T-S模糊模型之建構、強健穩定分析與H2/H∞控制

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林銘隆(Min-Long Lin) 查詢紙本館藏

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論文名稱

T-S模糊模型之建構、強健穩定分析與H2/H∞控制
(Model construction, robust stabilization and H2/H∞ control for T-S fuzzy models)

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★ 廣義H2模糊控制-連續系統線性分式轉換法	★ 廣義模糊控制-離散系統線性分式轉換法
★ H∞模糊控制－連續系統線性分式轉換法	★ H∞模糊控制—離散系統線性分式轉換法
★ 強健模糊動態輸出回饋控制-Circle 與 Popov 定理	★ 強健模糊觀測狀態回饋控制-Circle與Popov定理
★ H_infinity 取樣模糊系統的觀測型控制	★ H∞取樣模糊系統控制與觀測定理
★ H-ihfinity取樣模糊系統動態輸出回饋控制	★ H∞模糊系統控制-多凸面法
★ H∞模糊系統控制-寬鬆變數法	★ 時間延遲 T-S 模糊系統之強健 H2/H(Infinity) 控制與估測
★ 寬鬆耗散性模糊控制-波雅定理	★ 耗散性估測器-波雅定理

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摘要(中)

本篇論文主題有三：
1. 經由事先給定非線性系統的動態方程式，將此系統精確的轉換成 Takagi-Sugeno (T-S) 模糊模型。
2. 設計狀態回授控制器來穩定 T-S 模糊模型並推導出較不保守的穩定條件，使得該穩定條件可求解的程度比現有文獻中的穩定條件高，因而透過該穩定條件求解控制器增益值時可較求解其他穩定條件容易獲得有解。
3. 設計靜態輸出回授控制器來穩定 T-S 模糊模型並滿足 H2 或 H∞ 性能要求。主要在解決該控制器所導致的雙線性矩陣不等式 (BMI) 如何求解的問題。
T-S 模糊模型的優點在於它可充分運用以 Lyapunov 定理為基礎的系統分析與設計技巧，透過將非線性系統轉換成 T-S 模糊模型之後，可提供一套系統化的方法研究非線性系統的穩定分析與性能控制問題。當使用這套方法時，由於控制器是根據 T-S 模糊模型所設計而非直接針對非線性系統做設計，因此若非線性系統與 T-S 模糊模型之間存在誤差，則此誤差項將影響穩定與性能分析的結果。為確保系統與模型之間的誤差為零，本篇論文提供一個方法將誤差以不確定項 (norm bounded uncertainty) 來表示，而後，非線性系統即可精確的表達成具有不確定項的 T-S 模糊模型。
針對具有不確定項的 T-S 模糊模型，本篇論文根據平行分散式補償器 (PDC) 的概念設計控制器，分別討論採用狀態回授控制器的穩定分析問題及採用靜態輸出回授控制器的性能控制問題。
在狀態回授控制問題中，本篇採用的控制器並非一般傳統 PDC 形式的狀態回授控制器，而是一個更廣泛的設計架構，當中包含較多的自由參數，可提高穩定問題的可求解性。不同於以往採用單一正定矩陣的方式 (common $P$ approach) 驗證閉回路系統的穩定性，本篇採用一包含有模糊模型的激發強度及多個正定矩陣所構成的函數來做為Lyapunov函數的侯選。基於控制器的特殊設計及根據Lyapunov定理，本篇獲得一新的 T-S 模糊模型的狀態回授穩定條件，該條件是一個非二次 (non-quadratic) 穩定條件。經由證明發現本篇所提穩定條件比現有文獻當中以單一正定矩陣所推出的穩定條件較不保守，也就是說，
如果單一正定矩陣穩定條件成立時，本篇所提出的條件也一定會成立；但是當單一正定矩陣穩定條件不成立時，本篇所提的條件未必不會成立。因此，本篇所提的非二次穩定條件優於單一正定矩陣穩定條件，即採用此一新的條件
來判斷 T-S 模糊模型穩定性的可求解性比較高。
控制系統中，當系統狀態無法完全獲知時，則必須採用估測器獲得所需的資訊或直接以輸出回授做控制，本篇另一個方向所討論的即是一般較少研究的靜態輸出回授控制問題。在靜態輸出回授控制方面，最大的問題在於所推導出的穩定或性能條件並非線性矩陣不等式 (LMI) 而是以雙線性矩陣不等式 (BMI) 的形式呈現，而 BMI 無法如同 LMI 一般可輕易經由現有工具程式求解。因此，本篇將此部份的重點放在如何求解 BMI 的問題上。考慮靜態輸出回授的 H2 及 H∞ 性能控制問題，本篇發展三種方法來解決該性能條件當中的 BMI 求解問題，包括：
1. 系統狀態轉換，將原本會造成 BMI 的系統轉換成可獲得 LMI 形式性能條件的等效系統。
2. 引進一多餘變數(矩陣)，透過必要條件求解該變數，而後，其他未知矩陣可形成 LMI。
3. 利用時變的 Lyapunov 函數獲得時變性能條件，透過給定終值(矩陣)，該時變條件可形成 LMI。
透過這些方法可將輸出回授性能控制問題中的 BMI 條件轉換為 LMI 形式求解或者經由反覆求解某些 LMI 來達到求解 BMI 的目的。另外，本篇附錄當中說明了動態輸出回授控制與靜態輸出回授形式的關連，所以，以上的方法亦可應用於求解動態輸出回授控制器的增益值。
以上的穩定與性能問題乃是針對具有不確定項的 T-S 模糊模型所討論，不論是連續或離散時間系統皆有考慮到。由於具有不確定項的 T-S 模糊模型可精確的表達非線性系統，因此，所設計的控制器將可穩定該模型所表示的非線性系統並達到所要求的性能限制。

