Model-based Fault Diagnosis Techniques: Design Schemes, Algorithms, and Tools

Model-based Fault Diagnosis Techniques Steven X. Ding Model-based Fault Diagnosis Techniques Design Schemes, Algorit...

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Model-based Fault Diagnosis Techniques

Steven X. Ding

Model-based Fault Diagnosis Techniques Design Schemes, Algorithms, and Tools

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Prof. Dr. Steven X. Ding Head of Institute for Automatic Control and Complex Systems (AKS) Faculty of Engineering University of Duisburg-Essen Bismarckstr. 81 BB 47057 Duisburg Germany e-mail: [email protected]

ISBN 978-3-540-76303-1

e-ISBN 978-3-540-76304-8

DOI 10.1007/978-3-540-76304-8 Library of Congress Control Number: 2008921126 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data supplied by the author Coverdesign: eStudioCalamar S.L., F. Steinen-Broo, Pau/Girona, Spain Production: LE-TEX Jelonek, Schmidt & Voeckler GbR, Leipzig, Germany Printed on acid-free paper 987654321 springer.com

To My Parent and Eve Limin

Preface

The preparation of this book began nine years ago. As I was at the University of Applied Science Lausitz and planed my sabbatical in 1998, the idea of preparing a textbook on model-based fault diagnosis technique was born. I discussed with Prof. P. M. Frank about it and found a remarkable resonance. He invited me to spend my sabbatical in his institute and to work on the book. At that time, the model- and observer-based fault diagnosis technique became attractive and received enhanced attention both in the academic community and in industry. After the pioneering work in the 80’s, which led to the establishment of observer and parity space based fault diagnosis framework, the major topics in the 90’s focused on the advanced unknown input decoupling technique and robustness issues. Inspired by this trend and based on my Ph.D. work in Duisburg, I have, during March to September 1999 in Duisburg, provisionally completed the draft on the design of observer and parity relation based residual generators, the unknown input decoupling technique, fault isolation schemes and on the discussion about the robustness issues. They build the core of Chapters 5 - 7 and 13 of this book. Unfortunately, this work was interrupted by my engagement as vicepresident of the University of Applied Science Lausitz 1999 - 2000. Due to my move to the University of Duisburg in 2001 and the time consuming activity as the coordinator of the European research project IFATIS during 2002 - 2005, the break became longer and longer. On the other side, reviewing the progress in the model-based, in particular, in the observer-based fault diagnosis technique in the last years, I have to say that this break has also a unexpected positive side. In the past decade, the development of modelbased fault diagnosis technique was rapid and highly dynamic. Driven by the industrial demands for high reliability and safety on the one side and fully developed robust control theory on the other side, extensive and comprehensive research and development activities at universities and in industry have been dedicated to the model- and observer-based fault diagnosis technique. Advanced observer-based fault diagnosis schemes and new solutions to the robustness problems have been published in the leading journals in the ﬁeld

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of control theory and engineering, new research lines like the integrated design of control and fault diagnosis systems or the fault tolerant control have emerged, and successful applications in major industrial sectors have been reported. Today, model-based fault diagnosis is a part of control engineering and advanced control theory. A glance at the recent publications in journals and monographs on this topic reveals that it is one of the most vital research areas in the control community. Chapters 7 - 11 and 14 cover a wide range of the recent research topics of the observer-based fault diagnosis technique, including residual generator design with enhanced robustness against unknown inputs and model uncertainties, residual evaluation in the statistical and norm based frameworks and observer-based fault identiﬁcation schemes. A further positive aspect of the break is that the distance to my early work, the activity in the European project IFATIS and the recent cooperation with the automotive industry enable and motivate me to re-view the underlying ideas of the observer-based fault diagnosis technique and the associated design schemes under a dierent aspect. In this book, critical notes on the application of observer-based fault diagnosis technique are included and a new design strategy is proposed in Chapter 12. Thanks to the European project IFATIS and the industrial cooperation, my research group is involved in dierent benchmark studies. They enable me to include ﬁve benchmark systems in Chapter 3 and to use them in the subsequent chapters to illustrate the design schemes and algorithms. As a response to the increasing demands of industry for control engineers equipped with basic knowledge of model-based fault diagnosis and fault tolerant systems, a course entitled Fault Diagnosis and Fault Tolerant Systems is oered in the Department of Electrical Engineering and Information Technology at the University of Duisburg-Essen since 2002. It is a core course for the students of the master programs Automatic Control as well as Control and Information Systems. The draft of this book serves as the textbook for this course. It is also used in the seminar on Advanced Observer-based Fault Diagnosis Technique for the Ph.D. students in our institute. To help the students and the readers to understand the motivation and the original ideas of applying the advanced control theory to addressing the fault diagnosis problems, control theoretical preliminaries are integrated into the chapters where needed. If possible, they are described in the context of model-based fault diagnosis. It is remarkable that the main results and methods described in this book are presented in form of algorithms that enable the students and readers to check the theoretical results via short programs. Some of these algorithms are integrated into a MATLAB based FDI-Toolbox being available in our institute. This book is so structured that it can also be used as a self-study book for engineers working with automatic control and mechatronic systems. This book would not be possible without valuable support from many people. First, I would like to thank my wife and colleague, Eve Limin. It seems unusual. But, she is the person who inﬂuences my thinking at most, at least in the past two decades in working with fault diagnosis. As a holder of numerous

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patents on the model-based fault diagnosis systems in vehicles, she helps me to understand the practical side of the model-based fault diagnosis and to learn the link between the fault diagnosis theory and the engineering world. A lot of ideas and methods in this book are traced back to her contributions. I would especially like to thank Prof. Paul M. Frank, my respectful mentor. He paved me the way to the "fault diagnostic" world and opened me the door to a wonderful scientiﬁc community. I thank him for his inﬂuence on my research and his valuable support in preparing this book. I appreciate it very much to be able to work with wonderful colleagues in the dierent phases of my "fault diagnostic" life. During my Ph.D. study in Duisburg 1987 - 1992, I found in Jürgen Wünnenberg an excellent and most talented colleague who was full of new ideas and developed the ﬁrst unknown input observer scheme for the fault diagnosis purpose. In Senftenberg, at the University of Applied Science Lausitz, I have been successfully working with Torsten Jeinsch and Mario Sader in numerous industrial research projects, with Maiying Zhong on the robustness issues in the model-based fault diagnosis and with Hao Ye on the time-frequency domain properties of the observer and parity space based methods. In the past six years in Duisburg, I have found in Ping Zhang a valuable co-worker who is equipped with excellent mathematical and control theoretical skills. She has helped me to understand and solve some complex problems in dealing with model-based fault diagnosis. I am indebted to all of them for their great contributions to this book. I would like to thank my Ph.D. students for their valuable contribution to the benchmark study. They are Abdul Qayyum Khan and Yongqiang Wang (inverted pendulum), Muhammad Abid and Amol Naik (three-tank-system), Ibrahim Al-Salami, Jedsada Saijai, Wei Chen and Stefan Schneider (vehicle lateral dynamic system), Wei Li (DC motor), Alethya Salas and Alejandro Rodriguez (electrohydraulic servo-actuator). In addition, I would like to express my gratitude to Amol Naik for the extensive editorial corrections and Stefan Schneider for his valuable support in setting up the LATEX environment. I am also grateful to the technical stas and secretary for their support. Finally, I want to give an answer to one question that may arise (a typical formulation in such a book): Who has motivated me to continue the work on the book? It is Mrs. Hestermann-Beyerle from Springer-Verlag. On one occasion, she learned my previous work with lecture notes on model-based fault diagnosis and proposed the idea for this book. Thanks to her encouragement, I have re-started with this book project in May of this year. Without her constant support in the past months, it would be di!cult for me to complete this book. I am greatly indebted to her and her colleagues for the valuable help. Duisburg, December 2007

Steven X. Ding

Contents

Notation = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =XIX Part I Introduction, basic concepts and preliminaries 1

Introduction = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 3 1.1 Basic concepts of fault diagnosis technique . . . . . . . . . . . . . . . . . . 4 1.2 Historical development and some relevant issues . . . . . . . . . . . . . 8 1.3 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2

Basic ideas, major issues and tools in the observer-based FDI framework = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 2.1 On the observer-based residual generator framework . . . . . . . . 2.2 Unknown input decoupling and fault isolation issues . . . . . . . . . 2.3 Robustness issues in the observer-based FDI framework . . . . . . 2.4 On the parity space FDI framework . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Residual evaluation and threshold computation . . . . . . . . . . . . . . 2.6 FDI system synthesis and design . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 15 17 17 18 18

Modelling of technical systems = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 3.1 Description of nominal system behavior . . . . . . . . . . . . . . . . . . . . 3.2 Coprime factorization technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Representations of disturbed systems . . . . . . . . . . . . . . . . . . . . . . 3.4 Representations of system models with model uncertainties . . . 3.5 Modelling of faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Modelling of faults in closed loop feedback control systems . . . . 3.7 Benchmark examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Speed control of a DC motor . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Inverted pendulum control system . . . . . . . . . . . . . . . . . . 3.7.3 Three tank system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 23 25 25 27 30 31 31 34 38

3

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3.7.4 Vehicle lateral dynamic system . . . . . . . . . . . . . . . . . . . . . 42 3.7.5 Electrohydraulic Servo-actuator . . . . . . . . . . . . . . . . . . . . 46 3.8 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4

Structural fault detectability, isolability and identiﬁability = 4.1 Structural fault detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Excitations and su!ciently excited systems . . . . . . . . . . . . . . . . . 4.3 Structural fault isolability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Concept of structural fault isolability . . . . . . . . . . . . . . . . 4.3.2 Fault isolability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Structural fault identiﬁability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 56 57 57 58 65 67

Part II Residual generation 5

Basic residual generation methods = = = = = = = = = = = = = = = = = = = = = = = = = 71 5.1 Analytical redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2 Residuals and parameterization of residual generators . . . . . . . 75 5.3 Problems related to residual generator design and implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4 Fault detection ﬁlter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5 Diagnostic observer scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.5.1 Construction of diagnostic observer-based residual generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.5.2 Characterization of solutions . . . . . . . . . . . . . . . . . . . . . . . . 83 5.5.3 A numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.4 An algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.6 Parity space approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6.1 Construction of parity relation based residual generators 98 5.6.2 Characterization of parity space . . . . . . . . . . . . . . . . . . . . 100 5.6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.7 Interconnections, comparison and some remarks . . . . . . . . . . . . . 103 5.7.1 Parity space approach and diagnostic observer . . . . . . . . 103 5.7.2 Diagnostic observer and residual generator of general form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.7.3 Applications of the interconnections and some remarks . 110 5.7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.8 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6

Perfect unknown input decoupling = = = = = = = = = = = = = = = = = = = = = = = = = 115 6.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 Existence conditions of PUIDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2.1 A general existence condition . . . . . . . . . . . . . . . . . . . . . . . 117 6.2.2 A check condition via Rosenbrock system matrix . . . . . . 118

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6.2.3 Algebraic check conditions . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3 A frequency domain approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4 UIFDF design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.4.1 The eigenstructure assignment approach . . . . . . . . . . . . . 128 6.4.2 Geometric approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.5 UIDO design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.5.1 An algebraic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.5.2 Unknown input observer approach . . . . . . . . . . . . . . . . . . 142 6.5.3 A matrix pencil approach to the UIDO design . . . . . . . . 147 6.5.4 A numerical approach to the UIDO design . . . . . . . . . . . . 150 6.6 Unknown input parity space approach . . . . . . . . . . . . . . . . . . . . . . 153 6.7 An alternative scheme - null matrix approach . . . . . . . . . . . . . . . 153 6.8 Minimum order residual generator . . . . . . . . . . . . . . . . . . . . . . . . 154 6.8.1 Minimum order residual generator design by geometric approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.8.2 An alternative solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.9 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7

Residual generation with enhanced robustness against unknown inputs = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 161 7.1 Mathematical and control theoretical preliminaries . . . . . . . . . . 162 7.1.1 Signal norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.1.2 System norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1.3 Computation of H2 and H4 norms . . . . . . . . . . . . . . . . . . 168 7.1.4 Singular value decomposition . . . . . . . . . . . . . . . . . . . . . . . 169 7.1.5 Co-inner-outer factorization . . . . . . . . . . . . . . . . . . . . . . . . 170 7.1.6 Model matching problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.1.7 Essentials of the LMI technique . . . . . . . . . . . . . . . . . . . . . 173 7.2 Kalman ﬁlter based residual generation . . . . . . . . . . . . . . . . . . . . 174 7.3 Approximation of UI-distribution matrix . . . . . . . . . . . . . . . . . . . 178 7.3.1 Approximation of matrices Hg > Ig . . . . . . . . . . . . . . . . . . . 178 7.3.2 Approximation of matrices Kg>v . . . . . . . . . . . . . . . . . . . . . 180 7.3.3 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.4 Robustness, fault sensitivity and performance indices . . . . . . . . 184 7.4.1 Robustness and sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.4.2 Performance indices: robustness vs. sensitivity . . . . . . . . 185 7.4.3 Relations between the performance indices . . . . . . . . . . . . 186 7.5 Optimal selection of parity matrices and vectors . . . . . . . . . . . . 187 7.5.1 Vi>+ @Ug as performance index . . . . . . . . . . . . . . . . . . . . . . 188 7.5.2 Vi> @Ug as performance index . . . . . . . . . . . . . . . . . . . . . . 191 7.5.3 MVU as performance index . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.5.4 Optimization performance and system order . . . . . . . . . . 195 7.5.5 Summary and some remarks . . . . . . . . . . . . . . . . . . . . . . . . 197 7.6 H4 optimal fault identiﬁcation scheme . . . . . . . . . . . . . . . . . . . . 201 7.7 H2 @H2 design of residual generators . . . . . . . . . . . . . . . . . . . . . . . 202

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7.8 Relationship between H2 @H2 design and optimal selection of parity vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.9 LMI aided design of FDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.9.1 H2 to H2 trade-o design of FDF . . . . . . . . . . . . . . . . . . . 213 7.9.2 On H index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.9.3 H2 to H trade-o design of FDF . . . . . . . . . . . . . . . . . . . 225 7.9.4 H4 to H trade-o design of FDF . . . . . . . . . . . . . . . . . 227 7.9.5 An alternative H4 to H trade-o design of FDF . . . . 229 7.9.6 A brief summary and discussion . . . . . . . . . . . . . . . . . . . . . 232 7.10 The uniﬁed solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.10.1 Hl @H4 index and problem formulation . . . . . . . . . . . . . . 233 7.10.2 Hl @H4 optimal design of FDF: the standard form . . . . . 234 7.10.3 Discrete time version of the uniﬁed solution . . . . . . . . . . . 237 7.11 The general form of the uniﬁed solution . . . . . . . . . . . . . . . . . . . 238 7.11.1 Extended CIOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.11.2 Generalization of the uniﬁed solution . . . . . . . . . . . . . . . . 240 7.12 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 8

Residual generation with enhanced robustness against model uncertainties = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 247 8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 8.1.1 LMI aided computation for system bounds . . . . . . . . . . . 248 8.1.2 Stability of stochastically uncertain systems . . . . . . . . . . 249 8.2 Transforming model uncertainties into unknown inputs . . . . . . . 250 8.3 Reference model strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3.1 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.3.2 A reference model based solution for systems with norm bounded uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 252 8.4 Residual generation for systems with polytopic uncertainties . . 259 8.4.1 The reference model scheme based scheme . . . . . . . . . . . . 259 8.4.2 H_ to H4 design formulation . . . . . . . . . . . . . . . . . . . . . . 263 8.5 Residual generation for stochastically uncertain systems . . . . . 265 8.5.1 System dynamics and statistical properties . . . . . . . . . . . 266 8.5.2 Basic idea and problem formulation . . . . . . . . . . . . . . . . . . 266 8.5.3 An LMI solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 8.5.4 An alternative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 8.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

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Part III Residual evaluation and threshold computation 9

Norm based residual evaluation and threshold computation 281 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 9.3 Some standard evaluation functions . . . . . . . . . . . . . . . . . . . . . . . . 285 9.4 Basic ideas of threshold setting and problem formulation . . . . . 287 9.4.1 Dynamics of the residual generator . . . . . . . . . . . . . . . . . . 288 9.4.2 Deﬁnitions of thresholds and problem formulation . . . . . 289 9.5 Computation of Mwk>UPV>2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 9.5.1 Computation of Mwk>UPV>2 for the systems with the norm bounded uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.5.2 Computation of Mwk>UPV>2 for the systems with the polytopic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 9.6 Computation of Mwk>shdn>shdn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.6.1 Computation of Mwk>shdn>shdn for the systems with the norm bounded uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.6.2 Computation of Mwk>shdn>shdn for the systems with the polytopic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 9.7 Computation of Mwk>shdn>2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 9.7.1 Computation of Mwk>shdn>2 for the systems with the norm bounded uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 302 9.7.2 Computation of Mwk>shdn>2 for the systems with the polytopic uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 9.8 Threshold generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 9.9 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

10 Statistical methods based residual evaluation and threshold setting = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 311 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 10.2 Elementary statistical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 10.2.1 Basic hypothesis test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 10.2.2 Likelihood ratio and generalized likelihood ratio . . . . . . . 314 10.2.3 Vector-valued GLR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 10.2.4 Detection of change in variance . . . . . . . . . . . . . . . . . . . . . 317 10.2.5 Aspects of on-line realization . . . . . . . . . . . . . . . . . . . . . . . 318 10.3 Criteria for threshold computation . . . . . . . . . . . . . . . . . . . . . . . . . 320 10.3.1 The Neyman-Pearson criterion . . . . . . . . . . . . . . . . . . . . . . 320 10.3.2 Maximum a posteriori probability (MAP) criterion . . . . 321 10.3.3 Bayes’ criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 10.3.4 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 10.4 Application of GLR testing methods . . . . . . . . . . . . . . . . . . . . . . 324 10.4.1 Kalman ﬁlter based fault detection . . . . . . . . . . . . . . . . . . 324 10.4.2 Parity space based fault detection . . . . . . . . . . . . . . . . . . . 330

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10.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 11 Integration of norm based and statistical methods = = = = = = = = = 335 11.1 Residual evaluation in stochastic systems with deterministic disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 11.1.1 Residual generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 11.1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 11.1.3 GLR solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 11.1.4 Discussion and example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 11.2 Residual evaluation scheme for stochastically uncertain systems343 11.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 11.2.2 Solution and design algorithms . . . . . . . . . . . . . . . . . . . . . . 345 11.3 Probabilistic robustness technique aided threshold computation356 11.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 11.3.2 Outline of the basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 11.3.3 LMIs needed for the solutions . . . . . . . . . . . . . . . . . . . . . . . 359 11.3.4 Problem solutions in the probabilistic framework . . . . . . 360 11.3.5 An application example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 11.3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 11.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Part IV Fault detection, isolation and identiﬁcation schemes 12 Integrated design of fault detection systems = = = = = = = = = = = = = = = 369 12.1 FAR and FDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 12.2 Maximization of fault detectability by a given FAR . . . . . . . . . . 373 12.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 12.2.2 Essential form of the solution . . . . . . . . . . . . . . . . . . . . . . . 374 12.2.3 A general solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 12.2.4 Interconnections and comparison . . . . . . . . . . . . . . . . . . . . 378 12.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 12.3 Minimizing false alarm number by a given FDR . . . . . . . . . . . . 386 12.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 12.3.2 Essential form of the solution . . . . . . . . . . . . . . . . . . . . . . . 388 12.3.3 The state space form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 12.3.4 The extended form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 12.3.5 Interpretation of the solutions and discussion . . . . . . . . . 393 12.3.6 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 12.4 On the application to stochastic systems . . . . . . . . . . . . . . . . . . . 398 12.4.1 Application to maximizing FDR by a given FAR . . . . . . 398 12.4.2 Application to minimizing FAR by a given FDR . . . . . . . 399 12.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

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XVII

13 Fault isolation schemes = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 403 13.1 Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 13.1.1 Existence conditions for a perfect fault isolation . . . . . . . 404 13.1.2 PFIs and unknown input decoupling . . . . . . . . . . . . . . . . . 406 13.1.3 PFIs with unknown input decoupling (PFIUID) . . . . . . . 409 13.2 A frequency domain approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 13.3 Fault isolation ﬁlter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 13.3.1 A design approach based on the duality to decoupling control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 13.3.2 The geometric approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 13.3.3 A generalized design approach . . . . . . . . . . . . . . . . . . . . . . 417 13.4 An algebraic approach to fault isolation . . . . . . . . . . . . . . . . . . . . 426 13.5 Fault isolation using a bank of residual generators . . . . . . . . . . . 431 13.5.1 The dedicated observer scheme (DOS) . . . . . . . . . . . . . . . 432 13.5.2 The generalized observer scheme (GOS) . . . . . . . . . . . . . . 435 13.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 14 On fault identiﬁcation = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 441 14.1 Fault identiﬁcation ﬁlter and perfect fault identiﬁcation . . . . . . 442 14.2 FIF design with additional information . . . . . . . . . . . . . . . . . . . . 445 14.3 On the optimal fault identiﬁcation problem . . . . . . . . . . . . . . . . . 448 14.4 Study on the role of the weighting matrix . . . . . . . . . . . . . . . . . . 450 14.5 Approaches to the design of FIF . . . . . . . . . . . . . . . . . . . . . . . . . . 456 14.5.1 A general fault identiﬁcation scheme . . . . . . . . . . . . . . . . . 456 14.5.2 An alternative fault detection scheme . . . . . . . . . . . . . . . . 457 14.5.3 Identiﬁcation of the size of a fault . . . . . . . . . . . . . . . . . . . 458 14.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 References = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 463 Index = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 471

Notation

; 5 ^ _ := =, +, AA (??) max (min) sup (inf)

for all belong to subset union intersection identically equal approximately equal deﬁned as implies equivalent to much greater (less) than maximum (minimum) supremum (inﬁmum)

R and C C+ and C¯+ C and C¯ Cm$ C1 and C¯1 Rq Rq×p RH4 > RHq×p 4

ﬁeld of real and complex numbers open and closed right-half plane (RHP) open and closed left-half plane (LHP) imaginary axis open and closed plane with and outside of the unit circle space of real q-dimensional vectors space of q by p matrices denote the set of q by p stable transfer matrices, see [160] for deﬁnition denote the set of q by p stable, strictly proper transfer matrices, see [160] for deﬁnition denote the set of q by p transfer matrices, see [160] for deﬁnition

RH2 > RH2q×p LH4 > LHq×p 4

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Contents

[W [B udqn([) wudfh ([) det ([) ([) ¯ ([) ( max ([)) ([) ( min ([)) l ([) Im ([) Nhu([) gldj([1> · · · > [q )

transpose of [ orthogonal complement of [ rank of [ trace of [ determinant of [ eigenvalue of [ largest (maximum) singular value of [ least (minimum) singular value of [ the l-th singular value of [ image space of [ null space of [ block diagonal matrix formed with [1> · · · > [q

prob(d ? e) N (d> ) { 5 N (d> ) E ({) ydu ({)

probability that d ? e Gaussian distribution with mean d and variance { is distributed as N (d> ) mean of { variance of {

J(s)

transfer matrix, s is either v for a continuous time system or } for a discrete time system J (m$) = JW (m$) conjugate of J(m$) 1 (D> shorthand for G + F (sL D) E E> ¸F> G) DE shorthand for G + F (sL D)1 E FG udqn (J(s)) normal rank of J(s)> see [87] for deﬁnition

Part I

Introduction, basic concepts and preliminaries

1 Introduction

Associated with the increasing demands for higher system performance and product quality on the one side and more cost e!ciency on the other side, the complexity and the automation degree of technical processes are continuously growing. This development calls for more system safety and reliability. Today, one of the most critical issues surrounding the design of automatic systems is the system reliability and dependability. A traditional way to improve the system reliability and dependability is to enhance the quality, reliability and robustness of individual system components like sensors, actuators, controllers or computers. Even so, a fault-free system operation cannot be guaranteed. Process monitoring and fault diagnosis are hence becoming an ingredient of a modern automatic control system and often prescribed by authorities. Originated in the early 70’s, the model-based fault diagnosis technique has developed remarkably since then. Its e!ciency in detecting faults in a system has been demonstrated by a great number of successful applications in industrial processes and automatic control systems. Today, model-based fault diagnosis systems are fully integrated into vehicle control systems, robots, transport systems, power systems, manufacturing processes, process control systems, just to mention some of the application sectors. Although developed for dierent purposes by means of dierent techniques, all model-based fault diagnosis systems are common in the explicit use of a process model, based on which algorithms are implemented for processing data that are on-line collected and recorded during the system operation. The major dierence between the model-based fault diagnosis schemes lies in the form of the adopted process model and particular in the applied algorithms. There exists an intimate relationship between the model-based fault diagnosis technique and the modern control theory. Furthermore, due to the on-line requirements on the implementation of the diagnosis algorithms, powerful computer systems are usually needed for a successful fault diagnosis. Thus, besides the technological and economic demands, the rapid development of the computer technology and the control theory is another main reason why

4

1 Introduction

the model-based fault diagnosis technique is nowadays accepted as a powerful tool to solve fault diagnose problems in technical processes. Among the existing model-based fault diagnosis schemes, the so-called observer-based technique has received much attention since 90’s. This technique has been developed in the framework of the well-established advanced control theory, where powerful tools for designing observers, for e!cient and reliable algorithms for data processing aiming at reconstructing process variables, are available. The focus of this book is on the observer-based fault diagnosis technique and the related topics.

1.1 Basic concepts of fault diagnosis technique The overall concept of fault diagnosis consists in the following three essential tasks: • Fault detection: detection of the occurrence of faults in the functional units of the process, which lead to undesired or intolerable behavior of the whole system • Fault isolation: localization (classiﬁcation) of dierent faults • Fault analysis or identiﬁcation: determination of the type, magnitude and cause of the fault. A fault diagnosis system, depending on its performance, is called FD (for fault detection) or FDI (for fault detection and isolation) or FDIA (for fault detection, isolation and analysis) system, whose outputs are correspondingly alarm signals to indicate the occurrence of the faults or classiﬁed alarm signals to show which fault has occurred or data of deﬁned types providing the information about the type or magnitude of the occurred fault. The model-based fault diagnosis technique is a relatively young research ﬁeld in the classical engineering domain technical fault diagnosis, its development is rapid and currently receiving considerable attention. In order to explain the essential ideas behind the model-based fault diagnosis technique, we ﬁrst give a rough classiﬁcation of the technical fault diagnosis technique, as sketched in Fig.1.1, and brieﬂy review some traditional fault diagnosis schemes and their relationships to the model-based technique. • Hardware redundancy based fault diagnosis: The core of this scheme, as shown in Fig.1.2, consists in the reconstruction of the process components using the identical (redundant) hardware components. A fault in the process component is then detected if the output of the process component is dierent from the one of its redundancy. The main advantage of this scheme is its high reliability and the direct fault isolation. The use of redundant hardware results in, on the other hand, high costs and thus the application of this scheme is only restricted to a number of key components.

1.1 Basic concepts of fault diagnosis technique

5

Fig. 1.1 Classiﬁcation of fault diagnosis methods

Fig. 1.2 Schematic description of the hardware redundancy scheme

• Signal processing based fault diagnosis: On the assumption that certain process signals carry information about the faults of interest and this information is presented in form of symptoms, a fault diagnosis can be achieved by a suitable signal processing. Typical symptoms are time domain functions like magnitudes, arithmetic or quadratic mean values, limit values, trends, statistical moments of the amplitude distribution or envelope, or frequency domain functions like spectral power densities, frequency spectral lines, ceptrum, etc. The signal processing based schemes are mainly used for those processes in the steady state, and their e!ciency for the detection of faults in dynamic systems, which are of a wide operating range due to the possible variation of input signals, is considerably limited. Fig.1.3 illustrates the basic idea of the signal processing schemes.

6

1 Introduction

Fig. 1.3 Schematic description of the signal processing based scheme

• Plausibility test: As sketched in Fig.1.4, the plausibility test is based on the check of some simple physical laws under which a process component works. On the assumption that a fault will lead to the loss of the plausibility, checking the plausibility will then provide us with the information about the fault. The plausibility test is limited in its e!ciency for detecting faults in a complex process or for isolating faults.

Fig. 1.4 Schematic description of the plausibility test scheme

The intuitive idea of the model-based fault diagnosis technique is to replace the hardware redundancy by a process model which is implemented in the software form on a computer. A process model is a quantitative or a qualitative description of the process dynamic and steady behavior, which can be obtained using the well-established process modelling technique. In this way, we are able to reconstruct the process behavior on-line, which, associated with the concept of hardware redundancy, is called software redundancy concept. Software redundancies are also called analytical redundancies. Similar to the hardware redundancy schemes, in the framework of the software redundancy concept the process model will run in parallel to the process and be driven by the same process inputs. It is reasonable to expect that the re-constructed process variables delivered by the process model will well follow the corresponding real process variables in the fault-free operating states and show an evident derivation by a fault in the process. In order to receive this information, a comparison of the measured process variables

1.1 Basic concepts of fault diagnosis technique

7

(output signals) with their estimates delivered by the process model will then be made. The dierence between the measured process variables and their estimates is called residual. Roughly speaking, a residual signal carries the most important message for a successful fault diagnosis: if residual 6= 0 then fault, otherwise fault-free. The procedure of creating the estimates of the process outputs and building the dierence between the process outputs and their estimates is called residual generation. Correspondingly, the process model and the comparison unit build the so-called residual generator, as shown in Fig.1.5.

Fig. 1.5 Schematic description of the model-based fault diagnosis scheme

Residual generation can also be considered as an extended plausibility test, where the plausibility is understood as the process input-output behavior and modelled by an input-output process description. As a result, the plausibility check can be replaced by a comparison of the real process outputs with their estimates. Since no technical process can be modelled exactly and there often exist unknown disturbances, in the residual signal the fault message is corrupted with model uncertainties and unknown disturbances. Moreover, fault isolation and identiﬁcation require an additional analysis of the generated residual to distinguish the eects of dierent faults. A central problem with the application of model-based fault diagnosis technique can be expressed as ﬁltering/extracting the needed information about the faults of interests from the residual signals. To this end, two dierent strategies have been developed: • designing the residual generator to achieve a decoupling of the fault of interests from the other faults, unknown disturbances and model uncertainties

8

1 Introduction

• extracting the information about the fault of interests from the residual signals by means of post-processing of the residuals. This procedure is called residual evaluation. The ﬁrst strategy has been intensively followed by many of the research groups working on model-based fault diagnosis technique. One of the central schemes in this area is the so-called observer-based fault diagnosis technique, which is also the focus of this book. The basic idea behind the development of the observer-based fault diagnosis technique is to replace the process model by an observer which will deliver reliable estimates of the process outputs as well as to provide the designer with the needed design freedom to achieve the desired decoupling using the well-established observer theory. In the framework of residual evaluation, the application of the signal processing schemes is the state of the art. Among a number of evaluation schemes, the statistical methods and the so-called norm based evaluation are the most popular ones which are often applied to achieve optimal postprocessing of the residual generated by an observer. These two evaluation schemes are common in that both of them create a bound, the so-called threshold, regarding to all possible model uncertainties, unknown inputs and the faults of no interests. Exceeding the threshold indicates a fault in the process and will release an alarm signal. Integrated application of the both strategies, as shown in Fig.1.3 as well as in Fig.1.5, marks the state of the art of the model and observer-based fault diagnosis technique.

1.2 Historical development and some relevant issues The study on model-based fault diagnosis began in the early 1970s. Strongly stimulated by the newly established observer theory at that time, the ﬁrst model-based fault detection method, the so-called failure detection ﬁlter, was proposed by Beard and Jones. Since then, the model-based FDI theory and technique went through a dynamic and rapid development and is currently becoming an important ﬁeld of automatic control theory and engineering. As shown in Fig.1.6, in the ﬁrst twenty years, it was the control community that made the decisive contribution to the model-based FDI theory, while in the last decade, the trends in the FDI theory are marked by enhanced contributions from • the computer science community with knowledge and qualitative based methods as well as the computational intelligent techniques • the applications, mainly driven by the urgent demands for highly reliable and safe control systems in the automotive industry, in the aerospace area, in robotics as well as in large scale, networked and distributed plants and processes.

1.2 Historical development and some relevant issues

9

Fig. 1.6 Sketch of the historic development of model-based FDI theory

In the ﬁrst decade of the short history of the model-based FDI technique, various methods were developed. During that time the framework of the model-based FDI technique had been established step by step. In his celebrated survey paper in Automatica 1990, Frank summarized the major results achieved in the ﬁrst ﬁfteen years of the model-based FDI technique, clearly sketched its framework and classiﬁed the studies on model-based fault diagnosis into • observer-based methods • parity space methods and • parameter identiﬁcation based methods. In the early 90’s, great eorts have been made to establish relationships between the observer and parity relation based methods. Several authors from dierent research groups, in parallel and from dierent aspects, proven that the parity space methods lead to certain types of observer structures and are therefore structurally equivalent to the observer-based ones, even though the design procedures dier. From this viewpoint, it is reasonable to include the parity space methodology in the framework of the observer-based FDI technique. The interconnections between the observer and parity space based FDI residual generators and their useful application to the FDI system design and implementation build one of the central topics of this book. It is worth to point out that both observer-based and parity space methods only deal with residual generation problems. In the framework of the parameter identiﬁcation based methods, fault decision is performed by an on-line parameter estimation, as sketched in Fig.1.7.

10

1 Introduction

In the 90’s, there was an intensive discussion on the relationships between the observer and parameter estimation FDI schemes. Comparisons between these two schemes have been made on dierent benchmarks. These eorts lead to a now widely accepted point of view that both schemes have advantages and disadvantages in dierent respects, and there are arguments for and against each scheme.

Fig. 1.7 Schematic description of the parameter identiﬁcation scheme

It is interesting to notice that the discussion at that time was based on the comparison between an observer as residual generator and an parameter estimator. In fact, from the viewpoint of the FDI system structure, observer and parameter estimation FDI schemes are more or less common in residual generation but signiﬁcantly dierent in residual evaluation. The residual evaluation integrated into the observer-based FDI system is performed by a feedforward computation of the residual signals, as shown in Fig.1.5, while a recursive algorithm is used in the parameter estimation methods to process the residual signals aiming at a parameter identiﬁcation and the resulted parameter estimates are further fed back to the residual generator, as illustrated in Fig.1.8. Viewing from this aspect, the parameter identiﬁcation based fault diagnosis system is structured in a feedback closed-loop, and in against the observer-based FD system is open-loop structured.

Fig. 1.8 An alternative view of the parameter identiﬁcation scheme

The application of the well-developed adaptive observer theory to the fault detection and identiﬁcation in the recent decade is the result of a reasonable combination of the observer-based and parameter identiﬁcation FDI schemes.

1.3 Notes and references

11

The major dierence between the adaptive observer-based and parameter identiﬁcation FDI schemes lies in the residual generation. In other words, the adaptive observer-based FDI schemes dier from the regular observer-based ones in the way of residual evaluation. In this book, our focus in on the residual generation and evaluation issues in the framework of the observer and parity space based strategies. Besides of the introduction of basic ideas, special attention will be paid to those schemes and algorithms which are devoted to the analysis, design and synthesis of FDI systems.

