本論文中我們提出了一種改良的粒子群演算法，名為維度經驗重心分享粒子群演算法 SDGPSO(Standard Dimension Searching with Center of Gravity PSO algorithm)，其特殊的單維度搜索機制，讓其必須擁有較一般粒子群演法不同的經驗分享機制，且在前期能夠有優秀且快速的收斂能力，化簡的計算程序也能減少計算消耗時間，並且在多數函數都能夠有優良的最終收斂值，但在後期跳脫區域最優解的能力還是不如標準型粒子群演算法SPSO，所以在本論文又對SDGPSO作一個新的改良融合命名為SDGPSO-MSP (Standard Dime- nsion Searching with Center of Gravity PSO algorithm Mixing SPSO)，其中本文提出的銜接機制讓SDGPSO-MSP擷取SDGPSO前期快速收斂的能力和SPSO 後期優秀的跳脫能力，讓其優缺能夠達到良好的互補。最後我們使用測試函數對SDGPSO 和SDGPSO-MSP作性能測試並且與幾個已提出的擁有優良校能粒子群演算法作比較。經由模擬結果顯示，本文所提出的單維搜索與經驗重心分享機制在銜接SPSO後整體測試都均具有優越的表現，並且SDGPSO-MSP表現出兩邊所擷取的優點甚至更加優良。

In this thesis we have presented an improved algorithm for Particle Swarm Optimization (PSO) named Standard Dimension Searching with Center of Gravity PSO algorithm (SDGPSO). The SDGPSO algorithm needs to have a special experience-sharing mechanism to coordinate with single-dimensional searching mechanism. These mechanisms cause the faster convergence ability in the pre-convergence and have less computing time than SPSO. However, The ability to escape local optimal solution is worse than SPSO in the post-convergence. Because of this shortcoming, we further proposed a hybrid version named SDGPSO-MSP which takes advantages of fast and excellent convergence ability from SDGPSO and obtaining the better convergent solution from SPSO. The performances of SDGPSO and SDGPSO-MSP are fairly demonstrated by applying sixteen benchmark problems and compared it with several popular PSO algorithm. The simulations of results show that our proposed methods are effective and gain better performance than other compared PSO algorithms.

[1] J. Kennedy, R. C. Eberhart, and Y. H. Shi, “Swarm Intelligence,” Morgan Kaufmann division of Acadenic, 2001.

