Active Vibration & Noise Control: Design Towards Performance Limit: A New Paradigm to Active Control [1st ed. 2022] 9811941157, 9789811941153

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Jiqiang Wang

Active Vibration & Noise Control: Design Towards Performance Limit A New Paradigm to Active Control

Active Vibration & Noise Control: Design Towards Performance Limit

Jiqiang Wang

Active Vibration & Noise Control: Design Towards Performance Limit A New Paradigm to Active Control

Jiqiang Wang RC-SESA Ningbo Institute of Materials Technology and Engineering (NIMTE) Chinese Academy of Sciences (CAS) Ningbo, Zhejiang, China

ISBN 978-981-19-4115-3 ISBN 978-981-19-4116-0 (eBook) https://doi.org/10.1007/978-981-19-4116-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Modern technological systems have evolved towards high speed, heavy load, lightweight, flexible operation, and extreme condition as demonstrated in aerospace, marine, transportation, manufacturing industries, etc. The associated vibration and noise issues have become such problematic that may significantly confine the performance of the systems, to say the discomfort at least. For high-frequency vibration and noise, effective control can be implemented through dampers or mufflers. Yet, these passive devices, as is well known, are constrained by effective frequencies, weight, and size restrictions. Thus, they cannot provide effective solutions to many important vibration and noise problems. This can be remedied through active control providing a new avenue for structural vibration attenuation and noise reduction. Active control achieves vibration and noise attenuation by dissipating heat. Vibration-based energy harvesting aims to harness the dissipated energy, e.g. for powering wireless networks. In comparison with other forms of renewable energy, vibration energy is not on large scale yet has rich sources. For example, mechanical and even human body can be the vibration sources for harvesting energy. Consequently, vibration energy can provide an ideal resource for powering wireless networks, thus realizing condition monitoring and health management of equipment. Henceforth, vibration-based energy harvesting can be an enabling technology for future intelligent societies. However, any active control and energy harvesting system is always subject to constraints such as saturation, states, and physical parameter limits, etc. Thus, it is extremely important to take all the constraints into consideration for active control and energy harvesting implementations. Indeed, the fundamental philosophy for control is to excavate system performance by taking advantage of every piece of information, e.g. internal structural properties, topology and interactions, external noise, disturbance power spectral distributions, etc. Thus, underpinning the design philosophy lies in a pivotal issue: what is the performance limit of active control and energy harvesting? It is quickly realized that active control methodologies have been well established with feedforward approaches, feedback compensations such as H2/H∞ type of controls, linear parametric varying control, adaptive control, etc. Yet most of v

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the above approaches (including feedforward ones) utilize an optimization-based methodology, where a certain functional norm (or performance index) is defined and to be optimized against controller parameters. Detuning can be exercised later on through the commission phase to further improve or validate the system performance. However, this handle-turning methodology is of trial-and-error nature in the sense that the best achievable performance is not to be revealed. It is fair to claim that the optimization-based approaches are powerful in optimizing against performance indices but “weak” in determining performance limits. This monograph aims to develop systematic design methodologies with a “visualization technique” where performance limit can be readily determined solely based on visual inspections. Indeed, through the geometric representation of the performance specifications, fundamental issues such as (1) existence of feasible controllers; (2) optimality of controllers; (3) performance limit of controllers; (4) compromisablity among the performance specifications; (5) synthesis of controllers; (6) influence of constraints on optimal solutions, can all be resolved within the proposed framework. The state of the art is thus refined with a new approach complementary to those optimization-based routines, where extra effort would have to be exercised to disclose the compromisability of performance specifications. It is claimed that limit of performance is obtained where other methods can be benchmarked against the best achievable performance. It is also reasonable to expect that the proposed methodology can be applied to other engineering areas for vibration and acoustic applications. This forms the distinctive contribution of the monograph. Finally, it is worth pointing out that the proposed framework essentially results in a new design methodology—performanc -limit-oriented active control. It is initiated by the author with the project “Active Control for Performance Limit” (ACPL). A series of fundamental results have been obtained and will be disseminated in this monograph. Meanwhile, over the past decade, the research project has been funded by quite a several funding bodies, and the author would like to take this opportunity to express sincere thanks for the financial support from the Central Military Commission Foundation to Strengthen Program Technology Fund (No. 2019-JCJQ-JJ-347); the Central Military Commission Special Fund for Defense Science, Technology & Innovation (No. 20-163-00-TS-009-096-01); the State Foreign Affairs Bureau Fund for Introduction Plan of Foreign Experts (G20200010100); Aviation Science Fund of China-Xi’an 631 Research Institute (201919052001); and Natural Science Foundation of Jiangsu Province Project (BK20140829). The author would also like to thank Professor Steve Daley of the Institute of Sound & Vibration Research, University of Southampton for his fruitful discussions over the years; special thanks go to my wife Catherine for her loving support and patience during the preparation of the monograph. Ningbo, China

Jiqiang Wang

Contents

1 Introduction to Vibration and Noise Control Problems . . . . . . . . . . . . . 1.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Active Vibration and Noise Control: State-of-the-Art . . . . . . . . . . . . 1.2.1 H2 /H∞ Optimal Control for Linear Systems . . . . . . . . . . . . 1.2.2 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Variances of Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Nonlinear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 A New Paradigm: Global Control Towards Performance Limit . . . . 1.4 Contributions and Contents of the Monograph . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 7 10 12 13 16 17

2 Active Control for Performance Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Difficulties with Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Active Control for Performance Limit: A Geometric Method . . . . . 2.3.1 Performance-Limit-Oriented Solution to Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Constrained Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Active Control of Vehicle Suspension Structures . . . . . . . . . 2.3.4 Section Discussions and Conclusions . . . . . . . . . . . . . . . . . . 2.4 Active Pure Topology Feedback for Performance Limit . . . . . . . . . . 2.4.1 Problem Formulation: Pure Topology Feedback for Global Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Fundamental Solutions: Existence and Performance Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Realizable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Numerical Example with Real Time Simulation . . . . . . . . . 2.4.5 Section Conclusions and Discussions . . . . . . . . . . . . . . . . . . 2.5 Active Control and Design Optimization . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Design Optimization for Multiple-location Vibration Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 30 35 37 41 42 48 48 49 52 56 58 63 63 64

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2.5.2

Topology Optimization for Multiple-location Vibration Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Vibration Suppression for Vehicle Suspension at Two Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Section Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Active Control with Constraints and Uncertainty . . . . . . . . . . . . . . . . . . 3.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Generic Active Control Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Active Control with Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Sensing Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Actuation Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Components Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Controller Realization Constraints . . . . . . . . . . . . . . . . . . . . . 3.3.5 Other Forms of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Active Control with Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Robust Design for Unconstrained Case . . . . . . . . . . . . . . . . . 3.4.2 Robust Design for Constrained Cases . . . . . . . . . . . . . . . . . . 3.5 Active Control with Constraints and Uncertainty: Numerical Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Active Control Without Constraints . . . . . . . . . . . . . . . . . . . . 3.5.2 Active Control with Constraints . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Robust Active Control with Constraints . . . . . . . . . . . . . . . . 3.5.4 Summary for Robust Active Control with Constraints . . . . 3.6 Optimal Design of Tuned Mass Dampers: A Virtualization Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Introduction to Tuned Mass Dampers . . . . . . . . . . . . . . . . . . 3.6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Virtualization Technique for Optimal Solution . . . . . . . . . . 3.6.4 Optimal Design with Physical Constraints and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Optimal Design of Tuned Mass Dampers: Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.6 Summary for Optimal Design of Tuned Mass Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Frequency Dependence Performance Limit of Vibration Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Performance Limit: Lower Bound . . . . . . . . . . . . . . . . . . . . . 3.7.3 Performance Limit: Upper Bound . . . . . . . . . . . . . . . . . . . . . 3.7.4 Performance Limit: Refined Upper Bound . . . . . . . . . . . . . . 3.7.5 Summary for Frequency-Dependent Performance Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 70 74 76 76 81 81 82 86 87 88 90 91 93 94 94 95 97 98 98 101 103 104 104 105 107 112 115 119 121 121 122 126 128 131

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3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4 Active Control with Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Optimal Design for Broad-Band Control . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Stabilizing and Strong Stabilizing Controller . . . . . . . . . . . . 4.1.3 Optimal Broad-Band Control Design with N-P Interpolation and LMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Optimal Trajectory of αopt ( jω) . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Transfer Function Representation of αopt (s) . . . . . . . . . . . . 4.1.6 Real Rational Representation of αopt (s) . . . . . . . . . . . . . . . . 4.1.7 Performance Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.8 Out-Of-Band Performance Deterioration . . . . . . . . . . . . . . . 4.1.9 Broad-Band Control of Active Vibration Isolation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.10 Summary for Optimal Broad-band Control . . . . . . . . . . . . . 4.2 Performance Improvability Test for Harmonic Control . . . . . . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Improvability Test Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Summary and Discussions for Performance Improvability Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Experimental Design for Verification and Validation of Harmonic Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Experimental Design Procedures . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Summary and Discussions for V&V . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 138 139

5 Energy Harvesting for Performance Limit . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Active Control for Energy Harvesting with Performance Limit . . . . 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Performance of Energy Harvesters . . . . . . . . . . . . . . . . . . . . . 5.1.3 Graphical Approach to Active Control Performance . . . . . . 5.1.4 Performance Evaluation of Active Control of Harvesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Performance Limit of Active Control . . . . . . . . . . . . . . . . . . 5.1.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.7 Summary for Active Control of Energy Harvesters for Performance Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Optimal Design of Two-Degree-Of-Freedom Vibration Energy Harvesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 178 178 180

141 143 143 145 146 148 151 156 160 160 161 163 164 164 165 166 168 168 172 173 173

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5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 System Modeling and Performance Limit . . . . . . . . . . . . . . . 5.2.3 Optimal Design of Harvesters . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Optimal Design with Constraints . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Avoidance of Ineffective Designs . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Summary for Optimal Design of Energy Harvesters . . . . . . 5.3 Fundamental Problems in 1DOF and 2DOF Energy Harvesting Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Systems Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Performance Limit of Harvesters . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Performance Limit with Constraints . . . . . . . . . . . . . . . . . . . 5.3.5 Optimal Design Considerations . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Summary for Performance Limit of Energy Harvesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 192 194 200 202 204

6 Active Vibration Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction to Vibration Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Solution to Optimal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Optimal Solution Development . . . . . . . . . . . . . . . . . . . . . . . 6.4 Performance Limit with Stiffness and Damping Designs . . . . . . . . . 6.5 Numerical Examples for Optimal Vibration Distribution . . . . . . . . . 6.5.1 Optimal Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Optimal Performance with Stiffness or Damping Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Real Time Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Section Summary for Vibration Distribution . . . . . . . . . . . . 6.6 Vibration Distribution with Annihilation . . . . . . . . . . . . . . . . . . . . . . . 6.7 Active Vibration Distribution with Conventional Approaches . . . . . 6.7.1 Active Vibration Distribution Control Design . . . . . . . . . . . 6.7.2 Active Vibration Distribution with Constraints . . . . . . . . . . 6.7.3 Active Vibration Distribution: Fundamental Limit Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.5 Section Summary for Vibration Distribution with Conventional Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 221 222 224 224 226 229 231 232

205 205 206 208 213 215 217 217 217

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7 Conclusions & Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Active Control Towards Performance Limit for Higher-Order Hetero-Networks . . . . . . . . . . . . . . . . . . . . 7.2.2 Active Control Towards Performance Limit Through Successive Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Vibration Distribution for Energy Transfer and Military Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

Fig. 1.1

Fig. 1.2 Fig. 1.3

Fig. 1.4

Fig. 1.5

Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6

P is the generalized plant and K is the generalized controller. A general control design problem can be formulated as finding a controller K in such a way that some norm of the transfer function from d to z is minimized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gain scheduling adaptive control changes controller parameters in a pre-determined manner . . . . . . . . . . . . . . . . . . . . . Self-tuning regulator estimates the plant parameters from input/output data; plant parameters are then used in the control design algorithm to update the control law which can provide good closed-loop performance if the parameters of the plant model are estimated accurately . . . Block Diagram of MRAS. The controller is such designed to make the performance Y follow the desired model performance Y M for the same reference signal R. Thus is the name model-reference adaptive system . . . . . . . . . . . . . . . . . . Hyperstability design is a direct method of stability analysis of nonlinear models, considered as linear and nonlinear parts [61] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vibration/noise attenuation for a graph at all locations . . . . . . . . Topology design for global vibration/noise attenuation . . . . . . . . Geometric representation of performance, shaded area indicating the satisfaction of the performance index . . . . . . . . . . Non-compromise between performance variables . . . . . . . . . . . . Optimal constrained solution to active vibration control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric representation of performance, shaded area indicating simultaneous attenuation . . . . . . . . . . . . . . . . . . . . . . . .

4 4

8

9

9 33 34 39 40 42 43

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Fig. 2.7

Fig. 2.8

Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 2.15 Fig. 2.16 Fig. 2.17 Fig. 2.18 Fig. 2.19

Fig. 2.20

Fig. 2.21

Fig. 3.1

List of Figures

Performance limit for active vibration control: while (3, 3 dB) simultaneous reduction in both performance variable is achievable in (a), the performance reaches its limit for a (8, 3 dB) specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of constraint on solution existence: γ = 0.7 renders no existence of solution; while γ = 1.6 has no influence on solution existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Active vibration control: design for performance limit a X 1 ( jω) performance; b X 2 ( jω) performance . . . . . . . . . . . . . . . Graphical solution to solution existence . . . . . . . . . . . . . . . . . . . . No compromise can exist between performance variables . . . . . . Realizable solutions: constraints on realness of control parameter can significantly restrict the feasible solutions . . . . . . Graphical solution to solvability of global vibration suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Global vibration suppression through realizable topology feedback: a X 1 ( jω) performance; b X 2 ( jω) performance . . . . . Feasibility of specification is solved to the intersection of N circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance limit with visual inspections: vibration suppression limit for a Location-i; b Location-N . . . . . . . . . . . . . Performance does not necessarily compromise . . . . . . . . . . . . . . . Vehicle suspension model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feasibility of specification is solved to the intersection the two circles, henceforth the corresponding specification is solvable due to the existence of intersection as indicated in the shaded area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A relaxation of performance index in one variable would allow the total annihilation of the other variable, e.g. selecting the optimal design at the centre . . . . . . . . . . . . . . . . . . . Verification of optimal design through concept of real time simulation: a Performance in X 1 ( jω) b Performance in X 2 ( jω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mapping of |β( jω) + 1| < 1 on the complex plane with |α( jω) + 1| < 1, which are abbreviated as β−circle and α−circle for easy reference. The location of the centre of β−circle is prescribed by (–g) that is solely determined by the plant parameters. Hence a line connecting the two circle centers (denoted as αopt ( jω)) actually defines the fundamental performance compromise between y( jω) and z( jω), while the best achievable performance is also determined simultaneously . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

46 47 54 56 58 60 62 69 69 70 70

72

74

75

84

List of Figures

Fig. 3.2

Fig. 3.3

Fig. 3.4

Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9

Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13

Reduction in y( jω) and z i ( jω): each z i ( jω) is now corresponding to βi −circle, and henceforth solution existence problem must be answered by the existence of the “simultaneous” intersection over n circles . . . . . . . . . . . . . Mapping of |β( jω) + 1| ≤ δz on the complex plane with |α( jω) + 1| ≤ δ y . The situation in Fig. 3.1 for mapping of |β( jω) + 1| < 1 and |α( jω) + 1| < 1 is also re-produced here. It is seen that this can result in a significantly reduced area of intersection, consequently much constrained optimal controllers that can reduce the vibration in y( jω) and z( jω) . . . . . . . . . . . . . . . . . . . . . . . . . . Actuation constraint |α( jω)| ≤ γ , together with sensing constraints as represented by mapping of |β( jω) + 1| ≤ δz on the complex plane with |α( jω) + 1| ≤ δ y . . . . . . . . . . . . . . . . Realizable solutions: constraints on realness of control parameter can significantly restrict the feasible solutions . . . . . . Robust design of αopt ( jω) under structured and unstructured uncertainty without constraints . . . . . . . . . . . . . Active vibration isolation structure . . . . . . . . . . . . . . . . . . . . . . . . Active control without constraints: unit α−circle—solid line (black); β−circle—dashed line (red) . . . . . . . . . . . . . . . . . . . Active control with constraints: under sensing constraints the unit α−circle and β−circle become the scaled α−circle (solid black line) and scaled β−circle (dashed red line). For enforced actuation constraint, the length to point A indicates the minimum radius that can be specified for satisfying 12 dB reduction in y( jω); the length to point B indicates the minimum radius that can be specified for satisfying 12 dB reduction in y( jω) and 20 dB reduction in z( jω). The length to point C is a typical choice also associated with robustness consideration . . . . . . . . . . Performance of proposed active control with constraints . . . . . . . Performance of proposed active control with constraints and uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A TMD can be modeled by a 2-DOF mass-spring-damper system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three scenarios for the distance between the centre of |T1 ( jω)| = 1 (blue circle) and (−1, 0) as mapped on the complex α( jω)-plane: a greater than unity; b equal to unity; and c less than unity. Also note that the mapped circle always intersects the origin . . . . . . . . . . . . . . . . . . . . . . . . .

xv

86

88

89 93 96 97 98

99 102 103 106

109

xvi

Fig. 3.14

Fig. 3.15

Fig. 3.16 Fig. 3.17 Fig. 3.18 Fig. 3.19 Fig. 3.20 Fig. 3.21

Fig. 3.22

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4

Fig. 4.5

List of Figures

Graphical explanation for disturbance attenuation and enhancement: the corresponding attenuation and enhancement level can be “read” directly from the scaling (read dotted circle) with respect to |T1 ( jω)| = 1 (blue solid circle) . . . . . . . . . . . . . . . . . . . . . . . . . Graphical illustration for the scenario of | p1 | ≤ | p2 | where the maximum value of m 2max is shown. The feasible choice for m 2 is represented with the read solid segment. Also shown is the critical radius that needs to be satisfied for m 2min ≤ m 2max for the scenario of | p1 | > | p2 | as represented by the green dash-dotted circle and line . . . . . . . . An illustration of the maximum value of m 2 with respect to frequency ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of the parameter selection for stiffness and damping ratios for unity frequency ratio for | p2 | ≥ | p1 | . . . . Illustration of the parameter selection for stiffness and damping ratios for unity frequency ratio for | p2 | < | p1 | . . . . Simulation of performance of the designed damper for a mass ratio tuning and b stiffness and damping ratios tuning . . . . A TMD can be modeled by a 2-DOF mass-spring-damper system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Frequency-dependence of the maximum achievable performance; b mass-dependence of the maximum achievable performance; c stiffness/damping-dependence of the maximum achievable performance; d natural frequency-dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Feasible choices of parameters k2 , c2 , and ω within a 10 × 10 × 10 cubic unity grid; b upper lower bound for performance, and also shown is the lowest bound for the denoted parameter set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal solution αopt ( jω) on complex α − plane for broad band control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Active vibration isolation system . . . . . . . . . . . . . . . . . . . . . . . . . . a Disturbance response z( jω), b disturbance transmission . . . . . Mapping of |β + 1| = 1 on the complex α-plane over [0.06, 0.14] Hz with an obvious 0.02 Hz grid. Solid circle is the unit α-circle and dashed circles are the mappings of the unit β-circles; dotted lines are the corresponding optimal lines for each frequency and finally, the small circles are the optimal points . . . . . . . . . . . Open-loop and closed-loop FRFs with feasible and infeasible optimal choice. a Vibration z b vibration y . . . . . .

109

112 116 117 119 120 122

125

132 138 151 152

153 154

List of Figures

Fig. 4.6

Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4

Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8

Fig. 5.9 Fig. 5.10 Fig. 5.11

Fig. 6.1

Tuning of geometric broad band controllers for a vibration z b vibration y: comparison of closed-loop FRFs with different a and M pair. Dotted line is the original infeasible optimal choice and hence can be seen as a reference when a and M pair is tuned. It is seen that increasing M can improve the accuracy of approximation and hence the broad band controller performance in relation to the prescribed specification . . . . . . . . Whole-band geometric controller performance a vibration z, b vibration y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric controller performance a vibration z, b vibration y,c control signal u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On-line performance of geometric controller a vibration z, b vibration y, c control signal u . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-degree-of-freedom model for a tuned mass vibration absorber with a suspension configuration . . . . . . . . . . . . . . . . . . . Performance limit map over frequencies . . . . . . . . . . . . . . . . . . . . Performance limit map over stiffness . . . . . . . . . . . . . . . . . . . . . . General configuration for two-degree-of-freedom vibration energy harvesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of design freedom: a concentric scaling; b design freedom constraint . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of design freedom: a ω1 = 1.414 rad/s; b ω2 = 1 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Active control performance: a maximum relative magnitude with change of frequency; b performance limit with stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General configuration of 2DOF vibration energy harvesters . . . . Further optimized performance index function with c1 & k2 . . . . Optimal mass (10) is not attainable but can still be evaluated for best performance . . . . . . . . . . . . . . . . . . . . . . . . . . . Design parameters c1 and k1 are better not allocated within the area surrounded by the indicated line, two axes and the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General configuration for a 1DOF and b 2DOF vibration energy harvesters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal magnitude variation with a m2 ; b c1 ; and c c2 . . . . . . . . Ineffective designs should be avoided for optimizing harvesters: a avoid ineffective frequency ranges; and b avoid ineffective parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance specification can only be satisfied provided that the inner unit α-circle (or scaled α-circle) has intersection with the outer β2 -circle and β3 -circle (or scaled β2 -circle and scaled β3 -circle) . . . . . . . . . . . . . . . . . . .

xvii

155 157 158 159 169 171 172 179 183 188

189 193 198 201

204 207 213

216

227

xviii

Fig. 6.2

Fig. 6.3

Fig. 6.4

Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 6.9

Fig. 6.10 Fig. 6.11

Fig. 6.12

Fig. 6.13

Fig. 6.14 Fig. 6.15 Fig. 6.16

List of Figures

Performance limit by purely stiffness or damping design: a no feasible solutions would exist; b influence of performance limit by purely stiffness or damping design . . . . Geometry of the unit α-circle, β2 -circle and β3 -circle: shaded area  indicates that solutions exist where |X 1 ( jω)| AF |X 1 ( jω)|B E can be attenuated whileboth |X 2 ( jω)| AF |X 2 ( jω)| B E and |X 3 ( jω)| AF |X 31 ( jω)| B E can be enhanced . . . . . . . . . . . . . Geometry of the scaled α-circle scaling down by 8 dB: shaded area indicates that solutions exist  can be attenuated where |X 1 ( jω)| AF |X 1 ( jω)| B E  |X ( jω)| by 8 dB while both 2 AF |X 2 ( jω)| B E  and |X 3 ( jω)| AF |X 31 ( jω)| B E remain intact . . . . . . . . . . . . . . . . Optimal performance with a purely stiffness design and b purely damping design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A magnified view for stiffness design in Fig. 6.5 . . . . . . . . . . . . . Optimal design for vibration enhancement in a X 2 performance and b X 3 performance . . . . . . . . . . . . . . . . . . . . . . . Optimal design for vibration attenuation in X 1 performance . . . . Geometrical representation of performance indices with feasible solutions: the shaded area indicates the feasible design of α( jω) satisfying |α( jω) + 1| ≥ δ N > 1 and |βi ( jω) + 1| ≤ δi < 1; thus annihilation is achieved if and only if the centre (−G i ( jω)) locates outside of the scaled α-circle . . . . . . . . . . . . . If annihilation is not feasible, best achievable vibration attenuation is restricted to be the solution at Q mmin . . . . . . . . . . . Geometrical representation of performance indices with feasible solutions: the shaded area indicates the feasible design of α( jω) satisfying |α( jω) + 1| > 1 and |βi ( jω) + 1| ≤ δi < 1; thus annihilation can be achieved at node-2 yet not at node-1 . . . . . . . . . . . . . . . . . . . . . . . Annihilation is not feasible at node-1, and best achievable vibration attenuation is restricted to be the solution at Q mmin with (−9.6)dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the performance before and after design: a annihilation design in node-2; b performance limit design in node-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ranges of k1 and ω (with z-axis below zero) achieving prescribed performance specification . . . . . . . . . . . . . . . . . . . . . . Different feasible solutions of k1 for different ω: a ω = 0.08 rad/s; b ω = 0.1 rad/s . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional of LHS in (6.82) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

230

233

234 235 236 237 237

239 240

242

243

243 254 255 257

List of Figures

Fig. 6.17

Active vibration distribution through feedback control with performance detuning: a node 1 performance with 6 dB vibration attenuation requirement; b node 2 performance with 6 dB vibration enhancement requirement; c node 3 performance with 6 dB vibration enhancement requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

258

List of Tables

Table 2.1 Table 4.1

Parameters of the quarter-car suspension model . . . . . . . . . . . . . . Procedures for TF representation of αopt (s) . . . . . . . . . . . . . . . . .

72 144

xxi

Chapter 1

Introduction to Vibration and Noise Control Problems

Significant levels of vibration can cause a variety of problems that often lead to reductions in product quality in manufacturing systems, passenger and crew comfort and even structural fatigue failure in marine vessels, airplanes, automobiles and civil structures etc. Besides the well-known example of the collapse of Tacoma Narrows Bridge in 1940, a recent disaster is the failure of the Koror-Babeldaob Bridge that was build using the cantilever method in the late 1970s but collapsed suddenly in September 1996. Indeed, with increasing performance demand in modern technological systems towards high-speed, heavy-load, light-weight, flexible-operation, and extreme-condition as reflected in aerospace, marine, transportation, and manufacturing industries etc., the associated vibration and noise issues have become such problematic that may significantly restrict the performance improvement of the systems. As a result of these problems extensive efforts have been devoted to preventing the detrimental effects of vibration. Passive damping strategies including viscoelastic dampers, viscous dampers, metal dampers and friction dampers, tuned mass dampers and tuned liquid dampers have been and are still the dominated methods in tackling problematic vibrations. However the passive approach can be ineffective at low frequencies. It is also limited in the form of constraints on the choice of mass, damping and stiffness (static deflection) values [1]. Moreover, the response of the passive devices is static [2]: once a design is completed it is difficult to modify its design parameters. This is problematic since the system dynamics can change with age or environmental conditions. The performance can thus become unacceptable. As a result, active approach through feedback or feedforward control has to be utilized.

1.1 Historical Introduction The development of active vibration control has evolved from early simple applications to current structural vibration control. The evolution of active vibration control © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Wang, Active Vibration & Noise Control: Design Towards Performance Limit, https://doi.org/10.1007/978-981-19-4116-0_1

1

2

1 Introduction to Vibration and Noise Control Problems

reflects not only the development of actuating and sensing technologies but also the deepening of understanding into the physics of vibration and acoustics, which subsequently connects active vibration control with another somewhat independently developed field—active noise control. The early applications of active vibration control can be traced to the year 1905 when Mallock [3] reported on steam ship vibration reduction by synchronisation of two engines in opposite phase. At this stage applications are not massive but have appeared in many different areas: also in ship application Allan in 1945 [4] described the stabilisation of ships by activated fins, which is a technology that is still utilised today; in aircraft application, Vang in 1944 patented the active damping of aircraft skin vibrations to prevent fatigue damage [5]; in mechanical vibration, Mason in 1945 reported the damping of flexural wave mode by piezoelectric patches [6] etc. However the methods utilized in these early applications are insufficient to deal with structural vibration, where the transmission media has an infinite number of degrees of freedom. This poses a fundamental obstacle to the transmission from the early applications to the stage of active structural vibration control since it is impossible to control the entire infinity of modes. The system model has to be truncated in practice. But the modal truncation leads to both observation and control spillovers which degrade the system performance [7] while in the former case even system instability can result [8]. However this situation was demonstrated only in 1978 by Balas [9, 10]. Another barrier to active structural vibration control is that there are different wave types in flexible structures whose control need various types of actuators and sensors. In addition, adaptive control with digital signal processing that has been successfully implemented in the 1970s for the reduction of noise (see [11, 12]) was then confronted with an insurmountable difficulty: causality. This is because the propagation speed is generally higher in solids than in air and the signal processing speed is simply not fast enough to calculate the adaptive controller, particularly for the broad band case in the early 1980s. Stimulated by the above problems a considerable amount of effort has been devoted to the development of compound materials with embedded sensors and actuators (often known as smart materials, adaptive structures or bi-functional elements) as well as the algorithms for reduction of spillovers [13, 14] since the end of the 1980s. The availability of instrumentation and control strategies finally renders active structural vibration control possible, the result of which are summarized in the classical texts by Inman [15] and Meirovitch [16], more recently by Fuller et al. [17] and Preumont [18]. Recent development of active structural vibration control strikes out a very distinctive field—active structural vibration acoustic control (ASVA). The unique feature is that active vibration control makes an important connection with active noise control which has been developed independently since the “point of silence” experiment by Lord Rayleigh in 1878 [19]. Although active noise control (ANC) has a very different set of concepts and control strategies (mainly feedforward) which largely depend on the geometry of sound field, it does have similarities with active vibration control (AVC): wave feature of both vibration and sound field allows the

1.2 Active Vibration and Noise Control: State-of-the-Art

3

utilization of anti-phase destruction control method; many noise problems result from radiation of structure-borne sound etc. With the deepening of understanding of the physics into structural-acoustics interaction it is possible to control sound field through suppressing the structural vibration, although this is not necessarily the optimal solution. These aspects can be found in the texts by Fuller et al. [17]. The physics of the interaction between vibration and sound field can be found in [20]. Currently active structural vibration and acoustic control is still undergoing extensive investigation. From a control system design point of view, a typical active control design process consists of the following main steps, although more detailed steps can be found, e.g. in [18]: obtaining the system dynamics through physical principles or system identification methods; reduce the order or simplify the model if necessary in order to be tractable for design purpose; determine actuators and sensors type, number and placement in terms of the prescribed criteria; evaluate the effect of sensors and actuators and consequently incorporate the effect into the modified system dynamics1 ; design a controller to satisfy the performance requirement; system implementation and evaluation. There are numerous research topics within each step of the design procedures. However while generally recognized methods do exist for system modelling such as finite element analysis or for model reduction such as those standard model reduction techniques, the problem of advanced active controller design are currently still under close evaluation. The state-of-the-art solutions to these problems are reviewed in the following section.

1.2 Active Vibration and Noise Control: State-of-the-Art A variety of well-established methodologies have been developed for active controller design. These methods can be classified in terms of different criteria, e.g. model-based control and model-free control in terms of whether a pre-modelling process or accurate model is required before the controller design; centralized and decentralized control in terms of the subsystem structures and interconnection topologies etc. However most of the controller design problems can be discussed in a general framework as that shown in Fig. 1.1. This representation is known as the general control configuration, the standard representation or the linear fractional transformation representation. This general control configuration is introduced by Doyle [23, 24] as a generic method for formulating control problems. This formulation has also been widely recognized as the framework for controller synthesis problems resulting in a unified view towards H2 and H∞ control. The significance of the general control configuration is its generality since almost any linear control problem can be formulated in this way including the observer 1

An important principle for active vibration control is that sensors and actuators should not interfere with the system dynamics [21]. However it is rarely the case and the system dynamics are often altered due to the presence of sensors and actuators, especially in lightly damped space structures [22]. It is thus crucial to model the dynamics of sensors and actuators as an integral part of the system plant.

4

1 Introduction to Vibration and Noise Control Problems

Fig. 1.1 P is the generalized plant and K is the generalized controller. A general control design problem can be formulated as finding a controller K in such a way that some norm of the transfer function from d to z is minimized

design and most impressively, feedforward control design ([25], pp. 98 and 99). This is the motivation to approaching active control of vibration and noise via the general control configuration, unlike any other approaches found in the texts mentioned above where feedback control and feedforward control are thought of as two different control strategies to be selected depending on the nature of input or disturbance signals. With such an interpretation, the general active vibration and noise controller design problem can be formulated as follows. General Vibration and Noise Controller Design Problem: Find a realisable stabilising controller K which based on the information in the sensed output y generates a control signal u which counteracts the influence of the exogenous inputs d on the exogenous outputs z, with the consequence that the closed loop norm (H2 , H∞ or Hankel norm etc.) from d to z is minimized.

1.2.1

H2 /H∞ Optimal Control for Linear Systems

From Fig. 1.2, the linear system dynamics (incorporating sensors and actuators) can be represented as follows:

Fig. 1.2 Gain scheduling adaptive control changes controller parameters in a pre-determined manner

1.2 Active Vibration and Noise Control: State-of-the-Art

  z y

 =

P11 P12 P21 P22

5

  d u

,

(1.1)

and u = K y.

(1.2)

Therefore the closed loop transformation from the exogenous inputs d to the exogenous outputs z can be obtained: Fl (P, K ) = P11 + P12 K (I − P22 K )−1 P21 .

(1.3)

The standard H2 or H∞ optimal control problem can be stated as finding a realizable stabilising controller K such that the 2-norm Fl (P, K )2 or the ∞-norm Fl (P, K )∞ is minimized [26]. Standard algorithms exist for the above H2 and H∞ optimal control problems. The widely used approach is the state-space solution ([25], p. 363) that can be found in [27, 28]. In this approach, the algorithms for H2 optimal control are slightly different from those of H∞ optimal control (with the typical assumptions made on the system plant, see the original papers above) in that H∞ optimal control needs an iterative process called γ -iteration to find the optimal controller. However, in practice, the H∞ sub-optimal control problem is resorted to: let γmin be the minimum value of Fl (P, K )∞ over all realizable stabilising controllers K , find a γ > γmin such that: Fl (P, K )∞ ≤ γ .

(1.4)

With this consideration both the optimal controller for H2 control and the optimal controllers for H∞ control can be found from the solution of just two Riccati equations. It is in this sense that the formulation of general control configuration has resulted in a unified view towards H2 and H∞ control. However it should be noted that finding an optimal H∞ controller is much more complicated both numerically and theoretically. There are a lot of complications and tricks involving with the assumptions about the structure of the plants. For example H∞ theory can not be applied to systems with poles the imaginary axis therefore the integral action has  on 1 and ε must be very small. The resulting controller to be approximated by, e.g. s+ε will then also have a pole very close to the origin. These assumption-relaxation problems and tricks are widely discussed, even in the original formulation in the papers cited above. A discussion of using unstable and non-proper weights can be found in [29]; H∞ control for dead-time systems is discussed in [30].

6

1 Introduction to Vibration and Noise Control Problems

It is also noted that H∞ control is a vast area and includes a host of methodologies for H∞ controller design: H∞ loop-shaping design which is an open-loop shaping method that combines robust stabilization with classical loop shaping [31]; mixedsensitivity H∞ control aims to shape the closed-loop transfer functions such as the sensitivity function S along with the complementary sensitivity function T , although this method becomes increasingly difficult with more than two shaping functions; signal-based H∞ control provides a genuine framework for multivariable design, however, the design method called μ-synthesis is still not a fully resolved problem in the case of structured perturbation. The above discussions are mainly confined within state-space solutions to H∞ control. There are many other approaches to H∞ control, e.g. linear matrix inequality (LMI) approach [32, 33] emphasizes its ability to deal with non-regular plants and the utilization of non-central controller; operator-theoretic methods [34, 35] handle naturally infinite-dimensional H∞ control theory etc. Given so many facets of H2 and H∞ control, a word about the evolution of H2 and H∞ control is appropriate to establish their connection and clarify some of the concepts. H∞ control aims to address the problem of worst-case controller design for linear plants subject to unknown additive disturbances and plant uncertainties [36]. It is originally formulated in the frequency domain where operator and approximation theory as well as spectral factorization have been the main tools. The term “H2 control” is the generalization of linear quadratic Gaussian (LQG) control which is developed in the time domain. Since the solution to LQG is prescribed by the separation principle or certainty equivalence, it is thus known once the two algebraic Riccati equations are solved. So far H∞ and H2 control seem to be two totally different problems. The development of H∞ control in 1980s by Glover and Doyle, however, shows that there exist state-space solutions to H∞ control problems which are not different from those of H2 control problems. Thus the general H2 and H∞ control problems can be discussed on a same par, especially for the controller synthesis. More detailed stories can be found, e.g. in the book by Helton and Merino [37]. Finally H2 (LQG) and H∞ control theory has found a number of applications in active control of civil and mechanical engineering structures, with a long-standing application history for LQG control. Recent applications of LQG control can be found, for example, in [38–40]. H∞ optimal control is a more recent story but have found applications in active control of flexible structures: H∞ control of a plate structure can be found in [41]. Reference [42] presents a general approach toward the design of an H∞ controller for structural vibration suppression using an active mass damper (AMD) with structural uncertainties. More H∞ control applications can be found in [43–45]. Finally the extension of H∞ control to nonlinear systems have also been studied extensively and can be found, e.g. in the monograph by Helton and James [46].

1.2 Active Vibration and Noise Control: State-of-the-Art

7

1.2.2 Adaptive Control The H2 /H∞ control is “static” and the parameters of the plant are considered to be fixed, henceforth the parameters of the controller to be designed. However it is often the case that the process is time-varying or has large environmental disturbances, e.g. there are large parameter changes for aeroplane dynamics due to changes in the control environment, e.g. wind speed, air pressure etc. These problems motivate a control strategy that changes or adapts the design of the control algorithm to accommodate changes in the plant or its environment. This control strategy is called adaptive control. Several main approaches to adaptive control are gain-scheduling (GS), self-tuning regulators (STR), model-reference adaptive systems (MRAS) and adaptive duel control (ADC). GS is one of the simplest methods of adaptive controller design. Figure 1.2 shows the block diagram. GS changes the controller parameters in a predefined manner to compensate the changes in the plant. Therefore the controller can react quickly to changing conditions. However it achieves this through an open-loop way: the scheduled controller changes parameters without monitoring closed-loop performance (this is indicated in Fig. 1.2 by dashed lines). As a consequence, GS can not be used if the plant dynamics or the disturbances are not known accurately. Nowadays GS has been an important ingredient in modern flight control systems but its use (in its original formulation) in active vibration control is not widespread due to the above limitations. However the concepts of GS can be easily integrated with other control methods such as LQG and H∞ control: changing controller parameters according to a certain scheduling variable while designing controllers using LQG or H∞ techniques for a fixed scheduling value. Therefore it is seen that Hansson [47] describes an adaptive cab suspension controller using GS control where different values of the feedback gain vectors are pre-calculated using LQG technique. Saito et al. [48] describe a vibration control of tall buildings using a hybrid mass damper system by GS and linear optimal state feedback; Forrari and Kamiyama [49] demonstrate a robust gain-scheduled controller that is tested experimentally when the flexible structure is excited with a scaled historical earthquake record. Finally it is noted that GS, in its current form, has become one of the most popular approaches to nonlinear control design. This will be looked at in the review of nonlinear control to be presented in the Sect. 1.2.4. STR (Fig. 1.3) consists of three parts [50]: a feedback controller that can provide good closed-loop performance if the parameters of the plant model can be known accurately; a recursive estimator that estimates the plant parameters based on inputs and outputs measurements; a control design algorithm that utilizes the estimated parameters of the plant to update the control law. STR clearly distinguishes itself from GS in that it closely monitors the closed loop performance. One the other hand, since there are many methods for controller design as well as estimating parameters, the STR scheme is very flexible and there are a variety of combinations that have been explored [51]. This flexibility shows the great potential of STR in the application of active controls. For example, Shaw [52] adopts a recursive least-squares (RLS)

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1 Introduction to Vibration and Noise Control Problems

Fig. 1.3 Self-tuning regulator estimates the plant parameters from input/output data; plant parameters are then used in the control design algorithm to update the control law which can provide good closed-loop performance if the parameters of the plant model are estimated accurately

algorithm for system parameter estimation and the minimum variance controller for active vibration isolation; Johnson in his NASA research reports [53] describes a class of algorithms for the multi-cyclic control of helicopter vibration, e.g. identification of the helicopter model by least-squared-error or Kalman-filter methods and a minimum variance (MV) or quadratic performance function controller; Kumar et al. [54] presents a comparative study of self-tuning control of smart structures with a minimum variance controller, a pole-placement controller and a linear quadratic controller. However it is also noted that STR as a feedback control strategy has not been widely adopted in the area of active vibration control, albeit it possesses great potential. This is so due to the long “tradition” in vibration control community to assume that a measurement coherent with the disturbance is available and the tuning rate is slow enough and hence feedforward control is often utilized [17, 55]. But the essential concepts of control design algorithms and estimator of STR are still paramount in its feedforward “counterpart”. Adaptive feedforward control with online parameters estimation makes it popular for its relatively easy maintenance of stability. Filtered-x LMS, where a feedforward loop of the filtered input signal is introduced to the traditional LMS algorithm, together with its variations [56, 57] is the main control design algorithm. More on this approach and its widespread applications can be found in [17]. In MRAS (see Fig. 1.4), the output Y of the plant is required to follow the model output Y M for a given reference R. That is, the desired performance is specified by a reference model. Hence the key problem with MRAS is to determine the adjustment mechanism so that a stable system, which brings the error to zero, is obtained [51]. There are many different methods for the determination of the adjustment mechanism, three of which are most popular: the gradient approach based on MIT rule; Lyapunov’s second method and hyper stability design. In the gradient approach, the error E = Y − Y M and a cost function J (θ ) is defined where θ is the parameter (or

1.2 Active Vibration and Noise Control: State-of-the-Art

9

Fig. 1.4 Block Diagram of MRAS. The controller is such designed to make the performance Y follow the desired model performance Y M for the same reference signal R. Thus is the name model-reference adaptive system

Fig. 1.5 Hyperstability design is a direct method of stability analysis of nonlinear models, considered as linear and nonlinear parts [61]

parameters) to be designed. Then the MIT rule states that the parameter (or parameters) is changed in the direction of the negative gradient of J so that it is to be minimized: ∂J ∂ J ∂e dθ = −k = −k (k > 0). dt ∂θ ∂e ∂θ

(1.5)

 Therefore if the parameter change is assumed to be slow then the item ∂e ∂θ which is called sensitivity derivative, can be evaluated. Hence the controller adjustment mechanism can be determined by Eq. (1.5). Since the cost function can have many different forms (e.g. mean square error, absolute value error etc.) there are many alternatives to how the parameters are updated. The tuning of the adaptation gain k is crucial in the gradient approach because the MIT rule does not guarantee convergence or stability. Lyapunov’s second method is an alternative approach but with stability guarantee. In this approach, the adaptation mechanism for the controller is derived from the systems’ Lyapunov functions. Therefore the key problem is to define an appropriate Lyapunov function. However intuition and experience have to be exercised to find a Lyapunov function, although this can be done easily for linear stable systems.2 Many references on the control of dynamical systems can be made towards this purpose, e.g. [58–60]. Unlike the state-space view of the Lyapunov approach, hyper stability design employs an input–output view and decomposes the nonlinear model into a linear and a nonlinear part (Fig. 1.5).

 For an asymptotically linear system d x dt = Ax, it can be shown that for each symmetric Q > 0 there exists a unique P > 0 such that A T P + P A = −Q. Then V (x) = x T P x is a Lyapunov function.

2

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1 Introduction to Vibration and Noise Control Problems

Then the adjustment mechanism is to be derived by the utilization of Popov Integral Inequality (PII): t1 vwdt ≥ −λ20 ∀t1 ≥ 0.

(1.6)

0

Such designed controller will result in a hyper stable system if H (s) is positive real, or asymptotic hyper stable if H (s) is strictly positive real. The application of MRAS in active control is mainly focused on robotic vibration control and active suspension where a reference model can be defined. For example, Sunwoo et al. [62] describe an MRAS system for vehicle active suspension where the adjustment mechanism is derived from Lyapunov’s second method. Remarkably 30–50% performance improvement over passive system around and above the natural frequency of the sprung body is reported. Alleyne and Hedrick [63] present another active suspension of a quarter-car model where Lyapunov’s approach is also used to derive the adjustment mechanism, but for the enhancement of a sliding controller. It is shown that the build-in MRAS scheme improves the sliding controller that has been designed in a previous study. Experimental evaluation of an MRAS for a hydraulic robot can be found in [64], where the controller adaptation mechanism is recognized to be a variation of the MIT rule. More applications can be found in, e.g. [65, 66] In all the three main approaches to adaptive control discussed above, namely GS, STR and MRAS, an implicit assumption is the certainty-equivalence (CE) principle: parameter estimation is separated from controller design. That is, the identified parameters are used in the controller as if they were the true values of the unknown plant. The uncertainty of the estimation is not taken into account. This may lead to poor control during tuning phase [50]. Adaptive duel control (ADC) takes the task of treating the parameter uncertainties in a general optimization framework. However ADC is too complicated for practical problems [51, 67]. Nevertheless the development of ADC in the last 10 years, particularly the bicriterial approach, raises the possibility of application. A reference can be made to the monograph [68] where a survey of dual control methods is presented. In particular the bicriterial synthesis method for dual controllers is delineated in great detail by the authors who are also the originators of this approach.

1.2.3 Variances of Adaptive Control The idea of making controllers adaptive is very attractive to fighting against uncertainties and nonlinearities. With the dramatic development in signal processing this idea, together with the strong inputs from artificial intelligence (AI) has evolved into numerous control strategies. In particular the concepts of real time estimator and controller design algorithms/computing approaches have been “transplanted” into

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11

these new controller design strategies. There have also seen an increasing number of applications in active vibration control, although some of these design methods are originally motivated from a very different area of study and for a very different purpose. In this subsection several variances of adaptive control are briefly reviewed and many others are only mentioned. The purpose of this subsection is therefore not to be comprehensive but to point out that there are many adaptive methods as potential candidates for active vibration control applications. The first group of methods aim to control the plant to track a predetermined reference, as in the case of MRAS. Among these are model predictive control (MPC), repetitive control (RC) and iterative learning control (ILC). MPC differs from the conventional adaptive control methods discussed above in that it predicts the future outputs for a designated prediction horizon based on a mathematical model of the system dynamics. A control design algorithm produces a set of current and future control actions that optimize the cost function to keep the dynamics as close as possible to the set-point. However only the current control action is executed and the same process is repeated leading to an updated control action with corrections based on the latest measurements. This is the well-known receding horizon strategy. Formulations based on different system dynamics (impulse response model, transfer function model or state-space model) and different cost functions (Quadratic or NonQuadratic), together with constraints handling and robustness issues result in different types of MPC. Details can be found in, e.g. [69–71]. A survey of commercially available MPC technology is presented in [72]. Unlike MPC where a mathematical model of the system dynamics is required, RC and ILC can work with an uncertain or even unknown system that operates repetitively, with the same initial setting is returned in the case of ILC for each repetition. Therefore RC and ILC produce zero tracking error through a repetitive learning process. Hence one of the key problems is to ensure the error to decay monotonically since poor transient tracking is very problematic in practical applications [73]. Besides the theoretical efforts for monotonic convergence and robustness [74], it is also found that tuning of RC and ILC is even more straightforward than classical PID control. Longman discusses in [75] these tuning rules for engineering practice. The simplicity of the tuning of RC and ILC, together with the fact that vibration is often a periodic behaviour, makes them as good candidates for active vibration control, e.g. Daley et al. [76] use RC for active vibration isolation; Wassink et al. [77] demonstrate active vibration control of inkjet printheads using ILC. It is seen that the fundamental idea of RC and ILC is repetitive learning. Since learning is also characteristic of intelligence, there are a large number of control techniques that use various artificial intelligence (AI) computing approaches that can be classed as intelligent control (IC). However a precise definition of IC is difficult to make and it is very hard to draw a boundary between IC and other control strategies [78–80], e.g. learning control such as RC and ILC are sometimes classified as IC. A much more confined concept of IC can be defined as control techniques/computing approaches that are either biologically motivated such as neural networks (NN) and evolutionary algorithms (EA) or encoded for human knowledge as represented by fuzzy control (FC) and expert control (EC) [81]. These control methods achieve

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1 Introduction to Vibration and Noise Control Problems

automation via the emulation of biological intelligence. An overview of techniques in IC is referred to [82]. Some intelligent learning algorithms particularly applied in the area of active control (a flexible beam system in transverse vibration with the controller designed on the basis of optimal vibration suppression) are compared in [83]. However it is noted that the concepts and techniques of learning and optimization devised in IC can easily integrated with other adaptive control methods as well as those model-based H2 and H∞ optimal control methods. Thus there are literally numerous number of controller design approaches. In the area of active control, for example, in [84], Kumar et al. present a very careful study of two approaches to vibration control of flexible structures with quickly varying parameters: adaptive LQG and adaptive NN; a combination of LQG with an input recursive least squares (RLS) estimator for active structural vibration control is described by Ho and Ma [85]; in [86], Si and Li report the vibration control of a large flexible space truss using a self-adaptive fuzzy controller; even adaptive-fuzzy-neural control system for active vibration control has been discussed in [87]. While advertising the richness of controller design methods, it is even important to analyze the properties of the system (structural properties, disturbance properties etc.) to be controlled and the applicability of each controller design method. A final decision of a controller should retain the simplicity and easy maintenance of the active control system. See also the comment on control and system integration in the next section.

1.2.4 Nonlinear Control Adaptive control and its variances presented above are appealing approaches against model uncertainties and nonlinearities. For example, gain scheduling has become one of the most popular approaches to non-linear control design (see the survey article [88]); fuzzy control has long been argued to possess inherent robustness and ability to handle the nonlinear behaviour of structural vibration [89]; many nonlinear control problems are hard to solve analytically and evolutionary algorithms can play a significant role in tackling such problems etc. However besides these adaptive (intelligent) approaches to nonlinear control, there are also a rich collection of alternative model-based approaches to nonlinear controller design, e.g. generalization of H∞ control to nonlinear systems discussed in Sect. 1.2.1 above; feedback linearization (FL); sliding mode control (SMC) etc. SMC is also an important approach to robust control. Both SMC and FL possess two phases of design. For SMC in the first phase (reaching phase), the system is driven to a stable manifold; the system then slides to equilibrium in the second phase (sliding phase). In FL, the nonlinear system is first “transformed” into an equivalent linear system and then the standard linear controller design methods can be employed. Problems can occur in SMC during the sliding phase, e.g. chattering and large control authority [60]; in both SMC and FL for unstable zero dynamics (more details can be seen, e.g. in [90]). However there have been numerous applications in active vibration suspension and suppression using

1.3 A New Paradigm: Global Control Towards Performance Limit

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SMC, although less in the case of FL control. For example, for the control of active magnetic bearing (AMB) systems (which are inherently nonlinear) with a flexible rotor, Jang et al. [91] present a SMC approach while Lindlau and Knospe [92] utilize FL method with linear controller designed via μ-synthesis. Finally it should be noted that all the control approaches introduced so far are centralized control in the sense that a central controller is to be designed to achieve desirable system performance. Decentralized control is necessary where large scale system is interconnected by linear or nonlinear subsystems and each subsystem is actuated by a “local” controller. Hence decentralized control can be viewed as a control system structure design that is closely related with sensor and actuator strategies. In more complex systems, different control strategies are integrated forming distributed and networked systems, and this is the concept of control and system integration, e.g. a neural-fuzzy H∞ optimal controller is an example of control integration, while a full authority flight control combining with health monitoring and control reconfiguration is an example of system integration etc. This finishes the rather brief review of active controller design, even without mentioning pole placement design and the important concepts of vibration isolation and modal control. But before turning to the new methodology to be proposed in this monograph, some generic features of the current design methodologies are to be looked at from different perspectives; issues are raised that new approaches are indeed necessitated for addressing these problems. This is turned to in the next section.

