Dynamic Modeling and Boundary Control of Flexible Axially Moving System 9789811969409, 9789811969416

Table of contents :
Contents
1 Introduction
1.1 Control of the Axially Moving Structures
1.2 Outline of the Book
References
2 Mathematical Preliminaries
2.1 The Hamilton Principle
2.2 Functional and Variation
2.2.1 Functional Variation Rules
2.2.2 The Expansion of intt1 t2 δ( 12wt2 )dt
2.2.3 Definition of Variation
2.3 Discrete Simulation Method
2.3.1 Discretization of the Vibration Displacement w( x,t )
2.3.2 Several Discretization Methods
2.3.3 Discretization of Boundary Conditions
2.4 S-Curve Acceleration/Deceleration (Sc-A/D) Method
2.5 Preliminaries
References
3 PDE Modeling for the Axially Moving Structures
3.1 PDE Modeling
3.2 Simulation Example
Appendix 1: Simulation Program
References
4 Boundary Control of an Axially Moving System with High Acceleration/Deceleration
4.1 Introduction
4.2 PDE Dynamic Model
4.3 Boundary Controller Design
4.3.1 Boundary Control
4.3.2 Stability Analysis
4.4 Simulation Example
Appendix 1: Simulation Program
References
5 Robust Boundary Control of an Axially Moving System with High Acceleration/deceleration and Disturbance Observer
5.1 Introduction
5.2 PDE Dynamic Model
5.3 Boundary Controller Design
5.4 Stability Analysis
5.5 Simulation Example
Appendix 1: Simulation Program
References
6 Adaptive Boundary Control of an Axially Moving System with High Acceleration/Deceleration
6.1 Introduction
6.2 PDE Dynamic Model
6.3 Boundary Controller Design
6.4 Stability Analysis
6.5 Simulation Example
Appendix 1: Simulation Program
References
7 Adaptive Boundary Control of an Axially Moving System with High Acceleration/Deceleration and Disturbance Observer
7.1 Introduction
7.2 PDE Dynamic Model
7.3 Boundary Controller Design
7.4 Stability Analysis
7.5 Simulation Example
Appendix 1: Simulation Program
References
8 Boundary Control of an Axially Moving Accelerated/Decelerated Belt System
8.1 Introduction
8.2 PDE Dynamic Model
8.3 Boundary Controller Design
8.3.1 Step One
8.3.2 Step Two
8.4 Stability Analysis
8.5 Simulation Example
Appendix 1: Simulation Program
References
9 Stabilization of an Axially Moving Accelerated/Decelerated System via an Adaptive Boundary Control
9.1 Introduction
9.2 PDE Dynamic Model
9.3 Boundary Controller Design
9.3.1 Step One
9.3.2 Step Two
9.4 Stability Analysis
9.5 Simulation Example
Appendix 1: Simulation Program
References
10 Adaptive Output Feedback Boundary Control for a Class of Axially Moving System
10.1 Introduction
10.2 PDE Dynamic Model
10.3 Boundary Controller Design
10.3.1 Step One
10.3.2 Step Two
10.4 Stability Analysis
10.5 Simulation Example
Appendix 1: Simulation Program
References
11 Vibration Control and Boundary Tension Constraint of an Axially Moving String System
11.1 Introduction
11.2 PDE Dynamic Model
11.3 Boundary Controller Design
11.4 Stability Analysis
11.5 Simulation Example
Appendix 1: Simulation Program
References
12 Boundary Control for an Axially Moving System with Input Restriction Based on Disturbance Observers
12.1 Introduction
12.2 PDE Dynamic Model
12.3 Boundary Controller Design
12.3.1 Disturbance Observer Design
12.3.2 Backstepping Design
12.4 Stability Analysis
12.5 Simulation Example
Appendix 1: Simulation Program
References
13 Adaptive Neural Network Vibration Control for an Output-Tension-Constrained Axially Moving Belt System with Input Nonlinearity
13.1 Introduction
13.2 PDE Dynamic Model
13.3 Boundary Controller Design
13.3.1 Step One
13.3.2 Step Two
13.4 Stability Analysis
13.5 Simulation Example
Appendix 1: Simulation Program
References
14 Conclusions

Citation preview

Yu Liu · Fengjiao Liu · Yanfang Mei · Xiangqian Yao · Wei Zhao

Dynamic Modeling and Boundary Control of Flexible Axially Moving System

Dynamic Modeling and Boundary Control of Flexible Axially Moving System

Yu Liu · Fengjiao Liu · Yanfang Mei · Xiangqian Yao · Wei Zhao

Dynamic Modeling and Boundary Control of Flexible Axially Moving System

Yu Liu School of Automation Science and Engineering South China University of Technology Guangzhou, Guangdong, China

Fengjiao Liu School of Automation Science and Engineering South China University of Technology Guangzhou, Guangdong, China

Yanfang Mei School of Electronics and Information Guangdong Polytechnic Normal University Guangzhou, Guangdong, China

Xiangqian Yao School of Automation Science and Engineering South China University of Technology Guangzhou, Guangdong, China

Wei Zhao School of Automation Science and Engineering South China University of Technology Guangzhou, Guangdong, China

ISBN 978-981-19-6940-9 ISBN 978-981-19-6941-6 (eBook) https://doi.org/10.1007/978-981-19-6941-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 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

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Control of the Axially Moving Structures . . . . . . . . . . . . . . . . . . . . 1.2 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 6

2

Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Hamilton Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Functional and Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Functional Variation  tRules  .. . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Expansion of t12 δ 21 wt2 dt . . . . . . . . . . . . . . . . . . . . 2.2.3 Definition of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Discrete Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Discretization of the Vibration Displacement w(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Several Discretization Methods . . . . . . . . . . . . . . . . . . . . . 2.3.3 Discretization of Boundary Conditions . . . . . . . . . . . . . . . 2.4 S-Curve Acceleration/Deceleration (Sc-A/D) Method . . . . . . . . . . 2.5 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 8 8 9 9 10 10 11 11 12 13

3

PDE Modeling for the Axially Moving Structures . . . . . . . . . . . . . . . . 3.1 PDE Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 21 23 25

4

Boundary Control of an Axially Moving System with High Acceleration/Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Boundary Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 28 28

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vi

5

6

Contents

4.3.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 39 43

Robust Boundary Control of an Axially Moving System with High Acceleration/deceleration and Disturbance Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 47 48 55 60 65

Adaptive Boundary Control of an Axially Moving System with High Acceleration/Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 67 69 69 75 79 84

7

Adaptive Boundary Control of an Axially Moving System with High Acceleration/Deceleration and Disturbance Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8

Boundary Control of an Axially Moving Accelerated/Decelerated Belt System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Step One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Step Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 108 108 113 116 119 123 127

Contents

9

vii

Stabilization of an Axially Moving Accelerated/Decelerated System via an Adaptive Boundary Control . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Step One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Step Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 130 131 132 136 139 142 146 150

10 Adaptive Output Feedback Boundary Control for a Class of Axially Moving System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Step One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Step Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 152 152 153 156 159 163 167 171

11 Vibration Control and Boundary Tension Constraint of an Axially Moving String System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 174 176 177 183 188 193

12 Boundary Control for an Axially Moving System with Input Restriction Based on Disturbance Observers . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Disturbance Observer Design . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Backstepping Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 196 196 196 198 202 205 210 215

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13 Adaptive Neural Network Vibration Control for an Output-Tension-Constrained Axially Moving Belt System with Input Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 PDE Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Boundary Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Step One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Step Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1: Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217 217 218 218 219 222 225 228 234 243

14 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Chapter 1

Introduction

1.1 Control of the Axially Moving Structures Precision electronic manufacturing equipment spans many disciplines such as electronics, machinery, automation, optics, computers, and involves core technologies, such as precision visual inspection, precision machining, high-speed and highprecision control, and computer integrated manufacturing. It plays a vital role in the security of national economy, finance, and national defense information systems, and is also the core support of mainstream high-tech industries such as solar cells and optoelectronic devices. As an important part of high-speed and high-precision electronic manufacturing equipment, the vibration phenomenon caused by its elastic deformation will directly affect the production performance of the equipment and the production quality of products in terms of time and space scale [1, 2]. Therefore, how to quickly eliminate the vibration phenomenon of the axially moving structure in precision electronic manufacturing equipment has become a bottleneck problem that restricts precision electronic manufacturing, especially the packaging process of integrated circuits (IC). The axially moving structure, as an essentially typical distributed parameter system (DPS) with strong coupling, nonlinearity, and infinite dimensions, can be modeled by a nonlinear partial differential equation (PDE) and a set of ordinary differential equations (ODEs). This hybrid dynamic model makes it difficult to develop an effective control strategy due to its infinite-dimensionality. The conventional control methods for PDE system are based on truncated finite-dimensional modes by using the finite-element method, Galerkin’s method, or assumed modes method [3–5]. However, three main shortcomings exist in the control design of this truncated system [6–8]. One is that the control order needs to be increased and the number of flexible modes needs to be considered to achieve high-precision control performance. The other is that the distributed control force is required to overcome the difficulty of calculating the infinite-dimensional gain matrix. These two drawbacks make the control difficult to achieve from an engineering perspective. The third disadvantage is that the control design is restricted to a few key modes and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_1

1

2

1 Introduction

ignores the high frequency modes, which leads to the control spillover instability of the system. To overcome the aforementioned disadvantages, the boundary control design, which is to be implemented based on original infinite-dimensional model of the systems, has been developed for vibration suppression of flexible systems. Comparing with distributed control, boundary control is considered to be more economical and practical in vibration control of flexible structures because it needs fewer sensors and actuators. And the kinetic energy, the potential energy, and the work done by the nonconservative forces used for modeling can be directly used to construct a Lyapunov function for system stability analysis. In recent years, the boundary control integrated with other intelligent control methodologies, such as sliding-mode control, neural network control, robust adaptive control, iterative learning control, and fuzzy control, has obtained many achievements. But among these results, most of the research objects are about flexible manipulators. For axially moving systems, scholars have also made many attempts. In [9], a robust adaptive boundary control for a class of flexible axially moving string-type systems under unknown time-varying disturbance is developed to guarantee all the signals in the closed-loop system are uniformly ultimately bounded. In [10], an adaptive isolation scheme for an axially moving system is introduced to control the transverse vibration of the controlled span to zero asymptotically. In [11], by using Lyapunov’s method and energy approach, an energy-based robust adaptive boundary control is developed for an axially moving beam to suppress the vibration. In [12], robust and adaptive boundary control is developed to stabilize the vibration of a stretched string on a moving transporter. Simultaneous control of two-dimensional vibration, namely longitudinal and transverse, for vibration suppression of axially moving string is studied in [1]. In [13], an iterative learning boundary control technique is applied to an axially moving system to damp out any string oscillation during transportation. However, most of the research results above assume that the motion of the axially moving structure is a uniform motion, which obviously means only the simplest motion mode is considered. In practice, for high-precision electronics manufacturing equipments, the axially moving structures work with not only a variable speed motion, but usually also a high-acceleration/deceleration and highspeed movement for improving the efficiency under both the distributed disturbance and boundary disturbance in almost all cases. Therefore, the structure has strong geometric nonlinear characteristics, and the current dynamic models and active vibration control methods for axially moving structures at home and abroad are difficult to apply to the vibration control of large acceleration and deceleration axially moving structures [14]. The backstepping method is a recursive algorithm based on Lyapunov stability. The feedback controller is obtained by recursively constructing the Lyapunov function of the closed-loop system, which makes the derivative of the Lyapunov function along the trajectory of the closed-loop system have some performance, and

1.1 Control of the Axially Moving Structures

3

finally ensures that the trajectory of the closed-loop system converges to the equilibrium point. It was originally proposed for ODE systems, also known as finitedimensional backstepping (O-BKST). Until 2008, Krstic and Smyshlyaev proposed infinite-dimensional backstepping for PDE systems [15, 16], which was known as infinite-dimensional backstepping (P-BKST), then subsequent scholars began to deeply integrate backstepping with the control design of PDE systems. A PDE backstepping boundary control algorithm for the DPS boundary control problem of infinite-dimensional PDE model is presented in [17, 18]. The algorithm maps the source system to a stable target system based on trigonometric transformation, and finally a control law to stabilize the system by iterating Lyapunov functions is designed. The algorithm not only ensures the stability of the system, but also avoids control overflow and other problems. It has been successfully applied to static PDE systems such as beams, hyperbolas, wave equations, but the core problem of P-BKST boundary control algorithm is to solve the Volterra mapping kernel function. For some complex PDE systems, it becomes very difficult to solve the kernel function [19], which greatly limits the application of P-BKST boundary control method. In order to avoid solving the Volterra mapping kernel function, O-BKST for original nonlinear ODE system is used in the boundary control design. The O-BKST boundary control algorithm designed to realize the vibration control of flexible structures has achieved good control results. For spacecraft system, an adaptive control strategy based on neural network is proposed by using the backstepping technique to ensure the asymptotic stability of the closed-loop system in the case of unknown nonlinear function, actuator failure, and unknown disturbance [20]. In [21], through the combination of integral-barrier Lyapunov function (IBLF), backstepping technique, and adaptive technique, an adaptive Lyapunov-based barrier control with an auxiliary system is developed to achieve the control purposes of vibration suppression and constraint satisfaction. However, for the axially moving systems with large acceleration and deceleration, the active vibration control combined with backstepping technique is relatively few. Then we made many attempts and achieved some good results. In [22], an adaptive boundary control with disturbance observer is successfully designed based on the Lyapunov-based backstepping method and adaptive control for the axially moving acceleration/deceleration belt system to suppress the vibration of the belt. And in [23], applying the finite-dimensional backstepping control and Lyapunov’s direct method, a boundary controller is developed to stabilize the belt system at the small neighborhood of its equilibrium position and a disturbance observer is introduced to attenuate the effect of unknown external disturbance. These control schemes all show good control effect, but they only consider the influence of boundary interference. For the general axially moving system with large acceleration and deceleration, there are not only with disturbance uncertainties, but also with system parameters uncertainties, such as tension, stiffness, and mass. Therefore, in order to improve the quality of active vibration control of the structure and compensate for the uncertainty of the system, the design of the controller is required to be more robust and adaptive. After considering the H-A/D and distributed disturbance, the axially moving system is governed by a nonhomogeneous PDE with geometric nonlinearities which makes

4

1 Introduction

the system dynamic model and vibration control scheme design quite different from the existing works. The traditional linear/exponential acceleration/deceleration has a sudden change in acceleration when the system starts and ends, which is prone to shock and poor flexibility. The S-curve acceleration and deceleration ensure the performance of the motor and reduce the impact by attenuating the acceleration during the start-up phase. So it is widely used in high-speed and high-precision electronic manufacturing equipment due to its good flexibility. The axially moving structure in surface mount technology (SMT in Fig. 1.1) has typical characteristics of large acceleration and deceleration, and its maximum acceleration is usually more than 5g (g is the gravitational acceleration), and its structural vibration presents complex dynamic characteristics. The book takes the author’s team’s independent research and development of the axially moving structure of SMT as the research object, establishes its dynamic model including the geometric nonlinear characteristics of the structure, and uses the Lyapunov direct method and the S-curve acceleration and deceleration method based on the infinite-dimensional distributed parameter model. The boundary control algorithm is designed to actively control the vibration of the large acceleration and deceleration axial movement system, thereby suppressing its vibration and improving the processing accuracy of the equipment. Fig. 1.1 SMTs

1.2 Outline of the Book

5

1.2 Outline of the Book This book systematically proposes and summarizes a variety of different vibration control methods for axial motion systems, and comprehensively considers the control design problems of axial motion systems affected by boundary interference, distributed interference, inaccurate parameters, input nonlinearity and output constraints. The book starts with a brief introduction of control techniques for the axially moving structures in this chapter. Chapter 2 presents some preliminaries, basic lemmas and properties for subsequent deriving of dynamic model and further stability analysis. Chapter 3 establishes the PDE modeling for the axially moving structures of SMT. In Chap. 4, a boundary controller for an axially moving system with high acceleration/deceleration is presented to control vibration based on the PDE model. In Chap. 5, the disturbance observers are constructed to estimate the outside disturbance, and a robust boundary control based on disturbance observer is developed for SMT system subject to distributed and boundary disturbance to improve control performance, then the control-spillover phenomenon is avoided. In Chap. 6, an adaptive boundary control of SMT system subject to boundary disturbance is designed with Lyapunov’s synthesis method and adaptive technique. In Chap. 7, a boundary controller is presented with S-curve A/D and Lyapunov direct method to suppress vibration of SMT with high acceleration/deceleration and distributed disturbance. In Chap. 8, a boundary controller is proposed for the belt system with spatially varying tension and distributed disturbance by applying the finite-dimensional backstepping technique to guarantee the small neighborhood of an equilibrium position. In Chap. 9, an adaptive boundary controller is devised by utilizing adaptive technique and Lyapunov-based back-stepping method to realize vibration control of SMT system with system parameter uncertainties. In Chap. 10, a robust boundary adaptive output feedback control is presented to stabilize the SMT system by merging Lyapunov theory, observer backstepping technique, high-gain observers and robust adaptive control theory. In Chap. 11, a new boundary controller is designed for the SMT system with output constraint to guarantee the tension not violate the constraint based on barrier Lyapunov function, and a novel disturbance observer with a barrier term is introduced. In Chap. 12, a boundary controller strategy is constructed by employing the classical backstepping control, and the infinite-dimensional observer (IDO) and finite-dimensional observer (FDO) are, respectively, designed to estimate the unknown distributed and boundary disturbances, then the existence, uniqueness and convergence of the solution to the considered control system are proved based on Sobolev spaces. And in Chap. 13, for SMT system with parameter uncertainties and input–output constraint, the RBF neural networks are used to approximate system parameter terms and input error terms, and a barrier Lyapunov function is employed to guarantee the output restrictions, then a robust adaptive neural network boundary control is proposed successfully. The conclusions are summarized in Chap. 14.

6

1 Introduction

References 1. Q.C. Nguyen, K.S. Hong, Simultaneous control of longitudinal and transverse vibrations of an axially moving string with velocity tracking. J. Sound Vib. 331(13), 3006–3019 (2012) 2. L. Wang, H. Chen, X. He et al., Vibration control of an axially moving cantilever beam with varying length. J. Vib. Eng. 22(6), 565–570 (2009) 3. P.D. Christofifides, A. Armaou, Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control. Syst. Control Lett. 39(4), 283–294 (2000) 4. M. Kristic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design (Wiley, New York, 1995) 5. M.J. Balas, Feedback control of flexible systems. IEEE Trans. Autom. Control 23(4), 673–679 (1978) 6. W. He, S.S. Ge, B.V. How et al., Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47(4), 722–732 (2011) 7. K.D. Do, J. Pan, Boundary control of three-dimensional inextensible marine risers. J. Sound Vib. 327(3–5), 299–321 (2009) 8. L. Meirovitch, H. Baruh, On the problem of observation spillover in self-adjoint distributedparameter systems. J. Optim. Theory Appl. 39(2), 269–291 (1983) 9. W. He, S.S. Ge, Robust adaptive boundary control of a vibrating string under unknown timevarying disturbance. IEEE Trans. Control Syst. Technol. 20(1), 48–58 (2012) 10. Y. Li, D. Aron, C.D. Rahn, Adaptive vibration isolation for axially moving strings: theory and experiment. Automatica 38(3), 379–390 (2002) 11. K.J. Yang, K.S. Hong, F. Matsuno, Energy-based control of axially translating beams: varying tension, varying speed, and disturbance adaptation. IEEE Trans. Control Syst. Technol. 13(6), 1045–1054 (2005) 12. Z. Qu, Robust and adaptive boundary control of a stretched string on a moving transporter. IEEE Trans. Autom. Control 46(3), 470–476 (2001) 13. Z. Qu, An iterative learning algorithm for boundary control of a stretched moving string. Automatica 38(5), 821–827 (2002) 14. J. Chung, C. Han, K. Yi, Vibration of an axially moving string with geometric non-linearity and translating acceleration. J. Sound Vib. 240(4), 733–746 (2001) 15. M. Krstic, A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett. 57(9), 750–758 (2008) 16. A. Smyshlyaev, B.Z. Guo, M. Krstic, Arbitrary decay rate for Euler-Bernoulli beam by backstepping boundary feedback. IEEE Trans. Autom. Control 54(5), 1134–1140 (2009) 17. W.J. Liu, M. Krstic, Backstepping boundary control of Burgers’ equation with actuator dynamics. Syst. Control Lett. 41(4), 291–303 (2000) 18. T. Meurer, A. Kugi, Tracking control for boundary controlled parabolic PDEs with varying parameters: combining backstepping and differential fitness. Automatica 45(5), 1182–1194 (2009) 19. B.Z. Guo, F.F. Jin, Backstepping approach to the arbitrary decay rate for Euler-Bernoulli beam under boundary feedback. Int. J. Control 83(10), 2098–2106 (2010) 20. X. Cao, P. Shi, Z. Li et al., Neural-network-based adaptive backstepping control with application to spacecraft attitude regulation. IEEE Trans. Neural Netw. Learn. Syst. 29(3), 4303–4313 (2018) 21. F. Guo, Y. Liu, F. Luo, Adaptive stabilization of a flexible riser by using the Lyapunov-based barrier backstepping technique. IET Theory Appl. 11(14), 2252–2260 (2017) 22. Y. Liu, Z. Zhao, W. He, Stabilization of an axially moving accelerated/decelerated system via an adaptive boundary control. ISA Trans. 64, 394–404 (2016) 23. Y. Liu, Z. Zhao, W. He, Boundary control of an axially moving accelerated/decelerated belt system. Int. J. Robust Nonlinear Control 26(17), 3849–3866 (2016)

Chapter 2

Mathematical Preliminaries

In this chapter, we provide some mathematical preliminaries and useful technical lemmas.

2.1 The Hamilton Principle The Hamilton principle [1, 2] is a description of the minimum action, that is, in the process of an object moving from a specific position to another specific position, there are countless possible motion trajectories, but the real motion trajectory is the one with the minimum action of the system. In other words, the real motion trajectory satisfies that the functional composed of the corresponding energy function of the system takes the minimum value, which can be described mathematically as that the variational value of the functional is 0, and it is expressed as follows t2

  δ E k − E p + W dt = 0

(2.1)

t1

where t1 and t2 are two time instants, t1 < t < t2 is the operating interval, δ denotes the variational operator, E k and E p are the kinetic and potential energies of the system, respectively, and W denotes the work done by the non-conservative forces acting on the system, including internal tension, transverse load, linear structural damping, and external disturbance. The principle states that the variation of the kinetic and potential energies plus the variation of work done by loads during any time interval [t1 , t2 ] must equal zero. There are two obvious advantages to using the Hamilton principle to derive the mathematical model of the axially moving system. The first one is that the dynamic model of the system derived by this approach does not depend on the selection of starting point and end point, and the boundary conditions can be of the coordinates and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_2

7

8

2 Mathematical Preliminaries

automatically generated by this approach [3]. Anothor is that the kinetic energy, the potential energy, and the work done by the non-conservative forces in the Hamilton principle can be directly used to design a Lyapunov function to prove the stability of the closed-loop system.

2.2 Functional and Variation 2.2.1 Functional Variation Rules The variance of the function is a linear mapping, so its operation rules are similar to the linear operation of the function. Let L 1 and L 2 are the functions of x, x˙ and t, there are the following functional variation rules: (1) (2) (3) (4)

δ(L 1 + L 2 ) = δL 1 + δL 2 δ(L 2 δL 1  b1 L 2 ) = L 1 δL 2 + L b δ a L(x, x, ˙ t)dt = a δL(x, x, ˙ t)dt δ ddtx = dtd δx

2.2.2 The Expansion of

 t2  1 2  t1 δ 2 w t d t

From δ(L 1 L 2 ) = L 1 δL 2 + L 2 δL 1 , we have  t2  t2 1 δ wt2 dt = wt · δwt dt 2 t1

From δ ddtx =

d δx, dt

t1

we have t2

t2 wt · δwt dt =

t1

According to

b a

wt d(δw) t1

 b udv = (uv)tt21 − a vdu, we get

t2

t2 wt d(δwt ) =

t1

(wt δw)|tt2 1

−

wtt δwdt = − t1

where (wt δw)|tt2 = 0. 1

t2 wtt δwdt t1

2.3 Discrete Simulation Method

9

Then we get  t2  t2 1 2 δ wt dt = − wtt δwdt 2 t1

(2.2)

t1

2.2.3 Definition of Variation For the function x2 S=

  L f (x), f  (x), x d x

x1

Fixing two point x1 and x2 , define the extremum of functional S as function g(x), and define the function which is close to g(x) as h(x), i.e., h(x) = g(x) + δg(x), where δg(x) is a small value from x1 to x2 , also the following is satisfied: δg(x1 ) = δg(x2 ) = 0 Then δg(x) is the variation of g(x), which can be described in Fig. 2.1. According to the definition of variation, we have δw(t1 ) = δw(t2 ) = 0, then (wt δw)|tt2 = 0. 1

2.3 Discrete Simulation Method Define the sampling time as t = dt, and the axle spacing as x = d x. Consider x is set as i, t is set as j, PDE model as Fig. 2.1 Definition of variation

f (x )

h ( x)

δ ( x) g ( x)

o

x1

x2

x

10

2 Mathematical Preliminaries

mwtt −

3 1 E Awx3 − T0 wx x + mawx + 2mvwxt + mv 2 wx x + cwt + cvwx = 0 2 2 (2.3) ⎧ ⎨ w(0, t) = 0 (2.4) ⎩ 1 T0 wx (L , t) + 1 E Awx3 (L , t) = d + U 2 2

In this chapter, we discrete the model by using difference method.

2.3.1 Discretization of the Vibration Displacement w(x, t) For the Eq. (2.3), using forward differential method, then we have t (i, j−1) , then wtt (x, t)= wt (i, j)−w dt wtt (x, t)=

w(i, j)−w(i, j−1) dt

j−2) − w(i, j−1)−w(i, dt dt

(2.5)

Define the current time is j − 1, then wx x (x, t) can be expressed as wx x (i, j − 1), x (i, j−1) , then then wx x (x, t)= wx (i+1, j−1)−w dx wx x (x, t)=

w(i+1, j−1)−w(i, j−1) dx

j−1) − w(i, j−1)−w(i−1, dx dx

(2.6)

2.3.2 Several Discretization Methods Consider w(x, t) as w(i, j), x is set as i, t is set as j, there are three kinds of discrete methods to express w(x, t) as follows: j−1) | (1) Backward difference: ∂w = w(i, j)−w(i, ; ∂t t=i dt w(i, j+1)−w(i, j) ∂w (2) Forward difference: ∂t |t=i = ; dt w(i, j+1)−w(i, j−1) | (3) Central difference: ∂w = . t=i ∂t 2dt

In the simulation, one of the three methods can be used according to the requirements.

2.4 S-Curve Acceleration/Deceleration (Sc-A/D) Method

11

2.3.3 Discretization of Boundary Conditions Consider the time interval and the displacement interval as 1 ≤ j ≤ nt and 1 ≤ i ≤ nx, w(i, j) can be discretized as two conditions (1) Express w(i, j) at 1 ≤ i ≤ nx − 1 by boundary conditions

+ −

+

w(i, j) = 2w(i, j − 1) − w(i, j − 2)

dt 2 × 0.5T − mv 2 (1, j − 1) × (w(i + 1, j − 1) − 2w(i, j − 1) + w(i − 1, j − 1)) m × dx2 dt 2 × cv(1, j − 1) × (w(i, j − 1) − w(i − 1, j − 1))

dt × c(w(i, j − 1) − w(i, j − 2)) − m m × dx dt × 2mv(1, j − 1) × (w(i, j − 1) − w(i, j − 2) − w(i − 1, j − 1) + w(i − 1, j − 2)) − m × dx dt × m(v(1, j − 1) − v(1, j − 2)) × (w(i, j − 1) − w(i − 1, j − 1)) − m × dx

(2.7)

dt × 3E A(w(i, j − 1) − w(i − 1, j − 1))2 × (w(i + 1, j − 1) − 2w(i, j − 1) + w(i − 1, j − 1)) 2m × d x 4

(2) Express w(nx, j) at i = nx by boundary conditions w(nx, j) =

2 × d x × d(1, j) w(nx − 1, j) − E A(w(nx, j) − w(nx − 1, j))3 + T T × dx2

(2.8)

2.4 S-Curve Acceleration/Deceleration (Sc-A/D) Method Figure 2.2 shows a typical Sc-A/D process, which is divided into seven segments by time instants t1 to t7 [3]. Let v(t) be the belt’s axial speed, v(t) > 0 for all t, a(t) be the belt’s H-A/D, and the jerks J1 ∼ J4 be the time derivatives of the belt’s H-A/D, respectively. From Fig. 2.2, the a(t) can be expressed as: Fig. 2.2 A typical Sc-A/D process

12

2 Mathematical Preliminaries

⎛

J1 t 0  t < t1 ⎜a t 1  t < t2 ⎜ a ⎜ a − J (t − t ) t  t < t ⎜ a 2 2 2 3 ⎜ a(t) = ⎜ 0 t 3  t < t4 ⎜ ⎜ −J3 (t − t4 ) t4  t < t5 ⎜ ⎝ −ad t 5  t < t6 −ad + J4 (t − t6 ) t6  t < t7

(2.9)

where aa > 0 and ad > 0 are the maximum acceleration and maximum deceleration respectively, Ti = ti − ti−1 , i = 1, 2, 3, 4, 5, 6, 7 and t0 = 0. Hence the v = v0 + at can be obtained as: ⎛ v0 + 21 J1 t 2 0  t < t1 ⎜ v + 1 J T 2 + a (t − t ) t 0 1 1 a 1 1  t < t2 ⎜ ⎜ v + 21 J T 2 + a (T + t − t ) − 1 J (t − t )2 t  t < t ⎜ 0 2 1 1 a 2 2 2 2 3 2 2 ⎜ (2.10) v(t) = ⎜ vm t 3  t < t4 ⎜ 1 2 ⎜ vm − 2 J3 (t − t4 ) t 4  t < t5 ⎜ ⎝ vm − 21 J3 T5 2 − ad (t − t5 ) t 5  t < t6 vm − 21 J3 T5 2 − ad (T6 + t − t6 ) + 21 J4 (t − t6 )2 t6  t < t7 where vm = v0 + 21 J1 T12 + aa (T2 + T3 ) − 21 J2 T32 and v0 is the initial speed.

2.5 Preliminaries Lemma 2.1 Let ϒ1 (x, t), ϒ2 (x, t) ∈ R, σ > 0 with x ∈ [0, L] and t ∈ [0, +∞), the following inequalities hold [4]: ⎧ 2 2 ⎨ ϒ1 ϒ2 ≤ |ϒ1 ϒ2 | ≤ ϒ1 + ϒ2 , √ 1 1 ⎩ |ϒ1 ϒ2 | = |( √ ϒ1 )( σ ϒ2 )| ≤ ϒ12 + σ ϒ22 . σ σ

(2.11)

Lemma 2.2 Let ϒ(x, t) ∈ R be a function defined on (x, t) ∈ [0, L] × [0, +∞) satisfying the boundary condition [5] ϒ(0, t) = 0, then the following inequalities hold:

∀t ∈ [0, +∞),

(2.12)

References

13

⎧ L L ⎪ ⎪ ⎪ 2 2 ⎪ ϒ dx ≤ L ϒx2 d x ⎪ ⎪ ⎪ ⎨ 0

0

⎪ L ⎪ ⎪ ⎪ 2 ⎪ ≤ L ϒx2 d x ϒ ⎪ ⎪ ⎩

∀x ∈ [0, L]

(2.13)

0

Lemma 2.3 For any ϒ = [ϒ1 , · · · , ϒi , · · · , ϒn ]T with ϒi ∈ C1 [0, L], i = 1, · · · , n, the following inequalities hold [6] ⎧ L  L ⎪ ⎪ ⎪ 2 ⎪ ϒ · ϒd x ≤ 2Lϒ(0) · ϒ(0) + 4L ϒx · ϒx d x ⎪ ⎪ ⎪ ⎪0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ L  L ⎪ ⎪ ⎪ ⎪ 2 ⎪ ϒ · ϒd x ≤ 2Lϒ(L) · ϒ(L) + 4L ϒx · ϒx d x ⎪ ⎪ ⎪ ⎪ ⎨0 0    L  L ⎪   ⎪ ⎪   ⎪ ⎪  max (ϒ · ϒ) ≤ ϒ(0) · ϒ(0) + 2 ϒ · ϒd x ×  ϒx · ϒx d x ⎪ ⎪ x∈[0,L] ⎪ ⎪ ⎪ 0 0 ⎪ ⎪   ⎪ ⎪   ⎪ ⎪ L L ⎪ ⎪   ⎪ ⎪ max (ϒ · ϒ) ≤ ϒ(L) · ϒ(l) + 2 ϒ · ϒd x ×  ϒx · ϒx d x ⎪ ⎪ ⎩ x∈[0,L] 0

(2.14)

0

Lemma 2.4 For any function ϒ(x, t) that is continuously differentiable, the following inequalities hold [4] ⎧ L L ⎪ ⎪ 4L 2 ⎪ 2 2 ⎪ ϒ d x ≤ Lϒ (0, t) + ϒx2 d x ⎪ ⎪ 2 ⎪ π ⎨ 0

0

⎪ L ⎪ ⎪ ⎪ 2 2 ⎪ ϒ ≤ ϒ (0, t) + L ϒx2 d x ⎪ ⎪ ⎩

(2.15)

0

References 1. H. Goldstein, Classical Mechanics (Addison-Wesley, Massachusetts, 1951) 2. L. Meirovitch, Analytical Methods in Vibration (Macmillan, New York, 1967) 3. Y. Chen, X. Ji, Y. Tao, H. Wei, Look-ahead algorithm with whole S-curve acceleration and deceleration. Adv. Mech. Eng. 5, 974152 (2013)

14

2 Mathematical Preliminaries

4. C.D. Rahn, Mechantronic Control of Distributed Noise and Vibration (Springer, Berlin, 2001) 5. G. Hardy, J. Littlewood, G. Polya, Inequalities (Cambridge University Press, Cambridge, UK, 1952) 6. K.D. Do, J. Pan, Boundary control of three-dimensional in extensible marine risers. J. Sound Vib. 327(3–5), 299–321 (2009)

Chapter 3

PDE Modeling for the Axially Moving Structures

The axially moving flexible structures such as belt, string, and cable are widely applied in many mechanical systems. For these systems, the flexible structure exhibits vibration in the presence of disturbances. Since the undesirable or excessive vibration can degrade system performance [1, 2], the vibration suppression has become an important research area and gained more and more attention in recent decades. Most of the previous research was based on nonlinear ordinary differential equation (ODE) dynamic model. Although nonlinear ODE model is simple in form and convenient for controller design, it is not accurate for highly flexible manipulator and may cause spillover instability [3, 4]. Recently, many applications of nonlinear partial differential equation (PDE) have been reported for stationary (non-axially moving) systems [5–8], such as beams, hyperbolas, waves, and other equations. In fact, the flexible structure is a distributed parameter system and should be described by PDE model for accuracy.

3.1 PDE Modeling In this chapter, we mainly introduce the PDE modeling of the belt. From Fig. 3.1, the belt’s left boundary is fixed at the origin O of coordinate X OY . Let w(x, t) be the belt’s vibration displacement at the position x for time t, m c be the mass of the actuator, L be the length of the controlled span, d(t) be the unknown time-varying boundary disturbance, f (x, t) be the unknown spatiotemporally-varying distributed disturbance, and U (t) be the control input exerted at the right boundary of the belt. Remark 3.1 For clarity, notations (·)(t) = (·), (·)(x, t) = (·), (·)x = ∂(·)/∂ x and (·)t = ∂(·)/∂t are used throughout the whole book. The motion equations of the belt system with H-A/D and distributed disturbance can be derived by using the Hamilton’s principle, expressed as: © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_3

15

16

3 PDE Modeling for the Axially Moving Structures

Fig. 3.1 A typical axially moving system of SMT

t2 (δ E k − δ E p + δW )dt = 0

(3.1)

t1

where δ(·) represents the variation of (·), t1 and t2 (t1 < t < t2 ) are two-time instants. Consider the belt and its change rate at the origin at any time is zero, we have w(0, t) = 0, wx (0, t) = 0. The kinetic energy E k (t) of the belt system can be represented as: 1 Ek = m 2

L (wt +vwx )2 d x

(3.2)

0

where m is the mass per unit length of the belt and v is described in Chap. 2. The potential energy E p (t) of the belt system is expressed as: 1 Ep = 2

L T εd x

(3.3)

0

where the spatially varying tension T (x, t) is given as [3]: T = T0 +

1 E Awx2 2

(3.4)

where T0 is the initial tension of the belt, E is the coefficient of elasticity, and A is the cross-sectional area.

3.1 PDE Modeling

17

The displacement–strain relation ε is expressed as: ε=

1 2 w 2 x

(3.5)

The virtual work δWc (t) done by the non-conservative forces on the belt is computed as: L δWc = (U + d)δw(L , t) − c

L (wt +vwx )δwd x +

0

f δwd x

(3.6)

0

where c is viscous damping coefficient of the belt. The virtual momentum δWb (t) transport across the boundaries is computed by: δWb = mv[wt (L , t) + vwx (L , t)]δw(L , t) Firstly, the first item of (3.1) can be expanded as: t2

1 δ E k dt = mδ 2

t1

t2 L (wt +vwx )2 d xdt t1

0

t2 L =m

(wt +vwx )(δwt +vδwx )d xdt t1

0

L t2

t2 L (wt +vwx )dδwd x + m

=m 0

t1

v(wt +vwx )dδwdt t1

0

Since L t2

L (wt +vwx )dδwd x = m

m 0

t1

(wt +vwx )δw|tt21 d x 0

t2

L

−

m(wtt +vwxt + awx )δwd xdt t1

0

t2 L =−

m(wtt +vwxt + awx )δwd xdt t1

and

0

(3.7)

18

3 PDE Modeling for the Axially Moving Structures

t2 L

t2 v(wt +vwx )dδwdt =

m t1

mv(wt +vwx )δw|0L dt t1

0

t2 L −

mv(wxt +vwx x )δwd xdt t1

0

t2 =

mv[wt (L , t)+vwx (L , t)]δw(L , t)dt t1

t2 −

mv[wt (0, t)+vwx (0, t)]δw(0, t)dt t1

t2 L mv(wxt +vwx x )δwd xdt

− t1

0

then t2

t2 δ E k dt =

t1

mv[wt (L , t)+vwx (L , t)]δw(L , t)dt t1

t2 −

mv[wt (0, t)+vwx (0, t)]δw(0, t)dt t1

(3.8)

t2 L mv(wxt +vwx x )δwd xdt

− t1

0

t2 L m(wtt +vwxt + awx )δwd xdt

− t1

0

3.1 PDE Modeling

19

Then, expanding the second item of (3.1), we have t2

t2 L

1 T εd xdt 2

δ E p dt = δ t1

t1

0

 t2 L  1 1 1 2 =δ T0 + E Awx × wx2 d xdt 2 2 2 t1

0

t2 L =δ t1

1 T0 wx2 d xdt + δ 4

t2 L t1

0

1 E Awx4 d xdt 8

0

since t2 L δ t1

=

1 2

1 1 T0 wx2 d xdt = 4 2

t2

T0 wx δwx d xdt

0

t1

t2

t2 L

T0 wx δw|0L dt−

1 2

t1

1 = 2

t2 L 0

T0 wx x δwd xdt t1

0

1 T0 wx (L , t)δw(L , t)dt − 2

t1

t2 T0 wx (0, t)δw(0, t)dt t1

−

1 2

t2 L T0 wx x δwd xdt t1

0

and t2 L δ t1

=

1 2

t2

1 1 E Awx4 d xdt = 8 2

E Awx3 dδwdt t1

0

L 3 E Awx3 δw 0 dt − 2

t1

1 = 2

t2 L

E Awx2 wx x δwd xdt t1

t2 E Awx3 (L , t)δw(L , t)dt t1

0

t2 L 0

1 − 2

t2 E Awx3 (0, t)δw(0, t)dt t1

−

3 2

t2 L E Awx2 wx x δwd xdt t1

0

20

3 PDE Modeling for the Axially Moving Structures

then t2

1 δ E p dt = 2

t1

t2

1 T0 wx (L , t)δw(L , t)dt + 2

t1

−

1 2

1 2

E Awx3 (L , t)δw(L , t)dt t1

t2

t2

T0 wx (0, t)δw(0, t)dt −

1 2

t2 L

t2 L

t1

−

t2

E Awx3 (0, t)δw(0, t)dt

(3.9)

t1

T0 wx x δwd xdt − t1

3 2

E Awx2 wx x δwd xdt t1

0

0

Finally, the last item of (3.1) can be expanded as: t2

t2 δW dt =

t1

[δWc (t) − δWb (t)]dt t1

t2 =

t2 L [c(wt +vwx ) − f ]δwd xdt

(U + d)δw(L , t)dt − t1

t1

(3.10)

0

t2 −

mv[wt (L , t) + vwx (L , t)]δw(L , t)dt t1

Substituting Eqs. (3.8)–(3.10) and into Eq. (3.1) and integrating by parts, the governing equation of the presented belt system can be obtained as: mwtt −

3 1 E Awx2 wx x − T0 wx x + mawx + 2mvwxt + mv 2 wx x + cwt + cvwx = f 2 2 (3.11)

where ∀(x, t) ∈ (0, L) × [0, +∞). The boundary conditions of presented belt system are described as: ∀t ∈ [0, +∞) ⎧ ⎨ w(0, t) = 0 ⎩ 1 T0 wx (L , t) + 1 E Awx3 (L , t) = d + U 2 2 where ∀t ∈ [0, +∞).

(3.12)

3.2 Simulation Example

21

Assumption 3.1 For the speed v(t), H-A/D a(t), and boundary disturbance d(t), we assume that there exist constants a1 , a2 , a3 , a4 ∈ R+ , such that 0 < v(t) ≤ a1 , |a(t)| ≤ a2 , |d(t)| ≤ a3 , and | dt (t)| ≤ a4 , ∀t ∈ [0, +∞). This is a reasonable assumption since v(t), a(t), and d(t) have finite energy and uniformly continuous. Hence they are bounded [9, 10]. And in this chapter, we do not consider distributed interference, that is to say f = 0. Property 3.1 [11, 12] If the kinetic energy given by the belt system is bounded ∀t ∈ [0, +∞), then wt (L , t) and wxt (L , t) are bounded ∀(x, t) ∈ (0, L) × [0, +∞). Property 3.2 [11, 12] If the potential energy given by the belt system is bounded ∀t ∈ [0, +∞), then wx (L , t) and wx x (L , t) are bounded ∀(x, t) ∈ (0, L)×[0, +∞).

3.2 Simulation Example Consider the PDE model as Eq. (3.11), choose parameters as: E A = 1000 N, T0 = 10000 N, c = 1.0 Ns/m2 , L = 1.0 m, and m = 1.0 kg/m. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D in Fig. 2.2 are given as: aa = ad = 3.5g and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s (g = 9.8 m/s2 ). The initial conditions of the axially moving belt system are given as: w(x, 0) = wt (x, 0) = 0, U = 0.

(3.13)

The unknown time-varying boundary disturbance d(t) is described as: d(t) = 3 + 0.1 sin(0.1t) + 0.3 sin(0.3t) + 0.5 sin(0.5t). The simulation results are shown from Fig. 3.2 to Fig. 3.3.