1.3 Notes and references To author’s knowledge, the ﬁrst book on the model-based fault diagnosis technique with a strong focus on the observer and parity space based FDI schemes was published 1989 by Patton et al. [116]. For a long time, it was the only reference book in this area and has made a decisive contribution to the early development of the model-based FDI technique. The next two monographs, published by Gertler in 1998 [64] and by Chen and Patton in 1999 [21], address dierent issues of the model-based FDI technique. While [64] covers a wide spectrum of the model-based FDI technique, [21] is dedicated to the robustness issues in dealing with the observer-based FDI schemes. There are numerous books that deal with model-based FDI methods in part, for instance [10, 13, 69] or address a special topic in the framework of the model-based fault diagnosis technique like [100, 133]. In two recent books by Patton et al. [117] and Isermann [81], the latest results on model-based FDI technique achieved in the last decade are well presented. In the last three decades, numerous survey papers have been published. We divide them into three groups, corresponding to the dierent development phases of the model-based FDI technique, and give some representative ones from each group: • introduction and establishment of the observer, parity space and parameter identiﬁcation based FDI schemes [50, 67, 79, 146] • robustness issues [51, 52, 55, 114] • nonlinear, adaptive FDI schemes, application of computational intelligence [53, 90, 140]. Representative study on the relationships between the observer and parity relation based methods can be found, for instance, in [28, 62, 74]. For the comparison study on parameter identiﬁcation and observer-based FDI schemes the reader is referred to [1, 26, 63].

2 Basic ideas, major issues and tools in the observer-based FDI framework

In this chapter, we shall review the historical development of the observerbased FDI technique, the major issues and tools in its framework and roughly highlight the topics addressed in this book.

2.1 On the observer-based residual generator framework The core of the model-based fault diagnosis scheme shown in Fig.1.5 is a process model running parallel to the process. Today, it would be quite natural for anyone equipped with knowledge of the advanced control theory to replace the process model by an observer, in order to, for instance, increase the robustness against the model uncertainties, disturbances and deliver an optimal estimate of the process output. But, thirty years ago, the ﬁrst observer-based FDI system proposed by Beard and Jones marked a historical milestone in the development of the model-based fault diagnosis. The importance of their contribution lies not only in the application of observer theory, a hot research topic at that time in the area of the advanced control theory, to the residual generation, but also in the fact that their work built the fundament for the observer-based FDI framework and opened FDI community the door to the advanced control theory. Since that time, progress of the observer-based FDI technique is closely coupled with the development of the advanced control theory. Nowadays, the observer-based FDI technique is an active ﬁeld in the area of control theory and engineering. Due to the close relation to the observer study, the major topics for the observer-based residual generator design are quite similar to those concerning the observer design, including • observer/residual generator design approaches • reduced order observer/residual generator design and • minimum order observer/residual generator design.

14

2 Basic ideas, major issues and tools in the observer-based FDI framework

The major tools for the study of these topics are the linear system theory and linear observer theory. A special research focus is on the solution of the so-called Luenberger equations. In this book, Chapter 5 will address those topics. It is well-known that system observability is an important prerequisite for the design of a state observer. In the early development stage of the observer-based FDI technique, system observability was considered as a necessary structural condition for the observer construction. It has often been overlooked that diagnostic observers (i.e. observers for the residual generation or diagnostic purpose) are dierent from the well-known state observers and therefore deserve particular treatment. The wide use of the state observers for the diagnostic purpose misled some researchers to the erroneous opinion that for the application of the observer-based FDI schemes the state observability and knowledge of the state space theory would be indispensable. In fact, one of the essential dierences between the state observer and diagnostic observer is that the latter is primarily an output observer rather than a state observer often used for control purposes. Another misunderstanding of the observer-based FDI schemes is concerning the role of the observer. Often, the observer-based FDI system design is understood as the observer design and the FDI system performance is evaluated by the observer performance. It leads to an over-weighted research focus on the observer-based residual generation and less interests in studying the residual evaluation problems. In fact, the most important role of the observer in an FDI system is to make the generated residual signals independent of the process input signals and process initial conditions. The additional degree of design freedom can then be used, for instance, for the purpose of increasing system robustness.

2.2 Unknown input decoupling and fault isolation issues Several years after the ﬁrst observer-based FDI schemes have been proposed, it was recognized that such FDI schemes can only work satisfactorily if the model used describes the process perfectly. Motivated by it and coupled with the development of the unknown input decoupling control methods in the 80’s, study on the observer-based generation of the residuals decoupled from unknown inputs received strong attention in the second half of the 80’s. The idea behind the unknown input decoupling strategy is simple and clear: if the generated residual signals are independent of not only the inputs and initial conditions but also the unknown inputs, then they can be directly used as a fault indicator. Using the unknown input observer technique, which was still in its developing phase at that time, Wünnenberg and Frank proposed the ﬁrst unknown input residual generation scheme 1987. inspired and driven by this promising work, unknown input decoupling residual generation became one of the mostly addressed topics in the observer-based FDI framework in a

2.3 Robustness issues in the observer-based FDI framework

15

very short time. Since then, a great number of methods have been developed. Even today, this topic is still receiving considerable research attention. An important aspect of the study on unknown input decoupling is that it stimulated the study on the robustness issues in the model-based FDI. During the study on the unknown input decoupling FDI, it was recognized that the fault isolation problem can also be formulated as a number of unknown input decoupling problems. For this purpose, faults are, in dierent combinations, clustered into the faults of interests and faults of no interests which are then handled as unknown inputs. If it is possible to design a bank of residual generators that solves unknown input decoupling FDI for each possible combination, a fault isolation is then achieved. Due to its duality to the unknown input decoupling FDI in an extended sense, the decoupling technique developed in the advanced linear control theory in the 80’s oers one major tool for the FDI study. In this framework, there are numerous approaches, e.g. the eigenvalue and eigenstructure assignment scheme, matrix pencil method, geometric method, just to mention some of them. In this book, Chapter 6 is dedicated to the unknown input decoupling issues, while Chapter 13 to the fault isolation study. Already at this early stage, we would like to call reader’s attention to the dierence between the unknown input observer scheme and the unknown input residual generation scheme. As mentioned in the last section, the core of an observer-based residual generator is an output observer whose existence conditions are dierent (less strict) from the ones for a (state) unknown input observer. We would also like to give a critical comment on the original idea of the unknown input decoupling scheme. FDI problems deal, in their core, with a trade-o between the robustness against unknown inputs and the fault detectability. The unknown input decoupling scheme only focuses on the unknown inputs without explicitly considering the faults. As a result, the unknown input decoupling is generally achieved at the cost of the fault detectability. In Chapters 7 and 12, we shall discuss this problem and propose an alternative way of applying the unknown input decoupling solutions to achieve an optimal trade-o between the robustness and detectability.

2.3 Robustness issues in the observer-based FDI framework From today’s viewpoint, application of the robust control theory to the observer-based FDI should be a logical step following the study on the unknown input decoupling FDI. Historical development shows however a somewhat dierent picture. The ﬁrst work on the robustness issues was done in the parity space framework. In their pioneering work, Chow and Willsky as well as Lou et al. proposed a performance index for the optimal design of

16

2 Basic ideas, major issues and tools in the observer-based FDI framework

parity vectors if a perfect unknown input decoupling is not achievable due to the strict existence conditions. A couple of years later, in 1989 and 1991, Ding and Frank proposed the application of the H2 and H4 optimization technique, a central research topic in the area of control theory between the 80’s and early 90’s, to the observer-based FDI system design. Preceding to this work, a parametrization of (all) linear time invariant residual generators was achieved by Ding and Frank 1990, which builds, analogous to the well-known Youla-parametrization of all stabilization controllers, the basis of further study in the H4 framework. Having recognized that the H4 norm is not a suitable expression for the fault sensitivity, Ding and Frank in 1993 and Hou and Patton in 1996 proposed to use the minimum singular value of a transfer matrix to describe the fault sensitivity and gave the ﬁrst solutions in the H4 framework. Study on this topic builds one of the mainstreams in the robust FDI framework. Also in the H4 framework, transforming the robust FDI problems into the so-called Model-Matching-Problem (MMP), a standard problem formulation in the H4 framework, provides an alternative FDI system design scheme. This work has been particularly driven by the so-called integrated design of feedback controller and (observer-based) FDI system, and the achieved results have also been applied for the purpose of fault identiﬁcation, as described in Chapter 14. Stimulated by the recent research eorts on robust control of uncertain systems, study on the FDI in uncertain systems is receiving increasing attention in this decade. Remarkable progress in this study can be observed, since the so-called LMI (linear matrix inequality) technique is becoming more and more popular in the FDI community. For the study on the robustness issues in the observer-based FDI framework, H4 technique, including the so-called factorization technique, MMP solutions, and the LMI techniques are the most important tools. In this book, Chapters 7 and 8 are devoted to those topics. Although the above-mentioned studies lead generally to an optimal design of a residual generator under a cost function that expresses a trade-o between the robustness against unknown inputs and the fault detectability, the optimization is achieved regarding to some norm of the residual generator. In this design procedure, well known in the optimal design of feedback controllers, neither the residual evaluation nor the threshold computations are taken into account. As a result, the FDI performance of the overall system, i.e. the residual generator, evaluator and threshold, might be poor. This problem, which makes the FDI system design dierent from the controller design, will be addressed in Chapter 12.

2.5 Residual evaluation and threshold computation

17

2.4 On the parity space FDI framework Although they are based on the state space representation of dynamic systems, the parity space FDI schemes are signiﬁcantly dierent from the observerbased FDI methods in • the mathematical description of the FDI system dynamics, • and associated with it, also in the solution tools. In the parity space FDI framework, residual generation, the dynamics of the residual signals regarding to the faults and unknown inputs are presented in form of algebraic equations. Hence, most of the problem solutions are achieved in the framework of linear algebra. This not only brings with the advantages that (a) the FDI system designer is not required to have rich knowledge of the advanced control theory for the application of the parity space FDI methods (b) the most computations can be completed without complex and involved mathematical algorithms, but also provides the researchers with a valuable platform, at which new FDI ideas can be easily realized and tested. In fact, a great number of FDI methods and ideas have been ﬁrst presented in the parity space framework and later extended to the observer-based framework. The performance index based robust design of residual generators is a representative example. Motivated by these facts, we devote throughout this book much attention to the parity space FDI framework. The associated methods will be presented either parallel to or combined with the observer-based FDI methods. Comprehensive comparison studies build also a focus.

2.5 Residual evaluation and threshold computation Despite of the fact that an FDI system consists of a residual generator, a residual evaluator together with a threshold and a decision maker, in the observer-based FDI framework, studies on the residual evaluation and threshold computation have only been occasionally published. There exist two major residual evaluation strategies. The statistic testing is one of them, which is well established in the framework of statistical methods Another one is the so-called norm based residual evaluation. Besides of less on-line calculation, the norm based residual evaluation allows a systematic threshold computation using well-established robust control theory. The concept of norm based residual evaluation was initiated by Emaminaeini et al. in a very early development stage of the model-based fault diagnosis technique. In their pioneering work, Emami-naeini et al. proposed to use the root-mean-square (RMS) norm for the residual evaluation purpose and derived, based on the residual evaluation function, an adaptive threshold, also called threshold selector. This scheme has been applied to detect faults in dynamic systems with disturbances and model uncertainties. Encouraged by

18

2 Basic ideas, major issues and tools in the observer-based FDI framework

this promising idea, researchers have applied this concept to deal with residual evaluation problems in the H4 framework, where the L2 norm is adopted as the residual evaluation function. The original idea behind the residual evaluation is to create such a (physical) feature of the residual signal that allows a reliable detection of the fault. The L2 norm measures the energy level of a signal and can be used for the evaluation purpose. In practice, also other kinds of features are used for the same purpose, for instance the absolute value in the so-called limit monitoring scheme. In our study, we shall also consider various kinds of residual evaluation functions, besides of the L2 norm, and establish valuable relationships between those schemes widely used in practice, like limit monitoring, trends analysis etc. The mathematical tools for the statistic testing and norm based evaluation are dierent. The former is mainly based on the application of statistical methods, while for the latter the functional analysis and robust control theory are the mostly used tools. In this book, we shall in Chapters 9 and 10 address both the statistic testing and norm based residual evaluation and threshold computation methods. In addition, a combination of these two methods will be presented in Chapter 11.

2.6 FDI system synthesis and design In applications, an optimal trade-o between the false alarm rate (FAR) and fault detection rate (FDR), instead of the one between the robustness and sensitivity, is of primary interest in designing an FDI system. FAR and FDR are two concepts that are originally deﬁned in the statistic context. In their work in 2000, Ding et al. have extended these two concepts to characterize the FDI performance of an observer-based FDI system in the context of a norm based residual evaluation. In Chapter 12, we shall revise the FDI problems from the viewpoint of the trade-o between FAR and FDR. In this context, the FDI performance of the major residual generation methods presented in Chapters 6-8 will be checked. We shall concentrate ourselves on two design problems: (a) given an allowable FAR, ﬁnd an FDI system so that FDR is maximized (b) given an FDR, ﬁnd an FDI system to achieve the minimum FAR.

2.7 Notes and references As mentioned above, linear algebra and matrix theory, linear system theory, robust control theory, statistical methods and currently the LMI technique build the major tools for our study throughout this book. Among the great

2.7 Notes and references

19

number of available books on these topics, we would like to mention the following representative ones: • • • • •

matrix theory: [58] linear system theory: [19, 87] robust control theory: [49, 160] LMI technique: [14] statistical methods: [10, 93].

Below are the references for the pioneering works mentioned in this chapter: • the pioneering contributions by Beard and Jones that initiated the observerbased FDI study [11, 86] • the ﬁrst work of designing unknown input residual generator by Wünnenberg and Frank [149] • the ﬁrst contributions to the robustness issues in the parity space framework by Chow and Willsky, Lou et al., [23, 98], and in the observer-based FDI framework by Ding and Frank [37, 39, 43] as well as Hou and Patton [75] • the norm based residual evaluation initiated by Emami-naeini et al. [48] • the FDI system synthesis and design in the norm based residual evaluation framework by Ding et al. [31].

3 Modelling of technical systems

The objective of this chapter is to model a class of dynamic systems, which consist of a process, also known as plant, actuators and sensors for the control and supervision purposes, and may be, during their operation, disturbed, as schematically sketched in Fig. 3.1. Our major objective of addressing modelling issues is to describe nominal and faulty system behavior.

Fig. 3.1 Schematic description of the systems under consideration

We shall ﬁrst give a brief review of mathematical models for linear dynamic systems, including • input-output description • state space representation • dierent forms of models with disturbances and model uncertainties as well as • models that describe inﬂuences of faults. These model forms are essential for the subsequent studies in the latter chapters. Coprime factorization is a technique that bridges the system modelling and system analysis, synthesis in the advanced control theory. As one of the key tools for our study, coprime factorization will be frequently used throughout this book. This motivates us to address this topic in a separate section. We shall moreover deal with modelling of faults in a closed loop feedback control system, which is of a special interest for practical applications. A further focus of this chapter is on the introduction of ﬁve technical and

22

3 Modelling of technical systems

laboratory processes that will not only be used to illustrate the application of those model forms for the FDI purpose but also serve as benchmarks used throughout this book.

3.1 Description of nominal system behavior Depending on the process dynamics and modelling aims, dierent system model types can be used for the purpose of process description, among them the linear time invariant (LTI) system is the simplest and mostly used. In this book, we call disturbance-free and fault-free systems nominal and suppose that the nominal systems are LTI. There are two standard mathematical representations for LTI systems: the transfer matrix and the state space description. Below, they will be brieﬂy introduced. Roughly speaking, a transfer matrix is an input-output description of the dynamic behavior of an LTI system in the frequency domain. Throughout x will be used for presenting the transfer this book, notation J|x (s) 5 LHp×n 4 nx matrix from input vector x 5 R to output vector | 5 Rp > i.e. |(s) = J|x (s)x(s)=

(3.1)

It is assumed that J|x (s) is a proper real-rational matrix. We use s to denote either the complex variable v of Laplace transform for continuous time signals or the complex variable } of z-transform for discrete time signals. Remark 3.1 The results presented in this book generally hold for both continuous and discrete time systems except that the type of the system is speciﬁed. In that case, time variable w and complex variable v will be used for continuous time signal and systems, while time variable n and complex variable } for discrete time signals and systems. The standard form of the state description of a continuous time LTI system is given by {(w) ˙ = D{(w) + Ex(w)> {(0) = {0 |(w) = F{(w) + Gx(w)

(3.2) (3.3)

while for a discrete time LTI system we used {(n + 1) = D{(n) + Ex(n)> {(0) = {0 |(n) = F{(n) + Gx(n)

(3.4) (3.5)

where { 5 Rq is called the state vector, {0 the initial condition of the system, x 5 Rnx the input vector and | 5 Rp the output vector. Matrices D> E> F> G are appropriately dimensioned real constant matrices.

3.2 Coprime factorization technique

23

Remark 3.2 Considering that our subsequent study in the latter chapters will be carried out in the framework of linear system theory and thus be generally independent of the signal type, we shall use continuous time model to present the state space descriptions except that the signal type is speciﬁed. Also, for the sake of simplicity we shall drop out variable w so far no confusion is caused. State space model (3.2)-(3.3) can be either directly achieved by modelling or derived based on transfer matrix J|x (s)= The latter is called a state space realization of J|x (s) = F(vL D)1 E + G and denoted by ¸ DE = (3.6) J|x (s) = (D> E> F> G) or J|x (s) = FG In general, we assume that (D> E> F> G) is a minimal realization of J|x (s)=

3.2 Coprime factorization technique Coprime factorization of a transfer matrix gives a further system representation form which will be intensively used in our subsequent study. Roughly speaking, coprime factorization over RH4 is to factorize a transfer matrix into two stable and coprime transfer matrices. ˆ (s)> Q ˆ (s) in RH4 are called left Deﬁnition 3.1 Two transfer matrices P ˆ and \ˆ (s) in coprime over RH4 if there exist two transfer matrices [(s) RH4 such that ¸ £ ¤ ˆ ˆ (s) Q ˆ (s) [(s) = L= (3.7) P \ˆ (s) Similarly, two transfer matrices P (s)> Q (s) in RH4 are right coprime over RH4 if there exist two matrices \ (s)> [(s) such that ¸ £ ¤ P (s) [(s) \ (s) = L= (3.8) Q (s) Let J(s) be a proper real-rational transfer matrix. The so-called left coprime factorization (LCF) of J(s) is a factorization of J(s) into two stable and coprime matrices which will play a key role in designing residual generators. To complete the notation, we also introduce the right coprime factorization (RCF), which is however only occasionally applied in our study. ³ ´ ˆ (s) with the left coprime pair P ˆ (s)> Q ˆ (s) ˆ 1 (s)Q Deﬁnition 3.2 J(s) = P over RH4 is called LCF of J(s)= Similarly, RCF of J(s) is deﬁned by J(s) = Q (s)P 1 (s) with the right coprime pair (P (s)> Q (s)) over RH4 =

24

3 Modelling of technical systems

It follows from (3.7) and (3.8) that transfer matrices ¸ £ ¤ ˆ (s) Q ˆ (s) > P (s) P Q (s) are respectively right and left invertible in RH4 . Below, we present ³a lemma that ´ provides us with a state space comˆ ˆ putation algorithm of P (s)> Q (s) > (P (s)> Q (s)) and the associated pairs ³ ´ ˆ [(s)> \ˆ (s) and ([(s)> \ (s)) = Lemma 3.1 Suppose J(s) is a proper real-rational transfer matrix with state space realization (D> E> F> G)> and it is stabilizable and detectable. Let I and O be so that D + EI and D OF are both stable, and deﬁne ˆ (s) = (D OF> O> F> L) > Q ˆ (s) = (D OF> E OG> F> G) P (3.9) (3.10) P (s) = (D + EI> E> I> L) > Q (s) = (D + EI> E> F + GI> G) ˆ [(s) = (D + EI> O> F + GI> L) > \ˆ (s) = (D + EI> O> I> 0) (3.11) [(s) = (D OF> (E OG)> I> L) > \ (s) = (D OF> O> I> 0) = (3.12) Then ˆ (s) = Q (s)P 1 (s) ˆ 1 (s)Q J(s) = P

(3.13)

are the LCF and RCF of J(s), respectively. Moreover, the so-called Bezout identity given below is satisﬁed ¸ ¸ ¸ [(s) \ (s) L0 P (s) \ˆ (s) = = (3.14) ˆ (s) P ˆ (s) ˆ 0L Q Q (s)) [(s) In the textbooks on robust control theory, the reader can ﬁnd the feedback control interpretation of the RCF. For our purpose, we would like to give an observer interpretation of the LCF and the associated computation algorithm ³ ´ ˆ ˆ for P (s)> Q (s) = Introduce a state observer { ˆ˙ = Dˆ { + Ex + O (| |ˆ) > |ˆ = F { ˆ + Gx with an observer gain O that ensures the observer stability. Consider output estimation error u = | |ˆ= It turns out ´ ³ |(s) |ˆ(s) = F (sL D)1 E + G x(s) F (sL D)1 (O (|(s) |ˆ(s)) + Ex(s)) Gx(s) +, ³ ´ L + F (sL D)1 O (|(s) |ˆ(s)) = 0 +, |(s) |ˆ(s) = 0= On the other side,

3.4 Representations of system models with model uncertainties

25

³

´ |(s) |ˆ(s) = L F (sL D + OF)1 O |(s) ´ ³ 1 F (sL D + OF) (E OG) + G x(s)= It becomes evident that ˆ (s)x(s) ˆ (s)|(s) Q ˆ (s)x(s) = 0 +, |(s) = P ˆ 1 (s)Q P 1 ˆ (s)Q ˆ (s)= +, J(s) = P In fact, the output estimation error is the so-called residual signal.

3.3 Representations of disturbed systems In practice, environmental disturbances, unexpected changes within the technical process under observation as well as measurement and process noises are often modelled as unknown input vectors. We denote them by g> or and integrate them into input-output model (3.1) or state space model (3.2)-(3.3) as follows • input-output model |(s) = J|x (s)x(s) + J|g (s)g(s) + J| (s)(s)

(3.15)

where J|g (s) is known and called disturbance transfer matrix, g 5 Rng represents a deterministic unknown input vector, 5 Rn a steady stochastic process which is assumed to be, if no additional remark is made, a white, normal distributed noise vector with zero mean and variance matrix = gldj( 1 > · · · > n )= We use the notation 5 N (0> )= • state space representation {˙ = D{ + Ex + Hg g + > | = F{ + Gx + Ig g +

(3.16)

with Hg > Ig being constant matrices of compatible dimensions, g 5 Rng is again a deterministic unknown input vector, 5 N (0> )> 5 N (0> ).

3.4 Representations of system models with model uncertainties Model uncertainties refer to the dierence between the system model and the reality. It can be caused, for instance, by changes within the process or in the environment around the process. Representing model uncertainties is a research topic that is receiving more and more attention. In this book, we restrict ourselves to the following standard representations. Consider an extension of system model (3.1) given by

26

3 Modelling of technical systems

|(s) = J>|x (s)x(s) + J>|g (s)g(s)

(3.17)

where the subscript indicates model uncertainties. The model uncertainties can be represented either by an additive perturbation J>|x (s) = J|x (s) + Z1 (s)Z2 (s)

(3.18)

or in the multiplicative form J>|x (s) = (L + Z1 (s)Z2 (s)) J|x (s)

(3.19)

where Z1 (s)> Z2 (s) are some known transfer matrices and is unknown and ¯ (·) denotes the maximum singular value of bounded by ¯ () > where a matrix. Among a number of expressions for model uncertainties in the state space representations, we consider an extended form of (3.2)-(3.3) given by ¯ + Gx ¯ + I¯g g ¯ + Ex ¯ +H ¯g g> | = F{ {˙ = D{ ¯ = E + E> F¯ = F + F D¯ = D + D> E ¯ ¯g = Hg + H> I¯g = Ig + I G = G + G> H

(3.20) (3.21) (3.22)

where the model uncertainties D> E> F> G> H and I belong to one of the the following three types: • norm bounded type ¸ ¸ £ ¤ D E H H = (w) J K M F G I I

(3.23)

where H> I> J> K> M are known matrices of appropriate dimensions and (w) is unknown but bounded by ¯ () It is worth mentioning that (3.20)-(3.22) with norm bounded uncertainty (3.23) can also be written as {˙ = D{ + Ex + Hg g + Hs> | = F{ + Gx + Ig g + I s ˜ ˜ = (L + N)1 t = J{ + Kx + Mg + Ns> s = t> on the assumption that (L + N) is invertible. • polytopic type ¸ ¸¾ ¸ ½ Do Eo Ho D E H D1 E1 H1 >··· > = Fr F1 G1 I1 Fo Go Io F G I

(3.24) (3.25)

(3.26)

where Dl > El > Fl > Gl > Hl ,Il > l = 1> · · · > o> are known matrices of appropriate dimensions and Fr {·} denotes a convex set deﬁned by

3.5 Modelling of faults

¸

½

Do Eo Ho D1 E1 H1 >··· > F1 G1 I1 Fo Go Io ¸ o o X Dl El Hl X > l l = 1> l 0> l = 1> · · · > o= = Fl Gl Il Fr

l=1

27

¸¾

(3.27)

l=1

• stochastically uncertain matrices

D E H F G I

¸

¸ ¶ o μ X Dl El Hl s (n) = Fl Gl Il l

(3.28)

l=1

with known matrices £ Dl > El > Fl > Gl¤> Hl > Il > l = 1> · · · o> of appropriate dimensions. sW (n) = s1 (n) · · · so (n) represents model uncertainties and is expressed as a stochastic process with ¢ ¡ s¯(n) = E (s(n)) = 0> E s(n)sW (n) = gldj( 1 > · · · > o ) where E() = ¯ denotes the mean of variable and l > l = 1> · · · o> are known. It is further assumed that s(0)> s(1)> · · · > are independent and {(0)> x(n)> g(n) are independent of s(n)= Remark 3.3 Note that model (3.20)-(3.21) with polytopic uncertainty (3.27) can also be written as ! Ã o ! Ã o ! Ã o X X X {˙ = l (D + Dl ) { + l (E + El ) x + l (Hg + Hl ) g ol=1

Ã |=

o X

!

ol=1

Ã

l (F + Fl ) { +

ol=1

o X

ol=1

! l (G + Gl ) x +

ol=1

Ã

o X

! l (Ig + Il ) g=

ol=1

It is a polytopic system.

3.5 Modelling of faults There exist a number of ways to model faults, among them the extension of model (3.15) to |(s) = J|x (s)x(s) + J|g (s)g(s) + J|i (s)i (s)

(3.29)

is a widely used one, where i 5 Rni is a unknown vector that represents all possible faults and will be zero in the fault-free case, J|i (s) 5 LH4 is a known transfer matrix. Throughout this book, i is assumed to be a deterministic time function. No further assumption on it is made, provided that the type of the fault is not speciﬁed. Suppose that a minimal state space realization of (3.29) is given by

28

3 Modelling of technical systems

{˙ = D{ + Ex + Hg g + Hi i | = F{ + Gx + Ig g + Ii i

(3.30) (3.31)

with known matrices Hi > Ii = Then we have 1

J|i (s) = Ii + F (sL D)

Hi =

(3.32)

It becomes evident that Hi > Ii indicate the place where a fault occurs and its inﬂuence on the system components. As shown in Fig. 3.1, we divide the faults into three categories: • sensor faults iV : these are faults that directly act on the process measurement • actuator faults iD : these faults cause changes in the actuator • process faults iS : they are used to indicate malfunctions within the process. Sensor faults can be modelled by setting Ii = L> i.e. | = F{ + Gx + Ig g + iV

(3.33)

while actuator faults by setting Hi = E> Ii = G> i.e. {˙ = D{ + E (x + iD ) + Hg g> | = F{ + G (x + iD ) + Ig g=

(3.34)

Depending on their type and location, process faults can be modelled by Hi = HS and Ii = IS for some HS > IS = For a system with sensor, actuator and process faults, we deﬁne 5 6 iD £ £ ¤ ¤ (3.35) i = 7 iS 8 > Hi = E HS 0 > Ii = G IS L iV and apply (3.30)-(3.31) to represent the system dynamics. Due to the way how they aect the system dynamics, the faults described by (3.30)-(3.31) are called additive faults. It is very important to note that the occurrence of an additive fault will not aect the system stability, independent of if a feedback control loop is integrated into the system under observation. Typical additive faults met in practice are, for instance, oset in sensors and actuators or drift in sensors. The former can be described by a constant> while the latter by a ramp. In practice, malfunctions in the process or in the sensors and actuators often cause changes in the model parameters. They are called multiplicative faults and generally modelled in terms of parameter changes. They can be described by extending (3.20)-(3.22) to ¯ + EI )x + Hg g {˙ = (D¯ + DI ){ + (E ¯ + GI )x + Ig g | = (F¯ + FI ){ + (G

(3.36) (3.37)

3.5 Modelling of faults

29

where DI > EI > FI > GI represent the multiplicative faults in the plant, actuators and sensors, respectively. It is assumed that DI =

oD X

Dl Dl > EI =

l=1

FI =

oF X

oE X

El El

(3.38)

Gl Gl

(3.39)

l=1

Fl Fl > GI =

l=1

oG X l=1

where • Dl > l = 1> · · · > oD > El > l = 1> · · · > oE > Fl > l = 1> · · · > oF > and Gl > l = 1> · · · > oG > are known and of appropriate dimensions • Dl > l = 1> · · · > oD > El > l = 1> · · · > oE > Fl > l = 1> · · · > oF > and Gl > l = 1> · · · > oG > are unknown time functions Multiplicative faults are characterized by their (possible) direct inﬂuence on the system stability. This fact is evident for the faults described by DI = In case that state feedback or observer-based state feedback control laws are adopted, we can also see that EI > FI > GI would aect the system stability. Introducing tP = JI { + KI x> iP = I (w)tP 6 6 5 5 Lq×q 0 9 .. : 9 .. : 9 . : 9 . : : : 9 9 9 Lq×q : 9 0 : : : 9 9 JI = 9 : > KI = 9 Lnx ×nx : : 9 0 : 9 9 . : 9 . : 7 .. 8 7 .. 8 Lnx ×nx 0

(3.40)

³ ´ I (w) = gldj D1 Lq×q > · · · > DoD Lq×q > E1 Lnx ×nx > · · · > EoE Lnx ×nx ¤ ¤ £ £ HI = D1 · · · DoD E1 · · · EoE > II = F1 · · · FoF G1 · · · GoG we can rewrite (3.36)-(3.37) into ¯ + Ex ¯ + Hg g + HI iP {˙ = D{ ¯ + Gx ¯ + Ig g + II iP = | = F{

(3.41) (3.42)

In this way, the multiplicative faults are modelled as additive faults. Also for this reason, the major focus of our study in this book will be on the detection and identiﬁcation of additive faults. But, the reader should keep in mind that iP is a function of the state and input variable of the system and thus will aect the system stability.

30

3 Modelling of technical systems

3.6 Modelling of faults in closed loop feedback control systems Model-based fault diagnosis systems are often embedded in closed loop feedback control systems. Due to the closed loop structure with an integrated controller that brings the system in general robustness against changes in the system, special attention has been paid to the topic of fault detection in feedback control loops. In this section, we consider modelling issues for a standard control loop with sensor, actuator and process faults, as sketched in Fig. 3.2.

Fig. 3.2 Strcture of a standard control loop with faults

Suppose that the process with sensors and actuators is described by (3.16). Denote the control objective by }> reference signal by z> the preﬁlter by (v) and the control law by y(s) = N(s)|(s)= For the sake of simplifying the problem formulation, we only consider additive faults. The overall system model with sensor, actuator and process faults is then given by {˙ = D{ + E (x + iD ) + Hg g + HS iS | = F{ + G (x + iD ) + Ig g + iV + IS iS x(s) = N(s)|(s) + (s)z(s)=

(3.43) (3.44) (3.45)

Depending on the signal availability and requirements on the realization of FDI strategy, there are two dierent ways of modelling the system. In the framework of the so-called open loop FDI, it is assumed that input and output vectors x and | are available. For the FDI purpose, the so-called open loop model (3.43)-(3.44) can be used, which contains all information needed for detecting the faults. Note that this open loop model is identical with the one introduced in the last section. In practice, it is often the case that x is not available. For instance, if the control loop is a part of a large scaled system and located remotely from the supervision station, where the higher level controller and FDI unit are located, the reference signal z> instead of process input signal x, is usually available for the FDI purpose. In those cases, the so-called closed loop FDI strategy can

3.7 Benchmark examples

31

be applied. The closed loop FDI strategy is based on the closed loop model with z and | as input and output signals respectively. The nominal system behavior of the closed loop is described by |(s) = J|z (s)z(s)> J|z (s) = (L J|x (s)N(s))1 J|x (s) (s) J|x (s) = G + F (sL D)

1

(3.46)

E=

The overall system model with the faults and disturbances is given by |(s) = J|z (s)z(s) + J|g>fo (s)g(s) + J|iD >fo (s)iD (s) +J|iS >fo (s)iS (s) + J|iV >fo (s)iV (s) ³ ´ J|g>fo (s) = (L J|x (s)N(s))1 Ig + F (sL D)1 Hg ´ ³ 1 1 J|iD >fo (s) = (L J|x (s)N(s)) G + F (sL D) E ³ ´ J|iS >fo (s) = (L J|x (s)N(s))1 IS + F (sL D)1 HS J|iV >fo (s) = (L J|x (s)N(s))

1

(3.47)

=

From the viewpoint of residual generation, which utilizes the nominal model, it may be of additional advantage to adopt the closed loop FDI strategy. It is known in control theory that by means of some advanced control strategy the dynamics of the closed loop system, J|z (s)> may be well modelled in a form easy for further handling. For instance, using a decoupling controller will result in a diagonal J|z (s)> which may reduce an MIMO (multiple input, multiple output) system into a number of (decoupled) SISO (single input, single output) ones.