1.3 A New Paradigm: Global Control Towards Performance Limit Active control methods can be classified into two basic approaches, namely feedforward control [16–18, 61, 93, 94] and feedback control [95, 96]. Although it has been pointed out that the two approaches can be discussed within the general control configuration framework, the generic feature is to transform the control design parameters into the solution to appropriately defined optimization problems. This claim also applies to the famous H2 /H∞ design [97–99] while H∞ mainly for harmonic control and H2 mostly for multi-modal suppressions [100, 101]. However, vibration and noise problems generally occur either at specific discrete frequencies, caused by periodic disturbances such as out of balance forces in rotating machines, or in a narrow band, often associated with lightly damped structural modes. From a control-theoretic perspective, the problem of control over a frequency band must be tackled. Within the framework of H2 /H∞ control, this is addressed through two methods: the generalized KYP-lemma and the weighting functions. Yet, both methods are still relying on solving optimization problems and even requiring model reductions [102–105], and henceforth they are difficult to handle the following issues: (1)

Performance evaluation influenced by constraints and system parameters: the generalized KYP-lemma and the weighting functions methods can handle

14

(2)

(3)

(4)

1 Introduction to Vibration and Noise Control Problems

the finite frequency issue yet they are essentially of trial-and-error nature, and thus with system parameter variation and constraints, it becomes quickly cumbersome to determine the influence of such variation and constraints on performance. Non-global optimal solutions: for most of the current design methods, the optimum solution is often based solely on information local to the actuators and implementation can, in practice, result in increased levels of vibration or noise at other locations. That is, local optimal solution can lead to globally deteriorated performance. Performance compromise: for multivariable or multilocation control, as is generally the case for active control problems, the issue of how the performance variables compromise with each other has not been addressed to satisfaction. Indeed, the problem whether the performance variables really compromise with each other is still not totally solved in the literature. Performance limit: the fundamental performance limit as encoded in Bode’s Integral Relationships (or Waterbed Effect) is only a “negation theorem” and the more important question is, what is/are specific frequency/frequencies where all performance variables can be attenuated at the same time? A clear-cut answer must be provided with an easy-to-verify condition to tell the frequencies where simultaneous vibration attenuation for ALL performance variables is achieved.

To further comprehend the significance of the above issues, recall the general control configuration as shown in Fig. 1.1 as reproduced here. And now consider the following finite frequency control problems:   Problem 1.1 z2 d2 ≤ γ ω ∈ ω1 ω2 rad/s;

 z2 d2 ≤ γ   Problem 1.2: ω ∈ ω1 ω2 rad/s. y2 d2 ≤ β It is seen that Problem 1.1 is the conventional H∞ control problem. Problem 1.2 can be defined as global control problem in comparison with Problem 1.1 as the local one. This refers to the non-global optimal solutions in Issue (2). Essentially, both sensitivity function and complementary sensitivity function are required to be suppressed for active control problems. This is different from many control problem where complementary sensitivity function is necessarily to be attenuated. To obtain the solution to global control Problem 1.2, one might argue that a purely theoretical perspective can be suggested by an alternative method of “simultaneous shaping of sensitivity and co-sensitivity functions” of the system. That  is, a mixed sensitivity   Ws (s)S( jω)     W1 (s)R1 ( jω)    problem can be formulated as finding a controller so that   ≤γ ..   .    W (s)R ( jω)  n n ∞ is optimized, which is clearly a simultaneous sensitivity and co-sensitivity shaping problem. While there exist rule-of-thumbs to choose the weighting functions to

1.3 A New Paradigm: Global Control Towards Performance Limit

15

shape the sensitivities, it is still difficult to specify the performance or exploit performance limitations, nevertheless to tell if a performance specification in sensitivities is feasible or not. This brings us immediately to Issue (3) and Issue (4) where the performance compromise and performance limit are to be obtained. Obviously, the above questions cannot be answered based on the conventional H2 /H∞ design methods, where iterative routines are required to tell the detailed compromise scenario among the performance variables even for a discrete frequency ω. And it is even challenging to use conventional methods to tell the performance limit or figure out how the (globally defined) performance variables compromise with each other, if this is not possible. New method must be devised to provide insights into these difficult problems. To make the situation comprehensive, following the above Problems 1.1 and 1.2 for local and global

control  problems, further define: z2 d2 ≥ γ   Problem 1.3 ω ∈ ω1 ω2 rad/s; y2 d2 ≤ β

 γ1 ≤ z2 d2 ≤ γ2   Problem 1.4 ω ∈ ω1 ω2 rad/s. y2 d2 ≤ β Thus Problem 1.3 is a simultaneous active control and energy harvesting problem; and Problem 1.4 is the (global) envelope control problem with some performance variables are maintained within a performance envelope, and henceforth can be viewed as constrained active control and energy harvesting problem in comparison with Problem 1.3. It remains to demonstrate that both problems represent important classes of theoretical developments with practical significance, e.g. energyharvesting vibration absorbers [106–108]. Yet, the solutions are still based on optimization routines as those used in active controls (see the recent survey articles [109–112] and references therein). To summarize, for the Problems 1–4 defined above, the current methods have the difficult to answer questions such as: (1) (2) (3)

Whether all performance variables X i ( jω) ∀i ∈ [ 1, N ] can all be attenuated? What is the performance limit for each performance variable? How do the performance variables compromise with each other?

The aim of this monograph is to disseminate a new paradigm to active controls, namely a performance-limit oriented global design methodology. Specifically, a number of fundamental results upon significant issues such as (1) existence of feasible controllers; (2) optimality of controllers; (3) performance limit of controllers; (4) compromisablity among the performance specifications; (5) synthesis of controllers; (6) influence of constraints on optimal solutions, can all be resolved within the proposed framework for the four Problems 1–4. The state-of-art is thus refined with a new approach complementary to those optimization-based routines, where extra effort would have to be exercised to disclose the compromisability of performance specifications. Meanwhile, the proposed method can target the performance limit while quantitatively assessing the influence of constraints as imposed by physical parameters.

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Indeed, it is the author’s conviction that the fundamental philosophy for control is to excavate system performance by taking advantage of every piece of information, e.g. internal structural properties, topology and interactions, external noise and disturbance power spectral distributions etc. Thus underpinning the design philosophy lies in a pivotal issue: what is the performance limit of active control and energy harvesting? This must be answered within any design framework.

1.4 Contributions and Contents of the Monograph Such a design methodology is proposed in this monograph. It is claimed that limit of performance is obtained where other methods can be benchmarked against the best achievable performance. It is also reasonable to expect that the proposed methodology can be applied to other engineering areas for vibration and acoustic applications. This forms the distinctive contribution of the monograph. It is worth pointing out that the proposed framework essentially results in a new design methodology—performancelimit oriented active control. It is initiated by the author with the project “Active Control for Performance Limit” (ACPL). The project has been developed over the past decades with a series of publications [113–127] and patents [128–143]. This monograph is to summarize the results in a coherent manner for this performancelimit oriented active control design methodology with the following contents. Chapter 2: Difficulties with conventional methods are presented and illustrated with examples; new approach is thus necessitated alluding to the methodology of this monograph, namely active control for performance limit with a geometric approach. This will be illustrated through its resolution of fundamental issues such as the necessary and sufficient condition of solution existence, pushing the performance to limit, as well as reconciling the challenging problem of performance compromise. Furthermore, a link is established between active feedback control and parameter optimization design, making connection of the problem of noise and vibration control to that of multiple-location and multiple-variable control. Chapter 3: The basic design framework is then extended to handling constraints and robustness issues. Actuation, sensing, as well as states constraints are brought into the proposed framework. Robustness issues are also been integrated. Further important constraints of physical realizability is then taken into account. Essentially, this results in a unified and systematic methodology for active control of vibration and noise with constraints and uncertainties. Chapter 4: Several important extensions are discussed in this chapter. Of particular importance is the extension to broad-band situation where a universal design strategy is proposed providing performance tuning over an arbitrary frequency band. Another extension to performance improvability test is also alluded to. One more extension is considering the crucial task for verification & validation, and this follows from the development to test whether the control system has reached performance limit. Consequently, an experimental design for testing performance limit for harmonic vibration control systems is proposed.

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Chapter 5: Active control has been approached with the proposed geometric design method. Previous chapters have provided detailed exposition and demonstrated that the proposal can be a systematic design methodology to active control towards performance limit. This chapter aims to show that the proposed framework can be utilized to handle another important issue of energy harvesting, while another objective is to obtain the performance limit independent upon specific parameters. Chapter 6: The concept of active vibration distribution is novel yet important in both commercial and military applications. No approaches are available for tackling the active vibration distribution problem without significant modifications. This is addressed in a general formulation yet special attention is given to constrained solutions that are of great importance in engineering applications. Important results are obtained with extensive comments providing guidance for illustrating design procedures. The issue of improving performance towards achievable limit is also dealt with. Consequently a systematic design procedure is obtained with both control design and performance improvement. As an emerging concept, it is expected that active vibration distribution can lead to many innovative designs and smart structure applications. Chapter 7: Summary and contributions of the monograph are provided, with new perspectives being given for further research development.

References 1. D.J. Inman, Engineering Vibration, Prentice-Hall, Inc., New Jersey, 2001. 2. S. Daley, Active Control of Vibration, Lecture Notes ACS6109, University of Sheffield, 2004. 3. A. Mallock, A method of preventing vibration in certain classes of steamships, Trans. Inst. Naval Architects 47, (1905), 227-230. 4. J.F. Allan, The stabilization of ships by activated fins, Trans. Inst. Naval Architects, 87, (1945), 123-159. 5. A. Vang, Vibration dampening, U.S. Patent US 2,361,071, filed: Sept. 23, 1942, patented: Oct. 24, 1944. 6. W.P. Mason, Piezoelectric damping means for mechanical vibrations, U.S. Patent US 2,443,417, filed: March 29, 1945, patented: June 15, 1948. 7. L. Meirovitch, Some problems associated with the control of distributed structures, Journal of Optimization Theory and Applications, 54, (1987), pp. 1-20. 8. L. Meirovitch and H. Baruh, Effect of damping on observation spillover instability, Journal of Optimization Theory and Applications, 35, (1981), 31-44. 9. M.J. Balas, Feedback control of flexible systems, IEEE Transactions on Automatic Control, AC-23, 4, (1978), pp. 673–679. 10. M.J. Balas, Active control of flexible systems, Journal of Optimization Theory and Applications, 25, (1978), 415-436. 11. K. Kido, Reduction of noise by use of additional sound sources, Proceedings of Inter-Noise 75, Sendai, Japan, (1975), pp. 647–650. 12. G.B.B. Chaplin and R.A. Smith, Active methods of cancelling repetitive vibrations, UK Patent 19717/76, 1976. 13. C. Mei and B.R. Mace, Reduction of control spillover in active vibration control of distributed structures using multi-optimal schemes, Journal of Sound and Vibration, 251, (2002), pp. 184192.

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14. B.R. Mace and R.W. Jones, Feedback control of flexural waves in beams, Journal of Structural Control, 3, (1996), pp. 89-98. 15. D.J. Inman, Vibration with Control, Measurement and Stability, Prentice Hall, Englewood Cliffs, NJ., 1989. 16. L. Meirovitch, Dynamics and Control of Structures, Willey Interscience, 1990. 17. C.R. Fuller, S. J. Elliott and P. A. Nelson, Active Control of Vibration, Academic Press, London, 1996. 18. A. Preumont, Vibration Control of Active Structures: an Introduction, Kluwer Academic Publishers, 1997. 19. Lord Rayleigh, The Theory of Sound, MacMillan & Co, London, 1st ed. 1878, pp. 104–106. 20. F. Fahy, Sound and Structural Vibration: Radiation, Transmission and Response, Academic Press, London, 1985. 21. W.K. Gawronski, Dynamics and Control of Structures: a Modal Approach. Springer, New York, 1998. 22. H. Benaroya, Mechanical Vibration: Analysis, Uncertainties and Control (2nd Edition). Marcel Dekker, Inc., 2004. 23. J.C. Doyle, Synthesis of robust controllers and filters, Proceedings of IEEE Conference on Decision & Control, San Antonio, Texas, pp. 109–114, 1983. 24. J.C. Doyle, Lectures Notes on Advances in Multivariable Control, ONR/Honeywell Workshop, Minneapolis, October, 1984. 25. S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design. John Wiley & Sons, 1996. 26. J.M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, 1989, p267. 27. J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis, State-space solutions to standard H2 and H∞ control problems, IEEE Transactions on Automatic Control, 34(8), (1989), 831-847. 28. K. Glover and J.C. Doyle, State-space formulae for all stabilising controller that satisfy an H∞ norm bound and relations to risk sensitivity, Systems and Control Letters, 11, (1988), 167-172. 29. G. Meinsma, Unstable and nonproper weights in H∞ control, Automatica, 31, 1 (1995), pp. 1655-1658. 30. G. Meinsma and H. Zwart, On H∞ control for dead-time systems, IEEE Transactions on Automatic Control, vol. 45, no. 2 (2000), pp. 272-285. 31. D. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, Lecture Notes in Control and Information Sciences, vol. 138, SpringerVerlag, Berlin, 1990. 32. P. Gahinet and P. Apkarian, A linear matrix inequality approach to control, International Journal of Robust and Nonlinear Control, 1994. 33. P. Gahinet and P. Apkarian, An LMI-based parameterization of all controllers with applications, IEEE Conference on Decision and Control, 32nd, San Antonio, (1993), pp. 656–661. ˇ 34. C. Foias, H. Ozbay and A. Tannenbaum, Robust Control of Infinite Dimensional Systems, New York: Springer-Verlag, 1996. 35. J.A. Ball and J.W. Helton, Shift invariant subspace, passity and reproducing kernels and optimization, in Operator Theory: Advances and Applications, Basel: Birkhäuser, 1988. 36. T. Basar and P. Bernhard, H∞ Optimal Control and Related Minimax Design Problems: a Dynamic Game Approach, 2nd Edition, Birkhäuser, Boston, 1995. 37. J.W. Helton and O. Merino, Classical Control Using Methods: Theory, Optimization and Design. SIAM, 1998, Appendix A: History and Perspective, pp. 221–224. 38. J.W. Sohn, H.S. Kim and S.B. Choi, Active vibration control of smart hull structures using piezoelectric actuators, Journal of Mechanical Engineering Science, vol. 220, (2006), pp. 1329-1337. 39. C.K. Song, J.K. Hwang, J.M. Lee and J.K. Hedrick, Active vibration control for structuralacoustic coupling system of a 3-D vehicle cabin model, Journal of Sound and Vibration, vol. 267, (2003), pp. 851-865.

References

19

40. G. Cai, and J. Huang, Optimal control method with time delay in control, Journal of Sound and Vibration, vol. 251, no. 3, (2002), pp. 383–394. 41. K. Seto and I.N. Kar, A comparative study on H-infinity based vibration controller of a flexible structure system, Proceedings of the American Control Conference, Chicago, Illinois, June, 2000. 42. L. Huo, G. Song, H. Li and K. Grigoriadis, robust control design of active structural vibration suppression using an active mass damper, Smart Material and Structures, 17, (2008). 43. X. Zhang, C. Shao, S. Li, D. Xu and A.G. Erdman, Robust H∞ vibration control for flexible linkage mechanical systems with piezoelectric sensors and actuators, Journal of Sound and Vibration, vol. 243, (2001), pp. 145-155. 44. I.N. Kar, K. Seto and F. Doi, Multimode vibration control of a flexible structure using H∞ based robust control, IEEE/ASME Transactions on Mechatronics, vol. 5, (2000), pp. 23-31. 45. S. Wang, H.Y. Yeh and P.N. Roschke, Robust control of structural system with parametric and unstructured uncertainties, Journal of Vibration and Control, vol. 7, (2001), pp. 753-772. 46. J.W. Helton and M.R. James, Extending Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives, SIAM: Advances in Design and Control, 1999. 47. P.A. Hansson, Gain scheduling based controller for active cab suspension on agricultural tractors, Acta Hort. (ISHS) 406, (1996), pp. 249-256. 48. T. Saito, K. Shiba and K. Tamura, Vibration control characteristic of a hybrid mass damper system installed in tall buildings, Earthquake Engineering and Structural Dynamics, 30, (2001), pp. 1677-1696. 49. A. Forrai and K. Kamiyama, Robust gain-scheduled control for vibration suppression. Electrical Engineering, 87 (2005), pp. 151-162. 50. J. Hatonen, Adaptive and Self-Tuning Control: Part I. Lecture Notes, The University of Sheffield, 2005. 51. K.J. Åström and B. Wittenmark, Adaptive Control (2nd edition). Addison-Wesley, 1995. 52. J. Shaw, Active vibration isolation by adaptive control. Journal of Vibration and Control, 7 (2001), pp. 19-31. 53. W. Johnson, Self-Tuning Regulators for Multicyclic Control of Helicopter Vibration, NASA Technical Paper 1996, March, 1982. 54. R. Kumar, S.P. Singh and H.N. Chandrawat, Adaptive vibration control of smart structures: a comparative study, Smart Material and Structures, 15 (2006), pp. 1358-1369. 55. W.T. Baumann, An adaptive feedback approach to structural vibration suppression, Journal of Sound and Vibration, 205, 1 (1997), pp. 121-133. 56. B. Widrow and S.D. Stearns, Adaptive Signal Processing, Prentice-Hall, 1985. 57. S.J. Elliott, I.M. Stothers and P.A. Nelson, A multiple error LMS algorithm and its application to the active control of sound and vibration, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 35, (1987), pp. 1423-1434. 58. S. Shankar, Nonlinear Systems: Analysis, Stability and Control, Springer Verlag, New York, 1999. 59. M. Vidyasagar, Nonlinear Systems Analysis, 2nd Edition, Prentice Hall, Englewood Cliffs, New Jersey, 1993. 60. J.J.E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, New Jersey, 1991. 61. M.O. Tokhi, Adaptive, Self-Tuning Control, Lecture Notes, University of Sheffield, 2004. 62. M. Sunwoo, Ka. C. Cheok and N.J. Huang, Model reference adaptive control for vehicle active suspension systems, IEEE Transactions on Industrial Electronics, vol. 38, no. 3, (1991), pp. 217–222. 63. A. Alleyne and J.K. Hedrick, Nonlinear adaptive control of active suspensions, IEEE Transactions on Control Systems Technology, vol.3, 1995. 64. A. Kirecci, M. Topalbekiroglu and I. Eker, Experimental evaluation of a model reference adaptive control for a hydraulic robot: a case study, Robotica, vol. 21, (2003), pp. 71-78. 65. H.M. Schwartz, Model reference adaptive control for robotic manipulators without velocity measurements, International Journal of Adaptive Control and Signal Processing, vol. 8, (2007), pp. 279-285.

20

1 Introduction to Vibration and Noise Control Problems

66. S. Tzafestas, G. Stavrakakis and A. Zagorianos, Robot model reference adaptive control through lower/upper part dynamic decoupling, Journal of Intelligent and Robotic Systems, 1 (1988), pp. 163-184. 67. R. Isermann, K.H. Lachmann and D. Matko, Adaptive Control Systems, Prentice Hall, 1992. 68. N.M. Filatov and H. Unbehauen, Adaptive Dual Control: Theory and Applications, SpringerVerlag, Berlin Heidelberg, 2004. 69. E.F. Camacho and C. Bordons, Model Predictive Control, 2nd Edition, Springer, 2003. 70. J.A. Rossiter, Model Based Predictive Control: a Practical Approach, CRC Press, 2003. 71. J.M. Maciejowski, Predictive Control: With Constraints, Prentice Hall, 2002. 72. S.J. Qin and T.A. Badgwell, A survey of industrial model predictive control technology, IFAC Control Engineering Practice, 11 (2003), pp. 733-764. 73. D.H. Owens and J. Hätönen, Iterative learning control-an optimization paradigm, Annual Reviews in Control, 29 (2005), pp. 57-70. 74. H.S. Ahn, K.L. Moore and Y.Q. Chen, Iterative Learning Control: Robustness and Monotonic Convergence for Interval Systems, Springer, 2007. 75. R.W. Longman, Iterative learning control and repetitive control for engineering practice, International Journal of Control, vol. 73, 10 (2000), pp. 930-954. 76. S. Daley, J. Hätönen and D.H. Owens, Active vibration isolation in a “smart spring” mount using a repetitive control approach, IFAC Control Engineering Practice, vol. 14, 9 (2006), pp. 991-997. 77. M.B.G. Wassink, N.J.M. Bosch, O.H. Bosgra and S. Koekebakker, Enabling higher jet frequencies for an inkjet printhead using iterative learning control, Proceedings of the 2005 IEEE Conference on Control Applications, Toronto, Canada, 2005. 78. K.M. Hangos, R. Lakner and M. Gerzson, Intelligent Control Systems: an Introduction with Examples, Springer, 2001. 79. Z.X. Cai, Intelligent Control: Principles, Techniques and Applications, World Scientific, 1997. 80. P.J. Antsaklis, Defining intelligent control, IEEE Control Systems Magazine, 14, 3 (1994), pp. 4-66. 81. L.C. Westphal, Handbook of Control Systems Engineering, Springer, 2001. 82. K.M. Passino, Intelligent Control: an Overview of Techniques, Chapter in: T. Samad, ED., Perspectives in Control: New Concepts and Applications, IEEE Press, NJ, 2001. 83. A. Madkour, M.A. Hossain, K.P. Dahal and H. Yu, Intelligent learning algorithms for active vibration control, IEEE Transactions on Systems, Man, and Cybernetics-Part C: Applications and Review, vol. 37, 5 (2007). 84. R. Kumar, S.P. Singh and H.N. Chandrawat, MIMO adaptive vibration control of smart structures with quickly varying parameters: neural networks vs classical control approach, Journal of Sound and Vibration, vol. 307, (2007), pp. 639-661. 85. C.C. Ho and C.K. Ma, Active vibration control of structural systems by a combination of the linear quadratic Gaussian and input estimation approaches, Journal of Sound and Vibration, vol. 301, (2007), pp. 429-449. 86. H. Si and D. Li, Active control of vibration using a fuzzy control method based on scaling universes of discourse, Smart Materials and Structures, 16 (2007), pp. 555-560. 87. X. Dong, M. Yu, C. Liao, W. Chen, H. Zhang and S. Huang, Adaptive fuzzy neural network control for transient dynamics of magneto-rheological suspension with time delay, in Advances in Neural Networks, pp. 1048–1051, Springer, 2006. 88. D.J. Leith and W.E. Leithead, Survey of gain-scheduling analysis and design, International Journal of Control, 73, 11 (2000), pp. 1001-1025. 89. M. Battaini, F. Casciati and L. Faravelli, Fuzzy control of structural vibration: an active mass system driven by a fuzzy controller, Earthquake Engineering and Structural Dynamics, 27, (1998), pp. 1267-1276. 90. M.A. Henson and D.E. Seborg (editors), Nonlinear Process Control, Prentice Hall PTR, 1997. 91. M.J. Jang, C.L. Chen and Y.M. Tsao, Sliding mode control for active magnetic bearing system with flexible rotor, Journal of the Franklin Institute, vol. 342, 4 (2005), pp. 401-419.

References

21

92. J.D. Lindlau and C.R. Knospe, Feedback linearization of an active magnetic bearing with voltage control, IEEE Transactions on Control System Technology, vol. 10, 1 (2002), pp. 21-31. 93. C.R. Fuller and A.H. von Flotow, Active control of sound and vibration, IEEE Transactions on Control System Technology, December, 1995. 94. A. Palazzolo, Vibration Theory & Applications with Finite Elements & Active Vibration Control, John Wiley & Sons, 2016. ˇ 95. N. Alujevi´c, D.Cakmak, H. Wolf, M.Joki´c, Passive and active vibration isolation systems using inerter, Journal of Sound and Vibration 2018, 418: 163-183. 96. C. Liu, X. Jing, S. Daley, F. Li, Recent advances in micro-vibration isolation, Mechanical Systems and Signal Processing 2015, 56–57: 55-80. 97. C. Du, L. Xie, Modeling and Control of Vibration in Mechanical Systems, Taylor & Francis Group, LLC, 2010. 98. B. Erol, A. Deliba¸si, Proportional-integral-derivative type H∞ controller for quarter car active suspension system, Journal of Vibration & Control 2018, 24(10): 1951-1966. 99. N. Alujevi´c, G. Zhao, B. Depraetere, P. Sas, B. Pluymers, W. Desmet, H2 optimal vibration control using inertial actuators and a comparison with tuned mass dampers, Journal of Sound & Vibration 2014, 333: 4073-4083. 100. C. Hoffmann, H. Werner, A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations, IEEE Transactions on Control Systems Technology 2015, 23(2): 416-433. 101. D. Yang, J. Shin, H. Lee, S. Kim, M. Kwak, Active vibration control of structure by active mass damper and multi-modal negative acceleration feedback control algorithm, Journal of Sound & Vibration 2017, 392: 18-30. 102. A. Kazemy, J. Lam, X. Li, Finite-frequency H∞ control for offshore platforms subject to parametric model uncertainty and practical hard constraints, ISA Transactions 2018, 83: 53-65. 103. R. Canahuire, A. Serpa, Reduced order H∞ controller design for vibration control using genetic algorithms, Journal of Vibration & Control 2017, 23(10): 1693-1707. 104. W. Sun, H. Gao, O. Kaynak, Finite frequency H∞ control for vehicle active suspension systems, IEEE Transactions on Control Systems Technology 2010, 19(2): 416-422. 105. T. Iwasaki, S. Hara, Generalized KYP lemma: unified frequency domain inequalities with design applications, IEEE Transactions on Automatic Control 2005, 50(1): 41–59. 106. M. Yuan, K. Liu, A. Sadhu, Simultaneous vibration suppression and energy harvesting with a non-traditional vibration absorber, Journal of Intelligent Material Systems & Structures 2018, 29(8): 1748-1763. 107. A. Gonzalez-Buelga, L.R. Clare, A. Cammarano, S.A. Neild, S.G. Burrow, D.J. Inman, An optimized tuned mass damper/harvester device, Structural Control & Health Monitoring 2014, 21: 1154-1169. 108. X. Tang, L. Zuo, Simultaneous energy harvesting and vibration control of structures with tuned mass dampers. Journal of Intelligent Material Systems and Structures 2012, 23(18): 2117–2127. 109. N. Tran, M. Ghayesh, M. Arjomandi, Ambient vibration energy harvesters: a review on nonlinear techniques for performance enhancement, International Journal of Engineering Science 2018, 127: 162-185. 110. W. Tian, Z. Ling, W. Yu, J. Shi, A review of MEMS scale piezoelectric energy harvester, Applied Sciences 2018, 8(645): doi:https://doi.org/10.3390/app8040645 111. C. Wei, X. Jing, A comprehensive review on vibration energy harvesting: modeling and realization, Renewable & Sustainable Energy Reviews 2017, 74: 1-18. 112. S. Elias, V. Matsagar, Research developments in vibration control of structures using passive tuned mass dampers, Annual Reviews in Control 2017, 44: 129-156. 113. J. Wang, Vibration distribution in complex dynamical systems: local topology design for global performance? Mechanical Systems & Signal Processing 2021, 151: 107394. 114. J. Wang, Topology optimization for vibration suppression at multiple locations: a non-iterative approach, Applied Mathematics & Computation 2021, 398: 125990 (10pages).

22

1 Introduction to Vibration and Noise Control Problems

115. J. Wang, Optimal design for energy harvesting vibration absorbers, ASME Journal of Dynamic Systems, Measurement and Control 2021, 143: 051008 (6pages). 116. J. Wang, Active vibration distribution through restricted topology & single node control, Shock & Vibration 2020, Article ID 8823527, 12 pages. 117. J. Wang, Structural design for vibration distribution: limit of performance, International Journal of Structural Stability & Dynamics 2020, 20(5): 2050060 (19 pages). 118. J. Wang, Restricted control of complex systems with prevention of instability propagation, ISA Transactions 2020, 98: 284-291. 119. J. Wang, Active restricted control for harmonic vibration suppression, International Journal of Structural Stability & Dynamics 2019, 19(12): 1971007 (12 pages). 120. J. Wang, Simultaneous vibration suppression and energy harvesting: damping optimization for performance limit, Mechanical Systems & Signal Processing 2019, 132: 609-621. 121. J. Wang, Disturbance attenuation of complex dynamical systems through interaction topology design, Applied Mathematics & Computation 2019, 355: 576-584. 122. J. Wang, Active control for vibration distribution with performance specification and constraints, Mechanical Systems & Signal Processing 2019, 131: 112-125. 123. J. Wang, Active vibration control: a graphical approach for optimal distribution, Mechanical Systems & Signal Processing 2019, 124(1): 632-642. 124. J. Wang, H. Yue, G. Dimirovski, Disturbance attenuation in linear systems revisited, International Journal of Control, Automation & Systems 2017, 15(4): 1611-1621. 125. J. Wang, A single sensor & single actuator approach to performance tailoring over a prescribed frequency band, ISA Transactions 2016, 61: 329-336. 126. J. Wang, Active control of contact force for a pantograph-catenary system, Shock & Vibration 2016, vol. 2016, Article ID 2735297, 7 pages, 2016. 127. J. Wang, S. Daley, Broad band controller design for remote vibration using a geometric approach, Journal of Sound & Vibration, vol. 329, 19, 2010, pp. 3888-3897. 128. J. Wang, Vibration control and energy harvesting design for aeroengines, China, Application No.: 202110033749.6, Serial No.: 2021011201419210, Date: 2021.01.12. 129. J. Wang, Active noise control for multivariable systems, China, Application No.: 202011369780.9, Serial No.: 2020113002071070, Date: 2020.11.30. 130. J. Wang, A method for simultaneous vibration attenuation in both inner and outer cases of aeroengines over very low frequencies, China, Application No.: 202011345204.0, Serial No.: 2020112602021000, Date: 2020.11.26. 131. J. Wang, Optimal determination of loudspeakers for active noise control applications, China, Application No.: 202011328091.3, Serial No.: 2020112402577400, Date: 2020.11.24. 132. J. Wang, Protection design for aeroengine EEC and piping systems, China, Application No.: 202011328078.8, Serial No.: 2020112402577130, Date: 2020.11.24. 133. J. Wang, A method for optimal design of energy harvesting vibration absorbers, China, Application No.: 202011122530.5, Serial No.: 2020102001607200, Date: 2020.10.20. 134. J. Wang, A new method for vibration attenuation of Aeroengines, China, Application No.: 202010729614.9, Serial No.: 2020072701929500, Date: 2020.07.27. 135. J. Wang, A method for prevention of aeroengine high attitude/low Mach number oscillations, China, Application No.: 202010697863.4, Serial No.: 2020072002019300, Date: 2020.07.20. 136. J. Wang, Aeroengine & aircraft pylon coupled vibration suppression through stiffness feedback, China, Application No.: 202010386074.9, Serial No.: 2020050901812460, Date: 2020.05.09. 137. J. Wang, Isolation design for vehicle suspension systems, China, Application No.: 202010046032.0, Date: 2020. 01.16. 138. T. Fang, J. Wang, Z. Hu, An active vibration control device and design method, China, Application No.: 202010050597.6, Date: 2020. 01.17. 139. J. Wang, Active vibration control of rotor-blade and shaft systems, China, Application No.: 201911307938.7, Date: 2019. 12.18. 140. J. Wang, Force optimization for a pantograph-catenary structure, China, Application No.: 201910921273.2, Date: 2019.09.27.

References

23

141. J. Wang, Z. Hu, Design of dynamic vibration absorbers with zero damping, China, Application No.: 201910771096.4, Date: 2019.08.21. 142. J. Wang, Z. Hu, Optimization design for zero-stiffness vibration absorbers, China, Application No.: 201910639844.3, Date: 2019.07.16. (Licensed on 2021.02.08) 143. J. Wang, A new vibration absorber design method, China, Application No.: 201710628133.7, Date: 2017.07.28. (Licensed on 2020.08.05)

Chapter 2

Active Control for Performance Limit

Many vibration and noise problems occur at specific discrete frequencies and with a slightly abuse of concept, active control for discrete frequencies or harmonics is simply termed as active control. Performance-limit oriented active control is to be delineated in this chapter. Meanwhile, it is believed that the fundamental philosophy for control is to excavate system performance by taking advantage of every piece of information, e.g. internal structural properties, topology and interactions, external noise and disturbance power spectral distributions etc. It is reasonable to utilize a networked structural representation for system dynamics throughout the text.

2.1 Introduction and Preliminaries Active vibration and noise control has been recognized as an established field integrating vibration and noise with control theories [1–5]. It is thus no surprise that control-theoretic design methodologies have found extensive applications in recent years, e.g. from classical internal model control and Youla-Kucera parameterization methods [6, 7], to H2/H∞ designs [8–10], adaptive control [11–13], together with a variety of disturbance-observers [14] etc. Revealing examples can be found in [15, 16] where the performances of various design methodologies and adaptive schemes are applied, tested and evaluated in an international benchmark. These methods are powerful since they can guarantee not only the closed-loop stability but also prescribed performance (e.g. rising time, robustness etc.). However, it must also point out that these control-theoretic approaches still deal with active vibration control as a disturbance-attenuation or rejection problem, and they do not take full advantage of the distinctive feature that the vibration problems generally occur over a frequency band or even at a discrete frequency. This is particularly true for those H2/H∞ optimization-based methodologies. The finite-frequency-band issue does not look trivial as generally treated in the literature. Indeed, most of the control-theoretic √ ⌉ approaches treat the design problem over a full frequency band (ω ∈ 0 ∞ rad/s) © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Wang, Active Vibration & Noise Control: Design Towards Performance Limit, https://doi.org/10.1007/978-981-19-4116-0_2

25

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2 Active Control for Performance Limit

[17–19]. When applied to finite-frequency band or discrete harmonics for vibration control problems, the theory is such “tailored” through either weighting functions or generalized KYP-lemma (see [20, 21] and references therein). However, H∞ and linear matrix inequality techniques are required for both approaches leading to controllers with very high orders, which consequently necessitates even more complex optimization techniques such as generic algorithms for order reduction [22]. To summarize, the current state-of-the-art to vibration control problems has the following features: (1)

(2)

Full-frequency band: full frequency band is considered and finite-frequencyband is treated “passively” using weighting functions or small gain theorems. Consequently, the important problem whether performance can be further improved by exploiting the finite-frequency-band nature is ignored. Non-performance-limit Orientation: the state-of-art methods utilize the power of optimization techniques; yet these approaches are not performance-limit oriented in the sense that the important problem how the performance variables compromise is out of the question. Indeed, the problem whether the performance variables really compromise with each other is still not totally solved in the literature except qualitative evaluation based on Bode’s Integral Relationships, or “waterbed effects.”

The above not-fully-solved problems necessitate the consideration of moving from Control to Active Control. Treating active control as a disturbance attenuation problem for a discrete frequency or over a frequency band is not enough. The monograph aims to demonstrate that new methods are necessitated where the finitefrequency-band feature can be fully taken advantage of to excavate the potentials of the control system. The contributions can be listed as: (1)

(2) (3)

A graphical design framework is proposed where both the performance specifications and the system constraints are represented using geometric expressions; Important problems of performance limit and compromise are addressed based on visual inspections; A performance-limit-oriented approach is thus established providing a new design methodology for active vibration and noise control.

Indeed, it can be shown that conventional methods are conservative in the sense that the system performance can be further (even significantly) improved towards performance limit. As a consequence, a performance-limit oriented methodology is developed specially targeting the active control problems for a discrete frequency. Such an approach is proposed in this monograph. The feature of the proposed method is that it does not require any iterative solution for resolution to the above questions. Even more, clear-cut, and easy-to-verify conditions can be obtained for the questions in consideration. To put the discussions in the context, consider the following networked dynamical system represented by:

2.1 Introduction and Preliminaries

27

x˙i = f i (xi ) + Si (x1 , . . . , xi−1 , xi+1 , . . . , x N , t)Hi (x1 , . . . , xi−1 , xi+1 , . . . , x N , t), (2.1) where: i ∈ {1, . . . , N } with x1 , . . . , x N being state variables for agent 1, agent 2, …, agent N, respectively; the internal dynamics is captured by f i (xi ); Si (x1 , . . . , xi−1 , xi+1 , . . . , x N , t) denotes the topology and interaction strength, and Hi (x1 , . . . , xi−1 , xi+1 , . . . , x N , t) is the inner coupling over the network. For cases where Si (x1 , . . . , xi−1 , xi+1 , . . . , x N , t) is constant or slowly time-varying, it can be decomposed into Si ≡ λi ⋀i , or: x˙i = f i (xi ) + λi ⋀i Hi (x1 , . . . , xi−1 , xi+1 , . . . , x N , t),

(2.2)

where: λi is a vector with (possibly) uncertain elements, and it can be recognized as interaction strength while ⋀i is a diagonal matrix with zeros or unities identified as the topology of the complex networks. Equation (2.2) can be further expressed in elements: x˙i = f i (xi ) +

N ∑

si j (x1 , . . . , xi−1 , xi+1 , . . . , x N , t)h i j (x1 , . . . , xi−1 , xi+1 , . . . , x N , t),

j=1 j/=i

(2.3) Or dropping the variable dependent terms off for ease of representation, it becomes: x˙i = f i (xi ) +

N ∑

si j h i j

(2.4)

j=1 j/=i

Should the inner coupling h i j be set to identity, the above expression reads: x˙i = f i (xi ) +

N ∑

si j x j ,

(2.5)

j=1 j/=i

Now consider the situation where the networked system has non-identical yet linear nodes dynamics, and it is subject to exogenous vibration or noise: x˙i = ai xi + λi ⋀i [x1 , . . . , xi−1 , xi+1 , . . . , x N ]T + di (t),

(2.6)

Transform (2.6) into complex domain by taking Laplace transform on both sides, while assuming zero initial conditions:

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2 Active Control for Performance Limit

(s − a1 )X 1 = λ1 ⋀1 [X 2 , X 3 , . . . , X N ]T + D1 (s) (s − a2 )X 2 = λ2 ⋀2 [X 1 , X 3 , . . . , X N ]T + D2 (s) .. . , (s − ai )X i = λi ⋀i [X 1 , . . . , X i−1 , X i+1 , . . . , X N ]T + Di (s) .. . ,

(2.7)

(s − a N )X N = λ N ⋀ N [X 1 , X 2 , · · · , X N −1 ]T + D N (s) where: s is a complex number frequency parameter. For ease of presentation, the dependence of X i (s) on s has been dropped off. Now at a frequency ω0 , the frequency response of the vibration or noise can be expressed as Di ( jω0 ) = Ci D( jω0 ) and C i is a complex number (e.g. Ci = r < ϕ) representing the gain (r) and phase shift (ϕ) with respect to the exogenous frequency response signal D( jω0 ). Consequently at the frequencyω0 , Eq. (2.7) expresses an N-simultaneous algebraic equations and henceforth if the corresponding determinant is non-singular, X 1 ( jω0 ), X 2 ( jω0 ), …, X N ( jω0 ) can be solved for D( jω0 ) as follows: X 1 ( jω0 ) = G 1 ( jω0 , λ1 , ∆1 )C1 D( jω0 ) X 2 ( jω0 ) = G 2 ( jω0 , λ2 , ∆2 )C2 D( jω0 ) .. . , X N −1 ( jω0 ) = G N −1 ( jω0 , λ N −1 , ∆ N −1 )C N −1 D( jω0 ) X N ( jω0 ) = G N ( jω0 , λ N , ∆ N )C N D( jω0 )

(2.8)

where: G 1 ( jω0 , λ1 , ∆1 ), G 2 ( jω0 , λ2 , ∆2 ), …, G N ( jω0 , λ N , ∆ N ) denote the corresponding frequency response functions at ω0 with parameters in a1 , a2 , …, a N ; λ1 , λ2 , …, λ N ; and ⋀1 , ⋀2 , …, ⋀ N . ∆i s represent the additive increment to the corresponding topologies or dynamics to be designed, henceforth indicating the system parameters ready for either active control or passive design—the problem of design and feedback is henceforth integrated in one par. Equations (2.8) thus represents the vibration/noise response property in X i ( jω0 ) with respect to exogenous vibration/noise source Di ( jω0 ). Thus, should the tobe-designed topology or dynamics be able to provide harmonic vibration or noise attenuation for all agents (global control), the necessary and sufficient condition is: |G i ( jω0 , λi , ∆i )| ≤ 1 ∀i ∈ { 1, . . . , N } ,

(2.9)

This important result is summarized as: Proposition 2.1 For a networked system (2.6) with all agents of non-identical nodes dynamics, vibration/noise will be attenuated at a harmonic frequency ω0 for all agents if and only if the to-be-designed interaction topology or dynamics satisfies

2.1 Introduction and Preliminaries

29

condition (2.9); if condition (2.9) holds only for a fraction of the agents, then only this part of the agents can have attenuated vibration/noise levels with the resulting interaction topology or dynamics, while the remaining ones will lead to vibration/noise amplification. Remark 2.1 However, it must be noticed that the corresponding topological or dynamical parameters must also be real numbers, in contrast to transfer functions with complex number variables. The results can then be ready to generate into a frequency band [ω1 , ωN ] as below. Proposition 2.2 For a networked system (2.6) with all agents of non-identical node dynamics, the vibration/noise will be attenuated across a frequency band [ω1 , ωN ] for all agents if the to-be-designed interaction topology satisfies the following condition: |G i ( jω0 , λi , ∆i )| ≤ 1 ∀i ∈ { 1, . . . , N } over [ω1 , ωN ],

(2.10)

If condition (10) holds only for a fraction of the agents, then only this part of the agents can have attenuated disturbance with the resulting interaction topology over [ω1 , ωN ]. Remark 2.2 The loss of necessity in condition (10) is due to the constraint put forward by the requirement of transfer function realization. Thus an important issue is concerned with whether an interaction topology can be found such as G i ( jω0 , λi , ∆i ), ∀i ∈ { 1, . . . , N } can be interpolated with frequency response values over [ω1 , ωN ]. Remark 2.3 The above problem is essentially a real rational approximation problem where the transfer function G i (s) for i ∈ {1, . . . , N } is required to possess frequency response with designated G i ( jω0 , λi , ∆i ) values, subject to the realness of parameters in λi s and ∆i s. This problem can be addressed through a Neverlinna-Pick interpolation technique. Classically, the Neverlinna-Pick problem is stated as finding a transfer function H (s) ∈ H∞ such that H (si ) = Hi (∀i ∈ { 1, . . . ,N } ), with given N points (si )i∈{1,...,N } with ℜ(si ) > 0 and another series of N complex points (Hi ) satisfying sup(|Hi |) ≤ 1(∀i ∈ { 1, . . . , N } ). The solvability of the Neverlinna-Pick problem is given by the well-known Pick condition. Indeed, this can be further developed into a generalized Pick condition through linear matrix inequality optimization techniques, where suboptimal solutions can be found in case of non-solvability of original Pick condition. This will be delineated in the next chapter. Remark 2.4 Due to waterbed effect, it is known that condition (10) cannot be satisfied over arbitrarily large frequency band, indicating that the problem will eventually become non-solvable with the demanding requirement of vibration/noise suppression over a large frequency band. Thus a compromise between the level of vibration/noise suppression and the desirable frequency band must be made. This will also be discussed in detail in the next chapter.

30

2 Active Control for Performance Limit

2.2 Difficulties with Direct Method Now refer back to the formulation in the above section, it is seen that the networked structural representation relates perfectly well with the vibration and noise control problem, e.g. identifying each node as a variable or location, thus treating X i ( jω0 ) as a performance variable at a location. Therefore formulated as a multivariable or multilocation control problem, as is generally the case for active control problems, Proposition 2.1 can be re-stated as: Proposition 2.3 For a networked system (2.6) with all agents of non-identical nodes dynamics, multivariable/multi-location vibration/noise attenuation a harmonic frequency ω0 can be controlled for all locations if and only if the to-be-designed interaction topology or dynamics satisfies condition (2.9); if condition (2.9) holds only for a fraction of the locations, then only this part of the locations can have attenuated vibration/noise levels with the resulting interaction topology or dynamics, while the remaining ones will lead to vibration/noise amplification. This “seems” to have solved the active control problem for multiple variables or multiple locations. Yet before devoting to our proposed solution, it is necessary to see the difficulties associated with this conventional or direct method. This is illustrated in a numerical example below. Consider an interacting system with 3 agents: x˙i = ai xi +

3 ∑

si j x j , i ∈ {1, 2, 3}

(2.11)

j=1 j/=i

⎡

⎤ a1 s12 s13 with a graph G = ⎣ s21 a2 s23 ⎦. It is noted that ai s are the nodes dynamics while si j s31 s32 a3 captures both interaction strength and topological interaction. Now to see clearly the difficulties, we only consider the case for active control with topological parameters design of si j s. Now it is assumed that a harmonic vibration or noise signal propagates into the network Di ( jω0 ) = Ci D( jω0 ) (i ∈ {1, 2, 3}) and C i s represent gain and phase shift with respect to the signal sin ω0 . For simplicity, it is further assumed that disturbance enters into agent 1 while transmitting detrimental effects to other agents. The objective is thus to investigate the simultaneous disturbance attenuation properties/capabilities through topological design for all the three agents. To proceed, the disturbance attenuation transfer functions are calculated first:

2.2 Difficulties with Direct Method

31

⌉ √ (s − a2 ) (s − a2 )(s − a3 ) − s32 s23 ⌉√ ⌉ √ ⌉√ ⌉D X1 = √ (s − a1 )(s − a2 ) − s12 s21 (s − a2 )(s − a3 ) − s32 s23 − s13 (s − a2 ) + s12 s23 s31 (s − a2 ) + s32 s21 √ ⌉ √ ⌉ s21 (s − a2 )(s − a3 ) − s32 s23 + s23 s31 (s − a2 ) + s32 s21 ⌉√ ⌉ √ ⌉√ ⌉D X2 = √ (s − a1 )(s − a2 )−s12 s21 (s − a2 )(s − a3 ) − s32 s23 − s13 (s − a2 ) + s12 s23 s31 (s − a2 ) + s32 s21 ⌉ √ (s − a2 ) s31 (s − a2 ) + s32 s21 ⌉√ ⌉ √ ⌉√ ⌉D X3 = √ (s − a1 )(s − a2 ) − s12 s21 (s − a2 )(s − a3 ) − s32 s23 − s13 (s − a2 ) + s12 s23 s31 (s − a2 ) + s32 s21

(2.12) where: X 1 , X 2 , X 3 , and D are the Laplace transforms of x1 (t), x2 (t), x3 (t), and d(t), respectively, with dependence on complex variable s dropped off for convenience of expression. It is seen clearly that X i s are represented as functions of D with respect to the to-be-designed topology parameters. Now it is ready to calculate the topological conditions to provide global vibration or noise attenuation at all locatioins: at a frequency ω0 , for dynamical agents x˙1 = −x1 , x˙2 = −3x2 , x˙3 = −2x3 , the corresponding topological parameters must satisfy: G i ( jω0 , si j ) ≤ 1 ∀i ∈ {1, 2, 3},

(2.13)

That is: ⌉ √ (s − a2 ) (s − a2 )(s − a3 ) − s32 s23 ⌉√ ⌉ √ ⌉√ ⌉ ≤1 √ (s − a1 )(s − a2 ) − s12 s21 (s − a2 )(s − a3 ) − s32 s23 − s13 (s − a2 ) + s12 s23 s31 (s − a2 ) + s32 s21 s= jω0 √ ⌉ √ ⌉ s21 (s − a2 )(s − a3 ) − s32 s23 + s23 s31 (s − a2 ) + s32 s21 ⌉√ ⌉ √ ⌉√ ⌉ ≤1 √ (s − a1 )(s − a2 ) − s12 s21 (s − a2 )(s − a3 ) − s32 s23 − s13 (s − a2 ) + s12 s23 s31 (s − a2 ) + s32 s21 s= jω0 √ ⌉ (s − a2 ) s31 (s − a2 ) + s32 s21 ⌉√ ⌉ √ ⌉√ ⌉ ≤1 √ (s − a1 )(s − a2 ) − s12 s21 (s − a2 )(s − a3 ) − s32 s23 − s13 (s − a2 ) + s12 s23 s31 (s − a2 ) + s32 s21 s= jω0

(2.14)

Now evaluation of the above expressions at a harmonic frequency ω0 = 0.6526 rad/s (the reason is explained below) leads to three inequalities with 6 topological parameters. Considering the symmetry of graph G, there will be 3 indeterminate variables for 3 inequalities. After a very lengthy but straightforward calculation, condition (2.14) leads to:

 4 2 P(s12 , s13 , s23 ) − 9.4s23 − 105s23 + 393 ≥ 0, 

2 2 2 P(s12 , s13 , s23 ) − 42s12 + 38s12 s13 s23 + 9.4s13 s23 ≥ 0, 

2 2 2 2 4 P(s12 , s13 , s23 ) − 9s12 ≥ 0, s23 + 51.4s12 s13 s23 + 89s13 + 5.16s13 s23 + 0.43s23 (2.15) where: P(s12 , s13 , s23 ) is defined to be:

32

2 Active Control for Performance Limit

2 2 2 2 P(s12 , s13 , s23 ) ≡ 5.6s12 + 8.6s13 + 2.6s23 + 6s12 s13 s23 + 2

2 2 2 3.3s12 + 3.9s13 + 2.6s23 + 1.3s12 s13 s23 − 23 Solution to the above inequalities will provide conditions for the topological parameters for vibration or noise attenuation in all three agents. ⎡ ⎤ −1 1 0 Now a specific graph can be designated G = ⎣ 1 −3 1 ⎦, then G 1 (s, si j ) can 0 1 −2 be calculated to be: G 1 (s, si j ) =

(s + 3)(s2 + 5s + 5) , s4 + 9s3 + 27s2 + 30s + 9

(2.16)

From its Bode can be seen that the crossing frequency is ω0 = diagram, it 00.6526rad/s, or G 1 ( jω, si j ) ≤ 1 ∀ω ≥ ω0 . Henceforth the problem of disturbance attenuation transforms into the following conditions for ω0 = 0.6526rad/s: G 2 ( jω0 , si j ) ≤ 1 , G 3 ( jω0 , si j ) ≤ 1

(2.17)

This is equivalent to the last two equations in (2.15), and a straightforward evaluation with graph G shows that both inequalities are satisfied. It is thus concluded that disturbance attenuation at ω0 with given graph G at the three locations is achievable. A real time simulation is carried out as shown in Fig. 2.1, and it is seen clearly that the disturbance in both node 2 and node 3 (the two locations) are attenuated as predicted. Remark 2.5 Indeed, the level of attenuation can be calculated to be 8.6 dB and 15 dB, respectively. This can be used to verify the feasibility of specification with respect to a given graph, e.g. should the disturbance attenuation in node 2 be specified to be greater than 9 dB, the graph has to be re-designed. In many situations, simplistic topology is desirable either for easy implementation or cost-effectiveness. Now three scenarios can be considered with: Case I: Design s12 with s13 = s23 = 0; Case II: Design s13 with s12 = s23 = 0; Case III: Design s23 with s12 = s13 = 0; Three scenarios can be evaluated with substituting corresponding values into inequalities⎧in (2.15): 4 2 ⎪ ⎨ 42.25s12 − 217s12 + 170 ≥ 0 4 2 42.25s12 − 319s12 + 563 ≥ 0, ⎪ ⎩ 4 2 42.25s12 − 217s12 + 563 ≥ 0 ⎧ 4 2 ⎪ ⎨ 89s13 − 280s13 + 170 ≥ 0 4 2 Case II: 89s13 − 280s13 + 563 ≥ 0 , ⎪ ⎩ 4 2 89s13 − 369s13 + 563 ≥ 0

Case I:

2.2 Difficulties with Direct Method

33

Fig. 2.1 Vibration/noise attenuation for a graph at all locations

⎧ 4 2 ⎪ ⎨ 4.1s23 − 45s23 + 170 ≥ 0 4 2 Case III: 13.5s23 − 150s23 + 563 ≥ 0. ⎪ ⎩ 4 2 13.07s23 − 150s23 + 563 ≥ 0 A calculation to the above simultaneous inequalities leads to: Case I: |s12 | ≥ 2.18 or |s12 | ≤ 1.04, Case II: |s13 | ≥ 1.53 or |s13 | ≤ 0.9, Case III: Simultaneous inequalities are satisfied for any choice of s23 . It is seen clearly that a clear-cut topology design can be obtained for active control for all locations for various simplified yet useful situations. Take case I as an example, the original graph G indicates that node 2 is topologically connected with both node 1 and node 3; the above result shows that a simplified topology s13 = s23 = 0 can be deployed as long as s12 satisfies |s12 | ≥ 2.18 or |s12 | ≤ 1.04. This essentially means that node 2 and node 3 can be topologically disconnected while fulfilling vibration/noise attenuations. As now node 3 is isolated, it must check with node 1 and node 2. Areal time simulation with s12 = 0.5 is shown in Fig. 2.2, where it is seen clearly that now the vibration or noise level in both node 1 and node 2 (at the two locations) are attenuated confirming the claim made above. From the above development, it is seen that direct method can be utilized for a solution to the active control problem, even to the results for multiple variables or

34

2 Active Control for Performance Limit

Fig. 2.2 Topology design for global vibration/noise attenuation

multiple locations; essentially, global control can be achieved! Yet, the procedures are a bit lengthy although only with straightforward and not complicated calculations. Meanwhile, the important issue of performance limit is not revealed clearly. Indeed, even bringing the conventional H2 /H∞ solutions into considerations, the following issues still await for a solution: (1)

(2)

(3)

Performance Tradeoffs: multivariable or multilocation involves multiple variables or multiple locations, and how the performance variables compromise is still unsolved. Indeed, even the issue of whether a compromise always exists is not answered—“common sense” tells us that tradeoff always exists among the performance variables. Yet, it is proved that this is not always the case! Full-frequency band: In conventional H2 /H∞ design, full frequency band is considered and finite-frequency-band is treated “passively” using weighting functions or small gain theorems. Consequently, the important problem whether performance can be further improved by exploiting the finitefrequency-band nature is ignored. Non-performance-limit Orientation: the state-of-art methods utilize the power of optimization techniques; yet these approaches are not performance-limit oriented in the sense that the best achievable performance is not naturally tackled within the current framework.

