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Transactions of the Institute of
Measurement and Control
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DOI: 10.1177/0142331218794811
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Fuzzy adapting rate for a neural
emulator of nonlinear systems: real
application on a chemical process
Fatma Ezzahra Rhili, Asma Atig and Ridha Ben Abdennour
Abstract
This paper deals with a new fuzzy adapting rate for a neural emulator of nonlinear systems with unknown dynamics. This method is based on an online
intelligent adaptation by using a fuzzy supervisor. The satisfactory obtained simulation results are compared with those registered in the case of the
classical choice of adapting rate and show very good emulation performances. An experimental validation of the proposed fuzzy adapting rate on a
chemical reactor is also proposed to confirm the good performances in terms of speed of convergence and precision of representation.
Keywords
Fuzzy supervision, neural emulator, adapting rate, nonlinear systems, chemical reactor
Introduction
The study of real systems, whatever the scientific discipline
concerned (electronics, mechanics, thermal, chemistry, ecol-
ogy, biology, economics, physics, cosmology, etc.), for the
purpose of analysis, prediction, monitoring, control and/or
optimization, generally requires knowledge of a model of the
real system.
In the case of a weakly nonlinear system, a linear or linear-
ized model may be able to describe the system dynamics with
good accuracy (Ben Abdennour et al., 2001). However, in the
case of a relatively strong nonlinearity, the use of nonlinear
models becomes necessary. Neural networks are one possible
solution to the problem of complex systems thanks to their
approximation capabilities (Akhyar and Omatu, 1993;
Narendra and Parthasarathy, 1990; Williams and Zipser,
1989). Recurrent neural networks, which contain an internal
feedback loop, are known as an efficient tool to capture com-
plex, nonlinear system input–output mappings. Indeed, the
development of neural controllers requires efficient adapta-
tion and learning algorithms. The most popular algorithms
used for parameters adaptation are the gradient backpropa-
gation learning algorithm and numerous variants (Budik and
Elhanany, 2006; Williams and Zipser, 1989).
The Back Propagation Through Time (BPTT) and the
Real-Time Recurrent Learning (RTRL) algorithms are suit-
able for recurrent networks. The BPTT is an off-line training
algorithm. It is generally used for feed-forward networks.
Thus, the main drawback of BPTT is that it needs an exten-
sive memory and long training sequences (Werbos, 1990).
RTRL is based on the gradient backpropagation learning
algorithm in real time (Williams, 1990; Williams and Zipser,
1989). Weights are updated online thanks to the gradients of
the network states and outputs with regard to the parameters
(Atiya and Parlos, 2000). The major advantage of this algo-
rithm is its use for real-time applications (Atig et al., 2010a;
Zerkaoui et al., 2010).
In this context, indirect adaptive control schemes, formed
by a neural emulator (NE) and neural controller, were pro-
posed (Ge et al., 2004; Jeon and Lee, 1996; Narendra and
Parthasarathy, 1990; Zerkaoui et al., 2008). These schemes,
which adapt themselves thanks to a real-time recurrent adap-
tation based on the RTRL algorithm, are also successfully
applied in real time to chemical and thermal processes (Atig
et al., 2010b; Yeh et al., 2003; Zerkaoui et al., 2010; Zhang,
2008). The main advantage of these schemes is that they
require no dynamical model of the process, no particular
initialization and no a priori training of the neural networks.
The obtained performances depend on an emulator term
that is chosen arbitrarily. This parameter is used for initiation
and the autonomous evolution of the algorithm starting from
zero initial conditions (Zerkaoui et al., 2009). An intuitive and
bad choice of the emulator term can affect performances. An
important computing load is also observed in the NE.
