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.

目錄
中文摘要............................................................................................................I
英文摘要.........................................................................................................III
目錄..................................................................................................................V
圖目錄..........................................................................................................VIII
表目錄..............................................................................................................X
第一章 緒論	1
	1.1 文獻回顧與研究動機	1
	1.2 論文架構	5
第二章 背景知識、問題敘述與重要定理	6
	2.1 背景知識與問題敘述	6
	2.2 重要定理	11
	2.3 數值模擬	18
第三章 多邊形系統之指定頻段濾波器設計	23
	3.1 前言	23
	3.2 狀態空間多邊形之降階濾波器設計	23
3.2.1  問題描述	23
3.2.2  降階濾波器設計概念	25
3.2.3  LMI解法	28
	3.3 頻域多邊形系統之降階濾波器設計	41
3.3.1  問題敘述	41
3.3.2  降階濾波器設計概念	42
3.3.3  LMI解法	43
	3.4 研究方法之探討與推廣	55
3.4.1  研究方法之探討	55
3.4.2  多頻段的濾波問題	56
	3.5 數值模擬	57
第四章 LFT不確定系統之指定頻段濾波器設計	60
	4.1 前言	60
	4.2 強健濾波器設計	60
4.2.1  問題描述	60
4.2.2  問題轉換	65
4.2.3  濾波器與廣義乘數運算子之設計條件	71
4.2.4  濾波器之LMI解法	74
4.2.5  廣義乘數運算子之LMI解法	74
4.2.6  指定頻段μ合成演算法	83
第五章 三角積分調變電路設計	85
	5.1 前言與背景	85
	5.2 問題描述	86
	5.3 串疊2-1 ΔΣ調變電路設計	92
第六章 結論與未來研究方向	104
參考文獻	106
附錄...............................................................................................................114
附錄A.	定理2.1之證明	114
附錄B.	定理3.1之證明	116
附錄C.	定理3.2之證明	122
附錄D.	定理3.3之證明	127
附錄E.	定理3.4之證明	135
附錄F.	定理3.5之證明	138
附錄G.	定理3.6之證明	140
附錄H.	引理4.1之證明	144
附錄I.	引理4.2之證明	144
附錄J.	引理4.3之證明	145
附錄K.	引理4.4之證明	148
附錄L.	定理4.1之證明	148
附錄M.	定理4.2之證明	149
附錄N.	定理4.3之證明	151
附錄O.	三種不同廣義受控體模型之系統矩陣推導	154

圖目錄
圖2.1	濾波問題	8
圖2.2	標準 架構	9
圖2.3	系統之波德增益圖；δ = 0.45	21
圖3.1	濾波問題架構	23
圖3.2	濾波問題	25
圖3.3	濾波問題架構	41
圖3.4	條件(3.40)-(3.42)之系統架構	43
圖3.5	濾波問題	58
圖4.1	濾波系統架構	60
圖4.2	一般分析/合成架構	61
圖4.3	 迴路	63
圖4.4	 架構	66
圖4.5	架構轉換(一)	66
圖4.6	架構轉換(二)	66
圖4.7	架構轉換(三)	67
圖4.8	架構轉換(四)	67
圖4.9	架構轉換(五)	67
圖4.10	 架構	68
圖4.11	系統伴隨廣義乘數運算子	69
圖4.12	小增益架構	71
圖5.1	多級串疊架構ΔΣ調變器	86
圖5.2	一階與兩階之 功率譜密度圖	88
圖5.3	串疊2-1 ΔΣ調變電路	90
圖5.4	單頻段設計之效能	94
圖5.5	匹配誤差函數之波德增益圖	96
圖5.6	功率頻譜圖	97
圖5.7	放大器增益變動對照SNR效能	98
圖O.1	以乘法性誤差之架構轉換圖	158

表目錄
表2.1	xi之設定選擇	14
表2.2	各頻段之分析結果 (rad/s)； 	19
表2.3	各頻段之分析結果 (rad/s)； 	19
表2.4	各方法之矩陣P特徵值； 	20
表3.1	三種不同SA之設定所產生之 增益	58
表3.2	(3.22)式中SA之設定與其他兩種設定之比較	59
表5.1	設計方法總表	92
表5.2	設計參數	99
表5.3	所得濾波器、alpha_M與SNR值	102

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