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。