2.3 Active Control for Performance Limit: A Geometric Method

35

This motivates the development of new approach to performance-limit oriented active control for multi-variable or multi-location. This is turned to in the next section.

2.3 Active Control for Performance Limit: A Geometric Method Following the networked formulation, it is assumed that vibration or noise propagates through the dynamical system, together with active control as the following representation: x˙i (t) = ai xi (t) +

N ∑

si j x j (t) − si0 di (t) + u i (t)

j=1 j/=i

, i ∈ [1, N ] u i (t) =ki xi (t)

(2.18)

where: xi (t) represents the state variable and u i (t) the control action with ki being the control gain which is to be designed; ai and si j are the node dynamics and the topological interaction among the nodes, respectively; si0 denotes the interaction strength with the ambient vibration di (t). The system dynamics would therefore include both the lumped and the distributed structural systems [23–25]. The closed-loop system (2.18) can be written as: x˙i (t) = (ai + ki )xi (t) +

N ∑

si j x j (t)−si0 di (t), i ∈ [ 1, N ],

(2.19)

j=1 j/=i

For a frequency ω, Eq. (2.2–2.19) can be written into its frequency response representation: ⎡

⎤ ⎤−1 ⎡ (a1 + k1 − jω) s10 D1 s12 ··· s1N ⎢ ⎥ ⎢ ⎥ ⎢ s20 D2 ⎥ s21 (a2 + k2 − jω) · · · s2N ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥=⎢ ⎥ ⎢ . ⎥ .. .. . . . . . ⎣ ⎦ ⎣ ⎣ ⎦ . . . . ⎦ . sN 2 · · · (a N + k N − jω) XN sN 1 sN 0 DN (2.20) X1 X2 .. .

⎤

⎡

where: X i ( jω)s and Di ( jω)s are the frequency responses of xi (t)s and di (t)s, respectively, and the dependence on frequency is omitted for ease of reference. It

36

2 Active Control for Performance Limit

is also noted that the invertibility of the transfer matrix is implicitly assumed by a controllability and observability enforcement. For active control, the objective is to find ki such that the following condition is satisfied: √ ⌉ |X i ( jω)|C L ≤ 1 , i ∈ 1, N |X i ( jω)| O P

(2.21)

where: |X i ( jω)|C L and |X i ( jω)| O P denote the closed-loop and open-loop performances, respectively. Or, more generically, √ ⌉ |X i ( jω)|C L ≤ δi , i ∈ 1, N |X i ( jω)| O P

(2.22)

where: δi ≤ 1 is the performance index specified for each performance variable. To synthesize the control gains, certain forms of optimization methods have to be utilized, yet as commented in Introduction, the issues of performance limit and performance compromisability are difficult to resolve. To find an alternative approach to tackling this problem, now identify: ⎡

(a1 − jω) s12 ⎢ s (a 21 2 − jω) ⎢ ⎢ .. .. ⎣ . . sN 1

sN 2

··· ··· .. .

⎤−1

s1N s2N .. .

⎥ ⎥ ⎥ ⎦

· · · (a N − jω)

⎡

g11 g12 ⎢ g21 g22 ⎢ =⎢ . .. ⎣ .. . gN 1 gN 2

⎤ · · · g1N · · · g2N ⎥ ⎥ . . .. ⎥ . . ⎦ · · · gN N

(2.23)

Henceforth the system dynamics can now be expressed in frequency domain: X i ( jω) = Ui ( jω) +

N ∑

gi j s j0 D( jω) , i ∈ [ 1, N ]

(2.24)

j=1

To simplify the development, it is assumed that control is applied at the first performance variable yet with a generic form: U ( jω) = K ( jω)X 1 ( jω)

(2.25)

It is realized that control action (2.25) is more general than the purely gain feedback in (2–18), which can only represent a displacement feedback or proportional control. That is, the closed-loop system is: X i ( jω) = gi1 U ( jω) +

N ∑ j=2

gi j s j0 D( jω) , i ∈ [ 1, N ]

2.3 Active Control for Performance Limit: A Geometric Method

U ( jω) = K ( jω)X 1 ( jω)

37

(2.26)

Thus the problem of active control can be formulated as follows: Active Control Problem: for a frequency ω, finding a control K ( jω) such that the specification (2.22) is satisfied for all performance variables X i ( jω) , i ∈ [ 1, N ] ; either the specification is feasible or not, the compromise among the performance variables must be analyzed while further obtaining the performance limit. Remark 2.6 Obviously, the problem consists of the following “subproblems” such as (1) whether all performance variables X i ( jω) ∀i ∈ [ 1, N ] can be attenuated? (2) what is the performance limit for each performance variable? (3) it is imaginable that the performance variables will compromise with each other, but a pivotal question is: how do they compromise? Remark 2.7 In fact, the answer to the above questions, particularly question (1), must be negative, since not all performance variables can be attenuated across any frequencies, in terms of the Bode-integral theorems. But this is only a “negation theorem” and the more important question is, what is/are specific frequency/frequencies where all performance variables can be attenuated at the same time? A clear-cut answer must be provided with an easy-to-verify condition to tell the frequencies where simultaneous vibration attenuation for ALL performance variables is achieved. Remark 2.8 Obviously, the above questions cannot be answered based on the conventional H2/Hinf design methods, where iterative routines are required to tell the detailed compromise scenario among the performance variables even for a discrete frequency ω. Remark 2.9 Thus it is challenging to use conventional methods to tell the performance limit in δi s, or figure out how the δi s compromise with each other, if this is not possible. These issues will be handled in the following sections, where detailed information can be “extracted” for δi s with a clean number while illustrating the compromise among them.

2.3.1 Performance-Limit-Oriented Solution to Optimal Control To proceed, write (2.26) into: N √ ⌉−1 ∑ |X 1 ( jω)|C L = 1 − g11 K ( jω) g1k sk0 D( jω) k=2 N N ∑ ∑ √ ⌉ X i ( jω) gi1 K ( jω) 1 − g11 K ( jω) −1 , i ∈ [ 2, N ] = g s D( jω) + g s D( jω) 1k k0 i j j0 CL k=2 j=2

(2.27)

38

2 Active Control for Performance Limit

Thus one has: |X 1 ( jω)|C L = [1 − g11 K ( jω)]−1 |X 1 ( jω)| O P g K ( jω)[1 − g K ( jω)]−1  N g s |X i ( jω)|C L 11 1k k0 i1 k=2 = + 1 , i ∈ [ 2, N ] N |X i ( jω)| O P j=2 gi j s j0 (2.28) Consequently, the satisfaction of performance index (5) amounts to require: [1 − g11 K ( jω)]−1 ≤ δ1 g K ( jω)[1 − g K ( jω)]−1  N g s 11 i1 k=2 1k k0 + 1 ≤ δi , i ∈ [ 2, N ] N g s j=2 i j j0

(2.29)

That is, active vibration control boils down to the problem of finding K ( jω) such that the specification (2.29) is satisfied simultaneously; if the specification is not feasible, explanation must be provided for the failure; either the specification is feasible or not, compromisability and performance limit must be analyzed. To further develop the solution, define: α( jω) = [1 − g11 K ( jω)]−1 − 1 α( jω) G i ( jω)  g11 Nj=2 gi j s j0 G i ( jω) = N gi1 k=2 g1k sk0 βi ( jω) =

(2.30) (2.31)

(2.32)

Then the specification (2.29) can be manipulated into the following inequalities: |α( jω) + 1| ≤ δ1 |βi ( jω) + 1| ≤ δi , i ∈ [ 2, N ]

(2.33)

Henceforth, the active vibration control problem has been transformed into finding α( jω) such that the specification (2.29) is satisfied simultaneously. To develop the solution, optimization routines can be utilized yet it is relatively difficult to obtain the performance limits, if possible. Here a geometric method is proposed where the key step is to treat (2.33) as a Möbius transformation mapping the circle (and its interior) |βi ( jω) + 1| ≤ δi onto the complex plane expressed by |α( jω) + 1| ≤ δ1 . The situation becomes clear with the relationship between |α( jω) + 1| = 1 and |βi ( jω) + 1| = 1 where the former is a circle with unit radius while the latter is a circle with centre (−G i ( jω)) and radius |G i ( jω)|. Henceforth

2.3 Active Control for Performance Limit: A Geometric Method

39

Fig. 2.3 Geometric representation of performance, shaded area indicating the satisfaction of the performance index

the intersection of the two circles would indicate the simultaneous satisfaction of |α( jω) + 1| ≤ 1 and |βi ( jω) + 1| ≤ 1, which is illustrated in Fig. 2.3a Considering the fact that |α( jω) + 1| ≤ δ1 is a concentric-scaling to |α( jω) + 1| = 1 while |βi ( jω) + 1| ≤ δi is a concentric-scaling to |βi ( jω) + 1| = 1, it is immediately known that the intersection of |α( jω) + 1| ≤ δ1 and |βi ( jω) + 1| ≤ δi indicates that simultaneous satisfaction of the performance index in (2.33), or no intersection implies that the performance index (2.33) is infeasible. This is a remarkable development as shown in Fig. 2.3b, where the index has been defined by |α( jω) + 1| ≤ 0.707 and |βi ( jω) + 1| ≤ 0.707, or 3 dB reduction. Following Fig. 2.3, the following results can be obtained: Proposition 2.4 (Optimal Solution to Active Vibration Control Problem) The active vibration control problem as specified by (16) is solvable if and only if the scaled α−circle has an intersection with the (N - 1) scaled βi −circles simultaneously. Performance Limit The above development sets up the stage for handling the challenging problem of design for performance limit. Refer back to Fig. 2.3, (a) denotes the situation for performance index |α( jω) + 1| ≤ 1 and |βi ( jω) + 1| ≤ 1, while (b) illustrates the situation for performance index |α( jω) + 1| ≤ 0.707 and |βi ( jω) + 1| ≤ 0.707. In Fig. 2.3a, the shaded area indicates the feasible solution for simultaneous vibration attenuation in X 1 ( jω) and X i ( jω); in Fig. 2.3b, since the two circles |α( jω) + 1| = 0.707 and |βi ( jω) + 1| = 0.707 have no intersection, this implies that simultaneous 3 dB reduction in X 1 ( jω) and X i ( jω) is NOT possible. Thus it is realized that the performance limit is actually defined by the tangency of the two circles |α( jω) + 1| ≤ δ1 and |βi ( jω) + 1| ≤ δi . This geometric solution to performance limit is worth being summarized: Proposition 2.5 (Active Control for Performance Limit) The performance limit solution to active vibration control problem is defined by the tangency of the circles as specified by (2.33).

40

2 Active Control for Performance Limit

Remark 2.10 Once the performance limit solution is obtained for a discrete frequency, the problem of performance limit over a frequency band can be dealt with on a frequency-by-frequency base. However, care must be exercised with issues such as the grid of frequency spacing, the number of discrete controllers, as well as the implementation of controllers etc. Remark 2.11 An additional issue to be considered for broad-band frequency is the stability of closed-loop system, and it turns out that the above necessary and sufficient condition for discrete frequency will have to be sacrificed leading to the sufficiency for broad-band situation. Remark 2.12 All the above issues have been solved within the geometrical framework and it can also be demonstrated that conventional methods are indeed conservative, and the system performance can be further (even significantly) improved towards performance limit. Performance Compromisability Refer back to Fig. 2.3 and notice the line connecting the centers of the two circles, (-1, 0) and (−G i ( jω)), it is seen that at the two centers, the vibration will be totally suppressed for X 1 ( jω) and X i ( jω), respectively; while a compromise exists along the line moving from one centre to another. In Fig. 2.3a, since the line passes across the intersection area, it is immediately inferred that the vibration X 1 ( jω) and X i ( jω) will be attenuated simultaneously if the design freedom α( jω) is chosen as the points from R1 to R2 . Choosing the points in between compromises the performance between X 1 ( jω) and X i ( jω). Thus it seems that a compromise always exists yet this is not always true. Consider the situation where the centre (−G i ( jω)) locates within the unit α-circle as shown in Fig. 2.4, obviously both the line connecting the centers and their two extensions from (−1, 0) to R2 , and from (−G i ( jω)) to R1 locate within the intersection area (thus feasible solutions). Along the line connecting the centers, compromise certainly exists for the performance between X 1 ( jω) and X i ( jω); however, along the two extension lines, no compromise actually exists for the performance variables. This important result has not been revealed in the literature and thus is worth being summarized:

Fig. 2.4 Non-compromise between performance variables

2.3 Active Control for Performance Limit: A Geometric Method

41

Proposition 2.6 (Performance Compromisablity for Active Vibration Control) The performance does not necessarily compromise if either the centre (−G i ( jω)) or (−1, 0) locates within the intersection area.

2.3.2 Constrained Optimal Control Practical systems are always constrained in one form or another. Two commonly encountered constraints are sensing and actuation ones. For both situations, it is possible to enforce the constraints through the new design freedom α( jω). For example, consider the saturation by: |U ( jω)| = |K ( jω)X 1 ( jω)| ≤ γ

(2.34)

That is: N ∑ −1 g1k sk0 D( jω) ≤ γ K ( jω)[1 − g11 K ( jω)]

(2.35)

k=2

Manipulation leads to: N ∑ g1k sk0 D( jω)/g11 ( jω) ≤ γ α( jω)

(2.36)

k=2

N   g1k sk0 D( jω) , then the Eq. (2.36) results in: Define γ = γ g11 ( jω)/ k=2

|α( jω)| ≤ γ

(2.37)

Geometrically, this represents a circle centered at the Origin with a radius γ . Henceforth the active vibration control problem with constraints can be solved by taking the intersection of the feasible solutions with this γ -circle. The situation is illustrated in Fig. 2.5, which can be summarized as: Proposition 2.7 (Optimal Constrained Solution to Active Control Problem) The active vibration control problem as specified by (2.33) and constrained by (2.34) is solvable if and only if the feasible solutions further locate within the γ -circle.

42

2 Active Control for Performance Limit

Fig. 2.5 Optimal constrained solution to active vibration control problem

2.3.3 Active Control of Vehicle Suspension Structures A vehicle suspension structure is considered whose dynamics is represented by: x˙1 (t) = x2 (t) x˙2 (t) = −(q1 + q2 )x1 (t) − (c1 + c2 )x2 (t) + q2 x3 (t) + c2 x4 (t) + u(t) x˙3 (t) = x4 (t) x˙4 (t) = q2 x1 (t) + c2 x2 (t) − q2 x3 (t) − c2 x4 (t) + d(t)

(2.38)

With a slight abuse of notations, this can be transformed into the standard “representation” in (2.26): X 1 ( jω) = g11 U ( jω) + g12 D( jω) X 2 ( jω) = g21 U ( jω) + g22 D( jω) U ( jω) = K ( jω)X 1 ( jω)

(2.39)



   q2 + jc2 ω g11 g12 q2 − ω2 + jc2 ω 1 = det , and. g21 g22 −(q2 + jc2 ω) q1 + q2 − ω2 + j (c1 + c2 )ω √ ⌉

 det = q1 + q2 − ω2 + j (c1 + c2 )ω q2 − ω2 + jc2 ω + (q2 + jc2 ω)2 . The objective is thus to find a control K ( jω) such that the transmission of disturbance to both performance variables X 1 ( jω) and X 2 ( jω) are attenuated simultaneously for a frequency ω (if possible); and further obtain the performance limit while analyzing the compromise among the performance variables.

where:

Optimal Solution To proceed, the performance variables must be represented in geometry where the location of β-circle is determined through (2.32) at (−G( jω)):

2.3 Active Control for Performance Limit: A Geometric Method

−G( jω) =

43

√ ⌉

 q1 + q2 − ω2 + j (c1 + c2 )ω q2 − ω2 + jc2 ω

(2.40)

(q2 + jc2 ω)2

To be specific, it is assumed that the parameters are c1 = 2c2 = 1 N s/m, and q1 = 2q2 = 1 N/m; and the frequency of interest is at the natural harmonic frequency √ ω = ω1n = q1 = 1 rad/s. Now (2.40) reads −G( jω) = −1 + 2 j, the geometry of performance can be illustrated in Fig. 2.6. From Fig. 2.6, it is seen clearly that an intersection exists (indicated in the shaded area) and thus the active vibration control problem as specified by |α( jω) + 1| ≤ 1 and |β( jω) + 1| ≤ 1 is solvable. That is, a controller K ( jω) exists where the transmission of disturbance to both performance variables X 1 ( jω) and X 2 ( jω) can be attenuated simultaneously for the harmonic frequency. This illustrates the Optimal Solution to Active Control Problem in Proposition 2.4.

DESIGN FREEDOM 5 alfa-circle beta-circle (-G)

4

IMAG

3

2

R1 1

0 R2

-1 -3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

REAL

Fig. 2.6 Geometric representation of performance, shaded area indicating simultaneous attenuation

44

2 Active Control for Performance Limit

Performance Limit Now further scaling down the α-circle and β-circle will push the performance to limit: Fig. 2.7a shows the case for 3 dB reduction, or the performance is specified by: |α( jω) + 1| ≤ 0.707 |β( jω) + 1| ≤ 0.707

(2.41)

It is seen that the intersection marginally exists implying that performance is indeed improved towards limit. While remaining 3 dB reduction in performance variable X 2 ( jω), a “push” of 8 dB reduction in performance variable X 1 ( jω) leads to the tangency of the performance circles, see Fig. 2.7b. Thus a performance limit set is reached with the specification: |α( jω) + 1| = 0.4 |β( jω) + 1| = 0.707

(2.42)

This illustrates the Active Vibration Control for Performance Limit in Proposition 2.5. Performance Compromisability Refer back to Fig. 2.6, it is seen clearly that the center (−1, 0) locates within the intersection area thus following the performance compromisability result in Proposition 2.6, the performance does not always compromise. This is indeed so since a choice of points along (−1, 0) to R2 , a deterioration of performance in X 1 ( jω) always leads to the same performance decrease in X 2 ( jω); and vice versa, selection along the reverse direction gives the case where performance in both variables can be improved simultaneously without any compromise. This is in stark contrast with the conventional concept of performance compromise, although selection of designs along the points between the two centers does provide a compromising scenario. Optimal Constrained Solution To take the constraint into consideration, a further “constrained-circle” must be brought into the geometry of the performance. This is indicated by γ -circle where two scenarios have been considered as shown in Fig. 2.8. It is seen that for a 3 dB reduction specification in (2.41), if the γ -circle is constrained by 0.7, then the feasible solution would locates totally outside the γ -circle, indicating the constrained control problem is infeasible; on the other hand, if the constraint is relaxed by a radius of 1.6, then the feasible solution would locates totally inside the γ -circle, indicating the constraint would have no “effect” on the feasible solution at all. This illustrates the Optimal Constrained Solution to Active Vibration Control Problem in Proposition 2.7.

2.3 Active Control for Performance Limit: A Geometric Method

45

DESIGN FREEDOM

(a)

4 scaled alfa-circle: 3dB scaled beta-circle: 3dB

3.5

(-G)

3

2.5

IMAG

2

1.5

1

R1

0.5 R2 0

-0.5

-1 -3

(b)

-2.5

-2

-1.5

-1 REAL

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0

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DESIGN FREEDOM 4 scaled alfa-circle: 8dB scaled beta-circle: 3dB

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(-G)

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2

1.5 1 R

0.5 0

-0.5 -3

-2.5

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-1.5

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-0.5

0

0.5

1

Fig. 2.7 Performance limit for active vibration control: while (3, 3 dB) simultaneous reduction in both performance variable is achievable in (a), the performance reaches its limit for a (8, 3 dB) specification

46

2 Active Control for Performance Limit DESIGN FREEDOM 4 scaled alfa-circle: 3dB scaled beta-circle: 3dB y-circle:y=0.7 y-circle:y=1.6 (-G)

3

IMAG

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-0.5

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REAL

Fig. 2.8 Effects of constraint on solution existence: γ = 0.7 renders no existence of solution; while γ = 1.6 has no influence on solution existence

Real-Time Simulation The above theoretical results must be further confirmed by real-time simulation for verification. To be able to implement the optimal controller, the optimal design on α( jω) needs to be transformed back to K ( jω) through (2.30), namely: K ( jω) =

α( jω) [α( jω) + 1]g11

(2.43)

Of particular interest is the implementation of optimal control for performance limit, e.g. a (8, 3 dB) specification as in (2.42) is considered, hence |α( jω) + 1| = 0.4 and |β( jω) + 1| = 0.707. It is seen from Fig. 2.7b that α( jω) = −1 + 0.4 j, and the corresponding optimal control can be calculated through (2.43) as K ( jω) = 1.5 − 3.5 j. This can be implemented through either a recursive least-squares estimating the gain and phase shift, or an analog controller with gain and phase matching for the target frequency, or even a PID type of controller in time domain. The latter approach is taken and the results are shown in Fig. 2.9. It is seen clearly that the controller does provide the required vibration attenuation at the performance limit specification.

2.3 Active Control for Performance Limit: A Geometric Method

47

X1 Performance 1 0.8 0.6

Magnitude

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

100

200

300

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600

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(a) X2 Performance 2.5 2 1.5

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1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0

100

200

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600

700

800

900

1000

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(b) Fig. 2.9 Active vibration control: design for performance limit a X 1 ( jω) performance; b X 2 ( jω) performance

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2 Active Control for Performance Limit

2.3.4 Section Discussions and Conclusions One of the key objectives of control designs is to excavate the potentials of the system for performance improvement. Thus all available information must be fully taken advantage of. One such information is the properties of the disturbance to be attenuated. Conventionally, disturbance is treated either passively as in H2/Hinf control, or actively as in disturbance-observer-based methods. The fundamental philosophy for both approaches, however, is to handle disturbance over the whole frequency band. Indeed, active control has been generally regarded as a disturbance attenuation/rejection problem in control-theoretic communities. What has demonstrated in the section is that new methods are necessitated that can further “harness” performance by exploiting the finite-frequency-band feature as encountered in vibration problems. Simply treating active control as disturbance attenuation control can cause (significant) loss of performance due to the conservatism introduced by treating design over a full-frequency-range. Thinking of active control as a disturbance attenuation problem for a discrete frequency or over a frequency band is simply not enough. The shift from infinite-frequency band to finite-frequency band produces a challenging problem not fully solved in the state-of-the-art. This calls upon new approaches in developing the themes from CONTROL to ACTIVE CONTROL.

2.4 Active Pure Topology Feedback for Performance Limit The above section has deployed a strategy with feedback design through agents’ dynamics. Indeed, this is the most widely explored approach in the literature. And a variety of methods can be devised such as state feedback, output feedback, even adaptive designs can be considered. However, it is widely recognized that complex networks can represent many industrial, social, and biological systems [26–28], since the model can capture both the dynamics of the nodes and the interactions among the nodes. Thus it is motivating and beneficial to consider the pure topology feedback strategy and this is considered in this section. Indeed, with the background of complex networks, a variety of properties of complex networks have been delineated extensively over the past two decades such as network formations and propagations [29, 30], network controllability and observability [31, 32], network consensus and flocking [33, 34], and network synchronization [35, 36] etc. As a consequence, a number of important results have been archived on the conditions achieving these collective emergences. While consensus, flocking, and synchronization concern the “tracking” properties of the networks, scenarios also arise where vibration suppression is expected. Control-theoretically, vibration suppression has been conventionally and even routinely treated as a “disturbance attenuation problem,” and henceforth optimal H2/H∞ control [8–10] and adaptive control [11–13] are usually called upon for

2.4 Active Pure Topology Feedback for Performance Limit

49

rescue. Yet, it is the author’s conviction that treating vibration suppression as disturbance attenuation is NOT appropriate due to the following reasons (also delineated in the above section within an active control background of formulation): (1)

(2)

vibration problems generally occur either at discrete frequencies or over a limited frequency band, while disturbance attenuation often refers to the whole √ ⌉ frequency band, or ω ∈ 0 ∞ rad/s (see [17–19, 37] for example); vibration refers to specific locations which is usually not considered in disturbance attenuation problems.

It can be (surely reasonably) argued that vibration suppression can be reformulated as a “disturbance attenuation problem for a discrete frequency or over a frequency band,” thus one can still utilize H2/H∞ control to tackle vibration suppression problem through, e.g. intended choice of weighting functions or generalized KYP-lemma, as taken in [20, 21] and references therein. Yet such a tailored solution is really of “trial-and-error” nature since it is difficult to choose appropriate weighting functions to target specific discrete frequencies or frequency bands, not to mention that utilization of weighting functions can result in high-order controllers whose orders must be reduced [22]. This is a “disturbing” issue particularly in vibration suppression problems for many real-time and time-critical applications. The section is structured as follows: the problem formulation is given in Sect. 2.4.1; solutions and related fundamental results are presented in Sects. 2.4.2 and Sect. 2.4.3 delves further into the realizable solutions associated with physical implementations—this is pivotal since the resulting optimal controller must be physically realizable; examples for validating the proposed methods and results are given in Sect. 2.4.4, and finally the paper is concluded in Sect. 2.4.5.

2.4.1 Problem Formulation: Pure Topology Feedback for Global Performance It is again assumed that the control system is represented by an N-node heterogeneous network with pure topology feedback: x˙i (t) = ai xi (t) +

N ∑

si j x j (t) − si0 di (t) + u i (t)

j=1 j/=i

, i ∈ [ 1, N ] u i (t) =

N ∑ j=1 j/=i

ki j x j (t)

(2.44)

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2 Active Control for Performance Limit

where: xi (t) is the performance variable, u i (t) is the control action; ai is the node dynamics; si j s are the topological interaction coefficients among the nodes with si0 the interaction coefficient with the exogenous vibrational signal di (t); and ki j s are the control parameters to be designed. The closed-loop system (2.44) is simply: N ∑

 si j + ki j x j (t) − si0 di (t), i ∈ [ 1, N ] x˙i (t) = ai xi (t) +

(2.45)

j=1 j/=i

Thus at a discrete frequency ω, the corresponding frequency response representation is: ⎡ ⎤ N ⎢∑ ⎥

 si j + ki j X j ( jω)−si0 Di ( jω)⎥ X i ( jω) = ( jω − ai ) ⎣ ⎦, i ∈ [ 1, N ] (2.46) −1 ⎢

j=1 j/=i

where: X i ( jω) and Di ( jω) are the frequency response functions of xi (t) and di (t), respectively. Now since the vibrational signals have the same frequency, it can be assumed that Di ( jω) = D( jω). Then Eq. (2.46) can be further solved for the performance variables X i ( jω) s with respect to D( jω), so that the global vibration suppression can be formulated as finding the topological parameters ki j s so that the following performance index is satisfied:

 |X i ( jω)/D( jω)| 0) is a learning or convergence gain [4]. Recently a unified approach has emerged where certain norms of the disturbance transmission are optimized, e.g. H∞ norm [5, 6]. Indeed, by formulating the active control problem into a “general control configuration” which is a universal way of formulating control problems [7–9], many © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Wang, Active Vibration & Noise Control: Design Towards Performance Limit, https://doi.org/10.1007/978-981-19-4116-0_3

81

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well-established control design methodology can be utilized for vibration and noise control purpose, e.g. H2 /H∞ control [10–12], adaptive control [13], generalized KYP lemma-based loop shaping [14], and even the conventional proportional-derivative control [15]. As a general rule-of-thumb, H∞ control works efficiently for harmonic vibration suppression while H2 control is effective for multiple-modal attenuation, as verified from various experimentations [16–18]. Although significant progress has been made towards integrating control theory with active controls, it is still felt that a gap remains to be filled in. This is so since many control design methods do not take the nature of vibration problems fully into account. This is not surprising but reflects the fact that the development of active control and that of control theory itself, particularly feedback control theory, are historically detached. Specifically, in feedback control theory, disturbance attenuation is a primary goal for control, however, the disturbances considered are usually step signals or ramp signals [19]; in vibration and noise control, however, it is usually discrete frequencies or band limited signals (over a frequency band) that should be attenuated [20]. Therefore the key issue is to “tailor” control theory towards disturbance attenuation for discrete-frequency or band-limited signals. This has aroused attention over the past years [21, 22] and either weighting functions or generalized KYP-lemma is utilized to handle the band-limited control problem. For both approaches, however, H∞ and linear matrix inequality optimization techniques are required for the obtaining solutions; moreover, as the resulting controller possesses high orders, more complex optimization such as generic algorithms is needed for order reduction [23]. This necessitates new methods for attacking the discrete-frequency or band-limited disturbance attenuation/vibration control problems. This has been the motivation with the result as presented in the previous chapter. This is actually a continued effort to tackle such problems over the past decade as demonstrated in [20, 24–28]. But it is only recently that the author has been able to bring all the results into a unified framework. Before devoting to the extensions, previous development of the proposed solutions is briefly introduced in the next section.

3.2 Generic Active Control Solutions To make a clear presentation, the system is represented by a general control configuration directly in frequency domain as: ⌈

⌉⌈ ⌉ ⌈ ⌉ y( jω) g ( jω) g12 ( jω) u( jω) = 11 g21 ( jω) g22 ( jω) d( jω) z( jω)

(3.1)

where y( jω) and z( jω) are the performance variables while u( jω) and d( jω) are the control input and the exogenous disturbance respectively; gi j s are the corresponding

3.2 Generic Active Control Solutions

83

frequency response functions. The feedback control is defined as: u( jω) = −k( jω)y( jω)

(3.2)

Thus the problem is formulated as finding an optimal control k( jω), such that the performance variables y( jω) and z( jω) are optimized in terms of prescribed specification; should the specification be infeasible, find a suboptimal control which achieves the best performance (and thus limit of performance needs to be determined, e.g. existence and optimization problems). It is seen that the above system model and control are described exactly as the linear fractional transformation, which is used in H∞ and H2 control design formulations. However, in H∞ /H2 designs, performance specification cannot be prescribed before the optimization actually takes place; consequently limit of performance and the corresponding controller are not readily available. These hurdles have been overcome by a tailored method proposed by the authors specifically targeting disturbance attenuation for (any) discrete-frequency or over (an arbitrary) frequency band as presented in Chap. 2. A brief introduction is provided here to bring the state-of-the-art before attributing new results for constrained and robust solutions. Previous development has demonstrated that reduction in y( jω) and z( jω) for a frequency ω = ω0 is equivalent to satisfying the following conditions, respectively: |α( jω0 ) + 1| < 1

(3.3)

|β( jω0 ) + 1| < 1

(3.4)

and

where α( jω0 ) = S( jω0 ) − 1 with S( jω0 ) being the sensitivity and β(ω0 ) is defined by a Möbius transformation as: α( jω0 ) = β( jω0 )g( jω0 ), with g( jω0 ) =

g11 ( jω0 )g22 ( jω0 ) g12 ( jω0 )g21 ( jω0 )

(3.5)

The relationship of the reduction in y( jω) and z( jω) is “reflected” by that of α( jω0 ) and β( jω0 ), which is determined through Eq. (3.5). This observation and the corresponding geometrical interpretation have led to a number of important results answering such fundamental issues of solution existence, performance limit, and suboptimal solutions etc. These have been disseminated by the authors’ previous development, some of which are summarized below with reference to Fig. 3.1. Proposition 3.1 (Feasibility of Reduction) Attenuation in y( jω) and z( jω) for a frequency ω is feasible if and only if there exists intersection between β-circle and unity α-circle.

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Fig. 3.1 Mapping of |β( jω) + 1| < 1 on the complex plane with |α( jω) + 1| < 1, which are abbreviated as β-circle and α-circle for easy reference. The location of the centre of β-circle is prescribed by (–g) that is solely determined by the plant parameters. Hence a line connecting the two circle centers (denoted as αopt ( jω)) actually defines the fundamental performance compromise between y( jω) and z( jω), while the best achievable performance is also determined simultaneously

Proposition 3.2 (Level of Reduction) The level of reduction in y( jω) and z( jω) for a frequency ω is determined by the scaling with respect to the unit α-circle and β-circle, respectively. Proposition 3.3 (Distribution of Optimal Controllers) The optimal controller kopt ( jω) results from αopt ( jω), a point taken from the line on the complex α-plane that joints the point (0, 0) with the point (−g( jω)). Proposition 3.4 (Optimal Controller Synthesis) The optimal controller kopt ( jω) achieving the performance specified by αopt ( jω) can be obtained as follows: kopt ( jω) =

α( jω) [α( jω) + 1]g11 ( jω)

(3.6)

Proposition 3.5 (Higher Harmonic Control) If there exists intersection among the βi -circles of the higher harmonics to be controlled, then all the higher harmonics can be controlled by a single controller; if this intersection further intersects the unit α-circle, then this single controller will provide simultaneous reduction in all the performance variables. Proposition 3.6 (Annihilation Controller 1) A controller exists that can reduce z( jω) to zero without leading to an increase in y( jω) if and only if g( jω) is positive real and is located inside the unit circle having centre at (−1, 0) on the complex plane. Proposition 3.7 (Annihilation Controller 2) A controller exists that can reduce y( jω) to zero without leading to an increase in z( jω) if and only if Re(g( jω)) > 0.5.

3.2 Generic Active Control Solutions

85

This has been the state-of-the-art concerning the proposed solution to active control problems. However, one of the dissatisfying features is the concern upon constraints and robustness handling. Indeed, any sensible design methodology has to handle various constraints such as actuation constraints to prevent saturation, sensor constraints to avoid calibration error, as well as constraints from size, weight, energy consumption etc.; meanwhile, uncertainties should also be compensated, and structured uncertainty should be taken advantage of to alleviate and reduce the conservatism. This has bothered many conventional methods which often lead to very conservative designs. These problems will be dealt with in the next section. Before this, some comments follow below to further demonstrate the generality of the development. Remark 3.1 The proposed method can be readily utilized to tackle the active control problems in mechanical systems. For example, a conventional tuned mass damper system has the following equation of motion: M x¨ + C x˙ + K x = F

(3.7)

⌉ ⌈ ⌉ ⌈ ⌉ ⌈ ⌉ ⌈ c + c2 −c2 k + k2 −k2 d m1 0 ,C = 1 ,K = 1 ,F= and where M = u −c2 c2 −k2 k2 0 m2 ⌈ ⌉ x x = 1 . Then for a frequency ω, the above equation can be written as: x2 x( jω) = (K − Mω2 + jCω)−1 F( jω) ⌈ Identifying

(3.8)

⌉ ⌈ ⌉ y( jω) x2 ( jω) as thus leads to the “standard form” in Eq. (3.1), z( jω) x1 ( jω)

with: k1 + k2 − m 1 ω2 + jω(c1 + c2 ) , det(K − Mω2 + jCω) k2 + jωc2 = g21 = , det(K − Mω2 + jCω) k2 − m 2 ω2 + jωc2 = . det(K − Mω2 + jCω)

g11 = g12 g22

Remark 3.2 The proposed method is illustrated by two-input two-output systems, but it can also be generated into multivariable systems albeit restricted to scalar control input. To see this, consider the following representations with the understood notations:

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Fig. 3.2 Reduction in y( jω) and z i ( jω): each z i ( jω) is now corresponding to βi -circle, and henceforth solution existence problem must be answered by the existence of the “simultaneous” intersection over n circles

⎤ ⎡ ⎤⎡ ⎤ g11 ( jω) u( jω) y( jω) g12 ( jω) · · · g1(n+1) ( jω) ⎢ ⎢ z ( jω)⎥ ⎢ g ( jω) ⎥ g22 ( jω) · · · g2(n+1) ( jω) ⎥ ⎢ 1 ⎥ ⎢ 21 ⎥⎢ d1 ( jω)⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ z 2 ( jω)⎥ ⎢ g31 ( jω) ⎥ g32 ( jω) · · · g3(n+1) ( jω) ⎥ ⎢ ⎥=⎢ ⎥⎢ d2 ( jω)⎥ ⎢ ⎢ . ⎥ ⎢ ⎥ ⎥ .. .. .. .. ⎢ . ⎥ ⎢ ⎥⎢ .. ⎥ . . . . ⎣ . ⎦ ⎣ ⎦⎣ . ⎦ z n ( jω) g(n+1)1 ( jω) g(n+1)2 ( jω) · · · g(n+1)(n+1) ( jω) dn ( jω) ⎡

(3.9)

At a frequency, di ( jω) = Ci d( jω), where Ci is a complex number and represents the gain and phase shift with respect to d( jω). Then with the sensitivity defined as above, further defining:

g(i+1)1 ( jω) nj=1 C j g1( j+1) ( jω)

(i = 1 · · · n) βi ( jω) = (S( jω) − 1) g11 ( jω) nk=1 Ck g(i+1)(k+1) ( jω)

(3.10)

Then the condition that z i ( jω) is reduced can be expressed as:  |βi ( jω) + 1| < 1 (i ∈ 1 n )

(3.11)

Consequently, the prescribed procedures introduced above can still be utilized “as usual.” The only difference is that there are n circles at the frequency ω in corresponding to disturbance sources (Fig. 3.2).

3.3 Active Control with Constraints Control systems are subject to various constraints and these are typically from sensors, actuators, and components interface requirements. The common feature of these constraints is the restriction on relevant signal magnitudes. There are other constraints from product requirements such as weight, size, cost etc. This type of

3.3 Active Control with Constraints

87

constraints is relatively difficult to quantify but has to rely on a case-by-case analysis. Having said this, there are still situations where constraints can be handled as magnitudes restriction, e.g. the vibration magnitude is restricted from above due to limited size to contain the vibrating system. The relevant situations are looked at in detail in the following subsections.

3.3.1 Sensing Constraints Constraints from sensors can be formulated as restricting on the magnitudes of the system outputs. In current formulation, such constraints can be specified by: |y( jω)| ≤ δ y |z( jω)| ≤ δ z

(3.12)

To bring the constraints into the proposed framework, specification (3.12) must be defined in terms of the available design freedom. This can be achieved by substituting control (3.2) into system dynamics (3.1) and manipulating: y( jω) = S( jω)g12 ( jω)d( jω) z( jω) = [g22 ( jω) − g12 ( jω)k( jω)S( jω)g12 ( jω)]d( jω)

(3.13)

Utilization of the definition α( jω) = S( jω) − 1 and Eq. (3.5) leads to: y( jω) = [1 + α( jω)]g12 ( jω)d( jω) z( jω) = [1 + β( jω)]g22 ( jω)d( jω)

(3.14)

By “re-calibrating” the specifications δ y = δ y /|g12 ( jω)d( jω)| and δz = δ z /|g22 ( jω)d( jω)| constraints (3.12) can be expressed as: |1 + α( jω)| ≤ δ y |1 + β( jω)| ≤ δz

(3.15)

Now comparing (3.15) with (3.3) and (3.4), it is seen clearly that the original development can still be utilized, so long to the circles (and their inte as changing  riors) from unity (1, 1) specification to δy , δz specification correspondingly. From Proposition 3.2, this is a scaling of the circles illustrated in Fig. 3.1: the radius of the α-circle scales to δ y while β-circle to (δz |g|). The situation is shown in Fig. 3.3. As usually δy , δz specification is less than unity, this leads to a reduced area of intersection. This significantly restricts the optimal controllers and achievable performance in y( jω) and z( jω) according to the Propositions 3.1–3.7. Now refer to Fig. 3.3, it is seen that the intersection can finally disappear with gradual decrease of δy and δz . Henceforth the condition for vibration reduction in

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3 Active Control with Constraints and Uncertainty

Fig. 3.3 Mapping of |β( jω) + 1| ≤ δz on the complex plane with |α( jω) + 1| ≤ δ y . The situation in Fig. 3.1 for mapping of |β( jω) + 1| < 1 and |α( jω) + 1| < 1 is also re-produced here. It is seen that this can result in a significantly reduced area of intersection, consequently much constrained optimal controllers that can reduce the vibration in y( jω) and z( jω)

y( jω) and z( jω) under the sensing constraints is: δ y + δz |g| ≥ |g − 1|

(3.16)

11 g22 is solely determined by system parameters, henceforth condiNote that g = gg12 g21 tion (3.16) poses a fundamental restriction on the existence of optimal solutions under sensing

constraints, e.g. feasibility of vibration reduction in y( jω) and z( jω) under δy , δz specification has to satisfy (3.16), or equivalently, any specification that does not fulfill (3.16) would lead to no solution existence and henceforth the specification has to be violated. This is summarized below.

Proposition 3.8 (Feasibility of Reduction under Sensing Constraints) Attenuation in y( jω) and z( jω) for a frequency ω under sensing constraints (3.12) is feasible if and only if condition (3.16) is satisfied.

3.3.2 Actuation Constraints One of the most important constraints comes from actuation saturation, e.g. actuators will finally saturate. Hence actuation constraint must be taken into account. This constraint can be enforced by requiring: |u( jω)| = |−k( jω)y( jω)| ≤ γ

(3.17)

Substituting (3.6) and (3.14) and manipulating: |α( jω)g12 ( jω)d( jω)/g11 ( jω)| ≤ γ

(3.18)

3.3 Active Control with Constraints

89

Fig. 3.4 Actuation constraint |α( jω)| ≤ γ , together with sensing constraints as represented by mapping of |β( jω) + 1| ≤ δz on the complex plane with |α( jω) + 1| ≤ δ y

A re-calibration of γ = γ |g11 ( jω)/g12 ( jω)/d( jω)| leads to: |α( jω)| ≤ γ

(3.19)

Condition (3.19) can be expressed as a circle (and its interior) with origin as the centre and radius γ . Indeed, it is remarkable that the actuation constraint can exactly be represented by the design freedom α( jω), which is simply a circle bounded by the constraint specification γ . This situation is illustrated in Fig. 3.4. It is seen from Fig. 3.4 that a sensing constraint would further put a restriction on solution existence and optimal solutions. While the constraint specification γ decreases, as usually will be the case for preventing saturation, intersection among the three circles will gradually disease. Disappearance of intersection implies that no solution exists in reducing the vibration in y( jω) and z( jω) while subjecting to the sensing and actuation constraints. Thus the following result is claimed. Proposition 3.9 (Feasibility of Reduction under both Sensing and Actuation Constraints) Attenuation in y( jω) and z( jω) for a frequency ω under sensing constraints (3.12) and actuation constraint (3.17) is feasible if and only if the three circles have intersection. Meanwhile it is noted that condition (3.16) ensures that intersection exists between the scaled α-circle and scaled β-circle; similarly the condition: δy + γ ≥ 1

(3.20)

will ensure the intersection existence between the scaled α-circle and the circle |α( jω)| ≤ γ ; and the condition: δz |g| + γ ≥ |g|

(3.21)

ensures the intersection existence between the scaled β-circle and the circle |α( jω)| ≤ γ . But it is not true to state that simultaneous satisfaction of (3.16), (3.20) and (3.21) can provide attenuation in y( jω) and z( jω) under sensing and

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3 Active Control with Constraints and Uncertainty

actuation constraints. This is because mutual intersections among three circles do not guarantee the existence of intersection among the three circles. As a very good approximation, the three-circle-intersection condition can be stated by the middle point of intersection between the scaled α-circle and the scaled β-circle lies within the circle |α( jω)| ≤ γ . Should the centres of the scaled two circles do not lie within each other, this can be expressed as: γ = |s( jω)|, where s( jω) is an auxiliary variable satisfying: |g( jω) − 1| + δ y − δz |g( jω)| 2 |g( jω) − 1| − δ y + δz |g( jω)| |s( jω) + g( jω)| = 2 |s( jω) + 1| =

(3.22)

This important result associates the three specifications for constraints with system parameters and thus must be summarized. Proposition 3.10 (Reduction under both Sensing and Actuation Constraints) Attenuation in y( jω) and z( jω) for a frequency ω under sensing constraints (3.12) and actuation constraint (3.17) can be achieved if γ specified by γ = |s( jω)| with s( jω) solved from condition (3.22). Meanwhile conditions (3.20) and (3.21) solve the problems of solution existence for vibration reduction in either y( jω) or z( jω), and they are also worth while being listed. Proposition 3.11 (Feasibility of Reduction in y( jω) under both Sensing and Actuation Constraints) Attenuation in y( jω) for a frequency ω under sensing constraints (3.12) and actuation constraint (3.17) is feasible if and only if condition (3.20) is satisfied. Proposition 3.12 (Feasibility of Reduction in z( jω) under both Sensing and Actuation Constraints) Attenuation in z( jω) for a frequency ω under sensing constraints (3.12) and actuation constraint (3.17) is feasible if and only if condition (3.21) is satisfied.