摘要(英)

In this dissertation, construction of a Takagi-Sugeno (T-S) fuzzy model to exactly represent a given nonlinear system is investigated. Then, controller syntheses for stabilization of the resulting T-S fuzzy model or satisfaction of required H2 and H∞ performances are studied.
Since T-S fuzzy model renders itself naturally to a Lyapunov-based system analysis and design techniques, it provides an alternative way to study stabilization and performance problems for a nonlinear system which is formulated as a T-S fuzzy model. However, it should be noted that when there exist modeling errors between a model and a nonlinear system, they could influence the stability or performance of the nonlinear system when the underlying controller designed for the model is to be used in conjunction with the nonlinear system. To eliminate the modeling errors, we first show that a class of nonlinear systems can be exactly represented by a T-S fuzzy model with norm bounded uncertainties accountable for the modeling errors. Then, a state feedback controller or a static output feedback controller designed in the form of a parallel distributed compensator (PDC) is considered to stabilize the uncertain T-S fuzzy model.
For the state feedback case, a controller which is more general than the conventional PDC controller is designed to stabilize an uncertain T-S fuzzy model. To analyze the stability of the corresponding closed-loop system, the well known Lyapunov theory is used where a non-quadratic function via fuzzy blending of several quadratic functions is chosen as a Lyapunov candidate. Based on the designed state feedback controller and a given fuzzy Lyapunov candidate, a non-quadratic stabilization condition in terms of linear matrix inequalities (LMIs) is obtained. The new condition is shown to be less conservative than the existing results derived from the relaxed common $P$ approaches.
In the case of static output feedback control for the uncertain T-S fuzzy model, an H2 or H∞ control problem is considered. Based on Hamilton-Jacobi conditions as well as a common $P$ approach, a set of bilinear matrix inequalities (BMIs) is derived as a performance condition to assuring the required performance constraint. Note that the BMI condition is unlike the LMI conditions that can be directly and easily solved by convex techniques. To overcome the difficulty of solving the BMI condition, three approaches ranging from (1) coordinate transformation, (2) slack variable and (3) time-varying $P$ matrix are proposed. Furthermore, connection to static output feedback and dynamic output feedback is addressed. In Appendix B, a dynamic output feedback control scheme is developed and shown to have a similar structure to a static output feedback, indicating the proposed methods are applicable in such case as well.
In summary, the state or static/dynamic-output feedback controllers synthesized for the uncertain T-S fuzzy models can be equally applied to stabilize the nonlinear systems or to guarantee a desired performance, since a T-S fuzzy model with norm-bounded uncertainties is shown to exactly represent the nonlinear systems.

關鍵字(中)

★ H2/H∞ 控制
★ 雙線性矩陣不等式
★ 線性矩陣不等式
★ T-S模糊模型
★ 平行分散式補償器
★ 強健控制
★ 寬鬆穩定條件

關鍵字(英)

★ Bilinear matrix inequality (BMI)
★ Takagi-Sugeno (T-S) fuzzy model
★ Parallel distributed compensator (PDC)
★ Robust control
★ Relaxed stabilization condition
★ H2/H∞ control
★ Linear matrix inequality (LMI)

論文目次

{1}Introduction
{1.1}Overview
{1.2}Motivations and Main Contributions
{1.3}Organization of This Dissertation
{2}Preliminaries: Model Construction, Problem Formulations and Tools
{2.1}Construction of Uncertain T-S Fuzzy Models
{2.2}Control Objectives
{2.3}Useful Lemmas
{3}State Feedback PDC: Relaxed Stabilization Conditions
{3.1}Stabilizability of Continuous-Time T-S Models
{3.2}Stabilizability of Discrete-Time T-S Models
{3.3}Extensions to Robust Stabilizability
{3.4}Examples
{3.5}Summary
{4}Static Output Feedback PDC, Part I: Solving BMI for Continuous T-S
{4.1}H_2 Control for Noise-Free Uncertain T-S Models
{4.1.1}Method 1: System Coordinate Transformation
{4.1.2}Method 2: Iterative LMI Algorithm
{4.1.3}Method 3: Time-Varying $P$ Matrix
{4.2}H∞ Control for Uncertain T-S Models
{4.2.1}Method 1: System Coordinate Transformation
{4.2.2}Method 2: Iterative LMI Algorithm
{4.2.3}Method 3: Time-Varying $P$ Matrix
{4.3}Examples
{4.4}Summary
{5}Static Output Feedback PDC, Part II: Solving BMI for Discrete T-S
{5.1}H_2 Control for Noise-Free Uncertain T-S Models
{5.1.1}Method 1: System Coordinate Transformation
{5.1.2}Method 2: Iterative LMI Algorithm
{5.1.3}Method 3: Time-Varying $P$ Matrix
{5.2}H∞ Control for Uncertain T-S Models
{5.2.1}Method 1: System Coordinate Transformation
{5.2.2}Method 2: Iterative LMI Algorithm
{5.2.3}Method 3: Time-Varying $P$ Matrix
{5.3}An Example
{5.4}Summary
{6}Conclusion
References
Appendix {A} Proof of Lemmas
Appendix {B} Dynamic Output Feedback PDC

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指導教授

羅吉昌(Ji-Chang Lo)

審核日期

2003-6-22

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