On observability and controllability for a bioethanol dynamical model obtained from cocoa industrial waste

6 Pages·2018·0.556 MB·English

by  Pablo A. López Pérez, Dulce Jazmín Hernández Melchor, Teresa Romero Cortes, V. Peña Caballero, Omar Santos, F. Martínez Farías

On observability and controllability for a bioethanol
dynamical model obtained from cocoa industrial
waste

Teresa Romero Cortes Omar Santos
Pablo A. López Pérez
Escuela Superior de Apan CITIS, AACyE, ICBI
Escuela Superior de Apan
Universidad Autónoma del Estado de Universidad Autónoma del Estado de
Universidad Autónoma del Estado de
Hidalgo Hidalgo
Hidalgo
Apan, Hgo. Mexico Pachuca, Hidalgo Mexico
Apan, Hgo. Mexico
[email protected] [email protected]
[email protected]

V. Peña Caballero
Dulce Jazmín Hernández Melchor F. Martínez Farías
Departamento de Ingeniería
Colegio de Postgraduados Escuela Superior de Apan
Agroindustrial, División de Ciencias de
Estado de México, México Universidad Autónoma del Estado de
la Salud e Ingenierías
[email protected] Hidalgo
Universidad de Guanajuato
Apan, Hgo. Mexico
Gto.,Mexico
[email protected]
[email protected]
Abstract— The aim of this paper is to study the stability, mechanical extraction of cacao seeds, a predominance of
controllability and observability for a bioethanol production pulp and mucilage residues has been reported with an
dynamic system. Likewise, a novel model based on the average production of 727, 500.00 metric tons for the year
mechanism of multiple parallel coupled reactions was used to 2003/2004, of which in average 156, 333.33 m3 respectively
describe the kinetics of substrate, enzyme, and biomass and were derived from cocoa juice, the 50% of this is considered
product formation. This model has been extended to to be waste, with no further commercial application, but it is
continuous operation, which is employed as a virtual plant to now recognized that its components have potential as inputs
enable the implementation of the properties. The maximum
and energy for second-generation biofuels [7, 8].
ethanol production conditions are obtained by manipulating
the dilution rate with an optimal initial substrate However, it is difficult to develop and especially to
concentration. A nonlinear observer is implemented to show implement, advanced monitoring and control strategies for
the results of the observability analysis. Sufficient and real bioethanol production because of the absence of reliable
necessary conditions for state controllability and state instrumentation for measuring biological state variables, i.e.
observability of such system are established. Results indicated substrates, pH, biomass, metabolic enzymes and product
that the proposed model can be applied as a way of augmenting concentrations [9, 10]. For instance, the required quality of
bioethanol production, controlling and monitoring the process.
monitored data and frequency of sampling are functions of
the accuracy of the bio-sensors employed for process control.
Keywords—: phenomenological model, continuous operation,
In many cases, the state variables are not measurable on-line
parameters, matrix criterion
(in real-time) due to the high cost of sensors and extreme
operating conditions. These facts, together with the
I. INTRODUCTION nonlinearity and parameter uncertainty of the bioprocesses,
Scenarios’ research in energy systems have require an enhanced modeling effort, state estimation
demonstrated the momentum of new technologies and (observability) and control (controllability) strategies [11,
renewable energy generation results in the diversification of 12]. The test of a system’s observability and controllability is
primary energy (fossil). Energy efficiency and renewable a necessary prerequisite to the estimation and regulated of
energies processes will need to evolve before sustainable states. Because of the nonlinear aspects of their dynamics,
cities to enable successful integration of supportive stability and observability analysis is rather complex in (bio-
technologies to have direct effect the climate challenge [1- chemical) process applications. However, for nonlinear
2]. Currently, there are projects with economic support systems, the theory of observers and controllers are not
related to optimize integrated biorefineries. These projects nearly as neither complete nor successful as it is for linear
will work to solve critical encountered for the successful systems [13]. Controllability and observability are two
scale-up and reliable operations of integrated biorefineries important properties of state models which are to be studied
generating high-impact tools and techniques for increasing prior to designing a controller. The designs of observability
the productivity [3]. It has been reported that Brazil and the and controllability conditions for nonlinear systems are a
United States produced approximately 70% of the global challenging problem (even for accurately known systems)
biofuel supply in 2015, the above, equaled approximately 35 that has received a considerable amount of attention [14].
billion gallons consisting roughly of a 3:1 relation of ethanol
Therefore, a model was developed kinetic based on the
and biodiesel [4,5].
mechanism of multiple parallel coupled reactions was used
The development of bioethanol biotechnology is to describe the kinetics of substrate, enzyme, biomass and
advancing rapidly as a fuel of greater importance [6]. product formation, from this model experimentally adjusted
Considering the high theoretical yields of this product, in batch was extrapolated to a continuous operation for was
several researchers have begun exploring approaches to demonstrate the utility of the observability and controllability
increase bioethanol production. During the process of matrix as a metric in developing optimum measurement
XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE
strategies in a range of conditions of operations for ethanol acceptable simulation, whereas DCE > 0.9 represents a good
fermentation. simulation.

