延伸LQG設計法則於結構主動控制器設計

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姓名

陸立德(Li-teh Lu) 查詢紙本館藏

畢業系所

土木工程學系

論文名稱

延伸LQG設計法則於結構主動控制器設計
(Extended LQG Methodology for Active Structural Controller Design)

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摘要(中)

本論文針對結構主動控制文獻中最常被利用到LQG最佳控制理論作延伸性的拓展與探討。本文將LQG最佳控制器的應用拓展至LQG/LTR強健控制器，和現有的文獻相比可以看出LQG/LTR強健控制器應用在結構主動控制上可以對系統提供較佳的穩定裕度。
近來，H2 和 H¥ 最佳控制理論被應用在結構主動控制上，文獻記載應用H2 和 H¥ 最佳控制理論可有效的減低結構在強風與地震下的反應。本文將闡述在適當選擇權重函數下，定義於時間域的LQG最佳控制器和定義於頻率域的H2 最佳控制器可以在數學上全等。除了將LQG最佳控制器的應用拓展至LQG/LTR強健控制器外，本論文所探討另一個重點是嘗試將H¥ 最佳控制的理論精神融入LQG或說是H2 最佳控制器的設計考慮中。此時吾人不再緊守著LQG、H2 和H¥ 最佳控制理論中成本函數必須最佳化的要求，而是在這些成本函數求取適當的妥協。從本文的結果可以知道，不同最佳化控制理論中成本函數間的妥協可以使控制器的設計更有彈性。傳統的LQG、H2 和H¥ 最佳控制器可以由兩項聯立的Riccati Equation求得，但是不同最佳化控制理論中成本函數間的妥協問題卻不能由簡單的聯立的Riccati Equation求得。近來，interior-point method的出現與應用使得LMIs(Linear Matrix Inequalities)成為一套有用的數學工具。本文中，利用單一的Lyapunov函數將不同最佳化控制理論成本函數間的妥協問題轉換成LMIs並實際求解出控制器。吾人將此控制器設計法則應用到ASCE Committee on Structural Control提出的受風力影響高樓結構主動控制標準問題上可以得到滿意的結果。

摘要(英)

The LQG optimal control theory used extensively in the literatures for active structural control is extended in this dissertation. Since the LQG controller provides no guarantee for robust stability, we extend the LQG controller to the LQG/LTR controller. Simulations show that LQG/LTR controllers provide better stability margin than those presented in the published literatures.
Recently, H2 and H¥ optimal control techniques were introduced for active structural control problem, which results in effectual approaches in the design of controllers for seismic and wind excited buildings. Since LQG optimal control criteria defined in time domain can be numerically equivalent to the H2 optimal control criteria defined in frequency domain with appropriate selection of design weightings. In this dissertation, we present a control strategy that is the simultaneous treatment of both H2 and H¥ criteria and this control strategy quantitatively demonstrates design tradeoffs. Thus, we extend the popular LQG controller to the mixed LQG/H¥ or H2/H¥ controller. This mixed control problem can be formulated by linear matrix inequalities in terms of a common Lyapunov function. Solving linear matrix inequalities is a convex optimization problem. Simulation and design results demonstrates that decreasing H¥ attenuation constraint can be used to reduce the structural response under wind excitations at the expense of increasing H2 performance index and control efforts of the actuator.

關鍵字(中)

★ 結構主動控制
★ 最佳控制

關鍵字(英)

★ active structural control
★ optimal control law
★ h2 optimal control
★ hinf optimal control
★ mix h2 and hinf control law
★ LMIs

