (Acien Fernandez et al., 1997; Janssen
et al., 1999), or inside the photobioreactors
(Katsuda et al., 2000; Li et al., 2002; Sanchez
Miron et al., 2000). Two major types of quantum
sensors are commonly used in photobioreactor
applications, and they differ both in geometrical
appearance and in the physical characteristics of
irradiance sensed. The 2p sensors measure PPFD,
which is defined as the number of PAR photons
incident per unit time on a unit flat surface. The 4psensors with a spherical shape measure PPFFR,
which accounts for PAR photon flux radiances at
a point over all directions (Anon, 1990). Bothproperties have the same unit. The 2p sensors are
suitable to quantify the incident radiation on the
reactor surface, indicating the level of radiation
supplied to the photobioreactor. The 4p sensors
are suited for measuring the level of PAR inside
the reactor to indicate the photon flux available to
the cells from all directions at certain point inside
the photobioreactor.In our previous study (Li et al., 2002), we
developed a state estimator for stirred-tank photo-
bioreactor cultures using dissolved oxygen con-
centration as the measured state. While the
estimator worked well, signal drift from the
polarographic dissolved oxygen probe during
prolonged cultivation was observed. Quantum
sensors offer several advantages over electroche-mical type dissolved oxygen sensors in on-line
sensing of photobioreactor states. First and fore-
most, the signal from the electrochemical type
dissolved oxygen sensors is known to drift sub-
stantially during long-term cultivation. Drift is
commonly caused by accumulation of hydroxyl or
metal ions, chloride depletion, or external fouling
of the probe membrane surface (Bailey and Ollis,1986). This means that the sensor requires frequent
recalibration. However, frequent calibration could
be difficult to implement during the cultivation.
Moreover, the membrane body of the electroche-
mical type dissolved oxygen sensor is usually not
very durable, especially if the probe is exposed to
repeated autoclave cycles. In contrast, the quan-
tum sensors require no frequent calibrations andare very robust in their construction.
The stability of the quantum sensor was con-
firmed experimentally in this study. Quantum
sensor readings under fixed external illumination
were logged for a period of 3 days. The variation
of the data was less than 2% (data not shown).
This is true for both 2p and 4p sensors. If one
considers the irradiance fluctuation from thefluorescent lights, the sensor drift could be con-
sidered negligible. Because the 4p sensor was
inserted into the bioreactor culture to measure
the local PPFFR, it is possible that aeration and
agitation could affect the measurements. To ex-
amine these effects, sensor data were recorded at
different aeration and agitation rates with either
Fig. 1. Schematic drawing of the photobioreactor system (MC:
mass flow controller).
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178168
culture broth or cell-free media inside the bior-eactor, under fixed external illumination. The
result showed that within the operating conditions
comparable to those used in this study (i.e.
aeration from 0.1 to 0.8 vvm and agitation from
75 to 200 rpm), aeration and agitation have little,
if any, effect on the quantum sensor readings (data
not shown).
3.2. Measurement model
A measurement model connects the on-line
acquired measurement signals with the culture
states to be estimated. The dependence of local
irradiance (PPFFR) level in the photobioreactor
on biomass concentration forms the basis for state
estimation using the submersible quantum sensor.
Although mechanistic modeling of irradiance dis-
tribution in photobioreactors has been reported(Brucato and Rizzuti, 1997; Cornet et al., 1994;
Evers, 1991; Katsuda et al., 2000; Kurata and
Furusaki, 1993), use of such complex models in a
state estimator would greatly complicate computa-
tion. The inherent assumptions associated with
such models also inevitably limits their general
applicability. The measurement model was there-
fore established empirically in this research. Athree-parameter empirical model was developed
here to correlate the local irradiance data (IL, in
terms of PPFFR) with the biomass data (X ):
IL�I0E0
X n(1�e�E1X n
): (1)
This correlation was checked against the data
(online-measured IL vs. corresponding X ) col-
lected during the entire culture period of two
duplicated batch bioreactor experiments. These
experiments were conducted at an external illumi-
nating level of 450 mE m�2 s�1. Eq. (1) was used
to fit the data via Marquardt’s non-linear least
squares method (Marquardt, 1963). Values of themeasurement model parameters are listed in Table
1. As indicated in Fig. 2, the measurement model is
in good agreement with the data throughout the
entire biomass range tested. In addition, Eq. (1)
was found to provide a better fit of the data than
using a second-order polynomial equation that
also contains three fitting parameters (data notshown).