II. METHODOLOGY Analysis of Experimental Data
A model can be used to evaluate the behavior of the The fermentations were carried out in Erlenmeyer
biochemical reaction pathway and to determine the
flasks containing broth medium (30 g/L glucose, 20 g/L,
properties of the dynamic system such as the existence of an
peptone and 10 g/L yeast extract), inoculated with colonies
equilibrium point, local controllability and observability,
of P. kudriavzevii. Cocoa juice was extracted from cocoa
local or global asymptotic stability of equilibrium points
[15, 16]. fermentation baskets after two days. For experiments, cocoa
pulp juice was adjusted to 30 g/L of glucose with reducing
Several reports and reviews about the production of ethanol sugar. Samples were incubated at 30°C and 200 rpm for 18-
fermentation using microorganisms, as well as certain 22 h. Ethanol production was evaluated in mucilage juice
yeasts, bacteria, and fungi have been published. However,
samples inoculated with 1×107 cells 1/mL at 30°C, shaking
the models described here are complicated because (1) there
for 24 h [23]. Subsequently, 10 mL of re-suspended yeasts
are too many parameters, which cannot be individually
were transferred in 1 L of cocoa pulp juice and incubated at
determined (2) the models represent a higher order and
some of them are described by partial differential equations, 30°C for 24 h with 150 rpm in orbital agitation. This
(3) structured and unstructured processes models [17, 18]. experiment was performed in triplicate. Ethanol was
determined using a HPLC Waters [24].
On the other hand, in biochemical reaction networks, there
are multi-step processes which may follow either simple or B. Phenomenological model
complex rate laws, the rate law is a direct result of the Basically, expressions based on mass balance differential
sequence of elementary steps that constitute the multi-step equations terms can be employed to describe the models of
reaction mechanism. However, as such, it provides an microbial kinetics for growth and fermentation processes.
optimum tool for an unknown mechanism. Therefore, we The change of fermented product rate, substrate
need a new approach to solve these issues [19, 20]. The consumption and biomass were related to ethanol (P),
proposed methodology is summarized in three stages: 1) glucose (S) and biomass (X) concentrations in using
kinetic structure, 2) phenomenological model and 3) proposed reaction rate laws (see, eq. 2-5), that will serve to
assessment of the controllability and observability. predict the behavior of the overall system and identify the
optimal operating conditions for the inherently safe process
A. Kinetic structure
(see, Fig. 1). The proposed model makes the following
The first stage, a multiple coupled reaction mechanism assumptions: i) individual steps in any mechanism are n-
is proposed, where the substrate is consumed by the order unimolecular reactions ii) measured activity is directly
microorganism, while at the same time it produces the PG proportional to concentrations of active enzyme forms, iii)
enzyme for the degradation of the substrate and ethanol consider inhibition effects.
Equations:
production, in addition it is considered inhibition of
Substrate balance (S):
biomass by product (ethanol), the reactions kinetics are
dS
based on the power rate law. k XDS S (2)
dt 1 0

