Application of a GA-Optimized NNARX Controller to Nonlinear Chemical and
Bijan Medi, Ayyob Asadbeigi
Reference: HLY 7846
To appear in: HELIYON
Received Date: 1 May 2021
Revised Date: 9 August 2021
Accepted Date: 18 August 2021
Please cite this article as: B. Medi, A. Asadbeigi, Application of a GA-Optimized NNARX
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Application of a GA-Optimized NNARX Controller to Nonlinear
Chemical and Biochemical Processes
Bijan Media*, Ayyob Asadbeigib
a Department of Chemical Engineering, Hamedan University of Technology, P.O. Box 65155-
579, Hamedan, Iran.
b Department of Electrical Engineering, Islamic Azad University, Hamedan Branch,
*Correspondence to: email@example.com
Chemical and biochemical processes generally suffer from extreme nonlinearities with respect to
internal states, manipulated variables, and also disturbances. These processes have always
received special technical and scientific attention due to their importance as the means of large-
scale production of chemicals, pharmaceuticals, and biologically active agents. In this work, a
general-purpose genetic algorithm (GA)-optimized neural network (NNARX) controller is
introduced, which offers a very simple but efficient design. First, the proof of the controller
stability is presented, which indicates that the controller is bounded-input bounded-output
(BIBO) stable under simple conditions. Then the controller was tested for setpoint tracking,
handling modeling error, and disturbance rejection on two nonlinear processes that is, a
continuous fermentation and a continuous pH neutralization process. Compared to a
conventional proportional-integral (PI) controller, the results indicated better performance of the
controller for setpoint tracking and acceptable action for disturbance rejection. Hence, the GA-
optimized NNARX controller can be implemented for a variety of nonlinear multi-input multi-
output (MIMO) systems with minimal a-priori information of the process and the controller
Keywords: Chemical, Biochemical, Nonlinear, Controller, Neural Network, Genetic Algorithm.
Chemical and biochemical processes are among some of the most vital and yet nonlinear
modern industrial processes under technical and scientific considerations. A variety of chemicals
and active biological agents are produced in such systems [5, 6], while they are operated under
extreme working standards to comply with stringent production regulations and a competitive
Control of chemical and biochemical processes is a challenging task due to nonlinearities
associated with their internal states, manipulated variables, and also disturbances  as well as
their time variability , which is inherent to many chemical and biochemical systems [3, 27].
The controller design for nonlinear processes has been studied in numerous works. Fernández et
al.  studied a simple but efficient technique for tracking optimal profiles with error
minimization for nonlinear biochemical processes, which was based on linear algebra for the
calculation of control actions. The performance of the designed controller was tested through
simulations by adding parametric uncertainty and perturbations in the initial conditions.
Aguilar-López et al.  introduced an uncertainty-based observer with a polynomial structure
capable of estimating the unknown modeling error of a continuous bioreactor coupled to a linear
input-output controller. In Mailleret et al. , a nonlinear adaptive control and the global
asymptotic stability of closed-loop system were investigated for a bioreactor with unknown
kinetics. They verified their approach on a real-life wastewater treatment plant.
Artificial neural networks (ANNs) are general-purpose modeling tools that can be used for
various applications, including static and dynamic modeling, clustering, and pattern recognition
[8, 9, 30, 35]. ANN is useful in particular when modeling with fundamental governing equations
is costly, time-consuming or both [11, 22].
Naregalkar and Subbulekshmi  proposed a novel approach using NARX (nonlinear auto-
regressive with exogenous input) model and enhanced moth flame optimization (EMFO) for pH
neutralization of wastewater. They evaluated their method in terms of integral squared error
(ISE), integral absolute error (IAE), mean squared error (MSE), settling time, and peak
del Rio-Chanona et al.  utilized an ANN model for dynamic modeling and optimization of a
15-day fed-batch process for cyanobacterial C-phycocyanin production. To generate additional
datasets, they artificially introduced random noise to the original dataset. They also chose the
change of state variables as training data output.
The lack of online information on some bioprocess variables and the presence of model and
parametric uncertainties are important challenges for the control of such processes. To address
these issues, Rómoli et al.  proposed an online state estimator based on a Radial Basis
Function (RBF) neural network that feeds information to a controller, which was derived via a
linear algebra-based design strategy.