To get rid of this starting parameter, Bahri et al. (2012)
proposed the use of a multimodel emulator. This emulator is
obtained using a base of models that is the result of an offline
identification procedure for nonlinear systems based on a
decoupled multimodel approach. The main advantage of the
Research Unit of Numerical Control of Industrial Processes, National
Engineering School of Gabes, University of Gabes, Tunisia
Corresponding author:
Fatma Ezzahra Rhili, Research Unit of Numerical Control of Industrial
Processes,National Engineering School of Gabes, University of Gabes,
Tunisia.
Email: fatmarhili@gmail.com
multimodel emulator is a reduction in the equations of the
NE to a single expression of the output variation with respect
to the input. The main drawback of the multimodel emulator
is the selection of the number and the size of the local models.
To solve these problems, a new fuzzy adapting rate
approach can be used for NE. This approach is based on
online intelligent adaptation by using a fuzzy supervisor.
Fuzzy logic starts with a set of user-supplied human language
rules (Zadeh, 1965). The fuzzy systems convert these rules
into their mathematical equivalents. This simplifies the job of
the system designer and the computer, and results in much
more accurate representations of the way systems behave in
the real world. Additional benefits of fuzzy logic include its
simplicity and its flexibility. Fuzzy logic can handle problems
with imprecise and incomplete data, and it can model non-
linear functions of arbitrary complexity.
In this paper, we improve the performances of the neural
emulation thanks to the use of the fuzzy adapting rate
approach. The contributions of this work are as follows. (1)
The development of a new adaptation strategy based on fuzzy
rules. Performances of the proposed fuzzy adapting rate for
NE are evaluated and compared with the classical ones. (2)
The efficiency of the developed approach is validated with a
chemical reactor.
This rest of this paper is organized as follows. In the next
section, NE is developed for nonlinear systems. Fuzzy adapt-
ing rate for NE and its application are then cited. A simula-
tion example is also proposed, illustrating the performances
of the NE based on the fuzzy adapting rate compared with
the case of classical choice of adapting rate and the efficiency
of the proposed method is given. An experimental validation
of the proposed method to chemical reactor is then presented.
We end with our conclusions.
Neural emulator for nonlinear systems
Neural emulator
The NE was developed with fully connected recurrent neural
networks (Figure 1). Let us define NIN and NOUT as the inputs
and outputs, respectively, where IN and OUT denote the set
of inputs and outputs. We consider square systems
NIN =NOUT =N, where IN =1,...,Nand OUT =1,...,N.
The total number of neurons Neof NE is chosen equal to 2N.
To avoid perturbing any output signal with input ones, any
node is either an input or an output neuron but not both at
the same time.
In discrete time, the dynamics of the Neneurons of NE are
given by the following equation (Atig et al., 2010b; Bahri
et al., 2012; Zerkaoui et al., 2008):
sik+1ðÞ=ete
jjDTsikðÞ+1ete
jjDT
tanh X
Ne
j=1
wij kðÞsjkðÞ+xikðÞ
!
ð1Þ
where sikðÞ,wij kðÞ,1=tekðÞ
jj
and DTrepresent the ith neuron
state, the weights from the jth neuron to ith neuron, the NE
adaptive time parameter and the sampling period, respectively
(Druaux et al., 2004).
Here xikðÞ=uikðÞ if i21,...,N
fg
and xikðÞ=0if
i2N+1,...,2N
fg
, where uikðÞare the system inputs and
^
yikðÞ=si+NkðÞif i2N+1,...,2N
fg
, where ^
yikðÞrepresent
the estimation of the nonlinear system outputs.
In this work, we are interested only in emulating the
instantaneous outputs because the adaptive algorithm does
not memorize the dynamics of the system. Therefore, the
number of inputs and outputs is the only factor that affects
the size of such a network (Atig et al., 2010b; Leclercq et al.,
2005). The inputs and outputs signals for NE are normalized
in the range ½1,1. The adaptation of NE parameters and
weights are based on the RTRL algorithm. The major advan-
tage of this method is that it does not depend on any prelimi-
nary knowledge of the dynamics of the system (Williams,
1990).
