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 本論文主要是以可轉換連續型與散型控制系統的凱利變換為基礎，運用並根據保結構法(SDA)的概念，發展一套G-SDA法，來求解廣義連續型及離散型的黎卡迪方程。文中依各類型，分別列舉在矩陣E條件數偌大的情況下，本法的數值結果，以說明G-SDA法的有效性與特色。
 In this paper we extend the structure-preserving doubling algorithm (SDA) to compute the symmetric positive semi-definite solutions of the generalized continuous as well as discrete algebraic Riccati equations. Our main idea is to relate continuous and discrete time control systems based on the generalized Caley transformation.In the end, we select some examples to illustrate that the G-SDA performs better than the MATLAB commands in the control toolbox.
 1 Introduction...1 1.1 Generalized Continuous-Time Algebraic Riccati Equations 1.2 Generalized Discrete-Time Algebraic Riccati Equaitons2 G-SDA Algorithm for G-CAREs...63 G-SDA Algorithm for G-DAREs...104 Convergence of G-SDA Algorithm...185 Numerical Experiments for G-CAREs...206 Numerical Experiments for G-DAREs...287 Conclusions...32
 [1] E. K.-W. Chu, H. Y. Fan, W. W. Lin, and C.-S.Wang (2004). A Structure-PreservingAlgorithm for Periodic Discrete-Time Algebraic Riccati Equations, Int. J. Control,Vol. 77, no. 8, pp. 767-788.[2] E. K.-W. Chu, H. Y. Fan, and W. W. Lin (2005). A Structure-Preserving DoublingAlgorithm for Continuous-Time Algebraic Riccati Equations, Lin. Alg. Appl., Vol.396, pp. 50-80.[3] E. K.-W. Chu, H. Y. Fan, and W. W. Lin (2003). A Generalized Structure-PreservingDoubling Algorithm for Generalized Discrete-Time Algebraic Riccati Equations,preprint 2002-29, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan.[4] V. L. Mehrmann (1991). The Autonomous Linear Quadratic Control Problem: Theoryand Numerical Solution, Berlin, Springer-Verlag.[5] V. Mehrmann (1996). A Step Toward A Unified Treatment of Continuous and DiscreteTime Control Problems, Fakult?at f?ur Mathematik, TU Chemnitz-Zwickau,09107 Chemnitz, FRG.[6] W. W. Lin and S. F. Xu (2003). Convergence Analysis of Strucure-Preserving DoublingAlgorithms for Riccati-Type Matrix Equations, National Tsing Hua University,Hsinchu 300, Taiwan.[7] G. H. Golub and C. F. Van Loan (1996). Matrix Computations, 3rd ed., Baltimore,Johns Hopkins University Press.[8] K. Zhou, J. C. Doyle, and K. Glover (1996). Robust and Optimal Control, New Jersey,Prentice Hall.[9] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty (1993). Nonlinear Programming:Theory and Algorithms, 2nd ed., New York, Wiley.[10] N. J. Higham (1996). Accuracy and Stability of Numerical Algotirhms, Philadelphia,Society for Industrial and Applied Mathematics.[11] P. Benner, A. J. Laub, and V. Mehrmann (1995). A Collection of BenchmarkExamples for The Numerical Solution of Algebraic Riccati EquationsI: Continuous-Time Case, Tech. Rep. SPC 95 22, Fakult?at f?ur Mathematik,TU Chemnitz-Zwickau, 09107 Chemnitz, FRG. Retrieved formhttp://www.tu-chemnitz.de/sfb393/spc95pr.html.[12] P. Benner, A. J. Laub, and V. Mehrmann (1995). A Collection ofBenchmark Examples for The Numerical Solution of Algebraic Riccati Equa-tions II: Discrete-Time Case, Tech. Rep. SPC 95 23, Fakult?at f?ur Mathematik,TU Chemnitz-Zwickau, 09107 Chemnitz, FRG. Retrieved formhttp://www.tu-chemnitz.de/sfb393/spc95pr.html.
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