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    Title: 指定頻段H∞濾波問題分析與設計
    Other Titles: Studies of H∞ filtering problem over finite frequency interval
    Authors: 林俊辰;Lin, Chun-Chen
    Contributors: 淡江大學電機工程學系博士班
    周永山
    Keywords: 非最小實現;non-minimal realization;濾波問題;廣義KYP引理;指定頻段;線性矩陣不等式;串疊型三角積分調變電路;H∞ filtering problem;GKYP Lemma;finite frequency interval;LMI;cascaded delta-sigma modulators
    Date: 2012
    Abstract: 為了能有效處理特定頻段的雜訊抑制問題,普遍的做法是應用H∞ 濾波方法,並引入權重函數(weighting function)來輔助設計。雖然此舉能有效地抑制指定頻段內的雜訊,但是適合的權重函數並不容易選取,並且常會導致高階之濾波器。對此,Iwasaki等人提出的廣義KYP引理(Generalized Kalman–Yakubovic–Popov,GKYP)可用來處理這一類針對指定頻段H∞ 增益的設計。然而,該原始成果不適合用於不確定系統分析以及濾波器(或控制器)設計(例如無限脈衝響應(Infinite Impulse Response,IIR))。其後,雖然有另一類型的H∞ 增益性能條件問世,可用於上述問題,然而這條件僅為充分,而且現有降階濾波器設計成果乃沿用了舊有設計技巧,應用於狀態空間不確定系統。因此,本論文針對此種技術進行改良,提出新型指定頻段H∞ 增益分析條件,並且針對不同類型不確定系統推導出新型指定頻段H∞ 降階濾波器合成條件。
    本論文針對離散時間不確定系統之指定頻段分析與設計問題進行研究。首先,吾人藉由投影引理(projection lemma)推導出可判斷有限頻段H∞ 性能要求是否滿足的線性矩陣不等式(linear matrix inequality,LMI)條件,可用於系統性能分析問題。與現有成果相比,本文之條件具較低的保守性。應用所提出之條件,我們提出新型降階濾波器之設計方法,並應用至三種不確定性系統,包含狀態空間多邊形系統、頻域多邊形系統與線性分式轉換型系統。其中我們充分應用了非最小實現(non-minimal realization)之觀念來處理降階設計中關鍵的維度問題。最後,將所提出之設計方法用於串疊型三角積分調變電路(cascaded delta-sigma modulator)。由於製造誤差以及元件自然限制,造成了類比電路與數位電路中的不匹配的問題,導致量化雜訊的遺漏,使得訊號品質降低。從系統層面的探討可知,此問題可視為濾波問題中的特例,即模式匹配(model-matching)問題。因此,將本文所提出的設計方法用於電路中數位濾波器之設計。模擬結果顯示,雜訊的轉移函數之波德圖增益在訊號的頻帶內的確能被有效地抑制,進而改善了電路之訊號與雜訊比(signal-to-noise ratio,SNR)性能。
    In order to deal with the H∞ filtering problem, a common way is to introduce the weighting functions into the design procedure. Although it is efficient to suppress the noise over a specified frequency interval, it is difficult to choose a suitable weighting function and the consequence is the high-order filters. For easing the problem, Iwasaki et al. has proposed an important result, i.e. GKYP lemma, which can be used to analyze the H∞ gain of a filtering system without uncertainty by assigning the frequency interval(s). However, there are some limitations on their results, for example, to analyze state-space polytopic uncertain system or design IIR-type filters. Therefore, this dissertation has studied the problems.
    The dissertation investigates the problems of filtering over finite frequency interval, including analysis and synthesis problems. At first, we derive new LMI conditions for the requirements of GKYP performance via projection lemma. Next, based on the proposed analysis conditions, we have proposed new methods to design reduced-order filters under three kinds of uncertain systems (i.e. state-space polytopic uncertain system, frequency-domain polytopic uncertain systems and (linear-fractional- transformation type uncertain systems). The key design concept is non-minimal realization, which is applied to deal with the dimensions of system and filter.
    Finally, the proposed methods have been employed to design the digital filter for improving the performance of cascaded delta-sigma modulators. Because the fabrication error and natural limitation on components, it results low order noise shaping and poor signal-to-noise ratio (SNR). From the viewpoint of system level, this kind of quantization leakage problem can be regarded as a special case of the filtering problem, i.e. model-matching problem. Therefore, the proposed methods are also employed to redesign the digital filter of the modulator such that the H∞ gain of the noise transfer function is minimized in the signal bandwidth. Consequently, the signal-to-noise ratio (SNR) performance is improved. We compare the proposed method with other existing designs and establish its efficacy.
    Appears in Collections:[電機工程學系暨研究所] 學位論文

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