Notice that the FPSO algorithm can be regarded as a more generalized form of the PSO algorithm。 It can be used to control the real robot group。 It can also be viewed as an example such that researchers can separately design the decision algorithm and the control algorithm in terms of the characteristics of the optimization problem。 The partial version of this paper ap- pears in [26]。 In this paper, first, we will propose a continuous-time FPSO algorithm based on the continuous-time model of the PSO algorithm。 Since the introduction of a nonlinear damping item, the continuous-time FPSO algorithm can converge over a finite-time interval。 Furthermore, we will introduce a tuning parameter, which can enhance the exploration capability of the continuous-time FPSO algorithm。 We will employ the Lyapunov approach to analyze continuous-time FPSO algo- rithm’s finite-time convergence。 Second, we will derive a discrete-time version of the FPSO algorithm and analyze the cor- responding convergence by using an LMI approach。 Finally, we will illustrate the characteristics and performance capabilities of the FPSO algorithm based on two ill-posed functions and twenty-five benchmark functions, respectively。 In numerical simulation results, we will use the FPSO algorithm to deal with the problem of odor source localization。

Notation: lN   denotes the index set f1; 2; 。。。 ; Ng。 Let sigðr   a

function。

2。Background

2。1。Related works on particle swarm    optimization

¼ signðrÞjrj , where 0 < a < 1; r 2 R, and signð·Þ is a   sign

In the last decade, the PSO algorithm as a swarm intelligence technique has been widely studied [31]。 The various ver- sions of the PSO algorithm have been proposed to deal with different types of optimization problems, and empirical evi- dences indicate that the PSO algorithm is a useful tool for optimization problems [20]。 The widely used version of the PSO algorithm [36] is described  by

with

。 viðk þ 1Þ¼ xvi ðkÞþ ui ðkÞ

xiðk þ 1Þ¼ xiðkÞþ viðk þ 1Þ

ð1Þ

uiðkÞ¼ a1ðxlðkÞ— xiðkÞÞ þ a2ðxg ðkÞ— xi ðkÞÞ ð2Þ

where vi ðkÞ denotes the velocity vector while ui ðkÞ is the control vector; xl ðkÞ refers to the previously best position of the ith particle whereas xg ðkÞ is the globally best position of  the  swarm;  x is  the  inertia  factor;  and  aj  (j ¼ 1; 2)  are  called  acceler- ation   coefficients。   The   PSO   algorithm   provides   a   ‘‘decision-control   mechanism’’,   which   is   analyzed   in   the following。

Introducing the  following  oscillation  center  pi ðkÞ [14],

a1 xl ðkÞþ a2 xg ðkÞ

piðkÞ¼ 

we have

a1 þ a2

ð3Þ

uiðkÞ¼ ða1  þ a2Þðpi ðkÞ— xiðkÞÞ ð4Þ

From (4), one can see that ui ðkÞ can be regarded as a ‘‘P’’ (proportional) controller and keep the system (1) stable at the equi- librium   ð0; pi ðkÞÞ   under   several   conditions   [15]。   Therefore,   each   particle   uses   both   swarm   information   and       inpidual