3.3.3 Components Constraints For feedback control systems, sensing and actuation constraints are most frequently enforced. Another constraint that often requires being posed is from the controller’s input–output requirements. This is entailed through restriction upon the magnitude of the controller as below: |k( jω)| ≤ λ

(3.23)

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91

From (3.6), this constraint can be expressed with respect to the design freedom α( jω) by a redefinition λ = λ|g11 ( jω)| as:    1  1    α( jω) + 1 ≥ λ

(3.24)

Therefore the following result is readily obtained. Proposition 3.13 (Feasibility of Reduction under Sensing, Actuation and Components Constraints) Attenuation in y( jω) and z( jω) for a frequency ω under sensing constraints (3.12), actuation constraint (3.17), and component constraint (3.23) is feasible if and only if Proposition (3.9) holds while the referred intersection further intersect the circle expressed by (3.24). Several important results can be obtained by noting the relationship of (3.24) with the corresponding circle locations discussed above. However this is not developed here but turning to an important problem of feasibility of specifications—now inequality (3.24) represents another circle (and its exterior) centered at (−1, 0) with radius 1/λ. It is noted that this circle is concentric with the circle |α( jω) + 1| ≤ δ y (and its interior) as specified for the vibration reduction in y( jω). Hence, a necessary condition for feasibility of specification with δ y and λ is that the two circles with one interior and another exterior must have intersection, that is 1/λ ≤ δ y , or: δy λ ≥ 1

(3.25)

Condition (3.25) is a fundamental relationship as it does not even depend on system parameters but enforces restriction on feasibility of specification of δ y and λ. To give an example, if δ y is required a 6 dB reduction, or δ y = 0.5, then (3.25) dictates λ ≥ 2. That is, any specification of λ < 2 will imply that no solution exists under components constraint (3.23). Henceforth this is a far-reaching result. Proposition 3.14 (Reduction under Components Constraints) Attenuation in y( jω) under sensing constraint and component constraint as specified by δ y and λ respectively is feasible only if condition (3.25) is satisfied.

3.3.4 Controller Realization Constraints In the above subsections, constraints have been considered mainly associated with norms. However rarely noticed is the controller realization issue that has to be considered. Namely, an optimal selection α( jω), would result in an optimal design k1N . However, it is noticed that α( jω) is a point on complex plane henceforth a complex number, thus it will be (very likely) leading to a complex-valued k1N . Yet k1N is defined as topological interaction parameter to be designed by:

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u(t) = k1N x N (t)

(3.26)

Henceforth, k1N must be real other than complex. That is, the optimal design α( jω) is not only constrained by solvability, but also by the implementability of corresponding feedback controls. This constraint is enforced through the realness of α( jω) . Now assume that α( jω) = x + yi and g N 1 = a + bi, with x, y, a, and [1+α( jω)]g N 1 b being real variables. Then the requirement of

α( jω) [1+α( jω)]g N 1

being real is:

Im[(x + yi)(1 + x−yi)(a + bi)] = 0

(3.27)

where Im[•] indicates the imaginary part of the expression. Thus this leads to: 

b x 2 + x + y 2 − ay = 0

(3.28)

(1) b = 0 Should b = 0, then (3.28) results in ay = 0. Consequently, a nontrivial solution is y=0

(3.29)

This will constrain the optimal design α( jω) to be only real, or locating only on the real axis. (2) b /= 0 Then from (3.28), one has: x 2 + x + y2 −

a y=0 b

(3.30)

Or the optimal design α( jω) is constrained by: 

1 x+ 2

2

 a 2 a2 + 1 + y− = 2b 4b2

(3.31)

  This is a circle centered on (−1/2, a/2b) with a radius a 2 + 1 /4b2 , that is, the circle always centered on the line x = −0.5, while passing on the Origin and (−1, 0). This is illustrated on Fig. 3.5, where the circles have been indicated as R-circles. From Fig. 3.5, it is seen that while the realizable solution on R2 -circle still intersects with the feasible solution area, indicating that such a constraint will not influence the optimal designs. However, should the realizable solution locates on R1 -circle, this will render the optimal designs being infeasible, henceforth the global vibration suppression through pure topology feedback becomes insolvable. Remark 3.3 It is noticed that the realizable solutions must locate “exactly” on the circle (3.25), neither inside nor outside.

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Fig. 3.5 Realizable solutions: constraints on realness of control parameter can significantly restrict the feasible solutions

Remark 3.4 This realizable solution concerning the implementability of optimal control designs is often overlooked in the literature. It can clearly enforce significant restrictions on optimal designs henceforth must be given serious consideration.

3.3.5 Other Forms of Constraints There are other forms of constraints imposed either from product functional or from structural requirements. As mentioned in the introduction of this section, there are situations where constraints can be handled as magnitudes restriction such as vibration magnitude restricted by limited size or space available. This type of constraints can also be expressed similar with (3.12) as: |y( jω)| ≤ ρ y |z( jω)| ≤ ρz

(3.32)

Consequently, these constraints can be dealt with paralleling with the development in Sect. 4.1. This is not pursued here and it is included for the reason of completeness of exposition. Before finishing the discussion of this section, simple examples are provided to briefly illustrate the results. Example 3.1 (Component Constraint) Now start with considering the component constraint λ = 2, then Proposition 3.14 would dictate δ y ≥ 0.5, or any reduction less than 6 dB would become infeasible (bear in mind 6 dB reduction is often regarded as basic requirement for control). Example 3.2 (Component Constraint and Sensing Constraint) Follow Example 3.1 and further consider sensing constraints, since a 6 dB reduction is not possible, it is relaxed

to seek a 3 dB reduction, or δ y = 0.707 ≥ 0.5, then Proposition 3.8 requires δz ≥ |g − 1|−δ y /|g|. Assume g = −0.5 + 2i, then δz ≥ 0.87; or any intention of reducing δz less than 0.87 would doom the design to failure.

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Example 3.3 (Component Constraint, Sensing & Actuation Constraints) Follow Example 3.2 and further consider actuation constraint. Since a 6 dB reduction is not possible, it is again to seek a 3 dB reduction δ y = 0.707, then Proposition 3.11 requires γ ≥ 1−δ y = 0.293 for attenuation in y( jω) under sensing and actuation constraints; or the control system cannot be prevented from saturation as long as γ is specified by less than 0.293 (bearing in mind that a smaller γ is actually very desirable as it can also reduce energy consumption). Now suppose Proposition 3.11 is given satisfaction γ = 0.3 ≥ 0.293, then Proposition 3.12 requires δz ≥ 1−γ /|g| = 0.85 for z( jω) reduction, henceforth an significant reduction in z( jω) would not in y( jω) be achievable. Indeed, from Proposition 3.10, attenuation   and z( jω) under  1+γ −δ y  sensing and actuation constraints can be achieved if  2 + g  ≤ δz |g| is satisfied. This would lead to δz ≥ 0.97 to satisfy all the constraints, which is even demanding then Proposition 3.12 requirement of δz ≥ 0.85. Thus the above simple examples illustrates that performance is indeed severally restricted by system constraints. This also put strict bounds on performance specifications. Compromise has to be made to balance the corresponding designs. The power of the proposed approach provides direct visualization on the often-conflicting performance tradeoffs.

3.4 Active Control with Uncertainty Provided that optimal solutions do exist, or the specifications are feasible, robustness issues must be tackled for any systematic design methodology. Meanwhile, as a generic design method, it should be able to deal with different scenarios. In this section, different cases are considered but a key concept of robustness design is to choose solutions that are able to “resilient” against either structured or unstructured uncertainties.

3.4.1 Robust Design for Unconstrained Case If there is no constraint, then feasible solutions lie in the intersection between β-circle and α-circle, in terms of Propositions 3.1 and 3.3 (also refer to Fig. 3.1). Therefore for unstructured uncertainties, the most robust choice will be the centre of the intersection, or the mid-point of the optimal line in the intersection area, as shown in Fig. 3.5. If the uncertainties are structured, e.g. structured uncertainties associated with gi j ( jω), then the uncertainty information can be defined for g( jω) = g11 ( jω)g22 ( jω)/g12 ( jω)g21 ( jω). As β-circle locates at the centre (−g) with radius (|g|), this results in a series of β-circles“carrying” the structure of the uncertainty. A robust choice will also be the centre or average of these β-circles; and taking this value of the centre or average as the nominal parameters for β-circle will

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95

lead to the most robust location of the β-circles; once the location is determined, a further choice of the centre of the intersection of this nominal β-circle with α-circle will provide a robust controller counteracting for the structured uncertainties of the system. With this background and reference to Fig. 3.5, the robust choice can be calculated by noting the relative positions of the two circles as summarized below. Proposition 3.15 (Robust Design for Unconstrained System) Robust design for αopt ( jω) can be obtained by the following conditions: (1)

if the centre of β-circle is outside of α-circle, then: |α( jω) + g( jω)| =

(2)



|g( jω)|2 − |α( jω)|2 =

|g( jω) − 1| + |g( jω)| − 1 2 (3.33)

if the centre of α-circle is outside of β-circle, then:  |g( jω) − 1| − |g( jω)| + 1 |α( jω) + 1| = 1 − |α( jω)|2 = 2

(3.34)

Remark 3.5 In both (3.27) and (3.28), α( jω) is a complex number with two unknowns of real part and imaginary part, thus the two unknowns can be solved with two equations in each condition. Remark 3.6 For the case of both centers of α-circle and β-circle do not lie within each other, either (3.27) or (3.28) can be utilized and they result in equivalent αopt ( jω). Remark 3.7 Once αopt ( jω) is determined, the optimal controller is readily obtained through Proposition 3.4, where a nominal g11 ( jω) can be assumed to take its robustness into account.

3.4.2 Robust Design for Constrained Cases The above method for approaching structured and unstructured uncertainties can be adopted for cases considering sensing, actuation and component constraints. The difference lies in the detailed calculation to account for different cases. But it is warned that the detailed calculation can be complicated (but not difficult!). For example, consider the case for robust design with sensing constraints, and further assume the centre of scaled β-circle is outside of scaled α-circle, then with reference to Fig. 3.6 the following result can be obtained. Proposition 3.16 (Robust Design for Systems with Sensing Constraints) Consider the situation where the centre of scaled β-circle is outside of scaled α-circle, then robust design for αopt ( jω) can be obtained by the following conditions:

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Fig. 3.6 Robust design of αopt ( jω) under structured and unstructured uncertainty without constraints

|α( jω) + g( jω)| = =

 (δz |g( jω)|)2 − |α( jω) − s( jω)|2 |g( jω) − 1| + δz |g( jω)| − δ y 2

(3.35)

where s( jω) is an auxiliary variable to be calculated by the following two equations: |α( jω)|2 + |g( jω)|2 − |g( jω) − α( jω)|2 2|α( jω)||g( jω)| |α( jω)|2 + |s( jω)|2 − |s( jω) − α( jω)|2 = cos−1 2|α( jω)||s( jω)| 2 |s( jω)| + |g( jω)|2 − |g( jω) + s( jω)|2 + cos−1 2|s( jω)||g( jω)|

cos−1

|α( jω)|2 + |α( jω)−s( jω)|2 − |s( jω)|2 π + 2|α( jω)||α( jω)−s( jω)| 2 2 2 |α( jω)| + |α( jω) + g( jω)| − |g( jω)|2 = cos−1 2|α( jω)||α( jω) + g( jω)|

(3.36)

cos−1

(3.37)

Remark 3.8 The αopt ( jω) can be calculated for the case where the centre of α-circle locates outside β-circle similarly. In practical design, however, the location of αopt ( jω) can be found out by plotting the respective circles discussed in the last section, and hence it can be determined by graphical inspection. Complicated trigonometry and complex variable operations are thus avoided. Remark 3.9 Robust design for the cases with sensing, actuation, component, and even other forms of constraints mentioned in the last section can be dealt with similarly. The expressions are very lengthy and they are not reproduced here. However, as suggested in Remark 3.8, the corresponding robust designs can also be obtained through graphical inspections when tackling specific problems. The

3.5 Active Control with Constraints and Uncertainty: Numerical Validations

97

ease and convenience of obtaining robust optimal solutions will be illustrated in the next section.

3.5 Active Control with Constraints and Uncertainty: Numerical Validations To validate the proposed method, a three-degree-of-freedom active vibration isolation structure is utilized. As shown in Fig. 3.7, the disturbance d causes vibration transmission to the structure, however, in many applications such as suspension or satellite isolation systems, the vibration in m z needs to be minimized for payload protection, while the vibration in m y requires also being controlled to reduce the transmission force to the base. It is thus envisioned that design compromise must exist where solution existence and optimal control synthesis should be investigated for particular performance specifications for vibration reductions and various constraints. To proceed, the system is assumed to have the following matrices: ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 100 4 −1 −2 4 −1 −1 M = ⎣ 0 1 0 ⎦, C = ⎣ −1 1 0 ⎦ and K = ⎣ −1 2 −1 ⎦. 001 −2 0 2 −1 −1 2 Then a Fourier transform of the corresponding dynamic equation will bring the system model into the form of Eq. (3.1)—refer to Remark 3.1 and Eqs. (3.7) and (3.8). A plot of the frequency response of g22 ( jω) shows that there is a resonance at 0.1 Hz and this leads to a transmission peak in m y , as confirmed from g12 ( jω). Thus the control objective will be the design of harmonic vibration control for reduction Fig. 3.7 Active vibration isolation structure

98

3 Active Control with Constraints and Uncertainty Locations of the Two Circles

Fig. 3.8 Active control without constraints: unit α-circle—solid line (black); β-circle—dashed line (red)

1.5 1 0.5

Im

0 -0.5 -1 -1.5 -2 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Re

in m y (y( jω)) and m z (z( jω)), with particular emphasis on constrained control and robust designs.

3.5.1 Active Control Without Constraints To start with, first plot the unit α-circle and β-circle (hence for unconstrained situation), it is found that a significant area of intersection exists. With reference to both Fig. 3.1 and the corresponding Propositions 3.1–3.7, it can be concluded that the solutions from the shaded area can provide attenuations in both y( jω) and z( jω); furthermore, controllers actually exist that can annihilate vibrations in either y( jω) or z( jω); and annihilation of either of them can also guarantee that the other one is also significantly attenuated, see Fig. 3.8.

3.5.2 Active Control with Constraints Now we move on to the consideration of constraints. Sensing constraints are first considered. From Sect. 4.1, sensing constraints enforce themselves through Eq. (3.15), and this is the scaling to the unit α-circle and β-circle respectively (refer to Fig. 3.9). Now assume the harmonic input is of unity magnitude, then Proposition 3.8 providing a necessary and sufficient condition for attenuation in y( jω) and z( jω) now reads:

3.5 Active Control with Constraints and Uncertainty: Numerical Validations

99

0.4 0.3 0.2

Im

0.1 A

0 -0.1 C

-0.2

B

-0.3 -0.4 -1.4

-1.2

-1

-0.8

-0.6 Re

-0.4

-0.2

0

Fig. 3.9 Active control with constraints: under sensing constraints the unit α-circle and β-circle become the scaled α-circle (solid black line) and scaled β-circle (dashed red line). For enforced actuation constraint, the length to point A indicates the minimum radius that can be specified for satisfying 12 dB reduction in y( jω); the length to point B indicates the minimum radius that can be specified for satisfying 12 dB reduction in y( jω) and 20 dB reduction in z( jω). The length to point C is a typical choice also associated with robustness consideration

δ y + 1.31δz ≥ 0.31

(3.38)

As now δy = δ y /|g

12 d| = 1.324δ

y , δz = δ z /|g22 d| = 0.8405δ z , then (3.32) can be transformed from δy , δz to δ y , δ z specified for y( jω) and z( jω): 1.2δ y + δ z ≥ 0.28

(3.39)

Thus (3.33) is an “IFF” condition for the specification |y( jω)| ≤ δ y and |z( jω)| ≤ δ z . For example, if z( jω) is expected to be reduced by 20 dB, or δ z = 0.1, then (3.33) leads to δ y ≥ 0.15. Thus this will dictate: (1) (2)

the maximum reduction in y( jω) is 16.5 dB; and any specification requiring y( jω) reduced by more than 16.5 dB is infeasible. Before moving to sensing constraint, the case for reducing z( jω) by 20 dB and y( jω) by 12 dB is plotted in Fig. 3.8. It is seen that the intersection area is significantly reduced with the shaded area indicating the feasible solutions under the (12 dB, 20 dB) specification.

Now further consider actuation constraints, and this is enforced by a third circle located at origin with radius γ . Proposition 3.9 asserts that the necessary and sufficient condition is to have common intersection for the three circles. Refer to Fig. 3.9, it is

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3 Active Control with Constraints and Uncertainty

seen that the minimum radius is to have intersection with the scaled β-circle (point B), which is dictated from Proposition 3.12: 1.31δz + γ ≥ 1.31

(3.40)

 Following the δy , δz specification above, it is obtained: γ ≥ 1.2

(3.41)

γ = γ |g12 d/g11 | ≥ 1.73

(3.42)

Or:

Thus this will dictate: (3) (4)

the minimum magnitude required to satisfy both sensing and actuation constraints as specified must be greater than 1.73; and any specification with |u( jω)| < 1.73 will not achieve the required vibration reductions in y( jω) and z( jω) without saturation (or the specification is NOT achievable).

Finally in the above specification δ y +γ = 0.33+γ ≥ 1.53 > 1, condition (3.20) is satisfied and the correctness of Proposition 3.11 is obvious (point A in Fig. 3.9). Indeed, the (mild) sufficient condition as suggested in Proposition 3.10 is desirable, particularly with the consideration of robustness below. For this choice, it can be obtained (point C in Fig. 3.9): α( jω) = −1.218 − 0.228 j

(3.43)

Now the radius for actuation constraint is: γ = |α( jω)| = 1.24, or γ = 1.49

(3.44)

That is the specification |u( jω)| ≥ 1.49 will be feasible. Now further constraint from component controller requirement is enforced. From Proposition 3.13, it is seen that to satisfy the component constraint |k( jω)| ≤ λ, it must have:    1  1   (3.45)  α( jω) + 1 ≥ λ|g ( jω)| 11 Or:

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101

   1  1.6  ≥ + 1  α( jω)  λ

(3.46)

λ ≥ 6.3

(3.47)

Substituting (3.43) leads to:

Thus this will dictate: (5) (6)

the minimum magnitude required for the controller to satisfy both sensing and actuation constraints as specified must be greater than 6.3; and any specification with |k( jω)| < 6.3 will not achieve the required vibration reductions in y( jω) and z( jω) without violation of the expected specification (or the specification is infeasible).

Indeed, as Proposition 3.14 states that attenuation in y( jω) under sensing constraint and component constraint as specified by δ y and λ respectively is feasible only if condition (3.47) is satisfied, that is: δ y λ = 0.33λ ≥ 1, or: λ ≥ 3.03

(3.48)

This is compatible with condition (3.46) requiring λ ≥ 6.3 for satisfying the sensing, actuation, component constraints simultaneously, verifying the correctness of Proposition 3.14.

3.5.3 Robust Active Control with Constraints Finally robust design is considered and from Sect. 3.4, it has concluded that robust design for αopt ( jω) can be obtained by choosing the centre of the intersection where all sorts of constraints are satisfied. It is also commented in Remarks 3.8 and 3.9 that the value of αopt ( jω) can be obtained by plotting the respective circles defined by constraints, thus avoiding the complicated trigonometry and complex variable operations. The statement is indeed confirmed by the optimal selection given in (3.43) and its correctness can be observed from Fig. 3.9. Therefore obtaining robust optimal solutions is indeed easy and convenient from graphical inspections. Before finally subjecting the robust design for real time validation, the results are collected below. That is, with the optimal choice αopt ( jω) given in (3.43), it is expected that: (1) (2) (3)

Under sensing constraints, the specification for reducing z( jω) by 20 dB and y( jω) by 12 dB are achieved; Under actuation constraint, control signal will not violate the specification |u( jω)| ≥ 1.49; Under controller gain constraint, the specification |k( jω)| ≥ 6.3 will be feasible;

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3 Active Control with Constraints and Uncertainty

(4)

System performance under the above specifications is robust against uncertainties.

A real-time simulation is carried out and the results are shown in Fig. 3.10. It is seen that 11 dB reduction in y( jω) and 21 dB reduction in z( jω) are observed, where the slight deviation is due to the implementation of the transfer function representation of the optimal controller. Figure 3.10c shows that the control signal |u( jω)| = 1.51 is right on the verge of |u( jω)| ≥ 1.49; meanwhile |k( jω)| can be calculated from the magnitude of the control signal in Fig. 3.10c, divided by that of controller input signal from Fig. 3.10d, and this gives |k( jω)| = 6.9, which is also within the specification |k( jω)| ≥ 6.3. Figure 3.10c, d thus provide striking confirmation on the corresponding theoretical predictions. Finally uncertainties associated with gi j ( jω) are considered. For either structured or unstructured uncertainties, uncertainty information can be defined for g( jω) which will influence the location of β-circle through its centre (−g) and radius (|g|). To replicate the uncertainty propagation, a series of harmonic signals from [0.08, 0.12] Hz centered around the resonance frequency 0.1 Hz are injected, and the magnitudes are all of 0.1 representing a 10% exogenous disturbance. Thus, from the exposition in the above subsection, the proposed controller should be able to tolerate the y(jw) Performance

z(jw) Performance

0.8

1.5 Open-loop Performance Closed-loop Performance

0.6

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(c)

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Fig. 3.10 Performance of proposed active control with constraints

250

3.5 Active Control with Constraints and Uncertainty: Numerical Validations y(jw) Performance

103

z(jw) Performance

1

2 Open-loop Performance Closed-loop Performance

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(c)

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250

Fig. 3.11 Performance of proposed active control with constraints and uncertainties

uncertainty. That is, the above specifications should still be satisfied and the loss of performance should be marginal. The real-time simulation results are shown in Fig. 3.11. It is seen that 10 dB reduction in y( jω) is achieved while 21 dB reduction in z( jω) is retained. The magnitude of the control signal varies but remain |u( jω)| < 2.1; and a calculation with maximum magnitude of control signal over controller input signal gives |k( jω)| = 6.8. Both are not far away from that of undisturbed situation in Fig. 3.10. It is thus concluded that the controller is indeed robust against disturbances henceforth uncertainties.

3.5.4 Summary for Robust Active Control with Constraints This paper has brought together various results concerning solution existence, optimal controllers, constraints, and robust design issues into a unified framework. This has been a decade-long effort for attacking harmonic vibration control problems using graphical representation for solutions. Extensive numerical simulations have validated the proposed theoretical results. With the development of piezoelectric and magnetorheological self-sensing actuation devices for targeting active control (see

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3 Active Control with Constraints and Uncertainty

[29, 30] for example), it is believed that the proposed method provides a generic yet powerful design framework that can be applied to diverse area of engineering applications.

3.6 Optimal Design of Tuned Mass Dampers: A Virtualization Technique The above sections have delineated the active control with constraints as well as the robust design issues. This has been formulated in frequency domain and a link to physical parameters is established through numerical example in Sect. 3.5. To further demonstrate the power of the proposed method, the optimal design of tuned mass dampers is utilized as the application background. That is, once combining the method with physical parameters (either damping design or stiffness design), the damping or stiffness coefficients must be real and henceforth realizable constraints are enforced. Indeed, conventional methods for design of tuned mass dampers often employ optimization-based methods. These design methodologies are powerful in that optimal parameters can be synthesized by formulating optimization problems where well-established optimization algorithms exist for obtaining the solutions. Although robustness and parameter constraints issues can be tackled, important physical insight is often lost in the complicated optimization processes. This section utilizes the proposed methodology and coins a “virtualization technique” with graphical representations other than extensive optimization computation for parameter designs. Consequently, physical insight is retained during the design process. Both parametric constraints and uncertainties are considered leading to direct graphical interpretation within the proposed framework. Numerical examples are given to illustrate the design process while real time simulations are provided to validate the resulting designs.

3.6.1 Introduction to Tuned Mass Dampers Harmonic vibration is usually isolated through mounts with appropriately designed damping and stiffness parameters. These designs often employ heavily damped vibration isolators that are often demanding in size and weight [31–33]. To alleviate the situation, an additional mass-spring-damper system is added. The correctly tuned vibration absorber can provide much improved isolation property from harsh harmonic vibrations [34, 35]. This leads to a two-degree-of-freedom (2-DOF) isolation system named as Tuned Mass Damper (TMD). TMD is essentially acting as a secondary vibrating system that dissipates vibration energy of main structure. The efficiency of TMD thus depends on its parameters such as mass, stiffness and damping ratio.

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Due to its widespread applications in buildings, automobiles and power transmissions, many methods have been proposed for optimal selection of the TMD parameters. For example, in the classical formulation of Den Hartog, the target criterion is to adjust the invariant points to have equal height [36]. Modern development of optimal design often exploits numerical solutions through formulating nonlinear constraint optimization problems. Consequently, optimal parameters can be obtained by minimizing the performance functions [37–39]. For example, a Chebyshev’s min–max criterion can be used for optimal design of a damped dynamic vibration absorber [40]. This often leads to an iterative numerical search since the minimax amplitude is performed with the maximum amplitudes of response from the minimum values for fixed parameter selection [41, 42]. In fact, analytic solutions for optimal parameters can be obtained, should the H∞ and H2 performance indices be adopted, e.g. see [43] and references therein for a summary of results. As another example, hybrid optimization method can be utilized even with consideration of structure layout and components locations taken into account [44, 45]. However, it must be noticed that these optimizations generally result in a Pareto optimal set where infinite numbers of combination of optimal solutions exist. To reduce the number of selections, optimization can be performed with respect to one parameter variable such as stiffness ratio of linear [46, 47] or nonlinear nature [48, 49]. Indeed, there are inherent drawbacks associated with some multi-objective optimization methods, e.g. the homotopy techniques [50, 51] and goal programming [52, 53] do not generalize to the case of more than two objectives defined in a multi-objective optimization problem. This simplified approach is actually justified since it is often difficult to vary the effective mass of a solid state device while stiffness or damping can be ideally realized by piezoelectric ceramic material [54], shape memory alloy [55], electrorheological or magnetorheological materials [56]. In this section, a novel method using the proposed design approach is introduced for optimal parameter design of the dampers, where optimal conditions are obtained first with a Pareto set, and then specified to stiffness and damping coefficients. In comparison with the optimization-based approaches, the proposed method retains important physical insight. This is very useful when considering robustness and constraints issues inherently associated with system implementations.

3.6.2 Problem Formulation A tuned mass damper consists of one oscillator for damping the amplitude by coupling it to a second one. It models a damper attached to a structure, thus creating a two degree of freedom (2-DOF) system shown in Fig. 3.12. Different configurations exist but a conventional one can be defined as the primary system (m 1 , c1 , and k1 ) subject to a disturbance d; a secondary system, or an absorber (m 2 , c2 , and k2 ) is then attached to the primary system. Design of TMD aims to reduce the transmission force y¨ from the primary system to the base, through the optimal design of the absorber parameters m 2 , c2 , and k2 .

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Fig. 3.12 A TMD can be modeled by a 2-DOF mass-spring-damper system (with exogenous vibration u)

The model of the tuned mass damper can be written as: m 1 x¨1 + (c1 + c2 )x˙1 − c2 x˙2 + (k1 + k2 )x1 − k2 x2 = d m 2 x¨2 − c2 x˙1 + c2 x˙2 − k2 x1 + k2 x2 = 0

(3.49)

As the transmission force F = c1 x˙1 + k1 x1 , the frequency response function between the disturbance and the base can be obtained by taking Fourier transform to Eq. (3.39) and substitute the transmission force resulting in: F( jω) k1 + jc1 ω =   D( jω) −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω −m 2 ω2 + k2 + jc2 ω − (k2 + jc2 ω)2

(3.50)

where F( jω) and D( jω) are the Fourier transforms of F(t) and d(t), respectively. Therefore, should the transmission force be attenuated with respect to a harmonic disturbance, the magnitude of Eq. (3.40) must be optimized among the selection of the absorber parameters m 2 , c2 , and k2 . This problem can be solved by minimizing the magnitude of Eq. (3.40). However, the resulting optimal conditions can be difficult for investigating the following important issue: if there is uncertainty associated with damping or stiffness, or if the damping or stiffness is physically constrained, how the performance of the damper will be influenced? In the following section, a virtualization method is introduced where direct visualization of performance shift over both uncertainty and constraints can be achieved. This is turned to presently.

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3.6.3 Virtualization Technique for Optimal Solution The idea of virtualization amounts to augment a virtual force u(t) as shown in Fig. 3.12. This virtual force is further assumed to have a feedback form: U ( jω) = k( jω)X 2 ( jω)

(3.51)

Acute readers see immediately that the development will intend to utilize feedback control theory, yet for parameter optimization purpose. Consequently, the development must end with setting the virtual force back to zero, or equivalently: k( jω)X 2 ( jω) = 0

(3.52)

Equation (3.52) will be satisfied with either k( jω) = 0 or X 2 ( jω) = 0. The former simply reverts back to the conventional design approach explained in the last section; while the latter will result in a new design framework to be developed now. Before getting there, the story is preceded with writing in frequency domain representation: −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω X 1 ( jω) − (k2 + jc2 ω)X 2 ( jω) = D( jω)

 − (k2 + jc2 ω)X 1 ( jω) + −m 2 ω2 + k2 + jc2 ω X 2 ( jω) = U ( jω) (3.53)



where X 1 ( jω) and X 2 ( jω) are the Fourier transforms of x1 (t) and x2 (t), respectively. Simple manipulation leads to: ⌈

⌈ ⌉ ⌉ 1 −m 2 ω2 + k2 + jc2 ω X 1 ( jω) k2 + jc2 ω = −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω k2 + jc2 ω X 2 ( jω) Det (G) ⌈ ⌉ D( jω) (3.54) U ( jω) ⌈

⌉ −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω −(k2 + jc2 ω) where G = , and Det −m 2 ω2 + k2 + jc2 ω −(k2 + jc2 ω) takes the determinate of the transfer matrix.  2 −m 1 ω Now define g1 =  + k12+ k2 + j (c1 + c2 )ω /Det (G), g2 = −(k2 + jc2 ω)/Det (G), and g3 = −m 2 ω + k2 + jc2 ω /Det (G), then Eq. (3.54) is simply: ⌈

⌉ ⌈ ⌉⌈ ⌉ g1 ( jω) g2 ( jω) D( jω) X 1 ( jω) = X 2 ( jω) g2 ( jω) g3 ( jω) U ( jω)

(3.55)

Combining with Eq. (3.51) leads to:  X 1 ( jω) = 1 + g22 ( jω)k( jω)(g1 ( jω)(1 − g3 ( jω)k( jω))−1 g1 ( jω)D( jω)

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X 2 ( jω) = [1 − g3 ( jω)k( jω)]−1 g2 ( jω)D( jω)

(3.56)

A clear structure will be revealed by a re-definition of α( jω) ≡ [1 − g3 ( jω)k( jω)]−1 −1, and β( jω) ≡ α( jω)g22 ( jω)/(g1 ( jω)g3 ( jω)), as this gives: X 1 ( jω) = [β( jω) + 1]g1 ( jω)D( jω) X 2 ( jω) = [α( jω) + 1]g2 ( jω)D( jω)

(3.57)

Henceforth the disturbance attenuation properties of X 1 ( jω) and X 2 ( jω) with respect to the corresponding open loop systems, denoted by T1 ( jω) and T2 ( jω), can be written down: T1 ( jω) = β( jω) + 1 T2 ( jω) = α( jω) + 1

(3.58)

Thus if a certain α( jω) is chosen in such a way that |T1 ( jω)| = 0.5, then disturbance transmission at X 2 ( jω) is to be reduced by 6 dB; while a choice of α( jω) with |T1 ( jω)| = 2 will cause a 6 dB enhancement. However, as the force u(t) is virtual, whose zeroness can be guaranteed by X 2 ( jω) = 0, from Eq. (3.57), it can be set: [α( jω) + 1]g2 ( jω)D( jω) = 0

(3.59)

Therefore a choice of α( jω) ≡ −1 can fulfill the requirement. And this implies that the optimal parameters of m 2 , c2 , and k2 must be chosen as: |T1 ( jω)| < 1 for α( jω) ≡ −1

(3.60)

should the disturbance transmission at X 2 ( jω) be attenuated. Now as β( jω) ≡ α( jω)g22 ( jω)/(g1 ( jω)g3 ( jω)), hence the condition |T1 ( jω)| < 1 is defined as the “interior” of the unit circle |T1 ( jω)| = 1, bearing is actually a circle in mind that the unit circle, once mapped on α( jω) plane,  centered at −(g1 ( jω)g3 ( jω))/g22 ( jω), with a radius of (g1 ( jω)g3 ( jω))/g22 ( jω) (hence the circle always intersect the Origin). Meanwhile on the same α( jω) plane, |T2 ( jω)| = 1 is also a unit circle with centre (−1, 0) and radius unity. Thus α( jω) = −1 locates exactly at the origin of |T2 ( jω)| = 1. And the condition (3.60) is thus translated into the statement that “the point (−1,   0) must locate within the circle with centre −(g1 ( jω)g3 ( jω))/g22 ( jω) the radius (g1 ( jω)g3 ( jω))/g22 ( jω);” in practice, it is also desirable to require the radius less than unity. The two conditions will be fulfilled provided:   −(g1 ( jω)g3 ( jω))/g 2 ( jω) + 1 < 1 2

(3.61)

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Fig. 3.13 Three scenarios for the distance between the centre of |T1 ( jω)| = 1 (blue circle) and (−1, 0) as mapped on the complex α( jω)-plane: a greater than unity; b equal to unity; and c less than unity. Also note that the mapped circle always intersects the origin

The corresponding situation is shown in Fig. 3.13, also shown are the additional 2 scenarios where the distance between the centre −(g1 ( jω)g3 ( jω))/g  2 ( jω) and the point (−1, 0) either equals or is less than (g1 ( jω)g3 ( jω))/g22 ( jω) on α( jω) plane. As explained above, the value |T1 ( jω)| resulting from the choice of α( jω) indicates the disturbance transmission at X 2 ( jω) being reduced for |T1 ( jω)| < 1 while enhanced for |T1 ( jω)| > 1. Now the choice has been made for α( jω) = −1, consequently the disturbance transmission at X 2 ( jω) can be obtained directly by substituting α( jω) = −1 into the expression of |T1 ( jω)|. Graphically, this is only the scaling of |T1 ( jω)| = 1 on α( jω) plane so that the scaled circle crosses the point (−1, 0). As a consequence, disturbance transmission property at X 2 ( jω) can be “read” visually as shown in Fig. 3.14.

(a) Disturbance enhancement 2.3dB

(b) Disturbance attenuation 3.2dB

Fig. 3.14 Graphical explanation for disturbance attenuation and enhancement: the corresponding attenuation and enhancement level can be “read” directly from the scaling (read dotted circle) with respect to |T1 ( jω)| = 1 (blue solid circle)

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3 Active Control with Constraints and Uncertainty

Now should the optimal solution result in disturbance attenuation, the condition (3.61) must be met. Therefore the optimal solution for the absorber parameters m 2 , c2 , and k2 can be obtained by substituting g1 , g2 , and g3 into condition (3.61) leading to:    k + k − m ω2 + jω(c + c ) k − m ω2 + jωc  2 1 1 2 2 2 2  1  (3.62) −1 < 1  2   (k2 + jωc2 ) Now define:    k + k − m ω2 + jω(c + c ) k − m ω2 + jωc  2 1 1 2 2 2 2  1  −1 f (m 2 , c2 , k2 ) =    (k2 + jωc2 )2 (3.63) It is seen that (3.63) defines a function whose optimization with respect to the absorber parameters m 2 , c2 , and k2 determines the fundamental disturbance attenuation property of the damper. Consequently, the problem of optimal design of tuned mass dampers can be formulated as: min f (m 2 , c2 , k2 ) s.t. : m 2 > 0, c2 > 0, and k2 > 0

m 2 ,c2 ,k2

(3.64)

However, such an approach “reverts” back to that of conventional method of minimization the magnitude of Eq. (3.50). And the issue of uncertainty and constraints will be difficult to tackle. A different avenue will be explored in the following. Now re-group the parameters in (3.62) with the definition p1 = k1 −m 1 ω2 + jωc1 and p2 = k2 + jωc2 , then inequality (3.62) reads:     | p 2 |2 p1 p2 < m 2 −  ( p1 + p2 )ω2  | p1 + p2 |ω2

(3.65)

What inequality (3.65) says is that the distance from m 2 (a section of positive real 2 p1 p2 must be less than | p1|+p2p|2 |ω2 . To extract information from axis) to the point ( p1 + p2 )ω2 this formula, consider the circle interior defined by:     | p 2 |2 p1 p2  − <  ( p1 + p2 )ω2  | p1 + p2 |ω2

(3.66)

where is thought of as a variable on the complex plane. Now important results can be obtained by considering the following cases: (1) | p1 | ≤ | p2 | 2 p1 p2 | For this case, it is known | p1| + ≤ | p1|+p2p|2 |ω2 , henceforth inequality (3.66) reprep2 |ω2

p1 p2 | sents the interior of circle larger than the radius | p1| + . Considering m 2 is a section p2 |ω2 of positive real axis, therefore it is ready to derive the condition for m 2 to satisfy

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(3.66) is: 0 < m 2 < m 2max

(3.67)

where m 2 max is the intersection of the circle of radius | p1|+p2p|2 |ω2 with the real axis. It can be calculated to be:  2 2     | p 2 |2 p1 p2 p1 p2 imag(  ) + − ) m 2max = r eal(  2 2 2 | p1 + p2 |ω ( p1 + p2 )ω ( p1 + p2 )ω  (3.68) 2

(2) | p1 | > | p2 | For this case, it is known

| p1 p2 | | p1 + p2 |ω2

>

represents the interior of circle with radius | p1 p2 | | p1 + p2 |ω2 .

| p2 |2 | p1 + p2 |ω2 , henceforth inequality | p2 |2 | p1 + p2 |ω2 which is smaller than the

(3.69)

where m 2 min and m 2 max are the intersections of the circle of radius real axis. These can be written down explicitly as: 

m 2max

radius

Consequently the condition for m 2 will be: m 2min < m 2 < m 2max

m 2min

(3.66)

| p2 |2 | p1 + p2 |ω2

with the

2  p1 p2  ) ( p1 + p2 )ω2   2  2    | p 2 |2 p1 p2 p1 p2 imag(  = r eal( ) + − )  2 2 2 | p1 + p2 |ω ( p1 + p2 )ω ( p1 + p2 )ω  (3.70)

p1 p2 )− = r eal( ( p1 + p2 )ω2



| p2 |2 | p1 + p2 |ω2

2

  − imag(

However, the above calculation can only be proceeded with the fact the circle of 2 radius | p1|+p2p|2 |ω2 has intersection with the real axis. Therefore, a further constraint must be exercised to ensure the “existence of intersection.” This requirement can be expressed as m 2min ≤ m 2max leading to:     | p2 |2 p1 p2  ≥ imag( ) 2 2 | p1 + p2 |ω ( p1 + p2 )ω

(3.71)

As a final step, substitution of p1 = k1 − m 1 ω2 + jωc1 and p2 = k2 + jωc2 into the above inequalities provides a “clear-cut” answer to the optimal design of tuned mass dampers. This reveals the power of the proposed virtualization technique. Remark 3.10 In both cases, it must be noted that it is not necessary to take the p1 p2 ) in (3.68) and (3.70) since the non-positive of its absolute value of r eal( ( p1 + p2 )ω2

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Fig. 3.15 Graphical illustration for the scenario of | p1 | ≤ | p2 | where the maximum value of m 2 max is shown. The feasible choice for m 2 is represented with the read solid segment. Also shown is the critical radius that needs to be satisfied for m 2 min ≤ m 2 max for the scenario of | p1 | > | p2 | as represented by the green dash-dotted circle and line

value will result in no solution for optimal mass m 2 . That is, c2 and k2 must be 2 1 ω + jωc1 )(k2 + jωc2 ) is positive. This constraint by the requirement that real part of (kk11−m +k2 −m 1 ω2 + jω(c1 +c2 ) is indeed a fundamental constraint that would not be revealed by any other methods. A graphical illustration can be referred to Fig. 3.15.

3.6.4 Optimal Design with Physical Constraints and Uncertainty As claimed above, the proposed method can retain important physical insight particularly when dealing with robustness and constraints issues inherently associated with system implementations. In this section, we will bring the implication of the results up in Sect. 3.3 by jointing considering the various constraints and robustness enforced by physical parameters. Mass Constraints and Uncertainty A can be assumed where the mass m 2 is constrained in a sector m 2 ∈  scenario m 21, m 22 . From the development in the last section, it is known that: if | p1 | ≤ | p2 |, then the feasible condition is (3.67) 0 < m 2 < m 2max ; and if | p1 | > | p2 |, then the feasible condition is (3.69) m 2min < m 2 < m 2max .  Thus for both cases, should the mass m 2 be constrained m 2 ∈ m 21,m 22 , the constrained optimization problem can be resolved by the intersection of m 21, m 22 with m 2min < m 2 < m 2max (for | p1 | > | p2 |) or 0 < m 2 < m 2max (for | p1 | > | p2 |). If there is no intersection, it can be deduced that no optimal solution exists that would reduce the disturbance transmission at X 2 . (1) (2)

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The corresponding robustness can also be resolved in this setup: for either unconstrained or constrained case, a robust choice will be the mid-point value for achievable optimal solutions, e.g. for the constrained case, if the intersection between m 2min < m 2 < m 2max and m 21, m 22 leads to the feasible set m 2min < m 2 < m 22 , then a robust choice will be: m2 =

m 2min + m 22 2

(3.72)

This solves the difficult problem of constrained optimization against unstructured uncertainty. Stiffness and Damping Constraints with Uncertainty Now assume mass m 2 is known, yet it does not seem to be obvious to address the issue of stiffness and damping selection. This situation is indeed more involved but the story can nevertheless be developed by partitioning the space | p2 | < | p1 | and | p2 | ≥ | p1 |, thus lumping the absorber parameters c2 and k2 with the help of the definition p2 = k2 + jωc2 . (1) | p2 | ≥ | p1 | In this case, referring back to Fig. 3.14, as it is known that the mass m 2 should satisfy (3.67) 0 < m 2 < m 2max , but now it is m 2 that is known while m 2max is a variable (determined by to-be-selected c2 and k2 ). Henceforth to ensure the existence of m 2 , it is required from (3.68) that c2 and k2 should satisfy:  p1 p2 )+ m 2 ≤ r eal( ( p1 + p2 )ω2



| p 2 |2 | p1 + p2 |ω2

2

  − imag(

2  p1 p2 ) 2 ( p1 + p2 )ω (3.73)

 if the absorber parameters c2 and k2 are constrained, e.g. c2 ∈  Consequently c21, c22 , k2 ∈ k21, k22 , the existence of optimal solutions is determined by the simultaneous solution:  ⎧ c2 ∈ c21, c22 ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ k ∈ k21, k22 ⎪ ⎨ 2  2 2     | p2 |2 p p p1 p2 1 2 ⎪  ⎪ m 2 ≤ r eal( )+ − imag( ) ⎪ 2 2 2 ⎪ | p1 + p2 |ω ( p1 + p2 )ω ( p1 + p2 )ω ⎪ ⎪ ⎪   ⎩ 2   |k2 + jωc2 | ≥ k1 − m 1 ω + jωc1 (3.74) The last inequality is the condition | p2 | ≥ | p1 |. To deal with the robustness problem, an idea can be “borrowed” about mass robustness yet in a reversed manner: choose m 2max = 2m 2 , or a neighborhood m 2max = 2m 2 ± δ (where δ 0, ∀ω. Meanwhile, it can also p2 )ω2 be calculated that | p1 | > | p2 |, ∀ω, henceforth, from (3.69) and (3.70), it can be determined that the mass ratio should satisfy: m 2min < m 2 < m 2max

(3.69)

where √ 2ω4 − 4ω2 + 6− 16ω10 − 12ω8 + 57ω6 − 17ω4 + 51ω2 + 9 2ω2 (4ω4 − 3ω2 + 9) (3.78) √ 2ω4 − 4ω2 + 6 + 16ω10 − 12ω8 + 57ω6 − 17ω4 + 51ω2 + 9 = 2ω2 (4ω4 − 3ω2 + 9)

m 2min = m 2max

The mass m 2min and m 2max can be plotted against the frequency ratio as shown in Fig. 3.16. It is noted that the requirement m 2min ≤ m 2max from (3.70) requires frequency ratio ω < 0.7425. This is due to the specific parameter selections in this example, and in practice the frequency ratio should be centered on unity. Bearing this in mind, the figure thus shows the feasible mass set without constraints and robustness considerations.

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3 Active Control with Constraints and Uncertainty Frequency Ratio vs Maximum/Minimum Mass Ratio m2min 100

m2max

Maximum/Minimum Mass Ratio

80

60

40

20

0 0.1

0.2

0.3 0.4 0.5 Frequency Ratio w

0.6

0.7

Fig. 3.16 An illustration of the maximum value of m 2 with respect to frequency ratio

 Now suppose that the massm 2 is constrained m 2 ∈ m 21, m 22 , then it is inferred  immediately that the set m 2 | m 21, m 22 ∩ m 2 min, m 2max must not be null to have feasible solutions. Further suppose that above set is not null, and the intersection leads to a feasible set m 2min < m 2 < m 22 , then a robust choice can be m 2 = m 2min2+m 22 for solution to the robust design of constrained optimization against unstructured uncertainty. Stiffness and Damping Ratios Design, Constraints, and Robustness Now assume the mass ratio is unity, and the objective is to determine the optimal stiffness and damping ratios. Here we have p1 = 1 − ω2 + jω, p2 = k2 + jωc2 . To determine c2 and k2 will require considering the joint sets of the following two cases. (1) | p2 | ≥ | p1 | For the unconstrained case, it is known that the parameters c2 and k2 should satisfy (3.74) leading to:

2 2 2 2 c2 + k22 − ω4 − (c2 + 2k2 + 1)ω2 + k2 − ω3 (1 + 2c2 ) − (c2 + k2 )ω ≥ 0 (3.79)

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While | p2 | ≥ | p1 | leads to:

 c22 + k22 − 1−ω2 + ω4 ≥ 0

(3.80)

Thus optimal selection for unconstraint c2 and k2 can be obtained by solving (3.79) and (3.80) simultaneously. The situation for unity frequency ratio is plotted in Fig. 3.17. Obviously, combinations of c2 and k2 that result in negative values for the two planes are infeasible, which implies that the complementary sets are the feasible choices of c2 and k2 . Consequently, if stiffness they have to be constrained  and damping are  constrained, in such a way that c2 ∈ c21, c22 and k2 ∈ k21, k22 have intersection with the above unconstrained feasible sets. For robustness consideration, it is referred to (3.75), and it is noted that solution of (3.75) can be thought of increasing the “positiveness” of inequality (3.79). This is equivalent to parallel lifting the plane in Fig. 3.5 towards negative z-axis, thus further reducing the feasible choice of c2 and k2 . This can be a simple guidance for “robusification” design. (2) | p2 | < | p1 | Feasible Stiffness & Damping Ratios

10

Inequality (32)

5

0 Inequality (33)

-5 0 0.5 1 c2

1.5 2

0

1

0.5

1.5

2

k2

Fig. 3.17 Illustration of the parameter selection for stiffness and damping ratios for unity frequency ratio for | p2 | ≥ | p1 |

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3 Active Control with Constraints and Uncertainty

For this case, the parameters c2 and k2 should satisfy (3.76). It is seen that with p1 p2 ), then the third inequality for lower bound is satisfied while the m 2 ≥ r eal( ( p1 + p2 )ω2 fourth inequality is the one in (3.79). Consequently the solution for unconstrained case can be obtained by intersection of (3.79) with the following: ⎧ p1 p2 ⎪ r eal( )≤1 ⎪ ⎪ ⎪ ( p1 + p2 )ω2 ⎪ ⎨     | p2 |2 p1 p2  ≥ imag( ) ⎪ 2 2 ⎪ | p1 + p2 |ω ( p1 + p2 )ω ⎪ ⎪ ⎪   ⎩ 2  |k2 + jωc2 | < 1 − ω + jω

(3.81)

Calculation of the above inequality at unity frequency ratio leads to: c22 + 2c2 + 1 + k22 − k2 ≥ 0 2   3 2  2 k2 + (c22 + 2c2 )k + (c2 − 1)k22 + c23 + c22 − k22 + c22 + c2 ≥ 0 1 − c22 − k22 > 0

(3.82a) (3.82b) (3.82c)

The situation for unity frequency ratio is plotted in Fig. 3.18. From simplicity of illustration, inequality (3.79) is not shown here but it is clear that the combinations of c2 and k2 that result from the intersection of inequality (3.79) with those of Fig. 3.18 provide the feasible set. Again, constraints and robustness design can be handled similarly with the case of | p2 | ≥ | p1 |. Finally, as explained, the final feasible sets consist of the union of the results from the two cases of | p2 | ≥ | p1 | and | p2 | < | p1 |. As a final validation, a real time simulation is carried out for testing the performance of the tuned mass damper. Two experiments are conducted for validating (1) mass ratio design, and (2) damping and stiffness ratios design. For (1), refer to Fig. 3.16 and choose frequency ratio ω = 0.3 for example; then (3.78) gives the range of 1.26 < m 2 < 5.92. For robustness consideration, take the middle value of the mass m 2 = 3.59. The performance of the tuned mass damper is shown in Fig. 3.19a, also shown is the performance with a 20% amplitude corruption and a random frequency uncertainty centered on the intended frequency ratio. For (2), from either Figs. 3.17 and 3.18 or the corresponding inequalities, it can be verified that a simple choice of c2 = k2 = 1 is feasible (notice that c2 and k2 selections come from the union sets of | p2 | ≥ | p1 | and | p2 | < | p1 |, not from intersection set as mass ratio design). The performance is shown in Fig. 3.19b, where both amplitude and frequency uncertainty are injected. For both (1) and (2), it is seen that the proposed method results in successful design of the absorber parameters.

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Feasible Stiffness & Damping Ratios 10

Inequality(35-2)

5

Inequality(35-1)

0 Inequality(35-3)

-5 0

0.5

1

1.5

0 2

0.5

1

1.5

2

k2 c2

Fig. 3.18 Illustration of the parameter selection for stiffness and damping ratios for unity frequency ratio for | p2 | < | p1 |

3.6.6 Summary for Optimal Design of Tuned Mass Dampers A virtualization technique has been introduced for optimal design of tuned mass dampers. Both parametric uncertainty and constraints issues have been handled. It has been demonstrated that the proposed method utilizes graphical representations other than relying on extensive computation, should optimization-based methodologies be adopted. Numerical examples have been provided for illustrating the proposed method while real time simulations have been given presenting solid confirmation of the performance. The calculation process can be accused of being involved yet it is straightforward. More importantly, the feasible sets can be visualized and important physical insight can be retained during the design process, and this will become increasingly useful when dealing with constraint and robustness issues, particularly for structured parametric uncertainties. Finally, it is also pointed out that another beneficial advantage of the proposed method is that it will lead to simultaneous attenuation of disturbance transmission to both primary system and absorber system. This feature will become

120

3 Active Control with Constraints and Uncertainty With and Without Absorber System 1.5 Without Absorber Response With Absorber Response Absorber with Disturbance

1

Magnitude

0.5

0

-0.5

-1

-1.5

0

50

100

150

200

250 300 Time [t]

350

400

450

500

(a) With and Without Absorber System 1.5 Without Absorber Response With Absorber Response Absorber with Disturbance

1

Magnitude

0.5

0

-0.5

-1

-1.5

0

50

100

150

200

250

Time [t]

(b) Fig. 3.19 Simulation of performance of the designed damper for a mass ratio tuning and b stiffness and damping ratios tuning

3.7 Frequency Dependence Performance Limit of Vibration Absorbers

121

very important for cases where both payload and transmission force isolations are required.