Intelligent methods can also be used to control nonlinear chemical and biochemical processes.
These approaches can compensate modeling errors, tackle the occurrence of disturbances, and
advise optimal operation scenarios as control actions. Ünal et al.  reviewed different aspects
of using artificial intelligence and evolutionary algorithms i.e., genetic algorithm (GA) and ant
colony (AC) for PID controller tuning on real-time experimental setups. The performances of
these three techniques were compared with each other using the criteria of overshoot, rise time,
settling time, and root mean square (RMS) error of the trajectory. It was observed that the
performances of GA and AC are better than that of Ziegler-Nichols technique.
Latha et al.  used particle swarm optimization (PSO) algorithm for tuning of a proportional-
integral-derivative (PID) controller for a class of time-delayed stable and unstable process
models. The dimension of the search space was only three tuning parameters of conventional
PID controllers. They tested their approach in real-time on a nonlinear spherical tank system.
The real-time result with PSO-tuned PID offered better results for reference tracking, multiple
reference tracking, and disturbance rejection problems.
More recently, intensive studies have been conducted for designing the NARX network
architecture to enhance modeling accuracy and versatility. In this regard, various methods have
been proposed for activation function selection and network weights and biases tuning via
optimization by different algorithms. Liu et al.  considered NARX neural networks for
analysis and identification of noisy nonlinear magnetorheological (MR) damper systems. The
accuracy of their results supports the use of this modeling technique for identifying irregular
nonlinear models of MR dampers and similar devices. Rankovic et al.  developed a nonlinear
model predictive control (NMPC) scheme, with the assumption that the object model is
unknown. Therefore, they used a digital recurrent network (DRN) model instead to predict the
future evolution of the system, which is essential for model predictive control. From their
framework, one can infer that designing the network structure is crucial and complex.
Combination of NARX models with genetic algorithm has been used for forecasting, which
subsequently can be used for decision making. Han et al.  utilized a NARX network for
bitcoin price forecasting and concluded that genetic algorithm was effective to decide the
architecture of the NARX neural network better than some other information criteria.
In a similar study to our work, Hernández-Alvarado et al.  used a neural NARX network to
optimize PID controller gains for an underwater remotely operated vehicle in simulation and also
in a real-time manner. They took into consideration two criteria to assess the performance of the
controllers: position tracking error and energy consumption, leading to the conclusion that the
proposed method obtained the best performance with less energy.
In this work, we propose a general-purpose neural network-based NARX controller, which
does not rely on a predefined form of any conventional controller, but can emulate a PI action.
We will show that the controller can be tuned on some simple data sets, and hence might be
tuned in an online manner similar to Ziegler-Nichols closed-loop tuning procedure . We
evaluated the controller performance on a continuous fermentor model as a nonlinear single-
input single-output (SISO) process. Furthermore, as the importance of pH in chemical and
biochemical processes cannot be overemphasized, it has also been taken into consideration in
this study as a highly nonlinear multi-input multi-output (MIMO) system.
After introducing the controller structure and elaborating on its stability conditions, the tuning
procedure is described, which was done using genetic algorithm optimization of the network
weights. In the results section, the performance of the controller is tested for setpoint tracking,
modeling error, and disturbance rejection scenarios, and compared with a conventional PI
2 NARX formulation
The nonlinear autoregressive model with exogenous input (NARX) is a time series model,
which is represented as a function of the model output and one or more independent inputs all at
several past time steps. In the predictive form, a NARX model can be represented by:
( 1) ( ), , ( ), ( ), ( 1), ( )
( ) 0,1, ,
( ) 0,1, ,
y k f y k y k n u k u k u k m
y i y i n
u j u j m
where y is the network output, u is the network input, and f can be any linear or nonlinear
Fig. 1: General structure of a NNARX feedforward neural network with tapped delay lines.
