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| Title: | 離散時間鞅及其在卡爾曼濾波之應用研究
A Study of Discrete Time Martingales with Applications in the Kalman Filter |
| Authors: | 陳慬瑜
Chen, Cin-Yu |
| Contributors: | 許順吉
陳隆奇
Sheu, Shuenn-Jyi
Chen, Lung-Chi
陳慬瑜
Chen, Cin-Yu |
| Keywords: | 鞅理論
卡爾曼濾波器
遞迴估計
隨機過程
控制理論
線性二次高斯問題
應用機率
隨機控制
Martingale theory
Kalman filter
Recursive estimation
Stochastic processes
Control theory
Linear Quadratic Gaussian (LQG) problem
Applied probability
Stochastic control |
| Date: | 2024 |
| Issue Date: | 2024-09-04 16:11:52 (UTC+8) |
| Abstract: | 在此篇論文中介紹了鞅理論與卡爾曼濾波器的關聯，可以做為探索應用機率與隨機控制的入門。從一些關於鞅的定義以及基本性質開始，接著闡述這些機率的觀念如何應用在卡爾曼濾波上。當系統有噪音干擾時，卡爾曼濾波器是一個最基本的方法可以提供對於系統狀態的估計。接著介紹一些隨機控制的內容，特別是卡爾曼濾波提供的估計可以視作一個具體的應用在線性二次高斯問題（LQG)上 。此篇論文為初學者提供一個清晰、易懂的基礎，連結機率論的核心概念與濾波和控制的應用。
This thesis serves as an introductory guide for beginners in applied probability and stochastic control, focusing on the connection between martingale theory and the Kalman filter. By starting with the basics of martingales, the thesis explains how these probabilistic concepts can be applied to understand the Kalman filter, a fundamental tool for estimating the state of a system in the presence of noise. The thesis then introduces control theory, specifically the Linear Quadratic Gaussian (LQG) problem, to demonstrate the practical use of the Kalman filter in optimizing system performance. This work aims to provide a clear and accessible foundation for those new to these topics, linking key ideas in probability with their applications in filtering and control. |
| Reference: | [1] Masanao Aoki. Optimization of Stochastic Systems: Topics in Discrete-Time Systems. Academic Press, 1989.
[2] Karl J Åström. Introduction to Stochastic Control Theory. Courier Corporation, 2012.
[3] Krishna B Athreya and Soumendra N Lahiri. Measure Theory and Probability Theory, volume 19. Springer, 2006.
[4] Stephen P. Boyd. Ee363: Lecture slides 1. linear quadratic regulator: Discrete time finite horizon. https://web.stanford.edu/class/ee363/lectures/dlqr.pdf, 2008. Accessed: 2024-08-25.
[5] Stephen P. Boyd. Ee363: Lecture slides 10. linear quadratic stochastic control with partially observed states. https://web.stanford.edu/class/ee363/lectures/lqg.pdf, 2008. Accessed: 2024-08-25.
[6] Stephen P. Boyd. Ee363: Lecture slides 5. linear quadratic stochastic control. https://web.stanford.edu/class/ee363/lectures/stoch_lqr.pdf,
2008. Accessed: 2024-08-25.
[7] Stephen P. Boyd. Ee363: Lecture slides 8. the kalman filter. https://stanford.
edu/class/ee363/lectures/kf.pdf, 2008. Accessed: 2024-08-25.
[8] Peter E. Caines. Linear Stochastic Systems. John Wiley & Sons, 1988.
[9] Xu Chen and Masayoshi Tomizuka. Lecture notes for uc berkeley advanced control systems ii (me233). Accessed: 2024-08-25. http://www.me.berkeley.edu/ME233/sp14, 2014.
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[10] M.H.A. Davis and R.B. Vinter. Stochastic Modelling and Control. Chapman and Hall,
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[11] Rick Durrett. Probability: Theory and Examples. Self-published, 2019.
[12] SvanteJanson. Gaussian Hilbert Spaces. Number129.CambridgeUniversity Press, 1997.
[13] Jean-François Le Gall. Brownian Motion, Martingales, and Stochastic Calculus. Springer, 2016.
[14] Hamed Masnadi-Shirazi, Alireza Masnadi-Shirazi, and Mohammad-Amir Dastgheib. A step by step mathematical derivation and tutorial on kalman filters. arXiv preprint
arXiv:1910.03558, 2019.
[15] Ian R. Reid. Estimation ii. https://api.semanticscholar.org/CorpusID: 7460075, 2010. Accessed: 2024-08-25.
[16] Maria Isabel Ribeiro. Kalman and extended kalman filters: Concept, derivation and properties. 2004.
[17] Greg Welch, Gary Bishop, et al. An introduction to the kalman filter. 1995.
[18] David Williams. Probability with Martingales. Cambridge University Press, 1991. |
| Description: | 碩士
國立政治大學
應用數學系
111751005 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0111751005 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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