3.7 Frequency Dependence Performance Limit of Vibration Absorbers It is seen from the above section that optimal design of vibration absorbers results in a set of optimal parameters that are independent of exogenous forcing frequencies. In practical designs, however, it is often desirable to know the performance limits over a frequency band of interest. This fundamental problem is tackled in this section where both lower and upper bounds are obtained. A refined upper bound is also derived that can further provide a systematic design methodology. Extensive remarks are also given exploring different avenues useful for design. Numerical examples are given to validate the corresponding designs.

3.7.1 Problem Formulation In review of the available results in the literature, it is seen that the optimization of performance indices leads to the results that only optimize the pre-designated indices, yet performance limits, particularly the frequency-dependence performance bounding information cannot be obtained. Indeed, the optimization is usually taken over all frequencies, e.g. a mean square power spectral optimization is defined as +∞ I = −∞ | |2 dω. As a consequence, it is not known how the obtained results can be optimal for other frequencies or frequency bands. This necessitates the investigation of frequency-dependence performance bounds. These bounds will provide important information upon what is the best achievable performance to be expected. Thus it can also provide guidance on how to choose optimization indices, while making better understanding of the resulting optimal results, which can often be obscured in the complicated optimization processes. This frequency-dependence performance limit of vibration absorbers is considered. To proceed, a specific configuration is assumed to illustrate the key ideas and provide specific and clear-cut results, refer to the figure in the last section, also reproduced here in Fig. 3.20. The primary system with m 1 , c1 , and k1 is subject to a disturbance d; a secondary system named as an absorber with m 2 , c2 , and k2 is attached to the primary system. The objective of TMD design is to reduce the transmission force from the primary system to the base, through the optimal selection for the absorber parameters m 2 , c2 , and k2 .

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3 Active Control with Constraints and Uncertainty

Fig. 3.20 A TMD can be modeled by a 2-DOF mass-spring-damper system

The model can be represented as: m 1 x¨1 + (c1 + c2 )x˙1 − c2 x˙2 + (k1 + k2 )x1 − k2 x2 = d m 2 x¨2 − c2 x˙1 + c2 x˙2 − k2 x1 + k2 x2 = 0 F = c1 x˙1 + k1 x1

(3.83)

where the dependence on time has been omitted for easy reference. The frequency response property for the transmission force can then be written down:   (k1 + jc1 ω) −m 2 ω2 + k2 + jc2 ω F( jω) =   D( jω) −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω −m 2 ω2 + k2 + jc2 ω − (k2 + jc2 ω)2

(3.84)

where F( jω) and D( jω) are the Fourier transforms of F(t) and d(t), respectively. The aim of the TMD design is to design the absorber parameters m 2 , c2 , and k2 in such a way, so that certain appropriate performance indices such as energy/magnitude defined by H2 /H∞ norms are optimized. Although the optimization “routines” exist for obtaining feasible solutions to the corresponding optimization problems, it is still of great significance and interest to seek the boundary or limit of performance. As the limits will dictate the achievable performance, on the one hand, they are not to be conquered by any form of performance index to be optimized; on the other hand, they will actually provide guidance to the designers if a choice of performance index is suitable by knowing its “distance” to the limits. Henceforth, performance limits should be treated as a benchmarking that any design through any optimization method with any performance index must compare with. These issues are treated in the following sections.

3.7.2 Performance Limit: Lower Bound From Eq. (3.83), it is known that the following frequency responses relationship holds:

3.7 Frequency Dependence Performance Limit of Vibration Absorbers

123

F( jω) = (k1 + jωc1 )X 1 ( jω)

(3.85)

For any particular configuration, the primary system parameters c1 and k1 are known, hence optimizing F( jω)/D( jω) is equivalent to optimizing X 1 ( jω)/D( jω). From Eq. (3.84), it is known:   −m 2 ω2 + k2 + jc2 ω X 1 ( jω) =   D( jω) −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω −m 2 ω2 + k2 + jc2 ω − (k2 + jc2 ω)2

(3.86)

Thus the objective of TMD design can be re-stated to reduce the transmission magnitude |X 1 ( jω)/D( jω)| through the optimal selection of the absorber parameters m 2 , c2 , and k2 . In the following, the performance bounds for attenuation of the magnitude of X 1 ( jω)/D( jω) through the to-be-designed parameters of m 2 , c2 , and k2 will be sought. This is preceded by boldly stating the results and associated corollaries before a proof. Theorem 3.1 The performance of |X 1 ( jω)/D( jω)| is bounded from below by:    X 1 ( jω)     D( jω)  > 

1 k1 − m 1

ω2

2

+

c12 ω2

 +ω

m 2 (k22 +c22 ω2 ) 2

, ∀ω

(3.87)

The relationship is strictly “greater than” implying that the lower bound is absolute and never to be attained. Proof Define f (m 2 , c2 , k2 ) = 1/|X 1 ( jω)/D( jω)|, then from (3.86):      −m ω2 + k + k + j (c + c )ω −m ω2 + k + jc ω − (k + jc ω)2    1 1 2 1 2 2 2 2 2 2  

 f (m 2 , c2 , k2 ) =   −m 2 ω2 + k2 + jc2 ω  

(3.88)

Further define p1 = k1 + jc1 ω and p2 = k2 + jc2 ω, then the above equation can be written as:    p + p − m ω2  p − m ω2  − p 2  2 1 2 2  1 2 f (m 2 , c2 , k2 ) =  (3.89)  2   p2 − m 2 ω Manipulation leads to:    p2 m 2 ω2  f (m 2 , c2 , k2 ) =  p1 − m 1 ω2 − p2 − m 2 ω 2 

(3.90)

As the terms in the operator |•| are complex numbers forming a complete set, hence we have:   | p2 |m 2 ω2  f (m 2 , c2 , k2 ) ≤  p1 − m 1 ω2  +   p2 − m 2 ω 2 

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3 Active Control with Constraints and Uncertainty

  | p2 |m 2 ω2 =  p1 − m 1 ω 2  +  k22 + m 22 ω4 + c22 ω2 − 2k2 m 2 ω2

(3.91)

where p2 = k2 + jc2 ω has been substituted in the equality part. Now as m 2 and k2 are both positive, then (3.91) can be further manipulated leading to:   | p2 |m 2 ω2 f (m 2 , c2 , k2 ) <  p1 − m 1 ω2  +  | p2 |2 + m 22 ω4

(3.92)

Note that the relationship is strictly less than without attaining equality. Further notice that | p2 |2 + m 22 ω4 ≥ 2| p2 |m 2 ω2 , then (3.92) becomes:    | p2 |m 2 2 f (m 2 , c2 , k2 ) <  p1 − m 1 ω  + ω 2

(3.93)

Henceforth:    X 1 ( jω)  1      D( jω)  >   p1 − m 1 ω2  + ω | p2 |m 2 2

(3.94)

Substitutions of p1 and p2 and taking the absolute operation will recover the required result.  From the above result, a series of observations follow: (1)

(2)

For a frequency ω, the lower bound of |X 1 ( jω)/D( jω)| is inversely proportional to m 2 , c2 , and k2 . That is, the larger they are, the smaller the lower bound is, implying that the maximum achievable attenuation can be obtained. However, the to-be-designed parameters of m 2 , c2 , and k2 are physically constrained by m 2 ≤ M, c2 ≤ C, and k2 ≤ K , henceforth the performance will be limited by:    X 1 ( jω)     D( jω)  > 

1 k1 − m 1

ω2

2

+

c12 ω2

 +ω

M ( K 2 +C 2 ω2 ) 2

(3.95)

This means that it is not possible to achieve arbitrary level of attenuation. Indeed, for a specific primary system to be designed, the maximum attenuation is very limited. To see this, assume m 1 , c1 , and k1 are unities; suppose m 2 , c2 , and k2 are physically limited by M = 2m 1 , C = 4c1 and K = 4k1 , respectively. Then the bound-dependence on frequency is shown in Fig. √ 3.21a. It is seen that at the natural frequency of the primary system ω1 = k1 /m 1 , the bound is 16.4 dB; while at half the natural frequency, the bound quickly increases to 9.94 dB. This dictates that X 1 cannot be attenuated more than 16.4

3.7 Frequency Dependence Performance Limit of Vibration Absorbers Frequency Dependence of Lower Bound

125

Mass Dependence of Lower Bound

0

0 -2

-10 -4 -6

Magnitude [dB]

Magnitude [dB]

-20

-30

-40

-8 -10 -12 -14

-50 -16 -60

0

1

3

2

5 6 4 Frequency [rad/s]

7

9

8

-18

10

0

0.2

1 1.2 1.4 0.8 0.6 Maximum Mass up to 2m1

0.4

(a)

1.6

1.8

2

(b) Natural Frequency Dependence

Stiffness/Damping Dependence of Lower Bound -13.5

30

20

-14

10 Magnitude [dB]

Magnitude [dB]

-14.5

-15

0

-10

-15.5 -20 -16

-16.5

-30

0

0.5

1

3 2.5 2 1.5 Stiffness/Damping up to 4k1/4c1

3.5

(c)

4

-40

0

0.5

1

2.5 2 1.5 Natural Frequency

3

3.5

4

(d)

Fig. 3.21 a Frequency-dependence of the maximum achievable performance; b mass-dependence of the maximum achievable performance; c stiffness/damping-dependence of the maximum achievable performance; d natural frequency-dependence

(3)

and 9.94 dB at corresponding driving frequencies for any choice of m 2 , c2 , and k2 —the performance limit is fundamental and it is not conquered through any design. This is “frustrating” but it does provide guidance on achievable performance to be reasonably expected. √ At the natural frequency ω1 = k1 /m 1 , the above bound becomes:    X 1 ( jω)     D( jω)  >

1    M ( K 2 +C 2 ω12 ) ω1 c1 + 2

(3.96)

Thus at the harmonic disturbance with driving frequency ω1 = 1, the dependence of the lower bound for M, C/K can be plotted as shown in Fig. 3.21b

126

(4)

3 Active Control with Constraints and Uncertainty

and c, respectively. The corresponding plots can thus be used for evaluation of strategies for mass, stiffness and damping selections for the absorber parameters. From (3.96), it is seen that the lower bound is also inversely proportional to the natural frequency ω1 —to lower the bound, ω1 needs to be increased! A plot of ω1 dependence √ is shown in Fig. 3.21d. A particularly noticeable phenomena is that below k1 /m 1 = 0.197, it becomes never possible to attenuate X 1 and henceforth must be avoided.

3.7.3 Performance Limit: Upper Bound In the above section, it has been demonstrated that the performance is bounded from below. The lower bound provides extremely important information upon achievable performance, as well as rendering itself as a useful guidance to optimal design of absorber parameters. However, in this section, it will be shown that performance is also bounded from above. The existence of an upper bound possesses a sense of “things will not get too bad” philosophy since this implies that the vibration will only be enhanced by a limited extent. It is understandable from an energy conservation perspective, yet the upper bounding property has not been considered in the literature indeed. The result can be summarized below. Theorem 3.2 The performance of |X 1 ( jω)/D( jω)| is bounded from above by:   √ 2  X 1 ( jω)  K + M 2 ω4 + C 2 ω2   , ∀ω  D( jω)  < h(ω)

(3.97)

where h(ω) is a positive function dependent on frequency. Proof Define:  g(m 2 , c2 , k2 ) = −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω

 −m 2 ω2 + k2 + jc2 ω − (k2 + jc2 ω)2

(3.98)

Then the following inequality will hold for any frequency: |g(m 2 , c2 , k2 )| ≥ max( r eal(g(m 2 , c2 , k2 )), imag(g(m 2 , c2 , k2 )) ), ∀ω

(3.99)

where r eal(•) and imag(•) denote the real and imaginary parts of the operator; max(•) is another operator taking the maximum of the two functions. From (3.98), the real and imaginary parts can be calculated to be: r eal(g(m 2 , c2 , k2 )) = [m 1 k2 + m 2 (k1 + k2 ) + c1 c2 ]ω2 − m 1 m 2 ω4 − k1 k2

3.7 Frequency Dependence Performance Limit of Vibration Absorbers

imag(g(m 2 , c2 , k2 )) = [m 1 c2 + m 2 (c1 + c2 )]ω3 − (c1 k2 + k1 c2 )ω

127

(3.100)

Now from (3.4), it is known:      X 1 ( jω)   −m 2 ω2 + k2 + jc2 ω  =     D( jω)   g(m 2 , c2 , k2 )

(3.101)

Hence if |g(m 2 , c2 , k2 )| is bounded from below by a non-zero function, then |X 1 ( jω)/D( jω)| will have an upper bound. From inequality (3.99), it can be seen that the key is to prove that r eal(g(m 2 , c2 , k2 ) and imag(g(m 2 , c2 , k2 )) are NOT identically zero. We show this through proof by contradiction—assume that they are identically zero, then (3.100) leads to: [m 1 k2 + m 2 (k1 + k2 ) + c1 c2 ]ω2 = m 1 m 2 ω4 + k1 k2 [m 1 c2 + m 2 (c1 + c2 )]ω3 = (c1 k2 + k1 c2 )ω

(3.102)

Then it can be shown that m 2 , c2 , and k2 are located on an ellipse defined by: 2  2

ω2 2  2 m − k 2 2 c12 ω2 + m 1 ω2 − k1 2 m 1 ω − k1 m 22 ω2 m2 + = + c2 − 2c1 ω2 4 4c12 (3.103) This would imply that identical zeroness could be attained. Instead, it can be shown that the “ellipse” can be explicitly parameterized by m 2 as: c2 = − 

m 22 c1 ω4

2 (m 1 + m 2 )ω2 − k1 + c12 ω2    m 1 ω2 (m 1 + m 2 )ω2 − k1 m 1 ω2 − k1 + c12 ω2 k2 =  2 (m 1 + m 2 )ω2 − k1 + c12 ω2

(3.104)

However, it must bear in mind that the parameter c2 must be positive. And the solution (3.104) is thus not physically realizable. This simply implies that r eal(g(m 2 , c2 , k2 ) and imag(g(m 2 , c2 , k2 )) can not be identically zero, and thus max( r eal(g(m 2 , c2 , k2 )), imag(g(m 2 , c2 , k2 )) ) must be bounded below, e.g.: max( r eal(g(m 2 , c2 , k2 )), imag(g(m 2 , c2 , k2 )) ) ≥ h(ω) > 0

(3.105)

Or from (3.17): |g(m 2 , c2 , k2 )| ≥ h(ω) > 0, ∀ω

(3.106)

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3 Active Control with Constraints and Uncertainty



Substituting (3.106) into (3.101) thus leads to the desired result.

Remark It is noted that the upper bounding of performance has not been explicitly stated, albeit energy conservation consideration does provide physical intuition upon its correctness.

3.7.4 Performance Limit: Refined Upper Bound The result in the last section provides a “rough” estimate of upper bound. This is physically correct yet not satisfying since it is difficult to calculate the frequencydependence bounding function h(ω). It is thus extremely desirable to seek a refinement to the above upper bound, so that a clear-cut bounding property can be provided. This turns out to be a difficult problem and new techniques are required for derivations. Nevertheless, it can be shown that: Theorem 3.3 The performance of |X 1 ( jω)/D( jω)| is bounded from above by:    X 1 ( jω)     D( jω)  ≤ c1 ω +

1 (

, ∀ω

m 22 c2 ω5

(3.107)

)

2 k2 −m 2 ω2 +c22 ω2

The equality relationship is achieved for the following condition:   m 2 ω2 k2 k2 − m 2 ω2 + c22 ω2 k1 − m 1 ω =

2 k2 − m 2 ω2 + c22 ω2 2

(3.108)

Proof With the definition p1 = k1 + jc1 ω and p2 = k2 + jc2 ω as above, the following expression holds:      p2 − m 2 ω 2   X 1 ( jω)     D( jω)  =  p − m ω2  p − m ω2  − p m ω2  1 1 2 2 2 2

(3.109)

To derive the upper bound, the inverse is sought which is reproduced here:    p2 m 2 ω2  2  f (m 2 , c2 , k2 ) =  p1 − m 1 ω − p − m ω2  2

(3.110)

2

However, unlike the situation in the above section, the function f (m 2 , c2 , k2 ) must be frequency-dependently bounded from below. At a frequency  ω, 2 p denotes the distance between two complex numbers − m ω and Eq. (3.91) 1 1

 p2 m 2 ω2 /( p2 − m 2 ω2 ) . While the former depends solely on the primary system parameters, an “interwoven” technique must be sought for estimating this relative

3.7 Frequency Dependence Performance Limit of Vibration Absorbers

129

position problem. This can be done through a re-parameterization with dimensionless parameters: μ = m 2 /m 1 , v = ω2 /ω1 , ζ1 = c1 /(2m 1 ω1 ), ζ2 = c2 /(2m 2 ω2 )

(3.111)

where μ is the mass ratio, v is the frequency ratio, and λ is the forced frequency ratio; ζ1 and ζ2 are the primary and secondary damping ratios respectively, with ω1 and ω2 being the corresponding natural frequencies defined by: ω1 =

k1 /m 1 , ω2 =

k2 /m 2

(3.112)

Essentially m 1 , c1 , k1 , m 2 , c2 , k2 and ω are re-defined by the above set of seven parameters, but such a technique will allow an estimation to be made. Substitution leads to:   p1 − m 1 ω2 = m 1 ω12 1−λ2 + j(2ξ1 λ)

 p2 m 2 ω2 /( p2 − m 2 ω2 ) = μvλ2 [v + j(2ξ2 λ)]m 1 ω12 / v 2 − λ2 + j(2ξ2 λv) (3.113)

Therefore it can be shown that:    μvλ2 v(v 2 − λ2 ) + 4vλ2 ξ22 2 f (m 2 , c2 , k2 ) = m 1 ω1 1 − λ2 −  (v 2 − λ2 )2 + 4v 2 λ2 ξ22 ⌈ ⌉  2μvλ5 ξ2  + j 2λξ1 + 2 2 (v − λ2 )2 + 4v 2 λ2 ξ2 

(3.114)

Thus utilization of the result leading to (3.105) and (3.106) in the last section leads to:   ⌈ ⌉ 2   μvλ4 ξ2  p1 − m 1 ω2 − p2 m 2 ω  ≥ 2m 1 ω2 λ ξ1 + (3.115) 1  p2 − m 2 ω 2  (v 2 − λ2 )2 + 4v 2 λ2 ξ22 The equality is achieved while: ! m 1 ω12

1−λ − 2

 " μvλ2 v(v 2 − λ2 ) + 4vλ2 ξ22 (v 2 − λ2 )2 + 4v 2 λ2 ξ22

=0

(3.116)

Now transforming (3.115) and (3.116) back to the original set of parameters: ⌈ 2m 1 ω12 λ ξ1 +

⌉ k1 ω m 22 c2 ω4 μvλ4 ξ2 ω + = c 1

 (v 2 − λ2 )2 + 4v 2 λ2 ξ22 m 1 ω12 m 22 ω22 − ω2 2 + c22 ω2 (3.117)

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3 Active Control with Constraints and Uncertainty

and: ! m 1 ω12

1−λ − 2

 " μvλ2 v(v 2 − λ2 ) + 4vλ2 ξ22

(v 2 − λ2 )2 + 4v 2 λ2 ξ22   2 2 2 2 2 m 2 ω k2 k2 − m 2 ω + c2 ω = k1 − m 1 ω −

2 k2 − m 2 ω2 + c22 ω2

(3.118)

Further manipulation of (3.34) leads to:   2   m 22 c2 ω5  p1 − m 1 ω2 − p2 m 2 ω  ≥ c1 ω + 2  p2 − m 2 ω 2  k2 − m 2 ω2 + c22 ω2

(3.119)

Consideration of (3.110), (3.90), (3.116), (3.118) and (3.119) gives the desired result.  From the above result, a series of observations follow: (1)

(2)

It is noteworthy that the upper bound is independent of m 1 and k1 . Indeed, assertions that are not dependent on parameters are particularly desirable since they indicate fundamental limits to be respected for any design. While the minimum lower bound is attained at natural frequency of the primary √ upper bound is achieved at the natural system ω1 = k1 /m 1 , the minimum √ frequency of the absorber ω2 = k2 /m 2 with:     X 1 ( jω)  c2 m2 <  , ∀c2  D( jω)  c c + m k k2 1 2 2 2

(3)

(4)

(3.120)

Indeed, (3.120) can be written in the form of dependence on natural frequency ω2 as:    X 1 ( jω)  1    , ∀c2 (3.121)  D( jω)  <  m 22 ω22 c1 + c2 ω2 Henceforth just as the lower bound, the upper bound is also inversely proportional to the natural frequency ω2 —to increase the bound, ω2 needs to be decreased! Yet one of the most important applications for upper bound is the assertion that the performance |X 1 ( jω)/D( jω)| will always be attenuated over the frequency bands where the upper bound is less than unity. This can be developed into a very useful and powerful design methodology. For example, assume c1 and m 2 are unities, then the bound in (3.107) becomes: 1 ω+

c2 ω5

(k2 −ω2 )2 +c22 ω2

(3.122)

3.8 Summary

131

Now it is demanded that the bound should be less than unity: 1 ω+

c2 ω5 (k2 −ω2 )2 +c22 ω2

≤1

(3.123)

Then a calculation for cubic unities of k2 , c2 , and ω with a grid of 10 × 10 × 10 shows that a set of solutions exist for satisfying (3.123). This is shown in Fig. 3.22a. Feasible combinations of m 2 , c2 , and ω will lead to an upper bound that is shown in Fig. 3.22b. The performance |X 1 ( jω)/D( jω)| is bounded from above by this line, implying that all the choices of k2 , c2 will attenuate the disturbance with frequency ω. Indeed, while subject to unity frequency disturbance, the performance is even better than 21 dB for k2 = 1 and c2 = 0.1.

3.7.5 Summary for Frequency-Dependent Performance Limits Frequency-dependent performance limits of tuned mass dampers have been considered. It has been demonstrated that the performance is not only bounded from below, but also bounded from above. The existence of the lower bound is very useful providing guidance upon best performance to be expected. While the upper bound can be anticipated with physical intuition, important results on a refined upper bound have also been presented. Combining with the concept of attenuation by bounding from above, it has been shown that this can lead to design methods with such features as guaranteed performance. Finally, it is also worth pointing out that the existing results on vibration absorber designs do not reflect the frequency-dependence information. Should the exogenous forcing has changed its characteristics, it would be difficult to analyze the system performance. The results presented in this paper thus disclose the performance limits while rendering a close investigation to frequency-dependent parametric designs.

3.8 Summary The previous chapter has investigated the problems of dynamic and topology designs for both feedback control and design optimizations. This chapter has considered the constrained solutions where parameters constraints, sensing and actuation constraints, even system constraints are all treated as “restriction in system norms.” There is yet another important constraint of physical realizability. That is, implementation must be realizable since it is to be implemented through physical parameters. This has also been developed with robustness considerations. The proposed methodology has further been applied to tuned mass dampers resulting a novel

132

3 Active Control with Constraints and Uncertainty

(a) 0

X1/D Magnitude [dB]

-5

-10

-15

-20 k2=1 c2=0.1 w=1rad/s -25 0

50

100

150

200 k2/c2/w

250

300

350

400

(b) Fig. 3.22 a Feasible choices of parameters k2 , c2 , and ω within a 10 × 10 × 10 cubic unity grid; b upper lower bound for performance, and also shown is the lowest bound for the denoted parameter set

References

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optimal design approach over the state of the art. The important issue of frequencydependent performance limit has also been treated, forming important contribution of the monograph.

References 1. A. Palazzolo, Vibration Theory & Applications with Finite Elements & Active Vibration Control, John Wiley & Sons, 2016. 2. C. Fuller, S. Elliott, P. Nelson, Active Control of Vibration, Academic Press, London, 1996. 3. P. Nelson, S. Elliott, Active Control of Sound, Academic Press, London, 1992. 4. S. Daley, I. Zazas, J. Hatonen, Harmonic control of a ‘smart spring’ machinery vibration isolation system, Proceedings of IMechE, Part M: Journal of Engineering for the Maritime Environment, 222: 109–119 (2008). 5. P. Ballesteros, X. Shu, C. Bohn, A discrete-time MIMO LPV controller for the rejection of nonstationary harmonically related multisine disturbances, American Control Conference 2014, pp. 4464–4469. 6. Z. Emedi, A. Karimi, Fixed-order LPV controller design for rejection of a sinusoidal disturbance with time-varying frequency, IEEE International Conference on Control Applications 2012, pp. 837–842. 7. M.C. Arranz, W. Birk, G. Nikolakopoulos, A survey on control configuration selection and new challenges in relation to wireless sensor and actuator networks, IFAC-PapersOnline, 50(1): 8810–8825 (2017). 8. S. Pequito, S. Kar, A.P. Aguiar, A framework for structured input/output and control configuration selection in large-scale systems, arXiv, 1309.5868v3, (2014). https://arxiv.org/pdf/1309. 5868 9. S. Skogestad, I. Postlethwaite, Multivariable Feedback Control: Analysis & Design, Wiley, 2005. 10. S. Khot, N. Yelve, P. Kumar, A. Purohit, D. Singh, Implementation of H-infinity controller in experimental active vibration control of a cantilever beam, ICNTE Proceedings, 1–5.https:// doi.org/10.1109/ICNTE.2017.7947963, (2017). 11. P. Shukla, D. Ghodki, N. Manjarekar, P. Singru, A study of Hinf and H2 synthesis for active vibration control, IFAC-PapersOnline, 49(1): 623–628 (2016). 12. G. Pipeleers, B. Demeulenaere, J. Swevers, Optimal Controller Design for Periodic Inputs, Springer-Verlag Berlin Heidelberg, 2009. 13. G. Battistelli, D. Selvi, A. Tesi, Hierarchical switching for active disturbance attenuation with fine controller tuning, International Journal of Adaptive Control & Signal Processing, 31: 742–760 (2017). 14. C. Du, L. Xie, Modeling and Control of Vibration in Mechanical Systems, Taylor & Francis Group, LLC, 2010. 15. S. Wu, Remote vibration control for flexible beams subject to harmonic disturbances, Journal of Dynamical Systems, Measurement & Control, 126(1): 198–201 (2004). 16. D. Yang, J. Shin, H. Lee, S. Kim, M. Kwak, Active vibration control of structure by active mass damper and multi-modal negative acceleration feedback control algorithm, Journal of Sound & Vibration, 392: 18–30 (2017). 17. E. Omidi, N. Mahmoodi, Hybrid positive feedback control for active vibration attenuation of flexible structures, IEEE/ASME Transactions on Mechatronics, 20(4): 1790–1797 (2015). 18. C. Hoffmann, H. Werner, A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations, IEEE Transactions on Control Systems Technology, 23(2): 416–433 (2015). 19. K.J. Aström, Control System Design, Lecture Notes, University of California, 2002, chapter 5, p180.

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20. S. Daley, J. Wang, A geometric approach to the design of remotely located vibration control systems, Journal of Sound & Vibration, 318(4–5): 702–714 (2008). 21. W. Sun, H. Gao, O. Kaynak, Finite frequency H∞ control for vehicle active suspension systems, IEEE Transactions on Control Systems Technology, 19(2): 416–422 (2010). 22. T. Iwasaki, S. Hara, Generalized KYP lemma: unified frequency domain inequalities with design applications, IEEE Trans. Autom. Control, 50(1): 41–59 (2005). 23. R. Canahuire, A. Serpa, Reduced order H∞ controller design for vibration control using genetic algorithms, Journal of Vibration & Control, 23(10): 1693–1707 (2017). 24. J. Wang, H. Yue, G. Dimirovski, Disturbance attenuation in linear systems revisited, International Journal of Control, Automation & Systems, 15(4): 1611–1621 (2017). 25. U. Ubaid, S. Daley, S. Pope, I. Zazas, Design of stable and broadband remote vibration controllers for systems with local nonminimum phase dynamics. IEEE Transactions on Control Systems Technology, 24(2): 654–661 (2016). 26. U. Ubaid, S. Daley, S. Pope, Design of remotely located and multi-loop vibration controllers using a sequential loop closing approach. Control Engineering Practice, 38: 1–10 (2015). 27. J. Wang, S. Daley, Broad band controller design for remote vibration using a geometric approach. Journal of Sound and Vibration, 329(19), 3888–3897 (2010). 28. S. Daley, I. Zazas, A recursive least squares based control algorithm for the suppression of tonal disturbances, Journal of Sound & Vibration, 331: 1270–1290 (2012). 29. A. Pelletier, P. Micheau, A. Berry, Harmonic active vibration control using piezoelectric self-sensing actuation with complete digital compensation, Journal of Intelligent Material, Systems & Structures, published online: November 22, 2017. https://doi.org/10.1177/104538 9X17740978 30. W. Liao, C. Lai, Harmonic analysis of a magnetorheological damper for vibration control, Smart Materials & Structures, 11: 288–296 (2002). 31. M.J. Crocker, Handbook of Noise & Vibration Control, John Wiley & Sons, 2007. 32. M.Z. Kolovski, Nonlinear Dynamics of Active and Passive Systems of Vibration Protection, Berlin: Springer-Verlag, 1999. 33. W.T. Thomson, Theory of Vibration with Applications, 4th edition, Prentice Hall, 1993. 34. A.M. Veprik, V.I. Babitsky, Vibration protection of sensitive electronic equipment from harsh harmonic vibration, Journal of Sound and Vibration, 238(1): 19–30 (2000). 35. S.G. Kelly, Fundamentals of Mechanical Vibrations, 2nd edition, McGraw-Hill, 2000. 36. J.P. Den Hartog, Mechanical Vibrations, 4th edition, McGraw-Hall, New York, 1956. 37. S.T. Park, T.T. Luu, Techniques for optimizing parameters of negative stiffness, Proceeds of IMech Part C: J. Mech. Eng. Sci., 221(5): 505–510 (2007). 38. L. Kitis, B.P. Wang, W.D. Pilkey, Vibration reduction over a frequency range, Journal of Sound and Vibration, 89(4): 559–569 (1983). 39. A. Soom, Optimal design of linear and nonlinear vibration absorbers for damped systems, Journal of Vibration & Acoustics, 105(1): 112–119 (1983). 40. E. Pennestri, An application of Chebyshev’s min-max criterion to the optimal design of a damped dynamic vibration absorber, Journal of Sound and Vibration, 217(4): 757–765 (1998). 41. M. Murudi, S. Mane, Seismic effectiveness of tuned mass damper for different ground motion parameters, Proceedings of 13th World Conference on Earthquake Engineering, Vancouver, Canada, August 1–6, (2004), no. 2325. 42. H.C. Tsai, G.C. Lin, Optimum tuned mass damper for minimizing steady state response of support-excited and damped system, Journal of Earthquake Engineering and Structural Dynamics, 22: 957–973 (1993). 43. M. Zilletti, S.J. Elliott, E. Rustighi, Optimization of dynamic vibration absorbers to minimize kinetic energy and maximize internal power dissipation, Journal of Sound & Vibration, 331: 4093–4100 (2012). 44. P. Zhou, J. Du, Z. Lü, Hybrid optimization of a vibration isolation system considering layout of structure and locations of components, Structural and Multidisciplinary Optimization, 57(1): 1–15 (2018).

References

135

45. H. Liu, W. Zhang, T. Gao, A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations. Struct Multidiscip Optim 51(6):1321– 1333 (2015). 46. Y. Shen, H. Peng, X. Li, S. Yang, Analytically optimal parameters of dynamic vibration absorber with negative stiffness, Mechanical Systems & Signal Processing, 85: 193–203 (2017). 47. J. de Espíndola, C. Bavastri, E. Lopes, On the passive control of vibrations with viscoelastic dynamic absorbers of ordinary and pendulum types, Journal of the Franklin Institute, 347(1): 102–115 (2010). 48. B. Tang, M.J. Brennan, G. Gatti, N.S. Ferguson, Experimental characterization of a nonlinear vibration absorber using free vibration, Journal of Sound and Vibration, 367(14): 159–169 (2016). 49. R. Viguié, G. Kerschen, Nonlinear vibration absorber coupled to a nonlinear primary system: a tuning methodology, Journal of Sound and Vibration, 326(3–5): 780–793 (2009). 50. J. Rakowska, R.T. Haftka, L.T. Watson, Tracing the efficient curve for multi-objective controlstructure optimization, Computing Systems in Engineering, 2(6): 461–471 (1991). 51. J.R. Rao, P. Y. Papalambros, An on-linear programming continuation strategy for one parameter design optimization problems, Proceedings of ASME Design Automation Conference, Montreal, Quebec, Canada, Sept. 17–20, 1989, pp. 77–89. 52. D.F. Jones, M. Tamiz, Goal programming in the period 1990–2000, in Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys, M. Ehrgott and X. Gandibleux (Eds.), Kluwer, pp. 129–170. 53. M.J. Scniederjans, Goal Programming Methodology and Applications, Kluwer Publishers, Boston, 1995. 54. J. Erhart, P. Pulpán, M. Pustka, Piezoelectric Ceramic Resonators, Springer, Cham, 2017, pp. 11–27. 55. J. Jani, M. Leary, A. Subic, M. Gibson, A review of shape memory alloy research, applications, and opportunities, Materials & Design, 56: 1078–1113 (2014). 56. B. Dyniewicz, J. Bajkowski, C. Bajer, Semi-active control of a sandwich beam partially filled with magnetorheological elastomer, Mechanical Systems & Signal Processing, 60–61: 695–705 (2015).

Chapter 4

Active Control with Extensions

The previous chapters have formulated the active control problem and provided optimal solutions for both unconstrained and constrained situations. Several extensions are in immediate considerations such as broad-band control. More concepts and themes can be gravitated centered on the proposed methodology. These are delineated presently.

4.1 Optimal Design for Broad-Band Control Control of structural vibration becomes increasingly difficult when the system is excited by a sustainable broad band disturbance and the force transmission can then be dominated by structural resonances [1]. This can cause significant problems associated with premature ageing, structural failure and in the case of Naval vessels, severe detection hazards. As is already clear an approach to resolve the broad band control problems can be made by utilizing modern H2 /H∞ design procedures. However, although efficient methods of computing H2 /H∞ controllers exist ([2] or [3], for example), it is not obvious for the designer how to exploit certain additional freedom for design purposes. The “handle-turning” nature of the solution means that valuable physical insight into the existence of suitable solutions is often lost. This is particularly true in the case where the disturbance is known to be in a certain (broad) frequency band and it is hoped that the performance within this frequency band can be improved with the sacrifice or even ignorance of performance deterioration outside the targeted frequency band. In this chapter, it is shown that the broad band case can be resolved in the same geometric framework. Specifically the existence of an optimal broad band controller resulting from the geometric design methodology is examined by a Nevanlinna-Pick interpolation approach. The optimal broad band controller is then shown to be provided by the solution to a series of Linear Matrix Inequalities (LMIs). The real rational representation of the optimal controller can

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Wang, Active Vibration & Noise Control: Design Towards Performance Limit, https://doi.org/10.1007/978-981-19-4116-0_4

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then be obtained through the application of a systematic procedure. It will also be noted that the broad band design methodology can synthesize an optimal controller over any arbitrary frequency band.

4.1.1 Preliminaries Based on the fundamental results at the discrete frequency ω = ω0 presented in Chaps. 2 and 3, the broad-band case can be approached on a frequency-by-frequency basis. However, it is noted that the optimal choice αopt is to be varied from one frequency to the other over the desired frequency band ω ∈ [ω1 , ω N ] and this results in an optimal trajectory on the complex α-plane. These situations are illustrated in Fig. 4.1. Depending on the control objective it then becomes possible to express αopt ( jω) as a function of the elements of the transfer function matrix (4.3). For example, the contour that provides the optimum reduction in z( jω) without increasing y( jω) can be expressed as:  αopt ( jω) = where: Fig. 4.1 Optimal solution αopt ( jω) on complex α − plane for broad band control

−g( jω) |1 − g( jω)| ≤ 1 αr + jαt |1 − g( jω)| > 1

(4.1)

4.1 Optimal Design for Broad-Band Control

139

1 − Re(g( jω) αr =  −1 (1 − Re(g( jω)))2 + Im(g( jω))2 −Im(g( jω) αi =  (1 − Re(g( jω)))2 + Im(g( jω))2

(4.2)

Such a result would appear to imply that a compensator exists that would provide simultaneous reduction (or at least no enhancement) across an arbitrary wide range of frequencies. It turns out that this is not true because closed loop stability has not been considered. The design freedom αopt ( jω) is therefore constrained by the requirement of closed loop stability.

4.1.2 Stabilizing and Strong Stabilizing Controller In fact, the following main result can be proved for a stable but non-minimum phase plant: Proposition 4.1 If αopt ( jω) is a mapping of a stable function that also interpolates the unstable zeros of g11 , then the resulting compensator will internally stabilize the closed loop system for this stable but non-minimum phase g11 . For a stable and minimum phase plant, the only constraint on the trajectory of αopt ( jω) will be simply its being a mapping of a stable function. That is: Proposition 4.2 If αopt ( jω) is a mapping of a stable function, then the resulting compensator will internally stabilize the closed loop system for a stable and minimum phase g11 . Proof the proof follows from either the closed loop stability test or Youlaparameterization. ◼ However it is well known that a stabilizing controller can itself be unstable. In practice, it is often desirable to have a stable and stabilizing controller, e.g. if the feedback loop opens due to sensor or actuator failure, an unstable controller can be problematic. The following main result solves this strong stabilization problem. Proposition 4.3 If g11 ( jω) is both stable and minimum phase and αopt ( jω) is a mapping of a stable function and, in addition, Re(αopt ) > −1 when Im(αopt ) = 0, then the resulting compensator will not only internally stabilize the closed loop system and but also be itself stable. Proof The proof follows from noting that the loop gain can be described by: L = kg11 ( jω) =

−α( jω) 1 + α( jω)

(4.3)

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On inspection of the form of (4.3) (i.e. equivalent to a closed loop system with loop gain α( jω)), it is apparent that L will be stable if α( jω) is also stable and its mapping of the Nyquist D-contour does not enclose the (−1, 0) point. If L is stable then the closed loop system will also be stable if in addition the mapping of the Nyquist D-contour of L does not enclose the (−1, 0) point on the complex plane. It is straightforward to show that both of these stability conditions can simultaneously be met by ensuring that the α( jω) contour always crosses the real axis in the complex α-plane to the right of the (−1, 0) point. If g11 is stable and minimum phase, then for a stable L, k will be stable and there will be no unstable pole-zero cancellations ◼ between g11 and k. This completes the proof. The assumption that g11 is both stable and minimum phase is not unduly restrictive since given the problem outlined in the introduction, it is very likely that the sensor and actuator can be arranged to be collocated. As a consequence, the positive realness of g11 can be achieved. Remark: The stability results allow the compensator to be improper. Improper transfer functions are not considered to be problematic since on one hand, they can be arbitrarily accurately approximated by proper ones; on the other hand, and more importantly, there exist linear and even passive models with improper transfer functions such as an on-chip inductor model [4]. Indeed, identification of improper transfer functions has sparked intensive research, e.g. through subspace methods [5, 6] in recent years. However, direct admittance of improper transfer functions into control theory is still not generally recognized. Of course, if this is forbidden, then it can be shown that the only additional constraint on α(s) is that it possesses at least the poll-off rate of the plant g11 . Remark: It is very important to ensure the minimum-phase of g11 since in practical engineering even collocated control can result in a non-minimum phase system. However, it is seen from Proposition 1 that if g11 is stable but non-minimum phase, e.g. existence of right-hand-plane zeros, then the necessary and sufficient condition for the closed loop system to be stabilizing is that α is a mapping of a stable function that also interpolates the unstable zeros of g11 , although the controller itself is unstable. The exploration of the stabilizing properties shows the condition that αopt ( jω) is a mapping of a stable function is crucial. The scenario is that a stable function is required to interpolate the optimal points over certain frequency band. Interpolation by a stable transfer function is in fact the salient feature of the geometric broad band control problem since even for non-minimum phase and unstable plants, the specification on the optimal choice αopt ( jω) can be transformed into that on the optimal choice Q opt ( jω) (Q is the Youla-parameter). And the problem boils down to the interpolation of Q opt ( jω) by a stable transfer function. However we shall formulate the geometric broad band control problem in the domain of αopt and treat the transformation to the domain of Q opt as a trivial step in relation to the geometric framework. The issue of improper compensator will also be resolved by the analytic construction to be developed in the following subsections, which generally results in a real rational realization of αopt (s) with at least 2-pole roll-off, as will be consequently

4.1 Optimal Design for Broad-Band Control

141

demonstrated in the active vibration isolation system to be presented in Sect. 4.1.5 So if the optimal choice at frequency ωi ∈ [ω1 , ω N ] is denoted by α i , then the broad-band control problem can be stated as follows: Broad-band control problem: find a transfer function α(s) ∈ H∞ such that α( jωi ) = αi ∀i ∈ [1, N ] and sup|α(s)| ≤ M, where H∞ is the Hardy space of s

bounded analytic functions in the right half plane (s) ≥ 0 and M > 0 is a real number. Remark: It is obvious from the definition of sensitivity that the geometric broad α(s) . band controller is given by k(s) = − [α(s)+1]g 11 (s)

4.1.3 Optimal Broad-Band Control Design with N-P Interpolation and LMI Before embarking on the solution to the geometric broad-band control problem, one important question remains to be answered: given the optimal choice over the range of frequency point values, is it always possible to fit the data points by a stable transfer function (solution existence problem)? In the following subsections, it will be shown that this question is equivalent to the Nevanlinna-Pick interpolation problem (N-P problem) and the solution existence problem can therefore be answered by an equivalent Pick condition. The final optimal choice or design for αopt ( jω) can consequently be resolved using Linear Matrix Inequality (LMI) techniques. Even remarkably, it is to be observed that the optimal solution such obtained can accommodate naturally the modelling errors, and hence robustness issue is taken into consideration explicitly in this design framework. The Nevanlinna-Pick (N-P) and related interpolation and approximation problems have found a variety of applications in linear systems and control, e.g. H∞ control, signal processing, approximation and circuit theories [7, 8]. A mathematical treatment of N-P interpolation can be found in [9–12] and the references therein. Roughly speaking the classical N-P interpolation problem involves finding a function which takes prescribed values at prescribed points [11]. Thus the N-P interpolation problem resembles the geometric broad-band control problem formulated in the last subsection and this motivates to consider N-P interpolation as a suitable candidate. Linear Matrix Inequalities (LMIs) have been recognized as an effective way of solving systems and control problems since 1990s. Its success can largely be attributed to the fact that a wide variety of problems arising in systems and control theory can be reduced to convex or quasi-convex optimization over LMIs while general LMIs can be solved numerically but directly by efficient algorithms such as interior-point algorithms. For a good introduction to LMI concepts and applications in systems and control theory as well, see the influential book by Boyd et al. [13]. In the following subsections the interest is to show that the N-P mechanism can be utilized to handle the geometric broad band control problem and an optimal solution can consequently be obtained by the solution to a series of LMIs. It is also noted

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that the geometric broad band control problem can be identified as one of robust identification in H∞ . The N-P interpolation approach to model identification and validation in the frequency domain has been previously exploited in the references [14–17], but here it is to be combined with another essential ingredient, namely LMI, to show how the geometric broad band control problem can be resolved. The classical N-P problem can be stated as follows: given N complex points (si )i∈[1,N ] with (si ) > 0 and another series of N complex points (Hi ) satisfying sup(|Hi |) ≤ 1∀i ∈ [1, N ], does there exist a transfer function H (s) ∈ H∞ such that H (si ) = Hi ∀i ∈ [1, N ]? The answer to this question is the classical Pick condition: Pick Condition: The above N-P problem is solvable if and only if the Pick matrix ⎡ P=⎣

__ ⎤

1 − Hk H _

l

sk + sl

⎦

(4.4) 1≤k,l≤N

is positive definite. It should be noted that the points si above should strictly belong to the Right Half Plane. Therefore a transformation is needed since the original data takes the form si = jωi in the geometric broad band control problem. The following lemma (Lemma 2, [18]) explains how to transform the problem: Lemma There exists a transfer function H (s). 

(1)

which is analytic in the Half Plane (s) ≥ −a (a is the minimal degree of stability); which satisfies H ( jwi ) = H i ∀i ∈ [1, N ]; which satisfies sup H (s) ≤ M. 



(2) (3)



(s)≥−α

if and only if there exists a transfer function H (s). (1) (2) (3)

which is analytic in the Right Half Plane (s) ≥ 0; which satisfies H ( jω + a) = H ( jwi ) ; 

i

which satisfies sup |H (s)| ≤ 1.

M

(s)≥0

Proof The proof follows from the definition H (s + a) =

H (s) . M

◼

Following this lemma, the Pick condition can be obtained in terms of the original data in the geometric broad band control problem, and hence provides the answer to the question of solution existence: Pick condition for broad-band control (Existence of an optimal controller): the broad-band control problem is solvable if and only if the Pick matrix:

P= is positive definite.

___

1 − αk αl /M 2 j (ωk − ωl ) + 2a

 (4.5) 1≤k,l≤N

4.1 Optimal Design for Broad-Band Control

143

Remark: The minimal degree of stability a and M can also be considered as design freedom since they can influence the controller performance both inside and outside the frequency band [ω1 , ω N ]. Tuning of a and M is discussed in Sect. 4.1.7.

4.1.4 Optimal Trajectory of αopt ( j ω) The positive definiteness of the Pick matrix will ensure the solvability of the geometric broad band control problem. But the following question remains: How to modify the optimal trajectory in the case of non-positive-definiteness of the Pick matrix and hence to obtain the best approximation to the initial optimal choice? If an optimal solution is not feasible, a suboptimal one will have to be sought. The solution comes from using an LMI solver. However the Pick condition must be transformed to an LMI form to use the LMI solver. Rewrite the Pick matrix as follows:



  ___ 1 − αk αl /M 2 1 P= = j (ωk − ωl ) + 2a 1≤k,l≤N j (ωk − ωl ) + 2a 1≤k,l≤N

 ___ (αk /M) × ( αl /M) − j (ωk − ωl ) + 2a 1≤k,l≤N Let T0 =



1 ,Q j (ωk −ωl )+2a 1≤k,l≤N

= blockdiag( αM1 , ..., αMN ), then we have:

  P = T0 − QT0 Q ∗ ∗ is the Hermitian operator

(4.6)

Hence by applying the Schur complement to Eq. (4.11), where T0 > 0. We have: P is positive definite if and only if

T0 Q Q ∗ T0−1

 (4.7)

Remark: To provide a tight bound or best approximation for αopt ( jωi ) ∀i ∈ [1, N ], more LMIs defining the lower and upper bound around the initial points can be adopted. It is seen that the robustness to modelling errors can naturally be included in this geometric broad band controller design procedure.

4.1.5 Transfer Function Representation of αopt (s) The solution to the above LMIs produces a set of feasible optimal choice. An algorithm is required to construct a stable interpolating function αopt (s) out of the optimal

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data set αopt ( jωi ) ∀i ∈ [1, N ]. Suppose N complex data points (si ≡ a + jωi )i∈[1,N ] (a degree of stability) and another series of N complex points   > 0 is the minimal αopt ( jωi ) Hi ≡ M satisfying sup(|Hi |) ≤ 1∀i ∈ [1, N ], then it is clear from i∈[1,N ]

above that if H (s) is the interpolating function, then αopt (s) = M × H (s + a). The problem is thus solved if H (s) is obtained. The following strategy is exploited to obtain H (s): first solve the one-point N-P interpolation problem and then recursively solve the required N-point N-P interpolation problem. The solution to the one-point interpolation problem is provided by the following statement: s−s

Proposition The function G(s) =

G 1 (s) s+s1 +H 1 1 s−s

1+G 1 (s) s+s1 H 1

solves the one-point N-P

1

interpolation problem with the interpolation point (s1 , H1 ) and G 1 (s) ∞ ≤ 1. s−s1 Proof It is clear that z = G 1 (s) s+s is analytic in the right half plane and further 1 is a one–one mapping from right half plane to the unit disk. While the function z+H 1 defines another one–one mapping from unit disk to unit disk. Hence G(s) = 1+z H 1

G(s) =

s−s

G 1 (s) s+s1 +H 1 1 s−s

1+G 1 (s) s+s1 H 1

is analytic in the right half plane and satisfies G(s) ∞ ≤

1

1. It is also clear that G(s1 ) = H1 and hence G(s) is the required interpolating function. ◼ From this, the required N-point N-P interpolation problem can be solved recursively: Proposition: For functions G(s) and G 1 (s) above, if G 1 (s) solves the (N-1)-point N-P interpolation problem then G(s) solves the N-point N-P interpolation problem. Hence the following algorithm can be utilized to find the transfer function representation of αopt (s) (Table 4.1): However, a problem arises due to the fact that the data set (si , Hi )i∈[1,N ] is of complex values, which can result in a transfer function representation of αopt (s) with complex coefficients. A real rational representation of αopt (s) can be obtained  simply by augmenting the data set (si , Hi )i∈[1,N ] with its complex conjugates s i , H i i∈[1,N ] . The following subsection shows that this is true. Table 4.1 Procedures for TF representation of αopt (s) Step 1: Choose a > 0 the minimal degree of stability and M, then form N complex data points   α ( jω ) ; (si ≡ a + jωi )i∈[1,N ] and another series of N complex points Hi ≡ optM i i∈[1,N ]

Step 2: Start from the original N-points data set (si , Hi )i∈[1,N ] , recursively calculate the new     i-point data set si , Hi from the previous (i + 1)-point data set si , Hi through the inverse function G −1 (s) until i = 1; Step 3: Then start from i = 1, recursively solve the one-point N-P interpolation problems. After N iteration, the resulting transfer function H (s) will solve the original N-point N-P interpolation problem; Step 4: The transfer function representation of αopt (s) is then given by: αopt (s) = M × H (s + a)

4.1 Optimal Design for Broad-Band Control

145

4.1.6 Real Rational Representation of αopt (s) After augmenting the data  set,the problem is to solve the 2 N-point N-P interpolation problem with data set [ s i , H i i∈[1,N ] ,(si , Hi )i∈[1,N ] ]. Suppose the solution to this 2 Npoint N-P interpolation problem using the above algorithm results in an interpolation function α(s) then the real rational representation of αopt (s) is obtained from the following statement: Proposition: Partition α(s) = α R (s) + jα I (s), then α R (s) is the required real rational representation of αopt (s). Proof For simplicity, assume two data points conjugating with each other: (s1 , α1 ) and (s2 , α2 ) with s2 = s 1 and α2 = α 1 . From the interpolation requirement: α R (s1 ) + jα I (s1 ) = α1 &α R (s2 ) + jα I (s2 ) = α2

(4.8)

Further partition both α R (s) and α I (s) into real and imaginary parts, respectively: α R R (s1 ) + jα R I (s1 ) + jα I R (s1 ) − α I I (s1 ) = α1

(4.9a)

α R R (s2 ) + jα R I (s2 ) + jα I R (s2 ) − α I I (s2 ) = α2

(4.9b)

For real rational transfer functions we have: α R R (s1 ) = α R R (s2 )α R I (s1 ) = −α R I (s2 )α I R (s1 ) = α I R (s2 )α I I (s1 ) = −α I I (s2 ) Equation (4.9b) is now: α R R (s1 ) − jα R I (s1 ) + jα I R (s1 ) + α I I (s1 ) = α2

(4.10)

Conjugating the above equation and utilizing α2 = α 1 : α R R (s1 ) + jα R I (s1 ) − jα I R (s1 ) + α I I (s1 ) = α1

(4.11)

Adding Eq. (4.9a): α R R (s1 ) + jα R I (s1 ) = α1

(4.12)

α R (s1 ) = α1

(4.13)

That is:

This completes the proof.