In discrete time, the square outputs estimation error is
written as
eekðÞ=1
2X
N
l=1
(^ylkðÞylkðÞ)2ð2Þ
where ^
ylkðÞand ylkðÞrepresent the NE and the real nonlinear
system outputs, respectively.
The adaptation of the NE weights matrix is based on the
gradient of the instantaneous error given by
Dwij(k)=hekðÞ
jj
DT∂eek1ðÞ
∂wij
ð3Þ
where
∂eek1ðÞ
∂wij
=X
N
l=1
^
ylk1ðÞylk1ðÞðÞ
∂^
ylk1ðÞ
∂wij
where hekðÞis the NE adapting rate.
Let us define Plij =∂^
yl
∂wij as the network sensitivity function
updating according to the following equation:
Figure 1. Neural emulator with fully connected structure.
2Transactions of the Institute of Measurement and Control 00(0)
Plij k+1ðÞ=etekðÞjjDTPlij kðÞ+1etekðÞjjDT
3tanh0X
Ne
m=1
wlm kðÞsmkðÞ+xlkðÞ
! !
dl
isjkðÞ+X
Ne
m=1
wlm kðÞPmij kðÞ
!
ð4Þ
where dl
iis the Kronecker symbol.
To obtain an emulator able to represent the unknown
dynamics of different systems with zero initial conditions, the
adaptation of the NE parameters hekðÞand tekðÞis required.
Based on the same method used for NE weights matrix adap-
tation, an algorithm RTRL is used to adapt the parameters
he(k)andte(k) (Atig et al., 2010b; Bahri et al., 2012; Zerkaoui
et al., 2008):
Dhe(k)=DT∂eek1ðÞ
∂he
ð5Þ
and
Dte(k)=he(k)
jj
DT∂eek1ðÞ
∂te
ð6Þ
Starting from zero initial conditions, the functions
∂ee=∂heand ∂ee=∂teevolve in the same way. Consequently,
we obtain
∂eek1ðÞ
∂he
=∂eek1ðÞ
∂te
ð7Þ
These variations are due to small perturbations ∂heand ∂teof
the parameters heand te, respectively (Leclercq et al., 2005).
Using the equations (5) and (6), the adaptation of the para-
meter te(k) is given by the following formula:
tekðÞ=tek1ðÞ+hekðÞ
jj
DhekðÞ ð8Þ
Therefore, the NE adapting rate he(k) is the only factor that
affects the variation of the parameter te(k).
Numerical example
We consider the nonlinear single-input single-output (SISO)
system described by the following discrete equation to evalu-
ate the effectiveness of the NE (Atig et al., 2010a; Druaux
et al., 2004; Rhili et al., 2018):
ykðÞ=0:4u(k1)+0:12y(k1)
+0:4u(k1)y(k2)(u(k1)+2:5)
1+u2(k1)+y2(k2)
ð9Þ
The retained input is given by
ukðÞ=0:8sin 2p
250 k
ð10Þ
To evaluate emulator performances, two indic3es are used:
the mean squared error (MSE) and the variance accounted
for (VAF), given by (Orjuela, 2008)
MSE =1
NmX
Nm
k=1
^
ykðÞykðÞðÞ
2ð11Þ
VAF =max 1var YkðÞ
^
YkðÞ
var ^
YkðÞ
,0
()
3100%ð12Þ
where ^
YkðÞ=^
y1ðÞ,^
y2ðÞ,...,^
yN
m
ðÞðÞand YkðÞ=y1ðÞ,ð
y2ðÞ,...,yN
m
ðÞÞrepresent the NE and the nonlinear system
outputs vectors, respectively, with Nmthe number of
measurements.
According to the dynamics of the system, a sampling
period equal to 0.1 s is retained. The choice of the constant
value of the adapting rate hecleads to bad results in terms of
emulation performances. For example, the choice of hec=2
can affect the emulation results (Figure 2). This figure shows
that NE adapts itself to the variation of the plant output with
a relatively important emulation error. Figure 3 confirms that
performances are affected by using hec=2.
