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研究生:賴家朗
研究生(外文):Jia-Lang Lai
論文名稱:關於二次大型動態系統模型化簡的近來發展
論文名稱(外文):On Some Recent Developments in Model Reduction of Large-Scale Quadratic Dynamical System
指導教授:王辰樹
指導教授(外文):Chern-Shuh Wang
學位類別:碩士
校院名稱:國立成功大學
系所名稱:數學系應用數學碩博士班
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:43
外文關鍵詞:Low RankGramianLyapunov equationTransfer functionHankel norm
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Abstract
The purpose of this paper is to acquire model reductions of large-scale quadratic dynamical systems. The idea for solving the problem is based on the relation between a quadratic system and its related enlarged linear system. That is, we transfer the second-order system to an enlarged first-order systems and obtain the second-order model reduction by converting the first-order model reduction to a second-order one. Next, we give a brief introduction for numerical methods of the first-order model reduction. Particularly, these methods can be categorized to two groups: (a) SVD based methods; (b)Krylov based methods. Finally, some numerical experiments are performed. The results of numerical implementation show that some serious round-off errors may occur when the first-order model reduction converts to the second-order one.
So, we conclude that the propose methods in this paper are eventually not good enough for the second-order reduction. However, a reliable and ecient method for the model reduction of a quadratic dynamical system is still under investigation.
Contents
1 Introduction 2
1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Reduction via rst-order balance and truncate . . . . . . . . . 3
1.1.2 Transfer rst-order form in second-order form . . . . . . . . . 4
1.2 First-order form linear dynamical system . . . . . . . . . . . . . . . 5
1.3 Measures of the accuracy of the approximation . . . . . . . . . . . . 6
2 SVD based approximation methods 7
2.1 Proper orthogonal decomposition (POP) method . . . . . . . . . . . 9
2.2 Balanced truncation method . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 The minimal control and largest observation energies . . . . . 9
2.2.2 Hankel operator, Gramians and Lyapunov equations . . . . . 10
2.2.3 Balance and truncate . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Optimal Hankel norm approximation method . . . . . . . . . . . . . 13
3 Projection based method 14
3.1 Approximation by moment matching . . . . . . . . . . . . . . . . . . 14
3.1.1 Lanczos procedure . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 Arnoldi procedure . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.3 Implicitly restarted Arnoldi and Lanczos methods . . . . . . 18
3.1.4 Rational Krylov method . . . . . . . . . . . . . . . . . . . . . 19
3.2 Krylov subspace method . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Alternating direction implicity iteration (ADI) and LR-ADI . 21
3.2.2 Smith method . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.3 Low rank square root method (LRSRM) . . . . . . . . . . . . 22
3.2.4 Low rank Schur method (LRSM) . . . . . . . . . . . . . . . . 23
4 Some properties of the second-order system 24
4.1 Second-order transfer function . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Second-order Gramian . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Second-order singular values and balancing . . . . . . . . . . . . . . . 28
4.4 Direct second-order reduction method . . . . . . . . . . . . . . . . . 29
5 Numerical experiments 30
6 Conclusions and future work 36
References 37
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