Application of mathematical kinetic models in the wastewater treatment.

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Chemical Engineering Journal
journal homepage: www.elsevier.com/locate/cej
Model-based control of a fedbatch biodegradation process
by the control Lyapunov function approach
U. Ba˜nos-Rodrígueza,c
, O.J. Santos-Sánchezb
, R.I. Beltrán-Hernándeza
, G.A. Vázquez-Rodrígueza,∗
a
Centro de Investigaciones Químicas, Universidad Autónoma del Estado de Hidalgo, Carretera Pachuca-Tulancingo km. 4.5, 42184 Mineral de la Reforma, Hgo., Mexico
b
Centro de Investigación en Tecnologías de Información y Sistemas, Universidad Autónoma del Estado de Hidalgo, Carretera Pachuca-Tulancingo km. 4.5, 42184 Mineral de la Reforma,
Hgo., Mexico
c
Universidad Politécnica de Pachuca, Ex-Hacienda de Santa Bárbara, Carretera Pachuca-Cd. Sahagún km. 20, C.P. 43830, Zempoala, Hgo. Mexico
a r t i c l e i n f o
Article history:
Received 13 October 2011
Received in revised form 22 February 2012
Accepted 22 February 2012
Keywords:
Phenol
Inhibition kinetics
Modeling
Suboptimal control
Control Lyapunov function
Hill-climbing algorithm
a b s t r a c t
A strategy based on a kinetic model was developed to control a fedbatch process for phenol degra-
dation by means of the feeding pattern. Biodegradation of phenol by acclimated activated sludge was
modeled by the Haldane equation. This model was considered as the control system and the feed-
ing flow (Q) to the reactor as the input. The Control Lyapunov Function (CLF) approach was used to
synthesize the controller, which was applied in open loop. A feeding strategy and a fedbatch cycle,
consisting of 4 h of filling, 6 h of aeration, 0.58 h of settling and 0.03 h of drawing, were obtained
by simulation. The kinetic model and the control strategy were validated experimentally in a fed-
batch cycle, and the performance of the latter was compared with that of a non-controlled cycle. The
model allowed to predict adequately both the phenol and the biomass concentrations during the cul-
ture. The control of the process resulted in a lower hydraulic retention time (21.28 h) than that of
the non-controlled cycle (47.62 h). Higher applied and phenol mass loads (1.13 g phenol L−1
d−1
and
4.82 g phenol g−1
TSS d−1
, respectively) could also be obtained in the controlled cycle. Further, the phe-
nol mass load was higher than that reported in other processes even though any online monitoring
was carried out. This is the first reported use of the CLF approach to biodegradation processes, and it is
our view that it could be applied to other phenolic wastewater treatment systems lacking of real-time
monitoring.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Phenol is a naturally and industrially derived pollutant com-
monly found in industrial and domestic wastewater [1]. This
compound can be removed biologically from liquid streams
because there are numerous and widespread microorganisms that
can break it down. Indeed, aerobic prokaryotic and eukaryotic
microorganisms [1], as well as mixed cultures (i.e., activated sludge
[2]), are capable to oxidize phenol.
Nevertheless, even the microorganisms able of using phenol
as a substrate can only deal with relatively low concentrations of
this compound (of about 0.10 g L−1 [3]) due to its cell membrane
disrupting effect [1]. So, the Haldane inhibition model (Eq. (1)) is
frequently used to describe the relationship between the specific
∗ Corresponding author. Present address: NSERC Industrial Chair in Drinking
Water, École Polytechnique de Montréal, C.P. 6079 succ. Centre Ville, Montréal QC,
Canada H3C 3A7. Tel.: +1 514 340 4711x2334; fax: +1 514 340 5918.
E-mail addresses: gabriela.vazquez@polymtl.ca, g.a.vazquezr@gmail.com
(G.A. Vázquez-Rodríguez).
microbial growth rate ( ) and the phenol concentration (S) in batch,
fedbatch and continuous operating modes [2].
=
maxS
KS + S + S2
Ki
(1)
max, KS and Ki are constant parameters corresponding to the max-
imum specific growth rate, the half-saturation constant and the
Haldane-inhibition constant, respectively.