(3.14)

22

3 PDE Modeling for the Axially Moving Structures

Fig. 3.2 The displacement of the belt without control

Fig. 3.3 The displacement at x = 1 m and x = 0.5 m

Appendix 1: Simulation Program

23

Appendix 1: Simulation Program close all; clear all; clc; nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m=1; EA=1000; T0=10000; pi=3.1415926; J=3.5*9.8; V0=0; c=1; for i=1:nx for j=1:nt d(i,j)=3+0.1*sin(0.1*j*dt)+0.3*sin(0.3*j*dt)+0.5*sin(0.5*j*dt); end end V=zeros(nx,nt); for i=1:nx V(i,1)=V0; end for i=1:nx for j=2:10*10^3-1 V(i,j) = V(i,1)+0.5*J*(j*dt)^2; end V(i,10*10^3)=V(i,1)+0.5*J; for j=(10*10^3+1):20*10^3-1 V(i,j)=V(i,10*10^3)+0.5*J*(j*dt-1); end V(i,20*10^3)=V(i,10*10^3)+J; for j=(20*10^3+1):30*10^3-1 V(i,j) = V(i,20*10^3)+J*(j*dt-2)-0.5*J*(j*dt-2)^2;

24

3 PDE Modeling for the Axially Moving Structures end V(i,30*10^3)=V(i,20*10^3)+0.5*J; for j=(30*10^3+1):70*10^3-1 V(i,j)=V(i,30*10^3); end V(i,70*10^3)=V(i,30*10^3); for j=(70*10^3+1):80*10^3-1 V(i,j)= V(i,70*10^3)-0.5*J*(j*dt-7)^2; end V(i,80*10^3)=V(i,70*10^3)-0.5*J; for j=(80*10^3+1):90*10^3-1 V(i,j)=V(i,80*10^3)-J*(j*dt-8); end V(i,90*10^3)=V(i,80*10^3)-J; for j=(90*10^3+1):nt-1 V(i,j)=V(i,90*10^3)-J*(j*dt-9)+0.5*J*(j*dt-9)^2; end V(i,100*10^3)=V(i,90*10^3)-0.5*J;

end w1 = zeros(nx,nt); % ------- boundary condition + initial condition w1(1,:) = 0; for i=2:nx

% boundary condition w(0,t)=0 % initial condition

w1(i,1) = 0; w1(i,2) = 0; end for j=3:nt for i=2:nx-1 w1(i,j) = 2*w1(i,j-1) - w1(i,j-2) +(dt^2*(0.5*T0-m*V(1,j-1)*V(1,j-1)))/(m*dx^2)*(w1(i+1,j-1)-2*w1(i,j-1) +w1(i-1,j-1)) - (dt*c)/(m)*(w1(i,j-1)-w1(i,j-2)) -dt^2*c*V(1,j-1)/(m*dx)*(w1(i,j-1)-w1(i-1,j-1)) -(2*V(1,j-1)*m*dt)/(m*dx)*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1) +w1(i-1,j-2))- m*dt/(m*dx)*(V(1,j-1)-V(1,j-2))*(w1(i,j-1)-w1(i-1,j-1)) +1.5*dt^2*EA/(m*dx^4)*(w1(i,j-1)-w1(i-1,j-1))^2*(w1(i+1,j-1)-2*w1(i,j -1)+w1(i-1,j-1)); end w1(nx,j)=2*dx*d(1,j)/T0 +w1(nx-1,j)-EA*(w1(nx,j)-w1(nx-1,j))^3/(T0*dx^2); end

References

25

%for obtain the curve of the riser, short the point tshort = linspace(0,tmax,nt/400); xshort = linspace(0,L,nx); w1short=zeros(nx,nt/400); for j=1:nt/400 for i=1:nx

w1short(i,j)=w1(i,j*400); end end figure (1); surf(tshort,xshort,w1short); shading interp; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('w(x,t)[m]','interpreter','latex','Fontsize',14); set(gca,'YDir','reverse') figure (2); plot(tshort,w1short(25,:),'b',tshort,w1short(50,:),':r'); xlabel('t / s','Fontsize',14); ylabel('w(x,t)/ m','Fontsize',14); legend('x=0.5m','x=1m'); box on grid on

References 1. W. He, S.S. Ge, Robust adaptive boundary control of a vibrating string under unknown timevarying disturbance. IEEE Trans. Control Syst. Technol. 20(1), 48–58 (2012) 2. J.M. Sloss, J.C. Bruch Jr., I.S. Sadek et al., Boundary optimal control of structural vibrations in an annular plate. J. Frankl. Inst. 342(3), 295–309 (2005) 3. I. Kanellakopoulos, P. Kokotovic, A. Morse, A toolkit for nonlinear feedback design. Syst. Control Lett. 18(2), 83–92 (1992) 4. M. Krstic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design (Wiley, New York, 1995) 5. M. Krstic, A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett. 57(9), 750–758 (2008) 6. M. Krstic, A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs (Society for Industrial and Applied Mathematics, Philadelphia, 2008) 7. A. Smyshlyaev, B.Z. Guo, M. Krstic, Arbitrary decay rate for Euler-Bernoulli beam by backstepping boundary feedback. IEEE Trans. Autom. Control 54(5), 1134–1140 (2009) 8. T. Meurer, A. Kugi, Tracking control for boundary controlled parabolic PDEs with varying parameters: combining backstepping and differential fitness. Automatica 45(5), 1182–1194 (2009)

26

3 PDE Modeling for the Axially Moving Structures

9. W. He, S.S. Ge, S. Zhang, Adaptive boundary control of a flexible marine installation system. Automatica 47(12), 2728–2734 (2011) 10. Y. Liu, B.S. Xu, Y.L. Wu et al., Boundary control of an axially moving belt, in Proceeding of the 32th Chinese Control Conference (2013), pp. 1310–1315 11. M.S. Queiroz, D.M. Dawson, S.P. Nagarkatti, F. Zhang, Lyapunov Based Control of Mechanical Systems (Springer Science and Business Media, Boston, 2000) 12. M.S. Queiroz, D.M. Dawson, C.D. Rahn, F. Zhang, Adaptive vibration control of an axially moving string. J. Vib. Acoust. 121(1), 41–49 (1999)

Chapter 4

Boundary Control of an Axially Moving System with High Acceleration/Deceleration

4.1 Introduction Generally, the dynamics of an axially moving belt are modeled by a group of PDEs and a set of boundary conditions, which are described by ODEs. This hybrid dynamic model is difficult to control due to the infinite dimensionality of the system. The most conventional approach for the control design of axially moving belt system is the modal control approach based on truncated finite-dimensional model [1, 2]. But the vital disadvantage of the modal control approach is control spillover phenomenon because of the effects of residual mode and unmodeled mode, which are excluded in the truncated finite-dimensional model [3, 4]. The boundary control approach is much more practical than the modal control approach in sense that it excludes the spillover phenomenon. Thus, in recent years, the boundary control has been widely used to suppress vibration of the flexible structure systems, such as flexible riser [5– 11], string [12], beam [13–15], and especially in the area of axially moving structure [16, 17]. In this chapter, the boundary control design via Lyapunov’s direct method is discussed, in which the uniform boundedness of closed-loop system is guaranteed under disturbance. Finally, the numerical simulation with the finite difference method is presented to verify the validity of the proposed controller.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_4

27

28

4 Boundary Control of an Axially Moving System ...

4.2 PDE Dynamic Model In Chap. 3, we consider the PDE model as follows: The governing equation mwtt −

3 1 E Awx2 wx x − T0 wx x + mawx + 2mvwxt + mv 2 wx x + cwt + cvwx = f 2 2 (4.1)

where ∀(x, t) ∈ (0, L) × [0, +∞). Boundary conditions ⎧ ⎨ w(0, t) = 0 ⎩ 1 T0 wx (L , t) + 1 E Awx3 (L , t) = d + U 2 2

(4.2)

where ∀t ∈ [0, +∞). Assumption 4.1 We assume that there exist constants a1 , a2 , a3 , a4 ∈ R+ , such that 0 < v(t) ≤ a1 , |a(t)| ≤ a2 , |d(t)| ≤ a3 , and |dt (t)| ≤ a4 , ∀t ∈ [0, +∞). This is a reasonable assumption since v(t), a(t), and d(t) have finite energy and uniformly continuous. Hence, they are bounded. And in this chapter, we do not consider distributed interference, that is to say f = 0.

4.3 Boundary Controller Design To suppress the vibration of the belt system under the time-varying disturbance d(t), in this section, Lyapunov direct method is used to design a boundary control law U (L , t) at the right boundary of the flexible belt system and analyze the close-loop stability of the system.

4.3.1 Boundary Control Choose the Lyapunov function candidate as V (t) = V1 (t) + V2 (t) + V3 (t),

(4.3)

where the Lyapunov function V (t) is consisted of an energy term V1 (t), expressed as

4.3 Boundary Controller Design

γm V1 (t) = 2

29

L

γ (wt + vwx ) d x + 2

L T εd x

(4.4)

(wwt + 2vwt wx + 2v 2 wx2 )d x

(4.5)

2

0

0

and a small crossing term V2 (t), expressed as L V2 (t) = λm 0

and an auxiliary term V3 (t), expressed as λk2 + γ k3 2 w (L , t) 2

V3 (t) =

(4.6)

where γ , λ, k2 , k3 > 0. Lemma 4.1 The Lyapunov candidate function Eq. (4.3), can be upper and lower bounded as 0 ≤ ϑ1 [V1 (t) + V3 (t)] ≤ V (t) ≤ ϑ2 [V1 (t) + V3 (t)]

(4.7)

where ϑ1 and ϑ2 are two positive constants. Proof Using Eqs. (2.11)–(2.15) on Eq. (4.5), we have L V2 (t) ≤ λm L

L wx2 d x

2

+ 2λm

0

L wt2 d x

0

+ 2λm

v 2 wx2 d x 0

L wx2 d x + λm

0

vwt wx d x + λm 0

L ≤ λm L a

L

(wt + vwx )2 d x 0

≤ ξ V1 (t)

(4.8)

max(λm L a ,λm) . where L a = L 2 + a12 , ξ = min(0.25γ T0 ,0.5γ m) Hence the inequality Eq. (4.7) can be written as

−ξ V1 (t) ≤ V2 (t) ≤ ξ V1 (t)

(4.9)

Supposing ξ satisfied the following equations ξ1 = 1 − ξ > 0, ξ2 = 1 + ξ > 1 Applying the Eq. (4.10) on inequality (4.9), we have

(4.10)

30

4 Boundary Control of an Axially Moving System ...

0 < ξ1 V1 (t) ≤ V1 (t) + V2 (t) ≤ ξ2 V1 (t)

(4.11)

According to the Lyapunov functional candidate in Eq. (4.3), we have following inequality 0 ≤ ϑ1 [V1 (t) + V3 (t)] ≤ V (t) ≤ ϑ2 [V1 (t) + V3 (t)]

(4.12)

    where ϑ1 = min ξ1 , 0.5(λk2 + γ k3 ) > 0 and ϑ2 = max ξ2 , 0.5(λk2 + γ k3 ) > 0. Lemma 4.2 The time derivative of the Lyapunov’s function candidate Eq. (4.3) can be upper bounded with Vt (t) ≤ −ϑ V (t) + ε

(4.13)

where ϑ, ε > 0. Proof Differentiating Eq. (4.3) with time leads to Vt (t) = V1t (t) + V2t (t) + V3t (t)

(4.14)

According to Eq. (4.4), V1 (t) can be represented as V1t (t) = A1 + A2 + A3 + A4

(4.15)

L L where A1 = γ m 0 (wt wtt + vwx wtt )d x, A2 = γ m 0 (vwt wxt + v 2 wx wxt )d x L L A3 = γ2 0 (T0 wx wxt + E Awx3 wxt )d x, and A4 = γ m 0 (awt wx + avwx2 )d x. Substituting the governing Eq. (4.1) into A1 and integrating by parts, we obtain 3γ E Av 4 γ T0 v − 2γ mv 3 2 [wx (L , t) − wx4 (0, t)] + [wx (L , t) − wx2 (0, t)] 8 4 L − γ (cwt2 + cv 2 wx2 + 2cvwt wx − mv 2 wt wx x + mvawx2 + mawt wx )d x

A1 =

0

L +γ

1 3 T0 wt wx x + E Awx2 wt wx x d x − γ mv[2vwx (L , t) + wt (L , t)]wt (L , t) 2 2

0

(4.16) Integrating A2 and A3 by parts, we obtain

4.3 Boundary Controller Design

31

⎧ L ⎪ ⎪ γ mv 2 ⎪ 2 ⎪ ⎪ A = γ mv wx (L , t)wt (L , t) + wt (L , t) − γ m v 2 wt wx x d x ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ L ⎪  ⎨ γ T0 3γ E A wt (L , t)wx (L , t) − wt wx2 wx x d x A3 = ⎪ 2 2 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ L ⎪  ⎪ ⎪ γ T0 γEA ⎪ ⎪ wt (L , t)wx3 (L , t) − wt w x x d x + ⎪ ⎪ ⎩ 2 2

(4.17)

0

Substituting Eqs. (4.16), (4.17), and A4 into Eq. (4.14) and combining boundary conditions Eqs. (4.2) and (2.11)–(2.15), we can get γ E Av 4 v wx (L , t) V1t (t) ≤ γ [ wx (L , t) + wt (L , t)]U + γ dwt (L , t) + 2 8 γ mv 3 γ vd 3γ E Av 4 wx (0, t) − ( − γ δ1 mv 2 )wx2 (L , t) + wx (L , t) − 8 2 2 γ mv γ T0 v − 2γ mv 3 2 γ mv 2 2 )wt (L , t) − wx (0, t) − ( − 4 2 δ1 L L L 2 2 2 − γ c wt d x − γ c v wx d x − 2γ c vwt wx d x 0

0

(4.18)

0

where δ1 is a positive constant. Taking the time derivative of V2 (t), we obtain V2t (t) = B1 + B2 + B3

(4.19)

L L where B1 = 2λm 0 vwt wxt d x, B2 = λm 0 (wwtt + 2vwx wtt )d x, and B3 = L λm 0 (wt2 + 2awt wx + 4avwx2 + 4v 2 wx wxt )d x. Integrating B1 by parts, we obtain B1 = λmvwt2 (L , t) Substituting the governing Eq. (4.1) into B2 and integrating by parts, then

(4.20)

32

4 Boundary Control of an Axially Moving System ...

λE A λma + λcv 2 3λE Av 4 w(L , t)wx3 (L , t) − w (L , t) + [wx (L , t) − wx4 (0, t)] 2 2 4 λT0 λT0 v − 2λmv 3 2 [wx (L , t) − wx2 (0, t)] + ( − λmv 2 )w(L , t)wx (L , t) + 2 2 L 1 − λ (cwwt + E Awx4 + 2cv 2 wx2 + 2mavwx2 + 4mv 2 wx wxt − mv 2 wx2 + 2

B2 =

0

1 + T0 wx2 + 2cvwt wx − 2mvwt wx )d x − 2λmvw(L , t)wt (L , t) 2 (4.21) Substituting Eqs. (4.20)–(4.21) and A4 into Eq. (4.19) and combining boundary conditions Eqs. (4.2) and (2.11)–(2.15), we can get λT0 v − 2λmv 3 2 wx (0, t) 2 2mv λE Av 4 ma + cv mv 2 2 3λE Av 4 wx (0, t) + wx (L , t) − λ( − − )w (L , t) − 4 4 2 δ2 δ3 + λvdwx (L , t) + λ(mv + 2δ2 mv)wt2 (L , t) − λ(mv 3 − δ3 mv 2 )wx2 (L , t)

V2t (t) ≤ λ[vwx (L , t) + w(L , t)]U + λdw(L , t) −

cL 2 T0 2ma − λ( − 2mva − − ) 2 δ4 δ5 L v 2 wx2 d x −

− λ(2c − m) 0

λE A 2

L

L wx2 d x

− λ(−m − cδ4 − 2maδ5 )

0

wt2 d x 0

L

L wx4 d x − 2λ(c − m)

0

vwt wx d x 0

(4.22) where δ2 , δ3 , δ4 , δ5 are positive constants Taking the time derivative of Eq. (4.4), we obtain V3t (t) = (λk2 + γ k3 )w(L , t)wt (L , t) Then, substituting Eqs. (4.18), (4.22), and (4.23) into Eq. (4.14), we have

(4.23)

4.3 Boundary Controller Design

33

(2λ + γ )v wx (L , t) + γ wt (L , t) + λw(L , t)]U + (λk2 + γ k3 )w(L , t)wt (L , t) 2 (2λ + γ )vd (2λ + γ )E Av 4 wx (L , t) + wx (L , t) + γ dwt (L , t) + λdw(L , t) + 2 8 2λmv λma + λcv λmv 2 (2λ + γ )(T0 v − 2mv 3 ) 2 wx (0, t) − ( − − )w (L , t) − 4 2 δ2 δ3 γv γ γv − 2λδ2 )wt2 (L , t) − mv 2 ( + λv − γ δ1 − λδ3 )wx2 (L , t) − mv( − λ − 2 2 δ1 L L 3(2λ − γ )E Av 4 λE A − wx (0, t) − wx4 d x − 2(γ c + λc − λm) vwt wx d x 8 2

Vt (t) ≤ [

0

− λ(

2ma T0 cL 2 − ) − 2mav − 2 δ4 δ5

0

L

L wx2 d x − (2λc + γ c − λm)

0

v 2 wx2 d x 0

L − (γ c − λm − λcδ4 − 2λmaδ5 )

wt2 d x 0

(4.24) According to above analysis, we design the boundary controller as U = −k1 wx (L , t) − k2 wt (L , t) − k3 w(L , t)

(4.25)

where k1 > 0. Substituting Eq. (4.25) into Eq. (4.24) and combining Eqs. (2.11)–(2.15), we can get L Vt (t) ≤ −τ1 w (L , t) − 2

τ2 wt2 (L , t)

−

τ3 wx2 (L , t)

−

τ4 wx4 (0, t)

−

τ5 wx2 (0, t)

wt2 d x

− τ6 0

L

L v 2 wx2 d x − τ8

− τ7 0

L vwt wx d x − τ9

0

L wx2 d x − τ10

0

wx4 d x + ε 0

≤ −ϑ3 [V1 (t) + V3 (t)] + ε (4.26) where δ6 and δ7 are positive constants. Then the constants γ , λ, k1 ∼ k3 , and δ1 ∼ δ7 are chosen to satisfy the following conditions: τ1 = λk3 +

2λmv λma + λcv λmv 2 [2λk1 + (2λ + γ )vk3 ]δ7 − > 0, − − 2 δ2 δ3 2

34

4 Boundary Control of an Axially Moving System ...

γ mv 2 [2γ k1 + (2λ + γ )vk2 ]δ6 (γ − 2λ)mv − − 2λmvδ2 ≥ 0, − 2 δ1 2

(2λ + γ ) vk1 + mv 3 2λk1 + (2λ + γ )vk3 2γ k1 + (2λ + γ )vk2 τ3 = − − 2 2δ6 2δ7 2 − (λδ3 + γ δ1 )mv ≥ 0,

τ2 = γ k2 +

τ4 = τ5 =

3(2λ − γ )E Av ≥ 0, 8

(2λ + γ )(T0 v − 2mv 3 ) ≥ 0, 4

τ6 = γ c − λm − λcδ4 − 2λmaδ5 > 0, τ7 = 2λc + γ c − λm > 0, τ8 = 2γ c + 2λc − 2λm > 0, τ9 =

λcL 2 λT0 2λma − − − 2λmav > 0, 2 δ4 δ5 τ10 =

ε = max |

λE A > 0, 2

(2λ + γ )E Av 4 (2λ + γ )vd wx (L , t) + wx (L , t) + λdw(L , t) + γ dwt (L , t)|, 2 8

2τ1 2τ6 2τ7 τ8 4τ9 8τ10 , , , ϑ3 = min , , λk2 + γ k3 γ m γ m γ m γ T0 γ E A

Combining inequalities (4.11) and (4.25), we have Vt (t) ≤ −ϑ V (t) + ε

(4.27)

 where ϑ = ϑ3 ϑ2 and 0 ≤ ε < +∞. Remark 4.1 Both of the signals in the boundary control can be measured by sensors or obtained by a backward difference algorithm. w(L , t) and wx (L , t) can be measured by a laser displacement sensor and a tilt sensor at the right boundary of moving belt, respectively, and wt (L , t) can be obtained through a backwards difference algorithm of the values of w(L , t). This controller design does not need to know the exact model of the disturbance, so it is stable and robust to the changes of the system parameters.

4.3 Boundary Controller Design

35

4.3.2 Stability Analysis Theorem 4.1 The moving belt system described by Eqs. (4.1) and (4.2), under Assumption 4.1 and the proposed control law Eq. (4.25), the closed-loop system is uniformly bounded as (1)

uniform boundedness: the system state w(x, t) of the closed-loop belt system will remain in the compact set: 1 := { |w(x, t)| ≤ χ1 }

(4.28)

 where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 = γ T4L0 ϑ1 [V (0) + ϑε ]. (2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop belt system will eventually converge to the compact set: 2 := { lim |w(x, t)| ≤ χ2 }

(4.29)

t→∞

where ∀x ∈ [0, L], χ2 =



4Lε . γ T0 ϑ1 ϑ

Proof Multiplying Eq. (4.27) by eϑt yields: Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒

 ∂ V (t)eϑt ≤ εeϑt ∂t

(4.30)

Integrating the above inequality yields: V (t) ≤ [V (0) −

ε −ϑt ε ε ]e + ≤ V (0)e−ϑt + ϑ ϑ ϑ

(4.31)

Applying Eqs. (2.11)–(2.15) and (4.12), we have: γ T0 2 γ T0 w (x, t) ≤ 4L 4

L wx2 (x, t)d x ≤ V1 (t) ≤ V1 (t) + V3 (t) ≤ 0

1 V (t) ϑ1

(4.32)

Substituting Eq. (4.31) into Eq. (4.32) results in:  |w(x, t)| ≤

4L ε [V (0)e−ϑt + ] ≤ γ T0 ϑ1 ϑ



4L ε [V (0) + ] γ T0 ϑ1 ϑ

where ∀(x, t) ∈ [0, L] × [0, +∞), from above inequality, we have:

(4.33)

36

4 Boundary Control of an Axially Moving System ...

 lim |w(x, t)| ≤ lim

t→∞

t→∞

4L ε [V (0)e−ϑt + ] = γ T0 ϑ1 ϑ

 4Lε , ∀x ∈ [0, L] γ T0 ϑ1 ϑ (4.34)

4.4 Simulation Example Consider the PDE model as Eq. (4.1), choose parameters as: E A = 1000 N, T0 = 10000 N, c = 1.0 Ns/m2 , L = 1.0 m, and m = 1.0 kg/m. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D are given as: aa = ad = 3.5g and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s (g = 9.8 m/s2 ). The initial conditions of axially moving belt system are given as: w(x, 0) = wt (x, 0) = 0

(4.35)

The unknown time-varying boundary disturbance d(t) is described as: d(t) = 3 + 0.1 sin(0.1t) + 0.3 sin(0.3t) + 0.5 sin(0.5t)

(4.36)

Figure 4.1 is the simulation results with controller parameters k1 = k2 = k3 = 1 × 107 . The displacements comparison w(x, t) of the moving belt at x = 0.5 m and x = 1 m with control are shown in Fig. 4.2, the displacements comparison of the moving belt are examined at x = 0.5 m and x = 1 m (in Fig. 4.3) for controlled and uncontrolled responses (proposed in Chap. 3) and the control input is shown in Fig. 4.4.

4.4 Simulation Example

Fig. 4.1 The displacement of the belt with proposed boundary controller

Fig. 4.2 The enlarged view of the vibration displacement for controlled response

37

4 Boundary Control of an Axially Moving System ...

w(0.5,t) [m]

38

Fig. 4.3 Vibration displacement of the belt at: a x = 1 m, b x = 0.5 m

Fig. 4.4 Boundary control input U (t)

Appendix 1: Simulation Program

Appendix 1: Simulation Program

close all; clear all; clc; nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m=1; EA=1000; T=10000; pi=3.1415926; J=35; V0=0; c=1; d1=1;

39

40

4 Boundary Control of an Axially Moving System ...

for i=1:nx for j=1:nt d(i,j)=3+0.1*sin(0.1*j*dt)+0.3*sin(0.3*j*dt)+0.5*sin(0.5*j*dt); end end V= zeros(nx,nt); for i=1:nx V(i,1)=V0; end for i=1:nx for j=2:10*10^3-1 V(i,j) = V(i,1)+0.5*J*(j*dt)^2; end V(i,10*10^3)=V(i,1)+0.5*J; for j=(10*10^3+1):20*10^3-1 V(i,j)=V(i,10*10^3)+0.5*J*(j*dt-1); end V(i,20*10^3)=V(i,10*10^3)+J; for j=(20*10^3+1):30*10^3-1 V(i,j) = V(i,20*10^3)+J*(j*dt-2)-0.5*J*(j*dt-2)^2; end V(i,30*10^3)=V(i,20*10^3)+0.5*J; for j=(30*10^3+1):70*10^3-1 V(i,j)=V(i,30*10^3); end V(i,70*10^3)=V(i,30*10^3); for j=(70*10^3+1):80*10^3-1 V(i,j)= V(i,70*10^3)-0.5*J*(j*dt-7)^2; end V(i,80*10^3)=V(i,70*10^3)-0.5*J; for j=(80*10^3+1):90*10^3-1 V(i,j)=V(i,80*10^3)-J*(j*dt-8); end V(i,90*10^3)=V(i,80*10^3)-J; for j=(90*10^3+1):nt-1 V(i,j)=V(i,90*10^3)-J*(j*dt-9)+0.5*J*(j*dt-9)^2; end V(i,100*10^3)=V(i,90*10^3)-0.5*J; end

Appendix 1: Simulation Program

41

%******************************************************************** %

free vibration u_T=0 and u_L=0

%******************************************************************** w1 = zeros(nx,nt); % ------- boundary condition + initial condition w1(1,:) = 0;

% w(0,t)=0;

for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j=3:nt for i=2:nx-1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +(dt^2*(0.5*T-m*V(1,j-1)*V(1,j-1)))/(m*dx^2)*(w1(i+1,j-1)-2*w1(i,j-1) +w1(i-1,j-1))- (dt*c)/(m)*(w1(i,j-1)-w1(i,j-2)) dt^2*c*V(1,j-1)/(m*dx)*(w1(i,j-1)-w1(i-1,j-1)) -(2*V(1,j-1)*m*dt)/(m*dx)*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2 ))- m*dt/(m*dx)*(V(1,j-1)-V(1,j-2))*(w1(i,j-1)-w1(i-1,j-1))

+1.5*dt^2*EA/(m*dx^4)*(w1(i,j-1)-w1(i-1,j-1))^2*(w1(i+1,j-1)-2*w1(i,j -1)+w1(i-1,j-1)); end w1(nx,j)=2*dx*d(1,j)/T + w1(nx-1,j)-EA*(w1(nx,j)-w1(nx-1,j))^3/(T*dx^2); end %for obtain the curve of the riser, short the point tshort = linspace(0,tmax,nt/400); xshort = linspace(0,L,nx); w1short=zeros(nx,nt/400); for j=1:nt/400 for i=1:nx w1short(i,j)=w1(i,j*400); end end %******************************************************************** %

forced vibration u_T

%******************************************************************** w2 = zeros(nx,nt); k1=10^7; k2=10^7; k3=10^7;

42

4 Boundary Control of an Axially Moving System ...

% % ------- boundary condition + initial condition w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0; end for j=3:nt for i=2:nx-1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +(dt^2*(0.5*T-m*V(1,j-1)*V(1,j-1)))/(m*dx^2)*(w2(i+1,j-1)-2*w2(i,j-1) +w2(i-1,j-1))- (dt*c)/(m)*(w2(i,j-1)-w2(i,j-2)) dt^2*c*V(1,j-1)/(m*dx)*(w2(i,j-1)-w2(i-1,j-1)) -(2*V(1,j-1)*m*dt)/(m*dx)*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2 )) -m*dt/(m*dx)*(V(1,j-1)-V(1,j-2))*(w2(i,j-1)-w2(i-1,j-1)) +1.5*dt^2*EA/(m*dx^4)*(w2(i,j-1)-w2(i-1,j-1))^2*(w2(i+1,j-1)-2*w2(i,j -1)+w2(i-1,j-1)); end w2(nx,j)= (T/dx*w2(nx-1,j)+2*k2/dt*w2(nx,j-1)+2*k1/dx*w2(nx-1,j)+2*d(1,j))/(T/d x+2*k3+2*k1/dt+2*k2/dx)-EA*(w2(nx,j)-w2(nx-1,j))^3/(T*dx^2); end % %for obtain the curve of the riser, short the point tshort = linspace(0,tmax,nt/400); xshort = linspace(0,L,nx); w2short=zeros(nx,nt/400); for j=1:nt/400 for i=1:nx w2short(i,j)=w2(i,j*400); end end

figure (1); surf(tshort,xshort,w2short); shading interp; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('$\omega$(x,t)[m]','interpreter','latex','Fontsize',13); set(gca,'YDir','reverse') for j=2:nt u1(j)=-k3*w2(nx,j)k2*((w2(nx,j)-w2(nx,j-1)))/dt-k1*((w2(nx,j)-w2(nx-1,j)))/dx;

References

43

end figure (2); plot(tshort,w2short(25,:),'b',tshort,w2short(50,:),':r'); xlabel('t / s','Fontsize',14); ylabel('w(x,t)/ m','Fontsize',14); legend('x=0.5m','x=1m'); box on grid on figure (3); hold on subplot(2,1,1); plot(tshort,w1short(50,:),'b',tshort,w2short(50,:),':r'); title('(a)'); xlabel('t /s','Fontsize',14); ylabel('w(1,t)/ m','Fontsize',14); legend('without control','with control'); box on grid on

subplot(2,1,2); plot(tshort,w1short(25,:),'b',tshort,w2short(25,:),':r'); title('(b)'); xlabel('t / s','Fontsize',14); ylabel('w(0.5,t)/ m' ,'Fontsize',14); legend('without control','with control'); box on grid on hold off figure (4); plot(t,u1,'b'); xlabel('t / s','Fontsize',14); ylabel('U(t)/ N','Fontsize',14); box on grid on

References 1. J. Chung, C. Han, K. Yi, Vibration of an axially moving string with geometric non-linearity and translating acceleration. J. Sound Vib. 240(4), 733–746 (2001) 2. W. Zhang, L. Chen, Vibration control of an axially moving string system: wave cancellation method. Appl. Math. Comput. 175(1), 851–863 (2006) 3. L. Meirovitch, H. Baruh, On the problem of observation spillover in self-adjoint distributed systems. J. Optim. Theory Appl. 39(2), 269–291 (1983) 4. M.J. Balas, Active control of flexible systems. J. Optim. Theory Appl. 25(3), 415–436 (1978) 5. Y. Liu, H. Gao, Z. Zhao, X. Wu, Boundary control of a flexible marine riser, in Proceedings of the 31st Chinese Control Conference (2012), pp. 1228–1233 6. Y. Liu, H. Huang, H. Gao, X. Wu, Modeling and boundary control of a flexible marine riser coupled with internal fluid dynamics. J. Control Theory Appl. 11(2), 316–323 (2013)

44

4 Boundary Control of an Axially Moving System ...

7. Y. Wu, Y. Liu, X. Wu, Adaptive boundary control of a flexible riser coupled with time-varying internal fluid. Control Theory Appl. 30(5), 618–624 (2013) 8. X. Wu, L. Li, Y. Liu, H. Gao, Modeling and boundary control of a marine riser. J. South China Univ. Technol. (Nat. Sci. Ed.) 40(8), 32–38 (2012) 9. H. Gao, Z. Zhao, X. Wu, Y. Liu, Robust boundary control for flexible fluid-transporting marine riser based on internal fluid dynamics. Control Theory Appl. 29(6), 785–791 (2012) 10. W. He, S.S. Ge, B.V.E. How et al., Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47(4), 722–732 (2011) 11. W. He, S.S. Ge, S. Zhang, Adaptive boundary control of a flexible marine installation system. Automatica 47(12), 2728–2734 (2011) 12. W. He, S.S. Ge, Robust adaptive boundary control of a vibrating string under unknown timevarying disturbance. IEEE Trans. Control Syst. Technol. 20(1), 48–58 (2012) 13. Y. Liu, H. Huang, Y. Wu, X. Wu, Vibration control of flexible beam based on Lyapunov direct method. J. South China Univ. Technol. (Nat. Sci. Ed.) 41(2), 24–29 (2013) 14. Y. Wu, Y. Liu, Modeling and boundary vibration control of a distributed-parameter flexible beam. Acta Sci. Nat. Univ. Sunyatseni 52(3), 55–62 (2013) 15. X. Wu, J. Deng, Robust boundary control of a distributed parameter flexible manipulator with top unknown disturbance. Control Theory Appl. 28(4), 511–518 (2011) 16. J. Choi, K. Hong, Vibration control of an axially moving belt by a nonlinear boundary control, in Proceedings of the International Conference on Control, Automation and Systems (2001), pp. 1322–1325 17. K.S. Hong, C. Kim, K. Hong, Boundary control of an axially moving belt system in a thin-metal production line. Int. J. Control Autom. Syst. 2(1), 55–67 (2004)

Chapter 5

Robust Boundary Control of an Axially Moving System with High Acceleration/deceleration and Disturbance Observer

5.1 Introduction The axially moving flexible structures such as belt, string, and cable are widely applied in many mechanical systems. Since the undesirable or excessive vibration can degrade system performance [1, 2], how to weaken vibration of the flexible structure in motion is an urgent problem that needs to be solved. In [3], a boundary control law based on Lyapunov’s direct method is presented to reduce the vibration deflection of an axially moving belt. In [4], an adaptive isolation scheme is introduced for an axially moving system to regulate the controlled span. For these systems, the flexible structure exhibits vibration in the presence of disturbances. How to solve the impact of time-varying and unknown disturbances is the focus of this chapter. In this chapter, the adoption of the Sc-A/D method to regulate the axial speed of the axially moving belt system with H-A/D and SDD. Meanwhile, a boundary control with disturbance observer is proposed for boundary disturbance tracking to attenuate the effect of unknown external disturbance. What is more, using the Lyapunov’s synthesis and Sc-A/D methods to reduce the vibration of the belt, where the stability of the closed-loop system is guaranteed.

5.2 PDE Dynamic Model The motion equations of the belt system with H-A/D and distributed disturbance can be derived by using the Hamilton’s principle, expressed as [5]: t2 (δ E k − δ E p + δWc + δWd − δWb )dt = 0

(5.1)

t1

The kinetic energy E k (t) of the belt system can be represented as: © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_5

45

46

5 Robust Boundary Control of an Axially Moving System with High …

1 1 E k = m c wt2 (L , t) + m 2 2

L (wt +vwx )2 d x

(5.2)

0

where m c is the mass of the actuator. The potential energy E p (t) of the belt system is expressed as: 1 Ep = 2

L T εd x

(5.3)

0

where the spatially varying tension T (x, t) is given as: T = T0 +

1 E Awx2 2

(5.4)

where T0 is the initial tension of the belt, E is the coefficient of elasticity, and A is the cross-sectional area. The displacement-strain relation ε is expressed as: ε=

1 2 w 2 x

(5.5)

The virtual work δWc (t) done by the non-conservative forces on the belt is computed as: L δWc = [U − ds wt (L , t)]δw(L , t) − c

(wt +vwx )δwd x

(5.6)

0

where ds is damping coefficient of the actuator and c is viscous damping coefficient of the belt. The virtual work δWd (t) done by unknown boundary disturbance d(t) and distributed disturbance f (x, t) on the belt is given by: L δWd = dδw(L , t) +

f δwd x

(5.7)

0

The virtual momentum δWb (t) transport across the boundaries is computed by: δWb = mv[wt (L , t) + vwx (L , t)]δw(L , t)

(5.8)

Substituting Eqs. (5.2)–(5.8) into Eq. (5.1) and integrating by parts, the governing equation of the presented belt system can be obtained as:

5.3 Boundary Controller Design

mwtt + mawx + 2mvwxt + mv 2 wx x −

47

3 1 E Awx2 wx x − T0 wx x + cwt + cvwx = f 2 2 (5.9)

where ∀(x, t) ∈ (0, L) × [0, +∞). The boundary conditions of presented belt system are described as: ⎧ ⎨ w(0, t) = 0 ⎩ m c wtt (L , t) + 1 T0 wx (L , t) + 1 E Awx3 (L , t) + ds wt (L , t) = d + U 2 2

(5.10)

where ∀t ∈ [0, +∞). Assumption 5.1 For the speed v(t), H-A/D a(t), boundary disturbance d(t) and distributed disturbance f (x, t), we assume that there exist constants a1 , a2 , a3 , a4 , and a5 ∈ R+ , such that 0 < v(t) ≤ a1 , | a(t) | ≤ a2 , | d(t) | ≤ a3 , | dt (t) | ≤ a4 , ∀t ∈ [0, +∞), and | f (x, t) | ≤ a5 , ∀(x, t) ∈ (0, L) × [0, +∞). This is a reasonable assumption since v(t), a(t), d(t), and f (x, t) have finite energy and uniformly continuous. Hence, they are bounded.

5.3 Boundary Controller Design To stabilize the belt system given by Eqs. (5.9) and (5.10), we propose the following robust boundary control: 1 U (t) = −k[wt (L , t) + k1 wx (L , t)] − k1 m c wxt (L , t) + T0 wx (L , t) + ds wt (L , t) 2 ˆ −a3 sgn[wt (L , t) + k1 wx (L , t)] − d(t) (5.11) ˆ is the where k > 0 is the control gain, k1 > 0 is the weighting constant, and d(t) estimate of d(t), expressed as: ˆ = ϕ(t) + m c wt (L , t) d(t)

(5.12)

where we define: ϕt (t) = σ [wt (L , t) + k1 wx (L , t)] − m c wt (L , t) − U − ϕ(t) 1 1 + T0 wx (L , t) + E Awx3 (L , t) + ds wt (L , t) 2 2

(5.13)

where σ is a positive constant and the disturbance observer error is defined as:

48

5 Robust Boundary Control of an Axially Moving System with High …

Fig. 5.1 Block diagram of the robust boundary control

ˆ e(t) = d(t) − d(t)

(5.14)

Differentiating Eq. (5.14) with respect to time, and then substituting Eqs. (5.12) and (5.13) and boundary conditions Eq. (5.10), we have: et (t) = dt (t) − σ [wt (L , t) + k1 wx (L , t)] − e(t)

(5.15)

Remark 5.1 The block diagram of the proposed control Eq. (5.11) is shown in Fig. 5.1 and all signals in Eq. (5.11) can be measured by sensors or obtained by a backward difference algorithm. w(L , t) can be sensed by a laser displacement sensor and wx (L , t) can be measured by an inclinometer. wt (L , t) and wxt (L , t) can be calculated with a backwards difference algorithm to w(L , t) and wx (L , t), respectively.

5.4 Stability Analysis Consider the Lyapunov function candidate: V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t)

(5.16)

where the energy term V1 (t) is expressed as: γm V1 (t) = 2

L

γ (wt + vwx ) d x + 2

L T εd x

2

0

(5.17)

0

and a small crossing term V2 (t) is expressed as: L xwx (wt + vwx )d x

V2 (t) = 2λm 0

(5.18)

5.4 Stability Analysis

49

and an auxiliary term V3 (t) is expressed as: V3 (t) =

1 γ m c [wt (L , t) + k1 wx (L , t)]2 2

(5.19)

and an observer error term V4 (t) is expressed as: V4 (t) =

γ 2 e (t) 2σ

(5.20)

where γ and λ are two positive weighting constants. Applying Eqs. (2.11)–(2.15) to Eq. (5.17) results in: L |V2 (t)| ≤ λm L



 wx2 + (wt + vwx )2 d x ≤ ξ V1 (t)

(5.21)

0

where ξ =

4λm L . min(γ T0 ,2γ m)

According to Eq. (5.21),we have: −ξ V1 (t) ≤ V2 (t) ≤ ξ V1 (t)

(5.22)

Supposed ξ satisfies the following equations ξ1 = 1 − ξ > 0, ξ2 = 1 + ξ > 1

(5.23)

Applying the Eq. (5.23) on inequality (5.22), we have 0 < ξ1 V1 (t) ≤ V1 (t) + V2 (t) ≤ ξ2 V1 (t)

(5.24)

Combining Eq. (5.16), we obtain: 0 ≤ ϑ1 [V1 (t) + V3 (t) + V4 (t)] ≤ V (t) ≤ ϑ2 [V1 (t) + V3 (t) + V4 (t)]

(5.25)

where ϑ1 = min(ξ1 , 1) and ϑ2 = max(ξ2 , 1). Lemma 5.1 The time derivative of the Lyapunov function candidate Eq. (5.16) can be upper bounded with Vt (t) ≤ −ϑ V (t) + ε

(5.26)

where ϑ, ε > 0. Proof Differentiating Eq. (5.16) with time leads to Vt (t) = V1t (t) + V2t (t) + V3t (t) + V4t (t)

(5.27)

50

5 Robust Boundary Control of an Axially Moving System with High …

According to Eq. (5.17), V1 (t) can be represented as V1t (t) = A1 + A2 + A3 + A4 where A1 = γ m γ 2

A3 =

L

L

(wt wtt + vwx wtt )d x, A2 = γ m

0

L

(vwt wxt + v 2 wx wxt )d x

0

(T0 wx wxt +

E Awx3 wxt )d x,

and A4 = γ m

0

(5.28)

L

(awt wx + avwx2 )d x.

0

Substituting the governing Eq. (5.9) into A1 and integrating by parts, we obtain: 3γ E Av 4 γ T0 v − 2γ mv 3 2 [wx (L , t) − wx4 (0, t)] + [wx (L , t) − wx2 (0, t)] 8 4 L − γ (cwt2 + cv 2 wx2 + 2cvwt wx − mv 2 wt wx x + mvawx2 + mawt wx )d x

A1 =

0

L +γ

1 3 2 T0 wt wx x + E Awx wt wx x d x − γ mv[2vwx (L , t) + wt (L , t)]wt (L , t) 2 2

0

L

L wx f d x + γ

+ γv 0

wt f d x 0

(5.29) Integrating A2 and A3 by parts, we obtain ⎧ L ⎪ ⎪ γ mv 2 ⎪ 2 ⎪ ⎪ A w = γ mv w (L , t)w (L , t) + (L , t) − γ m v 2 wt wx x d x, 2 x t t ⎪ ⎪ 2 ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ L ⎪  ⎨ γ T0 3γ E A wt (L , t)wx (L , t) − wt wx2 wx x d x A3 = ⎪ 2 2 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ L ⎪ ⎪ ⎪ ⎪ + γ E A wt (L , t)w 3 (L , t) − γ T0 wt wx x d x ⎪ x ⎪ ⎩ 2 2

(5.30)

0

Substituting Eqs. (5.29)–(5.30) and A4 into Eq. (5.28) and combining boundary conditions Eq. (5.10) and Eqs. (2.11)–(2.15), we can get

5.4 Stability Analysis

51

γ T0 v − 2γ mv 3 2 γEA 3γ E Av 4 wx (0, t) − wx (0, t) + wt (L , t)wx3 (L , t) 8 4 2 γ mv 3γ E Av 4 γ T0 v 2 wx (L , t) − wx (L , t) + [wt (L , t) + vwx (L , t)]2 + 4 2 8 γ T0 2 γ T0 k1 2 γ T0 wt (L , t) − + wx (L , t) [wt (L , t) + k1 wx (L , t)]2 − 4k1 4k1 4 L L γ 2 − (cγ − γ δ1 ) (wt + vwx ) d x + f 2d x δ1

V1t (t) ≤ −

0

0

(5.31) where δ1 is a positive constant. Taking the time derivative of V2 (t), V2t (t) = B1 + B2 where B1 = 2λm

L

xwxt (wt + vwx )d x, B2 = 2λm

0

(5.32) L

xwx (wtt + awx + vwxt )d x.