3 Neural network structure
Artificial neural networks appear in a variety of architectures . Specifically, the feedforward
ANNs can be readily extended to the NARX networks with the introduction of tapped delay lines
at the network input or even between layers, as shown in Fig. 1. In this regard, the new
architecture is called a neural NARX or NNARX network. In addition, such networks may have
one or more feedback lines from outputs or hidden layers enclosing several layers of the
network, which are not shown in Fig. 1 since such feedback lines are not included in our
NNARX approach. This structure offers several interesting characteristics including time series
prediction and nonlinear input-output realization of dynamic systems.
It is worthy of attention that this network is fully connected, which means there is a connection
between every input and every neuron in each layer. The importance of these connections is
controlled by the weight parameters, while bias parameters shift the network output to a suitable
position. Typically, all weights and biases are regulated by any convenient optimization
It must be also noted that Fig. 1 refers to a specific structure of NARX networks that do not
directly receive y output(s) as their inputs. In our case, y as the controller output (manipulated
variable) is resolved in the closed-loop response of the process and does not directly appear as
the network input.
In fact, many control schemes are a stable subset of linear or nonlinear dynamic systems.
Therefore, a NNARX model can be utilized as a nonlinear parametric controller. As we will see
( 1, )nn
Inputs Hidden Layer 1 Output Layer
in the results section, this structure may add integral effect to the control action, while it acts as
an efficient nonlinear predictor. When combined together, these effects can be considered an
inverse model controller with integral effect.
3.1 Proof of BIBO stability
According to the Gronwall Lemma , if
n n n
f g h
are real nonnegative sequences for
n n k k
h f g h
n n k k j
k j k
h f g f g
Theorem 1. The system (1) is bounded-input bounded-output (BIBO) stable with the initial
y p y
,...,p k k n
if it is Lipchitz with the constants
y y k
u u k
for simplification. Hence:
11 ( ,..., , ,..., ) ( ,..., , ,..., )
k p k k n k k m p p n p p m
y y f y y u u f y y u u
0 0 0
k i p i k i ki pi
n m k
i i i i
y y u u hL L g f
0k i p
m j k
k i k i p i
h y y
1, 1 1k i k i p i
h y y
1 1,0 1 1k k k p
h h y y
,0k k k p
h h y y
,i k i k i p i
h h y y
From the Gronwall Lemma we have
0 0 1
11 0exp( )
k p k i p i k l
k k i k
i i i j
i i l i
y Ly u u LuuLL
Since we supposed that the initial condition is bounded, we easily obtain
0 0 1
11 0exp( )
k p k i p i k l
k k i k
i i i j
i i l i
y Ly u u LuuLL
which according to the assumptions all terms are bounded, and this yields the required result.
4 NNARX Controller tuning
Genetic algorithm (GA) is a heuristic optimization method inspired by the so-called evolution
process in nature [12, 23] as it mimics the evolutionary operators: selection, crossover, and
mutation to achieve a fitter population of solutions (individuals) over iterations (generations).
The main advantages of this algorithm are that GA does not require calculating any derivatives.
Hence in the case of the current problem, any sort of network topology and transfer function can
be used without a-priori assumption on their exact formulations, which otherwise were needed
for differentiation. On the other hand, as there are no derivates involved, the controller structure
is not significantly affected by the noise effects if implemented on an actual process. Moreover,
the error term(s) can be arbitrarily defined as a function of closed-loop response as utilized in
this work. It must be emphasized that for higher-order systems (more delays and/or MIMO
systems), the solution space grows tremendously, rendering classical gradient-based optimization
The objective function for tuning is defined as the weighted sum of the mean squared errors
(MSE) of the tracking tasks over time as:
w e k
( ) ( ) ( )
i spi mi
e k y k y k
where ei is the tracking error and ymi and yspi are the ith measured output and its respective
setpoint. wis are arbitrary weights, which regulate the importance of the error terms. They are set
to unity in this work.
It must be emphasized that ei is the input to the NNARX controller, while the manipulated
variables are the network outputs. On the other hand, the GA-optimizer receives MSE values for
every individual, and returns the NNARX weight (and bias) values (x in Fig. 2) to the network.
The overall scheme of the closed-loop diagram is shown in Fig. 2. Hence, the optimizer calls the
closed-loop system model for a sufficiently large number of times until it concludes that no
better solution can be found.