3.3. Process model and estimation equations
Several photobioreactor growth models havebeen reported in the literature (Acien Fernandez
et al., 1998; Baquerisse et al., 1999; Camacho
Rubio et al., 1999; Cornet et al., 1992; Cornet and
Albiol, 2000; Csogor et al., 1999; Rorrer and
Mullikin, 1999; Taya et al., 1995). In many of
these models, light-limited growth was implicated,
and cell growth rate was assumed to be governed
by the average irradiance in the photobioreactor.As such, accurate modeling of the radiant field
inside the photobioreactor is necessary to provide
a truthful estimation of the average irradiance.
This could be a very challenging task considering
that detailed mechanistic modeling of radiant
distribution in photobioreactors would entail con-
sideration of factors such as selective absorption of
photosynthetically active radiation, non-isotropiclight scattering, as well as size and shape of the
cells (Cornet et al., 1994). Furthermore, the
radiant absorption efficiency of the cells and the
Table 1
Model and filter parameters
Model
Rum 1147 mg g�1
Ro min 7.308 mg g�1 h�1
kla 13.50 h�1
qm 0.844 mg g�1 h�1
Km 5.100 mg l�1
O* 5.376 mg l�1
E0 0.749 g0.925 l�0.925
E1 3.873 l0.925 g�0.925
n 0.925
Filter
I0 time-invariant time-variant
pX 0�/Var X0 0.01 g2 l�2 10�6 g2 l�2
pm0�/Var m0 10�4 h�2 10�7 h�2
pg 0�/Var g0 10�6 h�4 10�6 h�4
pv 0�/Var v0 10�4 h�2 10�2 h�2
qX �/Var WX 10�3 g2 l�2 10�5 g2 l�2
qm �/Var Wm 10�5 h�2 10�7 h�2
qg �/Var Wg 10�8 h�4 10�7 h�4
qv �/Var Wv 10�6 h�2 10�3 h�2
R�/Var VI 130 mE m�2 s�1 700 mE m�2 s�1
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178 169
energy conversion of the absorbed radiances into
biomass vary substantially under different illumi-
nation conditions. This is partly due to the fact
that the antenna size of the cellular photosystems
changes with the irradiance levels (Baroli and
Melis, 1996). In addition to irradiance, one may
need to consider factors such as nutrient limitation
and presence of growth inhibitors. In light of these
challenges, it is desirable to establish alternative
approaches to model cell growth in photobioreac-
tors, rather than using constitutive models (Blanch
and Clark, 1997) that require prior knowledge of
how the growth response of a cell is related to its
physical-chemical and biological environment (e.g.
irradiance, nutrient, and inhibitor levels).
In the state estimator reported here, a precise
constitutive growth model is no longer required
because the biomass concentration is observable
through the measurement model. This situation
allows us to use an adaptive approach to model
the specific growth rate using EKF without the
need to relate the specific growth rate to the
environmental conditions. Such an approach
could greatly broaden the applicability and sim-
plify the design of the state estimator. Considering
that the specific growth rate m changes smoothly
during the cultivation process, and assuming that
within a small time interval around the currenttime instant (t � /[t , t�/Dt ]), m can be approximated
using a sinusoidal function with a proper set of
parameters, M , v , and 8 , then m can be expressed
as:
m�M sin(vt�8 ); (2)
where M , v , and 8 are the amplitude, angular
frequency, and phase angle, respectively. These
parameters can be made to evolve during the
culture process via Kalman filter so that Eq. (2)
is able to track the specific growth rate throughout
the bioreactor cultivation. If we introduce a new
Fig. 2. Empirical correlation between IL and X . The symbols represent data points taken from two repeated culture experiments
operated at I0�/450 mE m�2 s�1. The curve was obtained by fitting the experimental data with Eq. (1).