After several simulations of neural emulation, an optimal
value of hecis obtained. This optimum choice (hec=10) gen-
erates good neural emulation performances. We obtain the
results illustrated by Figure 4. The NE provides a satisfactory
estimation of the nonlinear system output. The emulation
error illustrated by Figure 5 confirms this result.
The two indices MSE and VAF are computed for some
values of hecin Table 1. We note that a good choice of the NE
adapting rate heccan improve the emulation performances.
Figure 2. Evolution of nonlinear system and NE outputs (hec=2).
Figure 3. Emulation error eekðÞon a logarithmic scale (hec=2).
Rhili et al. 3
Despite the fact that the emulator generates a major disad-
vantage that is summed up in the determination of its unique
parameter. Some simulations are necessary to obtain an ade-
quate value of the NE adapting rate making a compromise
between speed and accuracy. The search for an intelligent
method to solve this problem will be the objective of the next
section.
Fuzzy adapting rate for neural
emulator of nonlinear systems
The choice of the latter parameter is arbitrary and at the basis
of a compromise between the speed of convergence and the
precision of representation. To make an intelligent and online
selection of the adaptation factor, the use of ‘fuzzy logic’ is
required.
The methodology of a fuzzy supervision is of increasing
interest (Zadeh, 1965). It is considered an easy, effective and
simple way to solve identification problems and modelling
from ambiguous, unclear and wide conditions (Bayrak et al.,
2017; Ben Abdennour et al., 2001; Jnos, 2004). Many combi-
nations of ‘fuzzy logic’ with neural networks have been pro-
posed in the literature (Armendariz et al., 2012; Chen and
Richardson, 2012; Cheng, 2014; Jia et al., 2013; Tavakoli
et al., 2016; Treesatayapun, 2008). Then, a fuzzy supervisor is
considered (Messaoud et al., 2007; Mihoub et al., 2011).
In this paper, we propose to exploit this method to super-
vise the NE adapting rate. The performance indices e(k)and
De(k) are retained for the supervisor inputs. The output of the
proposed supervisor is the variation of the fuzzy adapting rate
DhefkðÞ, where e(k)andDe(k) represent the mean value of the
emulation error and its variance, respectively, and
ekðÞ=^
ykðÞykðÞ ð13Þ
DekðÞ=ekðÞek1ðÞ
jj
ð14Þ
The mean values e(k)andD
e(k) are calculated recursively on
a sliding window Nw(Ben Abdennour et al., 1998). If Sis the
mean value of a signal S,wehave
S(k)=
S(k1)+ 1
Nw
SkðÞSkNw
ðÞ½ð15Þ
The supervisor is illustrated by Figure 6. Each supervisor
input can be represented by three membership functions. The
supervisor procedure contains three stages: the fuzzyfication,
the fuzzy inference and the defuzzyfication (Ben Abdennour
et al., 1998; Ksouri-Lahmari, 1999).
Fuzzyfication
The fuzzyfication consists to convert the normalized real
inputs (
en(k), D
en(k)) to fuzzy inputs. Indeed, to each normal-
ized input of the supervisor, we associate nefuzzy sets
described by triangular membership functions centred on the
numbers Ncep.
The performance index e(k) is normalized in the interval
½1,1and we have
Figure 4. Evolution of nonlinear system and NE outputs (hec=10).
Figure 5. Emulation error eekðÞon a logarithmic scale (hec=10).
Table 1. The indices MSE and VAF calculated for some values of hec.
Neural emulator
hec=0:5hec=1hec=6hec=10 hec=40
MSE 0.15 4:731021:81038:64042:203
VAF 0 67.29% 98.84% 99.43% 98.46%
Figure 6. The fuzzy supervisor scheme.