Fig. 2: General closed-loop structure of the control scheme with the GA-optimized NNARX.
5 PI controller tuning
For the sake of comparison, a PI controller is selected and tuned by model parameters. In this
regard, a first-order plus time delay (FOPTD) model is fitted to the open-loop step response of
the selected processes. The tests data and the results are given in Table 1. The regression was
implemented via the sequential quadratic programming (SQP) approach to minimize MSE,
which is defined similar to Eq. 8 but without the weight parameters. It must be emphasized that
only process gain and time constant are fitted as the process time delay is considered to be equal
to one sample time, which is a reasonable assumption, considering the response time of the
Table 1: Step response test parameters, regression results, and PI controller tuning parameters.
-0.01 (Dilution rate)
+5 ml/s (Acid)
+5 ml/s (Base)
Process gain (kp)
Process time constant (
Process time delay (
Proportional gain (
Integral time (
Derivative time (
According to the Skogestad’s SIMC PID tuning rule , the following formulas are suggested
based on the FOPTD model parameters:
cp c D
I c D
where kp, τ, and tD are process gain, time constant, and time delay, respectively. Also, kc and τI
are controller proportional gain and integral time, respectively. It is apparent that the suggested
controller is in the PI mode. Hence the derivative mode is deactivated (
On the other hand, τc is the desired closed-loop time constant and the only tuning degree of
freedom. Skogestad  suggests that a good trade-off can be obtained by choosing τc equal to
process time delay. Hence, the values of 0.1 h and 1 s were initially used for the closed-loop time
constant of the continuous fermentation and pH neutralization processes, respectively. However,
these settings make the closed-loop response of both processes unstable. Hence, these values
were modified to 0.5 h and 3 s as the closed-loop time constants of the studied processes,
We have considered two typical nonlinear chemical and biochemical processes. The first process
is a continuous fermentor in which the biomass (cell-mass) concentration in g/l is the controlled
variable (plant output), while the manipulated variable (plant input) is the dilution rate,
representing a nonlinear SISO system.
A rather simplified but general form of the process is shown in Fig. 3 and represented by the
following equations :
X DX X
S D S S X
P DP X
P P S
K S S K
where D is the dilution ratio, X is the cell-mass concentration, S is the substrate concentration,
which is consumed by the microorganism, and Sf is the substrate concentration in the feed
stream. P is the product concentration and Pm is the product saturation constant.
is the cell-
mass yield. α and β are kinetic parameters of the fermentation reaction. µ is the growth rate and
µm is the maximum growth rate. Km and Ki are substrate saturation and inhibition constants,
respectively . The nominal parameter values are given in Table 2.
Fig. 3: Fermentation process diagram. X is the cell-mass concentration (measured variable), S is
the substrate concentration, and P is the product concentration.
Table 2: The nominal parameter values for the continuous fermentation process .
The second example is a neutralization process in which the liquid level of the tank and the
effluent pH are the controlled variables, while acid and base flow rates are the manipulated
variables. It is clear that this system is a MIMO process, as shown in Fig. 4.
The governing equations are given as follows:
1 2 3
h q q q C h
4 1 4 1 2 4 2 3 4 3
a a a a a a a
W W W q W W q W W q
4 1 4 1 2 4 2 3 4 3
b b b b b b b
W W W q W W q W W q
where h is the liquid level in the neutralization tank, and A is the tank cross-section. CV is the
outlet valve discharge coefficient, which is a constant in this work. q1, q2, and q3 are acid, buffer,
,,X S P
and base flow rates, respectively. Wa1 to Wa4 and Wb1 to Wb4 are reaction invariants as described
in Hu and Rangaiah .
The relation of pH with other variables is given by the following implicit algebraic equation
1 2 10
10 10 0
1 10 10
pK pH pH pK
However, to circumvent the solution of such a nonlinear algebraic equation, the first derivative
of Eq. 19 was calculated and simplified with respect to the derivative of pH. In this regard,
another differential equation was added to the set of state equations:
1( / )
pH N D
1 2 10 pH Pk
( ) ( )
1 10 10
Pk pH pH Pk
( ) ( )
( 14) ( )
2 10 10 10
log(10) 10 10
Pk pH pH Pk
W D N
The nominal parameters for the neutralization process are given in Table 3.