3.7 Benchmark examples In this section, ﬁve benchmark examples will be introduced. They will be used to illustrate the modelling schemes described in the previous sections and serve subsequently as benchmark systems in the forthcoming chapters. 3.7.1 Speed control of a DC motor DC (Direct Current) motor converts electrical energy into mechanical energy. Below, the laboratory DC motor control system DR300 will be brieﬂy described. Model of DC motor Fig. 3.3 gives a schematic description of a DC motor, which consists of an electrical part and a mechanical part. Deﬁne the loop current LD and the armature frequency as state variables, the terminal voltage XD as input

32

3 Modelling of technical systems

Fig. 3.3 Schematic description of a DC motor

and the (unknown) load PO as disturbance, we have the following state space description ¸ ¸ UD F ¸ ¸ 1 ¸ OD OD LD 0 L˙D O D XD + PO + (3.48) NP

M1 0

˙ 0 M as well as the transfer function

(v) =

³

³ F 1 +

1 MUD NP F v

+

MWD UD 2 NP F v

UD (1 + WD v)

NP F 1 +

MUD NP F v

+

MWD UD 2 NP F v

´ XD (v)

´ PO (v)> WD =

(3.49) OD UD

where the parameters given in (3.48) and (3.49) are summarized in Table 3.1. Table 3.1 Parameters of laboratory DC motor DR300 Parameter Symbol Value Total Inertia M 62=75 Voltage constant F 6=27 · 1033 Motor constant NP 0=06 Armature Inductance OD 0=003 Resistance UD 3=13 Tacho output voltage NW 5 · 1033 Tacho ﬁlter time constant WW 5

Unit Y @Xsp Y @Xsp Qp@D K Rkp Y @Xsp pv

Models of DC motor control system For the purpose of speed control, cascade control scheme is adopted with a speed control loop and a current control loop. As sketched in Fig. 3.4, the DC motor together with the current control loop will be considered as the plant that is regulated by a PI speed controller. The plant dynamics can be approximately described by

3.7 Benchmark examples

33

Fig. 3.4 Structure of DC motor control system

|(v) = J|x (v)x(v) + J|g (v)g(v) (3.50) 8=75 31=07 > J|g (v) = J|x (v) = (1 + 1=225v)(1 + 0=03v)(1 + 0=005v) v(1 + 0=005v) with | = Xphdv (voltage delivered by the Tacho) as output, x = Luhi as input and g = PO as disturbance. With a PI speed controller set to be x(v) = N(v)(z(v) |(v))> N(v) = 1=6

1 + 1=225v v

(3.51)

where z(v) = uhi (v)> the closed loop model is given by |(v) = J|z (v)z(v) + J|g>fo (v)g(v) 14=00 J|z (v) = v(1 + 0=03v)(1 + 0=005v) + 14=00 31=07(1 + 0=03v) = J|g>fo (v) = v(1 + 0=03v)(1 + 0=005v) + 14=00

(3.52)

Modelling of faults Three faults will be considered: • an additive actuator fault iD • an additive fault in Tacho iV1 and • a multiplicative fault in Tacho iV2 5 [1> 0]. Based on (3.50), we have the open loop structured overall system model |(v) = J|x (v)x(v) + J|g (v)g(v) + J|iD (v)iD + J|iV1 (v)iV1 + |(v)iV2 (3.53) J|iD (v) = J|x (v)> |(v) = (J|x (v)x(v) + J|g (v)g(v)) = The closed loop model can be achieved by extending (3.52) to |(v) = J|z (v)z(v) + J|g>fo (v)g(v) + J|iD >fo (v)iD (v) +J|iV1 >fo (v)iV1 (v) + |fo (v)

(3.54)

34

3 Modelling of technical systems

8=75v (1 + 1=225v) (v(1 + 0=03v)(1 + 0=005v) + 14=00) v(1 + 0=03v)(1 + 0=005v) J|iV1 >fo (v) = v(1 + 0=03v)(1 + 0=005v) + 14=00 J|iD >fo (v) =

1

|fo (v) = (1 + J|x (v)N(v)iV2 ) (J|z (v)z(v) + J|g>fo (v)g(v)) iV2 (J|z (v)z(v) + J|g>fo (v)g(v)) = 3.7.2 Inverted pendulum control system Inverted pendulum is a classical laboratory system that is widely used in the education of control theory and engineering. Below is a brief introduction to the laboratory pendulum system LIP100 that is schematically sketched in Fig.3.5.

y 7

r(t)

9 4

u = input voltage

8

Ø

2

6 1

x

3

5

Fig. 3.5 Schematic description of an inverted pendulum

The inverted pendulum system consists of a cart (pos. 6 in Fig.3.5) that moves along a metal guiding bar (pos.5). An aluminum rod (pos.9) with a cylindrical weight (pos.7) is ﬁxed to the cart by an axis. The cart is connected by a transmission belt (pos.4) to a drive wheel (pos.3). The wheel is driven by a current controlled direct current motor (pos.2) which delivers a torque proportional to the acting control voltage xv such that the cart is accelerated. This system is nonlinear and consists of four state variables: • • • •

the the the the

position of the cart u (marked by 6 in Fig.3.5) velocity of the cart u˙ angle of the pendulum as well as ˙ angle velocity =

Among the above state variables, u is measured by means of a circular coil potentiometer that is ﬁxed to the driving shaft of the motor, u˙ by means of the tacho generator that is also ﬁxed to the motor and by means of a layer

3.7 Benchmark examples

35

potentiometer ﬁxed to the pivot of the pendulum. The system input x is the acting control voltage xv that generates force I on the cart. Nonlinear system model The following nonlinear model describes the dynamics of the inverted pendulum: ´ ³ (3.55) u˙ = () d32 sin cos + d33 u˙ + d34 ˙ cos + d35 ˙ 2 sin + e3 I ´ ³ ˙ = () d42 sin + d43 u˙ cos + d44 ˙ + d45 ˙ 2 cos sin + e4 I cos (3.56) where ¶1 μ Q2 () = 1 + 2 sin Q01 2 Q Iu QF F PQ d32 = 2 j> d33 = 2 > d34 = 2 > d35 = 2 > d42 = 2 j Q01 Q01 Q01 Q01 Q01 2 Iu Q PF Q Q d43 = 2 > d44 = 2 > d45 = 2 > e3 = 2 > e4 = 2 = Q01 Q01 Q01 Q01 Q01 The parameters are given in Table 3.2. Disturbances There are two types of frictions in the system that may considerably affect the system dynamics. Theses are Coulomb friction and static friction, described by Coulomb friction : If = |If |sgn(u) ½ Iq > u = 0 static friction : IKU = = 0> u 6= 0 To include their eects in the system model, I is extended to Ivxp = I + g with g being a unknown input. Linear model After a linearization at operating point u = 1> u˙ = 0> = 0> ˙ = 0 and a normalization with 6 5 6 q11 u {1 9 {2 : 9 q22 : : 9 :=9 7 {3 8 7 q33 u˙ 8 {4 q44 ˙ 5

36

3 Modelling of technical systems

we have the following (linear) state space model of the inverted pendulum {˙ = D{ + Ex + Hg g> | = F{ + 5 6 5 6 0 0 1=95 0 0 90 9 : 0 0 1=0 : 0 :>E = 9 : D=9 7 0 0=12864 1=9148 0=0082 8 7 6=1343 8 0 21=4745 26=31 0=1362 84=303 6 5 1000 F = 7 0 1 0 0 8 > Hg = E> x = Ni I> 5 N (0> ) 0010

(3.57)

where denotes the measurement noise. It is worth noting that linear model (3.57) is valid under the following conditions: • |I | 20Q • |u| 0=5p • || 10 = Discrete time model By a discretization of model (3.57) with a sampling time W = 0=03 sec, we obtain the following discrete time model {(n + 1) = Dg {(n) + Eg x(n) + Hgg g(n)> |(n) = F{(n) + (n) (3.58) 5 6 5 6 1 0=0001125 0=05685 8=174h 006 0=005288 90 9 0=03723 : 1=01 0=01162 0=03003 : : 9 : Dg = 9 7 0 0=00384 0=9441 0=0002962 8 > Eg = Hgg = 7 0=1792 8 = 0 0=6434 0=768 1=005 2=461 LCF of the nominal model To illustrate the coprime factorization technique introduced in Subsection 3.2, we derive below an LCF for model (3.57). It follows from Lemma 3.1 that for the purpose of an LCF of (3.57) the so-called observer gain matrix O should be selected that ensures the stability of D OF= Using the pole assignment method with the desired poles v1 = 6=0> v2 = 6=5> v3 = 7=0> v4 = 7=5> O is chosen equal to 5 6 6=9983 0=0025 1=9544 9 0=2552 13=7523 13=2474 : : O=9 7 0=4027 0=0192 11=9981 8 3=5602 64=3490 158=1113 which gives 6 6=9983 0=0025 0=0044 0 9 0=2552 13=7523 13=2474 1=0000 : : D OF = 9 7 0=4027 0=1478 13=9131 0=0082 8 = 3=5602 42=8760 184=4213 0=1362 5

3.7 Benchmark examples

37

As a result, the LCF of system (3.57) is given by ˆ x1 (v)Q ˆx (v) J|x (v) = F (vL D)1 E = P 1 ˆx (v) = F (vL D + OF)1 E= ˆ x (v) = L F (vL D + OF) O> Q P

Table 3.2 Parameters of laboratory pendulum system LIP100 Constant Numerical value Nu 2.6 q11 14.9 q22 -52.27 q33 -7.64 q44 -52.27 P0 3.2 P1 0.329 P 3.529 ov 0.44 0.072 Q 0.1446 2 Q01 0.23315 2 Q 2 @Q01 0.0897 Iu 6.2 F 0.009

Unit Q@Y Y @p Y @udg Y v@p Y v@udg Nj Nj Nj p Njp2 Njp Nj 2 p2 Nj@v Njp2 @v

Model uncertainty Recall that linear model (3.57) has been achieved by a linearization at an operating point. The linearization error will cause uncertainties in the model parameters. Taking into account it, model (3.57) is extended to (3.59) {˙ = (D + D) { + (E + E)x + (Hg + H)g> | = F{ + £ ¤ £ ¤ D E H = H(w) J K K 5 6 6 6 5 5 00 ¸ 0 0100 90 0: d32 d33 d34 : 8 8 7 7 H=9 7 1 0 8 > (w) = d42 d44 d44 > J = 0 0 1 0 > K = 3=2 0 0001 01 ¯ ((w)) 1=7221= Modelling of faults Additive sensor and actuator faults are considered. To model them, (3.57) and (3.58) are respectively extended to

38

3 Modelling of technical systems

{˙ = D{ + Ex + Hg g + Hi i> | = F{ + Ii i + {(n + 1) = Dg {(n) + Eg x(n) + Hgg g(n) + Hgi i (n) |(n) = F{(n) + Ii i (v) + (n) £ ¤ £ ¤ Hi = E 0 0 0 > Hgi = Eg 0 0 0 5 6 5 6 6 5 i1 iD 0100 9 i2 : 9 iV1 : : 9 : Ii = 7 0 0 1 0 8 > i = 9 7 i3 8 = 7 iV2 8 = 0001 i4 iV3

(3.60) (3.61)

Closed loop model An observer-based state feedback controller with a disturbance compensation is integrated into LIP100 control system, which consists of • an observer

" # ¸ ¸ { ˆ˙ { ˆ D Hg + Ex + O (| F { ˆ) = ˙ 0 0 gˆ gˆ

(3.62)

which delivers an estimate for { and g respectively, • a state feedback controller with a disturbance compensator x = N { ˆ gˆ + Y z

(3.63)

where the observer, feedback gains O> N and the preﬁlter Y are respectively 5 6 6=9983 0=0025 1=9544 ¸ 9 0=2552 13=7523 13=2474 : 9 : O1 0=4027 0=0192 11=9981 : O= =9 9 : O2 7 3=5602 64=3490 158=1113 8 0=4596 0=2586 7=7164 £ ¤ N = 2=1000 2=2151 3=8604 0=4819 > Y = 2=1= The overall system dynamics is described by 65 6 5 6 5 6 5 { D EN EN Hg 0 {˙ 7 h˙ { 8 = 7 0 D O1 F Hg 8 7 h{ 8 + 7 0 8 g˙ + 1 h˙ g 0 O2 F 0 hg 5 6 5 6 5 6 E 0 Hi 7 0 8 Y z + 7 O1 8 y + 7 (Hi O1 Ii ) 8 i 0 O2 Ii O2 | = F{ + y + Ii i=

(3.64)

(3.65)

3.7.3 Three tank system Three tank system sketched in Fig.3.6 has typical characteristics of tanks, pipelines and pumps used in chemical industry and thus often serves as benchmark process in laboratories for process control. The three tank system introduced in here is a laboratory setup DTS200.

3.7 Benchmark examples Pump 1

39

Pump 2

A

A

Tank 1

Tank 2

H m ax

Tank 3

A

h1

h2 h3 sn

Fig. 3.6 DTS200 setup

Nonlinear model Applying the incoming and outgoing mass ﬂows under consideration of Torricellies law, the dynamics of DTS200 is modelled by Dk˙1 = T1 T13 > Dk˙2 = T2 + T32 T20 > Dk˙3 = T13 T32 p T13 = d1 v13 sgn(k1 k3 ) 2j|k1 k3 | p p T32 = d3 v23 sgn(k3 k2 ) 2j|k3 k2 |> T20 = d2 v0 2jk2 where • T1 > T2 are incoming mass ﬂow • Tlm is the mass ﬂow from the l-th tank to the m-th tank • kl (w)> l = 1> 2> 3> are the water level of each tank and measured. The parameters are given in Table 3.3. Linear model After a linearization at operating point k1 = 45fp, k2 = 15fp and k3 = 30fp, we have the following linear (nominal) model {˙ = D{ + Ex> | = F{ (3.66) 6 5 5 6 ¸ 0=0085 0 0=0085 k1 T1 0 0=00195 0=0084 8 >D = 7 { = | = 7 k2 8 > x = T2 0=0085 0=0084 0=0169 k3 5 6 5 6 0=0065 0 100 0=0065 8 > F = 7 0 1 0 8 = E = 70 0 0 001 Model uncertainty We consider the model uncertainty caused by the linearization and model it into

40

3 Modelling of technical systems Table 3.3 Parameters of DTS200 Parameters cross section area of tanks cross section area of pipes max. height of tanks max. ﬂow rate of pump 1 max. ﬂow rate of pump 2 coe. of ﬂow for pipe 1 coe. of ﬂow for pipe 2 coe. of ﬂow for pipe 3

Symbol D v12> v23 > v0 Kpd{ T1pd{ T2pd{ d1 d2 d3

Value 154 0.5 62 100 100 0.46 0.60 0.45

Unit fp2 fp2 fp2 fp3 @vhf fp3 @vhf

{˙ = (D + D) { + Ex> | = F{> D = (w)K (3.67) 6 5 6 1 (w) 0 0=0085 0 0=0085 0 0 0=00195 0=0084 8 (w) = 7 0 2 (w) 0 8 > K = 7 0=0085 0=0084 0=0169 0 0 3 (w) 5

((w)) 1=3620= Modelling of faults Three types of faults are considered in this benchmark system: • component faults: leaks in the three tanks, which can be modelled as additional mass ﬂows out of tanks, p p p D1 2jk1 > D2 2jk2 > D3 2jk3 where D1 > D2 and D3 are unknown and depend on the size of the leaks • component faults: pluggings between two tanks and in the letout pipe by tank 2, which cause changes in T13 > T32 and T20 and thus can be modelled by p p D4 d1 v13 sgn(k1 k3 ) 2j|k1 k3 |> D6 d3 v23 sgn(k3 k2 ) 2j|k3 k2 |> p D5 d2 v0 2jk2 where D4 > D5 > D6 5 [1> 0] and are unknown • sensor faults: three additive faults in the three sensors, denoted by i1 , i2 and i3 • actuator faults: faults in pumps, denoted by i4 and i5 . They are modelled as follows {˙ = (D + DI ) { + Ex + Hi i> | = F{ + Ii i 6 6 5 5 6 0=0214 0 0 0 0 0 X 0 0 0 8 > D2 = 7 0 0=0371 0 8 DI = Dl Dl > D1 7 l=1 0 00 0 0 0

(3.68)

3.7 Benchmark examples

5

6

5

41

6

00 0 0=0085 0 0=0085 8 > D4 = 7 8 0 0 0 0 D3 = 7 0 0 0 0 0=0262 0=0085 0 0=0085 5 6 6 6 5 5 i1 0 0 0 0 0 0 9 .. : 8 8 7 7 D5 = 0 0=0111 0 > D6 = 0 0=0084 0=0084 > i = 7 . 8 0 0 0 0 0=0084 0=0084 i5 £ ¤ £ ¤ 3×5 3×5 Hi = 0 E 5 R > Ii = L3×3 0 5 R = Closed loop model In DTS200, a nonlinear controller is implemented which leads to a full decoupling of the three tank system into • two linear subsystems of the ﬁrst order and • a nonlinear subsystem of the ﬁrst order. This controller can be schematically described as follows: x1 = T1 = T13 + D (d11 k1 + y1 (z1 k1 )) x2 = T2 = T20 T32 + D (d22 k2 + y2 (z2 k2 ))

(3.69) (3.70)

where d11 > d22 ? 0, y1 > y2 represent two preﬁlters and z1 > z2 are reference signals. The nominal closed loop model is 6 5 6 5 (d11 y1 ) {1 {˙ 1 : 7 {˙ 2 8 = 9 8 (3.71) 7 s (d22 y2 ) {2 s d1 v13 sgn({1 {3 ) 2j|{1 {3 |d3 v23 sgn({3 {2 ) 2j|{3 {2 | {˙ 3 D 5 6 ¸ y1 0 z1 + 7 0 y2 8 z2 0 0 while the linearized closed loop model with the faults is given by 6 6 5 5 y1 0 ¸ d11 y1 0 0 ¯i i 8 { + 7 0 y2 8 z1 + DI { + H 0 0 d22 y2 {˙ = 7 z2 0 0 0=0085 0=0084 0=0169 5 65 6 0=0085 0 0=0085 i1 + 7 0 0=0028 0=0084 8 7 i2 8 > | = F{ + Ii i (3.72) i3 0 0 0 5 6 d11 y1 0 0 0=0065 0 ¯i = 7 0 d22 y2 0 0 0=0065 8 = H 0 0 0 0 0

42

3 Modelling of technical systems

3.7.4 Vehicle lateral dynamic system In today’s vehicles, lateral dynamic models are widely integrated into control and monitoring systems. The so-called one-track model, also called bicycle model, is the simplest form amongst the existing lateral dynamic models, which is, due to its low demand for the on-line computation, mostly implemented in personal cars.

Fig. 3.7 Kinematics of one-track model

One-track model is derived on the assumption that the vehicle is simpliﬁed as a whole mass with the center of gravity on the ground, which can only move in { axis, | axis, and yaw around } axis. The kinematics of one-track model is schematically sketched in Fig. 3.7. It has been proven that one-track model can describe the vehicle dynamic behavior very well, when the lateral acceleration under 0=4j on normal dry asphalt roads. Further assumptions for one-track model are: • the height of center of gravity is zero, therefore the four wheels can be simpliﬁed as front axle and rear axle • small longitudinal acceleration, y˙{ 0, and no pitch and roll motion • the equations of motion are described according to the force balances and torque balances at the center of gravity • linear tire model, (3.73) I| = F where I| is the lateral force, F is the cornering stiness, is the side slip angle • small angles simpliﬁcation ( f K = + oK yuuhi Y

= + O oY

uf yuhi

3.7 Benchmark examples

43

The reader is referred to Table 3.4 for all variables and parameters used above and below. Nominal model Let vehicle side slip angle and yaw rate u be the state variables and steering angle O the input variable, the state space presentation of the onetrack model is given by ¸ ¸ {1 > x = O = (3.74) {˙ = D{ + Ex> { = u {2 5 6 0 " F0 # F 0 +FK oK FK oY FY Y Y 1 2 py py uhi uhi 8 > E = pyuhi D = 7 o F o F 0 = 0 2 0 2 o F oY FY +oK FK Y Y K K Y Y L} L} L} yuhi Typically, a lateral acceleration sensor (d| ) and a yaw rate sensor (u) are integrated in vehicles and available, for instance, in ESP (electric stabilization program). The sensor model is given by ¸ ¸ d | (3.75) | = F{ + Gx> | = 1 = | |2 u " # " # 0 0 0 FY +FK oK FK oY FY fY p pyuhi F= >G = p = 0 0 1 Below are the one-track model and the sensor model for yuhi = 50 p@v ¸ ¸ 3=0551 0=9750 1=12 D= >E = (3.76) 29=8597 3=4196 40=9397 ¸ ¸ 152=7568 1=2493 56 F= >G = = 0 1 0 By a sampling time of 0=1vhf., we have the following discrete time model {(n + 1) = Dg {(n) + Eg x(n)> |(n) = F{(n) + Gx(n) where for yuhi = 50 p@v ¸ ¸ 0=6333 0=0672 0=0653 > Eg = = Dg = 2=0570 0=6082 3=4462

(3.77)

Disturbances In model (3.74)-(3.75), the inﬂuences of road bank angle { > vehicle body roll angle !U and roll rate sf have not been taken into account. Moreover, sensor noises are inevitable. Generally, sensor noises can be modelled as steady stochastic process with zero mean Gaussian distribution. But, in vehicle systems, the variance or standard variance of sensor noises cannot be modelled as constant, since at dierent driving situations, the sensor noises are not only caused by the sensor own physical or electronic characteristic, but also

44

3 Modelling of technical systems Table 3.4 Parameter of the one-track model Physical constant j Vehicle parameters lO pU pQ U p oY oK L} yuhi u WO FY FK

Value 9=80665

Unit2 p@v

Explanation gravity constant

18=0 [3] steering transmission ratio 1630 [nj] rolling sprung mass 220 [nj] non-rolling sprung mass pU +pQ U [nj] total mass 1=52931 [p] distance from the CG to the front axle 1=53069 [p] distance from the CG to the rear axle 3870 [nj-p2 ] moment of inertia about the z-axis [np@k] vehicle longitude velocity [udg] vehicle side slip angle [udg@v] vehicle yaw rate [udg] vehicle steering angle 103600 [Q@udg] front tire cornering stiness 179000 rear tire cornering stiness

Table 3.5 Typical sensor noise of vehicle lateral dynamic control systems Sensor Yaw rate

lateral acceleration

Test condition

Unit Standard variation Nominal value [r @v] 0=2 Drive on the asphalt, even, dry road 0=2 surface Drive on the uneven road 0=3 Brake (ABS) on the uneven road 0=9 Nominal value [p@v2 ] 0=05 Drive on the asphalt, even, dry road surface Drive on the uneven road Brake (ABS) on the uneven road

0=2 1=0 2=4

strongly disturbed by the vibration of vehicle chassis. In Table 3.5, typical sensor data are listed. To include the inﬂuences of the above-mentioned disturbances, model (3.74)-(3.75) is extended to {˙ = D{ + Ex + Hg g> | = F{ + Gx + Ig g + 6 5 \! j pU k vlq{ + pyuhi !U + py s˙ yuhi ¸ uhi 100 : 9 Q! L {} g=7 8 > Hg = 0 1 0 L} !U L} s˙ \! p !U ¸ μ ¸¶ 001 0 > 5 N 0> d| Ig = = 000 0 u

(3.78)

3.7 Benchmark examples

45

Model uncertainties Below, major model parameter variations are summarized: • Vehicle reference velocity yuhi : the variation of longitudinal vehicle velocity is comparably slow, so it can be considered as a constant during one observation interval • Vehicle mass: when the load of vehicle varies, accordingly the vehicle spring mass and inertia will be changed. Especially the load variation are very large for the truck, but for the personal car, comparing to large total mass, the change caused by the number of passengers can be neglected normally • Vehicle cornering stiness F : Cornering stiness is the change in lateral force per unit slip angle change at a speciﬁed normal load in the linear range of tire. Remember that the derivation of one track model is based on (3.73). Actually, the tire cornering stiness F depends on road—tire friction coe!cient, wheel load, camber, toe-in, wheel pressure etc. In some studies, it is assumed, based on the stiness of steering mechanism (steering column, gear, etc.), that 0

FK = nFY =

(3.79)

In our benchmark study, we only consider the parameter changes caused by F and assume that 0 • FY = 103600 + FY > FY 5 [10000> 0] is a random number and 0 > n = 1=7278= • FK = nFY

As a result, we have the following system model: {˙ = (D + D) { + (E + E) x + Hg g | = (F + F) { + (G + G) x + Ig g + " 1+n noK oY # 1 pyuhi py2 £ ¤ pyuhi uhi D E = FY no o o2 +no2 K Y LY} yuhiK oLY} L} 1+n noK oY 1 ¸ £ ¤ p pyuhi p F G = FY = 0 0 0

(3.80)

Modelling of faults Three additive faults are considered in the benchmark: • fault in lateral acceleration sensor, which can also be a constant or a ramp and denoted by i1 • fault in yaw rate sensor, which can be a constant or a ramp and denoted by i2 • fault in steering angle measurement, which would be a constant and denoted by i3 . It is worth to remark that in practice a fault in the steering angle measurement is also called sensor fault.

46

3 Modelling of technical systems Table 3.6 Typical sensor faults Sensor

faults Oset Ramp Yaw rate ± 2 @v, ± 5 @v, ± 10 @v ± 10 @v@ min Lateral acceleration ± 2 p@v2 , ± 5 p@v2 ± 4 p@v2 @v, ± 10 p@v2 @v Steering angle ± 15 > ± 30 —

In Table 3.6, technical data of the above-mentioned faults are given. Based on (3.78), the one-track model with the above-mentioned sensor faults can be described by {˙ = D{ + Ex + Hg g + Hi i> | = F{ + Gx + Ig g + + Ii i 5 6 i1 £ ¤ £ ¤ Hi = 0 E 5 R2×3 > Ii = L2×2 G > i = 7 i2 8 = i3

(3.81)

3.7.5 Electrohydraulic Servo-actuator In this subsection, we brieﬂy introduce a linear model of electrohydraulic servo-actuator (EHSA) which is used in aileron control. Nominal model The state space representation of the EHSA considered in our benchmark study is given by 5 6 |˙ vy 9 {˙ s : ¸ 9 : s 9 : (3.82) {˙ = D{ + Ex> | = F{> { = 9 |vy : > x = lvy > | = {s 7 s 8 {s 6 5 5 6 2 0 $ vy 0 0 2gvy $ vy nvy $ 2vy Ds fd : iy golq 9 9 0 : ¸ 0 0 ps ps : 9 ps 9 : 00010 : 9 9 : D=9 1 0 0 0 0 :>E = 9 0 :>F = 00001 : 9 2F| 7 0 8 S 7 0 2D 0 0 8 FK FK 0 0 1 0 0 0 where the physical meanings as well as their values of the process variables and parameters are given in Table 3.7. Substituting these values into matrices D> E gives 6 5 0 0 884=67 0 3=06 × 105 9 0 12=19 × 104 26=28 : 0 3=6244 × 104 : 9 9 1 0 0 0 0 : D=9 : 7 0 3=29 × 1010 5=02 × 1012 0 0 8 0 1 0 0 0

3.7 Benchmark examples

47

Table 3.7 Technical data of the EHSA used in the benchmark study Symbol Ds Evy gvy gw H iy IO lpd{ lvy ns nvy Ph ps sD > sE sV sW sY uk YG $vy {g {pd{ {plq {s {˙ spd{ |pd{ |vy

Description Piston area Servovalve oriﬁce coe!cient Command input Damping factor Damping actuator Bulk modulus elasticity Viscose friction External air loads Max. input current Input current Controller gain Servovalve gain Aerodynamic hinge moment Piston mass Chamber A,B pressure Supply pressure Tank pressure System pressure Reduced moment arm Dead volume Cut-o frequency Desired piston position Max. extension movement Max. retraction movement Piston position Max. velocity Max. spool movement Servovalve position

Value 8=5348 × 1033 406=84 × 1036 [325> 25] [0=7> 0=8] 2=2 × 106 6895 × 105 11711 f() 0=008 [3lpd{ > lpd{ ] 2=88 0=111125 3680 7 [0> sV ] 205 × 105 5 × 105 200 × 105 0=09 3=2458 × 1035 [60> 88] [{plq > {pd{ ] 38=1 × 1033 337=96 × 1033 [{plq > {pd{ ] 0=11 0=889 × 1033 [3|pd{ > |pd{ ]

Unit p2 3 p I v Q

Q ·v2 p2 Q p2 Q ·v p

Q D D D p p D

Q·p nj Sd Sd Sd Sd p p3 K} p p p p p v

p p

6 3=3973 × 104 : 9 0 : 9 := 9 0 E=9 : 8 7 0 0 5

LCF of the nominal model Below, we brieﬂy describe the application of the LCF methods introduced in Subsection 3.2 to model (3.82). The eigenvalues of matrix D are v1 = 35101> v2 = 1143=1> v3 = 442=3 + 331=7m v4 = 442=3 331=7m> v5 = 0=0= ˆ (v)> Q ˆ (v)> an observer gain matrix O has been selected: To construct P

48

3 Modelling of technical systems

5

9=2418 × 105 9 1=6676 × 103 9 7 O=9 9 5=6 × 10 4 7 3=2451 × 10 1=8795 × 107

6 3=0326 × 103 7=1992 × 104 : : 19=116 : : 1=02 × 1012 8 1=262 × 103

which results in the following eigenvalues of matrix D OF : v1 = 38611> v2 = 1257=4> v3 = 486=5 + 364=9m v4 = 486=5 364=9m> v5 = 30000= Disturbance and model uncertainty Typical disturbance that aects the EHSA is the external air loads IO = The uncertainty due to the linearization can be described by 5 6 0 0 0 00 9 0 1 0 0 0 : 3¸ 9 : 4 3 p 9 : = D = 9 0 0 0 0 0 : > |1 | 1=0097 × 10 [Q ] > |2 | 1=144 × 10 v 7 0 0 2 0 0 8 0 0 0 00 This kind of uncertainty is of the polytopic type because they build a convex set depending on dierent operating points. In our benchmark study, it is assumed that D = Fr {D1 > · · · > D9 } corresponding to 9 operating points. The values of 1 > 2 in each operating point are given in Table 3.8. Table 3.8 Polytopic uncertainties i 1 2 3 4 5 6 7 8 9

1 493=25 7580=3 10097 6419=2 0 36419=2 310097 37580=3 3493=25

2 1=144 × 1033 0=858 × 1033 0=572 × 1033 0=286 × 1033 0 30=286 × 1033 30=572 × 1033 30=858 × 1033 31=144 × 1033

Integrating the inﬂuences of IO > D into (3.82) yields {˙ = (D + D) { + Ex + Hg g> | = F{ h iW £ ¤W = 0 0=1429 0 0 0 > g = IO = Hg = 0 p1s 0 0 0

(3.83)

3.8 Notes and references

49

Modelling of faults Dierent kinds of faults will be considered in the benchmark study. These include • Failures inside the EHSA: The faults that may occur in the EHSA are divided into two groups, — faults that degrade the dynamics of the EHSA. This kind of faults are multiplicative faults and will aect parameters in system matrices D and E. They are represented by 5 5 6 6 vy $ vy 0 $ vy 0 0 $ vy 9 0 9 0 : 0 0 0 0: 9 9 : : : > E I = 9 0 : E1 0 0 0 0 0 DI = D1 D1 > D1 = 9 9 9 : : F| 1 7 0 7 0 8 8 FK FK 0 0 0 0 0 0 00 E1 = D1 = — faults that cause undesired movement of the control surface. Denoted by i1 > this kind of faults can be represented as an additive fault aecting the system input lvy . • additive sensor faults, denoted by i2 and i3 = As a result, we have the following model to described the system dynamics when some of the above-mentioned faults occur: {˙ = (D + DI ) { + (E + E I ) x + Hi i + Hg g> | = F{ + Ii i (3.84) 5 6 i1 ¤ £ ¤ £ i = 7 i2 8 > Hi = E 0 0 > Ii = 0 L2×2 > i1 5 [1> 0] = i3

3.8 Notes and references In this chapter, we have introduced dierent model forms for the presentation of linear dynamic systems, which are fundamental for the subsequent study. We suppose that the nominal systems considered in this book are LTI. Modelling LTI systems by means of a state space representation or transfer matrices is standard in the modern control theory. The reader is referred to [19, 87] for more details. Modelling disturbances and system uncertainties is essential in the framework of robust control theory. In [49, 160, 161] the reader can ﬁnd excellent background discussion, basic modelling schemes as well as the needed mathematical knowledge and available tools for this purpose.

50

3 Modelling of technical systems

In the framework of model-based fault diagnosis, it is the state of the art that modelling of faults is realized in an analogous way to the modelling of disturbances and uncertainties. Coprime factorization technique is a standard tool in the framework of linear system and robust control theory for the system representation. In [49, 160, 161], the interested reader can ﬁnd well-structured and detailed description about this topic. To illustrate the application of the introduced system modelling technique, ﬁve laboratory and technical systems have been brieﬂy studied. The ﬁrst three systems, DC motor DR200, inverted pendulum LIP100 and three tank system DTS200, are laboratory set-ups produced by AMIRA. To author’s knowledge, they can be found in many laboratories for automatic control across the world. It is worth mentioning that three tank system DTS200 and inverted pendulum LIP100 are two benchmark processes that are frequently used in FDI study. There have been a number of invited sessions dedicated to the benchmark study on these two systems at some major international conferences. For the technical details of these three systems, the interested reader is referred to the practical instructions, [3] for DTS200, [4] for LIP100 and [5] for DR200. [104] is an excellent textbook for the study on vehicle lateral dynamics. The one-track model presented in this chapter is an extension of the standard one given in [104], which has been used for a benchmark study in the European project IFATIS [99]. A detailed description of EHSA is given in [128]. Another motivation for introducing these ﬁve systems is that they will serve as benchmarks for illustrating the applications of our study in the forthcoming chapters.

4 Structural fault detectability, isolability and identiﬁability

Corresponding to the major tasks in the FDI framework, the concepts of fault detectability, isolability and identiﬁability are introduced to describe the structural properties of a system from the FDI point of view. Generally speaking, we distinguish the structural fault detectability, isolability and identiﬁability from the performance based fault detectability, isolability and identiﬁability. For instance, the structural fault detectability is expressed in terms of the signature of the faults on the system without any reference to the FDI system used, while the performance based one refers to the conditions under which a fault can be detected using some kind of FDI systems. Study on structural fault detectability, isolability and identiﬁability plays a central role in the structural analysis for the construction of a technical process and for the design of an FDI system. In this chapter, we shall introduce the concepts of structural fault detectability, isolability and identiﬁability, study their checking criteria and illustrate the major results using the benchmark examples.

