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Eq. (
8.5
) such that the finite time horizon includes all transients of the fuzzy control
systems until
I
(ρ)
ρ
reaches the steady-state values, and set the feasible domain of
in Eq. (
8.5
) to include all constraints imposed to the elements of
.
Step 3
. Map the optimization problem from Eq. (
8.5
) onto the adaptive GSAs. The
relation between the objective function and the fitness function
f
, its value
f
j
(
ρ
k
)
of
the adaptive GSAs is
f
j
(
k
)
=
I
(ρ),
j
=
1
...
N
,
(8.13)
where
k
is the current iteration index,
j
is the index of a certain agent, and
N
is the
total number of agents. The relation between and parameter vector
ρ
of TS PI-FLCs
and agents' position vector
X
i
in the adaptive GSAs is
X
i
=
ρ,
i
=
1
...
N
,
(8.14)
where
i
is the index of a certain agent.
Step 4
. Apply the adaptive GSAs that give the optimal parameter vector
ρ
∗
and
the optimal parameters
ρ
∗
=[
ρ
1
ρ
2
ρ
3
]
T
,β
∗
=
ρ
1
,
B
e
=
ρ
2
,η
∗
=
ρ
3
,
(8.15)
and next Eq. (
8.11
).
8.3 Adaptive Gravitational Search Algorithms
The standardGSA functionality is inspired fromNewton's laws of motion and gravity
(Precup et al.
2011c
,
2013a
; David et al.
2012
; Precup et al.
2012b
; Precup et al.
2013b
). The present GSA implementation makes use of two depreciation laws of the
gravitational constant: a linear law
g
(
k
)
=
ψ(
1
−
k
/
k
max
)
g
0
,
(8.16)
and an exponential one
g
(
k
)
=
g
0
exp
(
−
ζ
k
/
k
max
),
(8.17)
where
g
0
0 influence GSA's convergence
and search accuracy, and
k
max
is the maximum number of iterations. For a
q
=
g
(
0
)
, the parameters
ψ>
0 and
ζ>
=
3-dimensional search space, the position of
i
th agent is the vector
x
i
x
i
x
i
T
R
q
X
i
=[
...
...
]
∈
,
i
=
1
...
N
(8.18)
where
x
i
is the position of
i
th agent in
d
th dimension,
d
=
1
...
q
. The acceleration
of
i
th agent in
d
th dimension is
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