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。

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.

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