4.1 Structural fault detectability In the literature, one can ﬁnd a number of deﬁnitions of fault detectability, introduced under dierent aspects. Moreover, there are some signiﬁcant dierences regarding to additive and multiplicative faults. One of these dierences is that a multiplicative fault may cause changes in the system structure. In order to give a uniﬁed deﬁnition which is valid both for additive and multiplicative faults, we ﬁrst specify our intension of introducing the concept of structural fault detectability. First, structural fault detectability should be understood as a structural property of the system under consideration, which describes how a fault affects the system behavior. It should be expressed independent of the system input variables, disturbances as well as model uncertainties. Secondly, fault

52

4 Structural fault detectability, isolability and identiﬁability

detectability should indicate if a fault would cause changes in the system output. Finally, the structural fault detectability should be independent of the type and the size of the fault under consideration. Bearing these in mind, we adopt an intuitive deﬁnition of fault detectability which says: a fault is detectable if its occurrence, independent of its size and type, would cause a change in the nominal behavior of the system output. To deﬁne it more precisely, we assume that • the following system model is under consideration {˙ = (D + DI ){ + (E + EI )x + Hi i | = (F + FI ){ + (G + GI )x + Ii i

(4.1) (4.2)

where, as introduced in Chapter 3, J|x (s) = G + F(sL D)1 E represents the nominal system dynamics, i 5 Rni the additive fault vector and DI > EI > FI > GI the multiplicative faults given by DI = FI =

oD X

Dl Dl > EI =

oE X

l=1

l=1

oF X

oG X

Fl Fl > GI =

l=1

El El

(4.3)

Gl Gl

(4.4)

l=1

• a fault is understood as a scalar variable, either l 5 {Dl > El > Fl > Gl } or il , and uniﬁedly denoted by l = Deﬁnition 4.1 Given system (4.1)-(4.2). A fault l is said structurally detectable if for some x C| | =0 g l 6 0= (4.5) C l l Equation (4.5) is the mathematical description of a change in the system output caused by the occurrence of a fault (from zero to a time function different from zero), independent of its size and type. A fault becomes detectable if this change is not constantly zero. In other words, it should dier from zero at least at some time instant and for some system input. The following theorem provides us with a necessary and su!cient condition for the detectability of additive and multiplicative faults. Theorem 4.1 Given system (4.1)-(4.2), then • an additive fault il is detectable if and only if F (sL D)1 Hil + Iil 6= 0

(4.6)

with Hil > Iil denoting the i-th column of matrices Hi > Ii respectively,

4.1 Structural fault detectability

53

• a multiplicative fault Dl is detectable if and only if F(sL D)1 Dl (sL D)1 E 6= 0

(4.7)

• a multiplicative fault El is detectable if and only if F (vL D)

1

El 6= 0

(4.8)

• a multiplicative fault Fl is detectable if and only if Fl (vL D)1 E 6= 0

(4.9)

• a multiplicative fault Gl is detectable if and only if Gl 6= 0=

(4.10)

Proof. While the proof of (4.6), (4.8)-(4.10) is straightforward and is thus omitted here, we just check (4.7). It turns out C| C{ C {˙ C{ =F > =D + Dl { CDl CDl CDl CDl It yields μ L

C| | =0 CDl Dl

¶ = F (sL D)

1

Dl (sL D)1 Ex(v)

with L denoting the Laplace transform (}-transform in the discrete time case) Hence, for some x> w> CC|D |Dl =0 6= 0 if and only if (4.7) holds. u t l

It can be easily seen from Theorem 4.1 that • an additive fault is structurally detectable as far as the transfer function from the fault to the system output is not zero • a multiplicative fault Gl is always structurally detectable • the structural detectability of multiplicative faults El and Fl can be interpreted as input observability and output controllability respectively • a multiplicative fault Dl will cause essential changes in the system structure. Also, it follows from Theorem 4.1 that we can estimate the changes in the system output caused by the dierent types of the faults. To this end, suppose that the faults occur at time instant w0 and their size is small at beginning, then • in case of additive fault il : |(w) = F{ + Iil il g{ = D{ + Hil il > { (w0 ) = 0 gw

(4.11)

54

4 Structural fault detectability, isolability and identiﬁability

• in case of multiplicative fault Dl C|(w) C{ |(w) |Dl =0 Dl = F D CDl CDl l μ ¶ C{ C{ C{ g =D + Dl { |Dl =0 > (w0 ) = 0 gw CDl CDl CDl where { |Dl =0 satisﬁes

(4.12)

{˙ = D{ + Ex

• in case of multiplicative fault El C|(w) C{ |(w) El = F E CEl CEl l μ ¶ C{ C{ C{ g =D + El x> (w0 ) = 0 gw CEl CEl CEl

(4.13)

• in case of a multiplicative fault Fl |(w)

C|(w) F = Fl {(w)Fl > {˙ = D{ + Ex CFl l

(4.14)

• in case of multiplicative fault Gl |(w)

C|(w) G = Gl x(w)Gl = CGl l

(4.15)

Comparing (4.11) with (4.12)-(4.15) makes it evident that • detecting additive faults can be realized independent of the system input, and • multiplicative faults can only be detected if x(w) 6= 0= In another word, exciting signal is needed for a successful fault detection. We see that transfer matrices F(sL D)1 Hil + Ii l > F(sL D)1 Dl (sL D)1 E> F(sL D)1 El > Fl (sL D)1 E> Gl give a structural description of the inﬂuences of the faults on the system output. For this reason and also for our subsequent study on fault isolability and idenﬁability, we introduce the following deﬁnition. Deﬁnition 4.2 Given system (4.1)-(4.2). Transfer matrices F(sL D)1 Hil + Ii l > F(sL D)1 Dl (sL D)1 E> F(sL D)1 El > Fl (sL D)1 E> Gl are called fault transfer matrices and denoted by Jil (s)> JDl (s)> JEl (s)> JFl (s) and JGl (s) respectively, or in general by Jl (s)=

4.1 Structural fault detectability

55

Example 4.1 To illustrate the results in this section, we consider three tank system DTS200 given in Subsection 3.7.3. The fault transfer matrices of the ﬁve additive faults are respectively 5 6 5 6 1 0 1 1 8 7 7 0 18 > F(vL D) H + I = H + I = F(vL D) i1 i1 i2 i2 0 0 5 6 5 0=0065v+0=0002 6 0 (v2 +0=0449v+0=0005) 8 0 F(vL D)1 Hi3 + Ii3 = 7 0 8 > F(vL D)1 Hi4 + Ii4 = 7 0=0001 1 (v2 +0=0449v+0=0005) 5 6 0 0=0065v+0=0002 1 7 F(vL D) Hi5 + Ii5 = (v2 +0=0449v+0=0005) 8 = 0=0001 (v2 +0=0449v+0=0005)

It is evident that these ﬁve faults are detectable. As to the multiplicative faults, we have the following fault transfer matrices F(vL D)1 D1 (vL D)1 E

5 0=0214(0=0065v2 + 0=0002v) 7 = (v3 + 0=0449v2 + 0=0005v)2

6 v2 + 0=0364v 0 0=0001 08 0=0085v + 0=0002 0

F(vL D)1 D2 (vL D)1 E 5 6 0 0=0001 0=0371(0=0065v2 + 0=0002v) 7 2 0 v + 0=0254v + 0=0001 8 = 3 (v + 0=0449v2 + 0=0005v)2 0 0=0084v + 0=0001 F(vL D)1 D3 (vL D)1 E 5 0=00000262v 7 = 3 (v + 0=0449v2 + 0=0005v)2

6 0=0085v + 0=0002 0=0085v + 0=0002 0=0084v + 0=0001 0=0084v + 0=0001 8 2 2 v + 0=0280v + 0=0002 v + 0=0280v + 0=0002

F(vL D)1 D4 (vL D)1 E 6 5 0=001v2 0=055v3 + 0=002v2 0=0000085 7 0=1v3 + 0=002v2 6=5v4 0=21v3 0=002v2 8 = 3 (v + 0=0449v2 + 0=0005v)2 0=1v3 0=002v2 6=5v4 + 0=227v3 + 0=003v F(vL D)1 D5 (vL D)1 E 5 6 0 0=0001 0=0111(0=0065v2 + 0=0002v) 7 2 0 v + 0=0254v + 0=0001 8 = 3 (v + 0=0449v2 + 0=0005v)2 0 0=0084v + 0=0001 F(vL D)1 D6 (vL D)1 E 6 5 0=001v2 0=055v3 + 0=002v2 0=0000084 7 0=1v3 + 0=002v2 6=5v4 0=21v3 0=002v2 8 = = 3 (v + 0=0449v2 + 0=0005v)2 0=1v3 0=002v2 6=5v4 + 0=227v3 + 0=003v As a result, all these multiplicative faults are detectable.

56

4 Structural fault detectability, isolability and identiﬁability

4.2 Excitations and su!ciently excited systems In this section, we brieﬂy address the issues with excitation signals, which are, as shown above, needed for detecting multiplicative faults. Let Jl (s) be the fault transfer matrix of a multiplicative fault and satisfy ¢ ¡ udqn Jl (s) = (A 0) then we can ﬁnd a -dimensional subspace Uh{f>l so that for all x 5 Uh{f>l Jl (s)x(s) 6= 0= From the viewpoint of fault detection, subspace Uh{f>l contains all possible input signals that can be used to excite a detection procedure. Deﬁnition 4.3 Let Jl (s) be the fault transfer matrix of multiplicative fault l= © ª (4.16) Uh{f>l = x | Jl (s)x(s) 6= 0 is called excitation subspace with respect to l = Mathematically, we can express the fact that detecting an additive fault, say l > is independent of exciting signals by deﬁning ª © Uh{f>l = x 5 Rnx = In this way, we generally say that Deﬁnition 4.4 System (4.1)-(4.2) is su!ciently excited regarding to a fault l if (4.17) x 5 Uh{f>l = With this deﬁnition, we can reformulate the deﬁnition of the fault detectability more precisely. Deﬁnition 4.5 Given system (4.1)-(4.2). A fault l is said structurally detectable if for x 5 Uh{f>l C| | =0 g l 6 0= (4.18) C l l Remark 4.1 In this book, the rank of a transfer matrix is understood as the so-called normal rank if no additional speciﬁcation is given.

4.3 Structural fault isolability

57

4.3 Structural fault isolability 4.3.1 Concept of structural fault isolability For the sake of simplicity, we ﬁrst study a simpliﬁed form of fault isolability problem, namely distinguishing the inﬂuences of two faults. An extension to the isolation of multiple faults will then be done in a straightforward manner. Consider system model (4.1)-(4.2) and suppose that the faults under consideration are detectable. We say any two faults, l > m > l 6= m> are isolable if the changes in the system output caused by these two faults are distinguishable. This fact can also be equivalently expressed as: any simultaneous occurrence of these two faults would lead to a change in the system output. Mathematically, we give the following deﬁnition. Deﬁnition 4.6 Given system (4.1)-(4.2). Any two detectable faults, = £ ¤W l m > l 6= m> are isolable, when for x 5 Uh{f>l _ Uh{f>m C| |=0 g 6 0 C

(4.19)

It is worth mentioning that detecting a fault in a disturbed system requires distinguishing the fault from the disturbances. This standard fault detection problem can also be similarly formulated as an isolation problem for two faults. In a general case, we say that a group of faults are isolable if any simultaneous occurrence of these faults would lead to a change in the system output. Deﬁne a fault vector ¤W £ (4.20) = 1 · · · o which includes o structurally detectable faults to be isolated. Deﬁnition 4.7 Given system (4.1)-(4.2). The faults in fault vector are o T isolable, when for all x 5 Uh{f>l l=1

C| |=0 g 6 0= C

(4.21)

We would like to call reader’s attention on the similarity between the isolability of additive faults and the so-called input observability which is widely used for the purpose of input reconstruction. Consider system {˙ = D{ + Hi i> | = F{ + Ii i> { (0) = 0= It is called input observable, when |(w) 0 implies i (w) 0. Except the assumption on initial condition { (0) > the physical meanings of the isolability of additive faults and input observability are equivalent.

58

4 Structural fault detectability, isolability and identiﬁability

4.3.2 Fault isolability conditions With the aid of the concept of fault transfer matrices, we now derive existence conditions for the structural fault isolability. Theorem 4.2 Given system (4.1)-(4.2), then any two faults with fault transfer matrices Jl (s)> Jm (s)> l 6= m> are structurally isolable if and only if ³ ´ ¢ ¤ ¡ £ udqn Jl (s) Jm (s) = udqn Jl (s) + udqn Jm (s) =

(4.22)

Proof. It follows from (4.11)-(4.15) that the changes in the output caused by l > m can be respectively written as ´ ³ ¡ ¢ L1 Jl (s)}l (s) > L1 Jm (s)}m (s) where }l (s) = L(gil ) for l = il or }l (s) = L (g l x(w)) for l 5 {Dl > El > Fl > Gl } with x 5 Uh{f>l _ Uh{f>m . Since C| C| C| |=0 g = |=0 g l + |=0 g m C C l C m it holds that if is not isolable, then ¶ μ ¶ μ C| C| C| |=0 g = 0 +, L |=0 g l + L |=0 g m = 0 ;w> C C l C m ¸ ¤ }l (s) £ +, Jl (s) Jm (s) = 0 +, }m (s) ³ ´ ¤ ¡ £ ¢ udqn Jl (s) Jm (s) ? udqn Jl (s) + udqn Jm (s) The theorem is thus proven. u t An extension of the above theorem to a more general case with a fault ¤W £ vector = 1 · · · o is straightforward and hence its proof is omitted. Corollary 4.1 Given system (4.1)-(4.2), then with fault transfer matrix £ ¤ J (s) = J1 (s) · · · Jo (s) is structurally isolable if and only if udqn (J (s)) =

o X l=1

¡ ¢ udqn Jl (s)

(4.23)

4.3 Structural fault isolability

59

In order to get a deeper insight into the results given in Theorem 4.2 and Corollary 4.1, we study some special cases often met in practice. ¤W £ are additive faults. Suppose that the faults in fault vector = 1 · · · o Then the following result is evident. Corollary 4.2 Given system (4.1)-(4.2) and assume that l > l = 1> · · · > o ni are additive faults. Then, these o faults are isolable if and only if udqn (J (s)) = o=

(4.24)

This result reveals that, to isolate o dierent faults, we need at least an odimensional subspace in the measurement space spanned by the fault transfer matrix. Considering that udqn (J (s)) min {p> o} with p as the number of the sensors, we have the following claim which is very easy to check and thus useful for the practical application. Claim. The additive faults are isolable only if the number of the faults is not larger than the number of the sensors. Denote the minimal state space realization of J (s) by J (s) = F (sL D)

1

H + I =

Check condition (4.24) can be equivalently expressed in terms of the matrices of the state space description. Corollary 4.3 Given system (4.1)-(4.2) and assume that l > l = 1> · · · > o ni , are additive faults. Then these o faults are isolable if and only if ¸ D sL H udqn = q + o= (4.25) F I Proof. The proof becomes evident by noting that ¸ ¸ D sL H (D sL)1 (sL D)1 H = F I 0 L ¸ ¸ L 0 D sL H =, udqn F I F (D sL)1 F (sL D)1 H + I μ ¸¶ ¸ 1 1 D sL H (D sL) (sL D) H = udqn F I 0 L ¸ L 0 = udqn 1 1 F (D sL) F (D sL) H + I ³ ´ = q + udqn F (D sL)1 H + I = t u

60

4 Structural fault detectability, isolability and identiﬁability

Recall that for additive faults the fault isolability introduced in Deﬁnition 4.7 is identical with the concept of input observability known and intensively studied in the literature, we would like to extend our study • to ﬁnd out alternative conditions for checking conditions (4.24) or (4.25) • to compare them with the results known in the literature and • to gain a deeper insight into the isolability of additive faults, which will be helpful for some subsequent studies in the latter chapters. To simplify our study, we ﬁrst consider J (s) = F (sL D)1 H = It follows from Cayley-Hamilton Theorem that Ã q Ã q ! ! X X 1 1 1 ql l1 F F Vl s l (s)D H = H F (sL D) H = !(s) !(s) l=1 l=1 (4.26) + d2 s + · · · + dq1 s + dq !(s) = det (sL D) = s + d1 s Vl = Vl1 D + dl1 L> V1 = L> l = 2> · · · > q q

q1

q2

1 (s) = sq1 + d1 sq2 + · · · + dq1 > · · · > q1 (s) = s + d1 > q (s) = 1 which can be rewritten into 5 F (sL D)

1

H =

¤9 1 £ 9 1 (s)L 2 (s)L · · · q (s)L 9 !(s) 7

FH FDH .. .

6 : : : = (4.27) 8

FDq1 H It is obvious that if

5 9 9 udqn 9 7

FH FDH .. .

6 : : :?o 8

FDq1 H then there exists a x which yields 6 5 FH 9 FDH : : 9 1 : x = 0 =, F (sL D) H x = 0= 9 .. 8 7 . FDq1 H Thus,

5 9 9 udqn 9 7

FH FDH .. . FDq1 H

6 : : :=o 8

(4.28)

4.3 Structural fault isolability

61

builds a necessary condition for the fault isolability. We would like to call reader’s attention that (4.28) is not a su!cient condition for the fault isolability. To see it, we consider a special case with p = 1> p ? o ? q and (F> D) being observable, i.e. 6 5 F 9 FD : : 9 udqn 9 : = q= .. 8 7 . q1 FD It immediately becomes clear that (4.28) is satisﬁed. But, the system is, due to p ? o> not isolable, as can be seen from Corollary 4.2. Remark 4.2 We would like to point out that (4.28) is claimed as a necessary and su!cient condition for the input observability in some publications, which is, as shown above, not correct. Below, we shall derive some su!cient conditions on the assumption that p o and (4.28) holds. Note that the orders (highest power) of l (s)> l = 1> · · · > q> given in (4.26) are dierent . If for some m 5 {1> · · · > q} ¢ ¡ (4.29) udqn FDm1 H = o then (4.27) can be rewritten into 3 4 q X 1 C 1 m (s)L + F (sL D) H = l (s)Tl D FDm1 H !(s) l=1>l6=m

where Tl 5 Rp×p > l = 1> · · · > q> l 6= m> are some matrices. Considering that 3 4 q X ¢ ¡ udqn Cm (s)L + l (s)Tl D = p o> udqn FDm1 H = o l=1>l6=m

we ﬁnally have

³ ´ 1 udqn F (sL D) H = o=

This proves the following theorem. p ¢o and Theorem 4.3 Given F (sL D)1 H as deﬁned in (4.26) with ¡ satisfying (4.28). Assume that for some m 5 {1> · · · > q} > udqn FDm1 H = o= Then ³ ´ udqn F (sL D)1 H = o= In the framework of linear system theory, FDl H > l = 0> 1> · · · > are called Markov matrices. Theorem 4.3 provides us with a su!cient condition for checking the isolability of additive faults by means of Markov matrices.

62

4 Structural fault detectability, isolability and identiﬁability

It is interesting to note that according to (4.26) F (sL D) be rewritten into 1

5

F (sL D)

H =

0 FH 9 £ ¤ 9 FDH FH dq1 L · · · d1 L L 9 9 .. .. 7 . . FDq1 H FDq2 H

65

1

H can also (4.30) 6

··· 0 L .. : 9 Ls : .. . . : : :9 .. : = :9 .. 7 . 8 . 0 8 q1 Ls · · · FH

This form is important in studying various algebraic properties of the so-called parity space methods. In a similar manner like the proof of Theorem 4.3, we are able to prove the following theorem that gives an alternative su!cient condition for the isolability. Theorem 4.4 Given F (sL D)1 H . Let l = FVl H > l = 1> · · · > q> and assume that for some m 5 {1> · · · > q} udqn (m ) = o ³

(4.31)

´

then udqn F (sL D)1 H = o. The above discussion and the results given in Theorems 4.3 and 4.4 can be easily extended to the general form of system model F (sL D)1 H + I = To this end, we extend the state space description as follows ¸ ¸ ¸ ¸ {˙ { 0 ˙ D H ¯{ + H ¯ ˙ + (4.32) := D¯ = 0 0 L ˙ ¸ ¸ ¤ { £ { := F¯ { ¯> { ¯= = (4.33) | = F I It is easy to prove that given F (sL D)1 H + I condition (4.28) can then be equivalently written as 6 5 ¯ F¯ H 6 5 ¯ : 9 F¯ D¯H I : 9 : 9 .. 9 FH : : 9 . : 9 : udqn 9 = udqn (4.34) :=o 9 .. ¯ : 9 F¯ D¯q H 8 7 . : 9 : 9 .. FDq1 H 8 7 . q+o1 ¯ F¯ D¯ H while conditions (4.29) and (4.31) respectively as ½ ¢ ¡ udqn (I ¯ = ¢ ) = o> if m = 1 ¡ udqn F¯ D¯m1 H udqn FDm2 H = o> if m 5 {2> · · · > q + 1}

(4.35)

4.3 Structural fault isolability

5 ¡ ¢ £ ¤9 9 udqn ¯m = o> m 5 {0> · · · > q} > ¯0 = dq L · · · d1 L L 9 7 5 9 9 9 9 £ ¤9 ¯ m = dq L dq1 L · · · d1 L L 9 9 9 9 9 7

0 .. . 0 I FH .. .

63

6

0 .. : . : : = I 0 8 I

(4.36)

6 : : : : : : > m 5 {1> · · · > q} = : : : : 8

(4.37)

FDl1 H We now review the conditions for the structural fault isolability of multiplicative faults. Although Corollary 4.1 holds for both additive and multiplicative faults, the forms of the faults matrices of multiplicative faults reveal that isolating multiplicative faults may demand more sensors. To illustrate it, we ﬁrst take a multiplicative process fault as an example. Remember that in this case the fault transfer matrix is F(sL D)1 Dl (sL D)1 E, which can be written as 5 6 D Dl 0 F(sL D)1 Dl (sL D)1 E = 7 0 D E 8 = F0 0 In the worst case, this multiplicative process fault can span a subspace with dimension equaling to ¢ ¡ udqn F(sL D)1 Dl (sL D)1 E = min {p> nx } := = To isolate such a (single) fault, we need at least sensors. As to multiplicative sensor and actuator faults, it seems that their fault transfer matrices, Fl (sL D)1 E> F(sL D)1 El > would span a lower dimensional subspace, for instance in case that udqn (Fl ) = 1> udqn (El ) = 1= On the other side, if those faulty sensors and actuators are embedded in a feedback control loop, for instance with x = N|> then they will cause change in the eigendynamics of the closed loop system. In another word, they will aect the system performance like a multiplicative process fault. Again, to isolate these faults, additional number of sensors are demanded. In practice, in particular in systems with integrated feedback control loops, it is often the case that the system input keeps constant or changes slowly over a relatively long time interval. On the assumption of a constant vector x, we introduce the concept of weak isolability of multiplicative faults.

64

4 Structural fault detectability, isolability and identiﬁability

Deﬁnition 4.8 Given system (4.1)-(4.2) and let 5 6 1 9 .. : =7 . 8 o with multiplicative faults l > l = 1> · · · > o= is called weakly isolable, if for all o T constant vector x 5 Uh{f>l l=1

C| |=0 g 6 0= C The theorem given below follows directly from Corollary 4.1 and the deﬁnition of weak isolability of multiplicative faults. Theorem 4.5 Given system (4.1)-(4.2) and let 5 6 1 9 .. : =7 . 8 o be a multiplicative fault vector with fault transfer matrix £ ¤ J (s) = J1 (s) · · · Jo (s) = Then, is weakly isolable if and only if for all constant vector x 5

o T l=1

Uh{f>l

£ ¤ udqn J1 (s)x · · · Jo (s)x = o= Comparing the results given in Corollary 4.1 and the above theorem makes it evident that the existence condition for a weak isolability of multiplicative faults can be remarkably released. Example 4.2 Consider again three tank system DTS200. It is evident that it is impossible to isolate all eleven faults, since we only have three sensors. However, if we are able to divide the faults into dierent groups and assume that faults from only one group can occur simultaneously, then a fault isolation becomes possible. For instance, if we divide the additive faults into two groups, a group with the sensor faults and a group with the actuator faults, then we have, using the fault transfer matrices given in the last section, ¸ D vL Hv =6 udqn F Iv and

4.4 Structural fault identiﬁability

udqn where

D vL Hd F Id

65

¸ =5

6 100 ¤ £ = 7 0 1 0 8 > Hd = Hi4 Hi5 > Id = 0= 001 5

Hv = 0> Iv

Thus, it follows from Corollary 4.2 that these additive faults are isolable on the above assumption. As for the multiplicative faults, it follows from Corollary 4.1 that a group of three faults is generally not isolable. In fact, if it is assumed that the six multiplicative faults are divided into three groups with (a) group 1: 1 > 2 (b) group 2: 3 > 4 (c) group 3: 5 > 6 > then using the fault transfer matrices given in the last section, we are able to prove that these faults are isolable.

4.4 Structural fault identiﬁability Roughly speaking, the concept of structural fault identiﬁability is understood as a characterization of system structure that is essential to reconstruct faults from the system output. From the mathematical viewpoint, fault identiﬁability characterizes the mapping from the system output to the faults under consideration. If this mapping is unique, then the faults are identiﬁable. Usually, we intend to express this mapping in terms of the model from the faults to the system output, then the structural fault identiﬁability is equivalent to the model invertibility. Motivated by this fact, we introduce the concept of structural fault identiﬁability in terms of, dierent from the structural fault detectability and isolability, fault transfer matrices. Deﬁnition 4.9 Given system (4.1)-(4.2) and let £ ¤ J (s) = J1 (s) · · · Jo (s) ¤W £ be the fault transfer matrix of fault vector = 1 · · · o = is called structurally identiﬁable if J (s) is invertible and its inverse is stable and causal. Note that the requirements on the stability and causality of the inverse of J (s) is an expression for the realizability of inversing J (s)= It is evident that without these two requirements, the structural fault identiﬁability would be equivalent to the structural fault isolability. In another word, the structural fault isolability is a necessary condition for the faults to be identiﬁable. To understand the idea behind the deﬁnition of structural fault identiﬁability, we now consider dierent types of faults respectively. Let i (s) be a vector of additive faults with fault transfer matrix Ji (s)= As shown in (4.11), the change of |(s) caused by i (s) can be written as

66

4 Structural fault detectability, isolability and identiﬁability

|(s) = Ji (s)i (s)= If Ji (s) is invertible and its inverse is stable and causal, then it is possible to reconstruct i (s) based on the relation i (s) = J1 i (s)|(s)=

(4.38)

Thus, fault vector i is structurally identiﬁable. For a multiplicative fault El > we have |(s) = JEl (s)L (x(w)El ) with JEl (s) = F (sL D)1 El = According to Deﬁnition 4.9, the structural idenﬁability of El means it is possible to reconstruct x(w)El based on L (x(w)El ) = J1 E (s)|(s) := E (s)= l

l

(4.39)

Since system input x(w) is generally on-line available, an identiﬁcation of the fault El can be achieved using the relation ¡ ¢1 W El = xW (w)x(w) x (w) E (w) for x(w) 6= 0= l

(4.40)

Analog to (4.39) and (4.40), we have the relations L (x(w)Fl ) = J1 Fl (s)|(s) := Fl (s) ¡ W ¢1 W x (w) F (w) for x(w) 6= 0 Fl = x (w)x(w)

(4.41)

L (x(w)Gl ) = Gl1 |(s) = G (s) l ¡ ¢1 W = xW (w)x(w) x (w) G (w) for x(w) 6= 0

(4.42)

l

Gl

l

for multiplicative faults Fl and Gl , respectively. Again, we can see that identifying a multiplicative fault requires not only the invertibility of the fault transfer matrix but also a su!cient excitation. As to a multiplicative fault Dl > remember that the change in | caused by Dl can only be approximated by |(s) JDl (s)L (x(w)Dl ) > JDl (s) = F(sL D)1 Dl (sL D)1 E in case of a small Dl = In general, we have μ ¶ C{ C{ C{ g = (D + Dl Dl ) + Dl {> (w0 ) = 0 gw CDl CDl CDl C{ D = {˙ = (D + Dl Dl ) { + Ex> |(w) = F CDl l It is evident that an identiﬁcation of Dl would become very di!cult.

(4.43)

4.5 Notes and references

67

Example 4.3 Consider three tank system DTS200 with the fault transfer matrices derived in Example 4.1. Since ;v ¸ D vL 0 =6 udqn F L3×3 the inverse of the transfer matrix of the sensor faults is stable and causal. According to Deﬁnition 4.9, these faults are identiﬁable. In against, the additive actuator faults and the multiplicative process faults are not structurally identiﬁable.

4.5 Notes and references Due to their important role in the FDI study, much attention has been devoted to the concepts of fault detectability and isolability. In the beginning phase, fault detectability and isolability have been often deﬁned in terms of the performance of the FDI systems used. Dierently, in most of the recent publications on this topic, fault detectability and isolability are expressed in terms of the structural properties of the system under consideration. In order to distinguish these two dierent ways of deﬁning fault detectability and isolability, we have adopted the notation structural fault detectability and isolability to underline the original idea behind the introduction of these two concepts. They are used to indicate the structural properties of the system under consideration from the FDI viewpoint. Deﬁnitions of (structural) fault detectability and isolability can be found in all recently published books, see for instance [13, 21, 64, 117]. The interested reader may wonder about many dierent deﬁnitions of (structural) fault detectability and isolability. One may also notice that most of these deﬁnitions are related to the additive faults. It is one of our motivations to deﬁne structural fault detectability and isolability both for additive and multiplicative faults uniﬁedly. A confusion by the deﬁnition of fault detectability and isolability is caused by way of deﬁning faults. In some publications, a fault is also understood as a vector. In this case, fault detectability requires a full (column) rank of the fault transfer matrix to ensure that the occurrence of any fault would cause changes in the system output. On the other side, this deﬁnition yields a conﬂict with the fault isolability deﬁned on the assumption that a fault is a scalar variable and a fault vector represents a number of faults. For this reason, it has been adopted in our study that a fault is understood as a scalar variable. In our view, this deﬁnition ﬁts real applications well. It also allows a uniﬁed handling of additive and multiplicative faults. In [76], the concept of input observability has been introduced, which has been, in its original study, motivated by the input identiﬁcation problem. Due to its close relation to the FDI problems, this concept has been lately

68

4 Structural fault detectability, isolability and identiﬁability

reformulated as fault detectability for additive faults, see for instance [96]. As pointed out above and shown in Subsection 4.3.2, the input observability is identical with the fault isolability deﬁned in our study. We would like to call attention of the interested reader that in Subsection 4.3.2 we have corrected some wrong results on the existence conditions for the input observability.

Part II

Residual generation

5 Basic residual generation methods

The objective of this chapter is to establish a framework and to lay foundations for the study on model-based residual generation. We shall address the concepts of analytical redundancy and residual generation on the assumption of a perfect system model, as sketched in Fig. 5.1, and introduce a general description form of model-based redundancy and residual generators. On this basis, tasks of designing and constructing model-based residual generators will be formulated.

Fig. 5.1 Schematic description of the object addressed in Chapter 5

Three types of residual generators including • fault detection ﬁlter (FDF) • diagnostic observer (DO) • parity relation based residual generator (PRRG) will be presented and studied. Main attention is paid to

72

5 Basic residual generation methods

• the implementation and design forms of these residual generators, • characterization of the solutions and • interconnections among the dierent types of residual generators.

5.1 Analytical redundancy The concept analytical redundancy stands generally for an analytical reconstruction of quantities or parts of the system under monitoring. For our purpose of residual generation, known as a comparison between system measurements and their redundancy, the analytical redundancy is understood as a reconstruction of the measured quantities of the system under consideration. Consider the following nominal model that describes the transfer behavior of the system or a part of the system under monitoring, |(s) = J|x (s)x(s)

(5.1)

where |(s) represents the measured variable, for which a redundancy will be established, and x(s) a process variable that may be the process input or even a measured variable. A natural and in practice often applied method to reconstruct |(s) is an on-line parallel simulation of input-output relationship (5.1) |ˆ(s) = J|x (s)x(s) where |ˆ(s) stands for an estimate of |(s) and is called analytical or software redundancy. Although this kind of redundancy promises a simple on-line implementation and seems easy to be understood, the scheme of generating redundancy is of a property that makes a direct application of this approach often impossible, at least theoretically and in many practical cases: The dynamics of the estimation error is identical with the one of the system, i.e. J|x (s). In order to show what this means and which consequence this property has, consider the inﬂuences of the system initial states and system model uncertainty on the residual signal. To this end, the system model (5.1) is extended to |(s) = J|x (s)x(s) + F(sL D)1 {(0) + |(s)

(5.2)

to include the process initial states {(0) and model uncertainty |(s), where the state space realization of J|x (s) is assumed to be (D> E> F> G). It turns out u(s) = |(s) |ˆ(s) = F(sL D)1 {(0) + |(s) (5.3) which means, in other words, • the variation of u(w) from zero caused by {(0) 6= 0 disappears only when the process x( s) is stable (i.e. D is stable), and even in this case the convergent rate exclusively depends on the position of the eigenvalues of D in the real part of the complex plane

5.1 Analytical redundancy

73

• the inﬂuence of the model uncertainty is not suppressed. As a result, the reconstructed variable may strongly dier from its original one (the measured one). From the viewpoint of control engineering, the reason for the abovementioned problems is evidently traced back to the so-called open-loop structure. A known solution is, therefore, to modify the structure of system (5.1) in such a way that a feedback loop is included. A reasonable and typical form of such a modiﬁcation is given by ¯ (|(s) |ˆ(s)) = |ˆ(s) = J|x (s)x(s) + O(s)

(5.4)

In comparison with the open-loop structured system (5.1), we see that the ¯ (|(s) |ˆ(s)) acts as a correction on |ˆ(s) that ensures a added term O(s) limited variation of |ˆ(s) from |(s). This system is closed-loop structured and is of, by a suitable choice of the feedback matrix O(s), the properties required for a redundancy system: L= u(s) = |(s) |ˆ(s) = 0 for all x(s) LL= lim (|(w) |ˆ(w)) for all {(0) w$4

LLL= The convergent rate is arbitrarily assignable LY= the inﬂuence of |(s) is suppressed.

(5.5) (5.6) (5.7) (5.8)

We now consider how to choose O(s). It follows from (5.2) and (5.4) that |(s) |ˆ(s) = J|x (s)x(s) + F(sL D)1 {(0) + |(s) ¯ J|x (s)x(s) O(s)(|(s) |ˆ(s))

(5.9)

and furthermore ¡ ¢ ¯ L + O(s) (|(s) |ˆ(s)) = F(sL D)1 {(0) + |(s)= Do a left coprime factorization of F(sL D)1 (see Section 3.2), ¡ ¢1 F(sL D)1 = L F(sL D + OF)1 O F(sL D + OF)1 ¯ with O ensuring D OF stable. Recall our task is to select O(s) so that (5.5)(5.8) are fulﬁlled. To this end, we have to, knowing from linear system theory, cancel the poles of transfer function matrix F(sL D)1 , which are obviously the zeros of matrix L F(sL D + OF)1 O. Setting ¡ ¢1 ¯ L + O(s) = L F(sL D + OF)1 O and noting the following equality ¡ ¢1 L F(sL D + OF)1 O = L + F(sL D)1 O

(5.10)

74

5 Basic residual generation methods

give ¯ ¯ = F(sL D)1 O= L + O(s) = L + F(sL D)1 O =, O(s)

(5.11)

Substituting (5.11) into (5.9) yields ¡ ¢ |(s) |ˆ(s) = F(sL D + OF)1 {(0) + L F(sL D + OF)1 O |(s)= On the assumption that (F> D) is observable, by choosing O suitably we can arbitrarily assign the poles of F(sL D + OF)1 and simultaneously suppress the inﬂuence of |(s). ¯ It is evident that system (5.4) with O(s) given by (5.11) satisﬁes conditions (5.4)-(5.8). However, a slight modiﬁcation is needed such that (5.4) is presented in a suitable form for the on-line implementation. We do the following calculations: |ˆ(s) = J|x (s)x(s) + F(sL D)1 O(|(s) |ˆ(s)) +, ¡ ¢ L + F(sL D)1 O |ˆ(s) = J|x (s)x(s) + F(sL D)1 O|(s) and thus ¢ ¡ |ˆ(s) = G + F(sL D + OF)1 (E OG) x(s) +F(sL D + OF)1 O|(s)=

(5.13)

During the above calculations, Lemma 3.1 and (5.10) have been used. With the aid of these relations, (5.13) can be brought into a compact form ˆ x (s) L)|(s) ˆx (s)x(s) (P |ˆ(s) = Q

(5.14)

ˆx (s) denoting an LCF of J|x (s)> i.e. J|x (s) = P ˆ x1 (s)Q ˆx (s)= ˆ x (s)> Q with P (5.14) describes a dynamic system whose input is x(s), |(s) and output an estimate of |(s). This system is stable and will converge to |(s), independent of x(s)> {(0), with an arbitrarily assignable velocity. Let’s transform (5.13) into the state space { ˆ˙ = Dˆ { + Ex + O (| F { ˆ Gx) |ˆ = F { ˆ + Gx=

(5.15) (5.16)

Its similarity to the well known state observer becomes evident. We call therefore system (5.14) or equivalently (5.15)-(5.16) output observer. As an estimate for |(s)> |ˆ(s) and the associated algorithm are also called soft- or virtual sensor. We summarize the main results of this section into a theorem. Theorem 5.1 Given a transfer function matrix J|x (s) 5 Rp×nx with the state space realization (D> E> F> G), then signal |ˆ(s) delivered by system (5.14) or equivalently (5.15)-(5.16) reconstructs |(s) in the sense of (5.5)-(5.8).

5.2 Residuals and parameterization of residual generators

75

The output observer builds the core of a residual generator. As will be shown in the next section, residual generator design can be reduced to the construction of an output observer. Remark 5.1 The original idea of using system model to construct redundancy and residual signals goes back to the works by Beard and Jones, in which a state observer in a quite similar form to (5.15)-(5.16) was used for the purpose of the output reconstruction. Since then, this approach is widely and successfully used in dealing with FDI problems under the name observerbased approach and has now become one of most powerful techniques in the ﬁeld of model-based fault diagnosis. Unfortunately, the expression observerbased approach often leads to the misunderstanding that a state observer is necessary. This is also the reason why we have paid much attention to the introduction of analytical redundancy construction using process input-output relationship. We would like to conclude this section with the following comments: • What we need for the residual generation is the input-output behaviors of the process under consideration. • The state observer form (5.15)-(5.16) provides us with a numerical solution for the purpose of creating analytical redundancy. It is not the only solution and, in some cases, also not the best one. • The use of the state observer form (5.15)-(5.16) is based on the assumption that J|x (s) has the state space realization (D> E> F> G). Known from the linear system theory, it means that only observable and controllable parts of the process are taken into account. From the viewpoint of residual generation, the system observability and controllability are in fact not necessary for the use of the so-called observer-based FDI scheme.

