In Table 10 and Table 11, compared with these methods like the PSO algorithm, the CPSO algorithm, the WUI-45 algo- rithm, the WUII algorithm, the PPSO-IM algorithm, and the LPSO algorithm, the CFPSO algorithm generates the better success rates and search time for different numbers of robots。 Correspondingly, the statistical results obtained by using the Wilcoxon rank sum test for each function and per pair of algorithms are also given in Table 10 and Table 11。 Moreover, the multiple problem analysis is used to present the statistical results given in Table 12 where the CFPSO algorithm shows a significant

improvement over the PSO algorithm, the CPSO algorithm, the WUI-45 algorithm, the WUII algorithm, the PSO-IM algorithm, and the LPSO algorithm with a level of significance a ¼ 0:05。 In addition, since the path length denotes the energy consumed by robots, the shorter path length means the better search performance。 From Figs。 14–17, one can also see that the robots coordinated by the CFPSO algorithm consume the lesser energy than those coordinated by the PSO algorithm, the CPSO algo- rithm, the WUI-45 algorithm, the WUII algorithm, the PPSO-IM algorithm, and the LPSO algorithm。

6。Conclusion

The problem of odor source localization has been investigated。 A continuous-time FPSO algorithm has been developed by introducing a nonlinear damping item and a parameter into the continuous-time model of the PSO algorithm such that the

CFPSO   PSO CPSO   WUI− 45   WUII  PPSO− IM LPSO

CFPSO   PSO CPSO   WUI− 45   WUII  PPSO− IM LPSO

CFPSO    PSO CPSO   WUI− 45    WUII   PPSO− IM LPSO

Fig。 17。 The consumed energy for case 7, case 8, and case 9 based on 5 robots with 50 runs。

continuous-time FPSO algorithm can converge over a finite-time interval and its exploration capability can be improved。 The Lyapunov approach has been employed to analyze the finite-time convergence of the continuous-time FPSO algorithm。 The discrete-time FPSO algorithm has been obtained by using a given dicretization scheme and the convergence of the discrete- time FPSO algorithm has been analyzed。 The characteristics and performance capabilities of the discrete-time FPSO algo- rithm has been shown through numerical simulations on two ill-posed functions and twenty-five benchmark functions, and the effectiveness of the continuous-time FPSO algorithm has been illustrated through the problem of odor source localization。

Acknowledgments

The authors thank the Editor and anonymous reviewers for providing valuable comments to improve the quality of the paper。 The research work of Q。 Lu and S。-R。 Liu was supported in part by the National Natural Science Foundation of China under Grants 61375104, 61203025, 61175093, and the Natural Science Foundation of Zhejiang Province under Grant LQ13F030014。 The research work of Q。-L。 Han was supported in part by the Australian Research Council Discovery Projects under Grant DP1096780, and the Research Advancement Awards Scheme Program (January 2010–December 2012) at Central Queensland University, Australia。摘要本文涉及的气味源定位的有限时间粒子群算法。首先，连续有限时间粒子群算法（FPSO）是基于粒子群优化（PSO）算法的连续模型的发展。由于引入了非线性阻尼项，建议的连续FPSO算法能够在一个有限的时间间隔收敛。此外，为了增强其勘探能力，调谐参数被引入构成连续FPSO算法。算法的有限时间收敛是通过使用李雅普诺夫方法进行分析。其次，离散时间FPSO算法是通过使用一个给定的离散方案获得。相应的收敛条件是通过使用线性矩阵不等式（LMI）方法而得。最后，所提出的FPSO算法的特征和性能通过使用两个病态功能和二十五基准函数，分别示出。在数值模拟结果中，来验证建议的FPSO算法的有效性气味源定位的问题。