Fault-Tolerant Attitude Control of Spacecraft 9780323898638, 0323898637

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Fault-Tolerant Attitude Control of Spacecraft

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Fault-Tolerant Attitude Control of Spacecraft

Qinglei Hu School of Automation Science and Electrical Engineering Beihang University Beijing, China

Bing Xiao School of Automation Northwestern Polytechnical University Xi’an, China

Bo Li Institute of Logistics Science and Engineering Shanghai Maritime University Shanghai, China

Youmin Zhang Department of Mechanical, Industrial, and Aerospace Engineering Concordia University Montreal, Canada

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-323-89863-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Glyn Jones Editorial Project Manager: Rafael G. Trombaco Production Project Manager: Sojan P. Pazhayattil Designer: Matthew Limbert Typeset by VTeX

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Contents

List of Figures List of Tables Biography Preface Acknowledgments

1.

Overview 1.1 Introduction 1.2 Fault analysis of spacecraft 1.2.1 General analysis 1.2.2 Analysis to faults in ACS 1.3 Fault-tolerant control systems 1.4 Review of FDD for spacecraft 1.4.1 Model-based FDD approaches 1.4.2 Data driven-based FDD schemes 1.5 Spacecraft attitude fault-tolerant control engineering 1.5.1 Engineering techniques for spacecraft FTC 1.5.2 Discussions 1.6 Review of spacecraft attitude fault-tolerant control 1.6.1 Attitude FTC design using adaptive control 1.6.2 Sliding mode-based attitude FTC methodologies 1.6.3 Control allocation-based attitude FTC 1.7 Open problems in spacecraft attitude fault-tolerant control 1.7.1 Without considering actuator nonlinearities 1.7.2 Having great conservativeness 1.7.3 Requiring angular velocity measurements 1.7.4 Without attitude fast slewing capability 1.8 Organization of this book

2.

xi xv xvii xix xxi

1 1 1 4 6 9 10 11 13 13 15 15 16 17 17 19 19 19 20 20 21

Preliminaries 2.1 Introduction 2.2 Mathematical notations 2.3 Definitions and preliminary lemmas

23 24 24 vii

viii Contents

2.4 Modeling of spacecraft attitude control system 2.4.1 Coordinate frames 2.4.2 Attitude kinematics 2.4.3 Spacecraft dynamics 2.5 Modeling of actuator faults 2.5.1 Reaction wheel faults 2.5.2 Mathematical model of RW faults 2.6 Summary

3.

Robust fault-tolerant attitude control 3.1 Introduction 3.2 Adaptive sliding-mode-based attitude FTC 3.2.1 Problem statement 3.2.2 Adaptive integral sliding-mode FTC law 3.2.3 Numerical example 3.3 Robust fault tolerant attitude stabilization control 3.3.1 Problem statement 3.3.2 Robust fault tolerant controller design 3.3.3 Robust fault tolerant controller design with actuator faults 3.3.4 Simulation example 3.4 Robust H∞ attitude tracking FTC 3.4.1 Attitude tracking control system and control problem 3.4.2 Adaptive sliding-mode FTC with H∞ performance 3.4.3 Simulation example 3.5 Summary

4.

37 38 38 39 46 53 53 54 57 59 63 65 67 74 80

Fault-tolerant attitude control with actuator saturation 4.1 Introduction 4.2 Sliding-mode attitude stabilization FTC 4.2.1 Problem formulation 4.2.2 Fault-tolerant sliding-mode controller design 4.2.3 Simulation results 4.3 Dynamic sliding-mode attitude stabilization FTC 4.3.1 Problem formulation 4.3.2 Main result 4.3.3 Simulation example 4.4 Fault estimation-based attitude FTC 4.4.1 Problem formulation 4.4.2 Active attitude FTC with loss of actuator effectiveness 4.4.3 Simulation example 4.5 Summary

5.

27 27 27 29 30 30 33 36

81 82 82 83 88 93 93 93 102 106 106 107 117 123

Fault-tolerant velocity-free attitude control 5.1 Introduction 5.2 Velocity-free attitude stabilization FTC

125 126

Contents ix

5.2.1 Sliding-mode observer-based FTC 5.2.2 Simulation results 5.3 Filter-based velocity-free attitude FTC 5.3.1 Attitude FTC without angular velocity magnitude 5.3.2 Numerical simulation 5.4 Attitude stabilization FTC with actuator saturation and partial loss of control effectiveness 5.4.1 Problem statement 5.4.2 Velocity filter design 5.4.3 Fault-tolerant attitude stabilization control design 5.4.4 Numerical example 5.5 Summary

6.

175 176 176 178 183 189 189 190 197 207

Active fault-tolerant attitude control 7.1 Introduction 7.2 Fault compensation attitude tracking control 7.2.1 Problem formulation 7.2.2 Attitude tracking compensation controller design 7.2.3 Numerical example 7.3 Active attitude stabilization FTC without rate sensors 7.3.1 Problem formulation 7.3.2 Attitude FTC without angular velocity measurements 7.3.3 Simulations results 7.4 Finite-time fault-tolerant attitude tracking control 7.4.1 Fault estimator design 7.4.2 Sliding-mode observer design 7.4.3 Attitude FTC law design 7.5 Summary

8.

154 154 155 156 167 172

Fault-tolerant finite-time attitude-tracking control 6.1 Introduction 6.2 Attitude tracking control with actuator misalignment and fault 6.2.1 Problem statement 6.2.2 Attitude-tracking FTC design 6.2.3 Numerical example 6.3 Finite-time attitude-tracking FTC 6.3.1 Problem formulation 6.3.2 Finite-time attitude-tracking control design 6.3.3 Numerical example 6.4 Summary

7.

126 137 144 144 151

209 210 210 212 218 224 224 228 237 243 244 248 251 260

Conclusions and future work 8.1 General conclusion 8.2 Future work

261 262

x Contents

References Index

265 273

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8 Fig. 1.9 Fig. 1.10 Fig. 1.11 Fig. 1.12 Fig. 1.13 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11

Fig. 3.12

The targets to be observed by a reconnaissance spacecraft. Subsystem faults. Fault types of spacecraft. Time of fault after spacecraft launch. Fault impact on aerospace mission. Fault types of ACS. Time of fault for ACS. ACS fault impact on task. Component fault of ACS. Schematic diagram for active FTC systems. Classification of FDD schemes for spacecraft attitude control system. The Far Ultraviolet Spectroscopic Explorer satellite (Far Ultraviolet Spectroscopic Explorer, n.d.). Collision between CERISE and space debris (n.d.). Definition of the coordinate reference frames for spacecraft attitude system. The stuck fault of RW. Failure to respond to signals. The increasing friction of wheel. The decreasing speed of wheel. An exploded structure of a reaction wheel. Time responses of attitude. Time responses of attitude quaternion. Time responses of velocity. Time responses of control input. Time responses of vibration displacements. Time responses of performance index I (t). Time responses of parameter β(t). Time responses of vibration displacements. Time responses of vibration displacements. Time responses of vibration displacements. Time histories for the attitude tracking control using the proposed methods. Case 1: proposed ASMFTC with faults (solid line); Case 2: proposed ASMFTC without faults (dashed line); Case 3: proposed MASMFTC with faults (dotted line); Case 4: proposed MASMFTC without faults (dot-dash line). Time histories for the attitude tracking control using PD control. Case 1: fault case (solid line); Case 2: fault-free case (dotted line).

1 2 3 3 4 4 5 5 5 8 9 14 14 27 32 32 32 33 35 49 49 50 50 51 52 52 61 62 64

77 78

xi

xii List of Figures

Fig. 3.13 Fig. 3.14 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. 5.12 Fig. 5.13 Fig. 5.14 Fig. 5.15 Fig. 5.16 Fig. 5.17 Fig. 5.18 Fig. 5.19 Fig. 5.20 Fig. 5.21 Fig. 5.22

Time histories for the attitude tracking control using CASMC. Case 1: fault case (solid line); Case 2: fault-free case (dotted line). Time histories for the attitude tracking control using the proposed methods with saturation limits. Time response of Fi (ui ). Response for the fault-free case under FTSMC (solid line), IAFTC (dashed line), and AWPID (dotted line). Response with partial loss of thruster effectiveness fault under FTSMC (solid line), IAFTC (dashed line), and AWPID (dotted line). Thruster fault information (time responses of e1 –e6 ). Time responses with healthy actuators. Time responses with fault actuators. Active fault-tolerant control for the flexible spacecraft attitude system. Fault reconstruction with the proposed FDD with constant loss of actuator effectiveness faults. Attitude control performance with constant loss of actuator effectiveness faults. Fault reconstruction by the proposed FDD with time-varying loss of actuator effectiveness faults. Attitude control performance with time-varying loss of actuator effectiveness faults. Definition of the coordinate reference frames for spacecraft attitude system. Definition of the coordinate reference frames for spacecraft attitude system. Time response of the observer error e1 in case of fault-free. Time response of the observer error e2 in case of fault-free. The attitude-tracking error σ − σ d in case of fault-free. The attitude-tracking error ω in case of fault-free. The commanded control in case of fault-free. Time response of the observer error e1 in case of actuator fault. Time response of the observer error e2 in case of actuator fault. The attitude tracking error σ − σ d in case of actuator fault. The attitude tracking error ω in case of actuator fault. The commanded control in case of actuator fault. Time response of attitude under normal controller equation (5.68) with the fault equations (5.92)–(5.93). Time response of angular velocity under normal controller equation (5.68) with the fault equations (5.92)–(5.93). Time response of control input. Case 1: normal controller equation (5.68). Time response of control input. Case 2: fault-tolerant controller equation (5.67). Failure to respond to control signals: attitude. Failure to respond to control signals: angular velocity. Spacecraft attitude orientation with VFFTC and UQOFC in the absence of faults and disturbances. Spacecraft angular velocity with VFFTC and UQOFC in the absence of faults and disturbances. Time response of τ (t) with VFFTC and UQOFC in the absence of faults and disturbances. Spacecraft attitude orientation with VFFTC and UQOFC in the presence of faults and disturbances.

79 79 82 90 92 103 104 105 116 119 120 121 123 132 136 138 139 139 139 140 140 141 141 141 142 152 153 153 154 154 154 168 169 169 170

List of Figures xiii

Fig. 5.23 Fig. 5.24 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 6.9 Fig. 6.10 Fig. 6.11 Fig. 6.12 Fig. 6.13 Fig. 6.14 Fig. 6.15 Fig. 6.16 Fig. 6.17 Fig. 6.18 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 7.7 Fig. 7.8 Fig. 7.9 Fig. 7.10 Fig. 7.11 Fig. 7.12 Fig. 7.13 Fig. 7.14 Fig. 7.15 Fig. 7.16 Fig. 7.17 Fig. 7.18 Fig. 7.19 Fig. 7.20 Fig. 7.21 Fig. 7.22 Fig. 7.23

Spacecraft angular velocity with VFFTC (solid line) and UQOFC (dashed line) in the presence of faults and disturbances. Time response with VFFTC (solid line) and UQOFC (dashed line) in the presence of faults and disturbances. Desired attitude and the distributed target. Attitude angle-tracking error. Angular velocity-tracking error ωe (deg/s). Commanded and the applied control power. Time response of switching surface. Performance index ASCCT. Performance index POSTI. Desired attitude and distributed target. Initial response of the sliding manifold S in fault-free case. Initial response of attitude angle-tracking errors in fault-free case. Angular velocity-tracking error ωe in fault-free case. Fault scenarios of reaction wheels. Response of the sliding manifold S in case of actuator fault. Initial response of attitude angle-tracking errors with actuator fault. Initial response of velocity-tracking error ωe with actuator fault. Initial response of commanded control inputs τ with actuator fault. Performance index POSTI in the fault-free and actuator fault cases. Performance index ASCCT in the fault-free and actuator fault cases. Structure of the attitude-tracking controller. Time sequence of the attitude maneuvers for the considered mission. External disturbances reconstruction error e2 with initial overshoot during the third orbital period. Angular velocity tracking error ωe with initial overshoot during the third orbital period. Attitude-tracking error σ e during the third orbital period. Commanded control torque τ c during the third orbital period. The x 2 (solid line) and its reconstruction (dashed line) during the tenth orbital period. Faults reconstruction error e2 during the tenth orbital period. Angular velocity-tracking error ωe during the tenth orbital period. Attitude-tracking error σ e during the tenth orbital period. Commanded control torque τ c during the tenth orbital period. Operation modes of the satellite. Time sequence of the attitude maneuvers for the scientific mission. The proposed fault-tolerant attitude control system of the microsatellite without rate sensor. The commanded control input. The observer error εo2 . The fault reconstruction error ε r2 . The angular velocity of the satellite. The unit attitude quaternion of the satellite. The indexes Pm and PE in the absence of reaction wheel faults. The indexes Pm and PE in case of reaction wheel faults. The closed-loop attitude-tracking system of the velocity-free fault-tolerant control. The estimation error e2 by using (5-59) in actuator fault-free case.

171 172 185 185 186 186 187 188 188 199 200 201 202 202 203 204 204 205 206 207 212 219 219 220 220 220 221 221 221 222 222 225 225 229 240 240 240 240 240 242 242 243 257

xiv List of Figures

Fig. 7.24 Fig. 7.25 Fig. 7.26 Fig. 7.27 Fig. 7.28 Fig. 7.29

The angular velocity-tracking error with the controller (5-59) in actuator fault-free case. The attitude-tracking error with the controller (5-59) in actuator fault-free case. The commanded control of the controller (5-59) in actuator fault-free case. The angular velocity tracking error with the controller (5-59) in case of actuator fault. The attitude-tracking error with the controller (5-59) in case of actuator fault. The commanded control of the controller (5-59) in case of actuator fault.

257 257 257 258 258 259

List of Tables

Table 1.1 Table 1.2 Table 1.3 Table 2.1 Table 4.1 Table 4.2 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 6.1 Table 7.1 Table 7.2

Failures classification of the past spacecrafts. Analysis to thruster faults in four spacecraft. Analysis to momentum wheel faults in five spacecrafts. The analysis of how reaction wheel faults occur. Performance comparison of maximum value of vibration and settling time of stabilization under the three control schemes. Control performance comparisons of the three controllers ( denotes stable, × denotes unstable). Main parameters of a rigid spacecraft. Performance summary on stabilization accuracy. Performance summary on attitude-tracking pointing accuracy. Performance summary on the time required to accomplish attitude-tracking maneuver. Main parameters of an on-orbit rigid spacecraft. Controller parameters chosen for numerical analysis. Performance summary under different control schemes. The numerical parameters of the simulated spacecraft. Control performance with RFTFTC and RIAFTC. Control performance with RFTFTC and RIAFTC.

2 6 7 34 91 106 137 143 143 143 167 167 172 184 223 223

xv

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Biography

Qinglei Hu Prof. Hu received the B.Eng. degree in electrical and electronic engineering from Zhengzhou University, Zhengzhou, China, in 2001, and the Ph.D. degree in guidance and control from the Harbin Institute of Technology, Harbin, China, in 2006. From 2003 to 2014, he had been with the Department of Control Science and Engineering at the Harbin Institute of Technology, Harbin, and was promoted to the rank of Professor with tenure in 2012. He joined Beihang University, Beijing, China, in 2014 as a full Professor. He worked as a Postdoctoral Research Fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, from 2006 to 2007, and from 2008 to 2009 he visited the University of Bristol, Bristol, U.K., as a Senior Research Fellow supported by Royal Society Fellowship. After that, he visited Concordia University supported by Natural Sciences and Engineering Research Council of Canada from 2010 to 2011. He has authored or coauthored over 100 technical papers. His research interests include variable structure control and applications and fault-tolerant control and applications. Dr. Hu is an Associate Fellow of American Institute of Aeronautics and Astronautics (AIAA).

Bing Xiao Dr. Xiao received the B.S. degree in mathematics from Tianjin Polytechnic University, Tianjin, China, in 2007, and the M.S. and Ph.D. degrees in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2010 and 2014, respectively. From May 2014 to 2017, he was a Professor with the College of Engineering, Bohai University, China. He was also a postdoctoral researcher with the School of Automation, Nanjing University of Science and Technology, Nanjing, China. Since 2018, he is an Associate Professor with the School of Automation, Northwestern Polytechnical University. His research interests include spacecraft attitude control and its fault-tolerant control design. xvii

xviii Biography

Bo Li Dr. Li received the B.Eng. degree in electrical and electronic engineering from the School of Computer and Information Engineering, Henan University, Kaifeng, China, in 2010, the M.Eng. degree in aerospace engineering with specialty in navigation, guidance, and control from the Civil Aviation University of China, Tianjin, China, in 2013, and the Ph.D. degree from the Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, China, in 2016. He is currently an Associate Professor with Shanghai Maritime University, Shanghai, China. His current research interests include spacecraft attitude control, control allocation, fault diagnosis and fault-tolerant control, optimal control and optimization methods, UAV-based port monitoring, and so on.

Youmin Zhang Prof. Zhang received the B.S., M.S., and Ph.D. degrees in automatic control from Northwestern Polytechnical University, Xi’an, China, in 1983, 1986, and 1995, respectively. He is currently a Professor with the Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, Quebec, Canada. His current research interests include fault diagnosis and fault-tolerant control systems and cooperative GNC of unmanned aerial/space/ground/surface vehicles, renewable energies, smart grids, and smart cities. He was awarded as a Concordia University Research Fellow in the Strategic Research Cluster “Technology, Industry and the Environment” in 2018 in recognition of his outstanding research works and contributions. Prof. Zhang is an Editorial Board Member, Editor-in-Chief, Editor-at-Large, Editor, or Associate Editor of several international journals, including an Editorat-Large and Regional Representative of North America for Journal of Intelligent and Robotic Systems, an Editor for Unmanned Systems, an Associate Editor for IEEE Transactions on Neural Networks and Learning Systems and IET Cyber-systems and Robotics, and so on. He has served as the General Chair, the Program Chair, and the IPC Member of several international conferences. He is the President of International Society of Intelligent Unmanned Systems and a member of the Technical Committee for several scientific societies. He is a fellow of CSME and a Senior Member of AIAA and IEEE. More detailed information can be found at http://users.encs.concordia.ca/~ymzhang/.

Preface

Recently, exploration of deep space has been extensively supported by many countries and companies such as the SpaceX Co., Ltd. The growing development of different exploration missions has motivated significant requirements for the guidance, navigation, and control of spacecrafts. The demand of reliable control system and autonomous control has been increased. The Attitude control system, as one of spacecraft subsystems, plays an important role in spacecraft design. Its reliability determines whether the planned missions can be accomplished or not. However, spacecraft challenging operating conditions increase the possibility of malfunctions in sensors, actuators, and controllers. Once a spacecraft is launched, it is highly unlikely that its hardware can be repaired. Hence any component or system fault/failure cannot be fixed with replacement parts. These issues can potentially cause a host of economic, environmental, and safety problems. This strongly motivates the development of attitude control systems that ensure an efficient and timely response to maintain stability, reliability, and required performance properties even when components fail. In the aerospace industry and academia, fault-tolerant control (FTC) is a widely used technique to accommodate or manage component failures. In contrast to most conventional control systems that the controllers are designed for fault-free case without considering the possibility of fault occurrence, the main characteristics of systems designed by FTC is that it can guarantee the desirable stability and performance properties even in the event of component fault/failure. This is quite important for spacecraft attitude control system. In such a system the consequence of a minor fault/failure in a component can be catastrophic. Therefore the demand on reliability, safety, and fault tolerance is generally high. To improve the reliability and availability while providing a desirable performance, it is necessary to design spacecraft attitude control systems with FTC capability, whereas potential faults are tolerated. These types of control systems are often known as fault-tolerant control systems (FTCSs). Applying the modern control theories, a number of fault tolerant control approaches have been reported for spacecraft attitude system. However, most of them can only guarantee that the closed-loop attitude control system is stable in the presence of component faults. Attitude control performance including the fast slewing maneuver, the attitude pointing-accuracy, and the attitude stability xix

xx Preface

may not be guaranteed. Moreover, most of the existing attitude fault-tolerant control methodologies are designed by using full-state feedback. It is known that full-state feedback would not be satisfied, since fault may occur in spacecraft sensors. Hence the attitude fault-tolerant control problem is still open. The objective of this book mainly focuses on the design of autonomous faulttolerant attitude control approaches for spacecraft. The main works of this book are as follows. Fitting passive fault-tolerant control methods conceptually into the autonomous attitude control system framework, improving pointing accuracy, and establishing precise attitude maneuvers even with the failure of control actuators onboard a spacecraft. From practical viewpoint, the main work is designing an attitude control system that can achieve high-precision pointing, fast slewing maneuvers in the presence of external disturbances, model uncertainties, and actuator faults. We approach the validation of the proposed control algorithms from two perspectives, conventional theoretical analysis and numerical simulation study. Analytical studies are performed using adequate nonlinear control theory to demonstrate attitude control performance and stability. Simulation studies are used as a design tool to demonstrate system performance and compliance with design requirements and to validate established theoretical framework. The book gives a systematically and almost self-contained description of the many facets of envisaging, designing, implementing or experimentally exploring fault-tolerant attitude control of spacecraft along with adequate designs of integrated modeling, dynamics, fault-tolerant attitude control, and fault reconstruction of spacecrafts. The book is specifically focusing on modeling, attitude control considering actuator fault, and fault diagnosis for spacecraft control systems involved in aerospace engineering. It intends to provide a unified platform for understanding and applicability of spacecraft fault-tolerant attitude control for different purposes. Qinglei Hu, Bing Xiao, Bo Li, and Youmin Zhang December 31, 2020

Acknowledgments

The contents included in this book are the outgrowth and summary of the authors’ academic researches to address the fault-tolerant control problem of spacecraft in the past several years. It was partially supported by National Natural Science Foundation of China under Grants This book was supported partially by the National Natural Science Foundation of China under Grants 61960206011, 61873207, and 61633003, Beijing Natural Science Foundation (JQ19017), and the Major Project of Science and Technology Innovation 2030–“New Generation Artificial Intelligence” under Grant 2018AAA0102700. The authors greatly appreciate the above financial support. The authors would like to thank the cooperators Prof. Guangfu Ma, Prof. Maruthi R. Akella, and the students Xiao Tan, Guanlin Niu, Yueyang Liu, Wenquan Gong, Wei Chen, et al., for their contributions and efforts dedicated to this book.

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Chapter 1

Overview 1.1 Introduction It is known that the components of any spacecraft may fail due to the harsh space environment and their aging. If the failure cannot be accommodated online and in real time, then the spacecraft attitude control performance may be deteriorated. This would lead the whole planned task to fail. Taking a spacecraft tasked with reconnaissance as an example, as shown in Fig. 1.1, if its component fails when observing the target point A and this failure is not handled in time, then the spacecraft may fail to observe the target points such as B, C, and D, and finally the entire reconnaissance mission will fail. Therefore the attitude fault-tolerant control system should be developed for spacecraft to guarantee its reliability. Motivated by this, the development of fault-tolerant attitude control methodologies has attracted more and more attention. Moreover, applying modern control theories, this book aims to establish new fault-tolerant attitude control approaches for spacecraft to improve its autonomy and reliability.

FIGURE 1.1 The targets to be observed by a reconnaissance spacecraft.

1.2 Fault analysis of spacecraft 1.2.1 General analysis During the design and manufacture of spacecraft on the ground, modern technology and quality management techniques are widely applied to avoid and eliminate potential faults. However, after long-time operating in the harsh space Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00011-3 Copyright © 2021 Elsevier Inc. All rights reserved.

1

2 Fault-Tolerant Attitude Control of Spacecraft

environment with varying temperatures and intense radiation, system failures or abnormal operation are inevitable for spacecraft. For example, the solar panel may be destroyed and the onboard software as well as the telemetry commands may be abnormal. We have collected and analyzed spacecraft data from databases such as the Satellite Encyclopedia (TSE), Satellite News Digest, Mission and Spacecraft Library, Airclaims Space Trak, Space Systems Engineering Database (SSED), and the Mission Failure Analysis for NASA AMES Research Center. Moreover, in this book, we will analyze the failures of all types of spacecraft launched by the United States, European countries, Canada, Japan, and other countries since 1990, and classify them according to the location of the failure and its impact on the spacecraft, as listed in Table 1.1. TABLE 1.1 Failures classification of the past spacecrafts. Categories

Values

The impact of the failures

Total lost of control, partial lost of control, mission termination, life span reduced, and control performance deterioration

Subsystem failures

Attitude Control System (ACS), Command/Data Processing System (CDH), Telemetry System (TTC), Energy System (Power), Mechanical System (MECH), and Others

FIGURE 1.2 Subsystem faults.

Based on the collected information from the faulty spacecraft provided by the above databases, the subsystem faults are further classified with the statistical results shown in Fig. 1.2. The failures of the attitude control and the energy subsystems account for 59% of the total failures. The types of spacecraft of faults can be further classified into four types failures: the mechanical failure, the electrical failure, the software failure, and other unknown failures. The mechanical failure mainly refers to the mechanical structure deformation caused by temperature changes, external force, friction, and pressure. The electrical failure is induced by power overload, short circuits, and abnormal battery power generation. The software failure mainly consists of incorrect computer instructions

Overview Chapter | 1

3

FIGURE 1.3 Fault types of spacecraft.

and onboard software abnormalities. Fig. 1.3 shows the percentage of various fault types in the total number of faulty spacecraft. It is seen that the electrical fault accounts for 45% of the total faults. It is much higher than the 32% mechanical faults. This is because the number of electronic devices in modern spacecraft is far larger than the mechanical parts. Therefore, the probability of electrical component fault is higher than the probability of mechanical component fault. In particular, the aging of some electrical equipment, such as solar cell arrays, causes them to fail easily. Simultaneously, the space environment such as solar radiation and electromagnetic storms may cause the spacecraft electrical equipment failure to a large extent.

FIGURE 1.4 Time of fault after spacecraft launch.

Further analysis is carried out to the faulty time after the spacecraft was launched into orbit. From Fig. 1.4, we see that almost 41% of the faults occurred in the orbital space missions within one year. The impact of faults on the space missions illustrated in Fig. 1.5 shows that nearly 40% of the faults are fatal. It lets the spacecraft be totally lost of control and its space missions fail. Nearly 65% of the faults result in partial loss of space missions. Those data indicate that the aerospace engineers cannot deal with all on-orbit faults. Therefore, the development of methods for handling on-orbit faults are particularly important.

4 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 1.5 Fault impact on aerospace mission.

1.2.2 Analysis to faults in ACS The attitude control system is one of the essential subsystems of any spacecraft. It consists of actuators, sensors, and controller. The electrical and mechanical parts of those devices may suffer from malfunctions after the long-time operation. We find in Fig. 1.2 that the faults of the attitude control system account for 32% of the total faults. Hence in this part, we focus on the analysis of the fault types of the attitude control system.

FIGURE 1.6 Fault types of ACS.

Comparing Figs. 1.3 and 1.6, we can see that fault types of the attitude control system are very different from those of other subsystems. 54% of the faults are mechanical faults. Only a small part of faults (about 20%) are electrical. From the statistical results of the fault occurrence time of the ACS shown in Fig. 1.7 we see that the fault occurrence time is about evenly distributed in all stages of the space mission. In particular, the proportion of faults in the ACS within one year of the spacecraft orbit is 22%, which is much smaller than the corresponding proportion of system-level faults. The latter accounts for about 41%, as shown in Fig. 1.4. Fig. 1.8 shows the effect of the ACS fault on the aerospace tasks. Compared with the influence of the corresponding system-level faults shown in Fig. 1.5, the total and partial mission faults caused by the ACS faults are all less than 10%.

Overview Chapter | 1

5

FIGURE 1.7 Time of fault for ACS.

FIGURE 1.8 ACS fault impact on task.

FIGURE 1.9 Component fault of ACS.

They accounts for about 30% and 55%, respectively. Those data indicate that the faults in the ACS are not as severe as the faults of other subsystems. It has better chance to achieve fault-tolerant control than other subsystem faults. The proportion of the component faults in the ACS is shown in Fig. 1.9. Due to the lack of sufficient information, nearly 18% of the faults cannot be accurately located. However, nearly 50% of the faults in the ACS are caused by the gyro-

6 Fault-Tolerant Attitude Control of Spacecraft

scope, the momentum wheel, and the thrusters including the XIPS (Xenon Ion Propulsion System). The following content focuses on the analysis of thruster fault and momentum wheel fault.

1.2.2.1 Thruster fault Thruster is one of the most important actuators in the attitude control system. It is mainly composed of a combustion chamber and a flow control valve. When the propellant passes through the combustion chamber, a chemical reaction occurs and generates thrust through its nozzle. The typical thruster faults are that the flow control valve is stuck-open or stuck-closed. The stuck-open fault causes the propellant to leak and eventually consume all fuel. The stuck-closed fault lets the thruster be unable to generate any thrust. Fig. 1.9 shows that the faults of thruster components account for about 22% of the total faults of the attitude control system. Table 1.2 lists four faulty spacecrafts due to thruster faults in recent years. TABLE 1.2 Analysis to thruster faults in four spacecraft. Spacecraft

Faults analysis

Occurrence time of fault

Fault impact

Eutelsat W3B

Thruster fuel leak

October 28, 2010

Total lost of control

JCSat-1B

Thruster abnormal

January 2, 2005

Mission interruption

Galaxy 8I Nozomi

3 thrusters failed valve stuck in open

September 1, 2000 December 20, 1998

Lifetime Reduction Mission interruption

1.2.2.2 Momentum wheel fault As shown in Fig. 1.9, momentum wheel (MW) faults and gyro faults account for 37% of the attitude control system faults. The momentum wheel faults are more difficult to deal with than the gyro faults. Even if the gyro fails, engineers can still design other methods to control the attitude of spacecraft such as BeppoSax and ERS 2. On the other hand, mechanical momentum wheel fault significantly reduces the attitude control performance of spacecraft. One or more momentum wheel faults enable the remaining momentum wheel or thruster to perform attitude control maneuvers and thereby shorten the spacecraft lifetime. Table 1.3 lists five spacecrafts undergoing momentum wheel faults.

1.3 Fault-tolerant control systems Over the last three decades, the growing demand for safety, reliability, maintainability, and survivability in aerospace systems and industrial processes has motivated significant research in fault-tolerant control (FTC) or fault-tolerant control system (FTCS). Survey papers by Blanke et al. (1997) and Zhang and Jiang (2008) provided excellent overviews of FTC. Generally speaking, the existing design techniques for FTC can be classified into two approaches, passive

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TABLE 1.3 Analysis to momentum wheel faults in five spacecrafts. Spacecraft

Faults analysis

Occurrence time of fault

Fault impact

GPS BII-07 Radarsat-1

One MW failed Large friction

May 21, 1996 September 15, 1999

Total Lost of control Performance deterioration

EchoStar V TOPEX

One MW failed One MW failed

July 1, 2001 October 9, 2005

Mission interruption Total Lost of control

Kepler

Two MWs failed

2013

Mission failed

FTC and active FTC. The controller in the passive FTC is designed to tolerate only a limited number of faults. Those faults are assumed to be known prior to the design of the controller. This approach does not require any online fault information. The passive FTC can be implemented easily as a fixed controller. It can compensate for the anticipated faults. However, the passive FTC has a very limited fault tolerance capability and has the following characteristics. 1) Greatly robust to anticipated or predetermined faults. 2) Utilize hardware redundancy such as multiple actuators and sensors, etc. 3) The controller has more conservativeness. 4) Achieving robustness to certain faults is only possible at the expense of decreased nominal performance. It should be stressed that most systems to be controlled may have a limited physical redundancy. It is not possible to increase or change the hardware configuration due to cost or physical restrictions. These may limit the applications of passive FTC to control those types of systems. Fortunately, another type of FTC approaches, that is, active FTC, could be designed for these cases using the available resources and employing both physical and analytic system redundancy to accommodate unanticipated faults. Fig. 1.10 shows a general schematic diagram for active FTC systems. In comparison with the passive FTC, the active FTC reacts to the fault events reactively by using a reconfiguration mechanism so that the stability and acceptable performance can be maintained. As such, the aforementioned drawbacks of the passive FTC are reduced. The active FTC compensates for the effects of faults either by selecting a precomputed control law or by synthesizing a new control law online in real time. Both need a fault detection and identification (FDI) (De Persis and Isidori, 2001) algorithm to provide the most up-to-date information about the true system status-induced changes and to reconfigure the control law online. In the literatures, fault detection and identification (FDI) (Wang et al., 2014) or fault detection and isolation (again, FDI) are often used. To avoid any confusion, in this paper, we adopt FDI to stand for fault detection and isolation, whereas we will use FDD when the fault identification function is also added to FDI. In the design of FTCS, fault identification is very important.

8 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 1.10 Schematic diagram for active FTC systems.

Therefore, FDD is mainly used to highlight the requirement of fault identification. Active FTC systems involve significant amount of online FDD Yin and Zhu (2015), real-time decision making, and controller reconfiguration; it accepts a graceful degradation in overall system performance in the case of faults. Generally, active FTC systems have the following characteristics: 1) Fully employ analytical redundancy. 2) Utilize FDD algorithm and reconfigurable controller. 3) Accept degraded performance in the presence of a fault. 4) Reduce conservativeness. Several survey papers and books are published prior to 2008, and different FTC design methodologies have been summarized in Patton (1997); Blanke et al. (2006). Since then, a large number of FTC related publications have appeared Zhang and Jiang (2008); Shui et al. (2009); Benosman (2010); Noura et al. (2006). Besides, some extra efforts have examined the relationship between feedback and FDD Esna Ashari et al. (2012), differences between passive and active FTC design strategies Jiang and Yu (2012), data driven-based FDD Yin et al. (2014a), Yin et al. (2014b), and FDD/FTC in flight control system Goupil (2011); Ducard (2013); Smaili et al. (2009); Isermann (2006). However, to the author’s best knowledge, there is no any review on the FTC including FDD design for a spacecraft attitude system. When we analyze recent spacecraft accident statistics Tafazoli (2009), it is not surprising to see that a significant portion is attributed to attitude control system failures. This category accounts for as much as 32%. Because this system regroups many critical systems, an attitude control system failure can heavily cripple a spacecraft. As a result, fault-tolerant attitude control is a key issue that needs to be addressed not only in practical aerospace engineering but also in academic community. In the following sections, we will provide a comprehensive review on the recent development of FTC technologies in spacecraft attitude control systems.

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Note that active FTC design incorporates FDD, a controller reconfiguration mechanism, and a reconfigurable feedback controller, whereas passive FTC design only consists of a fixed fault-tolerant controller design. Summarizing those two types of FTC approaches, FDD design and fault-tolerant controller design are two of the main works. For active FTC design, FDD and a fault-tolerant controller or control law including a controller reconfiguration mechanism should be designed. For passive FTC design, only a fault-tolerant controller needs to be developed. Fault-tolerant controller design is the common work to be carried out in both active and passive FTC approaches. Although a reconfiguration mechanism, a reconfigurable feedback controller, and others such as fault-tolerant estimation are of interest as well, the next section is to offer a review of FDD and fault-tolerant controller (or control law) design in both active and passive FTC for spacecraft attitude control systems.

1.4 Review of FDD for spacecraft The FDD designed for a spacecraft attitude control system or other industrial systems usually consists of three parts Huang et al. (2012). The first part is fault detection to detect that the system has underwent fault when it occurs. Then a fault isolation mechanism should be designed to determine the type and location of the fault, and which component is experiencing this type of fault. After that, fault diagnosis or identification should be carried out to obtain the detected fault value/magnitude. In an attempt to develop FDD approaches for spacecraft attitude control systems, extensive studies have been performed Patton et al. (2008, 2010). A refined classification of the existing FDD schemes for spacecraft attitude systems is shown in Fig. 1.11. Those approaches can be categorized into two classes, that is model-based FDD and data-based FDD.

FIGURE 1.11 Classification of FDD schemes for spacecraft attitude control system.

10 Fault-Tolerant Attitude Control of Spacecraft

1.4.1 Model-based FDD approaches Since most of control techniques are model-based, model-based FDD approaches have been well developed, and they have been applied to spacecraft research in last few decades Patton et al. (2010); Williamson et al. (2009); Lee (1999); Henry et al. (2010). In Patton et al. (2010) the problem of robust fault detection and isolation for faults affecting the thrusters of a satellite system Mars Express was addressed. The attitude measurement errors, uncertainty, and external disturbance were considered. In Henry (2008), two model-based FDD approaches were developed by H∞ /H2 filters. The fault diagnosis problem of Microscope thrusters in the presence of measurement noises, measurement delays, sensor misalignment, and even external disturbances, was addressed. In Tafazoli (2009) the design of robust model-based FDD systems for atmospheric reentry vehicles was investigated. Using H∞ /H− setting, a robust residual-based scheme was designed to diagnose faults on the vehicle wingflap actuators. In Gao et al. (2010) a state-space-based FDD was proposed for microsatellite to detect and reconstruct sensor faults. In parallel to modern spacecraft development, the number of faults considerably increases due to the growing complexity of equipments and systems. To achieve FDD and FTC design for spacecraft attitude control, a model-based FDD approach with simultaneous states/parameters estimation is becoming another widely applied method. For instance, a two-stage Kalman filtering algorithm was presented in Hou et al. (2008) to precisely estimate reaction flywheel faults. Those faults were modeled as a multiplicative factor. For a class of nonlinear systems, an FDD based on the interacting multiple model (IMM) extended Kalman filter (EKF) was presented Mehra et al. (1998). This strategy has been applied successfully to spacecraft FDD of sensor and actuator failures. For various faulty conditions due to changes and anomalies in the temperature, the power supply line voltage, the loss of effectiveness of the torque, and the current in each reaction wheel associated with the three-axes of the satellite, an IMM-based FDD was established in Tudoroiu and Khorasani (2007) for the detection of actuator faults. In Pirmoradi et al. (2009) a new EKF-based scheme was presented for FDD of spacecraft attitude sensors and gyro faults. In addition to the preceding model-based FDD approaches, the application of model-based observers for FDD design and spacecraft attitude system state estimation has been extensively discussed in aerospace academic community. In Boskovic et al. (2005), with application of an observer-based FDI, a theoretical framework for retrofitting reconfigurable flight control was explored. However, the FDI developed can only estimate constant damage of control effectors and also fail to achieve precise fault reconstruction. Based on actuator dynamics, Talebi Talebi et al. (2007) presented a nonlinear observer to achieve FDD of reaction wheel faults by the approximation capability of neural network. To detect faults in spacecraft actuators, an iterative neural network-based observer was explored in Wu and Saif (2006). Proportional, integral, and derivative information of fault estimation error was utilized as observer input, and then precise fault

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diagnosis was achieved. An iterative learning observer-based FDI was reported in Chen and Saif (2007) to achieve estimation of time-varying thruster faults. More recently, the problem of satellite FDD design was further investigated by combined geometric and neural network approach Baldi et al. (2013, 2015). On the other hand, in case of parametric uncertainties, an observer can be designed robust by applying sliding-mode control Pukdeboon et al. (2010); Hung et al. (1993) or adaptive control theory. These observers are often called as sliding-mode observers (SMO) Haskara et al. (1998); Veluvolu and Chai (2009); Yan and Edwards (2008) and adaptive observers Šimandl and Punˇcocháˇr (2009), respectively. The main advantage of using adaptive observers or SMO is that they are insensitive to the matched unknown inputs, parameter variations, and external disturbance. Up to date, a variety of adaptive observers and SMO approaches have been proposed for spacecraft FDD Šimandl and Punˇcocháˇr (2009); Alwi et al. (2009). In Gao et al. (2013) a robust actuator fault diagnosis scheme was investigated for satellite attitude control system. System uncertainties, disturbance, and gyro drifts were considered. The objective of FDD online was achieved by designing an adaptive observer with unknown input. For a satellite with its attitude control subsystem controlled by four reaction wheels in a tetrahedron configuration, an SMO algorithm was developed in Jiang and Khorasani (2007) to generate residual signals. An adaptive observer was then derived to detect actuator faults. Using reaction wheel dynamics, an SMO-based FDD algorithm was proposed in Wu and Saif (2007) as a residual generator to detect the anomalies and faults. Since two-order or high-order SMO has more robust capability to system uncertainties, a high-order SMO was designed to accomplish the work of FDD for spacecraft attitude control systems. The design of SMO for gyro and thruster fault detection and diagnosis in Mars Express satellite was studied in Alwi et al. (2010); Nagesh and Christopher (2011). In another recent work Zhang et al. (2013) the sensor fault detection and diagnosis of satellite attitude control system was addressed by developing an SMO and an adaptive observer. The sensor faults were precisely estimated with zero reconstruction error. The FDD approach is only applied and incorporated into the active FTC design. Hence, it is quite necessary to quantitatively evaluate the model-based FDD approaches. For the detailed results, we refer to Zhang and Jiang (2008). Moreover, we further find that none of the single FDD methods is able to satisfy all design criteria. Moreover, parameter estimation, simultaneous state/parameter estimation, and multiple-model-based schemes are more suitable to active FTC system design.

1.4.2 Data driven-based FDD schemes Although the existing model-based FDD approaches can detect and diagnose spacecraft component faults, they greatly depend on the physic model. Owing to the growing complexity of spacecrafts and their equipments, high-fidelity

12 Fault-Tolerant Attitude Control of Spacecraft

models of spacecraft and its component may not be available. The model-based FDD methodologies would be not able to perform. Hence FDD design without model is a key issue that needs to be addressed. To meet this challenge, the data-driven-based FDD techniques have been dominated in the applications of model-free FDD design. Survey papers by Yin et al. (2014a, 2015); Yin and Kaynak (2015) provide comprehensive overviews of data-driven-based FDD. Currently, investigations on data driven-based FDD for spacecraft attitude control system have been shown in literature. In Zhao and Zhang (2008), a data driven-based FDD scheme using support vector machine (SVM) classifier was proposed. Principal component analysis (PCA) method was first used to generate residual for classification, and then one-against-one SVM classifier was presented to detect sensor or actuator faults of satellite attitude control system. In Li et al. (2006), another data-driven-based FDD with dynamic neural network was designed. The goal of deciding whether a bus voltage fault, a current loss fault, or temperature fault has occurred in reaction wheel and further localizing which actuator was faulty were achieved. In Williamson et al. (2009) a set of fault detection filters were designed for deep space satellites to detect and identify faults in each of the sensors or actuators based on the Shiryayev sequential probability theory. In another related work Lieftucht et al. (2009) a regression-based fault reconstruction technique was developed to realize improved fault diagnosis in multivariable systems. In Valdes and Khorasani (2010) a dynamic neural network-based FDD scheme was developed for pulsed plasma thrusters employed in formation flying satellites. This strategy was designed in the framework of data-driven scheme, and online/offline running data of thrusters were applied to designing that FDD, which had satisfactory detection capability. However, its isolation capability was not adequate. The results demonstrated a high level of fault detection accuracy (99.79%) and precision (99.94%) with a misclassification rate that was quite negligible (less than 1%). The problem of FDD and reconstruction design for satellite was investigated in Li et al. (2011). A data-driven-based FDD was proposed by using independent component analysis technique. The fault occurring in gyroscope was precisely detected and reconstructed. For the satellite attitude control system with reaction wheel serving as its actuators, kernel fuzzy C-Means was applied in Hu et al. (2012a) to design a data-driven-based approach for fault diagnosis of both known and unknown faults. The aforementioned data-driven-based FDD can indeed solve the drawbacks of model-based FDD. However, the implementation of data driven-based FDD may need a quantity of online/offline data Wang and Yin (2015). This would be impractical for on-orbital spacecraft due to its limited data handling capability of onboard computer. Moreover, the problem of FDD design with low data quality should be further addressed for data-driven-based FDD. Actually, all measurements of all spacecraft components include noise, which can lead to the unreliability of data. Consequently, a lower effectiveness of diagnosis will be achieved by using unreliable data to design FDD approach.

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1.5 Spacecraft attitude fault-tolerant control engineering NASA Earth orbiting Lewis Spacecraft was launched on August 23, 1997. It lost contact with the spacecraft on August 26 and then reentered the atmosphere and was destroyed on September 28 Wander (2007). The investigation board report stated that the spacecraft failed mainly due to a technical flaw in the attitude control system design causing excessive thruster firings. This led to the shutdown of all thrusters and left the spacecraft in an uncontrolled attitude draining most of its battery charge. Motivated by this accident and to avoid such issue, fault management system is introduced in the stages of spacecraft design and its on-orbital running. This system is designed to detect, isolate, identify, and recover the status of on-orbital spacecraft. Consequently, reliability and capability of fault tolerance are guaranteed and increased.

1.5.1 Engineering techniques for spacecraft FTC Although many fault management systems or FTCSs have been developed in practical spacecrafts engineering, those systems or FTC actually can be summarized into unique engineering technique, that is, spacecraft missions have been deployed with the ability to execute a “safe” mode transition. In this mode a detected anomaly results in the spacecraft deactivating all thrusters and postponing science activities until the situation is resolved by engineers at the ground station. This solution has been sufficient to enable recovery from a variety of hardware and software failures so long as the spacecraft faces no immediate risks (e.g., collision) and provided that the mission can be continued following a potentially extended “safe mode” episode. To implement this technique, the following seven procedures are usually essential Nasir and Atkins (2010): Step 1) Engineers at ground station should analyze/monitor the telemetry data and evaluate the health status of spacecraft. Step 2) If it is analyzed from the telemetry data that the on-orbital spacecraft has experienced fault, then the command for switching into “safe” mode should be sent to make sure that the spacecraft is executing such a mode. For example, solar array rotates to the original point position. The solar array is thus guaranteed to facing sunlight for producing energy. Step 3) After that, engineers on ground should analyze the reason why that fault occurred. Step 4) Numerical simulation and experiments should be carried out to verify the analyzed reason and to locate and diagnose that fault. Step 5) A corresponding approach for handling that fault should be then proposed. Step 6) Ground tests should be done to demonstrate the effectiveness of the proposed approach. Step 7) If that proposed approach is effective, then this scheme will be uploaded to the spacecraft to recover the aerospace mission. Otherwise, it should redesign the scheme to handle the detected fault.

14 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 1.12 The Far Ultraviolet Spectroscopic Explorer satellite (Far Ultraviolet Spectroscopic Explorer, n.d.).

FIGURE 1.13 Collision between CERISE and space debris (n.d.).

Up to now, this technique has become almost the only one method applied in handling spacecraft accidents. For instance, Far Ultraviolet Spectroscopic Explorer satellite (as shown in Fig. 1.12) was at the peak of its scientific productivity when excessive friction between the rotors and wheel housings caused the loss of two of its four reaction wheels required for attitude control. However, engineers reprogrammed the control software by developing a new control law, integrating the magnetic torquer bars in the control loop with the remaining two reaction wheels, and fine pointing capability was reestablished Kruk et al. (2003). Another successful application of the preceding fault tolerance technique is CERISE microsatellite (as shown in Fig. 1.13). This satellite suffered a collision with a piece of space debris. A sudden change to the attitude dynamic was observed as it tumbled rapidly end-over-end in its 700 km polar Earth orbit. Fortunately, engineers at ground station successfully analyzed the collision dynamics. Then onboard computers were reprogrammed with novel magnetic control algorithms to restabilize the undamaged microsatellite and regain almost full operational mission capability Sweeting et al. (2004).

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1.5.2 Discussions With extended communication delays, increasingly complex science or maneuver schedules and improved sensing and computational capabilities, future missions will support increased onboard autonomy, including autonomous fault management. Repeated safe-mode transitions can be costly to the mission. As a result, “safe” mode as an engineering fault management technique is also not an option during critical mission phases such as one-time orbit insertions. As an Earth-orbiting example, consider a communication satellite tasked with providing coverage of a specific high-priority event. If this satellite goes to safe mode during the event (and there is no backup or the backup is already in “safe” mode), substantial loss of revenue could occur, or worse if providing communications for an area of unrest or disaster. However, it can learn from the recent fault management engineering techniques that: • Even limited to spacecraft cost and time schedule of design, component testing before launch is a necessary and important way to improve spacecraft reliability. • Having enough redundancy (both hardware and functional redundancy) is another key solution to improve spacecraft reliability. • The flexibility provided by the system design in the spacecraft software and hardware can allow the ground engineers/operators to reprogram different spacecraft systems. This is also a valuable asset in failure recovery, which is verified in satellite BeppoSAX Martella et al. (2002). The preceding stated engineering FTC technique for spacecraft is an active FTC. However, its reconfiguration mechanism, FDD, and controller are designed offline and at the ground station. The closed-loop attitude system consists of engineers at ground station and an on-orbital spacecraft. As a result, the resulted FTC attitude system is not autonomous. That is the reason why autonomous FTC design for satellite has received more and more consideration in academia communities.

1.6 Review of spacecraft attitude fault-tolerant control As summarized in Chapter 1.3, fault-tolerant controller design is the common work to be carried out in both active and passive FTC approaches. The attitude fault-tolerant control law is another main work in spacecraft attitude FTC system design Chen et al. (2006); Yang et al. (2013); Baldi et al. (2014). Therefore in this section, we review the recent works on the fault-tolerant controller design part in spacecraft attitude active and passive FTC Mirshams et al. (2014) system. With application of FTC law, spacecraft component faults should be tolerated with desirable performance guaranteed. Inspired by modern control theory, there have been a number of important developments in the design of attitude fault-tolerant controller. For instance, the problem of automated attitude recovery for rigid and flexible spacecraft was considered in Tafazoli (2009). The

16 Fault-Tolerant Attitude Control of Spacecraft

approach was based on feedback linearization control. Another fault-tolerant attitude control law design for handling reaction wheel faults was discussed in Baldi et al. (2010). The problem of FTC law design for compensation of lossof-effectiveness failures of reaction wheels was investigated in Jin et al. (2008). An FTC law was presented for satellite JC2Sat-FF subjecting to actuator faults De Ruiter (2011). For the Cassini spacecraft Macala et al. (2012), the feasibility study of two candidate reaction wheel/thruster hybrid controller design was presented for attitude FTC design. Based on Lyapunov theory and genetic algorithms, an autonomous FTC law was designed to achieve spacecraft rendezvous with thrust faults Li et al. (2013). To accomplish agile attitude control, a novel FTC law was developed in Noumi and Takahashi (2013) to tolerate faults in control moment gyros. When designing fault-tolerant attitude controller for spacecraft, the capability of the controller to handle system uncertainties and external disturbance should be guaranteed. Otherwise, it would lead to an inferior control performance. As a result, attitude FTC design with those two issues addressed has attracted more and more attention. Especially, adaptive control, sliding-mode control, and control allocation are three most widely techniques applied to designing attitude FTC law, which are reviewed as follows.

1.6.1 Attitude FTC design using adaptive control Adaptive control technique Krstic et al. (1995), as one of modern control theories, has been widely applied to handling uncertain system parameters, that is, owing to its capability to estimate those parameters online. Taking this advantage, many investigations on spacecraft have applied adaptive control to designing attitude FTC law. In Boskovic et al. (2005) an adaptive controller was designed for flight control system. An adaptive attitude FTC stabilization control law was proposed for a tethered satellite. The work in Cai et al. (2008) showed that adaptive control can be successfully applied to performing an attitude tracking maneuver for a rigid spacecraft with thrusters failures. Of particular interest to this paper, input constraint, uncertain inertia parameters and even external disturbance were explicitly addressed. The author in Jiang et al. (2010) derived an FTC strategy to follow the desired attitude for flexible spacecraft. The proposed scheme estimated the upper bounds on disturbances and model parameters online and used there estimates in the control law; more specifically, two types of reaction wheel faults were explicitly compensated by using adaptive technique. An adaptive control-based intelligent FTC law was derived in Chandrasekar et al. (2010) to achieve attitude control with high pointing accuracy, and actuator fault and misalignment were adaptively handled. In a recent work Zou and Kumar (2011), combing adaptive control with fuzzy control, an adaptive attitude controller was synthesized. System uncertainties, disturbances, and actuator faults were addressed, and two adaptive updating laws were presented to estimate uncertain parameters. For the spacecraft formation flying control subjecting to actuator fault, a passive FTC controller by using adaptive control was

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developed in Zou and Kumar (2012). The implementation of this controller did not require any FDD approach. To compensate for disturbances and actuator faults, an adaptive FTC law was designed to perform attitude tracking maneuver. For more recent development of FTC law design by using adaptive control technique, the works Zhang et al. (2004); Ma et al. (2014); Qiao et al. (2008); Bustan et al. (2013) can be further referred.

1.6.2 Sliding mode-based attitude FTC methodologies Sliding-Mode Control (SMC) Utkin (1977) has recently emerged to be an effective approach in controlling uncertain systems with highly coupled and nonlinear dynamics. One of the advantages is a rapid response, insensitive to uncertain parameters and external disturbances, and thus a variety of SMC algorithms have been developed in the literature and applied to spacecraft attitude FTC system design Xiao et al. (2013, 2011). Passive and active sliding-mode control laws were developed to solve attitude stabilization problems of spacecraft with actuator outage fault accommodated Liang et al. (2007). In Godard and Kumar (2010), terminal SMC was used to arrive at satellite formations flying, and actuator faults were taken into account in simulation. However, the stability analysis in the presence of degradation and stuck fault was not carried out. The result in that paper was further extended in Kumar et al. (2011) by considering the problem of rapid reorienting of a rigid spacecraft under external disturbances and uncertain nature of dynamics. The attitude was shown to be uniformly ultimately bounded for cases where there was no control available on either roll or yaw axis. An SMC-based attitude fault-tolerant controller was presented in Varma and Kumar (2010). The attitude was stabilized even in the presence of unknown, slow-varying satellite mass distribution and several fault scenarios of rotating solar flaps. Although a terminal SMC scheme was derived in Lee and Kim (2010) to perform the rest-to-rest maneuver with finite-time convergence, only the degradation of actuation effectiveness fault was addressed. In Post et al. (2012), another SMC-based attitude FTC controller was presented for nanosatellite. The effectiveness of that controller was tested on ground by using air bearing test bed. In a more recent work Cao et al. (2013), a nonsingular terminal SMC was employed to design an attitude FTC law for small satellite.

1.6.3 Control allocation-based attitude FTC To improve the reliability of spacecraft, attitude control system is constantly designed with redundancy, such as redundant actuators and sensors. When actuator undergoes faults, the remaining active actuators should be able to produce a combined and desirable control power sufficient enough to allow the spacecraft to perform given maneuvers. Due to actuator redundancy, the solution of control allocation to generate the desired control effort is not unique Johansen

18 Fault-Tolerant Attitude Control of Spacecraft

and Fossen (2013). However, the most applied allocation scheme is static allocation by using the pseudoinverse value of actuator distribution matrix Jin (2005). However, this approach is not an effective method to fully use the remaining control power. As a result, several effective dynamic control allocation schemes were proposed for overactuated control systems Härkegård (2004); Härkegård and Glad (2005). In combination with practical engineering, many control allocation approaches have been designed to tolerate actuator faults. In Servidia and Pena (2005) the problems of spacecraft thruster control allocation and attitude control were investigated. In Jin et al. (2006) a reliable attitude controller was proposed by solving a linear programming problem. For the algorithm, a cost function was defined as a quantity related to the fuel consumption for a maneuver. It was tested by applying to two attitude maneuvers. The thruster configuration design was considered in the fixed, single-gimbal, and double-gimbal thruster cases in Servidia (2010). This allocation-based attitude control was also presented with attitude maneuvers accomplished under nominal and single-point failure conditions. To solve the problem of trajectory tracking control from entry interface to parachute deployment, a nonlinear fault-tolerant adaptive controller was designed in Marwaha and Valasek (2011) for a special spacecraft, Mars entry vehicles. The proposed nonlinear control law provided a hybrid linear dynamic control allocation algorithm for generating an optimal control effort required to follow the desired trajectory. In Hu et al. (2013) a difficult problem of spacecraft fault-tolerant control design by compensating for the loss in effectiveness and time-varying faults using redundant actuators was studied. A novel observer-based fault diagnosis method incorporating an online control allocation scheme was proposed. A fault-tolerant control allocation scheme was presented in Shen et al. (2014) for overactuated spacecraft attitude tracking systems. Actuator faults were considered. A performance/robustness trade-off control allocation strategy was designed to redistribute the virtual control signals to the remaining actuators without reconfiguring the controller. Robust attitude control was achieved. In recent work Hu et al. (2015) a novel velocity-free nonlinear proportional-integral (PI) control allocation approach was proposed for fault-tolerant attitude control design of flexible spacecraft under thruster redundancy. It guaranteed the ultimately uniformly boundedness of the closedloop system even subject to disturbances and possible faults. In another work Zhang et al. (2014) a robust control allocation scheme was developed for rigid spacecraft attitude stabilization in the presence of actuator failure and actuator misalignment. In addition to the preceding fault-tolerant controller design methodologies, spacecraft attitude fault tolerant controller design can also be designed by the lessons learnt from FTC design for hypersonic flight vehicles Du et al. (2010); Shen et al. (2012, 2011); Zhao et al. (2014) and aircrafts Alwi et al. (2008); Boskovic et al. (2007) with similar attitude dynamics as that of spacecrafts.

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1.7 Open problems in spacecraft attitude fault-tolerant control The spacecraft attitude fault-tolerant control problem for the case of component fault/failures is well understood, although some works still remain for attitude control with high pointing accuracy. On the other hand, despite the recent advances in spacecraft attitude FTC system design with disturbance, uncertain parameters, and faults addressed, the problem of attitude reliable control is still not completely understood. More specifically, the following problems are needed to be solved further: • • • •

Actuator nonlinearities. The conservativeness of controller. Angular velocity measurements. The capability of attitude fast slewing maneuver.

1.7.1 Without considering actuator nonlinearities Almost all the existing attitude FTC schemes for spacecraft are designed based on the assumption that the actuator is with linear dynamics, that is, the actuator generates same output as the input. However, this assumption is rarely satisfied in practice. Almost all types of actuator are subject to nonlinear dynamics, such as actuator constraint/saturation, or actuator dead-zone. Taking actuator constraint for example, all actuators are subject to this problem due to physical limitations, that is, these actuators have an upper bound on the control effort that can be exerted on the system. When an unknown fault of actuator occurs, the control system continues issuing its maneuver that may no longer be achievable by the system in spite of the occurrence of the fault. Under this situation, the required control effort quickly saturates the actuators while striving to maintain the “healthy” maneuvering performance. It subsequently destabilizes the system due to saturation if the system is not equipped with a novel control methodology to dump the saturated actuators. This situation may quickly become mission critical. On the other hand, it may be difficult to establish the mathematical relationship between actuator saturation and reconfigurable attitude controller when actuator undergoes faults. As a consequence, actuator nonlinearities are still an open problem for further research and development in spacecraft attitude faulttolerant control system design.

1.7.2 Having great conservativeness As discussed, most of the existing FTC approaches for spacecraft attitude system are passive inherent. It is assumed that there exists an upper bound on faults. Adaptive control technique is applied to designing an adaptive law to estimate the upper bound. On the basis of this, that estimated bound is then used in the controller to tolerate faults. Therefore it will inevitably lead to the problem that the designed controller has certain conservativeness. This conservativeness will

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increase the consumption of on-orbital spacecraft energy. As a result, the age of spacecraft will be shortened. At present, there is no unified framework of designing an attitude FTC system without or at least with less conservativeness.

1.7.3 Requiring angular velocity measurements Most of the existing spacecraft attitude FTC schemes are assumed that fullstate measurements (i.e., attitude orientation and angular velocity) are available. However, not all signals of a spacecraft can be measured with high accuracy. For instance, angular velocity measurement may be unavailable for microsatellites. In practice, the availability of angular velocity measurements is not always satisfied due to either cost limitations or implementation constraints. Sometimes, to reduce the cost, the low-cost spacecraft, especially the microsatellites or nanosatellites, would be launched without any angular velocity sensor. Moreover, wrong and imprecise measurements of angular velocity may be introduced to some extent due to faults/failures in rate sensors. Although several FTC designs for spacecraft attitude system were observed without angular velocity measurements Xiao et al. (2014), they only have limited FTC capability. The design of efficient and low-cost attitude FTC system with angular velocity measurements still needs to be further studied.

1.7.4 Without attitude fast slewing capability Although the existing FTC approaches can accomplish attitude tracking maneuver with guaranteed performance even in the presence of faults/failures, they do not guarantee finite-time convergence of the tracking error. Actually, the attitude tracking problem considered in the existing studies is solved by the stability analysis, which implies that the attitude trajectories converge to the equilibrium with infinite settling time in the presence of actuator faults. Obviously, such infinite-time settling time is not an option during critical mission phases of some high real-time mission. As a result, those FTC schemes would not be able to supply the spacecraft with attitude fast-slewing capability. Consider a military satellite tasked with providing coverage of a specific high-priority area. Under actuator faults, if this spacecraft cannot accomplish attitude maneuver and handle faults in finite time, ground objectives could be totally lost. Consequently, the spacecraft attitude FTC system design with fast slewing capability has become one of future works. Overall, fault-tolerant control for spacecraft attitude system is a complex interdisciplinary research field. It covers a diverse range of engineering disciplines, such as modeling, signal processing, computing, etc. Moreover, the preceding four open problems are coupled each other. For example, large control torque will demanded to achieve attitude fast slewing. This would lead to the saturation of actuators. Therefore, when designing FTC for spacecraft attitude system, those four problems should be considered with a unique framework.

Overview Chapter | 1

21

1.8 Organization of this book The book is organized as eight chapters. Chapter 1 gives an overview of the spacecraft attitude control system and its fault-tolerant control design. It briefly introduces a literature review about the recent development of fault-tolerant control approaches for spacecraft attitude system. Chapter 2 begins with some basic mathematical notations and definitions. Then it introduces a mathematical model of rigid/flexible spacecraft attitude control system. Modeling of actuator faults is further carried out. Chapter 3 focuses on developing robust attitude fault-tolerant control approaches for spacecraft with actuator fault by using the adaptive control technique, the sliding-mode control theory, the backstepping control method, and the H∞ control theory. Chapter 4 presents three fault-tolerant control schemes for spacecraft attitude system subject to actuator saturation and partial loss of actuator effectiveness. Chapter 5 is concerned with developing attitude stabilization fault tolerant control approaches without angular velocity measurements. The partial loss of actuator effectiveness fault occurring in spacecraft is addressed. Chapter 6 is dedicated to solve the fault-tolerant attitude tracking control problem of spacecraft. Fast attitude slew maneuver is accomplished with the attitude tracking error having finite-time convergence. Chapter 7 focuses on developing active fault-tolerant control strategy for spacecraft attitude system. The designed controllers have less conservativeness. In Chapter 8 the book is concluded, and some future work is clarified.

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Chapter 2

Preliminaries 2.1 Introduction The mathematical model of any spacecraft attitude control system consists of the attitude dynamics and kinematics. It is the cornerstone of the attitude control design for any on-orbital spacecraft. In general, the attitude Euler angles are usually adopted to describe spacecraft attitude in engineering. The other methods such as the unit quaternion, the modified Rodriguez parameters (MRPs), and the attitude rotation matrix are utilized in theoretical analysis. Those attitude description methods have their own features (Sidi, 1997). Moreover, the attitude dynamics has great impact on the design of attitude controller and system performance. Therefore it is necessary to establish accurate attitude dynamics with external disturbances and uncertain inertia considered. The uncertainty of actuator is a critical factor in attitude model establishment. If it is ignored, then the control performance will be greatly degraded. Therefore it is urgent to analyze the mechanism of actuator uncertainties and influence on control precision. The dynamic model of the momentum wheel (MW) was demonstrated in (Bialke, 1998; Wang et al., 2003), but the fault model was not given. Based on the momentum wheel fault model affected by temperature and friction torque, a fault diagnosis algorithm based on interacting Kalman filters was proposed in (Tudoroiu and Khorasani, 2007). Several configurations based on three or four momentum wheels for a 3-axis satellite attitude control were investigated in (Ismail and Varatharajoo, 2010), whereas its mathematical model and momentum wheel misalignments were not taken into consideration. The theoretical validation of the fault-tolerance in control algorithm is essential for space applications. The spacecraft equipped with momentum reaction wheels consists of a configuration and faulty model system, and the actuator dynamics can be neglected in most cases. Thus, the explicit faulty model of momentum wheel in close loop is essential for fault detection and fault-tolerant controller design. In what follows, we will first give notations and lemmas for controller design and stability analysis. Then, we will establish mathematical models of a rigid spacecraft and a flexible one. Finally, we will give a mathematical model of momentum wheel fault model including installment deviation and uncertainties. Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00012-5 Copyright © 2021 Elsevier Inc. All rights reserved.

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24 Fault-Tolerant Attitude Control of Spacecraft

2.2 Mathematical notations For the purpose of control system design and stability analysis, the notations are defined as follows. R denotes the set of all real numbers, and R+ denotes the set of all positive real numbers; Rn represents the n-dimensional real space, and Rn×m represents the set of all n × m real matrices; I n ∈ Rn×n denotes the n × n identity matrix, 0n×m denotes the n × m zero matrix, and 0 represents the zero vector of appropriate dimension. For any matrix A ∈ Rn×n , we denote λmin (A) and λmax (A) as its minimal and maximal eigenvalues, respectively; A−1 is the inverse of a matrix A. For any matrix B ∈ Rn×m , B T and B † are denoted as its transpose and pseudoinverse. || · || and || · ||∞ are defined as the Euclidean and H∞ norms, respectively. The vector space L∞ is the space of bounded sequences. For any vector x = [x1 , x2 , . . . , xn ] ∈ Rn , we define the following vectors and diagonal matrices: diag(x) = diag[x1 , x2 , . . . , xn ] ∈ Rn×n , Tanh(x) = [tanh(x1 ), tanh(x2 ) . . . , tanh(xn )] ∈ Rn ,   q q q q [x] p = |x1 | p sign(x1 ), |x2 | p sign(x2 ), . . . , |xn | p sign(xn ) ∈ Rn , sgn(x) = [sign(x1 ), sign(x2 ), . . . , sign(xn )] ∈ Rn , Sat(x, μ0 ) = [sat(x1 ), sat(x2 ), . . . , sat(xn )] ∈ Rn , where p ∈ R+ , q ∈ R+ , μ0 ∈ R+ , sign(·) ∈ R, and tanh(·) ∈ R are the standard sign and hyperbolic tangent functions, respectively; sat(xi ) (i = 1, 2, . . . , n) is determined by ⎧ ⎪ if xi ≥ μ0 , ⎨ μ0 sat(xi ) = xi if −μ0 < xi < μ0 , ⎪ ⎩ −μ if x ≤ −μ . 0 i 0 Moreover, for any arbitrary vector a = [a1 , a2 , a3 ] ∈ R3 , we denote a × ∈ R3×3 as the skew-symmetric matrix ⎡ ⎤ 0 −a3 a2 ⎢ ⎥ a × = ⎣ a3 0 −a1 ⎦ . −a2 a1 0

2.3 Definitions and preliminary lemmas The stability analysis of all the fault-tolerant attitude schemes designed in this book are based on the Lyapunov stability theory. Therefore in this subsection, we give some relevant definitions and lemmas.

