多級解耦合方法應用於快速卡門估測法則推導之研究

Abstract

本論文探討多級卡門估測器一些重要的理論及應用。在第一部份，我們發展沒有系統限制的多級卡門估測器理論。首先，我們提出一個能恢復傳統雙級卡門估測器最佳效能之最佳雙級卡門估測器。其基本設計概念在修改傳統之 bias-free 濾波器結構，使其包含一個假設之外界輸入信號。藉由此一 modified bias-free 濾波器，我們可以得到一個具有最小均方根誤差之最佳雙級濾波器。我們提出一個一致性的雙級U-V轉換技巧來設計此一最佳雙級濾波器。接著，我們延伸此一最佳雙級濾波器結果至具有任意移轉矩陣之系統，而所設計出的雙級濾波器，我們稱為一般雙級卡門濾波器。此一濾波器等效於單級卡門濾波器，但運算效能上更具優勢。其次，我們證明所提出之雙級解耦合技巧也適用於描述系統。最後，我們證明雙級濾波器結果可以很直接的延伸至多級濾波器結果，同時依然保有運算效能上之優勢。在第二部份，我們發展雙級卡門濾波器理論的新應用方向；此包含最小 階數濾波器及未知輸入解耦合濾波器之設計問題。我們證明所提出的一般雙級卡門濾波器可作為設計用來解決無量測雜訊估測問題之最小階數觀測器或估測器之一致性結構。為證明此點，我們考慮三種別人已提出的最小階數濾波器。另外，我們也提出一個修正的統計Luenberger觀測器來恢復傳統統計Luenberger觀測器之最佳效能。另一方面，所提出之雙級解耦合技巧也適用在未知輸入解耦合濾波器設計問題上。文中，我們提出一個最佳三級描述卡門估測器來解決具有未知輸入信號之估測問題。

In this dissertation, some important issues in the area of multi-stage Kalman filtering have been considered. These issues are divided into two parts. In the first part, the development of the optimal multi-stage Kalman filtering theory where no system constraint is imposed on the obtained estimators is proposed. Firstly, we derive an optimal two-stage Kalman estimator to recover the optimal performance of the conventional two-stage filters. The basic idea is to modify the conventional bias-free filter structure to include a fictitious external input. With this modified bias-free filter, the optimal two-stage filter in the MMSE sense is obtained. We propose a unified two-stage U-V transformation technique to derive the optimal two-stage filter. Next, we consider the problem of extending the optimal two-stage solution to more general systems whose state transition matrices have arbitrary forms. The obtained filter is named as the general two-stage Kalman filter which is equivalent to, but more efficient than, the single-stage Kalman filter. Then, we show that the proposed two-stage decoupling technique is also applicable for descriptor systems. As a final attempt, we show that the generalization of the two-stage solution to a multi-stage solution is straightforward and the computational advantage of the obtained multi-stage filter is also guaranteed. In the second part, we propose to apply the two-stage Kalman filtering theory for designing minimal-order filters and unknown-input decoupled filters. We show that the proposed general two-stage Kalman filter may serve as a unified structure to derive minimal-order observers or estimators which are intended to solve the singular measurement noise estimation problem. In this regard, three previously proposed minimal-order filters are considered. Furthermore, we propose a modified stochastic Luenberger observer structure to recover the optimal performance of the conventional stochastic Luenberger observer when applied to stochastic systems. On the other hand, the proposed three-stage decoupling technique can also be used to derive unknown-input decoupled filters. In the sequel, the optimal three-stage descriptor Kalman estimator with the extra stage to account for the unknown inputs estimation problem is derived.