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| Title: | 運用狀態依賴跟黎卡提方程方法對非線性系統的最佳控制設計
Optimal Control Design for Nonlinear Systems Using State-Dependent and Riccati Equation Methods |
| Authors: | 陳仲安
Chen, Chung-An |
| Contributors: | 郭岳承
Kuo, Yueh-Cheng
陳仲安
Chen, Chung-An |
| Keywords: | 非線性系統
最佳控制
Riccati 微分方程
連續代數Riccati 方程
狀態依賴方法
線性化方法
倒立擺系統
Nonlinear systems
Optimal control
Riccati differential equation
Continuous algebraic Riccati equation
State-dependent method
Linearization method
Inverted pendulum system |
| Date: | 2025 |
| Issue Date: | 2025-08-04 13:10:52 (UTC+8) |
| Abstract: | 本研究旨在探討非線性動態系統之最佳控制器設計，並提出兩種方法進行比較：線性化方法與狀態依賴方法。透過解連續 Riccati 微分方程（RDE）與代數 Riccati 方程（CARE），可獲得使控制律 u(t)=−Kx(t) 最小化二次型成本函數並保證系統穩定之回授增益矩陣 K。本文以倒立擺推車系統為案例，建立完整動態模型後，分別使用線性化模型與狀態相依模型進行模擬實驗。模擬結果顯示，線性化方法於初始狀態接近平衡點時具備良好控制性能，但當初始狀態與平衡點偏差過大時，控制效果顯著下降。相較之下，狀態相依方法無須依賴近平衡點假設，能在大範圍初始條件下提供穩定且精確的控制，但計算負擔較重。因此，在實務應用上應依據系統需求在控制精度與計算成本間取得平衡。
This study investigates the design of optimal controllers for nonlinear dynamic systems by comparing two approaches: the linearization method and the state-dependent method. The Riccati differential equation (RDE) and the continuous algebraic Riccati equation (CARE) are utilized to compute the feedback gain matrix K , enabling the control law u(t)=-Kx(t) to stabilize the system while minimizing a quadratic cost function. An inverted pendulum on a cart serves as a case study. After modeling the full nonlinear dynamics, both linearized and state-dependent models are implemented for simulation. Results show that the linearized model performs well when the initial state is near the equilibrium, but degrades significantly under large deviations. In contrast, the state-dependent method provides reliable and accurate control across a wider range of initial conditions, though at the cost of increased computational complexity. Therefore, the choice between the two approaches should balance control accuracy and computational efficiency according to specific application requirements. |
| Reference: | [1] H. Abou-Kandil, ed. Matrix Riccati equations: in control and systems theory. Systems & control. Basel ; Boston: Birkha”user, 2003. isbn: 9783764300852.
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[5] S. B. Hsu and K. C. Chen. Ordinary Differential Equations With Applications. World Scientific Pub., 2023. isbn: 9789811250743.
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[7] Mao-Lin NI. “Existence Condition on Solutions to the Algebraic Riccati Equation”. In: Acta Automatica Sinica 34.1 (2008), pp. 85–87. issn: 1874-1029. doi: https://doi.org/10.3724/SP.J.1004.2008.00085. url: https://www.sciencedirect.com/science/article/pii/S187410290860004X.
[8] D. Ray et al. Linear Control of Inverted Pendulum. http://praveen.cfdlab.net/teaching/control2013. IFCAM Summer School on Numerics and Control of PDE, IISc, Bangalore. 2013.
[9] Bradley A. Steinfeldt and Panagiotis Tsiotras. “A State-Dependent Riccati Equation Approach to Atmospheric Entry Guidance”. In: AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics. 2010. doi: 10.2514/6.2010-8378. url: https://dcsl.gatech.edu/papers/aiaa10a.pdf.
[10] Kemin Zhou, John Comstock Doyle, and K. Glover. Robust and optimal control. Upper Saddle River, N.J: Prentice Hall, 1996. isbn: 9780134565675. |
| Description: | 碩士
國立政治大學
應用數學系
112751012 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0112751012 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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