Preliminaries Chapter | 2

Consider the nonlinear system  ˙ = f (x, t) + g(x, t)τ d , x(t) y = h(x, t),

25

(2.1)

where f (x, t) : Rn × R → Rn is locally Lipschitz and piecewise continuous in t, τ d ∈ Rs is an exogenous disturbance, whereas y ∈ Rm is the system output. Definition 2.1. (Khalil and Grizzle, 2002) The solution x(x 0 , t0 , t) of (2.1) is said to be uniformly ultimately bounded (UUB) if there exists a closed ball Bδ  {x ∈ Rn |||x|| ≤ δ} such that for all x 0 ∈ Bδ , there exist ς > 0 and a number T (ς) such that ||x(t)|| ≤ ς for all t ≥ t0 + T (ς). Definition 2.2. (Bhat and Bernstein, 2000) The origin is said to be a finitetime-stable equilibrium of (2.1) if there exist an open neighborhood U ⊂ D of the origin and a function T : U\{0} → R+ , called the settling-time function, such that the following statements hold: i) Finite-time convergence: For every x 0 ∈ U\{0}, ψ(t, x 0 ) is defined on [0, T (x)), ψ(t, x 0 ) ∈ U\{0} for all t ∈ [0, T (x 0 )), and lim ψ(t, x 0 ) = 0. t→T (x 0 )

ii) Lyapunov stability: For every open neighborhood Uε of 0, there exists an open subset Uδ of N containing 0 such that, for every x 0 ∈ Uδ , ψ(t, x 0 ) ∈ Uε for all t ∈ [0, T (x)). Moreover, the origin is said to be a globally finite-time-stable equilibrium if it is a finite-time-stable equilibrium with D = U = Rn . Definition 2.3. (Zhu et al., 2011b) The origin is said to be a practical finitetime-stable equilibrium of (2.1) if there exist  > 0 and T (x 0 ) ≤ ∞ such that ||ψ(t, x 0 )|| ≤ ε for all t ≥ T (x 0 ). Lemma 2.1. (Bhat and Bernstein, 2000) Suppose V (x) is a C 1 smooth positivedefinite function (defined on U ⊂ Rn ) and V˙ (x) + βV γ (x) < 0 is a negative semidefinite function on U ⊂ Rn for γ ∈ (0, 1) and β ∈ R+ . Then there exists a set U0 ⊂ Rn such that any V (x) that starts from U0 ⊂ Rn can reach V (x) ≡ 0 in finite time. Moreover, if Treach is the time needed to reach V (x) ≡ 0, then Treach ≤ t0 +

V 1−γ (x 0 ) , β(1 − γ )

(2.2)

where V (x 0 ) is the initial value of V (x). Lemma 2.2. (Zhu et al., 2011b) Suppose that there exist β ∈ R+ , γ ∈ (0, 1), and ϑ ∈ R+ such that a continuous function V (x) satisfies V˙ (x) ≤ −βV γ (x) + ϑ. Then the trajectory of system is practical finite-time stable.

(2.3)

26 Fault-Tolerant Attitude Control of Spacecraft

Lemma 2.3. (Bhat and Bernstein, 2000) Consider the scalar differential equation u˙ = f (t, u),

u (t0 ) = u0

(2.4)

where f (t, u) is continuous in t and locally Lipschitz in u for all t ≥ 0 and u ∈ J ⊂ R. Let [t0 , T ) (T may be infinity) be the maximal interval of existence of the solution u(t), and suppose u(t) ∈ J for all t ∈ [t0 , T ). Let v(t) be a continuous function whose upper right-hand derivative D + v(t) satisfies the differential inequality D + v(t) ≤ f (t, v(t)),

v (t0 ) ≤ u0 ,

(2.5)

with v(t) ∈ J for all t ∈ [t0 , T ). Then v(t) ≤ u(t) for all t ∈ [t0 , T ). Lemma 2.4. For any vector x = [x1 , x2 , . . . , xn ] ∈ Rn , we have ||x|| ≤ x  x. Lemma 2.5. For any vectors x, y ∈ Rn , we have ||x||||y|| ≤ ||x||∞ ||y||∞ . Lemma 2.6. (Su et al., 2010) For all x = [x1 , x2 , . . . , xn ] ∈ Rn , we have the following inequalities: 1 tanh2 (xi ) ≤ ln (cosh (xi )) , 2     Sech2 (x) ≤ 1,

(2.6) (2.7)

tanhT (x) tan h(x) ≤ x T tanh(x).

(2.8)

Lemma 2.7. For all x = [x1 , x2 , . . . , xn ] ∈ Rn , we have x ≤ x T sgn(x). Lemma 2.8. (Krstic et al., 1995) Let v and ρ be real-valued functions defined on R+ , and let b and c be positive constants. If they satisfy the differential inequality v˙ ≤ −cv + bρ(t)2 ,

v(0) ≥ 0,

(2.9)

and ρ ∈ L∞ [0, ∞), then v ∈ L∞ [0, ∞). Moreover, v(t) is bounded by b v(t) ≤ v(0)e−ct + ρ 2∞ . c

(2.10)

Lemma 2.9. (Yu et al., 2005) Let a continuous positive-definite function V (x) (defined on U ⊂ Rn ) satisfy V˙ (x) + α1 V (x) + α2 V m (x) ≤ 0,

t ≥ t0 ,

(2.11)

where α1 > 0, α2 > 0, and 0 < m < 1 are three constants. Then there exists a set U0 ⊆ n such that any V (x) starting from any given initial condition

Preliminaries Chapter | 2

27

x(t0 ) = x 0 ∈ U0 ⊆ Rn can reach V (x) ≡ 0 in finite time. Moreover, if T0 is the settling time needed to reach V (x) ≡ 0, then T0 ≤ t0 +

α1 V 1−m (x 0 ) + α2 1 ln . α1 (1 − m) α2

(2.12)

2.4 Modeling of spacecraft attitude control system 2.4.1 Coordinate frames The coordinate systems used for the rigid spacecraft attitude control are the inertial frame Fi (XI , YI , ZI ), the orbit reference frame Fo (XO , YO , ZO ), and the body-fixed frame Fb (XB , YB , ZB ). They are shown in Fig. 2.1. The frame Fi is with its origin at the center of the Earth, and it is used to determine the orbital position of the spacecraft. The frame Fo rotating about the YO axis with respect to Fi has its origin located in the mass center of the spacecraft. The axes of Fo are chosen such that the roll axis XO is in the flight direction, the pitch axis YO is perpendicular to the orbital plane, and the yaw axis ZO points toward the Earth. The frame Fb has the same origin as Fo , and its axes coincide with the principal axis of inertia.

FIGURE 2.1 Definition of the coordinate reference frames for spacecraft attitude system.

2.4.2 Attitude kinematics 2.4.2.1 The kinematics with attitude described by Euler angles The attitude kinematics described by Euler angles is governed by ω = ωbo − ωc (),

(2.13)

where  = [θ, φ, ψ] is the attitude Euler angles vector, ω ∈ R3 is the angular ˙ is the angular velocity velocity of the satellite given in Fb , and ωbo = R()

28 Fault-Tolerant Attitude Control of Spacecraft

in Fo with ⎡

1 0 ⎢ R() = ⎣ 0 cos θ 0 − sin θ

⎤ − sin φ ⎥ sin θ cos φ ⎦ , cos φ cos φ

⎤ cos φ sin ψ ⎢ ⎥ R() = ⎣ cos θ cos ψ + sin θ sin φ sin ψ ⎦ . − sin θ cos ψ + cos θ sin φ sin ψ

(2.14)

⎡

(2.15)

2.4.2.2 The kinematics with attitude described by unit quaternion The unit quaternion is adopted to describe the attitude of the spacecraft using a global representation without singularities (Sidi, 1997). The unit quaternion q is defined by     cos(/2) q0 , (2.16) q= = qv n sin(/2) where n ∈ R3 is the Euler axis,  ∈ R is the Euler angle, and q0 ∈ R and q v = [q1 , q2 , q3 ] ∈ R3 are the scalar and vector components of the unit quater2 nion, respectively. Moreover, the equation q  v q v + q0 = 1 is satisfied. Then the kinematics equation is  q˙ =

q˙0 q˙v



1 = 2



−q  vω q0 ω + q × vω

 .

(2.17)

2.4.2.3 The kinematics with attitude described by the modified Rodrigues parameters Given the Euler rotation angle  ∈ R about the Euler principle axis n ∈ R3 , the attitude of the spacecraft in the body-fixed frame Fb with respect to an inertial frame Fo can be represented in terms of the MRPs σ = [σ1 , σ2 , σ3 ] ∈ R3 given by (Schaub et al., 2001)   (t) , (t) ∈ [0o , 360o ). (2.18) σ = n tan 4 Then the differential equation for the kinematics described using MRPs can be summarized as (Shuster, 1993) 1 σ˙ = [(1 − σ  σ )I 3 + 2σ × + 2σ σ T ]ω = F (σ )ω. 4

(2.19)

Preliminaries Chapter | 2

29

Property 2.1. (Crassidis and Markley, 1996) The matrix F (σ ) satisfies the relationship F −1 (σ ) =

16 , (1 + σ  σ )2

F T (σ )F (σ ) =

(1 + σ  σ ) I 3. 4

(2.20)

This highlights that the inverse F −1 (σ ) always exists. Remark 2.1. (Schaub et al., 2001) It should be mentioned that as a complete revolution is performed (i.e., (t) → 360o ), this particular MRPs set goes singular. It is shown that it is possible to map the original MRPs vector σ to its corresponding shadow counterpart σ s = −σ /(σ  σ ). By choosing to switch the MRPs when σ > 1, the MRPs vector remains bounded within a unit sphere. Consequently, the global rotation representation without singularity can be guaranteed. Therefore, to avoid the singularity problem, MRPs σ and its shadow counterpart σ s are utilized to represent attitude rotation in this book.

2.4.3 Spacecraft dynamics 2.4.3.1 Dynamics of rigid spacecraft When all the actuators are fault-free, with the assumption of rigid body movement, the spacecraft dynamics can be found from Euler’s moment equation as (Bhat and Bernstein, 2000) J ω˙ + ω× J ω = τ + d,

(2.21)

where ω = [ω1 , ω2 , ω3 ] ∈ R3 denotes the angular velocity of the spacecraft with respect to an inertial frame Fi and expressed in the body frame Fb ; J ∈ R3×3 is the total inertia of the spacecraft; τ = [τ1 , τ2 , τ3 ] ∈ R3 is the control torque input; and d = [d1 , d2 , d3 ] ∈ R3 is the external disturbance. All internal and external torques are applied to the spacecraft aside from control torques, which may be due to onboard components, gravity gradients, solar pressure, atmospheric drag, or the ambient magnetic field.

2.4.3.2 Dynamics of flexible spacecraft Under the assumption of small elastic displacements, the mathematical model of a flexible spacecraft with flexible appendages is (Gennaro, 2003)   (2.22) J ω˙ + δ T χ¨ = −ω× J ω + δ T χ˙ + τ + d, χ¨ + C χ˙ + Kχ + δ ω˙ = 0,

(2.23)

where χ ∈ RN is the modal coordinate vector relative to the main body; δ ∈ RN ×3 denotes the coupling matrix between the flexible and rigid dynamics; C = diag{2ξi i , i = 1, 2, . . . , N} ∈ RN ×N and K = diag{2i , i = 1, 2, . . . , N } ∈

30 Fault-Tolerant Attitude Control of Spacecraft

RN ×N are the damping and stiffness matrices; N is the number of elastic modes considered; i are the natural frequencies; and ξi are the associated damping ratios.

2.5 Modeling of actuator faults In attitude maneuver or stabilization missions, control torques can be produced with the aid of momentum exchange devices, propulsion systems, magnetic torques, or even solar torque. The main sources of torques used for spacecraft attitude control can be classified as reaction wheel (RW), control moment gyros, and thruster systems. The thruster systems and control moment gyros are not discussed in this book because they are commonly used in large inhabited space structures to provide torques in the range of hundreds of Nm. As the effective actuator for fine pointing, the occurrence of RW faults will subsequently reduce performances and even destabilize the closed-loop system. Therefore in this section, we mainly focus on the analysis of the fault mechanism and mathematical model of RWs.

2.5.1 Reaction wheel faults Faults refer to any abnormal phenomenon of the control systems with undesired characteristics or the failure of some components, resulting in the deterioration of the control performance of the whole system. In general, the satellite attitude control system is composed of attitude sensor, control circuit, computer, actuator, and others. Its fault causes have the following several types: 1) Faults caused by the Sun and the Moon interference: when the Sun and the Moon enter the field of view of the Earth sensor, or the Sun shines directly on the Earth infrared sensor, certain interference will be generated to the sensor, resulting in measurement deviation ranging from 0.1 deg to 0.5 deg. So the sensor will fail to provide any measurement information during verification. 2) Faults caused by spacecraft pollution and environmental pollution: dust and gas from the spacecraft or space junk will contaminate the spacecraft mechanical parts. 3) Faults caused by temperature change: the harsh space environments with high or low temperatures may deform the material structure of the spacecraft, rendering that the relevant lubrication devices fail to operate normally. Thus the bearing friction torque of RW increases gradually, leading to motor stop stuck and other severe faults. 4) Faults caused by process problems: due to mechanical processing, circuit technology, welding technology, and other problems, the output error of the sensors may deviate from the error range specified in the design so that the system cannot work normally. 5) Faults caused by the damages of measuring elements during launching process: a spacecraft needs to go through a large overload launch process, during

Preliminaries Chapter | 2

31

which all components on the spacecraft are subjected to huge impact and strong vibration. As a result, the mechanical components that act as sensors or actuators may suffer different degrees of damage and cannot work normally. As above, the actual system may have different types of faults, so we need to classify the RW faults appropriately according to different classification criteria. The faults of spacecraft attitude control system can be divided into two types, single-machine level fault and system-level fault. Furthermore, threshold discrimination method sets different thresholds when the spacecraft is in different flight stages and discriminates the faults based on spacecraft attitude, angular velocity, and other system-level parameters and their rates of change. For the single-machine faults, different single-machine faults appear in different forms: 1) According to the fault location or unknown, the faults can be divided into the spacecraft body and its accessories fault, actuator fault, sensor fault, and onboard computer fault. 2) By the nature of the faults, they can be divided into the gradual fault, abrupt fault, and intermittent fault. 3) According to the relationship between the faults, the faults can be divided into the single fault, multiple fault (the fault involves multiple components), independent fault (the fault is caused by the fault of its own components and has no relationship with the fault of other components), and local fault. For example, power failure and measuring element failure are belong to the single fault. Through investigation and analysis of typical faults, we find that the fault of critical components, such as actuator, has a significant impact on the accuracy and stability of spacecraft attitude control. Note that RW faults account for 20% of the total actuator faults (Tafazoli, 2009). In what follows, we will illustrate the mechanism of RW faults in detail. The RW consists of a motor and a wheel body, which are mounted on the base by bearing. The RW model is governed by (Bialke, 1998) ˙ = KT i + τf , Jr 

(2.24)

where Jr ∈ R is the moment of inertia of the wheel body fixed to the motor rotor;  ∈ R is the rotational angular velocity relative to the spacecraft; i ∈ R is the current of the motor; KT ∈ R is the moment coefficient of the current; and τf ∈ R is the moment of friction. As the reaction flywheel is equipped with a rotating motor rotor and a wheel body, the rotor and the wheel body will continuously rotate during the wheel steady-state operation, which will lead to the following main faults: 1) Stuck: the RW will quickly stops rotating, and the actual output torque becomes zero after a huge disturbance; 2) Idle: the wheel body will maintain constant speed without generating any control torque;

32 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 2.2 The stuck fault of RW.

FIGURE 2.3 Failure to respond to signals.

FIGURE 2.4 The increasing friction of wheel.

3) Torque decrease: the friction torque between the bearing and the wheel body increases, which eventually leads to decrease of the RW output torque; 4) Deviation: the speed of the RW continues to drop for some reason, and the output torque is superimposed with certain deviation. Figs. 2.2–2.5 are the speed characteristics of the four kinds of RW faults. The solid lines in the figures represent the actual RW speed, whereas the dashed lines represent the expected output RW speed.

Preliminaries Chapter | 2

33

FIGURE 2.5 The decreasing speed of wheel.

According to the above fault characteristics, the RW stuck is a class of abrupt fault. If an appropriate diagnostic method is not designed to detect, diagnose, and deal with the fault, then it will have a great impact on the desired spacecraft attitude control performance. When the RW is used for attitude stabilization mission, the control torques required are generally small, and the RW speed is basically unchanged when it works normally. So it is difficult to detect whether the RW gets into the idling fault. However, when the spacecraft carries out attitude maneuver control command, the RW is required to change its body rotation quickly so as to output a large command control moment. Under such a circumstance, RW idling fault will have a serious impact on the attitude control performance. If the fault cannot be detected and handled in time, the attitude maneuver mission may be terminated unexpectedly. The latter two kinds of RW faults belong to the gradual fault. It is extremely difficult to diagnose these two faults in the initial stage due to the fact that the required output torque of the RW is small (the rotation speed of the flywheel does not change much at this stage), whereas the faults will accumulate and eventually affect the normal attitude control with the increasing of time, so if the corresponding fault handling method cannot be designed, then the fault will have a severe impact on attitude control. Finally, the fault mechanism of the RW can be summarized in Table 2.1.

2.5.2 Mathematical model of RW faults In spacecraft attitude control problems the active three-axis control algorithm is often designed to achieve high-precision control. The RW is one of the main actuators of spacecraft attitude control system due to its advantages such as low cost, simple structure, long life, and the ability to produce accurate and continuous changing control torque. The RW typically consists of a wheel body, a motor, and a bearing, as shown in Fig. 2.6. According to (Murugesan and Goel, 1987), the RW is a kind of precision equipment, which is extremely sensitive to its abnormal components, and there are mainly four kinds of faults: 1) Decreased reaction torque: the change rate of the RW rotor speed is affected by some factors, which include the increase of friction torque between

34 Fault-Tolerant Attitude Control of Spacecraft

TABLE 2.1 The analysis of how reaction wheel faults occur. Component name

Direct representation of the fault

Cause of the fault

Influence of the fault

RW

RW cannot start

Fault of the drive motor

RW reaction moment is 0

Fault of the photoelectric encoder Fault of the controller Stuck of the bearing RW

RW speed cannot be changed

Motor idling

RW reaction moment is 0

Drive motor

Drive motor cannot start

Stuck of the drive motor rotor and the stator

RW reaction moment is 0

Power down of the power devices Drive motor

RW speed remains constant

Motor idling

RW speed remains constant

Controller

No current output to the motor

Power down of the power devices

RW stops rotating

Abnormal output to the motor

Fault of the microprocessor

Fault of motor Controller

Fault of the power amplifier Hollow stator cup External rotor assembly Bearing

RW energy consumption increases

Deformation of the hollow cup Uncleaned iron particles

RW reaction moment is deviated from the normal value Reaction moment is reduced

Increase of bearing friction torque

the stator and the rotor, bearing fault, or decline of driving current. As a result, the actual torque is smaller than the command control torque, and the ratio of the former to the latter decreases. 2) Continuous generation of reaction torque: the RW body is accelerated or decelerated due to fault of voltage bus or intermittent fault of motor current, and thus an additional torque is generated based on the command control torque.

Preliminaries Chapter | 2

35

FIGURE 2.6 An exploded structure of a reaction wheel.

3) Increased bias torque: when the disturbance is ignored and the command torque of the attitude controller is zero, the RW should keep its wheel speed constant and generate no torque. However, the RW will still have a deviation moment due to the change of the bearing temperature or poor lubrication of the bearing. 4) Failure to respond to control signal: the RW may cannot respond to the control commands because of the fault of the driving circuit, the motor, or the power. Then the rotating speed of the wheel decreases slowly or remains almost unchanged, which results in zero torque. According to the classification of actuator faults in (Godard and Kumar, 2010), fault mode 1 (F1) corresponds to partial actuator fault in general sense; fault mode 2 (F2) corresponds to actuator float, fault mode 3 (F3) corresponds to actuator lock-in-place; and fault mode 4 (F4) corresponds to complete actuator fault. Therefore, for the ith RW (i = 1, 2, . . . , N), and N is the total number of RW installed), the above four faults can be modeled in the following mathematical form: ui = eii (t)uci + u¯ i ,

(2.25)

where ui ∈ R is the actual torque generated by the ith RW; uci ∈ R is the command control moment of the ith RW; u¯ i ∈ R is the float value of the ith RW; and 0 ≤ eii (t) ≤ 1 is the effective control factor of the ith RW. When the fault occurs to the RW, the values of each parameter in model (2.25) are as follows: 1) The ith RW works normally: eii (t) = 1 and u¯ i = 0; 2) The ith RW works in F1: 0 < eii (t) < 1 and u¯ i = 0; 3) The ith RW works in F2: eii (t) = 1 and u¯ i = 0; 4) The ith RW works in F3: eii (t) = 0 and u¯ i = 0; 5) The ith RW works in F4: eii (t) = 0 and u¯ i = 0.

36 Fault-Tolerant Attitude Control of Spacecraft

According to fault model (2.27) of the single RW, the actual control torque generated by total N RW is u = E(t)uc + u¯ c ,

(2.26)

where E(t) = diag([e11 , e22 , . . . , eN N ]) ∈ RN ×N is the fault factor matrix of total N RW; u = [u1 , u2 , . . . , uN ] ∈ RN is the actual control torque generated by total N RW; and u¯ c = [u¯ c1 , u¯ c2 , . . . , u¯ cN ] ∈ RN is the float moment of total N RW. For RW control systems, a unified actuator system consists of several RWs with different fixed directions. Therefore the actuator fault model can be established as τ = Du = D(E(t)uc + u¯ c ),

(2.27)

where D is the configuration matrix (or called control allocation matrix).

2.6 Summary In this chapter, we summarized the basic theory and attitude dynamics and kinematic equations of rigid and flexible spacecrafts. As the foundations, we introduced the basic notations and relevant lemmas. Then we gave the definitions of coordinate frames, spacecraft dynamics, and attitude kinematics described by Euler angle, the unit quaternion, and MRPs. In addition, we also established the flexible spacecraft dynamics. Finally, we developed a mathematical model of faulty in RWs with consideration of deviation and other uncertainties.

Chapter 3

Robust fault-tolerant attitude control 3.1 Introduction Actuator faults would deteriorate the control performance and even lead to catastrophic accidents of the spacecraft. In this chapter, we use robust faulttolerant control (FTC) technology to improve the reliability of the spacecraft when actuator failure happens. The intention of the technology is designing a control system that is insensitive to the faults and uncertainties by means of finding and tackling the worst situation of the faulty spacecraft. It also features that the structure of the controller does not change in the tuning stage, and no online fault detection and isolation mechanism, which are special algorithms to identify the fault information, are needed. Thus it can save computational power of the simple onboard processor. Numerous research results can be available for robust FTC with different approaches, such as linear matrix inequalities (Pujol, 2009), H∞ theory (Niemann and Stoustrup, 2005), sliding-mode control (Corradini et al., 2005), etc. However, the method is limited by the conservative design due to the unknown fault or disturbance. Hence an adaptive control method is incorporated in the robust fault-tolerant control method, which can estimate the boundary of the fault online to produce a nonconservative control system. Probably for these good characteristics, the robust FTC combined with adaptive control has attracted great attentions. For instance, a direct adaptive control scheme was developed for the time-variant linear systems for stuck fault, loss of effectiveness, and outage faults of actuator (Jin et al., 2010). An indirect adaptive fault-tolerant controller was developed for attitude tracking of a rigid spacecraft under thrusters faults, in which a bounded parameter of the lumped perturbations was introduced to be updated online (Cai et al., 2008). In the next section, we use the sliding-mode control method combined with the adaptive control technology to solve the spacecraft partial failure, stuck fault, and external disturbances. In Section 3.3, we design an FTC scheme based on adaptive backstepping method for nonlinear spacecraft dynamics and more general additive faults. As a foundation, we mainly considered the problem of attitude stabilization under fault. In Section 3.4, we further consider the problem of attitude tracking problem with model uncertainty and deliver an FTC method with H∞ performance index. Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00013-7 Copyright © 2021 Elsevier Inc. All rights reserved.

37

38 Fault-Tolerant Attitude Control of Spacecraft

3.2 Adaptive sliding-mode-based attitude FTC In this section, we first introduce the adaptive fault-tolerant control based on integral sliding-mode method to compensate the unknown time-invariant faults and external disturbances. The asymptotical stability of the closed-loop system is guaranteed. Then we extend the control method to tackle with the unknown time-varying faults. We achieve the L2 gain attenuation of external disturbance/parametric estimation error to system output.

3.2.1 Problem statement Consider the linear time-invariant system ˙ = Ax(t) + B 1 u(t) + B 2 w(t), x(t)

(3.1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the applied control input acting on the system, w(t) ∈ Rm is a continuous vector function denoting external disturbance torques and possible variations with respect to the nominal parameter values for the system, and A ∈ Rn×n , B 1 ∈ Rn×m , and B 2 ∈ Rn×p are constant matrices. We adopt the following multimodel fault model from (Jin and Yang, 2009): j

j

uFi,j (t) = ρi vi (t) + σi vsi (t)

(3.2)

for i = 1, 2, . . . , m and j = 1, 2, . . . , r, where uFi,j (t) represents the ith actuator j

j

j

that has failed in the j th faulty mode, 0 ≤ ρ i ≤ ρi ≤ ρ¯i is the unknown actuator j

efficiency factor for the ith actuator with known lower and upper bounds ρ i j

and ρ¯i , r is the total number of faulty modes and nonnegative constants, vi (t) is the ith commanded control input, vsi (t) is the bounded time-varying stuckactuator fault of the ith actuator, and σi is an unknown constant defined as ⎧ ⎨ j 0, ρi > 0, j σi = (3.3) ⎩ 0 or 1, ρ j = 0. i

Remark 3.1. Based upon the actuator fault model (3.2), we can note that the presented fault model can represent outage, loss of effectiveness, and stuck actuator faults. Based on the above analysis, for convenience of description, the following uniform actuator fault model is established for all possible faulty modes r: uF (t) = ρv(t) + σ v s (t),

(3.4)

where uF (t) = [uF1,j , uF2,j , . . . , uFm,j ] ∈ Rm , v s (t) = [vs1 , vs2 , . . . , vsm ] ∈ Rm , v(t) = [v1 , v2 , . . . , vm ] ∈ Rm , ρ = diag[ρ1 , ρ2 , . . . , ρm ] ∈ Rm×m , σ = j

j

j

Robust fault-tolerant attitude control Chapter | 3 j

j

39

j

diag[σ1 , σ2 , . . . , σm ] ∈ Rm×m , and j = 1, 2, . . . , r. Hence the linear system (3.1) incorporating actuator fault (3.4) can be rewritten in the following form: x˙ = Ax + β 1 ρv(t) + B 1 σ v s (t) + B 2 w(t).

(3.5)

To facilitate the controller development, we make the following assumptions with respect to faulty system (3.4). Assumption 3.1. All pairs (A, B 1 ρ) are uniformly completely controllable for any actuator fault mode ρ. Assumption 3.2. The considered actuator fault and external disturbance are piecewise continuous and bounded, that is, there exists unknown positive constants v¯s and w¯ such that v s (t) ≤ v¯s , w(t) ≤ w. ¯

(3.6)

Assumption 3.3. The equality Rank[B 1 ρ] = Rank[B 1 ] holds for any possible actuator fault mode ρ. Remark 3.2. It is worth mentioning that Assumption 3.1 is quite standard for feedback system design and denotes the internal stability of each normal and fault isolated system. Moreover, Assumption 3.2 is also generally used in the robust control in the existing literature. Therefore, Assumptions 3.1 and 3.2 are reasonable. Consider the faulty system (3.4) with actuator fault (3.4). The major objective of this section is developing an adaptive fault-tolerant control law such that the following goals are achieved in the presence of external disturbances: 1) When actuators experience constant fault all the time, the closed-loop system is asymptotically stable, that is, limt→∞ x(t) = 0. 2) In the case of time-varying actuator fault, all the signals of the resulting closed-loop system are uniformly ultimately bounded, and the controller allows L2 gain of the closed-loop system to be chosen arbitrarily small so as to achieve any level of L2 external disturbance attenuation.

3.2.2 Adaptive integral sliding-mode FTC law Note that the faulty system (3.4) contains mismatched uncertainty item w(t) and matched uncertainty item v s (t). As it is well known, SMCs have great robustness to the matched and mismatched uncertainties. Therefore, we develop the fault-tolerant controller based on SMC theory in this section. The selection of a sliding manifold and the synthesis of the control law are the main two steps involved in the SMC design, and the sliding-mode fault-tolerant attitude controller design will be finished in the subsequent part according to such two important steps.

40 Fault-Tolerant Attitude Control of Spacecraft

3.2.2.1 Sliding manifold design Due to the perfect benefit of the integral SMC, the system behavior is fully predictable from the initial time, and the generally slower and more oscillatory transient can be compensated for. In this subsection, we develop an integral sliding manifold following (Cao and Xu, 2004):  t L (A + B 1 K) x(τ )dτ, (3.7) S(x, t) = Lx(t) − Lx (t0 ) − t0

where the matrix L ∈ Rm×n is chosen by designer, and the matrix K ∈ Rm×n is designed such that the nominal fault-free system (3.1) is stable and some prescribed specifications are also satisfied via this nominal state feedback control. Remark 3.3. Note that the additional integral provides one more degree of freedom in design than the linear sliding surface. Moreover, the terms of −Lx (t0 ) achieve the nice property that S (x (t0 ) , t0 ) = 0, so that the reaching phase is eliminated. Since the sliding mode exists from the very beginning, the system is more robust to perturbations than the other SMC systems with reaching phase.

3.2.2.2 Adaptive sliding-mode controller under constant fault Once the sliding manifold is designed, according to the standard SMC design steps, the next stage is designing an appropriate control law such that the condition of the sliding mode is satisfied. In this subsection, we mainly tackle with constant fault problem, that is, ρ in the fault model (3.4) is an unknown but time-invariant constant matrix. According to the sliding manifold (3.7), the following SMC law is proposed to guarantee the stability of the considered system under actuator fault: ˆ 0 x(t) + K 1 (t), v(t) = K

(3.8)

ˆ 0 = [Kˆ 0,1 , Kˆ 0,2 , . . . , Kˆ 0,m ] ∈ Rm×n is the estimate of the unknown where K ˆ 0,i ∈ Rn (i = 1, 2, . . . , m), and K 0 is such that parameter matrix K 0 ∈ Rm×n , K B 1 ρK 0 − B 1 K = 0. Moreover, the nonlinear term K 1 (t) ∈ Rm in (3.8) is defined by    ˆ + λS2   μ1 S  L β(t) K 1 (t) = − S  LB 1 ,   2 μ2 S LB 1 

(3.9)

(3.10)

where λ is a positive constant, and the parameters μ1 and μ2 are chosen to satisfy the following inequality:  2      2 μ2 S  LB 1  ≤ μ1 S  LB 1 ρ  ,

(3.11)

Robust fault-tolerant attitude control Chapter | 3

41





j √ ˆ in (3.10) dewhere ρ = diag[ ρ j , ρ 2 , . . . , ρ jm ]. On the other hand, β(t) notes the estimate of an unknown constant parameter β, which is supposed to satisfy β ≥ B 1  σ v¯s + B 2  w. ¯ Remark 3.4. (Xiao-Zheng 2009) With the designed control   and Guang-Hong, ˆ law (3.8), we have x˙ = A + B 1 ρ K 0 x+ B 1 ρK 1 (t) + B 1 σ v s (t) + B 2 w(t). As a consequence, when the considered system is external disturbance free, equation B 1 ρK 1 (t) = −B 1 σ v s (t) must hold for compensation of the stuck fault v s (t). At this time, a necessary and sufficient condition of its solution is Rank[B 1 ρ] = Rank[B 1 ρ, −B 1 σ v s (t)], which leads to the equation Rank[B 1 ρ] = Rank[B 1 (ρ, −σ v s (t))], and then it follows that Rank[B 1 ρ] = Rank[B 1 ] according to the relations between ρ and σ . Consequently, Assumption 3.3 is a sufficient condition for the stability control of the system, that is, when some actuators fail to work, the remaining active actuator should be able to produce a sufficient actuating signals to perform the given maneuverer. More specifically, for the high-performance spacecraft dynamic systems, the total number of actuators mounted is usually greater than the number of states to be controlled. Hence Assumption 3.3 is feasible from the point of practical view. Using the designed control law (3.8), the derivative of S(x, t) with respect to time can be calculated as follows: ˙ ˙ − (LAx(t) + LB 1 Kx(t)) S(x, t) = Lx(t) = L (B 1 ρv(t) − B 1 Kx + B 1 σ v s (t) + B 2 w(t))  ˆ 0 x(t) + B 1 ρK 1 (t) − B 1 Kx(t) = L B 1ρK

(3.12)

+B 1 σ v s (t) + B 2 w(t)) . Accordingly, from (3.9) and (3.12) we easily obtain that ˜ 0 x(t) + S  LB 1 ρK 1 (t) S  S˙ = S  LB 1 ρ K + S  L (B 1 σ v s (t) + B 2 w(t)) ,

(3.13)

˜0=K ˆ 0 − K0 where K To this end, we are ready to summarize the control solution to the underlying control problem by the SMC action. Theorem 3.1. Consider the system with actuator faults given by (3.4) under Assumptions 3.1–3.3. Let the sliding-mode controller designed in (3.8) be implemented and updated by     β˙ˆ = α S  L , (3.14) ˆ 0,i (t) = −i S  LB 1,i x, i = 1, 2, . . . , m, K

(3.15)

42 Fault-Tolerant Attitude Control of Spacecraft

where α and i are two positive control parameters, and B 1,i is the ith column of B 1 . Then objective (1) stated in Section 3.2.1 can be achieved for any possible constant actuator fault ρ, that is, limt→∞ x(t) = 0. Proof. To examine the stability property of the resulting closed-loop system, consider the following Lyapunov function candidate:

 m 1 1 ˜ ˜ 1 2  ˜ V= ρi K 0,i K 0,i + β , (3.16) S S+ 2 i α i=1

ˆ 0,i , K 0,i is the ith column of K 0 , and K 0,i ˜ 0,i = K 0,i − K ˆ K where β˜ = β − β, is the estimate of K 0,i . Then inserting (3.14)–(3.15) into the time derivative of V yields m 1 ˜  ˙˜ 1 ρi K 0,i K 0,i + β˜ β˙˜ V˙ = S  S + i α i=1

˜ 0 x + S  LB 1 ρK 1 + S  LB 1 σ v s = S LB 1 ρ K m      ˜ ˜ S + S  LB 2 w − ρi S  LB 1,i K x − β L   0,i 

i=1

    ≤ S LB 1 ρK 1 + S  L B 1  σ  v s          + S  L B 2  w − β˜ S  L     ˆ ≤ − S  L β(t) + λS2         + β S  L − β˜ S  L = −λS2 , 

(3.17)

which implies that limt→∞ V (t) = V (∞) exists. By integrating V from 0 to ∞ we obtain  t 1 lim S(τ )2 dτ ≤ (V (0) − V (∞)) . (3.18) t→∞ 0 λ Because the term on the right-hand side is bounded, by the Barbalat lemma it follows that limt→∞ S(t) = 0. Moreover, from (3.7) we have S(x(0), 0) = 0, that is, a sliding mode occurs from the initial time t = 0. Thus the global asymptotical stability of the closed-loop system can be guaranteed, that is, limt→∞ x(t) = 0. Thereby the proof is completed. Remark 3.5. From above analysis we see that the controller designed here not only achieves the system stability, but also asymptotically stabilizes the system state even in the presence of disturbances and actuator faults, whose fault modes belong to the fault model defined in (3.4). Further, the actuator fault compensation adaptively adjusts the remaining actuators without the knowledge of the fault parameters.

Robust fault-tolerant attitude control Chapter | 3

43

3.2.2.3 Adaptive controller design under time-varying fault For the design of adaptive sliding-mode fault-tolerant control law (3.8), we assume that the fault mode ρ is a constant function of time. However, in practice, such a condition could not be satisfied, especially for the case in which the actuators partially lose their actuation power along the time, that is, fading actuation. In this sense, the fault mode is a time-varying function of time. In addition, to achieve the stabilization of the system states, particularly for the spacecraft control application, the tracking/regulation performance is also needed for the control system design in the presence of fault. In the following, we give an extended case of above designed control to treat the time-varying fault. Note that although ρ is time varying and unknown, it is still confined to requirement (3.4) during the entire operation of the system. It is therefore reasonable    j to assume that there exists a constant (unknown) such that ρi  is bounded for j = 1, 2, . . . , r and i = 1, 2, . . . , m. For convenience of controller development, we introduce the following assumption. Assumption  3.4. There exists a positive but unknown constant κ such that j ρ˙ maxi=1,2,...,m ii ≤ κ < ∞. Whereas the unknown constant parameter β is reassumed to such that β ≥ σ v¯s , we can state the following our second main result. Theorem 3.2. Consider the system with actuator faults given by (3.4) under Assumptions 3.1–3.4. We apply the sliding-mode fault-tolerant control (3.8) with the modified nonlinear term and update laws    ˆ + λS2   μ1 S  LB 1  β(t) K 1 (t) = − S  LB 1 , (3.19)   2 μ2 S LB 1      β˙ˆ = α S  LB 1  , ˙ˆ  K 0,i (t) = −i S LB 1,i x,

i = 1, 2, . . . , m.

(3.20) (3.21)

Suppose that the control parameter γ is chosen such that λ > LB 2 2 + γ , and ˆ be the estimate of β. Then objective (2) presented in Section 3.2.1 can let β(t) be achieved for any possible time-varying fault ρ, and the following objectives can be met: 1) The closed loop is globally stable in that all the states are bounded and continuous. 2) The L2 gain level of the disturbance attenuation can be guaranteed to be smaller than a given small value γ . 3) The system performance index

44 Fault-Tolerant Attitude Control of Spacecraft

 I (t) = lim

t

t→∞ t 0

S(τ )2 dτ

(3.22)

is bounded for any possible fault ρ. Proof. The stability and control performance can be analyzed by using the same Lyapunov candidate function given in (3.16), and following the same lines as in the proof of Theorem 3.1, we can detail it as m m 1 j ˜  ˙˜ 1 j ˜ ˙ ˙ ˜ 0,i (t) + 1 β˜ β˙˜ V =S S + ρi K 0,i K 0,i + ρ˙i K 0,i (t)K i i α i=1

i=1

˜ 0 x + S  LB 1 ρK 1 + S  LB 1 σ v s + S  LB 2 w = S  LB 1 ρ K m m   1 j ˜  j ˜ 0,i (t) − ˜ ˜  ρ˙i K 0,i (t)K ρi S  LB 1,i K + 0,i x − β S LB 1  i i=1 i=1       S LB 1 √ρ 2 μ1 S  LB 1  β(t) ˆ + λS2 ˜ 0x − ≤ S  LB 1 ρ K  2 μ2 S  LB 1              − β˜ S  LB 1  + β S  LB 1  + S  LB 2  w(t) −

m

j ˜ ρi S  LB 1,i K 0,i x

i=1

m 1 j ˜ ˜ 0,i (t) + ρ˙i K 0,i (t)K i i=1

m   1 j ˜   ˜ 0,i (t) ≤ − λS2 + S  LB 2  w(t) + ρ˙i K 0,i (t)K i i=1 

m  2   ˜  2 2 2 ≤ − λ − LB 2  − γ S + w(t) + κ K 0,i (t) − γ S2



m  2   ˜ 0,i (t) − γ S2 , ≤ w(t)2 + κ K

i=1

(3.23)

i=1





√ j j j where thematrix ρ = diag[ ρ1 , ρ2 , . . . , ρm ] and Assumption 3.4 are em   w(t)2 +κ

m 2 ˜ i=1 K0,i (t)

ployed in the derivation of (3.23). Clearly, if S > , y ˙ then V < 0, which implies that V (t) monotonically decreases. Therefore the sliding manifold signal is uniformly ultimately bounded by  ⎧  2  ⎫   ⎪   ⎪  m ⎪ ⎪ ⎪  w(t)2 + κ i=1 K˜ 0,i (t) ⎪ ⎬ ⎨  m , lim S ∈ S ∈ R | S ≤ t→∞ ⎪ ⎪ γ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ (3.24)

Robust fault-tolerant attitude control Chapter | 3

45

which is a small set containing the origin S = 0; moreover, the larger the selected γ , the better the control performance. Thus we can conclude from (3.23)–(3.24) that the resulting closed-loop system is globally stable. Integrating both sides of (3.23) results in  V (t) − V (t0 ) ≤ − γ

t

t0

+

 t t0

S2 dτ

m  2 ˜  w(τ ) + κ K 0,i (τ ) dτ.

(3.25)

2

i=1

Noting that V (t) is a nonnegative function, we have 

t

γ

S dτ ≤ V (t0 ) + 2

t0

 t t0

m  2  ˜ w(τ ) + κ K 0,i (τ ) dτ. 2

(3.26)

i=1

If the initial conditions are chosen to be zero, then the L2 gain 

t

γ t0

 t m  2   ˜ 0,i (τ ) dτ. S2 dτ ≤ w(τ )2 + κ K t0

(3.27)

i=1

In terms of (3.27) and the results of (Van der Schaft and Van Der Schaft, 2000), we can conclude that the L2 gain level of the disturbance attenuation can be guaranteed to be a given small value by adjusting γ , thereby completing the proof of achieving the control objectives stated in Section 3.2.1. Remark 3.6. Due to the discontinuity of K 1 (t) in (3.10) and (3.19), the developed fault-tolerant control law (3.8) is discontinuous across the switching manifold S = 0, which may result in chattering or singularity. This chattering is practically undesirable since it may excite unmodeled high-frequency dynamics. One approach to reduce the chattering is introducing a so-called boundary layer around the switching surface (Cao and Xu, 2004). Then (3.10) is revised as follows under boundary layer modification: ⎧    ⎪ S  L β(t) ˆ + λ2     μ ⎪ 1 ⎪    ⎪  ⎪ − S LB , S LB   ≥ ε, ⎪ 1 1   2 ⎨    μ2 S LB 1 K 1 (t) =    ⎪ ⎪ ˆ + λS2  ⎪    μ1 S  L β(t) ⎪ ⎪   ⎪ S  LB 1 , S  LB 1  < ε, ⎩− μ2 ε (3.28)

46 Fault-Tolerant Attitude Control of Spacecraft

where ε = 0.01, and also (3.19) is modified as ⎧ 2   ⎪ S  LB 1  β(t) ˆ + λS  ⎪    μ 1 ⎪ ⎪   ⎪ S  LB 1 , S  LB 1  ≥ ε, ⎪   2 ⎨ −   μ2 S LB 1 K 1 (t) =    ⎪ ⎪ S  LB 1  β(t) ⎪ ˆ + λS2     μ ⎪ 1 ⎪   ⎪ ⎩− S  LB 1 , S  LB 1  < ε. μ2 ε (3.29)

3.2.3 Numerical example 3.2.3.1 Flexible spacecraft attitude model and actuator fault modes We give an example of attitude control for a flexible linearized spacecraft system to verify the performance of the proposed scheme. After the model linearization, the dynamics and kinematics equations of a flexible spacecraft are described by (Hu, 2008c) φ˙ = ω0 ψ + ω1 ,

(3.30)

θ˙ = ω0 + ω2 ,

(3.31)

ψ˙ = −ω0 φ + ω3 ,

(3.32)

J ω˙ + δ  χ¨ = −ω× J ω + DT c + d,

(3.33)

¯ + Kχ ¯ + δ ω˙ = 0, χ¨ + Cχ

(3.34)

where [ψ, θ, φ] represents the attitude orientation of the spacecraft in the body frame Fb with respect to orbital frame Fo obtained by a yaw-roll-pitch sequence of rotations (φ, θ and ψ are, respectively, the roll, pitch, and yaw angles), ω = [ω1 , ω2 , ω3 ] denotes the angular velocity of the spacecraft with respect to the inertial frame I, and ω0 denotes the orbital rate for the considered spacecraft. Moreover, J = diag[Jx , Jy , Jz ] is the moment of inertia of the flexible spacecraft, Tc = [Tc1 , Tc2 , . . . , Tcl ] ∈ Rl denotes the control torque vector produced by l actuators, where the redundancy constraint l > 3 is required, L ∈ R3×l is the distribution matrix, Td ∈ R3 represents the external 1 ¯ = diag[i ] ∈ RN ×N are the disturbance, C¯ = diag[2ξi i2 ] ∈ RN ×N and K damping and stiffness matrices with modal damping ξi , and modal frequency 1

i2 (i = 1, 2, . . . , N, where N is the number of elastic modes considered), δ ∈ RN ×3 is the coupling matrix between the elastic structures and rigid body, and χ ∈ RN is the modal coordinate vector. From (3.33) it follows that   ω˙ = J −1 DT c + J −1 d − ω× J ω − δ  χ . (3.35)

Robust fault-tolerant attitude control Chapter | 3

47

Hence the flexible spacecraft attitude control system (3.30)–(3.34) can be summarized in the following form: x˙ = Ax + B 1 u(t) + B 2 w(t),

(3.36)

where x = [ϕ, θ, ψ, ω1 , ω2 , ω3 ] , w(t) = [ ω0 ⎡ 0 ⎢ 0 ⎢ ⎢ ⎢ −ω0 B1 = ⎢ ⎢ 0 ⎢ ⎣ 0 0

 d  ] , u(t) = T c , ⎤ 0 ω0 1 0 0 0 0 0 1 0 ⎥ ⎥ ⎥ 0 0 0 0 1 ⎥ ⎥. 0 0 0 0 0 ⎥ ⎥ 0 0 0 0 0 ⎦ 0 0 0 0 0

(3.37)

(3.38)

Moreover, we assume that the considered spacecraft plant is activated by four reaction wheels, that is, l = 4, and the wheel distribution matrix is given by ⎡ ⎤ 1 1 −1 −1 1⎢ 1 −1 ⎥ D = ⎣ −1 1 (3.39) ⎦. √ √ √ 2 √ 2 2 2 2 To illustrate the feasibility and effectiveness of the designed attitude control system, we introduce the following five possible fault modes for the four reaction wheels. 1) Normal mode 1: All the reaction wheels are normal, which means that ρi1 = 1 and σi1 = 0 (i = 1, 2, 3, and 4). 2) Fault mode 2: The first reaction wheel is stuck, and the remaining three reaction wheels may be normal or have lost effectiveness but still active, that is, the actuator efficiency factors are given by ρ12 = 0, σ12 = 1, 0 < ρi2 ≤ 1, and σi2 = 0 for i = 2, 3, and 4. 3) Fault mode 3: The second reaction wheel is stuck, and the others may be normal or have lost effectiveness. The actuator efficiency factors are given by ρ23 = 0, σ23 = 1, 0 < ρi3 ≤ 1, and σi3 = 0 for i = 1, 3, and 4. 4) Fault mode 4: The third reaction wheel is stuck, and the others may be normal or have lost effectiveness but still active. The actuator efficiency factors are given by ρ34 = 0, σ34 = 1, 0 < ρi4 ≤ 1, and 0 < ρi4 ≤ 1 for i = 1, 2 and 4. 5) Fault mode 5: The forth reaction wheel is stuck, and the others may be normal or have lost of effectiveness but still active. The actuator efficiency factors are given by 0 < ρi4 ≤ 1, σ45 = 1, 0 < ρi5 ≤ 1, and σi5 = 0 for i = 1, 2, and 3. The first four elastic modes have been taken into consideration in the model used for simulation at 1 = 0.7681, 2 = 1.1038, 3 = 1.8733 and

48 Fault-Tolerant Attitude Control of Spacecraft

4 = 2.5496 rad/s with dampings ξ1 = 0.0056, ξ2 = 0.0086, ξ3 = 0.013, and ξ4 = 0.013. The reaction wheels are assumed to experience the following faults during its operation: Between the 50th and 60th seconds, fault mode 2 has occurred with vs2 (t) = 0.5 + 0.2 sin(t), and ρi2 (i = 2, 3, 4) are random numbers at the interval [0.6, 1]; from the 90th to 100th seconds, fault mode 3 has been experienced with vs3 (t) = 0.5 + 0.25 cos(0.5t), and the other three wheels partially lose their control powers with ρ13 = 1 − 0.01t, ρ33 = 0.45, and ρ43 = 0.7 + 0.1 sin(0.5t); also, fault mode 4 will appear at 140 s and end at 150 s with vs4 (t) = 1 + 0.25 cos(0.8t), and the other three wheels partially lose theirs effectiveness with ρ14 = 0.1, ρ24 = 0.1, and ρ44 = 0.7 + 0.15 sin(0.5t + 2π/5). To implement the proposed controller (3.8), the design parameters used in the simulations have been chosen by trial-and-error until a good stabilization performance was obtained, and the following choices are made for the control gains in the proposed fault-tolerant controller (3.8): α = 10, λ = 10, μ1 = 10, ˆ μ2 = 10, i = 15 (i = 1, 2, 3, 4), β(0) = 0, and ⎡

8.6468 −7.0711 ⎢ ⎢ 8.6483 7.0711 K =−⎢ ⎣ −8.6721 7.0711 −8.6721 −7.0711 ⎡ 2 0 0 0 1 ⎢ ⎢ 0 3 0 0 0 L = 10 ⎢ ⎣ 0 0 3 0 2 0 0 0 2.5 0

10.0263 10.0263 9.9737 9.9737 ⎤ 0 ⎥ 1 ⎥ ⎥, 0 ⎦ 1

⎤ 55.3463 −44.4037 37.6878 ⎥ 55.3463 44.4037 37.6878 ⎥ ⎥, −55.217 44.4037 37.6382 ⎦ −55.217 −44.4037 37.6382

ˆ 0,1 = − [ 8.6483 −7.0711 10.0263 55.3463 −44.4037 37.6878 ] , K ˆ 0,i (0) = [ 0 0 0 0 0 0 ] K

(i = 2, 3, 4).

3.2.3.2 Simulation results In the simulation the physic parameters of the flexible spacecraft are given by ⎡

⎤ 350 0 0 ⎢ ⎥ J =⎣ 0 270 0 ⎦ kg · m2 0 0 190 and

⎡ ⎢ ⎢ δ=⎢ ⎣

⎤ 6.45637 1.27814 2.15629 ⎥ −1.25819 0.91756 −1.67264 ⎥ 1/2 2 2 ⎥ kg · m /s . 1.11687 2.48901 −0.83674 ⎦ 1.23637 −2.6581 −1.1253

Robust fault-tolerant attitude control Chapter | 3

49

The disturbance is d = (ω2 + 0.05)[sin 0.8t, cos 0.5t, cos 0.3t] Nm. Further, in the context of simulation, at time t = 0, the orientation of spacecraft is such that the initial attitude is described by φ(0) = 6◦ , θ (0) = 4◦ , and ψ(0) = −4◦ with zero initial body angular velocity, and the initial modal variables χ i (0) and their time derivatives χ˙ i (0) (i = 1, 2, 3, 4) are supposed to be χ i (0) = χ˙ i (0) = 0.

FIGURE 3.1 Time responses of attitude.

FIGURE 3.2 Time responses of attitude quaternion.

We first present the simulation results of the actuator fault-free case when applying controller (3.8). The time responses of attitude angle, angular velocity, modal displacement, and actuator output torque are shown in Figs. 3.1–3.5 (a, solid line), respectively. It is clear that the proposed controller managed to perform attitude stabilization maneuverer with good control performance. Especially, we see in Figs. 3.1 and 3.2 (a, solid line) that the control law (3.8) successfully stabilizes the attitude in 30 s with high pointing accuracy and high slew rate accuracy even in the presence of external disturbances. Moreover, the elastic vibrations are significantly attenuated by considering them in the design,

50 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 3.3 Time responses of velocity.

FIGURE 3.4 Time responses of control input.

and the oscillations settle within 40 s, as shown in Fig. 3.5 (a, solid line). In addition, the time responses of the performance index I (t) and the time-varying control parameter β(t) further verify the control performance of the proposed attitude controller, as we can see in Figs. 3.6 and 3.7 (dashed line); it is clear that the index I (t) converges to zero in 20 s. Considering the case that the actuator faults are introduced to the attitude system. When the proposed controller (3.8) is implemented to the attitude sys-

Robust fault-tolerant attitude control Chapter | 3

51

FIGURE 3.5 Time responses of vibration displacements.

tem, the time responses of attitude orientation and angular velocity are presented in Figs. 3.1–3.3 (b, solid line), respectively. The driving torque is shown in Fig. 3.4 (b, solid line). As expected, we clearly see that our proposed faulttolerant control scheme managed to compensate the actuator faults that high attitude pointing accuracy and slew rate accuracy are still met, and no significant amount of vibration occurred even if the faults were involved. Those results verify that our designed controller is capable of attenuating the external disturbances and unknown actuator faults while guaranteeing the capability of the spacecraft pointing precision. For comparison, we also applied the conventional proportional-derivative (PD) control method to the spacecraft attitude control system with and without actuator faults and designed the following PD controller: T c = D † (ω× J ω + δ  χ¨ − Kd ω − Kp q),

(3.40)

where D † is the pseudoinverse of D, that is, D † = D  (DD  )−1 , and the control gains Kp and Kd are chosen to be equal to 50 and 75, respectively. In the case of no actuator faults the application of PD controller (3.40) leads to the attitude orientation and angular velocity, as shown in Figs. 3.1 and 3.2 (a, dashed line) and Fig. 3.3 (a, dashed line), respectively. We clearly see that the attitude stabilization could be achieved in 20 s. The corresponding control torque is shown in Fig. 3.4 (a, dashed line). Moreover, no terrible elastic oscillation will be induced, as we see the time response of vibration displacement in Fig. 3.5 (a, dashed line). On the other hand, the PD controller (3.40) will significantly degrade the attitude control performance, the pitch angle cannot be stabilized,

52 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 3.6 Time responses of performance index I (t).

FIGURE 3.7 Time responses of parameter β(t).

as shown in Fig. 3.1 (b, dashed line), and severe oscillations are observed in Fig. 3.5 (b, dashed line). More specifically, as we can see in Fig. 3.2 (b, dashed line), the resulting closed-loop attitude system is unstable; this is because the PD controller has no any fault-tolerant effort to compensate the actuator faults. Summarizing all the cases (normal and fault cases), note that the proposed controller design method can significantly improve the attitude control performance than the PD method in both theory and simulations. We can also observe that as more and more faults case considered in the design, the proposed controllers can still guarantee stabilization performance as in the normal case. In addition, extensive simulations were also done using different control parameters, disturbance inputs, and even combination of the reaction wheel faults. These results show that in the closed-loop system, attitude control and vibration stabilization are accomplished in spite of these undesired effects in the system. Furthermore, the flexibility in the choice of control parameters can be utilized to obtain desirable performance while meeting the constraints on the control

Robust fault-tolerant attitude control Chapter | 3

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magnitude and elastic deflection. These control approaches provide a theoretical basis for the practical application of the advanced control theory to flexible spacecraft attitude control systems.

3.3 Robust fault tolerant attitude stabilization control In this section, we use adaptive backstepping method to design an asymptotically stable control law for the nominal system without failure. Subsequently, we consider a time-varying additive fault and partial failure fault and design an auxiliary compensation controller to realize the asymptotic stability of the closed-loop system.

3.3.1 Problem statement The MRPs are used to describe the attitude of the rigid body relative to the inertial frame with the kinematics given by (2.22) and the dynamics described by (2.21). The dynamic equation of a spacecraft in (2.21) assumes that all the actuators are fault-free, and it is called the nominal system. We now simultaneously consider two types of actuator faults, an additive fault f ∈ R 3 and the loss of effectiveness of the actuators represented by a multiplicative matrix δ ∈ R 3×3 . Hence the attitude dynamic model given by (2.21) can be rewritten as J ω˙ = −ω× J ω + (δu + f ) + d,

(3.41)

where δ = diag[δ1 , δ2 , δ3 ] is the actuator effectiveness matrix with 0 < εi < δi (t) ≤ 1, and εi is a known constant. The case δi (t) = 1 indicates that the ith actuator is working normally, and 0 < δi (t) < 1 corresponds to the case in which the ith actuator partially loses its effectiveness but still works all the time. Assumption 3.5. For the additive fault f , there exists a positive continuous function k(t) satisfying f  ≤ k(t).

(3.42)

Assumption 3.6. The disturbance d is assumed to be bounded, and hence there exists a positive unknown constant dmax such that d ≤ dmax .

(3.43)

Remark 3.7. The actual output torque generated is bounded due to practical physical limitations of the actuators, and thus the additive fault is also bounded. Therefore Assumption 3.5 is reasonable for an orbiting spacecraft. The external disturbance d, acting on the spacecraft, is time varying and incorporates gravitational perturbations, atmospheric drag, and solar radiation pressure forces. In practice, these forces are bounded, and hence Assumption 3.6 is reasonable.

54 Fault-Tolerant Attitude Control of Spacecraft

The control objective of the proposed method can now be stated as follows: Design a fault-tolerant attitude stabilization control scheme for the faulty attitude system given by (3.41) such that the following goals are met in the presence of external disturbances and actuator faults (in the form of additive and loss of effectiveness faults): 1) All the signals in the closed-loop system are bounded and continuous. 2) The attitude orientation and angular velocity asymptotically converge to zero, that is, limt→∞ σ = 0 and limt→∞ ω = 0.

3.3.2 Robust fault tolerant controller design We first develop a nominal controller using the standard adaptive backstepping technique in the case where the actuators are fault-free. Then we design an auxiliary controller and use it in addition to the nominal controller to compensate actuator faults. To design the desired control law, we introduce the following new variables:  x 1 = σ dt ∈ R3 , x 2 = σ ∈ R3 , x 3 = ω ∈ R3 . (3.44) Then (2.22) and (2.21) may be rewritten as x˙ 1 = x 2 ,

(3.45)

x˙ 2 = F (x 2 ) x 3 ,

(3.46)

J x˙ 3 = −x × 3 J x 3 + u + d,

(3.47)

which represents a set of general nonlinear equations of motion for the rigid spacecraft and will be used for the attitude controller design.

3.3.2.1 Nominal controller design for fault-free actuator Using (3.44), the spacecraft attitude system given by (2.22) and (2.21) may be written in triangular nonlinear form. The standard backstepping controller design (Kokotovi´c and Arcak, 2001) may be applied to the nominal plant given by (3.45)–(3.47). Consider the change of coordinates given by z1 = x 1 ,

(3.48)

z2 = x 2 − α 1 ,

(3.49)

z3 = x 3 − α 2 ,

(3.50)

where α 1 and α 2 are viewed as virtual controls and will be designed later. In the following theorem, we summarize our control solution to the underlying attitude stabilization problem by incorporating adaptive backstepping control.

Robust fault-tolerant attitude control Chapter | 3

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Theorem 3.3. Consider attitude control for the nominal system given by (2.22) and (2.21), with Assumptions 3.5 and 3.6. Suppose that the adaptive backstepping control law is implemented as  ˆ u = x× 3 J x 3 − c3 z3 − F (x 2 ) z2 − d max sgn (z3 )   −1 − J F˙ (x 2 ) (z1 + c1 x 2 + c2 z2 ) + F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z2 ) , (3.51) where

dˆ max = c4 z3 ,

(3.52)

the virtual control laws are given as α 1 = −c1 x 1 and α 2 = F −1 (x 2 ) (−z1 −c1 x 2 − c2 z2 ), ci (i = 1, 2, 3, 4) are positive control gains, and dˆ max is the estimate of d max · Then the closed-loop system is asymptotically stable, that is, the attitude orientation and angular velocity satisfy σ → 0 and ω → 0 as t → ∞. Proof. The proof uses elements of Lyapunov stability theory and is elaborated in three steps. Step 1. Consider (3.48) with x 2 as the control variable. The time derivative of z1 is z˙ 1 = x˙ 1 = x 2 = z2 + α 1 .

(3.53)

The objective is designing a virtual control law α 1 to ensure that z1 → 0. Choose the candidate Lyapunov function V 1 = 12 z 1 z1 . From the definition of the virtual control α 1 we have 2  V˙ 1 = z ˙ 1 = z 1z 1 (z2 − c1 z1 ) = −c1 z1  + z1 z2 .

(3.54)

Clearly, if z2 = 0, then V˙ 1 = −c1 z1 2 , and z1 is guaranteed to converge to zero asymptotically. Step 2. Differentiating (3.49), we have z˙ 2 = x˙ 2 − α˙ 1 = F (x 2 ) x 3 + c1 x 2 .

(3.55)

Choose the second candidate Lyapunov function V 2 as 1 z2 . V 2 = V 1 + z 2 2

(3.56)

Differentiating V 2 and using (3.50), (3.54), and (3.55) yield  V˙ 2 = c1 z1 2 + z 1 z2 + z2 (F (x 2 ) x 3 + c1 x 2 )  = c1 z1 2 + z 1 z2 + z2 (F (x 2 )    × z3 + F −1 (x 2 ) (−z1 − c1 x 2 − c1 z2 ) + c1 x 2

= − c1 z1 2 − c2 z2 2 + z 2 F (x 2 ) z3 .

(3.57)

56 Fault-Tolerant Attitude Control of Spacecraft

Clearly, V˙ 2 = −c1 z1 2 − c2 z2 2 if z3 = 0, and thus both z1 and z2 will converge to zero asymptotically. Step 3: Finally, from (3.50) we have " # z˙ 3 = x˙ 3 − α 2 = J −1 −x × 3 J x3 + u + d + F˙

−1

(x 2 ) (z1 + c1 x 2 + c2 z2 ) + F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z˙ 2 ) ,

(3.58)

where # " 1  −x 2 (F (x 2 ) x 3 ) I 3 + F (x 2 ) x × + F (x 2 ) x 3 x  F˙ (x 2 ) = 2 3 2   +x 2 x  3 F (x 2 )

(3.59)

and #2 # " "  16 1 + x 2 2 F˙ (x 2 ) − 64 1 + x 2 2 x  2 F (x 2 ) x 3 F (x 2 ) . F˙ (x x ) = " #4 1 + x 2 2 (3.60) Choose the third candidate Lyapunov function V 3 as −1

1 1 ˜2 d , J z3 + V 3 = V 2 + z 2 3 2c4 max

(3.61)

where d˜ max = d max − dˆ max . With the derived control law (3.51) and the updating law (3.52), the time derivative of V 3 may be calculated as 1  ˙ 3 − d˜ max d˙ˆ max V˙ 3 = V˙ 2 + z 3 Jz c4 1 ˜ ˙ = − c1 z1 2 − c2 z2 2 + z d d max 2 F (x 2 ) z3 − c4 max   × ˙ −1 (x 2 ) (z1 + c1 x 2 + c2 z2 ) + z 3 −x 3 J x 3 + u + d + J F  +F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z˙ 2 )   ˆ max sgn (z3 ) + d = − c1 z1 2 − c2 z2 2 − c3 z3 2 + z − d 3 −

1 ˜ ˙ d d max c4 max

  1  ≤ − c1 z1 2 − c2 z2 2 − c3 z3  −dˆ max + d max − d˜ max d˙ max c4 3 ≤− ci zi 2 ≤ 0, (3.62) i=

Robust fault-tolerant attitude control Chapter | 3

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which implies that limt→∞ V 3 (t) = V 3 (∞) exists. By integrating V˙ 3 from 0 to ∞ we have lim

 t 3

t→∞ 0

ci zi (τ )2 dτ ≤ V 3 (0) − V 3 (∞).

(3.63)

i=

Because the right-hand side is bounded, by Barbalat’s lemma it follows that z1 , z2 , z3 → 0 as t → ∞, and Lasalle’s theorem (Krstic et al., 1995) proves the global uniform boundedness of z1 , z2 , and z3 . Thus the global asymptotic stability of the closed-loop system is guaranteed. Moreover, since z1 = x 1 , x 1 is also bounded, and limt→∞ x 1 = 0. The boundedness of x 2 and limt→∞ x 2 = 0 are deduced from (3.48) and (3.49). Similarly, the boundedness of x 3 and x 3 → 0 follow from (3.50) and the definition of α 2 . Thus the proof is completed. Remark 3.8. The definition of the new variables in (3.44) explicitly incorporates integral feedback of the attitude into the adaptive backstepping control law (3.51). Thus constant external torques can be compensated, and the steady-state error can be reduced.

3.3.3 Robust fault tolerant controller design with actuator faults From the above analysis, the proposed control law in (3.51) can achieve the asymptotical stability of the closed-loop attitude system with fault-free actuators. This controller is called the nominal controller and denoted by unor . To perform attitude stabilization maneuvers with faults defined by (3.41), an auxiliary controller uaux is developed in addition to the nominal controller unor to compensate for the actuator faults. Consequently, the control input u to system (3.41) is u = unor + uaux ,

(3.64)

where unor is given by (3.51), and the fault-tolerant controller uaux is synthesized as   λ unor  k(t) + γ + , (3.65) uaux = −sgn (z3 ) unor  + δ¯ δ¯ where δ¯ = mini=1,2,3 εi , and λ and γ are positive constants with λ > 1. Theorem 3.4. Consider the rigid spacecraft attitude control problem with additive actuator faults and the partial loss of actuator effectiveness defined by (2.22) and (3.41). The control law defined in (3.64) and updated using (3.52) achieves the control objectives stated in Section 3.2.1. Proof. Since the actuator faults δ and f are only introduced into the attitude dynamic equations, the stability analysis for the first and second steps is identical to that for the nominal system. Hence in the analysis below, we concentrate on the third step.

58 Fault-Tolerant Attitude Control of Spacecraft

Substituting the fault-tolerant controller (3.64) into the equations of motion of the spacecraft with the actuator faults (3.41) gives ˙2 J z˙ 3 = −x × 3 J x 3 + (δu + f ) + d − J α

˙2 = −x × 3 J x 3 + (I 3 − η) u + f + d − J α

˙2 = −x × 3 J x 3 + unor + uaux − η (unor + uaux ) + f + d − J α

(3.66)

˙ 2, = −x × 3 J x 3 + unor + δuaux − ηunor + f + d − J α

where η = I 3 − δ. Hence the inequality 0 < η∞ ≤ 1 always holds. Here we also use the candidate Lyapunov function V 3 defined in Theorem 3.1. Substituting (3.66) into the time derivative of V 3 yields  ˙ −1 (x 2 ) (z1 + c1 x 2 + c2 z2 ) V˙ 3 = − c1 z1 2 − c2 z2 2 + z 3J F  +F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z˙ 2 ) + z 2 F (x 2 ) z3 " × # 1 ˙ ˆ ˜ D + z D 3 −x 3 J x 3 + unor + δuaux − ηunor + f + d − c4 ≤ − c1 z1 2 − c2 z2 2 − c3 z3 2 + z 3 (δuaux − ηunor + f ) 3         k(t) + ci zi 2 + z ≤− z3  η∞ unor   3 i=1

  λ unor  k(t) + γ unor  + + δ δ¯ 3         ≤− ci zi 2 + z 3  k(t) + z3  η∞ unor  − z 3 δsgn (z3 )

(3.67)

i=

  δ u  λ   min nor     u  δ − − z  z3   min nor 3 δ     − z 3  (k(t) + γ ) ≤−

3 i=

3     u  + (1 − λ) − ci zi 2 − γ z ci  2 .  nor 3 i=

Note that to derive (3.67), we used the inequalities 0 < δ¯ < δmin ≤ 1 and λ > 1. The analysis used for the fault-free attitude control system governed by the nominal control law (3.51) may also be used for (3.67) to conclude that the attitude orientation and the angular velocity are globally asymptotically stable. Thus σ → 0 and ω → 0 as t → 0. Hence we achieved the control objectives, and this completes the proof. Remark 3.9. Since the fault-tolerant control law designed in (3.64) does not involve the actuator effectiveness δ, FDD is not required to obtain knowledge of

Robust fault-tolerant attitude control Chapter | 3

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δ and f , and the actuator fault effect is compensated adaptively by the proposed controller. Furthermore, from (3.64) we see that the proposed controller can compensate the faults δ and f , which are either fixed constants or time varying. This will be demonstrated by the simulations. Remark 3.10. The control law given by (3.64) is discontinuous due to the sign function sign(·), which may lead to the well-known chattering effect. This chattering can excite unmodeled high-frequency dynamics and can make the system unstable. To overcome this problem, various smoothing functions may be substituted for sign(·); for example, (3.64) can be modified to (Hu, 2008c)   λ unor  k(t) + γ z3 unor  + , (3.68) + uaux = − z3  + ρ δ¯ δ¯ where ρ is time varying and calculated with the method given in (Hu, 2008c).

