H2迴路整型技術之研究

On the study of H2 loop shaping.

楊景富(Jing-Fu Yang)

指導教授 ： 周永山

淡江大學/工學院/電機工程學系碩士班/碩士(2014年)

https://doi.org/10.6846/TKU.2014.00973

摘要

迴路整型(loop shaping)技術為強健控制領域中的熱門研究主題， 而其中之一即為指定頻段H∞控制技術，其可應用廣義KYP (generalized Kalman-Yakubovich -Popov，GKYP)引理逕行設計，而毋需引用權重函數。相對地，指定頻段H2控制方法因受限於缺乏相關數學工具，並無這樣的直接設計方法。因此，引入權重函數的輔助設計仍為目前主要的近似解法。然而，權重函數的引入會造成兩個問題，其一為控制器階數遽增，增添了硬體實現之複雜度；其二會在擴增系統內增加無法移動的極點，進而可能會影響閉回路系統(不含權重函數)極點之配置。基於上述分析，本論文聚焦於傳統含權重函數之H2迴路整型技術的補強之道，特別是控制器階數以及極點是否可任意配置之問題。
本論文針對前述問題，提出三種設計方法。在設計條件有解的情況下，本論文的設計方法可使控制器的階數介於受控體階數與傳統方法所得階數(亦即受控體與權重函數階數之總和)之間。而在閉迴路極點的問題上，由於權重函數亦為設計的一部份，在可穩定與可偵測的前提之下，本論文所提方法可使閉迴路極點配置於指定區域(方法一)或任意指定位置(方法二、三)。本論文所提出的設計條件均為線性矩陣不等式，可運用現有數值軟體模擬迅速求解。最後，本論文針對一低頻雜訊問題，應用本文方法進行設計與模擬，結果證實我們所提出的方法的確有效。

關鍵字

H2控制 ； 迴路整型 ； 權重函數 ； 降階 ； 極點配置 ； 線性矩陣不等式

並列摘要

Loop shaping technique is a popular research topic in robust control, and one of them is finite frequency H∞ control technique, in which the generalized Kalman-Yakubovich-Popov (GKYP) lemma can be applied to synthesize controllers without introducing weighting functions. On the contrary there does not exist such a direct design method for finite frequency H2 control problems because there lacks mathematical tools like the GKYP lemma. Therefore, introducing the weighting functions into the design remains to be one of the major approaches that provide approximate solutions to the problems. However, introducing the weighting functions causes two problems. First, it increases the order of controllers, which in turn increases the implementation complexity in hardware. Second, the augmented system contains the poles of the weighting functions that can't be moved via any controller design. This could possibly affect the closed-loop poles design. Based on the above analysis, this thesis focuses on the reinforcement of the traditional weighted H2 loop shaping technique, in particular on the problems of controller order and closed-loop pole placement.
In this thesis three kinds of design methods are presented. When the solvability conditions are feasible, the order of controllers can be within two numbers, the order of the generalized plant without containing the weights and that of the generalized plant containing the weights. As for the pole placement problem, as the choice of weighting functions is a part of design, it will be shown that, under the stabilizability and detectability assumptions on the plant (without containing the weights), regional pole placement is possible by Method 1 and arbitrary pole placement is possible via Methods 2,3. The solvability conditions for computing the controllers are in terms of linear matrix inequalities (LMIs), which can be efficiently solved by computer software. Finally, the proposed methods are applied to a low-frequency noise rejection problem. The simulation results demonstrate the effectiveness of the proposed methods.

並列關鍵字

H2 control ； loop shaping ； weight functions ； reduced-order ； pole placement ； LMI

參考文獻

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