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178170
parameter, g , which represents the derivative of m ,the dynamics of g can be represented as:
g��v2 �m: (3)
By considering g and v as additional states and
assuming frequency v to be approximately con-stant, the following adaptive state equation for m
is obtained:
*m
g
v
24
35�
g
�v2m
0
24
35 �
Wm
Wg
Wv
24
35; (4)
where Wm , Wg and Wv represent system noise.Notice that by introducing g , the parameters M
and 8 in Eq. (2) are eliminated in the adaptive
state equation for m . By incorporating growth
dynamics, X ; into Eq. (4), an adaptive EKF
estimation model can be derived:
˙X˙m˙g˙v
2664
3775�
mX
g
�v2m
0
2664
3775 � K [Y � IL];
E
X*m
g*v
2664
3775
t�0
�
X 0
m0
g0
v0
2664
3775;
(5)
where Y is the on-line measured local irradiance(i.e. Y�/IL�/VI), and
IL�I0
E0
X n(1�e�E1X n
) (6)
is the estimated local irradiance, and
K�PCTR�1 (7)
is the Kalman filter gain matrix, and P is the
covariance matrix of filtering error satisfying thefollowing matrix Riccati equation:
P�AP�PAT�Q�KCP; Pjt�0�P0 (8)
Detailed expressions of P, C, R, A, and Q are
given in the Appendix A.
In addition to the four parameters/state vari-
ables included in the estimation model, Eq. (5),
dissolved oxygen concentration (O ) and nutrient
(phosphate) concentration (S ) were estimated
using a set of auxiliary model equations. It should
be noted that O and S are not observable from IL
measurement. This was concluded by examining
the observability criterion (Ramirez, 1994) using
matrices A and C with O and/or S included in the
state vector. In this case, we also noted that the
filtering simulation diverged. Therefore, O and S
were estimated using an auxiliary estimation
model with the EKF estimates of X and m . The
model for dissolved oxygen was based on adynamic mass balance as discussed elsewhere (Li
et al., 2002), and the phosphate model was derived
based on basic saturation-type uptake kinetics
commonly seen in microalgal species (Nyholm,
1977):
˙O˙S
� ��
(Rumm�Ro min)X �kla(O��O)
�qm
S
Km � SX
264
375;
O
S
� �t�0
�O0
S0
� �;
(9)
where Rum�/Ro max/mmax, Ro min, kla , qm, Km and
O* are constant parameters. In the oxygen bal-
ance, the specific photosynthetic oxygen evolution
rate was represented by the term Rum �m; which
implied a linear relationship with m , whereas thespecific respiration Ro min was approximated as a
constant since it is typically much lower than the
oxygen evolution rate (Li et al., 2002). As for the
balance of extracellular phosphate, we considered
the disappearance of phosphate from the medium
mainly as a result of cellular uptake. The values of
the model parameters were either calculated (O*,
based on oxygen solubility), measured (kla ), orfitted (Rum, Ro min, qm and Km) against two sets of
experimental culture data collected under 450 mE
m�2 s�1 incident irradiance, using Marquardt’s
non-linear least squares method (Marquardt,
1963). To enable model fitting, we used the
biomass data (X ) from the two culture experi-
ments, and calculated m based on a second-order
polynomial fit of the growth curves (i.e. X vs. t).To initiate the nonlinear model regression, initial
estimates of the parameters (Rum, Ro min, qm and
Km) were obtained from relevant published data
(Li et al., 2002; Nyholm, 1977). The fitting results
for O and S using Eq. (9) are given in Fig. 3. The
values of the auxiliary model parameters are listed
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178 171
in Table 1. As discussed in the following section,the poor model fitting of the dissolved oxygen data
seen in Fig. 3 could be substantially improved by
using the EKF.
3.4. Effect of Kalman filtering on state estimation
According to the measurement model, Eq. (1),ILis directly related to X , and therefore X and m
can be directly calculated from the measurement
model without using EKF, given that IL is con-
tinuously monitored on-line. To determine
whether state estimation can be improved by using
the EKF-based estimator over the direct calcula-
tion from Eq. (1), the two approaches were
compared and the estimation results are presentedin Fig. 4A and B, alongside the experimental
culture data obtained under 450 mE m�2 s�1
incident irradiance. The results revealed consider-
able improvement by using EKF over direct model
calculation on the state estimation, particularly for
m and O . The estimation results from direct model
calculation are too noisy to convey any clear trend
for m and O . The superior performance of EKFprimarily resulted from its filtering capacity. With
direct model calculation, estimation of all other
states was based on the estimation of X , which inturn was based on the IL signals and the measure-
ment model. Therefore the measurement noise for
IL was transmitted to the estimation of X and
subsequently propagated and amplified in the
estimation of m and O . Although simpler filtering
algorithms such as the moving-average filter could
also potentially reduce the estimation noise, they
generally are less effective than EKF (Stephano-poulos and San, 1984). As can be seen from Fig.
