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題 名 | 發展應用正交多項式于分析含有週期係數的動態系統之新方法 |
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作 者 | 吳德和; | 書刊名 | 國立屏東技術學院學報 |
卷 期 | 1 1992.06[民81.06] |
頁 次 | 頁62-72 |
分類號 | 310.14 |
關鍵詞 | 分析; 正交; 多項式; 係數; 動態系統; 週期; |
語 文 | 中文(Chinese) |
中文摘要 | 本研究提出一套革新而且又效率的數值分析技巧,針對於具有週期性 係數的動態系統作穩定性及反應變化之分析與預測。本方法基本上是根據將系統 之解及週期矩陣表示成Chebyshev正交多項式之展開式。如此的主要目的,乃是 將原常微分方程式化簡成為一簡單的線性代數方程組,於是可以輕易獲得解答。 然後,再與Floquet理論相結合而求得暫態矩陣在一週期末。經過分析Floquet暫 態矩陣,便能提供了此系統的穩定性以及任何時刻的反應變化的訊息。兩個公式 被提出,其一為適用於一次常微分系統,共二為可直接應用於二次常微分系統。 簡單的Mathieu方程式被用於示範。數值分析的結果包括了闡明本方法的準確性 及省時優點,尤其是直接二次積分方法。期則本方法於未來能成為一項有用的工 具,用於分析與預測週期系統之穩定性,反應變化以及週期性控制問題。 |
英文摘要 | This study proposes an mnovatiye and efficient numerical scheme for the stability analysis oflinear systems with periodically varying parameters. The approach is based on the idea thatthe state vector and the periodic matrix of the system can be expanded in terms of Chebyshevpolynomials over the principal period. Such an expansion reduces the original problem to a setof linear algebraic equations from which the solution in the interval of one period can be obtained. Further, the technique is combined with the Floquet theory to yield the transition matrix at theend of one period and provide the stability conditions via the eigen-analysis procedure. Twoformulations are presented. The first is suitable for a set of equations written in the state spaceform, while the second can be applied directly to a set of second order equations. Theapplication is demonstrated through Mathieu's equation. The numerical results thus obtainedare compared with those obtained from standard numerical codes available in the IMSL library.An error-bound analysis is also included. This study concludes that the suggested schemes notonly provide accurate results but are also computationally very efficient. In particular, the directformulation is found to be several times faster than the standard numerical codes. It isanticipated that the proposed technique would soon become a viable tool in the numerical analysisof large-scale periodic systems. |
本系統中英文摘要資訊取自各篇刊載內容。