3.3.4 Simulation example To demonstrate the effectiveness and performance of the proposed control scheme, numerical simulations have been performed using the model of a rigid spacecraft system given in (2.22) and (3.41) with the fault-tolerant control given by (3.64). The inertia matrix for the spacecraft is (Di Gennaro, 2003) ⎡ ⎤ 350 3 4 ⎢ ⎥ J =⎣ 3 270 10 ⎦ kg · m2 , 4 10 190 and the external torque disturbance is   d = ω2 + 0.05 [sin 0.8t, cos 0.5t, cos 0.3t] N · m. To demonstrate the superior performance of the proposed control scheme, we simulated three actuator fault cases: Case 1− All actuators are healthy; Case 2− Time-varying loss of actuator effectiveness; and Case 3− Time-varying loss of actuator effectiveness incorporating time-varying additive faults. For comparison, we also designed and implemented a conventional proportional-integral derivative (PID) controller. The PID controller is designed without reconfiguration and given by  (3.69) u = ω× J ω − Kd ω − Kp σ − KI σ dt, where Kp , Kd , and KI are the control parameters. To implement the controllers, we selected control and adaptation gains by trial-and-error until a good performance was obtained. The control gain parameters of the proposed controllers and the PID controller are

60 Fault-Tolerant Attitude Control of Spacecraft

c1 = c2 = 5, Kd = 15,

c3 = c4 = 10, KI = 10,

λ = γ = 1.5, Kp = 25.

(3.70) (3.71)

These parameters are fixed for all of the simulation cases to enable a fair comparison. At time t = 0, the orientation of spacecraft is given by σ (0) = [−0.3, −0.4, 0.2] with zero initial angular velocity. To make the simulations more realistic, we considered attitude sensor noise modeled as zero-mean Gaussian random variables with variance σp2 . In the attitude measurement and angular velocity equations, " # (3.72) σm,i (t) = σi (t) + N 0, σp , i = 1, 2, 3, # " (3.73) ωm,i (t) = ωi (t) + N 0, σp , i = 1, 2, 3, where σm,i (t) are the measured attitudes, and ωm,i (t) are the measured angular velocities. Note that σi (t) and ωm,i (t) are the real attitudes and angular # " velocities, respectively, obtained from the equations of motion, and N 0, σp represents a zero-mean Gaussian white noise with variance σp . In the simulations, σp = 0.01.

3.3.4.1 Case 1. Healthy actuators This case simulates the ideal situation where no actuator faults occur. Fig. 3.8 (a,c,e) shows the simulation results for the nominal controller (NC) given by (3.51) and demonstrates that high control precision and an acceptable system performance have been achieved. This illustrates that the designed controller is capable of maneuvering the spacecraft. For comparison, the nominal spacecraft attitude system is also controlled using the traditional PID controller given by (3.69), and the results are also shown in Fig. 3.8 (b,d,f). The attitude rotational maneuver is successfully performed using PID control, but severe oscillations are excited during the attitude maneuver process with a very large settling time. Although the performance could be improved by different design control parameters, the improvement in the attitude and velocity responses would be limited. 3.3.4.2 Case 2. Loss of actuator effectiveness only In this case, we consider the following partial loss of actuator effectiveness fault: $ 1, t < 10 s, δi = (3.74) 0.25 + 0.1 sin(0.5t + iπ/3), t ≥ 10 s. Fig. 3.9 shows the simulated results obtained by including the faulty actuators for three controllers, namely the designed FTC controller given by (3.64) (a,d,g), the NC controller given by (3.51) (b,e,h), and the PID controller given by (3.69) (c,f,i). The FTC controller compensates for the time-varying fault, although the system performance degrades to some degree and the settling time

Robust fault-tolerant attitude control Chapter | 3

61

FIGURE 3.8 Time responses of vibration displacements.

increases significantly once the failure is introduced. However, the system ultimately regulates the attitude to near zero within 20 s. Note that larger control forces are required than the NC controller even before the actuators have failed,

62 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 3.9 Time responses of vibration displacements.

and this is why the simulated attitude and angular velocity responses are different between the NC and FTC controllers for t < 10 s. When the NC controller is implemented, the partial loss of actuator effectiveness does not change the structure of the controller, but does decrease the control power. The attitude stabilization maneuver is still performed successfully due to the robustness of NC controller to external disturbances. However, the attitude control performance deteriorates severely due to the actuator fault, with severe overshoots in the attitude orientation and angular velocity, and an increased settling time. The attitude control performance with the PID controller is significantly degraded after faults are introduced. The oscillations are further excited after actuator failure, as demonstrated in the time responses of attitude and angular velocity with a significantly increased settling time. These simulations demonstrate the theoretical result that the desired performance of the system can be achieved by the proposed FTC even if the faults are unknown in advance.

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3.3.4.3 Case 3. Simultaneous faults This example represents the severe case in which both additive faults and the partial loss of actuator effectiveness occur. At t = 10 s, each actuator undergoes a partial loss of effectiveness, whereas at t = 15 s, the actuators also experience additive faults motivated by a stick-type fault. The nonlinearity summarized by (Jin et al., 2008; Cai et al., 2008; Chen and Saif, 2007) is used to generate the actuator fault scenario defined as $ 1, t < 10 s, δi = (3.75) 0.25 + 0.1 sin(0.5t + iπ/3), t ≥ 10 s, $ fi =

0, 1 + 0.05 sin(0.2πt),

t < 15 s, t ≥ 15 s.

(3.76)

Fig. 3.10 shows the results using the three different control laws based on the same simulation conditions. Clearly, the effects of the actuator faults propagate to the spacecraft dynamics and are not compensated by the NC and PID controllers. Significant degradation of the control performance and system instability after the faults are introduced can be observed. In contrast, the FTC controller can achieve the objective of fault-tolerant control and does succeed in simultaneously compensating for the two types of fault. Indeed, when the FTC controller in (3.64) is applied, the time-varying fault is immediately rejected from the attitude and angular velocity, as shown by the comparison of Figs. 3.10(a–d) with Figs. 3.8(a–c), due to the effect of the auxiliary controller unor in (3.64). In summary, for both normal and fault cases, the proposed controller significantly improves the normal control performance of the closed-loop attitude system, compared to the PID and NC approaches. For the cases with actuator faults, the proposed method gives better results than those of the conventional controllers. As the faults become more severe, the proposed controller still guarantees system stability. In addition, extensive simulations were performed using different control parameters, disturbance inputs, and even different combinations of actuator faults. These results show that closed-loop system attitude control stabilization is accomplished with the proposed control scheme in spite of these undesired effects on the system.

3.4 Robust H∞ attitude tracking FTC Aiming at the problem of flexible spacecraft attitude tracking with model uncertainties, we design an adaptive sliding-mode control method to achieve the uniformly ultimately boundedness of the closed-loop system and the disturbance suppression performance under the meaning of H∞ performance.

64 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 3.10 Time responses of vibration displacements.

Let tr(A) denote the trace of a matrix A. For all x, y ∈ R, we define the projection mapping operator Proj χ (y) as ⎧ ⎪ 0 if χ = χ max and y < 0, ⎪ ⎪ ⎪ ⎪ ⎨ −y if χ min < χ < χ max , Projχ (y) = −y if χ = χ max and y ≥ 0, ⎪ ⎪ ⎪ −y if χ = χ min and y ≤ 0, ⎪ ⎪ ⎩ 0 if χ = χ min and y > 0, where χ min and χ max are the known constants. For all b = [b1 , b2 , b3 ] ∈ R3 , we define the linear operator (Ahmed et al., 1998) L(b): R3 → R3×6 by ⎤ ⎡ b1 0 0 0 b 3 b 2 ⎥ ⎢ L(b) = ⎣ 0 b2 0 b3 0 b1 ⎦ . 0 0 b3 b2 b1 0

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3.4.1 Attitude tracking control system and control problem 3.4.1.1 Modeling of attitude tracking control system  Let q e = [qe0 , q  ev ] denote the relative attitude error from a desired reference frame to the body-fixed reference frame. Then   q e = q ⊗ q −1 d = [qe0 , q ev ] ,

(3.77)

where q is defined in (2.16), q −1 d denotes the inverse of the desired quaternion −1  q d with the definition q d = [qd0 , −q  d ] , and ⊗ is the operator for quaternion multiplication defined as % & qa0 qb0 − q  qb a (3.78) qa ⊗ qb = qa0 q b + qb0 q a − q × a qb for any given two quaternions q a and q b . Then the relative spacecraft attitude error dynamics is & % ' ( 1 q˙0e −q  ev = (3.79) ωe 2 qe0 I + q × q˙ e ev with the angular velocity error ωe given by ωe = ω − R d ωd ,

(3.80)

where R d is the rotation matrix from the desired reference frame to the bodyfixed reference frame, and ωd is the angular velocity of the desired reference frame with respect to the inertial reference frame. Under the assumption of small elastic displacements, the dynamic equations of a spacecraft with flexible appendages are given by (2.24)–(2.25). To facilitate the subsequent control formulation, we make the following two assumptions. Assumption 3.7. The external disturbance d(t) is unknown but has the property )∞ that d(t) ∈ L2 (0, ∞). Thus the disturbance energy 0 |d(t)|dt is bounded. Assumption 3.8. The elastic oscillation and its velocity are assumed to be ˙ bounded. Thus χ(t) and χ(t) are bounded during the whole attitude control process. Remark 3.11. The net disturbance force d(t) acting on the spacecraft consists mainly of gravitational perturbations, atmospheric drag, and solar radiation pressure preservative forces, and these disturbance torques are bounded. It is therefore reasonable to make Assumption 3.7 as for spacecraft. Remark 3.12. Assumption 3.8 is also satisfied for flexible spacecraft systems since damping always exists in flexible structures, even if it is small, such that the magnitudes of elastic vibration and its velocity are bounded.

66 Fault-Tolerant Attitude Control of Spacecraft

From the definition of ωe in (3.80) and the spacecraft dynamics given by (2.24) the attitude tracking error equation can be derived as (Cai et al., 2008)   # " ˙d J ω˙ e + δ  χ¨ = − ω× J ω + δ  χ˙ + J ω× e Rωd − R ω (3.81) + τ + d, where R is the" rotation matrix # related to the error× quaternion vector q e and 2 − q q given by R = qe0 ev ev I + 2q ev q ev − 2qe0 q ev . For the attitude tracking model given by (3.81), consider the situation where an actuator fault occurs, especially when the actuator loses total or partial control power. A simple model of the actuator faults is incorporated into the nonlinear spacecraft attitude error dynamics model and is described by   # " ˙d J ω˙ e + δ  χ¨ = − ω× J ω + δ  χ˙ + J ω× e Rωd − R ω (3.82) + DEu + d(t), where u ∈ Rl denotes the propulsion force vector produced by l actuators, and D ∈ R3×l is the actuator distribution matrix. For a given spacecraft, D is available and can be made full-row rank by properly placing the actuators at certain locations and directions on the spacecraft. The diagonal matrix E ∈ Rl×l characterizes the health of the actuators and is defined as E = diag[e11 , e22 , . . . , ell ]

(3.83)

with 0 ≤ eii ≤ 1 (i = 1, 2, . . . , l) indicating the actuator health for the ith actuator. Remark 3.13. In the actuation effectiveness matrix E, the case eii = 1 represents no fault in the ith actuator, whereas eii = 0 represents the complete failure of the ith actuator from which no force is generated. The case 0 < eii < 1 corresponds to the ith actuator partially losing actuation power. Hence the matrix E is an unknown or possibly even time-varying diagonal matrix.

3.4.1.2 Control problem statement   The signal z = [ρ1 q  ev , ρ2 σ ] is used to evaluate the attitude control performance, where σ defines a sliding surface, and ρi > 0 (i = 1, 2) are the weighting coefficients. Suppose the level of disturbance attenuation is given as γ > 0. The control objective is determining a control law u from the flexible spacecraft attitude tracking system given by (3.82) such that: 1) All the signals of the resulting closed-loop attitude tracking system are uniformly ultimately bounded; 2) Torque level disturbances and/or elastic vibration attenuation with respect to the attitude quaternion error, along with the angular velocity penalty, are ensured using the H∞ performance index.

Robust fault-tolerant attitude control Chapter | 3

67

Furthermore, these two objectives should be met despite unknown external disturbances and system uncertainties, severe actuator faults, and force limits on each actuator.

3.4.2 Adaptive sliding-mode FTC with H∞ performance 3.4.2.1 Adaptive sliding-mode FTC design We propose a robust adaptive SMC strategy for the attitude tracking control problem. By considering the error quaternion and the angular velocity vector we propose the following sliding surface: σ = ωe + βq ev ,

(3.84)

where β > 0 is a design parameter. Remark 3.14. To facilitate the controller design, let us consider d¯  d(t)− δ  χ¨ − ω× δ  χ˙ as the lumped disturbances. From Assumptions 3.7 and 3.8 it can be shown that the energy ) ∞ of d is bounded, that is, there exist a constant ¯ (unknown) μ > 0 such that 0 d(t)dt ≤ μ. From Remark 3.14, (3.82) can be simplified as # " ˙d J ω˙ e = − (ωe + Rωd )× J (ωe + Rωd ) + J ω× e Rωd − R ω ¯ + DEu + d(t). From (3.79), differentiating (3.84) and inserting (3.85) yield # " ˙d J σ˙ = − (ωe + Rωd )× J (ωe + Rωd ) + J ω× e Rωd − R ω # βJ " ¯ qe0 I + q × + ev ωe + DEu + d(t). 2

(3.85)

(3.86)

The elements of the unknown symmetric inertia matrix J = [J ij ] are represented by 1  [J 11

J 22 , J 33 , J 23 , J 13 , J 12 ] .

(3.87)

Then (3.86) can be rewritten as ¯ J σ˙ = M 1 1 + DEu + d,

(3.88)

where " # M 1 ωe , Rωd , R ω˙ d , q˙ e = − (ωe + Rωd )× L (ωe + Rωd )   " # # β" × ˙ q + L ω× + L ω Rω − R ω I + q d d e0 e . e ev 2 (3.89)

68 Fault-Tolerant Attitude Control of Spacecraft

For the controller derivation, consider the Lyapunov-like function   1 ˜˜ 1 2 tr ϒ ϒ + V = σ J σ + β q  ev q ev + (1 − qe0 ) 2 2α 1 ˜  −1 ˜ 1 ˜  −1 ˜ + 

1 + 

2 , 2 1 1 2 2 2

(3.90)

where i ∈ R6×6 (i = 1, 2) are positive definite symmetric constant matrices, α is a positive constant, and the elements of the unknown symmetric matrix (dJ /dt) are represented by '

dJ 11 dJ 22 dJ 33 dJ 23 dJ 13 dJ 12 2  , , , , , dt dt dt dt dt dt

( ,

(3.91)

ˆ and  ˆ i denote the estimates of the uncertain matrix ϒ  DED  and the ϒ uncertain vector i (i = 1, 2), respectively, and the estimate errors are defined ˜ ϒ −ϒ ˆ and  ˜ i  i −  ˆ i (i = 1, 2). as ϒ Differentiating V with respect to time and inserting (3.88) give V˙ = σ  M 1 1 + βσ  q ev + σ  DEu + σ  d¯ + σ  M 2 2 1  ˜  ˙  ˜  −1 ˙ ˜  −1 ˙˜ ˜ − β 2q  ev q ev + tr ϒ γ˜ + 1 1 1 + 2 2 2 , α

(3.92)

where dJ σ = L(σ )2 = 2M 2 2 . (3.93) dt We are now ready to summarize the first result for the attitude tracking problem. Theorem 3.5. Consider the flexible spacecraft attitude tracking system given by (3.79) and (3.85) with Assumptions 3.7 and 3.8. We apply the control law   ˆ −1 M 1  ˆ 1 + M 2 ˆ 2 + βq ev + κσ u = −D  ϒ (3.94) and update it by ˙ˆ = ασ u , ϒ

(3.95)

˙ˆ = M σ ,  1 1 1

(3.96)

˙ˆ = M σ .  2 2 2

(3.97)

Suppose that the control gains are chosen to satisfy κ−

1 − ρ22 > 0, 4γ 2

(3.98)

Robust fault-tolerant attitude control Chapter | 3

β > ρ1 > 0,

69

(3.99)

where κ is a positive control gain to be determined. Then the control objectives (a) and (b) stated in Section 3.4.1.2 are achieved. Remark 3.15. The conclusion of Theorem 3.5 is independent of the initial conditions q(0) and ω(0) and the reference trajectories q d and ωd . Therefore the proposed control law (3.94) can guarantee the spacecraft attitude to follow any desired attitude from any initial attitude with any angular velocity. In addition, for convenience, the initial conditions for the parameter updating laws in ˆ ˆ 1 (0) = 0, and  ˆ 2 (0) = 0. (3.95)–(3.97) are selected as ϒ(0) = I,  Remark 3.16. The actuation effectiveness matrix E is not used in the control scheme (3.94), and so there is no need to include a health monitoring unit to identify or estimate which actuator is unhealthy, and fault isolation is not required. The actuator fault accommodation/compensation is performed automatically and adaptively by the proposed control algorithm. There are l actuators (l > 3) properly mounted on the spacecraft, and the remaining active actuators are assumed to be able to produce a combined force sufficient to perform the required attitude manoeuvres. Moreover, the number of completely failed actuators is no more than l − 3 to guarantee that the attitude manoeuvres may be accomplished. If the number of failed actuators is more than l − 3, then the system will become underactuated, and the designed controller in (3.94) will not guarantee the stability of the resulting closed-loop attitude system. The underactuated system is not considered further. This feature is necessary to build affordable and effective fault-tolerant spacecraft control schemes. To stabilize the system, the rank of matrix D should be equal to 3, that is, the remaining active actuators are able to produce an efficient actuating torque vector for the spacecraft to perform the given mission. Remark 3.17. In the updating laws (3.95)–(3.97), it is known that even a small disturbance may lead to the divergence of the estimates of the parameters. To avoid this rare but possible case, we use the discontinuous projection for the adaptive laws (3.95)–(3.97) and rewrite the adaptation laws as follows:   ˙ˆ = Projϒ ˆ ασ u , ϒ

(3.100)

˙ˆ = Proj ( M σ ) ,  1 1 1 ˆ1 

(3.101)

˙ˆ = Proj ( M σ ) .  2 2 2 ˆ2 

(3.102)

Here we have slightly abused the notation by using Proj to stand for both scalarvalued and vector-valued projection operators.

70 Fault-Tolerant Attitude Control of Spacecraft

Proof of Theorem 3.5. In view of (3.92) and the proposed control law (3.94), we have V˙ ≤ σ  M 1 1 + βσ  q ev + σ  M 2 2 − β 2 q  ev q ev   2 σ  ¯ 2 − tr ϒ ˜  σ u + + γ 2 d 2 4γ  ˜  ˆ ˜ ˜ − 1 M 1 σ − 2 M 2 σ + σ ϒu + σ ϒu      1 2 = − κ − 2 − ρ22 σ 2 − β 2 − ρ12 q ev  4γ ¯ 2 − z2 + γ 2 d   2  ¯ 2 − z2 , ≤ −k1 σ 2 + q ev  + γ 2 d

(3.103)

# # " ## "" " where k1  min κ − 1/4γ 2 − ρ22 , β 2 − ρ12 , and the equation z2 =  2 ρ12 q ev  + ρ22 σ 2 is employed in the derivation of (3.103). Then integrating inequality (3.103) from t = 0 to ∞ yields  V (∞) +

∞

 z2 dt ≤ V (0) + γ 2

0

∞

¯ 2 dt. d

(3.104)

0

Clearly, we see in (3.104) that the L2 gain from the unknown bounded disturbance, caused by the external disturbances and elastic vibrations, to the penalty signal z is prescribed by the constant γ . (3.104) shows that the H∞ control performance is achieved. Thus the proof is completed.

3.4.2.2 Modified adaptive sliding mode FTC design In Section 3.4.2.1, we designed a controller for the attitude tracking manoeuvre for flexible spacecraft by an adaptive SMC law with an unknown inertia matrix and disturbance torques. However, the assumption of boundedness of the lumped perturbation must be satisfied in advance. To relax the assumption, we propose a modified adaptive SMC, in which we also consider the effect of elastic vibrations in the robustness design. Introduce the state ψ = χ˙ + δωe .

(3.105)

Note that ψ˙ = χ¨ + δ ω˙ e = [−K

' − C]

+ Cδωe + δω× e Rωd

χ ψ

− δR ω˙ d .

( (3.106)

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Thus from (3.81) we have J 0 ω˙ e = − (ωe + Rωd )× J 0 (ωe + Rωd ) − (ωe + Rωd )× δ  δRωd − (ωe + Rωd )× δ  ψ − δ  Cδωe " # ˙ d + DEu + d + J 0 ω× e Rωd − R ω ' ( χ + δ  [K C] , ψ

(3.107)

where J 0  J −δ  δ. Computing the derivative of σ defined in (3.84) and taking (3.107) into account, it follows that ' ( χ J 0 σ˙ = M 3 3 + H (ωe , Rωd ) − βq ev + δ  [K C] ψ (3.108) − (ωe + Rωd )× δ  ψ + DEu + d with " # M 3 ωe , Rωd , R ω˙ d , q˙ e

# " ˙d  (ωe + Rωd )× L (ωe + Rωd ) + L ω× e Rωd − R ω   # β" qe0 I + q × ω +L e , ev 2 H (ωe , Rωd )  − (ωe + Rωd )× δ  δRωd − δ  Cδωe + βq ev ,  ˆ −1 M 3  ˆ 3 + M 2 ˆ 2+H u = − Dϒ "  ⎞   # δ  [K C] + ω [0 δ  ] 2 + 1 σ + κσ ⎠ . + 4γ 2

(3.109)

(3.110)

(3.111)

Theorem 3.6. Consider the flexible spacecraft attitude tracking system governed by (3.79) and (3.107) with Assumption 3.7. Suppose that the adaptive SMC input u is determined by (3.111) (see below) and the updating law ˙ˆ = ασ u , ϒ

(3.112)

˙ˆ = M σ ,  2 2 2

(3.113)

˙ 3 = 3M 3σ , 

(3.114)

where i+1 (i = 1, 2) are positive definite symmetric matrices. The control parameters are chosen such that β > ρ1 > 0,

(3.115)

72 Fault-Tolerant Attitude Control of Spacecraft

β > ρ1 > 0.

(3.116)

Then, for all possible initial conditions q(0) and ω(0) and reference trajectories q d and ωd , the control objectives (a) and (b) stated in Section 3.4.1.2 are achieved. Proof. The starting point is again the choice of an appropriate Lyapunov function   1 2 V¯ = σ  J 0 σ + β q  ev q ev + (1 − qe0 ) 2 (3.117) 1  ˜  ˜  1 ˜  −1 ˜ 1 ˜  −1 ˜  +

. tr ϒ ϒ + 3 3 3 +  2 2α 2 2 2 2 By taking the time derivative of V¯ along trajectories generated from (3.108) to (3.116) we have   ˜ 3 + M 2 ˜2 V˙¯ ≤ −κσ 2 + σ  M 3  "    # δ  [K C] + ω [0 δ  ] 2 + 1 σ 2 + 4γ 2 '

(2   ˜ 2  χ 2  + d − β 2 q  q ev (3.118) + σ ϒu + γ  ev ψ       2 = − κ − ρ22 σ 2 − β 2 − ρ12 q ev  '

(2   2  χ 2  + d − z2 . +γ  ψ  # " ## "" If we choose k2  min κ − ρ22 , β 2 − ρ12 , then   2  V˙¯ ≤ −k2 σ 2 + q ev  + γ 2

'

(  χ 2 2 2    ψ  + d − z .

(3.119)

The validation of the specifications of cases (a) and (b) follows the same argument developed in Section 3.4.2.1. Note that to keep the estimated parameters bounded, the discontinuous projection is also applied to the adaptive law in (3.112)–(3.114). This completes the proof. Remark 3.18. In Theorems 3.5 and 3.6, there are many parameters that need to be determined, such as γ , ρ1 , ρ2 , β, κ, and i . Note that here the parameters ρ1 and ρ2 are the weighting coefficients of the penalty signal z and are usually selected as 1. The parameters β and i are easily determined according to the requirements of the closed-loop system.

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Remark 3.19. We can see from (3.104) and (3.119) that a high robustness to external disturbance/elastic vibrations is guaranteed by a prescribed level γ , and the smaller the γ , the better the resulting attitude control accuracy. However, small values of γ require large control inputs. Hence the value of γ will be determined according to the attitude pointing accuracy requirement of the considered spacecraft mission.

3.4.2.3 Modified adaptive sliding-mode FTC design with actuator constraint From the above analysis, controller (3.111) enables the spacecraft to track a desired trajectory in the presence of parameter uncertainties, disturbances, and even unknown actuator faults. From a practical perspective, one of the major issues in the attitude control system design is that the signal u in (3.111) generated by the control law might not be implemented because of physical constraints. A common example of such a constraint is actuator saturation, which imposes limitations on the magnitude of the achievable control input. When the actuator saturation is considered, the actual implemented adaptive SMC is different from (3.111) as follows:   (3.120) u = −umax Sat D  ut , "  # where ut is given by (3.111), and Sat D ut is defined as ⎧   ⎪ Dσ   ⎪ ⎪ , umax ≤ D  ut  ,   ⎨ Dσ  Sat D  ut = (3.121)   u ⎪ D ⎪    t ⎪ ⎩ , 0 ≤ D ut  < umax , umax # " |ui | ≤ uimax , where umax = min u1max , u2max , . . . , ulmax (i = 1, 2, . . . , l) denotes the minimal value of the maximal allowable actuator force uimax of each actuator. Note that here ui (i = 1, 2, . . . , l) denotes the components of the actuator control forces. Then we give the following stability analysis result. Theorem 3.7. Consider the flexible spacecraft attitude tracking system given by (3.79) and (3.107) with Assumption 3.7. Suppose that the functional actuators are able to produce a combined force sufficient to allow the spacecraft to follow a given target in the sense that there exists a constant ε such that the umax strictly dominates the unknown uncertainty, disturbance, and possible elastic vibration, that is,  ' ( D χ M 3 3 + M 2 2 + H + δ  [K C] umax ≥ ψ λ (3.122)  ×  −ω δ ψ + d + ε ,

74 Fault-Tolerant Attitude Control of Spacecraft

where λ is a design parameter less than the minimum eigenvalue of DED  . The we apply the control law in (3.120) with (3.121). For all possible initial conditions q(0) and ω(0) and reference trajectories q d and ωd , the control objectives (a) and (b) as stated in Section 3.4.1.2 are guaranteed. Proof. By (3.120) we need two cases to prove the stability of the closed-loop attitude system.   Case 1. For D  ut  ≥ umax , we consider a new Lyapunov function V˜ = # " 2 ˜ (1/2)σ  J 0 σ + β q  ev q ev + (1 − qe0 ) . Then the time derivative of V can be calculated as   ' ( λumax  χ ˙   ˜ + M 3 3 + M 2 2 + H + δ [K C] V ≤ σ  − ψ D  (3.123)  − (ωe + Rωd )× δ  ψ + d  − β 2 q  ev q ev ≤ − εσ  − β 2 q  ev q ev ≤ 0.   Case 2. For D  ut  < umax , we consider the same Lyapunov function V¯ as defined in (3.117). Then the Lyapunov derivative can be algebraically rearranged in steps identical to those employed in deriving (3.118), and the same argument as in Theorem 3.6 can be applied. This completes the proof. Remark 3.20. Based on the stability analysis in the previous section, the designed FTC law can be summarized as follows: Step 1: Check the amplitude value of actuator limit such that umax = min(u1max , u2max , . . . , ulmax ). Step 2: Choose the control gain β such that the sliding surface σ = 0 is stable and the system has the specified convergence speed. Step 3: Choose the scalar γ according to the attitude pointing accuracy of the considered spacecraft mission. Step 4: Based on Remarks 3.18 and 3.19, design the parameters κ, ρ1 , ρ2 , ε, α, and i (i = 1, 2, 3) such that conditions (3.98)–(3.99) or (3.115)–(3.116) are satisfied. Step 5: Compute the variables M i (i = 1, 2, 3) by using (3.89), (3.93), and (3.109), respectively. Step 6: Construct the designed control laws (3.94), (3.111), and (3.120) with the computed gain matrices and variables.

3.4.3 Simulation example We present a numerical application of the proposed control scheme for the attitude tracking control of a flexible spacecraft. The spacecraft parameters are chosen from (Di Gennaro, 2003) with the first four elastic modes considered at 1/2 1/2 1/2 1/2 1 = 0.7681, 2 = 1.1038, 3 = 1.8733, and 4 = 2.5496 rad/s and

Robust fault-tolerant attitude control Chapter | 3

75

damping ratios ξ1 = 0.0056, ξ2 = 0.0086, ξ3 = 0.013, and ξ4 = 0.025. There are six actuators to be distributed symmetrically on three axes of the spacecraft body frame, and every two actuators are mounted as a pair. For instance, actuators 1 and 2 are arranged to produce a torque about the X-axis, actuators 3 and 4 are arranged to generate a torque about the Y -axis, and actuators 5 and 6 are mounted to produce a torque about the Z-axis. In this case the propulsion force is perpendicular to the corresponding axis such that distribution matrix can be simply determined by the distance ri (i = 1, 2, . . . , 6) between the center of mass of spacecraft and the position of the actuator. In the following simulation, r1 = r2 = 0.7 r3 = r4 = 0.6, and r5 = r6 = 0.5, so that the distribution matrix D is given by ⎡

⎤ 0.7 −0.7 0 0 0 0 ⎢ ⎥ D=⎣ 0 0 0.6 −0.6 0 0 ⎦, 0 0 0 0 0.5 −0.5 ' √ ( q d = 0, 1 cos(0.2t), 1 sin(0.2t), 3 , 2 2 2 & % √ √ 1 3 3 cos(0.2t), − sin(0.2t), sin(0.2t) cos(0.2t) , ωd = − 40 40 20 d(t) = [0.3 cos(0.01t) + 0.1, 0.15 sin(0.02t) + 0.3, cos(0.025t) 0.3 sin(0.01t) + 0.1] .

(3.124)

(3.125)

(3.126)

(3.127)

To validate the controller performance with and without various combinations of actuator faults, we consider attitude tracking in the simulation. Suppose the desired attitude quaternion and angular velocity are selected as shown in (3.125) and (3.126) with the initial conditions set q0 (0) = 0.173648, q1 (0) = −0.263201, q2 (0) = 0.789603, q3 (0) = −0.526402, and, ωd (0) = [0, 0, 0] . In addition, the initial modal variables and their time derivatives χ i (0) and χ˙ i (0) (i = 1, 2, 3, 4) are assumed to be χ i (0) = χ˙ i (0) = 0, that is, the flexible appendages are initially undeformed. The simulated periodic disturbance torque is (3.127), and the time-varying part of the moment inertia matrix is assumed to be given by J = [1 + e−0.1t + 2ϑ(t − 10) − 4ϑ(t − 20)]diag[3, 2, 1], where ϑ(·) is defined as ϑ(t ≥ 0) = 1 and ϑ(t < 0) = 0. We also consider severe actuator fault scenarios, where some actuators lose partial power with randomly varying heath levels, and also some actuators totally fail. The elements of actuator health matrix are given by eii (t) = 0.6 + 0.25 rand (·) + 0.15 sin(0.5t + iπ/3),

(3.128)

76 Fault-Tolerant Attitude Control of Spacecraft

where rand (·) is a random number generator between −1 and 1. For illustration, we consider the following fault scenario: the first actuator totally fails after 6 s, the second actuator loses 60% of its control power after 10 s, the third actuator totally fails after 10 s, the fourth actuator loses 60% of its control power after 12 s, the fifth actuator totally fails after 15 s, and the sixth actuator lpses 40% of its control power after 16 s. In the context of the simulation, the gas jets (actuators) produce on–off control actions, whereas the control signals commanded by the SMC (3.94), (3.111), or (3.120) are continuous (the discontinuous switching only occurred on the sliding surface). Thus the control signals need to be implemented in conjunction with the on–off actuators. For discrete-type actuators, continuous signals are converted into equivalent discrete signals by PWPF modulation (Sidi, 1997). The PWPF modulator produces a pulse command sequence to the actuator by adjusting the pulse width and pulse frequency. In its linear range the average torque produced equals the demanded torque input. In this section, we do not give details of the characteristics and implementation of PWPF modulation (see (Sidi, 1997)). Furthermore, the simulations have been rendered more realistic by considering actuator limits, and we assumed that the maximum value of the control force for each actuator (gas jet) is 10 N, that is, umax = 10 N.

3.4.3.1 Simulation results in the absence of actuator constraint To show the effect of the proposed adaptive sliding-mode fault-tolerant controller (ASMFTC) in (3.94), simulations were performed with given initial conditions and the fault scenario in (3.128). The time histories of error quaternion, error angular velocity, and modal displacements of spacecraft are shown in Fig. 3.11(a–c) (solid line). In comparison with the nominal response with functioning actuators in Fig. 3.11(a–c) (dashed line), the tracking performance degrades to some degree once the actuator failure is introduced, and the settling time increases. However, the system ultimately regulates the tracking error to near zero within 30 s. In addition, the elastic vibrations are passively suppressed, and the oscillations settle within 30 s. The last plot in Fig. 3.11(c) shows the vibration energy response, which is described by E = χ˙  χ˙ + χ  Kχ , and this energy shows almost zero oscillations after 30 s. This illustrates that the designed controller is capable of reducing the system vibration while maintaining the tracking capability of the spacecraft. The attitude tracking system is also controlled by using the proposed modified adaptive sliding-mode fault-tolerant control (MASMFTC) law in (3.111), where the other controller parameters are fixed for a fair composition. The same simulation case is repeated, and the results are shown in Fig. 3.11(a–c) (dotted line). The simulation of both the nominal system and the case with failed gas jets also shows the ability of the controller to follow the desired reference signals. Moreover, the oscillations are further suppressed because the elastic vibrations are considered in the design during tracking. These results support the

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FIGURE 3.11 Time histories for the attitude tracking control using the proposed methods. Case 1: proposed ASMFTC with faults (solid line); Case 2: proposed ASMFTC without faults (dashed line); Case 3: proposed MASMFTC with faults (dotted line); Case 4: proposed MASMFTC without faults (dot-dash line).

theoretical result that the performance of the controller can be achieved with the parameter updating law even in the presence of an unknown inertia matrix. The same simulations of the attitude tracking manoeuvre were repeated with traditional PD controller, and the results are shown in Fig. 3.12 (solid line). The tracking performance of the PD controller is significantly degraded after

78 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 3.12 Time histories for the attitude tracking control using PD control. Case 1: fault case (solid line); Case 2: fault-free case (dotted line).

the actuator faults are introduced; severe oscillations also occur after the actuator failures, as demonstrated in the modal displacement and vibration energy responses shown in Fig. 3.12b (solid line). Furthermore, the closedloop system becomes unstable. Although some improvement may be possible with different design control parameter sets, there is little improvement in the attitude and velocity responses. For further comparison, the simulation was repeated using the conventional adaptive sliding-mode controller (CASMC) designed in (Ahmed et al., 1998) for the system. The results of the simulation are shown in Fig. 3.13 (solid line). The CASMC shows some tracking ability because of its robustness, although the tracking performance is degraded after the actuators fail. Figure 4 shows that the attitude responses can be improved significantly compared to the PD case but result in severe vibration compared with the proposed methods. Figs. 3.11–3.13 show that the two proposed designs perform better than the existing controller designs, even when existing designs adapt to the system parameters under external disturbances.

3.4.3.2 Simulation results in the presence of actuator constraint The proposed methods can achieve the desired tracking performance with different actuator faults when no actuator limits are explicitly considered. However, the controller parameters must be selected carefully, and in practice, there exist control output constraints. Here the limit of the actuator force is assumed to be 10 N. To overcome these constraints, we employed the modified controller design (3.120) for the system considered. To demonstrate the effectiveness of this controller, the simulations were repeated with the previous two fault scenarios

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79

FIGURE 3.13 Time histories for the attitude tracking control using CASMC. Case 1: fault case (solid line); Case 2: fault-free case (dotted line).

FIGURE 3.14 Time histories for the attitude tracking control using the proposed methods with saturation limits.

and the same initial conditions. The results are shown in Fig. 3.14 and demonstrate that the proposed controller given in (3.120) works effectively within the operational control limit, even when the actuators fail. Summarizing all the cases (normal and fault cases), note that the proposed controllers can significantly improve the tracking performance compared to the PD and CASMC methods, in

80 Fault-Tolerant Attitude Control of Spacecraft

both theory and simulations. Also, in the fault case the proposed controllers have better performance than the conventional controllers. In addition, extensive simulations were performed using different control parameters, disturbance inputs, and even combinations of actuator faults. These results show that closed-loop system attitude control and vibration stabilization are accomplished in spite of these undesired effects in the system. Moreover, the flexibility in the choice of the control parameters can be utilized to obtain desirable performance while meeting the constraints on the control magnitude and elastic deflection. These control approaches provide a theoretical basis for practical applications of the methods of advanced control theory to flexible spacecraft attitude control systems.

3.5 Summary In this chapter, we presented the basic method of spacecraft FTC, namely the robust FTC method, which does not require any fault detection, isolation, and identification processes. In spacecraft dynamics, spacecraft failures were expressed as additive or multiplicative time-varying or time-invariant uncertainties. For time-invariant faults, the asymptotic stability of the closed-loop system was obtained. For time-varying faults, the uniformly ultimately boundedness was achieved, where the level of disturbance attenuation from disturbance to system output was measured by L2 gain and H∞ performance. In the next chapters, on the basis of the robust FTC, we will consider more practical issues in spacecraft engineering, such as control input saturation, no angular velocity measurement, etc.

Chapter 4

Fault-tolerant attitude control with actuator saturation 4.1 Introduction Magnitude constraint/saturation on the actuator output is a major and unavoidable problem in practical spacecraft control system design due to the physical characteristics of actuators. When an actuator has reached its input limit, any effort to further increase the actuator output would result in no variation in the output. This may lead to system instability. Therefore actuator saturation is another practical issue that should be considered in the attitude controller design. Taking the actuator saturation into account, Hu (2008a), Boškovic et al. (2001), Boskovic et al. (2004), Tsiotras and Luo (2000) and the references therein have developed a range of controllers to effectively handle the limited actuator output. However, these controllers have not considered actuator faults and thus cannot be applied directly to the FTC of spacecraft attitude control system with limited actuator input. In light of this, several researchers simultaneously investigated actuator faults and actuator output constraints. Mhaskar et al. (2006), Benosman and Lum (2009) considered these two issues for a particular system model, and Guan and Yang (2008) provided an adaptive fault-tolerant controller for a system with actuator saturation. Nevertheless, note that the above-mentioned control schemes usually require large computation power and are difficult to implement. Consequently, based on the theoretical results of Chapter 3, we investigate more practical fault-tolerant control schemes for spacecraft attitude systems with actuator faults, saturation, parameter uncertainties, and external disturbances in this chapter. The remainder of this chapter is organized as follows. Chapter 4.2 addresses the fault-tolerant attitude control problem for a flexible spacecraft with actuator faults, uncertain inertia parameters, and external disturbances. Furthermore, we taken an input constraint into consideration. By means of dynamic sliding-mode technique, a similar fault-tolerant control problem is investigated for the flexible spacecraft in Chapter 4.3. Subsequently, considering that AFTC technique can react to fault events through FDD mechanism and adjusts the control effort online, which is beneficial to stability and performance optimization of the faulty system. In Chapter 4.4, we develop a theoretical framework for active faulttolerant attitude stabilization control scheme and apply it to flexible spacecrafts with actuation saturation. Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00014-9 Copyright © 2021 Elsevier Inc. All rights reserved.

81

82 Fault-Tolerant Attitude Control of Spacecraft

4.2 Sliding-mode attitude stabilization FTC In this section, we present a novel fault-tolerant attitude control synthesis for a flexible spacecraft subject to actuator faults and uncertain inertia parameters. Based on the sliding-mode control, we first derive a fault-tolerant control law for the attitude stabilization to protect against the partial loss of actuator effectiveness. Then we extend the result to address the problem that the actual output of the actuators is constrained. The associated stability proof is constructive and accomplished by the development of the Lyapunov function candidate. We also present numerical simulation results to highlight the effectiveness of the proposed control synthesis.

4.2.1 Problem formulation The flexible spacecraft attitude dynamics given by (2.15)–(2.16) is obtained in the actuator fault-free condition. In the situation where the actuator experiences a partial loss of effectiveness fault, the attitude dynamics can be rewritten as ˙ + δ χ¨ = F (u) + d J ω˙ + ω× (J ω + δ χ)

(4.1)

with F (u) = E(t)u, where E(t) = diag{e11 , e22 , . . . , eN N } is the effectiveness factor defined in (2.24) satisfying 0 < εi < eii ≤ 1. Thus, each element of F (u) is a nonlinear input due to eii , as shown in Fig. 4.1. Therefore, the control input

FIGURE 4.1 Time response of Fi (ui ).

satisfies the inequality uT F (u) ≥ εuT u,

(4.2)

with ε = min{εi , i = 1, 2, 3}. Considering the terms d, δ, and ωδ χ˙ as the lumped disturbances, (4.1) can be rewritten as J ω˙ = −ω× J ω + F (u) + d − ω× δ χ˙ − δ χ¨ .   

(4.3)

Td

Given any initial attitude and angular velocity, the control objective is to design a fault-tolerant controller for faulty attitude system such that the following

Fault-tolerant attitude control with actuator saturation Chapter | 4

83

goals are realized in the presence of external disturbances, uncertain moment inertia, actuator faults, and even input constraints: 1) The signals of the closed-loop system are bounded; 2) The attitude and angular velocity can globally asymptotically converge to zero, that is, limt→∞ q v = 0 and limt→∞ ω = 0. Assumption 4.1. During the whole attitude maneuvering process, under the effect of onboard payload motion and fuel leakage or jetting, the moment inertia J (·) is time varying and even unknown. However, it remains positive definite and bounded. It is therefore reasonable to assume that there exists a positive constant (unknown) c1 such that J  ≤ c1 < ∞. Also, it is practical to assume that  dJ dt  ≤ c2 < ∞ for unknown constant c2 > 0. Assumption 4.2. The external disturbance d considered in (4.1) incorporates gravitational perturbation, atmospheric drag, solar radiation pressure perturbation force, etc. Therefore, taking all those disturbances into account, it is practical to assume that the external disturbance is bounded by d ≤ c3 + c4 ω2 , where c3 and c4 are positive but unknown constants.

4.2.2 Fault-tolerant sliding-mode controller design The insensitivity and robustness properties of SMC to certain types of disturbances and uncertainties, especially to actuator faults, makes SMC attractive for fault-tolerant control, especially in the field of spacecraft attitude control. Therefore, in this section, SMC is employed in the fault-tolerant attitude control design to compensate for the effect of the partial loss of actuator effectiveness fault on the attitude system. In SMC design, it usually consists of two stages. The first stage is defining a sliding surface, and the second stage is developing a controller that satisfies the sliding condition, which dictates that the states remain on the sliding surface, and then the states converge to the desired equilibrium state. To this end, the details of our proposed control law design can be elaborated as discussed in the following sections.

4.2.2.1 Sliding manifold design The sliding mode manifold is defined as s = ω + kq v ,

(4.4)

where k is a positive definite matrix. Lemma 4.1. Hu (2008b) If an appropriate sliding-mode controller u satisfies the sliding-mode condition s T s˙ < 0, then s globally asymptotically converges to zero in finite time. As a result, the spacecraft attitude q v and angular velocity ω globally asymptotically converge to zero.

84 Fault-Tolerant Attitude Control of Spacecraft

4.2.2.2 Controller design under partial loss of actuator effectiveness fault Once the sliding mode surface is designed, the next stage is designing an appropriate control law such that the condition of the sliding mode is satisfied. Based on the designed sliding mode manifold in (4.4), we get  k  J s˙ = −ω× J ω + F (u) + T d + J q × v + q0 I 3 ω 2 1˙ = F (u) + J s + A(·), 2

(4.5)

 1 k  A(·) = −ω× J ω − J˙ s + T d + J q × v + q0 I 3 ω. 2 2

(4.6)

where

In view of Assumption 4.2, one can get the inequality T d  ≤ c3 + c4 ω2 + For the elastic vibration, there exist positive constants c5 and c6 such that

˙ + δ χ. ¨ ω× δ χ

T d  ≤ c3 + c4 ω2 + c5 ω + c6 .

(4.7)

2 According to the identity constraint q  v q v + q0 = 1, (4.4), and Assumption 4.1, we can easily establish the following inequalities:  ×  −ω J ω ≤ c1 ω2 ,    1  − J˙ s  ≤ c2 ω + kq v  ≤ kc2 + c2 ω,  2  (4.8)   k  ×       J q + q0 I 3 ω ≤ kc1  q × + q0 I 3  ω ≤ kc1 ω. v v 2 

From these inequalities it follows that there exist some unknown positive constants H i (i = 1, 2, 3) such that A(·) ≤ H1 + H2 ω + H3 ω2 = H ,

(4.9)

where H = [H1 , H2 , H3 ] and  = [1, ω, ω2 ] . Theorem 4.1. Consider the faulty attitude control system given by (2.15) and (4.1) with partial loss of actuator effectiveness. If the control law is designed as ˆ u = −λˆε H

s s

(4.10)

and updated by ˙ˆ = γ s , H

(4.11)

ˆ s, ε˙ˆ = λˆε 3 H

(4.12)

Fault-tolerant attitude control with actuator saturation Chapter | 4

85

where λ and γ are positive constants, then the system trajectories asymptotically converge to the sliding manifold of (4.4), and thus the control objectives stated in Section 4.2.1 can be achieved. Proof. Choose a new Lyapunov function candidate 1 1 ˜˜ 1 V2 = s  J s + H H + ε˜ 2 , 2 2γ 2 ˜ and ε˜ are defined as H ˜ =H −H ˆ and ε˜ = where H by (4.5) we can calculate the time derivative of V2 :

1 εˆ

(4.13)

− ε, respectively. Then

1 1 ˜ ˙ˆ  ε˜ εˆ V˙2 = s  J˙ s − H H − 2 + s  J s˙ 2 γ εˆ 1 ˙ˆ  − ε˜ εˆ . ˜H = s  F (u) + s  A(·) − H γ εˆ 2

(4.14)

In view of (4.9), this equation can be further simplified as 1 ˜ ˆ  ε˜ ε˙ˆ V˙2 ≤ sA(·) + s  F (u) − H H − 2 γ εˆ 1 ˜ ˆ˙  ε˜ εˆ˙ ≤ sH  + s  F (u) − H H − 2 γ εˆ ˜ +H ˆ ) + s  F (u) − = s(H

(4.15)

1 ˜ ˙ˆ  ε˜ ε˙ˆ HH − 2 . γ εˆ

By (4.2) and (4.10) we have   ˆ  s F (u) ≥ ελ2 εˆ 2 s  H ˆ H ˆ s . s  F (u) = −λˆε H s s s

(4.16)

This inequality can be rewritten as ˆ s. s  F (u) ≤ −ελˆεH

(4.17)

Substituting (4.11)–(4.12) and inequality (4.17) into (4.15) yields ˜ +H ˆ )s − ελˆεH ˆ s − H ˜ s − ε˜ λˆεH ˆ s V˙2 ≤ (H ˆ s = (1 − λ)H ˆ s. = [1 − (ελˆε + ε˜ λˆε )λ]H

(4.18)

Taking λ > 1, wet can easily confirm that V˙2 < 0.

(4.19)

According to the Lyapunov stability theorem Krstic et al. (1995), (4.19) guarantees that s → 0. Then by Lemma 4.1 we get that the attitude orientation

86 Fault-Tolerant Attitude Control of Spacecraft

limt→∞ q v = 0 and the angular velocity limt→∞ ω = 0, and thus we obtain the system asymptotic stability. This completes the proof. Remark 4.1. The designed controller (4.10) does not require the knowledge of the lower bound of the effectiveness factor of the actuators. Moreover, with our proposed control law (4.10), the globally asymptotic stability of the closed-loop system can be guaranteed.

4.2.2.3 Fault-tolerant sliding-mode controller design with input constraint From the preceding analysis, (4.10) enables the spacecraft to track a desired target in the presence of parameter uncertainties, disturbances, and even unknown actuator faults. However, from the standpoint of practical applications, one of the major issues in this attitude control system design is that the signal u(t) in (4.10) generated by the control law might not be implemented due to physical constraints. A common example of such a constraint is actuator saturation, which imposes limitations on the magnitude of the achievable control input. Let umax > 0 denote the maximum value of the actuator control torque, that is, |ui | < umax (i = 1, 2, 3). To this end, we make the following assumption. Assumption 4.3. Under a severe actuator fault of partial loss of effectiveness, the maximum output signal of the actuator can strictly dominate the unknown disturbance, that is, εumax > H .

(4.20)

Remark 4.2. For the considered space missions (the attitude control system design for the steady operation case), Assumption 4.1 is feasible. Otherwise, the actual maximum control torque εumax will be smaller than the unknown disturbance under a severe fault of the actuator. Hence the system instability or uncontrollability may be induced, and in such a situation, only actuator redundancy can be applied to stabilize the attitude system. Here we do not discuss it further. Based upon Assumption 4.1, we choose the actual control input as ⎧ s ⎨ −u ˆ  ≥ umax , if λˆεH max s u= ⎩ −λˆε H ˆ  ≤ u , ˆ s if λˆεH s

(4.21)

max

where the parameter updating laws for εˆ and Hˆ are given in (4.11) and (4.12), respectively. Then we can summarize the following theorem. Theorem 4.2. Consider the faulty attitude control system given by (4.1) with partial loss of actuator effectiveness satisfying Assumption 4.1. Under the actuator input constraint, with the application of the control law (4.21) and the

Fault-tolerant attitude control with actuator saturation Chapter | 4

87

updating laws (4.11)–(4.12), the globally asymptotic stability of the closed-loop attitude system can be guaranteed, that is, the attitude limt→∞ q v = 0 and also the angular velocity limt→∞ ω = 0. Proof. To show the stability, we address two cases. ˆ  ≥ umax , we choose the new Lyapunov function cana) In the case λˆε H didate 1 1 ˜ ˜ V3 = s  J s + HH . 2 2γ

(4.22)

With the updating law (4.11), the time derivative of the Lyapunov function of (4.22) is

1˙ 1 ˙ 1 ˜ ˙ˆ   ˙ F (u) + J s + A(·) V3 = s J s − H H + s 2 γ 2 s  ρ(t)s ˜ s + s  A(·) − H s ˜ s ≤ −εumax s + H s − H   ˆ  s < 0. = − εumax − H = −umax

(4.23)

s Moreover, the actuator constraint is ensured, that is, u =  − umax s ≤ umax . ˆ  ≤ umax , we consider the Lyapunov function candidate b) In the case λˆε H as in (4.13). Then we have

˙ˆ  − ε˜ ε˙ˆ ˜ +H ˆ ) + s  F (u) − 1 H ˜H V˙3 ≤s(H γ εˆ 2 ˜ +H ˆ )s − ελˆε H ˆ s − H ˜ s − ε˜ λˆε H ˆ s ≤(H

(4.24)

ˆ s =(1 − (ελˆε + ε˜ λˆε )λ)H ˆ s. =(1 − λ)H If λ > 1, then we can guarantee V˙3 < 0.

(4.25)

˙  s  ≤  − λˆε H ˆ  ≤ umax . We can also obtain the inequality u =  − λˆεH s Summarizing the analysis in the above two cases, we conclude that the designed controller (4.21) can guarantee the negative time derivative of the chosen Lyapunov function and simultaneously satisfy the magnitude constraint. Then, according to Lemma 4.1 and the same analysis as that used for Theorem 4.1, we can obtain Theorem 4.2. The proof is completed.

88 Fault-Tolerant Attitude Control of Spacecraft

4.2.3 Simulation results To verify the effectiveness and performance of our proposed control scheme, we numerically simulated the detailed response using the flexible spacecraft system with the developed control law (4.21). A complete set of physical parameters used in the numerical simulations is given by Di Gennaro (2003) ⎡ ⎤ 350 3 4 ⎢ ⎥ J = ⎣ 3 270 10 ⎦ kg · m2 and 4 10 190 ⎡ ⎤ 6.45637 1.27814 2.15629 ⎢ ⎥ ⎢ −1.25819 0.91756 −1.67264 ⎥ 1/2 δ=⎢ ⎥ kg m/s2 . ⎣ 1.11687 2.48901 −0.83674 ⎦ 1.23637 −2.6581 −1.1253 Here the first four elastic modes have been taken into account with natural frequencies E1 = 1.5362, E2 = 2.2076, E3 = 3.7466 rad/s and dampings ξ1 = 0.0056, ξ2 = 0.0086, ξ3 = 0.013, ξ4 = 0.013. Moreover, the external disturbance is chosen as (Cai et al., 2008) d = (|ω|2 + 0.05)[sin0.8t, cos0.5t, cos0.3t]T . For comparison, we carried out our proposed fault-tolerant controller of (4.21), a mature proportional integral-derivative (PID) control, and the indirect adaptive fault-tolerant controller of Cai et al. (2008) in the following simulations, dividing the performance evaluation of these three strategies into two cases: (1) all the thrusters are healthy but with input constraint, and (2) the thrusters experience loss of effectiveness, and the constraint input is also simultaneously considered. Further, the PID control without reconfigurable process is designed as  (4.26) u = ω× J ω − Kd ω − Kp q v − KI q v dt, where Kp , KI , and Kd are the designed parameters. To reduce the effect of constraint thruster outputs, the antiwindup approach Bang et al. (2003) is also introduced to the PID control (4.26), which can be redesigned as   (4.27) u = ω× J ω − Kd ω − Kp q − KI q v dt − Ka (usat − u) dt and

 usat =

umax sgn (usat ) usat

if |usat | > umax , if |usat | ≤ umax .

(4.28)

To implement the controllers while the spacecraft attitude is maneuvering, the parameters in (4.21) are chosen as k = 0.5, λ = 5, γ = 5, and the various design parameters in the antiwindup PID (AWPID) of (4.27) are chosen as KI =

Fault-tolerant attitude control with actuator saturation Chapter | 4

89

15, Kd = 25, Ka = 3.5, and Kp = 45. Further, in the context of simulation, the initial attitude orientation is set to be q v (0) = (−0.26, 0.78, −0.53) , the angular velocity is assumed to be ω(0) = (0, 0, 0) rad/s, and the initial modal displacement χi (0) = 0 with time derivative χ˙ i (0) = 0. To realistically simulate the partial loss of actuator effectiveness fault, the effectiveness factors are assumed to be combined with time-varying, constant, and random signals given by  1 if t ≤ 10, eii (t) = 0.55 + 0.25 rand(·) + 0.2 sin(0.25(i + 1)t + iπ/3) if t > 10, (4.29) where rand(·) ∈ [0, 1] is a random number generator. Remark 4.3. Note that the thruster is assumed to generate the continuous control force type. In this work, we do not go into the details of the characteristics of this kind of thruster and how it works. Furthermore, simulations were rendered more realistic by considering thruster limit, and we assumed that the maximum value of the control force of the thruster (gas jet) is 5 N, that is, umax = 5 N.

4.2.3.1 Simulation results of Case #1 In this case, all the thrusters are in healthy operation but with input constraint. Fig. 4.2 shows the time response of the attitude orientation, angular velocity, the control input signal, and the time response of vibration displacements. For the designed fault-tolerant sliding-mode controller (FTSMC) given in (4.21), we can clearly see that the desired performance can be achieved, and the attitude can regulate to near zero within 20 s even if the control input constraint is considered, as shown in Fig. 4.2(a) (solid line). Also, the elastic vibrations can be passively suppressed, and the oscillations settled within 20 s, as shown in Fig. 4.2(d) (solid line). These results completely support the theoretical result that the performance of the controller can also be achieved with the parameter updating law, even if no knowledge of the inertia matrix is given in advance. When the indirect adaptive fault-tolerant controller (IAFTC) of Cai et al. (2008) is applied to the nominal attitude system in this case, the response with this controller is shown in Fig. 4.2 (dashed line), and we clearly see that the IAFTC achieves the attitude stabilization control despite an uncertain inertia matrix and external disturbances on the spacecraft dynamics. However, it requires more than 60 s to stabilize the attitude, which is longer than the time for the FTSMC of (4.21). Moreover, we can observe severe elastic vibrations as shown in Fig. 4.2(d) (dashed line). For the AWPID control, although the desired position can be achieved and the control input constraint can be compensated by the antiwindup technique, there exist severe oscillations, which further excite the elastic vibrations, and this can converge to near zero within 100 s, as shown in Fig. 4.2 (dotted line). Despite the fact that there still exists some room for improvement with different design control parameter sets, there is no much improvement in the attitude and velocity responses.

90 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 4.2 Response for the fault-free case under FTSMC (solid line), IAFTC (dashed line), and AWPID (dotted line).

Fault-tolerant attitude control with actuator saturation Chapter | 4

91

4.2.3.2 Simulation results of Case #2 In this case, we also considered the partial loss of thruster effectiveness fault. Fig. 4.3 shows the time response of attitude orientation, angular velocity, the control input signal, and the time response of vibration displacement under the effect of the FTSMC, IAFTC, and AWPID, respectively. As we can see in Fig. 4.3 (solid line), the proposed control law manages to tolerate the loss of actuator effectiveness fault and achieve the attitude stability in 40 s when the actuator faults are introduced. Moreover, no significant vibration displacements are excited, and they are suppressed within 40 s. However, the application of the IAFTC leads to the time responses shown in Fig. 4.3 (dashed line). It is clear that although the server fault (4.29) can be compensated with the control objectives achieved, it needs almost 150 s to finish the attitude stabilization maneuver, which is almost four times longer than the time for our proposed control of (4.21). Moreover, the IAFTC will excite large vibration displacements during the attitude maneuver, as we can see in Fig. 4.3(d) (dashed line). We now report the results due to the AWPID. It is clear that the periodic fault effect propagated to the attitude system and that the AWPID, shown in Fig. 4.3(a) (dotted line), is unable to compensate for this fault. Indeed, as we can see in Fig. 4.3 (dotted line), the desired control objective is not achieved when the faults occur, which further excites the elastic vibration, which deteriorates the system performance, and the system becomes unstable. Summarizing all the cases (normal and fault cases), note that the proposed control design scheme can significantly improve the normal performance over the IAFTC and PID control in both theory and simulations. Also, in the fault case the proposed approach has better results than those of conventional methods, even in the presence of external disturbances, uncertain inertial parameters, and input constraints. Further, extensive simulations were also done using different control parameters and even disturbance inputs. The overall results on settling time of stabilization and maximum vibration displacement are summarized in Table 4.1. TABLE 4.1 Performance comparison of maximum value of vibration and settling time of stabilization under the three control schemes. Thrusters status

Control schemes

Settling time of stabilizations (s)

Settling time of vibration (s)

Fault-free

FTSM

20

0.001

Fault (4.29) occurs

IAFC

60

0.02

AWPID

100

0.01

FTSM

40

0.001

IAFC

150

0.015

AWPID

≥ 300

0.003

92 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 4.3 Response with partial loss of thruster effectiveness fault under FTSMC (solid line), IAFTC (dashed line), and AWPID (dotted line).

Fault-tolerant attitude control with actuator saturation Chapter | 4

93

4.3 Dynamic sliding-mode attitude stabilization FTC Based on the previous work, in this section, we investigate the robust faulttolerant attitude control problem of the flexible spacecraft with a combination of unknown actuator failure, input saturation, and external disturbances. We develop a fault-tolerant controller based on variable structure control, which is robust to the partial loss of actuator effectiveness. Then we extend the results to the case in which some of the actuators fail completely. Based on the Lyapunov theory, we provide the closed-loop system stability. Finally, we use simulation studies to evaluate the closed-loop performance of the proposed control solution and illustrate its robustness.

4.3.1 Problem formulation In this section, the moment inertia J is assumed to be a constant. The maximum output of the actuators still satisfies |ui | ≤ umax for all t > 0, i = 1, 2, 3. Remark 4.4. The composite external disturbances, T d in (4.3) include elastic vibration, gravitational perturbations, and atmospheric drag, and are also bounded (Yang and Sun, 2002). Thus it is reasonable to assume that there exists a constant T¯d such that T d  ≤ T¯d .

(4.30)

For a spacecraft control attitude dynamics system (2.9) and (4.3), in the presence of actuator saturation and the partial and/or complete loss of control effectiveness, we can state the control objectives as follows: 1) All of the internal signals in the closed-loop system are bounded. 2) The attitude q v and the angular velocity ω converge asymptotically to zero on an arbitrary specified bounded set. t 3) The performance index Ip = lim 1t 0 S2 dt is bounded, where S is the t→∞ auxiliary variable defined as (Boskovic et al., 2004) S = ω + k 2 (t)q v ,

(4.31)

and k (t) is a time-varying function that will be given later.

4.3.2 Main result 4.3.2.1 Controller design with partial loss of actuator effectiveness fault In this case, assume that the spacecraft is equipped with only three actuators, that is, one in each axis of the body frame. There is no actuator redundancy available, and each actuator may partially lose its effectiveness. Without going into the details of the possible nature of actuator faults, the nonlinear spacecraft attitude dynamics model with actuator faults is described in (4.3).

94 Fault-Tolerant Attitude Control of Spacecraft

To achieve our control objectives, we design the control law ui as ui = −

umax si |si | + k 2 δ

i = 1, 2, 3,

(4.32)

where si is the ith element of S defined in (4.31), and δ is a positive control constant. Moreover, we define k = k(t) for convenience. To establish stability of the attitude control system, consider the candidate Lyapunov function   1 1 2 2 V = ω J ω + k 2 q  k , + (4.33) v q v + (1 − q0 ) 2 2γ where γ is a positive constant to be determined. We can compute and simplify the derivative of (4.33) using (2.17), (4.31), 2 and (4.3) and the property of the unit quaternion q  v q v + q0 = 1. After some algebraic manipulation, we get 1 ˙ 2 ˙  V˙ = ω J ω˙ + k 2 (2q  v q v − 2(1 − q0 )q˙0 ) + 2k k(q v q v + (1 − q0 ) ) + k k γ    1  ×  ˙ = ω J ω˙ + k 2 q  v q v + q0 I3 ω + (1 − q0 ) q v + k k γ 1 = ω J ω˙ + k 2 ω q v + k k˙ + 4k k˙ (1 − q0 ) γ

1 = ω (Eu (t) + T d (t)) + k 2 ω q v + k k˙ 4 (1 − q0 ) + γ

1 ˙ − k4q  = ω (Eu (t) + T d (t)) + k 2 q  S + k k 4 − q + (1 ) 0 v v qv. γ (4.34) Suppose that εmin and εmax are two positive constants satisfying 0 < εmin ≤ min {Ei } and 1 ≥ εmax ≥ max {Ei }. Then

i=1,2,3

i=1,2,3

−Ei

umax ωi2 ωi2 ≤ − ε u min max |si | + k 2 δ |si | + k 2 δ ωi2 ≤ − εmin umax |ωi | + k 2 (δ + 1)

k 2 (δ + 1) = − εmin umax |ωi | 1 − . |ωi | + k 2 (δ + 1)

(4.35)

The updating law for k is designed to satisfy  k k˙ = − 1+4γγ(1−q 0)

3  i=1

εmin umax |ωi |(δ+1) |ωi |+k 2 (δ+1)

+

3  i=1

 εmin umax |qi ||ωi | |si |+k 2 δ

+ q vs

.

(4.36)

Fault-tolerant attitude control with actuator saturation Chapter | 4

95

Then from (4.32), (4.35), and (4.36), we can rewrite (4.34) as V˙ = −

3 

Ei

umax ωi si 1  2  ˙ + ω T q S + k k 4 − q + k + (1 (t) ) d 0 v |si | + k 2 δ γ

i=1 − k4q  v qv 3 

  umax ωi ωi + k 2 qi + ω T d (t) + k 2 q  vS |si | + k 2 δ i=1

1 − k4q  + k k˙ 4 (1 − q0 ) + v qv γ

=−

≤−

Ei

3  i=1

−

3  i=1

≤−

3  i=1

3    εmin umax |ωi | k 2 (δ + 1) |ωi | εmin umax − T¯d − k 4 q  |ωi | v qv + |ωi | + k 2 (δ + 1)



k2ω

i=1

umax 1 i qi ˙ + k2q  Ei v S + k k 4 (1 − q0 ) + |si | + k 2 δ γ

3    εmin umax |ωi | k 2 (δ + 1) |ωi | εmin umax − T¯d − k 4 q  |ωi | v qv + |ωi | + k 2 (δ + 1) i=1



umax 1 i qi 2  ˙ − + k q v S + k k 4 (1 − q0 ) + εmax |si | + k 2 δ γ i=1   4  = − |ω| εmin umax − T¯d − k q v q v . 3 

k2ω

(4.37)

To ensure V˙ ≤ 0 in (4.37), the following assumption is introduced. Assumption 4.4. The bound value T¯d of the disturbance T d satisfies the inequality T¯d < εmin umax .

(4.38)

Under Assumption 4.4, we can obtain V˙ ≤ 0. This implies that v and k are 2 bounded. With q  v q v + q0 = 1, q v and q0 are bounded. Hence V is bounded. Let εmin umax − T¯d = ci . Then, based on (4.37), we obtain  ∞    ∞ ¯ |ω|dt + V (0) − V (∞) ≥ εmin umax − Td k4q  v q v dt 0 0  ∞  ∞ |ω|dt + ≥ ci k4q  v q v dt 0 ∞

0 ∞ |ω|dt + ≥ c¯ k4q  q dt , (4.39) v v 0

0

96 Fault-Tolerant Attitude Control of Spacecraft

where c¯ = min {ci , 1}, and we know that c¯ > 0. Since V is bounded, ω ∈ L1 i=1,2,3

and k 2 q v ∈ L2 . We know that S is bounded because all its terms are bounded as shown above. Thus, since ω, S, and u are bounded, k˙ is bounded. Moreover, ω˙ is also bounded from the attitude dynamics in (2.21), and q˙ v is bounded from (2.17) because ω, q v , and q0 are bounded. By the Barbalat lemma (Popov, 1973) lim ω = lim k 2 q v = 0.

t→∞

t→∞

(4.40)

However, the fact that k 2 q v converges to zero does not ensure that q v will converge to zero. If k is bounded, then we can conclude that lim q v = 0. Next, we t→∞ will discuss whether k is bounded. Based on (4.36), it follows that  3  εmin umax |ωi | (δ + 1) γ k − k˙ ≥ |ωi | + k 2 (δ + 1) 1 + 4γ (1 − q0 ) i=1  3  εmax umax |qi | − q v S − |si | + k 2 δ i=1  3   εmin umax |ωi | (δ + 1) εmax umax |ωi | ≥ −γ k + + S . |ωi | + k 2 (δ + 1) |si | + k 2 δ

(4.41)

i=1

Consider the terms within the summation sign εmin umax |ωi | (δ + 1) εmax umax |ωi | + |ωi | + k 2 (δ + 1) |si | + k 2 δ εmin umax |ωi | (δ + 1) εmax umax |ωi | ≤ + |si | + k 2 δ |si | + k 2 δ (εmin (δ + 1) + εmax ) = umax |ωi | |si | + k 2 δ   si − k 2 q v  i = umax (εmin (δ + 1) + εmax ) |si | + k 2 δ |si | + k 2 ≤ umax (εmin (δ + 1) + εmax ) |s | + k 2 δ i

1 . ≤ umax (εmin (δ + 1) + εmax ) 1 + δ

(4.42)

From (4.40) we first note that lim S (t) = 0 due to its definition in (4.31). Theret→∞ fore there exist constants ω¯ and S¯ such that |ω (t)| ≤ ω¯ and |S (t)| ≤ S¯ for all t.