Biomass balance (X):
Determination of kinetic parameters using Model-
dX
Maker® software (based on Levenberg–Marquardt k Sk Ek Xk XDX  X
dt 1 3 t 2 3 0
optimization approach) was employed for the non-linear

fitting data. The performance of the proposed mathematical
(3)
model was statistically evaluated using the dimensionless
coefficient for efficiency (DCE) [21, 22]. Enzyme activity balance (Ez):
dE
z k XS DE (4)
dt 2 Z
N
 2 Bioethanol balance (E):
i i t
DCE 1.0 i1N 2 (1) d d E t t k3XS DEt (5)
i
i1 Here k : specific reaction rate for j concentrations, α, β, and
j
Where:  model simulated data at time t ,  the θ: Exponential term, i.e. the rate is proportional to the
i i i
concentrations of the reactants each raised to some power.
observed data at time ti , i is the mean value of the Model limitations are: the model is based on a
phenomenological rather than a mechanistic scheme, mass
observed variable, N is the data number, DCE varies
transfer limitations and conformation changes of the enzyme
between (0, 1], a positive value of DCE represents an
structure, as effect of pH and temperatures are not related.
assume that the origin x0is the system equilibrium of
cocoa processing bioreactor 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).
30 C 1L
D Output Measurements
Let x0 be an equilibrium point for the nonlinear
S
X
residpureosc fersosmin gco coa Signals Ez system (6), where f :Dncontinuously differentiable
D E
t and D is a neighborhood of the origin. Let
Controller Stateestimation f
Controllability Observability Signals J  x (9)
property property x x0
J be the system Jacobian evaluated at the zero equilibrium
Signals
of system. Then, the origin is:

• Asymptotically stable if Re λ < 0for all eigenvalues of J
Fig. 1. Diagram of the process i
• Unstable if Re λ > 0for at least one of the eigenvalues of
i
C. Assessment of the controllability and observability J
Stability • if at least one of the eigenvalues is on the J axis, (is
A continuous-time nonlinear control system (eq. 2-5) is marginally stable)[25].
generally described by a differential equation of the form
x fx,t;u  t 0, (6) Observability
0
The test of a system’s observability is a necessary
Where
prerequisite to the estimation of states. However, for
 x x(t) :state of the system belonging to a (usually
nonlinear systems, the theory of observers is not nearly as
bounded) region  n. neither complete nor successful as it is for linear systems.
x
Thus if a state is not observable then the controller will not
 f:nq n smooth nonlinear vector
be able to determine its behavior from the system output and
function and Lipschitz in x.
hence not be able to use that state to stabilize the system, we
 u is the control input vector belonging to another show the concept of observability, which is well understood
(usually bounded) region m, mn. for continuous processes operating around an equilibrium
u
point. Therefore, the system is observable if and only if the
To indicate the time evolution and the dependence on the
observability matrix of the system is full rank [26] if a
initial statex , the trajectory (or orbit) of a system state
0 system is unobservable, the current values of some of its
xtis sometimes denoted as x  states cannot be determined through output sensors.
t 0
As a background, consider the following lineal system
In control system (1), the initial time used is t0 0, unless representation:
otherwise indicated. The entire spacen , to which the x AxBu (10)
y Cx
system states belong, is called the state space. Associated
where A, B, C are the constant matrices with appropriate
with the control system (1), there usually is an observation
dimensions.
or measurement equation (observers)
ygx,t;u, 1mn (7) N C,CA,CA2,CA3,,CAn1T (11)
As can be seen, if the state vector can be determinate, the
 ym measured output vector, that is the vector of
matrix N (named as the observability matrix) must be
measured states. invertible (full rank) in order to obtain:
 nm1single-input/single-output (SISO) system X  N1Y
 n,m1multi-input/multi-output (MIMO) system
Such that, the state vector X is observable in respect to
Consider the general nonautonomous system (6) the measurable outputY . The local observability analysis
  is condensed as following:
x f x,t (8)
where the control input uthxt,t, if it exists [see
Theorem 2 Local Observability. A continuous time
system (6)], has been combined into the system function f 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].
for simplicity of discussion. Without loss of generality,
Controllability 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
Controllability is another geometric property of a system,
equilibrium state (see table 2).
describing the ability to \drive" the system states to arbitrary
1.0 1.0
values through the control input. Controllability deals with
the possibility of forcing the system to a particular state by )L 0.5 )L 0.5
/g /g
atOhpepnli ncaoBt ii,onAnp uBot ,fw Aai2l lB cbo,en atrbo,lleA itnnop 1cuBotn. tIrfo la t hsatta tset aties . uncontrolla(1b2le) ( E laudiseRt -00..50 ( E laudiseRz-00..50
-1.0 -1.0
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
Theorem 3 Dynamical system (2-5) is controllable if and Time (h) Time (h)
only if rank(O) = n. 1.0 2.0
1.5
experTimhee netfafle coItbIiIvs.ee rnRveEasStsiU ooLnfTs t Shc Aeo NnpcDreo DrpnIoSisnCegUd S bmSaItOocdNhe lf efromr ednetsactriiobnin igs )L/g( X laudiseR-000...550 )L/g( S laudiseR---011100......505050
presented in Fig. 2 (residual values), and quantified by -1.00 10 20 30 40 50 60 70 -2.00 10 20 30 40 50 60 70
applying RSD (Residual Standard Deviation), R2 Time (h) Time (h)
(correlation coefficient) and coefficient of efficiency DCE,
relating to the four state variables (see, Table 1). From these Fig. 2. Residual analysis for the proposed model including observed
criteria, it was concluded that the model presented versus simulated values
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, indicating the model’s good fit,
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 Fig. 3. Dynamic response of the bioreactor to different dilution rate.
Et (_) and productivity (…)
70 h in batch operation, and productivity was 0.18 g/L h
using mucilage juice residues from cocoa.

TABLE I. EFFECTIVENESS OF THE MODEL
The initial values for the state variables (S = 32 g/L; X =
0 0
0.60 g/L, E = 0.10 g/L, E = 0.02 g/L), where the parameter Variable R2 DSE RSD
z t
vector to be estimated is: Π: k = 0.13 ±0.09 [1/h], k = 0.03 Bioethanol 0.995 0.985 0.520
1 2
±0.01 [U L/g mL h], k = 0.003 ±0.001 [1/h], k = 0.0014 Biomass 0.996 0.993 0.519
3 4
Enzyme activity 0.998 0.990 0.402
±0.001 [1/h], k = 0.005 ±0.001 [1/h], k = 0.014 ±0.01 [1/h],
5 6 Substrate 0.991 0.970 0.787
k = 0.0067 ±0.001 [U L/g mL h], α=0.947±0.1, β=0.40±0.1,
7 Average 0.995 0.984 0.557
γ=0.08±0.01, δ=0.014±0.01, ε=0.9±0.1, values are
expressed as mean ± confidence intervals (p = 0.05). TABLE II. EVALUATION OF STATIONARY STATE STABILITY

Variable State Stability
The parameter values obtained in the present study fall
D=0.005 1/h
within the range of those reported in the literature, due to the Bioethanol 11.9863 -0.010
different operating conditions used in each case, i.e., Biomass 4.16474 -.0140
different carbon source, continuous or batch operation, Enzyme activity 5.47158 -0.014
Substrate 5.72509 -0.012
temperature, pH, among others [29, 30].

