a1 xl ðtÞþa2 xg ðtÞ

Remark 5。  p_ i ðtÞ describes  the  velocity  of  the  variable  piðtÞ ¼ a1 þa2 。  Moreover,  the  decision  process  is  a  discrete-time

decision   process,   which   results   in   that   piðtÞ  is   piece-wise   continuous。   Consequently,   the   direction   of   p_ iðtÞ  can   be

approximately calculated from piðk — DkÞ to piðkÞ (k — Dk < t 6 k) and the magnitude of that is decided by the maximum linear velocity of the robot。 Similarly, p€iðtÞ can also be approximately calculated based on p_ iðtÞ and the previous velocity information。

3。4。Convergence  analysis

In this subsection, we will prove the finite-time convergence of the continuous-time model of the FPSO algorithm。 The following Lyapunov analysis provides a global convergence  result。

Theorem 1。  Consider the continuous-time model of the FPSO algorithm (16) with ðx; a; a; c; bÞ 2 Xc  in (18) and a is a constant in

。 1—a 。

the deterministic case。 The continuous-time FPSO algorithm converges over a finite-time interval    0; ð1þaÞVð0Þ1þa

kð1—aÞ

where Vð0Þ

1—a

and k

  1þaÞVð0Þ1þa

can be  calculated according  to (20) and  (25),  respectively, i。e。  xiðtÞ ! piðtÞ and  viðtÞ ! p_ iðtÞ when t ! ð

kð1—aÞ  。

Proof。 Introduce ni ðtÞ ¼ xi ðtÞ— pi ðtÞ into (16), and set y1ðtÞ ¼ ni ðtÞ and y2ðtÞ ¼ n_ i ðtÞ。 As a result, the system (14) can be obtained。   Considering   the   deterministic  case,   we   write   the   system   (14) as

Fig。  4。  The  curve  of  the  average  oscillation  magnitude  for  the  parameter  c for  the  system  state  y2 ðtÞ  in  (14)  (x ¼ 0:8; a ¼ 6; a  ¼ 0:5; b  ¼ 0:1; y1 ð0Þ ¼ 5, and y2 ð0Þ ¼ —9)。

118 Q。 Lu et al。 / Information Sciences  277 (2014)     111–140

Fig。 5。  The curve  of  the  convergence  time  for  the parameter  b for  the  system  state  y2 ðtÞ in (14) (x ¼ 0:8;  a ¼ 6;  a  ¼ 0:5;  c ¼ 1;  y1 ð0Þ ¼ 5, and  y2 ð0Þ ¼   —9)。

。 y_ 1 ðtÞ¼ y2 ðtÞ

y_ 2 ðtÞ¼ —c/m  — bsigð/mÞ

ð19Þ

for  every a 2 ð0; 1Þ,  where /m   ¼ ð1 — xÞy2ðtÞþ ay1 ðtÞ。

If the origin of the system (19) is a finite-time-stable equilibrium, xiðtÞ and vi ðtÞ will reach piðtÞ and p_ iðtÞ in finite time, respectively。   Choose   a   Lyapunov   function   candidate   as