Information Technology Reference
In-Depth Information
Suppose that the projection of the load
m
on
X
−
Y
plane is
M
, its position
(
x
M
,
y
M
)
can be determined by the angle
θ
. The state of the system is defined as
T
T
. The dynamics of the system is: [13]
x
=[
x
1
,
x
2
,
x
3
,
x
4
]
=[
x
M
,
x
M
,
y
M
,
y
M
]
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
x
1
x
2
x
3
x
4
01 0 0
−
λ
x
1
x
2
x
3
x
4
0
⎣
⎦
=
⎣
⎦
⎣
⎦
+
⎣
⎦
2
000
00 0 1
00
2
r
cos
λ
θ
(7)
0
2
2
r
sin
−
λ
0
λ
θ
g
l
is the natural frequency,
g
8[m/s
2
where
is the acceleration of gravity.
It should be noted here that, in this rotary crane model, the control input is not the
torque of the crane, but the angle
λ
=
=
9
.
]
. Angle-command system is realizable when using
a pulse motor [15], the model (7) therefore can be used without the appearance of the
load mass
m
. However, it is necessary to consider the limit of the angular velocity of
the motor in order to avoid the loss of synchronism [15] (i.e.,
θ
˙
˙
θ
max
).
The task in the crane system control is to regulate the load mass, by rotating the
crane jib, from an arbitrary position
x
init
(determined by an initial angle
|
θ
|≤
θ
0
)toadesired
T
without loss of generality. This
rotation movement usually accompanies with the vibration or swinging of the load,
making the control problem more difficult as requiring to suppress the load sway.
position, which can be defined as
x
ref
=[
r
,
0
,
0
,
0
]
4
Performance Test and Control Simulations
4.1
Comparative Performance Test
In order to show that the RNN is a suitable choice, the RNN is compared to a three-layer
FNN. Under the same instances, the criteria for comparing the two networks are their
evolutionary performance and computation time.
i) Evolutionary performance:
The evolutionary performance of the two PSO-based
control systems (using the RNN and FNN) are evaluated by the mean success rate
S
m
,
i.e., the rate of successfully-evolved controllers over the total performed runs, and the
mean (
global best
) error value
E
m
obtained at each iteration. A controller is considered
to be successfully evolved if it can obtain an error smaller than a required accuracy
after running the
Design
by PSO, that is,
E
10
−
4
. The above two factors are
calculated over the replications of varying 20 different initial system positions, which
are random values uniformly distributed in the range
<
E
suc
=
, and 50 times of
randomly changing the initial population of the PSO for each of the initial positions.
In the PSO, the population size is set to be 20 particles and the number of iterations is
100. These values are selected to be not too large with respect to computational cost. As
suggested in [7], the constants in the constricted PSO should be set as
c
1
=
θ
0
∈
[
−
π
,
π
]
c
2
=
2
.
05,
hence
ε
=
4
.
1and
χ
≈
0
.
7298, for assuring the convergence. In our tests, the crane
˙
˙
system parameters are
l
=
2[m],
r
=
1[m], the constraint is
|
θ
|≤
θ
max
=
1
.
0
[
rad
/
s
]
,
and the fourth-order Runge-Kutta method with time step of 0.01 seconds is utilized.
To demonstrate the advantages of the RNN over FNN, we first tune the number of
hidden neurons and the range of PSO initial population (based on trial and error) such
that the FNN achieves the best performance. That is, the number of hidden neurons
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