In order to give the formal expression of the continuous-time FPSO algorithm, let y1 ðtÞ ¼ ni ðtÞ and y2 ðtÞ ¼ n_ i ðtÞ。 Conse- quently, the system (14) can be written as
a
ni ðtÞþ cð1 — xÞni ðtÞ¼ —cani ðtÞ— bsig ð1 — xÞniðtÞþ aniðtÞ
a1 xl ðtÞþa2 xg ðtÞ
ð15Þ
Further, by setting ni ðtÞ ¼ xi ðtÞ— pi ðtÞ, where pi ðtÞ ¼ a1 þa2 , the continuous-time FPSO algorithm can be described by
2
10
0
10
−2
10
−4
10
−6
10
−8
10
0 10 20 30 40 50 60 70 80
Time
2
10
0
10
−2
10
−4
10
−6
10
0 10 20 30 40 50 60 70 80
Time
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; y1 ð0Þ ¼ 5, and y2 ð0Þ ¼ —9)。
€x ðtÞ¼ —c/ ðtÞ— bsigð/ ðt a
where
þ p€i ðtÞ ð16Þ
/iðtÞ¼ ð1 — xÞðx_ i ðtÞ— p_ iðtÞÞ þ aðxi ðtÞ— pi ðtÞÞ
If we consider the dynamics model of the robot (5), we can derive the control input ui ðtÞ as
a
where
ui ðtÞ¼ —c/i ðtÞ— bsigð/iðtÞÞ
þ p€i ðtÞ ð17Þ
/iðtÞ¼ ð1 — xÞðvi ðtÞ— p_ iðtÞÞ þ aðxi ðtÞ— piðtÞÞ
If ðx; a; a; c; bÞ 2 Xc in (18), the continuous-time model of the FPSO algorithm (16) is globally finite-time stable, which will be proved in the next subsection。
Xc ¼ fðx; a; a; c; bÞj x < 1; a > 0; 0 < a < 1; 0 < c 6 1; b > 0g ð18Þ
It is worth mentioning that the parameters c and b are used to control the oscillation magnitude and the convergence speed of the state trajectory of the particle, respectively。 From Fig。 4 and Fig。 5, one can see that increasing the parameter c and b means the decrease of the average oscillation magnitude and the convergence time, respectively。 Hence, the pro- posed continuous-time FPSO algorithm provides a flexible mechanism to control ‘‘frequency’’ and ‘‘magnitude’’ such that we can adjust two parameters to deal with the problem of odor source localization in terms of its characteristics。 气味源定位的有限时间粒子群算法英文文献和中文翻译(7):http://www.youerw.com/fanyi/lunwen_101498.html