Min f(x) = {Rt(x), St(x), Os(x), Sse(x)}
s.t. a x ∈ D
D = { x ∈ R
n
}
(3)
Where x are membership functions and conclusion
table parameters.
Individuals of population in objective space have
its particular Rank (R) and crowding distance (cd).
Rank is equal to one if individual belongs to Pareto
Frontier (PF), later those are removed and the
sequential individuals continue whit rank two and so
on, this process discriminate several local PF. Rank
assignment is done by PF definition (Augusto et al,
2006), consider two solutions vectors x and y, x is
contained in the PF if.
⎪
⎩
⎪
⎨
⎧
<∈∃
≤∈∀
)()(:,...,2,1
)()(:,...,2,1
yfxfkj
and
yfxfki
ij
ii
(4)
In the case of (4) x dominates y in the R
k
objective
space and have Rank one.
Crowding distance is the distance between one
individual and two near it in the same PF (see eq. 5).
()
∑∑
=
=
=
=
−=
mp
p
nc
c
cicpi
XXcd
11
2
(5)
Where c is an objective space axis and n are the
number of the objectives; p is a particular point and
m are the total points in the same Pareto Frontier; i is
the individual.
Binary selection is carry out and tournament is
done first by Rank. individuals with minor Rank are
preferred, if both have equal R, cd is taken into
account, mayor cd wins the tournament to preserve
population diversity, two individuals are then
selected by this process for crossover and mutation.
Simulated binary crossover (Deb & Agrawal,
1995) makes information interchange, and to avoid
premature convergence polynomial mutation works
well (see eq. 6).
k
l
k
u
kkk
pppc
δ
−+=
(6)
where k is the vector k-component, c is the child, p
the parent δ an uniform random number u and l are
the upper and lower bounds in the search space.
New and old population are joined and selected via
tournament to conform the new generation, and then
survivors could appear. The process is repeated until
reach the maximum number of iterations.
In a previous work, population of the Initial
Individuals where created with restrictions in
membership functions (Reyes et al, 2008) in hope of
avoid overlapping or empty space but no restrictions
where imposed while NSGAII was running, thus
membership functions at the end shown empty space
in discourse universe, overlapping or both mixed
cases (Fig. 2,3).
4 PH REACTOR
The equations for the pH dynamic were developed
in (McAvoy et al, 1975). The main issue is to keep
the process around the neutral point, where the
system is very sensitive and highly non linear, then
pH control is regarded as a benchmark problem,
especially when the reference signal change from
pH=7 to a mayor value nearby. The interested reader
can easily verify this fact by the construction of the
neutralization or titration curve (TC). An
experimental method to obtain the TC is based on
holding the base concentration constant, slowly
adding the acid and then plotting the pH versus the
acid concentration. Three operating zones are
commonly considered: low, medium, high (see Fig
1).
pH is usually controlled by the mixture of two
solutions with different concentrations, one basic
and other acid. In this work, we validated our
SIMULINK® model by comparing the resulting TC
with the one presented in (Zhang, 2001).
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
4
5
6
7
8
9
10
11
12
13
acid flow cc/m in
pH
-:C2= 0.05 m oles/l; --:C2 = 0.04 m oles/l; ..:C 2 = 0.06 m oles/l
Figure 1: Titration curve, zones low, medium, high, pH
approximately 0~6, 6~11.5, 11.5~14, respectively.
The neutralization process takes place within a
Continuous Stirred Tank Reactor (CSTR). There are
two flows to the CSTR. One is acetic acid of
concentration C
1
at flow rate F
1
, and the other is
sodium hydroxide of concentration C
2
at flow rate
F
2
.
The mathematical equations of the CSTR are
shown in eq’s 7-12.
Table 2 shows the parameters and model variables.
ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics
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