4C, even when the moving-average interval was
taken as long as 10 min, the filtering result was still
very noisy. Further extending the averaging inter-
val will lead to a significant time lag that is not
ideal for timely control.
3.5. State estimation under constant external
illumination
From the growth experiments conducted at the
external illuminating level of 450 mE m�2 s�1, we
have estimated the model and filter parameters for
the state estimator. Using the same set of model/
filter parameters, the utility of the state estimatorwas further validated using data collected from
growth experiments under three additional exter-
Fig. 3. Model fitting of two repeated culture experiments operated at I0�/450 mE m�2 s�1. Legend: experimental data for biomass or
phosphate concentration (k); polynomial-fitted biomass curves (. . .); experimental dissolved oxygen data (*/); model (Eq. (9)) fitting
of dissolved oxygen and phosphate concentrations (----).
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178172
nal illumination conditions (I0�/70, 100 or 300 mE
m�2 s�1). In these experiments, the external
illumination level was fixed for the duration of
each cultivation (about 2 days). The results are
presented in Fig. 5. The estimation results indi-
cated effective tracking of all four culture states
examined (i.e. X , m , O , and S ). Among these
states, tracking of biomass X is most accurate.
Since X is included in the main state estimation
vector, it is anticipated that probable modeling
errors in depicting cell growth could be compen-
sated by the EKF. The accurate tracking of X also
indicated that it is appropriate to use Eq. (4) for
adaptive modeling of the specific growth rate m .
EKF also helped to filter out the strong noise
associated with the estimation of m , as indicated in
Fig. 4 and discussed in Section 3.4. The error
associated with the measurement model could be
partly compensated by the noise VI, even though
in theory VI indicates only the measurement noise
(cf. Eqs. (5) and (6)). By not including IL in the
state vector, the dimension of the state vector is
reduced, resulting in simplified estimator compu-
tations, while the accuracy of the estimator is not
compromised (this is discussed further in Section
3.7). The state estimator reported here does have
certain limitations. The observability of the system
is built on the relationship between IL and X , as
indicated in the measurement model. Since IL
decreases as X increases during the culture,
ILbecomes less sensitive to X at high biomass
concentrations (Fig. 2), and the system becomes
less observable through measurement of IL. The
practical limit of biomass concentration, within
which the state estimator would work, depends on
the cellular light absorption properties as well as
Fig. 4. Effect of Kalman filtering (I0�/450 mE m�2 s�1). (A) EKF estimates; (B) estimates from direct calculation using Eq. (1)) with
sampling period of 1 min; (C) estimates from direct calculation using Eq. (1)) with moving-average interval of 10 min. Legend:
experimental data for biomass or phosphate concentration (k), experimental dissolved oxygen or light intensity curves (*/),
estimation curves (*/).
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178 173
I0. As shown in Figs. 2 and 4, the state estimator
worked well at biomass concentration as high as 2
g l�1 (dry cell weight) and an I0 of 450 mE m�2
s�1, for the D. salina culture.Since O and S are not observable from IL
measurement, these states were estimated using an
auxiliary estimation model with the EKF estimates
of X and m . As seen in Fig. 5, the estimation
results of these two states generally agreed with the
experimental data and only began to deviate from
the data during the later stage of culture. This
suggested that the models for O and S are
reasonably accurate, considering the model errors
were not corrected by EKF as in the case of X .
However, as the culture progressed, errors existed
in the model structure/parameter and in the initial
state measurement/estimation accumulated, lead-
ing to the observed estimation discrepancy. Esti-
mation of these two states could be improved by
incorporating additional measurement variables
with which these two states become directly
observable.
Because there is only one on-line measurement
input (i.e. IL), and hence R�/R�/Var VI. In
principle, the variance corresponding to the noises
of the quantum sensor could be used to provide a
reasonable initial estimate of R . Based on this
information, and subsequent simulations of the
batch cultivation data acquired at 450 mE m�2 s�1
incident irradiance, we set R to 130 in this study.
Simulation results indicated no significant differ-
ence when R was set in the range from 30 to 900,
provided that other parameters were fixed at
appropriate values (data not shown). P0 and Q
were determined empirically by balancing fast
tracking with stable system response. In practice,
the values of the diagonal elements in the P0
matrix should be set close to the variances of the
respective initial state estimation errors. The
system noise covariance matrix Q can be set based
Fig. 5. Estimator performance under different I0. Legend: same as Fig. 4.