0

Integrating B1 by parts results in: L B1 =

λm Lwt2 (L , t)

− λm

L wt2 d x

+ 2λmv

0

xwx wxt d x

(5.33)

0

Substituting the governing Eq. (5.9) into B2 and integrating by parts, then 3λE AL 4 λL T0 2 λT0 wx (L , t) + wx (L , t) − λm Lv 2 wx2 (L , t) − B2 = 4 2 2

L wx2 d x 0

L

L xwx2 d x + λmv 2

− 2λcv 0

L − 2λmv 0

L wx2 d x − 2λc

0

3λE A xwx wxt d x − 4

xwt wx d x 0

L

L wx4 d x

0

+ 2λ

xwx f d x 0

(5.34)

52

5 Robust Boundary Control of an Axially Moving System with High …

Substituting Eqs. (5.33) and (5.34) into Eq. (5.32) and combining boundary conditions Eq. (5.10) and Eqs. (2.11)–(2.15), we can get

λL T0 3λE AL 4 2 2 wx (L , t) + λm Lwt (L , t) − λm Lv − wx2 (L , t) V2t (t) ≤ 4 2

L L λT0 2 2 − wx d x − λm wt2 d x − λma1 − 2λLδ2 − 2λcLδ3 2 0

+

2λcL δ3

L (wt + vwx )2 d x − 0

3λE A 4

0

L wx4 d x + 0

2λL δ2

L f 2d x 0

(5.35) where δ2 , δ3 are two positive constants. Taking the time derivative of Eq. (5.19), then substituting the boundary conditions Eq. (5.10) and control law Eq. (5.11), we obtain: V3t (t) ≤ −γ k[wt (L , t) + k1 wx (L , t)]2 + γ [wt (L , t) + k1 wx (L , t)]e(t) (5.36) γEA γ E Ak1 4 − wt (L , t)wx3 (L , t) − wx (L , t) 2 2 Differentiating Eq. (5.20) with respect to time and substituting the Eqs. (2.11)– (2.15), we have: γ γ e(t)dt (t) − γ [wt (L , t) + k1 wx (L , t)]e(t) − e2 (t) σ σ γ 2 γ 2 dt (t) − e (t) − γ [wt (L , t) + k1 wx (L , t)]e(t) ≤ 2σ 2σ

V4t (t) =

(5.37)

Substituting Eqs.(5.31), (5.35)–(5.37) into Eq. (5.27) and then using Eqs.(2.11)– (2.15), we obtain:

5.4 Stability Analysis

53

3λE AL 3γ E Av γ E Ak1 γ T0 v − 2γ mv 3 2 wx (0, t) − − − wx4 (L , t) Vt (t) ≤ − 4 2 4 8

− γ4kT10 − λm L wt2 (L , t) − γ T0 (k41 −v) + λm Lv 2 − λL2T0 wx2 (L , t)

γ T0 γ 2 γ 2 − γk − e (t) + d (t) [wt (L , t) + k1 wx (L , t)]2 − 4k1 2σ 2σ t L γ mv 3λE A 3γ E Av 4 2 wx (0, t) − wx4 d x − [wt (L , t) + vwx (L , t)] − 8 2 4 0



L wt2 d x −

− λm 0

− cγ − γ δ1 −

2λcL δ3

λT0 − λma12 − 2λLδ2 − 2λcLδ3 2

L

(wt + vwx )2 d x +

0



L wx2 d x 0

γ δ1

+

2λL δ2

L

f 2d x

0

(5.38) According to Eqs. (5.21), (5.23), and (5.38), the parameters λ, γ , k, k1 , δ1 ∼ δ3 are chosen to satisfy the following conditions: λ
0, 2 τ2 = γ c − γ δ1 − τ3 = γ k −

2λcL > 0, δ3

γ T0 > 0, 4k1

4τ1 2τ2 6λ k , , ϑ3 = min , ,1 . γ T0 γ m γ γ m c

54

5 Robust Boundary Control of an Axially Moving System with High …

Substituting Ineqs above into Eq. (5.38) and combining Eqs. (2.11)–(2.15), we can get L Vt (t) ≤ −τ3 [wt (L , t) + k1 wx (L , t)] − τ1

wx2 d x

2

0

−τ2

L

(wt + vwx )2 d x −

0

L

3λE A 4

wx4 d x −

0

γ 2 e (t) 2σ

+ε

(5.39)

≤ −ϑ3 [V1 (t) + V3 (t) + V4 (t)] + ε ≤ −ϑ V (t) + ε where 0 < ε = ϑ=



γ δ1

+

2λL δ2

L

ϑ3 . ϑ2

f 2d x +

0

γ 2σ

dt2 ≤

γ 2 a 2σ 4

+



γ δ1

+

2λL δ2

a52 L < +∞ and

Theorem 5.1 For the presented belt system described by Eqs. (5.9)–(5.10), under Assumption 5.1 and the proposed control law Eq. (5.11), given that the initial conditions are bounded, we can conclude that: (1) uniform boundedness: the system state w(x, t) of the closed-loop belt system will remain in the compact set: 1 := { w(x, t) ∈ R||w(x, t)| ≤ χ1 } where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 =



4L [V (0)e−ϑt γ ϑ1 T0

(5.40) + ϑε ].

(2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop belt system will eventually converge to the compact set: 2 := { w(x, t) ∈ R| lim |w(x, t)| ≤ χ2 } t→∞

where ∀x ∈ [0, L], χ2 =



(5.41)

4Lε . γ T0 ϑ1 ϑ

Proof Multiplying Eq. (5.39) by eϑt yields: Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒

 ∂ V (t)eϑt ≤ εeϑt ∂t

(5.42)

Integrating the above inequality yields: V (t) ≤ [V (0) −

ε −ϑt ε ε + ≤ V (0)e−ϑt + ]e ϑ ϑ ϑ

Applying Eqs. (2.11)–(2.15) and Eq. (5.17), we have:

(5.43)

5.5 Simulation Example

55

γ T0 2 γ T0 w (x, t) ≤ 4L 4

L wx2 (x, t)d x ≤ V1 (t) ≤ V1 (t) + V3 (t) + V4 (t) ≤ 0

1 V (t) ϑ1 (5.44)

Substituting Eq. (5.43) into Eq. (5.44) results in:  |w(x, t)| ≤

4L ε [V (0)e−ϑt + ] ≤ γ T0 ϑ1 ϑ



4L ε [V (0) + ] γ T0 ϑ1 ϑ

(5.45)

where ∀(x, t) ∈ [0, L] × [0, +∞), from above inequality, we have:  lim |w(x, t)| ≤ lim

t→∞

t→∞

4L ε [V (0)e−ϑt + ] = γ T0 ϑ1 ϑ

 4Lε , γ T0 ϑ1 ϑ

∀x ∈ [0, L] (5.46)

5.5 Simulation Example Consider the PDE model as Eq. (5.9), the parameters of the belt system are given in Table 5.1. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D are given as: aa = ad = 3.5g, and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s (g = 9.8m/s 2 ). The initial conditions of the axially moving belt system are given as: w(x, 0) = wt (x, 0) = 0

(5.47)

The unknown distributed disturbance f (x, t) is described as: f (x, t) = [1 + sin(xt) + sin(2xt) + sin(3xt) ]x × 10−4

(5.48)

The unknown time-varying boundary disturbance d(t) is described as: Table 5.1 Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

Value

m

1.0 kg/m

mc

5.0 kg

L

1.0 m

c

1.0 Ns/m2

EA

500 N

σ

1.0

ds

0.25 Ns/m

T0

9800 N

k1

1 × 102

a3

0.1

v0

0 m/s

k

1 × 104

56

5 Robust Boundary Control of an Axially Moving System with High …

Fig. 5.2 Vibration displacement of the belt without control

d(t) = 3 + 0.1 sin(t) + 0.2 sin(2t) + 0.3 sin(3t)

(5.49)

Figure 5.2 shows the three-dimensional (3D) vibration displacement of the axially moving belt for free vibration, i.e. U (t) = 0, with H-A/D under both distributed disturbance and boundary disturbance, while Fig. 5.3 shows the 3D vibration displacement with the proposed robust boundary control and disturbance observer. The vibration displacement of the belt is examined at x = 1m and x = 0.5m, and the simulation results for controlled and uncontrolled responses are shown in Fig. 5.4, respectively. Figure 5.5 shows the enlarged view of the vibration displacement for controlled response at x = 1m and x = 0.5m. The tracking of boundary disturbance is shown in Fig. 5.6 and the control input is shown in Fig. 5.7.

5.5 Simulation Example

Fig. 5.3 Vibration displacement of the belt with control

Fig. 5.4 Vibration displacement of the belt at: a x = 1m, b x = 0.5m

57

58

5 Robust Boundary Control of an Axially Moving System with High …

Fig. 5.5 The enlarged view of the vibration displacement for controlled response

Fig. 5.6 Tracking of boundary disturbance

5.5 Simulation Example

Fig. 5.7 Control input U (t)

59

60

5 Robust Boundary Control of an Axially Moving System with High …

Appendix 1: Simulation Program close all; clear all; clc; nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m = 1 ; EA = 500; T0 = 9800; pi = 3.1415926; V0=0; c = 1; V = zeros(1,nt); V(1,1) = V0; J=3.5*9.8;%J1=J2=J3 for j=2:10*10^3-1 V(1,j) = V(1,1)+0.5*J*(j*dt)^2; end V(1,10*10^3)=V(1,1)+0.5*J; for j=(10*10^3+1):20*10^3-1 V(1,j)=V(1,10*10^3)+0.5*J*(j*dt-1); end V(1,20*10^3)=V(1,10*10^3)+J; for j=(20*10^3+1):30*10^3-1 V(1,j) = V(1,20*10^3)+J*(j*dt-2)-0.5*J*(j*dt-2)^2; end V(1,30*10^3)=V(1,20*10^3)+0.5*J; for j=(30*10^3+1):70*10^3-1 V(1,j)=V(1,30*10^3); end V(1,70*10^3)=V(1,30*10^3); for j=(70*10^3+1):80*10^3-1 V(1,j)= V(1,70*10^3)-0.5*J*(j*dt-7)^2;

Appendix 1: Simulation Program

61

end V(1,80*10^3)=V(1,70*10^3)-0.5*J; for j=(80*10^3+1):90*10^3-1 V(1,j)=V(1,80*10^3)-J*(j*dt-8); end V(1,90*10^3)=V(1,80*10^3)-J; for j=(90*10^3+1):nt-1 V(1,j)=V(1,90*10^3)-J*(j*dt-9)+0.5*J*(j*dt-9)^2; end V(1,100*10^3)=V(1,90*10^3)-0.5*J; % -------------disburtance------------d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt d(1,j)=3+0.1*sin((j-1)*dt)+0.2*sin(2*(j-1)*dt)+0.3*sin(3*(j-1)*dt); end for j = 1 : nt for i = 1 : nx f(i,j)= (1+sin(1*(i-1)*dx*(j-1)*dt)+sin(2*(i-1)*dx*(j-1)*dt)+sin(3*(i-1)*dx*( j-1)*dt))*(i-1)*dx*10^-4; end end %******************************************************************** %

uncontrolled

%******************************************************************** ds=0.25; mc=5; w1 = zeros(nx,nt); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((0.5*T0+1.5*EA*(w1(i,j-1)-w1(i-1,j-1))^2/dx^2 -m*V(1,j-1)^2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dx^2 -c*(w1(i,j-1)-w1(i,j-2))/dt - c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt)

62

5 Robust Boundary Control of an Axially Moving System with High …

-m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j) ) / (m/dt^2); end w1(nx,j) = ( (2*mc/dt^2 + ds/dt )*w1(nx,j-1) - (mc/dt^2)*w1(nx,j-2) +d(1,j)+(0.5*T0/dx)*w1(nx-1,j-1)-0.5*EA*(w1(nx,j-1)-w1(nx-1,j-1))^3/( dx)^3 ) / (mc/dt^2 + ds/dt + 0.5*T0/dx); end tshort = linspace(0,tmax,(nt/400+1)); xshort = linspace(0,L,nx); for j=1:nt/400 for i=1:nx w1short(i,j)=w1(i,j*400); end end w1short=[w1(:,1),w1short]; figure (1); surf(tshort,xshort,w1short); shading interp;colormap jet; xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman' ); ylabel('x [m]','Fontsize',14,'Fontname','Times New Roman'); zlabel('w(x,t) [m]','Fontsize',14,'Fontname','Times New Roman'); zlim([0,1.2*10^-3]);set(gca,'xtick',[0:2:10]); set(gca,'YDir','reverse','Fontsize',14,'Fontname','Times New Roman') set(gca,'ztick',[0:0.2*(10^(-3)):1.2*(10^(-3))]); %******************************************************************** %

controlled + disturbance observer

%******************************************************************** k = 10^4; k1 = 10^2; sigma=1; a3=0.1; w2 = zeros(nx,nt); U = zeros(1,nt);%U(t) varphi = zeros(1,nt); u_a = zeros(1,nt); d_e = zeros(1,nt);%disturbance observer error = zeros(1,nt);%e(t),disturbance observer error w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0;

Appendix 1: Simulation Program

63

end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((0.5*T0+1.5*EA*(w2(i,j-1)-w2(i-1,j-1))^2/dx^2 -m*V(1,j-1)^2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1))/dx^2 -c*(w2(i,j-1)-w2(i,j-2))/dt - c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j) ) / (m/dt^2); end w2(nx,j) = ( mc*(2*w2(nx,j-1)-w2(nx,j-2))/dt^2 + k*w2(nx,j-1)/dt + k*k1*w2(nx-1,j)/dx + k1*mc*(w2(nx,j-1)+w2(nx-1,j)-w2(nx-1,j-1))/(dx*dt) - 0.5*EA*(w2(nx,j-1)-w2(nx-1,j-1))^3/dx^3- d_e(1,j-1) + d(1,j)) / (mc/dt^2 + k/dt + k*k1/dx + k1*mc/(dx*dt)); u_a(1,j) = (w2(nx,j)-w2(nx,j-1))/dt + k1*(w2(nx,j)-w2(nx-1,j))/dx ; U(1,j) = - k*u_a(1,j) -k1*mc*(w2(nx,j)-w2(nx,j-1)-w2(nx-1,j)+w2(nx-1,j-1))/(dx*dt) +0.5*T0*(w2(nx,j)-w2(nx-1,j))/dx+ ds*(w2(nx,j)-w2(nx,j-1))/dt -a3*sign(u_a(1,j))- d_e(1,j-1); varphi(1,j) = varphi(1,j-1) - dt*(varphi(1,j-1) -ds*(w2(nx,j)-w2(nx,j-1))/dt - T0/(2)*(w2(nx,j)-w2(nx-1,j))/dx -EA/(2)*(w2(nx,j)-w2(nx-1,j))^3/dx^3 + U(1,j) +mc*(w2(nx,j)-w2(nx,j-1))/dt - sigma*u_a(1,j) ); d_e(1,j) = varphi(1,j) + mc*(w2(nx,j) - w2(nx,j-1))/dt; error(1,j) = d(1,j) - d_e(1,j); end for j=1:nt/400 for i=1:nx w2short(i,j)=w2(i,j*400); end end w2short=[w2(:,1),w2short]; figure (2); surf(tshort,xshort,w2short); shading interp;colormap jet;set(gca,'xtick',[0:2:10]); set(gca,'YDir','reverse','Fontsize',14,'Fontname','Times New Roman'); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('x [m]','Fontsize',14,'Fontname','Times New Roman'); zlabel('w(x,t) [m]','Fontsize',14,'Fontname','Times New Roman'); figure (3);

64

5 Robust Boundary Control of an Axially Moving System with High …

hold on subplot(2,1,1); plot(tshort,w1short(50,:),'b',tshort,w2short(50,:),':r','LineWidth',2 ); set(gca,'Fontsize',14,'Fontname','Times New Roman'); title('(a)','Fontsize',14,'Fontname','Times New Roman'); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('w(1,t) [m]','Fontsize',14,'Fontname','Times New Roman'); h=legend('uncontrolled','controlled'); set(h,'Fontsize',14,'Fontname','Times New Roman'); box on;grid on; set(gca,'xtick',[0:1:10]) subplot(2,1,2); plot(tshort,w1short(25,:),'b',tshort,w2short(25,:),':r','LineWidth',2 ); set(gca,'Fontsize',14,'Fontname','Times New Roman'); set(gca,'xtick',[0:1:10]); ylim([-0.006*10^-4,6*10^-4]); title('(b)','Fontsize',14,'Fontname','Times New Roman'); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('w(0.5,t) [m]','Fontsize',14,'Fontname','Times New Roman'); h=legend('uncontrolled','controlled'); set(h,'Fontsize',14,'Fontname','Times New Roman'); box on;grid on;hold off; figure (4); plot(tshort,w2short(50,:),':r',tshort,w2short(25,:),'b','LineWidth',2 ); set(gca,'Fontsize',14,'Fontname','Times New Roman'); set(gca,'xtick',[0:1:10]); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('w(x,t) [m]','Fontsize',14,'Fontname','Times New Roman'); h=legend('x=1m','x=0.5m'); set(h,'Fontsize',14,'Fontname','Times New Roman'); figure(5); plot(t,d_e,'b',t,d,'--r','LineWidth',2); set(gca,'Fontsize',14,'Fontname','Times New Roman'); set(gca,'xtick',[0:1:10]); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('d(t) [N]','Fontsize',14,'Fontname','Times New Roman'); h=legend('estimate of disturbance','actual disturbance'); set(h,'Fontsize',14,'Fontname','Times New Roman');

References

65

figure(6); plot(t,U,'b','LineWidth',2); set(gca,'Fontsize',14,'Fontname','Times New Roman'); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('U(t) [N]','Fontsize',14,'Fontname','Times New Roman'); set(gca,'xtick',[0:1:10]);

References 1. W. He, S.S. Ge, Robust adaptive boundary control of a vibrating string under unknown timevarying disturbance. IEEE Trans. Control Syst. Technol. 20(1), 48–58 (2012) 2. J.M. Sloss, J.C.B. Jr., I.S. Sadek, et al., Boundary optimal control of structural vibrations in an annular plate. J. Frankl. Inst. 342(3), 295–309 (2005) 3. Y. Liu, B.S. Xu, Y.L. Wu, et al., Boundary control of an axially moving belt, in Proceeding of the 32th Chinese Control Conference, (2013), pp. 1310–1315 4. Y. Li, D. Aron, C.D. Rahn, Adaptive vibration isolation for axially moving strings: theory and experiment. Automatica 38(3), 379–390 (2002) 5. H. Goldstein, Classical Mechanics (Addison-Wesley, Massachusetts, 1951)

Chapter 6

Adaptive Boundary Control of an Axially Moving System with High Acceleration/Deceleration

6.1 Introduction In recent years, although there have been many studies on the vibration of axially moving systems, most of them ignore the uncertainties of parameters, such as [1, 2]. Usually, the high ac-/deceleration axial moving system has uncertainties and the system’s structural parameters, such as its tension and stiffness are unknown or cannot be measured accurately, and even some structural parameters may change with the change of its vibration offset. Therefore, to improve the control effect and quality of the axial moving structure control system, the designed control strategy is required to have good robustness and adaptive ability to compensate for the uncertainty of the axially moving system [3, 4]. In this chapter, to improve the performance of the vibration control for an axially moving system under high ac-/deceleration with the system parametric uncertainty, an adaptive boundary controller is designed to suppress the vibration displacement of the system by combining the Lyapunov theory, the adaptive control technique, and the S-curve ac-/deceleration method. With the proposed adaptive boundary control, the control-spillover phenomenon can be avoided, and the system parameter uncertainty can be compensated. The stability of the closed-loop system is demonstrated and the uniform ultimate boundedness of the closed-loop signals is ensured.

6.2 PDE Dynamic Model To obtain the nonlinear dynamic model of the presented belt system, we apply the extended Hamilton’s principle [5]: t2 (δ E k − δ E p + δWc − δWb )dt = 0

(6.1)

t1

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_6

67

68

6 Adaptive Boundary Control of an Axially Moving System with High…

The kinetic energy E k (t) of the belt system can be given as: 1 1 E k = m c wt2 (L , t) + m 2 2

L (wt +vwx )2 d x

(6.2)

0

The potential energy E p (t) of the belt system can be described by: 1 Ep = T 2

L wx2 d x

(6.3)

0

where T is the belt’s tension. The virtual work δWc (t) done by damping on the belt and the actuator is given as: L δWc = [U + d − ds wt (L , t)]δw(L , t) − c

L (wt +vwx )δwd x +

0

f δwd x 0

(6.4) where ds is damping coefficient of the actuator and c is viscous damping coefficient of the belt. The virtual momentum transport δWb (t) across the boundaries is computed by: δWb = mv[wt (L , t) + vwx (L , t)]δw(L , t)

(6.5)

Substituting Eqs. (6.2)–(6.5) into Eq. (6.1) and then integrating by parts, the governing equation of the presented belt system is derived as: mwtt + mawx + 2mvwxt + mv 2 wx x − T wx x + cwt + cvwx = f

(6.6)

where ∀(x, t) ∈ (0, L) × [0, +∞). The corresponding boundary conditions are expressed as: 

w(0, t) = 0 m c wtt (L , t) + T wx (L , t) + ds wt (L , t) = d + U

(6.7)

where ∀t ∈ [0, +∞). Assumption 6.1 For the speed v(t), H-A/D a(t), and boundary disturbance d(t), we assume that there exist constants a1 , a2 , d ∈ R+ , such that 0 < v(t) ≤ a1 , | a(t) | ≤ a2 , | d(t) | ≤ d, ∀t ∈ [0, +∞). And in this chapter, we do not consider distributed interference, that is to say f = 0.

6.4 Stability Analysis

69

6.3 Boundary Controller Design Because the presented belt system parameters T , m c , and ds are unknown or inaccurately measured, and the accurate measurement of unknown boundary disturbance d(t) is also difficult. To compensate for both parametric and disturbance uncertainties, here a nonlinear adaptive boundary control is proposed via parameter estimation U (t) = Tˆ wx (L , t) + dˆs wt (L , t) − dsgn(u i ) − mˆ c k1 wxt (L , t) − ku i

(6.8)

where u i = wt (L , t)+k1 wx (L , t), k and k1 are the control parameters, sgn(·) denotes symbolic function. Tˆ , mˆ c , dˆs are the estimations of T , m c , ds , respectively, and the corresponding estimation errors are T˜ = T − Tˆ

(6.9)

m˜ c = m c − mˆ c

(6.10)

d˜s = ds − dˆs

(6.11)

Then, the adaptive laws are designed as: ⎧ ˆ ˆ ⎪ ⎨ Tt = −γ1 ζ1 T − ζ1 wx (L , t)u i mˆ ct = −γ2 ζ2 mˆ c + ζ2 k1 wxt (L , t)u i ⎪ ⎩ˆ dst = −γ3 ζ3 dˆs − ζ3 wt (L , t)u i

(6.12)

where γ1 , γ2 , γ3 , ζ1 , ζ2 and ζ3 are positive constants. Remark 6.1 Since the signals w(L , t) and wx (L , t) can be measured by employing a laser displacement sensor and an inclinometer, then wt (L , t) and wxt (L , t) can be obtained by applying the backwards difference algorithm to w(L , t) and wx (L , t), respectively.

6.4 Stability Analysis Consider the Lyapunov function candidate as: V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t) where the energy term V1 (t) is expressed as:

(6.13)

70

6 Adaptive Boundary Control of an Axially Moving System with High…

λ V1 (t) = 2

L

 m(wt + vwx )2 + T wx2 d x

(6.14)

0

and a small crossing term V2 (t) is expressed as: L xwx (wt + vwx )d x

V2 (t) = 2βm

(6.15)

0

and an auxiliary term V3 (t) is expressed as: V3 (t) =

1 ηm c u i2 2

(6.16)

and an estimation error term V4 (t) is expressed as: V4 (t) =

η ˜2 η 2 η ˜2 m˜ (t) + T (t) + d (t) 2ζ1 2ζ2 c 2ζ3 s

(6.17)

where λ, β and η are positive weighting constants. Applying Eqs. (2.11)–(2.15) to Eq. (6.13) results in: L |V2 (t)| ≤ βm L

 2 wx + (wt + vwx )2 d x ≤ ξ V1 (t)

(6.18)

0

where ξ=

2βm L min(λT, λm)

(6.19)

The Eq. (6.18) can be rewritten as: −ξ V1 (t) ≤ V2 (t) ≤ ξ V1 (t)

(6.20)

By choosing ξ properly, we can obtain 

ξ1 = 1 − ξ > 0 ξ2 = 1 + ξ > 1

(6.21)

The Eq. (6.21) indicates 0 < ξ < 1 and substituting (6.21) into (6.20) yields 0 < ξ1 V1 (t) ≤ V1 (t) + V2 (t) ≤ ξ2 V1 (t)

(6.22)

6.4 Stability Analysis

71

Combining Eq. (6.13), we further obtain: 0 ≤ ϑ1 [V1 (t) + V3 (t) + V4 (t)] ≤ V (t) ≤ ϑ2 [V1 (t) + V3 (t) + V4 (t)]

(6.23)

where ϑ1 = min(ξ1 , 1) and ϑ2 = max(ξ2 , 1). As an important analytic tool to derive our main results, we state the following Lemma 6.1. Lemma 6.1 The time derivative of the Lyapunov function candidate Eq. (6.13) can be upper bounded with Vt (t) ≤ −ϑ V (t) + ε

(6.24)

where ϑ, ε > 0. Proof Differentiating Eq. (6.13) with respect to time yields: Vt (t) = V1t (t) + V2t (t) + V3t (t) + V4t (t)

(6.25)

According to Eq. (6.14), V1 (t) can be represented as V1t (t) = A1 + A2 + A3 + A4 where A1 = λm A3 = λm

L

L

(awt wx + avwx2 )d x, A2 = λm

0

L

(wt wtt + vwx wtt )d x,

0

(vwt wxt + v 2 wx wxt )d x and A4 = λT

0

(6.26)

L

wx wxt d x.

0

Substituting the governing Eq. (6.6) into A2 and integrating by parts, we obtain  λv T − mv 2 [wx2 (L , t) − wx2 (0, t)] − λmvwt2 (L , t)] A2 = 2 L 2 − 2λmv wx (L , t)wt (L , t) − λc (wt + vwx )2 d x 0

L wx2 d x

− λmva

+ λ T + mv

0

wt w x d x 0

2



L

(6.27) wt w x x d x

0

L − λma



72

6 Adaptive Boundary Control of an Axially Moving System with High…

Integrating A3 and A4 by parts, we obtain ⎧ L ⎪ ⎪ λmv ⎪ 2 2 ⎪ A3 = λmv wx (L , t)wt (L , t) + wt (L , t) − λm v 2 wt wx x d x, ⎪ ⎪ ⎪ 2 ⎨ 0 ⎡ ⎤ L ⎪  ⎪ ⎪ ⎪ ⎪ A = λT ⎣wt (L , t)wx (L , t) − wt wx x d x ⎦ ⎪ ⎪ ⎩ 4

(6.28)

0

Substituting A1 , Eqs. (6.27) and (6.28) into Eq. (6.26), we can get  λv T − mv 2 2 λT 2 λT 2 wx (0, t) − w (L , t) + u 2 2k1 t 2k1 i λT (k1 − v) 2 λmv wx (L , t) − [wt (L , t) + vwx (L , t)]2 − 2 2 L − λc (wt + vwx )2 d x

V1t (t) = −

(6.29)

0

Differentiating (6.15) with respect to time results in V2t (t) = B1 + B2 where B1 = 2βm

L

(6.30)

xwx (wtt + awx + vwxt )d x, B2 = 2βm

0

L

xwxt (wt + vwx )d x.

0

Integrating B1 by parts results in: L B1 =

βT Lwx2 (L , t)

− βm Lv

2

wx2 (L , t)

− 2βcv

xwx2 d x 0

L

L v 2 wx2 d x − 2βc

+ βm 0

L xwx wt d x − βT

0

wx2 d x

(6.31)

xwxt wx d x

(6.32)

0

L xwx wxt d x

− 2βmv 0

Integrating B2 by parts yields: L B2 =

βm Lwt2 (L , t)

− βm

L wt2 d x

0

+ 2βmv 0

6.4 Stability Analysis

73

Substituting Eqs. (6.31) and (6.32) into Eq. (6.30), we have  V2t (t) = − βm Lv 2 − βT L wx2 (L , t) − 2βcv

L xwx2 d x 0

L

L wt2 d x − 2βc

− βm 0

L xwx wt d x − βT

0

wx2 d x

(6.33)

0

L +

βm Lwt2 (L , t)

+ βm

v 2 wx2 d x 0

Differentiating (6.16) with respect to time, then substituting the boundary conditions (6.7) and the proposed control (6.8), we obtain   V3t (t) ≤ −ηku i2 + η m˜ c k1 wxt (L , t) − T˜ wx (L , t) − d˜s wt (L , t) u i

(6.34)

Differentiating Eq. (6.17) with respect to time and substituting the Eq. (6.12), we have:  η η V4t (t) ≤ − γ1 T˜ 2 + γ2 m˜ 2c + γ3 d˜s2 + γ1 T 2 + γ2 m 2c + γ3 ds2 2 2 (6.35)   ˜ ˜ +η ds wt (L , t) + T wx (L , t) − m˜ c k1 wxt (L , t) u i Substituting Eqs. (6.29), (6.33), and (6.35) into Eq. (6.25) and then using Eqs. (2.11)–(2.15), we obtain:     λT (k1 − v) 2 λT 2 2 u − βm Lv − βT L + wx (L , t) Vt (t) ≤ − ηk − 2k1 i 2   3 λT − λT v−λmv wx2 (0, t) − 2k − βm L wt2 (L , t) 2 1  − βT − βma12 − −βm

L 0

2βcL δ1

 L

wx2 d x −

0

λmv [wt (L , t) 2

+ vwx (L , t)]2

(6.36)

L wt2 d x − (λc − 2βcLδ1 ) (wt + vwx )2 d x − ϑ3 V4 (t) + ε 0

 where δ1 is a positive constant, ε = η2 γ1 T 2 + γ2 m 2c + γ3 ds2 . According to Eqs. (6.24) and (6.36), we choose parameters λ, η, k, k1 ,β, δ1 properly to satisfy the following conditions:

74

6 Adaptive Boundary Control of an Axially Moving System with High…

λT ≥ 0, 2k − βm L ≥ 0, βm Lv 2 − βT L + λT (k21 −v) ≥ 0,τ1 = 1 λT > 0, τ3 = ηk − 2k > 0, λc − 2βcLδ1 > 0, τ2 = βT − βma12 − 2βcL δ1 1   2τ1 2τ2 2τ3 ϑ4 = min λm , λT , ηm c . The Eq. (6.36) can be rewritten as: λT v−λmv 3 2

Vt (t) ≤ −ϑ4 [V1 (t) + V3 (t)] − ϑ3 V4 (t) + ε ≤ −ϑ5 [V1 (t) + V3 (t) + V4 (t)] + ε

(6.37)

≤ −ϑ V (t) + ε where ϑ5 = min(ϑ3 , ϑ4 ), ϑ = (ϑ5 /ϑ2 ). Theorem 6.1 For the presented belt system described by Eqs. (6.6) and (6.7), under Assumption 6.1 and the proposed control law Eq. (6.8), given that the initial conditions are bounded, we can conclude that: (1) uniform boundedness: the system state w(x, t) of the closed-loop belt system will remain in the compact set: 1 := { w(x, t) ∈ R||w(x, t)| ≤ χ1 } where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 =



2L [V (0) λT ϑ1

(6.38)

+ ϑε ].

(2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop belt system will eventually converge to the compact set: 2 := { w(x, t) ∈ R| lim |w(x, t)| ≤ χ2 } t→∞

where ∀x ∈ [0, L], χ2 =



2Lε γ ϑ1 ϑ T

(6.39)

.

Proof Multiplying Eq. (6.37) by eϑt yields: Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒

∂ V (t)eϑt ≤ εeϑt ∂t

(6.40)

Integrating the above inequality yields: V (t) ≤ [V (0) −

ε −ϑt ε ε ]e + ≤ V (0)e−ϑt + ϑ ϑ ϑ

Applying Eqs. (2.11)–(2.15) and Eq.(6.23), we have:

(6.41)

6.5 Simulation Example

λT 2 λT w (x, t) ≤ 2L 2

75

L wx2 (x, t)d x ≤ V1 (t) ≤ V1 (t) + V3 (t) + V4 (t) ≤ 0

1 V (t) ϑ1 (6.42)

Substituting Eq. (6.41) into Eq. (6.42) results in:  |w(x, t)| ≤

2L ε [V (0)e−ϑt + ] ≤ λT ϑ1 ϑ



2L ε [V (0) + ] λT ϑ1 ϑ

(6.43)

where ∀(x, t) ∈ [0, L] × [0, +∞), from above inequality, we have:  lim |w(x, t)| ≤ lim

t→∞

t→∞

2L ε [V (0)e−ϑt + ] = λT ϑ1 ϑ

 2Lε , λT ϑ1 ϑ

∀x ∈ [0, L] (6.44)

6.5 Simulation Example The system parameters are given in Table 6.1. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D are given as: aa = ad = 3.5g and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s (g = 9.8m/s 2 ). The control parameters are given as k = 1 × 105 , k1 = 100, ζ1 = ζ2 = ζ3 = 1.0, and γ1 = γ2 = γ3 = 0.001. The unknown boundary disturbance is given as:  1 (2i − 1)t , i = 1, 2, 3 sin d(t) = 3 + 10 i=1 n 



(6.45)

The initial conditions of the axially moving belt system are given as: w(x, 0) = wt (x, 0) = 0

(6.46)

Table 6.1 Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

Value

m

1.0 kg/m

T

5000 N

g

9.8 N/kg

ds

0.25 Ns/m

L

1.0 m

v0

0

d

6

Ns/m2

c

1.0

mc

5.0 kg

76

6 Adaptive Boundary Control of an Axially Moving System with High…

Fig. 6.1 Vibration displacement of the belt without control

Figure 6.1 shows the vibration displacement of the uncontrolled belt, i.e. U (t) = 0, with H-A/D under distributed disturbance and boundary disturbance. The vibration displacement of the belt under the same external conditions with the proposed adaptive boundary control is shown in Fig. 6.2. The vibration displacements of the belt are examined at the middle point x = 0.5m and boundary point x = 1m in Fig. 6.3, respectively. Figure 6.4 shows the enlarged view of the vibration displacement for controlled response at x = 1m and x = 0.5m, and the corresponding adaptive boundary control input is shown in Fig. 6.5.

6.5 Simulation Example

Fig. 6.2 Vibration displacement of the belt with control

Fig. 6.3 Vibration displacement of the belt at: (a)x = 1m, (b) x = 0.5m

77

78

6 Adaptive Boundary Control of an Axially Moving System with High…

Fig. 6.4 The enlarged view of the vibration displacement for controlled response

Fig. 6.5 The proposed adaptive boundary control input U (t)

Appendix 1: Simulation Program

Appendix 1: Simulation Program

close all; clear all; clc; nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m = 1 ; T =5000; pi = 3.1415926; V0=0; c = 1; ds=0.25; % -------------disburtance------------d = zeros(1,nt); f = zeros(1,nt); for j=1:nt d(1,j)=3+sin(0.1*(j)*dt)+sin(0.3*(j)*dt)+sin(0.5*(j)*dt); end % --------------velocity----------------V = zeros(1,nt); V(1,1) = V0; J=3.5*9.8;%J1=J2=J3 for j=2:10*10^3-1 V(1,j) = V(1,1)+0.5*J*(j*dt)^2; end V(1,10*10^3)=V(1,1)+0.5*J; for j=(10*10^3+1):20*10^3-1 V(1,j)=V(1,10*10^3)+0.5*J*(j*dt-1); end V(1,20*10^3)=V(1,10*10^3)+J; for j=(20*10^3+1):30*10^3-1 V(1,j) = V(1,20*10^3)+J*(j*dt-2)-0.5*J*(j*dt-2)^2;

79

80

6 Adaptive Boundary Control of an Axially Moving System with High…

end V(1,30*10^3)=V(1,20*10^3)+0.5*J; for j=(30*10^3+1):70*10^3-1 V(1,j)=V(1,30*10^3); end V(1,70*10^3)=V(1,30*10^3); for j=(70*10^3+1):80*10^3-1 V(1,j)= V(1,70*10^3)-0.5*J*(j*dt-7)^2; end V(1,80*10^3)=V(1,70*10^3)-0.5*J; for j=(80*10^3+1):90*10^3-1 V(1,j)=V(1,80*10^3)-J*(j*dt-8); end V(1,90*10^3)=V(1,80*10^3)-J; for j=(90*10^3+1):nt-1 V(1,j)=V(1,90*10^3)-J*(j*dt-9)+0.5*J*(j*dt-9)^2; end V(1,100*10^3)=V(1,90*10^3)-0.5*J; %******************************************************************** %

uncontrolled

%******************************************************************** mc=5; w1 = zeros(nx,nt); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((T-m*V(1,j-1)^2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dx^2 -c*(w1(i,j-1)-w1(i,j-2))/dt - c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx ) / (m/dt^2); end w1(nx,j) = ( 2*mc/dt^2*w1(nx,j-1) - (mc/dt^2)*w1(nx,j-2) + d(1,j) + (T/dx)*w1(nx-1,j)+ (ds/dt)*w1(nx,j-1)) / (mc/dt^2 + T/dx+ds/dt); end tshort = linspace(0,tmax,nt/1000);

Appendix 1: Simulation Program

81

xshort = linspace(0,L,nx/2); w1short=zeros(nx/2,nt/1000); for j=1:nt/1000 for i=1:nx/2 w1short(i,j)=w1(i*2,j*1000); end end figure (1); surf(tshort,xshort,w1short); shading interp;colormap jet; set(gca,'Fontsize',14,'Fontname','Times New Roman'); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('x [m]','Fontsize',14,'Fontname','Times New Roman'); zlabel('w(x,t) [m]','Fontsize',14,'Fontname','Times New Roman'); set(gca,'YDir','reverse'); set(gca,'xtick',[0:2:10]); %******************************************************************** %

controlled + disturbance observer

%******************************************************************** mc = 5; k = 10^5; k1 = 100; dbar = 6; zeta=1;%zeta1=zeta2=zeta3 gamma = 0.001;%gamma1=gamma2=gamma3 w2 = zeros(nx,nt); u = zeros(1,nt); u_i = zeros(1,nt); T_e = zeros(1,nt);%the estimated value of T mc_e = zeros(1,nt);%the estimated value of mc ds_e = zeros(1,nt);%the estimated value of ds err_T = zeros(1,nt); err_mc = zeros(1,nt); err_ds = zeros(1,nt); w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0; end for j = 1 : 2

82

6 Adaptive Boundary Control of an Axially Moving System with High… u_i(1,j) = 0; u(1,j) = 0; T_e(1,j) = 9000; mc_e(1,j) = 4; ds_e(1,j) = 0.2;

end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((T-m*V(1,j-1)^2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1))/dx^2 -c*(w2(i,j-1)-w2(i,j-2))/dt - c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx ) / (m/dt^2); end w2(nx,j) = ( 2*mc*w2(nx,j-1)/dt^2 - mc*w2(nx,j-2)/dt^2 + d(1,j-1)sign(u_i(1,j))*dbar + (T - T_e(1,j))*w2(nx-1,j)/dx + (dsds_e(1,j))*w2(nx,j-1)/dt+ k*(w2(nx,j-1)/dt + k1*w2(nx-1,j)/dx) + k1*mc_e(1,j)*(w2(nx-1,j) + w2(nx,j-1) - w2(nx-1,j-1))/(dx*dt)) / (mc/dt^2 + (T - T_e(1,j))/dx + (ds- ds_e(1,j))/dt+ k1*mc_e(1,j)/(dx*dt) + k/dt + k*k1/dx ); u_i(1,j) = (w2(nx,j) - w2(nx,j-1))/dt + k1*(w2(nx,j) w2(nx-1,j))/dx ; %ui T_e(1,j) = T_e(1,j-1) - dt*(gamma*zeta*T_e(1,j-1) + zeta*u_i(1,j)*(w2(nx,j)-w2(nx-1,j))/dx ); mc_e(1,j) = mc_e(1,j-1) - dt*(gamma*zeta*mc_e(1,j-1) zeta*u_i(1,j)*( k1*(w2(nx,j) - w2(nx-1,j) - w2(nx,j-1) + w2(nx-1,j-1))/(dx*dt))); ds_e(1,j) = ds_e(1,j-1) - dt*(gamma*zeta*ds_e(1,j-1) + zeta*u_i(1,j)*(w2(nx,j)-w2(nx,j-1))/dt ); err_T = T - T_e(1,j); err_mc = mc - mc_e(1,j); err_ds = ds - ds_e(1,j); u(1,j) = T_e(1,j)*(w2(nx,j)-w2(nx-1,j))/dx+ds_e(1,j)*(w2(nx,j)-w2(nx,j-1))/dt -k1*mc_e(1,j)*(w2(nx,j) - w2(nx-1,j) - w2(nx,j-1) +w2(nx-1,j-1))/(dx*dt) -k*u_i(1,j) - sign(u_i(1,j))*dbar; end tshort = linspace(0,tmax,nt/1000); xshort = linspace(0,L,nx/2); w2short=zeros(nx/2,nt/1000);

Appendix 1: Simulation Program for j=1:nt/1000 for i=1:nx/2 w2short(i,j)=w2(i*2,j*1000); end end figure (2); surf(tshort,xshort,w2short); shading interp;colormap jet; set(gca,'Fontsize',14,'Fontname','Times New Roman'); set(gca,'xtick',[0:2:10]); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('x [m]','Fontsize',14,'Fontname','Times New Roman'); zlabel('w(x,t) [m]','Fontsize',14,'Fontname','Times New Roman'); set(gca,'YDir','reverse'); zlim([0,5*10^-7]); figure (3); hold on subplot(2,1,1); plot(t,w1(50,:),'b',t,w2(50,:),':r','LineWidth',2); set(gca,'Fontsize',14,'Fontname','Times New Roman');set(gca,'xtick',[0:1:10]); title('(a)','Fontsize',14,'Fontname','Times New Roman'); xlabel('Time [s])','Fontsize',14,'Fontname','Times New Roman'); ylabel('w(1,t) [m]','Fontsize',14,'Fontname','Times New Roman'); h=legend('uncontrolled','controlled'); set(h,'Fontsize',14,'Fontname','Times New Roman') box on grid on subplot(2,1,2); plot(t,w1(25,:),'b',t,w2(25,:),':r','LineWidth',2); set(gca,'Fontsize',14,'Fontname','Times New Roman'); set(gca,'xtick',[0:1:10]); title('(b)','Fontsize',14,'Fontname','Times New Roman'); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('w(0.5,t) [m]','Fontsize',14,'Fontname','Times New Roman'); h=legend('uncontrolled','controlled'); set(h,'Fontsize',14,'Fontname','Times New Roman') box on;grid on;hold off; figure(4); plot(t,w2(25,:),':r',t,w2(50,:),'b','LineWidth',2);

83

84

6 Adaptive Boundary Control of an Axially Moving System with High… set(gca,'Fontsize',14,'Fontname','Times New Roman');set(gca,'xtick',[0:1:10]); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('w(x,t) [m]','Fontsize',14,'Fontname','Times New Roman'); h=legend('x=0.5m','x=1m'); set(h,'Fontsize',14,'Fontname','Times New Roman') figure(5); plot(t,u,'b','LineWidth',2); set(gca,'Fontsize',14,'Fontname','Times New Roman');set(gca,'xtick',[0:1:10]); xlabel('Time [s]','Fontsize',14,'Fontname','Times New Roman'); ylabel('U(t) [N]','Fontsize',14,'Fontname','Times New Roman');

References 1. Y. Liu, X. Weng, X. Wu et al., Vibration control of an axially moving system with high ac/deceleration. J. Vib. Eng. 10(6), 1299–1306 (2014) 2. Y. Liu, Y. Wu, Z. Zhao, Robust boundary control of an axially moving system with high acdeceleration. Control Decis 29(10), 1771–1776 (2014) 3. W. He, S.S. Ge, Robust adaptive boundary control of a vibrating string under unknown timevarying disturbance. IEEE Trans. Control Syst. Technol. 20(1), 48–58 (2012) 4. W. He, S.S. Ge, B.V. How et al., Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47(4), 722–732 (2011) 5. H. Goldstein, Classical Mechanics (Addison-Wesley, Massachusetts, 1951)

Chapter 7

Adaptive Boundary Control of an Axially Moving System with High Acceleration/Deceleration and Disturbance Observer

7.1 Introduction In this chapter, an adaptive boundary control is presented for vibration suppression of an axially moving belt system. Firstly, the infinite-dimensional model of the belt system including the dynamics of high acceleration/deceleration and distributed disturbance is derived by utilizing the extended Hamilton’s principle. Subsequently, by using the Lyapunov synthesis method and adaptive technique, an adaptive boundary control is developed to suppress the belt’s vibration and compensate for the system’s parametric uncertainties. Finally, the control performance of the closed-loop system is successfully demonstrated through simulations.

