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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 |
| Creator | 陳慬瑜 Chen, Cin-Yu |
| Contributor | 許順吉<br>陳隆奇 Sheu, Shuenn-Jyi<br>Chen, Lung-Chi 陳慬瑜 Chen, Cin-Yu |
| Key Words | 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 |
| Date Issued | 4-Sep-2024 16:11:52 (UTC+8) |
| Summary | 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. |
| 參考文獻 | [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. 75 [10] M.H.A. Davis and R.B. Vinter. Stochastic Modelling and Control. Chapman and Hall, 1985. [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 |
| 資料來源 | http://thesis.lib.nccu.edu.tw/record/#G0111751005 |
| Type | thesis |
| dc.contributor.advisor | 許順吉<br>陳隆奇 | zh_TW |
| dc.contributor.advisor | Sheu, Shuenn-Jyi<br>Chen, Lung-Chi | en_US |
| dc.contributor.author (Authors) | 陳慬瑜 | zh_TW |
| dc.contributor.author (Authors) | Chen, Cin-Yu | en_US |
| dc.creator (作者) | 陳慬瑜 | zh_TW |
| dc.creator (作者) | Chen, Cin-Yu | en_US |
| dc.date (日期) | 2024 | en_US |
| dc.date.accessioned | 4-Sep-2024 16:11:52 (UTC+8) | - |
| dc.date.available | 4-Sep-2024 16:11:52 (UTC+8) | - |
| dc.date.issued (上傳時間) | 4-Sep-2024 16:11:52 (UTC+8) | - |
| dc.identifier (Other Identifiers) | G0111751005 | en_US |
| dc.identifier.uri (URI) | https://nccur.lib.nccu.edu.tw/handle/140.119/153579 | - |
| dc.description (描述) | 碩士 | zh_TW |
| dc.description (描述) | 國立政治大學 | zh_TW |
| dc.description (描述) | 應用數學系 | zh_TW |
| dc.description (描述) | 111751005 | zh_TW |
| dc.description.abstract (摘要) | 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。 | zh_TW |
| dc.description.abstract (摘要) | 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. | en_US |
| dc.description.tableofcontents | 1 Introduction 1 2 Martingales 3 2.1 Martingales 3 2.2 Almost Sure Convergence of Martingales 7 2.2.1 Upcrossings 7 2.3 Uniformly Integrable Martingales 10 2.3.1 Uniformly integrable property 10 2.3.2 Convergence of UI martingale 14 3 Application: The Kalman Filter 17 3.1 State-Space Model 17 3.2 Filtering Problem 19 3.3 Recursive Estimation: Bayesian Case 20 3.3.1 Some Preparations 20 3.3.2 Noisy Observation of a Single Variable 28 3.3.3 Consistency in RLS 29 3.4 Kalman filter 32 3.4.1 Overview of the Calculation 33 3.4.2 Derivations 34 3.4.3 Summary of key equations 38 3.4.4 The Best Linear Filter: minimize the mean-squared error 39 4 The Use of Kalman Filter 46 4.1 Innovation Representation of Kalman Filter 46 4.2 Control problem 51 4.2.1 Linear Quadratic Regulator (LQR) 51 4.2.2 Duality 57 4.2.3 Stochastic Linear Quadratic Regulator (SLQR) 59 4.2.4 Linear Quadratic Gaussian Regulator (LQG) 64 5 Conclusion 69 A Appendix: Gaussian Space 72 A.1 Gaussian Space 72 A.1.1 Joint Normal 72 A.1.2 Gaussian Spaces from Joint Normal 73 Bibliography 75 | zh_TW |
| dc.format.extent | 718099 bytes | - |
| dc.format.mimetype | application/pdf | - |
| dc.source.uri (資料來源) | http://thesis.lib.nccu.edu.tw/record/#G0111751005 | en_US |
| dc.subject (關鍵詞) | 鞅理論 | zh_TW |
| dc.subject (關鍵詞) | 卡爾曼濾波器 | zh_TW |
| dc.subject (關鍵詞) | 遞迴估計 | zh_TW |
| dc.subject (關鍵詞) | 隨機過程 | zh_TW |
| dc.subject (關鍵詞) | 控制理論 | zh_TW |
| dc.subject (關鍵詞) | 線性二次高斯問題 | zh_TW |
| dc.subject (關鍵詞) | 應用機率 | zh_TW |
| dc.subject (關鍵詞) | 隨機控制 | zh_TW |
| dc.subject (關鍵詞) | Martingale theory | en_US |
| dc.subject (關鍵詞) | Kalman filter | en_US |
| dc.subject (關鍵詞) | Recursive estimation | en_US |
| dc.subject (關鍵詞) | Stochastic processes | en_US |
| dc.subject (關鍵詞) | Control theory | en_US |
| dc.subject (關鍵詞) | Linear Quadratic Gaussian (LQG) problem | en_US |
| dc.subject (關鍵詞) | Applied probability | en_US |
| dc.subject (關鍵詞) | Stochastic control | en_US |
| dc.title (題名) | 離散時間鞅及其在卡爾曼濾波之應用研究 | zh_TW |
| dc.title (題名) | A Study of Discrete Time Martingales with Applications in the Kalman Filter | en_US |
| dc.type (資料類型) | thesis | en_US |
| dc.relation.reference (參考文獻) | [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. 75 [10] M.H.A. Davis and R.B. Vinter. Stochastic Modelling and Control. Chapman and Hall, 1985. [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. | zh_TW |