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研究生： 凃毅瑋
Tu, Yi-Wei
論文名稱： 強健低階控制器之設計及其應用
Robust Low-order Controller Design and Its Applications
指導教授： 何明字
Ho, Ming-Tzu
學位類別： 博士
Doctor
系所名稱： 工學院 - 工程科學系
Department of Engineering Science
論文出版年： 2011
畢業學年度： 99
語文別： 英文
論文頁數： 143
中文關鍵詞： 強健控制 、低階控制器設計 、H∞模型匹配 、比例-積分-微分控制器
外文關鍵詞： robust control, low-order controller design, H∞ model matching, Proportional-Integral-Derivative controller
相關次數： 點閱：71  下載：1 
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    • 控制系統的設計通常需要受控體精確的數學模型，但事實上，人們很難使用精確的數學模型來描述一個物理系統，為了容易設計控制器，一般都會使用簡化過的數學模型以描述受控系統。此外，隨著工作條件或環境的變動，受控體的特性亦將隨之改變。因此，受控體的不確定性(uncertainty)是不可避免的，而且應該在控制系統設計時考慮其對系統的影響。因此，強健控制(robust control)已成為控制理論重要的研究主題之一。過去數十年來，H∞控制理論及其衍生的理論，如：μ合成(μ-synthesis)，其針對一個系統轉移函數做H∞範數(H∞ norm)最小化的控制器合成問題，提供一個設計架構與精確的解決方法，許多強健穩定(robust stability)與性能(performance)問題都可以被轉化成H∞架構，並可使用已發展完整且成熟的理論解決控制系統的合成問題。但是所得到H∞控制器的階數，往往都比受控系統的階數還高，而在實際應用上，基於實現簡單與易於維護的優點，一般都會希望使用低階控制器控制複雜的系統。因此，本論文旨在發展一套針對H∞模型匹配(H∞ model matching)問題之固定結構且低階控制器的合成方法，其H∞模型匹配架構可應用在大部分標準的H∞控制設計問題，而所選用的控制器架構能涵蓋大多數常用的低階控制器，例如：比例-積分-微分(Proportional-Integral-Derivative, PID)控制器及相位超前/落後補償器(phase lead/lag compensator)。其設計步驟首先將H∞模型匹配問題轉換成複數係數多項式(complex coefficient polynomials)的同步穩定化(simultaneous stabilization)問題，再利用generalized Hermite-Biehler theorem解決複數係數多項式的穩定化問題，最後可得到所有能達到H∞設計目標的固定結構且低階控制器參數集合。而所提出的方法將用來解決三個H∞控制問題，包含：(1)針對撓性臂的H∞強健性能(H∞ robust performance)控制問題；(2)針對具負載變動的伺服馬達控制之H∞模型匹配問題；(3)針對PID控制的反積分終結(anti-windup)補償，並利用模擬與實驗結果來驗證所設計控制系統的可行性。本論文所提出的合成方法能求得所有達到設計目標的H∞控制器參數空間集合，不像一般的H∞控制理論，只能設計出一組控制器，此特點對控制器參數的微調(fine-tuning)非常有用。因為產業界大多使用低階控制器，所以本論文的研究成果可望被廣泛地應用在實際的控制系統上。