7.2 PDE Dynamic Model To obtain the nonlinear dynamic model of presented belt system, we apply the extended Hamilton’s principle [1]: t2 (δ E k − δ E p + δW )dt = 0

(7.1)

t1

The kinetic energy E k (t) of the belt system can be given as: Ek =

1 1 m c wt2 (L , t) + m 2 2

L (wt +vwx )2 d x

(7.2)

0

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_7

85

86

7 Adaptive Boundary Control of an Axially Moving System…

The potential energy E p (t) of the belt system can be described by: 1 Ep = T 2

L wx2 d x

(7.3)

0

where T is the belt’s tension. The virtual work δWc (t) done by damping on the belt and the actuator is given as: L δWc = −ds wt (L , t)δw(L , t) − c

(wt +vwx )δwd x

(7.4)

0

where ds is damping coefficient of the actuator and c is viscous damping coefficient of the belt. The virtual work δW f (t) done by the boundary control and disturbances is calculated as: L δW f = (U + d)δw(L , t) +

f δwd x

(7.5)

0

The virtual momentum transport δWb (t) across the boundaries is computed by: δWb = mv[wt (L , t) + vwx (L , t)]δw(L , t)

(7.6)

Finally, the total work δW (t) on the belt system is summarized as: δW = δWc + δW f − δWb

(7.7)

Substituting Eqs. (7.2), (7.3), and (7.7) into Eq. (7.1) and then integrating by parts, the governing equation of the presented belt system is derived as: mwtt + mawx + 2mvwxt + mv 2 wx x − T wx x + cwt + cvwx = f

(7.8)

where ∀(x, t) ∈ (0, L) × [0, +∞). The corresponding boundary conditions are expressed as: 

w(0, t) = 0 m c wtt (L , t) + T wx (L , t) + ds wt (L , t) = d + U

where ∀t ∈ [0, +∞).

(7.9)

7.3 Boundary Controller Design

87

Assumption 7.1 For the speed v(t), H-A/D a(t), boundary disturbance d(t) and distributed disturbance f (x, t), we assume that there exist constants a1 , a2 , a3 , a4 ∈ R+ , such that 0 < v(t) ≤ a1 , | a(t) | ≤ a2 , | d(t) | ≤ a3 , ∀t ∈ [0, +∞) and | f (x, t) | ≤ a4 , ∀(x, t) ∈ (0, L) × [0, +∞). Assumption 7.2 For the time derivative of boundary disturbance dt (t), let us suppose that it is uniformly bounded and there exists a constant a5 ∈ R+ , such that | dt (t) | ≤ a5 , ∀t ∈ [0, +∞).

7.3 Boundary Controller Design In most cases, the presented belt system parameters T , m c , and ds are unknown or inaccurately measured, and the accurate measurement of unknown boundary disturbance d(t) is also difficult. To compensate for both parametric and disturbance uncertainties, in this section, a nonlinear adaptive boundary control with a disturbance observer is proposed via parameter estimation ˆ U (t) = −k[wt (L , t) + βwx (L , t)] − β mˆ c wxt (L , t) + Tˆ wx (L , t) + dˆs wt (L , t) − d(t) (7.10) where k > 0 is the control gain, β > 0 is the weighting constant, and mˆ c , Tˆ , ˆ are the estimation terms to estimate m c , T , ds , d(t), respectively, and the dˆs , d(t) corresponding estimation errors are T˜ = T − Tˆ , m˜ c = m c − mˆ c and d˜s = ds − dˆs . The disturbance observer is designed as dˆt (t) = σ [wt (L , t) + βwx (L , t)] − κσ dˆ

(7.11)

where σ and κ are positive constants. The boundary disturbance estimation error is defined as: d˜ = d − dˆ

(7.12)

Differentiating Eq. (7.12) and then substituting Eq. (7.11) into the resulting equation, we obtain: d˜t (t) = dt (t) − σ [wt (L , t) + βwx (L , t)] − σ κ dˆ

(7.13)

88

7 Adaptive Boundary Control of an Axially Moving System…

Then, the adaptive laws are designed as: ⎧ ˆ ˆ ⎪ ⎨ Tt = −γ1 ζ1 T − ζ1 wx (L , t)[wt (L , t) + βwx (L , t)] mˆ ct = −γ2 ζ2 mˆ c + ζ2 βwxt (L , t)[wt (L , t) + βwx (L , t)] ⎪ ⎩ˆ dst = −γ3 ζ3 dˆs − ζ3 wt (L , t)[wt (L , t) + βwx (L , t)]

(7.14)

where γ1 , γ2 , γ3 , ζ1 , ζ2 , and ζ3 are positive constants. Remark 7.1 Since the signals w(L , t) and wx (L , t) can be measured by employing a laser displacement sensor and an inclinometer, then wt (L , t) and wxt (L , t) can be obtained by applying the backwards difference algorithm to w(L , t) and wx (L , t), respectively. Assumption 7.3 [2]: For the axially moving belt system described by the governing Eq. (7.8) and boundary conditions (7.9) with the adaptive boundary control (7.10), we assume that the closed-loop belt system is well-posed.

7.4 Stability Analysis Consider the Lyapunov function candidate as: V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t)

(7.15)

where the energy term V1 (t) is expressed as: γm V1 (t) = 2

L

γT (wt + vwx ) d x + 2

L wx2 d x

2

0

(7.16)

0

and a small crossing term V2 (t) is expressed as: L xwx (wt + vwx )d x

V2 (t) = 2λm

(7.17)

0

and an auxiliary term V3 (t) is expressed as: V3 (t) =

1 ηm c [wt (L , t) + βwx (L , t)]2 2

and an estimation error term V4 (t) is expressed as:

(7.18)

7.4 Stability Analysis

89

V4 (t) =

η ˜2 η ˜2 η 2 η ˜2 m˜ (t) + T (t) + d (t) d (t) + 2σ 2ζ1 2ζ2 c 2ζ3 s

(7.19)

where γ , λ, and η are all positive weighting constants. Applying Eqs. (2.11)–(2.15) to Eq. (7.17) results in: L |V2 (t)| ≤ λm L

L (wt + vwx ) d x + λm L

wx2 d x ≤ ξ V1 (t)

2

0

(7.20)

0

where ξ=

2λm L min(γ T, γ m)

(7.21)

The Eq. (7.20) can be rewritten as: −ξ V1 (t) ≤ V2 (t) ≤ ξ V1 (t)

(7.22)

By choosing λ and γ properly, we can obtain ⎧ ⎪ ⎪ ⎨ ξ1 = 1 − ξ = 1 −

2λm L >0 min(γ T, γ m) 2λm L ⎪ ⎪ ⎩ ξ2 = 1 + ξ = 1 + >1 min(γ T, γ m)

(7.23)

The Eq. (7.23) indicates 0 < ξ < 1 and then combining (7.21), we have λ
0. Proof Differentiating Eq. (7.15) with respect to time yields: Vt (t) = V1t (t) + V2t (t) + V3t (t) + V4t (t)

(7.28)

According to Eq. (7.16), V1 (t) can be represented as V1t (t) = A1 + A2 + A3 + A4 where A1 = γ m γm

L

L

(awt wx + avwx2 )d x, A2 = γ m

0

L

(vwt wxt + v 2 wx wxt )d x and A4 = γ T

0

Substituting the governing Eq. (7.8) into

0 A2

L

(7.29)

(wt wtt + vwx wtt )d x,A3 =

0

wx wxt d x. and integrating by parts, we obtain

γ T v − γ mv 3 2 [wx (L , t) − wx2 (0, t)] − γ mvwt2 (L , t)] 2 L 2 − 2γ mv wx (L , t)wt (L , t) − γ c (wt + vwx )2 d x

A2 =

0

L wx2 d x

− γ mva

+ γ T + γ mv 2

0

L wt w x d x + γ

0

(7.30) wt w x x d x

0

L − γ ma

L

(wt + vwx ) f d x 0

Integrating A3 and A4 by parts, we obtain ⎧ L ⎪ ⎪ γ mv ⎪ 2 2 ⎪ A3 = γ mv wx (L , t)wt (L , t) + wt (L , t) − γ m v 2 wt wx x d x, ⎪ ⎪ ⎪ 2 ⎨ 0

⎪ L ⎪ ⎪ ⎪ ⎪ A = γ T wt (L , t)wx (L , t) − γ T wt wx x d x ⎪ ⎪ ⎩ 4 0

(7.31)

7.4 Stability Analysis

91

Substituting A1 , Eqs. (7.30) and (7.31) into Eq. (7.29) and applying Eq. (2.11), we can get γ T v − γ mv 3 2 γ mv γTv 2 wx (L , t) − wx (0, t) − [wt (L , t) + vwx (L , t)]2 2 2 2 γT γT 2 γ Tβ 2 [wt (L , t) + βwx (L , t)]2 − wt (L , t) − wx (L , t) + 2β 2β 2 L L γ 2 − (γ c − γ δ1 ) (wt + vwx ) d x + f 2d x δ1

V1t (t) ≤

0

0

(7.32) where δ1 is a positive constant. Differentiating (7.17) with respect to time results in: V2t (t) = B1 + B2 where B1 = 2λm

L

(7.33)

xwx (wtt + awx + vwxt )d x, B2 = 2λm

0

L

xwxt (wt + vwx )d x.

0

Integrating B1 by parts results in: L B1 = λT Lwx2 (L , t) − λm Lv 2 wx2 (L , t) − 2λc

xwx (wt + vwx )d x 0



− λ T − mv

2

L

L wx2 d x − 2λmv

0

(7.34)

L xwx wxt d x + 2λ

0

xwx f d x 0

Integrating B2 by parts yields: L B2 =

λm Lwt2 (L , t)

− λm

L wt2 d x

0

+ 2λmv

xwxt wx d x 0

(7.35)

92

7 Adaptive Boundary Control of an Axially Moving System…

Substituting Eqs. (7.34) and (7.35) into Eq. (7.33) and combining Eqs. (2.11)– (2.15), we have V2t (t) ≤

λm Lwt2 (L , t)



− λm Lv 2 − λT L wx2 (L , t) − λm

L wt2 d x 0



− λT − λma12 − 2λLδ2 − 2λcLδ3

L wx2 d x

(7.36)

0

2λcL + δ3

L

2λL (wt + vwx ) d x + δ2

L

2

0

f 2d x 0

where δ2 and δ3 are two positive constants. Differentiating (7.18) with respect to time, then substituting the boundary conditions (7.9) and the proposed control (7.10), we obtain: V3t (t) = η[wt (L , t) + βwx (L , t)]{−k[wt (L , t) + βwx (L , t)]  + m˜ c βwxt (L , t) − T˜ wx (L , t) − d˜s wt (L , t) + d˜

(7.37)

Differentiating Eq. (7.19) with respect to time and substituting the Eqs. (2.11)– (2.15), we have:



ηκ ηδ4 ˜ 2 1 2 η γ1 T 2 + γ2 m 2c + γ3 ds2 + − dt + κd 2 − d 2 σ δ4 2 2σ  γ1 η ˜ 2 γ2 η 2 γ3 η ˜ 2 m˜ c − − T − ds + η d˜s wt (L , t) + T˜ wx (L , t) 2 2  2 −m˜ c βwxt (L , t) − d˜ [wt (L , t) + βwx (L , t)]

V4t (t) ≤

where δ4 is a positive constant.

(7.38)

7.4 Stability Analysis

93

Substituting Eq. (7.32) and Eqs. (7.36)–(7.38) into Eq. (7.28) and then using Eqs. (2.11)–(2.15), we obtain:



γT γT Vt (t) ≤ − ηk − − λm L wt2 (L , t) [wt (L , t) + βwx (L , t)]2 − 2β 2β   3 γ T v − γ mv 2 γ T (β − v) 2 2 − wx (0, t) − λm Lv − λT L + wx (L , t) 2 2

L 2λcL γ1 η ˜ 2 γ 2 η 2 γ 3 η ˜ 2 − T − m˜ − d − γ c − γ δ1 − (wt + vwx )2 d x 2 2 c 2 s δ3 0

−

L

ηδ4 ˜ 2 ηκ 2 − wx2 d x d − λT − λma1 − 2λLδ2 − 2λcLδ3 2 2σ 0

+

−

2λL γ + δ1 δ2

L f 2d x + 0

η 1 2 γ1 T 2 + γ2 m 2c + γ3 ds2 + dt + κd 2 2 σ δ4

γ mv [wt (L , t) + vwx (L , t)]2 − λm 2

L wt2 d x 0

(7.39) According to Eqs. (7.24) and (7.39), we choose parameters λ, η, γ , k, β, κ, σ , δ1 ∼ δ4 properly to satisfy the following conditions: ⎧ min(γ m, γ T ) ⎪ ⎪ λ< ⎪ ⎪ 2m L ⎪ ⎪ ⎪ ⎪ γ T v − γ mv 3 ⎪ ⎪ ⎪ ≥0 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ γT ⎪ ⎪ − λm L ≥ 0 ⎪ ⎪ 2β ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ λm Lv 2 − λT L + γ T (β − v) ≥ 0 2 ⎪ ⎪ 2λcL ⎪ ⎪ >0 ⎪ τ1 = γ c − γ δ1 − ⎪ ⎪ δ3 ⎪ ⎪ ⎪ ⎪ τ2 = λT − λma12 − 2λLδ2 − 2λcLδ3 > 0 ⎪ ⎪ ⎪ ⎪ ⎪ γT ⎪ ⎪ >0 τ3 = ηk − ⎪ ⎪ ⎪ 2β ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ τ4 = ηκ − ηδ4 > 0 2 2σ

(7.40)

94

7 Adaptive Boundary Control of an Axially Moving System…

Substituting (7.40) into (7.39), and then combining (7.16)–(7.19), (7.26) and Assumptions 7.1 and 7.2, we further obtain γ1 η ˜ 2 γ2 η 2 γ3 η ˜ 2 Vt (t) ≤ − T − m˜ − d − τ1 2 2 c 2 s

L (wt + vwx )2 d x 0

−τ2

L

wx2 d x − τ3 [wt (L , t) + βwx (L , t)]2 − τ4 d˜ 2 + ε

(7.41)

0

≤ −ϑ3 [V1 (t) + V3 (t) + V4 (t)] + ε ≤ −ϑ V (t) + ε   2τ3 2σ τ4 where ϑ = (ϑ3 /ϑ2 ), ϑ3 = min γ2τm1 , γ2τT2 , ηm , and ε = , , ζ γ , ζ γ , ζ γ 1 1 2 2 3 3 η c     γ η 2λL 1 2 2 2 2 2 2 + δ2 a4 L + 2 γ1 T + γ2 m c + γ3 ds + κa3 + σ δ4 a5 < +∞. δ1 Theorem 7.1 For the presented belt system described by Eqs. (7.8) and (7.9), under Assumptions 7.1–7.3 and the proposed control law Eq. (7.10), given the initial conditions are bounded, we can conclude that: (1) uniform boundedness: the system state w(x, t) of the closed-loop belt system will remain in the compact set: 1 := { w(x, t) ∈ R||w(x, t)| ≤ χ1 }

(7.42)

 [V (0) + ϑε ]. where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 = γ 2L ϑ1 T (2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop belt system will eventually converge to the compact set: 2 := { w(x, t) ∈ R| lim |w(x, t)| ≤ χ2 } t→∞

where ∀x ∈ [0, L], χ2 =



2Lε γ ϑ1 ϑ T

(7.43)

.

Proof Multiplying Eq. (7.41) by eϑt yields: Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒

 ∂ V (t)eϑt ≤ εeϑt ∂t

(7.44)

Integrating the above inequality yields:  ε  −ϑt ε ε e V (t) ≤ V (0) − + ≤ V (0)e−ϑt + ϑ ϑ ϑ

(7.45)

7.5 Simulation Example

95

Table 7.1 Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

Value

m

1.0 kg/m

mc

5.0 kg

L

1.0 m

ds

0.25 Ns/m

g

9.8 N/kg

v0

0

Ns/m2

c

1.0

T

5000 N

Applying Eqs. (2.11)–(2.15) and Eq. (7.26), we have: γT 2 γT w (x, t) ≤ 2L 2

L wx2 (x, t)d x ≤ V1 (t) ≤ V1 (t) + V3 (t) + V4 (t) ≤ 0

1 V (t) ϑ1 (7.46)

Substituting Eq. (7.45) into Eq. (7.46) results in:  |w(x, t)| ≤

2L ε [V (0)e−ϑt + ] ≤ γ T ϑ1 ϑ



2L ε [V (0) + ] γ T ϑ1 ϑ

(7.47)

where ∀(x, t) ∈ [0, L] × [0, +∞), from above inequality, we have:  lim |w(x, t)| ≤ lim

t→∞

t→∞

2L ε [V (0)e−ϑt + ] = γ T ϑ1 ϑ

 2Lε , γ T ϑ1 ϑ

∀x ∈ [0, L] (7.48)

Remark 7.2 From the above analysis, we can see that the increase of k will lead to a larger τ3 , which will result in a greater ϑ3 . Then the value of ϑ will increase, which will reduce the size of the compact set 2 , and it will produce a better vibration control performance. However, increasing k will bring a high gain control problem. Hence, in practice, we should carefully adjust the design parameters to achieve suitable transient performance and control action.

7.5 Simulation Example The finite difference (FD) method is adopted to approximate the solution of the presented belt system [3–7]. The system parameters are given in Table 7.1. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D are given as: aa = ad = 3.5g and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s. The control parameters are given as k = 1 × 104 , β = σ = 100, ζ1 = ζ2 = ζ3 = 1.0, and γ1 = γ2 = γ3 = κ = 0.1.

96

7 Adaptive Boundary Control of an Axially Moving System…

The unknown distributed disturbance is described as:   n  x sin(iπ xt) , i = 1, 2, 3 f (x, t) = 1+ 10000 i=1

(7.49)

The unknown boundary disturbance is given as: 1  i sin(it), i = 1, 2, 3 10 i=1 n

d(t) = 3 +

(7.50)

The initial conditions of the axially moving belt system are given as: w(x, 0) = wt (x, 0) = 0

(7.51)

Figure 7.1 shows the vibration displacement of the uncontrolled belt, i.e. U (t) = 0, with H-A/D under distributed disturbance and boundary disturbance. The vibration displacement of the belt under the same external conditions with the proposed adaptive boundary control is shown in Fig. 7.2. The vibration displacements of the belt are examined at the middle point x = 0.5m and boundary point x = 1m in Fig. 7.3, respectively. Figure 7.4 shows the enlarged view of the vibration displacement for controlled response at x = 1 m and x = 0.5 m. The corresponding adaptive boundary control input is shown in Fig. 7.5.

Fig. 7.1 Vibration displacement of the belt without control

7.5 Simulation Example

Fig. 7.2 Vibration displacement of the belt with control

Fig. 7.3 Vibration displacement of the belt at: a x = 1 m, b x = 0.5 m

97

98

7 Adaptive Boundary Control of an Axially Moving System…

Fig. 7.4 The enlarged view of the vibration displacement for controlled response

Fig. 7.5 The proposed adaptive boundary control input U (t)

Appendix 1: Simulation Program

Appendix 1: Simulation Program close all, clear all, clc; nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m = 1 ; T = 5000; pi = 3.1415926; V0=0; c = 1; % --------------velocity----------------V = zeros(1,nt); V(1,1) = V0; J=3.5*9.8; for j=2:10*10^3-1 V(1,j) = V(1,1)+0.5*J*(j*dt)^2; end V(1,10*10^3)=V(1,1)+0.5*J; for j=(10*10^3+1):20*10^3-1 V(1,j)=V(1,10*10^3)+0.5*J*(j*dt-1); end V(1,20*10^3)=V(1,10*10^3)+J; for j=(20*10^3+1):30*10^3-1 V(1,j) = V(1,20*10^3)+J*(j*dt-2)-0.5*J*(j*dt-2)^2; end V(1,30*10^3)=V(1,20*10^3)+0.5*J; for j=(30*10^3+1):70*10^3-1 V(1,j)=V(1,30*10^3); end V(1,70*10^3)=V(1,30*10^3); for j=(70*10^3+1):80*10^3-1

99

100

7 Adaptive Boundary Control of an Axially Moving System… V(1,j)= V(1,70*10^3)-0.5*J*(j*dt-7)^2;

end V(1,80*10^3)=V(1,70*10^3)-0.5*J; for j=(80*10^3+1):90*10^3-1 V(1,j)=V(1,80*10^3)-J*(j*dt-8); end V(1,90*10^3)=V(1,80*10^3)-J; for j=(90*10^3+1):nt-1 V(1,j)=V(1,90*10^3)-J*(j*dt-9)+0.5*J*(j*dt-9)^2; end V(1,100*10^3)=V(1,90*10^3)-0.5*J; % -------------disburtance------------d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt d(1,j)=3+0.1*sin((j-1)*dt)+0.2*sin(2*(j-1)*dt)+0.3*sin(3*(j-1)*dt); end for j = 1 : nt for i = 1 : nx f(i,j)=0.0001*(1+ sin(pi*(i-1)*dx*(j-1)*dt) + sin(2*pi*(i-1)*dx*(j-1)*dt)+ sin(3*pi*(i-1)*dx*(j-1)*dt) )*(i-1)*dx; end end %******************************************************************** %

uncontrolled

%******************************************************************** ds=0.25; mc=5; w1 = zeros(nx,nt); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((T-m*V(1,j-1)^2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dx^2 -c*(w1(i,j-1)-w1(i,j-2))/dt - c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt

Appendix 1: Simulation Program

101

-m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j) ) / (m/dt^2); end w1(nx,j) = ( (2*mc/dt^2 + ds/dt )*w1(nx,j-1) - (mc/dt^2)*w1(nx,j-2) + d(1,j) + (T/dx)*w1(nx-1,j-1) ) / (mc/dt^2 + ds/dt + T/dx); end tshort = linspace(0,tmax,(nt/400+1)); xshort = linspace(0,L,nx); for j=1:nt/400 for i=1:nx w1short(i,j)=w1(i,j*400); end end w1short=[w1(:,1),w1short]; figure (1); surf(tshort,xshort,w1short); shading interp; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('$\omega$(x,t)[m]','interpreter','latex','Fontsize',14); set(gca,'YDir','reverse') %******************************************************************** %

adaptive control + disturbance observer

%******************************************************************** mc = 5; k = 10^4; beta = 10^2; sigma = 100; gamma = 0.1; kappa = 0.1; w3 = zeros(nx,nt); u2 = zeros(1,nt); u_b = zeros(1,nt); var1 = zeros(1,nt); T_e = zeros(1,nt); mc_e = zeros(1,nt); ds_e = zeros(1,nt); err_T = zeros(1,nt); err_mc = zeros(1,nt); err_ds = zeros(1,nt);

102

7 Adaptive Boundary Control of an Axially Moving System…

d_e2 = zeros(1,nt); error2 = zeros(1,nt); w3(1,:) = 0; for i=2:nx w3(i,1) = 0; w3(i,2) = 0; end for j = 1 : 2 u_b(1,j) = 0; u2(1,j) = 0; T_e(1,j) = 0; mc_e(1,j) = 0; ds_e(1,j) = 0; end for j = 3 : nt for i = 2 : nx - 1 w3(i,j) = 2*w3(i,j-1)-w3(i,j-2) +((T-m*V(1,j-1)^2)*(w3(i+1,j-1)-2*w3(i,j-1)+w3(i-1,j-1))/dx^2 - c*(w3(i,j-1)-w3(i,j-2))/dt -c*V(1,j-1)*(w3(i,j-1)-w3(i-1,j-1))/dx -2*V(1,j-1)*m*(w3(i,j-1)-w3(i,j-2)-w3(i-1,j-1)+w3(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w3(i,j-1)-w3(i-1,j-1))/dx + f(i,j-1) ) / (m/dt^2); end w3(nx,j) = ( 2*mc*w3(nx,j-1)/dt^2 - mc*w3(nx,j-2)/dt^2 + d(1,j-1)d_e2(1,j-1)+ (T - T_e(1,j))*w3(nx-1,j)/dx+ (ds- ds_e(1,j))*w3(nx,j-1)/dt + k*(w3(nx,j-1)/dt + beta*w3(nx-1,j)/dx) + beta*mc_e(1,j)*(w3(nx-1,j) + w3(nx,j-1) - w3(nx-1,j-1))/(dx*dt))/ (mc/dt^2 + (T - T_e(1,j))/dx + (dsds_e(1,j))/dt+ beta*mc_e(1,j)/(dx*dt) + k/dt + k*beta/dx ); u_b(1,j) = (w3(nx,j) - w3(nx,j-1))/dt + beta*(w3(nx,j) w3(nx-1,j))/dx ; T_e(1,j) = T_e(1,j-1) - dt*(gamma*T_e(1,j-1) + u_b(1,j)*(w3(nx,j)-w3(nx-1,j))/dx ); mc_e(1,j) = mc_e(1,j-1) - dt*(gamma*mc_e(1,j-1) u_b(1,j)*( beta*(w3(nx,j) - w3(nx-1,j) - w3(nx,j-1) + w3(nx-1,j-1))/(dx*dt))); ds_e(1,j) = ds_e(1,j-1) - dt*(gamma*ds_e(1,j-1) + u_b(1,j)*(w3(nx,j)-w3(nx,j-1))/dt ); err_T = T - T_e(1,j); err_mc = mc - mc_e(1,j); err_ds = ds - ds_e(1,j);

Appendix 1: Simulation Program

103

u2(1,j) = T_e(1,j)*(w3(nx,j)-w3(nx-1,j))/dx+ds_e(1,j)*(w3(nx,j)-w3(nx,j-1))/dt - beta*mc_e(1,j)*(w3(nx,j) - w3(nx-1,j) - w3(nx,j-1) + w3(nx-1,j-1))/(dx*dt)- k*u_b(1,j) - d_e2(1,j); d_e2(1,j) = d_e2(1,j-1) + dt*(sigma*u_b(1,j)-kappa*sigma*d_e2(1,j)); error2(1,j) = d(1,j) - d_e2(1,j); end for j=1:nt/400 for i=1:nx w3short(i,j)=w3(i,j*400); end end w3short=[w3(:,1),w3short]; figure (2); surf(tshort,xshort,w3short); shading interp; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('$\omega$(x,t)[m]','interpreter','latex','Fontsize',14); set(gca,'YDir','reverse') figure (3); hold on subplot(2,1,1); plot(t,w1(50,:),'b',t,w3(50,:),':r'); title('(a)'); xlabel('Time [s]','Fontsize',14); ylabel('w(1,t) [m]','Fontsize',14); legend('without control','with adaptive boundary control'); box on grid on subplot(2,1,2); plot(t,w1(25,:),'b',t,w3(25,:),':r'); title('(b)'); xlabel('Time [s]','Fontsize',14); ylabel('w(0.5,t)[m]','Fontsize',14); legend('without control','with adaptive boundary control'); box on grid on hold off

104

7 Adaptive Boundary Control of an Axially Moving System…

figure (4) plot(t,w3(50,:),'b',t,w3(25,:),':r'); xlabel('Time [s]','Fontsize',14); ylabel('w(x,t)[m]','Fontsize',14); legend('x=1m','x=0.5m'); figure (5) plot(t,u2(1,:),'b'); xlabel('Time [s]','Fontsize',14); ylabel('U(t) [N]','Fontsize',14);

References 1. H. Goldstein, Classical Mechanics (Addison-Wesley, Massachusetts, 1951) 2. W. He, S. Zhang, S.S. Ge, C. Liu, Robust adaptive control of a thruster assisted position mooring system. Automatica 50(7), 1843–1851 (2014) 3. M. Krstic, A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs (Society for Industrial and Applied Mathematics, Philadelphia, 2008) 4. W. He, S.S. Ge, Vibration control of a flexible beam with output constraint. IEEE Trans. Ind. Electron. 62(8), 5023–5030 (2015) 5. W. He, C. Sun, S.S. Ge, Top tension control of a flexible marine riser by using integral-barrier Lyapunov function. IEEE/ASME Trans. Mechatron. 20(2), 497–505 (2015) 6. G.D. Smith, Numerical solution of partial differential equations: finite difference methods (Oxford University Press, Oxford, UK, 1985) 7. W. He, S.S. Ge, Cooperative control of a nonuniform gantry crane with constrained tension. Automatica 66, 146–154 (2016)

Chapter 8

Boundary Control of an Axially Moving Accelerated/Decelerated Belt System

Abstract In this chapter, a boundary controller with disturbance observer is proposed for the vibration suppression of an axially moving belt system. The model of the belt system is described by a nonhomogeneous partial differential equation and a set of ordinary differential equations with consideration of the high acceleration/deceleration and unknown spatiotemporally varying distributed disturbance. Applying the finite-dimensional backstepping control and Lyapunov’s direct method, a boundary controller is developed to stabilize the belt system at the small neighborhood of its equilibrium position and a disturbance observer is introduced to attenuate the effect of unknown external disturbance. Moreover, another novelty of this chapter is the design of a boundary controller by utilizing the finite-dimensional backstepping control method and Lyapunov’s direct method based on the infinitedimensional model for vibration suppression of an axially moving system. The Scurve acceleration/deceleration method is adopted to plan the speed of the belt. With the proposed control scheme, the spillover instability problems are avoided, the uniform boundedness and the stability of the closed-loop belt system can be achieved. Simulations are provided to illustrate the effectiveness of the proposed control scheme.

8.1 Introduction In this chapter, a boundary controller with disturbance observer is proposed for the vibration suppression of an axially moving belt system. The model of the belt system is described by a nonhomogeneous partial differential equation and a set of ordinary differential equations with consideration of the high acceleration/deceleration and unknown spatiotemporally varying distributed disturbance. Applying the finitedimensional backstepping control [1–3] and Lyapunov’s direct method, a boundary controller is developed to stabilize the belt system at the small neighborhood of its equilibrium position and a disturbance observer is introduced to attenuate the effect of unknown external disturbance. Moreover, another novelty of this chapter is the design of a boundary controller by utilizing the finite-dimensional backstepping control method and Lyapunov’s direct method based on the infinite-dimensional © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_8

105

106

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

model for vibration suppression of an axially moving system. The S-curve acceleration/deceleration method is adopted to plan the speed of the belt. With the proposed control scheme, the spillover instability problems are avoided, the uniform boundedness and the stability of the closed-loop belt system can be achieved. Simulations are provided to illustrate the effectiveness of the proposed control scheme.

8.2 PDE Dynamic Model To obtain the dynamic model of the presented belt system, we apply the extended Hamilton’s principle [4]: t2 (δ E k − δ E p + δWc + δWd − δWb )dt = 0

(8.1)

t1

The kinetic energy E k (t) of the belt system can be represented as: 1 1 E k = m c wt2 (L , t) + m 2 2

L (wt +vwx )2 d x

(8.2)

0

where m c is the mass of the actuator. And the potential energy E p (t) of the belt system is expressed as: 1 Ep = 2

L T εd x

(8.3)

0

where the spatially varying tension T (x, t) is given as: T = T0 +

1 E Awx2 2

(8.4)

where T0 is the initial tension of the belt, E is the coefficient of elasticity, and A is the cross-sectional area. The displacement–strain relation ε is expressed as: ε=

1 2 w 2 x

(8.5)

The virtual work δWc (t) done by the non-conservative forces on the belt is computed as:

8.2 PDE Dynamic Model

107

L δWc = [U − ds wt (L , t)]δw(L , t) − c

(wt +vwx )δwd x

(8.6)

0

where ds is damping coefficient of the actuator and c is viscous damping coefficient of the belt. The virtual work δWd (t) done by unknown boundary disturbance d(t) and distributed disturbance f (x, t) on the belt is given by: L δWd = dδw(L , t) +

f δwd x

(8.7)

0

The virtual momentum δWb (t) transport across the boundaries is computed by: δWb = mv[wt (L , t) + vwx (L , t)]δw(L , t)

(8.8)

Substituting Eqs. (8.2)–(8.3), and (8.6)–(8.8) into Eq. (8.1) and integrating by parts, the governing equation of the presented belt system can be obtained as: mwtt + mawx + 2mvwxt + mv 2 wx x −

3 1 E Awx2 wx x − T0 wx x + cwt + cvwx = f 2 2 (8.9)

where ∀(x, t) ∈ (0, L) × [0, +∞). The boundary conditions of presented belt system are described as: ⎧ ⎨ w(0, t) = 0 ⎩ m c wtt (L , t) + 1 T0 wx (L , t) + 1 E Awx3 (L , t) + ds wt (L , t) = d + U 2 2

(8.10)

where ∀t ∈ [0, +∞). Assumption 8.1 For the speed v(t), H-A/D a(t), boundary disturbance d(t), and distributed disturbance f (x, t), we assume that there exist constants a1 , a2 , a3 , a4 ∈ R+ , such that 0 < v(t) ≤ a1 , | a(t) | ≤ a2 , | d(t) | ≤ a3 , ∀t ∈ [0, +∞) and | f (x, t) | ≤ a4 , ∀(x, t) ∈ (0, L) × [0, +∞). Assumption 8.2 We assume the time derivative of boundary disturbance dt (t) is uniformly bounded and there exists a constant a5 ∈ R+ , such that | dt (t) | ≤ a5 , ∀t ∈ [0, +∞).

108

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

8.3 Boundary Controller Design Transform the dynamic model (8.9)–(8.10) into a standard state-space form as ⎧ 1 3 ⎪ mwtt = f + T0 wx x + E Awx2 wx x − cwt − mawx − 2mvwxt − mv 2 wx x − cvwx ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ w(0, t) = 0 ⎪ ⎨ z 1 = w(L , t) ⎪ ⎪ ⎪ z 1t = z 2 ⎪ ⎪   ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎩ z 2t = d + U − ds z 2 − T0 wx (L , t) − E Awx3 (L , t) mc 2 2 (8.11) Remark 8.1 Equation (8.11) shows that the transformed belt system is a strictfeedback form. Hence, the backstepping technique can be employed to design the boundary control U (t) for vibration suppression of the belt.

8.3.1 Step One Choose the term wt (L , t), that is z 2 (t), as a control to design a boundary control law, and we define: ze = z2 − zv

(8.12)

where z v (t) is a virtual control of z 2 (t) and z e (t) is the corresponding error variable. The virtual control z v (t) can be designed by Lyapunov direct method. Consider the Lyapunov function candidate as V1 (t) = V3 (t) + V4 (t)

(8.13)

where the energy term V3 (t) is expressed as: γm V3 (t) = 2

L

γ (wt + vwx ) d x + 2

L T εd x

2

0

(8.14)

0

and a small crossing term V4 (t) is expressed as: L xwx (wt + vwx )d x

V4 (t) = λm 0

(8.15)

8.3 Boundary Controller Design

109

where γ and λ are positive weighting constants. Lemma 8.1 For the axially moving belt system described by (8.9)–(8.10) and the Lyapunov function candidate V1 (t) given by (8.13), the following conclusions hold: (1) the function V1 (t) is a positive definite function, and (2) the function V1 (t) is upper and lower bounded as: ξ1 V3 (t) ≤ V1 (t) ≤ ξ2 V3 (t)

(8.16)

where ξ1 and ξ2 are two positive constants. Proof Applying Eqs. (2.11)–(2.15) to Eq. (8.15) results in: λm L |V4 (t)| ≤ 2

L

2

wx + (wt + vwx )2 d x ≤ ξ V3 (t)

(8.17)

0

where ξ =

2λm L . min(γ T0 ,2γ m)

According to Eq. (8.17),we have: −ξ V3 (t) ≤ V4 (t) ≤ ξ V3 (t)

(8.18)

Choosing λ and γ properly, we can obtain ⎧ ⎪ ⎪ ⎨ ξ1 = 1 − ξ = 1 −

2λm L >0 min(γ T0 , 2γ m) 2λm L ⎪ ⎪ ⎩ ξ2 = 1 + ξ = 1 + >1 min(γ T0 , 2γ m)

(8.19)

Equation (8.19) indicates 0 < ξ < 1, then we have λ
0 τ1 = γ c − γ δ2 − δ1 ⎪ ⎪ ⎪ ⎪ λma12 λT0 ⎪ ⎪ ⎪ − − λLδ3 − λcLδ1 > 0 τ = 2 ⎪ ⎪ 4 2 ⎪ ⎪ ⎪ ⎪ k2 + ds − δ4 γ mv − λm L ⎪ ⎩ τ3 = + >0 2 mc

(8.43)

Substituting (8.43) into (8.42), then combining (8.14) and (8.21), we have 1 2 e + ε1 m c δ4 d 1 2 ≤ −ϑ2 V1 (t) − τ3 z e2 + e + ε1 m c δ4 d 1 2 ≤ −ϑ3 V2 (t) + e + ε1 m c δ4 d

V2t (t) ≤ −ϑ1 V3 (t) − τ3 z e2 +

(8.44)

116

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

  , ϑ2 = (ϑ1 /ξ2 ), ϑ3 = min(ϑ2 , 2τ3 ), and ε1 = where ϑ1 = min γ2τm1 , γ4τT20 , 3λ γ     L 2 γ γ λL λL + δ3 0 f d x ≤ δ2 + δ3 a42 L. δ2 Combining (8.12) and (8.32), the actual control described by (8.37) can be rewritten as    2  (γ m c − 1)T0 + mm c γ v − k1 γ v + k1 λL wx (L , t) U (t) = −k1 k2 − ds k1 − 2 E A(γ m c − 1) 3 wx (L , t) − de − k2 wt (L , t) − m c k1 wxt (L , t) − (8.45) 2 Remark 8.5 For the boundary control U (t) described by (8.45), the control design is based on the infinite-dimensional PDE (8.9), hence the spillover instability problem is avoided. w(L , t) can be measured by a position sensor, and wx (L , t) can be measured by an inclinometer. wt (L , t) and wxt (L , t) can be calculated with a backward difference algorithm to w(L , t) and wx (L , t), respectively. The observer is also introduced to handle the boundary disturbance d(t), which is arisen from the uncontrolled span of the belt.

8.4 Stability Analysis According to the previous analyses, a stability theorem for the closed-loop belt system can be presented as follows. Theorem 8.1 For the belt system described by (8.9)–(8.10), under the proposed control (8.45) and disturbance observer (8.38), given that the initial conditions are bounded and the new Lyapunov function is defined as 1 V (t) = V2 (t) + ed2 (t) 2

(8.46)

then the following conclusions hold. (1) The time derivative of the V (t) is upper bounded with Vt (t) ≤ −ϑ V (t) + ε

(8.47)

where ϑ and ε are both positive constants. (2) There exists a constant χ1 ∈ R+ , such that the closed-loop belt system state w(x, t) is bounded as |w(x, t)| ≤ χ1 , ∀(x, t) ∈ [0, L] × [0, +∞)

(8.48)

8.4 Stability Analysis

117

(3) There exists a constant χ2 ∈ R+ , such that the closed-loop belt system state w(x, t) is bounded as follows with time tending to infinity lim |w(x, t)| ≤ χ2 , ∀x ∈ [0, L]

t→∞

(8.49)

(4) The exponential stability of the closed-loop belt system under free vibration case, that is f (x, t) = d(t) = 0, can be achieved as lim |w(x, t)| = 0, ∀x ∈ [0, L]

t→∞

(8.50)

Proof Differentiating (8.46) and then substituting (8.41), we obtain Vt (t) = V2t (t) + dt ed −

k3 2 e mc d

(8.51)

Using Eqs. (2.11)–(2.15) and combining (8.44), we have: Vt (t) ≤ −ϑ3 V2 (t) −

k3 1 1 e2 + δ5 dt2 + ε1 − − mc δ5 m c δ4 d

≤ −ϑ V (t) + ε

(8.52)

where δ5 is a positive constant, and the constants λ, γ , k1 , k2 , k3 , δ1 , δ2 , δ3 , δ4 , and δ5 are designed to satisfy (8.19), (8.43), and the following conditions ⎧ k3 1 1 ⎪ ⎪ ⎪ ⎨ τ4 = m c − δ5 − m c δ4 > 0 ϑ = min(ϑ3 , 2τ4 ) ⎪ ⎪ ⎪ ⎩ ε = δ5 dt2 + ε1 ≤ δ5 a52 + ε1

(8.53)

Multiplying (8.52) by eϑt results in Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt

(8.54)

The afore-mentioned inequality can be rewritten as

∂ V (t)eϑt ≤ εeϑt ∂t

(8.55)

 ε ε ε  −ϑt e + ≤ V (0)e−ϑt + V (t) ≤ V (0) − ϑ ϑ ϑ

(8.56)

Integrating (8.55) yields

118

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

Using Eqs. (2.11)–(2.15) and combining (8.13), (8.14), (8.16), and (8.46), we have γ T0 γ T0 2 w (x, t) ≤ 4L 4

L wx2 (x, t)d x ≤ V3 (t) 0

1 1 2 1 1 2 ≤ V1 (t) + ze + ed ≤ V (t) ξ1 2ξ1 2ξ1 ξ1

(8.57)

Substituting Eq. (8.56) into Eq. (8.57), results in:  |w(x, t)| ≤

4L  ε = χ1 V (0)e−ϑt + γ T0 ξ1 ϑ

(8.58)

where ∀(x, t) ∈ [0, L] × [0, +∞). From above inequality, we have:  4Lε γ T0 ξ1 ϑ

lim |w(x, t)| ≤

t→∞

= χ2

(8.59)

where ∀x ∈ [0, L]. In the free vibration case, the following equations can be obtained f (x, t) = d(t) = 0 ⇒ ε = 0

(8.60)

Substituting (8.60) into (8.58) results in  lim |w(x, t)| ≤ lim

t→∞

t→∞

4L V (0) − ϑ t e 2 =0 γ T0 ξ1

(8.61)

where ∀x ∈ [0, L]. From (8.61), (8.50) can be obtained, and we can conclude that the free vibration belt system under the presented boundary control is exponentially stable with decay rate (ϑ/2). Remark 8.6 From (8.43), (8.53), and (8.59), k2 and k3 not only need to be satisfied the afore-mentioned conditions but also affect the vibration control performance. The increase of k2 and k3 will eventually yield a larger ϑ, which will reduce the compact set χ2 and bring a better control performance. However, the increase of k2 and k3 will bring the high-gain control. Hence, in practice, the system parameters should be designed carefully. Remark 8.7 From (8.56) and (8.57), we obtain w(x, t) is bounded ∀(x, t) ∈ [0, L]× [0, +∞) and V3 (t), z e (t), and ed (t) are all bounded ∀t ∈ [0, +∞). Because V3 (t) is bounded, the potential energy (8.3) of the belt system is bounded ∀t ∈ [0, +∞), and

8.5 Simulation Example

119

Table 8.1. Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

Value

m

1.0 kg/m

mc

5.0 kg

L

1.0 m

c

1.0 Ns/m2

EA

500 N

γ

0.1

ds

0.25 Ns/m

T0

9800 N

λ

0.001

k1

1000

k2

1000

k3

10

then, wx (x, t) and wx x (x, t) are also bounded ∀(x, t) ∈ [0, L]×[0, +∞). According to (8.11), (8.12), and (8.32), we have z e (t) = wt (x, t) + k1 wx (x, t), and then we can obtain wt (L , t) is also bounded ∀t ∈ [0, +∞). Because V3 (t) and wt (x, t) are bounded, the kinetic energy (8.2) of the belt system is bounded ∀t ∈ [0, +∞). Then we have wt (x, t) and wxt (x, t) are bounded ∀(x, t) ∈ [0, L] × [0, +∞). According to (8.40), the boundedness of ed (t) and Assumption 1, we obtain de (t) is bounded ∀t ∈ [0, +∞). Finally, because de (t) is bounded ∀t ∈ [0, +∞) and wx (x, t), wt (x, t), and wxt (x, t) are bounded ∀(x, t) ∈ [0, L] × [0, +∞), we can conclude that the proposed actual control (8.45) is bounded ∀t ∈ [0, +∞).

