On Observability and Controllability for a Bioethanol Dynamical Model Obtained from Cocoa Industrial Waste

Abstract

The aim of this paper is to study the stability,
controllability and observability for a bioethanol production
dynamic system. Likewise, a novel model based on the
mechanism of multiple parallel coupled reactions was used to
describe the kinetics of substrate, enzyme, and biomass and
product formation. This model has been extended to
continuous operation, which is employed as a virtual plant to
enable the implementation of the properties. The maximum
ethanol production conditions are obtained by manipulating
the dilution rate with an optimal initial substrate
concentration. A nonlinear observer is implemented to show
the results of the observability analysis. Sufficient and
necessary conditions for state controllability and state
observability of such system are established. Results indicated
that the proposed model can be applied as a way of augmenting
bioethanol production, controlling and monitoring the process

Full text

XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE
On observability and controllability for a bioethanol
dynamical model obtained from cocoa industrial
waste
Pablo A. López Pérez
Escuela Superior de Apan
Universidad Autónoma del Estado de
Hidalgo
Apan, Hgo. Mexico
save1991@yahoo.com.mx
Dulce Jazmín Hernández Melchor
Colegio de Postgraduados
Estado de México, México
dulcejazz@hotmail.com
Teresa Romero Cortes
Escuela Superior de Apan
Universidad Autónoma del Estado de
Hidalgo
Apan, Hgo. Mexico
tromerocortes@gmail.com
V. Peña Caballero
Departamento de Ingeniería
Agroindustrial, División de Ciencias de
la Salud e Ingenierías
Universidad de Guanajuato
Gto.,Mexico
vicente.caballero@ugto.mx
Omar Santos
CITIS, AACyE, ICBI
Universidad Autónoma del Estado de
Hidalgo
Pachuca, Hidalgo Mexico
omarj@uaeh.edu.mx
F. Martínez Farías
Escuela Superior de Apan
Universidad Autónoma del Estado de
Hidalgo
Apan, Hgo. Mexico
francisco_martinez@uaeh.edu.mx
Abstract— The aim of this paper is to study the stability,
controllability and observability for a bioethanol production
dynamic system. Likewise, a novel model based on the
mechanism of multiple parallel coupled reactions was used to
describe the kinetics of substrate, enzyme, and biomass and
product formation. This model has been extended to
continuous operation, which is employed as a virtual plant to
enable the implementation of the properties. The maximum
ethanol production conditions are obtained by manipulating
the dilution rate with an optimal initial substrate
concentration. A nonlinear observer is implemented to show
the results of the observability analysis. Sufficient and
necessary conditions for state controllability and state
observability of such system are established. Results indicated
that the proposed model can be applied as a way of augmenting
bioethanol production, controlling and monitoring the process.
Keywords—: phenomenological model, continuous operation,
parameters, matrix criterion
I. INTRODUCTION
       
demonstrated the momentum of new technologies and
renewable energy generation results in the diversification of
primary energy (fossil). Energy efficiency and renewable
energies processes will need to evolve before sustainable
cities to enable successful integration of supportive
technologies to have direct effect the climate challenge [1-
2]. Currently, there are projects with economic support
related to optimize integrated biorefineries. These projects
will work to solve critical encountered for the successful
scale-up and reliable operations of integrated biorefineries
generating high-impact tools and techniques for increasing
the productivity [3]. It has been reported that Brazil and the
United States produced approximately 70% of the global
biofuel supply in 2015, the above, equaled approximately 35
billion gallons consisting roughly of a 3:1 relation of ethanol
and biodiesel [4,5].
The development of bioethanol biotechnology is
advancing rapidly as a fuel of greater importance [6].
Considering the high theoretical yields of this product,
several researchers have begun exploring approaches to
increase bioethanol production. During the process of
mechanical extraction of cacao seeds, a predominance of
pulp and mucilage residues has been reported with an
average production of 727, 500.00 metric tons for the year
2003/2004, of which in average 156, 333.33 m3 respectively
were derived from cocoa juice, the 50% of this is considered
to be waste, with no further commercial application, but it is
now recognized that its components have potential as inputs
and energy for second-generation biofuels [7, 8].
However, it is difficult to develop and especially to
implement, advanced monitoring and control strategies for
real bioethanol production because of the absence of reliable
instrumentation for measuring biological state variables, i.e.
substrates, pH, biomass, metabolic enzymes and product
concentrations [9, 10]. For instance, the required quality of
monitored data and frequency of sampling are functions of
the accuracy of the bio-sensors employed for process control.
In many cases, the state variables are not measurable on-line
(in real-time) due to the high cost of sensors and extreme
operating conditions. These facts, together with the
nonlinearity and parameter uncertainty of the bioprocesses,
require an enhanced modeling effort, state estimation
(observability) and control (controllability) strategies [11,
12].  and controllability is
a necessary prerequisite to the estimation and regulated of
states. Because of the nonlinear aspects of their dynamics,
stability and observability analysis is rather complex in (bio-
chemical) process applications. However, for nonlinear
systems, the theory of observers and controllers are not
nearly as neither complete nor successful as it is for linear
systems [13]. Controllability and observability are two
important properties of state models which are to be studied
prior to designing a controller. The designs of observability
and controllability conditions for nonlinear systems are a
challenging problem (even for accurately known systems)
that has received a considerable amount of attention [14].
Therefore, a model was developed kinetic based on the
mechanism of multiple parallel coupled reactions was used
to describe the kinetics of substrate, enzyme, biomass and
product formation, from this model experimentally adjusted
in batch was extrapolated to a continuous operation for was
demonstrate the utility of the observability and controllability
matrix as a metric in developing optimum measurement

