本論文提出幾種混沌系統的同步控制器設計方法，其中有滑動模式控制器設計、使用進化演算法為基礎所設計的比例-積分-微分(PID)控制器以及利用粒子群優最佳化演算法所設計的比例-積分(PI)控制器；並且將主從混沌同步系統應用於通訊保密系統上。
(i)使用滑動模式控制技術，針對滑動模式上使用其比例-積分轉換面來簡化對於閉迴路系統所制定之性能指標。進而求得滑動模式控制器的等效輸出來保證主從聯合混沌系統同步化。
(ii)透過進化(EP)演算法，估測PID控制器Kp,Ki,Kd參數來實現最佳之PID控制器，使得以Sprott電路為基礎的混沌主從系統順利達到同步之要求，且其絕對誤差(IAE)的性能指數可收斂至最小。並且應用至以EP-based PID控制器為基礎的通訊保密系統上。
(iii)透過粒子群優(PSO)演算法，估測PI控制器Kp,Ki參數來實現最佳之PI控制器，使得以Lü電路為基礎的混沌主從系統順利達到同步之要求，且其平方誤差(ISE)的性能指數可收斂至最小。並且應用至以PSO-based PI控制器為基礎的通訊保密系統上。
另外，本論文進一步將所提出的EP-based PID控制器為基礎混沌主從同步系統應用於數位通訊保密系統，其方法為將類比混沌系統的狀態訊號利用電壓控制震盪器(VCO)轉換為數位型態(如一個可變頻率的方波)；在傳送器端(Transmitter)將欲加密的數位訊號和透過VCO轉換完成的主(Master)狀態訊號利用互斥或閘(XOR)將兩者結合(mask)後送至接收器端(Receiver)，並利用另一互斥或閘(XOR)與另一個VCO轉換完成的從(Slave)狀態訊號解密。
為了驗證上述各系統的可行性，進一步的運用一些包含OPA、電阻和電容等電子元件來實現混沌通訊安全系統。最後，經由電腦模擬及實際系統實現結果，說明此混沌通訊安全系統設計是有效而且可行的。

In this dissertation, several schemes such as sliding mode controller (SMC) design, the evolutionary programming (EP)-based proportional-integral-derivative (PID) control design, and particle swarm optimization (PSO)-based proportional-integral (PI) controller design have been proposed for the synchronization problem of a class of chaotic systems. Furthermore, these proposed synchronization systems are then applied to the secure communication.
A SMC technique-based control law is firstly established to guarantee synchronization of the master and slave unified systems. A proportional-integral (PI) switching surface is proposed to simplify the task of assigning the performance of the closed-loop error system in sliding mode. Then, extending the concept of equivalent control and using some basic electronic components, a secure communication system is constructed.
Secondly, a PID controller is developed via the EP algorithm. By using the EP algorithm, optimal control gains in PID controlled chaotic systems are derived such that a performance index of integrated absolute error (IAE) is as minimal as possible. Moreover, as an application, the proposed EP-based PID control scheme is then applied to a chaotic secure communication system. Then for the purpose of application to secure digital communication, a state signal of the analog chaotic system is converted to a digital train with variable pulse width (square wave with random frequency) by voltage controlled oscillator (VCO). The original digital message is masked by the digital train generated from VCO in the chaotic transmitter via a logical exclusive-OR (XOR) operation and it can be successfully recovered by another VCO and XOR operation at the chaotic receiver due to the synchronization.
Thirdly, by using the PSO algorithm, optimal control gains in PI controller are derived such that a performance index of integrated squared error (ISE) is as minimal as possible and synchronization can be achieved. Then a chaotic secure communication system based on synchronized coupled Lü systems is implemented using basic electronic components.
Furthermore, basic electronic components containing OPA, resistor and capacitor elements are used to implement a number of chaotic systems based on the SMC-based, EP-based PID, and PSO-based PI synchronization schemes. Finally, both simulation results and the circuit experiments demonstrate the proposed SMC schemes, EP-based PID scheme, and PSO-based PI scheme’s success in the communication application.

[1] E. N. Lorenz, “Deterministic non-periodic flows,” Journal of Atmospheric Sciences, vol. 20, pp. 130–141, 1963.
[2] G. Chen and T. Ueta, “Yet another chaotic attractor,” International Journal of Bifurcation and Chaos, vol. 9, pp. 1465–1466, 1999.
