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optimization ability of the PSO algorithm. The velocity of each particle in the swarm
is updated using the following equation [3]:
(
+
)=
(
)+
[
(
)
−
(
)] +
[
(
)
−
(
)]
v
i
t
1
wv
i
t
c
1
r
1
x
i
t
x
i
t
c
2
r
2
g
i
t
x
i
t
(9)
The index of the particle is represented by
i
. Thus,
v
i
(
t
)
is the velocity of particle
i
at time
t
and
x
i
(
is the position of particle
i
at time
t
. The parameters
w
,
c
1
,and
c
2
are user-supplied coefficients. The values
r
1
and
r
2
are random values regenerated
for each velocity update. The value
x
i
(
t
)
t
)
is the individual best candidate solution
for particle i at time t, and
g
(
t
)
is the swarms global best candidate solution at time
t
.Theterm
c
1
r
1
[
, called the
cognitive component
, acts as the particles
memory, causing it to tend to return to the regions of the search space in which it has
experienced high individual fitness and the term
c
2
r
2
[
x
i
(
t
)
−
x
i
(
t
)]
, called the
social
component
, causes the particle to move to the best region the swarm has found so far.
Once the velocity for each particle is calculated, each particles position is updated
by applying the new velocity to the particles previous position [3]:
g
i
(
t
)
−
x
i
(
t
)]
x
i
(
t
+
1
)=
x
i
(
t
)+
v
i
(
t
+
1
)
(10)
This process is repeated until some stopping condition is satisfied.
5.2.2
Particle Structure
As mentioned before, the goal is to find proper weights for stress generation, so
proposed particle structure is a vector of six components including
w
1
to
w
6
form
equation 8.
6
Simulation and Results
The proposed controller has been compared with the original basic controller, which
is a double PID. Note that without employing imitation phase, simulated BELBIC
controller did not learn the proper control signal in a reasonable time and the pen-
dulum fell down. Also a random voltage produced by a Gaussain distribution has
been applied to the test to evaluate the controller robustness in the presence of dis-
turbance. At the first step of disturbance evaluation, we applied the declared dis-
turbance in 5 times after imitation phase, and in the second step the disturbance
produced by new settings has been applied to the desired cart position whole train-
ing duration. Note that in optimization phase the parameters
w
,
c
1
,and
c
2
are set as
0.5, 1.5 and 1.5 respectively.
Fig. 4 and Fig. 5 show the result of controller in above evaluation tests respec-
tively. In both of them the imitation phase is in first 28.4 seconds. As it can be seen
the BELBIC can imitate the behavior of basic controller in a reasonable time and
the optimization phase is started in this time.
As mentioned before, in the first test we apply the disturbance with zero mean
and 0.2 of variance in some randomly selected times. In this step, the better solution
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