[2] J. Kennedy and R. C. Eberhart, “Particle Swarm Optimization,” In Proceedings of IEEE International Conference on Neural Networks, vol. IV, pp. 1942−1948, 1995.
[3] W. D. Chang and S. P. Shih, “PID controller design nonlinear systems using an improved particle swarm optimization approach,” Communication Nonlinear Science and Numerical Simulation, vol. 15, pp. 3632-3639, 2010.
[4] Y. Shi and R. C. Eberhart, “Particle Swarm Optimization: Development, Applications and Resource,” In Proceedings of the 2001 Congress on Evolutionary Computation, vol. 1, pp. 81-86, 2001.
[5] R. A. Krohling and L. S. Coelho, “Coevolutionary Particle Swarm Optimization Using Gaussian Distribution for Solving Constrained Optimization Problem,” In Proceedings of IEEE Transactions on Systems, Man and Cybernetics –Part B: Cybernetics, vol. 36, no. 6, pp. 1407-1416, 2006.
[6] D. Srinivasan , W. H. Loo and R. L. Cheu, “Traffic incident detection using particle swarm optimization,” In Proceedings of IEEE Swarm Intelligence Symposium, pp.144-151, 2003.
[7] S. –Y. Ho, H. –S. Lin, W. –H. Liauh and S. –J. Ho, “OPSO: Orthogonal particle swarm optimization and its application to task assignment problems,” In Proceedings of IEEE Transactions on Systems, Man and Cybernetics –Part A: Systems and Humans, vol. 38, no. 2, pp. 288-298, 2008.
[8] F. Xie, Y. Wang, Z. Zheng and C. Li, “Optimal Control of Switched Linear Systems Based on Migrant Particle Swarm Optimization Algorithm,” In Proceedings of the 2010 International Conference on Modelling, Identification and Control, pp. 237-241, 2010.
[9] Y. Shi, and R. C. Eberhart, “Parameter Selection in Particle Swarm Optimization,” Evolutionary Programming VII. Lecture Notes in Computer Science, vol. 1447, pp. 591–600, 1998.
[10] M. Clerc, “The Swarm and the Queen: Towards a Deterministic and Adaptive Particle Swarm Optimization,” Proceedings of the Congress on Evolutionary Computation, vol. 3, pp. 1951−1957, 1999.
[11] N. M. Kwok, D. K. Liu, K. C. Tan and Q. P. Ha, “An Empirical Study on the Settings of Control Coefficients in Particle Swarm Optimization,” In Proceedings of IEEE Congress on Evolutionary Computation, pp. 823-830, 2006.
[12] M. Clerc and J. Kennedy, “The Particle Swarm—Explosion, Stability, and Convergence in a Multidimensional Complex Space,” In Proceedings of IEEE Transaction on Evolutionary Computation, vol. 6, no. 1, pp. 58-73, 2002.
[13] J. Jie, J. zeng, C. Han and Q. Wang, “Knowledge-based cooperative particle swarm optimization,” Applied Mathematics and Computation, vol. 205, pp. 861-873, 2008.
[14] J. Wei, L. Guangbin and L. Dong, “Elite Particle Swarm Optimization with Mutation,” 2008 Asia Simulation Conference-7th International Conference on System Simulation and Scientific Computing, pp. 800-803, 2008.
[15] W. –P. Lee, W. –Y. Xian and C. –C. Wen, “Research on a Modified Particle Swarm Optimization Algorithm,” Journal of Engineering Technology, vol. 4, no. 2, pp. 51-62, 2008.
[16] Y. –T. Juang, S. –L. Tung and H. –C. Chiu, “Adaptive fuzzy particle swarm optimization for global optimization of multimodal functions,” International Journal of Information Sciences, vol. 181, pp. 4539-4549, 2011.
[17] 林柏勳,胡光復,沈哲緯,鄭錦桐,“最佳化方法於工程上之應用”中興工程季刊第103期，2009年4月.
[18] C. Liu and C. Ouyang, “An adaptive fuzzy weight PSO algorithm,” In Proceedings of IEEE Fourth International Conference on Genetic and Evolutionary Computing, pp. 8-10, 2010.
[19] 董聖龍，「粒子群演算法於二階時變系統穩定分析與穩定化設計」，國立中央大學，博士論文，民國100年。
[20] M. Dorigo, V. Maniezzo and A. Colorni, “The ant system: Optimization by a colony of cooperating agents,” In Proceedings of IEEE Transactions on Systems and Cybernetics - Part B, vol. 26, no. 1, pp. 29-41, 1996.
[21] D. Fogel and H. G. Beyer “A note on the empirical evaluation of intermediate recombination,” Evolutionary Computation, vol. 3, no. 4, pp. 491-495.
[22] Y. Shi, and R. C. Eberhart, “Comparison between genetic algorithms and particle swarm optimization,” Evolutionary Programming VII. Lecture Notes in Computer Science, vol. 1447, pp. 611-616, 1998.
[23] J. Kennedy and W. M. Spears, “Matching Algorithms to Problems: An Experimental Test of the Particle Swarm and Some Genetic Algorithms on the Multimodal Problem Generator,” In Proceeding of IEEE World Congress on Computational Intelligence, pp. 78-83, 1988.
[24] J. H. Holland, “Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence,” Cambridge, Mass: MIT Press, 1992.
[25] 王文俊，認識fuzzy，全華出版，第三版，2007。
[26] R. C. Eberhart and Y. H. Shi, “Particle swarm optimization: Developments, applications and resources,” In Proceeding of IEEE Congress on Evolutionary Computation, pp. 81-86, 2001.
[27] R. C. Eberhart and Y. H. Shi, “A modified particle swarm optimizer,” In Proceeding of IEEE World Congress On Computational Intelligence, pp. 69-73, 1988.
[28] R. C. Eberhart and Y. H. Shi, “Empirical study of particle swarm optimization,” In Proceeding of IEEE Congress on Evolutionary Computation, pp. 1945-1950, 1999.
[29] Z. -H. Zhan, J. Zhang, Y. Li and H. S. –H. Chung, “Adaptive Particle Swarm Optimization,” In Proceedings of IEEE Transactions on Systems ,Man, and Cybernetics－Part B: Cybernetics, vol. 39, no. 6,pp. 1362-1381, 2009.

[30] A. Ratnaweera, S. K. Halgamuge and H. C. Watson, “Self-Organizing Hierarchical Particle Swarm Optimizer With Time-Varying Acceleration Coefficients,” IEEE Transactions On Evolutionary Computation, pp. 240 - 255, 2004.
[31] Deepak Devicharan , Chilukuri K . Mohan, “Particle Swarm Optimization