Fedbatch reactors provide dynamic conditions of operation spe-
cially well-suited for the biodegradation of inhibitory compounds
as phenol. In these reactors the influent is fed to the treatment unit
following a specific pattern. Since the effluent is not drawn off until
the reactor is full or later, toxic compounds are diluted in the total
volume of the aeration tank, allowing their inhibitory effect to be
reduced [4]. For control purposes, the control signal (e.g., the feed-
ing flow of toxic influent to the reactor) has to be minimized to
diminish the inhibition of the biomass but without compromising
the performance of the process (i.e., the daily pollution load to the
system).
1385-8947/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.cej.2012.02.068

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Nomenclature
Ki Haldane-inhibition constant (g L−1)
KS half-saturation constant (g L−1)
Q feeding flow rate to the reactor (L h−1)
S phenol concentration (g L−1)
T time (h)
V volume of medium in the reactor (L)
X biomass concentration (g L−1)
Y observed growth yield (g g−1)
V(¯x) Control Lyapunov Function
VB(¯x) Bellman function
dV(¯x)
dt (∗)
derivative of the CLF along the trajectories of the
system (*)
x state of the control system
min
u
{.} minimum of all control functions u
u∗ optimal control
dVB(¯x)
dt
(u=u∗)
derivative of the Bellman equation when the
control is optimal
J performance index
Greek symbols
˛(¯x) penalty function of the state
specific growth rate (h−1)
max maximum specific growth rate (h−1)
Subscript
0 initial value
f final value
Superscript
in value at the entry of the reactor
Modeling and control of fedbatch reactors represent challenging
areas of research by two well-known factors. First, the biological
processes exhibit large nonlinearities, strongly coupled variables
and parameter uncertainties [5]. This can lead to large discrep-
ancies between the model predictions and the experimental data
when these are originated from batch runs and then used for model-
ing fedbatch cultures [6]. Second, the lack of reliable online sensors
limits the real-time monitoring of key variables such as the concen-
tration of substrate [5]. Thus, several strategies having as control
task the regulation of the biomass growth rate by means of the
feeding pattern, rather than the substrate concentration itself, have
been proposed. For instance, in a previous work [7], the dynamic
behavior of fedbatch processes with Haldane kinetics was inves-
tigated and some conditions for global stability and performance
improvement were presented. A stabilizing control law based on a
partial state feedback was presented too. The goal of this open-loop
control strategy was keeping the growth rate close to its maximal
value by providing an exponential feeding pattern. However, the
control efficiency was not validated experimentally.
Nonlinear systems can be successfully controlled by using the
Control Lyapunov Functions (CLF) [8,9]. The main advantage of this
approach is that the Bellman equation does not need to be solved
to minimize the control signal and to stabilize the process. Until
now, the CLF have not been used for controlling biodegradation
processes.
The purpose of this work was to enhance the performance of a
fedbatch process of phenol degradation by means of the control of
the feeding pattern. The control strategy was based on the kinetic
modeling of the inhibition of activated sludge by phenol and on the
CLF approach for synthesizing the controller. The kinetic model and
the control strategy were validated experimentally in a fedbatch
process of phenol biodegradation. A reference culture was also car-
ried out to compare its performance against that of the controlled
process.
2. Materials and methods
2.1. Organisms and chemicals
Samples of activated sludge were obtained from the aeration
tank of a plant treating municipal wastewater (Pachuca, Mexico).
Phenol was supplied by Sigma–Aldrich (Germany); all other chem-
icals were purchased from J.T. Baker (U.S.A.).
2.2. Acclimation of activated sludge – Reference fedbatch cycle
The acclimation of activated sludge to phenol was carried out by
cycles with the following sequence of operations: feeding, aeration,
settling and drawing. Each day, 1 L of mineral medium (Section 2.6)
supplied with a variable volume of concentrated phenol solution
(20 g L−1) was added to 1 L of activated sludge. After 23 h of aeration
and agitation and 0.58 h of settling, the supernatant was drained in
0.03 h. Following 0.36 h of idle time, fresh medium was fed (0.03 h).
This operation mode was considered the reference cycle; its over-
all duration was not optimized and was kept at 24 h. Acclimation
of activated sludge was accomplished in nine weeks by increasing
the initial phenol concentration in the medium (S0) from 0.005 to
0.45 g L−1 using 0.05 g L−1 steps on a weekly basis. This S0 value
(0.45 g L−1) and a total suspended solids (TSS) concentration in the
reactor of 4–6 g L−1 were maintained afterwards, in order to pro-
vide a biomass with similar characteristics for the assays. At least
once a week, the exhaustion of phenol was verified at the end of the
cycle. The phenol concentration in the effluent was always lower
than 2 × 10−4 g L−1.
