線性雙曲線分布參數系統之主動式控制器設計與應用

Abstract

本論文針對以偏微分方程式(Partial Differential Equations)所架構之分布參數系統(Distributed Parameter Systems)其主動式控制器之設計與應用，所探討之範圍以雙曲線式型態(Hyperbolic Type)為主。設計之方法乃直接對偏微分方程式進行分析，利用拉普拉斯轉換(Laplace Transform)的技巧從頻域(Frequency Domain)的觀點出發，結合物理系統的波動性質(波傳遞、反射、時間延遲特性)，設計系統之主動式控制器。所設計之控制器在工程應用上可用於吸收結構振動或阻絕噪音，其方法有別於以往傳統之"被動式"阻尼(Passive Damping)之方式，如添加減振器或是採用吸音材料等方式。 本論文之研究重點分為以下三項： 1. 探討廣義之一維二階雙曲線式型態偏微分方程式之主動式控制器設計與研究。針對工程應用上之需求，所設計之主動式控制器分為全反射式與全透射式兩種，所設計出之主動式控制器需加入Notch 濾波器之修正以符合系統穩定性之需求。 2. 對於Moving String-like物理系統之結構振動問題，首先以模式匹配法(Model Matching)推導控制法則，此控制法則可以借助一個或是兩個感測器(Sensr)來實現控制器，並且所設計之控制器與第1項以波動性質推導之全透射控制器作比較。致動器(Actuator)可置放於振動源之下游(Downstream)或是邊界(Boundary)，當置放於邊界時所實現之控制器與其他文獻之研究結果相同。使用兩個感測器所實現之主動控制器其最大之優點為具有與邊界條件(Boundary condition)無關之特性，可使系統較具強健性(Robustness)。 3. 將第1項所設計之全反射控制器應用於管路之前饋(Feedforward)主動式噪音控制問題(Active Noise Control)上，在寬頻消音之工程應用時，傳感器(Sensor/Actuator)特徵函數引入系統之效應必須要加以考慮。由於傳感器特徵函數通常屬於非最小相位(Nonminimum Phase)，導致特徵函數之反函數求取有誤差，所以第1項所設計之全反射控制器必須加以修正，文獻中並進一步探討全系統穩定性及參數限制範圍，最後並以實驗驗証在各種不同之邊界條件下控制器之強健性及管路寬頻消音效果。

A theoretical study and some applications in the area of deterministic control for systems governed by linear partial differential equations (PDE), especially for wave equations which are classified as of hyperbolic type, are investigated in this thesis. Instead of using model truncation or finite-dimensional approximation, the closed-form solution in the Laplace domain is applied to analyze this problem. The design algorithms are different from the classical one, the passive damping method, which adding porous or shock isolation material to attenuate the unwanted vibration or sound noise. The active control strategy, using wave propagation concepts, attempts to construct a destructive canceling wave corresponding to the unwanted noise by means of the superposition principle. Three related topics are considered in this thesis: 1. Theoretical study of active impedance control of stable, linear one-dimensional second order wave equation is investigated. The proposed algorithms are based on the concepts of wave propagation and impedance matching. Two control objectives, total reflection and impedance matching, is considered. Realizations of the designed controllers do not require the information of disturbance and the boundary conditions. However, the closed-loop system is marginally stable. A simple modification is added and the stability is analyzed. 2. Considering practical applications to axially moving string-like systems, a classical model matching technique is used to derive the control laws. Not only the in-domain and boundary but also non-collocated and collocated controller design are analyzed. Further, the results are compared with the control law derived using wave propagation concepts. 3. Feedforward active noise controller design in finite length duct systems is considered. The unidirectional sensor architecture is used to attenuate the effect of "acoustic feedback". Model uncertainty and sensor measurement errors are considered and the controller is modified to meet the stability requirement. Finally several experiments are conducted to illustrate the design procedure as well as verify the effectiveness of broadband noise reductions. 1.1 Overview 2 1.2 Main Results in this thesis 5 2. Active Impedance Control of Linear One-Dimensional Wave Equations 6 2.1 Introduction 7 2.2 The Dynamic Model 9 2.3 Active Control of Impedance 14 2.3.1 Total Reflection 15 2.3.2 Impedance Matching 17 2.4 Stability Analysis and Controller Modification 20 2.4.1 Total Reflection 20 2.4.2 Impedance Matching 27 2.5 Example 27 2.5.1 Finite-Length Duct with Moving Medium 27 2.6 Summary 31 Appendix 31 Appendix 2A 31 Appendix 2B 34 3. Vibration Attenuation of an Axially Moving String via Active In-Domain Control Methods 35 3.1 Introduction 36 3.2 Dynamic Modeling 38 3.3 Single-sensor Wave Absorbing Control Law 42 3.3.1 Control law derivation 42 3.3.2 Re-distributing the energy 44 3.3.3 Connection between in-domain and boundary control 46 3.3.4 Stability Analysis 48 3.4 Boundary-independent Control Law 50 3.4.1 Control Law Derivation 51 3.4.2 Stability Analysis and Controller Modification 52 3.5 Summary 58 Appendix 59 Appendix 3A 59 Appendix 3B 61 4. Feedforward Active Noise Controller Design in Ducts Without Independent Noise Measurements 62 4.1 Introduction 63 4.2 System Modeling 66 4.3 The Feedforward Control System 68 4.4 Stability Analysis and Controller Modification 70 4.5 Experimental Verification 79 4.5.1 Identification and Design 79 4.5.2 Experimental Results 85 4.6 Summary 89 5. Conclusions and Potential Research Topics 91 5.1 Conclusions 92 5.2 On-line calibration of the transducer's characteristics 93 5.2.1 Identification process 93 5.2.2 Adaptive control law 95 5.3 Active Control to Wave Equation in Existence of Non-plane Wave Mode 99 5.3.1 Acoustic Dynamic Modeling in a Finite-length Ducts 100 5.4 Active Control of Structural Vibrations 105 REFERENCES 106 VITA 114 PUBLICATION LIST 115