Fault-tolerant attitude control with actuator saturation Chapter | 4

Hence

where



1 ¯ k ≥ −γ k 3umax 1 + (εmin (δ + 1) + εmax ) + S , δ

97





1 ε = 3umax 1 + (εmin (δ + 1) + εmax ) + S¯ − γ kε. δ

(4.43)

(4.44)

Assume that k (0) = k0 > 0. Then integrating (4.43), we obtain k (t) ≥ k0 exp (−γ εt) .

(4.45)

Thus (4.45) shows that k (t) ≥ 0 for all t, and k (t) = 0 is possible only at t = ∞. On the other hand, since k (t) is bounded, for a given positive constant γ , there exists a positive function τ (γ ) satisfying k (t) ≤ τ (γ ). Then from (4.36) we have  3  εmin umax |ωi | (δ + 1) γ k k˙ ≥ − |ωi | + k 2 (δ + 1) 1 + 4γ (1 − q0 ) i=1  3  εmax umax |qi | ωi | − q v S − |si | + k 2 δ i=1  3  εmin umax |ωi | (δ + 1) εmax umax |ωi | ≥−γk + |ωi | + k 2 (δ + 1) |si | + k 2 δ i=1     + ω + k 2 q v   3

γ  εmin umax |ωi | (δ + 1) εmax umax |ωi | ≥− + k k 2 (δ + 1) k2δ i=1  +k 2 ω + k 4 q  v qv   εmax umax γ  εmin umax + + τ 2 (γ ) ω + k 4 q  q ≥− v v k δ γ cmax  ≥− ω + k 4 q  (4.46) v qv , k   where cmax = max εmin umax + (εmax umax /δ) + τ 2 (γ ), 1 . Thus   ˙ ≥ −γ cmax ω + k 4 q  kk (4.47) v qv . Integrating this inequality from 0 to ∞ gives  ∞  2 2 k (∞) ≥ k (0) − 2γ cmax ω(x) + k 4 q  v (x)q v (x) dx 0

98 Fault-Tolerant Attitude Control of Spacecraft

2γ cmax (V (∞) − V (0)) c¯ 2γ cmax ≥ k 2 (0) − V (0). c¯

≥ k 2 (0) +

If the initial value is chosen as k(0) > k 2 (∞) ≥

3γ cmax c¯ V (0),

(4.48) then from (4.48) we obtain

γ cmax V (0) > 0. c¯

(4.49)

Based on (4.45) and (4.49), k > 0 will hold for all time. Thus, with lim k 2 q v = t→∞ 0 in (4.40), we must have lim q v = 0. Since the convergence of ω and q v is t→∞ independent of their initial values, the global asymptotic stability of the closedloop system is demonstrated. Now we have the first result as follows. Theorem 4.3. Consider the faulty attitude control system given by (2.17) and (4.3), with partial loss of actuator effectiveness satisfying Eq. (4.38). With the application of the control law (4.32) and the updating law (4.36), all the signals in the closed-loop system are bounded and continuous, and the global asymptotic stability is guaranteed, that is, the attitude and angular velocity tend to zero: lim q v = 0 and lim ω = 0. t→∞

t→∞

Proof. The proof can be obtained from the above derivation and is omitted. Remark 4.5. In aerospace applications, difficulties in modeling the dynamics of the system may produce an inertia matrix of the whole spacecraft that is uncertain. However, the proposed control law in (4.32), together with the updating law in (4.36), does not require knowledge of the inertia matrix. Thus the designed control law is robust with respect to these parameter uncertainties, and these uncertainties will not cause the system performance to deteriorate. The control law is also robust to time-varying inertias, providing the timescales slow compared to the spacecraft dynamics. Remark 4.6. From the aforementioned analysis we know that k is bounded ¯ If the term k 2 δ is not contained in the control below by a positive constant k. law (4.32), then (4.32) can be rewritten as ui = −umax sign (si ). As a result, the chattering effect may be caused by the discontinuity of the sign function sign (·). This problem is well known in the sliding-mode control scheme (Slotine, 1984) and usually solved by approximating the sign function by a continuous function. Following !  this idea,  in this study, we substitute sign (·) with a continuous function si |si | + k 2 δ , as shown in (4.32). To approximate sign (·) with high precision, a small δ is required, leading to better approximation performance. Remark 4.7. The designed controller in (4.32) satisfies the actuator saturation limits, that is,

Fault-tolerant attitude control with actuator saturation Chapter | 4

       si umax si   ≤ umax   |ui | = − ≤ umax   2 2 |si | + k δ |si | + k δ 

99

for i = 1, 2, 3.

Remark 4.8. The actuation effectiveness matrix E is not used in the proposed control scheme or required for the stability analysis. Thus there is no need to include health monitoring to identify or estimate which actuator is faulty or to include a method for fault isolation. The only requirement is to specify the possible minimum and maximum values of the components of the fault matrix. The actuator fault accommodation and compensation is performed automatically and adaptively by the proposed control algorithm. This feature is necessary to build affordable and effective fault-tolerant spacecraft control schemes. Remark 4.9. Note that to stabilize the system, all the actuators are assumed to be active even if there exist faults in some actuators, that is, no actuator has failed completely for this case. If one or more actuators fail completely, then the system will become underactuated (Benosman and Lum, 2009), and the designed controller in (4.32) will not guarantee the system stability. The underactuated system is not considered further in this book. Assumption 4.4 requires the specification of εmin . In practice, it is often difficult to identify this value precisely, and hence the controller proposed in (4.32) may lack some practicality. To avoid this problem, in the following subsection, we propose an extended control scheme with actuator redundancy. Often in practice the spacecraft attitude control system is designed with actuator redundancy, giving an overactuated control system. This allows the control system to cope with complete actuator failure, as well as partial failure, and hence removes any required knowledge of the form of the actuator faults.

4.3.2.2 Controller design under total loss of actuator effectiveness fault In this section, we consider the case of redundant actuators, where some actuators may fail totally. With actuator redundancy, the attitude dynamics in (4.3) are modified to give J ω˙ + ω× J ω = DEu + T d ,

(4.50)

where D ∈ R3×n denotes the known control torque distribution matrix with fullrow rank, u ∈ Rn is the output of the n > 3 actuators, and E is a diagonal matrix characterizing the health condition of the actuators defined in (2.13). Then the control law is chosen as u=−

umax D  v, D

(4.51)

where v = [v1 , v2 , v3 ] , and vi = 

s , |si | + k 2 δ

i = 1, 2, 3.

(4.52)

100 Fault-Tolerant Attitude Control of Spacecraft

A candidate Lyapunov function is chosen as   1 1 2 2 V = ω J ω + q  k . + v q v + (1 − q0 ) 2 2γ

(4.53)

Then the time derivative of the Lyapunov function in (4.53), using (4.50), is 1 V˙ = ω J ω˙ + ω T q v + k k˙ γ = ω (DEu(t) + T d (t)) + q  vS+

1 ˙ kk − k2q  v qv. γ

(4.54)

By incorporating the controller in (4.51) we can rewrite (4.54) as 1 V˙ = ω (DEu(t) + T d (t)) + q v  S + k k˙ − k 2 q  v qv γ umax ω DED 1 ˙ 2  v + ω T d (t) + q  =− v S + kk − k q v q v D γ ≤

3 umax εmax  |ωi | |si | 1 ˙ 2  + ω T d (t) + q  v S + k k − k q v q v , (4.55) |si | + k 2 δ D γ i=1

where λ¯ max is a known positive constant greater than the maximum eigenvalue of the matrix DEDT . Choose the updating law for k according to the following equation: 3 1 ˙ umax λ¯ max umax λ¯ max  |ωi | |si | − q kk = − S vS− 2 |si | + k δ γ D D i=1

umax λ¯ max 2 k q v . − D

(4.56)

Then 1 ˙ 2  V˙ = ω (DEu(t) + T d (t)) + q  v S + kk − k q v q v γ umax ω DED 1 ˙ 2  v + ω T d (t) + q  =− v S + kk − k q v q v D γ umax λ¯ max 2 umax λ¯ max S − k q v  − k 2 q  ≤ ω T d (t) − v qv D D   umax λ¯ max umax λ¯ max 2   ≤ S − k 2 q v  T¯d − S − k q v  − k 2 q  v qv D D umax λ¯ max 2 umax λmax S − k q v  − k 2 q  ≤ ST¯d + k 2 q v T¯d − v qv D D

Fault-tolerant attitude control with actuator saturation Chapter | 4

=−

3  umax λ¯ max i=1

D

− T¯d



 S + k 2 q v  − k 2 q  v qv.

101

(4.57)

Then, we make the following assumption to ensure V˙ > 0. Assumption 4.5. The bound value T¯d of the disturbance T d satisfies the inequality umax λ¯ max T¯d < . D

(4.58)

Remark 4.10. Loosely speaking, Assumption 4.5 states that the actuators are able to produce sufficient torque, and therefore control authority, to perform a given attitude manoeuvre and reject any disturbance T d . This is a reasonable assumption in practice. The parameter λ¯ max must be greater than the maximum eigenvalue of the matrix DED ; since E is time varying and unknown, λ¯ max is selected to be greater than the maximum eigenvalue of DD . Using Assumption 4.5, we can guarantee V˙ ≤ 0 by (4.57). By an analysis identical to that of Theorem 4.3, we obtain lim S = 0 and lim k 2 q v = 0. Thus t→∞ t→∞ by (4.31) ω asymptotically converges to zero. From (4.53) we have 

3 umax λ¯ max  |ωi | |si | ˙ − q kk = γ − vS |si | + k 2 δ D i=1

umax λ¯ max umax λ¯ max 2 − S − k q v  D D  3 umax λ¯ max  |ωi | |si | ≥γ − |s | + k 2 δ D i=1 i    2umax λ¯ max + D umax λ¯ max S − ω − D D      3 umax λ¯ max  |si | + k 2 |si | ≥ −γ + 2 |si | + |ωi | + S |si | + k 2 δ D i=1  

3 umax λ¯ max  |si | + 3 |si | + |ωi | + S . (4.59) ≥ −γ D δ i=1

Using the same analysis, we can prove that k > 0 for all time. Therefore q v tends to zero since lim k 2 q v = 0. With the above analysis, we obtain the following t→∞ theorem.

102 Fault-Tolerant Attitude Control of Spacecraft

Theorem 4.4. Consider the faulty attitude control system given by (2.17) and (4.50) incorporating actuator failures and constraint actuator outputs. Based on Assumption 4.5, with the application of the control law in (4.51) and the updating law in (4.56), for all possible initial conditions q v (0) and ω (0), all the signals in the closed-loop system are bounded and continuous, and the global asymptotic stability can be guaranteed with actuator redundancy. That is, the attitude and the angular velocity tend to zero: lim q v = 0 and lim ω = 0. t→∞

t→∞

Proof. The proof can be obtained from above derivation and is omitted. Remark 4.11. Note that the controller designed in (4.51) is also robust to parametric uncertainties, and at the same time the actuator saturation limits are ensured, that is,      u D   D  S S  max       ≤ umax    ≤ umax . u = −    D  S + k 2 δ D S + k 2 δ  (4.60) Remark 4.12. In the above stability analysis the actuation effectiveness matrix E is not used in the control scheme, so that actuator health monitoring or fault isolation is not necessary. However, to stabilize the system, the number of active control inputs after failure should be greater than or equal to three, so that the remaining active thrusters are able to produce a sufficient control torque vector for the spacecraft to perform the given mission. Remark 4.13. Note that the developed controller given by (4.51)–(4.52) and the updating law in (4.56) are independent of the actuation effectiveness matrix E. Thus the control objectives can be achieved by our controller for faults occurring at any time and for any fault severity. Therefore the robustness of the proposed fault-tolerant scheme is achieved.

4.3.3 Simulation example We present a numerical application of the proposed control schemes to the attitude control of an orbiting spacecraft equipped with six thrusters. The simulation data including the inertia matrix and the actuator ⎡ distribution matrix ⎤ are ob20 0 0.9 ⎢ ⎥ tained from (Yang et al., 2001) and are given as J = ⎣ 0 17 0 ⎦ kg·m2 , 0.9 0 15 ⎡ ⎤ 0.8 −0.8 0 0 0 0 ⎢ ⎥ D=⎣ 0 0 0.7 −0.7 0 0 ⎦, and the external disturbance 0 0 0 0 0.7 −0.7   2 is assumed to be d = |ω| + 0.05 [sin 0.8t, cos 0.5t, cos 0.3t] Nm.

Fault-tolerant attitude control with actuator saturation Chapter | 4

103

FIGURE 4.4 Thruster fault information (time responses of e1 –e6 ).

Here we consider a severe failure scenario, where some thrusters partially lose their effectiveness, and some lose their power completely. Fig. 4.4 shows the effectiveness of the thrusters given by (Yang et al., 2001) eii = 0.7 + 0.15rand (ti ) + 0.1 sin(0.5t + iπ/3),

i = 1, . . . , 6,

(4.61)

where the definition of rand(ti ) is given in (Yang et al., 2001). Among the six thrusters, the third thruster only supplies 20 of its saturation value after 8 s, and the fourth and fifth thrusters fail completely after 10 and 12 s, respectively. The proposed fault-tolerant controller in (4.51) is compared to the conventional variable structure controller (CVSC) (Cai et al., 2008) in the following simulation. For the controller in (4.51), the control parameters are chosen as γ = 0.002, δ = 0.002, k (0) = 1.5, and λ¯ max = 1.26. Note that each thruster is assumed to generate a continuous control force, and we do not consider the detailed characteristics of this kind of actuator. The simulations have been rendered more realistic by considering the thruster limits, and the maximum control force available from the thrusters (gas jet) are assumed to be 4, 3.5, and 3.5 N for the x, y, and z directions, respectively. To verify the superior performance of the control law in (4.51), two cases are simulated; in the first case, all the actuators are functioning normally, and in the second case the actuator fault scenario given above occurs while the craft is manoeuvring. The initial attitude is set to q0 = 0.9631, q1 = −0.1, q2 = 0.15, and q3 = −0.2, and the angular velocity is assumed to be zero at t = 0, that is, ω (0) = 0.

104 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 4.5 Time responses with healthy actuators.

4.3.3.1 Response with healthy actuators Fig. 4.5 shows the time responses of the quaternion, angular velocities, and six thruster forces for the healthy actuator case. The controller proposed in (4.51) clearly has superior control performance compared with CVSC. Although both control schemes can achieve an acceptable orientation response, as shown in Figs. 4.5(a,b), the proposed scheme only need 10 s to complete the attitude manoeuvring, which is faster than CVSC, which requires almost 25 s. Fig. 4.5(c) shows the control input histories for the two control algorithms and highlights that the amplitude of the control force for both controllers is below the given saturation limits. Fig. 4.5(d) shows that the performance index Ip (t) for the controller in (4.51) is better than that for CVSC. 4.3.3.2 Response with actuator fault Fig. 4.6 shows the time responses for the case with actuator faults, and the actuator failures clearly propagate through to the spacecraft attitude. The CVSC

Fault-tolerant attitude control with actuator saturation Chapter | 4

105

FIGURE 4.6 Time responses with fault actuators.

shown in Fig. 4.6 is unable to compensate for these faults and leads to reduced system stability. However, the system with the proposed controller in (4.51) is stable and shows better performance, shown Fig. 4.6. Fig. 4.6 gives the attitude and angular velocity, and Fig. 4.6(c) emphasizes the changes in control input when the actuator faults occur. The controller in (4.51) is able to compensate for the severe fault. Indeed, when (4.51) is applied, the actuator faults in (4.61) are immediately rejected from the attitude and velocity trajectories. For further comparison, a conventional proportional–integral–derivative (PID) with antiwindup controller was also simulated for the case of perfect actuators and with the actuator faults given in (4.61). The antiwindup control is employed to reduce the effect of actuator saturation, and the results show that the actuator faults cannot be compensated by PID control. Table 4.2 summarizes the control performance in a quantitative manner, where ts (s) denotes the spacecraft attitude stabilization time. A comparison of all cases in Table 4.2 (healthy and faulty actuator cases, and different controllers) demonstrates that the proposed controller design method can significantly improve the perfor-

106 Fault-Tolerant Attitude Control of Spacecraft

TABLE 4.2 Control performance comparisons of the three controllers ( denotes stable, × denotes unstable). Controller

Healthy actuator case

Faulty actuator case

Eq. (4.51)

CVSC

PID

Eq. (4.51)

CVSC

PID

ts (s)

6

25

30

7

∞

∞

stability in roll axis stability in pitch axis

 

 

 

 

 ×

× ×

stability in yaw axis











×

mance over that for PID or CVSC methods in the healthy actuator case. For the faulty actuator case, the proposed method gives better results than those for the conventional controllers. In addition, extensive simulations were also performed using different control parameters, disturbance inputs, inertia parameters, and even combinations of the thruster faults. These results show that the closed-loop system attitude stabilization can be accomplished in spite of these undesired effects in the system. Moreover, the flexibility in the choice of control parameters can be utilized to obtain desirable performance while meeting the constraints on the control force magnitude. These control approaches provide the theoretical basis for the practical application of advanced control theory for the spacecraft.

4.4 Fault estimation-based attitude FTC In contrast to the passive and robust FTC in the previous sections, in this section, we study the active FTC technique for flexible spacecraft. We develop a theoretical framework for active fault-tolerant attitude stabilization control. By developing an observer-based fault detection and diagnosis mechanism we first reconstruct the partial loss of actuator effectiveness faults. Accordingly, we reconfigure a backstepping-based fault-tolerant control law using the reconstructed fault information. We show that the proposed design approach guarantees that all the signals of the closed-loop system are uniformly ultimately bounded. Finally, we extensively evaluate the closed-loop performance of the proposed control strategy through numerical simulations.

4.4.1 Problem formulation Consider the attitude dynamics system given in (2.17) and (4.3), where the composite disturbance satisfies Remark 4.4. Then the control objective can be stated as follows: Determine an FDD mechanism and an FTC law such that the following goals are achieved in the presence of external disturbances, possible partial loss of effectiveness fault E(t), and actuator input saturation: a) All the signals of the resulting closed-loop attitude system are bounded and continuous.

Fault-tolerant attitude control with actuator saturation Chapter | 4

107

b) The designed FDD can achieve precise fault reconstruction in the sense that the reconstruction signal can approximate the fault to any required accuracy. c) The attitude stabilization maneuver is accomplished with the attitude σ and angular velocity ω converging to an arbitrary small set containing the origin in finite time T ∗ , that is, σ  ≤ μ∗1 and ω ≤ μ∗2 for t ≤ T ∗ . t " # ˜ 2 dt is bounded d) The performance index Ip = limt→∞ 1t 0 ω˜ e 2 + p with ω˜ e = ωˆ − ω and p˜ = pˆ − p. Note that ωˆ and pˆ are the estimates of ω and p(t) = [p1 , p2 , p3 ] , respectively. Notably, in this section, let u(t) = u and T d (t) = T d .

4.4.2 Active attitude FTC with loss of actuator effectiveness We will present an active fault-tolerant attitude controller for flexible spacecraft, including the design of the observer-based FDD mechanism and of the fault-tolerant attitude controller. We firstly propose an FDD scheme to achieve precise fault reconstruction. Then we reconfigure a backstepping-based FTC law using the reconstructed faults from the FDD observer to perform the attitude stabilization maneuver.

4.4.2.1 Observer-based FDD design As E(t) is a diagonal matrix, the term E(t)u(t) in (4.3) can be rearranged as E(t)u = U p(t),

(4.62)

where U = diag{u1 , u2 , u3 }. Using (4.62), we can write the attitude dynamics (4.3) with faulty actuators as J ω˙ = −ω× J ω + U p(t) + T d (t).

(4.63)

We propose the following state observer-based FDD to detect and reconstruct the actuator fault vector p(t): ˆ − (ωˆ − ω) − l1 sgn(ωˆ − ω), J ω˙ˆ = −ωˆ × J ωˆ + U p(t)

(4.64)

ˆ − T ) + l3 (ωˆ − ω), ˆ = l2 p(t p(t)

(4.65)

where T is the updating interval, which is taken as the sampling time interval in this section,  ∈ R3×3 is a positive definite matrix, and li ∈ R (i = 1, 2, 3) are positive observer gains. To examine the performance of the proposed FDD scheme, we should develop an expression for the observer error dynamics. By defining ω˜ e = ωˆ − ω and p˜ = pˆ − p from (4.63) and (4.64) we can obtain the dynamics of the error as ˜ −  ω˜ e − l1 sgn(ω˜ e ) − T d . J ω˙˜ e = ω× J ω − ωˆ × J ωˆ + U p(t)

(4.66)

108 Fault-Tolerant Attitude Control of Spacecraft

Remark 4.14. As 0 < eii (t)  1, we always have l2 p(t − T ) − p(t)∞  (1 + l2 ). Remark 4.15. For a sufficiently small sampling time T , the assumption ˆ  κω˜ e  holds for some known constant κ > 0 (Chen and ω× J ω − ωˆ × J ω Saif, 2007). Remark 4.16. The term (ωˆ − ω) in (4.64) is introduced as a feedback term ˜ to guarantee the convergence of the observer errors ω˜ e (t) and p(t) as shown in (4.66). Furthermore, usually, the actuators operate normally initially, that is, p(0) = [1, 1, 1] ; in order to achieve a fast reconstruction of the actuator faults, ˆ the initial value of p(t) in (4.65) is set to p(0), that is, p(0) = [1, 1, 1] . Remark 4.17. Note that (4.64) is a continuous differential equation, whereas (4.65) is a recursive equation. In practice, the observer works in discrete time, and hence (4.64) should be transformed to its discrete-time equivalent. Alternatively, for small T , (4.65) may be considered as a discrete approximation to a ˆ first-order differential equation in p(t). Theorem 4.5. Consider the estimation error (4.66) and suppose that the observer gains are chosen such that

u2max a 1 = 1 − η1 + 1 + γ  0, (4.67) 2μ3 2

umax μ3 − η2 a2 = λmin () − κ − + 1 + γ > 0, (4.68) 2 2μ3 2

u a3 = l1 − T¯d − 2l3 (1 + l2 ) max + 1 + γ  0, (4.69) 2μ3   where η1 = l22 (1 + μ1 + μ2 ) and η2 = l32 1 + μ11 for positive constants μi (i = 1, 2, 3) and γ . Then the estimation errors ω˜ e and p˜ converge to a small set containing the origin, and the performance index I p is also bounded by a small positive constant. Thus the control objectives (b) and (d) are met with the designed FDD mechanism in (4.64) and (4.65). Proof. From (4.65) we have ˜ = l2 p(t ˜ − T ) + l3 ω˜ e + l2 p(t − T ) − p(t). p(t)

(4.70)

Define the new variable  = l2 p(t − T ) − p(t). Then it follows that ˜ e +   + 2l2 l3 p˜  (t − T )ω˜ e ˜ = l22 p˜  (t − T )p(t ˜ − T ) + l32 ω˜  p˜  (t)p(t) eω + 2l2 p˜  (t − T ) + 2l3 ω˜ Te .

(4.71)

Fault-tolerant attitude control with actuator saturation Chapter | 4

Employing the well-known Young inequality 2x  y  μi x  x + x, y ∈ R3 , we have, for arbitrary positive constants μi (i = 1, 2), l32  ω˜ ω˜ e , μ1 e 1  ˜ −T)+  . 2l2 p˜  (t − T )  μ2 l22 p˜  (t − T )p(t μ2

˜ −T)+ 2l2 l3 p˜  (t − T )ω˜ e  μ1 l22 p˜  (t − T )p(t

1  μi y y

109

for

(4.72) (4.73)

Substituting inequalities (4.72) and (4.73) into (4.71) gives

1 ˜e ˜ ˜ − T ) + l32 1 + p˜  (t)py(t) ω˜   l22 (1 + μ1 + μ2 ) p˜  (t − T )p(t e ω μ1

1   + 2l3 ω˜ e ∞ + 1+ μ2    η3 

˜ e + η3   + 2l3 (1 + l2 ) ω˜ e . ˜ − T ) + η2 ω˜  = η1 p˜ (t − T )p(t e ω (4.74) Consider the candidate Lyapunov function  t 1 ˜ ˜ p˜  (s)p(s)ds. J ω + V (t) = ω˜  e 2 e t−T u2

(4.75)

max ˜ e + 2μ ˜  p(t) ˜ ˜  μ23 ω˜  The inequality ω˜  p(t) holds for all μ3 > 0 due e U p(t) e ω 3 to |ui | ≤ umax . After straightforward algebra, followed by the application of (4.67), (4.69), and (4.74), the time derivative of V is   × × ˆ ˆ V˙ (t) = ω˜  ω J ω − ω J ω + U p(t) ˜ − e − l sgn (e) − T 1 d e

˜ − p˜  (t − T )p(t ˜ −T) + p˜  (t)p(t)   2 2 ˜  κω˜ e  − λmin ()ω˜ e  − l1 − T¯d ω˜ e  + ω˜  e U p(t) ˜ − p˜  (t − T )p(t ˜ −T) + p˜  (t)p(t)    μ3  ω˜ e 2 − l1 − T¯d ω˜ e   κ − λmin () + 2 2

umax ˜ − py ˜  (t − T )p(t ˜ −T) + + 1 p˜  (t)p(t) 2μ3 2

umax ˜ −T)  − 1 − η1 +1+γ p˜  (t − T )p(t 2μ3 2

umax μ3 − λmin () − κ − + γ + 1 ω˜ e 2 − η2 2 2μ3 2

umax + η3 + 1 + γ (1 + l2 )2 2μ3

110 Fault-Tolerant Attitude Control of Spacecraft



2 u 2 ˜ − l1 − T¯d − 2l3 (1 + l2 ) max + γ + 1 ω˜ e  − γ p(t) 2μ3 2 ˜  −a2 ω˜ e 2 − γ p(t) + a4 ,



(4.76)

 + 1 + γ (1 + l2 )2 , and κ is given √ √ ˜ in Remark 4.15. Clearly, if ω˜ e (t) > a4 /a2  μ¯ a and p(t) > a4 /γ  μ¯ b , then V˙ < 0, which implies that V (t) decreases monotonically. Using Theorem 4.18 (p. 172) in (Khalil, 2002), the estimated errors are ultimately uniformly bounded by ' &$ $ % %     ˜  μ¯ b , ∈ D1  ω˜ e (t), p˜ (t) ω˜ e   μ¯ a , p lim ω˜ e (t), p˜ (t)

where γ is a positive constant, a4 = η3

u2max 2μ3

t→∞

(4.77) $ % which defines a small set containing the origin ω˜  = [0, 0] . ˜  (t) e (t), p Moreover, the larger the value of a2 and/or the smaller the selected values of a4 and γ , the smaller the error set. Thus from (4.77) we can achieve a precise fault reconstruction with the designed observer-based FDD given by (4.64) and (4.65). In addition, (4.76) shows that the following inequality is achieved:

 % a4 V (0) a4 1 t$ 2 2 ˜ + = ω˜ e  + p ds  lim lim < ∞, (4.78) t→∞ t 0 t→∞ t γmin γmin where γmin = min {a2 , γ }. From (4.78) we can see that the larger γmin and the smaller a4 , the better the resulting performance. Thus the proof is completed. Remark 4.18. It is worth mentioning that the proposed observer-based FDD given by (4.64) and (4.65) is synthesized by using the previous state estimation error information, as we can see in (4.65). From the proof of Theorem 4.5, the actuator fault reconstruction error and the angular velocity estimation error may be driven to zero as closely as required by tuning the observer gains. Thus the proposed FDD can achieve a precise fault reconstruction, and it can also detect and reconstruct the partial loss of actuator effectiveness in the more general cases of constant, periodic time-varying, or even random signals. ˆ Remark 4.19. To guarantee that the reconstructed faults p(t) are bounded in the internal (0, 1], the fault reconstruction law in (4.65) can be modified to   ˆ − T ) + l3 (ωˆ − ω) , ˆ = Proj[pmin ,1] l2 p(t (4.79) p(t) where pmin is a small positive constant (such as 0.0001), and Proj {·} denotes the projection operator (Boskovic et al., 2005). Note that the role of the projection operator is to project the reconstructed faults onto the interval (0, 1].

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Remark 4.20. Although the control input u(t) is required to be bounded in the proof of Theorem 4.5, only the applied control action generated by the actuators is involved, and the output torque of each actuator is bounded by the physical limitations of the actuators. Therefore a precise fault reconstruction can always be met with the proposed FDD scheme. Remark 4.21. If large values of li (i = 1, 2, 3) are chosen, then the effect of measurement noise would be amplified in (4.64) and (4.65), and the fault estimates would be affected. Hence li in (4.67)–(4.69) should be made carefully and their magnitudes limited to avoid amplifying the effect of the measurement noise.

4.4.2.2 Fault-tolerant attitude stabilization controller design The designed FDD mechanism (4.64)–(4.65) requires the control signal u(t) to be within the saturation limit; otherwise, a precise reconstruction of the actuator faults would not be guaranteed. In this section, on the basis of the reconˆ structed faults p(t) from the observer-based FDD given by (4.64) and (4.65), a novel fault-tolerant attitude stabilization control law will be reconfigured for the spacecraft, where its magnitude is bounded by the maximum torque generated by actuator. Before deriving the specific control design, we introduce some new vari ables: x 1 = σ dt, x 2 = σ , and x 3 = ω. Accordingly, we may rewrite x˙ 1 = x 2 ,

(4.80)

x˙ 2 = F (x 2 )x 3 ,

(4.81)

ˆ ˜ J x˙ 3 = −x × 3 J x 3 + E(t)u − E(t)u + T d ,

(4.82)

ˆ ˜ where E(t) = diag{eˆ11 , eˆ22 , eˆ33 } and E(t) = diag{e¯11 , e¯22 , e¯33 }. From Theorem 4.5, e˜ii (i = 1, 2, 3) converges to an arbitrarily small value, and hence we can achieve the following inequality:     ˜ (4.83) −E(t)u + T d   umax + T¯d . By (4.80)–(4.82) we can write the spacecraft attitude system given by (2.17) and (4.63) in a triangular nonlinear form. Hence the standard backstepping controller design can be employed, and we perform the following state transformation to design the attitude controller to tolerate actuator faults: z1 = x 1 , z2 = x 2 − α 1 , z3 = x 3 − α 2 ,

(4.84)

where α 1 , α 2 ∈ R3 are virtual control inputs, which will be discussed later. On the basis of the preceding analysis, the design procedure of the active fault-tolerant controller can be elaborated as follows.

112 Fault-Tolerant Attitude Control of Spacecraft

Step 1: We start with (4.80) by considering x 2 as the control variable. Then the time derivative of z1 is given as z˙ 1 = x˙ 1 = x 2 = z2 + α 1 .

(4.85)

The task in this step is designing a virtual control law α 1 to make z1 → 0. Choose the candidate Lyapunov function V1 = 12 zT1 z1 , and select an appropriate virtual control α 1 as α 1 = c1 x 1 ,

(4.86)

where c1 is a positive constant. Then it follows that 2  ˙ 1 = z V˙1 = z 1z 1 (z2 − c1 z1 ) = −c1 z1  + z1 z2 .

(4.87)

Step 2: We now differentiate the second error, z2 using (4.81) to give z˙ 2 = x˙ 2 − α˙ 1 = F (x 2 ) x 3 + c1 x 2 .

(4.88)

Choose the new Lyapunov function V2 = V1 + 12 z 2 z2 . Let us select the appropriate virtual control law α 2 as α 2 = F −1 (x 2 ) (−z1 − c1 x 2 − c2 z2 ) ,

(4.89)

where c2 is a positive constant, and F −1 is explicitly given in (2.23). Differentiating both sides of V2 and inserting (4.89) yield V˙2 = −c1 z1 2 + z1 z2 + z2 (F (x 2 ) x 3 + c1 x 2 ) = −c1 z1 2 − c2 z2 2 + z 2 F (x 2 ) z3 .

(4.90)

Again, if z3 = 0, then we have V˙2 = −c1 z1 2 − c2 z2 2 , and thus both z1 and z2 converge to zero asymptotically. On the basis of the analysis in Steps 1 and 2, if z3 can be driven to zero by designing a suitable control law, then zi (i = 1, 2) will also be driven to zero, and thus the attitude orientation is stabilized according to (4.84). In the following theorem, we summarize our control solution to drive z3 to zero by incorporating a backstepping-based control action and the observer-based FDD. Theorem 4.6. Consider the closed-loop system consisting of the attitude dynamics system given by (2.26) and (4.63) under the designed FDD mechanism (4.64) and under (4.65). We implement the following FTC law: u = Sat (v, umax ) with v defined as  ˆ −1 {x × J x 3 − J v(t) = E(t) 3

dF −1 (x 2 ) (z1 + c1 x 2 + c2 z2 ) dt

(4.91)

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% +F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z˙ 2 ) − c3 z3 − K1 x a  z3 χT2d  −F (x 2 ) z2 − , (4.92) z3  χTd + μ exp(−βt) where χTd = umax + T¯d , μ is a sufficiently small positive scalar; K 1 , β, and c3 are positive control gains, and x a ∈ R3 is the output of the following auxiliary system: x˙ a = −K2 x a −

2 u2 x ˆ E(t) a ˆ − E(t)u, x a 2

(4.93)

where u = u−v, and K2 is a positive constant. Suppose that the control parameters are chosen such that c3 − 1 > 0, K2 −

K12 1 − > 0. 2 2

(4.94)

Then the closed-loop attitude system is stable in the sense that all the trajectories of the resulting closed-loop attitude system are uniformly ultimately bounded. Moreover, the attitude orientation and velocity converge to a small set containing the origin, that is, σ  ≤ μ∗1 and ω ≤ μ∗2 for t ∈ T ∗ . Thus the control objectives (a) and (c) are achieved. Proof. It is obvious from (4.82) and (4.89) that z˙ 3 =x˙ 3 − α˙ 2 " # dF −1 (x 2 ) =J −1 −x × (z1 + c1 x 2 + c2 z2 ) 3 J x 3 + E(t)u + T d + dt + F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z˙ 2 ) , (4.95) where dF −1 (x 2 ) dt  2  dF (x 2 )      16 1 + x  x − 64 1 + x  2 2 2 x 2 x 2 F (x 2 ) x 3 F (x 2 ) dt = ,  4 1 + x x 2 2 (4.96) dF (x 2 ) dt   ×    −x  2 (F (x 2 ) x 3 ) I 3 + (F (x 2 ) x 3 ) + F (x 2 ) x 3 x 2 + x 2 x 3 F (x 2 ) . = 2 (4.97)

114 Fault-Tolerant Attitude Control of Spacecraft

Consider another candidate Lyapunov function 1 1  V3 = V2 + z 3 J z3 + x a x a . 2 2

(4.98)

2 ˆ ˆ ˙ a = −K2 x a 2 − E(t) x u2 − x  ax a E(t)u.

(4.99)

From (4.93) we have

Differentiating (4.98) and inserting (4.95) and (4.99) lead to V˙3 = V˙2 + zT3 J z˙ 3 + x Ta x˙ a

2  ˆ  u2 = −c1 z1 2 − c2 z2 2 + zT2 F (x 2 ) z3 − K2 x a 2 − ρ(t) ˆ − x Ta E(t)u ⎛ ⎞ −x x3 J xy3 + * ρ (t)u − ρ(t)u + T d  −1 ⎠ +zT3 ⎝ +J dF dt(x 2 ) (z1 + c1 x 2 + c2 z2 ) + F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z˙ 2 ) 2  ˆ  u2 = −c1 z1 2 − c2 z2 2 + zT2 F (x 2 ) z3 − K2 x a 2 − ρ(t) − x Ta * ρ (t)u ⎛ +zT3 ⎝

+J



−x × ρ (t)v + * ρ (t)u − ρ(t)u + T d 3 J x3 + *

dF −1 (x 2 ) dt

⎞

⎠ . (z1 + c1 x 2 + c2 z2 ) + F −1 (x 2 ) (˙z1 + c1 x˙ 2 + c2 z˙ 2 ) (4.100)

Substituting the control law (4.91) into (4.100) results in V˙3 = − c1 z1 2 − c2 z2 2 − c3 z3 2 − K2 x a 2 2 ˆ ˆ − E(t) u2 − x  a E(t)u  ˜ ˆ + z 3 E(t)u − K1 x a − E(t)u + T d −

z3 χT2d



. z3  χTd + μ exp(−βt) (4.101)

By inequality (4.83) it follows that  z 3

˜ −E(t)u +Td −

 z3  χTd −

z3 χT2d



z3  χTd + μ exp(−βt) z3 2 χT2d

z3  χTd + μ exp(−βt)

 μ exp(−βt).

(4.102)

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Further, using the specific case of Young’s inequality x  y  12 x  x + 12 y  y for x, y ∈ R3 , we may establish the following inequalities: 1 1 ˆ 2 2 2 ˆ z 3 E(t)u  E(t) u + z3  , 2 2 −K1 z 3 xa 

K2 1 z3 2 + 1 x a 2 , 2 2

(4.103) (4.104)

1 1 ˆ 2 2 2 ˆ −x  a E(t)u  E(t) u + x a  . 2 2

(4.105)

Therefore, inserting inequalities (4.102)–(4.105) into (4.101) yields   K12 1 2 2 2 x a 2 − V˙3  −c1 z1  − c2 z2  − (c3 − 1) z3  − K2 − 2 2 + μ exp(−βt).

(4.106)

From (4.98), 2V3  z1 2 + z2 2 + Jmax z3 2 + x a 2 , where J max is the largest eigenvalue of J . Hence V˙3  −c1 z1 2 − c2 z2 2 − (c3 − 1) z3 2   K12 1 x a 2 + μ exp(−βt) − − K2 − 2 2    −m ¯ z1 2 + z2 2 + Jmax z3 2 + x a 2 + μ  −2mV ¯ 3 + μ, & −1 where m ¯ = min c1 , c2 , cJ3max , K2 −

K12 2

(4.107) ' − 12 , and all terms in the definition

of m ¯ are positive from the inequalities given by (4.94). Using Theorem 4.18 (p. 172) in (Khalil, 2002), inequality (4.107) means that V3 is uniformly ultimately bounded together with the states zi (i = 1, 2, 3) and x a . More precisely, there exists a finite time T ∗ > 0 such that for all μ∗ > zi  < μ∗ (i = 1, 2, 3) and

μ m

and T ∗ ≥ 0, we have

x a  < μ∗ .

In other words, the closed-loop attitude system converges in finite time to the ball . Dμ∗ = [z1 , z2 , z3 ] ∈ R9 | z1  < μ∗ , z2  < μ∗ , t ≥ T ∗ . Note that under the state transformation (4.84) and the definition of x i (i = 1, 2, 3), it is clear that x 1  = z1  < μ∗ ,

t  T ∗,

(4.108)

116 Fault-Tolerant Attitude Control of Spacecraft

and thus we have σ  = x 2  = z2 + α 1  = z1 − c1 z1  < (1 + c1 ) μ∗  μ∗1 ,

∀t  T ∗ . (4.109) We easily conclude from (4.71) that the attitude is ultimately uniformly bounded. Moreover, using (2.23), we have F −1 (x 2 )   16 2 F   8. 1+σ σ

Therefore ω = x 3  = z3 + α 2      < μ∗ + F −1 (x2 ) [1 + c1 (1 + c1 ) + c2 ] μ∗ < (9 + 8c1 (1 + c1 ) + 8c2 )μ∗  μ∗2 ,

∀t  T ∗ ,

(4.110)

which implies that the angular velocity is ultimately uniformly bounded. Thus the result stated in Theorem (4.6) is established, and the proof is completed. Remark 4.22. The structure of the corresponding new control law (4.91) is independent of the order of the truncated spacecraft model. This is important because theoretically flexible structures have an infinite number of elastic modes. Moreover, the proposed fault-tolerant controller (4.91) is also robust with respect to unmodeled dynamics describing the flexibility because modal vibration information is not needed. Therefore from the standpoint of disturbances and flexible vibration rejection the reconfigured control law has great stability robustness. Remark 4.23. A graphical representation of the whole attitude control plant with the proposed AFTC scheme is presented in Fig. 4.7. Note that the FDD in (4.64) and (4.65) is developed for the open-loop attitude system and does not require the stability of the closed-loop attitude system. As shown in Fig. 4.7, the FDD (4.64) and (4.65) and the fault-tolerant controller (4.91) are derived separately; the FDD is developed to reconstruct the actuator faults, whereas the controller is reconfigured to compensate for the effect of the actuator faults.

FIGURE 4.7 Active fault-tolerant control for the flexible spacecraft attitude system.

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Remark 4.24. Summarizing the analysis in Theorems (4.5) and (4.6), the observer and the control gains of the presented AFTC scheme can be chosen according to the following gain tuning procedure. Step 1: Choose small γ , l2 , l3 , μi (i = 1, 2), and also select a large μ3 . Step 2: Choose  and l1 such that (4.67)–(4.69) are satisfied. Step 3: Choose positive c1 and c2 , and then select c3 , K1 , K2 such that (4.94) is guaranteed.

4.4.3 Simulation example To verify the effectiveness and performance of the proposed attitude stabilization control scheme, we performed numerical simulations using the flexible spacecraft system. We used the same physical parameters considered in (Di Gennaro, 2003), which are ⎡ ⎤ 350 3 4 ⎢ ⎥ J = ⎣ 3 270 10 ⎦ kg · m2 , 4 10 190 ⎤ ⎡ 6.45637 1.27814 2.15629 ⎥ ⎢ ⎢ −1.25819 0.91756 −1.67264 ⎥ 1/2 δ=⎢ ⎥ kg m/s2 , ⎣ 1.11687 2.48901 −0.83674 ⎦ 1.23637 −2.6581 −1.12503 and the first four elastic modes were taken into account, that is, N = 4, the natural frequencies are e11 = 0.7681, e22 = 1.1038, e33 = 1.8733, and E4 = 2.5496 rad/s, and the damping ratios are ξ1 = 0.0056, ξ2 = 0.0086, ξ3 = 0.013, and ξ4 = 0.025. For an on-orbit spacecraft, the existence of external disturbances is inevitable. Therefore the external disturbances acting on the spacecraft are considered for allof the numerical examples presented in this section and are  given by d(t) = ω2 + 0.05 [sin 0.8t, cos 0.5t, cos 0.3t] N m. The control input saturation amplitude is assumed to be 5 Nm, that is, umax = 5 Nm. To better represent the engineering application, we added measurement noise to the attitude and angular velocity sensors outputs, modeled as a zero-mean Gaussian random variable with standard deviation 0.0001. An extended Kalman filter is used in the attitude determination subsystem. At present, there are few examples of the design of active fault-tolerant attitude controllers for spacecraft, and most of the existing literature on the design of FTCs for spacecraft attitude focus on PFTC control laws. Thus a comparison of the designed controller and existing AFTCs for spacecraft cannot be made. Hence, in this section, the proposed AFTC law (4.91) is compared with conventional control methods: a proportional–integral–derivative (PID) controller and the passive fault-tolerant controller proposed in (Cai et al., 2008). Using these three controllers, we simulate two different cases: 1) constant loss of actuator effectiveness and 2) time-varying loss of actuator effectiveness.

118 Fault-Tolerant Attitude Control of Spacecraft

To implement the developed control strategy according to the procedures in Remark 4.23, the observer gains of the presented FDD mechanism, (4.64) and (4.65), were selected by trial-and-error until a good performance was obtained. The observer gains were ultimately chosen as l1 = 50, l2 = 0.2, l3 = 2,  = diag(800, 850, 1000), and γ = 10, whereas the control gains of the control law were chosen as c1 = c2 = 0.5, c3 = 1.25, K1 = 0.55, K2 = 1.5, ε = 0.001, and β = 0.25. These parameters remain unchanged for all the simulation cases in order to allow a fair and meaningful comparison. Further, in the context of simulation, at time t = 0, the orientation of spacecraft is set to be $ % with zero initial body angular velocity, and the σ (0) = −0.3 −0.4 0.2 initial modal displacements and velocities are given by χi (0) = 0 and χ˙ i (0) = 0 for i = 1, 2, 3, 4.

4.4.3.1 Response under constant loss of actuator effectiveness fault In this case, we introduced a constant loss of actuator effectiveness fault to the flexible spacecraft attitude system, and the effectiveness factors are described by ⎧ ⎪ ⎨e11 (t) = 0.4, t  8 s, e22 (t) = 0.6, t  5 s, (4.111) ⎪ ⎩ e33 (t) = 0.7, t  10 s. We first present the simulation results when applying AFTC. Fig. 4.8 shows the time response of the reconstructed fault from the observer-based FDD mechanism (4.64) and (4.65) incorporated in the AFTC. It is shown from the response of the first 5 s in Fig. 4.8(a) that the proposed FDD can reconstruct the effectiveness factor in 3 s when the actuator is fault free. After the occurrence of the actuator fault at 5 s, within a short period, the actuator fault is successfully reconstructed. This reconstruction happens with pi − pˆ i   1.2 × 10−3 (i = 1, 2, 3) even in the presence of external disturbances. Fig. 4.8(b) shows that the observed velocity state ωˆ converges to the actual angular velocity ω in 15 s with high accuracy. Moreover, as shown in Fig. 4.8(c), the performance index Ip (t) is bounded for all time and ultimately converges to a small value. From the results obtained it is clear that the designed FDD can accurately reconstruct the fault values even if no knowledge of the faults is available in advance, despite the presence of disturbances. Hence the conclusions of Theorem 4.5 are verified. Fig. 4.9 (solid line) shows the attitude control performance with the application of the AFTC in the given fault case (4.111). We clearly see that the proposed controller has accomplished the attitude stabilization maneuver. Fig. 4.9(a, b, solid line) also shows that the desired system performance is achieved, and the attitude orientation is governed to near zero within 15 s, even in the presence of external disturbances. As the proposed FDD mechanism can reconstruct the

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FIGURE 4.8 Fault reconstruction with the proposed FDD with constant loss of actuator effectiveness faults.

actuator fault (4.111) in a short period, the associated control law (4.91) can tolerate the fault in 7 s when a constant loss of effectiveness fault occurs. Thus large and even maximum control torque is demanded to compensate for the lost power, as we see from the corresponding control input shown in Fig. 4.9(c, solid line). However, the control input is within its maximum allowable limit, and the modal displacements of flexible appendages are completely damped out within 70 s, as shown in Fig. 4.9(d, solid line). The application of the conventional PID control leads to the attitude control performance shown in Fig. 4.9 (dash-dot line). It is clear that the constant fault (4.111) is propagated to the attitude and that the controller shown in Fig. 4.9(c, dash-dot line) is unable to compensate for this fault within 100 s. As a result, a time-critical aerospace mission would not be performed. This control scheme not only fails to perform the attitude maneuver but also excites large oscillations in the flexible appendages, which may further deteriorate the system control performance. Although there exists some room for improvement with different control parameter sets, there is no much improvement in the attitude and velocity responses. The attitude control objectives can be achieved with the PFTC proposed in (Cai et al., 2008) because of its fault tolerance capability to certain faults. However, the time required to perform the attitude stabilization maneuver is much

120 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 4.9 Attitude control performance with constant loss of actuator effectiveness faults.

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FIGURE 4.10 Fault reconstruction by the proposed FDD with time-varying loss of actuator effectiveness faults.

longer than the proposed controller (4.91), and there also exists severe oscillations in the flexible appendages as shown in Fig. 4.9(d, dash line). Because of its great conservatives, the PFTC cannot compensate for the fault immediately once the partial loss of actuator effectiveness fault occurs. In contrast, the AFTC developed in (4.91) can accommodate the fault as soon as possible because of the effect of the incorporated FDD mechanism. Thus the PFTC in (Cai et al., 2008) requires more time than AFTC to stabilize the attitude in the presence of actuator faults. As the PFTC control in (Cai et al., 2008) always acts on the spacecraft whether faults occur or not, it usually requires more control power, as shown in Fig. 4.9(c, dash line), especially in the time interval 20–50 s. The AFTC and PFTC controllers can protect the control torque from actuator saturation, as shown in Fig. 4.9(c, solid and dashed line). These results completely support the theoretical result that the desired performance of the system can be achieved with the proposed controller and FDD even if the faults are unknown.

122 Fault-Tolerant Attitude Control of Spacecraft

4.4.3.2 Response under time-varying loss of actuator effectiveness fault To further verify the effectiveness of the proposed FTC, we consider a timevarying loss of actuator effectiveness fault, given by ⎧ ⎪ t  15 s, ⎨ e11 (t) = 0.2 + 0.1 sin(0.2πt), (4.112) e22 (t) = 0.2 + 0.1 cos(0.3πt), t  12 s, ⎪ ⎩ e (t) = 0.2 + 0.1 sin(0.4πt), t  10 s. 33 In this case the previous simulation is repeated for the fault (4.112). When the AFTC is implemented in the spacecraft attitude control system, under the effect of the incorporated FDD mechanism (4.64) and (4.65), the time response of the reconstructed fault, angular velocity estimation error, and the performance index are as shown in Fig. 4.10. After the occurrence of the time-varying fault (4.112), the proposed FDD can successfully estimate these faults in 5 s, as shown in Fig. 4.10(a, dashed line). Fig. 4.10(a–c) clearly shows that the proposed observer-based FDD successfully achieves a precise fault reconstruction with fairly good performance even for time-varying faults. Fig. 4.11 shows the attitude control performance under the effect of PID, PFTC, and AFTC controls. It is clear that the PID controller has no capability to accommodate the actuator fault (4.112) with inferior attitude pointing accuracy and also a lower slew rate, as shown in Fig. 4.11(a,b, dash-dot line). However, the application of AFTC and PFTC leads to better results (Fig. 4.11, solid and dash lines). Although both fault-tolerant controllers, AFTC and PFTC, managed to compensate for the severe time-varying fault in the presence of external disturbances and control input saturation, the PFTC control developed in (Cai et al., 2008) requires 150 s to stabilize the attitude, whereas the proposed AFTC needs just 20 s to perform the attitude maneuver Fig. 4.11(a, solid and dashed lines). This is because the developed AFTC includes an FDD mechanism given by (4.64) and (4.65) to precisely reconstruct the actual actuator faults online and in real time. Consequently, the control law (4.91) can react to the faults immediately when they occur, because the controller is synthesized by using knowledge of the reconstructed faults. The control input for the AFTC and PFTC is shown in Fig. 4.11(c, solid and dashed lines), and we clearly see that both control schemes have the ability to protect the control effort from actuator input saturation. Summarizing the two simulated cases, note that the proposed AFTC scheme has better control performance and also has greater fault tolerance capability than the PID control and PFTC. In addition, extensive simulations were performed using different control parameters, external disturbances, and even random loss of actuator effectiveness faults. The results show that the proposed controller performed very well and accomplishes the attitude stabilization maneuver despite these undesired effects in the closed-loop system. High attitude pointing accuracy and high attitude stability were achieved.

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FIGURE 4.11 Attitude control performance with time-varying loss of actuator effectiveness faults.

4.5 Summary In this chapter, we propose the different fault-tolerant control schemes to stabilize the attitude of the spacecraft with actuator faults and saturation. Both partial

124 Fault-Tolerant Attitude Control of Spacecraft

failures and total loss faults of the actuator are considered. Accordingly, passive and active FTC control algorithms are developed to handle these different faults. In addition, taking into actuator output limitation into consideration, the control approaches proposed are capable to guarantee the magnitude of the actuators always within the maximum output limitation. The numerical simulation results under different fault types further verify the effectiveness of the proposed control algorithms. It should be pointed out that the design of the controller in this chapter requires spacecraft attitude and angular velocity measurement information, so it is a full-state feedback control. Therefore, designing a gesture fault-tolerant controller that does not require angular velocity information feedback is the main research work in the subsequent chapters of this book.

Chapter 5

Fault-tolerant velocity-free attitude control 5.1 Introduction In the previous chapters, the complete information of both spacecraft attitudes and angular velocities are always required for the spacecraft attitude control strategies with/without fault-tolerant capability. This is achieved by assuming that the perfect measurement of angular rates is available by adopting onboard sensors (such as rate gyros). However, several practical engineering situations would expose great challenges on such an assumption. These situations mainly include: a) the malfunctions of sensors lead to incorrect measurement results of angular velocity; b) the noise effect of sensors introduces inaccurate angular rate measurement values; and c) some low-cost/small-size spacecrafts are not equipped with angular rate sensors. Therefore, a low-cost velocity-free and efficient spacecraft attitude control system is in great demand to address the lack of angular rate information. To tackle the issue of the unavailable angular velocity, several attempts of formulating velocity-free attitude controllers have been made to generate angular-rate-related signal from attitude information based on the passivity approach (Nicosia and Tomei, 1992; Lizarralde and Wen, 1996). For the attitude tracking control task, a control scheme with attitude-only measurement is proposed in (Akella, 2000), but its implementation requires accurate inertia information. Hence adaptive control strategies (Wong et al., 2000; Costic et al., 2001) without angular velocity feedback are further developed to address the uncertain inertia parameter while neglecting external disturbance. A novel angular velocity-free PI (proportional–integral) control law is proposed to achieve spacecraft attitude stabilization in the absence of disturbance based on the firstorder filtering technique. To cope with angular velocity unavailability, system uncertainties and external disturbance, a filter-based adaptive controller incorporated with Chebyshev neural network is formulated to achieve spacecraft attitude tracking. However, the computation burden of training networks hinders its practical applications. Another major solution to deal with the unavailable angular rate is achieved by constructing appropriate velocity observers. The designed observer is introduced to estimate the vibration mode of the flexible attachment and unmeasurable angular velocity for the attitude tracking mission (Di Gennaro, 2002). To achieve high-performance attitude tracking, an Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00015-0 Copyright © 2021 Elsevier Inc. All rights reserved.

125

126 Fault-Tolerant Attitude Control of Spacecraft

observer-based adaptive controller is formulated on the basis of separation principle (Kristiansen et al., 2009). This velocity-free controller relies on perfect inertia value and is inapplicable in the disturbance case. To eliminate this restriction literature (Tayebi, 2008), a model-independent adaptive controller is developed by employing the designed quaternion observer. Inspired by this, extensive observer-based velocity-free control strategies (Ding et al., 2008) depending on quaternion information only are formulated in the framework of Lyapunov stability. The aforementioned velocity-free controllers do not take actuator failures and saturation into account, which would degrade the control performance and system reliability. The aim of this chapter is to address the spacecraft velocityfree attitude control issue in consideration of actuator partial failures, actuator saturation, and external disturbances. This chapter first introduces a slidingmode observer-based control scheme to estimate the unavailable angular rate and handle the actuator saturation and partial loss of effectiveness in the presence of external disturbance. Note that the perfect model knowledge is needed for the observer-based control formulation. To release this restriction, a filterbased velocity-free control approach with fault-tolerant capability is further provided to achieve attitude stabilization while conquering actuator faults, actuator saturation, inertia uncertainties, and external disturbances simultaneously.

5.2 Velocity-free attitude stabilization FTC 5.2.1 Sliding-mode observer-based FTC 5.2.1.1 Problem statement For a spacecraft that is equipped with three reaction flywheels for attitude control (i.e., N = 3), and these reaction flywheels are respectively installed on the three body shafts of the spacecraft. In this chapter, we only consider the first reaction flywheel mode failure (i.e. partial failure), so far the control goal can be described as follows: design an attitude controller that does not require angular velocity feedback information and achieve attitude control in the case of external interference and partial failure of the actuator, and the controller can solve it when the actuator control input is restricted. In this section, we propose a new attitude tolerance control without angular velocity feedback based on the sliding-mode observer technology. This method first estimates the attitude and angular velocity of the spacecraft by designing a sliding-mode observer, and then a controller is designed based on the estimated information to handle the reaction flywheel failure and stabilize the spacecraft attitude. We can assume that l0 = mini=1,2,3 lii (t). Then the total control torque τ actually acting on the spacecraft body is τ = u.

(5.1)

Fault-tolerant velocity-free attitude control Chapter | 5

127

The spacecraft attitude dynamics under the partial failure of the reaction flywheel is given by J ω˙ + ω× J ω = E(t)uc + d.

(5.2)

Defining G(σ ) = F −1 (σ ) and applying the attitude kinematics and kinetic method described by MRPs, we can transform Eq. (5.2) into the following nonlinear kinetic equation: J ∗ (σ )σ¨ + C(σ , σ˙ )σ˙ = G (σ )E(t)uc + d,

(5.3)

  where J ∗ (σ ) = G J G, d = G d, and C(σ , σ˙ ) = −G J GF˙ + (J Gσ˙ )× G. Property 5.1. The matrix J ∗ (σ ) is positive definite, symmetric, and bounded, and there exist J min ∈ R+ and J max ∈ R+ that satisfy J min x2 ≤ x  J ∗ (σ )x ≤ J max x2 ,

x, σ ∈ R3 .

(5.4)

∗ Property 5.2. For any vector x ∈ R3 , J˙ (σ ) − 2C(σ , σ˙ ) is an antisymmetric matrix satisfying   ∗ (5.5) x  J˙ (σ ) − 2C(σ , σ˙ ) x = 0.

Property 5.3. The matrix C(σ , σ˙ ) is linear with respect to σ˙ , and there are bounds C min ∈ R+ and C max ∈ R+ such that (Su et al., 2010) C min σ˙  ≤ C(σ , σ ) ≤ C max σ˙ ,

σ ∈ R3 .

(5.6)

5.2.1.2 Design of terminal sliding-mode observer Define the variables x 1 = σ and x 2 = σ˙ . Then under the partial failure of the reaction flywheel, the mathematical model of spacecraft attitude transformation (5.3) can be rewritten as the nonlinear system x˙ 1 = x 2 ,

(5.7)

J ∗ (y)x˙ 2 = −C (y, x 2 ) x 2 + G (y)E(t)uc + G (y)d,

(5.8)

where y = x 1 is the system output. At the same time the attitude of the spacecraft in aerospace engineering can be directly measured and acquired through the installed attitude sensors (such as infrared horizons, star sensors, etc.), and therefore the output y of the attitude control system is measurable. Because the system designed by sliding-mode control theory is robust to system uncertainty and external interference, the system equivalent output can be obtained when the system state moves on the sliding-mode surface. Therefore, in this section, based on this technique, for nonlinear systems (5.7)–(5.8),

128 Fault-Tolerant Attitude Control of Spacecraft

the output sliding-mode observer is designed using output y, and the estimated value of the system equivalent output system state is obtained by obtaining the system equivalent. The design of this observer is as follows: x˙ˆ 1 = xˆ 2 − x v ,

(5.9)

  m J ∗ (y)x˙ˆ 2 = −C y, xˆ 2 xˆ 2 + G (y)uc − k2 x v − k3 |x v | n sgn (x v ) ,

(5.10)

value of x im, e1 = xmˆ 1 − x 1m, x v = where xˆ i ∈ R3 (i = 1, 2) is the estimated m [xv1 , xv2 , xv3 ] = k1 sgn (e1 ) ∈ R3 , |x v | n = diag{|xv1 | n , |xv2 | n , |xv3 | n ] } ∈ R3×3 , kj ∈ R+ (j = 1, 2, 3) are the observer gains, and m and n are positive integers such that m < n.   , e2 = xˆ 2 − x 2 . Then according Define the observation error e = e1 , e 2 to (5.7)–(5.8) and (5.9)–(5.10), e will satisfy the following equation: e˙ 1 = e2 − x v ,   J ∗ (y)˙e2 = − C y, xˆ 2 xˆ 2 + C (y, x 2 ) x 2 − G (y)d

(5.11) m

+ G (y) (I 3 − E(t)) uc − k2 x v − k3 |x v | n sgn (x v ) .

(5.12)

We assume that σ˙ is bounded, that is, there exists M0 ∈ R+ such that σ˙  = x 2  ≤ M0 . Since the spacecraft angular velocity is limited in aerospace engineering, this assumption is reasonable and is only used in the design of the viewer, but the controller design does not require this requirement. For specific analysis, we refer to the following theorem. Theorem 5.1. For the observer error dynamic equations (5.11)–(5.12), given a constant value of 0 < κ1 < 1, the observer gain kj ∈ R+ (j = 1, 2, 3) satisfies 

n k3 − κ1 − m k2 < − M0 , M1 Cmax

  n k3 − κ1 − m k1 > max , e2 (0) , M1 0
maxt∈[0,T ] e2 , where T is greater than or equal to the time required for e1 to reach the sliding-mode surface, then this fixed value of k1 will guarantee that V˙1 < 0 for any e1  = 0. √ Define the continuous function W1 = 2V1 = e1  according to the value k1 > e2  + λ1 . Then e1  V˙1 = −λ1 . ≤ −λ1 √ W˙ 1 = √ 2V1 (t) 2V1

(5.18)

Inequality (5.18) can be solved as W1 (t) ≤ W1 (e1 (0)) − λ1 t.

(5.19)

Since W1 (t) is nonnegative, according to formula (5.19), it can be proved that for any t ≥ e1 (0) /λ1  T0 , we have e1 (t) ≡ 0, that is, the estimated error e1 (t) will reach the sliding-mode surface e1 = 0 within a limited time T0 . Step 2. Stability analysis of the estimation error e2 : Since we have shown in step 1 that e1 (t) will reach the sliding-mode surface e1 = 0 within time T0 , the sliding-mode control theory can be obtained according to Eq. (5.11). The equivalent control term is (x v )eq = (k1 sgn (e1 ))eq = e2 . At this time, for t ≥ T0 , the observation error dynamics (5.11)–(5.12) can be written as e˙1 = 0,   J ∗ (y)˙e2 = − C y, xˆ 2 xˆ 2 + C (y, x 2 ) x 2 − G (y)d

(5.20) m

+ G (y) (I 3 − E(t)) uc − k2 e2 − k3 |e2 | n sgn (e2 ) .

(5.21)

In application Property 5.3, C(σ , σ˙ ) is linear with respect to σ˙ :   C (y, x 2 ) x 2 − C y, xˆ 2 xˆ 2 =C (y, x 2 ) x 2 + C (y, x 2 ) xˆ 2 − C (y, x 2 ) xˆ 2   − C y, xˆ 2 xˆ 2 = − C (y, x 2 ) e2 − C (y, e2 ) xˆ 2 . (5.22) Because the maximum output torque of the reaction flywheel is limited, and the maximum output torque of each flywheel is τmax , if the control input is

130 Fault-Tolerant Attitude Control of Spacecraft

√ designed to meet this amplitude limit, then uc  ≤ 3τmax . At this point, we ∗ choose another Lyapunov candidate function V2 (t) = 0.5e 2 J (y)e2 . According to (5.22) and Property 5.3, we have  ∗ ˙∗ e2 V˙2 =0.5e 2 J (y)e2 + e2 J (y)˙ ≤ e2  Cmax e2  xˆ 2 + (1 − l0 ) G(y) uc  + G(y)d m

− k2 e2 2 − k3 e2 1+ n

m

≤ − k2 e2 2 + Cmax (e2  + M0 ) e2 2 − k3 e2 1+ n  √ +4 3 (1 − l0 ) τmax + dmax e2   

(5.23)

M1

≤ − κ1 e2 

1+ m n

  m m − k3 − κ1 − M1 e2 − n e2 1+ n

− [k2 − Cmax (e2  + M0 )] e2 2 . 2 If e2  ≤ Ckmax − M0 , then selecting (5.13) according to the observer parameters, we can simplify Eq. (5.23) to

  m m m V˙2 ≤ −κ1 e2 1+ n − k3 − κ1 − M1 e2 − n e2 1+ n .

(5.24)

  − n  m k3 −κ1 3 When e2 is outside the region D1 = e2 ∈ R , e2  ≤ M1 , from Eq. (5.24) we get V˙2 ≤ −κ1 e2 1+ n . m

(5.25)

According to Property 5.1, V˙2 + κ1



2 Jmax

n+m 2n

n+m

V2 2n ≤ 0,

e2 ∈ / D1 .

(5.26)

Solving this inequality, we get n−m 2n

V2

n−m 2n

(t) ≤ V2

n−m (0) − κ1 2n



2 Jmax

n+m 2n

t,

e2 ∈ / D1 .

(5.27)

On the other hand, from Eqs. (5.13) and (5.15) we know that e2 starts from the set   k2 (5.28) D2 = e2 ∈ R3 , e2  ≤ − M0 , Cmax

Fault-tolerant velocity-free attitude control Chapter | 5

131

that is, e2 (0) ∈ D2 . Then, it follows that e2 (t) ∈ D2 \D1 for t ≥ T1 , where n−m

T1 =

V2 2n (0) −



Jmin 2

 n−m  2n



κ1 n−m 2n

k3 −κ1 M1

 n+m

− n−m m

.

(5.29)

2n

2 Jmax

Solving inequality (5.27), we get n−m

V2 2n (t) ≤



Jmin 2

n−m  2n

k3 − κ1 M1

− n−m m

,

e2 (0) ∈ D2 \D1 .

(5.30)

Further, applying Jmin e2 2 ≤ 2V2 given in Property 5.1, we have 

n−m n−m n−m   n−m 2n Jmin Jmin 2n k3 − κ1 − m 2 2n e2  ≤ V2 (t) ≤ , 2 2 M1 (5.31) e2 (0) ∈ D2 \D1 .

At this time, solving inequality (5.30), we obtain  e2  ≤

k3 − κ1 M1

− n

m

,

e2 (0) ∈ D2 \D1 .

(5.32)

The above formula shows that (5.16) is justified. When e2 (t) starts in the area D2 \D1 , according to the definition of the set D1 , we see that e2 (t) decreases and converges in the area D1 within a finite time T1 . Therefore D1 is an attractive area, and e2 (t) ≤ ε0 for t ≥ T1 , where

  n k3 − κ1 − m ε0  max , e2 (0) . (5.33) M1 According to this formula, we can prove that if k1 is selected to satisfy formula (5.14), then the inequality k1 > e2  + λ1 in Step 1 is established, so e1 (t) = 0 for any t ≥ T0 . So far, the theorem is proved according to the analysis given in Steps 1 and 2. In fact, the geometric relationship of each area in the proof of Theorem 5.3 can be described using Fig. 5.1. First, choose a large value of k2 to ensure that e2 (0) starts in the area D2 . If e2 (0) starts at point A (that is, e2 (0) starts at area D2 \D1 ), then e2 will enter area D1 within a limited time T1 . Since D1 is the attraction area, once e2 enters the area D1 , e2 will remain in D1 . If e2 (0) starts at point B (that is, e2 (0) starts at area D1 ), since D1 is an attractive area, then e2 will also move inside D1 . Combining the above two cases, we get the inequality

132 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 5.1 Definition of the coordinate reference frames for spacecraft attitude system.

e2  ≤ max



k3 −κ1 M1

− n

m

 , e2 (0) if k1 is carried out according to Eq. (5.14).

This choice ensures the inequality V˙1 < 0, so that e1 (t) ≡ 0 for t ≥ T0 .

