# 臺灣博碩士論文加值系統

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 本文說明如何最小化特徵值的幅度與相位靈敏度合成的量，再示以分析數位控制器封閉迴路系統的穩定性。控制器參數因四捨五入運算和計算誤差所造成的定點運算之不確定性可以表示為字長 (word length) ；我們可透過小增益定理以及 Bellman- Grownwall 定理於定點統計模型基礎上之封閉迴路穩態標準。此一結合了與控制器參數有關的封閉迴路特徵值的幅度與相位靈敏度之量測值，為最小化並建立在混合於matrix-2/ Frobenius 之規範中；接著我們便可以從建立於最小化量測的代數分析方法裡，取得最佳化的相似轉換；利用此相似轉換式以及穩態標準，我們可以得到最少的字長。
 This note presents an approach to analyze the stability of the closed-loop system for digital controller implementations by minimizing a measure synthesized by the magnitude and phase sensitivities of eigenvalues.First, uncertainties of the controller parameters using fixed-point arithmetic caused by roundoff and computational errors are expressed as a function of word length. Then, a stability criterion of the closed-loop system based on fixed-point statistical model is derived by means of small gain theorem and Bellman-Grownwall Lemma. Thus, a measure that combines the magnitude and phase sensitivities of the closed-loop system eigenvalues with respect to controller parameters is constructed and is minimized in the sense of mixed matrix-2/Frobenius norms.Then an optimal similarity transformation is obtained from an analytically algebraic method based on this minimum measure. Using this transformation as well as the stability criterion, a least word length can be obtained. Finally, an example is performed to illustrate the effectiveness of the proposed scheme.
 中文摘要 iAbstract i目錄 i表目錄 i第一章 緒論 11.1 前言 11.2 研究方法 11.3 論文架構 1第二章 問題公式化以及定點運算 1第三章 定點控制器執行穩態分析 1第四章 封閉迴路系統裡幅度與相位靈敏度特徵值 1第五章 最佳化控制器執行轉換式 1第六章 實際的數值範例 1第七章 結論 1參考文獻 1誌謝 1
 [1] J.H. Wilkinson, Rounding Errors in Algebraic Processes, Englewood Cliffs, NJ: Prentice Hall, 1963.[2] B.-S. Chen and C.-T. Kuo, ”Stability analysis of digital filters under finite word length effects,” IEE Proceedings, vol. 136, Pt. G, pp. 167-172, 1989.[3] L.M. Smith and Jr., M.E. Henderson, ”Roundoff noise reduction in cascade realizations of FIR digital filters,” IEEE Trans. Signal Processing, vol. 48, pp. 1196-1200, 2000.[4] T.H. Hinamoto, S. Yokoyama, T. Inoue, W. Zeng, and W.-S. Lu, ”Analysis and minimization of L2-sensitivity for linear systems and two-dimensional state-space filters using general controllability and observability grammians,” IEEE Trans.Circuits Syst. I, vol. 49, pp. 1279-1289, 2002.[5] D. Williamson and K. Kadiman, ”Optimal finite wordlength linear quadratic regulation,” IEEE Trans. Automat. Contr., vol. 34, pp. 1218-1228, 1989.[6] K. Liu, R.E. Skelton, and K. Grigoriadis, ”Optimal controllers for finite wordlength implementation,” IEEE Trans. Automat. Contr., vol. 37, pp. 1294-1304, 1992.[7] G. Li, ”On the structure of digital controllers with finite word length consideration,” IEEE Trans. Automat. Contr., vol. 43, pp.689-693, 1998.[8] J. Wu, S. Chen, G. Li, R.H.S. Istepanian, and J. Chu, ”Shift and delta operator realizations for digital controllers under finite word length considerations,” IEE Proc.-Control Theory Appl., vol. 147, pp. 664-672, 2000.[9] J. Wu, S. Chen, G. Li, R.H.S. Istepanian, and J. Chu, ”An improved closed-loop stability related measure for finiteprecision digital controller realizations,” IEEE Trans. Automat. Contr., vol. 46, pp. 1162-1166, 2001.[10] Jinxin Hao, Gang Li, and Chunru Wan, ”Two classes of efficient digital controller structures with stability consideration” IEEE Trans. Automat. Contr., vol. 51, pp. 164-170, 2006.[11] Jun Wu, Gang Li, Sheng Chen, and Jian Chu ” A μ-based optimal finite-word-length controller design,” Automatica, vol. 44, pp. 3093-3099, 2008.[12] Huijun Gao and Tongwen Chen, ”A new approach to quantized feedback control systems,” Automatica, vol. 44, pp. 534-542, 2008.[13] Wen-Shyong Yu and Hsien-Ju Ko, ”Improved eigenvalue sensitivity for finite-precision digital controller Realizations via Orthogonal Hermitian Transform,” IEE Proc.-Control Theory Appl., vol. 150, pp. 365-375, 2003.[14] Hsien-Ju Ko and Wen-Shyong Yu, ”Guaranteed robust stability of the closed-loop systems for digital controller implementations via orthogonal Hermitian transform,” IEEE Trans. Syst., Man, Cybern. B, vol. 34, pp. 1923-1931, 2004.[15] R.E. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor and Francis,1998.
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