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promising procedure. This tuning can be formulated as the following constrained
optimization problem:
8
<
minimize
fx
ðÞ
T
4
þ
x
¼
ð
K
e
;
K
d
;a;b
Þ
2S
R
subject
to
:
ð
1
Þ
max
g
1
x
ðÞ¼d d
0
:
ðÞ¼
t
s
t
max
s
g
2
x
0
ðÞ¼
E
ss
E
max
ss
g
3
x
0
x
up
the initial search
space, which is supposed containing the desired design parameters, and gl
l
4
4
þ
;
where f
:
R
!
R
the cost function,
S¼
x
2
R
x
low
x
4
:
R
!
the nonlinear problem
s constraints.
The optimization-based tuning problem (
1
) consists in
'
R
finding the optimal
decision variables, representing the scaling factors of a given PID-type FLC
structure, which minimizes the de
ned cost function, chosen as the Maximum
Overshoot (MO) and the Integral of Square Error (ISE) performance criteria. These
cost functions are minimized, using the proposed particular constrained metaheu-
ristics, under various time-domain control constraints such as overshoot
d
, steady
'
state error E
ss
, rise time t
r
and settling time t
s
of the system
s step response, as
shown in Eq. (
1
). Their speci
ed maximum values constrain the step response of
the tuned PID-type fuzzy controlled system, and can de
ne some time-domain
templates.
2.3 Proposed Constraints Handling Method
The considered metaheuristics in this study are originally formulated as an
unconstrained optimizer. Several techniques have been proposed to deal with
constraints. One useful approach is by augmenting the cost function of problem (
1
)
with penalties proportional to the degree of constraint infeasibility. This approach
leads to convert the constrained optimization problem into the unconstrained
optimization problem. In this paper, the following external static penalty technique
is used:
h
i
X
n
con
l
¼
1
k
l
max 0
2
u
ðÞ¼
ðÞþ
;
ðÞ
ð
Þ
x
fx
g
l
x
2
k
l
is a prescribed scaling penalty parameter, and n
con
is the number of
problem constraints gl
l
x
where
ðÞ
.
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