Information Technology Reference
In-Depth Information
From Eq. (
8.14
) it can be seen that g.x
i
x
j
/ Dg.x
j
x
i
/, i.e. g./ is an odd
function. Therefore, it holds that
0
1
M
X
1
M
@
A
D 0
g.x
i
x
j
/
(8.21)
jD1;j¤i
X
M
1
M
x D
Œr
x
i
V
i
.x
i
/
(8.22)
iD1
Denoting the goal position by x
, and the distance between the i-th particle and
the mean position of the multi-particle system by e
i
.t/ D x
i
.t/ x the objective
of the learning algorithm for can be summarized as follows: (i) lim
t!1
x D x
,
i.e. the center of the multi-particle system converges to the goal position, (ii)
lim
t!1
x
i
Dx, i.e. the i-th particle converges to the center of the multi-particle
system, (iii) lim
t!1
x D 0, i.e. the center of the multi-particle system stabilizes at
the goal position. If conditions (i) and (ii) hold then, lim
t!1
x
i
D x
. Furthermore,
if condition (iii) also holds, then all particle will stabilize close to the goal position
[
64
,
108
,
150
,
152
].
To prove the stability of the multi-particle system the following simple Lyapunov
function is considered for each particle:
1
1
2
jje
i
jj
2
e
i
T
e
i
2
V
i
D
) V
i
D
(8.23)
Thus, one gets
D e
i
T
V
i
) V
i
x/e
i
e
i
D .x
i
)
h
M
P
jD1
r
x
j
V
j
.x
j
/
i
e
i
:
r
x
i
V
i
.x
i
/
P
jD1;j¤i
g.x
i
V
i
x
j
/ C
1
D
Substituting g.x
i
x
j
/ from Eq. (
8.14
) yields
2
M
X
V
i
D
4
r
x
i
V
i
.x
i
/
.x
i
x
j
/a
jD1;j¤i
3
M
M
X
X
jD1
r
x
j
V
j
.x
j
/
1
M
5
e
i
.x
i
x
j
/g
r
.jjx
i
x
j
C
jj/ C
jD1;j¤i
which gives
