本論文中，主要探討含有不確定性因素的二階時變微分系統穩定度分析。由於在實際的環境中，不管儀器再精密或是實驗環境再理想，一定會有不可避免的擾動或是參數變動的問題，因此，這些不確定性因素的考量，對於系統本身而言，是不可或缺的。
本論文分別對系統的領導係數矩陣分別為奇異與非奇異矩陣做探討，利用矩陣以及不等式的性質，而推導出一個新的充份指數穩定條件，並與文獻的例子做比較，證明本論文所提出的定理比文獻中的結果更不具保守性。
基於此指數穩定條件，利用動態迴授來設計控制器，並結合最佳化方法來尋找控制器參數。最後以一個實際的二階系統為例，對此系統設計控制器，並分別對時間響應以及頻率響應做分析，討論補償前與補償後系統性能的差異，結果顯示補償後的性能優於未補償的系統，證實所設計的控制器是有效的。

This thesis is concerned with exponential stability analysis of second-order time-varying differential systems with nonlinear uncertainties bounded by Lipschitz constants. The systems with singular and non-singular leading coefficient matrices are both discussed. New sufficient conditions for exponential stability of the above mentioned uncertain systems are derived and the upper bounds of allowable nonlinear uncertainties are obtained. Furthermore, the proposed result is less conservative than the results of literatures. It is shown that proposed criterion is superior to the literatures and has better improved.
Then using proposed criterion combines with optimization method to design controller. Both singular and non-singular illustrative examples are shown to compare uncompensated system with compensated system. According to simulation results, after adding the controller can let the system become stable. Finally, a practical example is shown to demonstrate that using proposed criterion to design controller can improve the performances. In frequency domain analysis, after adding the controller, the resonant frequencies are eliminated. In time domain analysis, three experiments are implemented. All of these results show that the controller can improve the performance of system.

[1] M. Meisami-Azad, J. Mohammadpour and K. M. Grigoriadis, “An upper bound approach for control of collocated structural systems,” American Control Conference, July 11–13, New York City, USA, pp. 4631-4636, 2007.
[2] H. Tasso and G. N. Throumoulopoulos, “On Lyapunov stability of nonautonomous mechanical systems,” Phys. Lett. A, vol. 271, pp. 413–418, 2000.
[3] H. Tasso, “On Lyapunov stability of dissipative mechanical systems,” Phys. Lett. A, vol. 257, pp.309-311, 1999.
[4] J. Sun, Q. G. Wang and Q. C. Zhong, “A less conservative stability test for second-order linear time-varying vector differential equations,” Int. J. Contr., vol. 80, pp. 523-526, 2007.
[5] K. Inoue and T. Kato, “A stability condition for a time-varying system represented by a couple of a second- and a first-order differential equations”, 43rd IEEE Conference on Decision and Control, Dec. 14–17, Atlantis, Paradise Island, Bahamas, pp. 2934-2935, 2004.
[6] K. Inoue, S. Yamamoto, T. Ushio, and T. Hikihara, “Torque-based control of whirling motion in a rotating electric machine under mechanical resonance,” IEEE Trans. Automat. Contr., vol. 11, pp. 335-344, 2003.
[7] M. I. Gil’, “Stability of linear systems governed by second order vector differential equations,” Int. J. Contr., vol. 78, pp. 534–536, 2005.
[8] K. Inoue, S. Yamamoto, T. Ushio, and T. Hikihara, “Elimination of jump phenomena in a flexible rotor system via torque control,” Control of Oscillations and Chaos, 2000. Proceedings. 2000 2nd International Conference, vol. 1, pp. 58-61, 2000.
[9] Y. Fang and T. G. Kincaid, “Stability analysis of dynamical neural networks,” IEEE Trans. Neural Networks, vol. 7, pp. 996–1006, 1996.
[10] A. Zevin and M. Pinsky, “Exponential stability and solution bounds for systems with bounded nonlinearities,” IEEE Trans. Automat. Contr., vol. 48, pp. 1779-1804, 2003.
[11] A. A. Zevin and M. A. Pinsky, “Delay-independent stability conditions for time-varying nonlinear uncertain systems,” IEEE Trans. Automat. Contr., vol. 51, pp. 1482-1486, 2006.
[12] M. Vidyasagar, Nonlinear Systems Analysis. New Jersey: Prentice-Hall, Inc., 2000.
[13] H. Jeffreys and B. Jeffreys, ”Methods of mathematical physics,” 3rd edition, Cambridge, England: Cambridge University Press, pp. 53, 1988.
[14] P. Lancaster and M. Tismenetsky, “The theorem of matrices,” 2nd edition with applications, Academic Press, Inc., 1985.
[15] M. I. Gil, “Stability of linear systems governed by second order vector differential equations,” Int. J. Contr., vol. 78, pp. 534-536, 2005.