2.3. Batch cultures
Four shake-flask cultures were conducted at different S0
(0.20–0.60 g L−1) and at the same initial biomass concentration (X0,
0.03–0.04 g protein L−1) in order to estimate the Haldane parame-
ters of the acclimated cultures. For this, 1-L glass baffled Erlenmeyer
flasks containing 0.4 L of mineral medium and variable volumes
of a concentrated phenol solution (20 g L−1) were inoculated with
acclimated activated sludge. The flasks were incubated and main-
tained under agitation (120 rpm) over a thermostated water bath
(25 ± 1 ◦C). During the cultures, samples were obtained periodically
to assess the phenol and biomass concentrations.
2.4. Fedbatch reactor system
The bioreactor was a glass vessel of 2-L working volume with
a water jacket to control the operating temperature at 25 ± 1 ◦C
(Fig. 1). It was equipped with a magnetic stirrer and a pump pro-
viding about 3.2 L min−1 of air through a diffuser. The system was
conducted in fedbatch mode with the same sequence of opera-
tions than in the reference cycle (e.g., feeding, aeration, settling
and drawing). First, the medium feeding was started and simulta-
neously both the aeration and agitation of the culture were enabled.
After the feeding phase, the biomass was maintained under aera-
tion and agitation. At the end of this phase, the sludge was allowed
to settle. Finally, the supernatant was withdrawn from the reac-
tor. Feeding and drawing were performed by peristaltic pumps
(Masterflex 77200-50) configured in remote mode.
For the operation of the reactor, a Programmer Logic Controller
(PLC; Téléméchanique-Schneider SR3B261BD) was employed. The

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Fig. 1. Schematic setup of the fedbatch reactor system. 1: Programmer logic controller (PLC); 2: air pump; 3: feed tank; 4: controlled feeding pump; 5: magnet bar; 6:
magnetic stirrer; 7: air diffuser; 8: reactor vessel; 9: effluent tank; 10: sampling port; 11: controlled drawing pump; 12: monitoring computer. The discontinuous lines
represent control signals.
sequence of operations described above was programmed by
timers with delay to connection; e.g., following the digital signal
sent by the PLC, the first timer enabled the feeding into the reactor
by activating the feeding pump through a specific period. A Ladder
language software (Zelio Soft 4.3; Téléméchanique-Schneider) was
used for programming the PLC. The control law was implemented
by using both a timed-loop tool in Lab View 7 Express and a Mat-
lab Script function storing the values of the control law in a vector.
These values were converted to the corresponding volt values of
the pump by a calibration curve of the actuator. The timed loop
gave the appropriate time step to send the control signal value to
an analog channel of the card, corresponding to each simulation
time instant.
2.5. Fedbatch cultures
The aforementioned reactor system was used for the cultures
carried out in fedbatch mode. It was seeded with a sample of
acclimated activated sludge, which was previously centrifuged and
washed twice with phenol-free mineral medium. The biomass was
resuspended in 1 L of phenol-free mineral medium. The feeding
flow (Sin = 1 g L−1 of phenol) was supplied with no output flow until
the working volume of the reactor (2 L) was reached. During the
cultures, samples were taken at different times to assess the phe-
nol and biomass concentrations (in duplicate). The samples were
first centrifuged at 3500 rpm for 10 min and the supernatants were
frozen until analysis. After centrifugation, the precipitated biomass
was recovered for determining the protein content.
2.6. Mineral medium
Both fedbatch and batch cultures were carried out with a
medium having the following composition (mg L−1): K2HPO4 (404),
KH2PO4 (220), (NH4)2SO4 (50), MgSO4·7H2O (10), CaCl2·2H2O
(1.85), MnCl2·4H2O (1.5), FeCl3·6H2O (0.3). These concentrations
were balanced with a S0 of 0.20 g L−1; when different phenol
concentrations were used, the medium composition was propor-
tionally modified.
2.7. Analytical methods
Phenol concentration was determined by the 4-
aminoantipyrine method, which is fully described elsewhere
[2]. This method has a detection threshold limit of 7 × 10−5 g L−1
and a variation coefficient of 1.06% of the measured values (n = 9).
Biomass concentration was measured as protein by the Lowry
method [10] and gravimetrically as total suspended solids (TSS) by
drying 10 mL – samples for 24 h at 105 ◦C.