5.2 Residuals and parameterization of residual generators In the context of FDI study, a residual signal is understood as an indicator for the possible faults. The most important characteristic features of a residual, u(s), are L= lim u(w) = 0 for all x(w)> {(0) and |(w) = 0 w$4

LL= u(s) = Jui (s)i (s)> Jui (s) 6= 0=

(5.17) (5.18)

Using the output observer (5.14) we are able to generate a residual simply by a comparison of |ˆ(s) with |(s): ˆx (s)x(s)= ˆ x (s)|(s) Q u(s) = |(s) |ˆ(s) = P

(5.19)

76

5 Basic residual generation methods

On the other hand, we know that a signal constructed by e.g. U(s) (|(s) |ˆ(s)), where U(s) 6= 0 is some matrix or vector, is also a residual in the sense of (5.17)-(5.18). This motivates us to ask: What is the general form of a residual generator? It is reasonable to assume that all residual generators can be expressed in terms of u(s) = I (s)x(s) + K(s)|(s)> I (s)> K(s) 5 RH4

(5.20)

where I (s) and K(s) represent two stable systems with appropriate dimension. Thus, the answer to the above question can be concretely reformulated as a search for the existence conditions for I (s) and K(s) under which residual u(s) fulﬁlls conditions (5.17)-(5.18). Substituting (5.1) into (5.20) yields u(s) = I (s)x(s) + K(s)J|x (s)x(s) = (I (s) + K(s)J|x (s)) x(s)= We see that system (5.20) delivers a residual only if I (s) + K(s)J|x (s) = 0 which can be further written into I (s)Px (s) + K(s)Qx (s) = 0

(5.21)

with (Px (s)> Qx (s)) denoting a RCF pair of J|x (s). The following theorem shows under which conditions (5.21) holds. Theorem 5.2 Let ˆx (s)) and (Px (s)> Qx (s)) be left and right coprime factorization ˆ x (s)> Q • (P pair of transfer function matrix J|x (s) 5 LRp×nx , ˆ • \ (s)> [(s)> \ˆ (s)> [(s) be RH4 -matrices with appropriate dimensions that satisfy the Bezout identity (3.14) • N(s) be a nu × np -dimensional RH4 -matrix. Then, the set of RH4 -matrices I (s)> K(s) satisfying I (s)Px (s) + K(s)Qx (s) = N(s)

(5.22)

is given by ˆ x (s) ˆx (s)> K(s) = N(s)\ (s) + U(s)P I (s) = N(s)[(s) U(s)Q

(5.23)

where U(s) belongs to RH4 and is a nu × p-dimensional RH4 parameterization matrix. Furthermore, for every nu ×p-dimensional RH4 parameterization matrix U(s), I (s)> K(s) satisfying (5.23) ensure that (5.22) holds.

5.2 Residuals and parameterization of residual generators

77

Proof. Suppose I (s) and K(s) satisfy (5.22) and deﬁne ¸ £ ¤ \ˆ (s) U(s) = I (s) K(s) ˆ [(s) ˆ which, considering that I (s)> K(s)> [(s) and \ˆ (s) are RH4 matrices, belongs to RH4 . It results in ¸ £ ¤ £ ¤ Px (s) \ˆ (s) 1 I (s) K(s) = N(s) U(s) ˆ Qx (s) [(s) from which (5.23) follows readily. To prove that every I (s)> K(s) given by (5.23) satisfy (5.22) we use the double Bezout identity (3.14). Suppose I (s)> K(s) satisfy (5.23). Then I (s)Px (s) + K(s)Qx (s) = ¸ ¸ £ ¤ [(s) \ (s) Px (s) N(s) U(s) ˆx (s) P ˆ x (s) Qx (s) Q ¸ £ ¤ L = N(s) = N(s) U(s) R Hence, they ensure that (5.22) holds. u t Setting N(s) in Theorem 5.2 equal to null-matrix gives all solutions of (5.21) and thus a parameterization of all residual generators. Theorem 5.3 Given transfer function matrix J|x (s) 5 LRp×nx with a left ˆ x (s)> Q ˆx (s)), then coprime factorization pair (P ³ ´ ˆ x (s)|(s) Q ˆx (s)x(s) u(s) = U(s) P (5.24) represents a parameterization form of linear residual generators in the sense that • for every residual generator we can ﬁnd a RH4 -matrix U(s) such that the residual generator can be expressed in terms of (5.24), • for every U(s) 5 RH4 system (5.24) delivers a residual satisfying (5.17)(5.18). A comparison with (5.19) reveals that any residual generator can be considered as an extension of an output observer-based residual generator. They consist of two parts: an output observer and a dynamic system U(s). These two parts may take dierent functions: • the output observer builds the core of the residual generator and is used to reconstruct system behavior so that the original form of residual signal, |(s) |ˆ(s), provides us with the information about the variation of the system operation from its nominal value,

78

5 Basic residual generation methods

• the dynamic system U(s) acts in fact as a signal ﬁlter and can, by a suitable selection, help us to obtain signiﬁcant characteristics of faults, as will be discussed in the forthcoming chapters. Thus, U(s) is also called post-ﬁlter. Example 5.1 Consider the benchmark system EHSA given in Subsection 3.7.5. We would like to parameterize all residual generators for EHSA according to Theorem 5.3. To this end, we use the same LCF as given in Subsection 3.7.5 with ¸ ¸ ˆ ˆ ˆ ˆx (s) = Q11 (s) ˆ x (s) = P11 (s) P12 (s) > Q P ˆ 22 (s) ˆ 21 (s) P ˆ21 (s) P Q μ 5 ¶ 4 8 3 s + 38391s + 1=19 × 10 s + 1=36 × 1011 s2 +6=98 × 1013 s + 1=5 × 1016 ˆ 11 (s) = P F(s) μ 4 ¶ 3 s + 28256s 1=46 × 108 s2 7 1=88 × 10 1=6 × 1011 s 6=52 × 1013 ˆ 21 (s) = P F(s) μ 4 ¶ s + 39549s3 + 3=79 × 107 s2 1=02 × 1012 +1=43 × 1010 s + 3=17 × 108 ˆ 12 (s) = P F(s) μ 5 ¶ 4 4 s + 6=958 × 10 s + 1=22 × 109 s3 + 1=15 × 1012 s2 +4=28 × 1014 s 2=64 × 1011 ˆ 22 (s) = P F(s) ¶ μ 10 4 10 s 5=79 × 105 s3 5=1 × 1016 s2 3=34 1=91 × 1021 s 2=33 × 1024 ˆ11 (s) = Q F(s) μ 4 ¶ s + 40960s3 + 1=34 × 108 s2 5=82 × 1011 +5=51 × 1020 s + 1=86 × 1025 ˆ21 (s) = Q F(s) 5 4 4 s + 7=08 × 10 s + 1=31 × 109 s3 + 2=69 × 1012 s2 = F(s) = +1=87 × 1015 s + 5=38 × 1017 The parameterization form of the all residual generators is expressed by ´ ³ ˆx (s)x(s) > U(s) 5 RH4 = ˆ x (s)|(s) Q u(s) = U(s) P

5.3 Problems related to residual generator design and implementation Having addressed the parameterization form of residual generators, we are now faced with a practical task: how to design a residual generator described by

5.3 Problems related to residual generator design and implementation

79

ˆ x (s) and (5.24). Taking a look at (5.24) and recalling the meaning of U(s)> P ˆx (s) make it clear that there exist indeed two design parameters (parameter Q matrices): the observer gain matrix O and the post-ﬁlter U(s). The question arises: how to choose O and U(s)? Remember that the main objective of using a residual generator is to make the residual signal as sensitive to faults as possible and simultaneously as robust as possible against the model uncertainty. For this reason, we ﬁrst study the dynamics of residual generator (5.24). Let us consider system model of the form |(s) = J|x (s)x(s) + J|i (s)i (s) + |(s) and substitute it into (5.24). We immediately see that the dynamics of residual generator (5.24) is governed by ˆ x (s) (J|i (s)i (s) + |(s)) = u(s) = U(s)P

(5.25)

Obviously, the problem of residual generator design can be simply formulated as ﬁnding U(s) 5 RH4 and O ensuring the stability of matrix D OF such that ˆ x (s)Ji (s) as large as possible and simultaneously • U(s)P ˆ x (s)(s) as small as possible. • U(s)P In fact, the so-called observer-based residual generation approaches reported during the last three decades served only for one purpose, i.e. ﬁnding U(s) and O, although dierent mathematical and control theoretical tools have been applied, the structures of residual generators are various and the achieved results appear quite dierent. These approaches will be described in the subsequent sections of this chapter. We now have two dierent forms of residual generators, (5.24) and (5.25). (5.24) presents an explicit form that describes the structure and the possible algorithm for the on-line implementation. We call it implementation form of residual generators. In some references, it is also called computational form. Note that all variables and transfer function matrices used in (5.24) are known or measurable. In against, the variables given in (5.25) are unknown. Thus, (5.25) is an internal form that provides us with the dynamics of the FDI system and used for the purpose of residual generator design. For this reason, we call it design form of residual generators. Remark 5.2 There exist a variety of methods for the on-line realization of implementation form (5.24). We can use, for instance, the state space realization similar to (5.15)-(5.16) or transfer matrices. It is independent of which method is used for the determination of O and U(s). Our main attention in the following will be paid to the methods of residual generator design. The reader should keep in mind that the on-line implementation can be carried out independent of the design form used. One can use e.g. state space scheme for the on-line implementation even if O and U(s) are calculated by means of a frequency domain approach.

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5 Basic residual generation methods

Although (5.24) and (5.25) can be directly used for the residual generator design, the most important advantage of using them lies in their generality and the parameterization form, i.e. they represent the design and implementation forms of all linear residual generators. We shall in the following often make use of this property for the purposes of introducing some concept and or making system analysis. We call them therefore general forms of residual generators.

5.4 Fault detection ﬁlter Fault detection ﬁlter (FDF) is the ﬁrst kind of observer-based residual generators proposed by Beard and Jones in the early 70’s. Their work marked the beginning of a stormy development of model-based FDI techniques. Core of an FDF is a full-order state observer { ˆ˙ = Dˆ { + Ex + O (| F { ˆ Gx)

(5.26)

which is constructed on the basis of the nominal system model J|x (s) = F(sL D)1 E + G. Built upon (5.26), the residual is simply deﬁned by u = | |ˆ = | F { ˆ Gx=

(5.27)

Introducing variable h = { { ˆ yields h˙ = (D OF)h> u = Fh= It is evident that u possesses the characteristic features of a residual when the observer matrix O is so chosen that D OF is stable. In this case, { ˆ also provides a unbiased estimation for {, i.e. ˆ(w)) = 0= lim ({(w) {

w$4

The advantages of an FDF lie in its simple construction form (5.26)-(5.27) and, for the reader who is familiar with the state space control theory, in its intimate relationship with the state observer design, modern control theory and especially with the well established robust control theory by designing robust residual generators. We see that the design of an FDF is in fact the selection of the observer matrix O. To increase the degree of design freedom, we can switch a matrix to the output estimation error |(s) |ˆ(s)> i.e. u(s) = Y (|(s) |ˆ(s)) =

(5.28)

As discussed in the last section, (5.26)-(5.27) can be interpreted as a state ˆx (s)x(s). It thus turns out that an FDF ˆ x (s)|(s) Q space realization of P is indeed a special form of residual generator (5.24), namely the post-ﬁlter is a unit matrix for FDF given by (5.26)-(5.27) or a certain algebraic matrix

5.5 Diagnostic observer scheme

81

for FDF given by (5.26) and (5.28). A disadvantage of FDF scheme lies in the on-line implementation due to the full-order state observer, since in many practical cases a reduced order observer can provide us with the same or similar performance but with less on-line computation. This is one of the motivations for the development of Luenberger type residual generators, also called diagnostic observers. Example 5.2 Given benchmark system EHSA with model (3.82). For the residual generation purpose, an FDF of form (5.26)-(5.27) is designed with the same observer gain as used in the LCF, i.e. 6 5 9=2418 × 105 3=0326 × 103 9 1=6676 × 103 7=1992 × 104 : : 9 7 19=116 : O=9 : 9 5=6 × 10 4 7 3=2451 × 10 1=02 × 1012 8 1=8795 × 107 1=262 × 103 which ensures a stable FDF with poles v1 = 38611> v2 = 1257=4> v3 = 486=5 + 364=9m v4 = 486=5 364=9m> v5 = 30000=

5.5 Diagnostic observer scheme The diagnostic observer is one of mostly used and studied model-based residual generator forms. One reason for this popularity is its ﬂexible structure and its similarity to the Luenberger type observer. 5.5.1 Construction of diagnostic observer-based residual generators The core of a diagnostic observer is a Luenberger type (output) observer that is knowingly described by ¯ } + Y¯ | + Tx ¯ }˙ = J} + Kx + O|> |ˆ = Z

(5.29)

where } 5 Rv , v denotes the observer order and can be equal to or lower or higher than q, the system order. Although most contributions to the Luenberger type observer are focused on the ﬁrst case aiming at getting a reduced order observer, higher order observers will play an important role in optimization of FDI systems. ¯ Y¯ and Assume J|x (s) = F(sL D)1 E + G, then matrices J> K> O> T> ¯ Z together with a matrix W 5 Rv×q have to fulﬁll the so-called Luenberger conditions,

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5 Basic residual generation methods

L= J is stable LL= W D JW = OF> K = W E OG ¯ W + Y¯ F> T ¯ = Y¯ G + G LLL= F = Z

(5.30) (5.31) (5.32)

under which system (5.29) delivers a unbiased estimation for |, i.e. lim (|(w) |ˆ(w)) = 0=

w$4

(5.33)

To show it, we consider a dynamic system with h = W { } as its state vector and |(s) |ˆ(s) as its output. It turns out, according to (5.30)-(5.32), ¯h h˙ = Jh> | |ˆ = Z

(5.34)

which ensures (5.33). On account of (5.34), u = Y (| |ˆ) > Y 6= 0

(5.35)

builds a residual vector, whose dynamics is described by }˙ = J} + Kx + O| ¯ } Y Y¯ | Y Tx ¯ = Y | Z } Tx u = Y | Y Z where

(5.36) (5.37)

¡ ¢ ¯ > T = Y T= ¯ Y = Y L Y¯ > Z = Y Z

Thus, for the residual generator design condition III given by (5.32) should be replaced by LLL= Y F Z W = 0> T = Y G= (5.38) Remember that in the last section it has been claimed all residual generator design schemes can be formulated as the search for an observer matrix and a post-ﬁlter. It is therefore of practical and theoretical interest to reveal the relationships between matrices J> O> W> Y and Z solving Luenberger equations (5.30), (5.31), (5.38) and observer matrix as well as post-ﬁlter. A comparison with the FDF scheme makes it clear that • the diagnostic observer scheme may lead to a reduced order residual generator, which is desirable and useful for on-line implementation, • we have more degree of design freedom but, on the other hand, • more involved design. Having shown the importance of Luenberger equations (5.30)-(5.31), (5.38) in designing diagnostic observers, we concentrate our attention in the following on their solutions. Remark 5.3 On account of its importance in observer design, solution of Luenberger equations has received much attention in the 70’s and 80’s, and a large number of algorithms and studies have been published during this period.

5.5 Diagnostic observer scheme

83

On the other side, unlike most of observer design approaches, in which the observers are usually designed for the estimation of unmeasurable variables, the objective of using diagnostic observer is to reconstruct measured variable. This dierence, being observable by III condition (5.32), also motivated studies on characteristic properties of the special form of Luenberger conditions given by (5.30)- (5.31), (5.38). 5.5.2 Characterization of solutions In this subsection, a characterization of solutions of Luenberger equations (5.30), (5.31) and (5.38) will be provided. Some of results will be used later and help us get an insight into the structure of observer-based residual generators. We shall concentrate ourselves on the following topics • existence conditions, • minimum system order and • parameterization of solutions. Without loss of generality we ﬁrst make the following assumptions: • the pair (F> D) is given in the canonical observer form, i.e. 6 5 D¯11 · · · D¯1p 9 .. .. : 5 Rq×q > F = £ F¯ · · · F¯ ¤ 5 Rp×q D = 7 ... 1 p . . 8 D¯p1 · · · D¯pp 5 6 0 0 0 ··· 0 d ¯ll 1 91 0 0 ··· 0 d : ¯ll 2 9 : ll 90 1 0 ··· 0 d : ¯ 3 9 : D¯ll = 9 .. . . . . . . .. .. : 5 Rl ×l > l = 1> · · · > p 9. . . . . : . 9 : ll 70 ··· 0 1 0 d ¯(l 1) 8 0 ··· 0 0 1 d ¯ll l 6 5 0 ··· 0 d ¯lm 1 : 9 D¯lm = 7 ... ... ... ... 8 5 Rm ×l > p l A m> m = 1> · · · > p 1 0 ··· 0 d ¯lm m

6 d ¯lm 1 .. : 5 Rl ×m > p m A l> l = 1> · · · > p 1 . 8 lm 0 ··· 0 d ¯ l ¤ £ ¯ Fl = 0 · · · 0 h¯l 5 Rp×l > l = 1> · · · p ¤> £ ¯ll+1 · · · f¯lp h¯> 5 R1×p > l = 1> · · · > p l = 0 ··· 0 1 f 5

0 ··· 9. . ¯ Dlm = 7 .. ..

0 .. .

where 1 > · · · > p are the observability indices satisfying 1 > · · · > p 1> P p l=1 l = q. We denote the minimum as well as maximum observability indices with plq = minl l as well as pd{ = maxl l , respectively.

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5 Basic residual generation methods

• the residual is a scalar variable, i.e. u 5 R, and thus T> Y> Z will in the following be replaced by t> y> z, respectively. There are two reasons that explain why this assumption implies no restriction on the generality of the study: — A characterization of solutions will provide us with all possible solutions of y> z. Using linearly independent solutions we are able to construct a residual vector. On the other hand, a residual vector can equivalently be considered as a bank of scalar residuals. — More important, however, is the fact that in practical cases scalar residual signals are generally used. • matrices J> z take the following form 6 5 0 0 ··· 0 5 6 91 0 ··· 0: j1 : 9 ¤ £ 9 .. . . . . .. : 9 .. : v×(v1) > j = 7 . 8 5 Rv (5.39) J = Jr j > Jr = 9 . . . . : 5 R : 9 70 ··· 1 08 jv 0 ··· 0 1 £ ¤ z = 0 · · · 0 1 5 Rv = (5.40) Note that the dynamics of the residual generator is governed by h˙ = Jh> u = zh= It is reasonable to design the residual generator so that the pair (z> J) is observable. It is well known that by a suitable regular state transformation every observable pair can be transformed into the form (5.39)-(5.40). Therefore this assumption loses no generality. To begin with our study, we split D into two parts (5.41) D = Dr + Or F> Dr = gldj(Dr1 > · · · > Drp ) 6 5 0 0 0 ··· 0 91 0 0 ··· 0: : 9 : 9 Drl = 9 0 1 0 · · · 0 : 5 Rl ×l > l = 1> · · · > p> Or F 5 Rq×q 9 .. . . . . . . .. : 7. . . . .8 0 ··· 0 5

1 0 0 d ¯11 1 .. .. . . 0 d ¯11 1 .. .. . .

0 ··· .. .. . .

0d ¯p1 1 .. .. . . 0 ··· 0 d ¯p1 p

0 ··· .. .. . .

0 ··· 9 .. .. 9. . 9 90 ··· 9 9 Or F = 9 ... ... 9 90 ··· 9 9. . 7 .. ..

0 ··· .. .. . .

6 0 d ¯12 ¯1p 1 ··· 0 ··· 0 d 1 .. .. .. .. .. .. .. : . . . . . . . : : : 0 d ¯12 · · · 0 ··· 0 d ¯1p 1 1 : .. .. .. .. .. .. .. : = . . . . . . . : : p2 : 0d ¯1 · · · 0 · · · 0 d ¯pp 1 : .. .. .. .. .. .. .. : . . . . . . . 8

0 ··· 0 d ¯p2 ¯pp p · · · 0 · · · 0 d p

5.5 Diagnostic observer scheme

85

The pair (F> Dr ) is of an interesting property that is described by Lemma 5.1 and will play an important role in the following study. Lemma 5.1 Equation sr F + s1 FDr + · · · + sl FDlr + sv FDvr = 0> v 0

(5.42)

holds if and only if sl FDlr = 0> l = 0> · · · > v=

(5.43)

Furthermore, vectors sl > l = 0> · · · > v> satisfy sl = 0> l = 0> · · · > plq 1; sl FDlr = 0> l = plq > · · · > pd{ 1

(5.44)

and sl > l pd{ > are arbitrarily selectable. Proof. First note that Dlr = 0> l pd{ , hence we have sm FDmr = 0> for all sm > m pd{ and so sr F + s1 FDr + · · · + sv FDvr = 0 +, sr F + s1 FDr + · · · + so FDor = 0 ½ pd{ 1 for v pd{ with o = v for v ? pd{ We now prove (5.43) as well as£ (5.44). To¤ this end, we utilize the following fact: for a row vector t(6= 0) = t1 · · · tp 5 Rp we have ¤ £ tFDmr = t¯1 · · · t¯p > m 0 with the row vector t¯l 5 Rl satisfying for m l > t¯l = 0> and £ ¤ hl 0 · · · 0 for m ? l > t¯l = 0 · · · 0 t¯ where the entry t¯ hl lies in the ( l m)-th place. Thus, the non-zero entries of two row vectors sl FDlr and sm FDmr > l 6= m, are in dierent places. This ensures that (5.42) holds if and only if sm FDmr = 0> m = 0> · · · > o Note that udqn(FDmr ) = udqn(F) = p> m = 0> · · · > plq 1. Hence, we ﬁnally have: for m = 0> · · · > plq 1 sm FDmr = 0 +, sm = 0 The lemma is thus proven. u t

86

5 Basic residual generation methods

We now consider equation (5.31) and rewrite it into ¯ O ¯ = O W Or > Dr = D Or F W Dr JW = OF> and furthermore ¸ ¸ ¸ ¸ ¸ ¯v ¯ £ ¤ W¯v W¯v Dr O W¯v Dr O ¯ J j = ¯ F +, Jr Wv = ¯v F + jwv r wv Dr wv wv Dr ov ov (5.45) where 6 6 5 5 ¯o1 w1 ¸ ¸ ¯v O W¯v : 9 9 . . : ¯= ¯ > W¯v = 7 .. 8 > O W = ¯ov > Ov = 7 .. 8 = wv ¯ov1 wv1 Writing Jr as

5

0 0 9 ¸ 91 0 J1 0 9 Jr = > J1 = 9 ... . . . 0 1 9 70 ··· 0 ···

6 0 0: : .. : 5 R(v1)×(v2) .: : 1 08 0 1

··· ··· .. .

and considering the last row of (5.45) result in wv Dr wv1 = ¯ov F + jv wv +, wv1 = wv Dr (¯ov F + jv wv )=

(5.46)

Repeating this procedure leads to wv2 = wv1 Dr (¯ov1 F + jv1 wv ) = wv D2r (¯ov1 F + ¯ov FDr ) (jv1 wv + jv wv Dr ) ··· (5.47) w2 = w3 Dr (¯o3 F + j3 wv ) = wv Dv2 (¯o3 F + · · · + ¯ov FDrv3 ) (j3 wv + · · · + jv wv Drv3 ) r w1 = w2 Dr (¯o2 F + j2 wv ) = wv Dv1 (¯o2 F + · · · + ¯ov FDrv2 ) (j2 wv + · · · + jv wv Drv2 )= r Finally, from the ﬁrst row of (5.45) we have w1 Dr = ¯o1 F + j1 wv =

(5.48)

(5.46)-(5.48) give a kind of characterization of all solutions of (5.31), based on which we are going to derive the existence condition of residual generators. To this end, we ﬁrst consider (5.38). Since y 6= 0, it is evident that the equation ¸ £ ¤ W zW yF = 0 +, z y =0 F is true if and only if the last row of matrix W linearly depends on the rows of F. Based on this fact, the following existence condition can be derived.

5.5 Diagnostic observer scheme

87

Remark 5.4 It is worth pointing out that the above fact is contrary to the existence condition of a Luenberger type state observer which requires the linear independence of the rows of W from the ones of F. The reason for this is that state observers and observer-based residual generators are used for different purposes: state observers are used for the estimation of unmeasurable state variables, while the observers for the residual generation are used for the estimation of measurable state variables (output signal). Theorem 5.4 Equations (5.31) and (5.38) are solvable if and only if v plq =

(5.49)

Proof. Here, we only prove the necessity. The su!ciency will be provided below in form of an algorithm. The fact that the last row of matrix W is linearly dependent on the rows of F can be expressed by wv = y¯v F> for some y¯v 6= 0= This leads to wv1 = wv Dr (¯ov F + jv wv ) = y¯v FDr (¯on F + jn wv ) ··· v2 w1 = y¯v FDv1 (¯o2 F + · · · + ¯ov FDv2 r r ) (j2 wv + · · · + jv wv Dr ) Substituting w1 into (5.48) gives v1 y¯v FDvr (¯o1 F + ¯o2 FDr + · · · + ¯ov FDv1 r ) = j1 wv + j2 wv Dr + · · · + jv wv Dr

and further y¯v FDvr (¯o1 + j1 y¯v )F · · · (¯ov + jv y¯v )FDrv1 = 0 Following Lemma 5.1 we know that the above equation holds only if v plq Thus, the necessity is proven. u t Based on this theorem, we can immediately claim Corollary 5.1 Given system Jx (s) = F(sL D)1 E + G, the minimal order of residual generator (5.36)-(5.37) is plq . We now derive an algorithm for the solution of (5.30), (5.31) and (5.38), which also serves as the proof of the su!ciency of Theorem 5.4. We begin with the following assumption wv = y¯v F> y¯v 6= 0

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5 Basic residual generation methods

and suppose v plq . According to (5.46)-(5.47) we have wv1 = y¯v FDr (¯ov + jv y¯v )F ··· w1 = y¯v FDrv1 (¯o2 + j2 y¯v )F · · · (¯ov + jv y¯v )FDv2 r =

(5.50) (5.51)

Substituting w1 into (5.48) yields = 0= (5.52) y¯v FDvr (¯o1 + j1 y¯v )F (¯o2 + j2 y¯v )FDr · · · (¯ov + jv y¯v )FDv1 r Following Lemma 5.1, (5.52) is solvable if and only if (5.53) y¯v FDvr = 0> y¯v 6= 0 v1 plq ¯ ¯ (ov + jv y¯v )FDr = 0> · · · > (oplq +1 + jplq +1 y¯v )FDr = 0 (5.54) ¯o ¯ ¯v = 0> · · · > o1 + j1 y¯v = 0 (5.55) plq + j plq y and furthermore, since v plq , (5.53)-(5.55) are solvable. In order to simplify the notation, we introduce vectors y¯l > l = plq > · · · > v 1> deﬁned by y¯l = ¯ol+1 + jl+1 y¯v > (¯ol+1 + jl+1 y¯v )FDlr = 0= With the aid of these results the following theorem becomes evident. Theorem 5.5 Given v plq , then matrices O> W> y> z deﬁned by 5 6 65 0 ··· 0 y¯plq · · · y¯v F 9 0 : 9 · · · y¯plq · · · y¯v 0 : FDr 9 : :9 9 .. : .. .. .. .. : 9 9 . : :9 . . . . 9 : :9 v plq 1 : 9 0 y¯plq · · · : 9 y¯v · · · 0 : 9 FDr 9 : W =9 v plq : 9 y¯v 0 ··· 0 : 9 y¯plq · · · : : 9 FDr 9 . : 9 .. .. .. .. .. : 9 .. : : 9 . . . . . :9 9 : v2 7 y¯v1 y¯v 8 8 7 FDr 0 0 ··· 0 v1 0 0 0 ··· 0 FDr y¯v 5 6 j1 y¯v 9 : j2 y¯v 9 : 9 : .. 9 : . 9 : 9 : j y ¯ plq v ¯ 9 : ¯ O = O + W Or > O = 9 : ¯ y j y ¯ plq +1 v : 9 plq 9 : . .. 9 : 9 : 7 ¯ yv2 jv1 y¯v 8 ¯ yv1 jv y¯v £ ¤ z = 0 · · · 0 1 > y = y¯v

(5.56)

(5.57)

(5.58)

5.5 Diagnostic observer scheme

89

solve equations (5.31) and (5.32) for all j1 > j2 > · · · > jv that ensure the stability of J, where y¯pd{ 1 > · · · > y¯plq are the solution of the following equations y¯pd{ 1 FDrpd{ 1 = 0> · · · > y¯plq FDr plq = 0

(5.59)

y¯pd{ > · · · > y¯v are arbitrarily selectable and y¯v 6= 0. The proof follows directly from (5.50)-(5.55) as well as Lemma 5.1. Together with (5.59), Theorem 5.5 provides us with an algorithm for the solution of Luenberger equations for the residual generator design. We see that the solution of (5.31) and (5.38) is reduced to the solutions of equations given by (5.59). From pd{ up increasing the order v does not lead to an increase in computation. In fact, once equations (5.59) are solved for y¯pd{ 1 > · · · > y¯plq , we are able to design residual generators of arbitrary order without additional computation. From the above algorithm we know that the solution for (5.31) and (5.38) is usually not unique, since the solutions of equations given by (5.59) is not unique (see also below) and, if v pd{ , vectors y¯l > l = pd{ > · · · > v> are also arbitrarily selectable. It is just this degree of freedom that can be utilized for designing FDI systems. This also motivates the study on the parameterization of solutions, which builds the basis of a successful optimization. For our purpose, we ﬁrst re-arrange the matrix W given by (5.56) as a row vector: 5 6 w1 ¤ 9 .. : new arrangement £ w1 · · · wv := wˆ $ W =7 . 8 wv then we have, following Theorem 5.5, ¤ £ wˆ = y¯plq y¯plq +1 · · · y¯v T 5 FDr plq 1 · · · F 0 9 9 FDr plq · · · FDr F T=9 9 .. .. .. . .. 7 . . . v1 v plq v plq 1 FDr · · · FDr FDr

··· .. . .. . ···

6 0 .. : . : := : 08 F

Let us introduce the notation Qedvlv = gldj (Qplq > · · · > Qpd{ 1 > Lp×p > · · · > Lp×p ) where Ql 5 R(ppl )×p > l = plq > · · · > pd{ 1, stands for the basis matrix of left null space of matrix FDlr with ¢ ¡ pl = udqn FDlr = £ ¤ It is evident that any vector y¯plq y¯plq +1 · · · y¯v can be written as

90

5 Basic residual generation methods

£

¤ y¯plq y¯plq +1 · · · y¯v = y¯Qedvlv

where y¯ 6= 0 is a vector of appropriate dimension. This gives the following theorem. Theorem 5.6 Given v plq , then matrix W that solves (5.31) can be parameterized by 6 5 F 0 ··· 0 FDrplq 1 · · · 9 . . .. : 9 FDr plq · · · . .: FDr F := 9 ˆ (5.60) w = y¯Qedvlv 9 : .. .. .. . . . . 7 . 08 . . . . plq FDv1 · · · FDv FDrvplq 1 · · · F r r The proof is evident and therefore omitted. From Theorems 5.5 and 5.6 we know that • for every solution of (5.31) we are able to ﬁnd a vector y¯ 6= 0 such that this solution can be brought into the form given by (5.60) • on the other side, given a vector y¯ 6= 0 we have a W and further a solution for (5.31). In this sense, the vector y¯ 6= 0 is called the parameterization vector. Note that udqn (Qedvlv ) = number of the rows of Qedvlv = p(v pd{ + 1) +

pd{ X1

(p pl )

l=plq

p(v + 1 plq ) = number of the columns of Qedvlv

(5.61)

and moreover 5

FDr plq 1 · · ·

F

0

9 9 FDr plq · · · FDr F udqn 9 9 .. .. .. . .. 7 . . . v1 v plq v plq 1 FDr · · · FDr FDr

··· .. . .. . ···

6 0 .. : . : : : 08 F

= p(v + 1 plq ) = number of the rows.

(5.62)

Thus we have Corollary 5.2 Equation (5.31) has p(v pd{ + 1) + linearly independent solutions.

Ppd{ 1 l= plq

(p pl )

Remark 5.5 PvIf v ? pd{ , the number of the linearly independent solutions is given by l=plq (p pl ).

5.5 Diagnostic observer scheme

91

Remember that at the beginning of this subsection we have made the assumption that the observable pair (F> D) is presented in the canonical observer form. Note that all the results given Theorems 5.5 and 5.6 are expressed in terms of observability indices, matrices Dr > FDlr . It is known from the linear control theory that the observability indices, matrix Dr are structural characteristics of a system under consideration that are invariant to a regular state transformation. Moreover, for any regular state transformation, say Wvw , we have 1 DWvw JW Wvw = OFWvw W D JW = OF +, W Wvw Wvw yF zW = 0 +, yFWvw zW Wvw = 0 1 K = W E OG +, K = W Wvw Wvw E OG

i.e. the solutions J> K> O> t> y> z and so that the construction of the residual generator are invariant to the state transformation Wvw . This implies that the achieved results hold for every observable pair. 5.5.3 A numerical approach Based on the result achieved in the last subsection, we now present an approach to solving Luenberger equations (5.30)-(5.31) and (5.38). We ﬁrst consider Theorem 5.5, in which a solution is indeed provided except that knowledge of Dr and Or is needed. Although Dr and Or can be determined by (a) transforming (F> D) into observer canonical form (b) solving equation Or F = D Dr for Or > the required calculation is involved and in many cases too di!cult to be managed without a suitable CAD program. For this reason, further study is, on account of Theorem 5.5, carried out aiming at getting an explicit solution similar to the one given by Theorem 5.5 but expressed in terms of system matrices D> F. For our purpose, the following lemma is needed. Lemma 5.2 Given matrices Dr > E> F> H> I and Or with appropriate dimensions, then we have for l = 1> · · · > v 65 6 6 5 5 I I L 0 0 ··· 0 ¯ : 9 FO0 : 9 FH 9 L 0 ··· 0: : 9 FH : : 9 9 ¯ : 9 FDr O0 FO0 : 9 F D¯0 H : 9 FDr H L · · · 0 :9 : (5.63) :=9 9 9 .. : 9 .. : 9 .. : .. .. .. .. 8 7 . 8 7 7 . . . . . . 8 l ¯ l l1 ¯ FDr O0 FDr O0 · · · FO0 L F Dl0 H FDr H ¯ = H Or I> D¯r = Dr + Or F H 6 5 6 5 6 L 0 ··· 0 5 F F . : 9 F(D + O F) : .. 9 FDr : 9 r r . .. : L FOr 9 : 9 : :9 9 = (5.64) 9 .. : 9 := 9 . : .. . .. .. 7 . 8 7 8 7 . 8 . . 0 . FDlr F(Dr + Or F)l FDl1 r Or · · · FOr L

92

5 Basic residual generation methods

The proof is straightforward and thus omitted. Let’s introduce matrix K1 deﬁned by 5 FOr L FDr plq 1 Or · · · 9 9 FDr plq Or · · · FDr Or FOr K1 = 9 9 .. .. .. .. 7 . . . . v plq v plq 1 FDv1 O · · · FD O FD Or r r r r r Note that

5 9 9 K1 9 7

F FD .. .