Fig. 4: Continuous pH neutralization process diagram.
Table 3: The nominal parameter values for the continuous pH neutralization process .
The preliminary NNARX network structure was constructed by the Neural Network Toolbox-
Simulink code generation facility . The network was then modified in Simulink to suit the
design required as a MIMO controller. The overall network structural parameters are given in
Table 4. As shown in this table and earlier in Fig. 1, several delays were introduced at the input
layer. It is worthy of attention that this network is fully connected, which means there is a
connection between every input and every neuron in each layer. As mentioned earlier, the
importance of these connections is controlled by the weight parameters, which are regulated by
the GA-optimizer. On the other hand, as the controller is designed around the steady state
conditions for which
, and in order to reduce the number of optimizing parameters, all bias
parameters were permanently set to zero.
Using one hidden layer is conventional in working with artificial neural networks  unless
problem complexity necessitates adding one or more hidden layers . The number of neurons is
selected based on some trial and error. The number of delays is also selected based on the
complexity of the process. It must be emphasized that the poles and zeros of the controller in the
linear analogy are allocated by these delays, while they can generate integral and derivative
actions, for which at least three delayed instances of the error are required.
It must be noted that this network requires just a few neurons in the hidden layer to obtain
satisfactory results. This feature makes the optimization task faster and more efficient. The
completed closed-loop structure was implemented and simulated in Simulink.
Table 4: NNARX structural parameters.
Number of hidden layers
Number of input delays
Number of hidden layer neurons
Hidden layer transfer function
Output layer transfer function
The MATLAB Global Optimization Toolbox was used for the optimization task . The
optimizer parameters are given in Table 5. For optimization and simulation, MATLAB and
Simulink 2014a on a laptop with an Intel Core i5-3380 M (2.90 GHz) CPU with 6 GB RAM
were used. It is worthy of attention that the number of optimizing parameters equals the weights
of the designed NNARX controller. The lower and upper bounds on the optimizing parameters
were set by some trial and error.
The summary of the optimization results is given Fig. 5. For the continuous fermentation
process, the optimization has terminated in exactly 50 generations (iterations) with the criterion
that the average change in the fitness (objective function) values has fallen below a predefined
limit (10-6). Similarly, for the neutralization process, the variations in the fitness values have
reached a minimum in about 50 generations. However, the termination criterion (average change
in the fitness values) has been satisfied after 88 generations. The variations in most of the
weights have also dropped in about 50 generations. Careful tuning of genetic algorithm
parameters (e.g., population size, selection function, crossover, and mutation operators) helps in
finding at least the area in which the global optimum is expected to be found, and this is
sufficient as long as the controller performance is concerned.
Table 5: GA-Optimization parameters.
Number of optimizing parameters
Lower bound on weights
Upper bound on weights
In this section, the performance of the NNARX controller is assessed on the two nonlinear
systems mentioned earlier. As for the case studies, first, setpoint tracking is investigated on both
processes. Further, a modeling error is introduced to the formulation of the fermentation process.
Finally, a disturbance in the form of an unwanted change in the buffer flow rate (q2) is
introduced in the neutralization process. These changes are introduced at time
8.1 Setpoint tracking
For the first analysis of the controller performance on the continuous fermentor, a square pulse
train with a period of 50 h and amplitude of 0.5 g/l is introduced to the setpoint of cell-mass
concentration with respect to its steady state value, as shown in Fig. 6a. As can be seen, the
tracking error is remarkably small, and the output (cell-mass concentration) promptly follows the
setpoint with a slight overshoot incomparable to that of the PI controller.
A similar test was carried out on the neutralization process in which a square pulse train with a
period of 1000 s and amplitude of unity was introduced with respect to the steady state values of
level and pH (Fig. 6b). It is apparent that here also, the controller response is quite fast, and the
offset is very small, while the overshoot is much smaller compared to that of the PI controller.
Fig. 5: Evolution of weights, change in fitness (objective function) values, and change in average
distance between individuals (solutions) for the fermentation process (a, c, and e) and
neutralization process (b, d, and f).