2.8. Mathematical modeling and parameter estimation
The Haldane substrate-inhibition model (Eq. (1)) was used to
describe the specific cell growth rate ( ). A dynamic model (Eqs.
(2)–(4)) including the feeding flow (Q) as input to the reactor was
derived from mass-balance considerations:
˙X = X −
Q
V
X (2)
˙S =
Y
X +
Q
V
(Sin
− S) (3)
˙V = Q (4)
where V is the volume of the medium in the reactor (V(t0) /= 0), Y
is the observed growth yield (assumed as constant) and Sin is the
phenol concentration in the feeding medium.
The parameters were estimated by using the hill-climbing
method [11]. This algorithm is fully explained in Appendix A.
2.9. Implementation of the controller
The controller was simulated in closed loop in Simulink–Matlab
by using the fourth order Runge-Kutta method with a fixed step-
size of 0.001. After obtaining the controller signal (in units of flow,
u = Q), it was shifted to a voltage signal compatible with the actu-
ator (a peristaltic pump) by means of a data acquisition board
(National Instruments NI PCI-MIO-16XE-50). Due to the lack of
online sensors, the LabView 7.1 software was programmed to send
this control signal to the actuator, and a timed loop structure was

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Fig. 2. Phenol consumption (᭹) and biomass growth ( ) in a reference fedbatch
cycle.
used to guarantee that the signal of control was applied at the
correct instant of time.
3. Results and discussion
3.1. Acclimation of activated sludge – reference fedbatch cycle
The acclimation of activated sludge allowed a stable degrading-
phenol culture and a high average phenol removal (>99.9%) to be
obtained. Fig. 2 presents the typical biomass and phenol courses of
the reference cycle.
The mean applied phenol load to the reactor was
0.45 g phenol L−1 d−1, and the mean phenol mass load was
0.09 g phenol g−1 TSS d−1. As the reference cycle was not optimized
to minimize its duration, these load values are lower than those
reported usually. For instance, a load of 0.75 g phenol L−1 d−1
was applied to an aerobic granular SBR treating phenol in saline
wastewater [12]. In fixed biomass processes the applied phe-
nol load can be greater; for a packed-bed bioreactor, a load of
2.68 g phenol L−1 d−1 was reported [13]. Although the microbial
composition of the sludge was not analyzed in this study, Valivorax
paradoxus was found as the dominant bacterial strain in activated
sludge acclimated to phenol in a continuous system (with an
applied load of 0.4 g phenol L−1 d−1 and characterized by a low
residual phenol concentration) [14]. After a batch enrichment
(subject to higher phenol concentrations, as in the present study),
the dominant strains in the acclimated sludge were Pseudomonas
putida and Acinetobacter lwoffii [14].
3.2. Modeling of the phenol biodegradation in batch cultures
Four batch runs were conducted in shake flasks in order to deter-
mine the Haldane parameters. Initial phenol concentrations (S0)
were set in the range from 0.20 to 0.60 g L−1, and the phenol con-
sumption was followed during 12 h. The results are presented in
Fig. 3 in terms of percent biodegradation of phenol. No lag phases
were observed in any culture, due to the phenol concentration used
for the activated sludge acclimation (0.45 g L−1). S0 influenced the
biodegradation kinetics since phenol exhaustion needed different
periods to be accomplished (2 and 10 h for the lowest and the high-
est S0, respectively).
The kinetic parameters of the Haldane model determined by
the hill-climbing method are presented in Table 1 along with a
summary of values reported in other studies. The max value is
Fig. 3. Phenol consumption in batch cultures at different S0 (g L−1
):. (᭹) 0.20; ( )
0.29; ( ) 0.40; ( ) 0.60.
higher than the upper limit of the range of values reported for
phenol-degrading mixed cultures [19–22]. Actually, this value is
closer to those exhibited by axenic cultures as Cupriavidus tai-
wanensis [16] and Pseudomonas putida [17], which suggests the
absence of microbial competition for phenol [22]. The value found
for Ki is in the wide range of values reported for mixed cultures,
namely 0.072–1.199 g L−1 [19–22]. This value indicates that the
inhibition effect is observed rather at high phenol concentrations.