8.5 Simulation Example Consider the PDE model as Eq. (8.9), the parameters of the belt system are given in Table 8.1 and the initial conditions are v0 = w(x, 0) = wt (x, 0) = 0. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D are given as: aa = ad = 3.5 g and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10] s (g = 9.8 m/s2 ). The unknown distributed disturbance f (x, t) is described as:   3  1 sin(iπ xt) f (x, t) = 10x + 1000 i=1

(8.62)

The unknown time-varying boundary disturbance d(t) is described as: d(t) = 10 +

3 

cos(it)

(8.63)

i=1

Figure 8.1 shows the three-dimensional vibration displacement w(x, t) of the belt system for free vibration, that is, U (t) = 0, under both distributed disturbance and boundary disturbance. The three-dimensional vibration displacement of the belt system with the proposed boundary control and disturbance observer, by choosing control gains k1 = k2 = 1000 and observer gain k3 = 10, under the same disturbances is presented in Fig. 8.2.

120

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

Fig. 8.1. Vibration displacement of the belt without control

Fig. 8.2. Vibration displacement of the belt with control

The vibration displacement of the belt is examined at x = 1 m and x = 0.5 m, and the simulation results for controlled and uncontrolled responses are shown in Fig. 8.3, respectively. Figure 8.4 shows the enlarged view of the vibration displacement for controlled response at x = 1 m and x = 0.5 m. The trajectory tracking of the

8.5 Simulation Example

121

boundary disturbance d(t) is shown in Fig. 8.5, and the corresponding control input of the proposed boundary control U (t) is shown in Fig. 8.6.

Fig. 8.3. Vibration displacement of the belt at: a x = 1 m, b x = 0.5 m

Fig. 8.4. The enlarged view of the vibration displacement for controlled response

122

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

Fig. 8.5. Tracking of boundary disturbance

Fig. 8.6. Control input U (t)

Appendix 1: Simulation Program

Appendix 1: Simulation Program close all; clear all; clc; nx =50; nt =100*10ˆ3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m=1; EA = 500; T0 = 9800; pi = 3.1415926; V0=0; c = 1; % --------------velocity----------------V = zeros(1,nt); V(1,1) = V0; J=3.5*9.8; for j=2:10*10ˆ3-1 V(1,j) = V(1,1)+0.5*J*(j*dt)ˆ2; end V(1,10*10ˆ3)=V(1,1)+0.5*J; for j=(10*10ˆ3+1):20*10ˆ3-1 V(1,j)=V(1,10*10ˆ3)+0.5*J*(j*dt-1); end V(1,20*10ˆ3)=V(1,10*10ˆ3)+J; for j=(20*10ˆ3+1):30*10ˆ3-1 V(1,j)=V(1,20*10ˆ3)+J*(j*dt-2)-0.5*J*(j*dt-2)ˆ2; end V(1,30*10ˆ3)=V(1,20*10ˆ3)+0.5*J; for j=(30*10ˆ3+1):70*10ˆ3-1 V(1,j)=V(1,30*10ˆ3); end V(1,70*10ˆ3) = V(1,30*10ˆ3); for j=(70*10ˆ3+1):80*10ˆ3-1 V(1,j)=V(1,70*10ˆ3)-0.5*J*(j*dt-7)ˆ2; end V(1,80*10ˆ3)=V(1,70*10ˆ3)-0.5*J; for j=(80*10ˆ3+1):90*10ˆ3-1 V(1,j)=V(1,80*10ˆ3)-J*(j*dt-8); end V(1,90*10ˆ3)=V(1,80*10ˆ3)-J; for j=(90*10ˆ3+1):nt-1 V(1,j)=V(1,90*10ˆ3)-J*(j*dt-9)+0.5*J*(j*dt-9)ˆ2; end V(1,100*10ˆ3)=V(1,90*10ˆ3)-0.5*J;

123

124

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

% -------------disburtance------------d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt d(1,j)=10+cos(1*(j-1)*dt)+cos(2*(j-1)*dt)+cos(3*(j-1)*dt); end for j = 1 : nt for i = 1 : nx f(i,j) = 0.001*(10*(i-1)*dx+sin(pi*(i-1)*dx*(j-1)*dt)+sin(2*pi*(i1)*dx*(j-1)*dt)+sin(3*pi*(i-1)*dx*(j-1)*dt)); end end %****************************************** % uncontrolled %****************************************** ds=0.25; mc=5; w1 = zeros(nx,nt); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((0.5*T0+1.5*EA*(w1(i,j1)-w1(i-1,j-1))ˆ2/dxˆ2 -m*V(1,j-1)ˆ2)*(w1(i+1,j-1)-2*w1(i,j1)+w1(i-1,j-1))/dxˆ2 - c*(w1(i,j-1)-w1(i,j-2))/dt -c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j1))/dx + f(i,j) ) / (m/dtˆ2); end w1(nx,j) = ( (2*mc/dtˆ2 + ds/dt )*w1(nx,j-1) (mc/dtˆ2)*w1(nx,j-2) +d(1,j) + (0.5*T0/dx)*w1(nx-1,j-1) 0.5*EA*(w1(nx,j-1)-w1(nx-1,j-1))ˆ3/(dx)ˆ3 ) / (mc/dtˆ2 + ds/dt + 0.5*T0/dx); end tshort = linspace(0,tmax,(nt/500+1)); xshort = linspace(0,L,nx); for j=1:nt/500 for i=1:nx w1short(i,j)=w1(i,j*500); end end w1short=[w1(:,1),w1short]; figure (1); surf(tshort,xshort,w1short); shading interp; xlabel(’Time [s]’,’Fontsize’,14); ylabel(’x [m]’,’Fontsize’,14); zlabel(’$\omega$(x,t)[m]’,’interpreter’,’latex’,’Fontsize’,14); set(gca,’YDir’,’reverse’)

Appendix 1: Simulation Program

125

%**************************************** % controlled + disturbance observer %**************************************** ds = 0.25; mc = 5; k1 = 1000; k2 = 1000; guma = 0.1; lumda = 0.001; k3 = 1000; w2 = zeros(nx,nt); u = zeros(1,nt); d_e = zeros(1,nt); var = zeros(1,nt); beta=var; error = zeros(1,nt); w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((0.5*T0+1.5*EA*(w2(i,j1)-w2(i-1,j-1))ˆ2/dxˆ2 -m*V(1,j-1)ˆ2)*(w2(i+1,j-1)-2*w2(i,j1)+w2(i-1,j-1))/dxˆ2 -c*(w2(i,j-1)-w2(i,j-2))/dt - c*V(1,j1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j) ) / (m/dtˆ2); end w2(nx,j) = ( d(1,j) - d_e(1,j-1) + ds*w2(nx,j-1)/dt + ds*k1*w2(nx-1,j)/dx + mc*(2*w2(nx,j-1)-w2(nx,j-2))/dtˆ2 +mc*k1*(w2(nx,j-1)+w2(nx-1,j)-w2(nx-1,j-1))/(dx*dt) mc*( m*guma*V(1,j)ˆ2 - 0.5*guma*T0 - k1*m*guma*V(1,j) + k1*m*lumda*L)/dx*w2(nx-1,j) -0.5*mc*guma*EA*(w2(nx,j-1)-w2(nx1,j-1))ˆ3/dxˆ3 + mc*k2*w2(nx,j-1)/dt + mc*k1*k2*w2(nx-1,j)/dx )/ ( mc/dtˆ2 + ds/dt + ds*k1 + mc*k1/(dx*dt) - mc*( m*guma*V(1,j)ˆ2 - 0.5*guma*T0 - k1*m*guma*V(1,j)+ k1*m*lumda*L)/dx+ mc*k2/dt + mc*k1*k2/dx ); u(1,j) = -d_e(1,j-1) - ds*k1*(w2(nx,j)-w2(nx-1,j))/dx + 0.5*T0*(w2(nx,j)-w2(nx-1,j))/dx+0.5*EA*(w2(nx,j)-w2(nx1,j))ˆ3/dxˆ3 -mc*k1*(w2(nx,j)-w2(nx-1,j)-w2(nx,j-1)+w2(nx1,j-1))/(dt*dx) +mc*( m*guma*V(1,j)ˆ2 0.5*guma*T0 k1*m*guma*V(1,j) + k1*m*lumda*L)*(w2(nx,j)-w2(nx-1,j))/dx 0.5*mc*guma*EA*(w2(nx,j)-w2(nx-1,j))ˆ3/dxˆ3 -k2*(w2(nx,j)w2(nx,j-1))/dt-k1*k2*(w2(nx,j)-w2(nx-1,j))/dx; var(1,j) = var(1,j-1) - dt*(k3/mc*var(1,j-1) k3/mc*ds*(w2(nx,j)-w2(nx,j-1))/dt -k3*T0/(2*mc)*(w2(nx,j)w2(nx-1,j))/dx -k3*EA/(2*mc)*(w2(nx,j)-w2(nx-1,j))ˆ3/dxˆ3 + k3/mc*u(1,j) + k3ˆ2/mc*(w2(nx,j)-w2(nx,j-1))/dt ); d_e(1,j) = var(1,j) + k3*(w2(nx,j) - w2(nx,j-1))/dt; error(1,j) = d(1,j) - d_e(1,j); end

126

8 Boundary Control of an Axially Moving Accelerated/Decelerated …

for j=1:nt/500 for i=1:nx w2short(i,j)=w2(i,j*500); end end w2short=[w2(:,1),w2short]; figure (2); surf(tshort,xshort,w2short); shading interp; xlabel(’Time [s]’,’Fontsize’,14); ylabel(’x [m]’,’Fontsize’,14); zlabel(’$\omega$(x,t)[m]’,’interpreter’,’latex’,’Fontsize’,14); set(gca,’YDir’,’reverse’) figure (3); hold on subplot(2,1,1); plot(tshort,w1short(50,:),’b’,tshort,w2short(50,:),’:r’); title(’(a)’); xlabel(’Time (s)’,’Fontsize’,14); ylabel(’w(1,t) (m)’,’Fontsize’,14); legend(’uncontrolled’,’controlled’); box on grid on subplot(2,1,2); plot(tshort,w1short(25,:),’b’,tshort,w2short(25,:),’:r’); title(’(b)’); xlabel(’Time (s)’,’Fontsize’,14); ylabel(’w(0.5,t) (m)’,’Fontsize’,14); legend(’uncontrolled’,’controlled’); box on grid on hold off figure (4); plot(tshort,w2short(50,:),’:r’,tshort,w2short(25,:),’b’); legend(’x=1m’,’x=0.5m’); xlabel(’Time (s)’,’Fontsize’,14); ylabel(’w(x,t) (m)’,’Fontsize’,14); figure(5); plot(t,d_e,’b’,t,d,’--r’); title(’disturbance obsever’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’d(t) [N]’,’Fontsize’,14); legend(’estimate of disturbance’,’actual disturbance’); figure(6); plot(t,u,’b’); xlabel(’Time(s)’,’Fontsize’,14); ylabel(’U(t)(N)’,’Fontsize’,14);

References

127

References 1. K.D. Do, J. Pan, Boundary control of three-dimensional inextensible marine risers. J. Sound Vib. 327(3–5), 299–321 (2009) 2. I. Kanellakopoulos, P. Kokotovic, A. Morse, A toolkit for nonlinear feedback design. Syst. Control Lett. 18(2), 83–92 (1992) 3. M. Krstic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design (Wiley, New York, 1995) 4. H. Goldstein, Classical Mechanics (Addison-Wesley, Massachusetts, 1951)

Chapter 9

Stabilization of an Axially Moving Accelerated/Decelerated System via an Adaptive Boundary Control

9.1 Introduction For active vibration control studies of axially moving systems, the boundary control algorithms designed based on the infinite-dimensional distributed parameter system model have also achieved many important results [1–3]. It is worth noting that these existing studies are either limited to the case where the axial velocity is constant or only when the axial velocity is time-varying, and these studies have not considered unknown distributed disturbances. However, in practice, most axially moving systems use not only variable speed, but also high acceleration/deceleration (H-A/D) to improve efficiency, especially in high-speed precision mechanical systems. On the other hand, the boundary control integrated with the backstepping method has been displayed for non-axially moving systems such as beams, waves, and others. Yet, this control method for axially moving systems with both parametric and disturbance uncertainties has not yet been reported elsewhere. In this chapter, an adaptive boundary control is developed for vibration suppression of an axially moving accelerated/decelerated belt system. The dynamic model of the belt system is represented by partial-ordinary differential equations with consideration of the high acceleration/deceleration and unknown distributed disturbance. By utilizing adaptive technique and Lyapunov-based backstepping method, an adaptive boundary control is proposed for vibration suppression of the belt system, a disturbance observer is introduced to attenuate the effects of unknown boundary disturbance, the adaptive law is developed to handle parametric uncertainties and the S-curve acceleration/deceleration method is adopted to plan the belt’s speed. With the proposed control scheme, the well-posedness and stability of the closed-loop system are mathematically demonstrated.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_9

129

130

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

9.2 PDE Dynamic Model For deriving the mathematical model of the presented belt system, we apply the extended Hamilton’s principle [4]: t2

  δ E k − δ E p + δW dt = 0

(9.1)

t1

The kinetic energy E k (t) of the belt system can be represented as: 1 1 E k = m c wt2 (L , t) + m 2 2

L (wt +vwx )2 d x

(9.2)

0

where m c denotes the mass of the actuator. The potential energy E p (t) of the belt system is calculated as: 1 Ep = T 2

L wx2 d x

(9.3)

0

where T is the belt’s tension. The virtual work δW f (t) done by the non-conservative force on the belt is computed as: L δW f = [U + d − ds wt (L , t)]δw(L , t) +

[ f − c(wt + vwt )]δwd x

(9.4)

0

where ds is damping coefficient of the actuator and c is viscous damping coefficient of the belt. The virtual momentum transport δWb (t) across the boundaries is calculated by: δWb = mv[wt (L , t) + vwx (L , t)]δw(L , t)

(9.5)

The total virtual work δW (t) done on the belt system is given by: δW = δW f − δWb

(9.6)

9.3 Boundary Controller Design

131

Substituting Eqs. (9.2)–(9.3), (9.6) into Eq. (9.1) and integrating by parts, we obtain the belt’s governing equation as: mwtt + mawx + 2mvwxt + mv 2 wx x − T wx x + cwt + cvwx = f

(9.7)

where ∀(x, t) ∈ (0, L) × [0, +∞). The corresponding boundary conditions are expressed as: 

w(0, t) = 0 m c wtt (L , t) + T wx (L , t) + ds wt (L , t) − U − d = 0

(9.8)

where ∀t ∈ [0, +∞). Assumption 9.1 For the speed v(t), H-A/D a(t), boundary disturbance d(t) and distributed disturbance f (x, t), we assume that there exist constants a1 , a2 , a3 , a4 ∈ R+ , such that 0 < v(t) ≤ a1 , | a(t) | ≤ a2 , | d(t) | ≤ a3 , ∀t ∈ [0, +∞) and | f (x, t) | ≤ a4 , ∀(x, t) ∈ (0, L) × [0, +∞). Assumption 9.2 We assume the time derivative of boundary disturbance d(t), that is, dt (t), is uniformly bounded and there exists a constant a5 ∈ R+ , such that | dt (t) | ≤ a5 , ∀t ∈ [0, +∞).

9.3 Boundary Controller Design The mathematical model (9.7)–(9.8) of the presented belt can be transformed into a standard state-space form: ⎧ mwtt = f + T wx x − cwt − mawx − 2mvwxt + mv 2 wx x − cvwx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ w(0, t) = 0 z 1 = w(L , t) ⎪ ⎪ ⎪ z = z2 ⎪ ⎪ ⎪ 1t ⎩ z 2t = [U + d − ds wt (L , t) − T wx (L , t)]/m c

(9.9)

Remark 9.1 Closely checking (9.9), we note that this equation is a strict-feedback form. Hence we can use backstepping technique to design an adaptive boundary control for vibration suppression of the belt.

132

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

9.3.1 Step One Choose wt (L , t), i.e. z 2 (t), as a control to design an adaptive boundary control, then we define: ze = z2 − zv

(9.10)

where z v (t) is a virtual control of z 2 (t) and z e (t) is the corresponding error variable. The Lyapunov’s direction method is employed to design the virtual control z v (t), then consider the Lyapunov function candidate as V1 (t) = V3 (t) + V4 (t)

(9.11)

where the energy term V3 (t) is defined as: γm V3 (t) = 2

L

γT (wt + vwx ) d x + 2

L wx2 d x

2

0

(9.12)

0

and a small crossing term V4 (t) is defined as: L xwx (wt + vwx )d x

V4 (t) = λm

(9.13)

0

where γ and λ are both positive weighting constants. Lemma 9.2 For the belt system described by (9.7)–(9.8) and the Lyapunov function candidate V1 (t) given by (9.11), we can obtain (1) The function V1 (t) is a positive definite function, and it can be upper and lower bounded as: m(γ − λL) 2

L

(γ T − λm L) (wt + vwx ) d x + 2

L wx2 d x

2

0

0

m(γ + λL) ≤ V1 (t) ≤ 2

L

(γ T + λm L) (wt + vwx ) d x + 2

L wx2 d x

2

0

0

(9.14) (2) Furthermore, the function V1 (t) is upper and lower bounded as: −ξ1 V3 (t) ≤ V1 (t) ≤ ξ2 V3 (t)

(9.15)

9.3 Boundary Controller Design

133

where ξ1 and ξ2 are two positive constants. Proof Applying Eqs. (2.11)–(2.15) to Eq. (9.13) results in: λm L |V4 (t)| ≤ 2

L

2  wx + (wt + vwx )2 d x

(9.16)

0

The above inequality can be rewritten as: λm L − 2

L

λm L (wt + vwx ) d x − 2

L wx2 d x

2

0

0

λm L ≤ V4 (t) ≤ 2

L

λm L (wt + vwx ) d x + 2

L wx2 d x

2

0

(9.17)

0

It is obvious that the inequality (9.15) is the result of substituting (9.17) and (9.13) into (9.12). If we choose λ and γ to satisfy the following conditions 

γ − λL > 0 γ T − λm L > 0

(9.18)

hence, according to (9.15), the Lyapunov function candidate V1 (t) is a positive definite function. According to (9.13) and (9.16), we have −ξ V3 (t) ≤ V4 (t) ≤ ξ V3 (t)

(9.19)

λm L . where ξ = min(γ T,γ m) Choosing λ and γ properly, we can obtain

⎧ λm L ⎪ ⎪ ⎨ ξ1 = 1 − ξ = 1 − min(γ T, γ m) > 0 λm L ⎪ ⎪ ⎩ ξ2 = 1 + ξ = 1 + >1 min(γ T, γ m)

(9.20)

Equation (9.20) indicates 0 < ξ < 1, then we have λ
mλL/T when m ≥ T and γ > λL when m < T . Therefore, the inequalities (9.17) and (9.21) have different forms for the same constraints. Differentiating (9.11) with respect to time t yields V1t (t) = V3t (t) + V4t (t)

(9.23)

According to (9.12), we have V3t (t) = A1 + A2 + A3 + A4

(9.24)

L L  where A1 = γ m 0 awt wx + avwx2 d x, A2 = γ m 0 (wt wtt + vwx wtt )d x, L L  A3 = γ m 0 vwt wxt + v 2 wx wxt d x and A4 = γ T 0 wx wxt d x. Substituting the governing Eq. (9.7) into A2 and integrating by parts, we obtain γ T v − γ mv 3 2 [wx (L , t) − wx2 (0, t)] − 2γ mv 2 wx (L , t)wt (L , t) 2 L L 2 − γ mvwt (L , t) − 2γ c vwt wx d x − γ ma wt wx d x

A2 =

0

  + γ mv 2 + γ T

0

L

L wt w x x d x − γ c

0

L 0

+γ

0

L wx2 d x

− γ mva

L wt2 d x

− γc

wt f d x 0

L v

2

wx2 d x

+ γv

0

wx f d x

(9.25)

0

Integrating A3 and A4 by parts, we have ⎧ L ⎪ ⎪ γ mv 2 ⎪ 2 ⎪ A = γ mv wx (L , t)wt (L , t) + wt (L , t) − γ m v 2 wt wx x d x ⎪ ⎪ ⎪ 3 2 ⎨ 0

⎪ L ⎪ ⎪ ⎪ ⎪ ⎪ A4 = γ T wt (L , t)wx (L , t) − γ T wt wx x d ⎪ ⎩ 0

(9.26)

9.3 Boundary Controller Design

135

Substituting A1 and Eqs. (9.25)–(9.26) into Eq. (9.23), we can get    γ T v − γ mv 3 2 wx (L , t) − wx2 (0, t) + γ T − γ mv 2 wx (L , t)wt (L , t) 2 L L γ mv 2 2 − wt (L , t) − γ c (wt + vwx ) d x + γ (wt + vwx ) f d x (9.27) 2

V3t (t) =

0

0

According to Eq. (9.13), we have V4t (t) = B1 + B2

(9.28)

L L L where B1 = λm 0 xwx wtt d x + λm 0 xwt wxt d x, B2 = λm 0 axwx2 d x + L 2λm 0 xvwx wxt d x. Substituting (9.7) into B1 and then integrating by parts, we have: λT L − λm Lv 2 2 λm L 2 wx (L , t) + wt (L , t) − 2λm B1 = 2 2

L xvwx wxt d x 0

λm + 2

L

L v

2

wx2 d x−λc

0

−

λT 2

L xwt wx d x − λm

0

L 0

λm − 2

0

L wx2 d x − λc

axwx2 d x

L wt2 d x 0

L xvwx2 d x + λ

0

xwx f d x

(9.29)

0

Substituting B2 and (9.29) into (9.28), we have λm L 2 λT L − λm Lv 2 2 wx (L , t) + wt (L , t) − λc V4t (t) = 2 2

L xwx (wt + vwx )d x 0

λT − 2

L wx2 d x 0

λm − 2

L wt2 d x 0

λm + 2

L

L v

2

wx2 d x

+λ

0

xwx f d x 0

(9.30) Substituting (9.10), (9.27), and (9.30) into (9.23), we obtain   (γ v + λL) T − mv 2 2 γ T v − γ mv 3 2 wx (L , t) − wx (0, t) V1t (t) = 2 2

136

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

 λm γ mv − λm L  2 z e + z v2 − − m(γ v − λL)z e z v − 2 2

L

2

wt d x 0

  λT − γ mv 2 − γ T (z e + z v )wx (L , t) − 2

L wx2 d x 0

L

L (wt + vwx )2 d x − λc

− γc 0

λm + 2

xwx (wt + vwx )d x 0

L

L v

2

wx2 d x

+λ

0

L xwx f d x + γ

0

(wt + vwx ) f d x

(9.31)

0

According to (9.31), the virtual control z v (t) can be designed as z v = −k1 wx (L , t)

(9.32)

where k1 is a positive constant. Substituting (9.32) into (9.31) yields   γ mv − λm L 2 ze V1t (t) = − γ mv 2 + λk1 m L − γ k1 mv − γ T wx (L , t)z e − 2   mk12 (γ v − λL) + T − mv 2 (2γ k1 − γ v − λL) 2 wx (L , t) − 2 L γ T v − γ mv 3 2 wx (0, t) − λc xwx (wt + vwx )d x − 2 0

λm − 2

L

L wt2 d x

− γc

0

−

λT 2

0

L

v 2 wx2 d x

L (wt + vwx ) f d x + λ

0

L 0

L wx2 d x + γ

0

λm (wt + vwx ) d x + 2 2

xwx f d x

(9.33)

0

9.3.2 Step Two In this step, the actual control U (t) is designed to regulate z e (t) in a small neighborhood of the origin. Combining (9.32) and the last equation of (9.19), and then differentiating (9.10) with respect to time results in:

9.3 Boundary Controller Design

z et =

137

1 [U + d − ds wt (L , t) − T wx (L , t)] + k1 wxt (L , t) mc

(9.34)

ˆ dˆs , m, Let d, ˆ mˆ c and Tˆ be the estimates of d, ds , m, m c and T respectively, then the following vectors are defined as ⎧ θ (t) = [m, m c , ds , T ]T ⎪ ⎪ ⎪ ⎪

T ⎨ θˆ (t) = m, ˆ mˆ c , dˆs , Tˆ ⎪ ⎪

T ⎪ ⎪ ⎩ θ (t) = θ − θ = m, ˜ m˜ c , d˜s , T˜

(9.35)



Consider the following Lyapunov function candidate 1 V2 (t) = V1 (t) + m c z e2 (t) 2

(9.36)

Differentiating (9.36) with respect to time and then combining (9.33)–(9.35), we obtain γ mv − λm L 2 γ T v − γ mv 3 2 ze − wx (0, t) V2t (t) = [Qθ + U + d − ds z e ]z e − 2 2   mk12 (γ v − λL) + T − mv 2 (2γ k1 − γ v − λL) 2 wx (L , t) − 2 L L L λm 2 2 − wt d x − γ c (wt + vwx ) d x − λc xwx (wt + vwx )d x 2 0

λT − 2

0

L wx2 d x

λm + 2

0

0

L

L v

0

2

wx2 d x

+λ

L xwx f d x + γ

0

(wt + vwx ) f d x 0

(9.37) where the vector Q(t) is defined as Q(t) =

   k1 γ v − γ v 2 − k1 λL wx (L , t), k1 wxt (L , t), k1 wx (L , t), (γ − 1)wx (L , t) (9.38)

According to (9.37), the actual control U (t) can be designed as U (t) = −Q θˆ − k2 z e − dˆ where k2 is a positive constant.

(9.39)

138

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

The disturbance observer dˆt (t) is defined as ˆ = φ(t)+k3 m c wt (L , t), d(t) φt (t) = −k3 (φ(t) − ds wt (L , t) − T wx (L , t) + u(t) + k3 m c wt (L , t)),

(9.40)

where k3 is a positive constant. ˜ is defined as The disturbance estimate error d(t) d˜ = d − dˆ

(9.41)

Differentiating (9.41) and then combining (9.40), we obtain d˜t (t) = dt − k3 d˜ + k3 d

(9.42)

For the proposed control (9.39), the adaptive law can be designed as θˆt (t) = G Q T z e − ζ G θˆ

(9.43)

where G ∈ R4×4 is a diagonal positive definite matrix and ζ is a positive constant. According to (9.35) and (9.43), we have θ˜t (t) = −G Q T z e + ζ G θˆ

(9.44)

Combining (9.10) and (9.32), the actual control U (t) given by (9.39) can be rewritten as

  U (t) = − k1 k2 + dˆs k1 + Tˆ (γ − 1) + mˆ k1 γ v − γ v 2 − k1 λL wx (L , t) − k2 wt (L , t) − mˆ c k1 wxt (L , t) − dˆ

(9.45)

Remark 9.3 For the proposed control (9.45), the control design is based on the infinite-dimensional model of the system, hence the spillover instability problem is avoided. The adaptive law (9.43) is proposed to compensate for the system parametric uncertainties and the disturbance observer (9.40) is introduced to deal with the boundary disturbance d(t). The signals w(L , t) and wx (L , t) can be measured by position and inclinometer sensors at right boundary of the belt, then wt (L , t) and wxt (L , t) can be obtained through a backwards difference algorithm to w(L , t) and wx (L , t) respectively.

9.4 Stability Analysis

139

9.4 Stability Analysis Lemma 9.3 Consider the following Lyapunov function candidate 1 1 ˜ + d˜ 2 (t) V (t) = V2 (t) + θ˜ T (t)G −1 θ(t) 2 2

(9.46)

then the time derivative of the function V (t) can be upper bounded with Vt (t) ≤ −ϑ V (t) + ε

(9.47)

where ϑ and ε are both positive constants. Proof Differentiating (9.46), then combining (9.37), (9.39)–(9.44), we obtain   mk12 (γ v − λL) + T − mv 2 (2γ k1 − γ v − λL) 2 wx (L , t) Vt (t) ≤ − 2  L  λLc γ T v − γ mv 3 2 wx (0, t) − γ c − γ δ2 − − (wt + vwx )2 d x 2 δ1 0

 −

λma12 λT − − λLδ3 − λLcδ1 2 2

 L 0

−

λm 2

L

  1 k3 2 2 wx d x − k3 − − d δ5 δ4

   γ mv − λm L 2 ζ   2 ζ 2 z e − θ˜  + θ 2 wt d x − ds + k2 + 2 2 2

0

 +

δ5 dt2

+ δ4 k3 d + 2

γ λL + δ2 δ3

 L f 2d x

(9.48)

0

where δ1 , δ2 , δ3 , δ4 and δ5 are all positive constants. According to (9.21) and (9.48), the constants λ, γ , k1 , k2 , k3 , δ1 , δ2 , δ3 , δ4 and δ5 are chosen to satisfy (9.21) and the following conditions:

140

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

⎧ 2   mk1 (γ v − λL) + T − mv 2 (2γ k1 − γ v − λL) ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T − mv 2 ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ λcL ⎪ ⎪ >0 τ1 = γ c − γ δ2 − ⎪ ⎪ δ1 ⎪ ⎨ λT λma12 τ2 = − − λLδ3 − λcLδ1 > 0 ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ γ mv − λm L ⎪ ⎪ >0 τ3 = ds + k2 + ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ k 1 ⎪ ⎪ ⎩ τ4 = k3 − 3 − >0 δ4 δ5

(9.49)

Substituting (9.49) into (9.48), then combining (9.12), (9.22), and (9.36), we obtain L

L (wt + vwx ) d x − τ2

Vt (t) ≤ −τ1

2

0

wx2 d x − τ3 z e2 − τ4 d˜ 2 −

 ζ  ˜ 2 θ  + ε 2

0

 ζ  2 ≤ −ϑ1 V3 (t) − τ3 z e2 − τ4 d˜ 2 − θ˜  + ε 2  2 ζ   ≤ −ϑ2 V1 (t) − τ3 z e2 − τ4 d˜ 2 − θ˜  + ε 2  ζ  ˜ 2 2 ˜ ≤ −ϑ3 V2 (t) − τ4 d − θ  + ε 2

(9.50)

where ϑ1 = min(2τ1 /γ m, 2τ2 /γ T ),ϑ2 = (ϑ 1 /ξ2 ), ϑ3 = min(ϑ2 , 2τ3 /m c ), and L 2 ζ θ 2 +δ5 a52 +δ4 a32 + ε = max ζ2 θ 2 + δ5 dt2 + δ4 k3 d 2 + δγ2 + λL 0 f dx = 2 δ3   γ La42 . According to the properties of matrix G, we have + λL δ2 δ3   1  1   ˜ 2 1 ˜ T −1 ˜  ˜ 2 θ  ≤ θ G θ ≤ θ  2ςa 2 2ςi

(9.51)

where ςa and ςi are the maximum and minimum eigenvalues of matrix G respectively. Then we further obtain:  ζ ζ ςi T −1  2 − θ˜  ≤ − θ˜ G θ˜ 2 2

(9.52)

Substituting (9.52) into (9.50) and combining (9.46), we obtain: ζ ςi T −1 θ˜ G θ˜ + ε Vt (t) ≤ −ϑ3 V2 (t) − τ4 d˜ 2 − 2 ≤ −ϑ V (t) + ε

(9.53)

9.4 Stability Analysis

141

where ϑ = min(ϑ3 , ζ ςi , 2τ4 ). Theorem 9.1 For the belt system described by (9.7)–(9.8), under the proposed control (9.45), disturbance observer (9.40) and Assumptions 9.1–9.2, given that the initial conditions are bounded, we can conclude that: (1) uniform boundedness: the system state w(x, t) of the closed-loop belt system will remain in the compact set:

1 := { w(x, t) ∈ R||w(x, t)| ≤ χ1 }

(9.54)

 [V (0)e−ϑt + ϑε ]. where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 = γ2L ξ1 T (2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop belt system will eventually converge to the compact set:

2 := { w(x, t) ∈ R| lim |w(x, t)| ≤ χ2 } t→∞

where ∀(x, t) ∈ [0, L] × [0, +∞), χ2 =



2Lε γ ξ1 ϑ T

(9.55)

.

Proof Multiplying Eq. (9.53) by eϑt yields: Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒

 ∂

V (t)eϑt ≤ εeϑt ∂t

(9.56)

Integrating the above inequality yields:

ε  −ϑt ε ε e V (t) ≤ V (0) − + ≤ V (0)e−ϑt + ϑ ϑ ϑ

(9.57)

Applying Eqs. (2.11)–(2.15) and Eq. (9.19), we have: γT 2 γT w ≤ 2L 2

L wx2 d x ≤ V3 (t) 0

1 ˜2 1 1 T −1 1 mc 2 θ˜ G θ˜ + ≤ V1 (t) + ze + d = V (t) ξ1 2ξ1 2ξ1 2ξ1 ξ1

(9.58)

142

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

Substituting Eq. (9.57) into Eq. (9.58) results in:  |w(x, t)| ≤

2L ε ≤ V (0)e−ϑt + γ T ξ1 ϑ



ε 2L V (0) + γ T ξ1 ϑ

(9.59)

where ∀(x, t) ∈ [0, L] × [0, +∞), from above inequality, we have:  lim |w(x, t)| ≤ lim

t→∞

t→∞

2L ε = V (0)e−ϑt + γ T ξ1 ϑ

 2Lε γ T ξ1 ϑ

(9.60)

where ∀x ∈ [0, L]. Remark 9.4 From (9.49), (9.50), (9.53), and (9.60), the √ increase of k2 and k3 will eventually yield a larger ϑ, which will reduce the size of 2Lε/λT ξ1 ϑ and bring a better vibration control performance. However, the increase of k2 and k3 will bring a high-gain control. Hence, in practice, the design parameters should be adjusted carefully for achieving suitable transient performance and control action. Remark 9.5 From (9.57) and (9.58), we can obtain that w(x, t) is bounded  are all bounded ∀t ∈ ∀(x, t) ∈ [0, L] × [0, +∞) and V3 (t), z e (t),  θ (t), d(t) [0, +∞). Since V3 (t) is bounded, the potential energy (9.3) of the belt system is bounded ∀t ∈ [0, +∞), and then, wx (x, t) and wx x (x, t) are also bounded ∀(x, t) ∈ [0, L] × [0, +∞). According to (9.9)–(9.10) and (9.32), we have z e (t) = wt (x, t) + k1 wx (x, t), and then we can obtain wt (x, t) is also bounded ∀t ∈ [0, +∞). Since V3 (t) and wt (x, t) are bounded, the kinetic energy (9.2) of the belt system is bounded ∀t ∈ [0, +∞). Then we have wt (x, t) and wxt (x, t) are bounded ∀(x, t) ∈ [0, L] × [0, +∞). According to (9.39), the boundedness of ed (t) and Assumption 9.1, we obtain de (t) is bounded ∀t ∈ [0, +∞). Finally, because de (t) is bounded ∀t ∈ [0, +∞) and wx (x, t), wt (x, t), wxt (x, t) are bounded ∀(x, t) ∈ [0, L] × [0, +∞), we can conclude that the proposed actual control (9.44) is bounded ∀t ∈ [0, +∞).

9.5 Simulation Example Considering the PDE model as Eq. (9.7), the parameters of the belt system are given in Table 9.1 and the initial conditions are v0 = w(x, 0) = wt (x, 0) = 0. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D are given as: aa = ad = 3.5 g and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s (g = 9.8 N/kg). The unknown distributed disturbance f (x, t) is described as:   3  1 sin(iπ xt) f (x, t) = 10x + 1000 i=1

(9.61)

9.5 Simulation Example

143

Table 9.1. Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

Value

m

1.0 kg/m

mc

5.0 kg

L

1.0 m

c

1.0

ds

0.25 Ns/m

γ

0.1

ζ

1.0

T

4900 N

λ

0.001

k1

100

k2

100

k3

10

Ns/m2

Fig. 9.1. Vibration displacement of the belt without control

The unknown time-varying boundary disturbance d(t) is described as: d(t) = 10 +

3 

cos(it)

(9.62)

i=1

From Fig. 9.1, it can be seen that there is a large vibration displacement along the belt for the uncontrolled case. From Fig. 9.2, it can also be seen that the proposed adaptive control (9.45) can suppress the vibrations of the belt greatly when the corresponding adaptive law (9.43) is applied. As shown in Fig. 9.3, the vibration displacement of the belt is examined at x = 1 m and x = 0.5 m, and the simulation results for controlled and uncontrolled responses are shown. And Fig. 9.4 shows the enlarged view of the vibration displacement for controlled response at x = 1 m and x = 0.5 m. In Figs. 9.5 and 9.6, the trajectory tracking of the boundary disturbance d(t) and the corresponding control input of the proposed boundary control U (t) are shown respectively.

