在現代控制系統中，狀態觀測與估計是核心的基本問題。然而，現實世界的系統通常會受到程序雜訊與量測雜訊的影響，這使得直接基於量測值進行控制可能會產生偏差。卡爾曼濾波（Kalman filter）作為一種最佳線性狀態估計方法，已廣泛應用於工業控制、航空航天、機器人技術以及自動駕駛等領域。特別是當它與線性二次調節器（LQR）結合，組成線性二次高斯控制架構（LQG）時，即使在系統狀態無法完全觀測的情況下，也能提供最佳的控制性能。

在理論上，卡爾曼濾波存在連續時間（連續卡爾曼濾波，CKF）與離散時間（離散卡爾曼濾波，DKF）兩種形式，兩者各有其優點與局限性。連續時間濾波器通常能提供更高的估計精度，但計算成本也較高；而離散時間濾波器在計算上更具效率，不過其精度需要經過仔細的驗證。因此，比較連續與離散模型的性能、估計精度以及控制效果，具有重要的實際意義。

本研究以倒立擺系統作為說明範例。透過理論分析與數值驗證，探討了不同採樣間隔和雜訊特性對濾波性能與控制結果的影響。此外，本研究也檢視了數值穩定性與計算效率，旨在為實際的控制系統設計提供指引。本研究的主要貢獻包括：對控制系統中連續時間與離散時間卡爾曼濾波器進行了系統性的比較，並針對濾波器選擇與採樣間隔設計提出了建議。此外，本研究還將一維倒立擺模型擴展到了更高維度的情況。論文隨後的章節將介紹理論背景、模型推導、控制設計以及數值模擬結果。

In modern control systems, state observation and estimation are fundamental issues, while real-world systems are typically affected by process noise and measurement noise, making direct control based on measurements potentially inaccurate. The Kalman filter, as an optimal linear state estimation method, has been widely applied in industrial control, aerospace, robotics, and autonomous driving. In particular, when combined with a linear quadratic regulator (LQR) to form a linear quadratic Gaussian control framework
(LQG), it can provide optimal control performance even when the state of the system is not fully observable.
Theoretically, the Kalman filter exists in both continuous-time (Continuous Kalman Filter, CKF) and discrete-time (Discrete Kalman Filter, DKF) forms, each with its own advantages and limitations. The continuous-time filter generally offers higher estimation accuracy but at a greater computational cost, whereas the discrete-time filter is more computationally efficient, though its accuracy requires careful verification. Therefore, comparing the performance, accuracy of estimation and control effectiveness of continuous and discrete models is of significant practical importance.
This study uses the inverted pendulum system as an illustrative example. Through theoretical analysis and numerical validation, the effects of different sampling intervals and noise characteristics on filtering performance and control outcomes are investigated. In addition, numerical stability and computational efficiency are examined to provide guidance for practical control system design. The main contributions of this study include a systematic comparison between continuous-time and discrete-time Kalman filters in control systems, as well as recommendations for filter selection and sampling interval design. Furthermore, the one-dimensional inverted pendulum model is extended to higher-dimensional cases. The subsequent chapters of this thesis present the theoretical background, model derivation and control design and numerical simulation results