5.2.1.3 Design of fault-tolerant attitude controller Using the estimates xˆ 1 and xˆ 2 provided by observers (5.9)–(5.10), in this subsection, we design the attitude fault-tolerant controller to realize attitude control and solve the problem of limited control input. Firstly, define the following variables z1 = xˆ 1 − σ d ,

(5.34)

z2 = xˆ 2 + α 1 − χ 1 − σ˙ d ,

(5.35)

where σ d ∈ R3 is a twice continuously differentiable function representing the desired attitude of the spacecraft, σ˙ d ∈ R3 is the desired attitude differential, α 1 = c1 z1 , χ 1 = −0.5ηz1 , c1 ∈ R+ , and η ∈ R+ is a constant. At this time, if the design instruction controls uc ensuring that z1 is consistent and ultimately bounded and stable, then according to Theorem 5.3, xˆ 1 (t) ≡ x 1 (t) for t ≥ T0 , and we can prove that the attitude tracking error z1 is consistent and finally boundedly stable, and that is why we introduced transformation (5.34). We design the attitude controller uc = Sat (v, τmax ) ,

(5.36)

where v ∈ R3 is the controller input signal. From Eq. (5.36) we see that the desired attitude control torque uc has upper and lower bounds. To analyze the influence of this limit value on the stability of attitude control, we introduce the following auxiliary system:

x a − G (y)u if x a  ≥ δ, −K1 x a − g(y,u) x a 2 (5.37) x˙ a = 0 if x a  < δ,

Fault-tolerant velocity-free attitude control Chapter | 5

133

where u = uc − v, g(y, u) = G(y)2 u2 , K1 ∈ R+ is a constant, x a ∈ R3 is the state of the auxiliary system, and δ ∈ R+ is a constant selected by the designer. According to formula (5.37), when x a  ≥ δ, uc will be saturated, and if x a  < δ, then uc will not. Theorem 5.2. For the spacecraft attitude control system where the reaction flywheel has a mode failure, we use observers (5.9)–(5.10) to estimate the attitude and angular velocity of the spacecraft and select the initial value of the observer state to satisfy xˆ 1 (0) = x 1 (0), when controller (5.36) is used for attitude control, and the control input v is designed as    −1   v = G (y) C y, xˆ 2 χ 1 + σ˙ d − c1 z1     (5.38) − c1 + η2 J ∗ (y) z2 − c1 z1 + χ 1  +J ∗ (y)σ¨ d + χ 2 − z1 − K2 z2 − K3 x a ,   2 2 z 2 z , K , where χ 2 = −0.5η k22 + k32 + (c1 + 0.5η)2 J ∗ (y) + Cmax 2 2 2 K3 , and η ∈ R+ is the control gain. If the selected control gain satisfies K2 − 1 > 0,

K1 − 0.5K32 − 0.5 > 0,

(5.39)

then the closed-loop attitude control system is consistent, ultimately bounded, and stable, and there are T ∗ ∈ R+ and ε ∗ ∈ R+ such that for any t ≥ T ∗ , the attitude tracking error satisfies x 1 (t) − σ d (t) < ε ∗ , so that the control objectives are achieved. Proof. According to Eqs. (5.34)–(5.35), we have z˙ 1 = z2 − c1 z1 − e2 + χ 1 ,

(5.40)

  z˙ 2 = x˙ˆ 2 + (c1 + 0.5η) z2 − c1 z1 − e2 + χ 1 − σ¨ d .

(5.41)

Applying the definitions of e2 and C(σ , σ˙ ) provided in Property 5.3 regarding the linearity of σ˙ , we have        C y, xˆ 2 xˆ 2 = C y, xˆ 2 z2 + C y, xˆ 2 χ 1 + σ˙ d − c1 z1    = C (y, x 2 ) z2 + C (y, e2 ) z2 + C y, xˆ 2 χ 1 + σ˙ d − c1 z1 . (5.42) On the other hand, since the initial value of the observer state is selected to satisfy xˆ 1 (0) = x 1 (0), we know from the proof of Theorem 5.3 that e1 = e˙ 1 = 0 and (x v )eq = e2 for all t ≥ 0, and   m J ∗ (y)x˙ˆ 2 = −C y, xˆ 2 xˆ 2 + G (y)u − k2 e2 − k3 |e2 | n sgn (e2 ) ,

t ≥ 0. (5.43)

134 Fault-Tolerant Attitude Control of Spacecraft

So far, to analyze the stability of the closed-loop attitude control system, we separately discuss the following two cases. Case 1: If x a  ≥ δ, then we select the Lyapunov candidate function  ∗  V3 (t) = 0.5z 1 z1 + 0.5z2 J (y)z2 + 0.5x a x a .

(5.44)

Applying Property 5.1, differentiation of V3 (t), formulas (5.40)–(5.41), and control input (5.38), we obtain   2   V˙3 (t) = − c1 z 1 z1 − z1 e2 + z1 χ 1 − K1 x a  − g(y, u) − x a G (y)u   + z 2 −k2 e2 + G (y)u + χ 2 − K3 x a − C (y, e2 ) z2  m (5.45) −k3 |e2 | n sgn (e2 ) − (c1 + 0.5η) J ∗ (y)e2 − K2 z2 .

Using the inequality C(σ , σ ) ≤ Cmax σ˙  in Property 5.3, we get that the last term of formula (5.45) satisfies   ∗ z 2 −C (y, e2 ) z2 − (c1 + 0.5η) J (y)e2 − K2 z2 ≤ (c1 + 0.5η) J ∗ (y) e2  z2  + Cmax e2  z2 2 − K2 z2 2 (5.46) 2  2 2 η c +η ηC 2 ≤ 1 2 2 J ∗ (y) z2 2 + eη2  + 2max z2 4 − K2 z2 2 . On the other hand, put the following inequalities: 1 e2 2 , 2η

(5.47)

ηk22 1 z2 2 + e2 2 , 2 2η

(5.48)

2 −z 1 e2 ≤ 0.5η z1  +

k2 z  2 e2 ≤

K2 1 z2 2 + 3 x a 2 , 2 2 1 1  2 2 2 z 2 G (y)u ≤ G(y) u + z2  , 2 2 1 1  2 2 2 −x  a G (y)u ≤ G(y) u + x a  , 2 2 −K3 z 2 xa ≤

m

n −k3 z 2 |e2 | sgn (e2 ) ≤

ηk32 2m 3 z2 2 + e2  n . 2 2η

(5.49) (5.50) (5.51) (5.52)

Substituting (5.45), we have 3 2 e2  n + e2 2 − (K2 − 1) z2 2 V˙3 ≤ −c1 z 1 z1 + 2η η   K32 1 x a 2 . − − K1 − 2 2 2m

(5.53)

Fault-tolerant velocity-free attitude control Chapter | 5

135

At this time, selecting (5.39) according to the control parameters, from inequality (5.53) we have   2m 3 2  ∗ 2 e2  n + e2 2  x + ¯ 1 z z + z J (y)z + V˙3 ≤ −m 2 a 1 1 2 2η η     K2 −1 where m ¯ 1 = min c1 , KJ2max , K1 − 23 − 12 , Jmax = λmax J ∗ (y) .

(5.54)

Since by Theorem 5.3 e2 (t) ≤ ε0 for t ≥ T1 , we can rewrite (5.54) as follows: 2π 3 2 ε0  n + ε0 2 , ¯ 1 V3 + V˙3 (t) ≤ −2m 2η η

t ≥ T1

(5.55)

At this time, from Eq. (5.55) we can prove that V3 (t) with respect to zi , i = there is a finite 1, 2, is consistent with xa and finally bounded. More  precisely,   2m time T1∗ > T1 such that for any t ≥ T1∗ and ε1∗ > m11 η 1.5 ε0  n + 2 ε0 2 , we have z1 (t) ≤ ε1∗ and z2 (t) ≤ ε1∗ , that is, the  entire closed-loop attitude  control system will converge to the area Dε1∗ = z1 , z2 ∈ R6 z1  ≤ ε1∗ ,  z2  ≤ ε1∗ within time T1∗ , and x 1 (t) − σ d (t) = xˆ 1 (t) − σ d (t) = z1 (t) < ε1∗ ,

t > T1 .

(5.56)

Case 2: When x a  < δ, there is no saturation in the control uc , that is, u = 0 and uc = v. Select another Lyapunov candidate function V4 (t) = 0.5z 1 z1 + ∗  0.5z2 J (y)z2 . Then using the analysis steps in Case 1, we have K32 2m 3 2 2 2 n z e e x a 2    V˙4 (t) ≤ −c1 z z + + − − 1) + (K 2 2 2 2 1 1 2η η 2   K2 2m 3 2  ∗ e2  n + e2 2 + 3 δ 2 , ≤ −m ¯ 2 z 1 z1 + z2 J (y)z2 + 2η η 2 (5.57)   −1 . Since e2 (t) ≤ ε0 is always established in finite where m ¯ 2 = min c1 , KJ2max time T1 , we have K2 2m 3 2 e2  n + e2 2 + 3 δ 2 , ¯ 2 V4 + V˙4 (t) ≤ −m 2η η 2

t > T1 .

(5.58)

Similarly, from (5.58) we can prove that the closed-loop attitude system is consistent and ultimately boundedly stable and there is time T2∗ > T1 such   2m K 2 δ2 ∗ and t ≥ T2∗ , we have that for any ε2 > m12 η 1.5 ε0  n + 2 ε0 2 + η δ2

136 Fault-Tolerant Attitude Control of Spacecraft

z1 (t) ≤ ε2∗ and z2 (t) ≤ ε2∗ , that is, the closed-loop attitude control  sys    ∗ ∗ 6 ∗ tem converges to the area Dε2 = z1 , z2 ∈ R z1  ≤ ε2 , z2  ≤ ε2 within time T2∗ , and x 1 (t) − σ d (t) = xˆ 1 (t) − σ d (t) = z1 (t) ≤ ε2∗ , t > T1 . (5.59) By analysis given in Cases 1 and 2, we can prove that all signals in the entire closed-loop attitude control system are consistent and ultimately bounded and stable. According to Eqs. (5.56) and (5.59), we can further prove that if   ε ∗ > max ε1∗ , ε2∗ and t ≥ T ∗ = min T1∗ , T2∗ , then x 1 (t) − σ d (t) = xˆ 1 (t) − σ d (t) = z1 (t) ≤ ε ∗ , z2 (t) < ε ∗ . (5.60) We know that the attitude-tracking error is consistent and ultimately bounded and stable. In addition, because the function Sat(·, ·) is continuous and bounded, then |ui | ≤ |uci | ≤ τmax , i = 1, 2, 3, that is, the saturation problem of reaction flywheel control is solved at this time. The stated control objectives can be achieved, and the theorem is proved. When σ d = 0, controller (5.36) realizes attitude stability control, that is, the controller can achieve not only attitude stability control but also attitude tracking control, as well as the entire attitude closed-loop control under the action of the observer and controller. The system diagram is shown in Fig. 5.2. The attitudetracking error and command control signal uc formed by the spacecraft attitude σ and the expected state σ d are used to update the observers (5.9)–(5.10). State estimation signals xˆ 1 and xˆ 2 , attitude output feedback, and auxiliary system (5.37). The state x a is used to design controller (5.36). We can see that the controller does not require spacecraft angular velocity measurement information, so the controller can handle the problem of angular rate sensor failure or strong noise.

FIGURE 5.2 Definition of the coordinate reference frames for spacecraft attitude system.

Remark 5.1. According to Theorems 5.3 and 5.2 and their proofs, the parameters in the sliding-mode observer (5.9)–(5.10) and controller (5.36) can be selected as follows: Step 1: Select λ1 and 0 < κ1 < 1; Step 2: Select k3 such that k3 − κ1 > 0;

Fault-tolerant velocity-free attitude control Chapter | 5

Step 3: Select k2 to satisfy inequality (5.15);

k2 Cmax

137

− M0 > 0 and take e2 (0) ∈ D2 satisfying

− n  m Step 4: Take the value of parameter k1 satisfying k1 > max k3M−κ1 1 ,  e2 (0) + λ1 ; Step 5: Select the value according to the attitude control accuracy requirements; Step 6: Select K1 , K2 , and K3 satisfying inequality (5.39). Remark 5.2. The sliding-mode observers (5.9)–(5.10)) are designed based on the spacecraft attitude dynamics model (5.2). When the control system dynamics model has modeling bias or model uncertainty, the observer’s estimated performance on attitude and angular velocity has not been quantitatively analyzed, and therefore the observer’s design method needs to be improved.

5.2.2 Simulation results The spacecraft given in Table 5.1 of this subsection uses controller (5.36) for attitude control simulation to verify the effectiveness of the control strategy. The external disturbance torque suffered by the spacecraft is calculated. To detect and take high-definition photos of specific areas on the Earth, the spacecraft is required to be able to track a certain desired attitude and ensure that its payload can effectively work. The expected posture is σ d = [σd1 , σd2 , σd3 ] = [0.1151, −0.1506, 0.1249] . The initial attitude of the spacecraft is σ (0) = [0.8711, 0.8742, 0.8742] × 10−3 , and its initial attitude angular velocity is ω(0) = [0.02, 0.02, 0.02] deg/s. The parameter values of controller (5.36) are m/n = 0.35, k1 = 15, k2 = 7.5, k3 = 15, c1 = 0.05, η = 0.2, K1 = 1, K2 = 5, K3 = 7.5, and δ = 0.0001. TABLE 5.1 Main parameters of a rigid spacecraft. Parameters

Description or value

Task

Detecting specific areas on the Earth and taking high-resolution photos

Main inertia (kg·m2 )

J1 = 25, J2 = 22, J3 = 18

Product of inertia (kg·m2 )

0 and 2kx = ki . Then all the signals in the resulting closed-loop attitude system are bounded and continuous, and the global asymptotic stability is guaranteed in case of d(t) ≡ 0. That is, the attitude and the angular velocity converge to zero: lim q t→∞ v

=0

and lim ω = 0.

t→∞

Proof. Consider the Lyapunov function candidate   1 1 1 ˙  χ˙ − ki q  ˙ + ψ  P ψ, V = ω J 0 ω + kp (q0 − 1)2 + q  v qv + χ vχ 2 2 2 (5.71) where P is a positive definite matrix, and it is the solution of the following Lyapunov equation:     0 IN 0 IN (5.72) P + P  = −2Q, −K −C −K −C where Q is a positive definite symmetric matrix. Defining    ˙ − ki q v , ψ  ] , x = [(1 − q0 ) , q  v ,ω , χ V could be bounded by λmin (R)x2 ≤ V = x  Rx ≤ λmax (R)x2 , where R = diag{kp , (kp − 0.5ki2 ), 0.5J 0 , 0.5, 0.5P }.

(5.73)

Fault-tolerant velocity-free attitude control Chapter | 5

147

Here λmin () and λmax () denote the minimal and maximal eigenvalues of a pos itive matrix, respectively. Since the inequality kp − 0.5ki2 > 0 holds, R is positive definite, which implies that V is globally positive definite by (5.73). In the case of d(t) ≡ 0, the time derivative of the Lyapunov function (5.71) along the faulty spacecraft dynamics equation (5.62) can be calculated as   ˙  χ¨ − ki q˙  ˙ ˙ − ki q  ¨ V˙ =ω J 0 ω˙ + kp 2 (q0 − 1) q˙0 + 2q  q v v +χ vχ vχ    + ψ  P ψ˙ = ω αu + δ  [K, C]ψ + kp q v − χ˙ − ki q v     × χ˙ − ki q v + 2kx q × (−χ + (2kx − ki ) q v v + q0 I 3    ki  × q v + q0 I 3 [−χ + 2kx q v + ki q v (τ )dτ ) − (5.74) 2 0      0 IN +ki q v (τ )dτ ] − δ  Cδω + f + ψ  P ψ −K −C 0   −δ  ω. +ψ P Cδ Substituting the controller equation (5.67) into (5.74) yields   1 − μ0 1 V˙ =ω −α μ1 (t) sign(ω) (uN  + ε0 ) + μ0 μ0      − χ˙ − ki q v χ˙ − ki q v + f + δ  [K, C]ψ − δ  Cδω     0 IN −δ   ω − (I 3 − α) uN + ψ P ψ +ψ P −K −C Cδ

(5.75)

≤ (1 − μ0 ) ω uN  − (1 − μ0 ) (uN  + ε0 ) ω   ω      − χ˙ − ki q v χ˙ − ki q v − ω ψ   ψ    2 ω   , ≤ − (1 − μ0 ) ε0 ω − χ˙ − ki q v − ω , ψ  ψ where  is given by ⎡ =⎣



δ Cδ (K−P

C+CP ) δ 2

⎤

δ  (K−P

C+CP ) 2

⎦.

(5.76)

Q

By using the Schur complement lemma and appropriate choice of the matrix Q,  can be made a positive definite matrix. With μ0 < 1, from (5.75) we easily obtain that 2 V˙ ≤ − χ˙ − ki q v ≤ 0, (5.77)

148 Fault-Tolerant Attitude Control of Spacecraft

implying that the limit lim V (t) = V (∞)

t→∞

exists. We can further conclude that V ∈ L∞ , and then q0 , q v , ω ∈ L∞ . Thus from q˙ v ∈ L∞ we have (2.17). By integrating V˙ from 0 to ∞ we have 2 lim χ˙ − ki q v ≤ V (0) − V (∞).

t→∞

(5.78)

Because the term on the right-hand side of inequality (5.78) is bounded, it follows that χ˙ − ki q v ∈ L2 . From (5.66) we have χ¨ = −χ˙ + 2kx q˙ v + ki q v , which, together with q˙ v ∈ L∞ , implies that   χ¨ − ki q˙ v = − χ˙ − ki q v + (2kx − ki ) q˙ v L∞ . Hence we have χ˙ − ki q v ∈ L2 ∩ L∞ and χ¨ − ki q˙ v ∈ L∞ . Using Barbalat’s lemma, it follows that   (5.79) lim χ˙ − ki q v = 0. t→∞

Now consider the other variable 



y(t)  χ − ki o

q v (τ )dτ,

¨ which is uniformly bounded, and then we conclude the uniform continuity of y. Furthermore, since y˙ → 0 as t → ∞, it follows that 



lim

t→∞ 0

¨ )dτ + y(0) ˙ y(τ = 0.

(5.80)

By the alternate statement of Barbalat’s lemma, together with the uniform con ¨ this leads to y˙ → 0 as t → ∞. Note that when y¨ = − χ˙ − ki q v + tinuity of y, (2kx − ki ) q˙ v , we have lim q˙ (t) = 0 t→∞ v whenever   lim χ˙ − ki q v = 0

t→∞

and 2kx = ki . With lim q˙ (t) = 0, t→∞ v

Fault-tolerant velocity-free attitude control Chapter | 5

149

Then, we have lim q (t) = 0, t→∞ v

lim ω(t) = 0

(5.81)

t→∞

for any initial attitude and angular velocity. Thereby the globally asymptotic stability of the closed-loop system can be concluded. The proof is completed. When we take external disturbances d into consideration, the stability analysis of the closed-loop system can be stated by the following corollary. Corollary 5.1. Let the control parameters be chosen such that kp − 0.5ki2 > 0 and 2kx = ki ; the system in (2.17), (2.25), and (5.62) in the closed loop with the control law equation (5.67) is then ultimately uniformly bounded (UUB) in the presence of partial loss of actuator effectiveness and additive faults as well as external disturbances d(t). Proof. In the case where the spacecraft attitude system is affected by external disturbances, calculating the time derivative of V in (5.71) gives   V˙ =ω ατ + δ  [K, C]ψ − δ  Cδω + f + d + ψ  P ψ˙   (5.82) ˙ v + χ˙  χ¨ − ki q˙  ˙ − ki q  ¨ + kp 2 (q0 − 1) q˙0 + 2q  vq vχ v χ. Substituting (5.67) into (5.82) with the same derivation as in Theorem (5.3), it follows that ⎞ ⎛ ω    V˙ ≤ ω d − [ω , χ˙ − ki q v , ψ  ]  ⎝ χ˙ − ki q v ⎠ , (5.83) ψ where  is defined by ⎡ ⎢ =⎢ ⎣

δ  Cδ 0

0 I3

[K−P ,C+CP ] δ 2

0

δ  [K−P ,C+CP ] 2

0

⎤ ⎥ ⎥. ⎦

(5.84)

Q

Also, by using the Schur complement lemma  can be guaranteed to be pos2 itive definite by an appropriate choice of Q. Since q  v q v + q0 = 1 results in 2 2 |1 − q0 | ≤ (1 − q0 ) and q v  ≤ q v , we can easily obtain that      ˙ − ki q v , ψ   2 x2 = (1 − q0 ) , q  v ,ω , χ     ≤ (1 − q0 ) + q v  +  ω , χ˙ − ki q v , ψ  2 .

(5.85)

150 Fault-Tolerant Attitude Control of Spacecraft

Then   V˙ ≤ −πx2 + ω sup d(τ ) + π (1 − q0 ) + q v  0≤τ ≤t





(5.86)

≤ −πx + x 2π + sup d(τ ) , 2

0≤τ ≤t

where π = min() was introduced. Let 0 < θ < 1. Then (5.86) can be rewritten as    V˙ ≤ −πθ x2 + x 2π + sup d(τ ) − (1 − θ )πx . (5.87) 0≤τ ≤t

Clearly, if x >

π + sup0≤τ ≤t d(τ ) , (1 − θ )π

then we obtain V˙ < −πθ x2 ≤ −

πθ V. λmax (R)

(5.88)

Therefore the state is bounded ultimately by  x(t) ≤ max

λmax (R) λmin (R)

× 2x(0)e

−[πθ/λmax (R)]t

  2π + sup0≤τ ≤t d(τ ) ,2 , (1 − θ )π

(5.89)

which is a small set containing the origin x = 0. By (5.67) and (5.89), using the boundedness theorem in a nonlinear system, we can conclude that 1 − q0 ,    ˙ − ki q v , ψ  , and u are UUB. Thus all signals in the closed-loop q v,ω , χ system are UUB. This completes the proof. Remark 5.4. It is worth mentioning that although the proposed control law in (5.67) is independent on the magnitude measurements of spacecraft angular velocity, uF in (5.67) requires the knowledge of the direction of the angular velocity. As a result, the spacecraft still needs low-cost gyroscopes to obtain the direction information of the angular velocity. In addition to this hardware-based scheme, there exists an alternative analytical methodology to determine the sign of ω without any rate sensors. Actually, in practical aerospace engineering the controller is implemented with a digital computer; hence, the value of q˙ v in the

Fault-tolerant velocity-free attitude control Chapter | 5

151

time of (k + 1)T can be approximately estimated by using one-step previous information from attitude sensors as follows: q˙ v ((k + 1)T ) =

q v ((k + 1)T ) − q v (kT ) , T

(5.90)

where T is the sampling time. From (2.17) we have ⎡

⎤ ⎡ q0,k ω1 2 ⎣ ω2 ⎦ = ⎣ q3,k T ω3 −q2,k

−q3,k q0,k q1,k

⎤−1 ⎡ ⎤ q1,k+1 − q1,k q2,k −q1,k ⎦ ⎣ q2,k+1 − q2,k ⎦ , q0,k q3,k+1 − q3,k

(5.91)

where qi,k and qi,k+1 (i = 0, 1, 2, 3) are, respectively, the ith item of q at time moments kT and (k + 1)T . Based on (5.91), the sign of ω can be derived. Remark 5.5. It is worth mentioning that the chattering effect may be caused by the sign function of the controller equation (5.67). However, in practice, once the orbit and target are determined, the direction of angular velocity would not vary, and the positive direction of angular velocity is decided; in some sense, the sign of angular velocity will not change. Consequently, the proposed controller equation (5.67) is continuous; thus, vibration or instability will not be induced. This will be discussed further in the simulation study. Remark 5.6. Note that when actuators are fault free, we have μ0 ≡ 1 and μ1 (t) ≡ 1. Thus we can obtain u = uN for the controller equation (5.67); that is, the system can be stabilized by the controller equation (5.67).

5.3.2 Numerical simulation To verify the effectiveness and performance of the proposed control scheme in this section, numerical simulations were carried out using the flexible spacecraft system Eqs. (2.17), (2.25), and (5.62) with the developed control law equation (5.67). The same physical parameters as considered in (Di Gennaro, 2003) are used, which are given by ⎡ ⎤ 350 3 4 ⎢ ⎥ J = ⎣ 3 270 10 ⎦ kg · m2 4 10 190 and

⎡

⎤ 6.45637 1.27814 2.15629 ⎢ ⎥ δ = ⎣ −1.25819 0.91756 −1.67264 ⎦ kg1/2 · m/s2 . 1.11687 2.48901 −0.83674

The natural frequencies are 1 = 0.7681, 2 = 1.1038, and 3 = 1.8733 rad/s; and the damping ratios are ξ1 = 0.0056, ξ2 = 0.0086, and ξ3 = 0.013. Moreover, we also consider the bounded external disturbance d = [0.2, 0.1, −0.1] .

152 Fault-Tolerant Attitude Control of Spacecraft

In the context of simulation, we consider the following actuator fault scenario. At t = 40 s, each actuator undergoes a partial loss of effectiveness, whereas at t = 40 s, these actuators experience also an additive fault induced by a stuck type of actuator fault, and the following nonlinearity summarized from (Jin et al., 2008; Cai et al., 2008; Chen and Saif, 2007) is used to generate the actuator faults scenario:

αi (t) =

1, 0.25 + 0.05 sin(2πt),

t < 40 s, t ≥ 40 s,

(5.92)

0, 0.35 + 0.15 sin(10t),

t < 100 s, t ≥ 100 s.

(5.93)

fi (t) =

To implement the controller, the design parameters in (5.68) are chosen as kx = 6, ki = 0.1, kp = 3, μ0 = 0.15, μ1 = 1, and ε0 = 0.25 in (5.69). At time t = 0, the orientation of the spacecraft is set to be q v (0) = [−0.5, −0.26, 0.79] with zero initial body angular velocity, initial modal displacements ηi (0) = 0.001, and time derivative η˙ i (0) = 0.0005.

5.3.2.1 Response obtained from the nominal controller The fault scenario equations (5.92)–(5.93) were implemented in the case where the spacecraft attitude is governed by the normal controller equation (5.68). Because of the slowly time-varying additive fault and constant external disturbance, from Figs. 5.13 and 5.14 it is clear that the overall attitude system is stable. However, its attitude pointing accuracy is quite low, and it could not satisfy the requirement of the mission since such a controller does not have a mechanism to accommodate the actuator faults. This is due to the fact that once the actuators undergo partial loss of control effectiveness after 40 s, especially after the occurrence of additive fault in 100 s, the static value of control input, as shown in Figs. 5.15 and 5.16, is not large enough to compensate the fault. Therefore we can conclude that an FTC design is greatly needed to accommodate the effect of actuator faults.

FIGURE 5.13 Time response of attitude under normal controller equation (5.68) with the fault equations (5.92)–(5.93).

Fault-tolerant velocity-free attitude control Chapter | 5

153

FIGURE 5.14 Time response of angular velocity under normal controller equation (5.68) with the fault equations (5.92)–(5.93).

FIGURE 5.15 Time response of control input. Case 1: normal controller equation (5.68).

5.3.2.2 Response obtained from the velocity-free fault-tolerant controller In this case, we demonstrate the performance of the proposed strategy when the actuator fault equations (5.92)–(5.93) occur in the system. When the designed controller equation (5.67) is implemented to the flexible spacecraft in the actuator faults case, the quaternion and angular velocity responses of the attitude system are presented in Figs. 5.17 and 5.18. As expected, we can clearly see that the control law equation (5.67) managed to compensate the additive fault and partial loss of effectiveness, so that the closed-loop system is still stabilized within 150 seconds, and acceptable performance is also met despite of severe external disturbances. This is achieved by introducing the extra term uF in (5.69). However, compared with the control input in Case 1, a larger control effort is needed, as illustrated in Fig. 5.15 (solid line). Indeed, this is due to the fact that the fault-tolerant controller uF is always active whenever the actuator undergoes faults or not. Moreover, as shown in Fig. 5.17, the sign of angular velocity in each axis is invariable throughout the attitude maneuver. This further verified the analysis in Remark 3. Summarizing the results from Fig. 5.13 and 5.18, we can clearly see the fault tolerance capability of (5.69). If the controller equation (5.67) is implemented without the uF in (5.69), then the fault cannot be accommodated, as shown in Fig. 5.13. Otherwise, the actuator faults and external disturbances can be successfully compensated, as shown in Figs. 5.17 and 5.18.

154 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 5.16 Time response of control input. Case 2: fault-tolerant controller equation (5.67).

FIGURE 5.17 Failure to respond to control signals: attitude.

FIGURE 5.18 Failure to respond to control signals: angular velocity.

5.4 Attitude stabilization FTC with actuator saturation and partial loss of control effectiveness 5.4.1 Problem statement The actuator failures commonly encountered in spacecraft attitude system can be termed as: 1) partial loss of effectiveness (F1); 2) lock-in-place (F2); and 3) float (F3). In this section, we consider a spacecraft that has no actuator redundancy. If one of the actuators undergoes F2 or F3, then it will lead to the loss of three-axis attitude control. These two catastrophic faults are not the point of this investigation. Consequently, we mainly discuss partial loss of effectiveness fault F1. Consider actuator fault F1 represented by a multiplicative matrix E(t).

Fault-tolerant velocity-free attitude control Chapter | 5

155

The faulty attitude dynamics model is given by J ω˙ = −S(ω)J ω + E(t)τ + d,

(5.94)

where E(t) = diag{e11 (t), e22 (t), e33 (t)} ∈ Rt 3×3 with 0 < e0 ≤ eii (t) ≤ 1 being the actuator health indicator for the ith actuator. The case ei (t) = 1 implies that the ith actuator is healthy, and 0 < ei (t) < 1 corresponds to the case in which the ith actuator partially loses its actuating power but still works all the time. To facilitate the subsequent control formulation, we combine the faulty dynamics (5.94) and attitude kinematics (2.22) to form the following second-order nonlinear dynamics equation using appropriate procedures and definitions: ¯ J ∗ (σ )σ¨ + C(σ , σ˙ )σ˙ = G (σ )E(t)τ + d,

(5.95)

where the matrices G, J ∗ , C ∈ R3×3 and the lumped disturbance are defined as G(σ ) = T −1 (σ ),

(5.96)

J ∗ = G J G,

(5.97)

d¯ = G d,

(5.98)

˙ + G S(Gσ˙ )J G. C(σ , σ˙ ) = G J G

(5.99)

J∗

and C are bounded, provided that Note that by (2.22) and (5.96)–(5.99) their arguments are bounded. Then the control objective can be stated as follows. For the rigid spacecraft attitude system given by (5.94) and (5.95), determine a control law τ such that 1) the closed-loop attitude system is globally stable in that all the internal signal variables are bounded and 2) the attitude orientation converges to a small set containing the origin, that is, σ (t) ≤ ε ∗ for t ≥ T (ε ∗ ). Furthermore, this objective is to be met with angular velocity eliminated (i.e., ω is not required) and in the presence of 1) uncertain inertia parameters and unknown bounded external disturbances; 2) partial loss of actuator effectiveness fault E(t); and 3) actuator saturation |τ i | ≤ umax (i = 1, 2, 3), where the positive constant umax is the maximum control torque that each actuator can generate.

5.4.2 Velocity filter design Because direct or accurate measurement of angular velocity may be unavailable, we first introduce a passivity filter whose structure is motivated by the Lyapunov-like stability analysis to generate a velocity-related signal from attitude measurement only. The filter is defined as the following first-order dynamics: p˙ = −l1 p + l1 l2 σ ,

(5.100)

156 Fault-Tolerant Attitude Control of Spacecraft

where l1 , l2 ∈ R+ are filter gains. The output of the filter is defined as σ f = p − l2 σ .

(5.101)

To provide some insight on the above-designed filter, we now derive the dynamics of the filter output σ f . From (5.101) the time derivative of (5.102) can be calculated as σ˙ f = p˙ − l2 σ˙ = −l1 σ f − l2 σ˙ .

(5.102)

5.4.3 Fault-tolerant attitude stabilization control design In this section, we first show that when all the actuators are fault-free (i.e., E(t) ≡ I 3 for all t ≥ 0), the states σ and ω can be guaranteed to be stable through a novel control design with angular velocity ω eliminated. Then, based on the nominal controller, we design an auxiliary controller and add it to the nominal controller to compensate the considered actuator faults.

5.4.3.1 Nominal control law design The external disturbances d(t) incorporate gravitational perturbations, atmospheric drag, solar radiation pressure forces, magnetic forces, and aerodynamic drags. Those disturbances are bounded in practical aerospace engineering. Thus d(t) ∈ L∞ [0, ∞) by its definition in (5.96)–(5.99). Theorem 5.4. Consider the nominal attitude system governed by (5.94) and (2.22). The nominal control law is implemented by    −1 K Tanh(σ ) − K Tanh(χ) p I  τ = G(σ ) , (5.103)   +Kd Tanh σ f + KW σ f  t Tanh(σ (s))ds, (5.104) χ = γ 2σ − γ 0

where Kp , KI , Kd , and KW ∈ R+ are control gains. Suppose that the control parameters are appropriately chosen to satisfy Kp 1 − 2 J max > 0, 4 γ

3l2 KW J max − 4l1 γ

2γ l1 − 1 > 0, l2 3KW 1 > 0, − 4l1 l2 4η √  3KW 2 1 3C max − − η− > 0, 2l1 γ 4β1

(5.105) (5.106) (5.107) (5.108)

Fault-tolerant velocity-free attitude control Chapter | 5

1 1 l2 KW 1 KP − Kd − − 2 > 0, γ 4γ l1 γ 2 4γ β2

157

(5.109)

where β1 , β2 , and η ∈ R+ are prescribed constants specified by the designer. Then the nominal controller (5.103) guarantees that the attitude orientation σ is UUB. Proof. In the proof, we use elements of Lyapunov stability theory: we first design a candidate Lyapunov function, which is globally positive and radically unbounded in the states; then we prove that the time derivative of this Lyapunov candidate is negative definite along trajectories generated by (5.94) and (2.22); finally, we invoke Barbalat’s lemma to show that the closed-loop attitude system is stable. a) Lyapunov Function Candidate: Consider the Lyapunov candidate function of the form 3   1 Kd *  1 ln cosh σ fi V (t) = σ˙  J ∗ σ˙ − Tanh(σ ) J ∗ σ˙ + 2 γ l2 3 *

+ Kp

ln [cosh (σ i )] +

i=1

1 γ2



i=1

Tanh(χ)

s T KI cosh2 (χ)ds

(5.110)

0

KW  + σ σf , 2l2 f where 

Tanh(χ)



s KI cosh (χ)ds = 2

0

3  *

Tanh(χi )

KI cosh2 (χi ) si dsi .

(5.111)

i=1 0

Because KI and cosh2 (χi ) are positive, it follows that 

Tanh(χ)

s  KI cosh2 (χ)ds > 0

∀χ = 0 ∈ R3 .

(5.112)

0

Before proceeding with the time derivative computation of the Lyapunov function, we first show its positivity. From (5.105)–(5.109) we get 3 Kp * 1  ∗ 1 σ˙ J σ˙ − Tanh(σ ) J ∗ σ˙ + ln [cosh (σ i )] 4 γ 2 i=1 + + , , 2 2 1 ∗ σ˙ − Tanh(σ ) J σ˙ − Tanh(σ ) = 4 γ γ 3 Kp * 1  ∗ ln [cosh (σ i )] − 2 Tanh(σ ) J Tanh(σ ) + 2 γ i=1

(5.113)

158 Fault-Tolerant Attitude Control of Spacecraft

≥

3 Kp * 1 ln [cosh (σ i )] − 2 Tanh(σ ) J ∗ Tanh(σ ) 2 γ i=1

≥

3 + * i=1

, Kp J max − 2 Tanh2 (σ i ) > 0. 4 γ

In view of (5.112) and (5.113), rearranging (5.110) yields 3   KW  Kd *  1 ln cosh σ fi + σ σf V ≥ σ˙  J ∗ σ˙ + 4 l2 2l2 f i=1

 Tanh(χ) 3 Kp * 1 + ln [cosh (σ i )] + 2 s  KI cosh2 (χ)ds 2 γ 0 i=1

(5.114)

3 3   Kp * 1 Kd * ≥ σ˙  J ∗ σ˙ + Tanh2 σ fi + Tanh2 (σ i ) 4 2l2 4 i=1 i=1  Tanh(χ) KW  1 + σ σf + 2 s  KI cosh2 (χ)ds > 0 2l2 f γ 0

  for χ  σ˙  , σ  , σ  = 0. Hence we can conclude that the selected Lyapunov f function candidate V is continuously differentiable, radially unbounded, and positive definite in the states χ, σ˙ , σ , and σ f b) Stability Analysis: The time derivative of V can be calculated as 1 1 ∗ ∗ V˙ = σ˙  J˙ σ˙ + σ˙  J ∗ σ¨ − Tanh(σ ) J˙ σ˙ 2 γ  1 1 Sech2 (σ )σ˙ J ∗ σ˙ − Tanh(σ ) J ∗ σ¨ − γ γ   Kd  KW  + σ σ˙ f σ˙ Tanh σ f + Kp σ˙  Tanh(σ ) + l2 f l2 f KI d[Tanh(χ)] . + 2 Tanh(χ) cosh2 (χ) dt γ

(5.115)

Substituting (5.103) into (5.95) and using E(t) ≡ I 3 for the actuator fault-free case, we establish J ∗ (σ )σ¨ + C(σ , σ˙ )σ˙ =Kp Tanh(σ ) − KI Tanh(χ)   ¯ + Kd Tanh σ f + KW σ f + d.

(5.116)

Fault-tolerant velocity-free attitude control Chapter | 5

159

Through laborious yet relatively straightforward algebra followed by the application of (5.102) and (5.104), from (5.116) we get σ˙  J ∗ σ¨ = − σ˙  C(σ , σ˙ )σ˙ + Kp σ˙  Tanh(σ ) − KI σ˙  Tanh(χ)   + Kd σ˙  Tanh σ f + KW σ˙  σ f + σ˙  d¯ Tanh(χ) cosh2 (χ) d(Tanh(χ)) dt

= γ 2 Tanh(χ) σ˙ − γ Tanh(χ) Tanh(σ )        ˙ = − l σ Tanh σ f . Tanh σ σ + l σ˙  f 1 2 f f

(5.117)

(5.118) (5.119)

By the inequality   2   1 (5.120) Tanh(σ ) Tanh σ f ≤ Tanh(σ )2 + Tanh σ f 4 √ imposing the bound Tanh(σ ) ≤ 3, we simplify the time derivative of V in (5.115) as +   , 1  2 ˙ V =− Tanh(σ ) C(σ , σ˙ ) + Sech (σ )σ˙ J ∗ σ˙ γ   1 Kd − Tanh(σ ) d¯ − Tanh(σ ) Tanh σ f γ γ  Kp KW  − Tanh(σ ) Tanh(σ ) + σ˙  d¯ + KW σ˙  + σ˙ f σ f γ l2   l KW K d 1  Tanh(σ ) σ f − − σ Tanh σ f (5.121) γ l2 f   Kp Kd 1 √ ≤ − Tanh(σ )2 3C max + J max σ˙ 2 − γ γ 4γ    2 1 1 2γ l1 Kd  − Tanh(σ ) d − − Kd Tanh σ f γ γ l2  KW KW    ¯ σ f + σ˙  d. Tanh(σ ) σ f + KW σ˙ + − σ˙ γ l2 f In particular, by the Young inequality it follows that 1 σ˙ f 2 + ησ˙ 2 , 4η 1  1  σ˙  d¯ ≤ σ˙ σ˙ + β1 d¯ d¯ − Tanh(σ ) d¯ 4β1 γ 1 ¯  ≤ 2 Tanh(σ ) Tanh(σ ) + β2 d¯ d. 4γ β2

σ˙  f σ˙ ≤

(5.122) (5.123) (5.124)

160 Fault-Tolerant Attitude Control of Spacecraft

Hence, for the last three items on the right-hand side of (5.121), we can find that  KW KW    − σf Tanh(σ ) σ f + KW σ˙ + σ˙ γ l2 f l1 KW σ f 2 + l2 KW Tanh(σ )2 + KW σ˙  σ f ≤ 4l2 l1 γ 2  KW  l1 σ f + l2 σ˙ σ f − l2 3l1 KW  l2 KW =− σf σf + Tanh(σ )2 4l2 l1 γ 2 (5.125) 3KW σ f 2 − 3l2 KW σ˙ 2 − 3KW σ˙  σ˙ =− 4l1 l2 4l1 2l1 f l2 KW + Tanh(σ )2 l1 γ 2   3l2 Kw 3Kw 2 3Kw 1 2 σ˙ f 2 ≤− −( ) η σ˙  − − 4l1 2l1 4l1 l2 4η l2 Kw + Tanh(σ )2 . l1 γ 2 By using (5.105)–(5.133) and (5.123)–(5.125) we can further establish the total derivative of V along the closed-loop trajectories of (5.121) as   Kp Kd 1 √ V˙ ≤ − Tanh(σ )2 3C max + J max σ˙ 2 − γ γ 4γ    l2 KW 1 2γ l1 Kd 2 Tanh σ f 2 + Tanh(σ ) − − K d 2 2γ l2 l1 γ  2   3l2 KW 3KW 1 − − η σ˙ 2 + 2 Tanh(σ ) Tanh(σ ) 4l1 2l1 4γ β2  1 3KW σ˙ f + 1 σ˙  σ˙ + β1 d¯  d¯ + β2 d¯  d¯ − − 4l1 l2 4η 4β1  Kp Kd 1 l2 KW 2 ¯ − 2 Tanh(σ )2 (5.126) = (β1 + β2 ) d − − − γ 4γ l1 γ 2 4γ β2

  −

1 γ



−



m3



   2 2γ l1 Kd 3KW 1 σ˙ f 2 − Kd Tanh σ f − − l2 4l l 4η

1 2    m4

3l2 Kw J max − − 4l1 γ





m2

 √ 3Kw 1 3Cmax − η − σ˙ 2 2l1 γ 4β1   2

m1

Fault-tolerant velocity-free attitude control Chapter | 5

161

¯ 2 − m3 Tanh(σ )2 . ≤ (β1 + β2 ) d ¯ ≤ D0 , and thus V˙ is Since d¯ ∈ L∞ [0, ∞), there exists D0 > 0 such that d bounded: V˙ ≤ (β1 + β2 ) D02 − m3 Tanh(σ )2 .

(5.127)

From (5.127) we see that V˙ < 0 when σ is outside the set   (5.128) D = σ | Tanh(σ ) ≤ ε ∗ , √ where ε ∗ = D0 (β1 + β2 ) /m3 . Eq. (5.128) implies that V (t) decreases outside the set D. Hence all the signals in the closed-loop system are bounded. Moreover, we can choose ε0 small enough to guarantee that lim σ  ≈ lim Tanh(σ ) ∈ D.

t→∞

t→∞

(5.129)

From (5.129) we conclude that there exists T (ε ∗ ) > 0 such that σ (t) ≤ ε ∗ for t ≥ T (ε ∗ ). This shows that the attitude σ is UUB by Definition 2. The proof is completed. Remark 5.7. The control law (5.103) is independent of the precise knowledge of inertia matrix J (particularly, time varying and uncertain due to onboard payload motion, vibration of flexible appendages, or fuel consumption). Although the implementation of the controller needs upper bounds on J and C, which are used to determine the control gains in (5.103), those two bounds can be approximately estimated or be chosen larger before launch. Thus all the control gains can be determined. Therefore, from the standpoint of uncertainties and external disturbances rejection, the derived control law has great stability robustness.

5.4.3.2 Fault-tolerant control law design So far, all the actuators are assumed to be healthy. This is not a realistic assumption in practice. Due to aging of components, in general, actuator faults occur, especially, partial loss of effectiveness fault. This type of fault may deteriorate attitude control performance and even result in system instability. Taking partial loss of effectiveness fault into consideration, we propose the following fault-tolerant control law to perform attitude stabilization maneuver: τ = τN + τF,

(5.130)

where τ N is the nominal control (5.103) for the normal system, and τ F is the fault-tolerant control part designed and added to compensate possible actuator fault effects on the system, and it is given by  τ F = −sgn

  1 (1 − e0 ) κ  τ N  . σ˙ + Tanh(σ ) G γ e0

(5.131)

162 Fault-Tolerant Attitude Control of Spacecraft

The stability analysis of the closed-loop system under the effect of the faulttolerant control (5.130) can be stated as follows. Theorem 5.5. Consider the faulty attitude system governed by (5.94) and (2.22) under partial loss of actuator fault. With the application of the control law (5.130), suppose that the design parameters are chosen to hold (5.105)–(5.109). Then the attitude orientation σ is UUB if the nominal control (5.103) guarantees the nominal attitude system UUB. Proof. When partial loss of effectiveness fault E(t) occurs, from (5.95) and (5.130) we have that the effectiveness J ∗ (σ )σ¨ + C(σ , σ˙ )σ˙ =Kp Tanh(σ ) − KI Tanh(χ)   + KW σ f + d¯ + Kd Tanh σ f 

(5.132)



− G (I 3 − E(t)) τ + G τ F . Then we have σ˙  J ∗ σ¨ = − σ˙  C(σ , σ˙ )σ˙ + Kp σ˙  Tanh(σ ) − KI   × σ˙  Tanh(χ) + Kd σ˙  Tanh σ f + KW σ˙  σ f + σ˙  d¯ + σ˙  G τ F

(5.133)

− σ˙  G (I 3 − E(t)) τ , 1 1 Tanh(σ ) J ∗ σ¨ = Tanh(σ ) γ γ  × −C σ˙ + Kp Tanh(σ ) + d¯    −KI Tanh(χ) + Kd Tanh σ f + KW σ f 1 − Tanh(σ ) G (I 3 − E(t)) τ γ 1 + Tanh(σ ) G τ F . γ

(5.134)

With the same Lyapunov function candidate (5.4), substituting (5.133) and (5.134) into (5.115) and following the same lines as in (5.121)–(5.125) in Appendix result in ¯ 2 − m1 ||σ˙ V˙ ≤ (β1 + β2 ) ||d|| 2 − m2 σ˙ f ||2    2 1   − m4 ||Tanh σ f || + σ˙ + Tanh(σ ) G τ F γ  1 − σ˙  + Tanh(σ ) (I 3 − E(t)) τ − m3 ||Tanh(σ )2 . γ

(5.135)

Fault-tolerant velocity-free attitude control Chapter | 5

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Due to inequalities (5.105)–(5.109), from (5.130) and (5.131) we obtain that  1 2   ¯ ˙ V ≤ (β1 + β2 ) d + σ˙ + Tanh(σ ) E(t)τ F γ  1 − σ˙  + Tanh(σ ) (I 3 − E(t)) τ N − m3 Tanh(σ )2 γ 1 2    - (1 − e0 ) κ ¯ ≤ (β1 + β2 ) d − | σ˙ + Tanh(σ ) G - |τ N | | γ e0 - - (5.136) -1 + -- σ˙  + Tanh(σ ) G (1 − e0 )-- -τ N −m3 ||Tanh(σ )2 γ ¯ 2 − m3 Tanh(σ )2 = (β1 + β2 ) d - -  1  − - σ˙ + Tanh(σ ) G - (1 − e0 ) (1 − κ)||τ N || γ 1 ≤ (β1 + β2 ) ||d||2 − m3 ||Tanh(σ )2 , from which establish the result using the same argument as in the proof of Theorem 5.4. This completes the proof. Remark 5.8. The controller is implemented with digital computer in practical aerospace engineering. The value of σ f at the time (k + 1)T can be approximately estimated by using the previous one-step information from attitude sensors as σ f ((k + 1)T ) − σ f (kT ) , (5.137) σ˙ f (kT ) = T where T is the control update period. By using (5.101) we can calculate the value of σ˙ f by σ˙ f (kT ) =

p((k + 1)T ) − p(kT ) T σ ((k + 1)T ) − σ (kT ) . − l2 T

(5.138)

Accordingly, from (5.102) we obtain that 1 l1 σ˙ = − σ˙ f − σ f . l2 l2

(5.139)

Hence σ˙ can be numerically derived during the implementation of the control law (5.130), because σ˙ f is obtained from (5.138), and σ˙ f is given by the attitude sensors. We can summarize that (5.131) is independent on angular velocity measurements, although σ˙ is involved. On the other hand, τ N is also angular velocity-free. As a result, the proposed fault-tolerant control (5.130) can be implemented without the need of any rate sensor to measure the angular velocity.

164 Fault-Tolerant Attitude Control of Spacecraft

5.4.3.3 Analysis of the upper bound of the control effort From (5.100) we find that the state of the velocity filter is bounded by p˙ ≤ −l1 p + l1 l2 |σ |,

p(0) ≥ 0.

(5.140)

According to the analysis in the proof of Theorem 5.5, we have σ  ∈ L∞ [0, ∞). Hence inequality (5.140) ensures that p(t) ≤ p(0)e−l1 t + l2 |σ |2∞ .

(5.141)

The inequality |σ i | ≤ 1 (i = 1, 2, 3) holds by Remark 5.7. Thus √ by choosing p(0) = 0 we can evaluate p(t) ≤ l2 , which leads to p(t) ≤ 3l2 . Direct calculation shows that the matrix T (σ ) in (5.95) is such that 2 1 + σ σ T (σ )T (σ ) = I 3, 4  −1 T G(σ )T = T (σ ) = 1 + σ σ ≤ 1 . 4 2 



(5.142) (5.143)

On the basis of the preceding analysis, we summarize the upper bound of the proposed fault-tolerant control law (5.130) in the following theorem. Theorem 5.6. Consider the developed fault-tolerant control law (5.130). Choose the control gains satisfying (5.105)–(5.109) and the following inequality:   √  1 (1 − e0 ) κ √  1+ 3 Kp + KI + Kd + ( 3 + 1)KW l2 ≤ μmax . 2 e0 (5.144) Then the control output of each actuator is rigorously bounded by the actuator saturation value. Proof. Together with (5.144), the nominal control law (5.103) satisfies   1 √  τ  ≤ (5.145) 3 Kp + KI + Kd + KW σ f . 2 Next, by (5.101) σ f is bounded by √ σ f ≤ p + l2 σ  ≤ ( 3 + 1)l2 .

(5.146)

It follows that (1 − e0 ) κ |τ i | ≤ τ N  + |τ F i | ≤ τ N  + τ N  e0   1 (1 − e0 ) κ √  1+ ≤ 3 Kp + KI + Kd 2 e0  √ +( 3 + 1)KW l2

(5.147)

Fault-tolerant velocity-free attitude control Chapter | 5

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for i = 1, 2, 3, where τ F i is the ith argument of τ F . Thus by (5.133) and (5.145) we can demonstrate that |τ i | ≤ umax .

(5.148)

Hence the actuator constraints can be satisfied. From Theorem 5.5 we know that the smaller e0 in (5.131) is selected, the more severe actuator faults the controller (5.130) can tolerate, which results in much more fault-tolerant capability. In Theorem 5.6, it is proved that once the value of e0 is determined (no matter how small the value of e0 > 0 is), the control τ with the control gains chosen from (5.105)–(5.109) and (5.142) can always guarantee that the closed-loop system is UUB. We further theoretically analyze that the attitude orientation is guaranteed by the bound σ (t) ≤ ε ∗ for t ≥ T (ε ∗ ), and the control effort is such that |τi | ≤ umax , i = 1, 2, 3. Therefore, when implementing the fault-tolerant controller (5.130), the selection of the value e0 depends on the tolerable level of actuator faults, which is imposed on the spacecraft by the designers. For example, if the spacecraft needs to tolerate 90% loss of control, then we should choose 0 < e0 < 0.1. Moreover, the following two results should be pointed out.

5.4.3.4 Small value of e0 would not lead to “weak” control power A conservative selection of e0 may indeed lead to small control gains. Then a “weak” nominal control effort τ N may follow. However, as shown in (5.130), small e0 will result in large fault-tolerant control effort τ F . This is due to the term (1 − e0 ) /e0 in τ F . As a result, the total control power τ would not be weak, and the control objective can still be achieved. For instance, assume that τ N = [1, 1, 1] × 10−3 N · m at t = 50 s when e0 = 0.01 and κ = 1.001 are chosen. We will have τ F = [0.1683, 0.1683, 0.1683] N · m if the vector sgn(·) in (5.130) is equal to [−1, −1, −1]. This leads to the total control effort τ = [0.1673, 0.1673, 0.1673] N · m. This torque is not a weak control power in practical aerospace engineering. 5.4.3.5 Unacceptably long time would not be taken to stabilize attitude Because the conservative selection of control gains may not lead to weak control, as shown in the above analysis, the closed-loop system would be stabilized within a sufficiently reasonable amount of time. This is verified by using a numerical example, as presented in Section V. On the other hand, as shown in the proof of Theorem 5.5, the time T (ε ∗ ) within which the attitude is governed to be σ (t) ≤ ε ∗ depends on the attitude pointing accuracy ε ∗ . The higher pointing accuracy, that is, the smaller ε ∗ , the much more time T (ε ∗ ) will be spent to ensure the stabilization of the attitude system. Further, as shown in (5.128), we √ have ε ∗ = D0 (β1 + β2 ) /m3 . When small control gains result in small value of m3 , choosing small β1 and β2 can lead to small ε ∗ , whereas the selection of

166 Fault-Tolerant Attitude Control of Spacecraft

large β1 and β2 will introduce large ε ∗ . As a consequence, the attitude pointing accuracy ε ∗ is adjustable, and then the time T (ε ∗ ) is selectable for the designer. Consequently, the proposed control methodology will be practical and of real interest and used in real-life scenarios. Remark 5.9. A practical problem, namely, the chattering effect that may be induced by the use of the discontinuity function sgn(·) in (5.130) should be considered since it is impossible to switch the control at infinite time. This problem is practically undesirable because it may excite the neglected high-frequency dynamics. One approach to reduce the chattering is approximating the function sgn(·) by using a continuous function. Following this idea, a saturation function is used to approximate sgn(·), and thus (5.130) is modified as  τ F = −Sat

  1   (1 − e0 ) κ τ N  , σ˙ + Tanh(σ ) G γ e0

κ > 1,

(5.149) where we have slightly abused the notation by using Sat(·) to stand for both scalar-valued and vector-valued saturation functions. For x = [x1 , x2 , . . . , xn ] ∈ Rn , the argument of Sat(·) is defined by Sat(x)i = min {xi /ε, sgn (xi )}, i = 1, 2, 3, where ε is a small positive constant. Remark 5.10. The definition of saturation function Sat(·) leads to |Sat(x)i | ≤ 1, i = 1, 2, . . . , n. Hence, the modified fault-tolerant control power τ F given in (5.149) is still guaranteed to be such that |τ F i | ≤ (1 − e0 ) κ||τ N ||/e0 . With the control parameters chosen in (5.144), Theorem 5.6 is still valid, and thus the actuator upper bound condition is still ensured. Remark 5.11. Summarizing the analysis in Theorems 5.4 and 5.6, the filter and the control parameters can be chosen according to the following procedures. Step 1: Choose e0 > 0 as small as possible. The smaller the e0 , the much more severe partial loss of actuator effectiveness fault is tolerated. √Step  2: Choose κ > 1√as small as possible and calculate the value of π 3 Kp + KI + Kd + ( 3 + 1)KW l2 by using (5.144) Step 3: Estimate the values of J max and C max , choose large Step γ and small βi , i = 1, 2, 3. Step 4: Choose large η and l1 / l2 such that (5.105)–(5.109) are satisfied. Step 5: Choose the control gain KW such that (5.105)–(5.109) are satisfied. Step 6: Choose the control gains Kp and Kd such that (5.105)–(5.109) are satisfied. Step 7: Choose the control gain KI such that (5.144) is satisfied. Step 8: Check whether the chosen parameters satisfy the value of π in Step 2 or not. If the value of π is not met, then repeat Steps 3–7.

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TABLE 5.5 Main parameters of an on-orbit rigid spacecraft. Mass (kg)

75.6

Inertia moments (kg m2 ) Principal moments of inertia

J1 = 20, J2 = 20, J3 = 30

Products of inertia

Can be neglected

Orbit: Type

Circular

Altitude (km)

672

Inclination (degree)

97.4

Right ascension of the ascending node

10.30 am

Attitude control type

Three-axis control by three reaction wheels

Maximum torque (Nm) of the reaction wheel

umax = 0.6

TABLE 5.6 Controller parameters chosen for numerical analysis. Control schemes

Control gains KP = 0.15, KI = 0.01,

VFFTC

Kd = 0.1, KW = 0.0005, l1 = 0.5, l2 = 30, κ = 1.001, γ = 100

UQOFC

α1 = 0.15, α2 = 0.35, 1 = diag{0.25, 0.25, 0.25}

5.4.4 Numerical example To demonstrate the effectiveness of the proposed fault-tolerant control scheme, we numerically simulated a rigid spacecraft with the orbital parameters shown in Table 5.5 using the set of governing equations of motion (5.94) and (5.95) in conjunction with the control law (5.130). A time-varying moment inertia matrix as discussed is incorporated into the model J = 1 + e−0.1t + 2u(t − 10) − 4u(t − 20)diag{3, 2, 1},

(5.150)

where u(·) is defined as u(t ≥ 0) = 1 and u(t < 0) = 0. For comparison, we carried out the proposed fault-tolerant angular velocityfree control law (5.130) (here it is called VFFTC) and the unit quaternion output feedback control (UQOFC) developed. Both controllers are first compared for a fault and external disturbance-free case, and then for a faulty case in the presence of disturbances, the control parameters were chosen according to Remark 10 by trial-and-error until a good stabilization performance was obtained. The control gains are given in Table 5.6.

168 Fault-Tolerant Attitude Control of Spacecraft

At time t = 0, the attitude orientation is such that σ (0) = [0.3202, −0.4850, 0.3899] with zero initial body angular velocity. For the consideration of engineering application, attitude sensor noises   σ m,i (t) = σ i (t) + N 0, σ p , i = 1, 2, 3, (5.151) where σ m,i (t) is the measured from the attitude sensor, σ i (t) is the real   attitude attitude orientation, and N 0, σ p denotes a zero-mean Gaussian white noise with variance σ p .

5.4.4.1 Response with fault-free and disturbance-free case In this case, we simulated a relative ideal situation in which not only no actuator fault occurs but also there is not any external disturbance acting on the spacecraft. We first present the simulation results when applying VFFTC. We show in Figs. 5.19 and 5.20 that the velocity-free controller managed to perform attitude stabilization maneuver with good control performance. The attitude orientation is stabilized within 100 s with high pointing accuracy even in the presence of uncertain inertia parameters (5.150). Moreover, the results illustrated in Fig. 5.19 (solid line) further verify the conclusion that the proposed control law (5.130) can achieve the attitude control in the absence of actuator faults.

FIGURE 5.19 Spacecraft attitude orientation with VFFTC and UQOFC in the absence of faults and disturbances.

Although the application of UQOFC to the spacecraft can achieve almost the same attitude control accuracy and stability as VFFTC, as shown in Figs. 5.19 and 5.20 (dashed line), it requires 180 s to force the spacecraft attitude to a satisfied resolution. Much higher overshoot resulted with UQOFC than that by using VFFTC. This is due to the fact that the proposed control scheme can decrease the overshoot by tuning the proportional, integral, and derivative gains KP , KI , and Kd . It is also interesting to note that both controllers can protect the control

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FIGURE 5.20 Spacecraft angular velocity with VFFTC and UQOFC in the absence of faults and disturbances.

torque from actuator saturation magnitude, as we can see in Fig. 5.21 (solid and dashed lines). However, compared with the control power in Fig. 5.23, a larger control effort of VFFTC is observed than that of UQOFC. Indeed, this is due to the fact that τ f is always activated whether actuator fault occurs or not.

FIGURE 5.21 Time response of τ (t) with VFFTC and UQOFC in the absence of faults and disturbances.

5.4.4.2 Response with actuator fault and disturbances case On-orbit spacecraft is inevitably under the effect of external disturbances. Hence in this section, we impose a large constant external disturbance torque specified by d(t) = [0.02 − 0.010.01] . Moreover, we consider the following partial loss of actuator effectiveness fault.

170 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 5.22 Spacecraft attitude orientation with VFFTC and UQOFC in the presence of faults and disturbances.

1) Actuator Fault Scenario: The fault scenarios occur under the situation where the reaction wheel mounted in line with the roll axes decreases 50% of its normal value after 5 s; 2) the actuator mounted in line with the pitch axes loses its power of 40% in 10 s; 3) the reaction wheel mounted in line with the yaw axes undergoes 50% loss of effectiveness in 15 s. These are fairly severe faults, which, if not compensated for, will cause overall attitude system instability, as will be discussed later. When the VFFTC is implemented to the attitude system, the time responses of attitude orientation and angular velocity are presented in Fig. 5.20 (solid line) and Fig. 5.21 (solid line), respectively. The driving torque is shown in Fig. 5.22 (solid line). As expected, we clearly see that the proposed fault-tolerant control scheme managed to compensate the partial loss of effectiveness fault. High attitude pointing accuracy and attitude stability are still guaranteed without angular velocity measurements. As shown in Fig. 5.22 (solid line), the control torque of each reaction wheel is still within its maximum allowable limit even in the presence of external disturbances and uncertain inertia parameters. This is achieved by introducing the fault-tolerant part τ f in (5.131) Due to the limited control power of each reaction wheel, it demands longer time to stabilize the attitude in the presence of actuator fault. As we can see in Fig. 5.22 (solid line), the whole attitude stabilization maneuver is performed in nearly 500 s. The application of UQOFC leads to the attitude orientation and angular velocity shown in Fig. 5.22 and Fig. 5.23 (dashed line). As pointed out, that this control law can stabilize the attitude with angular velocity eliminated only in the absence of external disturbance and actuator fault. Therefore, when the dis-

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FIGURE 5.23 Spacecraft angular velocity with VFFTC (solid line) and UQOFC (dashed line) in the presence of faults and disturbances.

turbance d(t) and actuator fault are introduced in the system, UQOFC failed to achieve attitude stabilization control, as we can see in Fig. 5.24 (dashed line), although a relative higher slew rate accuracy and limited control effort were still observed in Figs. 5.23 and 5.24 (dashed line), respectively. On the basis of the above illustrated simulation results, the steady attitude stability and control accuracy under VFFTC and UQOFC are summarized in Table 5.7. 1) In the absence of actuator fault and external disturbance, both the proposed methodology and the scheme developed in (Tayebi, 2008) can accomplish the attitude stabilization maneuver without the measurements of angular velocity. Almost the same attitude pointing accuracy and attitude stability are achieved. However, the developed controller demands less time to govern the attitude; a comparison with (Tayebi, 2008) shows that our solution provides a faster response. 2) When actuator fault and external disturbances are considered, our presented control law can successfully perform the attitude stabilization maneuver, whereas controller (Tayebi, 2008) failed to stabilize the attitude, although a high slew rate accuracy is achieved. 3) When the developed strategy is implemented into the spacecraft, comparison with the results of the actuator normal case and the fault case shows that the slew rate accuracy is met with the same order of magnitude, and only the attitude control precision decreases one order of magnitude in the presence of actuator fault.

172 Fault-Tolerant Attitude Control of Spacecraft

TABLE 5.7 Performance summary under different control schemes. Control performance Roll Attitude Control Accuracy

Pitch Yaw

Slew Rate Accuracy (rad/s)

Roll Pitch Yaw

Attitude Stabilization time (s)

Actuator status healthy

Control schemes VFFTC

UQOFC

Normal

± 2.0 × 10−6

± 3.0 × 10−6

Fault

± 2.0 × 10−6

± 0.2

Normal

± 3.0 × 10−6

± 1.0 × 10−6

Fault

± 4.0 × 10−6

± 0.008

Normal

± 2.0 × 10−6

± 2.0 × 10−6

Fault

± 6.0 × 10−6

± 0.008

Normal

± 4.0 × 10−6

± 1.5 × 10−6

Fault

± 3.0 × 10−6

± 4.0 × 10−6

Normal

± 4.0 × 10−6

± 2.0 × 10−6

Fault

± 2.5 × 10−6

± 7.0 × 10−6

Normal

± 2.5 × 10−6

± 2.0 × 10−6

Fault

± 1.0 × 10−6

± 8.0 × 10−6

Normal

100

180

Fault

500

Infinity

FIGURE 5.24 Time response with VFFTC (solid line) and UQOFC (dashed line) in the presence of faults and disturbances.

5.5 Summary To address the attitude control issue of spacecraft without angular velocity, we introduced two main control strategies (namely, observer-based method and

Fault-tolerant velocity-free attitude control Chapter | 5

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filter-based method) to guarantee satisfying attitude maneuver performance in different practical situations. Firstly, we proposed a fault-tolerant control strategy for attitude tracking without angular velocity feedback based on a slidingmode observer. The sliding-mode observer is used to estimate the unmeasurable angular velocity of the spacecraft even in the presence of limited control input, partial actuator failures, and saturations and external disturbance. Note that the observer formulation needs a full knowledge of spacecraft inertia matrix. To release this specific restriction, we provided a new velocity-free control scheme with designed velocity filter to achieve attitude stabilization maneuver in the presence of partial loss of effectiveness fault, uncertainties in the inertia parameters, external disturbances, and actuator saturation. As some of future works, extension to the fault-tolerant attitude control with finite-time convergence should be investigated to perform time critical aerospace mission, in which the attitude maneuver is finished in finite time.

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Chapter 6

Fault-tolerant finite-time attitude-tracking control 6.1 Introduction As attitude tracking control is one of the fundamental maneuvers a spacecraft needs to perform during its operation, the past three decades have witnessed several important developments in the design of feedback control laws for spacecraft maneuver tracking objectives. However, there still remain certain open problems in this field that are of great theoretical and practical interest. In particular, from the standpoint of external disturbances rejection, there currently exists no unified framework for designing simple control structures. The main hindrance stems from the fact that the governing differential equations for the kinematics and dynamics of spacecraft attitude motion are nonlinear in nature. The attitude tracking problem is further complicated by the uncertainty of the spacecraft mass and inertia properties due to onboard payload motion and fuel consumptions. There have been several important developments in the design of feedback control laws for attitude tracking Costic et al. (2001); Sharma and Tewari (2004). Sliding-mode control (SMC) Edwards and Spurgeon (1998) is an effective approach for uncertain systems with highly coupled nonlinear dynamics. Some of its advantages are rapid response and insensitivity to uncertain parameters and disturbances. The first attempt of using SMC to achieve large-angle attitude maneuver of spacecraft was made in Dwyer and Sira-Ramirez (1988) and further pursued in Chen and Lo (1993). Another SMC-based attitude-tracking controller for spacecrafts with thrusters was presented in Yeh (2010). Uncertain inertia and disturbance were addressed by using adaptive technique. The attitude-tracking problem with uncertain inertia was addressed in Pukdeboon et al. (2010) by presenting a higher-order SMC law. An adaptive SMC law was synthesized in Xia et al. (2010) to accomplish attitude tracking in the presence of uncertain inertia. An SMC scheme was reported in Zhu et al. (2011a) to address attitude stabilization. Actuator saturation and external disturbance were considered, but the knowledge of inertia parameters was not needed to implement the controller. Extensive studies have been carried out on spacecraft attitude control to possess fault-tolerant capability Tafazoli and Khorasani (2006); Cai et al. (2008); Jin et al. (2008); Liang et al. (2007). Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00016-2 Copyright © 2021 Elsevier Inc. All rights reserved.

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176 Fault-Tolerant Attitude Control of Spacecraft

With an effort to tackle the aforementioned issues, we will investigate the feasibility of finite-time tracking of the desired attitude trajectory in the presence of actuator faults and misalignment, external disturbances, and constant but unknown inertia parameters. In practice, when the Earth-observing satellite actuators failure to work as normal, the observed ground target might be lost if the attitude controller is unable to track the moving target in finite time. Therefore for such a solution, we propose an adaptive nonlinear control technique-based SMC approach, which meets the finite-time requirement. Finally, we evaluate the attitude-tracking performance through numerical examples.