4Transactions of the Institute of Measurement and Control 00(0)
enk
ðÞ
=
ekðÞ
emax kðÞ+
emin kðÞ
2
emax kðÞemin kðÞ
2
ð16Þ
where
Ncep=1+p1
ne1
2
,1pne
The variation of the index performance De(k) is normalized in
the interval ½0,1and we have
D
enkðÞ=D
ekðÞD
emin kðÞ
D
emax kðÞD
emin kðÞ ð17Þ
where
Nceq=q1
ne1,1qne
with
emax and
emin are the maximum and the minimum values
of the mean value of the error, respectively, and D
emax and
D
emin represent the maximum and the minimum values of the
mean value of the variation of the error, respectively.
Three fuzzy sets (EFp,EFq) have been reserved for each
supervision input (ne=3) in the present case.
Fuzzy inference
The fuzzy inference phase consists of applying a set of linguis-
tic rules, provided by an a priori elaborated rules basis, to the
fuzzy inputs to evaluate the supervisor output DhefkðÞ. The
elaboration of the rule basis which manage the online supervi-
sion of the validities requires a phase of experimentation.
The evolution of the real and estimate outputs are illu-
strated by Figure 7. The evolution of mean value of error and
its variation are plotted in Figures 8 and 9.
According to the evolution of the performance indices
e(k)
and D
e(k), we can deduce the decision table of Dhefk
ðÞ
given
by Table 2.
The inference table associates nsfuzzy sets (SF1,...,SFns)
to the supervisor output ns=5ðÞ, which are also designated
by triangular membership functions. These functions are spec-
ified in the interval ½1,1and centred on the number Ncsr:
Ncsr=1+r1
ns1
2
ð18Þ
with 1rns:
The MIN–MAX inference method is used for the evalua-
tion of the fuzzy rules contribution (Flaus, 1994).
Consequently, the supervisor’s output is established by the
calculation of the coefficients Dr(k) given by the following
formula:
DrkðÞ=MAX
Rule r MIN
EFp,EFq!SFr
mEFp
enkðÞðÞ,mEFq
DenkðÞðÞ
hi
ð19Þ
where mEFp
enkðÞðÞand mEFqD
enkðÞðÞdenote the membership
degrees of enkðÞto fuzzy set EFpand DenkðÞto fuzzy set EFq,
respectively, with p,q=1,...,neand r=1,...,ns.
In fact, the terms Dr(k) correspond to the heights of the
trapezes obtained by levelling of the output triangular mem-
bership functions.
Defuzzyfication
A large number of techniques for implementing the defuzzyfi-
cation procedure are available nowadays (gravity centre, basic
defuzzyfication, bisector of area and adaptive integration,
etc.). Uzˇ ga-Rebrovs and Kulxes
ˇova (2017) presented a com-
parative analysis of some widespread methods of fuzzy set
defuzzyfication. In the case of symmetric membership func-
tions of the output linguistic variables, the most appropriate
method is the gravity centre. Alternatively, if relevant mem-
bership functions are not symmetric, the weighted average
Figure 7. Evolution of real and NE outputs for some values of hec.Figure 8. Evolution of the performance index
e(k)for some values of
hec.
Figure 9. Evolution of the variation of the performance index D
e(k)for
some values of hec.
Rhili et al. 5
method is an appropriate alternative. In our case, the gravity
centre method is retained. The numerical value of the supervi-
sor output DhefkðÞcan be calculated using the formula given
by (Ben Abdennour et al., 2001; Bu
¨hler, 1994; Tong, 1995)
DhefkðÞ=Pns
r=1DrkðÞNcsr
Pns
r=1Drk
ðÞ ð20Þ
The effective output hefkðÞcan be calculated, in each iteration
k, as follows:
hefkðÞ=hefk1ðÞ+Mo3DhefkðÞ ð21Þ
where Mois a positive weighting. This weighting affects only
the rapidity of the convergence to an optimum value of the
adapting rate.
