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However, developing precise nonlinear models from first principles may be a
difficult task in many complex processes. Another disadvantage is that the optimizer
solution, in non-linear Predictive Control, is a non-convex problem and a large com-
putational effort may be required to obtain the solution. This is especially relevant
when dealing with real time tasks.
TS Fuzzy models have been applied successfully in non-linear model based tech-
niques (Kiriakidis
1998
; Tatjewski
2007
). As we saw in the previous section, these
models may be formulated as an (ANFIS) (Jang et al.
1997
), allowing use learning
techniques of neural networks in order to obtain the parameters.
Moreover, fuzzy models permit explicit solutions of the optimization (without
restrictions) (Marusak and Tatjewski
2009
; Escaño et al.
2009
; Lu and Tsai
2007
),
with a low computational cost and getting better performance than linear MPC
schemes. It should be remarked that a NMPC procedure based on Fuzzy models
could be implemented on small hardware platforms like PLCs.
Fuzzy control has been applied successfully in many industrial processes (Bai
et al.
2006
), it has requested a special part of the IEC 1131 standard (CENELEC
2013
), which is about industrial PLC. Many groups have been involved in this part,
leading to IEC 1131-7 (CENELEC
2000
).
IEC 1131-7 define a set of function to program fuzzy control applications. This
set of functions is named FCL (
Fuzzy Control Language
). Function blocks defined
on FCL may be used in other languages established by IEC 1131-3.
Within the scope of the norm, programmers have a easy way to implement NMPC.
The problem occurs when, for a good description of the physical system, the number
of rules increases inordinately. Therefore, complexity reduction is an important issue.
A small number of rules, makes controllers suitable to implement into low cost
hardware or medium class PLC, reduces the programming time, decreasing thus the
price of carrying out in Industry.
3.4.1 Rules Reduction for Takagi-Sugeno Systems
After FPCA applied on Sect.
3.3
, we obtained a new subspace of functions
γ(
x
)
=
γ
1
(
)
T
x
)γ
2
(
x
)
···
γ
n
(
x
such that,
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
·
⎡
⎣
⎤
⎦
g
0
(
˜
x
)
h
10
h
20
...
h
R
0
ξ
0
(
x
)
g
1
(
˜
x
)
h
11
h
21
...
h
R
1
ξ
1
(
x
)
˜
(
)
=
g
x
(3.36)
.
.
h
1
n
h
2
n
...
.
ξ
R
(
g
n
(
˜
x
)
h
Rn
x
)
g
˜
(
x
)
=
H
·
ξ(
x
)
(3.37)
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