卡爾曼濾波器方法及其應用

Abstract

在動態系統中，參數估計是一直在研究的問題。假設我們知曉一些資訊，可以將參數估計得更準確嗎? 卡爾曼濾波器是一種解答。其背後的想法涉及了偏微分，統計方法，還有貝式法則下的後驗均值等。 本文中針對各種情況介紹了五種濾波器: 若模型是線性，卡爾曼濾波器是最優解；對低階非線性模型使用的擴展卡爾曼濾波器和無跡卡爾曼濾波器；針對高度非線性模型的粒子濾波器；解決極高維的集群卡爾曼濾波器。這五種濾波器的詳細推導都寫在各章中。 最後，在第七章，我們透過對牛頓系統，高度非線性系統，以及電阻抗斷層掃描反問題的數值模擬展現這些濾波器的成效。

In dynamic systems, parameter estimation is a persistent research challenge. Can we achieve more accurate parameter estimation if we have some prior information? The Kalman filter provides an answer to this question. Its underlying principles involve par tial differential, statistical techniques, and Bayesian inference, including posterior mean estimation. This paper introduces five types of filters for various scenarios: the Kalman filter is op timal for linear models; the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) are suitable for models with low-order nonlinearity; the Particle Filter (PF) is ap plicable to highly nonlinear models; and the Ensemble Kalman Filter (EnKF) is effective for addressing extremely high-dimensional systems. The derivation of these five filters is presented in detail in respective chapters. In the final chapter, Chapter 7, the effectiveness of these filters is demonstrated through numerical simulations of Newtonian systems, highly nonlinear systems, and Electrical Impedance Tomography (EIT) inverse problem.