    Control system design usually depends on a precise mathematical model of the plant. In practice, it is difficult to describe a physical system using a precise mathematical model. A simplified model is generally used for tractability. In addition, the plant dynamics vary due to changes in the operating conditions or environment. Thus, plant uncertainty is inevitable, and should be taken into account in the design of control systems. Consequently, robust control has become one of the most important research areas in control theory. Over the last decades, H∞ control theory and its offshoots, such as μ-synthesis, provide a precise formulation of and solution to the problem of synthesizing a controller which minimizes the H∞ norm of a prescribed system transfer function. Many robust stability and performance problems can be cast into the H∞ framework, for which there is a sophisticated and complete theory for control system synthesis. Nevertheless, the order of the obtained H∞ controller is almost always quite high, being comparable to that of the plant. In practice, it is usually desirable to control a complex system by using a low-order controller due to its simplicity in implementation and maintenance. Therefore, the objective of this dissertation is to develop a fixed-structure low-order controller synthesis method for the H∞ model matching problem, which can be applied to most standard H∞ control design problems. The selected controller structure encompasses most popular low-order controllers, such as Proportional-Integral-Derivative (PID) controllers and phase lead/lag compensators. It is shown that the H∞ model matching problem can be reduced into a simultaneous stabilization problem of complex coefficient polynomials. The generalized Hermite-Biehler theorem is then used to solve the resulting stabilization problems. Finally, the admissible parametric set of the fixed-structure low-order controller is obtained. The proposed method is used to solve three H∞ control problems, namely (1) the H∞ robust performance control of a flexible beam manipulator, (2) the H∞ model matching problem of servo motor control with load inertia variation, and (3) anti-windup compensation for PID control. Simulation and experimental results are used to verify the effectiveness of the designed control systems. Unlike the standard H∞ controller design method, the proposed synthesis method provides not just a single solution, but a set of admissible gain values of the controller. This feature is very useful for controller fine-tuning. The results given here should have a widespread impact on control applications because of the prevalence of low-order controllers in industry.

    Chinese Abstract I English Abstract III Acknowledgement V Contents VII List of Figures X List of Tables XIII Notation and Abbreviations XIV Chapter 1 Introduction 1 1.1 Motivation 1 1.2 Contributions 3 1.3 Organization 5 Chapter 2 The Hermite-Biehler Theorem and Its Generalization 8 2.1 The Hermite-Biehler Theorem for Hurwitz Polynomials 8 2.2 Generalizations of the Hermite-Biehler Theorem 12 2.2.1 No Imaginary Axis Roots 15 2.2.2 Roots Allowed on the Imaginary Axis Except at the Origin 17 2.2.3 No Restriction on Root Locations 22 2.3 Generalizations of the Hermite-Biehler Theorem for Complex Polynomials 24 Chapter 3 Synthesis of H∞ Second-order Controllers 36 3.1 H∞ Control Theory 36 3.1.1 Definition of H∞ Norm 36 3.1.2 Sensitivity Function and Complementary Sensitivity Function 37 3.1.3 Performance Specifications 39 3.1.4 Uncertainty 41 3.1.5 Small Gain Theorem 43 3.1.6 Robust Performance 44 3.2 Second-order Controllers 46 3.3 Synthesis of H∞ Second-order Controllers 47 Chapter 4 Position Control of a Single-link Flexible Manipulator System Using H∞ PID Control 65 4.1 Introduction 66 4.2 Description of Experimental Setup 67 4.3 PID Controller Design Using Ziegler-Nichols Tuning Methods 68 4.4 System Identification and Uncertainty Description 70 4.5 H∞ Second-order Controller Design for Robust Performance 75 4.6 Experimental Results 80 4.7 Summary 85 Chapter 5 Robust Second-order Controller Synthesis for Model Matching of Interval Plants and Its Application to Servo Motor Control 86 5.1 Introduction 87 5.2 Notation and Preliminaries 87 5.3 Problem Formulation 89 5.4 Robust Stabilization of Interval Plants Using Second-order Controllers 90 5.5 Design Algorithm for Model Matching of Interval Plants 99 5.6 Application to Servo Motor Control 104 5.7 Summary 114 Chapter 6 Synthesis of Anti-windup Low-order Compensators for PID Control 115 6.1 Introduction 115 6.2 Preliminaries 117 6.3 Anti-windup Configuration for PID Control and Problem Formulation 117 6.4 Anti-windup Second-order Controller Design Procedure 122 6.5 Summary 129 Chapter 7 Conclusions and Recommendations 130 7.1 Conclusions 130 7.2 Recommendations for Further Work 130 Bibliography 132 Publication List 141 Vita 143

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