Such a resistance to substrate inhibition is probably due to the
previous acclimation of the activated sludge. It has been reported
that acclimated bacteria can develop mechanisms to counteract
the increased fluidity of the membrane caused by phenol [1]. In
Pseudomonas putida, the resistance mechanism consists in the iso-
merization of cis-unsaturated fatty acids to the trans-configuration,
leading to more rigid cell membranes [1]. The acclimated sludge
showed a low affinity for phenol, as its KS value was higher
than the superior limit of the range of values signaled for mixed
cultures (0.005–0.266 g L−1 [19–22]). Besides the Haldane parame-
ters, the hill-climbing algorithm found Y = 0.8 g protein g−1 phenol.
This parameter cannot be compared with the bibliography values
because they are commonly expressed in terms of g of TSS, not of
protein, per g of phenol.
Simulations of the batch runs were made by using the dynamical
model given by Eqs. (1)–(4) and the aforementioned parame-
ters, and by setting Q = 0 for this mode of culture. Results are
given in Fig. 4. At least for the range of S0 that has been stud-
ied (0.20–0.60 g L−1), the model exhibited an adequate estimation
ability. This was evidenced by Pearson coefficients (R2) being com-
prised between 0.96 and 0.99.
3.3. Control strategy
The model presented in Eqs. (1)–(4) was considered the control
system, taking as input the feeding flow (Q) to the reactor. The state
of the system was described by the following variables: volume of
medium in the reactor (V), phenol concentration (S) and biomass
content (X). To compensate the lack of online measurements of the
two later variables, a state feedback controller was synthesized by
using a CLF [8,9,23]. However, it was applied in open loop by obtain-
ing the signal of control from a simulation and considering this
signal as the flow profile.

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Table 1
Summary of Haldane kinetic parameters obtained for phenol biodegradation.
Microorganism/culture Haldane-model constants Ref.
max (h−1
) Ks (g L−1
) Ki (g L−1
)
Bacillus brevis 0.026–0.078 0.002–0.029 0.868–2.434 [15]
Cupriavidus taiwanensis 0.50 0.061 0.280 [16]
P. putida ATCC 17484 0.534 <0.001 0.470 [17]
Ralstonia eutropha 0.410 0.002 0.350 [18]
Activated sludge 0.119 0.011 0.251 [19]
Activated sludge 0.131–0.363 0.005–0.266 0.142–1.199 [20]
Mixed culture 0.308 0.045 0.525 [21]
Activated sludge 0.438 0.029 0.072 [22]
Activated sludge 0.600 0.385 0.700 This study
The control system (Eqs. (1)–(4)) is nonlinear and affine. The
state ¯x = X S V
T
was defined and the input was set as u = Q.
Phenol concentration was controlled (S → 0) by using the input u.
The control system was rewritten as follows:
˙¯x = f0(¯x) + f1(¯x)u (5)
where
f0(¯x) = X Y
X 0
T
, f1(¯x) = − X
V
Sin−S
V
1
T
(6)
An equilibrium point of the control system (Eqs. (1)–(4)) is ¯x∗ =
Xf 0 Vf
T
, where Xf and Vf are constants, with u = 0. Without
loss of generality, the system (Eq. (5)) can be rewritten as a system
which the zero equilibrium is ¯x∗. The system was then transferred
to the equilibrium point ¯x∗. By using the CLF approach, a candidate
of Lyapunov function V(¯x) was proposed, which is positive-definite
and radially unbounded. V(¯x) was derived along the trajectories of
the system (Eq. (5)), and the following equation was obtained:
dV(¯x)
dt (5)
= ∇V(¯x) · f0(¯x) + ∇V(¯x) · f1(¯x)u (7)
Fig. 4. Kinetics of phenol consumption in batch cultures at different S0 (g L−1
): (᭹)
0.20; ( ) 0.29; ( ) 0.40; ( ) 0.60. The symbols represent the experimental results
and the continuous lines, the data resulting from the simulated kinetic model.
where ∇V(¯x) denotes the gradient of the scalar function V(¯x) and
(·) denotes the inner product. Now, consider the Bellman equation
[24].
dVB(¯x)
dt (u=u∗)
+ ˛(¯x, u∗
) = 0 (8)
where VB(¯x) is the Bellman functional equation and u is the optimal
control when the following performance index is considered:
J =
∞
0
˛(¯x, u)dt
where the function ˛(¯x, u) is positive-definite. As for nonlinear sys-
tems it is very difficult to construct or to propose a Bellman function
VB(¯x) satisfying the Eq. (8), a candidate Lyapunov function V(¯x) for
the system was proposed (Eq. (7)). With this approximation for
the Bellman function, a control u∗ that satisfies the Eq. (9) was
investigated.