The continuous operation can improve the efficiency of the
Otherwise, in a continuous culture was analyzed related to
fermentation by maintaining a constant substrate
dilution rate (D: 0.005 to 0.1 1/h), via a bifurcation diagram
concentration. Another advantage of this process over batch
is that there is no non-productive idle time for
6
resterilization-cleaning. The other disadvantage of this
process is that ethanol concentration becomes inhibitive
4
after a certain concentration. Therefore, is necessary to
L)
control and monitoring the system in continuous operation g/
[30]. (Ez2

0
6 5 5 6
0 10 20 30 40 50 60 70
5 5 Time (h)
4 4 15
Rank considering the output: X 4321 Rank considering the output: S 321 321 Rank considering the output: Et 4321 Rank considering the output: Ez (Eg/L)t10500 10 20 30 40 50 60 70
Time (h)
0 0 0 0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11
Dilution rate (1/h) Fig. 6. Temporal evolution ( ― ) vs Extended Luenberger Observer ( ˗ ˗),
enzyme and ethanol concentrations.
Fig. 4. Rank of the observability matrix analysis considering the starch The extended Luenberger observer provides a good state
concentration, biomass and glucose as real-time measurement for the
estimation (Fig. 5, 6), can be seen that the estimation error on
estimation of state variables.
average it is 3%, which allows to say that these variables can
It has been shown in this paper the concept of be measured in real time and considered as inputs for a
observability, which is well understood for continuous control law, the observer gain is k1 = 1/51 h. The trajectories
process operating around equilibrium point, as is illustrated of the extended Luenberger observer converge quickly to the
in Fig. 4. Therefore, the system is observable if and only if real trajectories.
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 Controllability deals with the possibility of forcing the
the system (2-5) is full observable for dilution rates: 0.005 system to a particular state by application of a control input.
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 If a state is uncontrollable then no input will be able to
only if enzyme concentration is the measurable output, is not control that state. On the other hand, whether or not the
full observable for dilution rate: 0.041/h, but there are several initial states can be observed from the output is determined
observable subspaces of different dimensions as can be seen using observability property. Thus if a state is not observable
in Fig. 4. Consequently, if substrate concentration is the then the controller will not be able to determine its behavior
measurable output, performs reconstruction is of three states: from the system output and hence not be able to use that state
substrate concentration, enzyme concentration, and ethanol to stabilize the system. Hence the system (2-5) is controllable
(Fig. 5) and the unobservable variable is: biomass for dilution rates: 0.005 1/h, 0.01 1/h, 0.02 1/h, 0.06 1/h, 0.08
concentration, for a dilution rate of 0.05 1/h (Fig. 6). 1/h and 0.1 1/h, and not controllable for dilution rate:
0.041/h.

6
IV. CONCLUSION
4

L)
g/
X ( 2 This paper presents results from the development and
testing of a novel model to estimate kinetic parameters in
0 batch fermentation, using mucilage juice residues from cocoa
0 10 20 30 40 50 60 70 industrial waste. This novel kinetic model was proposed in

Time (h) order to provide an experimental prediction for ethanol
40 production data under batch processes. This model allowed
exploring the behavior of the process in continuous operation
30
to obtain the maximum production of ethanol and
L)
S (g/ 20 pprroodpuercttiievsi tsyu, che vaas lustaatbinilgit y,a tc onthtriosl lapboiliintyt aonfd oebqsueirlvibarbiiulimty,
10
for possible large-scale production. Furthermore, this
0 experiment showed that maximum ethanol concentration was
0 10 20 30 40 50 60 70
about 13.0 g/L, with productivity 0.18 g/ L h. In continuous
Time (h)
operation the production of ethanol was about 11.98 g/L,
with productivity 0.17 g /L h, this state of equilibrium:
Fig. 5. Temporal evolution ( ― ) vs Extended Luenberger Observer ( ˗ ˗ ),
susbstrate and biomass concentrations. stable, observable and controllable. Several output
measurements combinations were proposed in order to show
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