information to predict a probable position with the optimal fitness, i。e。 pi ðkÞ, and then employs the ‘‘P’’ controller ui ðkÞ to enable the particle to move toward the position pi ðkÞ。 The PSO algorithm provides the decision pi ðkÞ and the controller ui ðkÞ。  The corresponding control block diagram is shown in Fig。 1 where ðvi ðkÞ; xi ðkÞÞ is the state of the ith particle and

xi ðkÞ is the output of the ith    particle。

From the perspective of the ‘‘decision-control mechanism’’, the study of PSO algorithms consists of two categories: optimi- zation performance improvement [18,44,45,39,47,6,42,32] and stability analysis [8,20,14]。 For optimization performance improvement, how to design a new pi ðkÞ based on the characteristics of the optimization problems is one research direction, such as the problem of odor source localization [28], the problem of disassembly sequencing [45], and the problem of vertical

electrical sounding [13]。 For stability analysis, how to analyze the convergence of a particle  swarm  [8,20,14]  under  a  given control law ui ðkÞ is another research direction。  Accordingly,  the  widely  used  analysis  tools  include  the  Lyapunov  approach and the passivity  approach。  The  existing  results  of  stability  analysis  [8,20,14]  indicate  that  the  particle  swarm  can  converge under several conditions when k ! 1。 It is worth mentioning that the convergence analysis of PSO algorithms is of practical significance because one can see that the better optimization results are obtained  only  in  the  convergence  region  of  PSO  algo- rithms   [13]。

2。2。Preliminaries  on  finite-time control

In this subsection, we will give several preliminaries that will be used in the following convergence analysis。 In order to deal with the problem of odor source localization, a continuous-time dynamic model of N identical robots is described by

。 x_ i  ¼ vi

v_ i  ¼ ui       i 2 lN

ð5Þ

where xi and vi denote the position and the velocity of the ith robot, respectively。 Since the dynamics of each dimension of robots is independent of others, we assume that the dimension number of robots n ¼ 1 without loss of generality。

We first give a definition of finite-time convergence   [4,3]。

Definition 1。 Consider the system x_ ¼ f ðxðtÞÞ, where f : Rn ! Rn is a map。 The origin is said to be a finite-time-stable equilibrium if there exist an open neighborhood N # D of the origin and a function T : N n f0g ! ð0; 1Þ, such that for every x0  2 N n f0g; xðtÞ is defined for t 2 ½0; Tðx0Þ];  xðtÞ 2 N n f0g, for t 2 ½0; Tðx0Þ], and limt   T  x   xðtÞ ¼ 0。 If D ¼ N ¼ Rn, the origin is   said   to   be   a   globally   finite-time-stable  equilibrium。

Then, we give the following lemma [4,3], which will be used in convergence analysis of the proposed FPSO algorithm。

Lemma 1 (Finite-Time Convergence)。 Suppose there exist a continuously differentiable function V : D ! R, the real numbers k > 0 and a 2 ð0; 1Þ, and a neighborhood U c D of the origin such that V is positive definite on U and V_ þ kVa is negative semidefinite on U。 Then, the origin is a finite-time-stable equilibrium of the system x_ ¼ f ðxðtÞÞ (f : Rn ! Rn is a map)。 Moreover, if T is  the  settling  time,  then  T             1        V  x    1—a   for  all  x    in  the  open  neighborhood  of  the origin。