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178174
on the variance of the model uncertainty of eachstate. For X , pX 0 was estimated by considering the
measurement error variance of the initial biomass
concentration X0, and qX was estimated by
calculating the mean-square deviation of the
biomass measurement data and the corresponding
least-squares fit for the entire cultivation period.
For m , g and v , there were no directly measurable
data available, and thus the initial filtering errorvariances pm0, pg0, pv0, and the system noises qm ,
qg and qv were estimated by trial and error based
on the batch cultivation data acquired at 450 mE
m�2 s�1 incident irradiance. The values of these
filter parameters are summarized in Table 1.
3.6. State estimation under time-variant external
illumination
Outdoor microalgal cultures are commercially
important and they are typically operated under
diurnal cycles. It is therefore valuable to test
whether the state estimator developed here can
function well under conditions of time-variant
incident radiation. Because the local irradiance
level changes synchronously with the level of
incident radiation as indicated in the measurementmodel, the same state estimator structure de-
scribed above can be used in conditions of time-
variant incident radiation. However, the filter
parameters need to be retuned. In this case,
compared with the respective values used under
fixed I0, the variances pX 0, pm0, qX , and qm were
reduced; pg0 remained unchanged; while pv0, qg ,
qv , and R were increased (Table 1). A typical setof state estimation results under time-varying I0 is
presented in Fig. 6. The estimator was able to
track the biomass and phosphate concentrations
and to predict the specific growth rate. Unlike the
estimation results under constant I0, v has to be
made adaptive throughout the culture under time-
varying I0 (Fig. 6), indicating the need to include v
in the state vector in this case. The estimation ofdissolved oxygen concentration deviated from the
measurement, although both shared a similar
trend. The less than satisfactory estimation per-
formance on O might be explained as follows.
First, the simple conservation balance model for O
may not be able to accurately express the actual
dynamic physiological response of the cells upon
rapid changes of incident light radiation. Further-
more, O was not directly observable from IL
measurement and its estimation was based on thedirect integration of the corresponding dynamic
mass balance with the filtered estimates of X and
m . As such, model errors could not be compen-
sated by the filter, leading to accumulation of
modeling errors as discussed in the preceding
sections. Estimation of O might be improved by
refining its dynamic model and/or incorporating
additional measurement variable(s) with which O
becomes directly observable.
3.7. Alternative state estimator designs
In this study, we developed a new internal model
for m , Eq. (4), and validated its utility in the state
estimation of microalgal photobioreactors. Other
internal models for m have been reported for the
on-line estimation of yeast and bacterial fermenta-
tions (Stephanopoulos and San, 1984; Bastin and
Fig. 6. Estimator performance under time-varying incident
irradiance. Legend: same as Fig. 4.
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178 175
Dochain, 1990; Takiguchi et al., 1997). A widelyadopted model that was first reported by Stepha-
nopoulos and San (1984) assumed that the dy-
namics of m could be represented by a second-
order system with certain random disturbances,
i.e.:
m�F�W1
F��d F�W2
;
(10)
where F is a variable representing the dynamics of
m , d is a tunable parameter equivalent to the
inverse of a characteristic time constant, and Wi
(i�/1, 2) represents system noises. Eq. (10) sug-gests that in case of low system noises, m satisfies
an exponential function with an invariable time
constant within a small time interval around the
current time instant. We tested Eq. (10) in this
study, and found that the estimator performance
under conditions of time-variant incident radiation
could be improved if estimation of the parameter
d was made adaptive, i.e.:
d�0�W3; (11)
where W3 represents a white noise. Upon optimal
tuning of the filter parameters, the state estimation
results based on Eq. (4) were found slightly moreaccurate than those based on Eqs. (10) and (11)
(on the basis of the sum of square errors; data not
shown).