6.2 Attitude tracking control with actuator misalignment and fault 6.2.1 Problem statement 6.2.1.1 Attitude tracking error dynamics The error kinematics of q e are given by (2.19), and the corresponding rotation matrix R(q e ) ∈ R3×3 is defined by R(q e ) = R(q)R(q d ) . The desired angular ˙ e ) = −ω× velocity is denoted by ωd ∈ R3 . Note that R(q e R(q e ) with velocity error ωe = ω − R(q e )ωd , and the open-loop attitude tracking error dynamics is described by ˙ d ) − ω× J ω˙ e = J (ω× e R(q e )ωd − R(q e )ω e J R(q e )ωd − (R(q e )ωd )× (J ωe + J R(q e )ωd ) − ω× e J ωe + u + d,

(6.1)

where u = [u1 , u2 , u3 ]T ∈ R3 is the control torque input, and d = [d1 , d2 , d3 ]T ∈ R3 is the external disturbance. Assumption 6.1. The inertia matrix J consists of two parts, the nominal inertia J 0 and the uncertain inertia J . Although unknown, J remains positive definite and bounded by an unknown scalar γ1 > 0, that is, J  ≤ J  ≤ γ1 . Assumption 6.2. The desired velocity ωd and its first-order derivative are bounded by ωd  ≤ γ2 and ω˙ d  ≤ γ3 , where γ2 > 0 and γ3 > 0 are unknown constants. Assumption 6.3. Disturbance is bounded by an unknown constant dmax > 0 such that d(t) ≤ dmax .

6.2.1.2 Reaction wheel misalignment We consider a spacecraft controlled by three reaction wheels. They are mounted orthogonally, aligned with the axes X, Y , and Z of Fb . The orthogonal configuration of the actuators is never perfect in practice. Due to finite-manufacturing tolerance or warping of spacecraft structure, alignment error may exist. As

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shown in Fig. 6.1, the reaction wheel mounted on the X axis is offset from the nominal direction with constant angles α1 and β1 . The reaction wheels mounted on the X and Y axes are assumed to be tilted away from nominal directions by α2 , β2 , α3 , and β3 . With alignment errors, the real control torque is expressed as the sum of the torque error terms in the body-fixed frame: ⎡

⎤ ⎡ ⎤ ⎡ ⎤ cosα1 cosα2 sinβ2 sinα3 cosβ3 ⎦ + τ3 ⎣ sinα3 sinβ3 ⎦ , u = τ1 ⎣sinα1 cosβ1 ⎦ + τ2 ⎣ cosα2 sinα1 sinβ1 sinα2 cosβ2 cosα3 (6.2) where τi , i ∈ {1, 2, 3}, are control torques generalized by reaction wheels. The misalignment angles αi are small, and βi ∈ [−π, π], i ∈ {1, 2, 3}. We have cos αi ≈ 1 and sin αi ≈ αi . Hence (6.2) can be approximated by u = Dτ = I3 τ + Dτ ,

(6.3)

where τ = [τ1 , τ2 , τ3 ] , the term Dτ denotes the error torque induced by reaction wheel misalignment, and ⎡ ⎤ 0 α2 sinβ2 α3 cosβ3 ⎢ ⎥ D = ⎣α1 cosβ1 (6.4) 0 α3 sinβ3 ⎦ . α1 sinβ1 α2 cosβ2 0

6.2.1.3 Reaction wheel fault As discussed in Murugesan and Goel (1987), reaction wheels are sensitive devices that are vulnerable to four main sources of fault: decreased reaction torque (F1), increased bias torque (F2), continuous generation of reaction torque (F3), and failure to respond to control signals (F4). Because the considered spacecraft has no redundant actuator, if one of the actuators undergoes F3–F4, it will lead to the loss of control in three axes. Those two catastrophic faults are not the point of this investigation. We mainly briefly discuss the faults F1–F2. For each reaction wheel, the faults F1 and F2 can be mathematically modeled by τ i = (1 − pi (t))τ ci + f i , i ∈ {1, 2, 3} ,

(6.5)

where τ ci is the desired torque, f i denotes additive fault, and 0 ≤ pi (t) < 1 is the actuator fault indicator. Taking reaction wheel misalignment (6.3) and fault (6.5) into consideration, the actual control torque u is u = (I3 − E(t))τ c + f + D((I3 − E(t))τ c + f ),

(6.6)

178 Fault-Tolerant Attitude Control of Spacecraft

where τ c = [τc1 , τc2 , τc3 ] , E(t) = diag{p1 (t), p2 (t), p3 (t)} is the actuator effectiveness matrix, and f = [f1 , f2 , f3 ] is the additive torque generated by the fault F2. Assumption 6.4. The bias torque f is bounded by f  ≤ γ4 .

6.2.1.4 Control objective Consider the rigid spacecraft attitude system described by (2.19) and (6.1) with reaction wheel misalignment (6.3), fault (6.5), external disturbance, and uncertain inertia parameters. For any given reference trajectories q d and ωd , design a control law τ c to guarantee that the tracking error q e and ωe are asymptotically stable in finite time. 6.2.2 Attitude-tracking FTC design Considering the presence of uncertain inertia parameters, external disturbances, reaction wheel misalignment, and actuator fault, we recognize that the variable structure control is an effectively robust control scheme for various applications. In this section, to eliminate all the variation influences of a rigid spacecraft during its attitude maneuver for the practical control law design, we develop a variable structure attitude-tracking control scheme, which can compensate the effect of the above-stated issues of the spacecraft in finite time.

6.2.2.1 Switching manifold design In general, there are two major steps when designing a variable structure controller. First, a suitable switching manifold is selected such that certain desired dynamic characteristics can be acquired once the dynamics of the system enters the switching manifold. The second step is designing a control such that it can drive the system trajectories into the synthesized switching manifold in finite time and stay thereafter. Before giving detailed switching manifold design, we first present a Lyapunov-type theorem, which we will utilize in the subsequent switching hyperplane development and analysis. For the attitude-tracking problem, we use the vector containing the attitude-tracking error q ev and angular velocity-tracking error ωe to design a switching manifold S given by S = ωe + α1 q ev + α2 sigm (q ev )

(6.7)

 with sigm (q ev ) = |qe1 |m sig(qe1 ), |qe2 |m sig(qe2 ), |qe3 |m sig(qe3 ) . Now we are ready to summarize the first result of this study. Theorem 6.1. Suppose that an appropriate control effort can be synthesized to guarantee that the kinematics system (2.19) reaches the designed manifold (6.7) and stays thereafter. Then it results in the attitude-tracking error reaching the equilibrium point q e = (1, 0) and angular velocity error ωe = 0 in finite time.

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Remark 6.1. It worth mentioning that there are two equilibrium points for the attitude-tracking system (2.19) and (6.1), q e = (1, 0) and q e = (−1, 0) . These equilibrium points represent the same equilibrium point in the physical space and yield the same attitude rotation matrix R(q e ). However, they denote a two-point set in the quaternion space. Strictly speaking, this topological obstruction does not allow us to claim any global property for the closed-loop attitude system using quaternion coordinates. Note that q e = (−1, 0) is not a stable equilibrium, as proved in Tayebi (2008), and therefore q e = (1, 0) is chosen as the equilibrium point to be stabilized to guarantee the almost global closed-loop stability. Afterward, the global asymptotic stability will be claimed. Proof. Once the control makes an effort to drive the system states into switching manifold and stay on it thereafter, we have S = ωe + α1 q ev + α2 sigm (q ev ) = 0.

(6.8)

Consider a Lyapunov function candidate of the form 2 V1 = q  ev q ev + (1 − qe0 )

(6.9)

with (2.19) and (6.7). The time derivative of the Lyapunov function can be calculated as m ˙ ev = −q  V˙1 = −2(1 − qe0 )q˙ ev + 2q  ev q ev (α1 q ev + α2 sig (q ev ))

2

m+1 ≤ −α1 q ev − α2 q ev

, (6.10)

where we used the inequality

3  i=1

|qei

|m+1

≥

3 

(m+1)/2 |qei

|2

for m ∈ (0, 1).

i=1

2 From (6.10) we clearly see that V˙1 ≤ −α1 q ev , and thus by the Barbalat lemma (Krstic et al., 1995) it is obvious that lim q ev (t) = 0. By the unity t→∞

2 2 = 1 it follows that lim q (t) = ±1. As shown in Reconstraint q ev + qe0 e0 t→∞

mark 6.1, it is not a stable equilibrium, and hence we have qe0 (t) → 1. Then there exists a finite time T¯ such that qe0 (t) ≥ 0 for all t ≥ T¯ . Since |qe0 (t)| ≤ 1, for t ≥ T¯ , we have

2

q ev = 1 − q 2 ≥ 1 − qe0 ≥ (1 − qe0 )2 . (6.11) e0 Thus

2 2 2 q ev ≥ q ev + (1 − qe0 )2 .

(6.12)

¯ Then V1 ≤ 2q  ev q ev for all t ≥ T , which leads to α1 1 m+1 m+1 V˙1 ≤ − V1 − α2 ( ) 2 V1 2 . 2 2

(6.13)

180 Fault-Tolerant Attitude Control of Spacecraft

< 1, q e ≡ (1, 0) and ωe ≡ 0 are reached in finite time T0 ≥   1−m 1−m 2 4 ln(1 + αα12 12 V 2 (t0 )). Thus we conmax{T¯ , T }, where T = t0 + α1 (1−m)

From 0
1 (i = 2, 3). Then: 1) controller (6.19) will drive the states on the switching manifold S in finite time T1 ; 2) the attitude-tracking error q e will asymptotically converge to its equilibrium point [1, 0] , and the velocity-tracking error ωe will converge to zero in finite time Tf . Proof. Consider another candidate Lyapunov function  (1 − κ)π˜ 12 (1 − η)π˜ 22 1  Y˜ Y˜ + + , V2 = S J S + 2 2λ1 2λ2 2λ3

(6.26)

182 Fault-Tolerant Attitude Control of Spacecraft

where Y˜ = Y − Yˆ and π˜ i = πi − πˆ i . Using (6.14), (6.19), (6.20), and (6.23), we can calculate the time derivative of V2 with respect to time for S = 0 as (1 − κ)π˜ 1 π˙ˆ 1 (1 − η)π˜ 2 π˙ˆ 2 V˙2 ≤ − K − − λ2 λ3  + S (D(I3 − E(t))τ c − E(t)τ c + τ c_mis (t) + τ c_fault (t)).

(6.27)

Because δi (t) (i = 1, 2) are positive, it follows from (6.24) and (6.25) that πi (t) (i = 1, 2) are increasing functions. Then πˆ i (t) for t ≥ t0 due to πˆ i (t) > 1. Imposing xy ≤ 3x∞ y∞ for x, y ∈ R3 and substituting (6.24) yield S  D(I3 − E(t))τ c + S  τ c_mis (t) ≤3D∞ S∞ δ1 (t) + S  D(I3 − E(t))τ c_mis (t) δ1 (t)S − S  (πˆ 1 (t) − 1) S∞ ≤κδ1 (t)S∞ + κδ1 (t)(πˆ 1 (t) − 1)S∞ − (πˆ 1 (t) − 1)δ1 (t)S∞ =(1 − κ)π˜ 1 (t)δ1 (t)S∞ . (6.28) Further, using π2 (1 − η) = 1 and (6.25), we have S  (−E(t)τ c + τ c_fault ) ≤ηSτ adp (t) + τ c_mis (t) − g + S  E(t)(πˆ 2 (t) − 1)

δ2 (t)S S

δ2 (t)S S ≤ηδ2 (t)S + η(πˆ 2 (t) − 1)δ2 (t)S − (πˆ 2 (t) − 1)δ2 (t)S =(1 − η)π˜ 2 (t)δ2 (t)S. − S  (πˆ 2 (t) − 1)

(6.29)

Hence, inserting (6.24), (6.25), and (6.28), (6.29) into (6.27) results in V˙2 ≤ −K < 0.

(6.30)

1) The obtained inequality (6.30) implies that V˙2 < 0 for S = 0. In partic ular, in case of S = 0, it leads to V2 = (Y˜ Y˜ )/(2λ1 ) + ((1 − κ)π˜ 1 )/(2λ2 ) + ((1 − η)π˜ 2 )/(2λ3 ). Differentiating V2 and inserting (6.23), (6.24), (6.25) result in V˙2 = 0. Combining thus with (6.30), we get that V2 is bounded for t ≥ t0 . Integrating (6.30) from t0 to t yields  t  t Kdt = −K (t − t0 ) , V˙2 ds = V2 (t) − V2 (t0 ) ≤ − (6.31) t0

t0

from which it follows that V2 (t) ≡ 0 for t ≥ t0 + ((V2 (t0 ))/K)T1 . Hence all the system states reach the switching manifold S = 0 by the finite time t = T1 .

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2) As proved in 1), the trajectory of the system reaches the switching manifold S = 0 in finite time T1 . At this time, all the system states will start from t = T1 . By Theorem 6.1 the attitude-tracking error q ev (t) and the angulartracking error signal ωe are asymptotically stable when the system states move on the switching manifold. The control objective q e (t) ≡ [1, 0] and ωe (t) ≡ 0   1−m 1−m 2 4 ln(1 + αα12 12 V1 2 (T1 ))T ∗ . We conclude are achieved for t ≥ T1 + α1 (1−m) that a finite time Tf = T ∗ − t0 is required to accomplish the attitude tracking maneuver, when the attitude maneuver is started from the given time t0 . Remark 6.2. In Theorem 6.1, 3D∞ = κ < 1 is a sufficient condition. Using (6.4), we has D∞ ≤ maxi=1,2,3 i , where ψ1 = |α2 | + |α3 |, ψ2 = |α1 | + |α3 |, ψ3 = |α1 | + |α2 |.

(6.32)

Additionally, the condition 3D∞ = κ < 1 leads to D∞ = κ < 1/3. Thus the sufficient condition can be restated as maxi=1,2,3 i < 1/3, that is, the sum of two arbitrary reaction-wheel misalignment angles in three body axes should be smaller than 1/3 rad, that is, 19.099 deg. Otherwise, controller (6.19) will be inefficient for the misalignment. That number is really rational for the practical engineering, because the misalignment angle of actuators has not exceeded 7 deg in previously launched spacecrafts. Remark 6.3. The condition η < 1 is required in the design of the controller. The proposed control is capable to compensate the decreased reaction torque fault F1, that is, 0 ≤ pi (t) ≤ 1, i = 1, 2, 3. Regarding to the bias torque fault f , the validation of Theorem 1 does not impose any restriction of the upper bound of f , that is, the bias torque fault f subject to Assumption 6.4 can be accommodated by controller (6.19). However, in practice, due to physical limitation of reaction wheel, the maximum torque generated by the wheel is limited. The bias torque f is actually constrained by that maximum torque. Remark 6.4. The implementation of control (6.19) needs values of the gains K, m, α1 , α2 , and λi , i = 1, 2, 3. It is shown from Theorem 6.1 and (6.31) that the choice of K, m, α1 , and α2 establishes the relationship between the parameters and finite time Tf . Larger K, m, and α1 (1 − m) or smaller α1 /α2 lead to a smaller Tf . The desired attitude can be followed in much shorter finite time. From (6.23), (6.24), and (6.25) we further obtain that larger λi , i = 1, 2, 3, yield a faster convergence of Yˆ (t), πˆ 1 (t), and πˆ 2 (t) to the values that guarantee that V˙2 (t) is negative semidefinite.

6.2.3 Numerical example To verify the effectiveness of the proposed control scheme, we numerically simulated a spacecraft. The planned mission requires the spacecraft to reori-

184 Fault-Tolerant Attitude Control of Spacecraft

ent toward some specific high-priority areas and take a series of images. The major parameters of numerical simulation are given in Table 6.1. The products of inertia are smaller than 0.5 kg · m2 and thus can be neglected. TABLE 6.1 The numerical parameters of the simulated spacecraft. Parameters

Circular orbit

Values Altitude

650 km

Inclination

95.4 deg

Orbital rate

ω0 = 0.0011 rad/s

Orbital period

nT = 5863.69 s J11 = 35.3 kg · m2

Inertial parameters

Principal moments of inertia

J22 = 42.5 kg · m2 J33 = 27.5 kg · m2

Products of inertia Attitude and velocity sensor noises Reaction wheel misalignment

≤ 0.5 kg · m2 σp2 = 0.0001 (1σ ) δαi ∈ [−8◦ , 8◦ ]

δβi ∈ [−180◦ , 180◦ ], i = 1, 2, 3

The disturbance d is calculated as in Sidi (1997). To accomplish the mission, the spacecraft must provide 0.01 deg attitude pointing accuracy with 0.0016 deg/s stability of the pointing. An extended Kalman filter is used in the attitude determination subsystem.

6.2.3.1 Desired attitude trajectory As discussed in Sidi (1997), the orientation of spacecraft body-fixed frame Fb with respect to orbital reference frame Fo can be obtained by a yaw-roll-pitch (ψ − φ − θ ) sequence of rotations, where ψ, φ, and θ are the yaw, roll, and pitch angles, respectively. To take images of specific high-priority areas, the desired attitude trajectory is ψd = 4 sin(ω0 t) degrees, φd = 8 sin(ω0 t) degrees, and θd = 12 sin(ω0 t) degrees. Fig. 6.1 shows the initial desired attitude of the first 20 seconds, and the circular points denote the desired target in the intervals of 2 seconds. 6.2.3.2 Tracking maneuver with reaction fault and misalignment During the tenth orbital period, the spacecraft is commanded to follow the desired attitude trajectory to perform the mission. The initial attitude angles are [0.009, −0.005, 0.0075] degrees, the initial velocity is [0.0012, 0.001, −0.0015] deg/s. To investigate the control performance, reaction wheels are assumed to be faulty after nine orbital periods. During the image taking mission, the following fault scenarios are introduced: 1) the reaction wheel mounted in line with the roll axes of Fb loses 70% of its normal power after 5 seconds; 2) the

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185

FIGURE 6.1 Desired attitude and the distributed target.

actuator mounted in line with the yaw axes of Fb experiences an increased-biastorque fault, that is, f2 = −0.1 Nm; moreover, this reaction wheel will lose its power of 60% after 8 seconds; 3) the reaction wheel fixed in line with the pitch axes of Fb always undergoes 0.15 Nm of increased bias torque fault. The control gains are chosen as K = 0.15, m = 0.7, α1 = α1 = 1.5, and λi = 0.05, i = 1, 2, 3. With the application of controller (6.19), time responses of the attitude angle-tracking error and the angular velocity-tracking error are shown in Figs. 6.2 and 6.3, respectively. The corresponding commanded control torque is shown in Fig. 6.4. The commanded control for the reaction wheel fixed in line with the pitch axes is almost equal to −0.15 Nm. As a result, the bias torque fault is successfully accommodated. The applied torque of that reaction

FIGURE 6.2 Attitude angle-tracking error.

186 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 6.3 Angular velocity-tracking error ωe (deg/s).

FIGURE 6.4 Commanded and the applied control power.

wheel is almost zero when the desired attitude is followed after 5000 s, as shown in the last row plot of Fig. 6.4.

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187

FIGURE 6.5 Time response of switching surface.

Fig. 6.5 shows that all the states are driven to the switching surface within 4.6 s and stay in thereafter. After a short period of time, roughly 0.4 s, the states are further governed to zero. Hence the attitude tracking maneuver is accomplished in five seconds even when actuator fault and misalignment occur. That is due to the fact that the actuator misalignment and fault are accommodated by the term τc_mis and τc_fault , respectively. In particular, 0.005 deg pointing accuracy with 0.0014 deg/s stability of the spacecraft is achieved, as we see in Figs. 6.2 and 6.3. Those obtained attitude stability and pointing accuracy satisfy a set of stringent pointing requirements to provide high-resolution images with the equipped CCD camera. The mission can be successfully accomplished.

6.2.3.3 Quantitative analysis of mission performing To investigate the performance of the control approach, we also simulated an attitude-tracking maneuver in the absence of actuator fault. We considered the following two indices that include the average of square of the commanded control torque (ASCCT) and the percentage proportion of the images successfully taken with respect to the areas that need to be imaged (POSTI):  nT τc (t)2 dt Nok ASCCT = 0 , POSTI = , (6.33) nT Nall where Nok is the number of areas that are imaged successfully, and Nall is the total number of areas that need to be imaged. As analyzed in 6.2.3.1, we have Nall = 2932. These two indices evaluate the efficiency of the controller. We compare the results under actuator fault-free and fault case in terms of ASCCT and POSTI. As shown in Figs. 6.6 and 6.7, it is interesting to find out that: 1) when all reaction wheels are fault-free, the control power is smaller than in faulty cases, as shown in Fig. 6.7. That is because more effort is required to compensate F1 and F2; 2) in the absence of fault, more images are taken than

188 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 6.6 Performance index ASCCT.

FIGURE 6.7 Performance index POSTI.

in faulty cases, as we see in Fig. 6.8. However, even in the presence of fault, the strategy can still accomplish 99.898% of the mission, and the index POSTI only decreases 0.068%. From the above analysis results, we come to the conclusion that whenever actuator fault occurs or not, the proposed solution provides a fast response and high-pointing accuracy to perform the mission.

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189

6.3 Finite-time attitude-tracking FTC 6.3.1 Problem formulation 6.3.1.1 Attitude-tracking error dynamics Based on the error dynamics obtained in the previous sections, manipulation of (2.19) and (6.1) results in     (6.34) M ∗ q e q¨ ev + C q e , q˙ ev q˙ ev + F  (H 1 + H 2 ) = F  (u + d),     where F = P −1 , C q e , q˙ ev = F  J F˙ − F  (J F q˙ ev )× F , M ∗ q e = F  J F , and      ×   H 2 =(F q˙ ev )× J R q e ωd + R q e ωd J R q e ωd   ˙ d ). + (R q e ωd )× J F q˙ ev − J (F q˙ × (6.35) ev R(q e )ωd − R(q e )ω Remark 6.5. Obtaining (6.34) requires that P is invertible. The condition det(P ) = 0.5qe0 (t) = 0 should be guaranteed to avoid the singularity of P for t ≥ 0. It thus requires the initial conditions such that qe0 (t) = 0 for t > 0. Regarding the restriction on the initial conditions, it is known that the desired trajectory can always be initialized to guarantee qe0 (t) = 0. Therefore the initial condition restriction is actually a very mild restriction on the desired trajectory. Remark 6.6. Note that there are two equilibrium points for the closed-loop attitude-tracking system, that is, q e = [1, 0] and q e = [−1, 0] . These two equilibrium points represent the same equilibrium point in the physical space, and they yield the same attitude rotation matrix R(q e ). However, only q e = [1, 0] is an attractive equilibrium point, q e = [−1, 0] is not an attractor, but a repeller equilibrium Tayebi (2008). Hence q e = [1, 0] is chosen as the equilibrium point to be stabilized. Property 6.1. The matrix M ∗ (q e ) is symmetric positive definite. ˙ ∗ (q e ) − 2C(q e , q˙ ev ) is skew-symmetric Lo and Property 6.2. The matrix M ∗  ˙ (q e ) − 2C(q e , q˙ ev )x = 0 for x ∈ R3 . Chen (1995), that is, x M Property 6.3. The first two terms on the left-hand side of (6.34) can be linearly parameterized as Costic et al. (2001)       (6.36) M ∗ q e q¨ ev + C q e , q˙ ev q˙ ev = F  W q e , q˙ ev η, where W (q e , q˙ ev ) ∈ R3×6 is the known regression matrix, and η ∈ R6 is the constant vector given by  ∗ ∗ ∗ ∗ ∗ ∗  η = J11 , J22 , J33 , J12 , J13 , J23 with Jij∗ being the ij th element of J .

(6.37)

190 Fault-Tolerant Attitude Control of Spacecraft

6.3.1.2 Control objective Given any angular velocity and the initial attitude-tracking error restricted by qe0 (0) = 0, the control objective can be stated as follows: Consider the attitudetracking error system described by (2.19) and (6.1) in the presence of actuator fault (6.5) to design a commanded control input τ to guarantee that q ev (t) ≡ 0, ωe (t) ≡ 0, and qe0 (t) ≡ 0 for t ≥ tf , where tf is a positive scalar, that is, the desired attitude q d and the desired velocity ωd can be followed in finite time tf . 6.3.2 Finite-time attitude-tracking control design To achieve attitude tracking in finite time even when actuator faults occur, we propose a control scheme using the SMC technique. We apply adaptive technique to handle constant but unknown parameters and disturbances. For the attitude-tracking error system (6.34), we design a novel sliding mode manifold as Lu et al. (2010) S = [s1 (t), s2 (t), s3 (t)] = q˙ ev + kq ev − f (t),

(6.38)

where f (t) = [s1 (t), s2 (t), s3 (t)] is the forcing function in sliding dynamics, and k is the positive scalar. We first introduce an auxiliary attitude-tracking error er = q˙ ev − S.

(6.39)

Using (6.36), we can define a desired linear parameterization       M ∗ q e e˙ r + C q e , q˙ ev er = F  W r q e , q˙ ev , er , e˙ r η.

(6.40)

Then, we can rewrite the open-loop dynamics of S as     M ∗ q e S˙ + C q e , q˙ ev S = F  (u + d − H 1 − H 2 − W r η) .

(6.41)

Because R(q) = 1 and ωoi  = ω0 , it follows that



(R(q)ωoi )× J R(q)ωoi ≤ Jmax ω2 , 0

(6.42)

×

J ω R(q)ωoi ≤ ω0 Jmax ωbo  ,

(6.43)

×

ω J R(q)ωoi ≤ ω0 Jmax ωbo  ,

(6.44)



(R(q)ωoi )× J ωbo ≤ ω0 Jmax ωbo  .

(6.45)

bo

bo

From (6.42)–(6.45) we obtain H 1  ≤ Jmax ω02 + 3ω0 Jmax

ωbo . The equal

ity R(q) = 1 and Assumption 6.2 ensure that ωe  = ωbo − R q e ωd ≤ ωbo  + c0 . Moreover, we can establish the following inequalities:

 

(F q˙ ev )× J R q e ωd ≤ c0 Jmax (ωbo  + c0 ) , (6.46)

Fault-tolerant finite-time attitude-tracking control Chapter | 6

191



×



(R q e )ωd J F q˙ ev ≤ c0 Jmax (ωbo  + c0 ) ,

(6.47)



×



(R q e )ωd J R(q e )ωd ≤ c02 Jmax ,

(6.48)



   

J (((F q˙ ev )× R q e ωd − R q e ω˙ d )) ≤ Jmax [(ωbo  + c0 )c0 + c1 ], (6.49) ˙ Thus H 2 is bounded by where ωe = F e. H 2  ≤ 3c0 Jmax ωbo  + 4c02 Jmax + c1 Jmax .

(6.50)

Additionally, using Assumptions 6.2, 6.3, and 6.4 and |hi (t)| ≤ 1, i = 1, 2, . . . , N, we have H 2 + H 1 + Dτ + d ≤ (3c0 Jmax + 3ω0 Jmax ) ωbo  + dmax   + c1 + ω02 Jmax + g0 D + 4c02 Jmax  =Y  1 χ 1 + Y 2 χ 2,

(6.51)

where Y 1 = [Jmax , g0 ] , Y 2 = [(4c02 + c1 )Jmax + dmax , (c0 + ω0 )Jmax ] , χ 1 = [ω02 , D] , and χ 2 = [1, 3ωbo ] . Obviously, χ 1 and χ 2 can be exactly obtained, whereas Y 1 and Y 2 are time-invariant but unknown vectors. For the mission tasked for the considered satellite, the attitude-tracking maf neuver is required to be accomplished in a finite time ti > 0. Yˆ i and ξˆ are the estimates of Y i and ξˆ = η, respectively. We present the following theorem to achieve attitude control goals. Theorem 6.3. Consider the attitude-tracking system described by (2.19) and (6.1) with reaction wheel fault defined in (6.5). Design τ as τ =−

2 W r  ξˆ D  F S Kc D  F S D F S  ˆ  − . Y i χi − F S F S F S2

(6.52)

i=1

Let Yˆ i and ξˆ be updated by Yˆˆ i = −βi2 Yˆ i + δi F Sχi ,

β˙i = −Ki βi ,

ξ˙ˆ = −β32 ξˆ + δ3 W r  F S,

Yˆ i (0) > 0,

β˙3 = −K3 β3 ,

i = 1, 2,

ξˆ (0) > 0,

(6.53) (6.54)

where ξi , Ki , i = 1, 2, 3, are positive scalars, and Kc is the positive control gain. Suppose that there exists a positive scalar σ such that 0 < σ < σmin (DE(t)D  ) f at any time. Choose the control gain Kc satisfying Kc ≥ V (0)/σ ti , where 0 <  < 1 is a scalar specified by the designer, and V (0) is the initial value of

192 Fault-Tolerant Attitude Control of Spacecraft

V (t) defined in (6.57). Then all the system states will reach the sliding manifold f f in a finite time tT ; moreover, it follows that tT ≤ ti ≤ ti . Define the forcing function f (t) as ⎧ f ⎪ ⎪ q˙ei0 + kqei0 , t < ti , ⎪ ⎪ ⎪ ⎪ f ⎪ f ⎪q˙ + kq − κ sin π(t−ti ) , ⎨ ti ≤ t ≤ tim , ei0 ei0 i f 2(tim −ti ) fi (t) = (6.55) ⎪ t−tim f 1 m ⎪ ⎪ ⎪ 2 (q˙ei0 + kqei0 − κi ) (1 + cos(π t f −t m )), ti < t ≤ ti , ⎪ ⎪ i i ⎪ ⎪ f ⎩0, t > ti , f

f

where qei0 = qei (ti ), q˙ei0 = q˙ei (ti ), i = 1, 2, 3, tim are scalars such that f f ti < tim < ti , and κi is given by        f f f T1i (q˙ei0 + kqei0 ) 1 + exp k ti − tim + 1k q˙ei0 exp −k ti − ti       κi = f f f T1i + T2i exp k ti − tim + T3i exp −k ti − ti (6.56) f

f

with T0i = 4t (tim − ti )2 /(4k 2 (tim − ti )2 + π 2 ), T1i = −(1/2k) + (k/2)T4i , f T2i = (1/2k) − kToi + (k/2)T4i , T3i = −(πToi /2tim ), and T4i = 4(ti − f tim )2 /(4k 2 (ti − tim )2 + π 2 ). Then the closed-loop attitude-tracking system is asymptotically stabilized with q ev (t) ≡ 0 and |qe0 (t)| ≡ 0 guaranteed for all f t > ti . Hence the spacecraft attitude q follows the desired trajectory q d in f finite time ti . Proof. We first prove that all the system states reach the sliding manifold S = 0 in finite time. Then, we demonstrate the finite-time convergence of the attitudetracking error e. Step 1 (Finite-time convergence of S). Consider a candidate Lyapunov function as follows 2  (Y i − σ Yˆ i ) (Y i − σ Yˆ i ) 1 V (t) = S  M ∗ (q e )S + 2 2σ δi i=1

+

2  i=1

Ki−1 βi2 Y  (ξ − σ ξˆ )2 K3−1 β32 ξ 2 i Yi + + . 8σ δi 2σ δ3 8σ δ3

(6.57)

With (6.6), it gives the error dynamics (6.41) as     M ∗ q e S˙ + C q e , q˙ ev S =F  D(E(t)τ + τ¯ ) + F  d − F  H 1 − F  H 2 − F  W r η.

(6.58)

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Applying Property 6.2, (6.51), and (6.57) yields V˙ = S  (F  D(E(t)τ + τ¯ ) + F  d − F  H 1 − F  H 2 )   ˆ 2 −1 2 Y − σ Y Y˙ˆi  ˙ i i K3 β3 β3 ξ   + − S F Wrη − 4σ δ3 δi i=1

2 (ξ − σ ξˆ )ξ˙ˆ  Ki−1 βi β˙i Y  i Yi − + δ3 4σ δi i=1

≤ S  F  DE(t)τ + F S

+

K3−1 β3 β˙3 ξ 2 4σ δ3



−

2 

Y i χ i + ξ W r  F S

i=1

2 Y i − σ Yˆ i  i=1



Y˙ˆi

δi

2 (ξ − σ ξˆ )ξ˙ˆ  Ki−1 βi β˙i Y  i Yi − + . δ3 4σ δi

(6.59)

i=1

Because Yˆ i (0) and ξˆ (0) are positive, from (6.53) and (6.54) we obtain that ξˆ (0) > 0 and Yˆ i (t) ≥ 0 for t ≥ 0, i = 1, 2. Substituting the control law (6.52) into (6.59) yields V˙ ≤ −S  F  DE(t)

2 W r  ξˆ D  F S Kc D  F S DF S  ˆ  + × Y i χi + F S|| F S F S2 i=1

+ F S

2 

Y i χ i + ξ W r  F S −

i=1

−

2  (Y i − σ Yˆ i ) Y˙ˆ i i=1

δi

2 K −1 β3 β˙3 ξ 2 (ξ − σ ξˆ )ξ˙ˆ  Ki−1 βi β˙i Y  i Yi + + 3 δ3 4σ δi 4σ δ3 i=1

≤ −Kc σ + F S

2  (Y i − σ Yˆ i ) χ i + (ξ − σ ξˆ ) W r  F S i=1



2 Ki−1 βi β˙i Y  (ξ − σ ξˆ )ξ˙ˆ  (Y i − σ Yˆ i ) Y˙ˆ i i Yi − − − − . 4σ δ3 δ3 δi 4σ δi β32 ξ 2

i=1

(6.60)

194 Fault-Tolerant Attitude Control of Spacecraft

Imposing the updating laws (6.53) and (6.54) results in V˙ ≤ −Kc σ −



2 β 2 (ξ − 2σ ξˆ )2



Yi − 2σ Yˆi − 3 4σ δ3

2  βi2 4σ δi i=1

≤ −Kc σ < 0.

(6.61)

Integrating (6.61) from 0 to t yields 

t

V˙ dγ = V (t) − V (0) ≤ −

0



t

Kc σ dγ = −Kc σ t,

(6.62)

0

which gives V (t) ≡ 0 for t ≥ (V (0)/Kc ) σ = tT . Thus from the definition of V (t) in (6.57) we get that S ≡ 0 for all t ≥ tT , that is, all the system states reach the sliding manifold S = 0 by the finite time t = tT . Furthermore, from the choice of the control gain Kc we get that tT = f f (V (0)/Kc ) σ ≤ ti . This ensures that S(t) ≡ 0 for all t ≥ ti . Thus the finitetime convergence of S can be concluded. Step 2 (Finite-time convergence of q ev ). As S is proved to be zero within f finite time ti in Step 1, it follows from (6.38) and (6.55) that

q˙ei + kqei =

⎧ ⎪ ⎪ si + q˙ei0 + kqei0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨q˙ + kq − κ sin ei0

ei0

i

f

t < ti , f

π(t−ti ) f 2(tim −ti )

,

⎪ t−tim 1 ⎪ ⎪ ⎪ 2 (q˙ei0 + kqei0 − κi ) (1 + cos(π t f −t m )), ⎪ ⎪ i i ⎪ ⎪ ⎩0,

f

ti ≤ t ≤ tim , f

tim < t ≤ ti , f

t > ti . (6.63)

To show the finite-time convergence of q ev , we only need to verify its ref sponse in the time interval ti ≤ t ≤ ∞, because all states absolutely reach the f sliding manifold S = 0 after time ti . To do this, we involve the following three procedures. f Procedure 1. We start with the first interval ti ≤ t ≤ tim . Given initial values f qei0 and q˙ei0 , solving (6.63) over ti ≤ t ≤ tim yields     f f π t − ti π t − ti  + C3 cos   ei (t) =C1 + C2 sin  f f 2 tim − ti 2 tim − ti    f + C4 exp −k t − ti ,

(6.64)

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where q˙ei0 + kqei0 , k C2 = −kκi T0i , πκi T0i , C3 =  f 2 tim − ti C1 =

C4 = −

q˙ei0 + kC3 . k

(6.65)

It is known that q˙ei always exists for t ≥ 0. This ensures the continuity of qei (t). Then from (6.64) we obtain that qei (tim ) =

f

limm qei (t) = C1 + C2 + C4 exp(−k(tim − ti )).

t→(ti

)+

(6.66) f

Procedure 2. Owing to the continuity of qei (t), solving (6.63) for tim < t ≤ ti with the initial value (6.66) results in

t − tim t − tim qei (t) =C5 + C6 sin π f + C7 cos π f ti − tim ti − tim    f + C8 exp −k t − ti , (6.67) f

where C5 = q˙ei0 + kqei0 − κi /(2k), C6 = πT4i (q˙ei,0 + kqei0 − κi /(2(ti − f tim )), C7 = k 2 T4i C5 , and C8 = C4 + (C1 − C5 + C2 − C7 ) exp(k(tim − ti )). Since qei (t) is continuous, using (6.56) and rearranging (6.67) lead to f

qei (ti ) =

lim qei (t) = 0. f

t→(ti )+

f

Procedure 3. For t ≥ ti , (6.63) can be rewritten as q˙ei + kqei = 0. Usf f ing qei (ti ) = 0, this shows that qei (t) = 0 for t ≥ ti . Further, solving for the f unity constraint for q e yields |qe0 (t)| = 1 for t ≥ ti . To this end, we obtain f that q e = [1, 0] or q e = [−1, 0] for t ≥ ti . As stated in Remark 6.6, only q e = 1, 0] is chosen as the equilibrium point for the attitude-tracking error system to be stabilized. Then we have q d = q d q e = q d (q˙ d )−1 q = q by using f quaternion multiplication, that is, q d and q coincide for all t ≥ ti . Hence the f attitude q follows the desired attitude q d in finite time ti . In Theorem 6.3, we imposed a sufficient condition for the control design, that is, δ = σmin (DE(t)D  ) > 0. Because D is of full-row rank, σ > 0 means that there are at most N − 3 reaction wheels undergoing F3 or F4 at any time.

196 Fault-Tolerant Attitude Control of Spacecraft

Otherwise, it would lead to δ = 0, and the controller would not be able to tolerate the corresponding faults. Furthermore, if N − 2 or more reaction wheels experience F3 or F4 at the same time, then the remaining active reaction wheels may not be able to produce a combined torque sufficient enough to compensate those faults. Consequently, the spacecraft attitude may not be stabilized. This results in a three-axis attitude maneuver failure owing to the lack of necessary hardware redundancy. In such a case the attitude control system becomes underactuated, which is not the main issue investigated in this paper. Therefore much more reaction wheels should be mounted to fully accomplish three-axis attitude tracking. Also, this is the reason why N should satisfy δ > 0 to achieve three-axis attitude control here. Now we have to consider a practical problem for SMC-based control approach, namely, the chattering effect. Because it is impossible to switch the control at infinite rate, the trajectory of an SMC system chatters with respect to the sliding manifold. This chattering phenomenon is practically undesirable as it may excite the neglected high-frequency dynamics. One practical approach to reduce the chattering is replacing the discontinuous function F S/F S in (6.52) by a continuous approximation such as F S/F S + ε, where ε is a small positive scalar Hung et al. (1993). Therefore controller (6.52) can be modified as the following finite-time FTC (FTFTC): 2 W r  ξˆ D  F S D F S  ˆ  Kc D  F S τ =− − . Y i χi − F S + ε F S + ε F S2 + ε

(6.68)

i=1

We see in Theorem 6.3 that when concerning the determination of controller f parameters, the terminal time ti can be explicitly and preliminarily specified in the proposed design, and the controller that achieves the desired terminal f time can be easily obtained using the proposed scheme. The choice of ti is f designer friendly in that with smaller ti , the attitude-tracking maneuver can be accomplished within shorter time. Summarizing the analysis in the proof of Theorem 6.3, we can choose all the gains in controller (6.68) according to the following procedure. • Step 1: Arbitrarily choose a positive k for the sliding manifold (6.38). f • Step 2: Choose 0 <  < 0 and ti , j = 1, 2, 3, according to the time requiref ment of the planned aerospace mission. If a smaller ti is selected, then the mission will be accomplished in a shorter time. f f • Step 3: Choose the value of tim such that ti < tim < ti , i = 1, 2, 3. However, there is no unique choice, and physical meaning for the time instant tim and decreasing tim will increase the required control effort during the initial period. Additionally, decreasing tim will increase the convergence rate during the initial period. Therefore a compromise between control effort and convergence rate should be made by selecting tim .

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• Step 4: Select the value of σ . Additionally, σ is also needed to be less than the value δ = σmin (DE(t)D  ). Because δ is unknown due to the unknown fault E(t), σ should be chosen as small as possible, say 0.01. f • Step 5: Choose a positive value of Kc such that Kc ≥ V (0)/(σ ti ). It should be pointed out that larger Kc will result in larger control effort. Therefore Kc should be chosen within the maximum torque generated by each actuator. • Step 6: Choose positive δi , Ki , i = 1, 2, 3. Larger δi and Ki will lead to faster convergence of Yˆ i and ξˆ . • Step 7: Choose a positive ε according to the set of requirements (such as attitude pointing accuracy and stability) imposed by the mission. If smaller ε is selected, then higher accuracy and more chattering for the control are obtained. Remark 6.7. Although the proof of Theorem 6.3 in the framework of Lyapunov stability analysis is standard, the main challenges arising from the issues of finite-time tracking control, fault-tolerance, external disturbances, and constant but unknown inertia in spacecraft attitude control are all addressed simultaneously in this paper. This is the main contribution of the paper in terms of practical engineering application or theoretical contribution. Additionally, the designed controller is implemented with digital computer in modern aerospace engineering. Hence controller (6.68) can be implemented and applied in practice. Remark 6.8. Although controller (6.52) is developed according to the sliding manifold presented in Lu et al. (2010), the controller has great fault-tolerant capabilities, but the controller in Lu et al. (2010) does not. This is another contribution of this paper when comparing with the result in Lu et al. (2010).

6.3.3 Numerical example A rigid spacecraft tasked with attitude tracking maneuver is numerically simulated under the proposed control schemes in this section. The orbit of the spacecraft is circular with an altitude of 750 km and an inclination of 95.4◦ . ∗ = 45 kg · m2 , J ∗ = 42 kg · m2 , and The principal moments of inertia are J11 22 ∗ 2 J33 = 37.5 kg · m . The products of inertia are smaller than 0.5 kg · m2 . Although in Theorem 6.3 we proved that the proposed controller can only handle constant but unknown inertia parameter, we simulated uncertain (timevarying) inertia as given in Cai et al. (2008) to further demonstrate its robustness against time-varying system uncertainties. Disturbances d are calculated as in Sidi (1997). To accomplish a particular aerospace mission, it initiates a tracking task commanding the spacecraft to reorient toward some specific high-priority areas and take a series of high-resolution images. To take images as much as possible, the attitude-tracking maneuver is required to be accomplished in 10 f seconds, that is, ti = 10, i = 1, 2, 3. Additionally, the mission further imposes

198 Fault-Tolerant Attitude Control of Spacecraft

a number of requirements on the attitude control system. It must provide 0.01◦ attitude pointing accuracy with 0.0055 deg/s stability. The spacecraft is developed as a fully redundant three-axis stabilized system with four reaction wheels. Three reaction wheels are fixed orthogonally and aligned with the axis of Fb , and the fourth redundant wheels are mounted at equal angle (54.7◦ ) to each of the body axes. As a common sensor, rateintegrating gyros are equipped to measure angular rates. Nongyroscopic attitude sensors are equipped to measure attitude angles. In practice, it is difficult for the attitude dynamics to provide a high-precision attitude rate reference. For the equipped gyros, we apply a widely used model Farrenkopf (1978) ω˜ bi = ωbi + β(t) + ηv (t),

(6.69)

˙ = ηu (t), β(t)

(6.70)

where ηv (t) and ηu (t) are independent zero-mean Gaussian white-noise pro2  2 cesses with E{ηv (t)η v (m)} = I 3 σv δ(t − m) and E{ηu (t)η u (m)} = I 3 σu δ(t − m), and δ(t − m) is the Dirac delta function. Attitude sensors are modeled by a zero-mean Gaussian white-noise process with standard deviation σST . To obtain high-accuracy attitude and rate measurement, we used an extended Kalman filter in attitude determination. The control gains for the controller are chosen as k = 0.25, δi = 7.5, Ki = 2, Kc = 150, σ = 0.01, ε = 0.01,  = 0.4, and tim = 7.5, i = 1, 2, 3. The initial values of the adaptive updating laws (6.53) and (6.54) are selected as Yˆ i = [0.05, 0.05] and ξˆ = 0.15, i = 1, 2. After attitude capture, damping, and stabilization maneuvers, the attitude is within 0.25◦ . Thus the initial attitude angles in the simulation are chosen as [ψ, φ, θ ] = [0.2◦ , 0.2◦ , 0.2◦ ] with initial velocity ωbo (0) = [0.02◦ , 0.02◦ , 0.02◦ ] deg/s. With the physical parameters of the gyros and attitude sensors equipped on the spacecraft, the gyros measurements are simulated with σu = 2.68 × 10−4 μrad/s3/2 , σv = 0.34μrad/s1/2 , and an initial bias 0.08 deg/h on each axis. The attitude measurement is simulated with σST = 35 arcsecond.

6.3.3.1 Desired attitude trajectory As stated in Sidi (1997), the attitude of Fb with respect to Fo can be obtained by a yaw-roll-pitch (ψ − φ − θ ) sequence of rotations, where ψ, φ, and θ are the yaw, roll, and pitch angle, respectively. To ensure that the CCD camera mounted on the spacecraft can successfully take images of specific high-priority areas, the desired attitude trajectory is planned as a smooth rotation about each axis by ψd = 5.4 sin(ω0 t), φd = 6.5 sin(ω0 t), and θd = 12 sin(ω0 t), corresponding to the desired unit-quaternion value q d . Fig. 6.8 shows the initial response of the first 20 s, the circular points denote the desired target in intervals of 2 s.

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FIGURE 6.8 Desired attitude and distributed target.

6.3.3.2 Control performance in reaction wheel fault-free case All reaction wheels are fault-free in this case, and only external disturbances act on the spacecraft. With the choice of gains, we have tT ≤ 4. With the application of the FTFTC controller (6.68), the time response of the sliding manifold S is shown in Fig. 6.9. We see that the states of the attitude-tracking error system will reach the sliding manifold within 4 s. This verifies the conclusion in Theorem 6.3 that all the system states will reach the sliding manifold in finite time. Figs. 6.10 and 6.11 show the control performance. The desired attitude trajectory is followed after a short period, roughly 5 s. More specifically, high-attitude pointing accuracy of 8.0 × 10−4 degree is obtained by the steady-state behavior as shown in Fig. 6.10. Fig. 6.11(b) indicates that the controller achieves the attitude stability with a value of 1.5 × 10−3 deg/s even in the presence of disturbances and unknown inertia parameters. Consequently, the attitude system by the presented control meets the set of stringent pointing requirements to provide precise geometric accuracy for globally observed high-resolution images. The considered images taking mission are thus successfully accomplished. 6.3.3.3 Control performance in case of reaction wheel faults In this case, we introduce and simulate the fault scenarios shown in Fig. 6.12.

200 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 6.9 Initial response of the sliding manifold S in fault-free case.

1) The reaction wheel mounted in line with the roll axis of Fb (No. 1 wheel) loses 20% of its normal power when the attitude maneuver is started. 2) The actuator mounted in line with the yaw axis of Fb (No. 2 wheel) loses 45% power in the time-interval between the fourth and tenth seconds. Moreover, it will experience F4 after 10 s, with τ¯i = −0.075 Nm. 3) The reaction wheel fixed in line with the pitch axis of Fb (No. 3 wheel) undergoes 30% loss of effectiveness in the first 15 s and then experiences F2 with τ¯i = −0.24 Nm. 4) The redundant wheel (No. 4 wheel) decreases 90% of its reaction torque after 8 s. When the controller is implemented to the spacecraft, the time response of the sliding manifold S is shown in Fig. 6.13. We see that although actuators are faulty, the proposed controller still guarantees that the states of the attitude tracking error system will reach the sliding manifold. The resultant attitude angle-tracking errors and angular-velocity tracking errors are shown in Figs. 6.14 and 6.15, respectively. The corresponding commanded control torque is shown in Fig. 6.18. To compensate the faults, especially F4 in No. 2 reaction wheel, extra torques are required. That is why the commanded signals of all actuators are nonzero even when the desired attitude is followed by the steadystate behavior of τ as shown in Fig. 6.16(b).

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FIGURE 6.10 Initial response of attitude angle-tracking errors in fault-free case.

Although inferior attitude-pointing accuracy and stability, that is, 2.5 × 10−3 deg/s and 5.0 × 10−3 deg/s, respectively, were obtained in comparison with the fault-free case (8.0 × 10−4 deg/s and 1.5 × 10−3 deg/s, respectively), the attitude stability and pointing accuracy still satisfy the stringent pointing requirements to provide high-resolution images with the CCD camera even subject to reaction wheel faults. Hence FTFTC still managed to compensate for actuator faults. Particularly, as we can see from the initial response of the velocity-tracking error ωe in Fig. 6.15(a), the desired attitude is followed in a finite time of 6 s F4 occurring in No. 2 reaction wheel after 10 s was also successfully accommodated. That is because the proposed scheme has a terminal f time of an explicit parameter ti = 10. The control objectives are thus fulfilled in 10 s after the actuator fault occurs. Such a feature with finite-time convergence is critical for designing practical FTC systems with respect to the real-time and hard deadline considerations as outlined in Zhang and Jiang (2008).

6.3.3.4 Quantitative analysis The control performance obtained from FTFTC is further compared with the indirect adaptive fault-tolerant control (IAFTC) Cai et al. (2008), the finite reaching time-based fault-tolerant control (FRFTC) Lee and Kim (2010), and the conventional finite-time attitude controller (CFTAC) Zhu et al. (2011b). In

202 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 6.11 Angular velocity-tracking error ωe in fault-free case.

FIGURE 6.12 Fault scenarios of reaction wheels.

assessing the effectiveness of these four control schemes, two important criterions should be considered, the average of square of the commanded control torque (ASCCT) and the percentage of the successfully taken images (POSTI) of the high-priority events with respect to the total high-priority events that need to be imaged:  T0 ASCCT =

0

τ (t)2 dt , T0

POSTI =

Nok . Nall

(6.71)

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FIGURE 6.13 Response of the sliding manifold S in case of actuator fault.

These two indexes state quantitatively the efficiency of the controller. Here T0 is the one orbital period, Nok is the number of high-priority events that are imaged successfully, and Nall is the total number of the high-priority events that need to be imaged. We can obtain from Section 6.3.3.1 that Nall = 2995. The performance index POSTI is compared as shown in Fig. 6.17. We found the following. 1) When all reaction wheels run normally, the resultant POSTI is shown in Fig. 6.17(a). Although FRFTC Lee and Kim (2010) and CFTAC Zhu et al. (2011b) can achieve attitude control with finite-time convergence, that finite time is not represented by an explicit parameter for user’s choice with both controllers, and it largely depends on the choice of the controller gains. If the controller gains are not appropriately selected, then FRFTC and CFTAC may lead to a quite large finite time, within which attitude maneuver is accomplished. However, the value of corresponding finite-time for FTFTC can be preliminarily set by the designer. That is why, as shown in Fig. 6.17(a), the POSTI resulted by FTFTC is much larger than by FRFTC and CFTAC. Additionally, because FRFTC can ensure finite reaching time for the sliding motion, whereas CFTAC may have an infinite-time reaching phase, the POSTI obtained from CFTAC is inferior to FRFTC. Additionally, IAFTC Cai et al. (2008) can only achieve attitude control, whereas the finite-time convergence may not be ensured. Hence

204 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 6.14 Initial response of attitude angle-tracking errors with actuator fault.

FIGURE 6.15 Initial response of velocity-tracking error ωe with actuator fault.

the POSTI resulted by IAFTC is inferior to other three schemes, although a relatively high value of 92.75% is obtained, as can be seen in Fig. 6.17(a).

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FIGURE 6.16 Initial response of commanded control inputs τ with actuator fault.

2) In the presence of actuator faults, Fig. 6.17(b) shows the POSTI resulted from the above four control schemes. Because CFTAC does not have fault-tolerant capability, it fails to take even one image of interest owing to the occurrence of F3 in No. 3 wheel when attitude maneuver starts, that is, POSTI is zero. Although FRFTC can tolerate F1, it is unable to accommodate F4 in No. 2 wheel. Thus the mission of image taking cannot be accomplished after 10 s, and POSTI equal to 2.00% is obtained. Not only IAFTC cannot achieve finite-time control, but it also cannot compensate F4 and thus leads POSTI to be 0.07%. For the proposed FTFTC, it can accomplish attitude-tracking maneuver within finite time even in the presence of faults. Consequently, it can accomplish 94.12% of the mission. The resultant POSTI decreases only 5.78% with respect to the fault-free case, as shown in Fig. 6.17(b). The resultant ASCCT by FTFTC, IAFTC, FRFTC, and CFTAC is compared (as shown in Fig. 6.18) and described as follows. 1) When reaction wheels are fault-free, FTFTC leads to a smaller ASCCT than other three controllers, as shown in Fig. 6.18(a). That is because FTFTC can accomplish attitude tracking in a short finite-time period. Once the desired attitude is followed, reaction wheels are only needed to generate torque to compensate disturbances. 2) Whether faults occur or not, ASCCT caused by FRFTC is larger than other three schemes, as shown in Fig. 6.18(a,b). That is because FRFTC is developed using terminal SMC, whereas the terminal SMC controller is characterized by larger control power.

206 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 6.17 Performance index POSTI in the fault-free and actuator fault cases.

3) Because IAFTC, FRFTC, and CFTAC are unable to tolerate F4 in No. 2 reaction wheel after 10 s, the attitude control system would continue issuing its maneuver in spite of F4. The required control effort will thus quickly saturate the actuator while striving to maintain the healthy attitude maneuvering performance. Subsequently, a larger control power is resumed. Thus larger ASCCT produced by IAFTC, FRFTC, and CFTAC compared with that of FTFTC is shown in Fig. 6.18(b). From POSTI and ASCCT shown in Figs. 6.17 and 6.18 we see that the proposed control strategy provides good performance compared with the controllers presented in Cai et al. (2008); Zhu et al. (2011b); Lee and Kim (2010),

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regardless of reaction wheel faults. The proposed solution provides a faster response and higher pointing accuracy to guarantee the spacecraft to take much more images and thus accomplish the planned mission as much as possible.

FIGURE 6.18 Performance index ASCCT in the fault-free and actuator fault cases.

6.4 Summary In this chapter, we study attitude-tracking control for a rigid spacecraft subject to external disturbance, uncertain inertia, actuator failures, and misalignment. We first developed a VSC-based compensation scheme to handle actuator misalignment and fault successfully while accomplishing attitude tracking maneuver

208 Fault-Tolerant Attitude Control of Spacecraft

within finite time. However, this strategy could only conquer decreased reaction torque fault and increased bias torque fault. To accommodate more types of actuator failures, we further proposed a globally attitude-tracking control approach based on a time-varying sliding-mode surface. Besides, the proposed controller accomplished attitude-tracking maneuver with attitude-tracking error and velocity-tracking error governed to zero in finite time. Therefore the practical requirement of rapid response to attitude-tracking maneuvers can be met. Note that it can handle all types of reaction wheel faults (F1–F4) without the actuator fault diagnosis information. Compared with existing SMC-based finitetime attitude controllers, the designed controller provides great fault-tolerant capability and explicit convergence time choice. Finally, we validated all these features by numerical simulations.

Chapter 7

Active fault-tolerant attitude control 7.1 Introduction The passive FTC methods can only tolerate limited predetermined faults, and have some certain degree of conservativeness. Moreover, achieving fault tolerance against certain faults is only possible at the expense of degraded nominal performance. On the contrary, the active FTC can react to actuator faults by using a reconfiguration mechanism so that consistent control performance can be maintained, whilst some inherent drawbacks of the passive FTC can be overcome. The application of the active FTC to spacecraft attitude control has nowadays attracted considerable interest. In Jiang and Khorasani (2007), an FDI mechanism was investigated for reaction wheel faults. An iterative learning observer was proposed in Chen and Nagarajaiah (2007) to estimate time-varying thruster faults. In Hou et al. (2008), a two-stage Kalman filtering algorithm was developed to estimate reaction wheel faults, and two reconfigurable faulttolerant controllers were designed to compensate the faults. Note that designing an active fault-tolerant controller without angular velocity measurement from gyro sensors is more desirable in practice Martella et al. (2001); Kruk et al. (2003). This is motivated by the fact that some small satellites or microsatellites do not carry gyro sensors due to design limitations in the satellite mass, size, power, etc., and that the gyro sensors may fail to work due to faults. Apart from this, the active FTC design should also take into account attitude finite-time control and input saturation. There is no previous study handling the above considerations all together in a well-balanced way. This chapter will investigate the active FTC design for spacecraft attitude system under multiple considerations. The remainder of this chapter is organized as follows. Chapter 7.2 addresses the finite-time active FTC problem for attitude tracking of spacecraft subject to external disturbances and two kinds of slight actuator faults. The active FTC problem for spacecraft attitude maneuvers without rate sensors is investigated in Chapter 7.3, where a velocity-free controller is first developed to asymptotically stabilize the attitude in the presence of four kinds of actuator faults, and then a finite-time velocity-free active fault-tolerant attitude-tracking controller is designed with additional consideration of input saturation. Finally, some concluding remarks are given in Chapter 7.5. Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00017-4 Copyright © 2021 Elsevier Inc. All rights reserved.

209

210 Fault-Tolerant Attitude Control of Spacecraft

7.2 Fault compensation attitude tracking control 7.2.1 Problem formulation 7.2.1.1 Open-loop attitude-tracking error dynamics The modified Rodriguez parameter (MRP) representation is used to represent the spacecraft attitude. Given an Euler rotation angle φ(t) ∈ R about the Euler principle axis n ∈ R, the spacecraft attitude in Fb with respect to Fi can be represented by the MRP vector σ = [σ1 , σ2 , σ3 ] = n tan[φ(t)/4] ∈ R3 . The corresponding rotation matrix R(σ ) ∈ R3×3 , which brings Fi into Fb , can be obtained by the Rodriguez formula   1 − σ σ 8 2 R (σ ) = I 3 + 4  (7.1) 2 S (σ ) +  2 S (σ ) ,   1+σ σ 1+σ σ where S(·) is a skew-symmetric matrix such that S(x)y = x × y for any vectors x, y ∈ R3 , with × being the vector cross product. Using the MRPs, the kinematics equation of a rigid spacecraft can be expressed by (2.19). Assume that the desired attitude orientation to be tracked is given by a desired body-fixed reference frame Fd , whose orientation with respect to Fi is specified by the desired MRPs σ d ∈ R3 . The desired MRPs are related to the desired angular velocity of the spacecraft, denoted by ωd ∈ R3 . We define the attitude-tracking error σ e as the discrepancy between the spacecraft attitude and its desired attitude. The corresponding rotation matrix related to σ e , denoted by R (σ e ) ∈ R3×3 , that brings Fd onto Fb is defined by R (σ e ) = R (σ ) R (σ d ) . 3 ˙ (σ e ) = −ω× Note that R e R (σ e ), where ωe = ω − R (σ e ) ωd ∈ R is the angular velocity error of Fb with respect to Fd . We can now obtain the governing equations for the open-loop tracking error dynamics as σ˙ e = G (σ e ) ωe ,

(7.2)

  ˙ d + τ + d. J ω˙ e = −ω× J ω + J ω× e R (σ e ) ωd − R (σ e ) ω

(7.3)

To facilitate the fault-reconstruction formulation, we combine (7.2) and (7.3) to form the following second-order nonlinear dynamic equation: J ∗ (σ e ) σ¨ e + C (σ e , σ˙ e , ωd ) σ˙ e + H (σ e , ωd , ω˙ d ) = P  (σ e ) τ + P  (σ e ) d,

(7.4)

where P (σ e ) = G−1 (σ e ), J ∗ (σ e ) = P  J P , C (σ e , σ˙ e , ωd ) ∈ R3×3 , and H (ωd , ω˙ d ) ∈ R3 are defined as    ˙ (σ e ) + (J P σ˙ e )× P + P  J (R(σ e )ωd )× C = − P JPG (7.5)  +(R(σ e )ωd )× J − (J R(σ e )ωd )× P ,

Active fault-tolerant attitude control Chapter | 7

  H = P  (R(σ e )ωd )× J R (σ e ) ωd + J R (σ e ) ω˙ d .

211

(7.6)

∗

˙ Note that J ∗ (σ e ) is a symmetric positive definite  matrix, and J (σ e )−2C(σ  e, σ˙ e , ωd ) is a skew-symmetric matrix, that is, x  J ∗ (σ e ) − 2C (σ e , σ˙ e , ωd ) x = 0 for all x ∈ R3 .

7.2.1.2 Reaction wheel faults Taking reaction wheel faults into consideration, the actual control torque τ generated by the reaction wheels is u = uc + B(t − T ) [(E(t) − I 3 ) uc + u¯ c ] ,

(7.7)

where the vector uc = [uc1 , uc2 , uc3 ] is the control torque commanded by the controller, E(t) is the reaction wheel effectiveness matrix E(t) = diag([μ1 (t), μ2 (t), μ3 (t)]), and u¯ c = [u¯ c1 , u¯ c2 , u¯ c3 ] represents the additive torque generated by the increased bias torque fault. The matrix function B(t − T ) ∈ R3×3 with T = [t1 , t2 , t3 ] ∈ R3 denotes the time profiles of faults, which is a diagonal matrix of the form B(t − T ) = diag{b1 (t − t1 ), b2 (t − t2 ), b3 (t − t3 )},

(7.8)

where ti , i ∈ {1, 2, 3}, denotes the unknown fault-occurrence time, and bi : R → R is a function representing the time profile of a fault affecting the ith reaction wheel. We consider faults with time profiles modeled by  bi (t − ti ) =

0

if t < ti ,

1 − e−ai (t−ti )

if t ≥ ti ,

(7.9)

where the scalar ai > 0 denotes the unknown fault evolution rate. Small values of ai characterize slowly developing faults, also known as “incipient” faults. For large values of ai , the time profile bi approaches a step function that models “abrupt” faults. Note that the fault time profile given by (7.9) represents only the developing speed of a fault, whereas all its other basic features are defined by the vector u = (E(t) − I 3 ) uc + u¯ c .

7.2.1.3 Problem statement Given any initial attitude and angular velocity, the control objective to be achieved can be stated as follows: Consider the rigid spacecraft attitude system described by (2.19) and (2.21) in the presence of external disturbances d and reaction wheel faults defined in (2.27) for any given reference trajectory σ d , ωd , and ω˙ d , which are bounded, and design a control scheme that guarantees that the tracking error σ e is globally asymptotically stable at σ e = 0 in finite time.

212 Fault-Tolerant Attitude Control of Spacecraft

7.2.2 Attitude tracking compensation controller design In this section, for the proposed control approach shown in Fig. 7.1, we present the nominal control power, compensation control effort, and fault reconstruction scheme. First, we propose a terminal sliding-mode observer to precisely reconstruct the reaction wheel faults and disturbances. Then, using the reconstructed information, we synthesize a compensation controller and add it to the nominal controller to guarantee the global asymptotic stability of the resulting closed-loop attitude-tracking system with finite time convergence.

FIGURE 7.1 Structure of the attitude-tracking controller.

7.2.2.1 Fault reconstruction scheme The external disturbance d is viewed as the system faults, and thus d and τ f can be treated as a lumped fault for the system, that is, d + B(t − T )u  uf . To reconstruct uf , the generalized moment inertia J m = J ∗ (σ e ) σ˙ e is adopted De Luca and Mattone (2005). Under the effect of faults in (7.7), we can rewrite the dynamics in (7.4) as ∗ J˙ m = J ∗ (σ e ) σ¨ e + J˙ (σ e ) σ˙ e

(7.10)

= P  (σ e ) τ − u¯ f (t) − N (σ e , σ˙ e , ωd , ω˙ d ) , ∗

∂J (σ e ) where u¯ f (t) = −P  (σ e ) uf , and N (σ e , σ˙ e , ωd , ω˙ d ) = − 12 σ˙  e ∂σ e σ˙ e + H (σ e , ωd , ω˙ d ). By (7.10) we define the residual vector



t

r(t) = − kJ m + k

[P  (σ e (s)) τ (s)

0

(7.11)

− N (σ e (s), σ˙ e (s), ωd (s), ω˙ d (s)) − r(s)]ds, where k is a positive scalar. Thus the residual signal r(t) satisfies the following dynamic equation: r˙ = −kr + k u¯ f .

(7.12)

Because the continuous function B(t − T ) in (7.8) is differentiable, u¯ f (t) can be assumed to be a differentiable function with time derivative v ∈ R3 . Then we can formulate the problem of reconstructing the signal u¯ f (t) as observing

Active fault-tolerant attitude control Chapter | 7

213

the states of a linear system driven by r(t) and by an unknown input. That linear system is expressed as x˙ 1 = −kx 1 + kx 2 ,

(7.13)

x˙ 2 = v,

(7.14)

y = x1,

(7.15)

where x 1 = r(t), x 2 = u¯ f (t), v is the unknown input, and y is the measured output. The initial vector x 2 (0) is chosen such that x 2 (0) = 0, which corresponds to the case where the attitude system is in a “healthy” condition. The problem of reconstructing u¯ f (t) can be changed into the observation of the state x 2 (t) with access to only the measurable output y. To reconstruct the plant states, a terminal sliding-mode observer is used: x˙ˆ 1 = −k xˆ 1 + k xˆ 2 − x v − l2 e1 ,

(7.16)

x˙ˆ 2 = −l3 e1 − l4 [x v ](q/p) − l5 sgn (x v ) ,

(7.17)

where x v = [xv1 , xv2 , xv3 ] = l1 sgn(e1 ), q

q

q

[x v ](q/p) = [|x v1 | p sign (x v1 ) |x v2 | p sign (x v2 ) |x v3 | p sign (x v3 )] , whereas e1 = xˆ 1 − y, li , i ∈ {1, 2, 3, 4, 5}, are positive observer gains, and p and q are odd integers such that p > q.   6 ˆ 2 − x 2 . Using Define the observer error e = [e 1 , e2 ] ∈ R with e2 = x (7.13), (7.14), (7.16), and (7.17), the dynamics of the error e between the actual states x 1 , x 2 and their estimates xˆ 1 , xˆ 2 is given by e˙ 1 = −ke1 + ke2 − x v − l2 e1 , q p

e˙ 2 = −l3 e1 − l4 [x v ] − l5 sgn (x v ) − v.

(7.18) (7.19)

Remark 7.1. Although the input v is unknown, the amplitude of the input signal v is bounded by a known constant π > 0 (i.e., v ≤ π). As shown in (7.7), the upper bound of v mainly depends on the time profiles B(t − T ) of the faults. The larger the ai , i ∈ {1, 2, 3, 4}, the larger the v.  Proof. Consider a candidate Lyapunov function V1 = 0.5e 1 e1 + 0.5e2 e2 . Differentiating V1 and using (7.18) and (7.19) result in     V˙1 = − ke 1 e1 + ke1 e2 − l1 e1 sgn (e1 ) − l2 e1 e1 − l3 e2 e1 q

  p − l4 e  2 [x v ] − l5 e2 sgn (x v ) − e2 v  q √ √ p  3l4 l1 + 3l5 + π e, ≤ − e Me − l1 e1  +

(7.20)

214 Fault-Tolerant Attitude Control of Spacecraft

where

M=

(k + l2 ) I 3 l3 I 3

−kI 3 03×3

 .

