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系統識別號 U0026-2102202019283900
論文名稱(中文) 卡爾曼濾波器之正則化綜述
論文名稱(英文) A Survey on the Regularisation of the Kalman Filtering
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 108
學期 1
出版年 109
研究生(中文) 鄭伯信
研究生(英文) Po-Hsin Cheng
學號 L16061030
學位類別 碩士
語文別 英文
論文頁數 99頁
口試委員 指導教授-王辰樹
口試委員-陳旻宏
口試委員-彭振昌
口試委員-蔣俊岳
中文關鍵字 卡爾曼濾波  病態系統  正則化  姿態和航向參考系統 
英文關鍵字 Kalman filter  Ill-condition  Regularisation  Attitude and heading reference system 
學科別分類
中文摘要 卡爾曼濾波因其為線性隨機動態系統進行最小方差估計而廣為人知,也因此被視為隨機訊號處理最重要的基礎工具之一,並在諸多實際應用上都有卓著的貢獻。然而在遭遇病態系統的過濾時,濾波估計值的發散與低度準確性常令人困擾不已。在本篇論文當中,我們研究了有關卡氏濾波的若干正則化方法,其中包含了吉洪諾夫正則化卡氏濾波以及脊型卡氏濾波。另外,受限於系統本身的低度觀測性而導致估計的窒礙難行,我們關注了以驅動響應同步化實現系統可觀測性的構想。這是一篇貫穿了卡爾曼濾波的理論到相關應用的獨立文章,在後半段中更為航空器的姿態和航向參考系統提供了初步的介紹,並展現了卡爾曼濾波在該系統上扮演的重要角色。期盼此篇論文能夠為有興趣的讀者帶來一些幫助與啟發。
英文摘要 As known as an optimal linear minimum mean-squared error estimator, Kalman filter is one of the most significant and fundamental approaches to incorporate stochastic state tracking for many practical applications. Ill-condition, nevertheless, in the filtering process challenges the convergence and accuracy of the estimate from the correct state. In this dissertation, we study several regularisation methods of the Kalman filter, including the ridge-type Kalman filter and the Tikhonov regularised Kalman filter, followed by the idea of drive-response synchronisation. This is a self-contained paper in which we go through the Kalman estimation theory and its related application, the attitude and heading reference system (AHRS). Hopefully it is beneficial to interested readers.
論文目次 1 Introduction 1

2 Literature Review 4

3 The Kalman Filtering and State Estimation Problems 7
3.1 State Estimation and Bayesian Filtering . . . . . . . . . . . . . . . . . . . 7
3.2 Kalman Filter as a Minimum Mean-Squared Error Estimator . . . . . . . . 9
3.3 Further Interpretation of the Kalman Filtering . . . . . . . . . . . . . . . . 15
3.4 Extensions of the Kalman Filter: EKF and UKF . . . . . . . . . . . . . . . 22

4 Stability and Convergence of the Kalman Filter 32
4.1 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 The Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Asymptotic Properties of the Steady-State Optimal Filter . . . . . . . . . . 47

5 Regularisation Methods of the Kalman Filter 53
5.1 Review of State Estimation Problems . . . . . . . . . . . . . . . . . . . . 53
5.2 Ill-conditioning in the Kalman Filtering . . . . . . . . . . . . . . . . . . . 54
5.3 Tikhonov Regularisation and Tikhonov Regularised Kalman Filter . . . . . 56
5.4 Ridge-Type Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.5 Drive-Response Systems: An Attempt of Partial Synchronisation . . . . . . 66
5.6 Drive-Response Synchronisation Scheme: With Dynamic Output Feedback 71

6 Attitude and Heading Reference System and its Numerical Simulations 78
6.1 Spatial Rotation and the Attitude and Heading Reference System . . . . . . 78
6.2 Technique of Sensor Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 AHRS Kinematic Model of a Helicopter . . . . . . . . . . . . . . . . . . . 80

7 Concluding Remarks 88

References 95
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