Local PPFFR (i.e. IL) was used in this study as
the on-line measured signal input for the state
estimator. Two approaches of incorporating IL
into the state estimator were examined in this
study. In one approach, as described in thepreceding sections, IL was not included as a state
in the system model, Eq. (5). Rather, IL was
estimated directly from the measurement model,
Eq. (6), using the optimal estimates of X . Since X
is observable through the measurement of IL, as
indicated in the measurement model, the difference
between the measured and estimated IL could be
used with the Kalman gain to generate optimalstate estimates according to Eq. (5). In a second
approach, IL was included as an additional state in
the system model. Here a dynamic state equation
for IL was conveniently derived based on the
empirical measurement model, Eq. (1). The mea-
surement state IL could then be estimated by the
EKF estimator. In this case, the state estimationequation becomes:
˙X˙m˙g˙v
˙IL
266664
377775�
mX
g
�v2m
0
I0m �I(X )�I0IL
I0
26666664
37777775
� K [Y � IL]; (12)
where
I(X )�n
�E0E1e�E1X n
�IL
I0
�: (13)
With this second approach, it is conceivable to
alleviate interferences such as modeling error
associated with the measurement equation, ran-
dom disturbance on I0 due to ambient lights, and
potential variation in the biomass light absorption
efficiency resulting from different levels of pig-
mentation during the cultivation process (Li et al.,
2002). We found, however, the state estimationresults obtained using the second approach were
not superior to those using the first approach (i.e.
not including IL in the state vector) under either
fixed or time-varying I0 conditions (data not
shown).
4. Conclusions
A state estimator was developed in this work
capable of effectively estimating a number of
culture states in microalgal photobioreactors, in-cluding specific growth rate and biomass, dis-
solved oxygen, and phosphate concentrations,
from the on-line measurement of local PPFFR
(i.e. IL) in the reactor, under constant or time-
variant incident radiation conditions. A submer-
sible quantum sensor was shown to give reliable
continuous on-line measurement of IL. The bio-
mass concentration was observable through themeasurement of IL, and an adaptive internal
model was shown to provide good estimates of
the specific growth rate. Eriksen et al. (1996)
reported improved biomass productivity in a
photobioreactor with on-line optimization of ex-
ternal artificial illumination. The intensity of
W.W. Su et al. / Journal of Biotechnology 105 (2003) 165�/178176
external illumination in this so-called ‘lumostat’reactor was automatically adjusted to give the
maximal photosynthetic activity, gauged by the
amount of carbon dioxide added to the aeration
gas to maintain the constant pH. Meireles et al.
(2002) reported a flow-injection-analysis system
for on-line monitoring of biomass in a microalgal
bioreactor. The state estimator developed in the
present study offers a cost-effective alternative anda more direct approach to estimating biomass
concentration, growth rates, and photosynthetic
rates in microalgal bioreactors. Accurate informa-
tion about these parameters will enable effective
manipulation of process parameters such as ex-
ternal illumination intensity and temperature to
improve photobioreactor performance, as exem-
plified in the work of Eriksen et al. (1996) andMeireles et al. (2002). With the general applic-
ability of submersible quantum sensors in micro-
algal bioreactor cultures, and the generic nature of
the state models, the state estimation system
developed here is expected to be highly useful for
monitoring a wide range of phototrophic micro-
algal processes.
Acknowledgements
The D. Salina strain was obtained from Dr
Anastasios Melis at the University of California,
Berkeley. This work was supported by the NSF
ERC Program. Contract grant number: EEC-
9731725.
Appendix A
For the estimation model based on the state
vector j� [X m g v]T; the filter matrices areexpressed as:
P�
pX pXm pXg pXv
pmX pm pmg pmv
pgX pgm pg pgv
pvX pvm pvg pv
2664
3775;
Pjt�0�P0�diag[pX0; pm0; pg0; pv0]
�diag[Var X0; Var m0; Var g0; Var v0] (A1)
Q�diag[qX ; qm; qg; qv]
�diag[Var WX ; Var Wm; Var Wg; Var Wv] (A2)
R�Var VI (A3)
A�
m X 0 0
0 0 1 0
0 �v2 0 �2vm
0 0 0 0
2664
3775 (A4)
C�@IL
@X jX
0 0 0
" #;
@IL
@X jX
�I0
nE0
X
�E1e�E1X n
�1
X n(1�e�E1X n
)
� (A5)
For the estimation model based on the state vector
j� [X m F d]T (i.e. using the alternative spe-
cific growth rate model, Eqs. (10) and (11)), the A
matrix becomes:
A�
m X 0 0
0 0 1 0
0 0 �d �F
0 0 0 0
2664
3775 (A6)
For the estimation model based on the state vector
j� [X m g v IL]T (i.e. Eq. (12)), the A and
C matrices are expressed as:
A�
m X 0 0 0
0 0 1 0 0
0 �v2 0 �2vm 00 0 0 0 0
I0m@I(X )
@XI0I(X ) 0 0
I0
I0
26666664
37777775
(A7)
C� [0 0 0 0 1] (A8)
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