strategies in a range of conditions of operations for ethanol
fermentation.
II. METHODOLOGY
A model can be used to evaluate the behavior of the
biochemical reaction pathway and to determine the
properties of the dynamic system such as the existence of an
equilibrium point, local controllability and observability,
local or global asymptotic stability of equilibrium points
[15, 16].
Several reports and reviews about the production of ethanol
fermentation using microorganisms, as well as certain
yeasts, bacteria, and fungi have been published. However,
the models described here are complicated because (1) there
are too many parameters, which cannot be individually
determined (2) the models represent a higher order and
some of them are described by partial differential equations,
(3) structured and unstructured processes models [17, 18].
On the other hand, in biochemical reaction networks, there
are multi-step processes which may follow either simple or
complex rate laws, the rate law is a direct result of the
sequence of elementary steps that constitute the multi-step
reaction mechanism. However, as such, it provides an
optimum tool for an unknown mechanism. Therefore, we
need a new approach to solve these issues [19, 20]. The
proposed methodology is summarized in three stages: 1)
kinetic structure, 2) phenomenological model and 3)
assessment of the controllability and observability.
A. Kinetic structure
The first stage, a multiple coupled reaction mechanism
is proposed, where the substrate is consumed by the
microorganism, while at the same time it produces the PG
enzyme for the degradation of the substrate and ethanol
production, in addition it is considered inhibition of
biomass by product (ethanol), the reactions kinetics are
based on the power rate law.
Determination of kinetic parameters using Model-
Maker® software (based on LevenbergMarquardt
optimization approach) was employed for the non-linear
fitting data. The performance of the proposed mathematical
model was statistically evaluated using the dimensionless
coefficient for efficiency (DCE) [21, 22].
 
 






 N
1i i
N
1i
2
ii
.DCE 2
01



(1)
Where:
i

model simulated data at time
i
t
,
i

the
observed data at time
i
t
,
i

is the mean value of the
observed variable, N is the data number, DCE varies
between (0, 1], a positive value of DCE represents an
acceptable simulation, whereas DCE > 0.9 represents a good
simulation.
Analysis of Experimental Data
The fermentations were carried out in Erlenmeyer
flasks containing broth medium (30 g/L glucose, 20 g/L,
peptone and 10 g/L yeast extract), inoculated with colonies
of P. kudriavzevii. Cocoa juice was extracted from cocoa
fermentation baskets after two days. For experiments, cocoa
pulp juice was adjusted to 30 g/L of glucose with reducing
sugar. Samples were incubated at 30°C and 200 rpm for 18-
22 h. Ethanol production was evaluated in mucilage juice
samples inoculated with 1×107 cells 1/mL at 30°C, shaking
for 24 h [23]. Subsequently, 10 mL of re-suspended yeasts
were transferred in 1 L of cocoa pulp juice and incubated at
30°C for 24 h with 150 rpm in orbital agitation. This
experiment was performed in triplicate. Ethanol was
determined using a HPLC Waters [24].
B. Phenomenological model
Basically, expressions based on mass balance differential
equations terms can be employed to describe the models of
microbial kinetics for growth and fermentation processes.
The change of fermented product rate, substrate
consumption and biomass were related to ethanol (P),
glucose (S) and biomass (X) concentrations in using
proposed reaction rate laws (see, eq. 2-5), that will serve to
predict the behavior of the overall system and identify the
optimal operating conditions for the inherently safe process
(see, Fig. 1). The proposed model makes the following
assumptions: i) individual steps in any mechanism are n-
order unimolecular reactions ii) measured activity is directly
proportional to concentrations of active enzyme forms, iii)
consider inhibition effects.
Equations:
Substrate balance (S):
 