◼

Remark: t is seen that the real rational representation of αopt (s) will have a high order with the increasing number of interpolating points, hence is the resulting

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4 Active Control with Extensions α

(s)

opt optimal compensator kopt (s) = − [αopt (s)+1]g . If a low order compensator is 11 (s) required, standard controller order reduction techniques, e.g. balanced truncation [19], balanced residualization [20] or optimal Hankel norm approximation [21] can be applied. Remark: It is also possible to construct a state-space representation from the interpolation data (See [22] for details).

4.1.7 Performance Tuning For any design methodology it can be very important to provide additional freedom to tackle objectives beyond the main design goal. But it is also important to be able to describe the effects of design freedom in terms of system performance. The optimal choice αi defines directly the levels of attenuation in y( jω) and z( jω). It is also noted that the Pick condition introduces additional design parameters, namely the minimal degree of stability a and the maximum modulus M of α(s) on the half plane (s) ≥ −a. Their introduction arises from the determination of the existence of a stable transfer function α(s) that interpolates exactly through the optimal choice α( jωi )∀i ∈ [1, N ]. The consequence is that a stable α(s) cannot be found to interpolate α( jωi ) when the Pick matrix fails to be positive definite. Hence an approximation to the optimal choice α( jωi ) has to be made to obtain a stable α(s) (to ensure internal stability). In the following it is shown that a and M can be used to increase the accuracy of the approximation and hence improve the performance of the broad band geometric controller kopt (s) within the targeted frequency band [ω1 , ω N ]. Denote the ith eigenvalue of P by λi , which is a real number following from the fact that P is Hermitian, then a fundamental theorem from linear algebra states that the sum of eigenvalues is equal to the matrix trace. That is: N 

N 

|αk |2

N − M2 λi = T race(P) = 2a i=1 k=1

(4.14)

N Thus i=1 λi is monotonic with respect to a or M. Hence increasing M or N decreasing a can increase the “positiveness” of i=1 λi . With a large enough M or small enough a, the Pick matrix may eventually become positive definite and therefore allows an exact interpolation with a stable α(s) to the optimal choice α( jωi )∀i ∈ [1, N ]. This leads to the following tuning rule: Tuning Rule: The performance of the geometric broad band controller kopt (s) within the target frequency band [ω1 , ω N ] can be refined by either decreasing a or increasing M. However caution is required since allowing the stability margin a to be small, the system becomes more vulnerable to instability and in addition, allowing the

4.1 Optimal Design for Broad-Band Control

147

maximum modulus M of α(s) on the half plane (s) ≥ −a to be large, the performance outside the targeted frequency band [ω1 , ω N ] can be “potentially” sacrificed. To see this, it is noted that the maximum deterioration in y( jω) and z( jω) is determined by the H∞ norm of α( jω) + 1 and β( jω) + 1. But:

α( jω) + 1 ∞ ≤ α( jω) ∞ + 1 < M + 1

(4.15)

This is so since from definition M ≥ sup|α(s)| and the maximum modulus (s)≥−a

principle we have: M ≥ sup|α(s)| > sup|α(s)| = sup|α( jω)| = α( jω) ∞ . (s)≥−a

(s)=0

(4.16)

The maximum deterioration in z( jω):

β( jω) + 1 ∞

      1   α( jω)  + 1 < M × N + 1,    + 1 ≤ α( jω) ∞ ×  = g( jω) g( jω) ∞ ∞ (4.17)

    where  g( 1jω)  ≡ N . ∞ From Eqs. (4.15) and (4.17) it is seen that the out of band deterioration in y( jω) and z( jω) for the worst case can be alleviated by decreasing M. However the tuning rule elaborated above states that the performance of the geometric broad band controller and hence the in-band performance can be improved by increasing M. It is strongly emphasized that this is not a compromise that has to be made between the in-band and out-of-band performance: increasing M can improve the in-band performance but not necessarily deteriorate the out-of-band performance. This is so since the improvement of in-band performance refers to the closeness with a desired specification while outof-band deterioration expresses the fact that disturbances can not be reduced over all frequencies. The latter is often termed as a “waterbed effect” associated with non-minimum-phase plants or more generally Bode’s sensitivity integral theorem [23]. By asserting that re-design M has no cause-and-effect relationship with the outof-band performance deterioration, it is immediately pointed out that out-of-band deterioration as embodied in Bode’s integral theorem is a fundamental design limitation that cannot be resolved by any advanced controller design method. An understanding of this limitation will be very helpful for the geometric broad band controller design and the phenomenon of out-of-band deterioration. This is examined in the next section and we conclude this section by pointing out that the above tuning rule retains its usefulness as a design guide.

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4 Active Control with Extensions

4.1.8 Out-Of-Band Performance Deterioration The problem of performance deterioration out of the target frequency band is closely related to a fundamental limitation imposed by feedback control structure, namely that expressed by Bode’s integral theorem. For a system whose loop transfer function is stable with at least 2-pole roll off, this theorem can be expressed as [24]: ∞ ln|S( jω)|dω = 0,

(4.18)

0

where S( jω) is the sensitivity function. As pointed out in the last section, this unity requirement puts a fundamental constraint on feedback design that is not conquered by any advanced feedback controller design method. Many control design puzzles can be explained through this unity constraint. In the geometric broad-band controller design methodology elaborated above, it is thus pondered that there might exist fundamental limitations on both the sensitivity S( jω) that is a measure of reduction in vibration y( jω) and the quantity (β( jω) + 1) that is a measure of reduction in vibration z( jω). Indeed, such limitations exist and can even be quantified by the design freedom α(s). Denote the set of solutions of α(s) = −1 in Res > 0 by {m i }, the set of solutions of α(s) = −g in Res > 0 by {n i } and the remote vibration reduction by R( jω). Further assume α(s) has at least 2-pole role off, then the following fundamental results hold. Proposition: The reduction in local vibration y( jω) is constrained by the following integral relationship: ∞  ln|S( jω)|dω = π Re(m i ).

(4.19)

0

 Proposition: If α g has at least 2-pole role off, then the reduction in remote vibration z( jω) is constrained by the following integral relationship: ∞  ln|R( jω)|dω = π Re(n i ).

(4.20)

0

Proof Denote Smp (s) and Sap (s) the minimum-phase and all-pass partitions of S(s), respectively, further define F(s) = ln(Smp (s))(hence F(s) is analytic in the right half plane) then for every point s0 = σ0 + jω0 with σ0 > 0 from Poisson integral formula: 1 F(s0 ) = π

∞ F( jω) −∞

σ02

σ0 dω. + (ω − ω0 )2

(4.21)

4.1 Optimal Design for Broad-Band Control

149

That is: r eal(F(s0 )) =

1 π

∞ r eal(F( jω)) −∞

σ0 dω. σ02 + (ω − ω0 )2

(4.22)

Or: 1 ln Smp (s0 ) = π

∞ ln|S( jω)| −∞

σ0 dω. σ02 + (ω − ω0 )2

(4.23)

σ0 dω. σ02 + ω2

(4.24)

Taking ω0 = 0 results in: 1 ln Smp (σ0 ) = π

∞ ln|S( jω)| −∞

That is: π σ0 ln Smp (σ0 ) = 2

∞ ln|S( jω)| 0

σ02

σ02 dω. + ω2

(4.25)

Taking the limit of the above equation as σ0 → ∞: ∞ π lim σ0 ln Smp (σ0 ) . ln|S( jω)|dω = 2 σ0 →∞

(4.26)

0

Noting that Smp (s) = S(s)/Sap (s), then Eq. (4.26) can be rewritten as: ∞ π π lim σ0 ln|S(σ0 )| − lim σ0 ln Sap (σ0 ) . ln|S( jω)|dω = 2 σ0 →∞ 2 σ0 →∞

(4.27)

0

By definition S(σ0 ) = 1 + α(σ0 ), also since α has at least 2-pole role off, then α(σ0 ) ≈ σAk for some constant A and k ≥ 2 as σ0 → ∞. Hence from the Maclaurin’s 0

2

series expansion we have ln(1 + α(σ0 )) = σAk − 2σA 2k + · · · . Therefore the first term o 0 of the right hand side of Eq. (4.26) converges to zero as σ0 → ∞. Equation (4.27) now reads: ∞ π lim σ0 ln Sap (σ0 ) . ln|S( jω)|dω = − 2 σ0 →∞ 0

(4.28)

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4 Active Control with Extensions

Since the zeros of the sensitivity function are the set of solutions of α = −1, Sap (σ0 ) in the above equation can therefore be expressed as: Sap (σ0 ) =

 σ0 − m i σ0 + m i

i

.

(4.29)

Equation (4.28) can thus be rewritten as: ∞ ln|S( jω)|dω 0

=−

 

2Re(m i ) |m i |2 2Re(m i ) |m i |2 π lim σ0 + ) − ln(1 + + ) . ln(1 − 4 σ0 →∞ σ0 σ0 σ02 σ02 i (4.30)

Applying again the Maclaurin’s series expansion leads to: 

∞ π 2Re(m i ) 2Re(m i ) ln|S( jω)|dω = − lim σ0 − − + ··· . 4 i σ0 →∞ σ0 σ0

(4.31)

0

That is: ∞  ln|S( jω)|dω = π Re(m i ).

(4.32)

i

0

This proves the result. On the other hand, from equation R( jω) = 1 + β( jω), which is simply a definition of the level of reduction in the vibration z( jω). One can show: R( jω) = 1 +

α( jω) . g( jω)

(4.33)

Therefore the set of unstable zeros of R(s) is {n i }, and hence all-pass part of R(s) can be expressed as: Rap (s) =

 s − ni i

s + ni

.

(4.34)

 Therefore if α g also has at least 2-pole role off, the procedures can be used to validate the above two propositions. ◼ In Eqs. (4.19) and (4.20), both Re(m i ) and Re(n i ) are positive, therefore the best situation for the reduction in vibration y( jω) and vibration z( jω) is expressed by:

4.1 Optimal Design for Broad-Band Control

∞ ln|S( jω)|dω = 0.

151

(4.35)

0

and ∞ ln|R( jω)|dω = 0.

(4.36)

0

Hence both vibration y( jω) and vibration z( jω) can not be suppressed over all frequencies and this is the ultimate reason for the out-of-band performance deterioration.

4.1.9 Broad-Band Control of Active Vibration Isolation System The active vibration isolation system as shown in Fig. 4.2 is utilized to validate the geometric broad-band design methodology. For better reference it is reproduced here and the same system parameters are adopted, e.g. mass, stiffness and damping matrices ⎡ as follows: ⎡ ⎡ ⎤ ⎤ ⎤ 100 4 −1 −2 4 −1 −1 M = ⎣ 0 1 0 ⎦, C = ⎣ −1 1 0 ⎦ and K = ⎣ −1 2 −1 ⎦. 001 −2 0 2 −1 −1 2 The system can be readily written as the standard form with y( jω) and z( jω) representing the locally measured vibration and the remote vibration, respectively. But now y( jω) and z( jω) will be displacements other than velocities. The purpose Fig. 4.2 Active vibration isolation system

152

4 Active Control with Extensions

is to demonstrate that the proposed algorithm is capable of producing a compensator with a proper other than improper transfer function even now g11 (s) is of 2-pole role-off, as promised in Sect. 4.1.7. Design Broad-Band Optimal Controller

10

5

5

0

0

-5 Magnitude dB

Magnitude dB

Figure 4.3a shows the frequency response of the vibration z to excitation of the disturbance d (the plot is therefore of |g22 ( jω)|). It can be seen that the first resonance occurs at 0.1 Hz and that this leads to a peak in the transmission path, as can be seen in the m y response to the same forcing (|g12 ( jω)|) in Fig. 4.3b. Therefore a control scenario can be considered where the objective is to provide optimum reduction at m z (z location) without increasing vibration at the m y (y location) over the frequency band, e.g. [0.06, 0.14] Hz centred at the resonance 0.1 Hz. The optimal choice (the data points over [0.06, 0.14] Hz) can be computed from elementary geometric arguments by considering the relative locations of α − cir cle and β − cir cle for each frequency over the same frequency band assuming, e.g. a 0.02 Hz grid. The optimal choice is illustrated in Fig. 4.4. It is seen from Fig. 4.4 the optimal points coincide with the centres of the β − cir cles representing the annihilation of the vibration z; meanwhile since all the optimal points locate inside the unit α − cir cle and are far away from the boundary representing a significant attenuation of the vibration y. Hence it will be anticipated that if the optimal choice is feasible then the vibration z will be completely attenuated while the vibration y will also have a significant amount of reduction. The feasibility

-5

-10

-10

-15

-15

-20

-20 0

0.1

0.2

0.3

0.4

-25

0

0.1

Frequency Hz

(a) Fig. 4.3 a Disturbance response z( jω), b disturbance transmission

0.2 Frequency Hz

(b)

0.3

0.4

4.1 Optimal Design for Broad-Band Control

153

2

1.5

1

0.5 -1

0

-0.5

-1

-1.5

-2 -4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Fig. 4.4 Mapping of |β + 1| = 1 on the complex α-plane over [0.06, 0.14] Hz with an obvious 0.02 Hz grid. Solid circle is the unit α-circle and dashed circles are the mappings of the unit β -circles; dotted lines are the corresponding optimal lines for each frequency and finally, the small circles are the optimal points

of the optimal choice is determined by the positive definiteness of the corresponding Pick matrix. With the assumption of a = 0.25 and M = 3, it is found that the Pick matrix is not positive definite and therefore the Pick condition is violated, and hence the initial optimal choice is infeasible. The application of the above optimization over LMIs, while setting the real and imaginary part of uncertainty for the initial data set to be uniformly bounded within 0.08 results in a “new” feasible optimal choice. With this new feasible optimal choice, the closed-loop as well as the open-loop FRFs is illustrated in Fig. 4.5. Also shown is the closed-loop FRFs that would have resulted from the original optimal choice, had this been feasible. The deviation between the two closed-loop FRFs reveals the accuracy of the approximation and hence the loss of theoretical performance. It is clear that such a deviation in achieving the feasible solution is very small. The design procedure proposed is therefore practically viable. Tuning Optimal Controller Performance Now increase the value M = 3 to M = 4 while still assuming a = 0.25 and the uncertainty bound to be 0.08 in the above example, it is found that the Pick condition is still violated, and hence the initial optimal choice is still infeasible. The application of the above optimization over LMIs results in another new set of

154

4 Active Control with Extensions 1.8 1.6

Displacement

1.4 1.2 1 0.8 0.6 Open-Loop FRF Feasible Closed-Loop FRF Infeasible Closed-Loop FRF

0.4 0.2 0 -0.2 0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

Frequency Hz

(a)

Displacement

1 0.8 0.6

0.4

Open-Loop FRF Feasible Closed-Loop FRF Infeasible Closed-Loop FRF

0.2 0 0.06

0.07

0.08

0.09 0.1 Frequency Hz

0.11

0.12

0.13

0.14

(b) Fig. 4.5 Open-loop and closed-loop FRFs with feasible and infeasible optimal choice. a Vibration z b vibration y

feasible optimal choice. The resulting closed-loop FRFs are compared as illustrated in Fig. 4.6. It is seen that the accuracy of approximation is increased. The loss of theoretical performance becomes smaller. Finally it is noted that when M increases to M = 6 or a decreases to a = 0.15, the Pick matrix becomes positive definite and this results in an exact interpolation and hence no loss of theoretical performance.

4.1 Optimal Design for Broad-Band Control

155

0.04

Displacement

0.03 0.02 0.01 0 Feasible Closed-Loop FRF: a=0.25, M=3 Feasible Closed-Loop FRF: a=0.25, M=4 Infeasible Closed-Loop FRF

-0.01 -0.02 0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.12

0.13

0.14

Frequency Hz

(a) 0.55

Displacement

0.5 0.45 0.4 Feasible Closed-Laoop FRF: a=0.25, M=3 Feasible Closed-Loop FRF: a=0.25, M=4 Infeasible Closed-Loop FRF

0.35

0.06

0.07

0.08

0.09

0.1

0.11

Frequency Hz

(b) Fig. 4.6 Tuning of geometric broad band controllers for a vibration z b vibration y: comparison of closed-loop FRFs with different a and M pair. Dotted line is the original infeasible optimal choice and hence can be seen as a reference when a and M pair is tuned. It is seen that increasing M can improve the accuracy of approximation and hence the broad band controller performance in relation to the prescribed specification

The tuning rule thereby indeed retains its usefulness as a design guide. This validates the results presented in Sect. 4.5. Out of Band Deterioration and Control Simulation Application of the algorithm presented in Sects. 4.1.5 and 4.1.6 results in a real rational representation of αopt (s) with an order of 18 but being of 2-pole roll-off. We shall use the original αopt (s) other than its order-reduction model when coming to real time simulation. Meanwhile the unstable solution (zero in right half plane) of α = −1 is 41.42, then from the Proposition:

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4 Active Control with Extensions

∞ ln|S( jω)|dω = 41.42π.

(4.37)

0

 On the other hand, α g has a 4-pole roll-off and there are 4 unstable solutions of α = −g at 0.01 ± 0.9448i and 0.014 ± 0.352i then from Proposition: ∞ ln|R( jω)|dω = 0.048π.

(4.38)

0

The particular numbers in the right hand side of the above equations are not the main concern since the area can be balanced over an infinitely wide frequency range. But they do reflect some design preferences, e.g. a large number can well indicate that there is either performance deterioration over a wide frequency band or a large peak over a certain narrow frequency band which can be problematic since this can cause saturation even instrumental damage. Nevertheless, Fig. 4.7 shows the open and closed loop performance over the whole frequency band. And it is seen clearly that performance deterioration exists both for the vibration y and the vibration z. Finally, the simulation results of the geometric broad band controller are shown in Figs. 4.8 and 4.9: Fig. 4.8 is the off-line simulation where signal variation is clearly illustrated and Fig. 4.9 uses a different time axis to demonstrate the online simulation. In both cases, the broad band disturbance is simulated as a random disturbance passing through a band-pass filter. With the reference to the theoretical performance shown in Fig. 4.5, it is seen that the method has therefore successfully enabled the extension of the geometric design approach to the control of remotely located vibrating systems to the broad band case. Remark: The simulation results demonstrate clearly the power of the geometric design methodology. However although not shown in Fig. 4.9c it is observed that the control signal u has a large overshoot before settling down, due to its instability nature. This is certainly not a problem for the off-line control as can be seen from Fig. 4.8c. But it is indeed shown the importance of the utilization of strong stabilizing controller, hence the importance of the proposed result.

4.1.10 Summary for Optimal Broad-band Control Broad-band control has been presented and the existence of an optimal geometric controller has been examined based on the N-P interpolation method. A calculation of feasible controllers is carried out via convex optimization over LMIs. Algorithm has been developed for the finding of a real rational representation of the interpolating function. Further design freedom has been exploited to further improve the performance of the geometric controller over the target frequency band. Out-of-band

4.1 Optimal Design for Broad-Band Control

157

Bode Diagram 10 0

Magnitude (dB)

-10 -20 -30 -40 Open-Loop Performance

-50 Closed-Loop Performance

-60 -70 -80 -2 10

-1

10

2

1

0

10

10

10 Bode Diagram

(a) Frequency (rad/sec)

20 0 -20

Magnitude (dB)

-40 -60

Open-Loop Performance

-80

Closed-Loop Performance

-100 -120 -140 -160 -180 -2 10

-1

10

0

1

10 10 Frequency (rad/sec)

2

10

3

10

(b) Fig. 4.7 Whole-band geometric controller performance a vibration z, b vibration y

performance deterioration has also been investigated. These theoretical considerations have been validated through their application to the active vibration isolation system.

158

4 Active Control with Extensions 1.5

1

Displacement

0.5

0

-0.5

-1 Open-Loop Response -1.5

0

50

100

150

200

Closed-Loop Response

250 Time (s)

300

350

400

450

500

(a) 0.8 0.6 0.4

Displacement

0.2 0 -0.2 -0.4 -0.6 -0.8 Open-Loop Response -1

0

50

100

150

200

Closed-Loop Response 250 Time (s)

300

350

400

450

500

(b) 0.8 0.6 0.4

Force

0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

50

100

150

200

250 Time (s)

300

350

400

450

500

(c) Fig. 4.8 Geometric controller performance a vibration z, b vibration y, c control signal u

4.1 Optimal Design for Broad-Band Control

159

2 1.5 1

Displacement

0.5 0 -0.5 -1 -1.5 -2

0

1000

2000

3000

4000

6000 5000 Time (s)

7000

8000

9000

10000

(a) 1.5

1

Displacement

0.5

0

-0.5

-1

-1.5

0

1000

2000

3000

4000

5000 6000 Time (s)

7000

8000

9000

10000

(b) 1 0.8 0.6

0.2 0

p

Force

0.4

-0.2 -0.4 -0.6 -0.8 -1

0

1000

2000

3000

4000

6000 5000 Time (s)

7000

8000

9000

10000

(c) Fig. 4.9 On-line performance of geometric controller a vibration z, b vibration y, c control signal u

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4 Active Control with Extensions

4.2 Performance Improvability Test for Harmonic Control For both active control and passive design, it is implicitly assumed that vibration and noise can be attenuated through appropriate designs. The right problem of performance improvability for vibration or noise attenuation is not considered. However, the investigation of the improvability issue should really be necessitated as a first step towards design due to its intimate relationship with constraints upon system parameters. This section presents the necessary and sufficient condition for performance improvability for harmonic control. It is seen that the proposed methodology can be extended to derive a clear-cut, and easily verifiable condition. Its implications to system design and optimization are discussed. A numerical example is also provided to facilitate the explanations.

4.2.1 Introduction Vibration or noise can be attenuated through either passive design or active control approaches. For both methodologies, a variety of methods have been proposed where the detrimental effect of vibration or noise can be ameliorated. Most of the methods utilize the optimization-based routines for optimal design. Successful applications of H2 /H∞ optimization, harmony search, particle swarm and ant colony optimization, genetic algorithms etc. have all been reported. Indeed, by combining these optimization routines with feedback controls, a variety of active control methods such as generalized predictor-based approach, error-driven designs etc. can be “devised.” It is thus seen clearly that both passive and active design methods have implicitly assumed that vibration or noise can be attenuated through appropriate designs. The right question of vibration attenuability is “disregarded.” This pertains to an important issue of performance improvability for harmonic control—if the performance of the system cannot be improved at all, there must be some reason to be explained for the failure. This represents a new type of fundamental design limit: a negative answer to performance improvability either implies that the performance cannot be improved with any design method, be it active or passive; or indicates method of modification to make the performance improvable. In either case, the current performance reaches limit and the performance improvability test will provide important design guidance. To probe into this fundamental problem, it is anticipated that the “capability” of vibration attenuation must depend on system parameters, and it is thus strongly felt that a clear-cut answer can be obtained with an easily-verifiable condition. This will be sought in this section.

4.2 Performance Improvability Test for Harmonic Control

161

4.2.2 Improvability Test Theorem To put the development into stage, a general configuration is assumed for the system dynamics:





 y( jω) g11 ( jω) g12 ( jω) u( jω) = g21 ( jω) g22 ( jω) d( jω) z( jω)

(4.39)

where y( jω) and z( jω) are the performance variables while u( jω) and d( jω) are the general input and the exogenous disturbance respectively; gi j s are the corresponding frequency response functions. Thus the harmonic vibration attenuation problem is to design appropriate signal u( jω) such that the vibration transmission from d( jω) to y( jω), and from d( jω) to z( jω) can both be attenuated for the frequency ω. One word about the general input u( jω) must be explained: (1) (2) (3)

Should u( jω) = k( jω)y( jω) be utilized, this leads to active control methodology; Should u( jω) = gi j ( jω)y( jω) be utilized, this leads to passive design methodology; Optimal control or optimal design is initiated if an appropriately defined performance index is optimized against.

Thus the general configuration representation (4.39) would include both active and passive control methodologies, together with conventional optimization-based approaches under the same roof; the corresponding claim made to (4.39) should possess generic validity to harmonic vibration attenuation problems. This claim is stated as: Performance Improvability Test Theorem: For harmonic vibration attenuation problem with dynamics (1), the vibration reduction in both y( jω) and z( jω) is ( jω)g22 ( jω) . achievable if and only if G is not a negative real number defined by G = gg11 12 ( jω)g21 ( jω) Proof To prove the claim, use u( jω) = K ( jω)y( jω) to denote the design action for both active and passive control. Substitution of the design action into the dynamics (4.39) leads to: y( jω) = [1 − g11 ( jω)K ( jω)]−1 g12 ( jω)d( jω)   z( jω) = g21 ( jω)K ( jω)[1 − g11 ( jω)K ( jω)]−1 g12 ( jω) + g22 ( jω) d( jω) (4.40) This is the vibration reduction in performance variables after the design; while before the design: y( jω) = g12 ( jω)d( jω) z( jω) = g22 ( jω)d( jω)

(4.41)

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4 Active Control with Extensions

Henceforth the performance ratios before the design and after the design are: T y( jω) = [1 − g11 ( jω)K ( jω)]−1 T z( jω) = g21 ( jω)K ( jω)[1 − g11 ( jω)K ( jω)]−1 g12 ( jω)/g22 ( jω) + 1

(4.42)

where: T y( jω) and T z( jω) represent the corresponding performance ratios in performance variables. Thus should the design u( jω) = K ( jω)y( jω) result in performance improvement, or lead to further vibration reduction in y( jω) and z( jω), the necessary and sufficient condition will be: 

|T y( jω)| < 1 |T z( jω)| < 1

(4.43)

That is, a design K ( jω) must exist such that the two inequalities are satisfied simultaneously:  [1 − g11 ( jω)K ( jω)]−1 < 1 g21 ( jω)K ( jω)[1 − g11 ( jω)K ( jω)]−1 g12 ( jω)/g22 ( jω) + 1 < 1

(4.44)

To simplify (4.44), a crucial step comes from a re-definition to the design K ( jω) α( jω) , then inequalities (4.44) become: with K ( jω) = [α( jω)+1]g 11 ( jω) ⎧ ⎪ jω) + 1| < 1 ⎨ |α( α( jω) ⎪ + 1 < 1 ⎩ G( jω)

(4.45)

That is, the necessary and sufficient condition for harmonic vibration attenuation is the existence of design α( jω) such that the inequalities in (4.45) are satisfied simultaneously. Now comes another crucial step: |α(  jω) + 1| = 1 represents a circle with centre (−1, 0) and unity radius while α( jω) G( jω) + 1 = 1 can represent another circle (through Möbius transformation) on the same complex plane with a radius |G( jω)| centered at (-G( jω)). The third crucial step comes from the observation that the inequalities in (4.45) will be satisfied if and only if the two circles intersect. The last crucial step emerges with the observation that the two circles always intersect at the Origin and henceforth the for the inequalities in (4.45) NOT satisfied  only condition is that the circle α( jω) G( jω) + 1 = 1 locates on the right-half plane centered on the real axis. That is, (−G( jω)) becomes a positive real number. This proves the claim. ◼

4.2 Performance Improvability Test for Harmonic Control

163

4.2.3 Examples Consider a generic two-degree-of-freedom suspension configuration vibration absorber with the primary system having parameters m 1 = 1 kg, c1 = 2 N s/m, and k1 = 10 N/m while the absorber system’s parameters m 2 = 1 kg, c2 = 1 N s/m, k2 = 1 N/m. As a novel type of vibration absorber where the vibration transmission to both primary and absorber systems needs to be attenuated, the objective   is to determine the performance improvability at the harmonic frequency ω = k1 m 1 = 3.16 rad/s. Label the frequency response functions for the primary system and absorber system with y( jω) and z( jω), respectively; then with manipulation, one can have the expression (4.1) and further derive: !  k1 + k2 − m 1 ω2 + j (c1 + c2 )ω k2 − m 2 ω2 + jc2 ω g11 ( jω)g22 ( jω) = G= g12 ( jω)g21 ( jω) (k2 + jc2 ω)2 (4.46) Substitution leads to G = −1.4+8.2 j, henceforth the performance is improvable where solutions exist for simultaneous vibration attenuation in both y( jω) and z( jω). On the other hand, consider the absorber parameters c2 and k2 are to be designed. Substitution of the known parameters values into (4.46) leads to:

  √ √ k2 + 10(2 + c2 ) j k2 −10 + 10c2 j G=  2 √ k2 + 10c2 j

(4.47)

The condition that G becomes a negative real number is thus enforced by the condition: r eal(G) < 0 & imag(G) = 0. This would represent the values to be “prohibited” for the absorber parameters since they will result in performance nonimprovability. In fact, the result can be greatly generalized by avoiding G becoming “nearly” a negative real number: 

r eal(G) < 0 |imag(G)| < δ ∼ =0

(4.48)

The resulting parameters will also result in almost non-improvability and henceforth “ineffective designs” to be avoided. Such a generalization is believed to be able to explain those designs failing to improve performance.

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4.2.4 Summary and Discussions for Performance Improvability Test The “capability” of vibration or noise attenuation must depend on system parameters, and it is necessitated to tell if the performance can be improved, through either active control or passive control. This is important since, on one hand, if the performance is not improvable, actions must be taken to remedy the loss of performance; on the other hand, non-improvability conditions must provide stringent conditions for parameters to be avoided of designations. The analysis of performance improvability thus provides important insights into the physical properties of the system dynamical parameters. Following this rationale, a necessary and sufficient improvability test has been derived. The theorem turns out that vibration attenuation through either active or passive design is generally feasible only with the exception of negativity of G. This justifies the direct adoption of active and passive design techniques without concerning upon the issue of performance improvability. Yet, the performance improvability test does provide important insight into the mechanism of nonimprovability, and more importantly, the approach to improving the performance such as: (1) (2)

(3)

(4)

If G is not a negative real number, solutions always exist where vibration reduction in y( jω) and z( jω) can be achieved; If G is a negative real number, the system achieves optimal performance, henceforth appropriate designs (active or passive) must be conducted to relocate G for performance improvement to be feasible. Consequently, vibration in y( jω) and z( jω) can be “continuously” attenuated until G becomes a negative real number—in a certain sense, the “negativity” of G can be regarded as a measure of optimality of performance, which provides a useful guidance to system design. Indeed, as a measure of improvability of performance, condition (4.10) on G not only defines the parameters that must be avoided, but also provide an explanation to the failure or at least ineffectiveness of some designs, be it active or passive.

4.3 Experimental Design for Verification and Validation of Harmonic Control Systems Verification and validation represents an important procedure for model-based systems engineering design processes. One of the crucial tasks for verification & validation is to test whether the control system has reached performance limit. This is challenging since complicated theories and complex steps are often involved to achieve such an objective; meanwhile, the state-of-art for testing performance limit requires iterative procedures. A simple and one-off experimental design for telling

4.3 Experimental Design for Verification and Validation …

165

whether a control system reaches its performance limit is thus necessitated. This note introduces a remarkable test criterion for fulfilling the requirement. Both theoretical foundation and experiment design procedures are presented. Numerical examples are illustrated for the proposed method, where it is also shown that the simple method can be generalized to determining performance limit maps over both frequencies and physical parameters.

4.3.1 Introduction Model-based systems engineering (MBSE) has been gradually recognized as a key enabling technology for transition from traditional document-based and code-centric processes to more efficient and effective model-based processes [25–28]. One of the important phases among MBSE procedures is “verification & validation (V&V),” where the system design is checked to meet a set of design specifications. Indeed, the verification procedures involve performing special tests or regularly repeating tests devised specifically to ensure that the system continues to meet the initial design requirements and specifications as time progresses [28]. Many verification and validation methods have been proposed (see [29, 30] and references therein]. These methods usually utilize iterative procedures for reconciling whether the desired specification is satisfied. For (either passive or active) harmonic vibration control systems, one of the key questions for V&V to answer is whether the performance is optimal, and thus an appropriate (set of) experiments must be designed for testing the control system performance limit. A variety of well-established approaches can be routinely deployed for this purpose, e.g. animation, degenerate tests, event validity, multistage validation, sensitivity analysis, predictive validation, and even Turing tests etc. Yet it is seen that all the approaches would require iterative procedures for determining the optimality of control system performance, not to mention detecting the performance limits [31, 32]. Indeed, determining performance limit turns out to be an extremely challenging issue, and there is even lack of theoretical predictions—many optimization methodologies have been proposed such as H2/H∞ optimization [33], convex and combinatory optimizations [24–38], heuristic optimization [39, 40] etc. Yet the implementation of these approaches depends on certain design freedom in one way or another, such that determining performance limit still involve with (often complicated) iterative procedures, and significant modifications may have to be implanted to the formulations. It is thus necessitated to design an experiment that can tell the performance limit. It turns out that a remarkable criteria exists that is amenable to an experimental design to determine whether the performance limit has reached. It is the intention of this section to disseminate the theoretical foundations and experimental design procedures for this remarkable approach.

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4.3.2 Theoretical Foundations The potential of vibration attenuation must depend on the system parameters. This is the conviction from which the theory is developed. To start with, consider the following system dynamics (reproduced from above):





 y( jω) g11 ( jω) g12 ( jω) u( jω) = g21 ( jω) g22 ( jω) d( jω) z( jω)

(4.49)

where y( jω) and z( jω) are the vibrations to be attenuated; d( jω) is the ambient vibration injected into the system while u( jω) is the control variable; gi j ( jω)s are the corresponding frequency response functions representing the dynamical properties of the system. Thus the harmonic vibration control problem is formulated as designing u( jω) such that the vibration transmission from d( jω) to y( jω), and from d( jω) to z( jω) can be attenuated for a frequency ω. To proceed, define: u( jω) = K ( jω)y( jω)

(4.50)

where: K ( jω) denotes the feedback controller to be designed. However, for the purpose of determining performance limit, K ( jω) must not be involved. To develop the story, substitute (4.50) into (4.49) and manipulate: y( jω) = [1 − g11 ( jω)K ( jω)]−1 g12 ( jω)d( jω)   z( jω) = g21 ( jω)K ( jω)[1 − g11 ( jω)K ( jω)]−1 g12 ( jω) + g22 ( jω) d( jω) (4.51) Without design, or K ( jω) = 0, the corresponding vibrations are: y( jω) = g12 ( jω)d( jω) z( jω) = g22 ( jω)d( jω)

(4.52)

Consequently the vibration attenuation ratios as defined by the corresponding responses in y( jω) and z( jω) with and without design K ( jω) result in: T y( jω) = [1 − g11 ( jω)K ( jω)]−1 T z( jω) = g21 ( jω)K ( jω)[1 − g11 ( jω)K ( jω)]−1 g12 ( jω)/g22 ( jω) + 1

(4.53)

where: T y( jω) and T z( jω) are the vibration attenuation ratios. Therefore, if the design u( jω) = K ( jω)y( jω) achieves the performance limit, there must be: 

|T y( jω)| = 1 |T z( jω)| = 1

(4.54)

4.3 Experimental Design for Verification and Validation …

Or:  [1 − g11 ( jω)K ( jω)]−1 = 1 g21 ( jω)K ( jω)[1 − g11 ( jω)K ( jω)]−1 g12 ( jω)/g22 ( jω) + 1 = 1

167

(4.55)

Now shift the design K ( jω) to another parameter α( jω) with relationship: K ( jω) =

α( jω) [α( jω) + 1]g11 ( jω)

(4.56)

Then the condition (4.55) can be simplified to be: ⎧ ⎪ jω) + 1| = 1 ⎨ |α( α( jω) ⎪ + 1 = 1 ⎩ G( jω)

(4.57)

( jω)g22 ( jω) where: G = gg11 , depending solely on the system parameters. Henceforth, 12 ( jω)g21 ( jω) the necessary and sufficient condition for harmonic vibration control reaching its performance limit is the existence of α( jω) such that the two conditions in (4.57) must be satisfied simultaneously. |α( To further develop, it is known that on the complex plane,  jω) + 1| = 1 represents a circle with centre (−1, 0) and unity radius while α( jω) G( jω) + 1 = 1 can be mapped onto the same complex plane (through a Möbius transformation) with center (−G( jω)) and radius |G( jω)|. Therefore, the necessary and sufficient condition for harmonic vibration control reaching its performance limit will be the situation where the two circles are tangent. Now comes a crucial observation: the two circles always intersect at the Origin and thus the only situation for them to become tangent is that the circle  α( jω) G( jω) + 1 = 1 locates on the right-half plane centered on the real axis. That is, (-G( jω)) must be a positive real number. This important result is summarized below: Fundamental Theorem on Performance Limit: The necessary and sufficient condition for harmonic vibration control system reaching its performance limit is that the system parameters have the following property: 11 ( jω)g22 ( jω) becomes a negative real number. G = gg12 ( jω)g21 ( jω) Remark: It is noted that the above theorem is strict in the sense that satisfaction of the condition dictates the performance strictly reaching the limit; in practice, due to uncertainties, it is worth to define a performance limit index:

|Im(G)| ≤ ε

(4.58)

where: ε is a small positive number. Thus (4.58) “measures” the distance from performance limit.

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4.3.3 Experimental Design Procedures The above theoretical exposition lays a foundation upon experimental design for testing the performance limit in verification & validation of harmonic vibration control systems. This can be simply summarized as: (1) (2) (3) (4)

(5) (1) (2)

(3)

set up the experiments for the harmonic vibration control system to be tested; designate the performance variables y( jω) and z( jω) to be attenuated; define the control variable u( jω) together with the ambient vibration d( jω); identify the vibration transmission paths between (y( jω) & z( jω)) and (u( jω) & d( jω)), which will define the dynamical properties gi j s as represented by the corresponding frequency response functions; ( jω)g22 ( jω) calculate G = gg11 , with the following conclusion to be drawn: 12 ( jω)g21 ( jω) if G is negative real, the harmonic vibration control system reaches its performance limit; if Re(G) < 0 while |Im(G)| ≤ ε, the performance of the harmonic vibration control system “almost” reaches its performance limit, indicating the performance improvement is marginal; neither case (1) or case (2) implies that the performance of the harmonic vibration control system can still be improved, thus the system needs to be further detuned/calibrated.

Remark: Steps 1–4 are preliminary, and the corresponding variables would have been clearly defined when coming to the V& V process. Remark: Thus the testing essentially involves only with Step 5, which is easy ! to calculate. Indeed, repeating calculating G across a frequency band ω1 ω2 rad/s will result in a total characterization of performance improvability/limit map for the harmonic vibration control system. This is a remarkable result that has not been explored in the literature.

4.3.4 Numerical Examples System dynamics can be represented in either state-space or frequency response function representations in a practical V&V process. For example, state-space representation is usually presented based on first principle modeling; while frequency response function representation is given following from system identification procedures. Yet it is clear that the two representations are equivalent and in the following we shall illustrate the example based on a first-principle model for a tuned mass vibration absorber. A two-degree-of freedom model with a suspension configuration as shown in Fig. 4.10 can be written as:

4.3 Experimental Design for Verification and Validation …

169

Fig. 4.10 Two-degree-of-freedom model for a tuned mass vibration absorber with a suspension configuration

m 1 [x¨1 (t) + p(t)] ¨ + k1 x1 (t) + c1 x˙1 (t) + k2 [x1 (t) − x2 (t)] + c2 [x˙1 (t) − x˙2 (t)] = 0 m 2 [x¨2 (t) + p(t)] ¨ − k2 [x1 (t) − x2 (t)] − c2 [x˙1 (t) − x˙2 (t)] = 0

(4.59) where: m 1 , k1 , and c1 are the primary system parameters; m 2 , k2 , and c2 are the absorber system parameters, respectively; p(t), x1 (t), and x2 (t) are displacements ¨ representing the disturbance force excitation. Now for either active or with p(t) passive control, a generic design c(t) can be augmented as: m 1 [x¨1 (t) + p(t)] ¨ + k1 x1 (t) + c1 x˙1 (t) + k2 [x1 (t) − x2 (t)] + c2 [x˙1 (t) − x˙2 (t)] = −c(t) m 2 [x¨2 (t) + p(t)] ¨ − k2 [x1 (t) − x2 (t)] − c2 [x˙1 (t) − x˙2 (t)] = c(t)

(4.60) For a frequency ω, the above dynamics can be represented as: "

X 1 ( jω) X 2 ( jω)

# =

" #" # 1 m 2 ω2 −m 2 ω2 + 2(k2 + jc2 ω) u( jω)  −m 1 ω2 + (k1 + jc1 ω) −m 1 ω2 + (k1 + jc1 ω) + 2(k2 + jc2 ω) d( jω)

(4.61) where: X 1 ( jω), X 2 ( jω) and u( jω) are the Fourier transforms of x1 (t), x1 (t) and c(t), respectively; and d( jω) = ω2 P( jω), with P( jω) being the Fourier transform of p(t);  is the determinant that will become a dumb variable unnecessary to compute. Finally, an identification of X 1 ( jω) & X 2 ( jω) with y( jω) & z( jω), respectively, would lead to the “standard” representation in (4.49). Now following the expositions in Sects. 4.2 and 4.3, the V&V process will only involve with the determination of G:

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4 Active Control with Extensions

! m 2 ω2 −m 1 ω2 + (k1 + jc1 ω) + 2(k2 + jc2 ω) g11 ( jω)g22 ( jω) ! ! G= = g12 ( jω)g21 ( jω) −m 2 ω2 + 2(k2 + jc2 ω) −m 1 ω2 + (k1 + jc1 ω) (4.62) To be specific, now it is assumed that the vibration absorber has a primary system with parameters m 1 = 1 kg, c1 = 2 N s/m, and k1 = 10 N/m while the absorber system’s parameters  m 2 = 1 kg, c2 = 1 N s/m, k2 = 1 N/m. At the harmonic frequency ω = k1 m 1 = 3.16 rad/s, the above expression can be calculated:   ω2 12−ω2 + 4 jω   = −1.73 - 0.98 j G= 2−ω2 + 2 jω 10−ω2 + 2 jω

(4.63)

It is seen that Re(G) < 0 yet the condition |Im(G)| ≤ ε is not satisfied. According to the assertions in the above section, this indicates that the performance of the harmonic vibration control system can still be improved—further detuning/calibration is thus required. The above conclusion does not seem to be trivial and the exposed experimental design has a much wider implication to control system design. (1)

Performance Limit Map over Frequencies

ω2 12−ω2 +4 jω) Indeed, the expression G = 2−ω2 +2( jω 10−ω2 +2 can be evaluated over frequenjω) ( )( cies, and this will lead to a performance limit map for a frequency range as shown in Fig. 4.11. It is seen clearly that over the frequency range ω ≥ 1.6 rad/s, Re(G) < 0 while ω ≥ 2.4 rad/s, Im(G) begins approaching zero; when the frequency reaches ω = 5 rad/s (and beyond), |Im(G)| ≤ ε, henceforth the harmonic vibration control system reaches its performance limit. Figure 4.10 thus illustrates clearly the performance limit characteristics for the control system to be validated.

(2)

Performance Limit Map over Parameters

Meanwhile, for vibration absorber design, the primary structure parameters are predetermined. To be specific, it is still assumed to be m 1 = 1 kg, c1 = 2 N s/m, and k1 = 10 N/m, thus one has: ! m 2 ω2 10−ω2 + 2 jω + 2(k2 + jc2 ω) !  G= (4.64) −m 2 ω2 + 2(k2 + jc2 ω) 10−ω2 + 2 jω This would have defined the “performance limit map” for the harmonic vibration control system over parameters. For example, designation of m 2 = 1 kg and c2 = 1  N s/m for harmonic frequency ω = k1 m 1 = 3.16 rad/s leads to: G=

5k2 + 31.6 j (3.16 j)(k2 − 5 + 3.16 j)

(4.65)

4.3 Experimental Design for Verification and Validation …

171

Performance Limit Map over Frequencies 0.5

Real(G)

0 -0.5 -1 -1.5 -2 0

1

2

3

4

5

6

7

5

6

7

Frequency [rad/s]

Imaginary(G)

0.5 0 -0.5 -1 -1.5 0

1

2

3

4

Frequency [rad/s]

Fig. 4.11 Performance limit map over frequencies

Evaluation of stiffness k2 over a specified range will lead to a characterization of performance limit for stiffness design, as shown in Fig. 4.12. It is seen clearly that at the harmonic frequency, it is not possible to approach the performance limit through only the design of stiffness, unless it is constrained by physical parameter restriction. Remark: Similarly, performance limit map can be obtained for both mass and damping designs illustrating the performance limit characteristics for physical parameters. Remark: The analysis can be further carried out for simultaneous consideration of both frequency and parameters. Although the resulting maps are multi-dimensional, they do provide a straightforward “visualization” feature to performance limit characteristics for the control system to be validated. Remark: Finally, it is noted that the presented model can also be used for vehicle suspension systems, tuned mass dampers etc. Thus the simple experimental design exposed here can be readily utilized for vibration control V&V for other realistic systems.

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Performance Limit Map over Stiffness k2 1

Real(G)

0 -1 -2 -3 0

5

10

15

10

15

Stiffness k2 [N/m]

Imaginary(G)

0 -1 -2 -3 -4 0

5 Stiffness k2 [N/m]

Fig. 4.12 Performance limit map over stiffness

4.3.5 Summary and Discussions for V&V An experimental design for testing performance limit for harmonic control systems has been proposed in the note. Theoretical foundation together with detailed experimental design procedures has been presented. This has been further illustrated through numerical examples, which shows clearly that the proposal is simple and does not require complicated iterative procedures. This is remarkable particularly considering the challenges in determining the performance limit using the state-ofart approaches. This should be recognized as a contribution of the current proposal; meanwhile further discussions are worth being forged: (1)

(2)

The proposed experimental design method can be utilized to determining performance limit maps over both frequencies and physical parameters. This is an important extension since the resulting maps characterize the performance limits of the control system to be validated, and thus they can facilitate the designer for system detuning towards optimal performance. A further generalization can be considered for “intentional” design for achieving specific features. For example, if locating (−G) to be near (−1, 0) through appropriate parameter selections, the resulting design will possess

References

(3)

173

the distinctive feature of simultaneous and same level of attenuation. Thus no performance has to be comprised between the performance variables y( jω) and z( jω). This would open an avenue for exploring smart structures using the proposed method. Indeed, the analysis of performance limit as illustrated by performance limit maps can provide important insights into the physical properties of the vibration control system to be designed. This brings new elements to the current MBSE practice where formal procedures have been the focus while the attention to the physics of the system needs to be paid.

4.4 Summary Several important extensions have been discussed in this chapter. Of particular importance is the extension to broad-band situation where a universal design strategy has been proposed providing performance tuning over an arbitrary frequency band. Indeed, further directions can be pursued such as tailoring performance over and out of frequency band for military purpose, e.g. stealth etc. Another extension to performance improvability test has also been alluded to. This is motivated from the fact that the “capability” of vibration or noise attenuation must depend on system parameters, and it is necessitated to tell if the performance can be improved, through either active control or passive control. Following this rationale, a necessary and sufficient improvability test has been derived with the help of design methodology from previous chapters. Another extension considering the crucial tasks for verification & validation, and this follows from the development in Sect. 4.2 to test whether the control system has reached performance limit. Consequently, an experimental design for testing performance limit for harmonic vibration control systems has been proposed. Finally, it is pointed out that the presentations in Sects. 4.2 and 4.3 are “intentionally” re-derived from the procedures from previous chapters, to demonstrate how “effortless” it is to adopt the derivations for approaching and solving new problems. Henceforth the power of the proposed geometric design methodology is clearly revealed.

References 1. S. Daley, J. Hätönen and D.H. Owens, Active vibration isolation in a “smart spring” mount using a repetitive control approach, IFAC Control Engineering Practice, vol. 14, 9 (2006), pp. 991-997. 2. J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis, State-space solutions to standard H2 and H∞ control problems, IEEE Transactions on Automatic Control, 34(8), (1989), 831-847. 3. P. Gahinet and P. Apkarian, A linear matrix inequality approach to control, International Journal of Robust and Nonlinear Control, 1994.

