指定頻段μ合成定階控制器設計

On The Design of Fixed-Order μ Synthesis in Finite Frequency Domain

廖文翔(Wen-Shiang Liao)

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.2014.00874

摘要

強健控制器設計是控制理論中一個重要的研究領域，主要討論系統中具有不確定的參數(或動態模式誤差)或外部環境干擾的情況下，如何設計控制器使此擾動系統保持穩定，而且有良好的強健性能。μ合成理論是解決這方面問題的利器。然而，其會導致高階數的控制器，增添硬體實現之困難。而且當雜訊來自指定頻段，有時效果不佳。因此，本論文將聚焦於閉回路系統特定頻段的強健H∞性能，提出固定階數的控制器設計方法。
本論文研究連續時間、線性、非時變、含不確定參數的擾動系統的強健性能控制問題，其中性能意指特定頻段的H∞性能。本論文結合Iwasaki等人提出的廣義KYP (Generalized Kalman- Yakubovich-Popov, GKYP) 引理與μ理論，提出兩種類型的強健控制器設計方法，使得閉迴路系統具有適度的穩定邊界，並且儘量最小化擾動系統在指定頻段的 範數。我們發現所謂的尺度因子扮演著如同權重函數之角色，可用來居中協調穩定邊界與指定頻段的強健H∞性能。不同於現存μ合成方法，在我們所研擬的新型μ合成方法中，控制器與廣義乘數運算子的階數在求解過程皆可維持固定。而且，相關條件之求解等同求解線性矩陣不等式(Linear Matrix Inequalities，LMIs)，其可由現存軟體有效解得。

關鍵字

強健控制 ； μ合成 ； 參數擾動 ； GKYP引理 ； 尺度因子 ； 線性矩陣不等式 ； 穩定邊界

並列摘要

The design of robustly stabilizing controllers is an important issue in the control area, which focuses on synthesizing a controller to keep the closed-loop system stable and possess a robust performance in the presence of real parametric uncertainty (or un-modeled dynamics). μ synthesis is a powerful tool to solve this problem. However, the method would lead to high-order controllers, which increases the implementation complexity in hardware. In addition, the resulting performance may not be satisfactory, for example when the method is applied to a disturbance attenuation problem in which the disturbances were known coming from a particular finite frequency range. Therefore, this thesis aims at providing a fixed-order controller design that improves the robust finite frequency H∞ performance of the perturbed system.
Specifically, this thesis investigates the robust performance control problem for continuous-time linear time-invariant systems with uncertain real parameters. We combine the GKYP Lemma and μ theory to derive two methods, each of which renders the closed-loop system a desired stable margin and in addition, minimizes the robust H∞ norm in a finite frequency range. We found that the so-called scaling factor acts like a weighting function. It tradeoffs between stability margin and robust finite frequency H∞ performance. In contrast with the existing μ synthesis methods, the new method has the feature that the order of the controllers and that of the generalized multipliers were kept fixed during the process of design. On the other hand, the relevant solvability conditions are all in terms of linear matrix inequalities (LMIs) which can be efficiently solved by computer software.

並列關鍵字

Robust control ； μ Synthesis ； real parametric uncertainty ； GKYP Lemma ； Scaling factor ； LMI ； Stability Margin

參考文獻

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