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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
= ρ 1 ,
B e
= ρ 2 = 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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