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4. G. Vandersteen, R. Pintelon, D. Linten and S. Donnay. Extended subspace identification of improper linear systems. Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE’04). 2004. 5. J. Maciejowski. Guaranteed stability with subspace methods. Systems & Control Letters, 26 (2), 1995, pp. 153-156. 6. T. McKelvey, H. Akcay and L. Ljung, Subspace-based multivariable system identification from frequency response data, IEEE Transactions on Automatic Control, vol. 41, no. 7, (1996). 7. A. Bultheel and D.De Moor, Rational approximation in linear systems and control, Journal of Computational and Applied Mathematics, vol. 121, (2000), pp. 355-378. 8. J. Partington, Interpolation, Identification and Sampling, vol. 17, London Mathematical Society Monographs, Oxford University Press, 1997. 9. T.T. Georgiou, The interpolation problem with a degree constraint, IEEE Transactions on Automatic Control, vol.44, (1999), pp.631-635. 10. C.I. Byrnes, A. Lindquist, S.V. Gusev and A.S. Matveev, A complete parameterization of all positive rational extensions of a covariance sequence, IEEE Transactions on Automatic Control, vol.40, (1995), pp.1841-1857. 11. J.A. Ball and J.W. Helton, Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parameterization of the set of all solutions, Integral Equations and Operator Theory, vol.9, (1986), pp.155-203. 12. J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Colloquium Publications, Volume XX, 1935, pp.281–309. 13. S. Boyd, L. EL Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM books, Philadelphia, 1994. 14. D.S. Bayard, Y. Yam and E. Mettler, A criterion for joint optimization of identification and robust control, IEEE Transactions on Automatic Control, vol.37, (1992), pp.986-991. 15. J. Chen, C.N. Nett and M.K.H. Fan, Worst case system identification in : validation of a priori information, essentially optimal algorithms, and error bounds, in Proc. American Control Conference, pp.251–257, 1992. 16. G. Gu, D. Xiong and K. Zhou, Identification in H ∞ using Pick’s interpolation, Systems and Control Letters, vol.20, (1993), pp.263-272. 17. B. Boulet and B.A. Francis, Consistency of open-loop experimental frequency-response data with coprime factor plant models, IEEE Transactions on Automatic Control, vol.43, no.12, (1998). 18. G. Ferreres and G. Puyou, Feasibility of H design specifications: an interpolation method, International Journal of Control, vol.78, no.12, (2005), pp.927–936. 19. B.C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Transactions on Automatic Control, AC-26(1), (1981), 17–32. 20. K.V. Fernado and H. Nicholsan, Singular perturbational model reduction of balanced systems, IEEE Transactions on Automatic Control, AC-27(2), (1982), 466–468. 21. K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds, International Journal of Control, 39(6), (1984), 1115-93. 22. C. Coelho, L. Silveira and J. Philips, Passive constrained rational approximation algorithm using Nevanlinna-Pick interpolation, in Proceedings of the Conference on Design, Automation and Test in Europe, 2002, pp.923–931. 23. A. Megretski, Multivariable Control Systems, MIT Lectures Notes, 2004. 24. C. Mohtadi, Bode’s integral theorem for discrete-time systems, IEE Proceedings, Vol. 137, Pt. D, No. 2, March, 1990. 25. D. Gianni, A. D’Ambrogio, A. Tolk, Modeling and Simulation-Based Systems Engineering Handbook (1st ed.), USA: CRC Press, December 2014, ISBN 9781466571457. 26. M. Patrice, Model Based Systems Engineering: Fundamentals and Methods, Wiley-ISTE, October, 2014. 27. A. Ramos, V.F. José, B. Jaume, Model-based systems engineering: an emerging approach for modern systems, IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews 2012, 42(1): 101-111.

References

175

28. "Systems and software engineering - Vocabulary," ISO/IEC/IEEE std 24765: 2010(E), 2010. 29. R.G. Sargent, Verification and validation of simulation models, Proceedings of 2010 Winter Simulation Conference, 5–8 Dec, 2010, Baltimore, MD, USA. 30. Y. Liu, L. Zhang, Y. Yang, L. Zhou, L. Ren, F. Wang, R. Liu, Z. Pang, M. Jamal Deen, A novel cloud-based framework for the elderly healthcare services using digital twin, IEEE Access 2019, 7: 49088–49101. 31. M. Lutovska, I. Pavkov, V. Mijakovski, Z. Stamenkoviˇc, A basic approach to the verification and validation of sorption isotherm models, Journal on Processing and Energy in Agriculture, vol. 22, pp. 95, 2018. 32. J. Zumberge, M. Boyd, R. Ordonez, Validation of a DC-DC Boost Circuit Control Algorithm, SAE International Journal of Aerospace, vol. 9, pp. 112, 2016. 33. S. Skogestad, I. Postlethwaite, Multivariable Feedback Control: Analysis and Design (2nd ed.), Wiley, 2005, ISBN 978–0–470–01167–6. 34. S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge: Cambridge University Press, 2004. ISBN 0-521-83378-7 35. P.E. Gill, W. Murray, M.H. Wright, Practical Optimization, London: Academic Press, 1982. ISBN 0-12-283952-8 36. J. Lee, A First Course in Combinatorial Optimization, Cambridge University Press, 2004. ISBN 0–521–01012–8. 37. J. Nocedal, S.J. Wright, Numerical Optimization (2nd ed.), Berlin: Springer, 2006. ISBN 0387-30303-0 38. J.A. Snyman, D. N. Wilke, Practical Mathematical Optimization: Basic Optimization Theory and Gradient-Based Algorithms (2nd ed.), Berlin: Springer, 2018. ISBN 978-3-319-77585-2 39. F. Rothlauf, Optimization methods, In: Design of Modern Heuristics, Natural Computing Series. Springer, Berlin, Heidelberg, 2011. 40. J. Evans, Optimization Algorithms for Networks and Graphs (2nd edition), CRC Press, 1992.

Chapter 5

Energy Harvesting for Performance Limit

Active control has been approached with the proposed geometric design method. Previous chapters have provided detailed exposition and demonstrated that the proposal can be a systematic design methodology to active control towards performance limit. This chapter aims to show that the proposed framework can be utilized to handle another important issue of energy harvesting, while another objective is to obtain the performance limit independent upon specific parameters.

5.1 Active Control for Energy Harvesting with Performance Limit The concept of active control for energy harvesting is not widely recognized due to the inherent energy consumption with sensing and actuation devices in active solutions. It is only necessitated if the energy produced is greater than that consumed. This raises the issue of quantifying how much performance improvement can be achieved with active controls. This challenging problem upsets many conventional methods since it requires the obtainment of performance limit and the corresponding controller. In this section, a graphical method is proposed which provides appealing features of determining performance limit from simple visual inspections. A number of important results can be obtained regarding solution existence, best achievable performance, optimal controller synthesis etc. Thus the feasibility of implementing active controls can be determined at least for initial estimation and evaluation. This confidence is consolidated with obtainment of active control performance limit that will not be conquered by any other method. These results and related issues are illustrated through numerical examples.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Wang, Active Vibration & Noise Control: Design Towards Performance Limit, https://doi.org/10.1007/978-981-19-4116-0_5

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5 Energy Harvesting for Performance Limit

5.1.1 Introduction Vibration-based energy harvesters have been recognized as one of the promising candidates for renewable power sources [1–5]. Comparing with other renewable energies, vibration-based energy harvesting is usually not of large-scale yet with rich sources ranging from civil structures [6, 7] to transportation [8] and aerospace [9], even human motions can be scavenged for power generation [10–13] etc. As a consequence, vibration harvesters can be ideal for powering wireless sensor networks. This opens a new world of applications particularly in internet of things from smart city to intelligent healthcare, all of which are of fundamental significance to future society. Due to such wide and important applications, extensive investigation has been carried out for optimal design of energy harvesters (e.g. see [14–19] and references therein). A variety of approaches have also been proposed for further enhancement of harvester performance such as modal design [20]; performance improvement with 2 DOF [21–23] and multiple DOF [24]; nonlinear designs [25–30] etc. All the optimal design and performance enhancement techniques proposed so far are referring to the passive design methodology to energy harvesting, in the sense that active controls with sensors and actuators are not deployed. Indeed, active control for energy harvesting has not been considered in the literature. However, the fundamental idea of implementing active control for energy harvesting purpose should be carefully scrutinized—any active control will consume energy, thus it becomes unnecessary so long as the consumed energy surpasses the energy that can be produced. Certainly, for many industrial systems, active control can be so costly that the energy consumption may be more than the harvester can generate! Consequently, it must be justified that the benefit gained from active control should exceed the loss without it. Following this rationale, the following avenues should be explored: (1) (2) (3)

For a given energy harvester, can the performance be always improved with active control? How much performance improvement can be “gained” from active control? Can the benefit of implementing active control be outperformed against the cost?

If active controls can provide a positive answer to the first question, and furthermore a performance index can be relatively easily defined and evaluated for feasibility consideration, preferably with the obtainment of performance limit for quick estimation, a strong motivation for commissioning active controls can stand up worthwhile being considered, albeit subject to careful evaluation due to material, structure, and other implantation issues.

5.1.2 Performance of Energy Harvesters A general configuration of two-degree-of-freedom harvesters is considered as shown in Fig. 5.1 with dynamics:

5.1 Active Control for Energy Harvesting with Performance Limit

179

Fig. 5.1 General configuration for two-degree-of-freedom vibration energy harvesters

m 1 [x¨1 (t) + y¨ (t)] + k1 x1 (t) + c1 x˙1 (t) + k2 [x1 (t) − x2 (t)] + c2 [x˙1 (t) − x˙2 (t)] = 0 m 2 [x¨2 (t) + y¨ (t)] − k2 [x1 (t) − x2 (t)] − c2 [x˙1 (t) − x˙2 (t)] = 0 (5.1) where: m 1 and m 2 are the masses; k1 and k2 are the stiffness with c1 and c2 as damping coefficients, respectively; y(t), x1 (t), and x2 (t) are displacements in Fig. 5.1. As the parameters include the contribution from both mechanical and electrical devices, the model (5.1) can be a prototype where generic discussions can be carried out. From Eq. (5.1), it can be obtained: 

   1 −m 2 ω2 + k2 + jc2 ω X 1 ( j ω) k2 + jc2 ω = −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω k2 + jc2 ω X 2 ( j ω) Det (G)   m 1 ω2 Y ( j ω) (5.2) m 2 ω2 Y ( jω)

where: Y ( j ω), X 1 ( j ω), and X 2 ( j ω) are the Fourier trans(t), and x (t), respectively, with G = forms of y(t), x 1 2   2 −m 1 ω + k1 + k2 + j (c1 + c2 )ω −(k2 + jc2 ω) , and Det denotes −m 2 ω2 + k2 + jc2 ω −(k2 + jc2 ω) the determinate of the transfer matrix. From (5.2), the relative magnitude can be obtained: X 1 ( j ω) − X 2 ( jω) Y ( j ω) −m 2 ω2 (k1 + jc1 ω)   = −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω −m 2 ω2 + k2 + jc2 ω − (k2 + jc2 ω)2 (5.3)

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5 Energy Harvesting for Performance Limit

For energy harvesting purpose, the natural frequency of the harvester should match with the ambient vibration frequency. Once frequency matching is achieved, the relative magnitude of the harvester should be maximized for more efficient harvesting. This is usually done by solving the following optimization problem:    X 1 ( jω) − X 2 ( j ω)    max  Y ( j ω)

(5.4)

while subjecting to frequency matching conditions and other forms of constraints. The resulting solution will provide optimal performance of energy harvesters. However, one question can still be raised by asking how much performance improvement can be gained from active control. It is expected that such an active control solution should utilize minimum of sensing and actuating equipment, yet achieving much improved performance over the original design. It is thus reasonable to assuming that active control is applied through one of the mass: u(t) = k(t)x1 (t)

(5.5)

Then the question remaining to be answered is how much performance can be gained through this active control solution, albeit the feasibility of its application is still to be assessed. This will be approached from a novel graphical method to be delineated below.

5.1.3 Graphical Approach to Active Control Performance Come back to Eq. (5.2), define the disturbance force with D( j ω) = ω2 Y ( j ω) and then assume that an active control is applied at the first mass, then Eq. (5.2) becomes: 

X 1 ( j ω) X 2 ( j ω)



    1 m 1 −m 2 ω2 + k2 + jc2 ω m 2 (k2 + jc2 ω)   m 2 −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω m 1 (k2 + jc2 ω) Det (G)   D( jω) + U ( j ω) (5.6) D( j ω) =

with: U ( j ω) = K ( j ω)X 1 ( j ω) Equation (5.6) can be further manipulated to be:

(5.7)

5.1 Active Control for Energy Harvesting with Performance Limit



    X 1 ( j ω) g11 ( j ω) g12 ( j ω) U ( j ω) = X 1 ( j ω) − X 2 ( jω) g21 ( j ω) g22 ( j ω) D( j ω)

181

(5.8)

where: g11 = m1 (−m2 ω2 + k 2 + jc2 ω)/Det(G); g22 = −m2 (k 1 + jc1 ω)/Det(G); g12 = [m1 (−m2 ω2 + k 2 + jc2 ω) + m2 (k 2 + jc2 ω]/Det(G); g21 = −m1 m2 ω2 /Det(G). Thus if there is no control force, it can be readily read to be (5.3), or equivalently: X 1 ( j ω) − X 2 ( j ω) = g22 D( j ω)

(5.9)

Should the control force applied, the problem becomes how much the magnitude can be enhanced in comparison with the open-loop expression (5.9). As now the closed-loop expression from (5.8) is: X 1 ( j ω) − X 2 ( j ω) = g21 U ( j ω) + g22 D( jω)

(5.10)

Thus the enhancement over open-loop case is: X 1 ( j ω) − X 2 ( jω) g21 U ( j ω) = +1 g22 D( j ω) g22 D( jω)

(5.11)

Now define the above quantity as:    X 1 ( jω) − X 2 ( j ω)  M AC  =   M N AC g22 D( jω)

(5.12)

with M N AC denotes the relative magnitude without active control and M AC the relative magnitude with active control. Then, the feasibility of implementing active control can be evaluated against how much the magnitude can be increased from its implementation. That is: max

U ( jω)

M AC , subject to (5.7) M N AC

(5.13)

Indeed, problem (5.13) is often to be solved through H2/H∞ optimization. However, H2 optimization tackles the problem by optimizing the energy over the whole frequency band; while H∞ optimization would optimize the peak magnitude. The former can be adopted for broad band application and the latter is ideal for discrete frequency situations. For harmonic design of energy harvesters, H∞ norm is obviously an ideal candidate. Yet, the application of H∞ optimization algorithm requires weighting functions for specified discrete frequencies, and the choice of weighting functions can be of trial-and-error nature. In the following, a novel graphical approach is proposed, which can clearly illustrate the design freedom while revealing the limit of performance simultaneously. To proceed, from (5.7), (5.8), (5.11) and (5.12), it can be derived:

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5 Energy Harvesting for Performance Limit

   g21 K ( jω)[1 − g11 K ( j ω)]−1 g12  M AC  = + 1 M N AC g22

(5.14)

Define another quantity with: α( j ω) = [1 − g11 K ( j ω)]−1 − 1

(5.15)

Then (5.14) assumes a very simple form as:     M AC g21 g12 = α( j ω) + 1 M N AC g11 g22

(5.16)

Now α( jω) can be thought of as design freedom since the active controller is to be obtained as: K ( j ω) =

α( j ω) [1 + α( j ω)]g11

(5.17)

Henceforth the original problem is transformed to be finding α( j ω) so that M AC /M N AC is maximized. However, as the controller cannot be designed “arbitrarily,” henceforth α( j ω) must also be constrained. This constraint is often imposed by restricted control signal to prevent saturation. That is: |U ( j ω)/D( j ω)| ≤ β

(5.18)

Thus from (5.7) and (5.8), this constraint will enforce the controller to satisfy:    K ( j ω)[1 − g11 K ( jω)]−1 g12  ≤ β

(5.19)

Substitution of (5.17) leads to constraint on α( jω):    g11    |α( j ω)| ≤ β  g 

(5.20)

12

Essentially, the problem now becomes: max

α( jω)

M AC with subject to (5.16) and (5.20) M N AC

(5.21)

To solve the optimization problem (5.21), first note that active control only makes sense with M AC /M N AC greater than unity. From (5.16), this is to say:     α( jω) g21 g12 + 1 > 1   g11 g22

(5.22)

5.1 Active Control for Energy Harvesting with Performance Limit

183

Assume a constant γ (γ > 1), and let:     α( jω) g21 g12 + 1 ≥ γ   g g

(5.23)

11 22

then optimization problem (5.21) is equivalent to the following problem: max γ with subject to (5.20) and (5.23)

α( jω)

(5.24)

Now let g = g11 g22 /g12 g21 , and observe (5.22) while considering |α( jω)/g + 1| = 1, it represents a circle with centre (−g) and radius |g| (hence the circle always intersects with the origin). Inequality (5.23) with equality would be another circle concentric with |α( j ω)/g + 1| = 1 yet scaled γ times. This is illustrated in Fig. 5.2a. Now the constraint (5.20) can be brought in which is simply a circle centered at the Origin with radius β|g11 /g12 |, see Fig. 5.2b. Here comes the crucial observation: the solution to the optimization problem (5.24) is obtained by maximizing γ subject to constraints (5.20) and (5.23); constraint (5.20) is the circle |α( jω)| ≤ β|g11 /g12 | and its interior, while constraint (5.23) is the scaled concentric circle with respect to |α( jω)/g + 1| = 1. Considering that γ should be maximized, it is required γ > 1, henceforth is the outer scaling circle (refer to the blue dash-dotted circle). Thus the feasible selection for the design freedom α( j ω) will be the shaded area in Fig. 5.2b. The original optimization problem is to seek feasible α( j ω) that would maximize γ > 1. The optimal selection will be the intersection of the extension line (connecting the centre (−g) with the origin) with the circle |α( j ω)| = β|g11 /g12 |. That is, the point P in Fig. 5.2b. As the length of |O P| is the radius β|g11 /g12 |, then obviously the optimization result will be: γmax

  β g11 g12  =1+ |g|

(5.25)

A simple manipulation leads to:

Fig. 5.2 Graphical representation of design freedom: a concentric scaling; b design freedom constraint

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5 Energy Harvesting for Performance Limit

  γmax = 1 + β g21 g22 

(5.26)

This graphical solution is remarkable due to its intuition appealing. The complicated optimization process is thus avoided. The feasibility of implementing active control can thus be evaluated with the value of (1 + β|g21 /g22 |), against the cost associated with implantation. Before this, the following important results can be obtained by inspecting  the relative locations of the two circles |α( jω)/g + 1| = 1 and |α( j ω)| ≤ β g11 g12 , with the crucial observation that the former circle always intersects the origin while the latter circle has center at the origin. Proposition 1 Feasible solution α( j ω) always exists as shown in the shaded area, implying that active control solution can always be found for relative magnitude enhancement over open-loop.       Proposition 2 If (k1 + jc1 ω) m 1 ω2  ≤ β/2, then the circle α( jω) g + 1 = 1 will be totally located within the circle |α( jω)| = β g11 g12 , henceforth any    to the circle |α( j ω)| = β g11 g12 and exterior to the circle selection interior α( j ω) g + 1 = 1 will be a feasible solution. Proposition 3 The relative magnitude enhancement over open-loop is  maximum γmax = 1 + β g21 g22 .

5.1.4 Performance Evaluation of Active Control of Harvesters   Now the value of β g21 g22  in γmax should be determined. It is pointed out that a performance improvement is always achievable with active control. Thus the remaining question is whether the improvement of performance can overcome the cost associated with its implementation.   From the result of Proposition 3, the key point is to further maximize β g21 g22 . A direct observation is that γmax is proportional to β, meaning that control with a large saturation envelope will provide a better performance. This complies with the naïve thinking that the more the control energy inputs, the better the performance can possibly be. This point can be further pursued by specifying the controller K ( j ω), e.g. assuming damping and stiffness feedback, then: K ( j ω) = k + jcω

(5.27)

where: k and c are the corresponding stiffness and damping coefficients. Then from (5.17), it follows: α( j ω) = k + jcω [1 + α( j ω)]g11

(5.28)

5.1 Active Control for Energy Harvesting with Performance Limit

185

Solving for α( j ω) gives: α( j ω) =

(k + jcω)g11 1 − (k + jcω)g11

(5.29)

With the help of (5.7), (5.8), and (5.18), one has: |U ( jω)/D( j ω)| = |(k + jcω)[1 + α( j ω)]g12 | Substitution of (5.29) into (5.30) leads to:    U ( j ω)  |g12 |     D( j ω)  =  1   k+ jcω − g11 

(5.30)

(5.31)

As it is required |U ( j ω)/D( j ω)| ≤ β, thus the value β is dependent on k and c. For fixed system parameters g11 and g12 , the design of controller parameters k and c should be conducted by optimizing (5.30), which will result in a larger β, henceforth a better performance. While in practice, β is often to be determined from the operational envelope of actuation devices, whose physical constraints is prescribed  a prior. Thus one is left g21 g22  totally specified γ = 1 + β with a maximum magnitude enhancement max     by g21 g22 . Specifically:    g21  m 1 ω2 =   g  |k + jc ω| 22 1 1

(5.32)

Consequently, the maximum enhancement of relative magnitude is finally defined feasi- by the harvester parameters. Given these parameters, at a certain frequencyω0 , g21 g22  e.g. whether bility of active control can then be evaluated through (5.32),   is large enough so that γmax = 1 + β g21 g22  can provide an improved performance which is beneficial over its cost of implementation, or simply over the dissatisfaction of open-loop performance.

5.1.5 Performance Limit of Active Control From the exposition of above sections, combining Proposition 3 and (5.32), it is seen that the enhancement of the maximum relative magnitude is given by: γmax = 1 + β

m 1 ω2 |k1 + jc1 ω|

(5.33)

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5 Energy Harvesting for Performance Limit

Considering the harvester works efficiently when its natural frequency matches with the ambient vibration frequency. The following two scenarios are considered.

(1) Harvester working at the natural frequency ω1 = k1 m 1 Then it is obtained: γmax

 = 1 + β

1 1+

(5.34)

c12 m 1 k1

Hence for fixed β, small c1 and large m 1 &k1 are desired. Considering mass is usually left for frequency matching with general design guidelines such as large parameters for mass-1 system (c1 and k1 ), one is left with choosing a large k1 for further maximizing γmax . However, k1 cannot be arbitrarily large and must be constrained by its maximum value k1 ≤ K 1 , a decision can then be made by evaluating the gain of active control with this performance limit: γmax

(2)

 = 1 + β

1 1+

(5.35)

c12 m1 K1

Harvester working at the natural frequency ω2 =

k2 m 2

Similarly, now it is obtained: m 1 ω22 γmax = 1 + β

k12 + c12 ω22

(5.36)

Hence for fixed β, small c1 and large m 1 will be desirable, which is the same as situation (5.1); however, contrary to the above situation, a small k1 is required. Again considering the general design guidelines with large c1 and k1 , this puts the situation into a dilemma where the active control performance is much compromised. Fortunately now it is the mass m 2 that is left for frequency matching and a large m 1 is still available for further optimize γmax . As again, m 1 cannot be arbitrarily large and must be constrained by its maximum value m 1 ≤ M1 , a decision can then be made by evaluating the active control performance limit: M1 ω22 γmax = 1 + β

k12 + c12 ω22

(5.37)

Indeed, one is left with the hope that large parameters of m 1 , c1 and k1 still provide beneficial gain of active control with large enough γmax .

5.1 Active Control for Energy Harvesting with Performance Limit

187

Remark 1 The above performance limit is very important since it provides the best achievable performance with active control solution. Remark 2 Thus (5.35) and (5.37) can be utilized as a benchmark for evaluating feasibility analysis. Although detailed implementation depends on structure, size, material, and many other issues, the maximum relatively magnitude at the matching frequency is still an important index for quick assessment. Remark 3 The above analysis also reflects the fact that harvester performance depends crucially on its operating frequency.

5.1.6 Numerical Examples To illustrate the proposed graphical method and the procedures for evaluating active control solutions, consider a harvester with parameters m 1 = m 2 = 1, c1 = 1, c2 = 0.1, k1 = 2, and k2 = 1. Then it is readily  calculated with:   3 − ω2 + j1.1ω −(1 + 0.1 j ω) G= , g11 = 1 − ω2 + 0.1 j ω Det (G); 2 −(1 + 0.1 jω) 1 − ω + 0.1 j ω   g22 = −(2 + j ω) Det (G); g12 = 2 − ω2 + 0.2 j ω Det (G); g21 = −ω2 Det     (G).    1−ω2 +0.1 j ω  (1−ω2 +0.1 jω)(2+ j ω) . g11 g22 = Thus  gg1211  =  2−ω 2 +0.2 j ω , and g = g g 21 12 (2−ω2 +0.2 j ω)ω2 Graphical Approach to Active Control Performance As a first step, At the natural frequen it is preceded with plotting the design

freedom. cies ω1 = k1 m 1 = 1.414 rad/s and ω2 = k2 m 2 = 1 rad/s, the graphical representation of design freedom is illustrated in Fig. 5.3. Also shown is the different selection of saturation values: specifically, for ω1 = 1.414 rad/s, β = 1 and β = 0.5; while for ω2 = 1 rad/s, β = 1 and β = 5. Thus: (1)

(2)

(3)

According to Propositions 1, feasible  α( jω) always exists in the inter-  solution section area for each pair of α( j ω) g + 1 = 1 and |α( jω)| = β g11 g12  circles in Fig. 5.3;    + jc1 ω) m 1 ω2  = 2.24, thus for the case β = 5, the Refer to Fig.  5.3b, (k1  2   jc1 ω) m 1 ω ≤ β/2 is satisfied. Consequently, the circle condition  (k1 + α( j ω) g + 1 = 1 is totally located within the circle |α( j ω)| = β g11 g12 ,     thus any selection interior to  the circle |α( j ω)| = β g11 g12 and exterior    to the circle α( j ω) g + 1 = 1 is a feasible solution. This confirms with Proposition 2. According to Proposition 3, the maximum  relative magnitude enhancement over open-loop is γmax = 1 + β g21 g22 . Thus γmax for the four cases illustrated in Fig. 5.3 can be calculated, or even direct read from the figure to be: γmax = 1.82 or 5.2 dB (for ω1 = 1.414 rad/s and β = 1), γmax = 1.41 or 3 dB

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5 Energy Harvesting for Performance Limit

Fig. 5.3 Graphical representation of design freedom: a ω1 = 1.414 rad/s; b ω2 = 1 rad/s

(4)

(for ω1 = 1.414 rad/s and β = 0.5), γmax = 1.45 or 3.2 dB (for ω2 = 1 rad/s and β = 1), and γmax = 3.24 or 10.2 dB (for ω2 = 1 rad/s and β = 5). The corresponding optimal selections for α( j ω) are simply the values corresponding to the points P1 and P2 in Fig. 5.3, which can be directly written down as: α( j ω) = −0.9 + 1.55 j (with γmax = 1.82), α( j ω) = −1.7 + 3.16 j (with γmax = 1.41), α( jω) = −0.02 + 0.1 j (with γmax = 1.45), and α( j ω) = −0.13 + 0.48 j (with γmax = 3.24). Consequently the optimal controller K ( j ω) can be readily obtained from (5.17).

5.1 Active Control for Energy Harvesting with Performance Limit

189

Thus fundamentally important results regarding with optimal controller design and optimal performance can be easily determined from the graphical inspections. Active Control Performance Evaluation and Performance Limit As stated in Sect. 4, β is often determined from the operational envelope of actuation devices, and thus it is prescribed  prior to the design. With the maximum magnitude enhancement γmax = 1 + β g 21 g22 , further performance improvement can only be proceeded with the term g21 g22 . Now assume β = 1, still with the above parameters m 1 = 1, c1 = 1, and k1 = 2: γmax = 1 +

ω2 |2 + j ω|

(5.38)

A plot of γmax with frequency is shown in Fig. 5.4a. Obviously a monotonic increase of γmax with frequency is observed. Thus for increased efficiency of harvester, the first national frequency of ω1 = 1.414 rad/s should be “matched” with increased value of γmax confirmed in Sect. 6.1. Meanwhile, suppose for matched frequency ω1 = 1.414 rad/s with “matching” mass m 1 = 1, and c1 = 1, then from (5.34) we have:  γmax = 1 +

k1 1 + k1

(5.39)

A plot of γmax with stiffness is shown in Fig. 5.4b. It is seen that γmax is not only constrained by non-arbitrarily large k1 ≤ K 1 , but also it is actually limited by γmax < 2. (

)

1.9

3.5 1.8

Gamma-max

Gamma-max

3 2.5 2

1.7 1.6 1.5 1.4

1.5

1.3

1 1.2

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

Frequency [rad/s]

k1

(a)

(b)

2

2.5

3

Fig. 5.4 Active control performance: a maximum relative magnitude with change of frequency; b performance limit with stiffness

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5 Energy Harvesting for Performance Limit

Thus for frequency matching at ω1 , even one has the design freedom of specifying a large parameter of k1 for further maximizing γmax , it is still having the performance limit: γmax < 2 or 6 dB

(5.40)

Thus active control will provide best achievable 6 dB improvement over open-loop performance. And one is left with the decision whether this active control solution is “good” enough over its cost of implementation. Of course, in certain situations, there can be a sufficient reason for dissatisfaction with the available harvester design, and one might ponder upon that 6 dB improvement is significant and thus worth with the implementation.

5.1.7 Summary for Active Control of Energy Harvesters for Performance Limit The concept of active control for energy harvesting has been proposed. This has been tackled by a graphical approach for performance improvement of harvesters with active controls. This approach has been demonstrated to have intuitively appealing features, with direct obtainment of optimal design and limit of performance. It is noteworthy that such capabilities have rarely been found in the existing methods. Particularly with the obtainment of performance limit, the problem of implementing active control can be assessed and evaluated. Yet, it must be pointed out immediately that this is only a crude evaluation in the sense that there are many other issues for determination of active control schemes, such as requirement from size, weight, maintenance considerations etc. Indeed, the design proposed here should not be treated as a “stand-alone” result for active control to energy harvesting: on the one hand, it points out that active control can always be found for performance improvement; on the other hand, active controls themselves are constrained whose feasibility of implementation needs to be evaluated. But situations do exist where the proposed results can find promising applications, e.g. in vibration absorbing systems such as tuned mass dampers, vehicle suspension systems, or military surveillance & attack systems, active controls are often necessitated; part of the resulting control can thus be assigned for energy harvesting while completing the task of vibration attenuation. Consequently the results in this section provide the opening up of a new avenue for energy harvesting through active control. While the feasibility of such a concept is still an open debate, the proposed design methodology should really be thought of as a benchmarking for initial estimate and evaluation. This also furnishes the contribution of the monograph.

5.2 Optimal Design of Two-Degree-Of-Freedom Vibration …

191

5.2 Optimal Design of Two-Degree-Of-Freedom Vibration Energy Harvesters Section 5.1 has presented the active control of energy harvesters with the proposed design methodology. This alluded to the general question of optimal design of energy harvesters. Indeed, vibration-based energy harvesting has received extensive investigation due to its promising applications particularly for self-sustained wireless sensor networks. Many methods have been proposed for optimal design of harvesters. Most of the approaches utilize H2/H∞ optimization to obtain optimal design parameters. However, such optimization usually leads to a Pareto set, and the questions associated with optimal selection within the optimal set have rarely been considered. This paper thus proposes a successive optimization method, and continuous improvement of harvester performance is ensured by gradually incorporating the available design freedom. The proposed design method can guarantee frequency matching while operating towards the limit of the harvester performance. A series of design guidelines are proposed during the development with sound theoretical foundation yet for straightforward applications. Optimal design with constraints is considered. The rarely noticed yet important issue of avoiding ineffective designs is also addressed. A variety of numerical examples are provided for illustration of design concepts and claims. As a consequence, a systematic treatment towards optimal design of harvesters is presented.

5.2.1 Introduction Vibration-based energy harvesters have been widely recognized as promising candidates for renewable power sources [31–33]. A variety of vibration energy harvesting configurations has been proposed, among which mass-damper-spring systems receive general acceptance as prototypes for analysis and optimal design of harvester performances. Indeed, a conventional energy harvester is often modeled by a single DOF mass-damper-spring system [1, 8, 34]. However, it is also known that the 2 DOF mass-damper-spring harvester is more energy-efficient [21, 23]. Indeed, even the local maximum power of 2DOF harvesters can be larger than the maximum power of single DOF ones [22, 27]. And this has provided strong motivation to the investigation into optimal design of vibration energy harvesters with dual mass or multiple DOF designs [26, 35]. Most of the design approaches are preceded with H∞/H2 optimizations [14–19]. Essentially, optimal design of harvesters is achieved through optimization of the corresponding performance indices. However, optimization usually leads to a Pareto set, indicating that all the combinations within the set provide optimal selections for parameters. Then a series of questions can be posed such as: (1)

How to choose one combination of parameters within the set as the optimal selection? Or:

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5 Energy Harvesting for Performance Limit

(2)

Within the Pareto set, why is one combination of parameters better than another? Is the resulting design can be further optimized with this additional design freedom?

(3)

These questions are rarely asked but it is believed that they should be answered by any systematic optimal design method. Besides, optimization methods do not usually consider the contribution of ambient vibration frequency in the first place. For example, for H2 optimizations, the whole frequency range is “integrated off” from performance indices, resulting in the loss of frequency information for harvester design; for H∞ optimizations, however, they do provide optimal results at the natural frequency, yet it is not necessarily true for the natural frequency to match with the ambient vibration frequency—bear in mind that an optimal harvester must match its natural frequency with the ambient vibration frequency. Indeed, one of the most prominent issues in optimal design of harvesters is the mismatching between the harvester natural frequency and the ambient vibration frequency. Both theoretical and experimental results demonstrate that the harvester performance deteriorates seriously once the ambient vibration frequency moves out of the effective range around the natural frequency band [2, 3, 5, 25]. Therefore, the harvester must be designed to have a natural frequency as the ambient vibration frequency, not the other way around. Indeed, frequency matching should be considered before any performance enhancement technique can be applied; even after the frequency match, a methodology is necessitated for avoiding those ineffective designs with mismatched frequencies. This effort is attempted in this paper. The proposed methodology possesses the following features: (1) (2) (3) (4)

Performance limit is obtained and used as the starting point for design; Frequency match is guaranteed through the design process; Best achievable harvester performance is maintained over the to-be-designed parameters; Ineffective designs can be avoided that may seriously deteriorate the performance.

The above features form the contribution of the current section. It is thus emphasized that design should be based on knowing the performance limit, so that the resulting optimal design can achieve best available performance.

5.2.2 System Modeling and Performance Limit To illustrate the optimal design procedures, the general configuration for the 2DOF harvesters is considered as shown in Fig. 5.5. And the dynamics of the systems can be written as:

5.2 Optimal Design of Two-Degree-Of-Freedom Vibration …

193

Fig. 5.5 General configuration of 2DOF vibration energy harvesters

m 1 [x¨1 (t) + y¨ (t)] + k1 x1 (t) + c1 x˙1 (t) + k2 [x1 (t) − x2 (t)] + c2 [x˙1 (t) − x˙2 (t)] = 0 m 2 [x¨2 (t) + y¨ (t)] − k2 [x1 (t) − x2 (t)] − c2 [x˙1 (t) − x˙2 (t)] = 0 (5.41) where: m 1 and m 2 are the masses; k1 and k2 are the stiffness with c1 and c2 being damping coefficients, respectively; y(t), x1 (t), and x2 (t) are displacements as indicated. The corresponding parameters include both mechanical and electrical contributions, rendering the model to be a prototype for discussion of generic properties and illustration of general design procedures. With these preliminaries, the harmonic frequency response can be written down: X 1 ( j ω) − X 2 ( j ω) Y ( jω) −m 2 ω2 (k1 + jc1 ω)   = −m 1 ω2 + k1 + k2 + j (c1 + c2 )ω −m 2 ω2 + k2 + jc2 ω − (k2 + jc2 ω)2 (5.42) where: Y ( j ω), X 1 ( j ω), and X 2 ( j ω) are the Fourier transforms of y(t), x1 (t), and x2 (t), respectively. In a previous result, it has been able to show the following result: Theorem 1 The magnitude of the 2DOF energy harvesters at a frequency ω is bounded by:    X 1 ( j ω) − X 2 ( j ω)      Y ( j ω)

194

5 Energy Harvesting for Performance Limit

≤

m2ω c1 +

m 22 c2 ω4 2 k2 −m 2 ω2 +c22 ω2

(





)

k12 + c12 ω2 , ∀ω 2 k2 − m 2 ω2 + c22 ω2

(5.43)

with the equality condition:     m 2 ω2 k2 k2 − m 2 ω2 + c22 ω2 k1 − m 1 ω =  2 k2 − m 2 ω2 + c22 ω2 2

At the natural frequency ω2 =

(5.44)

k2 m 2 , the magnitude is bounded by:

   X 1 ( j ω) − X 2 ( j ω)  ≤    Y ( j ω)

m 2 (m 2 k12 + k2 c12 ) c1 c2 + m 2 k2

(5.45)

with the equality is achieved with condition: m1 + m2 k1 = k2 m2

(5.46)

The latter part of the result will be taken as the starting point for optimal design of harvesters, while the former part can be shown for avoidance of ineffective designs that has rarely been taken into consideration in the literature.

5.2.3 Optimal Design of Harvesters While most of the design proposals start from defining performance index, moving to optimization of the performance index, ending with a set of optimal selection for design parameters. As commented in the introduction, additional design freedom has been utilized in an efficient manner. In current proposal, optimization result is used as the starting point for design, henceforth, e.g. it is known that at the second mass natural frequency, the magnitude has an upper bound from (5.5); then a good idea for design is to choose k2 m 2 so that the natural frequency ω2 = k2 m 2 matches with the ambient vibration frequency. Thus the resulting harvester will have best efficiency, should it work with maximum magnitude. That is:    X 1 ( jω) − X 2 ( j ω)  =    Y ( j ω)



m 2 (m 2 k12 + k2 c12 ) c1 c2 + m 2 k2

(5.47)

5.2 Optimal Design of Two-Degree-Of-Freedom Vibration …

195

The upper bound of the magnitude can indeed be achieved provided that the parameters are such chosen to satisfy condition (5.46). To recap, the general design procedures can be summarized as:

Step 1: Choose k2 m 2 so that the natural frequency ω2 = k2 m 2 matches with the ambient vibration frequency ω0 —this guarantees “frequency matching;” 2 Step 2: Choose m 1 , k1 so that kk21 = m 1m+m —this guarantees the attainment of upper 2 magnitude bound (5.47); Step 3: Choose the remaining available design freedom to further maximize the upper bound (5.47)—this provides further performance improvement within the selections from “Pareto set.” Step 4: The performance of the harvester over other frequencies is evaluated through the general bound as expressed in (5.43). It is seen from the procedures that optimality is guaranteed in the first place; and rather the key to design has been transformed to the quest to the question of how to choose parameters within the Pareto set as the optimal design? To say the least, it is seen that the resulting design will be further optimized through the utilization of remaining available design freedom. By doing this, the concept of optimal design has been refined from simply optimization to performance improvement upon optimization. Essentially, this can be regarded as another optimization problem stated as:

m 2 (m 2 k12 + k2 c12 ) optimi ze f = (5.48) c1 c2 + m 2 k2 subject to :

k2 m2

= ω02 &

k1 k2

=

m 1 +m 2 m2

Now notice that the frequency matching condition is only a ratio of k2 and m 2 , while the equality condition can be manipulated to be: k1 = k2 + m 1 ω02

(5.49)

Thus rather than proceeding with solving the optimization problem, it is illuminating to analyze the upper bound with consideration of each design freedom; the constrains are then fulfilled through remaining design variables. Performance Improvement with Mass Design Freedom Observe the upper bound (5.7), it is seen that its value is independent on m 1 . This implies that the bound is respected for any choice of m 1 . Of course, the value m 1 must be prescribed and henceforth enforced through the equality condition (5.9). While for m 2 , it can be derived that the maximum of (5.7) is achieved by setting d f dm 2 = 0 leading to:

196

5 Energy Harvesting for Performance Limit

m2 =

k2 c12 c2 − 2k12 c2

(5.50)

k22 c1

Satisfaction of (5.50) surely implies that:  2 c1 k1 >2 c2 k2

(5.51)

due to the positive definiteness of mass; otherwise the maximum is not achieved yet the magnitude is still upper-bounded on the boundary (determined from the specific system parameters). Now assume (5.50) is satisfied, then the upper bound becomes:

f =

m 2 (m 2 k12 + k2 c12 ) c1 c2 + m 2 k2

k2 c1 =   2 c2 k22 c1 − k12 c2

(5.52)

The magnitude of the harvester now achieves its maximum value (5.52), provided with the satisfaction of frequency matching k2 = m 2 ω02 and conditions (5.49)– (5.51). Meanwhile this maximum magnitude is totally indepedent on the mass of the harvester. The fact that design freedom is still available implies that the performance can be further improved through stiffness and damping design of the harvester. This can be illustrated with the following numerical examples. Example 1 Assume that k1 = 2k2 and c1 = 9c2 , thus inequality (5.51) is satis81c2 fied; the upper bound is calculated to be f = 2√9 5 at m 2 = k2 2 ; the corresponding frequency matching and equality conditions are k2 = m 2 ω02 and k2 = m 1 ω02 , respectively. This can be re-stated as: the maximum magnitude is achieved with f = 2√9 5 for conditions:

(i) m 1 =

k2 k2 ; (ii) m 2 = 2 ; (iii) c2 = ω02 ω0

√

m 2 k2 9

(5.53)

Thus for any ambient vibration frequency ω0 , although the maximum magnitude is fixed, a design freedom k2 is still available for exploitation, e.g. to accommodate further design constraints from other requirements. The above example demonstrates that additional design freedom exists albeit the maximum value is fixed. Indeed, delicious manipulations can be made for further performance improvement through transforming into another optimization problem (with respect to optimize the maximum magnitude (5.52)) by exploiting the available design freedom as:

5.2 Optimal Design of Two-Degree-Of-Freedom Vibration …

197

k2 c1 optimi ze f =   2 c2 k22 c1 − k12 c2 subject to : (i ) m 1 =

k1 −k2 ; ω02

(5.54)

(ii) m 2 =

k2 ; ω02

(iii) c2 =

k22 c1 . c12 ω02 +2k12

The constraints in (5.54) are intentionally manipulated to “exhibit” design freedoms, and they are equivalent to the combined frequency matching condition, equality condition (9), and mass optimum condition (10). It is noteworthy that from the third constraint, it is obtained:  2 c2 k1 c1 = 1 +2 c2 m 2 k2 k2

(5.55)

Henceforth this guarantees that the inequality (5.51) is “automatically” satisfied as well. Exploiting further design freedom for performance improvement is handled in the next subsection, after a numerical example is given below to illustrate the concept. Example 2 Still assume k1 = 2k2 , then the first two constraints in (5.54) again lead to the same ones as in (5.53), namely:

m1 = m2 =

k2 ω02

(5.56)

The third constraint is: c2 =

k22 c1 2 2 c1 ω0 + 8k22

(5.57)

While the performance index function f becomes: c1 f = √ 2 c2 (c1 − 4c2 )

(5.58)

Substituting (5.57) into the optimization function leads to: f =

c12 ω02 + 8k22

2k2 c12 ω02 + 4k22

(5.59)

Thus the optimization problem in (5.54) becomes the optimization of (5.59), subject to (5.56) and (5.57). Now assume that the ambient vibration frequency ω0 = 1rad/s, then a plot of (5.59) with respect to c1 and k2 is shown in Fig. 5.6. This shows clearly that the harvester performance can be further improved with mass design

198

5 Energy Harvesting for Performance Limit 60

Fig. 5.6 Further optimized performance index function with c1 & k2

c1=0.1 c1=0.5 c1=0.8 c1=1.0

Performance Index f

50 40

30 20 10

0 0.01

0.02 0.03

0.04

0.05

0.06 0.07

0.08

0.09

0.1

k2

freedom. Even more motivating is that design freedom is still available indicating that the performance can be enhanced. And this is looked at in the next subsection. Performance Improvement with Stiffness Design Freedom The numerical examples in the last subsection demonstrate that the performance function can be further optimized since extra design freedom is still available. Performance improvement through stiffness design is analyzed. To proceed, it is seen that the performance index function in (5.54) can be manipulated to be: c1 f =   2 c2 c1 −

k12 c k22 2



(5.60)

From (5.60), it is seen that the performance index function is monotonically proportional (albeit nonlinear) to k1 and monotonically inverse-proportional (albeit nonlinear) to k2 . Henceforth, the following design guidelines can be alluded to for improvement of the harvester performance: Design Guideline 1: With performance improvement through mass design in (5.54), the harvester performance can be further enhanced by choosing large k1 and small k2 . Performance Improvement with Damping Design Freedom Now consider the problem of improving performance by damping design. A straightforward derivation of d f /dc1 = 0 and d f /dc2 = 0 leads to the identical optimality condition:  2 c1 k1 =2 c2 k2

(5.61)

5.2 Optimal Design of Two-Degree-Of-Freedom Vibration …

199

However, condition (5.51) dictates that the equality condition (5.61) can not be attained for positive mass requirement. Thus inequality (5.51) implies that the performance function in (5.54) not only cannot achieve its maximum, but also monotonic in damping. This is so since it can be shown that: f =  2

k24 4k12

k2 

−k12

k22 − cc21 2k12

2

(5.62)

From condition (5.51), it is required: c2 k22 > 2 c1 2k1

(5.63)

Thus the performance function in (5.62) is indeed monotonically inverseproportional (albeit nonlinear) to cc21 . Hence the following design guidelines follow for improvement of the harvester performance: Design Guideline 2: With performance improvement through mass design in (5.54), the harvester performance can be further enhanced by choosing large c1 and small c2 . Meanwhile, from (5.62) and (5.63), it is seen that: f
1 , i ∈ [ 2, N ]  N   g s j=2 i j j0

(6.13)

Re-parameterize K ( jω) with α( jω) as defined by: α( jω) = [1 − g11 K ( jω)]−1 − 1

(6.14)

And further define: α( jω) G i ( jω) N g11 j=2 gi j s j0 G i ( jω) = N gi1 k=2 g1k sk0 βi ( jω) =

(6.15)

(6.16)

6.3 Solution to Optimal Performance

227

Then the performance specification (6.12) can be simplified to be: |α( jω) + 1| ≤ δ1 < 1 |βi ( jω) + 1| ≥ δi > 1 , i ∈ [ 2, N ]

(6.17)

That is, the original design problem has been transformed to the design of α( jω) such that (6.17) is satisfied. Remark 6.3 α( jω) is now the design freedom, and once optimal value is designated, the optimal design K ( jω) is readily available from (6.14): K ( jω) =

α( jω) [1 + α( jω)]g11

(6.18)

Remark 6.4 The performance specification for vibration distribution in X i ( jω) (i ∈ [ 2, N ]) is determined by the design freedom α( jω) through βi ( jω) in (6.15). Remark 6.5 It is worth pointing out that G i ( jω) relates α( jω) with βi ( jω), yet it is solely determined by the system dynamics (see (6.16)). This is a crucial observation since it lays down the foundation for later development towards best achievable performance or performance limit. To proceed, it is noted that |α( jω) + 1| = 1 is a unit circle on the complex plane and |βi ( jω) + 1| = 1 can be mapped onto the same complex plane through (6.15), which is also a circle with centre (−G i ( jω)) and radius |G i ( jω)|. Thus the two circles always intersect at the origin. The situation with N = 3 is illustrated in Fig. 6.1a, where the referred circles are indicated by unit α-circle, β2 -circle, and β3 -circle, respectively. Considering the fact that |α( jω) + 1| = δ1 < 1 is the inner scaling with respect to the unit α-circle (called scaled α-circle) while |βi ( jω) + 1| = δi > 1 is the outer scaling with respect to βi -circle (called scaled βi -circle), then it is seen

Fig. 6.1 Performance specification can only be satisfied provided that the inner unit α-circle (or scaled α-circle) has intersection with the outer β2 -circle and β3 -circle (or scaled β2 -circle and scaled β3 -circle)

228

6 Active Vibration Distribution

that the performance specification as designated in (6.17) will be satisfied if and only if an intersection exists between the inner unit α-circle (or scaled α-circle) and the outer βi -circle (or scaled βi -circle); the feasible solutions are indicated in the shaded area in Fig. 6.1. With these observations, the following important results can be deduced: Proposition 6.1 (Existence of Solution) The vibration distribution problem with performance specification (6.4) is solvable if and only if the scaled α-circle intersects with the exterior of the (N-1) scaled βi -circles. Remark 6.6 As a consequence of Proposition 6.1, if there is no intersection, then the performance specification is too stringent and henceforth must be relaxed for existence of solution. Proposition 6.2 (Uniqueness of Solution) If the common section of the exterior of the (N-1) scaled βi -circles is tangent to the scaled α-circle, then the vibration distribution has a unique solution at the point of tangency. This claim follows from the observation in Fig. 6.1b, where the increase of performance specification gradually reduces the feasible area of solutions until the point of tangency is reached, where further improvement of performance will no longer be possible since the vibration distribution problem with such a specification becomes infeasible. Such an observation is crucial since it dictates the best achievable performance as detailed in the following results: Proposition 6.3 (Optimality of Solution for Vibration Attenuation) Given the spec|X i ( jω)| AF ≥ δi > 1 , i ∈ [ 2, N ] , the best achievable performance for ification |X i ( jω)| BE

|X 1 ( jω)| AF vibration attenuation in |X is δ1opt , which is the radius of the scaled α-circle 1 ( jω)| B E crossing the point of tangency.

Proposition 6.4 (Optimality of Solution for Vibration Enhancement) Given the |X 1 ( jω)| AF ≤ δ1 < 1, the best achievable performances for vibration specification |X 1 ( jω)| BE

|X i ( jω)| AF enhancement in |X (i ∈ [ 2, N ]) are δiopt s, which are a set of radii of the i ( jω)| B E (N-1) scaled βi -circles with common section of the exterior crossing the point of tangency.