Simulation example
The application of the proposed method on the nonlinear sys-
tem, given by Equation (9), is illustrated. Figure 10 gives the
evolution of the input signal. To get rid of problems caused
by the choice of the starting term eeand the constant value of
NE adaptive rate hec, a NE based on the fuzzy adapting rate
is considered. A comparative study with the NE based on a
constant value of adaptive rate (Rhili et al., 2018), given by
Figures 11 and 12, shows the contribution in precision of the
new proposed method. We remark that the NE output, based
on the fuzzy adapting rate, follows the real output with rela-
tive precision. To prove the efficiency of the developed strat-
egy, results are compared with those based on the starting
term (Atig et al., 2010a), given by Figure 13. The evolution of
the fuzzy NE adapting rate hefkðÞis presented in Figure 14.
These results demonstrate the efficiency of the proposed
method based on a fuzzy supervisor, in terms of emulation
performances with regards to the case of classical choice of
adapting rate hecand to the case of classical choice of starting
term ee.
Table 3 summarizes the MSE and VAF performances
indices calculated in neural emulation case for both arbitrary
choice of hecand fuzzy adapting rate hefkðÞ. In Table 3, we
note that emulation performances obtained using a fuzzy
adapting rate of the emulator are better than those obtained
for the case of classical choice of adapting rate.
Performances of the proposed approach are evaluated by
computing the MSE and VAF for the classical emulator based
on the starting term ee(Table 4).
Table 2. Inference table for the fuzzy supervisor (S, small; M, medium;
L, large; N, negative; Z, zero; P, positive; NL, negative large; NS, negative
small; PS, positive small; PL, positive large).
en(k)
DhefkðÞ NZP
D
en(k)SPLZPL
MPSZNS
LNLZNL
Figure 10. Input signal.
Figure 11. The evolution of real and NE outputs (cases of hec=1 and
fuzzy adapting rate hefkðÞ).
Figure 12. The evolution of real and NE outputs (cases of hec=75
and fuzzy adapting rate hefkðÞ).
Figure 13. The evolution of real and NE outputs (cases of ee=0:1,
ee=10 and fuzzy adapting rate hefkðÞ).
6Transactions of the Institute of Measurement and Control 00(0)
Application to a chemical reactor
Process description
The chemical process considered in this application is used to
esterify olive oil (Figure 15). The reaction carried out in this
reactor is a chemical esterification of the crude acid of olive
oil by an alcohol, such as butane, to extract a high-quality
ester. The produced ester is widely used for the manufacture
of cosmetic products (Messaoud et al., 2009). The esterifica-
tion reaction is given as follows:
Acid + Alcohol $Ester + Water
The ester’s proportion can be increased by vaporization of
water. We set the acid and the ester ebullition temperatures as
approximately 3008C. The butane is characterized by an ebul-
lition temperature of 1188C. Consequently, the temperature of
the reactor is over 1008C to get rid of water only (Mihoub
et al., 2009). The process temperature is regulated by means
of a fluid circulation through the reactor jacket. This fluid is
heated by three resistors whose electric power can be varied
from 0 to 2500 W and located in the heat exchanger. It is also
cooled in a tubular cooler whose cooling rate is changed by
varying the external water (Messaoud et al., 2009).
The variables Tede and Tsde are the input and output tem-
perature of the double envelope, respectively, Tris the reac-
tor’s temperature and Qis the heating power. Then,
temperature must follow a specific trajectory consisting of
three stages.
(i) Heating stage: the reactor’s temperature Tris
increased to 1058C.
(ii) Reaction stage: the reactor’s temperature Tris main-
tained constant during the reaction (when no more
water is dripping out of the condenser).
(iii) Cooling stage: the reactor’s temperature is decreased.