論文目次

Contents pages
1. Introduction1
1.1 Background and Motivations 1
1.2 Main Contributions5
1.3 Organization of This Dissertation6
2. Optimal Control Theory7
2.1 The Linear Quadratic Regulator8
2.2 The Kalman Filter9
2.3 LQG Optimal Controller: combined optimal state estimation and optimal
state feedback10
2.4 H2 optimal controller12
2.5 LQG: a special H2 optimal controller14
3. LQG/LTR Robust Controller for Active Structural Control16
3.1 Equations of Motion of Structural Systems16 3.2 Control Actions16
3.3 Collocated Velocity Feedback17
3.4 Model Reduction18
3.5 Robustness Requirement and Design Specifications22
3.6 LQG/LTR Robust Controllers25
3.7 Numerical Simulation29
4. Controller Synthesis: LMIs (Linear Matrix Inequalities) Approach39
4.1 Linear Matrix Inequalities39
4.2 Performance measurement40
4.2.1 H¥ system norm40
4.2.2 H2 system norm43
4.3 Full order output feedback controller construction with LMIs44
4.3.1 H¥ optimal controller synthesis45
4.3.2 H2 optimal controller synthesis 48
4.3.3 Controller construction with multiple objectives49
5. Application to Active Structural Control Benchmark Problems52
5.1 76-Story Building and Model53
5.2 Wind Excitations56
5.2.1 Davenport wind load spectrum for along-wind excitations57
5.2.2 Across-Wind Excitations from the Results of Wind Tunnel Tests 59
5.3 Design Constraints60
5.4 Performance Criteria61
5.4.1 Performance Criteria for Davenport wind load spectrum 61
5.4.2 Performance Criteria for Wind Tunnel Tests64
5.5 Statements of the Control Objectives66
5.6 LMI Formulation of the Design Specifications68
5.7 Numerical Analysis and Simulation Results71
5.7.1 the Benchmark Structure under Davenport Wind Load Spectrum71
5.7.2 the Benchmark Structure under Across-Wind Excitations from
the Results of Wind Tunnel Tests80
6. Conclusion Remarks87
Bibliography88
Appendix92
List of Tables pages
3.1. Maximum magnitudes of response quantities for a base-isolated building 103
3.2. Maximum magnitudes of response quantities for a fixed-base full-scaled building 103
5.1. Peak Response Quantities of the Benchmark Building Installed AMD System
with Different Controllers 104
5.2 RMS Response Quantities of the Benchmark Building Installed AMD
System with Different Controllers 105
5.3 H¥ Attenuation Constraints, Actual H¥ Attenuation, H2 Performance Bound
and H2 Actual Performance with Different Controllers 106
5.4 Evaluation Criteria for the sequential H2 Controllers 107
5.5 Damping ratios of the First Five Vibrational Modes with Different Controllers
and the Uncertainty of Building Stiffness 108
5.6 Peak Response Quantities of the Benchmark Building Installed AMD
System with Different Controllers 109
5.7 RMS Response Quantities of the Benchmark Building Installed AMD
System with Different Controllers 110
5.8 H¥ Attenuation Constraints, Actual H¥ Attenuation, H2 Performance Bound
and H2 Actual Performance with Different Controllers 111
5.9 Evaluation Criteria for the sequential H2 Controllers
case : 112
case : 113
case : 114
List of Figures pages
2.1 LQG controlled plant 116
3.1 LQG/LTR controller block diagram 117
3.2 Block Diagram of a General Compensated Plant with Uncertainty 117
3.3 Block Diagram of LQ Regulator 118
3.4 Block Diagram of Kalman Filter 118
3.5 Block Diagram of the Target Feedback Loop for the Augmented Plant 118
3.6 Block Diagram of LQG/LTR Robust Controller 119
3.7 (a) Base-Isolated Structural Model 119
3.7 (b)Structure Model with Active Bracing 119
3.8 EL-CENTRO Earthquake 120
3.9 Deformation of First and Fourth Floor Units 120
3.10 Absolute Acceleration of First and Fourth Floors 121
3.11 Bode Plots of Transfer Functions: Actuator to the First Floor Deformation 121
3.12 Two-State Reduced Order Model and Additive Modeling Error with Robustness 122
3.13 LQG/LTR Robust Controller Design Result 122
3.14 Frequency response: From earthquake excitations to first floor deformation 123
3.15 Control force (solid) and reference input (dashed) 123
5.1(a): Plan View of the 76-Story 124
5.1(b): Elevation View of the Benchmark Building. 124
5.2 Time Histories of Response Quantities of the 75th Floor Using LQG Control:
(a) Displacement Responses (b) Acceleration Responses (c) Required Control Force. 125
5.2 Time Histories of Response Quantities of the 75th Floor Using K2i(s):
(d) Displacement Responses (e) Acceleration Responses (f) Required Control Force. 125
5.3 Power Spectral Densities of the 75th Floor Using Different Controllers: 126
(a) Displacement PSD for “No Control”, “LQG Controller”, and “ K2i Controller”;
(b) Acceleration PSD for “No Control”, “LQG Controller”, and “ K2i Controller”;
(c) Acceleration PSD for “LQG Controller”, “ K2i Controller” and “K25 “ Controller.
5.4 Trade-off Lines Between the and performances 127
5.5 Additive Uncertainty Represents the Spillover Effects 128
5.6 vs. for Controllers and 128
5.7 Evaluation Criteria vs. Attenuation Constraints with the Uncertainty of
Building Stiffness 129
5.8 Time Histories of Response Quantities of the 75th Floor Using LQG Control:
(a) Displacement Responses (b) Acceleration Responses (c) Required Control Force. 130
5.8 Time Histories of Response Quantities of the 75th Floor Using K2i’’:
(d) Displacement Responses (e) Acceleration Responses (f) Required Control Force. 130
5.9 (a) The maximal singular values for wind excitations to the displacement of the
75th floor with the LQG controller; the controller and No Control case.
(c) The maximal singular values for wind excitations to the acceleration of the
75th floor with the LQG controller; the controller and No Control case. 131
5.10 Trade-off Lines Between the and performances 132
5.11 vs. for Controllers and 133

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指導教授

蔣偉寧、唐治平
(Wei-Ling Chiang、Jhy-Pyng Tang)

審核日期

2002-1-12

推文