[16] J. Sun, Q. G. Wang and Q. C. Zhong, “A less conservative stability test for second-order linear tome-varying vector differential equations,” Int. J. Contr., vol. 80, pp. 523-526, 2007.
[17] A. Zevin and M. A. Pinsky, “Exponential stability and solution bounds for system with bounded nonlinearities,” IEEE Trans. Automat. Contr., vol. 48, pp. 1779-1804, 2003.
[18] W. J. Rugh, Linear system theorem, 2nd edition, New Jersey: Prentice-Hall, Inc.
[19] S. Z. Abdul-Rahman and G. Ahmadi, “Stability analysis of a non-autonomous linear systems by a matrix decomposition method,” Int. J. Systems Sci, vol. 17, pp. 1645-1660, 1986.
[20] R. K. Yedavalli and S. R. Kolla, “Stability analysis of linear time-varying systems,” Int. J. Systems Sci, vol. 19, pp. 1853-1858, 1988.
[21] S. Wolfram, The Mathematica Book, 5th Edition. Champaign: Wolfram Media, Inc., 2003.
[22] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. IEEE Int. Conf. Neural Networks, vol. IV, Perth, Australia, pp. 1942-1948, 1995.
[23] Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” in Proc. IEEE Int. Conf. Evol. Comput., Anchorage, AK, pp. 69-73, 1998.
[24] Y. Shi and R. C. Eberhart, “Empirical study of particle swarm optimization,” in Proc. IEEE Int. Conf. Evol. Comput., Washington, DC, pp. 1945-1950, 1999.
[25] Z. L. Gaing, “A particle swarm optimization approach for optimum design of pid controller in avr system,” in IEEE Tans. On Energy Conversion, vol. 19, pp.384-391, 2004.
[26] J. Kennedy and R. Eberhart, Swarm Intelligence, Morgan Kaufmann Publishers, 2001.
[27] P. Angeline, ”Using selection to improve particle swarm optimization,” in Proc. IEEE Int. Conf. Evolutionary Computation, pp.84-89, 1998.
[28] S. Naka, T. Genji, T. Yura, and Y. Fukuyama, “Hybrid particle swarm optimization based distribution state estimation using constriction factor approach,” in Proc. Int. Conf. SCIS & ISIS, vol. 2, pp. 1083-1088, 2002.
[29] V. Miranda and N. Fonseca, “EPSO-best-of-two-worlds meta-heuristic applied to power system problems,” in Proc. Congr. Evolutionary Computation, vol. 2, pp. 12-17, 2002.
[30] V. Miranda and N. Fonseca, “EPSO-evolutionary particle swarm optimization, a new algorithm with applications in power systems,” in Proc. IEEE Trans. Distribution Conf. Exhibition, vol. 2, pp. 6-10, 2002.
[31] V. Miranda, “Evolutionary algorithm with particle swarm movements,” in IEEE 13th Int. Conf. ISAP, Nov. 6-10, pp. 6-21, 2005.
[32] J. S. Kwang, Y. L. Lee and R. G. Ramirez, “Multiobjective control of power plants using particle swarm optimization techniques,” in IEEE Tans. On Energy Conversion, vol. 21, pp.552-561, 2006.
[33] Baozhan Lu; Sihong Zhu and Ying Zhang ,“Research on parameter matching of front axle suspension vehicle based on matlab” International Conference on Automation and Logistics, pp. 2441-2445, 2007.
[34] P. Y. Sun and H. Chen, “Multi-objective output-feedback suspension control on a half-car model,” in Proceedings of IEEE Conference Control Applications, pp. 290-295, 2003.
[35] D. Hrovat, “Survey of advanced suspension development and related optimal control applications,” Automatica, vol. 33, pp. 1781-1817, 1997.
[36] J. Cao, H. Liu, P. Li, and D. J. Brown, “State of the art in vehicle active suspension adaptive control systems based on intelligent methodologies,” IEEE Trans. Intell. Transp. Syst., vol. 9, pp. 392-405, 2008.
[37] M. Sunwoo, K. C. Cheok and N. J. Huang, “Model reference adaptive control for vehicle active suspension systems,” IEEE Trans. Ind. Electronics, vol. 38, pp. 217-222, 1991.
[38] M. Nagai, “Recent researches on active suspensions for ground vehicles,” JSME International Journal Series C, vol. 36, pp. 161-170, 1993.
[39] A. Alleyne and J. K. Hedrick, “Nonlinear adaptive control of suspensions,” IEEE Trans. Con. Sys. Technology, vol. 3, pp. 94-101, 1995.
[40] N. Giorgetti, A. Bemporad, H. E. Tseng and D. Hrovat, “Hybrid model predictive control application towards optimal semi-active suspension,” Int. J. Contr., vol. 79, pp. 521-533, 2008