Reactor emulation
Experimental studies show that the process is nonlinear
(M’sahli et al., 2001). The reactor is considered as a SISO
system. The output is the reactor’s temperature Tr.The
input is the heating power Q. Then, a two-neuron neural
network is capable of emulating the chemical reactor input–
output mapping. The input and the output number is the
only one that can affect the number of neurons. The pro-
posed network does not learn the plant dynamics, but only
adapts its parameters.
The process sampling time DTis chosen equal to 3 min
according to the step response of the system. A pseudo-
random binary input signal (PRBS) is applied to the real sys-
tem (Figure 16). The amplitudes of the signal are chosen so
that they focus on the three reaction stages (Messaoud et al.,
2009; Mihoub et al., 2009).
Results of the proposed fuzzy adapting rate for NE of the
reactor temperature are given by Figures 17 and 18. The pro-
posed emulator using a fuzzy supervision of the NE adapting
rate (continuous line) provides a satisfactory estimation of
the process output with regard to the classical NE using a
constant value of the adapting rate (dotted line). Figure 19
illustrates the estimation error. The evolution of the output
error proves the efficiency of the proposed method emulation.
The evolution of the fuzzy adapting rate hefkðÞ, in this case,
is given by Figure 20. This last figure illustrates a satisfactory
adaptation of the NE adapting rate hefkðÞ.
Figure 14. Evolution of the fuzzy adapting rate hefkðÞ.
Table 3. MSE and VAF emulation performances indices for fuzzy
adapting rate hefkðÞand arbitrary choice of hec.
Fuzzy adapting rate hefkðÞ Arbitrary choice of hec
MSE 4:21044:3103(hec=1)
5:05104(hec=75)
VAF 96.14% 69.82% (hec=1)
95.31% (hec=75)
Table 4. MSE and VAF emulation performance indices for fuzzy adapting
rate hefkðÞand arbitrary choice of ee.
Fuzzy adapting rate hefkðÞ Arbitrary choice of ee
MSE 4:21046:7103(ee=0:1)
1:1103(ee=10)
VAF 96.14% 52.09% (ee=0:1)
91.74% (ee=10)
Figure 15. Synoptic scheme of the reactor.
Rhili et al. 7
The obtained results prove, once again, the efficiency of
the fuzzy supervision of adapting rate in the case of a real-
time emulation of a chemical reactor.
Conclusion
A fuzzy adapting rate for NE of nonlinear systems has been
proposed in this paper and compared with the classical choice
of adapting rate for an emulator. The main advantage in using
a proposed fuzzy adapting rate is to avoid the effort required
for searching an optimal choice of NE adapting rate. This
type of fuzzy supervisor does not require any initialization
parameter. Consequently, we have solved the problem related
to the choice of the NE adapting rate. The simulation results
demonstrate the efficiency of the proposed online fuzzy adapt-
ing rate. A real-time emulation of a chemical reactor based on
the proposed fuzzy adapting rate has been realized to confirm
good precision and rapidity of the real output prediction. In
our future work, an extension of the developed fuzzy adapting
rate for multivariable nonlinear systems will be studied. An
adaptive neural control scheme based on the NE structure will
also attract our interest.
Declaration of conflicting interests
The authors declare that there is no conflict of interest.
Funding
This research received no specific grant from any funding
agency in the public, commercial, or not-for-profit sectors.
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Figure 16. Input signal.
Figure 17. Real and estimated temperatures (cases of hec=1 (dotted
line) and of the fuzzy adapting rate hefkðÞ(continuous line)).
Figure 18. Real and estimated temperatures (cases of hec=50
(dotted line) and of the fuzzy adapting rate hefkðÞ(continuous line)).
Figure 19. Evolution of the emulation error eekðÞin logarithmic scale
(cases hec=1 (dashed line), hec=50 (dotted line) and the fuzzy
adapting rate hefkðÞ(continuous line)).
Figure 20. Evolution of the fuzzy adapting rate hefkðÞ.
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