臺灣博碩士論文加值系統

本篇論文的內容主要在描述：在一個非對稱代數黎卡迪方程中，在適當的限制條件下，我們將可得到一個唯一的最小非負解，利用保持結構平方演算法(Structure-Preserving Doubling Algorithm)，我們可以找出此解。
並且我們知道經由此演算法所求得最小非負解的過程為二次收歛，再比較其他現有的演算法解非對稱代數黎卡迪方程，保持結構平方演算法大幅縮短計算機求解的所需時間。

In this paper we consider the nonsymmetric algebraic Riccati
equation (NARE) for which the four coefficient matrices form an M-matrix. Nonsymmetric algebraic Riccati equations of this type appear in applied probability and transport theory. The minimal nonnegative solution of these equations can be found by a structure-preserving doubling algorithm (SDA).

Contents
1.Introduction.................................1
1.1 Overview.................................1
1.2 Preliminaries............................2
2.Doubling Transformation......................4
3.SDA Algorithm................................7
4.Convergence Analysis of Algorithm SDA.......13
5.Parameter r.................................16
6.Numerical Examples..........................18
Referecnes....................................27

[1] B. De Moor and J. David, Total linear least squares and the algebraic Riccati equation, Systems Control Lett., 18:5(1992), 329-337.
[2] F.F. Clancey and I. Gohberg, Factorization of matrix functions and singular integral operators, In Operator Theory: Advances and Applications, 3 Birkh‥auser Verlag, Basel, 1981.
[3] I. Gohberg and M.A. Kaashoek, An inverse spectral problem for rational matrix functions and minimal divisbility , Integral Equations Operator Theory, 10:3(1987), 437-465.
[4] I.R. Petersen, Disturbance attenuation and H∞-optimization: Adesign method based on the algebraic Riccati equation, IEEE Trans. Automat. Control, 32:5(1987), 427-429.
[5] K.-M. Zhou and P.P. Khargonekar, An algebraic Riccati equation approach to H∞-optimization, Systems Control Lett., 11:2(1988), 85-91.
[6] D.S Bernstein and W.M. Haddad, LQG control with an H∞ performance bound: a Riccati equation approach, IEEE Trans. Automat. Control, 34:3(1989), 293-305.
[7] G. Tadmor, Worst-case design in the time domain: The maximum principle and the standard H∞ problem, Math. Control Signals Systems, 3:4(1990), 301-324.
[8] A. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, 2nd Edition, Springer-Verlag, London, 2000.
[9] P. Lancaster and L. Rodman, Solutions of the continuous and discrete-time algebraic Riccati equations: A review, In The Riccati Equation,S. Bittanti, A.J. Lamb and J.C. Willems eds., Springer-Verlag, Berlin, 1991.
[10] I. Gohberg and S. Rubinstein, Proper contractions and their unitary minimal completions, In Operator Theory: Advances and Applications, 33 Birkh‥auser Verlag, Basel, 1988.
[11] D. Hinrichsen, B.Kelb and A. Linnemann, An algorithm for the computation of the structured complex stability radius, Automatica J. IFAC, 25:5(1989), 771-775. [12] J. Juang and W.-W. Lin, Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices, SIAM J. Matrix Anal. Appl., 20:1(1999), 228-243.
[13] D. Williams, A ”potential-theoretic” note on the quadratic Wiener-Hopf equation for Q-matrices,In Seminar on Probability XV I , Lecture Notes in Math. 920, Springer-Verlag, Berlin, 1982, pages 91-94.
[14] J.D. Roberts, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Intern. J. Control, 32:4(1980), 677-687.
[15] A.J. Laub, Invariant subspace methods for the numerical solution of Riccati equations, In The Riccati Equation, S. Bittanti, A.J. Lamb and J.C. Willems eds., Springer-Verlag, Berlin, 1991.
[16] P. Lancaster and L. Rodman, Algebraic Riccati Equations, The Clarendon Press, Oxford, 1995.
[17] V. Mehrmann, The Autonomous Linear Quadraic Control Problem, Lecture Notes in Control and Information Sciences, 163, Springer-Verlag, Berlin,1991.
[18] C.-H. Guo, Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices, SIAM J. Matrix Anal. Appl., 23:1(2001), 225-242.
[19] C.-H. Guo and A.J. Laub, On the iterative solution of a class of nonsymmetric algebraic Riccati equations, SIAM J. Matrix Anal. Appl., 22:2(2000), 376-391.
[20] X.-X. Guo and Z.-Z. Bai , On the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, J. Comput. Math., (2005), In press.
[21] E.K.-W. Chu, H.-Y. Fan, and W.-W. Lin, A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations, to appear in Lin. Alg. Appl.
[22] E.K.-W. Chu, H.-Y. Fan, W.-W. Lin, and C.-S. Wang, A structure-preserving doubling algorithm for periodic discrete-time algebraic Riccati equations, preprint 2002-18,NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003.
[23] E.K.-W. Chu, H.-Y. Fan, and W.-W. Lin, A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations, preprint 2002-29, NCTS, National Tsing Hua University, Hsinchu 300, Taiwan, 2003.
[24] W.-W. Lin and S.-F. Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-Type matrix equations, preprint 2002-29,NCTS, National Tsing Hua University, Hsinchu 300,Taiwan,2004,http://math.cts.nthu.edu.tw/Mathematics/preprints/prep2004-10-012.pdf
[25] Horn, Roger A.Johnson, Charles R. Matrix analysis, Cambridge University Press,1985.
[26] Horn, Roger A.Johnson, Charles R. Topic in Matrix analysis, Cambridge University Press, 1991.
[27] A. Bennman and R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979.
[28] R.H. Bartels and G.W. Stewart, Solution of the matrix equation AX + XB = C , Comm. ACM, 15:9(1972), 820-826.
[29] J.A. Meijerink and H.A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput., 31:13(1977), 148-162.
[30] N.J. Higham, Accuracy and Stability of Numerical Algorithm, Society for Industrial and Applied Mathematics, 1996.
[31] M.S. Bazaraa, H.D. Sheraii, and C.M. Shetty, Nonlinear Programming, John Wiley and Sons, 1993.
[32] R.H. Bartels and G.W. Stewart, Solution of the matrix equation AX + XB = C, Comm. ACM, 15:9(1972), 820-826.
[33] G.H. Golub, S. Nash and C.F. Van Loan, A Hessenberg-Schur method for the problem AX + XB = C, IEEE Trans. Automat. Control, 24:6(1979), 909-913.
[34] Z.Z. Bai, X.X. Guo and S.F. Xu, Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations,2005.