min
u
dV(¯x)
dt (5)
+ ˛(¯x) (9)
In a previous work [23], a suboptimal controller for an elec-
tromechanical system was presented, where the function ˛(.)
depends on both the state and the controller. For simplicity, in this
work ˛(¯x) is proposed only as a state function. The function ˛(¯x)
penalizes the state variables in order to obtain a fast or a slow con-
vergence to the state ¯x. Simulation results showed that the control
of the biomass X is very complex, but the variable S can be con-
trolled in such a way that S → 0 in a finite time. By substituting Eq.
(7) in Eq. (9), the following expression was obtained:
min
u
∇V(¯x) · f0(¯x) + ∇V(¯x) · f1(¯x)u + ˛(¯x)
Following a notation previously used [8,9], if we define:
0(¯x) = ∇V(¯x) · f0(¯x) + ˛(¯x)
1(¯x) = [∇V(¯x) · f1(¯x)]
T
D(x, u) = 0(¯x) + 1(¯x)u,
then, we have that:
min
u
D(x, u) .
Note that if the scalar term D(x, u) < 0, one can conclude that the
system is stable in closed loop following the sufficient conditions of
the stability Lyapunov theorem [9]. By the projection theorem, the
controller was finally obtained (Eq. (10)). As the CLF is not a Bellman
function, the controller proposed herein is of suboptimal-type.
u(x) =
− 0(¯x) 1(¯x)
T
1
(¯x) 1(¯x)
, when 0(¯x) > 0
0, when 0(¯x) ≤ 0
(10)

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Table 2
Controlled fedbatch cycle.
Phase/actuator Feeding Aeration Settling Draw
Duration [h] 4 6 0.58 0.03
Feed pump On Off Off Off
Magnetic stirrer On On Off Off
Air pump On On Off Off
Drawing pump Off Off Off On
The function V(¯x) was proposed to apply in open loop the con-
troller to the reactor system:
V(¯x) = ¯xT
¯x
whilst a penalty function ˛(¯x) was proposed as follows:
˛(¯x) = 0.1X2
+ 5S2
+ 1.4V2
The function ˛(¯x) penalizes the convergence of the state ¯x, as
it gives negativity to the derivative of the CLF V(¯x). This is a
positive-definite function and the three scalars (0.1, 5 and 1.4)
were heuristically proposed by considering that the biomass con-
tent (X) could be relatively high and that the phenol concentration
(S) should be depleted as soon as possible to minimize the inhi-
bition phenomena. Note that function V(¯x) is a CLF for the system
(Eqs. (5)–(6)), because 1(¯x) = 0 if ¯x = 0.
3.4. Experimental validation of the model and the control strategy
As mentioned previously, the aim of the control strategy was to
minimize the length of the cycle by maintaining the phenol con-
centration at not-inhibitory levels. The simulation of the controller
(Eq. (10)) allowed to obtain a feeding pattern (Fig. 5a) to the reactor
and an initial flow rate (Q0) of 20.11 L h−1. Fig. 5b shows the vol-
ume profile obtained by simulation of the control strategy from an
initial value (V0) of 1 L. According to this profile, the reactor volume
reaches 1.99 L after 4 h of feeding. A fedbatch cycle was designed
from this pattern (Table 2).
The sequence of operations shown in Table 2 was programmed
in the PLC and conducted in open loop in the reactor system
(Fig. 1) with the same initial conditions used in the simulation
(Q0 = 20.11 L h−1, V0 = 1 L). The phenol concentration in the medium
entering to the reactor (Sin) was 1 g L−1, and the initial biomass
Fig. 6. Phenol consumption (᭹) and biomass growth ( ) during the controlled fed-
batch cycle. The symbols represent the experimental results and the continuous
lines, the data resulting from the simulated kinetic model.
concentration (X0) was set at 0.19 g protein L−1 (measured after
suspension of the inoculum in the medium and equivalent to
0.23 g SST L−1). Experimental and simulated results of the fedbatch
culture are shown in Fig. 6. At the end of the controlled cycle,
the contents of phenol and protein in the medium were 0.01 and
0.47 g L−1, respectively. Thus, an overall amount of 0.98 g of phenol
was degraded in 10 h.
