鎖相迴路之濾波器設計:LMI方法

Abstract

本論文提出了一個新型的迴路濾波器設計。在考慮/不考慮時間延遲因素之下，應用多目標的控制技術來設計濾波器，以同時達成多種目標。設計目標包含了降低雜訊頻寬、好的暫態響應(小的安定時間、小的最大超越量)以及大的增益邊限與相位邊限。 在濾波器的設計中，利用近年來發展的凸最佳化技巧(亦即線性矩陣不等式)將設計參數作適當的調整，以協調各項互相衝突的性能指標。我們的方法另有一項特色是可預先設定若干迴路濾波器的極點與零點，因此可以設計別具意義的PI形式迴路濾波器，其極點全部在原點。此外，我們提出的方法適用於任意階數的鎖相迴路濾波器設計。 在數值模擬的部分，本論文首先使用非線性的模型來模擬設計完成的迴路。在不考慮時間延遲因素的情形，將本論文所提出的PI形式濾波器設計，與一般的設計作比較。而在考慮時間延遲因素的情形，則比較本論文所提出的兩種濾波器設計的效果。最後，利用更逼真的模型來進一步驗證本論文所提出的設計之實用性。

A new loop filter design method for phase locked loops (PLLs) is presented subject to loop with/without pure delay, which employs multi-objective control technique to deal with the various design objectives: small noise bandwidth, good transient response (small settling time, small overshoot), and large gain and phase margins. Trade-off among the conflicting objectives is made via recently developed convex optimization skill (i.e., linear matrix inequality, LMI) in conjunction with appropriate adjustment of certain design parameters. One salient feature of the proposed method is that it allows one to specify some filter poles and zeros in advance, including the special case of PI form filter of which all the filter poles are at the origin. Moreover, the proposed method is applicable to PLL of any order. Numerical simulation on nonlinear PLL model is performed. For the case without considering delay, comparisons of our results with that by generic PI form loop filter design are made. As for the case taking time delay into account, comparisons between two proposed methods are made. Finally, more realistic model is employed to simulate the resulting PLLs for demonstration of the effectiveness of the proposed method in practice.