SSDXk
dt
dS  01

(2)
Biomass balance (X):
 
XXDXkXkEkSk
dt
dX t 03231



(3)
Enzyme activity balance (Ez):
Z
zDESXk
dt
dE 

2
(4)
Bioethanol balance (Et):
t
tDESXk
dt
dE 

3
(5)
Here kj : specific reaction rate for j conce
: Exponential term, i.e. the rate is proportional to the
concentrations of the reactants each raised to some power.
Model limitations are: the model is based on a
phenomenological rather than a mechanistic scheme, mass
transfer limitations and conformation changes of the enzyme
structure, as effect of pH and temperatures are not related.

Fig. 1. Diagram of the process
C. Assessment of the controllability and observability
Stability
A continuous-time nonlinear control system (eq. 2-5) is
generally described by a differential equation of the form
 


 ,t u;t,xfx 00

(6)
Where

)t(xx 
:state of the system belonging to a (usually
bounded) region
n
x

.

 
nqn
:f  

smooth nonlinear vector
function and Lipschitz in
x
.

u
is the control input vector belonging to another
(usually bounded) region
nm ,
m
u

.
To indicate the time evolution and the dependence on the
initial state
0
x
, the trajectory (or orbit) of a system state
 
tx
is sometimes denoted as
 
0
x
t

In control system (1), the initial time used is
0
0t
, unless
otherwise indicated. The entire space
n

, to which the
system states belong, is called the state space. Associated
with the control system (1), there usually is an observation
or measurement equation (observers)
 
nm ,u;t,xgy  1
(7)

m
y
measured output vector, that is the vector of
measured states.

1 mn
single-input/single-output (SISO) system

1m,n
multi-input/multi-output (MIMO) system
Consider the general nonautonomous system (6)
 
t,xfx 

(8)
where the control input
    
t,txhtu 
, if it exists [see
system (6)], has been combined into the system function
f
for simplicity of discussion. Without loss of generality,
assume that the origin
0x
is the system equilibrium of
interest. Lyapunov stability theory concerns various
stabilities of the system orbits with respect to this
equilibrium.
Theorem 1 (First Method of Lyapunov: Continuous-Time
Autonomous Systems).
Let
0x
be an equilibrium point for the nonlinear
system (6), where
n
D:f 
continuously differentiable
and
D
is a neighborhood of the origin. Let
 
0


x
x
x
f
J
(9)
J
be the system Jacobian evaluated at the zero equilibrium
of system. Then, the origin is:
Asymptotically stable if Re i < 0for all eigenvalues of
J
Unstable if Re i > 0for at least one of the eigenvalues of
J
          

J
axis, (is
marginally stable)[25].
Observability
        
prerequisite to the estimation of states. However, for
nonlinear systems, the theory of observers is not nearly as
neither complete nor successful as it is for linear systems.
Thus if a state is not observable then the controller will not
be able to determine its behavior from the system output and
hence not be able to use that state to stabilize the system, we
show the concept of observability, which is well understood
for continuous processes operating around an equilibrium
point. Therefore, the system is observable if and only if the
observability matrix of the system is full rank [26] if a
system is unobservable, the current values of some of its
states cannot be determined through output sensors.
As a background, consider the following lineal system
representation:
BuAxx

(10)
Cxy
where A, B, C are the constant matrices with appropriate
dimensions.
 