[3] J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos, vol. 12, no. 3, pp. 659–661, 2002.
[4] J. Lü, G. Chen, D. Z. Cheng and S. Celikovsky, “Bridge the gap between the Lorenz system and the Chen system,” International Journal of Bifurcation and Chaos, vol. 12, pp. 2917–2926, 2002.
[5] G. Chen and J. Lü, “Dynamics of the Lorenz system family: analysis, control and synchronization,” Beijing: Science Press, 2003. [In Chinese].
[6] L. M. Pecora, T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, pp. 821-824, 1990.
[7] C. K. Huang, S. C. Tsay and Y. R. Wu, “Implementation of chaotic secure communication systems based on OPA circuits,” Chaos, Solitons & Fractals, vol. 23, pp. 589-600, 2005.
[8] G. Alvarez, L. Hernández, J. Muñoz, F. Montoya and S. J. Li, “Security analysis of communication system based on the synchronization of different order chaotic systems,” Physics Letters A, vol. 345, no. 4-6, pp. 245-250, 2005.
[9] H. T. Yau, C. L. Kuo and J. J. Yan, “Fuzzy sliding mode control for a class of chaos synchronization with uncertainties,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, pp. 333-338, 2006.
[10] R. Martinez-Guerra and W. Yu, “Chaotic synchronization and secure communication via sliding-mode observer,” International Journal of Bifurcation and Chaos, vol. 18, no. 1, pp. 235–243, 2008.
[11] M. Hu and Z. Xu, “Adaptive feedback controller for projective synchronization,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 1253-1260, 2008.
[12] T. L. Liao and S. H. Tsai, “Adaptive synchronization of chaotic systems and its application to secure communications,” Chaos, Solitions & Fractal, vol. 11, no. 9, pp. 1387-1396, 2000.
[13] Z. Li, K. Li, C. Wen and Y. C. Soh, “A new chaotic secure communication system,” IEEE Transactions on Communications, vol. 51, no. 8, pp. 1306-1312, 2003.
[14] Z. Li and D. Xu, “A secure communication scheme using projective chaos synchronization,” Chaos, Solitions & Fractal, vol. 22, no. 2, pp. 477-481, 2004.
[15] D. I. R. Almeida, J. Alvarez and J. G. Barajas, “Robust synchronization of Sprott circuits using sliding mode control,” Chaos, Solitons & Fractals, vol. 30, no. 1, pp. 11-18, 2006.
[16] J. Zhou, H.B. Huang, G. X. Qi, P. Yang and X. Xie, “Communication with spatial periodic chaos synchronization,” Physics Letters A, vol. 335, no. 2-3, pp. 191-196, 2005.
[17] G. Wen, Q. G. Wang, C. Lin, X. Han and G. Li, “Synthesis for robust synchronization of chaotic systems under output feedback control with multiple random delays,” Chaos, Solitons & Fractals, vol. 29, no. 5, 1142-1146, 2006.
[18] D. B. Fogel, Evolutionary Computation: Toward A New Philosophy of Machine Intelligence. IEEE Press, Piscataway, NJ, USA, 1995.
[19] Y. J. Cao, “Eigenvalue optimization problems via evolutionary programming,” Electronics Letters, vol. 33, pp. 642-643, 1997.
[20] J. J. Yan, S. H. Tsai and I. E. Sheen, “Stability analysis of large-scale time-varying systems via evolutionary programming algorithm,” Journal of Dynamic Systems, Measurement, and Control, vol. 123, pp. 293-296, 2001.
[21] M. C. Chen and S. H. Huang, “Credit scoring and rejected instances reassigning through evolutionary computation techniques,” Expert Systems with Applications, vol. 24, no. 4, pp. 433-441, 2003.
[22] M. J. Kim, S. H. Min and I. Han, “An evolutionary approach to the combination of multiple classifiers to predict a stock price index,” Expert Systems with Applications, vol. 31, no. 2, pp. 241-247, 2006.
[23] J. J. Huang, G. H. Tzeng and C. S. Ong, “Optimal fuzzy multi-criteria expansion of competence sets using multi-objectives evolutionary algorithms,” Expert Systems with Applications, vol. 30, no. 4, pp. 739-745, 2006.
[24] J. Kenny and R. C. Eberhart, “Particle swarm optimization,” Proceedings of the IEEE International Joint Conference on Neural Networks, Vol. IV, pp. 1942-1948, 1995.