6

0 L .. . ···

··· .. . .. . FOr

6 0 .. : .: := : 08 L

6 FDr plq : 9 FDr plq +1 : : 9 : :=9 : .. 8 7 8 . 5

FDl

FDvr

whose proof can readily be obtained by using equality (5.64). It turns out 5 6 5 6 FDr plq F plq +1 : 9 FD : ¤9 £ 9 FDr : 9 : y¯plq y¯plq +1 · · · y¯v 9 = y ˜ K : .. v 19 . :=0 . 7 8 7 . . 8 FDvr

FDv

where y¯plq > y¯plq +1 > · · · > y¯v satisfy (5.59). We now deﬁne a new vector ¤ £ yv = yv>0 yv>1 · · · yv>v = y˜v K1 and then apply (5.64) to (5.56)-(5.57). As a result we obtain 5 6 6 6 5 5 F yv>1 yv>2 · · · yv>v1 yv>v 9 yv>0 : FD : 9 yv>2 · · · · · · yv>v 0 : 9 9 yv>1 : : 9 .. : : 9 9 W =9 . > O = : 9 : 9 .. : jyv>v . . . . . . 9 : 7 . ··· ··· 7 . 8 . . 87 FDv2 8 yv>v1 0 yv>v 0 · · · · · · FDv1 5 6 6 6 5 5 F yv>1 yv>2 · · · yv>v1 yv>v 9 yv>0 : 9 yv>2 · · · · · · yv>v 0 : 9 FD : 9 yv>1 : : 9 .. : : 9 9 K=9 . E + 9 . : G + jyv>v G : 9 : . . . .. 8 .. .. 8 9 : 7 .. · · · · · · 7 7 FDv2 8 yv>v1 0 yv>v 0 · · · · · · FDv1 6 5 G 6 5 : yv>0 + j1 yv>v yv>1 yv>2 · · · yv>v1 yv>v 9 9 FE : 9 yv>1 + j2 yv>v yv>2 · · · · · · yv>v 0 : 9 FDE : :9 9 : =9 : .. .. .. .. : 9 .. : 9 8 7 . . ··· ··· . . . : 9 v2 8 7 yv>v1 + jv yv>v yv>v 0 · · · · · · 0 FD E FDv1 E y = yv>v > t = yG = yv>v G=

5.5 Diagnostic observer scheme

93

We now remove the assumption that (F> D) is given in the observer canonical form, under which Theorem 5.5 has been derived. To this end, we suppose the original system matrices are given by S DS 1 > FS 1 > S E with S denoting a regular state transformation. Note that W D JW = OF +, W S 1 S DS 1 JW S 1 = OFS 1 K = W E OG +, K = W S 1 S E OG yF zW = 0 +, yFS 1 zW S 1 = 0 6 5 FS 1 6 5 yv>1 yv>2 · · · yv>v1 yv>v 9 : FS 1 S DS 1 : 9 yv>2 · · · · · · yv>v 0 : 9 : .. :9 9 : = W S 1 9 . 9 .. .. .. : 9 : 8 7 . ··· ··· . . : 9 .. 8 7 . 0 yv>v 0 · · · · · · FS 1 (S DS 1 )v1 S 1 5 6 5 6 F FS 1 9 FD : 9 FS 1 S DS 1 : 9 : 9 : yv 9 . : = 0 +, yv 9 : = 0= .. 7 .. 8 7 8 . v 1 v FS (S DS ) FD

(5.65) (5.66) (5.67)

(5.68)

(5.69)

We ﬁnally have the following theorem. Theorem 5.7 Given system model J|x (s) = F(sL D)1 E +G and suppose that v plq and 5 6 F 9 FD : ¤ £ 9 : (5.70) yv 9 . : = 0> yv = yv>0 yv>1 · · · yv>v . 7 . 8 FDv then matrices O> W> K> t> y> z deﬁned by 6 F 69 : yv>v 9 FD : 9 . .. : 0 : : :9 9 .. : 9 . : : . 8 9 .. : : 9 0 7 FDv2 8 FDv1

(5.71)

: £ ¤ : : jyv>v > z = 0 · · · 0 1 > y = yv>v 8

(5.72)

5

5

yv>1 9 yv>2 9 W =9 . 7 .. yv>v 5 9 9 O = 9 7

yv>2 · · · yv>v1 · · · · · · yv>v .. ··· ··· . 0 ···

yv>0 yv>1 .. . yv>v1

6

···

94

5 Basic residual generation methods

5

yv>0 + j1 yv>v yv>1 9 ¸ 9 yv>1 + j2 yv>v yv>2 K 9 .. .. =9 . . t 9 7 yv>v1 + jv yv>v yv>v 0 yv>v

yv>2 · · · yv>v1 · · · · · · yv>v .. ··· ··· . 0 ··· 0 ···

0 0

6 5 6 G yv>v 9 : 9 FE : 0 : : 9 FDE : : .. : 9 : 9 .. . : :9 : . : 9 0 87 v2 8 FD E 0 FDv1 E

(5.73)

solve the Luenberger equations (5.30)-(5.31) and (5.38), where vector j should be so chosen that the matrix J is stable. It is clear that once the system matrices D> F are given we are able to calculate the solution of Luenberger equations (5.30)-(5.31) and (5.38) using (5.71)-(5.72). To this end, we provide the following algorithm. Algorithm 5.1 Solution of Luenberger equations (5.30)-(5.31) and (5.38) Step Step Step Step

1. 2. 3: 4.

Set v plq Solve (5.70) for yv>0 > · · · > yv>v Select j such that J given in (5.39) is stable Calculate O> W> K> t> y> z according to (5.71)-(5.73).

We see that the major computation of the above approach consists in solving (5.70). It reminds us of the so-called parity space approach. In fact, the main advantage of this approach, as will be shown in the next sections, is its intimate connection to the parity space approach and to parameterization form presented in the last subsection, which are useful for such applications like robust FDI, analysis and optimization of FDI systems. Example 5.3 Given benchmark system EHSA with model (3.82). We now design a diagnostic observer based residual generator using Algorithm 5.1. Below is the design procedure with the achieved result: Step 1: Set v = 2 Step 2: Solve (5.70), which results in ¤ £ yv = 3=36 × 108 7=25 × 104 1=08 × 1019 1 0 2=76h × 105 Step 3: Set j

j=

¸ ¸ 20000 0 20000 =, J = 300 1 300

which results in two poles at 100 and 200 respectively Step 4: Calculate O> W> K> t> y> z> which gives ¸ ¸ 0 3=36 × 108 0=551 K= >O = 4=6 × 1021 1 0 ¸ 0 2=76 × 105 0 1=08 × 1019 1 W = 0 0 0 0 2=76 × 105 ¤ £ ¤ £ y = 0 2=76105 > z = 0 1 > t = 0=

5.5 Diagnostic observer scheme

95

Example 5.4 We now design a minimum order diagnostic observer for the inverted pendulum system LIP100 that is described in Subsection 3.7.2. It follows from Corollary 5.1 that the minimum order of a DO is the minimum observability index of the system under consideration. For LIP100 whose model can be found in (3.57), the minimum observability index is 1= Below is the design procedure for a minimum order DO: Step 1: Set v = 1 Step 2: Solve (5.70), which results in £ ¤ v = 0 0=0645 0=0051 0=4947 0=0041 0=5011 Step 3: Select j = 3= Note that for v = 1 J = j = 3 Step 4: Calculate O> W> K> t> y> z> which gives £ ¤ K = 3=0738> O = 1=4841 0=0521 1=5083 > t = 0> z = 1 £ ¤ £ ¤ W = 0=4947 0=0041 0=5011 0 > y = 0=4947 0=0041 0=5011 = To make an impression on the reader how a residual signal responds to the occurrence of a fault, we show in Fig.5.2 the response of the generated residual signal to a unit step fault occurred in the sensor measuring the angular position of the inverted pendulum at 20 vhf. We can see that due to the initial condition the residual generator needs a couple of minutes before delivering a zero residual signal in the fault-free situation. Mathematically, it is described by the requirement (5.17), i.e. lim u(w) = 0 for all x(w)> {(0)=

w$4

In practice, such a time interval is considered as the calibration time and is a part of a measurement or monitoring process. In this context, in our subsequent study, we generally do not take into account the inﬂuence of the initial conditions. From Fig.5.2, we can further see that the residual signal has a strong response to the fault. 5.5.4 An algebraic approach The original version of the approach presented in this subsection was published by Ge and Fang in their pioneer work in the late 80’s. In a modiﬁed form, the key points of this approach are summarized in the following theorem. Theorem 5.8 Given system model J|x (s) = F(sL D)1 E + G and v plq , then matrices O> W> Y> Z deﬁned by O = f(J)[> W = \ M ¡ ¢1 W Y = Z W F > FF > > Z W FQ =

(5.74) (5.75)

96

5 Basic residual generation methods 1 0.15

0.8 0.6

Residual signal

0.1

0.4 0.2

0.05

0 −0.2 0

0

10

20

30

40

50

−0.05

−0.1 0

5

10

15

20

25

30

35

40

45

50

Time [s]

Fig. 5.2 Response of the residual signal to a sensor fault

solve the Luenberger equations (5.30)-(5.31) and (5.32), where matrix G should be chosen stable, X ∈ Rs×m is an arbitrary matrix, and ¸ ∙ C (n−m)×n T = n, CCN and rank =0 (5.76) CN ∈ R CN ¤ £ (5.77) Y = X GX · · · Gn−1 X c(p) = det(pI − A) = an pn + an−1 pn−1 + · · · + a1 p + a0 ⎤ ⎡ an CAn−1 + an−1 CAn−2 + · · · + a2 CA + a1 C ⎥ ⎢ an CAn−2 + an−1 CAn−3 + · · · + a2 C ⎥ ⎢ J =⎢ ⎥ .. ⎦ ⎣ .

(5.78)

(5.79)

an C

Proof. Substituting (5.74) into the left side of (5.31) yields T A − F T = Y JA − GY J = X Since a0 C +

n X

n X i=1

i

ai CA −

n X

ai Gi XC

i=1

ai CAi = 0

i=1

we obtain T A − F T = −a0 XC −

n X i=1

ai Gi XC = −c(G)XC = LC.

That (5.74) solves (5.31) is thus proven. Note V C = W T given by (5.31) means W T belongs to the range of C, which, considering CN T spans the nullspace of C, equivalently implies W T CN T = 0. Furthermore, multiplying the both sides of V C = W T by C T gives ¡ ¢−1 V CC > = W T C T ⇐⇒ V = W T C T CC T

5.6 Parity space approach

97

Hence, the theorem is proven. u t It is evident that the design freedom is provided by the arbitrary selection W of matrix [, possible solutions of equation Z W FQ = 0 that are generally not unique. We summarize the main results in the following algorithm. Algorithm 5.2 Solution of Luenberger Equations by Ge and Fang Step Step Step Step

1. 2. 3. 4.

Calculate f(s) = ghw(sL D) for d0 > d1 > · · · > dq Calculate O> W according to (5.74) W = 0 for Z Solve Z W FQ Set Y subject to (5.75).

Example 5.5 We now design a DO for LIP100 using Algorithm 5.2. To this end, model (3.57) is used. Below is the design procedure: Step 1: Calculate f(s) = ghw(sL D) for d0 > d1 > · · · > dq , which results in d4 = 1=0> d3 = 2=0512> d2 = 20=9964> d1 = 37=7364> d0 = 0 Step 2: Calculate O> W according to (5.74): 5 6 43=7363 20=6009 25=6136 5=2725 W = 7 31=9964 32=0988 5=1094 10=0610 8 3=9488 11=1150 66=6030 3=5182 5 6 23=6928 49=8218 297=1340 O = 7 0=2995 115=0328 494=9239 8 8=3036 49=5223 182=1012 W Step 3: Solve Z W FQ = 0 for Z : 6 5 0=8461 0=5040 0=1734 Z = 7 1=6922 1=0081 0=3469 8 2=5383 1=5121 0=5203

Step 4: Set Y subject to (5.75): 6 5 21=5620 3=1786 30=6468 Y = 7 43=1240 6=3572 61=2936 8 = 64=6860 9=5358 91=9404

5.6 Parity space approach In this section, we describe the parity space approach, initiated by Chow and Willsky in their pioneering work in the early 80’s. Although a state space model is used for the purpose of residual generation, the so-called parity relation, instead of an observer, builds the core of this approach. This is also the reason why the parity space approach is generally recognized as one of the important model-based residual generation approaches, parallel to the observer-based and the parameter estimation.

98

5 Basic residual generation methods

5.6.1 Construction of parity relation based residual generators A number of dierent forms of parity space approach have, since the work by Chow and Willsky, been introduced. We consider in the following only the original one that is based on the assumption of a state space model of a linear discrete time system described by {(n + 1) = D{(n) + Ex(n) + Hg g(n) + Hi i (n) |(n) = F{(n) + Gx(n) + Ig g(n) + Ii i (n)=

(5.80) (5.81)

It is further assumed that (F> D) is observable and udqn(F) = p. For the purpose of constructing residual generator, we ﬁrst suppose i (n) = 0> g(n) = 0. Following (5.80)-(5.81), |(n v)> v 0, can be expressed in terms of {(n v)> x(n v) and |(n v + 1) in terms of {(n v)> x(n v + 1)> x(n v), |(n v) = F{(n v) + Gx(n v) |(n v + 1) = F{(n v + 1) + Gx(n v + 1) = FD{(n v) + FEx(n v) + Gx(n v + 1)=

(5.82)

Repeating this procedure yields |(n v + 2) = FD2 {(n v) + FDEx(n v) + FEx(n v + 1) +Gx(n v + 2)> · · · > |(n) = FDv {(n v) + FDv1 Ex(n v) + · · · + FEx(n + 1) + Gx(n)= (5.83) Introducing the notations 5 5 6 6 |(n v) x(n v) 9 |(n v + 1) : 9 x(n v + 1) : 9 9 : : |v (n) = 9 : > xv (n) = 9 : .. .. 7 7 8 8 . . 5 9 9 Kr>v = 9 7

|(n) 6

F FD .. .

FDv

: : : > Kx>v 8

5

(5.84)

x(n)

6 0 9 .. : 9 FE . : 9 : =9 : .. 7 08 . FDv1 E · · · FE G G

0 ··· . G .. .. .. . .

(5.85)

allows us to rewrite (5.82)-(5.83) into the following compact form |v (n) = Kr>v {(n v) + Kx>v xv (n)=

(5.86)

Note that (5.86), the so-called parity relation, describes the input and output relationship in dependence on the past state variable {(n v). It is expressed in an explicit form, in which

5.6 Parity space approach

99

• |v (n) and xv (n) consist of the temporal and past outputs and inputs respectively and are known • matrices Kr>v and Kx>v are composite of system matrices D> E> F> G and also known • the only unknown variable is {(n v)= The underlying idea of the parity relation based residual generation lies in the utilization of the fact, known from the linear control theory, that for v q the following rank condition holds: udqn (Kr>v ) = q ? the number of the rows of matrix Kr>v = This ensures that for v q there exists at least a (row) vector yv (6= 0) 5 R(v+1)p such that (5.87) yv Kr>v = 0= Hence, a parity relation based residual generator is constructed by u(n) = yv (|v (n) Kx>v xv (n))

(5.88)

whose dynamics is governed by, in case of i (n) = 0, u(n) = yv (|v (n) Kx>v xv (n)) = yv Kr>v {(n v) = 0= Vectors satisfying (5.87) are called parity vectors, the set of which, Sv = {yv | yv Kr>v = 0}

(5.89)

is called the parity space of the v-th order. In order to study the inﬂuence of i> g on residual generator (5.88), the assumption that i (n) = 0> g(n) = 0 is now removed. Let us repeat procedure (5.82)-(5.83), which gives |v (n) = Kr>v {(n v) + Kx>v xv (n) + Ki>v iv (n) + Kg>v gv (n) where 5 6 6 0 ··· 0 Ii i (n v) 9 . : . 9 i (n v + 1) : 9 FHi Ii . . .. : 9 : 9 : iv (n) = 9 : > Ki>v = 9 .. : .. .. .. 7 8 . 7 . 0 8 . . i (n) FDv1 Hi · · · FHi Ii 5 6 5 6 0 ··· 0 Ig g(n v) 9 . : . 9 g(n v + 1) : 9 FHg Ig . . .. : 9 : 9 := gv (n) = 9 : > Kg>v = 9 .. : .. . . 7 8 . . . 7 . 0 8 . . g(n) FDv1 Hg · · · FHg Ig 5

Constructing a residual generator according to (5.88) ﬁnally results in

(5.90)

(5.91)

100

5 Basic residual generation methods

uv (n) = yv (Ki>v iv (n) + Kg>v gv (n)) > yv 5 Sv =

(5.92)

We see that the design parameter of the parity relation based residual generator is the parity vector whose selection decisively eects the performance of the residual generator. Remark 5.6 One of the signiﬁcant properties of parity relation based residual generators, also widely viewed as the main advantage over the observer-based approaches, is that the design can be carried out in a straightforward manner. In fact, it only deals with solutions of linear equations or linear optimization problems. In against, the implementation form (5.88) is surely not ideal for an on-line realization, since it is presented in an explicit form, and thus not only the temporal but also the past measurement and input data are needed and have to be recorded. Remark 5.7 The requirement on the past measurement and input data is one of the reasons why the parity space approach is mainly applied to the discrete time dynamic systems. 5.6.2 Characterization of parity space Due to its simple form as solution of (5.87) a characterization of the parity space seems unnecessary. However, some essential questions remain to be solved: • What is the minimum order of a parity space? Remember that v q presents a su!cient condition for (5.87). This implies that the order of the designed residual generator is at least as high as the one of the system under consideration. Should it be? Dose there exist a lower order residual generator? • How to parameterize the parity space for a given v? As will be shown in the forthcoming chapters, parameterization of the parity space plays an important role in optimization of parity relation based FDI systems • How to select the order of the parity space? • Are there close relationships between the parity space approach and the observer-based approaches? Finding out suitable answers to these questions motivates a study on the characterization of parity space. To begin with, we introduce the following notation for yv ¤ £ yv = y0>v y1>v · · · yv>v > yl>v 5 Rp > l = 0> · · · > v= Notice that (5.87) is identical with (5.70) given in Theorem 5.7, which is necessary and su!cient for solving Luenberger equations (5.30)-(5.31) and (5.38). This relationship reveals

5.6 Parity space approach

101

Theorem 5.9 The minimum order of the parity space is plq . ¤ £ Theorem 5.10 Given v plq , then yv = y0>v · · · yv1>v yv>v 5 Sv can be written as ¤ £ (5.93) yv = y¯K1 > y¯ = y¯plq y¯plq +1 · · · y¯v1 y¯v where y¯m 5 Tm > Tm = {t | tFDmr = 0}> plq m v 5 FOr L FDrplq 1 Or · · · 9 9 FDr plq Or · · · FDr Or FOr K1 = 9 9 .. .. .. .. 7 . . . . v plq 1 plq FDrv1 Or · · · FDv O FD Or r r r

0 L ..

. ···

(5.94) 6 ··· 0 .: .. . .. : : : .. . 08 FOr L

D0 is deﬁned in (5.41) Theorem 5.11 Assume that v plq and let ¢ ¡ udqn FDmr = pm > Qm FDmr = 0> m = plq > · · · > v Then the base matrix of parity space Sv , denoted by Tedvh>v , can be described by ¯ edvh>v K1 > T ¯ edvh>v = gldj(Qplq > · · · > Qpd{ 1 > Qpd{ > · · · > Qv ) Tedvh>v = T Qpd{ = Qpd{ +1 = · · · = Qv = Lp×p (5.95) and the dimension of parity space y¯v is given by glp(¯ yv ) =

v X

(p pl )>

for plq v ? pd{

l=plq

= p × (v pd{ + 1) +

pd{ X1

(p pl )> for v pd{ = (5.96)

l=plq

Theorem 5.10 gives another way to write the parity vectors deﬁned by (5.87). It shows that all parity vectors yv can be characterized by vectors y¯m > m = plq > · · · > v, which belong to the subspaces Tm deﬁned by (5.94). In other words: the selection of parity vectors only depends on the solution of equations y¯m FDmr = 0> plq m v. Theorem 5.11 provides us with an explicit expression for the base matrix of parity space y¯v and shows that the degree of freedom for the selection of a parity vector is the sum of the dimensions of subspaces Tm > m = plq > · · · > v. The results presented in Theorems 5.9-5.11 have not only answered the questions concerning the structure of the parity space but also shown an intimate relationships between the observer-based and the parity relation based approaches.

102

5 Basic residual generation methods

5.6.3 Examples Example 5.6 Consider nominal system model |(s) =

eq sq + eq1 sq1 + · · · e1 s + e0 x(s)= sq + dq1 sq1 + · · · + d1 s + d0

(5.97)

A trivial way to construct a parity space based residual generator for (5.97) is (a) to rewrite the system into its minimum state space realization form and (b) to solve (5.87) for yv design the residual generator and ﬁnally (c) to construct the residual generator according to (5.88). On the other side, it follows from Cayley-Hamilton Theorem that 6 5 f : £ ¤9 9 fD : Dq + dq1 Dq1 + · · · + d1 D + d0 D = 0 =, d0 · · · dq1 1 9 . : = 0 7 .. 8 fDq where D> f denote the system matrices of the minimum state space realization of J|x (s)= That means £ ¤ (5.98) yv = d0 · · · dq1 1 is a parity space vector of system (5.97). To construct the residual generator based on yv given by (5.98), (5.88) is used, which yields u(n) = yv |v (n) yv Kx>v xv (n) = |(n) + · · · + d1 |(n v + 1) + d0 |(n v) yv Kx>v xv (n)= It follows from (5.97) that yv Kx>v should satisfy ¤ £ yv Kx>v = e0 · · · eq1 eq = As a result, the residual generator is given by £ ¤ £ ¤ u(n) = d0 · · · dq1 1 |v (n) e0 · · · eq1 eq xv (n)=

(5.99)

It is interesting to note that residual generator (5.99) can be directly derived from the nominal transfer function without a state space realization. In fact, (5.99) can be instinctively achieved by moving the characteristic polynomial sq + dq1 sq1 + · · · + d1 s + d0 to the left side of equation (5.97). Study on this example will be continued in the next section, which will show an interesting application of this result. Example 5.7 We now design a PRRG for the inverted pendulum system LIP100. For our purpose, we set v = 4 and compute a parity vector using matrices D and F given in discrete time model (3.58), which leads to ¤ £ £ ¤ yv = yv>0 yv>1 yv>2 yv>3 yv>4 > yv>0 = 0=0643 0=0756 0=1674 £ ¤ £ ¤ yv>1 = 0=1418 0=0841 0=0391 > yv>2 = 0=0700 0=0440 0=0425 £ ¤ £ ¤ yv>3 = 0=0686 0=0030 0=0467 > yv4 = 0=0674 0=0397 0=0518 =

5.7 Interconnections, comparison and some remarks

103

5.7 Interconnections, comparison and some remarks In the early 90’s, study on interconnections and comparison among the residual generation approaches has increasingly received attention. In this section, we focus our study on the interconnections between the design parameters as well as the comparison of dynamics of the residual generator schemes presented in the last sections. We shall also make some remarks on the implementation and design forms of these residual generation approaches. 5.7.1 Parity space approach and diagnostic observer We ﬁrst study the interconnections between the design parameters of the parity space and diagnostic observer approaches, i.e. interconnections between O> W> K> t> y> z and parity vector yv = The following two theorems give an explicit expression for these connections. system model (5.80)-(5.81) and a parity vector yv = £Theorem 5.12 Given ¤ yv>0 yv>1 · · · yv>v , then matrices O> W> K> t> y> z deﬁned by 6 5 F 5 69 : yv>1 yv>2 · · · yv>v1 yv>v 9 FD : .. : 9 yv>2 · · · · · · yv>v 0 : 9 : 9 :9 9 . : W =9 . (5.100) : . . 9 . .. .. 8 9 . : 7 .. · · · · · · : . : 9 0 7 FDv2 8 yv>v 0 · · · · · · FDv1 6 5 6 5 G yv>0 + j1 yv>v yv>1 yv>2 · · · yv>v1 yv>v 9 : : 9 FE : ¸ 9 9 yv>1 + j2 yv>v yv>2 · · · · · · yv>v 0 : 9 FDE : K : 9 .. .. .. .. : 9 =9 : (5.101) 9 .. . . ··· ··· . . : t :9 : 9 . : 7 yv>v1 + jv yv>v yv>v 0 · · · 0 0 89 7 FDv2 E 8 0 0 ··· 0 0 yv>v FDv1 E 6 5 yv>0 9 yv>1 : £ ¤ : 9 (5.102) O = 9 . : jyv>v > z = 0 · · · 0 1 > y = yv>v . 7 . 8 yv>v1 solve the Luenberger equations (5.30)-(5.31), (5.38), where matrix J is given in the form of (5.39) with j ensuring the stability of matrix J. Theorem 5.13 Given system model (5.80)-(5.81) and observer-based residual generator (5.36)-(5.37) with matrices O> W> y> z solving the Luenberger equations (5.30)-(5.31), (5.38) and J satisfying (5.39), then vector yv = £ ¤ yv>0 yv>1 · · · yv>v with

104

5 Basic residual generation methods

5 9 9 yv>v = y> 9 7

yv>0 yv>1 .. .

6 : : : = O jy 8

yv>v1 belongs to the parity space Sv . These two theorems are in fact a reformulation of Theorem 5.7 and the proof is thus omitted. It is interesting to notice the relationship between yv Kx>v and K> t as deﬁned in (5.88) and in (5.36)-(5.37) respectively. Suppose that j = 0> then 6 5 G 0 ··· 0 9 . : . ¤ 9 FE ¤ £ £ G . . .. : : := ky>0 ky>2 · · · ky>v 9 yv Kx>v = yv>0 yv>1 · · · yv>v 9 : .. . . .. .. 0 8 7 . v1 FD E · · · FE G 65 6 5 6 5 G yv>0 yv>2 · · · yv>v ky>0 ¸ ¸ 9y ··· y : 9 FE : 9 ky>2 : v>v 0 : 9 K W E OG : : 9 9 v>1 = =9 . :9 : = 9 .. : = . .. . . t yv>v G 8 7 . 8 7 . ··· ··· . 87 . ky>v yv>v 0 · · · 0 FDv1 E (5.103) That means we can determine K> t> as far as yv Kx>v is known, by just rearranging row vector yv Kx>v into a column vector without any additional computation, and vice versa. Theorems 5.12 - 5.13 reveal an one-to-one mapping between the design parameters of observer and parity relation based residual generators. While Theorem 5.12 implies that for a given parity relation based residual generator there exists a set of corresponding observer-based residual generators with j being a parameter vector, Theorem 5.13 shows how to calculate the corresponding parity vector when an observer-based residual generator is provided. Now, questions may arise: Is there a dierence between the residuals delivered respectively by a diagnostic observer and its corresponding parity relation based residual generator? Under which conditions can we get two identical residuals delivered respectively by these two kinds of residual generators? To answer these questions, we bring the diagnostic observer }(n + 1) = J}(n) + Kx(n) + O|(n)> u(n) = y|(n) z}(n) tx(n) into a similar form like the parity relation based residual generator given by (5.88) u(n) = y|(n) + tx(n) z}(n) = y|(n) + tx(n) z (J}(n 1) + Kx(n 1) + O|(n 1)) = y|(n) + tx(n) zJv }(n v) zKx(n 1) · · · zJv1 Kx(n v) zO|(n 1) · · · zJv1 O|(n v)=

5.7 Interconnections, comparison and some remarks

105

Recalling (5.71)-(5.72) in Theorem 5.7 and noting that £ ¤ zJl = 0 · · · 0 1 zj · · · zJl1 j it turns out 35 9 £ ¤E E9 zJl O = 0 · · · 0 1 zj · · · zJl1 j E9 C7 5

yv>0 yv>1 .. . yv>v1 0 .. .

6

4

: F : F : + jyv>v F 8 D 6

9 : 9 : 9 : 9 : 0 9 : : ¤9 £ L p×p 9 : = yv>0 yv>1 · · · yv>v1 yv>v 9 : zjL p×p 9 : 9 : . .. 9 : 9 : 7 zJl1 jLp×p 8 zJl jLp×p £ ¤ zJl K = zJl (W E OG) = 0 · · · 0 1 zj · · · zJl1 j 6 5 4 3 F 69 6 5 : F E5 yv>0 F E yv>1 yv>2 · · · yv>v1 yv>v 9 FD : 9 .. : F E9 : : 9 y F E9 yv>2 · · · · · · yv>v 0 : 9 . : v>1 : 9 : 9 F E9 . G E + G + jy : : 9 .. .. .. v>v F : 9 E7 . . 7 . 8 . . 8 9 .. : F E . ··· ··· : 9 F E yv>v1 0 7 FDv2 8 D C yv>v 0 · · · · · · FDv1 5

6

9 : : 69 : 0 9 9 : 0 9 : .. : 9 : 9 : ¤ 9 FE . :9 Lnx ×nx : yv>v 9 : 9 zjLnx ×nx : 9 .. : 7 0 89 . 9 : .. : FDv1 E · · · FE G 9 . 9 : 7 zJl1 jLn ×n 8 x x zJl jLnx ×nx 5

£ = yv>0 yv>1 · · · yv>v1

0 .. .

G

0 ··· . G .. .. .. . .

which ﬁnally results in ¡ ¢ u(n) = zJv }(n v) + yv L¯|v |v (n) Kr>v L¯xv xv (n) where

(5.104)

106

5 Basic residual generation methods

5 9 9 L¯|v = 9 9 7 5 9 9 L¯xv = 9 9 7

Lp×p zjLp×p .. . zJv1 jLp×p Lnx ×nx zjLnx ×nx .. . zJv1 jLnx ×nx

6 ··· 0 .. : .. . . : : : .. .. . . 0 8 · · · zjLp×p Lp×p 0 .. .

6 ··· 0 .. : .. . . : := : .. .. . . 0 8 · · · zjLnx ×nx Lnx ×nx 0 .. .

Comparing (5.104) with (5.88) evidently shows the dierences between these two types of residual generators: • in against to the parity relation based residual generators, the diagnostic observer does not possess the s-step dead-beat property, i.e. the residual u(n) depends on }(n v)> · · · > }(0)> if j 6= 0 • the construction of the diagnostic observer depends on the selection j, and in this sense, we can also say that the diagnostic observer possesses more degree of design freedom. On the other side, setting j = 0 leads to zJv = 0> L¯|v = Lp(v+1)×p(v+1) > L¯xv = Lnx (v+1)×nx (v+1) = Thus, under condition j = 0 the both types of residual generators are identical. It is interesting to note that in this case 6 5 yv>0 : ¸ 9 9 yv>1 : O : 9 . .. =9 (5.105) := y : 9 7 yv>v1 8 yv>v Remember that a residual signal is originally deﬁned as the dierence between the measurement or a combination of the measurements and its estimation. This can, however, not directly recognized from the deﬁnition of the parity relation based residual signal, (5.88). The above comparison study reveals that u(n) = yv (|v (n) Kx>v xv (n)) can be equivalently written as }(n + 1) = J}(n) + Kx(n) + O|(n)> u(n) = y|(n) z}(n) tx(n)= It is straightforward to demonstrate that z}(n) + tx(n) is in fact an estimate for y|(n)= Thus, a parity relation based residual signal can also be interpreted as a comparison between y|(n) = yv>v |(n) and its estimation.

5.7 Interconnections, comparison and some remarks

107

5.7.2 Diagnostic observer and residual generator of general form Our next task is to ﬁnd out the relationships between the design parameters of the diagnostic observer and the ones given by the general residual generator ³ ´ ˆ x (s)|(s) Q ˆx (s)x(s) u(s) = U(s) P (5.106) whose design parameters are observer matrix O and post-ﬁlter U(s). We study two cases: v q and v A q. Firstly v q: We only need to demonstrate that for v ? q the diagnostic observer (5.36)(5.37) satisfying (5.30)-(5.31), (5.38) can be equivalently written into form (5.106). Let us deﬁne ¸ W 5 Rq×q > W1 5 R(qv)×q > udqn(W ) = q (5.107) W = W1 ¸ J 0 W1 D J1 W1 = O1 F> J1 5 R(qv)×(qv) is stable> J = (5.108) 0 J1 ¸ £ ¤ O K = W E O G 5 Rq×nx > O = > Z = Z 0 5 Rp×q (5.109) O1 and extend (5.31) and (5.38) as follows W D JW = OF =, W D J W = O F Y F Z W = 0 =, Y F Z W = 0 K = W E OG =, K = W E O G=

(5.110) (5.111) (5.112)

Note that choosing, for instance, W1 as a composite of the eigenvectors of DOr F and O1 = W1 Or guarantees the existence of (5.108), where Or denotes some matrix that ensures the stability of matrix D Or F. Since Z (sL J)1 (Kx(s) + O|(s)) = Z (sL J )1 (K x(s) + O |(s)) = Z W (sL D + W 1 O F)1 W 1 (K x(s) + O |(s)) ´ ³ = Y F(sL D + W 1 O F)1 (E W 1 O G)x(s) + W 1 O |(s) the residual generator u(s) = Y |(s) + Tx(s) Z (sL J)1 (Kx(s) + O|(s)) can be equivalently written as ³ ´ ˆ x (s)|(s) Q ˆx (s)x(s) u(s) = Y P with ˆ x (s) = L F(sL D + W 1 O F)1 W 1 O P ˆx (s) = G + F(sL D + W 1 O F)1 (E W 1 O G)= Q We thus have the following theorem.