The prediction capability of the model was low for the first five
hours of culture, as it was evidenced by the Pearson coefficient
(R2 = 0.73). During this period the biomass was not so inhibited by
phenol as the model predicted, and so the measured phenol con-
centrations were lower than the simulated ones. This discrepancy
could arise from the use of kinetic parameters obtained from batch
runs for modeling a fedbatch culture [6]. However, the model could
depict adequately the phenol consumption during the last phase
of the culture (R2 = 0.99), as well as the length of the culture per-
mitting the phenol exhaustion (e.g., 10 h). Moreover, the simulated
Fig. 5. Simulation of the controller. (a) Feeding flow rate (Q) to the reactor [L h−1
]; (b) resulting volume (V) profile of the reactor medium [L].

7. Please cite this article in press as: U. Ba˜nos-Rodríguez, et al., Model-based control of a fedbatch biodegradation process by the control Lyapunov
function approach, Chem. Eng. J. (2012), doi:10.1016/j.cej.2012.02.068
ARTICLE IN PRESS
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CEJ-9019; No.of Pages8
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Table 3
Operating parameters of the controlled and reference cycles of the fedbatch reactor.
Parameter Unit Controlled cycle Reference cycle
Cycle duration h 10.61 24
Hydraulic retention time h 21.28 47.62
Treated flow L h−1
0.09 0.04
X0 g TSS L−1
0.23 6.10
Applied phenol load g phenol L−1
d−1
1.13 0.45
Phenol mass load g phenol g−1
TSS d−1
4.82 0.09
values of the biomass content tracked well the experimental results
throughout the culture (R2 = 0.94).
Table 3 presents a comparison between the operating param-
eters of the controlled and the reference fedbatch cycles. The
CLF approach did improve the performance of the biodegradation
process because the cycle length diminished from 24 h in the ref-
erence cycle to 10.61 h in the controlled cycle. Consequently, the
hydraulic retention time diminished in more than 55% of its origi-
nal value. The applied phenol load in the controlled cycle was more
than doubled compared to that of the reference cycle. In spite of
this augmentation, the phenol load of the controlled cycle is still
lower than those reported for other processes conducted at higher
biomass concentrations (e.g., 2.68 [13] and 8.15 g phenol L−1 d−1
[22] in packed-bed and membrane bioreactors, respectively). In
a SBR controlled by online respirometric measurements [25], a
higher phenol load (3.12 g phenol L−1 d−1) was applied too. Leonard
et al. [18] reported an even higher value of the applied phenol
load (19 g phenol L−1 d−1) for a fedbatch process provided with a
closed-loop control strategy based on online colorimetric monitor-
ing of an inhibitory metabolite of phenol degradation, namely the
2-hydroxymuconic semialdehyde.
The major enhancement due to the CLF approach was related
to the phenol mass load. As the model predicted that only a low
biomass concentration was needed for the controlled feeding strat-
egy, a 50-fold increase was observed for this parameter. The value
of the phenol mass load (4.82 g phenol g−1 TSS d−1) was higher than
that reported for a membrane bioreactor (0.72 g phenol g−1 TSS d−1
[22]). Also, the mass load obtained in this study was more than
6-fold higher than that reported for the SBR controlled by online
respirometry (0.88 g phenol g−1 TSS d−1 [25]). The control of the
phenol concentration in the reactor at non inhibitory levels avoids
the hydrophobic perturbation of the cell membrane, the decrease
in phenol hydroxylase activity [18] and the production of inhibitory
metabolites as 2-hydroxymuconic semialdehyde [2,6]. A high pro-
ductivity of a phenol-degrading culture can thus be achieved by
appropriate control of the feeding rate even without online moni-
toring of the process variables.
3.5. Parameter sensitivity analysis
A parameter sensitivity analysis was performed to know how
the predictions of the model are affected by variations of the values
of the parameters. The model ability to depict the evolution of both
biomass and phenol concentrations was studied for varying values
of the kinetic parameters max, KS and Ki (Table 4). The minimum
and maximum levels of the parameters were obtained arbitrarily
by subtracting and adding respectively 50% of the nominal value
of each parameter. The model (Eqs. (1)–(4)) was solved by vary-
ing one parameter at a time while the remaining parameters were
held constant at the nominal values shown in Table 4. The obtained
concentrations of phenol and biomass were compared with those
calculated with the set of nominal values of the parameters. The
absolute error between those values was calculated at regular time
intervals (0.1 h) all throughout the simulation period (10 h), and
then the mean was obtained. The results of the sensitivity analysis
are summarized in Table 4.
