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| 研究生： |
陳仲安 Chen, Chung-An |
|---|---|
| 論文名稱： |
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 |
| 指導教授： |
郭岳承
Kuo, Yueh-Cheng |
| 口試委員: |
郭岳承
Kuo, Yueh-Cheng 黃聰明 Huang, Tsung-Ming 謝世峰 Shieh, Shih-Feng |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2025 |
| 畢業學年度： | 113 |
| 語文別： | 英文 |
| 論文頁數： | 52 |
| 中文關鍵詞： | 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 |
| 外文關鍵詞： | Nonlinear systems, Optimal control, Riccati differential equation, Continuous algebraic Riccati equation, State-dependent method, Linearization method, Inverted pendulum system |
| 相關次數： | 點閱：22 下載：6 |
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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.
致謝 i
中文摘要 ii
Abstract iii
Contents iv
List of Figure vi
1. Introduction 1
2. Prerequisite Knowledge 2
2.1 Reachability and Controllability 2
2.2 Stabilizability and Detectability 6
3. Riccati Differential Equation and Riccati Equation 7
3.1 Riccati Differential Equation 7
3.2 Continuous Algebra Riccati Equation 10
4. Case Study: Inverted Pendulum on a Cart 15
4.1 Modeling 15
4.2 Linearized Model 18
4.3 State-Dependent Model 19
5. Controllability of Linearized and State-Dependent Model 20
5.1 The Controllability of Linearized Model 20
5.2 The Controllability of State-Dependent Model 22
6. Numerical Experiments 27
6.1 Numerical Experiments for Linearized Model 27
6.2 Numerical Experiments for State-Dependent Model 29
6.3 Comparison 34
6.4 Conclusion and Discussion 39
References 40
Appendix A Proof from Section 2 41
A.1 Proof of Theorem 2.1 41
A.2 Proof of Theorem 2.2 43
A.3 Proof of Theorem 2.3 44
A.4 Proof of Theorem 2.4 46
Appendix B Proof from Section 3 48
B.1 Proof of Theorem 3.2 48
B.2 Proof of Theorem 3.3 48
B.3 Proof of Theorem 3.4 49
B.4 Proof of Theorem 3.5 49
B.5 Proof of Theorem 3.6 51
B.6 Proof of Theorem 3.7 52
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