(5.113) (5.114)

108

5 Basic residual generation methods

Theorem 5.14 Every diagnostic observer (5.36)-(5.37) of order v ? q can be considered as a composite of a fault detection ﬁlter and post-ﬁlter Y . Remark 5.8 Theorem 5.14 implies that the performance of any diagnostic observer (5.36)-(5.37) of order v ? q can be reached by an FDF together with an algebraic post-ﬁlter. Now v A q: We ﬁrst demonstrate that for v A q the diagnostic observer (5.36)-(5.37) satisfying (5.30)-(5.31), (5.38) can be equivalently written into form (5.106). To this end, we introduce following matrices £ ¤ (5.115) W = Wr W 5 Rv×v > Wr 5 Rv×(vq) > udqn(W ) = v ¸ Du 0 (5.116) Wr Du JWr = 0> Du 5 R(vq)×(vq) is stable> D = 0 D ¸ £ ¤ 0 5 Rv×nx > F = 0 F 5 Rp×v E = (5.117) E ¸ O1 O = W 1 O = > O1 5 R(vq)×p > O2 5 Rq×p (5.118) O2 and extend (5.31) and (5.38) to W D JW = OF =, W D W 1 J = OF W 1 £ ¤ Y F Z W = 0 =, Y F Z W + Z Wr 0 = 0 K = W E OG =, K = W E OG=

(5.119) (5.120) (5.121)

Since J is stable, there does exist Wr satisfying (5.116). Applying (5.115)(5.118) to the diagnostic observer u(s) = Y |(s) + Tx(s) Z (sL J)1 (Kx(s) + O|(s)) results in Z (sL J)1 (Kx(s) + O|(s)) = Z W (sL D + O F )1 W 1 (Kx(s) + O|(s)) £ ¤ = Z Wr Y F (sL D + O F )1 ((E O G)x(s) + O |(s)) and furthermore ¢ ¡ u(s) = Y L F(sW D + O2 F)1 O2 |(s) ¡ ¢ Z Wr (sL Du )1 O1 L F(sL D + O2 F)1 O2 |(s) ¡ ¢ +Z Wr (sL Du )1 O1 G + F(sL D + O2 F)1 (E O2 G) x(s) ¢ ¡ (5.122) Y F(sW D + O2 F)1 (E O2 G) + G x(s) which, by setting

5.7 Interconnections, comparison and some remarks

ˆ x (s) = L F(sL D + O2 F)1 O2 P ˆx (s) = G + F(sL D + O2 F)1 (E O2 G) Q U(s) = Y Z Wr (sL Du )1 O1

109

(5.123) (5.124) (5.125)

ﬁnally gives u(s) = Y |(s) + Tx(s) Z (sL J)1 (Kx(s) + O|(s)) ³ ´ ˆ x (s)|(s) Q ˆx (s)x(s) = = U(s) P

(5.126)

We see that for v A q the diagnostic observer (5.36)-(5.37) can be equivalently written into form (5.106), in which the post-ﬁlter is a dynamic system. Solve equation ¸ ¸ ¤ L(vq)×(vq) 0 Wr £ W W = r W 0 Lq×q for Wr > Wr > W , then we obtain O = Wr O1 + W O2 =, O1 = Wr O> O2 = W O= The following theorem is thus proven. Theorem 5.15 Given diagnostic observer (5.36)-(5.37) of order v A q with J> O> W> Y> Z solving the Luenberger equations (5.30)-(5.31) and (5.38), then it can be equivalently written into ´ ³ ˆx (s)x(s) ˆ x (s)|(s) Q (5.127) u(s) = U(s) P ˆ x (s) = L F(sL D + W OF)1 W O P ˆx (s) = G + F(sL D + W OF)1 (E W OG) Q U(s) = Y Z Wr (sL Du )1 Wr O=

(5.128) (5.129) (5.130)

We are now going to show that for a given residual generator of form (5.106) we are able to ﬁnd a corresponding diagnostic observer (5.36)-(5.37). For this purpose, we denote the state space realization of U(s) with Gu + Fu (sL Du )1 Eu . Since μ ¸¶1 ¸ ¤ £ DO 0 O ˆ sL Fu (sL Du ) Eu Px (s) = 0 Fu Eu Eu F Du μ ¸¶1 ¸ ¤ £ DO 0 EO ˆx (s) = 0 Fu sL Fu (sL Du )1 Eu Q Eu G Eu F Du 1 ˆ ˆ Gu Px (s) = Gu Gu F(sL DO ) O> Gu Qx (s) = Gu G + Gu F(sL DO )1 EO 1

it is reasonable to deﬁne

110

5 Basic residual generation methods

¸ ¸ ¤ £ O D OF 0 ¯ >O = > Z = Gu F Fu J= Eu Eu F Du ¸ E OG Y = Gu > K = > T = Gu G= Eu G

Note that

¸ ¸ ¸ L L ¯ Z L =YF DJ = OF> 0 0 0 ¸ L ¯ T = Y G K= E OG> 0

(5.131) (5.132)

(5.133) (5.134)

ensure that residual generator ¡ ¢ ¯ u(s) = Y |(s) + Tx(s) Z (sL J)1 Kx(s) + O|(s) satisﬁes Luenberger conditions (5.30)-(5.31), (5.38). The discussion on the possible applications of the interconnections revealed in this subsection will be continued in the next subsections. 5.7.3 Applications of the interconnections and some remarks In literature, parity relation based residual generators are often called openloop structured, while the observer-based residual generators closed-loop structured. This view may cause some confusion, since, as known in the control theory, closed-loop and open loop structured systems have dierent dynamic behavior. The discussion carried out above, however, reveals that this is not the case for the parity relation and observer-based residual generators: They have the identical dynamics (under the condition that the eigenvalues are zero), also regarding to the unknown inputs and faults, as will be shown later. A further result achieved by the above study indicates that the selection of a parity space vector is equivalent with the selection of the observer matrix, the feedback matrix (i.e. feedback of system output |) of an s-step deadbeat observer. In other words, all design approaches for the parity relation based residual generation can be used for designing observer-based residual generators, and vice-versa. What is then the prime dierence between the parity relation based and the observer-based residual generators? The answer can be found by taking a look at the implementation forms of the both types of residual generators: the implementation of the parity relation based residual generator uses a non-recursive form, while the observer-based residual generator represents a recursive form. A similar fact can also be observed by the observer-based approaches. Under certain conditions the design parameters of a residual generator can be equivalently converted to the ones of another type of residual generator, also the same performance can be reached by dierent residual generators.

5.7 Interconnections, comparison and some remarks

111

This observation makes it clear that designing a residual generator can be carried out independent of the implementation form adopted later. We can use, for instance, parity space approach for the residual generator design, then transform the parameters achieved to the parameters needed for the construction of a diagnostic observer and ﬁnally realize the diagnostic observer for the on-line implementation. The decision for a certain type of design form and implementation form should be made on account of • the requirements on the on-line implementation, • which approach can be readily used to design a residual generator that fulﬁlls the performance requirements on the FDI system, • and of course, in many practical cases, the available design tools and designer’s knowledge of design approaches. Recall that parity space based system design is characterized by its simple mathematical handling. It only deals with matrix- and vector-valued operations. This fact attracts attention from industry for the application of parity space based methods. Moreover, the one-to-one mapping between the parity space approach and the observer-based approach described in Theorems 5.12 and 5.13 allows an observer-based residual generator construction for a given a parity vector. Based on this result, a strategy called parity space design, observer-based implementation has been developed, which makes use of the computational advantage of parity space approaches for the system design (selection of a parity vector or matrix) and then realizes the solution in the observer form to ensure a numerically stable and less consuming on-line computation. This strategy has been for instance successfully used in the sensor fault detection in vehicles and highly evaluated by engineers in industry. It is worth mentioning that the strategy of parity space design, observer-based implementation can also be applied to continuous time systems. Table 5.1 summarizes some of important properties of the residual generators described in this section, which may be useful for the decision on the selection of design and implementation forms. In this table, • "solution form" implies the required knowledge and methods for solving the related design problems. LTI stands for the needed knowledge of linear system theory, while algebra means for the solution only algebraic computation, in most cases solution of linear equations, is needed. • "dynamics" is referred to the dynamics of LTI residual generator (5.24). OEE + y implies a composite of output estimation error and an algebraic post-ﬁlter, OEE + U(s) a composite of output estimation error and a dynamic post-ﬁlter.

112

5 Basic residual generation methods Table 5.1 Comparison of dierent residual generation schemes Type Order Design parameters

FDF v=q

v$q

y> O

yv

Design freedom

y> O

Solution form

LTI

PRRG vAq

DO v$q

vAq

yv

J> O> y> z

J> O> y> z

yv

yv

yv > J

yv > J

algebra

algebra

LTI or algebra

LTI or algebra

Implement. form recursive non-recurs. non-recurs. recursive

recursive

Dynamics OEE + y OEE + y OEE + U(s) OEE + y OEE + U(s)

5.7.4 Examples Example 5.8 We now extend the results achieved in Example 5.6 to the construction of an observer-based residual generator. Suppose that (5.97) is a discrete time system. It follows from Theorem 5.12 and (5.103) that }(n + 1) = J}(n) + Kx(n) + O|(n)> u(n) = y|(n) z}(n) tx(n) with 5

0 91 9 J=9. 7 ..

0 0 .. .

··· ··· .. .

6 5 0 : 0: 9 .. : > K = W E OG = 7 .8

0 ··· 1 0

£ ¤ t = eq > y = 1> z = 0 · · · 0 1

5 6 6 d0 e0 9 : .. : > O = 9 d1 : 9 : 8 . . 7 .. 8 eq1 dq1

builds a residual generator. If we are interesting in achieving a residual residual generator whose dynamics is governed by F(s) = sq jq1 sq1 · · · j1 s j0 then the observer gain matrix O should be extended to 6 5 6 5 j0 d0 : 9 : 9 O = 7 ... 8 7 ... 8 dq1

jq1

5.7 Interconnections, comparison and some remarks

113

Note that in this case the above achieved results can also be used for continuous time systems. In summary, we have some interesting conclusions: • given a transfer function, we are able to design a parity space based residual generator without any involved computation and knowledge of state space realization • the designed residual generator can be extended to the observer-based one. Once again, no involved computation is needed for this purpose • the observer-based form can be applied both for discrete and continuous time systems. We would like to mention that the above achieved results can also be extended to MIMO systems. Example 5.9 We now apply the above result to the residual generator design for our benchmark DC motor DR300 given in Subsection 3.7.1. It follows from (3.50) that e0 + d2 + d1 s2 + d0 d2 = 234=0136> d1 = 6857=1> d0 = 5442=2> e0 = 47619 J|x (s) =

which yields

s3

s2

£ ¤ yv = d0 d1 d2 1 =

(5.135)

Now, we design an observer-based residual generator of the form }˙ = J} + Kx + O|> u = y| z} tx

(5.136)

without the knowledge of the state space representation of the system. To this end, using Theorem 5.12 and (5.103) with yv given in (5.135) results in 6 5 6 5 6 5 6 5 e0 j1 d0 0 0 j1 J = 7 1 0 j2 8 > K = 7 0 8 > O = 7 d1 8 7 j2 8 0 1 j3 d2 j3 0 £ ¤ t = 0> y = 1> z = 0 0 1 = To ensure a good dynamic behavior, the eigenvalues of matrix J are set to be 10> 10> 10> which leads to 5 6 5 6 j1 1000 7 j2 8 = 7 300 8 j3 30 and further

5

6 4442=2 O = 7 6557=1 8 = 204

114

5 Basic residual generation methods

5.8 Notes and references The general form and parameterization of all LTI stable residual generators were ﬁrst derived by Ding and Frank [38]. The FDF scheme was proposed by Beard [11] and Jones [86]. These works are recognized as marking the beginning of the model-based FDI theory and technique. Both FDF and DO techniques have been developed on the basis of linear observer theory, to which O’Reilly’s book [111] gives an excellent introduction. Only few references concerned characterization of DO and parity space approaches can be found in the literature. For this reason, an extensive and systematic study on this topic has been included in this chapter. The most signiﬁcant results are • the necessary and su!cient condition for solving Luenberger equations (5.30)-(5.31), (5.38) and its expression in terms of the solution of parity equation (5.87) • the one-to-one mapping between the parity space and the solutions of the Luenberger equations • the minimum order of diagnostic observers and parity vectors and • the characterization of the solutions of the Luenberger equations and the parity space. Some of these results are achieved based on the works by Ding et al. [28] (on DO) and [44] (on the parity space approach). They will also be used in the forthcoming chapters. The original versions of numerical approaches proposed by Ge and Fang as well as Ding at al. have been published in [61] and [28], respectively. Accompanied with the establishment of the framework of the model-based fault detection approaches, comparison among dierent model-based residual generation schemes has increasingly received attention. Most of studies have been devoted to the interconnections between FDF, DO on the one side and parity space approaches on the other side, see for instance, the signiﬁcant work by Wuenneberg [148]. Only a few of them have been dedicated to the comparison between DO and factorization or frequency approach. A part of the results described in the last section of this chapter was achieved by Ding and his co-worker [42]. An interesting application of the comparison study is the strategy of parity space design, observer-based implementation, which can be applied both for discrete and continuous time systems and allows an easy design of observerbased residual generators. In [131], an application of this strategy in practice has been reported. It is worth emphasizing that this strategy also enables an observer-based residual generator design based on the system transfer function, instead of the state space representation, as demonstrated in Example 5.9.

6 Perfect unknown input decoupling

In this chapter, we address the problems of generating residual signals which are decoupled from the disturbances (unknown inputs). That means the generated residual signals will only be inﬂuenced by the faults. In this sense, such a residual generator also acts as a fault indicator. It is often called unknown input residual generator. Fig.6.1 sketches the objective of this chapter schematically.

Fig. 6.1 Schematical description of unknown input decoupled residual generation

6.1 Problem formulation Consider system model (3.29) and its minimal state space realization (3.30)(3.31). It is straightforward that applying a residual generator of the general form (5.24) to (3.29) yields ˆ x (s) (J|g (s)g(s) + J|i (s)i (s)) = u(s) = U(s)P

(6.1)

116

6 Perfect unknown input decoupling

Remember that for the state space realization (3.30)-(3.31), residual generator (5.24) can be realized as a composition of a state observer and a post-ﬁlter, { ˆ˙ = Dˆ { + Ex + O (| F { ˆ Gx) > u(s) = U(s) (|(s) F { ˆ(s) Gx(s)) = It turns out, by setting h = { { ˆ> h˙ = (D OF) h + (Hg OIg )g + (Hi OIi )i u(s) = U(s) (Fh(s) + Ig g(s) + Ii i (s)) which can be rewritten into, by noting Lemma 3.1, ³ ´ ˆi (s)i (s) + Q ˆg (s)g(s) u(s) = U(s) Q

(6.2)

1

ˆi (s) = F (sL D + OF) (Hi OIi ) + Ii Q ˆg (s) = F (sL D + OF)1 (Hg OIg ) + Ig Q ˆ 1 (s)Q ˆi (s) and J|g (s) = P ˆ 1 (s)Q ˆg (s)= It is with an LCF of J|i (s) = P i g interesting to notice that ˆ g (s) = P ˆ i (s) = L F (sL D + OF)1 O= ˆ x (s) = P P Hence, we assume in our subsequent study, without loss of generality, that ˆ x (s)J|i (s) 5 RH4 = ˆ x (s)J|g (s)> P P For the fault detection purpose, an ideal residual generation would be a residual signal that only depends on the faults and is simultaneously independent of disturbances. It follows from (6.1) that this is the case for all possible disturbances and faults if and only if ˆ x (s)J|g (s) = 0= ˆ x (s)J|i (s) 6= 0 and U(s)P U(s)P

(6.3)

Finding a residual generator which satisﬁes condition (6.3) is one of the mostly studied topics in the FDI area and is known as, among a number of expressions, perfect unknown input decoupling. Deﬁnition 6.1 Given system (3.29). Residual generator (5.24) is called perfectly decoupled from the unknown input g if condition (6.3) is satisﬁed. The design of such a residual generator is called . In the following of this chapter, we shall approach PUIDP. Our main tasks consist in • • • • •

the study on the solvability of (6.3), presentation of a frequency domain approach to PUIDP design of unknown input fault detection ﬁlter (UIFDF) design of unknown input diagnostic observer (UIDO) and design of unknown input parity relation based residual generator.

6.2 Existence conditions of PUIDP

117

6.2 Existence conditions of PUIDP In this section, we study • under which conditions (6.3) is solvable and • how to check those existence conditions. 6.2.1 A general existence condition We begin with a reformulation of (6.3) as £ ¤ £ ¤ ˆ x (s) J|i (s) J|g (s) = 0 U(s)P

(6.4)

with 6= 0 as some RH4 transfer matrix. Since ³ ´ ˆ x (s) = p udqn P and U(s) is arbitrarily selectable in RH4 > the following theorem is obvious. Theorem 6.1 Given system (3.29), then there exists a residual generator ³ ´ ˆ x (s)|(s) Q ˆx (s)x(s) u(s) = U(s) P such that (6.3) holds if and only if ¤ £ udqn J|i (s) J|g (s) A udqn (J|g (s)) =

(6.5)

Proof. If (6.5) holds, then there exists a U(s) such that ˆ x (s)J|i (s) 6= 0 ˆ x (s)J|g (s) = 0 and U(s)P U(s)P This proves the su!ciency. Suppose that (6.5) does not hold, i.e. ¤ £ udqn J|i (s) J|g (s) = udqn (J|g (s)) = ˆ x (s) one can always ﬁnd a transfer matrix As a result, for all possible U(s)P W (s) such that ˆ x (s)J|g (s)W (s) ˆ x (s)J|i (s) = U(s)P U(s)P ˆ x (s)J|g (s) = 0 would lead to Thus, U(s)P ˆ x (s)J|i (s) = 0 U(s)P i.e. (6.3) can never be satisﬁed. This proves that condition (6.5) is necessary for (6.3). u t

118

6 Perfect unknown input decoupling

The geometric interpretation of (6.5) is that the subspace spanned by J|i (s) is dierent from the subspace spanned by J|g (s)> i.e. Lp (J|i (s)) 6 Lp (J|g (s)) = Note that

£ ¤ udqn J|i (s) J|g (s) p

(6.5) also means udqn (J|g (s)) ? p= In other words, the subspace spanned by J|g (s) should be smaller than the pdimensional measurement space. From the viewpoint of system structure, this can be understood as: the number of the unknown inputs that have inﬂuence on |(s) (output controllability) or equivalently that are observable from |(s) (input observability) should be smaller than the number of sensors. For the purpose of residual generation, those unknown inputs that have no inﬂuence on the measurements and those measurements that are decoupled from the faults are of no interest. Bering it in mind, below we continue our study on the assumption £ ¤ (6.6) ng ? p> udqn J|i (s) J|g (s) = p= Although (6.5) sounds compact, simple and has a logic physical interpretation, its check, due to the rank computation of the involved transfer matrices, may become di!cult. This motivates the derivation of alternative check conditions which are equivalent to (6.5) but may require less special mathematical computation or knowledge. Example 6.1 Consider the inverted pendulum system LIP100 described in Subsection 3.7.2. Suppose that we are interested in achieving a perfect decoupling from the friction g. It is easy to ﬁnd out ¤ £ udqn J|i (v) J|g (v) = 3 A udqn (J|g (v)) = 1= Thus, following Theorem (6.1), for this system the PUIDP is solvable. 6.2.2 A check condition via Rosenbrock system matrix We now consider minimal state space realization (3.30)-(3.31), i.e. J|i (s) = (D> Hi > F> Ii )> J|g (s) = (D> Hg > F> Ig ). Let us do the following calculation ¸ ¸ D sL Hg (sL D)1 (sL D)1 Hg F Ig 0 Lng ×ng ¸ Lq×q 0 = F(sL D)1 F(sL D)1 Hg + Ig 5 6 ¸ (sL D)1 (sL D)1 Hi (sL D)1 Hg D sL Hi Hg 7 8 0 Lni ×ni 0 F Ii Ig 0 0 Lng ×ng ¸ Lq×q 0 0 = F(sL D)1 F(sL D)1 Hi + Ii F(sL D)1 Hg + Ig

6.2 Existence conditions of PUIDP

119

from which we immediately know ¸ ¸ Lq×q R D sL Hg = udqn udqn F Ig F(sL D)1 J|g (s) ¢ ¡ = q + udqn F(sL D)1 Hg + Ig ¸ ¸ Lq×q D sL Hi Hg 0 0 = udqn udqn F Ii Ig F(sL D)1 J|i (s) J|g (s) ¤ £ = q + udqn J|i (s) J|g (s) =

(6.7)

(6.8)

Thus, we have Theorem 6.2 Given J|i (s) = F(sL D)1 Hi + Ii and J|g (s) = F(sL D)1 Hg + Ig , then (6.3) holds if and only if ¸ ¸ D sL Hi Hg D sL Hg ? udqn q + p= (6.9) udqn F Ig F Ii Ig Given a transfer function matrix J(s) = F(sL D)1 + G, matrix ¸ D sL E F G is called Rosenbrock system matrix of J(s). Due to the importance of the concept Rosenbrock system matrix in linear system theory, there exist a number of algorithms and CAD programs for the computation related to properties of a Rosenbrock system matrix. This is in fact one of the advantages of check condition (6.9) over the one given by (6.5). Nevertheless, keep it in mind that a computation with operator s is still needed. A check of existence condition (6.9) can be carried out following the algorithm given below. Algorithm 6.1 Solvability check of PUIDP via Rosenbrock system matrix Step 1. Calculate udqn

¸ ¸ D sL Hg D sL Hi Hg > udqn F Ii Ig F Ig

Step 2. Prove (6.9). If it holds, the PUIDP is solvable, otherwise unsolvable. Example 6.2 Given benchmark system EHSA with model (3.83) and suppose that the model uncertainty is zero and three additive faults (two sensor and one actuator faults) are considered. We now check the solvability of the PUIDP. To this end, Algorithm 6.1 is applied, which leads to Step 1:

udqn

¸ ¸ D sL Hg D sL Hi Hg = 7> udqn =6 F Ii Ig F Ig

120

6 Perfect unknown input decoupling

Step 2:

D sL Hi Hg udqn F Ii Ig

¸

¸ D sL Hg A udqn = F Ig

Thus, the PUIDP is solvable. 6.2.3 Algebraic check conditions As mentioned in the last chapter, the parity space approach provides us with a design form of residual generators, which is expressed in terms of an algebraic equation, u(n) = yv (Ki>v iv (n) + Kg>v gv (n)) > yv 5 Sv (6.10) From (6.10) we immediately see that a parity relation based residual generator delivers a residual decoupled from the unknown input gv (n) if and only if there exists a parity vector such that yv Ki>v 6= 0 and yv Kg>v = 0=

(6.11)

Taking into account the deﬁnition of parity vectors, (6.11) can be equivalently rewritten into £ ¤ £ ¤ yv Ki>v Kr>v Kg>v = 0 0 with vector 6= 0, from which it becomes clear that residual u(n) is decoupled from gv (n) if and only if ¤ £ ¤ £ (6.12) udqn Ki>v Kr>v Kg>v A udqn Kr>v Kg>v = Comparing (6.12) with (6.9) or (6.5) evidently shows the major advantage of check condition (6.12), namely the needed computation only concerns determining matrix rank, which can be done using a standard mathematical program. Another advantage of using (6.12) consists in the possibility to get the knowledge of whether a residual generator of order v can deliver a residual decoupled from the unknown inputs. On the other side, the reader may ask: Should I prove condition (6.12) for all possible v in order to ensure whether the decoupling problem is solvable? The following result gives an answer to this question. It follows from the parameterization of parity vectors presented in Theorem 5.10 and equality (5.64) given in Lemma 5.2 that yv can be rewritten into yv = y¯K1>v > yv 5 Sv

(6.13)

¯ g>v > yv Ki>v = y¯K ¯ i>v > yv 5 Sv yv Kg>v = y¯K

(6.14)

and moreover where

6.2 Existence conditions of PUIDP

£

y¯ = y¯plq y¯plq +1 · · · y¯v1 y¯v

121

¤

y¯m 5 Tm > Tm = {t | tFDmr = 0}> plq m v 5 6 FOr L 0 ··· 0 FDrmin 1 Or · · · 9 : .. 9 FDr min Or · · · : . FDr Or FOr L : K1>v = 9 9 : .. .. .. .. 7 . . 08 . . v min FDv1 Or · · · · · · FOr L r Or · · · FDr p(v plq +1)×ng (v+1) ¯ ¯ > Ki>v 5 Rp(vplq +1)×ni (v+1) Kg>v 5 R 5

¯ g>v K

¯g ··· FDrplq 1 H 9 .. .. 9 . . 9 ¯g FDrpd{ 2 H ¯g 9 FDr pd{ 1 H =9 pd{ 1 ¯ 9 0 FD H g r 9 9 .. .. 7 . . 5

¯ i>v K

0

···

¯i ··· FDrplq 1 H 9 .. .. 9 . . 9 ¯i FDrpd{ 2 H ¯i 9 FDr pd{ 1 H =9 pd{ 1 ¯ 9 0 FDr Hi 9 9 .. .. 7 . . 0

···

0 ··· .. .. . . Ig 0 · · · Ig .. . . . . ¯g · · · · · · FH

¯g FH .. . ··· ··· .. .

Ig .. . ¯g FH ··· .. .

¯g FDrpd{ 1 H ¯i FH .. . ··· ··· .. . pd{ 1 ¯ FDr Hi

Ii .. . ¯ F Hi ··· .. .

6 0 .. : . : : ···: : 0 : : .. : . 8 Ig 6 0 .. : . : : ···: : 0 : : .. : . 8

··· .. . 0 Ii .. . ¯i Ii · · · · · · FH 0 .. . Ii ··· .. .

¯i = Hi Or Ii ¯g = Hg Or Ig > H H with Dr and Or as deﬁned in (5.41). We are now able to prove the following theorem. Theorem 6.3 Given v = pd{ + plq as well as ¤ £ ¯ g>v = p( pd{ + 1) = the row number of K ¯ r>v K ¯ g>v udqn K

(6.15)

then for v = pd{ + plq + 1 ¤ £ ¯ g>v = p( pd{ + 2) = the row number of K ¯ r>v K ¯ g>v = udqn K

(6.16)

Proof. Following Theorem 5.11 we rewrite y¯ as ¤ £ ¯ edvh>v > zv = zplq >v · · · zv>v 6= 0 y¯ = zv T ¯ edvh>v = gldj(Q > · · · > Q 1 > Q T plq pd{ pd{ > · · · > Qv ) l Ql FDr = 0> l = 1> · · · > pd{1 > Qpd{ = Qpd{ +1 = · · · = Qv = Lp×p = For v = pd{ + plq , (6.15) holds if and only if for all vector zv 6= 0

122

6 Perfect unknown input decoupling

¯ edvh>v K ¯ g>v 6= 0 zv T We now check the case v = pd{ + plq + 1. To this end, we write ¯ edvh> + +1 into ¯ g> + +1 > T K pd{ plq pd{ plq ¸ ¯ g>pd{ +plq +1 = Kg>pd{ +plq 0 K ˜ g> + +1 Ig K pd{ plq ¤ £ ¯g ¯g · · · F H ˜ Kg>pd{ +plq +1 = 0 · · · 0 FDr pd{1 H ¯ edvh>pd{ +plq +1 = gldj(T ¯ edvh>pd{ +plq > L) T respectively. It is important to notice that the rows of matrix ¤ £ ˜ g> + +1 Ig K pd{ plq is linearly independent of the rows of £ ¤ Kg>pd{ +plq 0 and so matrix

Kg>pd{ +plq 0 ˜ g>pd{ +plq +1 Ig K

¸

is of full row rank. Thus, for any £ ¤ zpd{ +plq +1 = zpd{ +plq z 6= 0 we have ¯ edvh>pd{ +plq +1 K ¯ g>pd{ +plq +1 = zpd{ +plq +1 T £ ¤ ¤ £ ¯ edvh>pd{ +plq Kg>pd{ +plq R + z K ˜ g>pd{ +plq +1 Ig 6= 0= zpd{ +plq T This implies (6.16) holds. The theorem is thus proven. u t It follows from Theorem 6.3 that v = pd{ + plq sets a up-bound for the check of condition £ ¤ yv Kr>v Kg>v = 0= This means if for v = pd{ + plq the above equation is not solvable, then it remains unsolvable for all v A pd{ + plq . Remember that a diagnostic observer can also be brought into a similar form like a parity relation based residual generator, as demonstrated in the last chapter. It is thus reasonable to prove the applicability of condition (6.12) to the design of diagnostic observers decoupled from the unknown inputs. We begin with transforming the design form of diagnostic observer h(w) ˙ = Jh(w) + (W Hi OIi )i (w) + (W Hg OIg )g(w) u(w) = zh(w) + yIi i (w) + yIg g(w)

(6.17) (6.18)

6.2 Existence conditions of PUIDP

123

into a non-recursive form using the similar computation procedure like the one given in Theorem 5.7, in which K = W E OG is replaced by W Hg OIg as well as W Hi OIi . It turns out ¡ ¢ (6.19) u(s) = zJv sv h(s) + yv Ki>v L¯i v iv (s) + Kg>v L¯gv gv (s) where ¤ £ £ ¤ yv = yv>0 yv>1 · · · yv>v 5 Sv > z = 0 · · · 0 1 5 6 R ··· R Lni ×ni 9 .. : .. .. 9 zjLni ×ni . . . : 9 : ¯ Li v = 9 : .. . . . . 7 . . R 8 . zJv1 jLni ×ni · · · zjLni ×ni Lni ×ni 5

6 ··· R 9 .. : .. 9 zjLng ×ng . . : 9 : ¯ Lgv = 9 : .. . . . . 7 . . R 8 . zJv1 jLng ×ng · · · zjLng ×ng Lng ×ng 6 6 5 5 i (s)sv g(s)sv : : 9 9 .. .. : : 9 9 . . iv (s) = 9 : > gv (s) = 9 : 7 i (s)s1 8 7 g(s)s1 8 i (s) g(s) Lng ×ng

R .. .

We immediately see that choosing yv satisfying (6.11) yields u(s) = zJv sv h(s) + yv Ki>v L¯i v iv (s) i.e. the residual signal is decoupled from the unknown input. Remark 6.1 We would like to emphasize that in the above derivation operator s instead of n is consciously used for the purpose of indicating the applicability of the achieved results for both discrete and continuous time processes. We have seen that (6.11) is a su!cient condition for the construction of a diagnostic observer decoupled from the unknown input. Moreover, (6.19) shows the dependence of the residual dynamics on j. Does the selection of j inﬂuence the solvability of PUIDP? Is (6.11) also a necessary condition? A clear answer to these questions will be given by the following study. Note that 6 5 z 9 zJ : : 9 udqn 9 . : = v 7 .. 8 zJv1

124

6 Perfect unknown input decoupling

thus, a decoupling from g only becomes possible if W Hi OIi 6= 0 or yIi 6= 0 W Hg OIg = 0 and yIg = 0=

(6.20) (6.21)

Remember that (Theorem 5.7) 5

yv>0 yv>1 .. .

9 9 O = 9 7

6 : : : jyv>v 8

yv>v1 (6.20)-(6.21) can be rewritten into ¸ ¸ ¸ ¸ Lj W y¯v1 Hi Hg 1 0 = =, 01 0 yv>v Ii Ig 2 0 ¸ ¸ ¸ W y¯v1 Hi Hg 3 0 = 0 yv>v Ii Ig 4 0 where

5 9 9 y¯v1 = 9 7

yv>0 yv>1 .. .

(6.22)

6 : : : > 3 6= 0 or 4 6= 0= 8

yv>v1 Since

6 F 69 : yv>v 9 FD : 9 .. : : 0 :9 . : : .. : 9 : 9 . 8 9 ... : : 9 0 7 FDv2 8 FDv1 5

5

yv>1 9 yv>2 9 W =9 . 7 .. yv>v

yv>2 · · · yv>v1 · · · · · · yv>v .. ··· ··· . 0 ···

···

equation (6.22) is, after an arrangement of matrix W into a vector, equivalent to yv Kg>v = 0 and yv Ki>v 6= 0> yv 5 Sv = It is evident that the solvability of the above equations is independent of the choice of j and so the eigenvalues of J. We have proven the following theorem. Theorem 6.4 Given J|i (s) = F(sL D)1 Hi + Ii and J|g (s) = F(sL D)1 Hg +Ig , then a diagnostic observer of order v delivers a residual decoupled from the unknown input if and only (6.11) or equivalently (6.12) holds. Furthermore, the eigenvalues of the diagnostic observer are arbitrarily assignable.

6.3 A frequency domain approach

125

An important message of Theorem 6.4 is that • the parity relation based residual generator and the diagnostic observer have the same solvability conditions for the PUIDP and furthermore, • upon account of the discussion on the relationships between the dierent types of residual generators, the algebraic check conditions expressed in terms of (6.11) or equivalently (6.12) are applicable for all kinds of residual generators. We now summarize the main results of this subsection into an algorithm. Algorithm 6.2 An algebraic check of the solvability of PUIDP Step 1. Form Kr>v > Ki>v > Kg>v Step 2. Prove (6.12). If it holds for some v, the PUIDP is solvable, otherwise unsolvable. Example 6.3 We consider again benchmark system EHSA with model (3.83) and suppose that three additive faults are considered. We now check the solvability of the PUIDP by means of Algorithm 6.2. We have ﬁrst formed In the second step, Kr>v > £Ki>v and Kg>v ¤for v = 2>£ 3 and 4 respectively. ¤ udqn Ki>v Kr>v Kg>v and udqn Kr>v Kg>v have been computed for dierent values of v= The results are: for v = 2 ¤ £ ¤ £ udqn Ki>v Kr>v Kg>v = 6 = udqn Kr>v Kg>v for v = 3 ¤ £ ¤ £ udqn Ki>v Kr>v Kg>v = 8 A udqn Kr>v Kg>v = 7 for v = 4 ¤ £ ¤ £ udqn Ki>v Kr>v Kg>v = 10 A udqn Kr>v Kg>v = 8= Thus, the PUIDP is solvable.

6.3 A frequency domain approach The approach presented in this section provides a so-called frequency domain solution for the residual generator design problem: given general residual generator in the design form ˆ x (s) (J|g (s)g(s) + J|i (s)i (s)) 5 R u(s) = U(s)P

(6.23)

ﬁnd such a post-ﬁlter U(s) 5 RH4 that ensures ˆ x (s)J|i (s) 6= 0= ˆ (s)J|g (s) = 0 and U(s)P U(s)P

(6.24)

126

6 Perfect unknown input decoupling

In order to illustrate the underlying idea, we ﬁrst consider a simple case ¸ ˆ x (s)J|g (s) = j1 (s) P j2 (s) with j1 (s)> j2 (s) 5 R and stable. Set ¤ £ U(s) = j2 (s) j1 (s) gives ˆ x (s)J|g (s) = j2 (s)j1 (s) j1 (s)j2 (s) = 0= U(s)P We see from this example that the solution is based on a simple multiplication and an addition of transfer functions. No knowledge of modern control theory, the state space equations and associated calculations are required. We now present an algorithm to approach the design problem stated by (6.23)-(6.24). We suppose p A ng and ¤ £ udqn J|i (s) J|g (s) A udqn (J|g (s)) and denote 6 j¯11 (s) · · · j¯1ng (s) : . .. .. p×n ¯ g (s) = 9 ˆ x (s)J|g (s) = J P 8 5 RH4 g = 7 .. . . j¯p1 (s) · · · j¯png (s) 5

Algorithm 6.3 A frequency domain approach Step 1: Set initial matrix W (s) = Lp×p Step 2: Start a loop: l = 1 to l = ng Step 2-1: When j¯ll (s) = 0, Step 2-1-1: set nl = l + 1 and check j¯nl l (s) = 0? Step 2-1-2: If it is true, set nl = nl + 1 and go back to Step 2-2-1, otherwise Step 2-1-3: set 6 5 w11 · · · w1p ½ 1 : n 6= l and nl : 9 .. .. . .. 8 > wnn = Wnl = 7 . . 0 : n = l or nl wp1 · · · wpp ½ 0 : (n 6= o) and (n 6= nl and o 6= l or n 6= l and o 6= nl ) wno = 1 : (n 6= o) and (n = nl and o = l or n = l and o = nl ) ¯ g (s) = Wn J ¯ g (s)> W (s) = Wn W (s) J l

l

Step 2-2: Start a loop: m = l + 1 to m = p:

6.3 A frequency domain approach

5 9 Wlm (s) = 7 ½ wnn (s) = ½

127

6

· · · w1p (s) : .. .. 8 . . wp1 (s) · · · wpp (s) w11 (s) .. .

1 : n 6= m or n = m and j¯ml (s) = 0 j¯ll (s) : n = m and j¯ml (s) 6= 0

0:n= 6 o and n 6= m and o = l ¯ jml (s) : n = m and o = l ¯ g (s)> W (s) = Wlm (s)W (s) ¯ g (s) = Wlm (s)J J wno (s) =

Step 3: Set

¤ £ U(s) = R(png )×ng L(png )×(png ) W (s)=

To explain how this algorithm works, we make the following remark. Remark 6.2 All calculations in the above algorithm are multiplications and additions of two transfer functions, in details • Step 2-1 serves as ﬁnding j¯ll (s) 6= 0 by a row exchange j¯ll (s) (6= 0). • After completing Step 2-2 we have j¯mm (s) 6= 0> m = 1> · · · > l> j¯nm (s) = 0> n A m l= • When the loop in Step 2 is ﬁnished, we obtain 5 jˆ11 (s) 9 0 jˆ22 (s) 9 Y Y . .. ¯ g (s) = 9 Wnl Wlm (s)J 9 .. . 9 l>m nl 7 0 ··· 0 ···

6 : : : : . : 0 jˆng ng (s) 8 ··· 0

..

where jˆll (s) 6= 0> l = 1> · · · > ng and denotes some transfer matrix of no interest. Since Wnl and Wlm (s) are regular transformations, the above results are ensured. • It is clear that Y Y ¤ £ Wnl Wlm (s) and so U(s) = 0(png )×ng L(png )×(png ) W (s) 5

£ ¯ g (s) = 0(pn )×n U(s)J g g

= 0=

nl

l>m

6 jˆ11 (s) 9 0 jˆ22 (s) : : ¤9 9 .. : . . .. .. L(png )×(png ) 9 . : 9 : 7 0 ··· 0 jˆng ng (s) 8 0 ··· ··· 0

128

6 Perfect unknown input decoupling

We have seen that the above algorithm ensures that the residual generator of the form ³ ´ ˆ x (s)|(s) Q ˆx (s)x(s) u(s) = U(s) P delivers a residual perfectly decoupled from the unknown input g.