For predicting both the phenol and biomass concentrations, the
model is mostly sensitive to changes of the max value. In con-
trast, Ki is the parameter modifying in the least degree the model
predictions. Decreases in max and Ki modify the model ability to
depict the phenol and biomass concentrations more than increases
in these parameters. Particularly, the predictions of the model
are relatively unaffected when the highest Ki value is employed.
These results are in agreement with those reported previously for
a Haldane-based model depicting a fedbatch phenol biodegrada-
tion process [6]. Although that model was based on the inhibitory
metabolite production, max was considered as the critical param-
eter too. In contrast, the model predictions were almost insensitive
to changes in Ks and Ki [6].
4. Conclusions
The control of the feeding pattern to a fedbatch reactor allowed
the performance of a phenol degradation process to be enhanced.
The control system was a model based on mass balance consider-
ations and on Haldane-type kinetics. Although the kinetic model
was generated from batch cultures, it depicted adequately the evo-
lution of both the phenol and the biomass concentrations in a
fedbatch cycle. The CLF approach was used to synthesize the con-
troller and afterwards to obtain a suboptimal feeding strategy in
relation to the convergence time necessary for phenol depletion. To
the best knowledge of the authors, this is the first attempt to use the
Lyapunov functions to control biodegradation processes. The feed-
ing strategy was applied in open loop and validated in a lab-scale
fedbatch process. During the controlled cycle, the biodegradation
activity of the biomass was increased by mitigating inhibition
phenomena due to high phenol concentrations. The proposed feed-
ing strategy improved the performance of the process, notably
concerning the phenol mass load, even without online monitoring
of the process variables. As the control of wastewater treatment
processes is often limited by the availability and the employ of
reliable online sensors, this open-loop methodology is a low-cost
Table 4
Results of the parameter sensitivity analysis.
Parameter Nominal value Minimum and
maximum values
Mean absolute
error, S [g L−1
]
Mean absolute
error, X [g L−1
]
max 0.600 0.300/0.900 0.116/0.039 0.093/0.031
KS 0.385 0.193/0.578 0.029/0.028 0.023/0.022
Ki 0.700 0.350/1.050 0.020/0.006 0.016/0.005

8. Please cite this article in press as: U. Ba˜nos-Rodríguez, et al., Model-based control of a fedbatch biodegradation process by the control Lyapunov
function approach, Chem. Eng. J. (2012), doi:10.1016/j.cej.2012.02.068
ARTICLE IN PRESS
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CEJ-9019; No.of Pages8
8 U. Ba˜nos-Rodríguez et al. / Chemical Engineering Journal xxx (2012) xxx–xxx
alternative for those biological systems dealing with toxic pollu-
tants.
Appendix A. Appendix A–Hill-climbing algorithm
The hill-climbing technique is a local search method with a
stochastic component that uses generally a bit string to represent a
set of prototypes or, in some experiments, a collection of features.
From the experimental results of the batch cultures (and by setting
Q = 0 in the simulation), the hill-climbing method was used in order
to minimize the following index:
e = Sexperimental − Ssimulated + Xexperimental − Xsimulated (A.1)
For prototype representation, a real-matrix code was utilized.
The hill-climbing algorithm was thus designed as follows [11]:
1. Prototype representation. The parameters max, KS, Ki and Y were
encoded in a 4-dimensional vector:
= max Ks Ki Y
2. Vector initialization. The components i (i = 1,. . ., 4) were gener-
ated randomly. This vector was called best-evaluated.
3. Mutation. The i of the best-evaluated vector were mutated
according to a random variable, as:
i = i + ı
where ı is an uniform random variable [11].
4. Fitness calculation. The index given by Eq. (A.1) depends on
the components i; then, the fitness of the mutated vector was
defined as:
f ( ) =
1
e( )
5. If the fitness was strictly greater than the fitness of the best-
evaluated vector, then the best-evaluated vector was set as the
mutated vector.
6. If the maximum number of iterations has been performed, then
return to the best-evaluated vector. Otherwise, go to step 3.
The simulation was made for each iteration in order to obtain
the values Ssimulated and Xsimulated, needed for the calculation of e( ).
The number of iterations was fixed at 1000.
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