It is important to note that this offset-free response may suggest the integral action of the
NNARX controller considering that an integral action in the discrete form is very similar to the
NNARX structure with a linear transfer function.
The variations in the manipulated variables (controller outputs) for the setpoint tracking problem
are shown in Fig. 7. The prompt setpoint tracking action of the NNARX controller for the
fermentation process has come at the cost of aggressive changes in the manipulated variable
(dilution rate in Fig. 7a). However, for the neutralization process, the NNARX controller offers
even smoother controlling actions compared to the PI controller, as can be seen in Fig. 7b.
Moreover, there is no sustained fluctuation in the manipulated variables, which is a desired
Fig. 6: Setpoint tracking results for: (a) fermentation process, (b) neutralization process.
020 40 60 80 100
Cell Mass (g/l)
0500 1000 1500 2000
Fig. 7: Variations in the controller outputs for the setpoint tracking problem for: (a) fermentation
process, (b) neutralization process.
8.2 Modeling error
For a test of modeling error, it is here assumed that the growth rate (
) is wrongfully calculated
from the following equation instead of Eq. 15 based on which the NNARX controller was
The simulation results are given in Fig. 8. From this figure, one can infer that although the
controller quickly reduces the tracking error, there has remained a small offset over time. Hence,
for this problem, the PI controller is obviously superior.
020 40 60 80 100
0500 1000 1500 2000
Fig. 8: Results of including a modeling error in the fermentation process model.
8.3 Disturbance rejection
For another investigation, a disturbance in the form of an unmeasured change in the buffer flow
rate (q2) is applied at time
. At this moment, the buffer flow rate is reduced to 50% of its
nominal value. It is noteworthy that a decrease in buffer value in any neutralization process
increases process sensitivity, causing the controlling task more tedious. The results are shown in
Fig. 9. It is apparent from Fig. 9a that the deviation from the setpoint value is small, but it must
be admitted that the integral effect is slightly compromised. The changes in the manipulated
variables (acid and base flow rates) are shown in Fig. 9b, which shows mild fluctuations. Based
on these results, the NNARX controller is able to tackle the relatively large unmeasured
010 20 30 40
Cell Mass (g/l)
Fig. 9: Results of including a disturbance as an unmeasured change in the buffer flow rate (-50%
of the nominal value) for the neutralization process: (a) process variables, (b) controller outputs.
8.4 Quantitative analysis
A quantitative analysis was carried out based on the integral absolute error (IAE) criterion:
IAE e t dt
As given in Table 6, the IAE values for the proposed NNARX controller are significantly smaller
compared to the PI controller for setpoint tracking, especially for the neutralization process.
However, the IAE values for modeling error and disturbance rejection are smaller for the PI
controller due to the perfect integral action of this controller.
020 40 60 80 100
020 40 60 80 100
Table 6: Integral absolute error (IAE) values for all the case studies.
In this work, a neural network-based nonlinear controller was tested on two nonlinear chemical
and biochemical processes. The controller was tuned using genetic algorithm by running the
closed-loop models for a sufficiently large number of times. There are at least two advantages of
using GA as the training algorithm: 1) it does not require any knowledge of the neuron transfer
function properties as opposed to gradient-based methods, which require exact derivatives of the
transfer functions for back propagation, 2) GA is heuristically a global optimization method, and
as we have seen in this problem, it is efficient in training dynamic neural networks, which are
generally hard to train to an acceptable level of accuracy in a limited time.
The results indicated that the proposed NNARX controller enjoys a relatively simple but
versatile structure. Moreover, it can be readily and quickly tuned with the minimum degree of
richness in the information provided to the tuning algorithm. In this regard, the authors believe
that the proposed method can be implemented in an online manner as well [24, 34]. In other
words, it is possible that the process is started up without a-priori knowledge of the process, but
the controller is tuned as the real-time process is in operation.
The NNARX controller action was not perfect for disturbance rejection for significant modeling
errors. However, we expect that with the introduction of the measured disturbances as
independent inputs to the controller, and also real-time optimization of the controller as it “flies”,
the above-mentioned problems be alleviated.
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