144

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

Fig. 9.2. Vibration displacement of the belt with control

Fig. 9.3. Vibration displacement of the belt at: ax = 1 m, b x = 0.5 m

9.5 Simulation Example

Fig. 9.4. The enlarged view of the vibration displacement for controlled response

Fig. 9.5. Tracking of boundary disturbance

145

146

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

Fig. 9.6. Control input U (t)

Appendix 1: Simulation Program close all; clear all; clc; nx=50; nt=100*10ˆ3; L= 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m=1; T = 4900; pi = 3.1415926; V0=0; c = 1; % --------------velocity----------------V = zeros(1,nt); V(1,1) = V0; J=3.5*9.8; for j=2:10*10ˆ3-1 V(1,j) = V(1,1)+0.5*J*(j*dt)ˆ2; end V(1,10*10ˆ3)=V(1,1)+0.5*J;

Appendix 1: Simulation Program

147

for j=(10*10ˆ3+1):20*10ˆ3-1 V(1,j)=V(1,10*10ˆ3)+0.5*J*(j*dt-1); end V(1,20*10ˆ3)=V(1,10*10ˆ3)+J; for j=(20*10ˆ3+1):30*10ˆ3-1 V(1,j) = V(1,20*10ˆ3)+J*(j*dt-2)-0.5*J*(j*dt-2)ˆ2; end V(1,30*10ˆ3)=V(1,20*10ˆ3)+0.5*J; for j=(30*10ˆ3+1):70*10ˆ3-1 V(1,j)=V(1,30*10ˆ3); end V(1,70*10ˆ3)=V(1,30*10ˆ3); for j=(70*10ˆ3+1):80*10ˆ3-1 V(1,j)=V(1,70*10ˆ3)-0.5*J*(j*dt-7)ˆ2; end V(1,80*10ˆ3)=V(1,70*10ˆ3)-0.5*J; for j=(80*10ˆ3+1):90*10ˆ3-1 V(1,j)=V(1,80*10ˆ3)-J*(j*dt-8); end V(1,90*10ˆ3)=V(1,80*10ˆ3)-J; for j=(90*10ˆ3+1):nt-1 V(1,j)=V(1,90*10ˆ3)-J*(j*dt-9)+0.5*J*(j*dt-9)ˆ2; end V(1,100*10ˆ3)=V(1,90*10ˆ3)-0.5*J; % -------------disburtance------------d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt d(1,j)=10+cos(1*(j-1)*dt)+cos(2*(j-1)*dt)+cos(3*(j-1)*dt); end for j = 1 : nt for i = 1 : nx f(i,j) = 0.001*(10*(i-1)*dx+sin(pi*(i-1)*dx*(j1)*dt)+sin(2*pi*(i-1)*dx*(j-1)*dt)+sin(3*pi*(i-1)*dx*(j1)*dt)); end end %**************************************** % uncontrolled %**************************************** ds=0.25; mc=5; w1 = zeros(nx,nt); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((T-m*V(1,j-1)ˆ2)*(w1(i+1,j1)-2*w1(i,j-1)+w1(i-1,j-1))/dxˆ2 -c*(w1(i,j-1)-w1(i,j-2))/dt c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt) -m*(V(1,j1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j) ) / (m/dtˆ2);

148

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

end w1(nx,j) = ( (2*mc/dtˆ2 + ds/dt )*w1(nx,j-1) - (mc/dtˆ2)*w1(nx,j2) + d(1,j) + (T/dx)*w1(nx-1,j-1) )/ (mc/dtˆ2 + ds/dt + T/dx); end tshort = linspace(0,tmax,(nt/400+1)); xshort = linspace(0,L,nx); for j=1:nt/400 for i=1:nx w1short(i,j)=w1(i,j*400); end end w1short=[w1(:,1),w1short]; figure (1); surf(tshort,xshort,w1short); colormap(’jet’) ; xlabel(’Time [s]’,’Fontsize’,14); ylabel(’x [m]’,’Fontsize’,14); zlabel(’w (x,t) [m]’,’Fontsize’,14); set(gca,’YDir’,’reverse’) %**************************************** % backstepping adaptive + controlled + disturbance observer %**************************************** ds = 0.25; mc = 5; k1 = 100; k2 = 100; k3 = 10; gamma = 0.1; lambda = 0.001; G=diag([1 1 1 1]); w2 = zeros(nx,nt); u1 = zeros(1,nt); z_e = zeros(1,nt); d_e1 = zeros(1,nt); phi_1 = zeros(1,nt); error_1 = zeros(1,nt); T_e = zeros(1,nt); mc_e = zeros(1,nt); m_e = zeros(1,nt); ds_e = zeros(1,nt); err_T = zeros(1,nt); err_mc = zeros(1,nt); err_ds = zeros(1,nt); err_m = zeros(1,nt); zeta=1; for j = 1 : 2 T_e(1,j) = 0; mc_e(1,j) = 0; ds_e(1,j) = 0; m_e(1,j) = 0; end w2(1,:) = 0; for i=2:nx w2(i,1) = 0;

Appendix 1: Simulation Program

149

w2(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((T-m*V(1,j-1)ˆ2)*(w2(i+1,j1)-2*w2(i,j-1)+w2(i-1,j-1))/dxˆ2 -c*(w2(i,j-1)-w2(i,j-2))/dt c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j) ) / (m/dtˆ2); end w2(nx,j) = 2*w2(nx,j-1)-w2(nx,j-2)+dtˆ2/mc*(u1(1,j-1)+d(1,j)T*(w2(nx,j-1)-w2(nx-1,j-1))/dx-ds*(w2(nx,j-1)-w2(nx,j-2))/dt); z_e(1,j) = (w2(nx,j)-w2(nx,j-1))/dt + k1*(w2(nx,j) - w2(nx1,j))/dx ; m_e(1,j) = m_e(1,j-1)+ dt*(-zeta*G(1,1)*m_e(1,j1) + z_e(1,j)*G(1,1)*(k1*gamma*V(1,j)-gamma*V(1,j)ˆ2k1*lambda*L)*(w2(nx,j)-w2(nx-1,j))/dx ); mc_e(1,j) = mc_e(1,j-1)+dt*(-zeta*G(2,2)*mc_e(1,j1)+z_e(1,j)*G(2,2)*k1*((w2(nx,j) - w2(nx-1,j) - w2(nx,j-1) + w2(nx-1,j-1))/(dx*dt))); ds_e(1,j) = ds_e(1,j-1) +dt*(-zeta*G(3,3)*ds_e(1,j1)+z_e(1,j)*G(3,3)*k1*(w2(nx,j) - w2(nx-1,j))/dx); T_e(1,j) = T_e(1,j-1) +dt*(-zeta*G(4,4)*T_e(1,j1)+z_e(1,j)*G(4,4)*(gamma-1)*(w2(nx,j)-w2(nx-1,j))/dx ); err_T = T - T_e(1,j); err_mc = mc - mc_e(1,j); err_m = m - m_e(1,j); err_ds = ds - ds_e(1,j); u1(1,j) = -d_e1(1,j-1) - ds_e(1,j)*k1*(w2(nx,j)-w2(nx1,j))/dx -(gamma-1)*T_e(1,j)*(w2(nx,j)-w2(nx-1,j))/dxmc_e(1,j)*k1*(w2(nx,j) -w2(nx-1,j)-w2(nx,j-1)+w2(nx-1,j1))/(dt*dx)+m_e(1,j)*( gamma*V(1,j)ˆ2 - k1*gamma*V(1,j) + k1*lambda*L)*(w2(nx,j)-w2(nx-1,j))/dx -k2*(w2(nx,j)-w2(nx,j1))/dt - k1*k2*(w2(nx,j)-w2(nx-1,j))/dx; phi_1(1,j) = phi_1(1,j-1) - dt*(k3*phi_1(1,j-1) k3*ds*(w2(nx,j)-w2(nx,j-1))/dt - k3*T*(w2(nx,j)-w2(nx-1,j))/dx + k3*u1(1,j) + (k3)ˆ2*mc*(w2(nx,j)-w2(nx,j-1))/dt ); d_e1(1,j) = phi_1(1,j) + k3*mc*(w2(nx,j) - w2(nx,j-1))/dt; error_1(1,j) = d(1,j) - d_e1(1,j); end for j=1:nt/400 for i=1:nx w2short(i,j)=w2(i,j*400); end end w2short=[w2(:,1),w2short]; figure (2); surf(tshort,xshort,w2short); colormap(’jet’) ; xlabel(’Time [s]’,’Fontsize’,14); ylabel(’x [m]’,’Fontsize’,14); zlabel(’w(x,t) [m]’,’Fontsize’,14); set(gca,’YDir’,’reverse’)

150

9 Stabilization of an Axially Moving Accelerated/Decelerated System …

figure (3); hold on subplot(2,1,1); plot(tshort,w1short(50,:),’b’,tshort,w2short(50,:),’:r’); title(’(a)’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’w(1,t) [m]’,’Fontsize’,14); legend(’uncontrolled’,’controlled’); box on grid on subplot(2,1,2); plot(tshort,w1short(25,:),’b’,tshort,w2short(25,:),’:r’); title(’(b)’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’w(0.5,t) [m]’,’Fontsize’,14); legend(’uncontrolled’,’controlled’); box on grid on hold off figure (4); plot(tshort,w2short(50,:),’:r’,tshort,w2short(25,:),’b’); legend(’x = 1m’,’x = 0.5m’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’w(x,t) [m]’,’Fontsize’,14); figure(5); plot(t,d_e1,’b’,t,d,’--r’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’d(t) [N]’,’Fontsize’,14); legend(’estimate of disturbance’,’actual disturbance’); box on grid on figure(6); plot(t,u1,’b’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’U(t) [N]’,’Fontsize’,14); box on grid on

References 1. K.J. Yang, K.S. Hong, F. Matsuno, Boundary control of a translating tensioned beams with varying speed. IEEE/ASME Trans. Mechatron. 10(5), 594–597 (2005) 2. W. He, S.S. Ge, D. Huang, Modeling and vibration control for a nonlinear moving string with output constraint. IEEE/ASME Trans. Mechatron. 20(4), 1886–1897 (2014) 3. Y. Liu, B.S. Xu, Y.L. Wu, et al., Boundary control of an axially moving belt, in Proceeding of the 32th Chinese Control Conference, (2013), pp. 1310–1315 4. H. Goldstein, Classical Mechanics (Addison-Wesley, Massachusetts, 1951)

Chapter 10

Adaptive Output Feedback Boundary Control for a Class of Axially Moving System

10.1 Introduction Based on Lyapunov’s direct method, the active boundary control of axially moving systems has made great progress in recent years [1–5]. Among the literatures, it is worth noting that the control design is restricted to assuming that all boundary information can be accurately measured, especially some control signals need to be obtained by the differential method. From the point of view of practical engineering, the effects of noise constraints exist in almost all the systems due to the inherent physical constraints of the systems. By using the differential method, this adverse effect will be further amplified. To solve this problem, we apply some high-gain observers in this chapter to estimate the unmeasured boundary state signals. In this chapter, the main concern lies in the development of a boundary control scheme for suppressing the vibration of an axially moving system under the influence of unknown disturbances. To that end, a boundary adaptive output feedback control is presented by merging Lyapunov theory, observer backstepping technique, highgain observers, and robust adaptive control theory to globally stabilize the vibration and compensate for the uncertainties of the system. Besides, the disturbance observer dynamics is developed to handle unknown boundary disturbance and the σ-modification is adopted to adjust the robustness of the system. Under the designed control, the uniformly bound stability of the controlled system is achieved, and the state observer error exponentially converges to zero as time tends to infinity.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_10

151

152

10 Adaptive Output Feedback Boundary Control for a Class …

10.2 PDE Dynamic Model The belt system dynamics of this paper is presented in Chap. 7 in the following: mwtt + mawx + 2mvwxt + mv 2 wx x − T wx x + cwt + cvwx = f

(10.1)

where ∀(x, t) ∈ (0, L) × [0, +∞). The corresponding boundary conditions are expressed as: 

w(0, t) = 0 m c wtt (L , t) + T wx (L , t) + βwt (L , t) = d + U

(10.2)

where ∀t ∈ [0, +∞). Assumption 10.1 For the speed v(t), H-A/D a(t), boundary disturbance d(t) and distributed disturbance f (x, t), we assume that there exist constants a1 , a2 , a3 , a4 ∈ R+ , such that 0 < v(t) ≤ a1 , | a(t) | ≤ a2 , | d(t) | ≤ a3 , ∀t ∈ [0, +∞) and | f (x, t) | ≤ a4 , ∀(x, t) ∈ (0, L) × [0, +∞).

10.3 Boundary Controller Design Before proceeding further, we conduct the following transformation of coordinates:

z 2t =

z 1 = w(L , t)

(10.3)

z 1t = z 2 = wt (L , t)

(10.4)

1 [U + d − βz 2 − T wx (L , t)] mc

(10.5)

As shown in (10.3)–(10.5), it is worth pointing out that the system state z 2 = wt (L , t) available for feedback is obtained by applying the backward difference algorithm to the boundary signal w(L , t), which is measured by a displacement sensor [6]. However, in practice, the measurement noise will be amplified in the backward differential process and then the value of the state wt (L , t) will be more inaccurate. To resolve this problem, the observer backstepping technique [7] can be employed to refactor system states available for feedback and guarantee both boundedness of the closed-loop states and convergence of the state errors to zero. Then we estimate the state z 2 of (10.5) by zˆ 2 : zˆ 2t =

 1  U + d − β zˆ 2 − T wx (L , t) mc

(10.6)

10.3 Boundary Controller Design

153

The state observer error z˜ 2 is defined as z˜ 2 = z 2 − zˆ 2

(10.7)

Differentiating (10.7), and substituting (10.5) and (10.6), we obtain z˜ 2t = −

β z˜ 2 mc

(10.8)

From (10.8), we further have β

z˜ 2 = z˜ 2 (0)e− mc

(10.9)

Combining (10.7) and (10.8), the transformed coordinates are rewritten as ⎧ z 1 = w(L , t) ⎪ ⎪ ⎪ ⎪ ⎪ z 1t = zˆ 2 + z˜ 2 ⎪ ⎪ ⎨  1  U + d − β zˆ 2 − T wx (L , t) zˆ 2t = ⎪ m ⎪ c ⎪ ⎪ ⎪ ⎪ β ⎪ ⎩ z˜ 2t = − z˜ 2 mc

(10.10)

10.3.1 Step One Choose zˆ 2 as a control to develop an adaptive boundary control scheme for vibration reduction of the belt, then we define e = zˆ 2 − α

(10.11)

where α is a virtual control of zˆ 2 , and e is the corresponding error variable. Consider the Lyapunov function candidate as V1 (t) = V3 (t) + V4 (t)

(10.12)

where the energy term V3 (t) is defined as: γm V3 (t) = 2

L

γT (wt + vwx ) d x + 2

L wx2 d x

2

0

and a small crossing term V4 (t) is defined as:

0

(10.13)

154

10 Adaptive Output Feedback Boundary Control for a Class …

L xwx (wt + vwx )d x

V4 (t) = λm

(10.14)

0

where γ and λ are both positive weighting constants. Lemma 10.1 For the belt system described by (10.1)–(10.2) and the Lyapunov function candidate V1 (t) given by (10.12), we can obtain that the function V1 (t) is a positive definite function, and it can be upper and lower bounded as: −ξ1 V3 (t) ≤ V1 (t) ≤ ξ2 V3 (t)

(10.15)

where ξ1 and ξ2 are two positive constants. Proof Applying Eqs. (2.11)–(2.15) to Eq. (10.14) results in: |V4 (t)| ≤

λm L 2

L

 2  wx + (wt + vwx )2 d x ≤ ξ V3 (t)

(10.16)

0

where ξ =

λm L . min(γ m,γ T )

Then the above inequality can be rewritten as: −ξ V3 (t) ≤ V4 (t) ≤ ξ V3 (t)

(10.17)

Choosing ξ properly, we can obtain 

ξ1 = 1 − ξ > 0 ξ2 = 1 + ξ > 1

(10.18)

Equation (10.18) indicates 0 < ξ < 1, then we have λ
0. Substituting control Eq. (10.1) into V4 (t) and then integrating by parts, we have: λm Lv 2 − λT L 2 λm λm L 2 wt (L , t) − wx (L , t) − V4t (t) ≤ 2 2 2

L wt2 d x 0

−

λma12 λT − − λLδ2 − λcLδ3 2 2

L wx2 d x 0

+

λcL δ3

L (wt + vwx )2 d x + 0

L

λL δ2

f 2d x

(10.23)

0

Substituting (10.22)–(10.23) into (10.21), we obtain λm L − γ mv 2 γ T v + λT L − γ mv 3 − λm Lv 2 2 wx (L , t) + z2 2 2 

γ T v − γ mv 3 2 wx (0, t) + γ T − γ mv 2 wx (L , t)z 2 − 2

L L λma12 λm λT 2 − − λLδ2 − λcLδ3 wx d x − wt2 d x − 2 2 2

V1t (t) ≤

0

0

L

L λcL γ λL 2 + f 2d x − γ c − γ δ1 − (wt + vwx ) d x + δ3 δ1 δ2 0

0

(10.24) Combining (10.7) and (10.11), we derive z 2 = e + α + z˜ 2

(10.25)

156

10 Adaptive Output Feedback Boundary Control for a Class …

Substituting (10.25) into (10.24), we obtain γ T v + λT L − γ mv 3 − λm Lv 2 2 γ T v − γ mv 3 2 wx (0, t) + wx (L , t) 2 2

L λm L − γ mv λcL 2 + (wt + vwx )2 d x (e + α + z˜ 2 ) − γ c − γ δ1 − 2 δ3

V1t (t) ≤ −

0

−

λma12 λT − − λLδ2 − λcLδ3 2 2

L

wx2 d x

+

0

−

λm 2

L

γ λL + δ1 δ2

L f 2d x 0



wt2 d x + γ T − γ mv 2 wx (L , t)(e + α + z˜ 2 )

(10.26)

0

To stabilize the presented system, from (10.26), the virtual control α is designed as α = −k1 wx (L , t)

(10.27)

where k1 > 0. Substituting (10.27) into (10.26) results in 

 λm L − γ mv 2 e + z˜ 22 + 2e˜z 2 + γ T − γ mv 2 wx (L , t)(e + z˜ 2 ) 2  

 L T − mv 2 (γ v + λL) mk12 (λL − γ v) λm 2 + wt2 d x + wx (L , t) − 2 2 2

V1t (t) ≤

0

−

λma12 λT − − λLδ2 − λcLδ3 2 2

L wx2 d x −

γ T v − γ mv 3 2 wx (0, t) 2

0

L

L λcL γ λL 2 + f 2d x − γ c − γ δ1 − (wt + vwx ) d x + δ3 δ1 δ2 0

− mk1 (λL − γ v)wx (L , t)˜z 2 − mk1 (λL − γ v)wx (L , t)e

0

(10.28)

10.3.2 Step Two In this step, a boundary adaptive control U (t) is developed to stabilize e at a small neighborhood of the origin.

10.3 Boundary Controller Design

157

Differentiating (10.11) and then combining (10.6) and (10.27), we have et =

 1  U + d − β zˆ 2 − T wx (L , t) + k1 wxt (L , t) mc

(10.29)

Let mˆ and Tˆ be the estimates of m and T respectively, then the following vectors are defined as  T˜ = Tˆ − T (10.30) m˜ = mˆ − m Consider the following Lyapunov function candidate γ1 γ2 1 ζ V2 (t) = V1 (t) + m c e2 (t) + z˜ 22 + T˜ 2 + m˜ 2 2 2 2 2

(10.31)

where ζ, γ1 , γ2 > 0. Differentiating (10.31) yields: V2t = V1t + m c eet + ζ z˜ 2 z˜ 2t + γ1 T˜ T˜t + γ2 m˜ m˜ t

(10.32)

Substituting (10.8), (10.28), (10.29) into (10.32) results in   γ T v − γ mv 3 2 wx (0, t) V2t (t) ≤ U + d − β zˆ 2 − T wx (L , t) + m c k1 z xt (L , t) e − 2 

 λm L − γ mv 2 e + z˜ 22 + 2e˜z 2 + γ T − γ mv 2 wx (L , t)(e + z˜ 2 ) + 2  

 2 T − mv 2 (γ v + λL) mk1 (λL − γ v) ζβ 2 + z˜ + wx2 (L , t) − 2 2 mc 2 −

λma12 λT − − λLδ2 − λcLδ3 2 2

L wx2 d x 0

λm − 2

L wt2 d x 0

L

L λcL γ λL 2 + f 2d x − γ c − γ δ1 − (wt + vwx ) d x + δ3 δ1 δ2 0

0

− mk1 (λL − γ v)wx (L , t)˜z 2 − mk1 (λL − γ v)wx (L , t)e

 − γ k1 T − mv 2 wx2 (L , t)+γ1 T˜ T˜t + γ2 m˜ m˜ t

(10.33)

According to (10.33), the actual control U (t) is developed as U (t) = −k2 e + Tˆ (1 − γ )wx (L , t) − m c k1 wxt (L , t) 

+mˆ γ v 2 + λLk1 − γ vk1 wx (L , t) − dˆ + β zˆ 2

(10.34)

158

10 Adaptive Output Feedback Boundary Control for a Class …

where k2 > 0, dˆ is the estimate of d and the disturbance observer dt is proposed as dˆt = e − k3 dˆ

(10.35)

where k3 > 0. The disturbance estimate error d˜ is defined as d˜ = dˆ − d

(10.36)

Taking the derivative of (10.36) and combining (10.35), we further have   d˜t = e − k3 d˜ + d − dt

(10.37)

For the proposed control (10.34), the adaptive laws are designed as 

Tˆt = −γ1−1 (1 − γ )wx (L , t)e − γ1−1 σ1 Tˆ

 mˆ t = −γ2−1 γ v 2 + k1 λL − k1 γ v wx (L , t)e − γ2−1 σ2 mˆ

(10.38)

where σ1 , σ2 > 0 are σ-modification parameters. Combining (10.11) and (10.27), the actual control U (t) described by (10.34) is rewritten as: U (t) = −(k2 − β)ˆz 2 − k1 k2 wx (L , t) − dˆ − m c k1 wxt (L , t) 

+ Tˆ (1 − γ )wx (L , t)+mˆ γ v 2 + λLk1 − γ vk1 wx (L , t)

(10.39)

Remark 10.1 In (10.39), the signal wx (L , t) can be measured by employing an inclinometer and then the signal wxt (L , t) can be obtained by applying the backward difference algorithm to wx (L , t). However, the measurement noise will be magnified in the backward differential process to obtain wxt (L , t), which will affect the designed control (10.39) implementation. Therefore, to resolve this difficulty, the high-gain observers are adopted to estimate the unmeasurable system state wxt (L , t). According to the lemma in [8], it is evident that πk+1 /κ k converges to x1(k) , which is the k-th derivative of x1 , namely, ξk converges to zero due to the high gain 1/κ when x1 and its k-th derivatives are bounded. Therefore, it is reasonable to choose πk+1 /κ k as an observer to estimate the output signals up to the n-th order derivative. Define x1 = wx (L , t) and x2 = wxt (L , t). The observer for the presented belt system is discussed with n = 2 and then the estimate of the state x2 is designed as xˆ2 =

π2 κ

(10.40)

where the dynamics of π2 (t) and the error  x2 (t) are respectively designed as:

10.4 Stability Analysis

159

⎧ ⎪ ⎨ κπ1t = π2 κπ2t = −λ1 π2 − π1 + x1 ⎪ ⎩ x˜2 = xˆ2 − x2

(10.41)

Based on the above analysis, the proposed control (10.34) is updated as U (t) = −k2 e − dˆ + β zˆ 2 − m c k1 xˆ2 + Tˆ (1 − γ )x1 

+mˆ γ v 2 + λLk1 − γ vk1 wx (L , t)

(10.42)

Substituting (10.38) and (10.42) into (10.33), then using Eqs. (2.11)–(2.15), we obtain   |λm L − γ mv| λm L − γ mv − − m c k 1 δ6 e 2 V2t (t) ≤ − k2 − β − 2 δ4  

2

 mk1 (λL − γ v) + T − mv 2 (γ v + λL) 2 − γ k1 T − mv − 2  

  2  ζβ 2   − γ T − mv − mk1 (λL − γ v) δ5 x1 − − |λm L − γ mv|δ4 mc   

 γ T − mv 2 − mk1 (λL − γ v) λm L − γ mv − − z˜ 22 δ5 2 σ1 ˜ 2 σ2 2 σ1 2 σ2 2 m c k1 2 x˜ − ed˜ T − m˜ + T + m + 2 2 2 2 δ6 2

L λcL γ T v − γ mv 3 2 wx (0, t) − γ c − γ δ1 − − (wt + vwx )2 d x 2 δ3

−

0

λm − 2



L wt2 d x

−

λma12 λT − − λLδ2 − λcLδ3 2 2

0

+

λL γ + δ1 δ2

L wx2 d x 0

L f 2d x 0

where δ4 , δ5 , δ6 > 0.

10.4 Stability Analysis Lemma 10.2 Consider the following Lyapunov function candidate

(10.43)

160

10 Adaptive Output Feedback Boundary Control for a Class …

1 V (t) = V2 (t) + d˜ 2 2

(10.44)

then the time derivative of the function V (t) can be upper bounded with Vt (t) ≤ −ϑ V (t) + ε

(10.45)

where ϑ and ε are both positive constants. Proof According to (10.12), (10.20), (10.31) and (10.44), we can derive that V (t) is upper and lower bounded as  0 < ϑ1 V3 (t) +  ≤ ϑ2 V3 (t) +

1 m c e2 (t) + 2 1 m c e2 (t) + 2

ζ 2 w˜ + 2 2 ζ 2 w˜ + 2 2

γ1 ˜ 2 T + 2 γ1 ˜ 2 T + 2

 γ2 2 m˜ ≤ V (t) 2  γ2 2 m˜ 2

(10.46)

where ϑ1 = min(ξ1 , 1) > 0 and ϑ2 = max(ξ2 , 1) > 0. Differentiating (10.44), substituting (10.37) and (10.43), and then using Eqs. (2.11)–(2.15), we have   |λm L − γ mv| λm L − γ mv Vt (t) ≤ − k2 − β − − m c k 1 δ6 e 2 − 2 δ4  



 mk12 (λL − γ v) + T − mv 2 (γ v + λL) 2 − γ k1 T − mv − 2  

   ζβ −γ T − mv 2 − mk1 (λL − γ v)δ5 x12 − − |λm L − γ mv|δ4 mc   

 γ T − mv 2 − mk1 (λL − γ v) λm L − γ mv − − z˜ 22 δ5 2 γ T v − γ mv 3 2 σ1 σ2 σ1 σ2 wx (0, t) − T˜ 2 − m˜ 2 + T 2 + m 2 2 2 2 2 2

1 ˜ 2 m c k1 2 k3 2 2 − k3 − δ7 k3 − x˜ + d + δ8 dt d + δ8 δ6 2 δ7

L

L λcL γ λL 2 − γ c − γ δ1 − + f 2d x (wt + vwx ) d x + δ3 δ1 δ2 −

0

−

λm 2

wt2 d x − 0

where δ7 , δ8 > 0.



L

λma12 λT − − λLδ2 − λcLδ3 2 2

0

L wx2 d x 0

(10.47)

10.4 Stability Analysis

161

According to (10.18) and (10.47), the positive constants k1 ∼ k3 , λ, γ , ζ and δ1 ∼ δ8 should be chosen to satisfy the following conditions: ⎧ min(γ T, γ m) ⎪ ⎪ ⎪λ < ⎪ mL ⎪ ⎪ ⎪ ⎪ ⎪ τ = T − mv 2 ≥ 0 ⎪ ⎪ 1 

⎪ ⎪ ⎪

 mk12 (λL − γ v) + T − mv 2 (γ v + λL) ⎪ 2 ⎪ ⎪ τ2 = γ k1 T − mv − ⎪ ⎪ 2 ⎪ 

 ⎪  ⎪ γ T − mv 2 − mk1 (λL − γ v)δ5 ≥ 0 ⎪ − ⎪ ⎪ ⎪ ⎪ ⎪ λcL ⎪ ⎪ >0 ⎪ τ3 = γ c − γ δ1 − ⎪ ⎪ δ3 ⎪ ⎨ λma12 λT − − λLδ2 − λcLδ3 τ4 = ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ |λm L − γ mv| λm L − γ mv ⎪ ⎪ − τ5 = k2 − β − − m c k 1 δ6 ⎪ ⎪ 2 δ4 ⎪ ⎪  

 ⎪ ⎪ γ T − mv 2 − mk1 (λL − γ v) ⎪ ζβ ⎪ ⎪ ⎪ τ = − 6 ⎪ ⎪ mc δ5 ⎪ ⎪ ⎪ ⎪ λm L − γ mv ⎪ ⎪ >0 − |λm L − γ mv|δ4 − ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ τ7 = k3 − δ7 k3 − >0 δ8

(10.48)

Substituting (10.48) into (10.47) yields: L

L (wt + vwx ) d x − τ4

Vt (t) ≤ −τ3

wx2 d x − τ5 e2

2

0

0

σ1 σ2 − − τ7 d˜ 2 − T˜ 2 − m˜ 2 + ε 2 2   γ1 γ2 1 1 ζ ≤ −ϑ3 V3 (t) + m c e2 (t) + z˜ 22 + T˜ 2 + m˜ 2 + d˜ 2 + ε 2 2 2 2 2 τ6 z˜ 22

  where ε = δγ1 + λL La52 + δ8 a42 + kδ73 a32 + mδc6k1 x˜22 + δ2   2τ6 σ1 σ2 5 ϑ3 = min γ2τm3 , γ2τT4 , 2τ , , , , 2τ 7 . mc ζ γ1 γ2 Combining (10.46) and (10.49), we further have Vt (t) ≤ −ϑ V (t) + ε where ϑ = ϑ3 /ϑ2 .

σ1 2 T 2

+

σ2 2 m 2

(10.49)

< +∞ and

(10.50)

162

10 Adaptive Output Feedback Boundary Control for a Class …

Lemma 10.3 For the belt system described by (10.1)–(10.2), under the proposed control (10.42) and Assumptions 10.1–10.2, given that the initial conditions are bounded, we can conclude that: (1) uniform boundedness: the system state w(x, t) of the closed-loop belt system will remain in the compact set: 1 := { w(x, t) ∈ R||w(x, t)| ≤ χ1 }

(10.51)

 2L where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 = γ ϑξ [V (0)e−ϑt + ϑε ]. 1T (2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop belt system will eventually converge to the compact set:   2 := w(x, t) ∈ R| lim |w(x, t)| ≤ χ2 t→∞

where ∀(x, t) ∈ [0, L] × [0, +∞), χ2 =



2Lε γ ϑ1 ϑ T

(10.52)

.

Proof Multiplying Eq. (10.52) by eϑt yields:  ∂ V (t)eϑt ≤ εeϑt ∂t

(10.53)

 ε  −ϑt ε ε e V (t) ≤ V (0) − + ≤ V (0)e−ϑt + ϑ ϑ ϑ

(10.54)

Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒ Integrating the above inequality yields:

Applying Eqs. (2.11)–(2.15) and Eq. (10.20), we have: γT 2 γT w ≤ 2L 2

L

1 V (t) ϑ1

(10.55)

ε 2L  V (0) + γ T ϑ1 ϑ

(10.56)

wx2 d x ≤ V3 (t) = 0

Substituting Eq. (10.54) into Eq. (10.55) results in:  |w(x, t)| ≤

2L  ε ≤ V (0)e−ϑt + γ T ϑ1 ϑ



where ∀(x, t) ∈ [0, L] × [0, +∞), from above inequality, we have:  lim |w(x, t)| ≤ lim

t→∞

t→∞

2L  ε = V (0)e−ϑt + γ T ϑ1 ϑ

 2Lε γ T ϑ1 ϑ

(10.57)

10.5 Simulation Example

163

where ∀(x, t) ∈ [0, L] × [0, +∞). Combining with (10.11) and (10.27), the proposed control U(t) described by (10.42) can be further updated as U (t) = −(k2 − β)ˆz 2 − k1 k2 x1 − dˆ − m c k1 xˆ2 

+ Tˆ (1 − γ )x1 + mˆ γ v 2 + λLk1 − γ vk1 x1

(10.58)

Accordingly, the disturbance observer dˆt is rewritten as dˆt = zˆ 2 + k1 x1 − k3 dˆ

(10.59)

and the adaptive laws described by (10.38) are rewritten as 

 Tˆt = −γ1−1 (1 − γ )x1 zˆ 2 + k1 x1 − γ1−1 σ1 Tˆ

 

mˆ t = −γ2−1 γ v 2 + k1 λL − k1 γ v x1 zˆ 2 + k1 x1 − γ2−1 σ2 mˆ

(10.60)

10.5 Simulation Example Considering the PDE model as Eq. (10.1), the parameters of the belt system are given in Table 10.1 and the initial conditions are v0 = w(x, 0) = wt (x, 0) = 0. The maximum acceleration, maximum deceleration, and time coordinate of the Sc-A/D are given as: aa = ad = 3.5 g and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s (g = 9.8 m/s2 ). The unknown distributed disturbance f (x, t) is described as:   3  x i cos(i xt) f (x, t) = 3+ 10000 i=1

(10.61)

The unknown time-varying boundary disturbance d(t) is described as: Tab 10.1 Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

Value

m

1.0 Kg/m2

mc

5.0 Kg

L

1.0 m

c

1.0

Ns/m2

β

0.25 Ns/m

T

4900 N

k1

100

k2

1000

k3

100

γ

0.1

λ

0.001

ζ

100

κ

0.005

λ1

10.0

γ1

1.0

γ2

1.0

σ1

0.001

σ2

0.001

164

10 Adaptive Output Feedback Boundary Control for a Class …

Fig. 10.1. Vibration displacement of the belt without control

d(t) = 10 +

3 

i sin(it)

(10.62)

i=1

As it’s shown in the following figures, from Fig. 10.1, it can be seen that there is a large vibration displacement along the belt for the uncontrolled case. From Fig. 10.2, it can also be seen that the proposed adaptive control (10.58) can suppress the vibrations of the belt greatly when the corresponding update laws (10.59)–(10.60) are applied. As shown in Fig. 10.3, the vibration displacement of the belt is examined at x = 1 m and x = 0.5 m, and the simulation results for controlled and uncontrolled responses are shown. And Fig. 10.4 shows the enlarged view of the vibration displacement for controlled response at x = 1 m and x = 0.5 m. And in Fig. 10.5, the corresponding control input of the proposed boundary control U (t) is shown.

10.5 Simulation Example

Fig. 10.2. Vibration displacement of the belt with control

Fig. 10.3. Vibration displacement of the belt at: a x = 1 m, b x = 0.5 m

165

166

10 Adaptive Output Feedback Boundary Control for a Class …

Fig. 10.4. The enlarged view of the vibration displacement for controlled response

Fig. 10.5. Control input U (t)

Appendix 1: Simulation Program

Appendix 1: Simulation Program close all; clear all; clc; nx =50; nt =100*10ˆ3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m=1; T = 4900; pi = 3.1415926; V0=0; c = 1; % --------------velocity----------------V = zeros(1,nt); V(1,1) = V0; J=35; for j=2:10*10ˆ3-1 V(1,j) = V(1,1)+0.5*J*(j*dt)ˆ2; end V(1,10*10ˆ3)=V(1,1)+0.5*J; for j=(10*10ˆ3+1):20*10ˆ3-1 V(1,j)=V(1,10*10ˆ3)+0.5*J*(j*dt-1); end V(1,20*10ˆ3)=V(1,10*10ˆ3)+J; for j=(20*10ˆ3+1):30*10ˆ3-1 V(1,j) = V(1,20*10ˆ3)+J*(j*dt-2)-0.5*J*(j*dt-2)ˆ2; end V(1,30*10ˆ3)=V(1,20*10ˆ3)+0.5*J; for j=(30*10ˆ3+1):70*10ˆ3-1 V(1,j)=V(1,30*10ˆ3); end V(1,70*10ˆ3)=V(1,30*10ˆ3); for j=(70*10ˆ3+1):80*10ˆ3-1 V(1,j)= V(1,70*10ˆ3)-0.5*J*(j*dt-7)ˆ2; end V(1,80*10ˆ3)=V(1,70*10ˆ3)-0.5*J; for j=(80*10ˆ3+1):90*10ˆ3-1 V(1,j)=V(1,80*10ˆ3)-J*(j*dt-8); end V(1,90*10ˆ3)=V(1,80*10ˆ3)-J; for j=(90*10ˆ3+1):nt-1 V(1,j)=V(1,90*10ˆ3)-J*(j*dt-9)+0.5*J*(j*dt-9)ˆ2; end V(1,100*10ˆ3)=V(1,90*10ˆ3)-0.5*J; % -------------disburtance-------------

167

168

10 Adaptive Output Feedback Boundary Control for a Class …

d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt d(1,j)=10+1*sin(1*(j-1)*dt)+2*sin(2*(j-1)*dt)+3*sin(3*(j1)*dt); end for j = 1 : nt for i = 1 : nx f(i,j) = (3+cos((i-1)*dx*(j-1)*dt)+2*cos(2*(i-1)*dx*(j1)*dt)+3*cos(3*(i-1)*dx*(j-1)*dt))*(i-1)*dx/10000; end end %********************************************** % uncontrolled %********************************************** ds=0.25; mc=5; w1 = zeros(nx,nt); z_e0 = zeros(1,nt); e_z0 = zeros(1,nt); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((T-m*V(1,j1)ˆ2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dxˆ2 -c*(w1(i,j-1)w1(i,j-2))/dt - c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt) m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j) ) / (m/dtˆ2); end z_e0(1,j)= z_e0(1,j-1)+dt/mc*(d(1,j)-ds*z_e0(1,j-1)(T/dx)*(w1(nx,j-1)-w1(nx-1,j-1))); e_z0(1,j)=((w1(nx,4) - w1(nx,3))/dt)*exp(-ds/mc*j*dt); w1(nx,j) = w1(nx-1,j)+dt*(z_e0(1,j)+e_z0(1,j)); end tshort = linspace(0,tmax,nt/800); xshort = linspace(0,L,nx); w1short=zeros(nx,nt/800); for j=1:nt/800 for i=1:nx w1short(i,j)=w1(i,j*800); end end for j=1 for i=1:nx w1short(i,j)=0; end end figure (1); surf(tshort,xshort,w1short);

Appendix 1: Simulation Program

169

colormap(’jet’) ; xlabel(’Time [s]’,’Fontsize’,14); ylabel(’x [m]’,’Fontsize’,14); zlabel(’w(x,t) [m]’,’Fontsize’,14); set(gca,’YDir’,’reverse’) % %********************************************** % %observer backstepping adaptive controlled + disturbance observer % %********************************************** k2 = 1*10ˆ3; k1 = 1*10ˆ2; gamma = 1*10ˆ-1; lambda = 1*10ˆ-3; k3 = 100; kappa=0.005; lambda_1=10; w2 = zeros(nx,nt); u1 = zeros(1,nt); d_e = zeros(1,nt); beta = zeros(1,nt); error = zeros(1,nt); w0 = zeros(1,nt); z_2e = zeros(1,nt); z_e = zeros(1,nt); e_z2 = zeros(1,nt); x1 = zeros(1,nt); x2 = zeros(1,nt); pi_1=zeros(1,nt); pi_2=zeros(1,nt); x2_e=zeros(1,nt); e_x2=zeros(1,nt); T_e = zeros(1,nt); m_e = zeros(1,nt); err_T = zeros(1,nt); err_m = zeros(1,nt); gamma_1=1; gamma_2 = 1; sigma_1=1*10ˆ-3; sigma_2 = 1*10ˆ-3; for j = 1 : 2 T_e(1,j) = 4000; m_e(1,j) = 0.8; end w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((T-m*V(1,j1)ˆ2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1))/dxˆ2 -c*(w2(i,j-1)w2(i,j-2))/dt - c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j) ) / (m/dtˆ2);

170

10 Adaptive Output Feedback Boundary Control for a Class …

end x1(1,j-1)=(w2(nx,j-1)-w2(nx-1,j-1))/dx; z_2e(1,j)= z_2e(1,j-1)+dt/mc*(-k2*z_2e(1,j-1)-k2*k1*(w2(nx,j1)-w2(nx-1,j-1))/dx -(T-T_e(1,j-1))*(w2(nx,j-1)-w2(nx1,j-1))/dx-gamma*T_e(1,j-1)*(w2(nx,j-1)-w2(nx-1,j-1))/dxmc*k1*x2_e(1,j-1) +m_e(1,j-1)*(gamma*V(1,j)ˆ2 - k1*gamma*V(1,j) + k1*lambda*L)*(w2(nx,j-1)-w2(nx-1,j-1))/dx +d(1,j)-d_e(1,j1)); x2_e(1,j)=pi_2(1,j)/kappa; pi_1(1,j)=pi_1(1,j-1)+dt*x2_e(1,j); pi_2(1,j)= pi_2(1,j-1)+dt/kappa*(-lambda_1*pi_2(1,j)pi_1(1,j)+x1(1,j)); e_x2(1,j)=x2_e(1,j)-x2(1,j); z_e(1,j) = z_2e(1,j) + k1*(w2(nx,j-1) - w2(nx-1,j-1))/dx ; %e T_e(1,j) = T_e(1,j-1) - dt*(1/gamma_1*sigma_1*T_e(1,j-1) + 1/gamma_1*z_e(1,j)*(1-gamma)*(w2(nx,j-1)-w2(nx-1,j-1))/dx ); m_e(1,j) = m_e(1,j-1) - dt*(1/gamma_2*sigma_2*m_e(1,j1) + 1/gamma_2*z_e(1,j)*( gamma*V(1,j)ˆ2 - k1*gamma*V(1,j) + k1*lambda*L)*(w2(nx,j-1)-w2(nx-1,j-1))/dx ); err_T(1,j) =T_e(1,j) -T; err_m(1,j)= m_e(1,j) - m; d_e(1,j)=d_e(1,j-1)+dt*(z_2e(1,j-1)+k1*x1(1,j-1)-k3*d_e(1,j1)); u1(1,j) = -d_e(1,j-1) -k2*z_2e(1,j-1)-k2*k1*(w2(nx,j-1)-w2(nx1,j-1))/dx + ds*z_2e(1,j-1) +(1-gamma)*T_e(1,j)*(w2(nx,j-1)w2(nx-1,j-1))/dx-mc*k1*x2_e(1,j-1) + m_e(1,j)*( gamma*V(1,j)ˆ2 -k1*gamma*V(1,j) + k1*lambda*L )/dx*(w2(nx,j-1)-w2(nx-1,j-1)); error(1,j) =d_e(1,j) - d(1,j); e_z2(1,j)=((w2(nx,4) - w2(nx,3))/dt)*exp(-ds/mc*j*dt); w2(nx,j) = w2(nx-1,j)+dt*(z_2e(1,j)+e_z2(1,j)); end for j=1:nt w0(1,j)= z_2e(1,j)+e_z2(1,j); end for j=1:4 e_z2(1,j)= e_z2(1,5); end w2short=zeros(nx,nt/800); for j=1:nt/800 for i=1:nx w2short(i,j)=w2(i,j*800); end end for j=1 for i=1:nx w2short(i,j)=0; end end figure (2); surf(tshort,xshort,w2short); colormap(’jet’) ; xlabel(’Time [s]’,’Fontsize’,14); ylabel(’x [m]’,’Fontsize’,14); zlabel(’w(x,t) [m]’,’Fontsize’,14);

References

171

set(gca,’YDir’,’reverse’) figure (3); hold on; subplot(2,1,1); plot(tshort,w1short(50,:),’b’,tshort,w2short(50,:),’:r’); title(’(a)’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’w(1,t) [m]’,’Fontsize’,14); legend(’uncontrolled’,’controlled’); box on grid on subplot(2,1,2); plot(tshort,w1short(25,:),’b’,tshort,w2short(25,:),’:r’); title(’(b)’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’w(0.5,t) [m]’,’Fontsize’,14); legend(’uncontrolled’,’controlled’); box on grid on hold off figure (4); plot(tshort,w2short(50,:),’:r’,tshort,w2short(25,:),’b’); legend(’x=1m’,’x=0.5m’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’w(x,t) [m]’,’Fontsize’,14); figure(5); plot(t,u1,’b’); xlabel(’Time [s]’,’Fontsize’,14); ylabel(’U(t) [N]’,’Fontsize’,14); box on grid on

References 1. S.Y. Lee, C. Mote, Vibration control of an axially moving string by boundary control. J. Dyn. Syst. Meas. Control 118(1), 66–74 (1996) 2. J.Y. Choi, K.S. Hong, K.J. Yang, Exponential stabilization of an axially moving tensioned strip by passive damping and boundary control. J. Vib. Control 10(5), 661–682 (2004) 3. Q.C. Nguyen, K.S. Hong, Simultaneous control of longitudinal and transverse vibrations of an axially moving string with velocity tracking. J. Sound Vib. 331(13), 3006–3019 (2012) 4. Y. Liu, Z. Zhao, W. He, Boundary control of an axially moving accelerated/decelerated belt system. Int. J. Robust Nonlinear Control 26(17), 3849–3866 (2016) 5. Y. Liu, Z. Zhao, W. He, Stabilization of an axially moving accelerated/decelerated system via adaptive boundary control. ISA Trans. 64, 394–404 (2016) 6. Z. Zhao, Y. Liu, W. He et al., Adaptive boundary control of an axially moving belt system with high acceleration/deceleration. IET Control Theory Appl. 10(11), 1299–1306 (2016) 7. M. Kristic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and adaptive control design (Wiley, New York, 1995) 8. S. Behtash, Robust output tracking for non-linear system. Int. J. Control 51(6), 1381–1407 (1990)

Chapter 11

Vibration Control and Boundary Tension Constraint of an Axially Moving String System

11.1 Introduction In recent decades, the control community has paid more and more attention to the boundary control of axially moving systems. In [1], the authors propose a boundary control for controlling the vibration of axially translating system under unknown boundary disturbance stemming from uncontrolled span based on the energy approach. In [2], the exact model-based and adaptive control for a class of axially moving accelerated/decelerated system is presented by the merging adaptive technique, Lyapunov theory, and backstepping method. In [3], the asymptotical convergence of the transverse vibration of the axially moving system is guaranteed via adaptive isolation controllers, in which the actual experimental verification is provided. However, in the aforementioned literatures, the control design is limited to the vibration suppression of the systems, and constraint on the boundary tension is not taken into consideration, such as degrading the performance of the control system, causing damage to the equipment, leading to mechanical stoppages and even causing production safety accidents [4]. The consideration of the effects of the tension constraint will make the control design more difficult and challenging, which drives us to do this research. In this chapter, it is concerned with vibration control and boundary tension constraint of an axially moving string system. The main control objectives are to suppress the vibration of the string system and ensure the boundary tension in a constrained region. To this end, a boundary control strategy is developed via constructing a proper barrier Lyapunov function. In addition, the disturbance observer including the barrier term is introduced to tackle the effects of unknown boundary disturbance. With the proposed control strategy, the states of the closed-loop system are proven to be uniformly ultimately bounded through rigorous Lyapunov analysis and the boundary tension constraint is not violated.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_11

173

174

11 Vibration Control and Boundary Tension Constraint …

11.2 PDE Dynamic Model To obtain the nonlinear dynamic model of the presented string system, we apply the extended Hamilton’s principle [5]: t2 (δ E k − δ E p + δW )dt = 0

(11.1)

t1

The kinetic energy E k (t) of the string system is given as: 1 1 E k = m c wt2 (L , t) + m 2 2

L (wt +vwx )2 d x

(11.2)

0

The potential energy E p (t) of the string system can be described by: 1 Ep = 2

L T (x, t)wx2 d x

(11.3)

0

where the spatiotemporally varying tension T (x, t) is given as [6]: T (x, t) = T0 (x) + κ(x)wx2 (x, t)

(11.4)

with T0 (x) > 0 being the initial tension and the scalar function κ(x) ≥ 0 being the nonlinear elastic modulus. The virtual work δW1 (t) done by the nonconservative force is calculated as:   δW1 = U − γ wt (L , t) δw(L , t) − c

L (wt +vwx )δwd x 0

L

(11.5)

f δwd x

+d(t)δw(L , t) + 0

where γ is damping coefficient of the actuator and c is viscous damping coefficient of the string. The virtual momentum transport δW2 (t) across the boundaries is computed by: δW2 = mv[wt (L , t) + vwx (L , t)]δw(L , t)

(11.6)

11.2 PDE Dynamic Model

175

Finally, the total work δW (t) on the string system is summarized as: δW = δW1 − δW2

(11.7)

Substituting Eqs. (11.2), (11.3), and (11.7) into Eq. (11.1) and then integrating by parts, the governing equation of the presented string system is derived as:   m wtt + awx + 2vwxt + v 2 wx x − 6κwx2 wx x −2κx wx3 − T0 wx x − T0x wx + c(wt + vwx ) = f

(11.8)

where ∀(x, t) ∈ (0, L) × [0, +∞). And the corresponding boundary conditions of the considered string system are expressed as: 

w(0, t) = 0 m c wtt (L , t) + T0 (L)wx (L , t) + 2κ(L)wx3 (L , t) + γ wt (L , t) = d + U

(11.9)

where ∀t ∈ [0, +∞). Assumption 11.1 We assume that the distributed disturbance f (x, t), speed v(t), boundary disturbance d(t) and its time derivative dt (t) are uniformly continuous and there exist constants a1 , a2 , a3 , a4 ∈ R+ , such that | f (x, t) | ≤ a1 , ∀(x, t) ∈ (0, L) × [0, +∞) and 0 < v(t) ≤ a2 , | d(t) | ≤ a3 , | dt (t) | ≤ a4 , ∀t ∈ [0, +∞). Assumption 11.2 For the scalar function T0 (x) and κ(x), we assume that they are uniformly bounded and there exist nonnegative constants a5 , a6 , a7 , a8 , a9 and a10 such that a5 ≤ T0 (x) ≤ a6 , a7 ≤ κ(x) ≤ a8 , | T0x (x) | ≤ a9 , | κx (x) | ≤ a10 , ∀x ∈ [0, +∞). Remark 11.1 After taking the unknown distributed disturbance f (x, t) into account, the governing Eq. (11.8) of the string system is described by a nonhomogeneous PDE, which makes this model differ from the string system governed by a homogeneous PDE in [7]. Consequently, the control schemes in [7] may not be suitable for our system. In this chapter, a boundary control based on the original nonhomogeneous PDE will be developed to stabilize the string system and ensure the boundary tension constraint satisfaction. Remark 11.2 Due to the existence of the nonlinear terms wx2 (x, t)wx x (x, t) and wx3 (x, t) in the dynamical Eq. (11.8), which makes the string’s dynamical Eq. (13.8) governed by a nonlinear PDE, then the model differs from the string system governed by a linear PDE in [8], where a proper gain kernel needs to be found to design the backstepping boundary controller and observer. As the consequence, the control schemes in [4] cannot be used for our system.

