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3.3. The continuous-time model of finite-time particle swarm optimization
In this subsection, based on the continuous-time model of the PSO algorithm (6), we will first give a continuous-time
FPSO algorithm. Then, we will present some remarks about parameters introduced by the continuous-time FPSO algorithm.
In what follows, we propose a continuous-time FPSO algorithm described by (14), where a nonlinear damping item and a
parameter
c
are introduced into the system (13) such that the system (13) can be stable in finite time and the magnitude of
state oscillation is controlled.
_
y
1
ðtÞ¼y
2
ðtÞ
_
y
2
ðtÞ¼
c
ð1
x
Þy
2
ðtÞþ
a
y
1
ðtÞðÞbsig ð1
x
Þy
2
ðtÞþ
a
y
1
ðtÞðÞ
a
ð14Þ
where 0 < a < 1; b > 0, and 0 <
c
6 1.
It is worth noting that the system (14) becomes the linear system (13), which was studied by Fernández Martínez et al.
[15], if we set b ¼ 0 and
c
¼ 1. Hence, the system (13) can be regarded as a special case of the system (14). The finite-time
convergence of the system states in (14) under the same parameter values
x
,
a
and initial states y
1
ð0Þ; y
2
ð0Þ is elaborated in
Fig. 3.
In order to give the formal expression of the continuous-time FPSO algorithm, let y
1
ðtÞ¼n
i
ðtÞ and y
2
ðtÞ¼
_
n
i
ðtÞ. Conse-
quently, the system (14) can be written as
€
n
i
ðtÞþ
c
ð1
x
Þ
_
n
i
ðtÞ¼
ca
n
i
ðtÞbsig ð1
x
Þ
_
n
i
ðtÞþ
a
n
i
ðtÞ
a
ð15Þ
Further, by setting n
i
ðtÞ¼x
i
ðtÞp
i
ðtÞ, where p
i
ðtÞ¼
a
1
x
l
ðtÞþ
a
2
x
g
ðtÞ
a
1
þ
a
2
, the continuous-time FPSO algorithm can be described by
0 10 20 30 40 50 60 70 80
10
−8
10
−6
10
−4
10
−2
10
0
10
2
Time
Log(|y1|)
0 10 20 30 40 50 60 70 80
10
−6
10
−4
10
−2
10
0
10
2
Time
Log(|y2|)
Fig. 3. The convergence curves of the system states in (14) (x ¼ 0:8; a ¼ 6; a ¼ 0:5; b ¼ 1:9; c ¼ 0:5; y
1
ð0Þ¼5, and y
2
ð0Þ¼9).
116 Q. Lu et al. / Information Sciences 277 (2014) 111–140