本篇論文提出新的方法以求得廣義代數李亞普諾夫方程式解之上、下界值，並且用模擬結果證明，此方法在遞迴時具有收斂性。針對求上界值時所做假設的合理性也進一步分析。除了與文獻的結果比較外，並討論其在擾動系統上的應用。在各個章節的最後都有數值範例，說明所推得的結果。

This thesis proposes a new method to compute the lower and upper bounds of solution to the generalized Lyapunov matrix equation. Convergence of the iteratively computed bounds is illustrated by several numerical examples. Moreover, regularization of the condition required in computing the upper bound is also investigated. Finally, the method is employed to derive a less conservative bound on the magnitude of perturbation without destroying the stability of the perturbed system. All the theoretical results are verified by the numerical examples.

摘要 i
符號表 iv
第一章 序論 1
1-1 文獻回顧與研究動機 1
1-2 論文綱要 2

第二章 基本性質 3

第三章 廣義代數李亞普諾夫方程式解之界值 5
3-1 廣義代數李亞普諾夫解之界值 5
3-2 界值之遞迴性 9
3-3 合理化的假設 11
3-4 數值模擬與文獻結果比較 12

第四章 廣義代數李亞普諾夫方程式解之界值的延伸 18
4-1 連續時間系統的李亞普諾夫方程式解之界值與文獻結果比較 18
4-2 系統強韌性的分析與數值例證 27

第五章 結論 32

參考文獻 33

[1] S. Gutman and E. I. Jury, “A general theory for matrix root-clustering in subregions of the complex plane,” IEEE Trans. on Automat. Contr., vol. 26, pp. 853-863, 1981.
[2] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
[3] A. R. Amir-Moez, “Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations,” Duke Math. J., vol. 23, pp. 463-476, 1956.
[4] C. H. Lee, and S. T. Lee, “On the estimation of solution bounds of the generalized Lyapunov equations and the robust root clustering for the linear perturbed systems,” Int. J. Control, vol.74, pp. 996-1008, 2001.
[5] C. H. Lee, “Upper and Lower Matrix Bounds of the Solutions for the Continuous and Discrete Lyapunov Equation, ” J. Franklin Inst., vol. 334b, pp. 539-546, 1997.
[6] H. H. Choi and T. Y. Kuc, “Lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations,” Automat., vol. 38, pp. 1147-1152, 2002.
[7] C. H. Lee, “New Results for the Bounds of the Solution for the Continuous Riccati and Lyapunov equations,” IEEE Trans. Automat. Contr., vol. 42, pp. 118-123, 1997.
[8] N. Komaroff, “Upper summation and product bounds for solution eigenvalues of the Lyapunov matrix equation,” IEEE Trans. Automat. Contr., vol. 37, pp. 1040-1042, 1992.
[9] Y. Fang, K. A. Loparo, and X. Feng, “New estimates for solutions of Lyapunov equations,” IEEE Trans. Automat. Contr. vol. 42, pp. 408-411, 1997.
[10] C. H. Lee, “Upper and lower matrix bounds of the solution for the discrete Lyapunov equation,” IEEE Trans. Automat. Contr. vol. 41, pp. 1338-1341, 1996.
[11] D. G. Lee, G. H. Heo, and J. M. Woo, “New bounds using the solution of the discrete Lyapunov matrix equation,” International Journal of Control, Automation, and Systems, vol. 1, pp. 459-463, 2003.
[12] W. H. Kwon, Y. S. Moon, and S. C. Ahn, “Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results.” Int. J. Contr., vol. 64, pp. 377-389, 1996.
[13] N. Komaroff, “Upper bounds for the eigenvalues of the solution of the Lyapunov matrix equation,” IEEE Trans. Automat. Contr., vol. 35, pp. 737-739, 1990.
[14] N. Komaroff, and B. Shahian, “Lower summation bounds for the discrete Riccati and Lyapunov equations.” IEEE Trans. Automat. Contr., vol. 37, pp. 1078-1080, 1992.
[15] N. Komaroff, “Lower bounds for the solution of the discrete algebraic Lyapunov equation,” IEEE Trans. Automat. Contr., vol. 37, pp. 1017-1018, 1992.
[16] C.H. Lee, “Solution bounds of the continuous and discrete Lyapunov matrix equations,” Journal of Optimization Theory and Applications, vol. 120, pp. 559-578, 2004.
[17] C. H. Lee, “Upper and lower bounds of the solutions of the discrete algebraic Riccati and Lyapunov matrix equations,” Int. J. Contr., vol. 68, pp. 579-598, 1997.