This paper is concerned with a finite-time particle swarm optimization algorithm for odor source localization。 First, a continuous-time finite-time particle swarm optimization (FPSO) algorithm is developed based on the continuous-time model of the particle swarm optimi- zation (PSO) algorithm。 Since the introduction of a nonlinear damping item, the proposed continuous-time FPSO algorithm can converge over a finite-time interval。 Furthermore, in order to enhance its exploration capability, a tuning parameter is introduced into the pro- posed continuous-time FPSO algorithm。 The algorithm’s finite-time convergence is ana- lyzed by using the Lyapunov approach。 Second, the discrete-time FPSO algorithm is obtained by using a given dicretization scheme。 The corresponding convergence condition is derived by using a linear matrix inequality (LMI) approach。 Finally, the features and per- formance of the proposed FPSO algorithm are illustrated by using two ill-posed functions and twenty-five benchmark functions, respectively。 In numerical simulation results, the problem of odor source localization is presented to validate the effectiveness of the pro- posed FPSO algorithm。85054

1。Introduction

This paper is motivated by our collaborations with an environmental protection institute in order to reduce the localiza- tion time of the dangerous gas source based on a cooperative multi-robot system, which can be formulated as the problem of odor source localization。 In the light of the studies [12,34], this problem is an ill-posed and dynamical optimization problem [11,16,41,35,29], and its main characteristics can be summarized   as

●The maximum odor concentration occurs in the vicinity of the position of the odor source and there exist multiple local odor concentration maxima along the   plume。

●The positions of local concentration maxima change with time and the local odor concentration maxima are also time-varying。

●The odor concentration can be only detected within the plume except which the odor concentration approaches to zero。

q The material in this paper was partially presented at the IEEE World Congress on Computational Intelligence, June 10–15, 2012, in Brisbane, Australia。

⇑ Corresponding authors。 Address: School of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, PR China。 Tel。: +86 571 86878535;

fax: +86 571 86878535, Centre for Intelligent and Networked Systems, Central Queensland University, Rockhampton, Qld 4702, Australia (Q。 Lu)。 Tel。: +61 7 49309270; fax: +61 7 49306886 (Q。-L。    Han)。

E-mail  addresses: lvqiang@hdu。edu。cn  (Q。 Lu),  q。han@cqu。edu。au (Q。-L。  Han), liushirong@hdu。edu。cn  (S。 Liu)。

http://dx。doi。org/10。1016/j。ins。2014。02。010

0020-0255/© 2014 Elsevier Inc。 All rights reserved。

On the basis of the characteristics of the odor source localization problem, the PSO algorithm and its variants, such as PSO [30], CPSO [19], PPSO-IM [25], and LPSO [27], which have the attribute of the ‘‘decision and control mechanism’’ [22–24], can been used to make a decision on the positions of the odor source and to control the robots to move toward the positions。 It is worth mentioning that the aforementioned approaches focus on how to design a new decision algorithm to make a decision on the probable position of the odor source in terms of concentration information and wind information detected by the ro- bot group。 However, cooperative control algorithms, which can control the robot to reach the predicted position of the odor source, have not paid enough attention in the existing    literature。

In fact, cooperative control algorithms are significant in the problem of odor source localization。 From the control engi- neering point of view, since the robots are used to search for the odor source, a cooperative controller should be of the better performance in disturbance rejection and the robustness against uncertainties。 Moreover, since the predicted position of the odor source is time-varying, the cooperative controller should be of the better tracking performance。 Due to the character- istics of the odor source localization problem, in order to adapt the dynamical search environment, the cooperative controller should enable the system to be stable over a finite-time interval。 Furthermore, the knowledge about the dynamical search environment is only from the current and previous information of the robot group。 As a consequence, the robot is required to quickly move to the predicted position of the odor source such that new information can be obtained to update the predicted position of the odor source [40]。 Because of several advantages including higher control accuracy, better disturbance rejec- tion, and robustness against uncertainties [4], finite-time controller design has received a growing interest from researchers and engineers [9,43]。 In order to meet the aforementioned requirements, it is of practical significance to develop a finite-time cooperative controller, which is in conjunction with the decision algorithm, to form an FPSO algorithm, which is the moti- vation of the current  study。