Because k, l2 , and l3 are positive and M is positive definite, the minimum singular value of M, denoted by λmin , is positive. Then we can obtain from (7.20) that  q √ √ p 2 ˙ V1 ≤ −λmin e + 3l4 l1 + 3l5 + π e  (7.21) q √ √ p 3l4 l1 + 3l5 + π . = −e λmin e − Thus, V˙1 < 0 if the bracketed terms in (7.21) are positive, that is,  q √ √ 1 3l4 l1p + 3l5 + π  λR . e > λmin Thus V˙1 is negative when e is outside the compact set D  {e : e ≤ λR }, which implies that V1 (t) decreases monotonically. A decreasing value of V1 (t) eventually drives e into the set D, and then it cannot go out of D, that is, the set D is attractive. With the chosen initial estimates, e(0) = 0 is ensured. According to standard Lyapunov theory and the LaSalle extension Khalil and Grizzle (2002), all the states are confined in D, and this also demonstrates the ultimately uniform boundedness of e. Theorem 7.1. Consider the observer error dynamics in (7.18) and (7.19) obtained from the linear system (7.13)–(7.15) and the terminal sliding-mode observer in (7.16) and (7.17). Choose the observer gains li i = 1, 2, 3, 4, 5, such that ⎧ p/(p−q) ⎨ √3kl + √3l + π 4 5 l1 > max + ε0 , ⎩ λmin  (7.22) ⎡ √ ⎤p/q ⎫ ⎪ ⎬ k 3l5 + π + ε0 λmin ⎣ √ ⎦ , ⎪ 3l5 + π + ε0 λmin ⎭ l5 k − π > 0,

(7.23)

where ε0 > 0 is a scalar. Then terminal sliding motion is achieved in finite time on e1 = 0 and e2 = 0, and u¯ f (t) can be precisely estimated in finite time. Further, the lumped fault uf can be exactly reconstructed within finite time. Proof. The proof uses Lyapunov stability theory and is separated into two parts.

Active fault-tolerant attitude control Chapter | 7

215

S1) Finite-time convergence of e1 : We start with (7.18) by considering a candidate Lyapunov function for e1 as V2 (t) = 0.5e 1 e1 . Then V˙2 = e 1 (−ke1 + ke2 − x v − l2 e1 ) ≤ − (k + l2 ) e1  − (l1 − k e2 ) e1  . √ Using l1 >

√ 3kl4 + 3l5 +π λmin

+ ε0



p/(p−q)

k

and l1 >

√ √

(7.24)

p/q  3l5 +π +ε0 λmin

3l5 +π+ε0 λmin

in

(7.22), we have

l1 >

>

q √  q √ 3kl4 l1p + 3l5 + π l1l

λmin q √  √ p 3kl4 l1 + k 3l5 + π λmin

q

+ ε0 l1p (7.25) + ε0

=kλR + ε0 . Since ke ≤ kλR < l1 is obtained from Lemma 1 for all t ≥ 0, V˙2 < −ε0 e1  < 0 is guaranteed by (7.24) and (7.25) for e1 (t) = 0. Furthermore, we use the comparison lemma to obtain e1 (t) ≡ 0 for t ≥ e1 (0) /ε0  tF 1 . Therefore by the time t = tF 1 the sliding motion takes place on e1 = e˙ 1 = 0. Solving for the equivalent output injection yields (x v )eq = (l1 sgn (e1 ))eq = ke2 . S2) Finite-time convergence of e2 : After time tF 1 , the error dynamics are e˙ 1 = 0 and q

q

e˙ 2 = −l4 k p [e2 ] p − l5 e 2 sgn (ke2 ) − v.

(7.26)

Let us consider another candidate Lyapunov function V3 (t) = 0.5e 2 e2 . It follows that   q q  p p V˙3 =e 2 −l4 k [e2 ] − l5 e2 sgn (ke2 ) − v q

≤ −l4 k p e2  q

< −l4 k p 2

p+q 2p

p+q p

− (l5 k − π) e2 

(V3 )

p+q 2p

(7.27)

.

Because q 0 and 0 < γ < 1 are tuning parameters. Then we propose the following control solution for the underlying attitude-tracking problem: τ c = τ c_nom + τ c_fault ,

(7.31)

where τ c_nom is used to eliminate the nominal portion of the derivative of S, and τ c_fault is used to compensate the effect of reaction wheel faults and disturbances. These two components are designed as sgn(S) + αC (σ e , σ˙ e , ωd ) [σ e ]γ S − H (σ e , ωd , ω˙ d ) + αγ J ∗ ϑ(t)σ˙ e ]

τ c_nom = − G (σ e ) [K

τ c_fault = G (σ e ) xˆ 2 (t),

(7.32)

(7.33)

where ϑ(t) = diag([|σe1 |γ −1 , |σe2 |γ −1 , |σe3 |γ −1 ]), and K is a positive gain. Theorem 7.2. Consider a rigid spacecraft involving reaction wheel faults (2.25) and external disturbances with the attitude dynamics governed by (2.19) and (2.21). With the application of observer (7.16)–(7.17) and controller (7.33), the closed-loop attitude tracking system is asymptotically stable with finite-time convergence, and there exists a finite time Tc > 0 such that σ b (t) ≡ σ d (t) and ωb (t) ≡ ωd (t) for t ≥ Tc . Proof. To show the stability with finite-time convergence, we first prove the finite-time convergence of S and then show a corresponding proof for σ e and ωe . 1) Finite-time convergence of S: Consider a candidate Lyapunov function as V4 = 0.5S  J ∗ (σ e ) S. With reaction wheel faults (2.25), differentiating V4 with

Active fault-tolerant attitude control Chapter | 7

217

respect to time for S = 0 and inserting (7.31)–(7.33) yield ∗ V˙4 = 0.5S  J˙ (σ e ) S + S  J ∗ (σ e ) S˙ ∗ = 0.5S  J˙ (σ e ) S + S  J ∗ (σ e ) [σ¨ e + αγ ϑ(t)σ˙ e ]  sgn(S) + P  (σ e ) τ c fault + P  (σ e ) uf . = S  −K S

(7.34)

Recall that by Theorem 7.1 uf ≡ −G (σ e ) xˆ 2 (t) for t ≥ tF 2 . From (7.34) it follows that, after finite time tF 2 , S  sgn(S) ≤ −K, V˙4 = −K S

(7.35)

which implies that V˙4 < 0 for S = 0. In the case of S = 0, we have V˙4 = 0. Therefore from (7.35) it follows that V4 is a bounded function for all t ≥ 0. Integrating (7.35) from tF 2 to t yields V4 (t) − V4 (tF 2 ) ≤ −K (t − tF 2 ). Because V4 (t) ≥ 0 for all t ≥ 0, we obtain V4 (t) ≡ 0 for t ≥ tF 2 + V4 (tF 2 ) /K  tc1 . As a result, the state trajectories reach the switching manifold S = 0 in finite time t = tc1 . 2) Finite-time convergence of σ e and ωe : When all the states are driven onto S = 0 after finite time tc1 , we have σ˙ e = −α [σ e ]γ . Define a candidate Lyapunov function V5 = 0.5σ  e σ e . Given 0 < γ < 1, it leaves V5 as γ γ +1 V˙5 = −ασ  = −α2 e [σ e ] ≤ −α σ e 

γ +1 2

γ +1 2

V5

,

t ≥ tc1 .

(7.36)

Noting that 0 < 0.5(γ + 1) < 1 and using the comparison lemma in De Luca and Mattone (2005), we can find the unique solution of (7.36) as 1−γ 2

V5

1−γ 2

(t) ≤ V5

(tc1 ) − 0.5α(1 − γ )2

γ +1 2

(t − tc1 ) ,

t ≥ tc1 ,

(7.37)

and V5 (t) ≡ 0 for t ≥ Tc  tc1 +

2

1−γ 2

1−γ

V5 2 (tc1 ) . α(1 − γ )

We further obtain from S = 0 and (7.2) that σ e (t) ≡ 0,

ωe (t) ≡ 0,

t ≥ Tc .

(7.38)

Thus we conclude that the arguments stated in Theorem 7.2 hold, and hence the proof is completed. As we see in the proof of Theorem 7.2, the finite-time convergences of S, σ e , and ωe occur because the lumped fault uf is exactly reconstructed in finite time tF 2 . For t < tF 2 , uf = −G (σ e ) xˆ 2 (t) is not guaranteed, and (7.35)

218 Fault-Tolerant Attitude Control of Spacecraft

cannot be deduced from (7.34). Additionally, according to the definition of Tc , within which the attitude-tracking error is driven to zero, we know that a smaller tF 2 leads to a smaller Tc . That is, if the lumped fault uf can be exactly reconstructed within a shorter time, then the reaction wheel faults and disturbances will be compensated in a shorter time. Thus the attitude-tracking maneuver can be accomplished within a shorter time. If uf cannot be reconstructed in finite time, then the faults and disturbances will not be compensated in finite time, which may lead to Tc = ∞. As stated in the Introduction, such an infinite settling time is not an option during critical phases of the mission. That is why in this work, we propose a terminal siding-mode observer to exactly reconstruct the lumped faults in finite time.

7.2.3 Numerical example To demonstrate the effectiveness of the proposed fault compensation and disturbance rejection scheme, we numerically simulate a spacecraft using (2.19) and (2.21) in conjunction with the control law in (7.31). We carry out the simulation using parameters of a rigid spacecraft currently being developed. The mission simulated requires the spacecraft to reorient toward some specific high-priority areas and take a series of high-resolution images. The orbit of that spacecraft is circular with an altitude of 650 km and inclination of 95.4 deg. Its orbital rate is ω0 = 0.00109 rad/s with orbital period nT = 5893.69 s. The spacecraft mass is about 45 kg with principal moments of inertia J11 = 20, J22 = 22.5, and J33 = 25.2 kg m2 . The products of inertia are smaller than 0.5 kg m2 and thus can be neglected. External disturbances d in (2.21) are calculated as in Sidi (1997). The science objectives of the mission impose a number of requirements on the attitude control system. The spacecraft must provide 0.005 deg attitude pointing accuracy with 0.0012 deg/s stability. Attitude and angular velocity sensor noises (σ 2p ) : 0.0001(1σ ) are modeled as zero-mean Gaussian random variables with variance σ 2p . An extended Kalman filter is used in the attitude determination. When the spacecraft is initially separated from the launch vehicle at t = t0 , a rate damping control is accomplished in the first orbital period nT . The attitude capturing and rotation spin-up of the reaction wheels are finished during the second orbital period. Starting in the third orbital period, attitude stabilization maneuvers are conducted to obtain high pointing accuracy and stability. Attitude-tracking maneuvers are conducted starting in the tenth period. The maneuvers are used to adjust the spacecraft attitude so that it can take images of specific high-priority areas. The time sequence of the attitude maneuvers used to accomplish the considered mission is shown in Fig. 7.2. In the third orbital period an attitude stabilization maneuver is performed. The desired angular velocity to be followed is ωd = [0, −ω0 , 0] rad/s. Because of the short time operation after separation from the launch vehicle, reaction wheel faults are unlikely to occur during this orbital period. Thus the only external disturbances act

Active fault-tolerant attitude control Chapter | 7

219

FIGURE 7.2 Time sequence of the attitude maneuvers for the considered mission.

on the spacecraft. After the attitude capture maneuver, the attitude orientation of the spacecraft in this stage is such that σ b (0) = [0.01569, 0.01707, −0.01168] with initial velocity ωb (0) = [0.2, 0.1, −0.15] deg/s. The control gains for the controller are chosen as α = 0.2, γ = 0.8, and K = 0.75 The following parameters are suitable choices for the observer in (7.16) and (7.17): q = 19, p = 21, l1 = 0.15, l1 = 0.05, l3 = 0.5, l4 = l5 = 0.00005, and k = 0.15. In the absence of reaction wheel faults, we have d = uf . Fig. 7.3 shows the errors in the reconstruction of external disturbances obtained from the observer in (7.16) and (7.17). Fig. 7.3 shows that the finite-time convergence of the reconstruction error

FIGURE 7.3 External disturbances reconstruction error e2 with initial overshoot during the third orbital period.

e2 is obtained after a short period, roughly 110 s. Moreover, high reconstruction accuracy (< 6.0e − 7) for e2 is guaranteed. From the results obtained it is clear that the estimated states xˆ 2 globally asymptotically converge to the external disturbances d. These results verify the conclusion in Remark 7.2 that the fault reconstruction scheme can precisely reconstruct external disturbances even in the absence of actuator faults. Because the reconstruction scheme can provide precise estimation of external disturbances, the compensation control τ c fault is always active and can compensate the effect of disturbances. Thus, as shown in Fig. 7.4, the controller asymptotically drives the angular velocity tracking error to zero with finite-time convergence. It guarantees the attitude stability of 1.0e-3 deg/s. Moreover, after changing the value of MRPs into Euler angles, the results in Fig. 7.5 indicate an attitude pointing accuracy of

220 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 7.4 Angular velocity tracking error ωe with initial overshoot during the third orbital period.

FIGURE 7.5 Attitude-tracking error σ e during the third orbital period.

FIGURE 7.6 Commanded control torque τ c during the third orbital period.

0.0045 deg. These results satisfy the attitude requirements imposed by the science objectives of the mission, and thus the attitude stabilization maneuver is successfully accomplished with the proposed control scheme. The associated control input is shown in Fig. 7.6. Because the term τ c fault (t) always actively compensates external disturbances, the commanded torque is never zero. During the tenth orbital period, the spacecraft is commanded to follow a desired attitude trajectory to perform an Earth imaging. The desired motion is ωd = 0.03[sin(πt/200) sin(πt/300) sin(πt/250)] rad/s. The initial attitude and angular velocity are assumed to be σ b (0) = e−4 [−0.174530.08727 − 0.15272] and ωb (0) = [−0.001, 0.0012, −0.0009] deg/s. During the image-taking mission, we introduce and simulate the following fault scenarios. 1) The reaction wheel mounted in line with the roll axes of Fb experiences F2, which is described by the fault indicator μ1 (t) ≡ 1 and ⎧ 0 ⎪ ⎪ ⎪ ⎪ 0.03(t−80) ⎪ ⎪ ⎨ 70 τ¯ c1 (t) = sign(t − 220) 0.03(t−220) 70 ⎪ ⎪ ⎪ 0.03(t−360) ⎪ sign(t − 360) ⎪ ⎪ 70 ⎩ 0.001

if t ≤ 80, if 80 ≤ t ≤ 150, if 150 ≤ t ≤ 290, if 290 ≤ t ≤ 460, if t ≥ 460.

Active fault-tolerant attitude control Chapter | 7

221

2) Failure F1 with μ2 = 0.45 occurs in the actuator mounted in line with the yaw axes of Fb after 100 s. 3) The reaction wheel fixed in line with the pitch axes of Fb loses 40% of control effectiveness after 50 s.

FIGURE 7.7 The x 2 (solid line) and its reconstruction (dashed line) during the tenth orbital period.

FIGURE 7.8 Faults reconstruction error e2 during the tenth orbital period.

When the controller is implemented within the attitude system, Fig. 7.7 shows a successful fault reconstruction using the incorporated sliding-mode observer. As shown by the steady-state behavior of the fault reconstruction error e2 in Fig. 7.8, the accuracy of the fault and external disturbance reconstructions are smaller than 6e-7 Nm. Because of the precise reconstruction information of the reaction wheel faults and external disturbances supplied by the reconstruction scheme in (7.16) and (7.17), controller (7.31) can completely compensate the effect of actuator faults and reject external disturbances. Thus the controller produces the asymptotical convergence of the velocity-tracking r and attitudetracking errors.

FIGURE 7.9 Angular velocity-tracking error ωe during the tenth orbital period.

The angular velocity tracking error is shown in Fig. 7.9, and the attitudetracking error is presented in Fig. 7.10. As the steady-state behavior is clearly shown in the bottom of Figs. 7.9 and 7.10, the attitude pointing accuracy is within 0.0045 deg and attitude stability is within 0.0012 deg/s. Those attitude stability and pointing accuracies satisfy a set of stringent pointing requirements

222 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 7.10 Attitude-tracking error σ e during the tenth orbital period.

FIGURE 7.11 Commanded control torque τ c during the tenth orbital period.

that provide high-resolution images with the mounted charge-coupled device camera, even in the face of reaction wheel faults and external disturbances. Consequently, the images taken during the mission can be acquired with sufficient accuracy. It is interesting to see the corresponding commanded control torque in Fig. 7.11. When reaction wheel faults occur, a larger control torque is needed to compensate faults F1 and F2. Based on the attitude performance obtained from the proposed terminal sliding observer-based fault-tolerant control scheme, we consider the following two important criterions to assess the control performance: Pm =

Tsuccessful × 100% PE = Ttotal



Ttotal

τ c (ϑ)2 dϑ,

0

where Tsuccessful is the time period within which the satellite attitude is brought within required attitude pointing accuracy. The time Ttotal is the total time for the attitude maneuver. The criterion PE denotes the total energy consumed during the entire attitude maneuver. Note that Ttotal = 5700 s is selected for all simulations. If a larger Pm is obtained (which implies a larger Tsuccessful ), then many more images can be taken during the mission. Furthermore, a larger PE implies that much more power is consumed to accomplish attitude control. To demonstrate the advantage of the terminal sliding-mode-based reconstruction module in (7.16) and (7.17), we compared it with the linear observerbased reconstruction scheme presented in Murugesan and Goel (1987). Although different observers are used, the same control law (7.31) is applied to perform the attitude maneuver. As shown in Table 7.1, the attitude control results from the linear observer are inferior to our proposed approach. That is because the linear observer in Murugesan and Goel (1987) is not able to achieve finite-time reconstruction of the faults. Moreover, the fault reconstruction errors using the linear observer only converge to a small value rather than to zero. Hence the reaction wheel faults cannot be compensated in finite time with the control law (7.31). Consequently, a smaller Pm and a larger PE result from the

Active fault-tolerant attitude control Chapter | 7

223

TABLE 7.1 Control performance with RFTFTC and RIAFTC. Attitude maneuver type

Control scheme

Stabilization

Tracking

Control performance Pm , %

PE

Control law + observer and

99.35

0.0582

Control law + linear observer in De Luca and Mattone (2005)

94.84

0.0745

Control law + observer and

99.35

0.2055

Control law + linear observer in De Luca and Mattone (2005)

94.84

0.3210

TABLE 7.2 Control performance with RFTFTC and RIAFTC. Attitude maneuver type

Control scheme

Control performance Pm , %

PE

99.35

0.1027

Stabilization

RFTFTC RIAFTC

93.13

0.1729

Tracking

RFTFTC

99.35

0.3618

RIAFTC

86.46

0.4352

linear observer in comparison with the terminal sliding-mode observer, no matter whether faults occur or not. The performance of the proposed control scheme was also compared with the robust indirect adaptive fault-tolerant control (RIAFTC) Cai et al. (2008) and the robust finite time fault-tolerant control (RFTFTC) Hu et al. (2012b). The resulting values of Pm and PE are listed in Table 7.2. Comparing Table 7.1 with Table 7.2, we find the following: 1) Whether reaction wheel faults occur or not by tuning control parameters, RFFTC can achieve the same Pm as the proposed control scheme because RFTFTC can accomplish the attitude maneuver in finite time. 2) No matter whether actuator faults occur or not, a smaller Pm is obtained from RIAFTC in comparison with the proposed approach. That is because the RIAFTC only guarantees the attitude-tracking error to be within a small set rather than zero. It would take much more time to satisfy the required attitude pointing accuracy and stability. Consequently, the value of Pm from RIAFTC is inferior to the proposed methodology. 3) When performing attitude stabilization and tracking maneuvers, the proposed methodology consumed less energy than both RAFTC and RFTFTC (i.e., a smaller PE was obtained from the designed scheme than from RAFTC and RFTFTC). This occurs because RAFTC and RFTFTC are inherently passive FTC methods.

224 Fault-Tolerant Attitude Control of Spacecraft

4) Although the proposed control is an active FTC, it can reconstruct and compensate reaction wheel faults in real time. The RAFTC and RFTFTC are much more conservative than the proposed control scheme.

7.3 Active attitude stabilization FTC without rate sensors In this subsection, we present a fault-tolerant control approach without rate sensors for the attitude stabilization of a satellite being developed. External disturbances, reaction wheel faults, actuator saturation, and unavailable angular velocity are addressed. We propose a sliding-mode observer by using attitude feedback only and estimate the unavailable angular velocity by this observer in finite time. Using the attitude and the estimated velocity, we propose another sliding-mode observer to reconstruct actuator faults and disturbances. We prove that reconstruction with zero observer error is achieved in finite time. With the reconstructed value, we then develop a velocity-free controller to asymptotically stabilize the attitude. We also provide simulation results to verify the effectiveness of the proposed approach.

7.3.1 Problem formulation 7.3.1.1 Attitude maneuvers sequence in the planned missions Fig. 7.12 illustrates the fundamental operation modes of the satellite. Among these modes, taking images in sun light condition and downloading image data during eclipse period are two main and challenging missions. Attitude stabilization maneuver is started at the point A, where the sun light is tangent to the orbit. This maneuver ensures the satellite to be oriented toward the Earth during the eclipse period. Data such as images can thus be downloaded to the ground station. When the satellite is at the point B, large angle attitude maneuver is going to be performed. The mounted CCD camera is ensured to be toward specific high-priority areas, and high-resolution images are taken. In any imaging or communication mission, the satellite undergoes a number of attitude maneuvers during each pass above the horizon. The time sequence of the attitude maneuvers used to accomplish the planned two missions is illustrated in Fig. 7.13. After the satellite separates from the launch vehicle at t = t0 , rate damping control maneuver is performed in the first orbital period OT . Thereafter, attitude acquisition maneuver is finished during the second orbital period. It then follows rotation spin-up of the reaction wheels in the third orbital period. Starting in the fourth orbital period, attitude stabilization maneuver is conducted to establish high pointing accuracy and stability. Suppose that those two planned missions are started at t = t1 (t1 > t0 + 4OT ). Then large angle attitude maneuver and attitude stabilization are alternately conducted in one orbital period. After rate damping and attitude acquisition maneuvers, the angular velocity and the attitude of the satellite are maintained within 1 deg/s and 0.5 deg,

Active fault-tolerant attitude control Chapter | 7

225

FIGURE 7.12 Operation modes of the satellite.

FIGURE 7.13 Time sequence of the attitude maneuvers for the scientific mission.

respectively. Moreover, every time, at the end of the large angle attitude maneuver for image taking mission, the angular velocity and the attitude are planned to be within 2 deg/s and 1 deg by using thrusters, respectively. That is, the accomplishment of image taking mission will supply the attitude stabilization maneuver for downloading data with small initial velocity and small initial attitude angle.

226 Fault-Tolerant Attitude Control of Spacecraft

7.3.1.2 Satellite model description The orientation of the satellite with respect to Fo is obtained by a yaw-pitch-roll sequence of rotations. For the considered rigid satellite, its model can be given by Xiao et al. (2011) ω = ωbo − ωc (),

(7.39)

×

(7.40)

J ω˙ = −ω J ω + τ + d,

where  = [θ, φ, ψ] is the attitude Euler angles vector, J = diag{J1 , J2 , J3 } is the inertia matrix, τ ∈ R3 is the total torque generated by all reaction wheels, d ∈ R3 is the external disturbance, ω ∈ R3 is the angular velocity of the satellite ˙ is the angular velocity in Fo with given in Fb , and ωbo = R() ⎡ ⎤ 1 0 − sin φ ⎢ ⎥ R() = ⎣ 0 cos θ (7.41) sin θ cos φ ⎦ , 0 − sin θ cos θ cos φ ⎡ ⎤ cos φ sin ψ ⎢ ⎥ ωc () = ω0 ⎣ cos θ cos ψ + sin θ sin φ sin ψ ⎦ . (7.42) − sin θ cos ψ + cos θ sin φ sin ψ Using (7.41)–(7.42), we can rewrite the attitude kinematics (7.39) as ˙ ω = χ + g 1 (ω0 , , )

(7.43)

˙ − ωc − χ. Substiwith χ = [θ˙ − ω0 ψ, φ˙ − ω0 , ψ˙ + ω0 θ ] and g 1 = R() tuting (7.43) into (7.40) gives the model for microsatellite attitude dynamics: ¨ + C ˙ + K + g 2 + g 3 = τ + d, J

(7.44)

where g 2 = g × 1 J (χ + g 1 ) + J

d(g 1 ) , K = ω02 diag{J2 − J3 , 0, J2 − J1 }, dt ⎡

⎤ 0 0 −1 ⎢ ⎥ C = ω0 (J1 − J2 + J3 ) ⎣0 0 0 ⎦ , 1 0 0 and ⎡

⎤ ˙ ψ˙ + ω0 θ ) −(J2 − J3 )φ( ⎢ ⎥ g 3 = ⎣(J1 − J3 )(θ˙ ψ˙ + ω0 θ θ˙ − ω0 ψ ψ˙ − ω02 θ ψ)⎦ . ˙ θ˙ − ω0 ψ) ˙ −(J1 − J2 )φ(

Active fault-tolerant attitude control Chapter | 7

227

As the small attitude deviation in Euler angles and angular velocity are guaranteed after attitude acquisition or image taking mission, the term g1 is thus a vector with small values. The angular acceleration of the satellite is also bounded. Hence g 2 and g 3 are bounded. In addition to Euler angles representation, the unit-quaternion can also be used to represent attitude. The corresponding quaternion of the Euler angles   4 2  is denoted by q = [q0 , q  v ] ∈ R , q0 + q v q v = 1, and ⎡

cos θ¯ cos φ¯ cos ψ¯ + sin θ¯ sin φ¯ sin ψ¯ ⎢ ⎢ sin θ¯ cos φ¯ cos ψ¯ − cos θ¯ sin φ¯ sin ψ¯ q =⎢ ⎢ ⎣ cos θ¯ sin φ¯ cos ψ¯ + sin θ¯ cos φ¯ sin ψ¯ − sin θ¯ sin φ¯ cos ψ¯ + cos θ¯ cos φ¯ sin ψ¯

⎤ ⎥ ⎥ ⎥ = f (), ⎥ ⎦

(7.45)

where θ¯ = 0.5θ , φ¯ = 0.5φ, and ψ¯ = 0.5ψ. Using the unit-quaternion q to represent satellite attitude, another nonlinear model of satellite attitude system can be described by Xiao et al. (2013) q˙0 = −0.5e ωbo , ×

q˙v = 0.5(q v + q0 I 3 )ωbo , J ω˙ bo = −ω× bo J ωbo

+ H 1 + τ + d,

(7.46) (7.47) (7.48)

× × where H 1 = (J ω× bo − ωbo J )R(q)ωoi − (R(q)ωoi ) J (ωe + R(q)ωoi ), ωoi =  × [0, −ω0 , 0] , and R(q) = (q02 − q  v q v )I 3 + 2q v q v − 2q0 q v . Actually, the term  [q0 , q v ] in (7.46) and (7.47) denotes the satellite relative attitude between Fb and Fo . For the considered satellite, aerodynamic torque, gravity-gradient torque, the solar radiation torque, and Earth magnetic torque are the primary disturbances for d in (7.40). Using the calculation in Xiao et al. (2011) and the main satellite physical parameters including altitude and structure, we can estimate the upper bound of d can be estimated as dmax = 2.6 × 10−4 Nm, that is, d(t) ≤ dmax .

7.3.1.3 Actuator fault Attitude stabilization maneuver is planned to be performed using four reaction wheels. Thus, taking reaction wheel faults into account, the total control torque τ has the form τ = Du, u = un + B(t − Tfo )uf ,

(7.49)

where D is the reaction wheel distribution matrix, u ∈ R4 is the actual torque generated by four wheels, un ∈ R4 denotes the nominal/commanded torque, and the term B(t − Tfo )uf ∈ R4 denotes the deviation in output torque due to the occurrence of a fault. The matrix B(t − Tfo ) ∈ R4 denotes the time profiles of a

228 Fault-Tolerant Attitude Control of Spacecraft

fault that occurs at some unknown time Tfo = [Tfo1 , Tfo2 , Tfo3 , Tfo4 ] ∈ R4 , and uf is the nonlinear fault function. Let the fault time profile B(t − Tfo ) be a diagonal matrix of the form B(t − Tfo ) = diag(b1 (t − Tfo1 ), b2 (t − Tfo2 ), b3 (t − Tfo3 ), b4 (t − Tfo4 )), (7.50) where bi : R → R is a function representing the time profile of a fault affecting the ith reaction wheel, i = 1, 2, 3, 4, and this profile is modeled by ⎧ ⎨0 if t < Tfoi , (7.51) bi (t − Tfoi ) = o ⎩1 − e−ai (t−Tf ) if t ≥ T o , fi where the scalar ai > 0 denotes the unknown fault evolution rate. Small values of ai characterize slowly developing faults, also known as incipient fault. For large values of ai , the time profile of bi approaches a step function that models abrupt fault.

7.3.1.4 Problem statement To provide theoretical support for next generation microsatellite attitude control subsystem design without rate sensors, the objective of this work is designing a commanded control un to asymptotically stabilize the attitude, that is , q v → 0 q0 → 1,  → 0, and ωbo → 0, even in the presence of external disturbances, fault (11), and reaction wheel saturation umax The controller should be designed with angular velocity eliminated, that is, ωe and  are not required. 7.3.2 Attitude FTC without angular velocity measurements Define the matrix P = 0.5(q v × + q0 I 3 ). From (7.47) we obtain that ωbo = P −1 (q e )q˙v . Substituting this equation into (7.48) and premultiplying both sides of the resulting expression by (P −1 ) J P −1 lead the nonlinear attitude model (7.46)–(7.48) with faulty input (7.49) to J ∗ q¨v + C q˙v + F  H 2 = F  [Dun + DB(t − T0 )uf + d],

(7.52)

where F = P −1 , C = F  J F˙ − F  (J F e˙ )× F , J ∗ = F  J F , and H 2 = [J (P e˙ )× − (P e˙ )× J ]R(q e )ωoi − (R(q e )ωoi )× J (P e˙ + R(q e )ωoi ). Note that J ∗ is symmetric positive definite and J˙∗ − 2C is skew-symmetric such that ∗ a  (J˙ − 2C)a = 0 for all a ∈ R3 . Remark 7.3. Obtaining (7.52) requires that P is invertible. The condition det(P ) = 0.5q0 (t) = 0 should be guaranteed to avoid the singularity of P for t ≥ 0. It thus requires the initial conditions such that q0 (0) = 0 and the subsequently designed controller ensuring q0 (t) = 0 for t > 0. Regarding the restriction on the initial conditions, we know from q02 + q  v q v = 1 that the desired

Active fault-tolerant attitude control Chapter | 7

229

trajectory can always be initialized to meet q0 (0) = 0. Therefore the initial condition restriction is actually a very mild restriction on the desired trajectory. To handle faults and disturbances, DB(t − Tfo )uf and d are lumped into a single fault vector ul , that is, ul = DB(t − Tfo )uf + d. In this section, we propose a fault-tolerant attitude control without angular velocity measurement to stabilize attitude of the satellite. The structure of the closed-loop system is shown in Fig. 7.14. This control structure includes three modules: a sliding˙ with zero observer error in finite time, mode observer designed to estimate  a reconstruction module synthesized to reconstruct the lumped fault in finite time, and a velocity-free controller proposed by using the reconstructed fault information and the satellite attitude. The detailed time-sequence of implementing the proposed control scheme is described as follows. Starting at t = t0 + 2OT : The measurable attitude Euler angles  are fed ˙ At the back to the sliding-mode observer module to obtain the estimate of . ˙ same time, q and q˙ are calculated by using , the estimation of , and (7.45). Starting at t = t0 + 3OT : Injecting q and q˙ into the reconstruction module, precise information of the lumped fault ul is obtained. Suppose that the stabilization maneuver is required to start at ts (ts ≥ t0 + 3OT ): Using the attitude q and the reconstructed fault, the controller module should be implemented. Assumption 1: After attitude acquisition maneuver and rotation spin-up of reaction wheel, the satellite attitude satisfies the constraint F  (t0 + 3OT )dmax  umax . Assumption 2: When rotation spin-up of reaction wheels is finished at t = t0 + 3OT , all reaction wheels are healthy, that is, uf (t0 + 3OT ) = 0.

FIGURE 7.14 The proposed fault-tolerant attitude control system of the microsatellite without rate sensor.

230 Fault-Tolerant Attitude Control of Spacecraft

7.3.2.1 Sliding-mode observer design ˙ system (7.44) with reaction wheel Defining two new states x 1 = , x 2 = , fault (7.49) can be rewritten as 



0 0 I3 x1 x˙ 1 = + x˙ 2 J −1 D(un + B(t − Tfo )uf ) −J −1 K −J −1 C x 2

 0 + . (7.53) −1 J (d − g 2 − g 3 ) Because attitude sensors are equipped to measure satellite attitude, the attitude Euler angles  are available. The system output defined by yo =  is thus available. To estimate the states x 1 and x 2 in finite time, we propose an equivalent output injection sliding-mode observer: x˙ˆ 1 = xˆ 2 + λo1 sgn(y o − xˆ 1 ), x˙ˆ 2 = −J −1 K xˆ 1 − J −1 C xˆ 2 + δo (λo2 sgn(x˜ 2 − xˆ 2 )) + J −1 Dun ,

(7.54) (7.55)

where x˜ 2 = xˆ 2 +(λo1 sgn(y o − xˆ 1 ))eq , xˆ i , i = 1, 2, is the estimate of xi ·λo1 , and λo2 are positive scalars. The term δo = 0 if x 2 − xˆ 2 = 0 and δo = 1 otherwise. Theorem 7.3. For the plant (7.44) subject to reaction wheel faults (7.49) and with the observer (7.54)–(7.55) applied, there exist λ01 λo2 such that the estimated states xˆ i , i = 1, 2, converge to its actual states xi with zero errors in finite time, respectively. Proof. Defining the observer errors ε01 = xˆ 1 − x 1 and εo2 = xˆ 2 − x 2 , from (7.53)–(7.55) we can obtain the observer error dynamics: ε˙ o1 = εo2 − λo1 sgn(ε o1 ), −1

ε˙ o2 = g 4 (x 2 , xˆ 2 ) − J Kε o1 − J + δo (λo2 sgn(x˜ 2 − xˆ 2 )),

(7.56) −1

DB(t − T0 )uf (7.57)

where g 4 (x 2 , xˆ 2 ) = J −1 [Cx 2 − C xˆ 2 − (d − g 2 − g 3 )]. Consider the candidate Lyapunov function Vo1 = 0.5ε o1 ε o1 . It then follows from (7.56) that V˙o1 ≤ −εo1 (λo1 − εo2 ). We can choose λo1 > εo2  + o1 with positive scalar o1 . Then # V˙˙o1 ≤ −o1 εo1  = −o1 2Vo1 .

(7.58)

This ensures that V˙o1 < 0 for all εo1  = 0 and also the finite-time convergence of ε o1 to the sliding surface ε o1 = 0. Alternatively, choosing λo1 > maxt∈[t0 +2OT ,T01 ] εo2 (t), where To1 denotes the time it takes for εo1 to converge to the sliding surface, allows a fixed value for λo1 , and ensures that

Active fault-tolerant attitude control Chapter | 7

231

V˙o1 < 0 for all εo1  = 0. To show that the convergence of εo1 to zero is achieved in To1 , integrating both sides of (A3) from t0 + 2OT and t yields εo1 (t) ≤ εo1 (t0 + 2OT ) − o1 [t − (t0 + 2OT )]. Then, for all t ≥ To1 = t0 + 2OT + εo1 (t0 + 2OT )/o1 , we obtain εo1 (t) ≡ 0; εo1 will reach the sliding surface εo1 = 0 by the time To1 . On that surface, ε o1 = ε˙ o1 = 0, and solving for the equivalent output injection yields (λo1 sgn(ε o1 )) = εo2 . This leads to x˜ 2 = x 2 . After time To1 , the error dynamics (7.56)–(7.57) has the form ε˙ ρ1 = 0, ε˙ o2 = g 4 (x2 , xˆ2 ) − λo2 sgn(ε o2 ) − J

(7.59) −1

D[B(t − T0 )uf ].

(7.60)

ε + At this time, define another candidate Lyapunov function Vo2 = 0.5εo1 o1 0.5ε o2 ε o2 . From (7.59)–(7.60) it follows that −1 ˆ 2 ) − ε V˙o2 ≤ −λo2 εo2  + ε D[B(t − T0 )uf ] o2 g 4 (x 2 , x o2 J

≤ −εo2 {λo2 − g 4 (x2 , xˆ2 ) − 2J −1 Dumax }.

(7.61)

Choosing λo2 > g 4 (x 2 , xˆ 2 ) + 2J −1 Dumax + o2 , o2 > 0, results in V˙o2 ≤ −o2 εo2  < 0 for all εo2  = 0. This guarantees that ε o2 reaches the sliding surface ε o2 = 0 in finite time. We can alternatively choose a fixed value for λo2 as λo2 > maxt∈[T01 ,T02 ] [g 4 (x2 , xˆ2 ) + 2J −1 Dumax ], where To2 is the time required for εo2 to reach the surface εo2 = 0. Integrating both sides of V˙o2 ≤ −o2 εo2  from To1 and t yields εo2 (t) ≤ εo2 (To1 ) − o2 (t − To1 ).

(7.62)

Hence solving (7.62) yields To2 = To1 + εo2 (To1 )/o2 and ε o2 (t) ≡ 0 for all t ≥ To2 . This completes the proof. From Theorem 7.3 and its proof we obtain that xˆ 2 converges to x2 in finite time To2 by only using the measurable attitude , that is, xˆ 2 (t) ≡ x 2 (t) for all t ≥ To2 . According to (7.45), q is differentiable. Furthermore, q˙e can be calculated by differentiating both sides of (7.45). Denote q˙ as q˙ = f˙ () = f1 (, θ˙ ). Then by the time To2 we have q˙ = [ q˙0

 q˙v  ] = f 1 (, xˆ 2 ).

(7.63)

7.3.2.2 Fault reconstruction module design By introducing an generalized inertia J g = J ∗ e˙ and applying the skewsymmetric matrix J ∗ − 2C and (7.52) we get J˙ g = F  Dun − τ¯ l − H 3 with ∗ ˙ + F  H 2 and τ¯ l = −F  ul . Now the residual vector is deH 3 = − 12 e˙  ∂J ∂e e

232 Fault-Tolerant Attitude Control of Spacecraft

fined by

 r(t) = kr

t

t0 +3OT

[F  Dun − H 3 − r(s)]ds − kr J g

(7.64)

with scalar kr > 0. This residual satisfies the dynamics r˙ = −kr r + kr τ¯ l .

(7.65)

Based on (7.45) and (7.63), we can see that r can be obtained with only attitude measurements . Furthermore, because the continuous function B(t − T of ) in (7.50) is differentiable, τ¯ l can be assumed to be a differentiable function with time derivative ur ∈ R3 . The problem of reconstructing the evolution of τ¯ l can be formulated as observing the state of a linear system driven by the residual r and an unknown input. The linear system is expressed as ξ˙ 1 = −kr ξ 1 + kr ξ 2 , ξ˙ 2 = ur ,

(7.66) (7.67) (7.68)

yr = ξ 1,

where ξ 1 = r, ξ 2 = τ¯ l , ur is the unknown input, and y r is the measurable output. According to the operating principle of reaction wheel, the faulty torque B(t − Tfo )uf is a function of the rotor speed of reaction wheel. Thus the time derivative of B(t − Tfo )uf is a function of the acceleration of the rotor. Due to physical limitation, this acceleration is bounded and can be obtained when purchasing reaction wheel. According to the calculation of d in [27], the time derivative of d is also bounded. It is thus reasonable to make following assumption. Assumption 7.1. Although the input ur is unknown, its amplitude is bounded by a constant ζ > 0, that is, ur  ≤ ζ . To reconstruct the states of system (7.66)–(7.68), a sliding-mode observerbased reconstruction scheme is designed as ξ˙ˆ 1 = −kr ξˆ 1 + kr ξˆ 2 − r1 sgn(ε r1 ) − r2 εr1 , ξ˙ˆ = − ε −  ξ m1 −  sgn(ξ ), 2

r3 r1

r4

v r1

r5

v

(7.69) (7.70)

where ξ v = 1 sgn(ε r1 ), εr1 = ξˆ 1 − y r , ri , i ∈ {1, 2, 3, 4, 5}, are gains, and m1 and n1 are positive odd integers such that m1 < n1 .   ˆ The reconstruction error εr = [ε r1 , ε r2 ] with ε r2 = ξ 2 −ξ 2 is defined to examine the performance of that reconstruction module. Subtracting (7.66)–(7.67) from (7.69)–(7.70) results in the dynamics of the error ε r : ε˙ r1 = − kr εr1 + kr εr2 − r1 sgn(ε r1 ) − r2 εr1 ,

(7.71)

Active fault-tolerant attitude control Chapter | 7

ε˙ r2 = − r3 εr1 − r4 ξ v 

m1 − r5 sgn(ξ v ) − ur . n1

233

(7.72)

Because the fault reconstruction module is started at t = t0 + 3OT while it is not applied for t < t0 + 3OT , it follows from (7.64) that ξ1 (t0 + 3OT ) = −kr Jg (t0 + 3OT ), which leads to ξ2 (t0 + 3OT ) = −F  (t0 + 3OT )d(t0 + 3OT ). Lemma 7.1. Choose the initial values of observer (7.69)–(7.70) as ξˆ 1 (t0 + 3OT ) = −kr J g (t0 + 3OT ) and ξˆ 2 (t0 + 3OT ) = 0. Then the signal εr in the reconstruction error system (7.71)–(7.72) is uniformly ultimately bounded.  Proof. Choosing a candidate Lyapunov function Vr1 = 0.5ε r1 ε r1 + 0.5ε r2 ε r2 for the (7.71)–(7.72), we have    V˙r1 = − kr ε  r1 ε r1 + kr ε r1 ε r2 − r1 ε r1 sgn(ε r1 ) − r2 ε r1 ε r1  m1 /n1  − r3 ε − r5 ε r2 ε r1 − r4 ε r2 ξ v  r2 sgn(ξ v ) − ε r2 ur √ √ m /n ≤ ( 3r4 r11 1 + 3r5 + ζ )ε r  − ε  (7.73) r Mε r − r1 ε r1 ,

 (kr + r2 )I 3 −kr I 3 where M = . Because kr , r2 , and r3 are positive, M r3 I 3 0 is positive definite, and λmin (M) > 0. Further, √ √ m /n V˙r1 ≤ −λmin (M)ε r 2 + ( 3r4 r11 1 + 3r5 + ζ )εr  √ √ m /n (7.74) = −εr [λmin (M)ε r  − ( 3r4 r11 1 + 3r5 + ζ )].

If εr is outside the compact set D = {εr : εr  ≤ πr }, where πr = √ √ m /n ( 3r4 r11 1 + 3r5 + ζ )/λmin (M), then V˙r1 < 0, and εr  will decrease monotonically. A decreasing value of Vr1 eventually drives εr into the set D in finite time, and it will never go outside of D. Thus the set D is attractive. According to Lyapunov theory and the LaSalle extension, εr is uniformly ultimately bounded. Further, from the initial values of ξˆ1 and ξˆ2 we obtain that εr (t0 + 3OT ) = ξ 2 (t0 + 3OT ) = F  (t0 + 3OT )d(t0 + 3OT ) ≤ F  (t0 + 3OT )dmax .

(7.75)

Then ε r  ≤ max{πr , F  (t0 + 3OT )dmax } is guaranteed for all t ≥ t0 + 3OT . Theorem 7.4. Starting at t0 + 3OT , with application of the fault reconstruction scheme (7.69)–(7.70), choose kr and ri i = 1, 2, 3, 4, 5, such that ⎧ √ n1 ((n1 −m1 ) ⎫ √ ⎪ ⎪ 3kr r4 + 3r5 +ζ ⎪ + δ1 , ⎪ ⎪ ⎪ ⎬ ⎨ λmin (M)  √ n1 /m1  (7.76) r1 > max kr 3r5 +ζ +δ1 λmin (M) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎭ ⎩ 3r5 +ζ +δ1 λmin (M)

234 Fault-Tolerant Attitude Control of Spacecraft

r5 kr − ζ > 0,

(7.77)



πr ≤ F (t0 + 3OT )dmax ,

(7.78)

where δ1 is a positive scalar. Then εr1 ≡ 0 and εr2 ≡ 0 will be achieved in finite time. Moreover, the lumped fault ul is reconstructed by −P  ξˆ 2 in finite time. Proof. The proof uses Lyapunov stability theory and is separated into two parts. P1) Finite-time convergence of εr1 : Consider a Lyapunov function Vr2 =  ε . From (7.71) we obtain that 0.5εr1 r1 (7.79) V˙r2 ≤ −(kr + r2 )ε r1 2 − (r1 − kr εr2 )ε r1 . √ √ Rearranging r1 > [δ1 + ( 3kr r4 + 3r5 + ζ )/λmin (M)]n1 /(n1 −m1 ) and r1 > √ ( 3r5 +ζ )+δ1 λmin (M) n1 /m1 ) in (7.76) yields ( kr √ 3r5 +ζ +δ1 λmin (M)

√ √ m /n m /n 3kr r4 r11 1 + ( 3r5 + ζ )r11 1 m /n + δ1 r11 1 r1 > λmin (M) √ √ kr r4 m1 /n1 3 + kr ( 3r5 + ζ ) + δ1 = kr πr + δ1 . > λmin (M)

(7.80)

From Lemma 7.1 and (7.80) we have r1 − kr εr2  > δ1 in finite time and # (7.81) V˙r2 < −δ1 εr1  = −δ1 2Vr2 < 0 for εr1  = 0. Integrating both sides of (7.81) from t0 + 3OT to t yields εr1 (t) ≤ εr1 (t0 + 3OT ) − δ1 [t − (t0 + 3OT )] and εr1 (t) ≡ 0 for t ≥ Tr1 = t0 + 3OT + εr1 (t0 + 3OT )/δ1 . Thus we conclude that the error state εr1 reaches the sliding surface εr1 = ε˙ r1 = 0 in finite time Tr1 . Moreover, by the time t = Tr1 solving for the equivalent output injection yields (ξ v )eq = (r1 sgn(ε r1 ))eq = kr εr2 .

(7.82)

P2) Finite-time convergence of εr2 : After time Tr1 , substituting εr1 = ε˙ r1 = 0 and (7.82) into (7.72) yields m1

ε˙ r2 = −r4 kr ε r2  n1 − r5 sgn(kr εr2 ) − ur .

(7.83)

 ε . With (7.83) and (7.77), Define another Lyapunov function as Vr3 = 0.5εr2 r2 we have m /n  V˙r3 = εr2 (−r4 kr 1 1 εr2 m1 /n1 − r5 sgn(kr εr2 ) − ur ) m1 /n1

≤ −r4 kr

εr2 (m1 +n1 )/n1 − (r5 kr − ζ )εr2 

m1 /n1 (m1 +n1 )/(2n1 )

< −r4 kr

2

(m1 +n1 )/(2n1 )

Vr3

.

(7.84)

Active fault-tolerant attitude control Chapter | 7 m1 +n1 2n1

Because m1 and n1 are odd, m1 + n1 is always even. Using Vr2 solving (7.84) lead to Vr2 (t) ≡ 0 for t ≥ Tr2 =

235

> 0 and

(n −m1 )/(2n1 ) (Tr1 ) m1 /n1 r4 kr (n1 −m1 )

2(n1 −m1 )/(2n1 ) n1 Vr2 1

+

Tr1 · Accordingly, we find that εr2 (t) ≡ 0 for all t ≥ Tr2 . With the definition of εr2 and τ¯l , we have τ¯l (t) ≡ ξˆ2 (t), and ul (t) ≡ −P  τ¯ l (t) ≡ −P  ξˆ 2 (t) for t ≥ Tr2 · ul is thus reconstructed by P  ξˆ 2 in finite time Tr2 .

7.3.2.3 Velocity-free fault tolerant controller design In Theorem 7.4 the lumped fault ul is reconstructed by −P  ξˆ 2 in finite time Tr2 . Thus we present the following theorem to accomplish attitude stabilization maneuver. Theorem 7.5. Consider the satellite attitude model (7.46)–(7.48) with actuator faults (7.49), observer (7.54)–(7.55), and the fault reconstruction scheme (7.69)–(7.70). Let the commanded control un be designed as un = − kc1 D † [q v + kc2 P  Tanh(μ + kc2 q v )] + D † (R(q)ωoi )× J R(q)ωoi + D † P  ξˆ 2 , μ = − kc3 Tanh(μ + kc2 q v ),

(7.85) (7.86)

where μ = [μ1 , μ2 , μ3 ] ∈ R3 , kci , i = 1, 2, 3, are positive gains. Then the satellite attitude is asymptotically stabilized, that is, q v → 0, q0 → 1, ωbo → 0, and  → 0. Further, reaction wheel saturation problem is solved with the control torque bounded by |ui | ≤ umax , i = 1, 2, 3, 4, provided that the control gains kc1 and kc2 are chosen such that D i [kc1 (1 + kc2 ) + ω02 J  + dmax + F  (t0 + 3OT )dmax ] ≤ umax . (7.87) Proof. The proof is separated into two parts. P1) Stability analysis of the closed-loop system: Consider the following radially unbounded positive-definite Lyapunov function candidate for the system: $ ω bo J ωbo + q v  q v + (1 − q0 )2 + ln(cosh(μi + kc2 q v i)). 2kc1 3

Vc =

(7.88)

i=1

Using (7.46)–(7.48) and (7.85)–(7.86), we obtain V˙c =

ω bo ˙  Tanh(μ/k ˙ c3 ). (P  ξˆ 2 + ul ) − μ kc1

(7.89)

Applying the reconstruction module leads to ul ≡ −P  ξˆ 2 for all t ≥ tr2 . Thereby after the time Tr2 , from (7.89) we have ˙  Tanh(μ/k ˙ c3 ) ≤ −kc3 Tanh(μ/k ˙ c3 )2 ≤ 0, V˙c = −μ

(7.90)

236 Fault-Tolerant Attitude Control of Spacecraft

which is a negative semidefinite decreasing function. Because the attitude stabi%∞ lization maneuver starts at ts , (7.90) leads to ts V˙c dt ≤ Vc (ts ) − Vc (∞) < ∞, and since Vc is radially unbounded, all the solutions are bounded. Hence, noting that V˙c is uniformly continuous, by invoking Barbalat’s lemma we have lim V˙c (t) = 0. Using the uniform continuity of V˙c results in lim V¨c (t) = 0. t→∞

t→∞

˙ ¨ = 0 and lim q˙ v (t) = 0. Then from (7.86) it follows that lim μ(t) = lim μ(t) t→∞

t→∞

t→∞

Thus from (7.47) we obtain lim ωe (t) = 0, because the nonsingularity of t→∞

P (q e ) is guaranteed. Finally, finally from those results, ul (t) ≡ −P  ξˆ 2 (t) for t ≥ Tr2 , (7.48), and (7.85) we calculate that lim q v (t) = 0. This implies that t→∞

lim q0 (t) = ±1.

t→∞

Since q = [1, 0] and q = [−1, 0] denote the same equilibrium point in the physical space, they yield the same attitude rotation matrix R(q). However, only q = [1, 0] is an attractor but a repeller equilibrium [17]. We can conclude that the equilibrium point of the closed-loop system is asymptotically stable. Using the algebraic relationship as shown in (7.45), lim q(t) = [1, 0] it leads to lim (t) = 0.

t→∞

t→∞

P2) Analysis of the designed control torque upper bound: From (7.49) and controller (7.85) we have u = − D † d − kc1 D † [q v + kc2 P  Tanh(μ + kc2 q v )] + D † ul + D † (R(q)ωoi )× J R(q)ωoi + D † P  ξˆ 2 .

(7.91)

Using R(q e ) = 1, P  = 1, and Tanh(a) ≤ 1 for all a, the actual output torque of each reaction wheel can be calculated as |ui | = − kc1 D i [e + kc2 P  Tanh(μ + kc2 q v )] + D i (R(q)ωxoi J R(q)ωoi + D i P  ξˆ 2 + D i ul − D i d. ≤ kc1 (1 + kc2 )D i  + ω02 J D i  + D i dmax + D i (P  ξˆ 2 + ul ), (7.92) where D i , i = 1, 2, 3, 4, is the ith row of the matrix D † . Because from the reconstruction module we can obtain that ul (t) ≡ −P  ξˆ 2 (t) for t ≥ Tr2 , this yields P  ξˆ 2 + ul = 0 for t ≥ Tr2 and P  ξˆ 2 + ul = P  εr2 (t) for t < Tr2 . From the proof of Lemma 1 and (30) we have εr  ≤ F  (t0 + 3OT )dmax This leads to P  ξˆ 2 + ul  ≤ P  ε r2  = F  (t0 + 3OT )dmax , t ≥ ts .

(7.93)

Then we have |ui | ≤ D i [kc1 (1 + kc2 ) + ω02 J  + dmax + F  (t0 + 3OT )dmax ]. (7.94)

Active fault-tolerant attitude control Chapter | 7

237

By proper selection of the controller gains kc1 and kc2 such that (7.87) we can ensure that |ui | remains less than a maximum allowable torque, that is, |ui (t)| < umax , t ≥ ts . From the paragraph following (7.90) in the proof of Theorem 7.5 we see that μ is used to guarantee the stability of closed-loop system by the proposed control but without angular velocity measurement. By application of μ and the ˙ → 0, and μ ¨ → 0. Then from proposed controller we obtain that V¨c → 0, μ (7.86) and (7.47) it follows that e˙ → 0 and ωe → 0, respectively. That is why μ is incorporated in controller (7.85). In fact, a sufficient condition is implicitly imposed in Theorem 7.5 for the proposed attitude controller, that is, when a reaction wheel undergoes faults, the combined torque generated by the remaining reaction wheels is sufficient enough to compensate those faults and external disturbances, and also to achieve three-axis attitude stabilization control. This means that the combined torque is still able to achieve robust control with respect to disturbances and reliable control with respect to reaction wheel faults. Otherwise, three-axis attitude stabilization maneuver will never be accomplished with any control approach in the literature. At this time, the satellite becomes underactuated, but it is not the main issue to be investigated in this work.

7.3.3 Simulations results As the initial step toward final experimental verification, we first numerically simulate the performance of the proposed control scheme. Nongyroscopic attitude sensors are equipped measure the attitude angle . Attitude sensors are modeled by a zero-mean Gaussian white-noise process with standard deviation σST = 35. To obtain high-accuracy attitude measurement, in attitude determination, we use an extended Kalman filter (EKF). Because observers (7.54)–(7.55) and (7.69)–(7.70) and controller (7.85) are inherently nonlinear, there does not exist a systematic or standard procedure to choose their gain values like in the case of traditional PID controller. Hence these gains are chosen by trial-and-error until good performance is obtained. On the other hand, as we see in the proof of Theorem 7.3, larger λo1 and λo2 will lead to smaller To1 and To2 obtained from observer (7.54)–(7.55). As shown in the proof of Theorem 7.4, different gains in the reconstruction scheme (7.69)–(7.70) will result in different finite times Tr1 and Tr2 . In addition, as shown in the proof of Theorem 7.5, different values of the control gains in the controller (7.85) will lead to different convergence times of the closedloop attitude system and may also result in different behavior of the control torque. Consequently, the gains for controller (7.85) are chosen as kc1 = 0.01, kc2 = 0.05, and kc3 = 1.5. The gains of observer (7.54)–(7.55) and the slidingmode observer (7.69)–(7.70) are chosen as λo1 = 0.75, λo2 = 6.5 r1 = 0.15, r2 = 0.05, r3 = 0.5, r4 = r5 = 0.00005, m1 = 19, n1 = 21, and kr = 0.15.

238 Fault-Tolerant Attitude Control of Spacecraft

The initial attitude angles in simulation are chosen randomly in the intervals −0.5◦ ≤ ψ(0) ≤ 0.5◦ , −0.5◦ ≤ φ(0) ≤ 0.5◦ , and −0.5◦ ≤ θ (0) ≤ 0.5◦ . The corresponding quaternion is e = [−0.000966, 0.000552, −0.000801] , whereas the initial velocity is ωbo (0) = [0.5702, 0.2467, −0.4887] deg/s.

7.3.3.1 Main parameters and hardware resources A microsatellite being developed in the detailed design phase consists of a cubic structure with a size of 500 mm × 500 mm × 500 mm. Its payload includes a push-broom type remote sensing camera and an antenna. Taking highresolution images of specific high-priority areas is the main mission of the satellite. The antenna is used to communicate with ground stations for receiving commands and downloading data. The orbit of the satellite is circular, with an altitude of 650 km and an inclination of 94.5 degrees. Its orbital rate is ω0 = 0.00109 rad/s with orbital period OT = 5893.69 s. The satellite mass is about 45 kg. The principal moments of inertia are 20.0 kg m2 , 22.5 kg m2 , and 25.2 kg m2 , respectively. The products of inertia are smaller than 0.2 kg m2 and thus can be neglected. Because of the requirement of high pointing accuracy with high-quality images taking by the satellite, a sophisticated attitude determination and control system (ADCS) for three-axis stabilization is necessary. The pointing accuracy of 0.005 degrees with 0.0012 deg/s stability of the platform has to be achieved for quality images and stored image data transmission. Due to its small size and with cost consideration, it is desirable to be without gyro/rate sensors installed on the satellite. Hence, stabilizing attitude by using attitude sensors only is investigated for this satellite. Consequently, only one three-axis magnetometer, two solid-state horizon sensors, and one two-axis coarse sun sensor are equipped to measure the satellite attitude. There is no rate gyro mounted to measure the angular velocity of the satellite. To increase attitude control reliability, the satellite uses four reaction wheels for attitude stabilization. Three wheels are mounted orthogonally, aligned with the satellite body axes +, +Y , and +Z, respectively, and a fourth, redundant, wheel is mounted skewed at equal angles (54.7 deg) to each of the body axes, aligned diagonally in the +, +Y, +Z quadrant. To solve the potential saturation problem of reaction wheels, three magnetic torque rods are additionally equipped to unloading the saturated control power. The maximum torque umax generated by each reaction wheel is 0.1 Nm, that is, umax = 0.1 Nm. 7.3.3.2 Reaction wheel fault scenarios The reaction wheel mounted in line with + axis loses 20% power when the attitude maneuver begins, thats is, Tfo1 = 0 a1 = 20, and uf 1 = −0.2un1 . The actuator mounted in line with the +Y axis (wheel No. 2) loses 45% of its power in the time interval between 4 and 10 seconds. Moreover, continuous generation of reaction torque fault with a value of −0.025 Nm will immediately occur,

Active fault-tolerant attitude control Chapter | 7

that is, a4 = ∞, and

 uf 2 =

−0.45un2 −0.025

if 4 ≤ t ≤ 10, if t > 10.

239

(7.95)

The wheel fixed in line with +Z axis (wheel No. 3) fails totally after 15 seconds, that is, uf 3 = −un3 , Tfo3 = 15, and a3 = 150, whereas the redundant wheel (wheel No. 4) decreases 90% of its torque after 8 seconds, that is, uf 4 = −0.9un4 , Tfo4 = 8, and a3 = 75.

7.3.3.3 System responses When reaction wheels undergo the above faults, with application of the proposed velocity-free fault-tolerant attitude stabilization control scheme (named as VFTASC), Figs. 7.15–7.19 illustrate its attitude control performance. It is interesting to see that the actuator saturation problem is addressed by the proposed scheme as the commanded control shown in Fig. 7.15. As shown in Fig. 7.16, the estimation error εo2 of sliding-mode observer for the immeasurable angu˙ will converge to zero in a finite time of 4.4 s, that is, lar velocity ω (or ) TO2 = 4.4. This successfully verifies the conclusion in Theorem 7.1 that the sliding observer state xˆ 2 can estimate the state x 2 with zero error in finite time. Fig. 7.17 shows the successful fault reconstruction using the incorporated reconstruction scheme (7.71)–(7.72). Accurate reconstruction is achieved within 100 seconds. This validates the result in Theorem 7.4. Moreover, as shown by the steady-state behavior of εr2 in Fig. 7.17, the accuracy of the reconstruction for ul is smaller than 1.2 × 10−6 Nm. Due to the accurate reconstruction of the reaction wheel faults and external disturbances supplied by scheme (7.71)–(7.72), controller (7.85) can completely compensate the effects of actuator faults and reject external disturbances. Thus thee controller produces an asymptotic stabilization of the angular velocity and the attitude. Although the angular velocity measurement is not used to implement the proposed controller, the actual satellite angular velocity is still shown in Fig. 7.18, whereas the unit attitude quaternion is presented in Fig. 7.19. As the steady-state behavior shown in Figs. 7.18 and 7.19, the attitude pointing accuracy achieves ei |≤ 4.0 × 10−5 level, which corresponds to the attitude angles  less than 0.005 degrees, and the attitude stability is within 0.001 deg/s. Therefore the attitude stability and pointing accuracy satisfy a set of stringent pointing requirements for downloading image data to ground station in the face of faults and disturbances, and thus the considered communication mission is considered as successfully accomplished. 7.3.3.4 Quantitative analysis The results are further compared with the unit quaternion-based output feedback control (UqOFC) Tayebi (2008) and the terminal sliding-mode-based velocity-free controller (TSMOBVFC) Xiao et al. (2014). Both TSMOBVFC

240 Fault-Tolerant Attitude Control of Spacecraft

FIGURE 7.15 The commanded control input.

FIGURE 7.16 The observer error εo2 .

FIGURE 7.17 The fault reconstruction error εr2 .

FIGURE 7.18 The angular velocity of the satellite.

FIGURE 7.19 The unit attitude quaternion of the satellite.

and UqOFC do not require the angular velocity measurement to implement their controllers. The simulation in the absence of actuator faults is also carried out to further validate the effectiveness of the proposed approach. In assessing the performance of these three schemes, two important criterions should be considered,

Active fault-tolerant attitude control Chapter | 7

241

the average consumption of energy (PE ) and the percentage of successfully accomplished mission (Pm ) with respect to the planned mission: PE =

1 Ttotal

 0

Ttotal

un (t)2 dt,

Pm =

Tsuccessfully × 100%. Ttotal

These two indexes quantitatively state how efficient the controller is. Here Ttotal denotes the attitude maneuver time period. It is chosen as Ttotal = 0.5OT = 2946.8 s according to the attitude stabilization mission. Tsuccessfully denotes the sum of the time period during which the attitude pointing accuracy is within 0.005 degrees, and its stability is within 0.0012 deg/s. Tsuccessfully is very crucial and is to be minimized so that data downloading is able to be carried out in a few tens of seconds during the period of a typical satellite passing over the ground tracking station. The larger Tsuccessfully is, the more image data can be downloaded. In addition, the larger PE is, the more energy is consumed during attitude stabilization maneuver. These two indexes are affected the worst during actuator failures and thus degrade the performance of the mission to a significant extent. In the absence or presence of reaction wheel faults, the corresponding performances of the indexes PE and Pm are shown in Figs. 7.20 and 7.21, respectively. It is found that: • When all actuators run normally, although TSMOBVFC can achieve the same value Pm (as shown in Fig. 7.20(a)) as VFTASC by tuning the gains of TSMOBVFC, the index value PE obtained from TSMOBVFC is larger than VFTASC. That is because although TSMOBVFC is able to handle actuator saturation and external disturbance as VFTASC, TSMOBVFC is actually a robust control scheme, and hence its controller has certain conservativeness. Due to the control term D † P  ξ 2 in (7.85), VFTASC can compensate the external disturbance in real time, and thus its conservativeness is much weaker. On the other hand, when reaction wheels undergo faults, the index value Pm from TSMOBVFC is less than VFTASC. This can be seen in Fig. 7.21(a). That is because TSMOBVFC can only handle partial loss of control power, but it is unable to handle the fault type occurring in the No. 2 wheel. • Because UQOFC is only able to solve the attitude stabilization problem of satellite subject to actuator saturation, the satellite should be free of external disturbance. As a result, when reaction wheels are healthy, due to the external disturbances, UqOFC will achieve a smaller Pm than VFTASC and TSMOBVFC, that is, Pm = 74.5%. That is because VFTASC and TSMOBVFC are capable to eliminate the effect of external disturbance on the control accuracy. Additionally, because UqOFC has no capability to tolerate actuator faults, a quite small Pm = 1.64% is obtained in the presence of faults. This can be seen in Fig. 7.21(a). • In the presence of actuator faults, both TSMOBVFC and UqOFC consumed more energy than VTFASC, as can be seen in Fig. 7.21(b). This is because

242 Fault-Tolerant Attitude Control of Spacecraft

TSMOBVFC and UqOFC were not able to handle the fault in the No. 2 reaction wheel. When that type of fault occurred, TSMOBVFC and UqOFC continued issuing the maneuvering control commend that may no longer be achievable by the system, in spite of the occurrence of the fault. Under this situation, the required control effort quickly saturated the actuators while striving to maintain the “healthy” maneuvering performance. Consequently, much more energy was consumed.

FIGURE 7.20 The indexes Pm and PE in the absence of reaction wheel faults.

FIGURE 7.21 The indexes Pm and PE in case of reaction wheel faults.

Based on the comparisons of PE and Pm in Figs. 7.20–7.21, we see that whether faults occur or not, the proposed control approach provides better performance than the controllers presented in Tayebi (2008) and Xiao et al. (2014). The proposed solution maintains good attitude control under both normal and fault conditions and provides a higher pointing accuracy to guarantee the satellite to download much more image data and thus accomplishes the planned mission as much as possible.

Active fault-tolerant attitude control Chapter | 7

243

7.4 Finite-time fault-tolerant attitude tracking control To achieve spacecraft attitude control under reaction flywheel failure, in this subsection, we propose a compensation control strategy. The schematic diagram of its closed-loop attitude control system is shown in Fig. 7.22. As shown in the figure, this system mainly includes three parts: nominal control quantity uc_nom , compensation control quantity uc_fault , and fault estimator. Specifically, we firstly design the terminal sliding-mode observer to accurately estimate the reaction flywheel fault and partial interference value, and then we design the compensation controller uc_fault based on this estimated value and add it to the nominal control uc_nom to form the total control amount uc . Compensation control for external disturbances and faults, and limited time control to ensure attitude tracking is guaranteed.

FIGURE 7.22 The closed-loop attitude-tracking system of the velocity-free fault-tolerant control.

Based on the above analysis, in this section, we propose a fault-tolerant attitude tracking control strategy without angular velocity feedback. The closedloop system diagram is shown in Fig. 7.22. This control method consists of three parts: One is designing a sliding-mode observer to accurately estimate the ˙ then according to this attiattitude Euler angle  and its derivative value ; tude estimation value, we use formulas (7.124) and (7.125) to calculate q and ˙ then we use this calculated value to design an estimator to accurately estiq; mate the generalized fault uf ; finally, we use the fault estimated value to design a feedback controller without angular velocity to achieve attitude-tracking control. According to the posture operation time sequence shown in Fig. 7.2, the time sequence for achieving this posture control method can be summarized as follows: 1) After the attitude acquisition is completed, it starts at time t = t0 + 2nT , the measurable attitude Euler angle  is fed back to the sliding-mode observer ˙ and the q and q˙ values are calculated to obtain the estimated values of  and , according to Eqs. (7.123) and (7.125); 2) Start at time t = t0 + 3nT , apply the q and q˙ values obtained according to the calculation, and run the fault estimator to obtain an accurate estimate of the generalized fault uf ;

244 Fault-Tolerant Attitude Control of Spacecraft

3) Assuming that the attitude-tracking operation starts at time ts ≥ t0 + 3nT , execute the feedback controller without angular velocity to achieve attitudetracking control.

7.4.1 Fault estimator design If the external interference d is regarded as the failure of the attitude-tracking system, then the failure of d and the reaction flywheel can be regarded as a generalized failure of the spacecraft attitude control system, that is, d + DB(t − 

T )ub = uf , and then the corresponding conversion model of the tracking system (7.46)–(7.48) at this time can be given as follows as similarly done in (5.3): M ∗ (q e )q¨ e + C ∗ (q e , q˙ e )q˙ e + Z  H 1 = Z  Duc + Z  uf .