176

11 Vibration Control and Boundary Tension Constraint …

11.3 Boundary Controller Design The control objective is to develop a boundary control for regulating the vibration of the string system in a small range around zero and simultaneously ensuring the boundary tension constraint is satisfied, i.e., | T (L , t) | ≤ Tm . To stabilize the considered string system given by (11.8) and (11.9), we propose the following boundary control  2φ 2 m c wx (L , t)wxt (L , t) / ln 2 U (t) = −[wt (L , t) + k2 wx (L , t)] k1 + 2 2 φ − wx (L , t) φ − wx2 (L , t) ˆ −k2 m c wxt (L , t) + T0 (L)wx (L , t) + 2κ(L)wx3 (L , t) + γ wt (L , t) − d(t) −k3 [wt (L , t) + k2 wx (L , t)] − 2κ(L)wx3 (L , t)/ ln

2φ 2 φ 2 − wx2 (L , t) (11.10)

ˆ is the estimate of d(t), and φ > 0 is where k1 , k2 , k3 > 0 are the control gains, d(t) √ related to the boundary tension constraint Tm such that φ = [Tm − T0 (L)]/κ(L). ˆ is defined as: Moreover, the disturbance estimate d(t) ˆ = α(t) + m c wt (L , t) d(t)

(11.11)

with α(t) being a function designed in the following. Differentiating (11.11) and then substituting (11.9) into the resulting equation, we obtain the disturbance observer dynamics dˆt (t) as   dˆt (t) = αt (t) + d + U − γ wt (L , t) − T0 (L)wx (L , t) − 2κ(L)wx3 (L , t) (11.12) Then, αt (t) can be designed as 2φ 2 − m c wt (L , t) − U − wx2 (L , t) (11.13) 3 −α(t) + T0 (L)wx (L , t) + 2κ(L)wx (L , t) + γ wt (L , t)

αt (t) = β[wt (L , t) + k2 wx (L , t)] ln

φ2

where β > 0. Substituting (11.13) into (11.12), the disturbance observer dynamics dˆt (t) is modified as: dˆt (t) = β[wt (L , t) + k2 wx (L , t)] ln

2φ 2 ˆ + d(t) − d(t) φ 2 − wx2 (L , t)

(11.14)

11.4 Stability Analysis

177

˜ is defined as: The boundary disturbance estimation error d(t) ˜ = d(t)−d(t) ˆ d(t)

(11.15)

Differentiating Eq. (11.15) and then substituting Eq. (11.14) into the resulting equation, we obtain: d˜t (t) = β[wt (L , t) + k2 wx (L , t)] ln

2φ 2 ˜ − dt (t) − d(t) φ 2 − wx2 (L , t)

(11.16)

Remark 11.3 The signals wx (L , t), wt (L , t) and wxt (L , t) used to make up the control law (11.10) can be measured by sensors directly or obtained by the backward difference algorithm, i.e., w(L , t) and wx (L , t) can be measured by employing a laser displacement sensor and an inclinometer, and then, wt (L , t) and wxt (L , t) can be obtained by applying the backward difference algorithm to w(L , t) and wx (L , t), respectively.

11.4 Stability Analysis Consider the Lyapunov function candidate as: V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t)

(11.17)

where the energy term V1 (t) is expressed as: ηm V1 (t) = 2

L

η (wt + vwx ) d x + 2

L T wx2 d x

2

0

(11.18)

0

and a small crossing term V2 (t) is expressed as: L xwx (wt + vwx )d x

V2 (t) = 2μm

(11.19)

0

and an auxiliary term V3 (t) is expressed as: V3 (t) =

2φ 2 1 ηm c [wt (L , t) + k2 wx (L , t)]2 ln 2 2 φ − wx2 (L , t)

and an estimation error term V4 (t) is expressed as:

(11.20)

178

11 Vibration Control and Boundary Tension Constraint …

V4 (t) =

η ˜2 d (t) 2β

(11.21)

where η, μ > 0. Applying Eqs. (2.11)–(2.15) to Eq. (11.19) results in: L |V2 (t)| ≤ μm L

 2  wx + (wt + vwx )2 d x ≤ ξ V1 (t)

(11.22)

0

where ξ=

2μm L min(ηh 5 , ηm)

(11.23)

The Eq. (11.22) can be rewritten as: −ξ V1 (t) ≤ V2 (t) ≤ ξ V1 (t)

(11.24)

By choosing η and μ properly, we can obtain ⎧ ⎪ ⎪ ⎨ ξ1 = 1 − ξ = 1 −

2μm L >0 min(ηh 5 , ηm) 2μm L ⎪ ⎪ ⎩ ξ2 = 1 + ξ = 1 + >1 min(ηh 5 , ηm)

(11.25)

The Eq. (11.25) indicates 0 < ξ < 1, and then combining (11.23), we have μ
0 and ϑ2 = max(ξ2 , 1) > 0 As an important analytic tool to derive our main results, we state the following Lemma 11.1. Lemma 11.1 The time derivative of the Lyapunov’s function candidate Eq. (11.17) can be upper bounded with

11.4 Stability Analysis

179

Vt (t) ≤ −ϑ V (t) + ε

(11.29)

where ϑ, ε > 0. Proof Differentiating Eq. (11.17) with respect to time yields: Vt (t) = V1t (t) + V2t (t) + V3t (t) + V4t (t)

(11.30)

Differentiating Eq. (11.18), substituting (11.8), integrating by parts and combining boundary conditions Eq. (11.9) and Eqs. (2.11)–(2.15), we can get   ηv T0 (0) − mv 2 2 ηmv wx (0, t) − [wt (L , t) + vwx (L , t)]2 V1t (t) ≤ − 2 2 ηT0 (L)[k2 − v] 2 ηT0 (L) wx (L , t) + − [wt (L , t) + k2 wx (L , t)]2 2 2k2  ηT0 (L) 2 3ηv  κ(L)wx4 (L , t) − κ(0)wx4 (0, t) − wt (L , t) + 2k2 2 L + 2ηκ(L)wt (L , t)wx (L , t) − (cη − ηδ1 ) (wt + vwx )2 d x

(11.31)

0

+

η 2

L vκx wx4 d x + 0

η δ1

L f 2d x 0

where δ1 is a positive constant. Taking the derivative of (11.19), then substituting (11.8), integrating by parts and using Eqs. (2.11)–(2.15) result in:

V2t (t) ≤

μm Lwt2 (L , t)



 2μL − μm Lv − μT0 L wx2 (L , t) + δ2

L

2

f 2d x 0

  − μ min(T0 ) − μ min(T0x )L − μmv 2 − 2μLδ2 − 2μcLδ3

L wx2 d x 0

+

2λcL δ3

L (wt + vwx )2 d x + 3μLκ(L)wx4 (L , t) 0

L

L wt2 d x

− μm 0

[3κ − xκx ]wx4 d x

−μ 0

(11.32)

180

11 Vibration Control and Boundary Tension Constraint …

where δ2 , δ3 are two positive constants Differentiating (11.20) and substituting the boundary conditions (11.9) along with the control law (11.10) into the consequence yield V3t (t) = −ηk1 [wt (L , t) + k2 wx (L , t)]2 − 2ηk2 κ(L)wx4 (L , t) −η(d˜ + k3 )[wt (L , t) + k2 wx (L , t)] ln

2φ 2 φ 2 − wx2 (L , t)

(11.33)

−2ηκ(L)wt (L , t)wx3 (L , t) Substituting (11.16) into the resulting equation and substituting the Eqs. (2.11)– (2.15) lead to: ˜ t (L , t) + k2 wx (L , t)] ln V4t (t) ≤ ηd[w

η ˜2 2φ 2 η 2 − d (11.34) d + 2 2 φ − wx (L , t) 2β 2β t

Substituting (11.31)–(11.34) into Eq. (11.30), we obtain:   ηT0 (L) η ˜2 η 2 d Vt (t) ≤ − ηk1 − d + [wt (L , t) + k2 wx (L , t)]2 − 2k2 2β 2β t    ηv T0 (0) − mv 2 2 ηT0 (L) 2 wx (0, t) − − μm L wt (L , t) − 2k2 2 2φ 2 − ηk3 [wt (L , t) + k2 wx (L , t)]2 ln 2 φ − wx2 (L , t)  ηT0 (L)[k2 − v] 2 wx (L , t) − μm Lv 2 − μT0 L + 2  3ηv 3ηv κ(L)wx4 (L , t) − κ(0)wx4 (0, t) − 2ηk2 − 3μL − 2 2  L  2λcL − cη − ηδ1 − (wt + vwx )2 d x δ3 0

  − μ min(T0 ) − μ min(T0x )L − μmv 2 − 2μLδ2 − 2μcLδ3

L wx2 d x 0

L  L  η 4 3κ − xκx − vκx wx d x − μm wt2 d x −μ 2 0

ηmv [wt (L , t) + vwx (L , t)]2 + − 2

0



η 2μL + δ1 δ2

 L f 2d x 0

(11.35)

11.4 Stability Analysis

181

The intermediate parameters μ, η, k1 , k2 , k3 , δ1 , δ2 and δ3 are selected to satisfy the following conditions: ⎧ min(ηm, ηh 5 ) ⎪ ⎪ μ< ⎪ ⎪ 2m L  ⎪  ⎪ ⎪ ⎪ ηv T − mv 2 (0) ⎪ 0 ⎪ ⎪ ≥0 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ηT0 (L) ⎪ ⎪ ≥0 ηk1 − ⎪ ⎪ 2k2 ⎪ ⎪ ⎪ ⎪ ηT0 (L) ⎪ ⎪ ⎪ − μm L ≥ 0 ⎪ ⎪ ⎨ 2k2 ηT0 (L)[k2 − v] ⎪ ≥0 μm Lv 2 − μT0 L + ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 2λcL ⎪ ⎪ τ1 = cη − ηδ1 − >0 ⎪ ⎪ δ3 ⎪ ⎪   ⎪ ⎪ ⎪ τ2 = min μT0 − μT0x L − μmv 2 − 2μLδ2 − 2μcLδ3 > 0 ⎪ ⎪ ⎪   ⎪ η ⎪ ⎪ vκ τ >0 = min 3κ − xκ − ⎪ 3 x x ⎪ ⎪ 2 ⎪   ⎪ ⎪ ⎪ η 2μL η 2 ⎪ ⎩ε = La12 + a + δ1 δ2 2β 4

(11.36)

Substituting (11.36) into (11.35) yields: η Vt (t) ≤ − d˜ 2 − τ1 2β

L

L (wt + vwx ) d x − τ2

0

L wx2 d x

2

− τ3

0

−ηk3 [wt (L , t) + k2 wx (L , t)]2 ln

wx4 d x 0

2

2φ +ε φ 2 − wx2 (L , t)

(11.37)

≤ −ϑ3 [V1 (t) + V3 (t) + V4 (t)] + ε ≤ −ϑ V (t) + ε where ϑ = (ϑ3 /ϑ2 ) and ϑ3 = min



2τ1 2τ2 2τ3 2k3 , , , ,1 ηm ηa6 ηa8 m c

 .

Theorem 11.1 For the presented string system described by Eqs. (11.8)–(11.9), under Assumption 11.1 and the proposed control law Eq. (11.10), given that the initial conditions are bounded, we can conclude that: (1) uniform boundedness: the system state w(x, t) of the closed-loop string system will remain in the compact set: 1 := { w(x, t) ∈ R||w(x, t)| ≤ χ1 }

(11.38)

182

11 Vibration Control and Boundary Tension Constraint …

 where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 = ηϑ2L1 a5 [V (0)e−ϑt + ϑε ]. (2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop string system will eventually converge to the compact set: 2 := { w(x, t) ∈ R| lim |w(x, t)| ≤ χ2 }

(11.39)

t→∞

where ∀x ∈ [0, L], χ2 =



2Lε . ηϑ1 ϑa5

Proof Multiplying In Eq. (11.37) by eϑt yields:  ∂ V (t)eϑt ≤ εeϑt ∂t

(11.40)

ε −ϑt ε ε ]e + ≤ V (0)e−ϑt + ϑ ϑ ϑ

(11.41)

Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒ Integrating the above inequality yields: V (t) ≤ [V (0) −

Applying Eqs. (2.11)–(2.15) and Eq. (11.28), we have: ηh 5 2 η w ≤ 2L 2

L T0 wx2 d x ≤ V1 (t) ≤ V1 (t) + V3 (t) + V4 (t) ≤ 0

1 V (t) ϑ1

(11.42)

Substituting Eq. (11.41) into Eq. (11.42) results in:  |w(x, t)| ≤

2L ε [V (0)e−ϑt + ] ≤ ηϑ1 a5 ϑ



2L ε [V (0) + ] ηϑ1 a5 ϑ

(11.43)

where ∀(x, t) ∈ [0, L] × [0, +∞), from above inequality, we have:  lim |w(x, t)| ≤ lim

t→∞

t→∞

2L ε [V (0)e−ϑt + ] = ηϑ1 a5 ϑ

 2Lε , ηϑ1 ϑa5

∀x ∈ [0, L] (11.44)

11.5 Simulation Example

183

Remark 11.4 From (11.41) and (11.42), it can be derived that V (t) is bounded, ∀t ∈ [0, +∞). Thus, we can obtain that V1 (t), V3 (t) and d(t) are bounded, ∀t ∈ [0, +∞). Since V1 (t) is bounded, we reach a conclusion that wx (x, t), wt (x, t), wxt (x, t), and wx x (x, t) are bounded, ∀(x, t) ∈ [0, L] × [0, +∞). Combining Assumption 1, the governing Eq. (11.8) and the above analysis, we conclude that wtt (x, t) is bounded, ∀(x, t) ∈ [0, L] × [0, +∞), and the proposed control U (t) described by (11.10) is also bounded.   In addition, itis obvious that V3 (t) = 1/2ηm c [wt (L , t) + k2 wx (L , t)]2 ln 2φ 2 / φ 2 − wx2 (L , t) → ∞, as |wx (L , t)| → φ. Consequently, it can be known that |wx (L , t)| = φ. Given that −φ < wx (L , 0) < φ; thus, it can be deduced that wx (x, t) remains in the set −φ < wx (L , t) < φ, ∀t ∈ [0, +∞). From the above inequality, it can be obtained that −Tm < T (L , t) < Tm , ∀t ∈ [0, +∞).

11.5 Simulation Example The system parameters are given as follows: m = 1.0 kg/m, m c = 5.0 kg, L = 1.0 m, c = 1.0 Ns/m2 , γ = 1 Ns/m, v(t) = (2 + sin t)m/s, T0 (x) = 100(1 + 0.1x)N , and κ(x) = 5000(1 + 0.1x). The corresponding initial conditions are w(x, 0) = wt (x, 0) = 0. The control parameters are given as k1 = 5000, k2 = 300, k3 = 1000, β = 3000, and φ = 0.1. The unknown distributed disturbance is described as:   n  x i sin(iπ xt) , i = 1, 2, 3 (11.45) f (x, t) = 1+ 10 i=1 The unknown boundary disturbance is given as: d(t) = 10 +

n 

(2i − 1) sin[(2i − 1)t], i = 1, 2, 3

(11.46)

i=1

Figure 11.1 shows the vibration displacement of the uncontrolled string, i.e. U (t) = 0, with H-A/D under distributed disturbance and boundary disturbance. The vibration displacement of the string under the same external conditions with the proposed adaptive boundary control is shown in Fig. 11.2. The vibration displacements of the string are examined at the middle point x = 0.5 m and boundary point x = 1 m in Fig. 11.3 respectively. ˜ Figure 11.4 shows the estimation error d(t) of the disturbance observer. Figure 11.5 shows the spatiotemporally varying tension T (x, t) without control while in Fig. 11.6, we can see the boundary tension T (L , t) of the string can be well controlled. And the corresponding boundary control input is shown in Fig. 11.7.

184

11 Vibration Control and Boundary Tension Constraint …

Fig. 11.1 Vibration displacement of the string without control

Fig. 11.2 Vibration displacement of the string with proposed control

11.5 Simulation Example

Fig. 11.3 Vibration displacement of the string at: ax = 1m, b x = 0.5 m

Fig. 11.4 The estimation error of the disturbance observer

185

186

11 Vibration Control and Boundary Tension Constraint …

Fig. 11.5 The spatiotemporally varying tension T (x, t) without control

Fig. 11.6 Boundary tension T (L , t) of the string

11.5 Simulation Example

Fig. 11.7 The proposed boundary control input U (t)

187

188

11 Vibration Control and Boundary Tension Constraint …

Appendix 1: Simulation Program

close all, clear all, clc; nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m=1; c = 1; % -------------disburtance------------d = zeros(1,nt); f = zeros(nx,nt); kx= zeros(nx,1); V = zeros(1,nt); T0= zeros(nx,1); T= zeros(nx,nt); phi=0.1; for j=1:nt d(1,j)=10+1*sin(1*(j)*dt)+3*sin(3*(j)*dt)+5*sin(5*(j)*dt); V(1,j)=2+1*sin(1*(j)*dt); end for i=1:nx kx(i,1)=5000*(1+0.1*(i)*dx); T0(i,1)=100*(1+0.1*(i)*dx); end for j = 1 : nt for i = 1 : nx f(i,j) = 0.1*(1+ sin(pi*(i)*dx*(j)*dt) + 2*sin(2*pi*(i)*dx*(j)*dt) + 3*sin(3*pi*(i)*dx*(j)*dt) )*(i)*dx; end end %******************************************************************** %

uncontrolled

Appendix 1: Simulation Program

189

%******************************************************************** gamma=1; mc=5; w1 = zeros(nx,nt); w0 = zeros(1,nt); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((T0(i,1)-m*V(1,j-1)^2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dx^2 -c*(w1(i,j-1)-w1(i,j-2))/dt - c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j-1) +6*kx(i,1)*((w1(i,j-1)-w1(i-1,j-1))/dx )^2*(w1(i+1,j-1)-2*w1(i,j-1)+w 1(i-1,j-1))/dx^2+2*(kx(i,1)-kx(i-1,1))/dx*((w1(i,j-1)-w1(i-1,j-1))/dx )^3) / ( m/dt^2); end w1(nx,j) = 2*w1(nx,j-1)-w1(nx,j-2)-dt^2/mc*(T0(nx,1)/dx*(w1(nx,j-1)-w1(nx-1,j-1) )+2*kx(nx,1)*((w1(nx,j-1)-w1(nx,j-2))/dt )^3+guma/dt*(w1(nx,j-1)-w1(n x,j-2))-d(1,j-1)); end for j=1:nt for i=2:nx T(i,j)= T0(i,1)+ kx(i,1)*((w1(i,j)-w1(i-1,j))/dx)^2; end end t1short = linspace(1,tmax,nt/1000); tshort = [0,t1short]; xshort = linspace(0,L,(nx/2)+1); w1short=zeros((nx/2)+1,nt/1000+1); for j=1:nt/1000 for i=1:nx/2 w1short(i+1,j+1)=w1(i*2,j*1000); Tshort(i+1,j+1)=T(i*2,j*1000); end

190

11 Vibration Control and Boundary Tension Constraint …

end figure (1); surf(tshort,xshort,w1short); shading interp; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('$\omega$(x,t)[m]','interpreter','latex','Fontsize',14); set(gca,'YDir','reverse') for j=1:nt w0(1,j)=(w1(nx,j)-w1(nx-1,j))/dx; end % %****************************************************************** % %

controlled + tension constraint + disturbance observer

% %****************************************************************** k1 = 5*10^3; k2 =3*10^2; k3=1000; beta=3000; w2 = zeros(nx,nt); u1 = zeros(1,nt); u_d = zeros(1,nt); w = zeros(1,nt); dm=max(d); alpha = zeros(1,nt); d_e = zeros(1,nt); error = zeros(1,nt); w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((T0(i,1)-m*V(1,j-1)^2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1))/dx^2 -c*(w2(i,j-1)-w2(i,j-2))/dt - c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j-1) +6*kx(i,1)*((w2(i,j-1)-w2(i-1,j-1))/dx )^2*(w2(i+1,j-1)-2*w2(i,j-1)+w 2(i-1,j-1))/dx^2+2*(kx(i,1)-kx(i-1,1))/dx*((w2(i,j-1)-w2(i-1,j-1))/dx )^3 )/ ( m/dt^2);

Appendix 1: Simulation Program

191

end u_d(1,j-1) = (w2(nx,j-1)-w2(nx,j-2))/dt + k2*(w2(nx,j-1)-w2(nx-1,j-1))/dx; w2(nx,j) = 2*w2(nx,j-1)-w2(nx,j-2)- dt^2/mc*(d_e(1,j-1)- d(1,j-1) +2*kx(nx,1)*((w2(nx,j-1)-w2(nx-1,j-1))/dx)^3/log(2*phi^2/(phi^2-((w2( nx,j-1)-w2(nx-1,j-1))/dx)^2))+u_d(1,j-1)*mc*(w2(nx,j-1)-w2(nx-1,j-1)w2(nx,j-2)+w2(nx-1,j-2))/(dx*dt)*((w2(nx,j-1)-w2(nx-1,j-1))/dx) /((phi^2-((w2(nx,j-1)-w2(nx-1,j-1))/dx)^2)*log(2*phi^2/(phi^2-((w2(nx ,j-1)-w2(nx-1,j-1))/dx)^2)))+k1*u_d(1,j-1)/log(2*phi^2/(phi^2-((w2(nx ,j-1)-w2(nx-1,j-1))/dx)^2))+k2*mc*(w2(nx,j-1)-w2(nx-1,j-1)-w2(nx,j-2) +w2(nx-1,j-2))/(dx*dt)+k3*u_d(1,j-1) ); u_d(1,j) = (w2(nx,j)-w2(nx,j-1))/dt + k2*(w2(nx,j)-w2(nx-1,j))/dx; u1(1,j) = - k3*u_d(1,j)k1*u_d(1,j)/log(2*phi^2/(phi^2-((w2(nx,j)-w2(nx-1,j))/dx)^2)) + T0(nx,1)*(w2(nx,j)-w2(nx-1,j))/dx + gamma*(w2(nx,j)-w2(nx,j-1))/dt u_d(1,j-1)*mc*(w2(nx,j)-w2(nx,j-1)-w2(nx-1,j)+w2(nx-1,j-1))/(dx*dt)*( (w2(nx,j)-w2(nx-1,j))/dx)/((phi^2-((w2(nx,j)-w2(nx-1,j))/dx)^2)*log(2 *phi^2/(phi^2-((w2(nx,j)-w2(nx-1,j))/dx)^2)))-k2*mc*(w2(nx,j)-w2(nx-1 ,j)-w2(nx,j-1)+w2(nx-1,j-1))/(dx*dt)-d_e(1,j)-2*kx(nx,1)*((w2(nx,j-1) -w2(nx-1,j-1))/dx)^3/log(2*phi^2/(phi^2-((w2(nx,j)-w2(nx-1,j))/dx)^2) )+ 2*kx(nx,1)*((w2(nx,j)-w2(nx-1,j))/dx)^3; alpha(1,j) = alpha(1,j-1) - dt*(alpha(1,j-1) gamma*(w2(nx,j)-w2(nx,j-1))/dt - T0(nx,1)*(w2(nx,j)-w2(nx-1,j))/dx 2*kx(nx,1)*((w2(nx,j)-w2(nx,j-1))/dt )^3+u1(1,j)+mc*(w2(nx,j)-w2(nx,j -1))/dt-beta*u_d(1,j))*log(2*phi^2/(phi^2-((w2(nx,j-1)-w2(nx-1,j-1))/ dx)^2)); d_e(1,j) = alpha(1,j) + mc*(w2(nx,j) - w2(nx,j-1))/dt; error(1,j) = d_e(1,j) -d(1,j); end for j=1:nt w(1,j)=(w2(nx,j)-w2(nx-1,j))/dx; end for j=1:nt for i=2:nx Tn(i,j)= T0(i,1)+ kx(i,1)*((w2(i,j)-w2(i-1,j))/dx)^2; end end Tm=165; Tnshort=zeros((nx),nt/1000); w2short=zeros((nx/2)+1,nt/1000+1);

192

11 Vibration Control and Boundary Tension Constraint … for j=1:nt/1000 for i=1:nx/2 w2short(i+1,j+1)=w2(i*2,j*1000); Tnshort(i,j)=Tn(i,j*1000); end end Tnshort=[Tn(:,1),Tnshort]; TO=zeros(1,nt/1000); for j=1:nt/1000 TO(1,j)=Tm; end TO=[TO(1,1),TO]; figure (2); surf(tshort,xshort,w2short); shading interp; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('$\omega$(x,t)[m]','interpreter','latex','Fontsize',14); set(gca,'YDir','reverse') figure (3); hold on subplot(2,1,1); plot(t,w1(nx,:),'b',t,w2(nx,:),':r'); title('£¨a£©'); xlabel('Time (s)','Fontsize',14); ylabel('w(1,t) (m)','Fontsize',14); legend('uncontrolled','controlled'); box on grid on subplot(2,1,2); plot(t,w1(nx/2,:),'b',t,w2(nx/2,:),':r'); title('£¨b£©'); xlabel('Time (s)','Fontsize',14); ylabel('w(0.5,t) (m)','Fontsize',14); legend('uncontrolled','controlled'); box on grid on hold off figure(4); plot(t,d_e,'b',t,d,'--r'); title('disturbance obsever');

References

193

xlabel('Time [s]','Fontsize',14); ylabel('d(t)) [N]','Fontsize',14); legend('estimate of disturbance','actual disturbance'); figure (5); surf(tshort,xshort,Tshort);shading interp xlabel('Time (s)','Fontsize',14); ylabel('x (m)','Fontsize',14); zlabel('T(x,t) (m)','Fontsize',14) figure (6); plot(t,T(nx,:),'b',t,Tn(nx,:),'k',tshort,TO(1,:),'r'); xlabel('Time (s)','Fontsize',14); ylabel('T(L,t) (N)','Fontsize',14); legend('without control','with proposed control','maximum boundary tension'); M1=1; M2=4*10^1+1; axes('position', [0.2,0.55,0.2,0.2]); hold on plot(tshort(M1:M2),Tnshort(nx,M1:M2),'g') axis tight figure(7); plot(t,u1,'b'); xlabel('Time(s)','Fontsize',14); ylabel('U(t)(N)','Fontsize',14);

References 1. K.J. Yang, K.S. Hong, F. Matsuno, Energy-based control of axially translating beams: varying tension, varying speed, and disturbance adaptation. IEEE Trans. Control Syst. Technol. 13(6), 1045–1054 (2005) 2. Y. Liu, Z. Zhao, W. He, Stabilization of an axially moving accelerated/decelerated system via adaptive boundary control. ISA Trans. 64, 394–404 (2016) 3. Y. Li, D. Aron, C.D. Rahn, Adaptive vibration isolation for axially moving strings: theory and experiment. Automatica 38(3), 379–390 (2002) 4. W. He, C. Sun, S.S. Ge, Top tension control of a flexible marine riser by using integral-barrier Lyapunov function. IEEE/ASME Trans. Mechatron. 20(2), 497–505 (2015) 5. H. Goldstein, Classical Mechanics (Addison-Wesley, Massachusetts, 1951) 6. Z. Qu, Robust and adaptive boundary control of a stretched string on a moving transporter. IEEE Trans. Autom. Control 46(3), 470–476 (2001) 7. Q.C. Nguyen, K.S. Hong, Asymptotic stabilization of a nonlinear axially moving string by adaptive boundary control. J. Sound Vib. 329(22), 4588–4603 (2010) 8. A. Smyshlyaev, M. Krstic, Adaptive Control of Parabolic PDEs (Princeton University Press, Princeton, 2010)

Chapter 12

Boundary Control for an Axially Moving System with Input Restriction Based on Disturbance Observers

12.1 Introduction In the mathematical sense, the dynamics of flexible structure systems is an infinitedimensional distributed parameter system, which leads to the vibration reduction strategy design much more difficult than the rigid structure systems. For stabilizing a class of axially moving flexible structure systems, boundary control synthesis based on infinite-dimensional models has also achieved remarkable progress [1– 3]. In [1], the exact model-based and adaptive controllers are proposed for damping the controlled span offsets of an axially moving string, where the experimental verification is provided. In [2], a robust vibration controller is presented to stabilize the axially translating beams by constructing appropriate Lyapunov function candidates. In [3], a stabilizing robust control is constructed for suppressing transverse oscillation of a stretched string on the basis of a straightforward Lyapunov argument. However, in all the above-mentioned research results regarding axially moving systems, the actuator’s input saturation constraint problems are ignored in boundary control design. In practical work processes, the input saturation constraint is almost an inevitable problem in all physical systems, due to the physical restrictions of the system, which often severely limits system performance or even makes the system unstable. Therefore, for an industrial axially moving accelerated system, the input saturation effects should be integrated into the control design process. In this chapter, the vibration offsets reduction is studied for an industrial axially moving accelerated belt subjected to the external disturbances and input saturation. A boundary control strategy is put forward to reduce the vibration offsets and an auxiliary term is introduced to handle the restriction of the input saturation. In addition, both infinite-dimensional observer (IDO) and finite-dimensional observer (FDO) are, respectively, designed to track unknown disturbances.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu et al., Dynamic Modeling and Boundary Control of Flexible Axially Moving System, https://doi.org/10.1007/978-981-19-6941-6_12

195

196

12 Boundary Control for an Axially Moving System with Input Restriction …

12.2 PDE Dynamic Model The belt system dynamics of this paper is presented in Chap. 7 in the following: mwtt + mawx + 2mvwxt + mv 2 wx x − T wx x + cwt + cvwx = f

(12.1)

where ∀(x, t) ∈ (0, L) × [0, +∞). The corresponding boundary conditions are expressed as: 

w(0, t) = 0 m c wtt (L , t) + T wx (L , t) + ds wt (L , t) − d = u

(12.2)

where ∀t ∈ [0, +∞). For the considered belt system with input saturation restriction, the saturation curve can be mathematically expressed as:  u(t) =

sgn[u c (t)]u, |u c (t)| ≥ u u c (t), |u c (t)| < u

(12.3)

where u represents the saturation limit and u c (t) represents the control command. Assumption 12.1 For the speed v(t), H-A/D a(t), boundary disturbance d(t), and distributed disturbance f (x, t), we assume that there exist constants a1 , a2 , a3 , a4 , a5 , a6 ∈ R+ , such that 0 < v(t) ≤ a1 , | a(t) | ≤ a2 , | d(t) | ≤ a3 , | dt (t) | ≤ a4 , ∀t ∈ [0, +∞), and | f (x, t) | ≤ a5 , | f t (x, t) | ≤ a6 , ∀(x, t) ∈ (0, L) × [0, +∞).

12.3 Boundary Controller Design 12.3.1 Disturbance Observer Design Design the following FDO (i.e., boundary disturbance observer): dˆ = φ + ς1 m c wt (L , t)

(12.4)

ˆ is the estimate of d(t), and φ(t) is given as follows: where ς1 > 0, d(t) φt = −ς1 φ + ς1 [T wx (L , t) + (ds − ς1 m c )wt (L , t) − u]

(12.5)

Define the estimation error of d(t) as d˜ = d − dˆ

(12.6)

12.3 Boundary Controller Design

197

Moreover, design the following IDO (i.e., infinite-dimensional disturbance observer) as: fˆ = ν + ς2 mwt

(12.7)

where ς2 > 0, fˆ(x, t) is the estimate of f (x, t) and ν(x, t) is given as     νt = −ς2 ν + ς2 (ma + cv)wx + (c − mς2 )wt + 2mvwxt + mv 2 − T wx x (12.8) The estimation error of f (x, t) as f˜ = f − fˆ

(12.9)

Choose the Lyapunov function candidate as Vo (t) = V1 (t) + V2 (t) where V1 (t) = 1/2d˜2 (t) and V2 (t) = 1/2 ultimately bounded. Differentiating (12.10) gives:

L 0

(12.10)

f˜2 (x, t)d x then Vo (t) is uniformly

Vot (t) = V1t (t) + V2t (t)

(12.11)

Differentiating (12.6) and then substituting (12.2) and (12.7) yield d˜t = dt − ς1 d˜

(12.12)

Combining (12.12) and Assumption 12.1, then using Eqs. (2.11)–(2.15), we obtain  1 ˜2 1 2 V1t ≤ − ς1 − d + a4 2 2

(12.13)

Differentiating (12.9) and then substituting (12.1) and (12.8) give f˜t = f t − ς2 f˜

(12.14)

Similarly, combining (12.14) and Assumption 12.1, then using Eqs. (2.11)–(2.15), we derive: L  1 1 f˜2 (x, t)d x + La62 V2t ≤ − ς2 − 2 2 0

(12.15)

198

12 Boundary Control for an Axially Moving System with Input Restriction …

Substituting (12.13) and (12.15) into (12.11) leads to L   1 ˜2 1 1 1 Vot ≤ − ς1 − f˜2 (x, t)d x + a42 + La62 d − ς2 − 2 2 2 2

(12.16)

0

≤ −ϑ0 Vo + ε0 where ς1− 1/2 > 0, ς2 − 1/2 > 0, ϑ0 = min[2(ς1 − 1/2), 2(ς2 − 1/2)], and ε0 = 1/2 a42 + La62 < +∞. Integrating the above equation yields Vo ≤ [Vo (0) −

ε0 −ϑ0 t ε0 ε0 ]e + ≤ Vo (0)e−ϑ0 t + ϑ0 ϑ0 ϑ0

(12.17)

Then, according to the above analysis, it can be concluded that Vo (t) converges to a ball of radius ε0 /ϑ0 as time approaches infinity.

12.3.2 Backstepping Design Step one First, we define the following coordinate transformation:

z 2t =

z 1 = w(L , t)

(12.18)

z 1t = z 2 = wt (L , t)

(12.19)

1 [u + d − ds z 2 − T wx (L , t)] mc

(12.20)

In this step, we define e = z 2 − αv

(12.21)

where e is the error variable, and αv is the corresponding virtual control. Consider the Lyapunov function candidate as Va (t) = V3 (t) + V4 (t) where the energy term V3 (t) is defined as:

(12.22)

12.3 Boundary Controller Design

ηm V3 (t) = 2

199

L

ηT (wt + vwx ) d x + 2

L wx2 d x

2

0

(12.23)

0

and a small crossing term V4 (t) is defined as:

L xwx (wt + vwx )d x

V4 (t) = μm

(12.24)

0

where η, μ > 0. Applying Eqs. (2.11)–(2.15) to Eq. (12.24) results in: μm L |V4 (t)| ≤ 2

L

 2  wx + (wt + vwx )2 d x ≤ ξ V3 (t)

(12.25)

0

where ξ=

μm L min(ηm, ηT )

(12.26)

Then (12.25) can be rewritten as: −ξ V3 (t) ≤ V4 (t) ≤ ξ V3 (t)

(12.27)

Choosing ξ properly, we can obtain 

ξ1 = 1 − ξ > 0 ξ2 = 1 + ξ > 1

(12.28)

Equation (12.28) indicates 0 < ξ < 1, then we have μ
0. Substituting (12.21) into (12.31) gives:   ηv T − mv 2 2 μm L ηT v + μT L − μm Lv 2 2 wx (L , t) − wx (0, t) + Vat (t) ≤ (e + αv )2 2 2 2 L  μmv 2 μL μT − − μcLδ1 − + ηT wx (L , t)(e + αv ) − wx2 d x 2 2 δ2 0

L 

L η μcL − f 2d x − ηc − (wt + vwx )2 d x + (ηδ3 + μLδ2 ) δ1 δ3 0

0

(12.32) From (12.32), the virtual control αv is designed as αv = −k1 wx (L , t)

(12.33)

where k1 > 0. Substituting (12.33) into (12.32) results in  ηT v + μT L − μm Lv 2 2 wx (L , t) + μm Le2 Vat (t) ≤ − ηT k1 − μm Lk12 − 2   L  ηv T − mv 2 2 μcL η wx (0, t) − ηc − − − (wt + vwx )2 d x 2 δ1 δ3 0

 −

μmv 2 μL μT − − μcLδ1 − 2 2 δ2

L wx2 d x + ηT wx (L , t)e 0

L + (ηδ3 + μLδ2 )

f 2d x 0

(12.34)

12.3 Boundary Controller Design

201

Step two In this step, the auxiliary term is developed to satisfy the input saturation restriction and then the control command u c (t) is proposed to stabilize error e around zero. Then, the auxiliary term is first given as ⎧ 2 ⎪ ⎨ u − k (t) − 2eu + u , |(t)| ≥  2 0 2(t) t (t) = ⎪ ⎩ 0, |(t)| <  0

(12.35)

where k2 , 0 > 0, u = u(t) − u c (t), and t (t) is the state variable. Differentiating (12.21) and then combining (12.20) and (12.33), we have et =

1 [u + d − ds e + ds k1 wx (L , t) − T wx (L , t)] + k1 wxt (L , t) mc

(12.36)

Redefine the Lyapunov function candidate as: 1 1 Vb (t) = Va (t) + m c e2 (t) + 2 (t) 2 2

(12.37)

Differentiating (12.37) yields: Vbt = Vat + m c eet + t

(12.38)

Substituting (12.34)–(12.36) into (12.38) results in  ηT v + μT L − μm Lv 2 2 2 wx (L , t) + μm Le2 Vbt (t) ≤ − ηT k1 − μm Lk1 − 2 + e[u c + d − T wx (L , t) − ds e + ds k1 wx (L , t) + m c k1 wxt (L , t)]   L  ηv T − mv 2 2 μcL η wx (0, t) − ηc − − − (wt + vwx )2 d x 2 δ1 δ3 0

 −

μmv 2 μL μT − − μcLδ1 − 2 2 δ2

L

+ (ηδ3 + μLδ2 )

L wx2 d x + ηT wx (L , t)e 0

 1 2 2  f d x − k2 − 2

0

(12.39) Remark 12.1 It is worth mentioning that the first case [i.e., |(t)| ≥ 0 ] in (12.35) is discussed in this chapter, that is to say, the considered belt system exists input

202

12 Boundary Control for an Axially Moving System with Input Restriction …

saturation and the designed auxiliary term is used for minimizing the input saturation effects. Another case [i.e., |(t)| < 0 ] in (12.35) represents that there is no input saturation in the considered system. However, for the case of ignoring input saturation, the previous design methods can be used to develop the boundary control schemes [4–8]. From (12.39), the control command is put forward as u c (t) = −k3 e − dˆ − ds k1 wx (L , t) + T wx (L , t) − m c k1 wxt (L , t) − ηT wx (L , t) + k4  (12.40) where k3 , k4 > 0. Substituting (12.40) into (12.39) and then applying Eqs. (2.11)–(2.15), we derive  ηT v + μT L − μm Lv 2 2 1 2 wx (L , t) + d˜ 2 Vbt (t) ≤ − ηT k1 − μm Lk1 − 2 2    k4 1 2 ηv T − mv 2 2 − k3 + ds − μm L − − e − wx (0, t) 2 2 2   L μT μL k4 μmv 2 1 2 2 wx d x − k2 − − − − μcLδ1 − −  2 2 δ2 2 2 0

L 

L μcL η 2 − ηc − − f 2d x (wt + vwx ) d x + (ηδ3 + μLδ2 ) δ1 δ3 0

0

(12.41)

12.4 Stability Analysis Lemma 12.1 Consider the following Lyapunov function candidate: V (t) = Vb (t) + Vo (t)

(12.42)

then its time derivative satisfies Vt (t) ≤ −ϑ V (t) + ε

(12.43)

where ϑ and ε are both positive constants. Proof Differentiating (12.42), substituting (12.41) and (12.16), and then using Eqs. (2.11)–(2.15), we have

12.4 Stability Analysis

203

 ηT v + μT L − μm Lv 2 2 1 2 wx (L , t) + d˜ 2 Vt (t) ≤ − ηT k1 − μm Lk1 − 2 2    1 2 ηv T − mv 2 2 k4 − e − wx (0, t) − k3 + ds − μm L − 2 2 2 L   μmv 2 1 2 μL k4 μT 2 − − μcLδ1 − −  wx d x − k2 − − 2 2 δ2 2 2 0

 L

L μcL η 2 − ηc − − f 2d x (wt + vwx ) d x + (ηδ3 + μLδ2 ) δ1 δ3 0

(12.44)

0

L   1 ˜2 1 1 1 f˜2 (x, t)d x + a42 + La62 − ς1 − d − ς2 − 2 2 2 2 0 ⎡ ⎤

L 1 1 1 m c e2 + 2 + d˜ 2 + ≤ −ϑ1 ⎣V3 (t) + f˜2 (x, t)d x ⎦ + ε 2 2 2 2 0

  where ε = (ηδ3 + μLδ2 )La52 + 1/2 a42 + La62 < +∞, and the constants k1 ∼ k4 , η, μ, ς1 , ς2 , and δ1 ∼ δ3 should be chosen to satisfy the following conditions: ⎧ min(ηT, ηm) ⎪ ⎪ ⎪μ < ⎪ ⎪ m L ⎪  ⎪ ⎪ 2 ⎪ ηv T − mv ⎪ ⎪ ≥0 ⎪ ⎪ ⎪ 2 ⎪  ⎪ ⎪ ⎪ ηT v + μT L − μm Lv 2 ⎪ 2 ⎪ ≥0 ⎪ ⎨ ηT k1 − μm Lk1 − 2  (12.45) 1 μcL η k4 1 ⎪ ⎪ − , ς , /(ηm), k ϑ = 2 min ηc − − − − 1, ς − ⎪ 1 2 1 2 ⎪ ⎪ δ1 δ3 2 2 2 ⎪ ⎪  ⎪ 2 ⎪ ⎪ μT μmv μL ⎪ ⎪ /(ηT ), − − μcLδ1 − ⎪ ⎪ 2 2 δ2 ⎪ ⎪ ⎪  ⎪ ⎪ 1 k4 ⎪ ⎪ − /m c > 0 ⎩ k3 + ds − μm L − 2 2 Combining (12.22), (12.30), (12.31), and (12.42), we can obtain

204

12 Boundary Control for an Axially Moving System with Input Restriction …

⎡

⎤

L 1 1 1 m c 2 e + 2 + d˜ 2 + 0 ≤ ϑ2 ⎣V3 (t) + f˜2 (x, t)d x ⎦ 2 2 2 2 0 ⎡ ⎤

L m c 2 1 2 1 ˜2 1 ≤ V (t) ≤ ϑ3 ⎣V3 (t) + e +  + d + f˜2 (x, t)d x ⎦ 2 2 2 2

(12.46)

0

where ϑ2 = min(ξ1 , 1) > 0 and ϑ3 = max(ξ2 , 1) > 0. Combining (12.44) and (12.46), we further have Vt (t) ≤ −ϑ V (t) + ε

(12.47)

where ϑ = ϑ1 /ϑ3 . Combining (12.21) and (12.33), the control command designed for u c (t) in (12.40) is rewritten as u c (t) = k4  − (k1 k3 + ds k1 − T + ηT )wx (L , t) − k3 wt (L , t) − m c k1 wxt (L , t) − dˆ (12.48) Theorem 12.1 For the belt system described by (12.1)–(12.2), under the proposed control (12.44) and Assumption 12.1, given that the initial conditions are bounded, we can conclude that: (1) uniform boundedness: the system state w(x, t) of the closed-loop belt system will remain in the compact set: 1 := { w(x, t) ∈ R||w(x, t)| ≤ χ1 }

(12.49)

 where ∀(x, t) ∈ [0, L] × [0, +∞), χ1 = ηϑ2L2 T [V (0)e−ϑt + ϑε ]. (2) uniform ultimate boundedness: the system state w(x, t) of the closed-loop belt system will eventually converge to the compact set: 2 := { w(x, t) ∈ R| lim |w(x, t)| ≤ χ2 } t→∞

where ∀(x, t) ∈ [0, L] × [0, +∞), χ2 =



2Lε ηϑ2 ϑ T

(12.50)

.