with Adaptive Linkage Learning” Dept of EECS, 2-177 Center for Science and

Technology Syracuse University .
[32] J. J. Liang, A. K. Qin, P. N. Suganthan and S. Baskar, “Comprehensive learning particle swarm optimizer for global optimization of multimodal functions,” IEEE Transactions on Evolutionary Computation, pp. 281–296, 2006.
[33] Zhi-Hui Zhan, Jun Zhang ,Yun Li, Yu-Hui Shi, “ Orthogonal Learning Particle Swarm Optimization” IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 15, NO. 6, DECEMBER 2011
[34] R.Venkata Rao, Vivek Patel, “An elitist teaching-learning-based optimization algorithm for solving complex constrained optimization problems”, International Journal of Industrial Engineering Computations 3 ,(2012)
[35] Sheng-Ta Hsieh, Tsung-Ying Sun, Chan-Cheng Liu, Shang-Jeng Tsai,“Efficient Population Utilization Strategy for Particle Swarm Optimizer” IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009
[36] N. Iwasaki, K. Yasuda and G. Ueno, “Dynamic parameter tuning of particle swarm optimization,” IEEE Transactions on Electrical and Electronic Engineering, pp. 353-363, 2006.
[37] M. A. Montes de Oca, J. Pena, T. Stutzle, C. Pinciroli and M. Dorigo, “Heterogeneous particle swarm optimizers,” Proceedings of IEEE congress on Evolutionary Computation, pp. 698–705, 2009.
[38] P. N. Suganthan, N. Hansen, J. J. Liang and K. Deb, Y. -P. Chen, A. Auger & S. Tiwari, “Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization,” Technical report of Nanyang Technological University, 2005.
[39] 陳珈妤，「快速平衡粒子群最佳化方法」，國立中央大學，碩士論文，民國100年。
[40] 蔡憲文，「以時變學習因子策略改良粒子群演算法」，國立中央大學，碩士論文，民國99年。

[41] J. Hu, J. Zeng, Y. Yang, “A two-order Particle Swarm Optimization Model and the Selection of its Parameters,” In Proceedings of the 6th Congress on Intelligent Control and Automation, June 21-23, 2006.
[42] M. Pant, T. Radha and V. P. Singh, “A New Particle Swarm Optimization with Quadratic Interpolation,” In Proceedings of IEEE International Conference on Computational Intelligence and Multimedia Applications, pp. 55-60, 2007.
[43] K. E. Parsopoulos and M. N. Vrahatis, “UPSO: a unified particle swarm scheme,” In Lecture series on Computer and Computational Sciences, vol. 1, pp. 868-873, 2004.
[44] R. Mendes, J. Kennedy and J.Neves, “The fully informed particle swarm: simpler, maybe better,” IEEE Transactions on Evolutionary Computation, vol. 8, pp. 204-210, 2004.
[45] J. J. Liang and P. N. Suganthan, “Dynamic multi-swarm particle swarm optimizer,” In Proceedings of IEEE on Swarm Intelligence Symposium, pp. 124-129, 2005.
[46] 吳 讚 展，「自調整非線性慣性權重粒子群演算法」，國立中央大學，碩士論文，民國101年。

[47] A. Chatterjee and P. Siarry, “Nonlinear inertia weight variation for dynamic adaptation in particle swarm optimization,” Computers and Operations Research, vol. 33, no. 3, pp. 859–871, 2004.

[48] Y. Tang, M. Cui, C. Hua, L. Li and Y. Yang, “Optimum design of fractional order controller for AVR system using chaotic ant swarm,” Expert Systems with Applications, vol. 39, pp. 6689-6896, 2012.
[49] A. Biswas, S. Das, A. Abraham and S. Dasgupta, “Design of fractional-order controllers with an improved differential evolution,” Engineering Applications of Artificial Intelligence, vol. 22, pp. 243-350,2009.
[50] J. Y. Cao and B. G. Cao, “Design of Fractional Order Controller Based on Particle Swarm Optimization,” International Journal of Control, Automation and Systems, vol. 4, pp. 775-781,2006.
[51] A. A. Jalali and A. Khosravi, “Tuning of FOPID Controller Using Taylor Series Expansion,” International Journal of Scientific and Engineering Research, vol. 2, 2011.
[52] 薛定宇,趙春娜 , “分數階系統的分數階PID 控制器設計” 控制器理論與應用 Control Theory & Applications Vol. 24 No. 5 2007.

[53] I. Podlubny, “Fractional-Order Systems and Controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208-214, 1999.
[54] A. Oustaloup , “La commande CRONE: commande robuste d’order non entire,” Herme’s, Paris, 1991.
[55] A. Oustaloup, X. Moreau, M. Nouillant, “The CRONE suspension,” Control Engineering Practice, vol. 4, pp. 1101-1108, 1996.

[56] G. Huang and S. Lee, “PC based PID speed control in DC motor,” IEEE International Conference on Audio, Language and Image Processing, pp. 400-407, 2008.
[57] K. A. Naik and P. Srikanth, “Stability Enhancement of DC Motor using IMC Tuned PID Controller,” International Journal of Advanced Engineering Sciences and Technologies, vol. 4, pp. 92-96, 2011.