H∞控制器與狀態回授估測器設計-齊次多項式平方和檢測法

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姓名

謝宜哲(Yi-Che Hsieh) 查詢紙本館藏

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機械工程學系

論文名稱

H∞控制器與狀態回授估測器設計-齊次多項式平方和檢測法
(H∞ Controller and State Feedback Observer Design based on Homogeneous SOS)

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摘要(中)

本論文主要探討在連續時間下，考慮一加入干擾的非線性系統，再利用H∞性能指標來探討非線性系統的穩定性與性能影響，而我們藉由尤拉齊次定理分別推導出H∞控制系統與H∞狀態回授估測系統的李亞普諾夫檢測條件，因利用尤拉齊次多項式定理的關係，我們可以避免了李亞普諾夫函數V(x)對時間t微分所產生的Q(x)之微分項。

　　最後，再將所得之李亞普諾夫函數經平方和檢測方法改寫為純量形式，以平方和檢測法去檢驗其系統之穩定性，藉此確保我們的閉迴路系統的穩定性與狀態回授估測器追蹤狀態的性能，在論文的第五章我們分別提供控制系統與狀態回授估測系統各2個數值分析的例子，來證明其有效性。

摘要(英)

In this thesis, a polynomial nonlinear system, modelled by T-S fuzzy model with added disturbances, is studied. Based on non-quadratic, homogeneous Lyapunov function, both controller and observer are considered in the analysis where Euler′s homogeneous polynomial theorem is used to avoid the derivative term dot Q(x) that is seen in the existing papers.

After some background reviewed, we started with fuzzy system models established by Taylor series. To tackle the derivative Lyapunov dot Q(x) terms and the zero row structure in the input matrix B(x) in the existing papers, Euler homogeneous polynomial theory is applied to derive the stabilization condition in LMI formulation and then converted into SOS form so that SOSTOOLS is used to for synthesis analysis.

　Finally, Sum of Square is applied to solve for the Lyapunov Q(x) and controller/observer gains, thereby ensuring the stability of the closed-loop feedback system as well as the observed-state feedback control system. Several examples are provided in Chapter 5 to demonstrate the analysis is effective.

關鍵字(中)

★ 平方和
★ Takagi-Sugeno模糊系統
★ H∞ 控制
★ H∞觀測
★ 尤拉齊次多項式定理
★ 參數相依齊次多項式

關鍵字(英)

★ sum of squares
★ T-S fuzzy systems
★ H∞ control
★ H∞ observer
★ Euler’s theorem for homogeneous function
★ HPPD

論文目次

一、背景介紹..................................................................... 1
1.1 文獻回顧 . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 研究動機 . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 論文結構 . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 符號標記 . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 預備定理 . . . . . . . . . . . . . . . . . . . . . . . 6
二、基礎定理介紹.............................................................. 8
2.1 H1性能優化定理 . . . . . . . . . . . . . . . . . . . 8
2.2 尤拉齊次多項式定理(Euler′s homogeneity theorem) 9
2.3 李亞普諾夫定理(Lyapunov theorem) . . . . . . . . 11
2.4 蕭轉換定理(Schur complement) . . . . . . . . . . . 13
2.5 建模技巧 . . . . . . . . . . . . . . . . . . . . . . . 14
三、連續系統架構與檢測條件............................................ 17
3.1 連續系統架構介紹 . . . . . . . . . . . . . . . . . . 17
3.2 H1連續模糊閉迴路控制系統之檢測條件 . . . . . . 18
3.3 H1連續模糊閉迴路觀測系統之檢測條件 . . . . . . 21
四、平方和檢測條件 .......................................................... 27
4.1 平方和檢驗法 . . . . . . . . . . . . . . . . . . . . . 27
4.2 平方和檢驗法之連續模糊閉迴路控制系統檢測條件 . 30
ix
4.3 平方和檢驗法之H1控制連續模糊系統檢測條件 . . . 31
4.4 平方和檢驗法之連續模糊閉迴路觀測系統檢測條件 . 32
4.5 平方和檢驗法之H1觀測連續模糊系統檢測條件 . . . 33
五、 Matlab電腦模擬 ......................................................... 35
5.1 例題一 . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 例題二 . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 例題三 . . . . . . . . . . . . . . . . . . . . . . . . 46
5.4 例題四 . . . . . . . . . . . . . . . . . . . . . . . . 53
六、結論與未來方向 .......................................................... 60
6.1 結論 . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 未來研究方向 . . . . . . . . . . . . . . . . . . . . . 61
文獻 ................................................................................................. 63

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指導教授

羅吉昌(Lo, Ji-Chang)

審核日期

2015-7-30

推文