541637468546

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

mailto:[email protected]

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).


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


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).


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


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 (----).


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 (*/).


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.


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.


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


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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