(7.96)

To accurately estimate the broad fault uf , define the broad inertia moment matrix J m = M ∗ (q e )q˙ e , and the applied property is ˙ ∗ (q e )q˙ e = Z  Duc − u¯ f (t) − N(q e , q˙ e , ωd , ω˙ d ). J˙ m = M ∗ (q e )q¨ e + M (7.97) Among them, 1 ∂J · (q α ) u¯ f (t) = −Z  uf , N (q e , q˙ e , ωd , ω˙ d ) = − q˙  q˙ e + Z  H 1 . 2 e iq c According to Eq. (7.97), define the following residual vector: r(t) = − kJ m  t + k [Z (q e (s))Duc (s) − N (qe (s), q e (s), ωd (s), ω˙ d (s)) − r(s)]ds, 0

(7.98) where k ∈ R+ is a constant. At this time, according to Eq. (7.97), we can see that the residual r(t) satisfies r˙ = −kr + k u¯ f .

(7.99)

Since B(t − T ) and u¯ f is continuously differentiable, the derivative of the latter can be set as ν ∈ R3 . At this point, the estimate u¯ f can be transformed into an estimate of the state of the input linear system as follows: x˙ 1 = −kx 1 + kx 2 ,

(7.100)

x˙ 2 = ν,

(7.101)

y = x1,

(7.102)

Active fault-tolerant attitude control Chapter | 7

245

where x 1 = r(t), x 2 = u¯ f (t), ν is the unknown input of the system, y is the output of the measurable system, and the initial value x 2 (0) of the state is selected to satisfy x 2 (0) = 0 to ensure attitude control. The system is normal at the initial time. According to system (7.100)–(7.102), the estimated u¯ f is converted so that only the application system can measure the output y to estimate the system state x 2 (t). To this end, we design the following terminal sliding-mode observer: x˙ˆ 1 = −k xˆ 1 + k xˆ 2 − x ν − l2 e1 ,

(7.103)

q

x˙ˆ 2 = −l3 e1 − l4 [x ν ] p − l5 sgn(x ν ),

(7.104)

where xˆ 1 and xˆ 2 are the estimated values of the states x 1 and x 2 , respectively, x ν = [xν1 , xν2 , xν3 ] = l1 sgn(e1 ), e1 = xˆ 1 − y, li ∈ R+ , i ∈ {1, 2, 3, 4, 5}, are observer gains, and p ∈ R+ and q ∈ R+ are integers such that p > q.   6 ˆ 2 − x 2 , according to Define the observation error e = [e 1 , e2 ] ∈ R , e2 = x Eqs. (7.100)–(7.102) and (7.103)-(7.104). The following dynamic equation will be satisfied: e˙ 1 = −ke1 + ke2 − x ν − l2 e1 ,

(7.105)

q

e˙ 2 = −l3 e1 − l4 [x ν ] p − l5 sgn(x ν ) − ν.

(7.106)

Although the input ν is unknown, it is bounded by a constant π ∈ R+ , that is, ν ≤ π. In fact, the upper bound of ν mainly depends on the time characteristic function B ( t − T ); a larger value of ai will result in a larger value of ν, i ∈ {1, 2, 3, 4}. Lemma 7.2. For linear systems (7.100)–(7.102), we apply terminal slidingmode observers (7.103)–(7.104). If the initial value of the observer is chosen so that xˆ 1 (0) = x 1 (0) and xˆ 2 (0) = x 2 (0), then the observation error e is consistent and ultimately bounded stable (UUB).  Proof. Select a Lyapunov candidate function V1 = 0.5e 1 e1 + 0.5e2 e2 . According to Lemma 2.1 and formulas (7.105)–(7.106),    V˙1 = −(k + l2 )e 1 e1 + ke1 e2 − l1 e1 sgn(e1 ) − l3 e2 e1 q

  p − l4 e  2 [x v ] − l5 e2 sgn(x v ) − e2 v q √ √ ≤ −e Me − l1 e1  + ( 3l4 l1p + 3l5 + π)e,

(7.107)

(k + l2 )I3 −kI 3 ]. Since k, l2 , and l3 are all normal numbers, l3 I 3 03×3 the matrix M is positive definite, and λmin (M) > 0. Then from Eq. (7.107) we get where M = [

246 Fault-Tolerant Attitude Control of Spacecraft q √ √ V˙1 ≤ −λmin (M)e2 + ( 3l4 l1p + 3l5 + π)e q √ √ = −e[λmin (M)e − ( 3l4 l1p + 3l5 + π)].

If e >

√

1 λmin (M) (

q

3l4 l1p +

(7.108)

√ 3l5 + π)  λR , then V˙1 < 0. Therefore, when e

is outside the area D  {e|e ≤ λR }, V1 (t) is strictly decreasing, and finally it is guaranteed that e enters the area D. Once e enters the area D, according to V˙1 < 0, we see that e will always stay in the area D, so D is an attractive area. On the other hand, we know that e(0) = 0 is selected according to the initial value of the observer state. Therefore, according to the Lyapunov stability theory and LaSalle extension Khalil and Grizzle (2002), we can prove that the observation error e will eventually converge to the region D, so by Definition 2.1 e is uniformly ultimately bounded and stable (UUB), and the theorem is proved. Theorem 7.6. For the observation error system (7.105)–(7.106), if the observer gain is selected to satisfy ⎧ p/(p−q) ⎨ √3kl + √3l + π 4 5 + ε0 l1 > max , ⎩ λmin (M)  (7.109) ⎡ √ ⎤p/q ⎫ ⎪ ⎬ k 3l5 + π + ε0 λmin (M) ⎣ √ ⎦ , ⎪ 3l5 + π + ε0 λmin (M) ⎭ l5 k − π > 0,

(7.110)

where ε0 ∈ R+ is a constant, then the observation error system states e1 and e2 will reach the sliding-mode surface e1 = 0 and e2 = 0 in finite time, and u¯ f (t) will be accurately estimated in finite time, so that the generalized fault uf will also be accurately estimated within limited time. Proof. According to the Lyapunov stability theory, the proof of the theorem can be divided into two parts. Step 1. Analysis of the time-limited convergence of the observation error e1 : According to Eq. (7.105), we select the Lyapunov candidate function V2 (t) = 0.5e 1 e1 . Then V˙2 = e 1 (−ke1 + ke2 − v v − l2 e1 ) ≤ −(k + l2 )e1 2 − (l1 − ke2 )e1 . According to formula (7.109), we get q g √ √ q 3kl4 l1p + ( 3l5 + π)l1p + ε0 l1p l1 > λmin (M)

(7.111)

Active fault-tolerant attitude control Chapter | 7 q √ √ 3kl4 l1p + k( 3l5 + π) + ε0 > λmin (M) = kλR + ε 0 .

247

(7.112)

According to Lemma 7.1, ke ≤ kλR < l1 for any t ≥ 0, so according to formulas (7.111)–(7.112), we obtain that V˙2 < −ε0 e1  < 0 with e1 (t) = 0 s. At this time, according to comparative Lemma 2.2, we can prove that when t ≥ e1 (0)/ε0  tF 1 , we will have e1 (t) ≡ 0. Therefore, within a limited time t = tF 1 , the sliding-mode motion e1 = e˙ 1 = 0 will be generated, and the corresponding equivalent output (x ν )eq = (l1 sgn(e1 ))eq = ke2 will be obtained. Step 2. Convergence analysis for the limited time of observation error e2 : after a finite time tF 1 , e˙1 = 0, and q

q

e˙ 2 = −l4 k p [e2 ] p − l5 e 2 sgn(ke2 ) − ν.

(7.113)

Choosing another Lyapunov candidate function V3 (t) = 0.5e 2 e2 and applying Lemma 2.1, we have g

q

 p p V˙3 = e 2 {(−l4 k [e2 ] − l5 e2 sgn(ke2 ) − ν} q p

≤ −l4 k e2  g p

< −l4 k 2

p+q 2p

p+q p

V3 (t) ≡ 0 for any t ≥ tF 2  l4

− (l5 k − π)e2 

(V3 )

According to the solution of q < p →

(7.114)

p+q 2p

p+q 2p

p−q 2p 2pV3 (0) q p+q k p 2 2p (p−q)

. < 1, solving Eq. (7.114), we get that . Therefore e2 (t) ≡ 0 after the finite

time tF 2 , and the sliding-mode motion e2 = 0 will be realized within the time finite time tF 2 , so uf (t) = x 2 (t) = xˆ 2 (t) + e2 (t) for t ≥ tF 2 , that is, uf (t) can be accurately estimated by eˆ 2 (t) in a finite time tF 2 . According to the definition of u¯ f (t), we can obtain uf = −P  (q e )u¯ f (t). Therefore, when t ≥ tF 2 , the generalized fault uf can be accurately estimated by the formula uf ≡ −P  (q e )xˆ 2 (t),

(7.115)

and the estimation error is zero. So the theorem is proved. Remark 7.4. When the reaction flywheel is working normally, the generalized fault uf is the external interference suffered by the spacecraft, that is, d = uf . At this time, the estimator is still given by (7.103)–(7.104), and using formula (7.115), we accurately estimate the external interference.

248 Fault-Tolerant Attitude Control of Spacecraft

7.4.2 Sliding-mode observer design Although the controller designed in Section 7.3.3.1 can realize the finite-time control of attitude tracking, we do not consider the problems of unpredictable angular velocity and saturation of actuator control input. To this end, in this section, we propose a new attitude-tracking control method to explicitly solve the above problems. Neglecting the product of inertia of the spacecraft, we can write its moment of inertia as J = diag{J1 , J2 , J3 }. According to the contents in Sidi (1997) and Section 4.6.1, the attitude of the orbital system Fo relative to the own system Fb of the spacecraft can be rotated around the yaw, roll, and pitch axes, respectively, by ψ (yaw angle), φ (Rolling angle), and θ (pitch angle). At this time, the mathematical model of the spacecraft attitude can be expressed as Chen et al. (2000) ω = ωbo − ωc (),

(7.116)

J ω˙ = −ω× J ω + τ + d,

(7.117)

˙ where  = [φ, θ, ψ] is the attitude Euler angle vector, and ωbo = R() represents the angular velocity of the spacecraft on the orbit system F o , ⎡ ⎤ cos θ 0 − sin θ cos φ ⎢ ⎥ R() = ⎣ 0 (7.118) 1 sin φ ⎦, sin θ 0 cos θ cos φ ⎤ cos θ sin ψ + sin θ sin φ cos ψ ⎦, ωc () = ω0 ⎣ cos φ cos ψ sin θ sin ψ − cos θ sin φ cos ψ ⎡

(7.119)

where ω0 ∈ R+ is the orbital angular rate of the spacecraft. By Eqs. (7.118) and (7.119) the spacecraft attitude kinematics Eq. (7.116) can be rewritten as ˙ ω = χ + g 1 (ω0 , , ),

(7.120)

˙ − ωc () − χ. where χ = [φ˙ − ω0 ψ, θ˙ − ω0 , ψ˙ + ω0 φ] and g 1 = R() Substituting formula (7.120) into the attitude dynamic Eq. (7.117), we get ¨ + C¯  ˙ + K ¯ + g 2 + g 3 = τ + d, J where     d g 1 , χ + g 1 + J dt   ¯ = ω02 diag [J2 − J3 0 J2 − J1 ] , K g 2 = g × 1J

(7.121)

Active fault-tolerant attitude control Chapter | 7

249

⎡

⎤ 0 0 −1 ⎢ ⎥ C¯ = ω0 (J1 − J2 + J3 ) ⎣ 0 0 0 ⎦ , 1 0 0 ⎤   − (J2 − J3 ) φ˙ ψ˙ + ω0 θ  ⎥  ⎢ g 3 = ⎣ (J1 − J3 ) θ˙ ψ˙ + ω0 θ θ˙ − ω0 ψ ψ˙ − ω02 θ ψ ⎦ .   − (J1 − J2 ) φ˙ θ˙ − ω0 ψ˙ ⎡

After the attitude is captured and stabilized, the initial attitude and initial angular velocity of the spacecraft during attitude tracking are both small values, so the value of g 1 will be small and bounded. On the other hand, due to the finite and bounded angular acceleration of spacecraft attitude in aerospace engineering, g 2 and g 3 in Eq. (7.121) are bounded. On the other hand, according to the attitude Euler rotation angles ψ, φ, θ , applying formula (7.45), we can calculate the attitude quaternion q ob (ψ, φ, θ ) of the spacecraft orbit system Fo relative to the own system Fb of the spacecraft as follows: q ob (ψ, φ, θ )  f 1 ().

(7.122)

At this time, can be obtain the attitude quaternion q between the spacecraft body coordinate system Fb and the inertial coordinate system Fi by the formula q = q io  q ob (ψ, φ, θ ),

(7.123)

where q io = [cos( ω20 t ), 0, − sin( ω20 t ) 0] is the posture quaternion of the coordinate system Fi to Fo . ˙ = [φ, ˙ θ˙ , ψ] ˙  can be obtained, then q˙ ob (ψ, φ, θ ) = [q˙ bo , q˙ bo , q˙ bo , q˙ bo ] If  0 1 2 3 can be calculated according to Eq. (7.124). We denote ˙ q˙ ob (ψ, φ, θ )  f2 (, ).

(7.124)

Since qio can be differentiated, q˙ can be calculated according to Eqs. (7.123) and (7.124): ˙ q˙ = q˙ io  q ob (ψ, φ, θ ) + q io  f2 (, ).

(7.125)

˙ Then the attitude control system Define the state variable ξ 1 = , ξ 2 = . (7.121) can be rewritten as follows under the action of flywheel failure (2.27) and external interference:







 03×3 I3 0 ξ˙ 1 ξ1 = + ¯ −J −1 C¯ ξ2 J −1 D (uc + u¯ c ) −J −1 K ξ˙ 2

250 Fault-Tolerant Attitude Control of Spacecraft

+

0

  J −1 d − g 2 − g 3

 .

(7.126)

Since the spacecraft is equipped with an attitude sensor, the attitude Euler angle  is measurable. At this time, if the output of the spacecraft attitude control system is defined as ξ y = , then ξ y can be measured. To realize the accurate ˙ of the attitude within a limited time, we estimation of the Euler angle  and  design the following sliding-mode observer: ξ˙ˆ 1 = ξˆ 2 + λo1 sgn(ξ y − ξˆ 1 ), ¯ ξˆ 1 − J −1 C¯ ξˆ 2 + δo [λo2 sgn(ξ˜ 2 − ξˆ 2 )] + J −1 Duc , ξˆ 2 = −J −1 K

(7.127) (7.128)

where ξ˜ 2 = ξˆ 2 + [λo1 sgn(ξ y − ξˆ 1 )]eq , ξˆ i are the estimated values of ξ i (i = 1, 2), and λo1 ∈ R+ and λo2 ∈ R+ are the observer gains. When ξ 2 − ξˆ 2 = 0, δo = 0; otherwise, δo = 1. The equivalent output term [λo1 sgn(ξ y − ξˆ 1 )]eq is the semaphore λo1 sgn(ξ y − ξˆ 1 ), which is obtained through a low-pass filter Haskara (1998). Theorem 7.7. For the attitude control system (7.125) affected by external interference and reaction flywheel fault (2.27), if the sliding-mode observer (7.127)–(7.128) is used for state estimation, then the gains λo1 and λo2 ensure that the estimator ξˆ i converges to the estimated value of its true value ξ i (i = 1, 2) within finite time. Proof. It is proved that the definition state estimation error ε o1 = ξˆ 1 − ξ 1 and εo2 = ξˆ 2 − ξ 2 . Then according to formulas (7.126)–(7.128), the estimated error can be obtained to satisfy the dynamics equation ε˙ o1 = εo2 − λo1 sgn(ε o1 ),

(7.129)

¯ o1 + g 4 (ξ 2 , ξˆ 2 ) − J −1 Dub + δo [λo2 sgn(ξ˜ 2 − ξˆ 2 )], ε˙ o2 = −J −1 Kε (7.130) where g 4 (ξ 2 , ξˆ 2 ) = J −1 [Cξ 2 − C ξˆ 2 − (d − g 2 − g 3 )] Choose the Lyapunov candidate function Vo1 = 0.5ε o1 ε o1 . Then according to formula (7.129), we get V˙o1 ≤ −εo1 (λo1 − εo2 ). If λo1 satisfies λo1 > εo2  + o1 , where λo1 ∈ R+ is a constant, then # (7.131) V˙o1 ≤ −o1 εo1  = −o1 2Vo1 . This formula guarantees that for any εo1  = 0, we have V˙o1 < 0, so bmεo1 will reach the sliding surface, and thus bmεo1 = bm0 in finite time. Especially, if we choose λo1 > maxt∈[t0 +2OT ,To1 ] εo2 (t), where To1 is greater than or equal

Active fault-tolerant attitude control Chapter | 7

251

to the time required for εo1 to reach the sliding surface, then this fixed λo1 value will ensure that V˙o1 < 0 for any εo1  = 0. To prove that ε o1 converges in finite time, we can obtain ε o1 (t) ≤ εo1 (t0 + 2nT ) − o1 [t − (t0 + 2nT )] by integrating both sides of Eq. (7.131) from t0 + 2nT to t. Therefore the equation εo1 (t) = 0 is constant for any   t ≥ To1  εo |t0+2n + t0 + 2nT , that is, bmεo1 will reach the sliding surface o1 ε o1 = 0 within limited time To1 . At this time, according to the theory of slidingmode control, the equivalent control quantity can be calculated from Eq. (7.129) as [λo1 sgn(ε o1 )]eq = ε o2 , so ξ˜ 2 = ξ 2 . After the time To1 , the observation error kinetic Eqs. (7.129) and (7.130) have the following form: ε˙ o1 = 0,

(7.132)

ε˙ o2 = g 4 (ξ 2 , ξˆ 2 ) − J −1 Dub − λo2 sgn(ε o2 ).

(7.133)

Although the reaction flywheel has faults, according to its physical characteristics, we know that the fault value of each flywheel will not exceed its √ maximum output amplitude torque τ max , so we can obtain ub  ≤ Nτ max . ε + At this time, choose another Lyapunov candidate function Vo2 = 0.5εo1 o1  0.5εo2 εo2 . Then  −1 ˆ Dub V˙o2 ≤ −λo2 ||ε o2  + ε o2 g 4 (ξ 2 , ξ 2 ) − ε o2 J √ −1 ˆ ≤ − | ε o2 [λo2 − g 4 (ξ 2 , ξ 2 ) − NJ Dτmax ].

(7.134)

√ If we select λo2 > g 4 (ξ 2 , ξˆ 2 )− NJ −1 Dτ max +o2 , where lo2 ∈ R+ is a constant, then for any εo2  = 0, V˙o2 ≤ −o2 ||ε o2  < 0.

(7.135)

This formula guarantees that εo2 will reach the sliding-mode surface ε o2 = 0 within finite time. At this time, select the fixed parameter λo2 to satisfy λo2 > √ ˆ maxt∈[Tol ,To2 ] [g 4 (ξ 2 , ξ 2 ) − NJ −1 Dτmax ], where To2 is the time for ε o2 to reach the sliding surface ε o2 = 0. Therefore by integrating both sides of Eq. (7.135) from To2 to t we obtain εo2 (t) ≤ εo2 (To1 ) − o2 (t − To1 ).

(7.136)

Solving this inequality, we can prove that εo2 (t) = 0 is constant for any ol ) + To1 . The theorem is proved. t ≥ To2  εo2(T o2

7.4.3 Attitude FTC law design According to Theorem 7.3 and its proof, we can only use the measurable attitude output information ξ y of the spacecraft to ensure that the estimator ξˆ 2

252 Fault-Tolerant Attitude Control of Spacecraft

completely converges to ξ 2 in the finite time To2 , and the estimation error is εo2 (t) = 0, that is, ξ 2 (t) ≡ ξˆ 2 (t), t ≥ To2 . Therefore, according to formula (7.125), we can accurately obtain  ˙ ˙ io  q ob (ψ, φ, θ ) + q io  f 2 (, ) q˙e = [q˙e0 , q  ev ] = q

= q io  q ob (ψ, φ, θ ) + q io  f 2 (, ξˆ 2 (t)).

(7.137)

Although the angular velocity of the spacecraft cannot be measured, according to Eq. (7.137), we know that q˙ e v, q e , and H 1 in the attitude conversion model (7.96) can be obtained without angular velocity information. On the other hand, because the fault estimation module starts to run at time t = t0 + 3nT , and the fault does not occur at this time, from Eq. (7.98) we get x 1 (t0 + 3nT ) = −kJ m (t0 + 3nT ) and x 2 (t0 + 3nT ) = −Z (t0 + 3nT )D(t0 + 3nT ). Lemma 7.3. If the initial value of the state of the terminal sliding-mode observer (7.103)–(7.104) is selected as xˆ1 (t0 + 3nT ) = −kJ m (t0 + 3nT ), xˆ1 (t0 + 3nT ) = 0, then the observation error e is consistent, ultimately bounded, and stable (UUB). Proof. According to the proof of Lemma 7.1, we know that the observation error e eventually converge in the area D  {e | e ≤ λR }. On the other hand, according to the initial choice given by Lemma 7.2, we get e(t0 + 3nT ) = x 2 (t0 + 3nT ) = Z  (t0 + 3nT )d(t0 + 3nT ) ≤ Z  (t0 + 3nT )dmax .

(7.138)

Therefore, according to this formula and the definition of D, we can prove that e ≤ max{λR , Z  (t0 + 3nT )dmax } for any t ≥ t0 + 3nT . Theorem 7.8. Starting at t0 + 3nT , we use the terminal sliding-mode observer to estimate the fault (7.103)–(7.104). If the gain of this observer is selected, Eqs. (7.109) and (7.110), and satisfies λR ≤ Z (t0 + 3nT )dmax ,

(7.139)

then the observation errors e1 and e2 will reach the sliding-mode surface e1 = 0 and e2 = 0 in finite time, and u¯ f (t) will be estimated in finite time. The generalized fault uf can be estimated by Eq. (7.115) in finite time. Proof. According to Theorem 7.4, the generalized fault uf can be accurately estimated in limited time. Therefore, based on this estimated information, an attitude controller can be designed to realize attitude-tracking control. The specific design results are as follows. Theorem 7.9. Suppose that for the spacecraft attitude tracking control system (7.46)–(7.48) with reaction flywheel failure and quaternion description, the

Active fault-tolerant attitude control Chapter | 7

253

sliding-mode observer (7.127)–(7.128) is applied to estimate the attitude, and terminal sliding mode observer (7.103)–(7.104) is used to estimate actuator failure and external interference. Design the following attitude controller without angular velocity feedback: uc = uc_nom + uc_f ault ,

(7.140)

where uc_nom = − kc1 D † [q e + kc2 P  (q e )T anh(μ + kc 2q e )] + D † J R(q e )ω˙ d + D ∗ [R(q e )ωd ]× J R(q e )ωd ,

(7.141)

uc_f ault = D † P  (q e )xˆ 2 ,

(7.142)

μ = −kc3 (μ + kc2 q e ),

(7.143)

˙ = [μ1 , μ2 , μ3 ] ∈ R3 , and kcj ∈ R+ (j = 1, 2, 3) are the control parameters. μ If the design control parameters satisfy √ D i [kc1 (1 + 3kc2 ) + J (ω˙ d  + ωd 2 ). . + (1 + Z  (t0 + 3nT ))dmax ] ≤ τmax ,

(7.144)

where D i represents the ith row vector of the matrix D † , i = 1, 2, . . . , N . Then the closed-loop attitude-tracking system is asymptotically stable, limt→∞ q e (t) = 0, limt→∞ ωe (t) = 0, and the problem of the input limitation of the actuator control can be solved. Proof. Applying the Lyapunov stability theory, this theorem can be proved in two parts. Step 1. Closed-loop attitude control system stability analysis: choose the following Lyapunov candidate function: $ ω e J ωe + 2q  ln(cosh(μi + kc2 qei )). e q e + 2(1 − qe0 )q˙e0 + kc1 3

Vc =

i=1

(7.145) Considering the actuator failure (2.27), according to Eqs. (7.46)–(7.48) and (7.46)–(7.48), we get $ ω J ω˙ e ˙ e − 2(1 − qe0 )q˙e0 + + 2q  (μ˙ i + kc2 q˙ei )Tanh(μi + kc2 qei ) V˙c = e eq kc1 3

i=1

ω ˙ d ] + Duc + uf } = e {−ω× J ω + J [ω× e R(q e )ωd − R(q e )ω kc1

254 Fault-Tolerant Attitude Control of Spacecraft

˙ + kc2 q˙ e  Tanh(μ + kc2 q e ) + q e ωe + μ ω e × × [J ω× e R(q e )ωd − ω J ω + (R(q e )ωd ) J R(q e )ωd ] kc1 ˙  Tanh(μ + kc2 q e ) +μ =

=

ω e ˙  Tanh(μ/k ˙ c3 ). [P  (q e )xˆ2 + uf ] − μ kc1

(7.146)

× × × Since ω e [J ωe R(q e )ωd − ω J ω + (R(q e )ωd ) J R(q e )ωd ] = 0, and the application of terminal sliding-mode observers (7.103)–(7.104) can accurately estimate actuator failure and external interference, according to Eq. (7.115), for any t ≥ tF 2 , we have

˙  Tanh(μ/k ˙ c3 ) ≤ −kc3 Tanh(μ/k ˙ c3 )2 ≤ 0. V˙c = −μ

(7.147)

On the other hand, since the spacecraft attitude-tracking operation starts at time ts %, according to the negative semidefinite V˙c shown in Eq. (7.147), we ∞ obtain ts V˙c dt ≤ Vc (ts ) − Vc (∞) < ∞, and we can prove that Vc is a radial unsolvable function. All solutions are bounded. Therefore, by Barbalat’s lemma (Pirmoradi et al., 2009), limt→∞ V˙c = 0 according to the uniform continuity of V˙c , and we can further prove that limt→∞ V¨c = 0 by the uniform continuity ˙ ¨ = limt→∞ μ(t) = 0 and limt→∞ q˙ e = 0 accordof V˙c . Therefore limt→∞ μ(t) ing to (7.143). At this time, from the nonsingularity of P (q e ) we can obtain limt→∞ ωe (t) = 0, and the asymptotic stability of the closed-loop attitudetracking system is proved. Step 2. Analysis of the limit of the output torque amplitude of the flywheel for the reaction: According to formulas (7.46)–(7.48) and (7.142), we get u =uc + B(t − T )ub

(7.148) 

= − kc1 D [q e + kc2 P (q e )Tanh(μ + kc2 q e )] †

+ D † J R(q e )ω˙ d + D † (R(q e )ωd )× J R(qe )ωd + D † P  (q e )xˆ 2 + D † uf − D † d. (7.149) √ Since Tanh(a) ≤ 3 for any a ∈ R3 , applying R(q e ) = 1, q e  ≤ 1, and P (q e ) = 1, we can prove that the actual output torque of each flywheel satisfies |ui | =| − kc1 D i [q e + kc2 P  (q e )Tanh(μ + kc2 q e )] + D i J R(q e )ω˙ d + D i (R(q e )ωd )× J R(q e )ω˙ d + D i P  (q e )xˆ 2 + D i uf − D i d | √ ≤kc1 D i (1 + 3kc2 ) + D i J ωd  + D i J ωd 2 + D i dmax + Di [P  (q e )xˆ 2 + uf ] for i = 1, 2, . . . , N.

(7.150)

Active fault-tolerant attitude control Chapter | 7

255

In addition, according to the fault estimator (7.103)–(7.104), we know that uf (t) = −P  (q e )xˆ 2 (t) for any t ≥ tF 2 , so we get    0 if t ≥ tF 2 , P  q e xˆ2 + uf = (7.151)    P q e e2 (t) if t < tF 2 . At the same time, according to Lemma 7.2 and formula (7.139), we have e ≤ Z  (t0 + 3nT )dmax , so according to formula (7.151), we have the inequality P  (q e )xˆ 2 + uf  ≤ P  (q e )e2 (t) ≤ Z (t0 + 3nT )dmax .

(7.152)

It is constant for all t ≥ ts . Based on the above analysis, we can prtove the following inequality: √ |ui | ≤kc1 D i (1 + 3kc2 ) + D i J k d  + D i J k d 2 + Di dmax + D i [Z  (t0 + 3nT )dmax .

(7.153)

Therefore if the control parameters kcj (j = 1, 2, 3) are selected to satisfy inequality (7.144), according to inequality (7.97), we can prove that |ui | ≤ τmax , that is, the actual output amplitude and torque of each reaction flywheel remain within their maximum output amplitude range. So the problem of limited actuator control input is solved. According to the analysis in Steps 1 and 2, the conclusion of Theorem 7.5 is always valid, and the theorem is proved. Remark 7.5. We can see from Theorem 7.5 that controller (7.142) does not require angular velocity measurement information and the problem of limited actuator control input is also solved. Therefore this control method can solve the problem of attitude-tracking fault-tolerant control without angular velocity.

7.4.3.1 Numerical simulation To verify the effectiveness of the attitude-tracking control method designed in this section, we use a certain type of rigid-body spacecraft physical parameters and established tasks to perform simulation analysis. The spacecraft payload is required to be able to point to certain hot spots and take highdefinition reconnaissance photos. The orbit of this spacecraft is an approximately circular orbit with an ascending node of 95.4 deg and an orbital height of 650 km, an orbital angular rate of ω0 = 0.06245 deg/s, and an orbital period of nT = 5893.69 s. The main inertia of the aerospace vehicle is 20 kg m2 , 22.5 kg m2 , and 25.2 kg m2 , and the product of inertia does not exceed 0.5 kg m2 . The attitude of the spacecraft is controlled by the four reaction flywheels shown in Fig. 7.2. At this time, the saturation control torque of each reaction flywheel is τmax = 0.2 Nm. The external interference received by the spacecraft is calculated according to the literature. To complete the established space mission,

256 Fault-Tolerant Attitude Control of Spacecraft

its attitude control system is required to provide the payload with an attitude pointing accuracy better than 0.005 deg and an attitude stability better than 0.001 deg/s. Assume that the spacecraft separates the spacecraft at time t = t0 , completes the rate damping operation in the first orbital period nT , and completes the attitude capture and the reaction flywheel spin-up operation in the second orbital period. Then, in the third period, attitude stabilization operations are completed within the state to establish a high-precision and high-stability state, the spacecraft attitude is adjusted in the 10th orbital period, and the attitudetracking operation is performed to detect the target and take high-definition photos. The attitude operation sequence required for the entire mission is shown in Fig. 7.2. To verify the effectiveness of the designed attitude-tracking control method without angular velocity, in this section, we select the spacecraft physical parameters given in Section 7.3.2 for numerical simulation. When implementing this control strategy, the gain of the terminal sliding-mode observer (7.103)–(7.104) is consistent with the gain selected in Section 7.3.3 while using the sliding-mode observer (7.127). The gains λo1 and λo2 of (7.128) are λo1 = 0.75, λo2 = 6.5; the control parameters of controller (7.142) are selected as kc1 = 0.01, kc2 = 0.05, and kc2 = 1.5.

7.4.3.2 Simulation results under healthy actuators When the actuator is normal, the simulation result of attitude stability is in the third orbit period, and the reaction flywheel failure is not considered. If controller (7.142) is used for attitude stabilization operation, then the terminal sliding-mode observer (7.103)–(7.104) can realize the fault estimation performance as shown in Fig. 7.23. In fact, at this time, the observer estimates the external disturbance torque. According to the initial response of the estimation error, the external interference is accurately estimated in about 120 seconds. From the steady-state response of the estimation error we can see that although different controller design methods are adopted, the terminal sliding-mode observers (7.103)–(7.104) can still achieve a relatively high fault estimation accuracy |e2i | ≤ 1.2 × 10−6 Nm, i = 1, 2, 3. The attitude-tracking control performance obtained by the application controller (7.142) is shown in the attitude-tracking error curve shown in Fig. 7.24, and the angular velocity-tracking error curve is shown in Fig. 7.25. We can see from the steady-state response of the two that controller (7.142) can still complete the attitude-tracking operation without the angular velocity feedback information, and the command control torque response is shown in Fig. 7.26. Obviously, the output torque of each flywheel is within the range of its maximum control capability, which further verifies the ability of the control algorithm designed in this section to deal with the limitation of actuator control input.

Active fault-tolerant attitude control Chapter | 7

257

FIGURE 7.23 The estimation error e2 by using (5-59) in actuator fault-free case.

FIGURE 7.24 The angular velocity-tracking error with the controller (5-59) in actuator fault-free case.

FIGURE 7.25 The attitude-tracking error with the controller (5-59) in actuator fault-free case.

FIGURE 7.26 The commanded control of the controller (5-59) in actuator fault-free case.

7.4.3.3 Simulation results under faulty actuators When the actuator fails, the simulation result of attitude tracking is in the 10th orbital period. Considering the actuator failure described in Section 7.3.3.2, controller (7.142) is used to control the attitude tracking of the spacecraft body as the expected attitude trajectory given in Section 7.3.3.2. The simulation study found that the sliding-mode terminal observers (7.103)–(7.104) included in this control algorithm can accurately estimate the sum of the reactive flywheel fault and external disturbance torque, and estimate the error e2 steady-state response and use controller (7.140). The e2 response during attitude-tracking control is basically the same and has the same order of magnitude estimation accuracy, so its response curve will not be given in this section. In fact, the reason for this

258 Fault-Tolerant Attitude Control of Spacecraft

phenomenon is that the performance of estimator (7.103)–(7.104) has little correlation with the commanded control torque. The steady-state response curve of the angular velocity tracking error shown in Fig. 7.27 and the steady-state response curve of the attitude tracking error shown in Fig. 7.28 show that controller (7.140) can act on limited actuator control input, external interference, and actuator failure under the control of attitude tracking without any angular velocity feedback information. In particular, the command control torque shown in Fig. 7.29 further verifies that controller (7.140) can solve the problem of limited actuator control input.

FIGURE 7.27 The angular velocity tracking error with the controller (5-59) in case of actuator fault.

FIGURE 7.28 The attitude-tracking error with the controller (5-59) in case of actuator fault.

7.4.3.4 Quantitative analysis By simulation analysis we further verify the effectiveness of the attitude control method designed in this section without angular velocity feedback. In addition to the above simulation research, in this section, we compare it with the attitude control algorithm without angular velocity feedback (TSMOBVFC) designed in Chapter 3 and in Sidi (1997). The attitude output feedback control method (UqOFC) compares and analyzes attitude stabilization and tracking operations. Based on the simulation results of these three control methods, we use the evaluation indexes PE and Pm given by equations employed in Section 7.2.3 to quantitatively analyze the respective attitude control performance, and still select the attitude operation time Ttotal as a track Period nT . When the above three control methods are used for attitude operation, the achieved indexes PE and Pm are shown in Table 7.2. When the attitude is stabilized and the actuator is working normally, although the TSMOBVFC control parameters can ensure that it and controller

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FIGURE 7.29 The commanded control of the controller (5-59) in case of actuator fault.

(7.142) can have the same task completion rate (that is, the Pm values are equal), the energy consumption PE of TSMOBVFC is higher that the energy consumption of controller (7.142). This is because although TSMOBVFC can handle actuator input limitation and external interference like controller (7.142), TSMOBVFC is essentially a robust control strategy, and its actuator has a certain conservative nature, whereas controller (7.142) can compensate the estimated external interference information in real time, so its conservativeness is weak. On the other hand, when performing attitude-tracking operation, because the reaction flywheel has a mode two failure, and TSMOBVFC can only deal with the partial failure of the actuator but cannot deal with the reaction flywheel mode to failure, resulting in its task completion rate far lower than the original. The completion rate of the section controller (7.142), that is, the Pm value realized by the former is much smaller than the Pm value realized by controller (7.142). Since the controller method can only solve the problem of attitude tracking with limited actuator control and does not consider the robust control problem against external interference, the existence of external interference has a certain impact on the accuracy of its attitude control, resulting in its task rate lower than the completion rate of controller (7.142). The Pm value of UqOFC is about 22% smaller than that of controller (7.142). When an actuator failure occurs, because UqOFC does not have fault-tolerant control capabilities, the existence of the failure verifies the smooth execution of its tasks and finally can only complete 1.64% of the total task, that is, the Pm value is 1.64%. When the actuator fails, the energy consumption of both TSMOBVFC and UQOFC is much higher than that of controller (7.142). This is because TSMOBVFC and UQOFC do not have the second fault of the flywheel mode. Therefore,

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when this fault occurs, their actuators will control the task attitude of the system to be normal, so they continue to execute the control commands, resulting in their actuators outputting a larger control torque, which in turn leads to higher energy consumption. Comparing the control performance of controllers (7.140) and (7.142) shown in Tables 7.1 and 7.2, we can further find that no matter what attitude operation is performed, although the Pm value of controller (7.142) that can be achieved by attitude control is less than the Pm value of controller (7.140), but the difference between the two is small. On the other hand, from the perspective of attitude operation energy consumption, we can find that the energy consumption of controller (7.142) is much smaller than that of controller (7.140). This is because controller (7.142) explicitly solves the problem of actuator control input saturation, but controller (7.140) cannot solve this problem, which causes controller (7.140) to be prone to actuator saturation during the attitude control process, as shown in Figs. 7.17 and 7.21. From the above quantitative analysis results we can see that the fault-tolerant control method proposed in this section without angular velocity measurement feedback can achieve attitude-tracking operations with low power consumption, high reliability, and high task completion rate.

7.5 Summary Aiming at the problem of spacecraft attitude control with fly wheel faults, in this chapter, we proposed three active fault-tolerant control schemes to achieve consistent control performance under different practical scenarios. First, we proposed a compensation scheme for reaction wheel faults and external disturbances for rigid spacecraft attitude-tracking systems, which, by the method of a terminal sliding-mode observer, reconstructs the actuator faults and external disturbances in finite time. Subsequently, we presented a fault-tolerant attitude stabilization control without angular velocity measurement to achieve asymptotic attitude stabilization in the presence of external disturbances and reaction wheel faults. Moreover, the problem of actuator saturation was solved. Then we further presented a velocity-free active fault-tolerant controller to achieve finitetime attitude tracking. This method takes into account actuator faults, no angular velocity measurements, and input saturation.

Chapter 8

Conclusions and future work This chapter ends the book with some concluding remarks and discussions, and some future work is presented.

8.1 General conclusion For any in-orbital spacecraft, there exist unpredictable abnormalities or actuator failures during its mission. With the development of hardware and software technology, spacecraft controllers can be designed to handle some failures autonomously, thus having certain failure recovery capability. Therefore, based on the existing research and cutting-edge nonlinear control technology, a faulttolerant attitude controller can be proposed to handle actuator failures and ensure the stability of the spacecraft attitude control system with acceptable attitude control performance. The main contribution of this book is that faulttolerant control algorithms are designed and applied to the spacecraft attitude control system to ensure that the spacecraft can successfully complete the mission even when actuators fail. According to the attitude fault-tolerant control methods proposed in the previous chapters, the main results and innovations of this book are summarized into the following four aspects. (1) A fault estimation-based control strategy for attitude stabilization is proposed, which can simultaneously solve actuator control input saturation, partial failures, and external disturbances. The design of the fault estimator helps to ensure that the controller can adjust and reconstruct the control parameters online according to the estimated information within the maximum control capacity of the actuator, and thus a less conservative attitude control can be achieved. This method can overcome the conservative shortcomings of attitude fault-tolerant controllers designed based on robust control ideas. (2) Considering the partial failures of the actuator and external disturbances, a fault-tolerant control scheme without angular velocities is proposed to achieve attitude-tracking mission. Compared with most existing fault-tolerant control strategies based on full-state feedback (attitude and angular velocity) of the attitude control system, the controller in this chapter only needs attitude measurement information. Therefore this method can solve the two major problems. One is that it is difficult for a small spacecraft or microsatellite to provide highprecision angular velocity measurement information. The other is that when the angular rate sensors fail, its fault measurement information will degrade the control performance of the controller. Meanwhile, this method explicitly considers Fault-Tolerant Attitude Control of Spacecraft. https://doi.org/10.1016/B978-0-32-389863-8.00018-6 Copyright © 2021 Elsevier Inc. All rights reserved.

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the limitation of actuator control input. Therefore this method does not require installing magnetic torques to unload the reaction wheel saturation torques. (3) To solve the challenges to the reliability of the spacecraft attitude control system caused by diversified types of the actuator failures, a finite-time attitudetracking control method is proposed for rigid spacecraft equipped with reaction wheels. Based on the Lyapunov theory, it is proved that the proposed method can guarantee the attitude and angular velocity tracking errors to zero in finite time under external disturbances, inertia moment uncertainties, and reaction wheel failures. Moreover, the convergence time can be tuned by the designer. In particular, this control method does not require any online FDD algorithm and can fill the gap of fault-tolerant capabilities of most existing rapid attitude maneuver control methods. (4) Considering actuator failures, input saturation, external disturbances, and unavailability of angular velocities, a fault-tolerant attitude-tracking controller based on fault estimation is proposed. A nonlinear observer is designed to estimate the synthesis of actuator faults and external disturbances. Based on this estimated information, the controller is designed to autonomously compensate the influence of disturbances and faults on attitude control performance online. Although this control method uses observers to estimate faults, it does not need to explicitly use fault detection, isolation, and diagnosis techniques. The time of fault occurrence, fault type, and fault value can be completely unknown to the designer, so it can effectively avoid problems aroused by the application of FDD technology, such as fault false alarms, and so on. On the other hand, the controller can explicitly solve input saturation issues and thus requires no additional actuators to generate unloading torque. Meanwhile, this control method can achieve attitude-tracking control without angular velocity measurements such as gyro.

8.2 Future work The proposed approaches in this book have shown significant advantages over the existing literature; nevertheless, there still remain some challenges, which can be concluded as follows. The angular velocity observers are designed based on accurate spacecraft attitude dynamics. When the dynamics have modeling deviations, the estimation performance of the angular velocity observers is not quantitatively analyzed. Therefore the angular velocity observer should be extended as a future research topic by considering the modeling errors of spacecraft attitude dynamics. The fault tolerance of the controllers is realized by the redundant actuators. The proposed methods are based on the assumption that the control torque generated by the remaining actuators can still achieve three-axis attitude control when the actuators fail. Hence it is necessary to determine whether the spacecraft attitudes are controllable in three axes when the actuator fails. If the three-axis attitudes are uncontrollable, then the attitude control system degenerates to an underactuated system, whereas the underactuated spacecraft attitude

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controller design poses a great challenge for fault-tolerant control and still needs significant efforts to be solved in the future. Although the fault estimator can accurately estimate the total actuator faults of the spacecraft, it cannot locate and estimate a single actuator fault among multiple actuator faults. Therefore, aiming to achieve more detailed fault tolerance processing, it is necessary to further improve the fault estimation algorithm to complete the fault location and identification of a single actuator. The fault-tolerant attitude controller designed in this book based on redundant actuators only guarantees the realization of the desired attitude operation without considering energy consumption optimization. This problem will directly influence the service life of spacecrafts. Therefore the design of the control allocation strategy can be further studied to achieve the attitude faulttolerant control with the optimal energy consumption under the actuator failure. In addition, to further verify the effectiveness of the proposed control method, experimental validation should be conducted in the future work.

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Index

A Abrupt fault, 31, 33, 228 Acceptable performance, 7, 153 Actuation effectiveness fault, 17 effectiveness matrix, 66, 69, 99, 102 saturation, 81 Actuator constraint, 19, 73, 76, 78, 87, 165 control, 253, 259, 260 forces, 73 input, 126, 256, 261, 262 torque, 86 distribution, 18, 66, 102 dynamics, 10, 23 effectiveness, 53, 58, 178 fault, 99 fault loss, 91, 118, 122 fault partial loss, 21, 60, 83, 84, 89, 93, 106, 121, 144, 155, 166, 169 loss, 59, 60, 107, 117, 149 partial loss, 21, 57, 62, 63, 82, 84, 86, 93, 98, 110, 146 efficiency factors, 47 failures, 10, 18, 37, 62, 76, 99, 102, 104, 126, 140–142, 154, 241, 253, 254, 257, 261, 262 fault, 10, 11, 16, 30, 31, 37–40, 81–83, 86, 91, 144, 145, 152, 153, 176–178, 180, 209, 221, 223, 224, 227, 262 accommodation/compensation, 69, 99 diagnosis, 11 partial loss, 162

health, 66, 75 health indicator, 155 input, 86, 106, 259 limits, 74, 76, 78 misalignment, 18, 176, 181, 187, 207 nonlinearities, 19 output, 81 output constraints, 81 output torque, 49 partial failures, 126 redundancy, 17, 86, 93, 99, 102, 154 saturation, 19–21, 73, 81, 86, 93, 98, 126, 154, 155, 164, 175, 224, 239, 241, 260 saturation control input, 248 uncertainties, 23 Adaptive attitude controller, 16 backstepping control, 54 backstepping control law, 55, 57 control, 11, 16, 17, 19, 37 control scheme, 37 controller, 16, 126 controller design, 43 nonlinear control, 176 observers, 11 Adaptive fault-tolerant controller (AFTC), 118, 121, 122 application, 122 controls, 122 law, 17 technique, 81 Advanced control theory, 53, 80, 106 Aerodynamic torque, 227 273

274 Index

Aerospace applications, 98 mission, 13, 197 Antiwindup control, 105 controller, 105 Antiwindup PID (AWPID), 88, 89 Application AFTC, 122 controller, 185, 256 passive FTC, 7 spacecraft control, 43 UQOFC, 168 Asymptotical stability, 38, 57 Atmospheric drag, 29, 53, 65, 83, 93, 156 Attitude acquisition, 227, 243 acquisition maneuver, 224, 229 angles, 49, 184, 198, 237–239 capture, 198, 256 capture maneuver, 219 compensation control, 180 compensation control law design, 216 control, 2, 11, 14, 16, 33, 46, 52, 55, 62, 102, 116, 132, 133, 137, 191, 222, 238, 242, 261 design, 23 maneuvers, 6 mission, 140 performance, 33, 51, 52, 62, 66, 118, 119, 122, 139, 142–144, 239, 258, 261 stability, 132 stabilization, 63 controller, 18, 23, 35, 50, 111, 126, 132, 139, 142, 176, 237, 252, 253 controller design, 39, 54, 81 description, 23 determination, 117, 184, 198, 218, 237, 238 deviation, 227 dynamic equations, 57 dynamic model, 53 dynamics, 14, 18, 23, 36, 82, 96, 99, 107, 180, 198, 248 dynamics system, 106, 112 Euler angle, 243, 248–250

fast slewing capability, 20 fast slewing maneuver, 19 FTC, 15, 20, 107, 144, 228, 251 design, 16 law, 16, 17 schemes for spacecraft, 19 stabilization control law, 16 kinematics, 23, 27, 36, 127, 155, 226 maneuver, 18, 20, 30, 60, 69, 91, 101, 119, 153, 178, 183, 200, 218, 222, 224 control command, 33 control methods, 262 mission, 33 performance, 173 maneuvering, 104 maneuvering process, 83 measurement, 60, 138, 155, 198, 232, 261 measurement errors, 10 model, 23 operation, 142, 256, 258, 260 operation time, 258 orientation, 20, 46, 51, 54, 55, 58, 85, 89, 91, 112, 113, 155, 157, 162, 165, 168, 170, 219 output information, 251 performance, 222 pointing accuracy, 51, 73, 74, 122, 142, 143, 152, 165, 184, 197, 198, 218, 219, 221–223, 239 quaternion, 66, 75, 145, 239, 249 rotation, 29 rotation matrix, 179, 189, 236 rotational maneuver, 60 satellite, 222, 227, 229, 230, 235, 238 sensor, 30, 138, 151, 163, 168, 198, 230, 237, 238, 250 sensor noises, 60, 168 slew maneuver, 21 spacecraft, 6, 8, 9, 23, 31, 54, 60, 69, 81, 83, 125–127, 192, 196, 210, 218 stability, 91, 122, 139, 141–143, 170, 171, 187, 199, 201, 219, 221, 239, 256

Index 275

stability control, 136 stabilization, 17, 37, 51, 54, 82, 106, 126, 175, 223, 224, 241, 256, 258, 260, 261 control, 89, 171 control scheme, 117, 239 FTC, 21, 93, 154, 224 maneuver, 57, 62, 91, 107, 118, 119, 122, 144, 145, 161, 168, 170, 171, 218, 220, 224, 225, 227, 235, 236 mission, 33, 241 problem, 146 system, 50, 83, 86, 91, 152, 153, 165, 170, 199, 213, 221 tolerance control, 126 tracking, 37, 75, 125, 139, 140, 173, 175, 205, 243, 248, 249, 257 compensation controller design, 212 control, 74, 125, 136, 176, 258 control problem, 67 control system, 65 error, 21, 66, 132, 133, 139, 258 error dynamics, 176 error system, 200 FTC, 63 maneuver, 16, 17, 20, 70, 77, 183, 187, 197, 207 mission, 125 model, 66 problem, 20, 37, 68, 175 system, 66, 76 trajectory, 257 Attitude control system (ACS), 4, 6, 8, 17, 47, 127, 142, 196, 198, 206, 218, 256, 261, 262 design, 13, 86 failures, 8 faults, 4, 6

B Backstepping control method, 21 controller design, 111 Bias torque fault, 183, 185

C Chattering, 45, 59, 166, 196, 197 effect, 151, 166, 196 phenomenon, 196 Command control moment, 33, 35 signal, 136 torque, 34, 142, 256, 258 Command torque, 35 Commanded control, 185, 228, 235, 239 input, 38, 190 torque, 140, 185, 187, 200, 202, 222, 258 Commanded torque, 220, 227 Compensation control, 219, 243 attitude, 180 effort, 212 strategy, 243 Compensation controller, 53, 212, 243 Component faults spacecraft, 11, 15 Control accuracy, 171, 241 actuator, 253, 259, 260 adaptive, 11, 16, 17, 19, 37 algorithm, 14, 23, 104, 138, 257 allocation, 16–18 allocation strategy, 18 approaches, 53, 80, 106, 187, 237 attitude, 2, 11, 14, 16, 33, 46, 52, 55, 62, 102, 116, 132, 133, 137, 191, 222, 238, 242, 261 attitude stabilization, 89, 171 attitude tracking, 74, 125, 136, 176, 258 authority, 101 capability, 138, 256 circuit, 30 commands, 35, 260 design, 111, 156, 195 effectiveness, 221 effectiveness loss, 93, 144, 152 effectiveness partial loss, 154 effectors, 10 effort, 17, 19, 81, 153, 164, 165, 171, 178, 196, 197 force, 61, 76, 89, 103, 104 force magnitude, 106

276 Index

gains, 48, 51, 55, 59, 68, 69, 113, 117, 118, 133, 146, 156, 161, 164–166, 185, 191, 194, 198, 219, 235, 237 input, 38, 73, 82, 86, 89, 104, 105, 129, 132, 134, 152, 220 saturation, 80, 117, 122 signal, 89, 91 law, 7, 9, 14–16, 39–41, 49, 53, 54, 56, 57, 59, 83, 84, 86, 88, 91, 98, 125, 144–146, 149, 178, 180, 193, 218, 222 loop, 14 loss, 165, 177 maneuver, 224 method, 38, 142, 243, 255, 258, 262 objective, 54, 58, 66, 69, 72, 74, 82, 93, 94, 102, 106, 108, 133, 136, 144, 178, 183, 190, 211 output, 78, 164 parameter, 42, 43, 52, 59, 63, 91, 103, 106, 135, 146, 149, 156, 253, 256, 261 parameter sets, 119 performance, 16, 23, 30, 37, 44, 45, 50, 63, 105, 126, 141, 142, 184, 199, 209, 222, 260, 261 power, 18, 62, 66, 76, 121, 165, 169, 187, 205, 206, 238 problem, 41, 65 problem statement, 66 reaction flywheel, 136 saturation, 140 scheme, 54, 69, 81, 99, 102, 104, 125, 126, 181, 190, 202, 211, 222 signals, 35, 76, 111, 177 spacecraft attitude, 10, 30, 83, 175, 209, 243, 260 stability, 41 strategy, 118, 137, 172, 256 structure, 229 system, 19, 30, 36, 37, 99, 238 system design, 24, 43 system dynamics, 137 techniques, 10 theory, 21

torque, 20, 29–31, 33, 51, 86, 99, 119, 155, 170, 177, 222, 227, 235, 237, 262 Controller adaptive, 16, 126 application, 185 attitude, 18, 23, 35, 50, 111, 126, 132, 139, 142, 176, 237, 252, 253 control performance, 260 derivation, 68 design, 23, 67, 78, 84, 93, 99, 107, 111, 128, 235, 256 development, 39, 43 FTC, 63 gains, 203, 237 module, 229 parameters, 76, 78, 196 passive FTC, 16 performance, 75 reconfiguration, 8, 9 reconfiguration mechanism, 9 Conventional control methods, 117 controllers, 63, 80, 106 PID control, 119 Conventional finite-time attitude controller (CFTAC), 201, 203, 205, 206 Conventional variable structure controller (CVSC), 103, 104

D Degraded nominal performance, 209 Demanded torque input, 76 Disturbances attenuation, 43, 45, 66, 80 energy, 65 inputs, 52, 63, 80 lumped, 67, 82, 155 suppression performance, 63 torques, 65, 70, 75

E Earth magnetic torque, 227 Effectiveness actuator, 53, 58, 178 control, 221 factor, 82, 86, 118

Index 277

fault loss, 54, 82, 119, 161 fault partial loss, 106, 154, 161, 162, 170 loss, 10, 37, 38, 53, 63, 88, 152, 170, 200 partial loss, 86, 126, 153, 154 Electrical component fault, 3 Electrical fault, 2 Error dynamics, 129, 189, 192, 215, 231 Error dynamics spacecraft attitude, 65, 66 Exogenous disturbance, 25 Extended Kalman filter (EKF), 10, 237

F Fault detection and diagnosis (FDD), 7, 8 design, 8–10, 12, 107 design satellite, 11 mechanism, 107 observer, 107 Fault detection and identification (FDI), 7 Fault-tolerant control (FTC), 6 attitude, 15, 20, 107, 144, 228, 251 attitude stabilization, 93, 154, 224 attitude tracking, 63 controller, 60, 63 law, 15, 16, 106, 112 law design, 16, 17 spacecraft, 13, 80 spacecraft attitude, 15, 17, 19, 20 technology, 37 Fault-tolerant control system (FTCS), 6 Fault-tolerant sliding-mode controller (FTSMC), 89, 91 Faults actuator, 10, 11, 16, 30, 31, 37–40, 81–83, 86, 91, 144, 145, 152, 153, 176–178, 180, 209, 221, 223, 224, 227, 262 actuator effectiveness, 99 attitude control system, 4, 6, 84, 86, 98, 102, 144 dynamics, 155 system, 54, 82 cases, 52, 79, 91, 187 causes, 30

characteristics, 33 compensation, 218 compensation attitude tracking control, 210 conditions, 10 detection, 7, 9, 10, 12, 23, 37, 80, 262 detection accuracy, 12 diagnosis, 9–12, 23 dynamics, 145, 155 estimates, 111 estimation, 252, 262 accuracy, 256 algorithm, 263 performance, 256 estimator, 243, 255, 261, 263 estimator design, 244 events, 81 factor matrix, 36 handling method, 33 identification, 7, 8 information, 37, 106, 229 input, 228 isolation, 9, 69, 99, 102 location, 31, 263 lumped, 212, 214, 216–218, 229, 234, 235 management, 13, 15 matrix, 99 measurement information, 261 mechanical, 2–4 mechanism, 33 mode, 35, 38, 42, 43, 47 model, 23, 36, 40 momentum wheel, 6 nonlinear, 228 occurrence, 4, 262 parameters, 42 problem, 40 reaction flywheel, 10, 140, 243, 250 reaction wheel, 10, 16, 30–33, 52, 177, 191, 199, 201, 207, 209, 211, 212, 216, 218, 219, 221–224, 227 reconstruction, 10, 107, 110, 111, 122, 221, 231, 233, 239 error, 221, 222 scheme, 212, 219

278 Index

scenarios, 76, 78, 152, 170, 184, 199, 220 severity, 102 spacecraft, 2, 3, 6, 37 spacecraft dynamics, 147 system, 81 thrusters, 6, 11, 37, 106, 209 time profile, 211, 228 tolerance, 7, 13, 14, 119, 122, 140, 153, 209, 262, 263 tolerant controller design, 54, 57 torque, 142, 232 types, 4, 241 value, 118, 138 wheel, 260 Finite reaching time-based fault-tolerant control (FRFTC), 201, 203, 205, 206 Flight control, 10 Flight control system, 8, 16 Flow control valve, 6 Flywheel, 33, 251, 254, 256 failure, 249 mode, 259 reaction, 31, 126, 127, 129, 133, 138, 140–143, 244, 247, 251, 255, 259 Following finite-time FTC (FTFTC), 196, 199, 201, 203, 205, 206 Formation flying satellites, 12 Friction torque, 23, 32, 33

Indirect adaptive fault-tolerant control (IAFTC), 89, 91, 201, 204–206 Inertia in spacecraft attitude control, 197 matrix, 59, 70, 77, 89, 98, 102, 145, 161, 176, 226 moment matrix, 244 parameters, 173, 175 properties, 175 uncertainties, 126 value, 126 Inertial coordinate system, 249 Inertial frame, 27–29, 46, 53 Interacting multiple model (IMM), 10 Intermittent fault, 31, 34

L

Healthy actuators, 60, 104, 106, 256 attitude maneuvering performance, 206 operation, 89

Linear observer, 222, 223 Loss actuator effectiveness, 59, 60, 107, 117, 149 actuator effectiveness fault, 91, 118, 122 control, 165, 177 effectiveness, 10, 37, 38, 53, 63, 152, 170, 200 effectiveness fault, 82, 161 Lumped disturbances, 67, 82, 155 fault, 212, 214, 216–218, 229, 234, 235 fault in finite time, 218, 229 Lyapunov candidate function, 42, 44, 55, 56, 58, 82, 85, 87, 94, 100, 109, 112, 114, 129, 130, 134, 135, 146, 157, 158, 162, 179, 181, 192, 213, 215–217, 230, 231, 245–247, 250, 251, 253 stability, 24, 25, 55, 85, 126, 129, 157, 214, 234, 246, 253 stability analysis, 197

I

M

Idling fault, 33 Image data transmission, 238 Incipient fault, 211, 228

Magnetic torque, 30 torque rods, 238

G Globally asymptotic stability, 86, 87, 149 Gradual faults, 31, 33 Gyro faults, 6, 10

H

Index 279

Maneuvering control commend, 242 performance, 19, 242 Matrix fault, 99 inertia, 59, 70, 77, 89, 98, 102, 145, 176, 226 Maximum allowable torque, 237 output torque, 129, 140 torque, 111 Mechanical component fault, 3 faults, 2–4 momentum wheel fault, 6 Microsatellites, 10, 20, 209, 238 attitude control, 228 attitude dynamics, 226 Microscope thrusters, 10 Mission attitude control, 140 maneuver, 33 stabilization, 33, 241 tracking, 125 failure analysis, 2 performing, 187 spacecraft, 13, 73, 74 Modified Rodriguez parameter (MRP), 23, 28, 29, 36, 53, 127, 210, 219 set, 29 vector, 29 Momentum wheel (MW), 6, 23 fault, 6 fault model, 23 faulty model, 23 misalignments, 23 Multimodel fault model, 38

N Nanosatellites, 17, 20 Net disturbance force, 65 Nominal controller (NC), 60–63 Nongyroscopic attitude sensors, 198, 237 Nonlinear control law, 18 fault, 228

observer, 10, 262 spacecraft dynamics, 37 Numerical simulation results, 82

O Observer error, 108, 128, 224, 230 dynamics, 107, 214, 230 in finite time, 229 gains, 107, 108, 110, 118, 128, 213, 214, 245, 246, 250 input, 10 nonlinear, 10, 262 parameters, 130 state, 133, 246 state estimation, 128 Operational control limit, 79 mission capability, 14 Orbital period, 184, 203, 218, 220, 224, 255–257 space missions, 3 Orbiting spacecraft, 53, 102 Overactuated control systems, 18, 99 spacecraft attitude tracking systems, 18

P Parameter updating law, 77, 89 Partial failure, 99, 126, 127, 259 fault, 53 spacecraft, 37 Passive FTC, 7, 15, 209, 223 application, 7 approaches, 9, 15 controller, 16 design, 9 Performance attitude, 222 attitude control, 33, 51, 52, 62, 66, 118, 119, 122, 139, 142–144, 239, 258, 261 attitude maneuver, 173 control, 16, 23, 30, 37, 44, 45, 50, 63, 105, 126, 141, 142, 184, 199, 209, 222, 260, 261 controller, 75 evaluation, 88

280 Index

fault estimation, 256 index, 37, 43, 66, 122 optimization, 81 spacecraft attitude control, 1, 6, 33 PFTC, 121, 122 control, 121, 122 control laws, 117 controllers, 121 Pitch axes, 170, 185, 221, 248 Planned mission, 183, 207, 224, 241, 242 Posture control method, 243 Predetermined faults, 7, 209 Principal component analysis (PCA), 12 Problem statement, 38, 53, 126, 144, 154, 176, 211, 228 Proportional-derivative (PD) control method, 51 controller, 51, 52, 77 controller application, 51 Proportional-integral derivative (PID) control, 60, 88, 91, 105, 122 control conventional, 119 controller, 59, 60, 62, 63, 122, 237 Pulsed plasma thrusters, 12

R Reaction flywheel, 31, 126, 127, 129, 133, 138, 140–143, 244, 247, 251, 255, 259 control, 136 failure, 126, 141, 243, 252, 256 failure scenarios, 138 fault, 10, 140, 243, 250 for attitude control, 126 torque, 34 torque fault, 238 Reaction wheel (RW), 10–12, 14, 30, 47, 48, 170, 176, 177, 183–185, 187, 195, 198, 199, 211, 216, 218, 221, 224, 226–229, 262 effectiveness matrix, 211 failures, 262 fault, 10, 16, 30–33, 52, 177, 191, 199, 201, 207, 209, 211, 212, 216, 218, 219, 221–224, 227 fault mechanism, 30 fault scenarios, 138, 238

misalignment, 176–178 output torque, 32 saturation, 228, 235 Reconfigurable attitude controller, 19 controller, 8 Reconnaissance mission, 1, 140 Redundant actuators, 17, 18, 99, 177, 262, 263 wheel, 198, 200, 239 Relative attitude error, 65 Robust finite time fault-tolerant control (RFTFTC), 223, 224 Robust indirect adaptive fault-tolerant control (RIAFTC), 223

S Satellite attitude, 222, 227, 229, 230, 235, 238 attitude control, 23 attitude control system, 11, 12, 30 body axes, 238 FDD design, 11 mass, 209, 238 model description, 226 relative attitude, 227 Saturation actuator, 19–21, 73, 81, 86, 93, 98, 126, 154, 155, 164, 175, 224, 239, 241, 260 control, 140 control input, 80, 117, 122 control torque, 255 function, 166 limits, 104, 111 reaction wheel, 235 value, 103 Settling time, 20, 27, 60, 62, 76, 91, 218 Simulation results, 48, 49, 60, 76, 78, 88, 89, 91, 118, 137, 138, 140, 168, 171, 224, 256–258 Single fault, 31, 229 Sliding manifold, 39, 40, 44, 83, 85, 190, 192, 196, 197, 199, 200 design, 40, 83 in finite time, 199

Index 281

observer state, 239 surface, 40, 66, 67, 74, 76, 83, 84, 230, 231, 234, 250, 251 Sliding-mode control (SMC), 17, 39, 40, 83, 175, 176, 190, 196, 205 Sliding-mode observer (SMO), 11 Software fault, 2 Solar torque, 30 Space missions, 3, 4 Space Systems Engineering Database (SSED), 2 Spacecraft actuators, 10 attitude, 6, 8, 9, 23, 31, 54, 60, 69, 81, 83, 125–127, 192, 196, 210, 218 control, 10, 30, 83, 175, 209, 243, 260 control performance, 1, 6, 33 control problems, 33 control stability, 31 control strategies, 125 control system, 8, 9, 11, 12, 23, 27, 51, 99, 122, 125, 133, 244, 250, 261, 262 control system actuators, 33 control system faults, 31 dynamics, 93, 127, 137, 144, 262 error dynamics, 65, 66 FTC, 15, 17, 19, 20 FTC schemes, 20 kinematics, 248 maneuvers, 209 motion, 175 stabilization, 105, 125 system, 20, 21, 209 system state estimation, 10 tracking, 125, 209, 252 body, 31, 126, 249, 257 body frame, 75 component, 12 component faults, 11, 15 control application, 43 attitude dynamics, 93 problem, 21 system design, 81

controllers, 261 data, 2 design, 13 dynamics, 29, 36, 63, 66, 80, 89, 98, 144, 145 engineering, 80 failures, 80 fault, 2, 3, 6, 37 fault analysis, 1 FDD, 10, 11 formation flying control subjecting, 16 FTC, 13, 80 inertia matrix, 173 lifetime, 6 maneuver tracking, 175 mass, 144, 218 mission, 13, 73, 74 model, 116 orbit, 4, 249 partial failure, 37 payload, 138, 140, 142, 143, 255 pointing precision, 51 reliability, 15 structure, 176 thruster control allocation, 18 whether faults, 121 Stability analysis, 17, 20, 23, 24, 57, 73, 74, 99, 102, 129, 149, 158, 162, 235, 253 attitude, 91, 122, 139, 141–143, 170, 171, 187, 199, 201, 219, 221, 239, 256 attitude control, 132 control, 41 Lyapunov, 24, 25, 55, 85, 126, 129, 157, 214, 234, 246, 253 property, 42 robustness, 116, 161 spacecraft attitude control, 31 Stabilizing attitude, 238 Stringent pointing requirements, 187, 199, 201, 221, 239 Stuck actuator faults, 38 fault, 17, 37, 41 Subsystem faults, 2, 5

282 Index

Superior performance, 59, 103 Support vector machine (SVM), 12 System instability, 63, 81, 86, 161

T Terminal sliding-mode-based velocity-free controller (TSMOBVFC), 142, 143, 239, 241, 242, 258, 259 attitude stability performance, 142 control parameters, 258 Tethered satellite, 16 The Satellite Encyclopedia (TSE), 2 Thee controller, 239 Thrust faults, 16 Thrusters components faults, 6 configuration design, 18 effectiveness fault partial loss, 91 failures, 16 fault, 6, 11, 37, 106, 209 fault detection, 11 firings, 13 forces, 104 hybrid controller design, 16 limits, 103 outputs, 88 redundancy, 18 systems, 30 Torque actuator control, 86 commanded control, 140, 185, 187, 200, 202, 222, 258 control, 20, 29–31, 33, 51, 86, 99, 119, 155, 170, 177, 222, 227, 235, 237, 262 decrease, 32 fault, 142, 232

level disturbances, 66 maximum, 111 reaction, 34 Trajectory tracking control, 18

U Ultimately uniformly bounded (UUB), 25, 128, 149, 150, 157 Uncontrollability, 86 Underactuated spacecraft attitude controller design, 263 Unit quaternion output feedback control (UQOFC), 142, 143, 167–169, 171, 239, 241, 242, 258, 259 application, 168, 170 control strategy, 142 Updating law, 56, 69, 71, 86, 87, 94, 98, 100, 102

V Variable structure control, 93, 178, 180 controller, 178 Velocity-free fault-tolerant attitude stabilization control scheme (VFTASC), 239, 241 VFFTC, 167–171 Virtual control inputs, 111 law, 55, 112 signals, 18

W Wheel body, 31–33 faults, 260 speed constant, 35