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Input
layer
Hidden
layer
Output
layer
x
1
ref
- x
1
u
1
Limiter
z
-1
z
-1
z
-1
PSO
x
ref
+
y
u
RNN
Plant
x
I
ref
- x
I
u
O
−
O
I
H
^
Estimation (PF)
z
-1
z
-1
z
-1
Fig. 1.
The proposed control system.
Fig. 2.
Recurrent neurocontroller.
Fig. 2 shows the structure of the RNN used herein. The network consists of three
layers: the input layer with
I
neurons, the hidden layer with
H
neurons, and the output
layer with
O
neurons (i.e., it is a
I
O
structured RNN). The input layer contains
only feedforward connections and there is no recurrent/feedback loop in its own neurons
or from/to other layers. In contrast, the hidden and output layers are fully recurrent.
The PSO and PF are employed in two following operators:
−
H
−
i) Design:
Used to design a feedback controller in offline manner. That is, a PSO
is utilized to evolve the RNN controller. A sum-squared error is used as the evaluation
function for the PSO, which is:
i
=
1
Q
i
x
ref
2
m
=
1
Q
m
u
ref
u
en
m
2
I
O
x
end
i
E
=
−
+
−
(3)
i
m
end
(
.
)
where
Q
(
.
)
are the weight coefficients,
represents the values at the final state (at
ref
the end of control simulation), and
(
.
)
represents the desired values. In most cases,
such as in this paper,
u
ref
0.
In this study we utilize the constricted version of PSO proposed in [7]. After
finding the
local best PB
n
=[
=
T
pb
n
1
,
pb
n
2
, ···,
pb
nD
]
and the
global best gb
=
T
[
gb
1
,
gb
2
, ···,
gb
D
]
in the
j
-th iteration, the
n
-th particle updates its velocity
V
n
=
T
T
[
v
n
1
,
v
n
2
, ···,
v
nD
]
and position
W
n
=[
w
n
1
,
w
n
2
, ···,
w
nD
]
in the next iteration
(
j
+
1
)
with the following equations:
v
j
c
1
r
1
pb
j
nd
c
2
r
2
gb
j
nd
v
j
+
1
nd
w
j
w
j
=
χ
nd
+
nd
−
+
d
−
(4)
w
j
+
1
nd
w
nd
+
v
j
+
1
nd
=
(5)
where
d
=
1
,
2
, ···,
D
is the dimension number;
n
=
1
,
2
, ···,
N
(
N
is the swarm popula-
tion size);
j
J
is the iteration number;
c
1
and
c
2
are positive constants;
r
1
and
r
2
are random values in the range
=
1
,
2
, ···,
[
0
,
1
]
;and
χ
is the constriction coefficient defined as:
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