Remark 6.7 Solutions are non-unique since different combinations of δi s (i ∈ [ 2, N ]) exist that can fulfill the optimality condition in Proposition 6.4. Indeed, the optimality conditions can lead to very specific results, e.g. consider a |X 1 ( jω)| AF |X 2 ( jω)| AF ≤ δ1 < 1 and |X ≥ δ2 > 1, 2-nodes performance specification with |X 1 ( jω)| B E 2 ( jω)| B E then Proposition 6.3 will give the following result: |X 2 ( jω)| AF Claim 6.1 For a 2-node network, if the specification |X ≥ δ2 > 1 is desig2 ( jω)| B E nated while the point (−1, 0) is inside the scaled β2 -circle, then the best achievable |X 1 ( jω)| AF is δ1opt = δ2 |G 2 | − |1 − G 2 |. performance for vibration attenuation in |X 1 ( jω)| BE

6.4 Performance Limit with Stiffness and Damping Designs

229

Similarly, Proposition 6.4 will testify the following claim: Claim 6.2 For a 2-node network, if the specification

|X 1 ( jω)| AF |X 1 ( jω)| B E

≤ δ1 < 1 is desig-

nated, then the best achievable performances for vibration enhancement in is δ2opt =

|1−G 2 |+δ1 . |G 2 |

|X 2 ( jω)| AF |X 2 ( jω)| B E

Remark 6.8 From the above claims, it is seen that the corresponding best achievable performances depend only on the prescribed performance specification and the system dynamics G 2 . Thus the proposed design reaches the performance limit in an “once for all” manner without, e.g. iterative procedures.

6.4 Performance Limit with Stiffness and Damping Designs Optimal performance or performance limit has been disseminated in the above section. The resulting solutions have implicitly assumed that there is no constraint put on the design freedom α( jω), that is, the optimal design K ( jω) resulting from the selection of α( jω) through (6.18) has been assumed to be implementable. This is not always true since the optimal design K ( jω) is usually a complex number; while K ( jω) = −(k1 + jk2 ω), thus it would require both k1 = 0 and k2 = 0 for its implementation. This is the case (3) discussed in Sect. 6.3.1 where both stiffness and damping must be designed simultaneously for optimal performance. For cases (1) and (2) where only stiffness or damping can be designed, the best achievable performance will be constrained. To see this, recall Eq. (6.18) and identify with: α( jω) = −(k1 + jk2 ω) [1 + α( jω)]g11

(6.19)

Now define the complex numbers α( jω) = μ + iν and g11 = a + ib, where μ, ν, a, and b are real numbers; substitution into (6.19) leads to:

aμ2 + aν 2 + aμ + bν − i bμ2 + bν 2 + bμ − aν α( jω)  = [1 + α( jω)]g11 (aμ + a − bν)2 + (aν + bμ + b)2 (1)

(6.20)

Performance Limit with Stiffness Design

If k1 = 0 and k2 = 0, then φ(t) = −k1 x1 with displacement or stiffness design, then the imaginary part of (6.20) must be zero leading to: bμ2 + bν 2 + bμ − aν = 0 If Im(g11 ) = 0 or b = 0, then (6.21) will be a circle:

(6.21)

230

6 Active Vibration Distribution

  1 2  a 2 1 a2 μ+ + ν− = + 2 2 2b 4 4b (2)

(6.22)

Performance Limit with Damping Design

Similarly if k1 = 0 and k2 = 0, or φ(t) = −k2 x˙1 with velocity or damping design, then the real part of (6.20) must be zero leading to: aμ2 + aν 2 + aμ + bν = 0

(6.23)

If Re(g11 ) = 0 or a = 0, then (6.21) will be a circle:     1 2 b 2 1 b2 μ+ + ν+ = + 2 2 2a 4 4a

(6.24)

It is seen clearly that purely stiffness or damping design would be significantly constrained: unlike the optimal solutions delineated in the above section where they can be located within or outside of the corresponding performance circles, the design freedom is exactly confined on the circle as dictated by (6.22) or (6.24). This is obviously a very strict constraint that may easily render the vibration distribution problem insolvable. To see this, it is further observed that the circle in (6.22) is centered at (−0.5, a/2b) while the circle in (6.24) is centered at (−0.5, −b/2a)—both circles have centers on the line with vertical axis μ = −0.5 while obviously passing through the Origin and the centre of α-circle (−1, 0). Therefore the scenario as assumed in Fig. 6.2a results in non-feasibility of vibration distribution problem since the circle does not intersect the area of feasible solutions. The influence on solution existence and optimality is further illustrated in Fig. 6.2b where all cases confirm with the statement that purely

Fig. 6.2 Performance limit by purely stiffness or damping design: a no feasible solutions would exist; b influence of performance limit by purely stiffness or damping design

6.5 Numerical Examples for Optimal Vibration Distribution

231

stiffness or damping design will enforce a severe constraint on feasible solutions, henceforth on best achievable performance. This is summarized below: Proposition 6.5 (Existence of Solution with Purely Stiffness or Damping Design1) The vibration distribution problem with performance specification (6.4) under purely stiffness or damping design is solvable if and only if the intersection among the scaled α-circle and the exterior of the (N-1) scaled βi -circles further intersects with the circle defined by (6.22) or (6.24). A further contemplation will also confirm the following result: Proposition 6.6 (Existence of Solution with Purely Stiffness or Damping Design2) If there exists a scaled βi -circle that coincides with circle (6.22) or (6.24), then no further performance improvement can be achieved for the corresponding performance specification δi . Proposition 6.7 (Loss of Performance with Purely Stiffness or Damping Design-2) Suppose solutions exist with purely stiffness or  damping   design, then loss  of performance due to this constraint can be up to δ1 G i  for |X i ( jω)| AF |X i ( jω)| B E (i ∈ [ 2, N ]). Remark 6.9 The last claim is based on the fact that the stiffness or damping circle passes through (−1, 0) and the Origin, thus its existence can cause the scaled βi -circle being (maximally) scaled back  by a distance of δ1 ; and this distance   corresponds to the performance in |X i ( jω)| AF |X i ( jω)| B E being exactly δ1 G i . Remark 6.10 Finally, it is worth pointing out that these fundamental constraints as imposed by implementability of design parameters do not seem to be paid enough attention in the literature, yet it is seen clearly that significant performance can be compromised simply due to these constraints.

6.5 Numerical Examples for Optimal Vibration Distribution A scenario is considered where prescribed vibration distribution is targeted across a structure at three locations. It is further assumed that the scenario can be described by a 3-node network as: x˙1 (t) = a1 x1 (t) +

3 

s1k xk (t) − φ(t)

k=2

x˙i (t) = ai xi (t) +

3  j=1 j=i

si j x j (t) − si0 d(t), i ∈ [ 2, 3 ]

(6.25)

232

6 Active Vibration Distribution

where φ(t) = −k1 x1 − k2 x˙1 as defined in (6.5) and the objective is to design k1 and k2 such that the designated specification is satisfied (if possible) and further elaborate on the best achievable performance at certain specified frequency: |X 1 ( jω)| AF ≤ δ1 < 1 |X 1 ( jω)| B E |X i ( jω)| AF ≥ δi > 1 , i ∈ [ 2, 3 ] |X i ( jω)| B E

(6.26)

Following the development in Sect. 6.3, it is known that (6.26) is equivalent to: |α( jω) + 1| ≤ δ1 < 1 |β2 ( jω) + 1| ≥ δ2 > 1 |β3 ( jω) + 1| ≥ δ3 > 1

(6.27)

where β2 ( jω) =

α( jω) g11 (g22 s20 + g23 s30 ) , with G 2 ( jω) = G 2 ( jω) g21 (g12 s20 + g13 s30 )

(6.28a)

β3 ( jω) =

α( jω) g11 (g32 s20 + g33 s30 ) , with G 3 ( jω) = G 3 ( jω) g31 (g12 s20 + g13 s30 )

(6.28b)

The corresponding gi j s are totally determined by the system dynamics: ⎡ 1 ⎢ G= ⎣ Det (G)

⎤ s13 s32 − s12 (a3 − jω) s12 s23 − s13 (a2 − jω) (a2 − jω)(a3 − jω) − s23 s32 ⎥ s23 s31 − s21 (a3 − jω) (a1 − jω)(a3 − jω) − s13 s31 s13 s21 − s23 (a1 − jω) ⎦ s12 s31 − s32 (a1 − jω) (a1 − jω)(a2 − jω) − s12 s21 s21 s32 − s31 (a2 − jω)

(6.29)

where

Det (G) = (a1 − jω)(a2 − jω)(a3 − jω) − s23 s32 (a1 − jω) − s13 s31 (a2 − jω) . −s12 s21 (a3 − jω) + s12 s23 s31 + s13 s21 s32

6.5.1 Optimal Performance Optimal performance can be analyzed by prescribing specific topological graph and targeted frequency, e.g. nodes dynamics and topological interactions are defined by ⎤ ⎡ −1 1 0 ⎣ 1 −3 1 ⎦ while assuming the targeted frequency ω, as well as s20 and s30 being 0 1 −2 unities. Then the parameters in (6.28a, 6.28b) and (6.29) can be readily computed, and the corresponding unit α-circle, β2 -circle, and β3 -circle can be plotted in Fig. 6.3.

6.5 Numerical Examples for Optimal Vibration Distribution

233

DESIGN FREEDOM

5 scaled a-circle scaled b2-circle scaled b3-circle

B A

0

IM A G

-5

-10

-15

-20

-25 -10

-5

0

5

10

15

REAL Fig. 6.3 Geometry of the  unit α-circle, β2 -circle and β3 -circle: shaded area indicates  that solutions exist where |X 1 ( jω)| AF |X 1 ( jω)| B E can be attenuated while both |X 2 ( jω)| AF |X 2 ( jω)| B E and  |X 3 ( jω)| AF |X 31 ( jω)| B E can be enhanced

From Fig. 6.3, it isthen immediately known that optimal solutions  exist that will attenuate |X 1 ( jω)| AF |X 1 ( jω)| B E while enhancing |X 2 ( jω)| AF |X 2 ( jω)| B E and |X 3 ( jω)| AF |X 3 ( jω)| B E . However, it is also noted that the shaded section is very and β3 -circle, and thus if a design is small when comparing with the radii of β2 -circle  |X |X ( jω)| ( jω)| B E while optimally enhancing to maintain the attenuation of 1 1 AF   |X 2 ( jω)| AF |X 2 ( jω)| |X |X or ( jω)| ( jω)| 3 3 B E AF B E , then the maximal enhance be quickly estimated through observation at ment for |X 2 ( jω)| AF |X 2 ( jω)| B E can  point A to be 1.5 dB; for |X 3 ( jω)| AF |X 3 ( jω)| B E at point B to be 1.0 dB. Consequently, the specification δ2 = 1.19 and δ3 = 1.12 will be the best achievable performance that can not be conquered through any design. On the other hand, if a design is to maintain the enhancement of |X 2 ( jω)| AF  |X 2 ( jω)| B E or |X 3 ( jω)| AF |X 3 ( jω)| B E while optimally attenuating |X 1 ( jω)| AF |X 1 ( jω)| B E , then due to the shaded  section is comparable with the radius of the unit α-circle, then |X 1 ( jω)| AF |X 1 ( jω)| B E can be significantly attenuated. The situation with an attenuation of 8 dB is illustrated in Fig. 6.4.

234

6 Active Vibration Distribution DESIGN FREEDOM

5 scaled a-circle scaled b2-circle scaled b3-circle

0

IM A G

-5

-10

-15

-20

-25 -10

-5

0

5

10

15

REAL Fig. 6.4 Geometry of the scaled α-circle scaling down by 8 dB: shaded area indicates  that solutions exist where |X 1 ( jω)| AF |X 1 ( jω)| B E can be attenuated by 8 dB while both   |X 2 ( jω)| AF |X 2 ( jω)| B E and |X 3 ( jω)| AF |X 31 ( jω)| B E remain intact

Consequently, following Propositions 6.1, 6.2, 6.3 and 6.4, the following results can be claimed: Existence of Solution: The vibration distribution problem with performance specification δ1 < 1 and δ2 > 1 and δ3 > 1 is solvable due to the existence of the shaded area. Uniqueness of Solution: When δ2 = 1.19 or δ3 = 1.12, the common section of the exterior of the 2 scaled βi -circles is tangent to the unit α-circle, then the vibration distribution has a unique solution at the point of tangency at A or B. Optimality  of Solution for Vibration Attenuation: Given the specification |X i ( jω)| AF |X i ( jω)| B E ≥ δi > 1 , i ∈ [ 2, 3 ] , the best achievable performance  for vibration attenuation in |X 1 ( jω)| AF |X 1 ( jω)| B E is δ1opt = 0.4 (or −8 dB). Optimality  of Solution for Vibration Enhancement: Given the specification |X 1 ( jω)| AF |X 1 ( jω)| B E ≤ δ1 <  1, the best achievable performances for vibra|X ( jω)| tion enhancement in 2 AF |X 2 ( jω)| B E is δ2opt = 1.19 (or +1.5 dB) while  in |X 3 ( jω)| AF |X 3 ( jω)| B E being δ3opt = 1.12 (or +1.0 dB).

6.5 Numerical Examples for Optimal Vibration Distribution

DESIGN FREEDOM

5 0

-5

IM A G

IM A G

scaled a-circle scaled b2-circle scaled b3-circle damping-circle

0

-5 -10

-10

-15

-15

-20

-20

-25 -10

DESIGN FREEDOM

5

scaled a-circle scaled b2-circle scaled b3-circle stiffness-circle

C

235

-5

0

5

10

15

-25 -10

-5

0

5

REAL

REAL

(a)

(b)

10

15

Fig. 6.5 Optimal performance with a purely stiffness design and b purely damping design

6.5.2 Optimal Performance with Stiffness or Damping Design For all the optimal selections in the above subsection, the resulting optimal selection for α( jω) is a complex number leading to simultaneous stiffness and damping design. For example, at the point A in Fig. 6.3, α( jω) = −0.85 + 0.99 j, then from (6.18) it can be calculated that the optimal design is K ( jω) = 0.2034 − 0.0074 j; the definition K ( jω) = −(k1 + jk2 ω) leads immediately to the optimal parameters: k1 = −0.2034 and k2 = 0.0074

(6.30)

Should only stiffness or damping design is targeted (or permissible), the optimal designs in the above subsections would become infeasible. First consider the case with stiffness design, thus k1 = 0 and k2 = 0, then it is known that the feasible solutions have to be further constrained on the circle as dictated by (6.22): (μ + 0.5)2 + (ν − 0.4)2 = 0.41

(6.31)

then consider the case with damping design, thus k1 = 0 and k2 = 0, then the feasible solutions are on the circle as dictated by (6.24): (μ + 0.5)2 + (ν + 0.625)2 = 0.6406

(6.32)

Both situations are shown in Fig. 6.5, and it is then known from the result expounded in Sect. 6.4.

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6 Active Vibration Distribution

Existence of Solution with Purely Stiffness Design: The vibration distribution problem with performance specification δ1 < 1 and δ2 > 1 and δ3 > 1 under purely stiffness design is solvable due to the existence of the shaded area further intersecting with the circle defined by (6.32). Meanwhile, the stiffness-circle crosses entirely the shaded area indicating no loss of performance will result with purely stiffness design. Existence of Solution with Purely Damping Design: The vibration distribution problem with performance specification δ1 < 1 and δ2 > 1 and δ3 > 1 under purely stiffness design is only marginally solvable due to damping-circle defined by (6.32) only having a very small fraction of crossing with the shaded area. Indeed, purely damping design would almost render the vibration distribution problem infeasible, thus imposing a significant constraint on achievable performance. As a comparison, purely stiffness design would still be able to maintain a [−8 dB, 1.5 dB, 1.0 dB] performance without significant deterioration.

6.5.3 Real Time Validation To further confirm the theoretical results, choose a purely stiffness design with φ(t) = −k1 x1 and magnify the section near point C in Fig. 6.5a now illustrated in Fig. 6.6: it is known from the above subsection that a choice of α( jω) = the best achievable performances −0.78 + 0.975 j at point C1 would maintain  for vibration enhancement in|X 2 ( jω)| AF |X 2 ( jω)| B E with δ2opt = 1.15 (or + 1.2 dB) while in |X 3 ( jω)| AF |X 3 ( jω)| B E being δ3opt = 1.04 (or +0.4 dB); the selection α( jω) = −0.78 + 0.975 j results in the optimal design K ( jω) = 0.1951, or k1 = −0.1951. A simulation for the optimal design is then shown in Fig. 6.7 DESIGN FREEDOM 1.5

scaled a-circle scaled b2-circle scaled b3-circle stiffness-circle

C1

1

IMAG

0.5 C2

0

-0.5

-1

-3

-2.5

-2

-1.5

-1

-0.5

REAL

Fig. 6.6 A magnified view for stiffness design in Fig. 6.5

0

0.5

1

6.5 Numerical Examples for Optimal Vibration Distribution

237

Optimal Design in X2

Optimal Design in X3

0.6

0.6 0.4

0.2

M agnitude

M agnitude

0.4

0 -0.2

0 -0.2 -0.4

-0.4

-0.6

-0.6 -0.8 0

0.2

Before Design After Design

Before Design After Design

-0.8

20 40 60 80 100 120 140 160 180 200

0

20 40 60 80 100 120 140 160 180

Time [s]

Time[s]

(a)

(b)

Fig. 6.7 Optimal design for vibration enhancement in a X 2 performance and b X 3 performance Optimal Design in X1

0.4 0.3

Magnitude

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

Before Design After Design

20

40

60

80

100

120

140

160

180

200

Time [s]

Fig. 6.8 Optimal design for vibration attenuation in X 1 performance

confirming the claim, although it is admitted that there is a marginal deviation from the theoretical prediction, and this is believed to be due to the solver tolerance. Another design is chosen at point C2 and  it is known that it will |X |X |X 3 ( jω)| B E while attenuating ( jω)| and ( jω)| retain |X 2 ( jω)| 2 3 AF B E AF  |X 1 ( jω)| AF |X 1 ( jω)| B E by −8 dB. The point leads to K ( jω) = 0.6667, or k1 = −0.6667. A real time simulation for this optimal design is shown in Fig. 6.8 clearly confirming the claim.

238

6 Active Vibration Distribution

6.5.4 Section Summary for Vibration Distribution The problem of vibration distribution across a network structure has been approached with a geometric design method. A number of fundamental results upon solution existence, uniqueness, and optimality have been obtained. The unique feature of the method is its visibility of the results, which would otherwise be relatively difficult to evaluate through conventional optimization-based approaches. Meanwhile, the proposed method can target the performance limit while quantitatively assessing the influence of constraints as imposed by physical parameters. Indeed, it has been demonstrated that purely stiffness or damping designs can enforce significant restriction on best achievable performance. And this should be paid enough attention in design optimizations. It is also worth pointing out that the proposed design has been restricted to one node, and there is actually no loss of generality since parameter design over other nodes can be carried out within the framework resulting in successive optimization of performance. In this sense, multiple nodes optimal design can be regarded as an iterative process. Yet for each node, the proposed design methodology does not require any iterative or trial-and-error procedures to approaching the performance limit. The proposal is thus of “once for all” feature to obtain the best achievable performance that is not to be conquered by any further modification.

6.6 Vibration Distribution with Annihilation The generic problem of vibration distribution has been discussed and validated through examples. From the development, it is seen that complete annihilation at nodes 1-M can be achieved by designing the newly defined parameter α( jω) to be: α( jω) ≡ −G i ( jω), i ∈ [ 1, M ]

(6.33)

while further satisfying the restriction |α( jω) + 1| ≥ δ N > 1. This result is not as “naive” as it looks, but reveals important insight since G i ( jω) is fully “prescribed” by the system parameters henceforth feasibility of annihilation is determined by the system’s own “potentiality” with: |1 − G i ( jω)| ≥ δ N > 1, i ∈ [ 1, M ]

(6.34)

The result can be summarized below: Theorem 6.1 (Analytical Solution) Complete harmonic vibration annihilation at the locations of nodes 1-M can be achieved if and only if condition (6.34) is satisfied. Remark 6.11 Sufficient conditions can be refined by noting that |1 − G i ( jω)| ≥ |1 − |G i ( jω)||, henceforth sufficiency retains with |1 − |G i ( jω)|| ≥ δ N >

6.6 Vibration Distribution with Annihilation

239

1 , i ∈ [ 1, M ] . However, this line is not pursued since condition (6.34) is easily verified albeit with both sufficiency and necessity. Remark 6.12 Consequently the problem of feasibility of complete harmonic vibration annihilation can be answered by a simple test with condition (6.34). Remark 6.13 Finally, the design parameter N can be synthesized to be:

N = α( jω)[(1 + α( jω))g N 1 ]−1

(6.35)

while complete vibration annihilation is feasible, it becomes:

N = −G i ( jω)[(1 − G i ( jω))g N 1 ]−1

(6.36)

A geometrical methodology can be developed parallel with the analytical approach above. To proceed, define: βi ( jω) =

α( jω) G i ( jω)

(6.37)

Then the performance specification can be written as: |βi ( jω) + 1| ≤ δi ≈ 0, i ∈ [ 1, M ] |α( jω) + 1| ≥ δ N > 1

(6.38)

The harmonic vibration annihilation problem can thus be re-formulated as designing the parameter α( jω) such that the inequalities (6.38) can be satisfied simultaneously. Obviously, direct optimization of (6.38) is not an easy task due to the two seemingly conflicting specifications. To provide a “visualized” version of the analytical design, the following steps are followed with reference to Fig. 6.9. Step 1: For an index i, consider |βi ( jω) + 1| = 1 and |α( jω) + 1| = 1, they can represent two unit circles (βi -circle and α-circle) on the complex planes, respectively. Fig. 6.9 Geometrical representation of performance indices with feasible solutions: the shaded area indicates the feasible design of α( jω) satisfying |α( jω) + 1| ≥ δ N > 1 and |βi ( jω) + 1| ≤ δi < 1; thus annihilation is achieved if and only if the centre (−G i ( jω)) locates outside of the scaled α-circle

240

6 Active Vibration Distribution

Step 2: Now mapping the circle |βi ( jω) + 1| = 1 onto the complex α-plane through the Möbius transformation (6.37) will result in another circle centered at (−G i ( jω)) with radius |G i ( jω)|. Step 3: On complex α-plane, |α( jω) + 1| ≥ δ N > 1 and |βi ( jω) + 1| ≤ δi < 1 will indicate concentric circles scaled outward and inward with respect to α-circle and βi -circle, respectively. Naming the two circles as scaled α-circle and scaled βi -circle. Step 4: Thus |βi ( jω) + 1| ≤ δi < 1 and |α( jω) + 1| ≥ δ N > 1 can be satisfied simultaneously if and only if the interior of scaled βi -circle has intersection with the exterior of the scaled α-circle. Step 5: Consequently, |βi ( jω) + 1| ≤ δi ≈ 0 and |α( jω) + 1| ≥ δ N > 1 can be satisfied simultaneously if and only if the centre (−G i ( jω)) locates outside of the scaled α-circle. The above procedures have demonstrated the following important result: Theorem 6.2 (Geometrical Solution) Complete harmonic vibration annihilation at the locations of nodes 1-M can be achieved if and only if the corresponding M centers (−G i ( jω)) all locate outside the scaled α-circle. Remark 6.14 The geometrical solution is remarkable in that it solves the solvability, uniqueness, and optimality problems within one unified framework, yet all with simple geometric observations. Theorem 6.2 implies that harmonic vibration annihilation is not possible should the centre (−G i ( jω)) locates inside the scaled α-circle. However the above exposition can be utilized to probe the best achievable performance or performance limit with optimal solution at Q min in Fig. 6.10. This solution is thus unique and optimal, i| . That is: which leads to vibration attenuation limit δi = δ N −|1−G |G i | Theorem 6.3 (Performance Limit) If the centre (−G i ( jω)) locates inside the scaled α-circle, complete harmonic vibration annihilation is not feasible and the optimal i| . vibration attenuation is restricted by δi = δ N −|1−G |G i | Fig. 6.10 If annihilation is not feasible, best achievable vibration attenuation is restricted to be the solution at Q min

6.6 Vibration Distribution with Annihilation

241

To recap, the analytical solution to harmonic vibration annihilation can be developed into a parallel geometric framework, where the problems of solution existence, uniqueness, and performance limit can all be resolved. The proposed geometric approach essentially provides a method for “visualizing” the performance specifications, and optimal solutions can be “read” directly from graphical inspections. These claims will be verified with the examples below. The following 3-node network is assumed where the objective is to design 3 for harmonic vibration annihilation at specific locations: x˙1 (t) = a1 x1 (t) +

N 

s1k xk (t) − u(t)

k=2

x˙i (t) = ai xi (t) +

N 

si j x j (t) − si0 d(t) , i ∈ [ 2, 3 ]

(6.39)

j=1 j=i

u(t) = 3 x3 (t)

(6.40)

⎡

⎤ −1 2 1 The network has a graph ⎣ 1 −2 1 ⎦, and the annihilation design problem is −1 0 −4 to be addressed for frequency ω = 0.1 rad/s with performance specification: |X 1 ( jω)| AF ≈0 |X 1 ( jω)| B E |X 2 ( jω)| AF ≈0 |X 2 ( jω)| B E |X 3 ( jω)| AF ≥1 |X 3 ( jω)| B E

(6.41)

The locations of centers of the scaled βi -circles are firstly determined: −G 1 ( jω) = −

g31 (g12 + g13 ) = −1.46 − 0.22i g11 (g32 + g33 )

(6.42a)

−G 2 ( jω) = −

g31 (g22 + g23 ) = −2.23 − 0.57i g21 (g32 + g33 )

(6.42b)

where the si0 s are assumed to be unity indicating that the vibration enters directly into node-1 and node-2. With reference to Fig. 6.11, now since (−G 2 ( jω)) locates outside the α-circle, henceforth Theorem 6.2 dictates that complete harmonic vibration annihilation at the location of node-2 can be achieved; while (−G 1 ( jω)) locates inside the α-circle, then Theorem 6.3 predicts that complete harmonic vibration annihilation at the location of node-1 is not feasible yet the best achievable vibration suppression comparing with the performance before the design is:

242

6 Active Vibration Distribution DESIGN FREEDOM 2 a-circle b1-circle b2-circle

1.5 1 0.5

IMAG

0 -0.5 -1 -1.5 -2 -2.5 -3 -5

-4

-3

-2

-1

0

1

REAL

Fig. 6.11 Geometrical representation of performance indices with feasible solutions: the shaded area indicates the feasible design of α( jω) satisfying |α( jω) + 1| > 1 and |βi ( jω) + 1| ≤ δi < 1; thus annihilation can be achieved at node-2 yet not at node-1

 δ2 = (δ3 − |1 − G 1 |) |G 1 | = 0.33, or −9.6 dB

(6.43)

This claim is illustrated in Fig. 6.12 where a scaling of (−9.6 dB) also confirms the prediction. To finally verify the theoretical predictions, real time simulations must be carried out. Now for annihilation design in node-2, α( jω) = −G 2 ( jω), then N can be calculated from (6.36) to be N = 5.994; for performance limit design in node1, α( jω) is chosen to be Q min = −1.89 − 0.45 j, calculation from (6.35) gives

N = 4.03. The corresponding results are presented in Fig. 6.13, which clearly confirms with the theoretical predictions. Finally, it is worth stressing again that the geometric method becomes important with increasing number of nodes. Optimization-based methods would then have to rely on extensive computation plus trials-and-errors for annihilation design and approaching the performance limit. With the capability of providing direct visualization to performance variables, the proposed method can retain physical insight with non-iterative solutions to optimal performances.

6.6 Vibration Distribution with Annihilation

243

DESIGN FREEDOM 1.5 a-circle b1-circle scaled b1-circle

1

IM A G

0.5

0 Qmin

-0.5

-1

-1.5

-2 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

REAL Fig. 6.12 Annihilation is not feasible at node-1, and best achievable vibration attenuation is restricted to be the solution at Q min with (−9.6)dB Optimal Design Performance: Node-2

Optimal Design Performance: Node-1 15

Before Design After Design

2

10

1

5

Magnitude

Magnitude

3

0

0

-1

-5

-2

-10

-3 0

200

400

600

800 1000 1200 1400 1600 1800 2000

Before Design After Design

-15 0

200

400

600

800 1000 1200 1400 1600 1800 2000

Time (s)

Time (s)

(a)

(b)

Fig. 6.13 Comparison of the performance before and after design: a annihilation design in node-2; b performance limit design in node-1

244

6 Active Vibration Distribution

6.7 Active Vibration Distribution with Conventional Approaches The above sections have demonstrated the power of the proposed geometric method. However, to further reveal the advantage and benefit of the design methodology, the last section concerns upon tackling the vibration distribution problem using conventional approaches. To proceed, consider the following control system with a general N-nodes heterogeneous dynamics subject to exogenous vibration: x˙i (t) = ai xi (t) + u i (t) = ki xi (t)

N 

si j x j (t) − si0 di (t) + u i (t)

j=1 j=i

, i ∈ [ 1, N ]

(6.44)

where xi (t) represents the state variable and u i (t) the control action with ki being the to-be-designed control gain; ai is the node dynamics, si j is the topological interaction among the nodes, and si0 is the interaction strength with the exogenous vibration di (t). The closed-loop system (6.44) can be written as: x˙i (t) = (ai + ki )xi (t) +

N 

si j x j (t) − si0 di (t), i ∈ [ 1, N ]

(6.45)

j=1 j=i

At a frequency ω, Eq. (6.45) can be written into its frequency response representation: ⎡ ⎢ ⎢ ⎢ ⎣

X1 X2 .. . XN

⎤

⎡

(a1 + k1 − jω) s12 ··· ⎥ ⎢ s (a + k − jω) · ·· 21 2 2 ⎥ ⎢ ⎥=⎢ .. .. . .. ⎦ ⎣ . . sN 1

sN 2

s1N s2N .. .

· · · (a N + k N − jω)

⎤−1 ⎡ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎣

s10 D1 s20 D2 .. .

⎤ ⎥ ⎥ ⎥ ⎦

sN 0 DN (6.46)

where X i ( jω)s and Di ( jω)s are the frequency responses of xi (t)s and di (t)s, respectively, and the dependence on frequency is omitted for ease of reference. It is also noted that the invertibility of the transfer matrix is implicitly assumed by a controllability and observability requirement. As the vibration signals can often be defined by Di ( jω) = Cˆ i D( jω), with Cˆ i being a complex number representing the gain and phase shift over D( jω), where D( jω) is a referencing vibrational signal. Thus a redefinition Ci = si0 Cˆ i leads to:

6.7 Active Vibration Distribution with Conventional Approaches

⎡

⎤ ⎤−1 ⎡ (a1 + k1 − jω) C1 s12 ··· s1N ⎢ ⎥ ⎢ ⎥ ⎢ C2 ⎥ s21 (a2 + k2 − jω) · · · s2N ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥=⎢ ⎥ ⎢ . ⎥D .. .. . .. .. ⎣ ⎦ ⎣ ⎦ ⎣ .. ⎦ . . . sN 2 · · · (a N + k N − jω) XN sN 1 CN (6.47) X1 X2 .. .

⎤

245

⎡

Now without loss of generality, for X 1 ( jω), X 2 ( jω), …, X N ( jω), it is assumed that the first M (M < N) variables are required to be attenuated for vibration attenuation; while the remaining variables are to be enhanced for vibration harvesting. That is, the following performance index is required:       X i ( jω) D  ≤ 1, i ∈ 1, M       X j ( jω) D  ≥ 1, j ∈ M + 1, N

(6.48)

While (6.48) can be referred as unitary specification, in practice, it is often required to have different specification for each performance variable leading to the general specification requirement:       X i ( jω) D  ≤ δi ≤ 1, i ∈ 1, M       X j ( jω) D  ≥ δ j ≥ 1, j ∈ M + 1, N

(6.49)

Thus the generic problem of active vibration distribution is to find feedback controls u i (t) = ki xi (t) (i ∈ [ 1, N ]), such that conditions in (6.49) are satisfied simultaneously. It is seen clearly that the aforementioned problems of distributability, performance limit and constrained handling must all be addressed for this generic active vibration distribution design.

6.7.1 Active Vibration Distribution Control Design One of the challenging issues to resolve this active vibration distribution problem is the “inconsistent” specification for performance variable sets (X 1 ( jω), …, X M ( jω)) and (X M+1 ( jω), …, X N ( jω)). To circumvent this trouble, first write (6.47) into matrix form: X = ( + S)−1 C D ⎡

X1 X2 .. .

⎤

⎡

C1 ⎢ C2 ⎥ ⎢ ⎥ ⎥, C = ⎢ . ⎣ .. ⎦

⎤

(6.50) ⎡

0 s12 ⎢ ⎢ s21 0 ⎥ ⎢ ⎢ ⎥ where X = ⎢ ⎥, S = ⎢ . . ⎣ ⎣ .. .. ⎦ XN CN sN 1 sN 2

diag (a1 + k1 − jω) · · · (a N + k N − jω) .

··· ··· .. .

⎤ s1N s2N ⎥ ⎥ .. ⎥,  = . ⎦

··· 0

246

6 Active Vibration Distribution

Then partition performance variables into attenuating variables and harvesting variables as:      X va P11 P12 Cva = D (6.51) X ve P21 P22 Cve where X va denotes the performance variables being attenuated and X ve being harvested. Obviously there have the following definitions:  X≡

     Cva P11 P12 X va ,C ≡ , and ≡ ( + S)−1 X ve Cve P21 P22

(6.52)

with appropriate dimensions compatible with specification for X 1 ( jω), X 2 ( jω), …, X N ( jω). As a consequence of such manipulations, the generic problem of active vibration distribution can be reformulated as finding state feedback controls such that the following inequalities are satisfied: P11 Cva + P12 Cve ew ≤ 1 P21 Cva + P22 Cve ew ≥ 2

(6.53)

T  where · ew denotes an element-wise operation with 1 = δ1 · · · δM and T 

2 = δM+1 · · · δN specifying the performance. For example, the first inequality actually means that the performance variables X 1 ( jω), …, X M ( jω) are all attenuated with the corresponding specification δi . To proceed, define the following partition: 

A11 A12 A21 A22

 ≡+S

(6.54)

Thus from the definition in (6.52), one can obtain: 

A11 A12 A21 A22



P11 P12 P21 P22

 =I

(6.55)

Now consider the following “identity”: 

 A11 A12 A21 A22

−1 −1 −1 −1 −1 −1 −1 A−1 11 + A11 A12 (A22 − A21 A11 A12 ) A21 A11 −A11 A12 (A22 − A21 A11 A12 ) −1 −1 −1 −1 −1 −(A22 − A21 A11 A12 ) A21 A11 (A22 − A21 A11 A12 )

 =I

(6.56)

6.7 Active Vibration Distribution with Conventional Approaches

247

Thus it can be obtained: 

 P11 P12 P21 P22

 =

−1 −1 −1 −1 −1 −1 −1 A−1 11 + A11 A12 (A22 − A21 A11 A12 ) A21 A11 −A11 A12 (A22 − A21 A11 A12 ) −1 −1 −1 −1 −1 −(A22 − A21 A11 A12 ) A21 A11 (A22 − A21 A11 A12 )



(6.57)

Substitution of (6.57) into the (6.53) would transform the original formulation into:  −1 

  A Cva − A12 P22 Cve − A21 A−1 Cva  ≤ 1 11 11 ew 

  P22 Cve − A21 A−1 Cva  ≥ 2 11 ew

(6.58)

Thus the following result can be obtained: T  Proposition 6.8 For a given specification 1 = δ1 · · · δM and 2 = T  δM+1 · · · δN , state feedback control for active vibration distribution can be obtained by solving inequalities (6.58) for ki s. Proof The result is clearly true through the above derivations.

◼

Remark 6.15 This result provides a check to the feasibility of performance specifications, thus aiming to address the vibration distributability problem explained above. Remark 6.16 If a solution to the N-inequalities in (6.58) exists, the corresponding state feedback controls in ki s can be obtained; while non-feasibility implies that the specification is not achievable and thus to be relaxed. The above result is the general condition for active state feedback control of vibration distribution. It is obtained from mathematical consideration, while physical considerations can be exercised for further development. In practice, it is often the case where vibration enters the system through (part of) the to-be-attenuated nodes, propagating to those required to be enhanced; control is then required at the to-beattenuated nodes in such a way that they achieve attenuation (or for further vibration suppression, e.g. from 6 dB reduction to 12 dB reduction etc.) while the remaining nodes achieve enhancement with prescribed specification (e.g. above 10 dB enhancement etc.). This accomplishes the concept of vibration distribution in physical setup. Thus control allocation implies that k j = 0 for j ∈ [ M + 1, N ] while Cva and Cve  T can be assigned to be Cva = I M,1 and Cve = I N −M,0 , where I M,1 = 1, · · · 1 and  T I N −M,0 = 0, · · · 0 . Substitution Cva and Cve into (6.58) leads to:  −1 

 A I + A12 P22 A21 A−1 I M,1  ≤ 1 11 11 ew    P22 A21 A−1 I M,1  ≥ 2 11 ew

(6.59)

Considering the result from control allocation, the terms A pq s in (6.59) can be written out as:

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A11 = S M×M +  M ; A12 = S M×(N −M) ; A21 = S(N −M)×M ; A22 = S(N −M)×(N −M) (6.60) where S M×M is the first M rows and M columns from matrix S and S M×(N −M) refers to the first M rows but the remaining (N-M) columns; similarly definition

applies for A21 and A22 with  M = diag (a1 + k1 − jω) · · · (a M + k M − jω) . Now from (6.57), it is seen: −1 P22 = (A22 − A21 A−1 11 A12 )

(6.61)

Substitute (6.61) into (6.59) for vibration distribution design conditions:  −1    A + A−1 A12 (A22 − A21 A−1 A12 )−1 A21 A−1 I M,1  ≤ 1 11 11 11 11 ew   (A22 − A21 A−1 A12 )−1 A21 A−1 I M,1  ≥ 2 11 11 ew

(6.62)

As S is known, it is seen from (6.60) that A12 , A21 , and A22 (with its inversion) can be pre-computed for speeding up numerical implementation. The control design parameters ki s for i ∈ [ 1, M ] are all associated with A11 . Now consider the following decomposition:

−1 −1 −1 −1 A−1 = −1 S M×M −1 11 = (S M×M +  M ) M −  M I + S M×M  M M

(6.63)

As −1 M is simply:

−1 M = diag 1/(a1 + k1 − jω) · · · 1/(a M + k M − jω)

(6.64)

It can also be computed efficiently. Summarizing the above collections gives the result below: T  Proposition 6.9 For a given specification 1 = δ1 · · · δM and 2 = T  δM+1 · · · δN , state feedback control for active vibration distribution can be obtained by the following steps:   A11 A12 Step 1: Partition  + S = as A11 = S M×M +  M ; A12 = S M×(N −M) ; A21 A22 A21 = S(N −M)×M ; A22 = S(N −M)×(N −M) ; −1 Step 2: Solve −1 M and A11 in terms of (6.64) and (6.63), respectively; Step 3: Substituting the calculated matrices into (6.62) forming inequalities with specification; Step 4: Solve the inequalities for ki s. Remark 6.17 The second inequality in (6.62) is included in the first one; hence the computation of the latter can be simplified by first carrying out the calculation of the second inequality. Meanwhile, it is also noted that multiplication by I M,1 is simply the sum of the row elements for each matrix.

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Remark 6.18 If the solution in step (6.4) exists, this leads to a Pareto set for feasible ki s; if no solution exists, performance specification is too stringent and has to be relaxed. Remark 6.19 However, relaxation of performance specification for obtaining feasible solution can be difficult, particularly for vibration distribution over large numbers of nodes. Trials-and-errors in addition to physical insights are required for a judicious combination of specification indices to permit the existence of feasible solution.

6.7.2 Active Vibration Distribution with Constraints In practice, constraints always exist in one form or another, e.g. system states or outputs are constrained that can be represented by feedback control gains as: ←

k i ≤ ki ≤ k i ∀i ∈ [ 1, M ]

(6.65)

←

where · and · denote the lower and upper limits, respectively. Thus state feedback control for active vibration distribution must take the constraints into consideration leading to: T  Proposition 6.10 For a given specification 1 = δ1 · · · δM and 2 = T  δM+1 · · · δN , state feedback control for active vibration distribution can be obtained by solving the steps in Proposition 6.9, with the consideration of inequalities (6.65) for ki s. Proof The result is obvious following the above argument.

◼

Remark 6.20 This result provides the solution to the active vibration distribution design problem under the constraints (6.65). Such constraints are very common in engineering applications, e.g. sensing or actuation saturations. The results up to now assume that the states xi (t) (i ∈ [ 1, M ]) are all available for sensing, meanwhile there are equal number of actuators for implementing the feedback controls u i (t) = ki xi (t) (i ∈ [ 1, M ]). However, large distribution of sensing and actuation is often either not feasible or cost-effective. This is particularly true for actuation since actuators are usually large and thus to be avoided in design. Thus a seemingly special yet practically important constraint is often enforced where a single actuator is implemented for “global” vibration distribution. That is, vibration distribution conditions in (6.49) are to be fulfilled through feedback control: u 1 (t) = k1 x1 (t)

(6.66)

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noting that there is no loss of generality in assuming performance variable x1 (t) is selected out. To develop this important solution, first note that now one has: A11 = a1 + k1 − jω

(6.67)

while A12 = S1×(N −1) , A21 = S(N −1)×1 , and A22 = S(N −1)×(N −1) are all with known parameters henceforth can be pre-computed. Substitution of (6.67) with I M,1 = I1,1 = 1 into (6.62) results in:   1 + A12 [(a1 + k1 − jω)A22 − A21 A12 ]−1 A21   ≤ δ1 (a1 + k1 )2 + ω2   [(a1 + k1 − jω)A22 − A21 A12 ]−1 A21  ≥ 2 ew

(6.68)

Indeed, the values for A12 = S1×(N −1) , A21 = S(N −1)×1 , and A22 = S(N −1)×(N −1) can be written out explicitly to be: ⎡ 

A12 = s12

 T  ⎢ · · · s1N ; A21 = s21 · · · s N 1 ; A22 = ⎣

⎤ (a2 − jω) · · · s2N ⎥ .. .. .. ⎦ . . . sN 2 · · · (a N − jω) (6.69)

Thus: ⎡

A21 A12

s21 s12 s21 s13 ⎢ s31 s12 s31 s13 ⎢ =⎢ . .. ⎣ .. . s N 1 s12 s N 1 s13

⎤ · · · s21 s1N · · · s31 s1N ⎥ ⎥ .. ⎥ .. . . ⎦ · · · s N 1 s1N

(6.70)

This can be inserted into (6.68) to facilitate the calculations. This simple yet important result is summarized below: Proposition 6.11 Active vibration distribution through single sensor and single actuator feedback control can be obtained by (6.68) with the help of (6.69) and (6.70) for solution of k1 . Existence of solution for a specification 1 = δ1 and T 

2 = δ2 · · · δN indicates the feasibility of single control for global distribution. Proof The proof follows from collecting the argument leading to (6.70).

◼

Remark 6.21 This result provides a single sensor and single actuator solution to the active vibration distribution design problem. This is an important extension relevant to applications in harsh environment

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Remark 6.22 Thus the implementation of state feedback control (6.66) can significantly reduce the complexity and cost for implementing the scheme. Remark 6.23 However, it is also reminded that modifications can be made for application to the case where the number of to-be-attenuated variables is still M, albeit only one variable is utilized for feedback control. Remark 6.24 For all the situations above, it is implicitly assumed that the system is controllable, henceforth a controllability test is necessitated prior to the design procedures. Remark 6.25 Finally it is also worth pointing out that the above scheme is of great engineering relevance, which is also the subject of quite a several patents from marine systems, civil structural control as well as vibration isolation system designs.

6.7.3 Active Vibration Distribution: Fundamental Limit Discussion One of the most important issues in designing a control system is to approach the performance limit. Indeed, it is always easy to design a feedback controller for stabilization or even performance improvement. But it has been challenging for optimal control as one needs to know what the best performance is. Optimal control certainly provides an avenue through optimization of certain performance indices. But it remains an open issue to actually find the performance limit—bearing in mind that optimizing a performance index does not mean the best performance, or even optimal performance. An inappropriately chosen performance index can even lead to much deteriorated performance. Superficially, performance limit can be approached through the corresponding vibration distribution conditions listed in the above sections, e.g. if no feasible control T T   and 2 = δM+1 · · · δN , relaxation of solution exists for 1 = δ1 · · · δM performance specification is conducted and the emergence of feasible solution thus would hit the performance limit with 1 and 2 . Similarly, for a given 1 and 2 , gradually imposing constraints on control u i (t) = ki xi (t) (i ∈ [ 1, N ]) for general design, or u 1 (t) = k1 x1 (t) for single sensor and single actuator design, the emergence of infeasible solution would hit the performance limit. However, it must be pointed out that the Pareto set can involve many variables, even for the single sensor and single actuator control case. The proposal is thus of trial-and-error nature, yet it does provide guidance for successive improvement to approach the performance limit. This is illustrated in the example in the next section.

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6.7.4 Numerical Examples To illustrate the theoretical results as well as the design procedures and performance improvement, the following 3-node system is considered with vibration entering the first node: x˙i = ai xi +

3 

si j x j − dk (t), i ∈ [ 1, 3 ], k = 1

(6.71)

j=1 j=i

The nodes dynamics and topology can be written together as a graph G = ⎡ ⎤ ⎤ −10 1 0 a1 s12 s13 ⎣ s21 a2 s23 ⎦. To be more specific, G is given as G = ⎣ 1 −1 1 ⎦. A s31 s32 a3 0 1 −1.15 design scenario is assumed where the first node is to be attenuated by at least 6 dB, while the remaining two nodes to be enhanced by at least 3 dB and 6 dB, respecT  tively. That is, the design objective is to achieve the specification 1 2 though   1.414 . the design of u 1 (t) = k1 x1 (t) with 1 = 0.5 and 2 = 2 ⎡

State Feedback Control for Active Vibration Distribution The required control design can be obtained from Proposition 6.8, yet now the answer is requested by a single sensor and a single actuator feedback, thus simplification can be made to facilitating the design through the application of Proposition 6.10. That is, condition (6.68) needs to be checked for solution of k1 with the help of (6.69) and (6.70), which can now be written down as:   1 + A12 [(k1 −10 − jω)A22 − A21 A12 ]−1 A21   ≤ 0.5 (k1 −10)2 + ω2     [(k1 −10 − jω)A22 − A21 A12 ]−1 A21  ≥ 1.414 ew 2

(6.72)

with    T A12 = 1 0 ; A21 = 1 0 ; A22 =



   −1 − jω 1 1 1 ; and A21 A12 = 1 −1.15 − jω 0 0

(6.73)

Substituting (6.73) into (6.72) and manipulation leads to: |(1.15 + jω)(1 + jω)(k1 − 10 − jω) − (k1 − 10 − jω) + 1| ≤ 0.5 |(1.15 + jω)[(1 + jω)(k1 − 10 − jω) + 1] − (k1 − 10 − jω) + 1||k1 − 10 − jω| ⎤  ⎡  |1.15+ jω|     ⎣ (1.15+ jω)[(1+ jω)(k1 −10− jω)+1]−(k1 −10− jω)+1 ⎦ ≥ 1.414 1   2   (1.15+ jω)[(1+ jω)(k1 −10− jω)+1]−(k1 −10− jω)+1

(6.74)

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Solving for k1 will lead to the required feedback control solution. Solving (6.74) is actually not complicated as it looks, and simplification can always be made for analysis. For example, define the frequency response transfer: |T (k1 , jω)| = |(1.15 + jω)[(1 + jω)(k1 − 10 − jω) + 1] − (k1 − 10 − jω) + 1| (6.75) then it is noted that (6.74) can be transformed to be: |(1.15 + jω)(1 + jω)(k1 − 10 − jω) − (k1 − 10 − jω) + 1| 0.5|k1 − 10 − jω| |1.15 + jω| |T (k1 , jω)| ≤ 1.414 1 |T (k1 , jω)| ≤ (6.76) 2

|T (k1 , jω)| ≥

Further note that the second inequality will always be satisfied so long as the third one holds. This observation will become important for performance limit analysis to be presented in the next subsection. For now, it is seen clearly that (6.76) will be satisfied if k1 and ω satisfy the condition: |(1.15 + jω)(1 + jω)(k1 − 10 − jω) − (k1 − 10 − jω) + 1| 1 − ≤0 0.5|k1 − 10 − jω| 2

(6.77)

The left hand of the inequality can be plotted as in Fig. 6.14, where the sets of k1 and ω leading to values below zero are all feasible solutions. It is seen that a range of k1 and ω exist, and this renders the prescribed performance specification to be achievable. Active Vibration Distribution with Constraints From Fig. 6.14, it is seen that feedback control gains will possess different feasible set for different vibration frequency. Thus, investigation of the effect of constraints on system performance must take the exogenous vibration into account. However, in practice, the frequency of vibration can be estimated, thus specific result can be easily obtained for feasible control gains. For example, two scenarios are considered in Fig. 6.15 where it is seen that the feasible sets even take markedly different forms: k1 ∈ (−∞, 6.8] ∪ [36.5, +∞) , f or ω = 0.08 rad/s   k1 ∈ −70 6.05 , f or ω = 0.1 rad/s

(6.78)

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Fig. 6.14 Ranges of k1 and ω (with z-axis below zero) achieving prescribed performance specification

Thus should any constraint be imposed upon the control gain, it must be evaluated against the feasible solutions. This is the result expressed in Proposition 6.10, for example, for ω = 0.1 rad/s, if the constraint −10 ≤ k1 ≤ 10 is in force, e.g. for preventing actuator from saturation, this constraint must take an intersection with feasible set in (6.78), namely −10 ≤ k1 ≤ 6.05. Yet, detailed analysis should also be carried out for verification of system performance. The joint issue with performance improvement is looked at in the next subsection. State Feedback Control for Active Vibration Distribution: Performance Limit While acknowledging that the important issue in control design is to approach the performance limit, “synthesis” of feasible sets would indicate that design is only a starting point. Indeed, it is the author’s conviction that it can be easy to design a feedback controller for stabilization or even performance improvement, but the more pertinent thing is to gradually approach the best achievable performance.

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Feasible Set: Magnitude