具Vidyasagar結構補償器之穩健設計

Abstract

對於有限維度的線性系統而言，目前所有穩定補償器皆可表示成眾所皆知的Youla- Kucera參數式。系統的穩定性要滿足Diophantine方程式，然而Youla-Kucera參數式只滿足Diophantine identity，但是Vidyasagar’s structure滿足Diophantine方程式，且這兩種補償器只有一個共同解，因此這兩種補償器被擴充且被擴充的兩者皆滿足Diophantine方程式，另外會用一條方程式來說明Youla-Kucera參數式、Vidyasagar’s structure與被擴充的兩者之間的關係。Vidyasagar’s structure是一擁有良好性質的觀測控制補償器，就回授的性質而言，它可以轉換成較複雜的Youla-Kucera參數式且它裡面的觀測器等效於Youla-Kucera參數式裡的中央控制器的左互質分式，後面的性質使得Vidyasagar’s structure裡的觀測器設計可繼承迴路成型設計程序的所有性質，前面的性質說明了Vidyasagar’s structure是一個富有彈性的架構，即Vidyasagar’s structure的控制器可用代數的方法來設計，以應付追蹤控制、輸入與輸出步階干擾或輸入與輸出已知弦波干擾。Vidyasagar’s structure設計程序會被提出，有三個步驟，首先根據基本的開迴路成型原則，用權重矩陣W1與W2來成型名目受控體的奇異值，再者求解正規化互質式 的問題得到穩定成型受控體的中央穩健控制器，Vidyasagar’s structure的觀測器則可由此中央控制器的左互質分式而得，最後，Vidyasagar’s structure的控制器可用來協調穩健的穩定性與效能性，如此一來，Vidyasagar’s structure可同時處理輸入與輸出的步階干擾及輸入與輸出已知弦波的干擾。另外會提出由Vidyasagar’s structure擴充而來的補償器的設計程序，使應付追蹤系統，此擴充的補償器除了有Vidyasagar’s structure的H參數，還有Qv 參數，對於最小相位且方正形的受控體，H與Qv可用反動態來設計;對於最小相位且寬胖形的受控體，H 用代數方法設計，Qv用右反動態來設計; 對於非最小相位且方正或寬胖形的受控體，H 用代數方法設計，Qv的問題被轉化模型匹配的問題，模型匹配的問題化簡成次最佳化的Nehari問題，最後求得Qv.

All stabilizing controllers known to date involve well-known Youla-Kucera parameterization for finite-dimensional linear systems. Stability is guaranteed when the Diophantine equation is satisfied. However, the Youla-Kucera parameterization satisfies the Diophantine identity, but the Vidyasagar’s structure satisfies the Diophantine equation. They have only one common solution. Hence, they are expanded so that the two expanded compensators satisfy the Diophantine equation. An equation is introduced to describe the relationships among the Youla-Kucera parameterization, the Vidyasagar’s structure and the two expanded compensators. The Vidyasagar’s structure, an observer-controller compensator, has advantageous properties. It can be in the form of the more complicated Youla-Kucera parameterization in terms of feedback property, and the observer in the Vidyasagar’s structure equals the left coprime factorization of the central controller in Youla-Kucera parameterization. The latter property yields that the observer design in the Vidyasagar’s structure can inherit all properties in the loop shaping design procedure. The former property causes the Vidyasagar’s structure to be a flexible control scheme. That is, the controller in the Vidyasagar’s structure can be designed algebraically for tracking control or input-output step or known sinusoidal disturbances. The Vidyasagar’s structure design procedure is proposed and has three stages: 1) using weighting matrices W1 and W2 to shape the singular values of the nominal plant follows the elementary open-loop shaping principles; 2) the normalized coprime factor problem is used to find a robust central controller stabilizing this shaped plant, and the observer is obtained from the left coprimeness of the central controller; 3) the H parameter in the controller of the VS is used as a tradeoff between robust stability and performance. Then the input-output step and known sinusoidal disturbances can be handled simultaneously. Moreover, the design approach in the expansion of Vidyasagar’s structure is proposed for a tracking system. The expansion of Vidyasagar’s structure has an extra parameter Qv in addition to the parameter H also possessed by Vidyasagar’s structure. For the minimum-phase square plant, the inverse dynamic method can be applied to H and Qv. For the minimum-phase wide plants, H and Qv are designed using the algebraic method and the right inverse dynamic method, respectively. For the non-minimum-phase square and wide plants, the H is designed by the same algebraic method and Qv is solved as the model matching problem, which is then reduced to a suboptimal Nehari problem.