[1] Brian DO Anderson and John B Moore. Optimal control: linear quadratic methods. Courier Corporation, 2007.
[2] Rudolph Emil Kalman. “A new approach to linear filtering and prediction problems”. In: Journal of Basic Engineering (1960).
[3] Greg Welch, Gary Bishop, et al. “An introduction to the Kalman filter”. In: (1995).
[4] Chung-An Chen. “Optimal Control Design for Nonlinear Systems Using State-Dependent and Riccati Equation Methods”. MA thesis. Department of Applied Mathematics, National Chengchi University, 2026.
[5] Bang Xin Jiang, Yang Liu, Kit Ian Kou, and Zhen Wang. “Controllability and observability of linear quaternion-valued systems”. In: Acta Mathematica Sinica, English Series 36.11 (2020), pp. 1299–1314.
[6] Rudolf E Kalman et al. “On the general theory of control systems”. In: Proceedings first international conference on automatic control, Moscow, USSR. 1960, pp. 481–492.
[7] MLJ Hautus. “Stabilization controllability and observability of linear autonomous systems”. In: Indagationes mathematicae (proceedings). Vol. 73. North-Holland. 1970, pp. 448–455.
[8] S. B. Hsu and K. C. Chen. Ordinary Differential Equations With Applications. World Scientific Pub., 2023.
[9] Malo LJ Hautus. “Controllability and observability conditions of linear autonomous systems”. In: Indagationes mathematicae (proceedings). Vol. 72. 5. 1969, pp. 443–448.
[10] Bijoy Kumar Ghosh and Joachim Rosenthal. “A generalized Popov-Belevitch-Hautus test of observability”. In: IEEE transactions on automatic control 40.1 (2002), pp. 176–180.
[11] Steven L. Brunton and J. Nathan Kutz. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, 2019.
[12] Carlos Olalla, Ramon Leyva, Abdelali El Aroudi, and Isabelle Queinnec. “Robust LQR Control for PWM Converters: An LMI Approach”. In: IEEE Transactions on Industrial Electronics (2009).
[13] Russ Tedrake. Underactuated Robotics. Course Notes for MIT 6.832, 2023.
[14] Greg Welch1 and Gary Bishop. An Introduction to the Kalman Filter. Tech. rep. University of North Carolina at Chapel Hill, Department of Computer Science, 2006.
[15] Masoud Khodarahmi and Vafa Maihami. “A review on Kalman filter models”. In: Archives of Computational Methods in Engineering (2023).
[16] Rudolph E Kalman and Richard S Bucy. “New results in linear filtering and prediction theory”. In: Journal of Basic Engineering (1961).
[17] Harold W Sorenson. “Least-squares estimation: from Gauss to Kalman”. In: IEEE spectrum (2009).
[18] Yan Pei, Swarnendu Biswas, Donald S. Fussell, and Keshav Pingali. “An Elementary Introduction to Kalman Filtering”. In: arXiv preprint arXiv:1907.11883 (2019).
[19] Hui-Hsiung Kuo. White Noise Distribution Theory. CRC Press, 1996.
[20] M Darouach, M Boutayeb, and M Zasadzinski. “Kalman filtering for continuous descriptor systems”. In: Proceedings of the 1997 American Control Conference (Cat.No. 97CH36041). Vol. 3. IEEE. 1997, pp. 2108–2112.
[21] Wook Hyun Kwon, Pyung Soo Kim, and PooGyeon Park. “A receding horizon Kalman FIR filter for linear continuous-time systems”. In: IEEE Transactions on Automatic Control 44.11 (1999), pp. 2115–2120.
[22] Hui-Hsiung Kuo. White noise distribution theory. CRC press, 2018.
[23] Daniel Alazard. “Introduction to Kalman filtering”. In: France. ISAE-SUPAERO (FRANCE) (2005), pp. 1–74.
[24] Robert Grover Brown and Patrick Y. C. Hwang. Introduction to Random Signals and Applied Kalman Filtering. John Wiley & Sons, 1997.
[25] Robert Grover Brown and Patrick YC Hwang. “Introduction to random signals and applied Kalman filtering: with MATLAB exercises and solutions”. In: Introduction to random signals and applied Kalman filtering: with MATLAB exercises and solutions (1997).
[26] Shuuji Kajita, Fumio Kanehiro, Kenji Kaneko, Kazuhito Yokoi, and Hirohisa Hirukawa. “The 3D Linear Inverted Pendulum Mode: A simple modeling for a biped walking pattern generation”. In: Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). 2001.
[27] Thai Van Nguyen, Nguyen Huu Thai, Hai Tuan Pham, Tuan Anh Phan, Linh Nguyen, Hai Xuan Le, and Hiep Duc Nguyen. “Adaptive Neural Network-Based Backstepping Sliding Mode Control Approach for Dual-Arm Robots”. In: Journal of Control, Automation and Electrical Systems (2019).
[28] Richard M Murray, Zexiang Li, and S Shankar Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994.
[29] Mohinder S. Grewal and Angus P. Andrews. Kalman Filtering: Theory and Practice Using MATLAB. 3rd. Wiley, 2008.
[30] Azra Redzovic and Adnan Tahirovic. “State Estimation and Control for Continuous-Time Nonlinear Systems: A Unified SDRE-Based Approach”. In: Proceedings of the 2025 IEEE International Conference on Control, Decision and Information Technologies (CoDIT). 2025.
[31] Richard Bellman. “Dynamic programming”. In: science (1966).
[32] Bernt Øksendal. “Stochastic differential equations”. In: Stochastic differential equations: an introduction with applications. Springer, 2003.