[25] B. Liu, L. Wang, Y. H. Jin, F. Tang and D. X. Huang, “Improved particle swarm optimization combined with chaos,” Chaos, Solitons & Fractals, vol. 25, pp. 1261-1271, 2005.
[26] Y. Shi and R. C. Eberhart, “A modified particle swarm optimization,” Proceedings of the IEEE Congress on Evolutionary Computation, Piscataway, pp. 69-73, 1998.
[27] Y. Shi and R.C. Eberhart, “Empirical study of particle swarm optimization,” Proceedings of the IEEE Congress on Evolutionary Computation., Washington, DC, pp. 1945-1950, 1999.
[28] S. P. Ghoshal, “Optimizations of PID gains by particle swarm optimizations in fuzzy based automatic generation control,” Electric Power Systems Research., vol. 72, pp. 203-212, 2004.
[29] Y. J. Cao, “Eigenvalue optimization problems via evolutionary programming,” Electronics Letters, vol. 33, pp. 642-643, 1997.
[30] S. Li, G. Alvarez, Z. Li and W. A. Halang, “Analog chaos-based secure communication and cryptanalysis: a brief survey,” arXiv: 0710.5455v1, 2007.
[31] C. Y. Soong, W. T. Huang and F. P. Lin, “Chaos control on autonomous and non-autonomous systems with various types of genetic algorithm-optimized weak perturbations,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1519-1537, 2007.
[32] Q. He, L. Wang and B. Liu, “Parameter estimation for chaotic systems by particle swarm optimization,” Chaos, Solitons & Fractals, vol. 34, no. 2, pp. 654-661, 2007.
[33] J. J. Yan, M. L. Hung, “A novel stability criterion for interval time-delay chaotic systems via the evolutionary programming approach,” Chaos, Solitons & Fractals, vol. 29, no. 5, pp. 1079-1084, 2006.
[34] J. J. Yan, S. H. Tsai and I. E. Sheen, “Stability analysis of large-scale time-varying systems via evolutionary programming algorithm,” Journal of Dynamic Systems, Measurement, and Control, vol. 123, pp. 293-296, 2001.
[35] M. J. Kim, S. H. Min and I. Han, “An evolutionary approach to the combination of multiple classifiers to predict a stock price index,” Expert Systems with Applications, vol. 31, no. 2, pp. 241-247, 2006.
[36] U. Itkis, Control System of Variable Structure. New York, Wiley, 1976.
[37] V. I. Utkin, Sliding Mode and Their Application in Variable Structure Systems. Moscow, Mir Editors, 1978.
[38] C. P. Tan and C. Edwards, “Sliding mode observers for detection and reconstruction of sensor faults,” Automatica, vol. 38, pp. 1815-1821, 2002.
[39] C. Edwards, S. K. Spurgeon and R. J. Patton, “Sliding mode observers for fault detection and isolation”, Automatica, vol. 36, pp. 541-548, 2000.
[40] K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz, “Synchronization of Lorenz-Based Chaotic Circuits with Applications to Communications,” IEEE Transactions on circuits and systems-11: analog and digital signal processing, vol. 40, no. 10, pp. 626-633, 1993.
[41] S. Oancea, “Synchronization of chaotic electronic Sprott's circuits,” Journal of Optoelectronics and Advanced Materials, vol. 7, no. 6, 2919-2923, 2005.
[42] M. T. Yassen, “Adaptive control and synchronization of a modified Chua’s circuit system,” Applied Mathematics and Computation, vol. 135, pp. 113-128, 2003.
[43] H. H. Tsai, C. C. Fuh and C. N. Chang, “A robust controller for chaotic systems under external excitation,” Chaos, Solitions & Fractal, vol. 14, no. 4, pp. 627-632, 2002.
[44] S. K. Yang, C. L. Chen and H. T. Yau, “Control of chaos in Lorenz system,” Chaos, Solitions & Fractal, vol. 13, no. 4, pp. 767-780, 2002.
[45] J. Lü and J. Lu, “Controlling uncertain Lü system using linear feedback,”Chaos, Solitions & Fractal, vol. 17, no. 1, pp. 127-133, 2003.
[46] H. C. Chen, J. F. Chang, J. J. Yan and T. L. Liao, “EP-based PID control design for chaotic synchronization with application in secure communication,” Expert Systems with Applications, vol. 34, pp. 1169-1177, 2008.