T
n
CACACACACN 132 ,,,,, 

(11)
As can be seen, if the state vector can be determinate, the
matrix N (named as the observability matrix) must be
invertible (full rank) in order to obtain:
YNX 1

Such that, the state vector
X
is observable in respect to
the measurable output
Y
. The local observability analysis
is condensed as following:
Theorem 2 Local Observability. A continuous time
linear (or linearized) system (2-5) is observable if and only
if rank(N) = n, where n is the order of the system [27, 28].
cocoa processing
1L
residues from cocoa
processing
bioreactor
30C
S
Et
EzX
Observability
property
Controllability
property
D
D
Output Measurements
State estimation
Controller
Signals
Signals
Signals

Controllability
Controllability is another geometric property of a system,
describing the ability to \drive" the system states to arbitrary
values through the control input. Controllability deals with
the possibility of forcing the system to a particular state by
application of a control input. If a state is uncontrollable
then no input will be able to control that state.
 
BA,,BA,AB,BO n12 

(12)
Theorem 3 Dynamical system (2-5) is controllable if and
only if rank(O) = n.
III. RESULTS AND DISCUSSION
The effectiveness of the proposed model for describing
experimental observations concerning batch fermentation is
presented in Fig. 2 (residual values), and quantified by
applying RSD (Residual Standard Deviation), R2
(correlation coefficient) and coefficient of efficiency DCE,
relating to the four state variables (see, Table 1). From these
criteria, it was concluded that the model presented
experimental data accurately, as evaluated by RSD. The
smaller the residual standard deviation, the closer it fits to
the data. In effect, the smaller the residual standard
deviation, compared to the sample standard deviation, the
more predictive or adequate, the model. Likewise, in all
cases R2 was close to unity, indi
as apparent in Table 1. Moreover, results indicate that it is
possible to accurately infer concentration in batch
fermentation DCE. From this criterion, it was concluded that
the model accurately portrayed experimental data, evaluated
by applying E (average) = 0.995. Likewise in all cases, DCE
was close to unity, indicating that the model represents a
good fit; as evident in Table 1 where for example, a value of
0.991 for DCE corresponding to substrate indicates that the
mean square error (i.e., the squared differences between the
observed and model simulated values) is 0.9 % of the
variance in the observed data (see equation (6)). In this
work, maximum ethanol concentration was 13.00 g/L after
70 h in batch operation, and productivity was 0.18 g/L h
using mucilage juice residues from cocoa.
The initial values for the state variables (S0 = 32 g/L; X0 =
0.60 g/L, Ez = 0.10 g/L, Et = 0.02 g/L), where the parameter
vector to be estimated is: 1= 0.13 ±0.09 [1/h], k2= 0.03
±0.01 [U L/g mL h], k3= 0.003 ±0.001 [1/h], k4= 0.0014
±0.001 [1/h], k5= 0.005 ±0.001 [1/h], k6= 0.014 ±0.01 [1/h],
k7= 0.0067 ±0.001 [U L/g mL h], 
   values are
expressed as mean ± confidence intervals (p = 0.05).
The parameter values obtained in the present study fall
within the range of those reported in the literature, due to the
different operating conditions used in each case, i.e.,
different carbon source, continuous or batch operation,
temperature, pH, among others [29, 30].
Otherwise, in a continuous culture was analyzed related to
dilution rate (D: 0.005 to 0.1 1/h), via a bifurcation diagram
and productivity (Fig. 3). The maximum ethanol production
by manipulating the dilution rate was 11.98 g/L and
productivity was 0.17 g/L h to D: 0.005 1/h with stable
equilibrium state (see table 2).
010 20 30 40 50 60 70
-1.0
-0.5
0.0
0.5
1.0
Residual Et (g/L)
Time (h) 010 20 30 40 50 60 70
-1.0
-0.5
0.0
0.5
1.0
Residual Ez (g/L)
Time (h)
010 20 30 40 50 60 70
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Residual S (g/L)
Time (h)
010 20 30 40 50 60 70
-1.0
-0.5
0.0
0.5
1.0
Residual X (g/L)
Time (h)
Fig. 2. Residual analysis for the proposed model including observed
versus simulated values
Fig. 3. Dynamic response of the bioreactor to different dilution rate.
Et (_) and productivity (…)
TABLE I. EFFECTIVENESS OF THE MODEL
Variable
R2
DSE
RSD
Bioethanol
0.995
0.985
0.520
Biomass
0.996
0.993
0.519
Enzyme activity
0.998
0.990
0.402
Substrate
0.991
0.970
0.787
Average
0.995
0.984
0.557
TABLE II. EVALUATION OF STATIONARY STATE STABILITY
Variable
D=0.005 1/h
State
Stability
Bioethanol
11.9863
-0.010
Biomass
4.16474
-.0140
Enzyme activity
5.47158
-0.014
Substrate
5.72509
-0.012
The continuous operation can improve the efficiency of the
fermentation by maintaining a constant substrate
concentration. Another advantage of this process over batch

is that there is no non-productive idle time for
resterilization-cleaning. The other disadvantage of this
process is that ethanol concentration becomes inhibitive
after a certain concentration. Therefore, is necessary to
control and monitoring the system in continuous operation
[30].
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
0
1
2
3
4
5
Rank considering the output: S
Dilution rate (1/h)