Proof Multiplying Eq. (12.47) by eϑt yields: Vt (t)eϑt ≤ −ϑ V (t)eϑt + εeϑt ⇒

 ∂ V (t)eϑt ≤ εeϑt ∂t

(12.51)

12.5 Simulation Example

205

Integrating the above inequality yields: V (t) ≤ [V (0) −

ε −ϑt ε ε ]e + ≤ V (0)e−ϑt + ϑ ϑ ϑ

(12.52)

Applying Eqs. (2.11)–(2.15), we have: ηT 2 ηT w ≤ 2L 2

L wx2 d x ≤ V3 (t) ≤ 0

1 V (t) ϑ2

1 2 1  (t) ≤ V (t) 2 ϑ2

(12.53)

(12.54)

Substituting Eq. (12.52) into Eqs. (12.53) and (12.54) results in:  |w(x, t)| ≤  |(t)| ≤

2L ε [V (0) + ] ηT ϑ2 ϑ

(12.55)

2 ε [V (0) + ] ϑ2 ϑ

(12.56)

where ∀(x, t) ∈ [0, L] × [0, +∞) and ∀t ∈ [0, +∞), from above inequalities, we have:   2L 2Lε ε , ∀x ∈ [0, L] (12.57) lim |w(x, t)| ≤ lim [V (0)e−ϑt + ] = t→∞ t→∞ ηT ϑ2 ϑ ηT ϑ2 ϑ   2 2ε ε lim |(t)| ≤ lim (12.58) [V (0) + ] = t→∞ t→∞ ϑ2 ϑ ϑ2 ϑ

12.5 Simulation Example Considering the PDE model as Eq. (12.1), the parameters of the belt system are given in Table 12.1. Here, the time step size is given as t = 10−4 s and the space step size is given as x = 0.02 m.

206

12 Boundary Control for an Axially Moving System with Input Restriction …

Table 12.1 Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

m

1.0kg/m

mc

5.0kg

L

1.0m

Value

c

1.0N s/m 2

ds

1N s/m

T

100N

k1

100

k2

100

k3

100

k4

10

ς1

20

ς2

50

η

0.1

0

0.001

u

2N

The following are the initial conditions and axial velocity: w(x, 0) = wt (x, 0) = 0

(12.59)

v(t) = 1 + 0.5 sin(2t)

(12.60)

The unknown distributed disturbance f (x, t) is described as   3  x sin(iπ xt) f (x, t) = 10 + 100 i=1

(12.61)

The unknown time-varying boundary disturbance d(t) is described as: d(t) =

3 

i cos(it)

(12.62)

i=1

Figure 12.1 shows the three-dimensional (3D) vibration displacement of the axially moving belt for free vibration, i.e. U (t) = 0, with H-A/D under both distributed disturbance and boundary disturbance, while Fig. 12.2 shows the 3D vibration displacement with the proposed robust boundary control and disturbance observer. The vibration displacement of the belt is examined at x = 1 m and x = 0.5 m, and the simulation results for controlled and uncontrolled responses are shown in Fig. 12.3 respectively. Figure 12.4 shows the tracking of boundary disturbance, and the developed control command u c (t) and the saturated control input u(t) are shown in Fig. 12.5 and in Fig. 12.6 respectively.

12.5 Simulation Example

Fig. 12.1 Vibration displacement of the belt without control

Fig. 12.2 Vibration displacement of the belt with control

207

208

12 Boundary Control for an Axially Moving System with Input Restriction …

Fig. 12.3 Vibration displacement of the belt at: a x = 1 m, b x = 0.5 m

Fig. 12.4 Trajectory tracking of boundary disturbance

12.5 Simulation Example

Fig. 12.5 Developed control command u c (t)

Fig. 12.6 Saturated control input u(t)

209

210

12 Boundary Control for an Axially Moving System with Input Restriction …

Appendix 1: Simulation Program close all; clear all; clc; nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m = 1 ; T = 100; ds=1; pi = 3.1415926; V0=0; c = 1; % -------------disburtance------------V = zeros(1,nt); d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt d(1,j)=1*cos(1*(j-1)*dt)+2*cos(2*(j-1)*dt)+3*cos(3*(j-1)*dt); V(1,j)=1+0.5*sin(2*(j)*dt); end for j = 1 : nt for i = 1 : nx f(i,j) = (10+sin(pi*(i-1)*dx*(j-1)*dt)+sin(2*pi*(i-1)*dx*(j-1)*dt)+sin(3*pi*(i -1)*dx*(j-1)*dt) )*(i-1)*dx*10^-2; end end %******************************************************************** %

uncontrolled

%******************************************************************** mc=5; w1 = zeros(nx,nt); w1(1,:) = 0;

Appendix 1: Simulation Program

211

for i=2:nx w1(i,1) = 0; w1(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((T-m*V(1,j-1)^2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dx^2 -c*(w1(i,j-1)-w1(i,j-2))/dt-c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j) ) / (m/dt^2); end w1(nx,j) = ( (2*mc/dt^2 + ds/dt )*w1(nx,j-1) - (mc/dt^2)*w1(nx,j-2) + d(1,j)+ (T/dx)*w1(nx-1,j-1)) / (mc/dt^2 + ds/dt + T/dx); end tshort = linspace(0,tmax,(nt/500+1)); xshort = linspace(0,L,nx); for j=1:nt/500 for i=1:nx w1short(i,j)=w1(i,j*500); end end w1short=[w1(:,1),w1short]; %******************************************************************** %

backstepping adaptive + controlled + disturbance observer

%******************************************************************** huah = zeros(1,nt); du = zeros(1,nt); bar_u=2; eta = 0.1; huah_0 = 0.001; k1 = 100; k2=100; k3 = 100; k4=10; w2 = zeros(nx,nt); u = zeros(1,nt); u0 = zeros(1,nt); u_b = zeros(1,nt);%e_z

212

12 Boundary Control for an Axially Moving System with Input Restriction …

d_e = zeros(1,nt); phi = zeros(1,nt); Error_d = zeros(1,nt); nu = zeros(nx,nt); f_e = zeros(nx,nt); e_f = zeros(nx,nt); varsigma1=20; varsigma2=50; w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((T-m*V(1,j-1)^2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1))/dx^2 -c*(w2(i,j-1)-w2(i,j-2))/dt - c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j) ) / (m/dt^2); nu(i,j) = nu(i,j-1) +varsigma2*dt*(-(T-m*V(1,j-1)^2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1) )/dx^2+c*(w2(i,j-1)-w2(i,j-2))/dt+c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/ dx +2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt)+ m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx -f_e(i, j-1)); f_e(i, j) =nu(i,j) + varsigma2*m*(w2(i,j) - w2(i,j-1))/dt; e_f(i, j-1)=f(i,j-1)-f_e(i, j-1); e_f(i,j) = e_f(i,j-1) - dt*varsigma2*e_f(i,j-1); end w2(nx,j) = (k4*huah(1,j-1)+ du(1,j)+ d(1,j) - d_e(1,j-1) + ds*w2(nx,j-1)/dt+ds*k1*w2(nx-1,j)/dx+mc*(2*w2(nx,j-1)-w2(nx,j-2))/dt^ 2+mc*k1*(w2(nx,j-1)+w2(nx-1,j)-w2(nx-1,j-1))/(dx*dt)+eta*T*w2(nx-1,j) /dx+ k3*w2(nx,j-1)/dt + k1*k3*w2(nx-1,j)/dx ) /( mc/dt^2 + ds/dt + ds*k1/dx + mc*k1/(dx*dt)+ eta*T/dx+ k3/dt + k1*k3/dx ); u_b(1,j) = (w2(nx,j) - w2(nx,j-1))/dt + k1*(w2(nx,j) w2(nx-1,j))/dx ; if abs(huah(1,j-1))>=huah_0 huah(1,j)=huah(1,j-1)+dt*(-k2*huah(1,j-1)-(u_b(1,j)*du(1,j)

Appendix 1: Simulation Program

213

+0.5*(du(1,j))^2)/huah(1,j-1)+du(1,j)); else if abs(huah(1,j-1))=bar_u u(1,j)=sign(u0(1,j))*bar_u; elseif abs(u0(1,j)) 0 is the boundary tension constraint and M1 , M2 > 0

13.3 Boundary Controller Design

219

are the output constraints. In order to use the backstepping technique, the following transformations are introduced. ⎧ 1 3 ⎪ mwtt = f + T0 wx x + E Awx2 wx x − cwt − cvwx − mawx − 2mvwxt − mv 2 wx x ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ w(0, t) = 0 ⎪ ⎨ z 1 = w(L , t) ⎪ ⎪ ⎪ ⎪ z 1t = z 2 = wt (L , t) ⎪   ⎪ ⎪ ⎪ 1 1 1 ⎪ 3 ⎩ z 2t = u + d − ds z 2 − T0 wx (L , t) − E Awx (L , t) mc 2 2 (13.4) Define ze = z2 − zv

(13.5)

where z e is the error variable, and z v is the virtual control.

13.3.1 Step One Consider the Lyapunov function candidate as V1 (t) = V3 (t) + V4 (t)

(13.6)

where the energy term V3 (t) is defined as: βm V3 (t) = 2

L

β (wt + vwx ) d x + 4

L T wx2 d x

2

0

(13.7)

0

and a small crossing term V4 (t) is defined as: L xwx (wt + vwx )d x

V4 (t) = ηm

(13.8)

0

where η, β > 0. Applying Eqs. (2.11)–(2.15) to Eq. (13.8) results in: ηm L |V4 (t)| ≤ 2

L 0

2  wx + (wt + vwx )2 d x ≤ λ1 V3 (t)

(13.9)

220

13 Adaptive Neural Network Vibration Control …

where λ1 =

2μm L min(2βm, βT0 )

(13.10)

Adding V3 (t) to both sides of (13.9) yields (1 − λ1 )V3 (t) ≤ V1 (t) ≤ (1 + λ1 )V3 (t)

(13.11)

where the corresponding parameters should be chosen to satisfy (1 − λ1 ) > 0. Differentiating (13.6) yields: V1t = V3t + V4t

(13.12)

Differentiating (13.7) and substituting (13.1), we obtain L V3t = −βc

L (wt + vwx ) d x + β

(wt + vwx ) f d x −

2

0

βmv 2 wt (L , t) 2

0

 βE A βT0 v − 2βmv 3 2 wt (L , t)wx3 (L , t) + wx (L , t) − wx2 (0, t) + 2 4  βT0 − 2βmv 2 3β E Av 4 4 wx (L , t) − wx (0, t) + wt (L , t)wx (L , t) + 8 2

(13.13)

Differentiating (13.8) and substituting (13.1), we obtain ηT0 V4t = − 4

L wx2 d x

3ηE A − 8

0

−

ηm 2

L wt2 d x + 0

+

L

L wx4 d x

− ηc

0

ηm 2

xwx (wt + vwx )d x 0

L

L v 2 wx2 d x + η

0

xwx f d x −

3ηE AL 4 wx (L , t) 8

0

ηm L 2 ηT0 L − 2ηm Lv 2 wt (L , t) + wx (L , t) 2 4 2

(13.14) Substituting (13.13) and (13.14) into (13.12) gives:

13.3 Boundary Controller Design

L V1t = −βc

221

ηT0 (wt + vwx ) d x − 4

0

−

ηm 2

L wt2 d x − 0

L

L wx2 d x

2

0

3ηE A 8

L

− ηc

xwx (wt + vwx )d x 0

L wx4 d x + η

0

xwx f d x + 0

ηm 2

L v 2 wx2 d x 0

L

3β E Av 4 3E A(βv + ηL) 4 wx (L , t) − wx (0, t) 8 8 0   2βmv 2 − βT0 βE A 3 βmv − ηm L 2 − wx (L , t) − wx (L , t) z e − ze 2 2 2   2βmv 2 − βT0 βE A 3 βmv − ηm L 2 wx (L , t) − wx (L , t) z v − zv − 2 2 2 βT0 v − 2βmv 3 2 − (βmv − ηm L)z e z v − wx (0, t) 4

(βv + ηL) T0 − 2mv 2 2 wx (L , t) + 4 (13.15) +β

(wt + vwx ) f d x +

Design the virtual control z v is designed as z v = −k1 wx (L , t) where k1 > 0. Substituting (13.16) into (13.15) results in:

(13.16)

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13 Adaptive Neural Network Vibration Control …

   βT0 βE A 3 wx (L , t) − wx (L , t) z e V1t = − βmv 2 + k1 ηm L − k1 βmv − 2 2  

2 2 T0 − 2mv (2βk1 − βv − ηL) mk1 (βv − ηL) + − wx2 (L , t) 4 2 βT0 v − 2βmv 3 2 βmv − ηm L 2 3β E Av 4 se − wx (0, t) − wx (0, t) 2 8 4 L E A(4βk1 − 3βv − 3ηL) 4 wx (L , t) − ηc xwx (wt + vwx )d x − 8

−

0

+

ηm 2

L v 2 wx2 d x − 0

3ηE A 8

L wx4 d x − 0

L

L wt2 d x −

0

ηT0 4

0

L (wt + vwx )2 d x + β

− βc

ηm 2

L wx2 d x 0

L (wt + vwx ) f d x + η

0

xwx f d x 0

(13.17)

13.3.2 Step Two Consider the barrier Lyapunov function as 1 V2 = V1 + βm c z e2 φ 2

(13.18)

where φ = ln

M22 M32 + J , t)] ln [w(L M32 − wx2 (L , t) M22 − w(L , t)

M12 +{1 − J [w(L , t)]} ln 2 M1 − w(L , t)

(13.19)

where M1 , M2 , M3 > 0. −M1 and M2 are the lower bound and the upper bound of w(L , t)√respectively. M3 denotes the limit of wx (L , t), which can be calculated as M3 = 2(TM − T0 )/E A. The J (·) is defined as  J (ξ ) = The derivative of z e is

1, ξ > 0 0, ξ ≤ 0

(13.20)

13.3 Boundary Controller Design

z et =

223

  1 1 1 u + d − ds z 2 − T0 wx (L , t) − E Awx3 (L , t) + k1 wxt (L , t) mc 2 2 (13.21)

Differentiating (13.18) and substituting (13.17) and (13.21) yield   βT0 βE A 3 V2t = − βmv 2 + k1 ηm L − k1 βmv − wx (L , t) − wx (L , t) + [βds z 2 2 2   βT0 βE A 3 wx (L , t) + wx (L , t) − βk1 wxt (L , t) − βu − βd − βu 0 φ z e + 2 2  

2 T0 − 2mv (2βk1 − βv − ηL) mk12 (βv − ηL) + − wx2 (L , t) 4 2 βmv − ηm L 2 E A(4βk1 − 3βv − 3ηL) 4 z e + βmφt z e2 − wx (L , t) 2 8 L 3β E Av 4 βT0 v − 2βmv 3 2 ηT0 − wx2 d x wx (0, t) − wx (0, t) − 8 4 4

−

0

+

ηm 2

L

L v 2 wx2 d x − βc

0

ηm − 2

(wt + vwx )2 d x − ηc 0

L wt2 d x 0

L

3ηE A − 8

xwx (wt + vwx )d x 0

L

L wx4 d x

0

+β

L (wt + vwx ) f d x + η

0

xwx f d x 0

(13.22) where u = u − u 0 . To stabilize the system, we design the control law u 0 as T0 EA 3 wx (L , t) + wx (L , t) − m c k1 wxt (L , t) − k3 z e + ds z 2 − u − d 2 2  L mv 2 + k1 ηm − k1 mv − T20 wx (L , t) − E2A wx3 (L , t) β (k2 + mφt )z e + − φ φ (13.23)

u0 =

It should be noted that m, T0 , E A, ds , u and the boundary disturbance d in the designed control law are uncertain or hard to measure precisely, so we adopt the RBF neural networks to estimate the uncertainties of the system and develop a disturbance adaptation law to eliminate the effect of external boundary disturbance:

224

13 Adaptive Neural Network Vibration Control …

W1∗T H (X 1 ) + ε1 =



 T0 EA 3 wx (L , t) + wx (L , t) − m c k1 wxt (L , t) + ds z 2 φ 2 2 (13.24) W2∗T H (X 2 ) + ε2 = φu

W3∗T

(13.25)

  k1 ηm L T0 EA 3 2 wx (L , t) − H (X 3 ) + ε3 = mv + − k1 mv − w (L , t) β 2 2 x (13.26) W4∗T H (X 4 ) + ε4 = mφt z e

(13.27)

where W1∗ , W2∗ , W3∗ and W4∗ are the ideal weights, ε1 , ε2 , ε3 and ε4 are approximation errors satisfying the bound |εi | ≤ εi∗ , i = 1, 2, 3, 4. Besides, T

X 1 = z 1 , z e , wx (L , t), wx3 (L , t), wxt (L , t), z 2 , X 2 = [z 1 , z e , u 0 ]T , X 3 =

T z 1 , z e , wx (L , t), vwx (L , t), v 2 wx (L , t), wx3 (L , t) , X 4 = [z 1 , z e , φt z e ]T , and the RBF is defined as H (χ ) = [h i (χ )]T

(13.28)

  χ − ζi 2 h i (χ ) = exp − , i = 1, 2, ...n ξi2

(13.29)

Therefore, the control law is revised as k2 z e − Dˆ φ Wˆ T H (X 1 ) Wˆ 2T H (X 2 ) Wˆ 3T H (X 3 ) Wˆ 4T H (X 4 ) − − − − 1 φ φ φ φ

u 0 = −k3 z e −

(13.30)

where Wˆ i are estimates of Wi∗ , i = 1, 2, 3, 4, and the updating laws are defined as

where ι1 , ι2 , ι3 , ι4 > 0.

Wˆ 1t = β H (X 1 )se − ι1 Wˆ 1

(13.31)

Wˆ 2t = β H (X 2 )se − ι2 Wˆ 2

(13.32)

Wˆ 3t = β H (X 3 )se − ι3 Wˆ 3

(13.33)

Wˆ 4t = β H (X 4 )se − ι4 Wˆ 4

(13.34)

13.4 Stability Analysis

225

Dˆ is the estimate of d, and its updating law is Dˆ t = −k4 Dˆ + βφz e

(13.35)

Substituting (13.30) and (13.35) into (13.22) yields V2t = βz e

  4    ˜ e − βk2 + βmv − ηm L z e2 εi − W˜ iT H (X i ) − β Dφz 2 i=1

βT0 v − 2βmv 3 2 3β E Av 4 wx (0, t) − wx (0, t) − βk3 z e2 φ − 4 8  

T0 − 2mv 2 (2βk1 − βv − ηL) mk12 (βv − ηL) + − wx2 (L , t) 4 2 E A(4βk1 − 3βv − 3ηL) 4 wx (L , t) − ηc − 8

L xwx (wt + vwx )d x

(13.36)

0

3ηE A − 8

L wx4 d x

ηT0 − 4

0

L wx2 d x 0

L

wt2 d x

ηm + 2

0

L (wt + vwx )2 d x + β

− βc

ηm − 2

L

0

L v 2 wx2 d x 0

L (wt + vwx ) f d x + η

0

xwx f d x 0





i = W i − Wi∗ , i = 1, 2, 3, 4 and D  = D − d. where W

13.4 Stability Analysis Consider the following positive definite Lyapunov function: 1 ˜ ˜T 1 Wi Wi + D˜ 2 2 i=1 2 4

V = V2 +

Differentiating (13.37), substituting (13.36) yields:

(13.37)

226

13 Adaptive Neural Network Vibration Control …

    4 1 ˜2 βmv − ηm L  2 Vt ≤ − k4 − δi z e2 D − βk3 z e φ − βk2 + − 2 2 2 i=1 i=1  

2 2 T0 − 2mv (2βk1 − βv − ηL) mk1 (βv − ηL) − + wx2 (L , t) 4 2 4  ιi

W˜ iT W˜ i

E A(4βk1 − 3βv − 3ηL) 4 βT0 v − 2βmv 3 2 wx (0, t) − wx (L , t) 4 8  L  L 3β E Av 4 ηm Lηc 2 wx (0, t) − − wt d x − βc − βδ5 − (wt + vwx )2 d x 8 2 δ6 −

0

 −

0

ηT0 ηmvm2 − − Lηcδ6 − Lηδ7 4 2

 L wx2 d x −

3ηE A 8

0

 +

L wx4 d x 0

 L  4  

ιi ∗T ∗ β 2 1 Lη β 2 Wi Wi + εi + a12 + a22 β + f dx + δ5 δ7 2 δ 2 i i=1 0

(13.38) It is noted that |εi | ≤ εi∗ , I = 1, 2, 3, 4. Combined with Assumption 13.1 and (13.11), we can obtain L Vt ≤

−τ1 z e2

−

τ2 wx2 (L , t)

−

τ3 wx4 (L , t)

−

τ4 wx2 (0, t)

− τ5

wx2 d x 0

L −τ6

(wt + vwx )2 d x − τ7 D˜ 2 − βk3 z e2 φ −

0

−

3ηE A 8

ηm 2

L wt2 d x 0

L 0

 ιi 3β E Av 4 wx (0, t) − W˜ iT W˜ i + C 8 2 i=1 4

wx4 d x −

≤ −λ2 V3 − βk3 z e2 φ −

4  ιi i=1

≤ −λ3 V1 − βk3 z e2 φ −

2

4  ιi i=1

2

(13.39)

W˜ iT W˜ i − τ7 D˜ 2 + C W˜ iT W˜ i − τ7 D˜ 2 + C

≤ −λ4 V + C  

4  ιi ∗T ∗ β 2  1 2 La 2f + i=1 W Wi + δi εi + 2 a1 + a22 , λ2 = where C = δβ5 + Lη δ7 2 i  

2τ6 4τ5 3η λ2 min βm , βT0 , β , λ3 = 1+λ , λ4 = min λ3 , 2km3 , ι1 , ι2 , ι3 , ι4 , 2τ7 , and the 1

13.4 Stability Analysis

227

corresponding parameter values are chosen to satisfy the following conditions: ⎧ βmv − ηm L ⎪ ⎪ τ1 = βk2 + − δ1 − δ2 − δ3 − δ4 ≥ 0 ⎪ ⎪ 2 ⎪ ⎪

⎪ ⎪ T0 − 2mv 2 (2βk1 − βv − ηL) mk12 (βv − ηL) ⎪ ⎪ ⎪ = τ + ≥0 ⎪ 2 ⎪ 4 2 ⎪ ⎪ ⎪ ⎪ E A(4βk1 − 3βv − 3ηL) ⎪ ⎪ τ3 = ≥0 ⎪ ⎪ ⎪ 8 ⎪ ⎪ ⎪ 3 ⎪ ⎨ τ = βT0 v − 2βmv ≥ 0 4 4 ⎪ ⎪ ⎪ ηmvm2 ηT0 ⎪ ⎪ − − Lηcδ6 − Lηδ7 > 0 τ = 5 ⎪ ⎪ 4 2 ⎪ ⎪ ⎪ ⎪ Lηc ⎪ ⎪ >0 τ6 = βc − βδ5 − ⎪ ⎪ ⎪ δ6 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ τ7 = k4 − > 0 ⎪ ⎪ 2 ⎪ ⎪ ⎩ 1 − λ1 > 0

(13.40)

Theorem 13.1 For the axially moving accelerated/decelerated belt system under the designed adaptive neural network boundary control law (13.30), given the initial condition −M1 < w(L , 0) < M2 , we can make the following conclusions. The axially moving accelerated/decelerated belt system is uniformly ultimately bounded. Proof: Multiplying (13.39) by eλ4 t and integrating the resulting equation result in.  ∂

V (t)eλ4 t ≤ Ceλ4 t ∂t

(13.41)

C −λ4 t C C ]e + ≤ V (0)e−λ4 t + λ4 λ4 λ4

(13.42)

Vt (t)eλ4 t ≤ −λ4 V (t)eλ4 t + Ceλ4 t ⇒ Integrating the above inequality yields: V (t) ≤ [V (0) −

From (13.1) and (13.7), using Eq. (2.11)–(2.15), we can obtain: L w ≤L

wx2 d x ≤

2

0

4L 4L V (t) V3 (t) ≤ βT0 βT0 (1 − λ1 ) D˜ 2 ≤ 2V (t)

Form (13.42) to (13.44), we have:

(13.43)

(13.44)

228

13 Adaptive Neural Network Vibration Control …



  4L C V (0)e−λ4 t + βT0 (1 − λ1 ) λ4      C  ˜ −λ t 4 +  D  ≤ 2 V (0)e λ4

|w| ≤

(13.45)

(13.46)

where ∀(x, t) ∈ [0, L] × [0, +∞) and ∀t ∈ [0, +∞), from above inequalities, we have:  4LC (13.47) lim |w| ≤ t→∞ βT0 λ4 (1 − λ1 )    2C  ˜ lim  D  ≤ (13.48) t→∞ λ4 Theorem 13.2 From (13.42), we can know that V (t) is bounded, so V2 (t) is bounded. Hence V2 (t) → ∞ as w(L , t) → −M1 or w(L , t) → M2 or wx (L , t) → M3 , given w(L , 0) ∈ (−M1 , M2 ) and wx (L , 0) ∈ (−M3 , M3 ), it can be concluded that w(L , t) ∈ (−M1 , M2 ) and T (L , t) ≤ TM .

13.5 Simulation Example By using the finite difference method (FDM) [7], numerical simulations are carried out to demonstrate the effectiveness of the proposed control scheme. The system parameters of the moving accelerated/decelerated belt system are given as in Table 13.1. The maximum acceleration and deceleration aa = ad = 0.5g(g = 9.8 N/kg) and [t1 , t2 , t3 , t4 , t5 , t6 , t7 ] = [1, 2, 3, 7, 8, 9, 10]s. The boundary disturbance and distributed disturbance are given as Table 13.1 Parameters of the axially moving belt system Parameter

Value

Parameter

Value

Parameter

Value

m

1.0 kg/m

mc

c

1.0 Ns/m2

ds

5.0 kg

L

1.0 m

0.25 Ns/m

T0

200 N

EA

5000 N

β

200

TM

225 N

M1

0.005

M2

0.03

M3

0.01

v0

0

uM

500 N

k1

1000

k2

500

k3

10

k4

10

13.5 Simulation Example

229

⎧   3  ⎪ 1 ⎪ ⎪ f (x, t) = sin(iπ xt) 10x + i ⎪ ⎪ ⎨ 10 i=1 3 ⎪  ⎪ ⎪ ⎪ d(t) = 10 + 2i sin(2it) ⎪ ⎩

(13.49)

i=1

The initial conditions are given as w(x, 0) = wt (x, 0) = 0

(13.50)

From Fig. 13.1, it is obvious that the belt, without control, vibrates irregularly under the effects of boundary disturbance and distributed disturbance. After the controller is implemented, the vibration of the belt is well suppressed as shown in Fig. 13.2. Figure 13.3 shows the details of the control performance, the vibration displacements at x = L with and without control are compared, and it can be known that the displacement reduces significantly under the proposed control, and the maximum of w(L , t) is less than 0.05 m. Thus, we can conclude that the vibration problem is well dealt with. As for the output constraints, the boundary displacement w(L , t) remains in the bound as shown in Fig. 13.4. Besides, Fig. 13.5 shows the trend of tension with and without control. The tension exceeds the limit significantly in the uncontrolled situation, but after the controller is executed, the tension is maintained near T0 and never break the constraint. The trends of neural network weights while approaching unknown terms are shown in Fig. 13.6. The control input is effected by input nonlinearity as shown in Fig. 13.7, but the proposed controller can still maintain the stability of the axially moving belt system.

230

13 Adaptive Neural Network Vibration Control …

Fig. 13.1 Vibration displacement of the belt without control

Fig. 13.2 Vibration displacement of the belt with control

13.5 Simulation Example

Fig. 13.3 Vibration displacement of the belt at x = L

Fig. 13.4 Constraint of w(L , t)

231

232

13 Adaptive Neural Network Vibration Control …

Fig. 13.5 Constraint of tension

Fig. 13.6 Norm of the neural network weights W1 , W2 , W3 and W4

13.5 Simulation Example

Fig. 13.7 Control input with input nonlinearity

233

234

13 Adaptive Neural Network Vibration Control …

Appendix 1: Simulation Program close all; clear all; clc; q=@(x)1-stepfun(sign(-x),0); nx =50; nt =100*10^3; L = 1; tmax =10; % --- Compute mesh spacing and time step dx = L/(nx-1); dt = tmax/(nt-1); % --- Create arrays to save data for export x = linspace(0,L,nx); t = linspace(0,tmax,nt); % --- parameters m = 1;%%rho EA = 5000; T0 = 200; pi = 3.1415926; V0=0; c = 1; ds=0.25; mc=5; %%m % --------------velocity----------------V = zeros(1,nt); V(1,1) = V0; J=0.5*9.8; for j=2:10*10^3-1 V(1,j) = V(1,1)+0.5*J*(j*dt)^2; end V(1,10*10^3)=V(1,1)+0.5*J; for j=(10*10^3+1):20*10^3-1 V(1,j)=V(1,10*10^3)+J*(j*dt-1); end V(1,20*10^3)=V(1,10*10^3)+J; for j=(20*10^3+1):30*10^3-1 V(1,j) = V(1,20*10^3)+J*(j*dt-2)-0.5*J*(j*dt-2)^2; end V(1,30*10^3)=V(1,20*10^3)+0.5*J; for j=(30*10^3+1):70*10^3-1 V(1,j)=V(1,30*10^3);

Appendix 1: Simulation Program

235

end V(1,70*10^3)=V(1,30*10^3); for j=(70*10^3+1):80*10^3-1 V(1,j)= V(1,70*10^3)-0.5*J*(j*dt-7)^2; end V(1,80*10^3)=V(1,70*10^3)-0.5*J; for j=(80*10^3+1):90*10^3-1 V(1,j)=V(1,80*10^3)-J*(j*dt-8); end V(1,90*10^3)=V(1,80*10^3)-J; for j=(90*10^3+1):nt-1 V(1,j)=V(1,90*10^3)-J*(j*dt-9)+0.5*J*(j*dt-9)^2; end V(1,100*10^3)=V(1,90*10^3)-0.5*J; % -------------disburtance------------d = zeros(1,nt); f = zeros(nx,nt); for j=1:nt d(1,j)=10+2*sin(2*(j-1)*dt)+4*sin(4*(j-1)*dt)+6*sin(6*(j-1)*dt); end for j = 1 : nt for i = 1 : nx f(i,j) = 0.1*(10*(i-1)*dx+sin(pi*(i-1)*dx*(j-1)*dt)+2*sin(2*pi*(i-1)*dx*(j-1)* dt)+3*sin(3*pi*(i-1)*dx*(j-1)*dt)); end end %******************************************************************** %

uncontrolled

%******************************************************************** w1 = zeros(nx,nt); w1lx = zeros(1,nt); T1 = zeros(1,nt); TM = 225*ones(1,nt); T1(1) = T0; T1(2) = T1(1); w1(1,:) = 0; for i=2:nx w1(i,1) = 0; w1(i,2) = 0;

236

13 Adaptive Neural Network Vibration Control …

end for j = 3 : nt for i = 2 : nx - 1 w1(i,j) = 2*w1(i,j-1)-w1(i,j-2) +((0.5*T0+1.5*EA*(w1(i,j-1)-w1(i-1,j-1))^2/dx^2 -m*V(1,j-1)^2)*(w1(i+1,j-1)-2*w1(i,j-1)+w1(i-1,j-1))/dx^2 -c*(w1(i,j-1)-w1(i,j-2))/dt - c*V(1,j-1)*(w1(i,j-1)-w1(i-1,j-1))/dx -2*V(1,j-1)*m*(w1(i,j-1)-w1(i,j-2)-w1(i-1,j-1)+w1(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w1(i,j-1)-w1(i-1,j-1))/dx + f(i,j) ) / (m/dt^2); end w1(nx,j) = ( (2*mc/dt^2 + ds/dt )*w1(nx,j-1) - (mc/dt^2)*w1(nx,j-2) +d(1,j)+(0.5*T0/dx)*w1(nx-1,j-1)-0.5*EA*(w1(nx,j-1)-w1(nx-1,j-1))^3/( dx)^3 ) / (mc/dt^2 + ds/dt + 0.5*T0/dx); w1lx(j)=(w1(nx,j)-w1(nx-1,j))/dx; %w_x(L,t) T1(j) = T0 + 0.5*EA*w1lx(j)^2; end tshort = linspace(0,tmax,(nt/500+1)); xshort = linspace(0,L,nx); for j=1:nt/500 for i=1:nx w1short(i,j)=w1(i,j*500); end end w1short=[w1(:,1),w1short]; figure (1); surf(tshort,xshort,w1short); shading interp; xlabel('Time [s]','Fontsize',14); ylabel('x [m]','Fontsize',14); zlabel('$\omega$(x,t)[m]','interpreter','latex','Fontsize',14); set(gca,'YDir','reverse') %******************************************************************** %

controlled + disturbance observer

%******************************************************************** k1 = 1000; k2 = 500; k3 = 10; k4 = 10; beta = 200;

Appendix 1: Simulation Program M1 = 0.005; M2 = 0.03; M3 = 0.01; m1 = -M1*ones(1,(nt/500+1)); m2 = M2*ones(1,(nt/500+1)); m3 = M3*ones(1,(nt/500+1)); um = 500; umax = 500*ones(1,(nt/500+1)); w2 = zeros(nx,nt); w2lx = zeros(1,nt); u = zeros(1,nt); tu = zeros(1,nt); du = zeros(1,nt); de = zeros(1,nt); net1=zeros(1,nt); net2=zeros(1,nt); net3=zeros(1,nt); net4=zeros(1,nt); tar1=zeros(1,nt); tar2=zeros(1,nt); tar3=zeros(1,nt); tar4=zeros(1,nt); W1_0=zeros(9,1); W1=zeros(9,1); dW1=zeros(9,1); Wn1=zeros(1,nt); W2_0=zeros(9,1); W2=zeros(9,1); dW2=zeros(9,1); Wn2=zeros(1,nt); W3_0=zeros(9,1); W3=zeros(9,1); dW3=zeros(9,1); Wn3=zeros(1,nt); W4_0=zeros(9,1); W4=zeros(9,1); dW4=zeros(9,1); Wn4=zeros(1,nt); T2 = zeros(1,nt); T2(1) = T0;

237

238

13 Adaptive Neural Network Vibration Control …

T2(2) = T0; w2(1,:) = 0; for i=2:nx w2(i,1) = 0; w2(i,2) = 0; end for j = 3 : nt for i = 2 : nx - 1 w2(i,j) = 2*w2(i,j-1)-w2(i,j-2) +((0.5*T0+1.5*EA*(w2(i,j-1)-w2(i-1,j-1))^2/dx^2 -m*V(1,j-1)^2)*(w2(i+1,j-1)-2*w2(i,j-1)+w2(i-1,j-1))/dx^2 -c*(w2(i,j-1)-w2(i,j-2))/dt - c*V(1,j-1)*(w2(i,j-1)-w2(i-1,j-1))/dx -2*V(1,j-1)*m*(w2(i,j-1)-w2(i,j-2)-w2(i-1,j-1)+w2(i-1,j-2))/(dx*dt) -m*(V(1,j-1)-V(1,j-2))/dt*(w2(i,j-1)-w2(i-1,j-1))/dx + f(i,j) ) / (m/dt^2); end w2(nx,j) = ( tu(1,j-1) + (2*mc/dt^2 + ds/dt )*w2(nx,j-1) (mc/dt^2)*w2(nx,j-2) + d(1,j) + (0.5*T0/dx)*w2(nx-1,j-1) -0.5*EA*(w2(nx,j-1)-w2(nx-1,j-1))^3/(dx)^3 ) / (mc/dt^2 + ds/dt + 0.5*T0/dx); se=(w2(nx,j)-w2(nx,j-1))/dt+k1*(w2(nx,j)-w2(nx-1,j))/dx; w2lx(j)=(w2(nx,j)-w2(nx-1,j))/dx; %w_x(L,t) phi=q(w2(nx,j))*log(M2^2/((M2)^2-(w2(nx,j))^2)) +(1-q(w2(nx,j)))*log(M1^2/((M1)^2-(w2(nx,j))^2))+log(M3^2/((M3)^2-(w2 lx(j))^2)); phit= q(w2(nx,j))*w2(nx,j)*(w2(nx,j)-w2(nx,j-1))/(dt*(M2^2-w2(nx,j)^2)) +(1-q(w2(nx,j)))*w2(nx,j)*(w2(nx,j)-w2(nx,j-1))/(dt*(M1^2-w2(nx,j)^2) )+w2lx(j)*(w2lx(j)-w2lx(j-1))/(dt*(M3^2-w2lx(j)^2)); %%%%%%%%%%%%%%%%%%%%%%%RBF1%%%%%%%%%%%%%%%%%% zeta1=10*[-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2; -3 -2 -1 -0.5 0 0.5 1 2 3; -0.3 -0.2 -0.1 -0.05 0 0.05 0.1 0.2 0.3; -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002; -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2; -0.05 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.05; ]; chi1=[w2(nx,j) se (w2(nx,j)-w2(nx-1,j))/dx (w2(nx,j)-w2(nx-1,j))^3/dx^3 (w2(nx,j)-w2(nx-1,j)-w2(nx,j-1)+w2(nx-1,j-1))/(dt*dx)

Appendix 1: Simulation Program

239

(w2(nx,j)-w2(nx,j-1))/dt]'; h1=zeros(9,1); ksi1=10; for jj=1:1:9 h1(jj)=exp(-norm(chi1- zeta1(:,jj))^2/( ksi1^2)); end iota1=0.2; H1=[h1(1) h1(2) h1(3) h1(4) h1(5) h1(6) h1(7) h1(8) h1(9)]'; W1=W1_0+dW1*dt; W1_0=W1; net1(j)=W1'*H1; Wn1(j)=norm(W1); dW1=beta*H1*se-iota1*W1; %%%%%%%%%%%%%%%%%%%%%END1%%%%% %%%%%%%%%%%%%%%%%%%%%%%RBF2%%%%%%%%%%%%%%%%%% zeta2=1*[-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2; -3 -2 -1

-0.5 0 0.5 1 2 3;

-10 -5 -2 -1 0 1 2 5 10;]; chi2=[w2(nx,j) se du(1,j-1)]'; h2=zeros(9,1); ksi2=5; for jj=1:1:9 h2(jj)=exp(-norm(chi2- zeta2(:,jj))^2/( ksi2^2)); end iota2=2; H2=[h2(1) h2(2) h2(3) h2(4) h2(5) h2(6) h2(7) h2(8) h2(9)]'; W2=W2_0+dW2*dt; W2_0=W2; net2(j)=W2'*H2; Wn2(j)=norm(W2); dW2=beta*H2*se-iota2*W2; %%%%%%%%%%%%%%%%%%%%%END%%%%% %%%%%%%%%%%%%%%%%%%%%%%RBF3%%%%%%%%%%%%%%%%%% zeta3=1*[-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2;%20 -3 -2 -1

-0.5 0 0.5 1 2 3;%30

-0.3 -0.2 -0.1 -0.05 0 0.05 0.1 0.2 0.3; -30 -20 -10 -5 0 5 10 20 30; -300 -200 -100 50 0 50 100 200 300; -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002;]; chi3=[w2(nx,j) se (w2(nx,j)-w2(nx-1,j))/dx

240

13 Adaptive Neural Network Vibration Control … V(1,j)*(w2(nx,j)-w2(nx-1,j))/dx V(1,j)^2*(w2(nx,j)-w2(nx-1,j))/dx (w2(nx,j)-w2(nx-1,j))^3/dx^3]'; h3=zeros(9,1); ksi3=2; for jj=1:1:9 h3(jj)=exp(-norm(chi3- zeta3(:,jj))^2/( ksi3^2)); end iota3=2; H3=[h3(1) h3(2) h3(3) h3(4) h3(5) h3(6) h3(7) h3(8) h3(9)]'; W3=W3_0+dW3*dt; W3_0=W3; net3(j)=W3'*H3; Wn3(j)=norm(W3); dW3=beta*H3*se-iota3*W3; %%%%%%%%%%%%%%%%%%%%%END%%%%% %%%%%%%%%%%%%%%%%%%%%%%RBF4%%%%%%%%%%%%%%%%%% zeta4=1*[-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2; -3 -2 -1

-0.5 0 0.5 1 2 3;

-50 -20 -10 -5 0 5 10 20 50;]; chi4=[w2(nx,j) se phit*se]'; h4=zeros(9,1); ksi4=5; for jj=1:1:9 h4(jj)=exp(-norm(chi4- zeta4(:,jj))^2/( ksi4^2)); end iota4=10; H4=[h4(1) h4(2) h4(3) h4(4) h4(5) h4(6) h4(7) h4(8) h4(9)]'; W4=W4_0+dW4*dt; W4_0=W4; net4(j)=W4'*H4; Wn4(j)=norm(W4); dW4=beta*H4*se-iota4*W4; %%%%%%%%%%%%%%%%%%%%%END%%%%% de(j)=de(j-1)+dt*(-k4*de(j-1)+beta*phi*se); u(1,j) = -k3*se-k2*se/phi-(net1(j)+net2(j)+net3(j)+net4(j))/phi-de(j); if(u(1,j)>=um||u(1,j)