0
1
2
3
4
5
6
Rank considering the output: X
0
1
2
3
4
5
Rank considering the output: Et
0
1
2
3
4
5
6
Rank considering the output: Ez
Fig. 4. Rank of the observability matrix analysis considering the starch
concentration, biomass and glucose as real-time measurement for the
estimation of state variables.
It has been shown in this paper the concept of
observability, which is well understood for continuous
process operating around equilibrium point, as is illustrated
in Fig. 4. Therefore, the system is observable if and only if
the observability matrix of the system is full rank, if a system
is unobservable; this means the current values of some of its
states cannot be determined through output sensors. Hence
the system (2-5) is full observable for dilution rates: 0.005
1/h, 0.01 1/h, 0.02 1/h, 0.06 1/h, 0.08 1/h and 0.1 1/h, if and
only if enzyme concentration is the measurable output, is not
full observable for dilution rate: 0.041/h, but there are several
observable subspaces of different dimensions as can be seen
in Fig. 4. Consequently, if substrate concentration is the
measurable output, performs reconstruction is of three states:
substrate concentration, enzyme concentration, and ethanol
(Fig. 5) and the unobservable variable is: biomass
concentration, for a dilution rate of 0.05 1/h (Fig. 6).
010 20 30 40 50 60 70
0
10
20
30
40
S (g/L)
Time (h)
010 20 30 40 50 60 70
0
2
4
6
X (g/L)
Time (h)
Fig. 5. Temporal evolution (  vs Extended Luenberger Observer ,
susbstrate and biomass concentrations.
010 20 30 40 50 60 70
0
5
10
15
Et (g/L)
Time (h)
010 20 30 40 50 60 70
0
2
4
6
Ez (g/L)
Time (h)
Fig. 6. Temporal e Extended Luenberger ,
enzyme and ethanol concentrations.
The extended Luenberger observer provides a good state
estimation (Fig. 5, 6), can be seen that the estimation error on
average it is 3%, which allows to say that these variables can
be measured in real time and considered as inputs for a
control law, the observer gain is k1 = 1/51 h. The trajectories
of the extended Luenberger observer converge quickly to the
real trajectories.
Controllability deals with the possibility of forcing the
system to a particular state by application of a control input.
If a state is uncontrollable then no input will be able to
control that state. On the other hand, whether or not the
initial states can be observed from the output is determined
using observability property. Thus if a state is not observable
then the controller will not be able to determine its behavior
from the system output and hence not be able to use that state
to stabilize the system. Hence the system (2-5) is controllable
for dilution rates: 0.005 1/h, 0.01 1/h, 0.02 1/h, 0.06 1/h, 0.08
1/h and 0.1 1/h, and not controllable for dilution rate:
0.041/h.
IV. CONCLUSION
This paper presents results from the development and
testing of a novel model to estimate kinetic parameters in
batch fermentation, using mucilage juice residues from cocoa
industrial waste. This novel kinetic model was proposed in
order to provide an experimental prediction for ethanol
production data under batch processes. This model allowed
exploring the behavior of the process in continuous operation
to obtain the maximum production of ethanol and
productivity, evaluating at this point of equilibrium,
properties such as stability, controllability and observability
for possible large-scale production. Furthermore, this
experiment showed that maximum ethanol concentration was
about 13.0 g/L, with productivity 0.18 g/ L h. In continuous
operation the production of ethanol was about 11.98 g/L,
with productivity 0.17 g /L h, this state of equilibrium:
stable, observable and controllable. Several output
measurements combinations were proposed in order to show

the dimensions of the corresponding observable subspaces
employing the observability matrix criterion. Finally, this
model is observable for dilution rates: 0.005 1/h, 0.01 1/h,
0.02 1/h, 0.061/h, 0.08 1/h and 0.1 1/h, consequently, is not
observable and controllable for dilution rates: 0.04 1/h, thus
creating a firm basis for advanced control strategies to
bioethanol production process.
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