6.4 UIFDF design The problem to be solved in this section is the design of UIFDF that is formulated as: given system model {(w) ˙ = D{(w) + Ex(w) + Hi i (w) + Hg g(w) 5 Rq |(w) = F{(w) + Gx(w) + Ii i (w) 5 Rp

(6.25) (6.26)

{ ˆ˙ = Dˆ {(w) + Ex(w) + O(|(w) |ˆ(w)) u(w) = y (|(w) |ˆ(w)) > |ˆ(w) = F { ˆ(w) + Gx(w)

(6.27) (6.28)

and an FDF

ﬁnd O> y such that residual generator (6.27)-(6.28) is stable and yF(sL D + OF)1 Hg = 0 ¢ ¡ y F(sL D + OF)1 (Hi OIi ) + Ii 6= 0= We shall present two approaches, • the eigenstructure assignment approach and • the geometric approach. 6.4.1 The eigenstructure assignment approach Eigenstructure assignment is a powerful approach to the design of linear state space feedback system, as it can be shown easily that the closed-loop system structure like (sL D + OF) depends entirely on the eigenvalues and the left and right eigenvectors of D OF. The eigenstructure approach proposed by Patton and co-worker is dedicated to the solution of equation yF(sL D + OF)1 Hg = 0

(6.29)

for O> y. To this end, the following two well-known lemmas are needed. Lemma 6.1 Suppose matrix DOF 5 Rq×q has eigenvalues l > l = 1> · · · > q and the associated left and right eigenvectors l 5 R1×q > l 5 Rq×1 , then we have l m = 0> l = 1> · · · > q> m 6= l> m = 1> · · · > q=

6.4 UIFDF design

129

Lemma 6.2 Suppose matrix DOF 5 Rq×q can be diagonalized by similarity transformations and has eigenvalues l > l = 1> · · · > q and the associated left and right eigenvectors l 5 R1×q > l 5 Rq×1 , then the resolvent of D OF can be expressed by (sL D + OF)1 =

1 1 q + ··· + q = s 1 s q

Below are two su!cient conditions for the solution of (6.29). Theorem 6.5 If there exists a left eigenvector of matrix D OF, l , satisfying l = yF and l Hg = 0 then (6.29) is solvable. Proof. The proof becomes evident by choosing y so that yF = l and considering Lemmas 6.1-6.2 : μ ¶ 1 1 q l l Hg = + ··· + q l Hg = 0= yF(sL D + OF)1 Hg = l s 1 s q s l t u Theorem 6.6 Given Hg whose columns are the right eigenvectors of DOF, then (6.29) is solvable if there exists a vector y so that yFHg = 0= The proof is similar to Theorem 6.5 and is thus omitted. Upon account of Theorem 6.5 Patton et al. have proposed an algorithm for the eigenstructure assignment approach. Algorithm 6.4 Eigenstructure assignment approach by Patton and Kangethe Step 1: Compute the null space of FHg , Q , i.e. Q FHg = 0 Step 2: Determine the eigenstructure of the observer Step 3: Compute the observer matrix O using an assignment algorithm and set y = zQ> z 6= 0= Theorem 6.7 follows directly from Lemmas 6.1-6.2 and provides us with a necessary and su!cient condition. Theorem 6.7 Suppose that matrix D OF 5 Rq×q can be diagonalized by similarity transformations and has eigenvalues l > l = 1> · · · > q and the associated left and right eigenvectors l 5 R1×q > l 5 Rq×1 , then (6.29) holds if and only if there exist y> O such that yF 1 1 Hg = 0> · · · > yF q q Hg = 0=

(6.30)

130

6 Perfect unknown input decoupling

Note that (6.30) holds if and only if yF l = 0 or l Hg = 0 and there exists an index n> 0 ? n ? q> so that yF l = 0> l = 1> · · · > n> l Hg = 0> l = n + 1> · · · > q= Let the m-th column of Hg , hgm , be expressed by hgm =

q X

nml l

l=1

then we have l Hg = 0> l = n + 1> · · · > q =, nml = 0> l = n + 1> · · · > q> m = 1> · · · > ng =, hgm =

n X

nml l > m = 1> · · · > ng =

l=1

This veriﬁes the following theorem. Theorem 6.8 Equation (6.29) holds if and only if Hg can be expressed by ¤ ¤ £ £ Hg = 1 · · · n H > H 5 Rn×ng > yF 1 · · · n = 0= From the viewpoint of linear control theory, this means the controllable eigenvalues of (D OF> Hg ), 1 > · · · > n , are unobservable by yF. Since y 6= 0 is arbitrarily selectable, Theorem 6.8 can be reformulated as Corollary 6.1 (6.29) holds if and only if Hg can be expressed by ¤ ¤¢ £ ¡ £ Hg = 1 · · · n H > H 5 Rn×ng > udqn F 1 · · · n ? p= Now, we introduce some well-known deﬁnitions and facts from the linear system theory and the well-established eigenstructure assignment technique: • }l is called invariant or transmission zero of system (D> Hg > F)> when ¸ D }l L Hg udqn ? q + min{p> ng }= F 0 • Vectors l and l satisfying £

¸ ¤ D }l L Hg l l =0 F 0

are called state and input direction associated to }l respectively. • Observer matrix O deﬁned by 5 61 5 6 1 s1 9 .. : 9 .. : O = 7 . 8 7 . 8 > l = sl F(D l L)1 q

sq

ensures l (l L D + OF) = 0> l = 1> · · · > q i.e. l is the eigenvalue of D OF and yl the left eigenvector.

6.4 UIFDF design

131

• Let l = }l > sl = l > l = l then we have l (D l L) sl F = 0> l Hg = 0= Following algorithm is developed on the basis of Corollary 6.1 and the above-mentioned facts . Algorithm 6.5 An eigenstructure assignment approach Step 1: Determine the invariant zeros of system (D> Hg > F) deﬁned by ¸ D }l L Hg ? q + min{p> ng }> l = n + 1> · · · > q udqn F 0 Step 2: Solve

£

l l

¸ ¤ D }l L Hg = 0> l = n + 1> · · · > q F 0

for l > l Step 3: Set l = }l > sl = l > l = l > > l = n + 1> · · · > q Step 4: Deﬁne l > l > sl > l = 1> · · · > n satisfying 6 1 9 : l = sl F(D l L)1 > l = 1> · · · > n> udqn 7 ... 8 = q 5

q Step 5: Set 61 5 6 s1 1 9 : 9 : O = 7 ... 8 7 ... 8 5

q Step 6: Solve

5

6 1 ¤ 9 .. : £ 7 . 8 1 · · · n = q

for 1 > · · · > n Step 7: If then solve for y=

sq

Ln×n 0(qn)×n

£ ¤ udqnF 1 · · · n ? p ¤ £ yF 1 · · · n = 0

¸

132

6 Perfect unknown input decoupling

Note that the condition that there exists a vector y so that yFHg = 0 can equivalently be reformulated as the solution of equations yF = zW> W Hg = 0> W 5 R(qng )×q = Furthermore, the requirement that the rows of W are the left eigenvectors of matrix D OF leads to 6 5 n+1 0 0 : 9 W D W 7 0 . . . 0 8 = W OF= 0 0 q This veriﬁes that Luenberger conditions (5.30)-(5.32) are necessary for the use of the eigenstructure assignment approach provided above. 6.4.2 Geometric approach The so-called geometric approach is one of the ﬁelds in the control theory, where elegant tools for the design and synthesis of control systems are available. On the other side, the application of the geometric approach requires a profound mathematical knowledge. The pioneering work of approaching the design and synthesis of FDF by geometric approach was done by Massoumnia, in which an elegant solution to the FDF design has been derived. In this subsection, we shall brieﬂy describe the geometric approach to the FDF design without elaborate handling of its mathematical background. The core of the geometric approach is the search for an observer matrix O that makes (D OF> Hg > F) maximally uncontrollable by g. It is the dual form of the geometric solution to the disturbance decoupling (control) problem (DDP) by means of a state feedback controller. Below, we brieﬂy describe an algorithm for this purpose, which is presented as the dual form of the algorithm proposed by Wonham for the DDP-controller design. The addressed problem is formulated as follows: given system {˙ = (D OF) { + Hg g> | = F{

(6.31)

ﬁnd O such that the pair (DOF> Hg ) becomes maximally uncontrollable. The terminology maximally uncontrollable is used to express the uncontrollable subspace with the maximal dimension. We shall also use maximal solution to denote the maximally dimensional solution [ of an equation P [ = 0 (or [P = 0) for a given P= Algorithm 6.6 Computation of observer gain O for generating maximally uncontrollable subspace

6.4 UIFDF design

133

Step 0: Setting initial condition: ﬁnd a maximal solution of HgW Y0 = 0 for Y0 Step 1: Find a maximal solution of £ ¤ Zl F W Yl1 = 0> l = 1> 2> · · · for Zl Step 2: Find a maximal solution of ¸ HgW Yl = 0> l = 1> 2> · · · Zl DW for Yl Step 3: Check udqn (Yl ) = udqn (Yl1 ) If no, increase l = l + 1 and go to Step 1, otherwise set Y¯ = Yl Step 4: Find a solution of ¸ £ ¤ N DW Y¯ = F W Y¯ S for (N> S ) Step 5: Solve N = OW Y¯ for the observer gain O= Remark 6.3 Step 0 to Step 3 are the algebraic version of the algorithm proposed by Wonham for the computation of the supremal (DW > F W )-invariant subspace contained in the null-space of HgW = As a result, the dual representation of system (6.31) becomes maximally unobservable. The following lemma is known in the geometric control framework, based upon which a UIFDF can be designed. Lemma 6.3 Suppose O makes (D OF> Hg ) maximally uncontrollable by g, i.e. ((D OF)W > HgW ) is maximally unobservable. Then by a suitable choice of output and state bases, Y and W , the resulting realization can be described by ¸ ¸ ¸ ¯g1 D¯11 D¯12 H F¯1 0 1 ¯ > W H > F = Y FW = = W (D OF) W 1 = g 0 F¯2 0 D¯22 0 (6.32) ¯g1 > F¯1 ) is perfectly controllable. where the realization (D¯11 > H

134

6 Perfect unknown input decoupling

Remark 6.4 A system (D> E> F) is called perfectly controllable if ¸ D L E ;> has full row rank. F 0 Let Omax be the observer gain that makes (D Omax F> Hg ) maximally uncontrollable by g. When F¯2 6= 0, we construct, according to Lemma 6.3, the following FDF ¸ ¸ ¸ }1 }˙1 D¯11 O11 F¯1 D¯12 O12 F¯2 = + W Ex + W (Omax + O0 Y ) | }˙2 0 D22 O22 F¯2 }2 μ ¸ ¸¶ £ ¤ }1 F¯1 0 u = 0 y2 Y| > y2 6= 0 (6.33) }2 0 F¯2 with

O11 O12 W O0 = 0 O22

¸ (6.34)

and O11 , O22 ensuring the stability of D¯11 O11 F¯1 and D22 O22 F¯2 = Introducing ¸ ¸ h } } = 1 >h = W{ } = 1 }2 h2 gives

h˙ 1 h˙ 2

¸

¸ ¸ ¸ ¯g1 h1 D¯11 O11 F¯1 D12 O12 F¯2 H = + g 0 D22 O22 F¯2 h2 0 ¸ ¸ £ ¤ F¯1 0 h1 u = 0 y2 = y2 F¯2 h2 = h2 0 F2

(6.35) (6.36)

It is evident that residual signal u is perfectly decoupled from g. It is straightforward to rewrite (6.33) into the original FDF form (6.27)(6.28) with ¤ £ (6.37) O = Omax + O0 Y> y = 0 y2 Y as well as { ˆ = W 1 }= Below is a summary of the above results in the form of an algorithm. Algorithm 6.7 The geometric approach based UIFDF design Step 1: Determine Omax that makes (D Omax F> Hg > F) maximally uncontrollable by using Algorithm 6.6 Step 2: Transform (D Omax F> Hg > F) into (6.32) by a state (W ) and an output (Y ) transformation (controllability and observability decomposition) Step 3: Select O0 satisfying (6.34) and ensuring the stability of D¯11 O11 F¯1 and D¯22 O22 F¯2

6.4 UIFDF design

135

Step 4: Construct FDF (6.33) or in the original form (6.27)-(6.28) with O> y satisfying (6.37). Remember that the construction of residual generator (6.33) is based on the assumption that F¯2 6= 0= Without proof we introduce the following necessary and su!cient condition for F¯2 6= 0, which is known from the geometric control theory. Theorem 6.9 Under the same conditions as given in Lemma 6.3, we have • F¯2 6= 0 if and only if udqn (F) A udqn (Hg ) • (D¯22 > F¯2 ) is equivalent to ¸ μ ¶ £ ¤ D¯221 0 ¯21 0 F > (D¯22 > F¯2 ) D¯222 D¯223 where (D¯221 > F¯21 ) is perfectly observable, the eigenvalues of matrix D¯223 are the invariant zeros of (D> Hg > F) and they are unobservable. An immediate result of the above theorem is Corollary 6.2 Given system model |(s) = J|x (s)x(s) + J|g (s)g(s) with J|x (s) = (D> E> F) and J|g (s) = (D> Hg > F)> then there exists an FDF that is decoupled from g if and only ¸ D sL Hg ?q+p udqn F 0 and J|g (s) = (D> Hg > F) has no unstable invariant zero. We know from Theorem 6.9 that there exists an observer matrix O such that (D OF> Hg > F) can be brought into (6.32) with F¯2 6= 0 if and only if ¸ D sL Hg ? q + p= udqn F 0 Moreover, if udqn

¸ ¸ D sL Hi Hg D sL Hg ? udqn q+p F 0 F 0 0

then by suitably choosing output and state bases, Y and W , the resulting realization can be described by equations of the form ¸ ¸ ¯ D¯11 D¯12 ¯ = Y FW 1 = F1 0 > F W (D OF) W 1 = 0 D¯22 0 F¯2 ¸ ¸ ¯g1 ¯ H H W Hg = > W Hi = ¯i 1 0 Hi 2

136

6 Perfect unknown input decoupling

¯i 2 6= 0. As a result, constructing an FDF according to (6.33) yields where H ¸ ¸ ¸ ¸ ¯g1 ¯i 1 h1 D¯11 O11 F¯1 D12 O12 F¯2 H H = + i g+ ¯ 0 D22 O22 F¯2 h2 0 Hi 2 ¸ ¸ £ ¤ F¯1 0 h1 u = 0 y2 =, h2 0 F2 ¯i 2 i (s) u(s) = y2 F2 (sL D22 + O22 F2 )1 H

h˙ 1 h˙ 2

¸

i.e. a fault detection is achievable. Recall Corollary 6.2, we have Corollary 6.3 Given system model |(s) = J|x (s)x(s) + J|i (s)i (s) + J|g (s)g(s) with J|x (s) = (D> E> F)> J|i (s) = (D> Hi > F) and J|g (s) = (D> Hg > F)> then there exists an FDF that solves PUIDP if ¸ ¸ D sL Hi Hg D sL Hg ? udqn q+p (6.38) udqn F 0 F 0 0 and the invariant zeros of J|g (s) are stable. It is interesting to notice the fact that, if there exists a UIFDF> then we are also able to construct a reduced order residual generator decoupled from g. To this end, we consider FDF (6.33). Instead of constructing a full order observer, we now deﬁne the subsystem regarding to }2 > i.e. ¢ ¡ }˙2 = D22 O22 F¯2 }2 + W2 Ex + W2 (Omax + O0 Y ) | ¡ ¢ (6.39) u = y2 Y2 | F¯2 }2 with

W :=

¸ ¸ Y1 W1 > Y := = W2 Y2

It is evident that (6.39) is a reduced order residual generator which is decoupled from g= Recall that residual generator (6.33) becomes unstable if system (D> Hg > F) has unstable invariant zeros. This problem can be solved by constructing a reduced order residual generator. Without loss of generality, suppose that after applying Algorithm 6.7 (D¯22 > F¯2 ) is of the form ¸ £ ¤ D¯ 0 > F¯2 = F¯21 0 (6.40) D¯22 = ¯221 ¯ D222 D223 as described in Theorem 6.9, i.e.

6.4 UIFDF design

5

137

6

D¯11 D¯121 D¯122 7 0 D¯221 0 8 W (D OF) W = 0 D¯222 D¯223 5 6 ¯g1 ¸ H ¯ 0 0 F 1 > W Hg = 7 0 8 = F¯ = Y FW 1 = 0 F¯21 0 0 1

(6.41)

Corresponding to the decomposition given in (6.41), we now further split }2 > O22 and W2 into ¸ ¸ ¸ }21 O221 W21 > O22 = > W2 = }2 = }22 O222 W22 and construct the following residual generator ¢ ¡ }˙21 = D¯221 O121 F¯21 }21 + W21 Ex + W21 (Omax + O0 Y ) | ¡ ¢ u = y2 Y2 | F¯21 }21 =

(6.42)

It is straightforward to prove that for h21 = W21 { }21 ¢ ¡ h˙ 21 = D¯221 O121 F¯21 h21 > u = y2 F¯21 h21 = That means residual generator (6.42) is stable and perfectly decoupled from g= Corollary 6.4 Given system model |(s) = J|x (s)x(s) + J|g (s)g(s) with J|x (s) = (D> E> F) and J|g (s) = (D> Hg > F) and suppose that ¸ D sL Hg ? q + p= udqn F 0 Then residual generator (6.42) delivers a residual signal decoupled from g. A very useful by-product of the above discussion is that residual generator (6.42) can be designed to be of the minimum order and decoupled from g= This will be handled at the end of this chapter. Algorithm 6.8 The geometric approach based design of reduced order residual generator Step 1: Determine Omax that makes (D Omax F> Hg > F) maximally uncontrollable by using Algorithm 6.6 Step 2: Transform (D Omax F> Hg > F) into (6.32) by a state and an output transformation (controllability and observability decomposition) Step 3: Transform (D¯22 > F¯2 ) into (6.40) by a state transformation (observability decomposition)

138

6 Perfect unknown input decoupling

Step 4 Select O221 ensuring the stability of D¯221 O221 F¯21 Step 5: Construct residual generator (6.42). The results achieved in this section can be easily extended to the systems described by (3.30)-(3.31) with Ig 6= 0= To this end, we can, as done in the former chapters, rewrite (3.30)-(3.31) into 65 6 5 6 5 6 5 6 5 6 5 {˙ { E 0 0 D Hg Hi 7 g˙ 8 = 7 0 0 0 8 7 g 8 + 7 0 8 x + 7 L 8 g˙ + 7 0 8 i˙ (6.43) 0 0 0 i 0 0 L i˙ 5 6 ¤ { £ (6.44) | = F Ig Ii 7 g 8 + Gx= i Note that 5

D sL Hg Hi 0 0 9 0 sL 0 L 0 udqn 9 7 0 0 sL 0 L F Ig Ii 0 0 5 D sL Hg 0 sL L udqn 7 0 F Ig 0

6

¸ : : = udqn D sL Hg Hi + ng + ni 8 F Ig Ii 6

¸ 8 = udqn D sL Hg + ng F Ig

it holds 5

6 D sL Hg Hi 0 0 9 0 sL 0 L 0 : : q + ng + ni + p udqn 9 7 0 0 sL 0 L 8 F Ig Ii 0 0 ¸ D sL Hg Hi q+p +, udqn F Ig Ii 6 5 ¸ D sL Hg 0 D sL Hg sL L 8 ? q + ng + p +, udqn udqn 7 0 ? q + p= F Ig F Ig 0 Recall further the deﬁnition of invariant zeros, the results given in Theorem 6.9 and Corollary 6.2 can be extended to Corollary 6.5 Given system model (3.30)-(3.31), then there exists an FDF that ensures a perfect unknown input decoupling if ¸ ¸ D sL Hg Hi D sL Hg ? udqn q+p (6.45) udqn F Ig F Ig Ii and the invariant zeros of J|g (s) = Ig + F (sL D)1 Hg are stable.

6.5 UIDO design

139

Example 6.4 We now apply Algorithm 6.7 to the design of a full order UIFDF for the benchmark system LIP100 described by (3.57). This UIFDF should deliver residual signals decoupled from the unknown input g. Using Algorithm 6.6, we obtain 5 6 0 0=0000 1=9500 9 0 0=0000 13=7429 : : Omax = 9 7 0 0=7131 0=2470 8 = 0 0=0519 0=0180 It is followed by the computation of the state and output transformation matrices, which results in 5 6 5 6 0 0 0=1630 0 001 9 0=0055 0=9473 0=3193 0=0232 : : 7 8 W =9 7 0=0676 0=3191 0=9428 0=0686 8 > Y = 0 1 0 = 100 0=9977 0=0269 0=0621 0=0045 To ensure the desired dynamics, O11 and O22 are selected as follows, from which O0 is computed 5 6 1=0000 0=5000 0=5000 9 0 11=0500 11=8438 : := W O0 = 9 7 0 0=8796 5=1779 8 0 0=1223 7=0497 Finally, set

£

0 y2

¤

0 0=500 1=000 = 0 0=750 1=500

¸

and based on which y as well as O are determined= Having designed O> y> a residual generator of the form (6.33) is constructed. In Fig.6.2, the response of the two residual signals to dierent faults is sketched. These faults occurred after the 10-th second. It can be seen that in the fault-free time interval (before the 10-th second) the residual signals are almost zero. It veriﬁes a perfect decoupling.

6.5 UIDO design The UIDO design addressed in this section is formulated as: given system model (3.30)-(3.31) and diagnostic observer }(w) ˙ = J}(w) + Kx(w) + O|(w) u(w) = y|(w) z}(w) tx(w)

(6.46) (6.47)

that satisﬁes Luenberger conditions (5.30)-(5.31) and (5.38), and thus whose design form is described by

140

6 Perfect unknown input decoupling 2 1.5

Residual signal

1 0.5 0 Ŧ0.5 Ŧ1 Ŧ1.5 Ŧ2 0

5

10

15 Time [s]

20

25

30

Fig. 6.2 Response of the residual signals to faults

h(w) ˙ = Jh(w) + (W Hi OIi )i (w) + (W Hg OIg )g(w) u(w) = zh(w) + yIi i (w) + yIg g(w)

(6.48) (6.49)

ﬁnd J> O> W> y> z such that residual generator (6.46)-(6.47) is stable and zF(sL J)1 (Hg OIg ) + yIg = 0 zF(sL J)1 (Hi OIi ) + yIi 6= 0=

(6.50) (6.51)

6.5.1 An algebraic approach In this subsection, the approach by Ge and Fang to the DO design is extended to the construction of UIDO. Suppose that ¤ £ Ig = 0> Ii = 0 and Hg Hi = Lq×q > ng ? p= Then, there exists an UIDO if and only if ¸ Lng ×ng =0 W Hg = W 0 which is, by denoting the l-th column of W with wl , equivalent to wl = 0> l = 1> · · · > ng = Based on the method introduced in Chapter 5, Ge and Fang have proposed a recursive algorithm to the design of a UIDO satisfying (6.50). To this end, they have proven the following theorem. Theorem 6.10 Let

6.5 UIDO design

5

6 t 0 W1 91 t 9 9 : W = 7 ... 8 > J = 9 . . 7 .. . . Wv 0 ··· 5

··· ··· .. .

141

6

0 0: : .. : .8

(6.52)

1 t

where W 5 Rv×q > J 5 Rv×v and v is the order of the DO, then l1 X gn 1 Yln F n T(t)> l = 1> · · · > v n! gt n=0 6 5 [1 m1 q X X : 9 dm t n Dm1n > [ = 7 ... 8 T(t) = m=1 n=0 [v

Wl =

(6.53)

(6.54)

provide an equivalent solution with the one given in Theorem (5.8) for the Luenberger equations, where t denotes the eigenvalue being arbitrarily selectable. The proof is straightforward and thus omitted. Following algorithm, developed on the basis of Theorem 6.10, can be used to approach the design of UIDO satisfying (6.50). Algorithm 6.9 The approach to UIDO design by Ge and Fang Step Step Step Step

1: 2: 3: 4:

Calculate ghw(sL D) = dq sq + · · · + d1 s + d0 Solve (5.76) for FQ Set v = 1 Calculate T(t) by (6.54) and set P0 = FT(t)

Step 5: Denote the i-th column of P0 (s) with P0>l and form ¤ £ V0 = P0>1 · · · P0>ng

(6.55)

(6.56)

Step 6: Solve [1 V0 = 0> [1 6= 0 for [1 and set W1 = [1 P0 Step 7: Form W according to (6.52) and prove if udqn (W FQ ) ? v Step 8: If (6.57) holds, then go to Step 11

(6.57)

142

6 Perfect unknown input decoupling

Step 9: Increase the observer order by one: v = v + 1, and calculate Pv1 = then set

1 FTv1 (t) (v 1)!

¤ £ Vv1 = Pv1>1 · · · Pv1>ng

Step 10: Solve [v V0 +

v1 X

[vn Vn = 0

(6.58)

n=1

for [v , set Wv =

v1 X

[vn Pn = 0

(6.59)

n=0

and go to Step 7 Step 11: Solve (5.75) for z(6= 0), set J according to (6.52) and calculate f(J) by (5.78) Step 12: Set [> O> y subject to (6.54), (5.74), (5.75) respectively. Remark 6.5 The above algorithm is a recursive realization of Theorem 6.10, and thus the achieved UIDO is of minimal order. Example 6.5 We now apply Algorithm 6.9 to design a minimum order UIDO for the benchmark system LIP100 described by (3.57). We start with v = 1 and set J = 1= It follows £ ¤ W = 92=2308 0=0000 0=0002 0=0000 £ ¤ £ ¤ O = 92=2308 0=0003 179=8499 > y = 46=1154 0=0000 0=0001 based on which, we can also determine K> z= Thus, it can be concluded that using Algorithm 6.9 we are able to design a minimum order UIDO for LIP100. 6.5.2 Unknown input observer approach In the early 80’s, the so-called unknown input observer (UIO) design received much attention due to its importance in robust state estimation and observerbased robust control. Consider system model {(w) ˙ = D{(w) + Ex(w) + Hg g(w)> |(w) = F{(w)=

(6.60)

A UIO is a Luenberger type observer that delivers a state estimation { ˆ independent of unknown input g in the sense that

6.5 UIDO design

lim ({(w) { ˆ(w)) = 0 for all x(w)> g(w)> {0 =

w$4

143

(6.61)

Making use of { ˆ> a residual signal can be constructed as follows u(w) = |(w) F { ˆ(w)= This is the way that is widely used to design UIDO, also for the reason that the technique of designing UIO is well established. It is worth pointing out that the primary objective of using a UIO is to reconstruct the state variables. It is dierent from the one of residual generation, where only measurements have to be reconstructed. In the next subsections, we shall present some approaches to the design of UIO only for the residual generation purpose. We now outline the underlying idea of the UIO design technique. It follows from (6.60) that |(w) ˙ FD{(w) Fx(w) = FHg g(w)=

(6.62)

udqn (FHg ) = udqn (Hg ) = ng

(6.63)

Assume that then there exists a matrix Pfh satisfying Pfh FHg = Lng ×ng =

(6.64)

Multiplying the both sides of (6.62) by Pfh gives ˙ FD{(w) Fx(w)) = g(w)= Pfh (|(w) This means, using |˙ (|(n + 1) for discrete time systems), an estimation of the state vector { ˆ and the input vector x> the unknown input vector g can be constructed by ˆ = Pfh (|(w) ˙ FDˆ {(w) Fx(w)) = g(w) ˆ we are able to construct a full order state observer, on the On account of g> assumption that |˙ is available, as follows { ˆ˙ = Dˆ { + Ex + Hg (FHg )1 (|˙ FDˆ { FEx) + O(| F { ˆ)

(6.65)

whose estimation error is evidently governed by ˆ= h˙ = (D OF Hg Pfh FD)h> h = { { In case that there exists an observer matrix O such that matrix D OF Hg Pfg FD is stabilizable, observer (6.65) fulﬁlls (6.61). Note that observer (6.65) requires knowledge of |, ˙ which may cause troubles by the on-line implementation. To overcome this di!culty, modiﬁcation is made. Introduce a new state vector

144

6 Perfect unknown input decoupling

}(w) = { ˆ(w) Hg Pfh |(w) and a matrix W = L Hg Pfh F

(6.66)

}˙ = (W D OF)} + W Ex + ((W D OF)Hg Pfh + O) | { ˆ = } + Hg Pfh |=

(6.67) (6.68)

then it turns out

It is clear that for all g> x> {0 lim (}(w) W {(w)) = 0> lim ({(w) { ˆ(w)) = 0

w$4

w$4

and furthermore, setting J = W D OF and after some calculations, we have W D JW = ((W D OF)Hg Pfh + O) F> K = W E= This means system (6.67)-(6.68) is a Luenberger type unknown input observer, and by setting (6.69) u = y ((L FHg Pfh )| F}) > y 6= 0 we get a UIDO. Algorithm 6.10 UIO based residual generation Step 0: Check the existence conditions given in Corollary 6.6. If they are satisﬁed, go to the next step, otherwise stop. Step 1: Compute Pfh according to (6.64) and further W according to (6.66) Step 2: Selection of O that ensures the stability of D OF Hg Pfh FD Step 3: Construct residual generator following (6.67) and (6.69). It can be seen that the core of UIO technique is the reconstruction of the unknown input g, which requires condition (6.63) or equivalently (6.64). Furthermore, to ensure the stability of observer (6.65) or equivalently (6.67), the pair (F> W D) should be observable or at least detectable. In summary, we have the following theorem. Theorem 6.11 Given system model (6.60) and suppose Condition I: udqn (FHg ) = udqn(Hg ) = ng Condition II: (F> W D) is detectable, where W = L Hg Pfh F then there exists a UIO in the sense of (6.61).

6.5 UIDO design

145

Remark 6.6 It can be demonstrated that Condition I and II are also necessary conditions for the existence of a UIO. It is interesting to notice that matrix W is singular. This can be readily seen by observing the fact W Hg = Hg Hg Pfh FHg = 0= Thus, by a suitable transformation we are able to ﬁnd a low order UIO. Notice the following equality ¸ μ ¸ ¸¶ L D Hi L 0 L D Hi = udqn udqn F 0 F 0 Pfh FD L ¸ ¸ L W D Hi L D + Hi Pfh FD Hi = udqn = = udqn F 0 F 0 This means if (F> W D) is undetectable, then (D> Hi > F) has at least one unstable transmission zero, i.e. there exists at least one r 5 C+ such that ¸ L D Hi ? q + ni udqn r F 0 since the fact (F> W D) is undetectable implies there exists at least one r 5 RHP such that ¸ r L D ?q udqn F Hence, Theorem 6.11 can be reformulated as Corollary 6.6 Given system model (6.60), then there exists a UIO in the sense of (6.61) if • udqn (FHg ) = ng • (D> Hi > F) has no unstable transmission zero. In a number of publications, it has been claimed that Conditions I and II stated in Theorem 6.11 are necessary for the construction of UIDO in the form of (6.67) and (6.69). It should be pointed out that these two conditions are not equivalent to the solvability conditions of the PUIDP described at the beginning of this chapter. To illustrate it, we only need to consider the case udqn (FHg ) ? udqn (Hg ) and p A ng which does not satisfy Condition I in Theorem 6.11. In against, following Theorem 6.3 the PUIDP is solvable in this case, i.e. we should be able to ﬁnd a residual generator that is decoupled from the unknown input vector g. We now check a special case: p = ng . Since Pfh FHg = L +, FHg Pfh = L =, FW = F(L Hg Pfh F) = 0

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6 Perfect unknown input decoupling

we claim that (F> W D) is not observable for p = ng . In other words, we are able to construct a UIDO of form (6.67) and (6.69) whose eigenvalues are arbitrarily assignable only if p A ng . Indeed, following (6.69) we have for p = ng u = y ((L FHg Pfh )| F}) = yF}= Multiplying the both sides of (6.67) by F gives F }˙ = FOF}= Moreover, notice that lim (}(w) W {(w)) = 0 =, lim F (}(w) W {(w)) = lim F}(w) = 0=

w$4

w$4

w$4

Thus, it is evident that for p = ng the residual u is independent of the fault vector i (w) and therefore it cannot be used for the purpose of fault detection. The above-mentioned two cases reveal that approaching the UIDO design using the UIO technique may restrict the solvability of the problem. The reason lies in the fact that UIDO and UIO have dierent design aims. While a UIO is in fact used to reconstruct the state variables, the design objective of a UIDO is to reconstruct measurable state variables for the purpose of generating analytical redundancy. The realization of these dierent aims follows dierent strategies. By the design of UIO an exact estimation of the unknown input is required such that the inﬂuence of the unknown input can totally be compensated. In comparison, an exact compensation of the unknown input is not necessary by a UIDO. Therefore, the existence conditions of UIO are, generally speaking, stronger than the ones of UIDO. In the following subsections, two approaches to the design of UIDO will be presented. Example 6.6 In this example, we design a UIO for the vehicle lateral dynamic system aiming at generating a residual signal decoupled from the (unknown) road bank angle. As described in Section 3.7.4, the linearized model of this system is described by {(w) ˙ = D{(w) + Ex(w) + Hg g(w)> |˜(w) = |(w) Gx(w) = F{(w) with the system matrices given in (3.76). For our purpose, Algorithm 6.10 is used. After checking the existence conditions, which are satisﬁed, Pfh > W ,O and y are determined respectively: ¸ ¸ £ ¤ £ ¤ 0 0=0082 10 >y = 1 1 = >W = Pfh = 0=0065 0 > O = 0 1 01 Finally, UIO (6.67) is constructed, which delivers a residual signal on account of (6.69).

6.5 UIDO design

147

6.5.3 A matrix pencil approach to the UIDO design Using matrix pencil to approach the design of UIDO was initiated by Wuennenberg in the middle 80’s and lately considerably developed by Hou and Patton. The core of the matrix pencil approach consists in a transformation of an arbitrary matrix pencil to its Kronecker canonical form. For the required knowledge of matrix pencil, matrix pencil decomposition and Kronecker canonical form, we refer the reader to the references given at the end of this chapter. We introduce the following lemma. Lemma 6.4 An arbitrary matrix pencil sH + D can be transformed to the Kronecker canonical form by a regular transformation, i.e. there exist regular constant matrices S and T such that S (sH +D)T = gldj(sL +Mi > sMlqi +L> sHu +Du > sHf +Df > 0) (6.70) where • sL + Mi is the ﬁnite part of the Kronecker form, Mi contains the Jordan blocks Mil ; • sMlqi + L is the inﬁnite part of the Kronecker form, Mlqi contains the Jordan blocks Mlqil with 6 5 0 1 9 .. .. : 9 . . : : Mlqil = 9 9 .. : 7 . 18 0 • sHu + Du is the row part of the Kronecker form. It is a block diagonal matrix pencil with blocks in the form ¤ £ ¤ £ sHul + Dul = s Lul ×ul 0 + 0 Lul ×ul of the dimension ul × (ul + 1); • sHf + Df is the column part of the Kronecker form. It is a block diagonal matrix pencil with blocks in the form ¸ ¸ 0 Lfl ×fl sHfl + Dfl = s + 0 Lfl ×fl of the dimension (fl + 1) × fl ; • 0 denotes the zero matrix of appropriate dimension. There exists a number of numerically stable matrix pencil decomposition methods for the computation of the regular transformation described above, for instance we can use the one proposed by Van Dooren.

148

6 Perfect unknown input decoupling

Remark 6.7 It is evident that Kronecker blocks sL +Mi > sMlqi +L> sHu + Du have full row rank. Corresponding to system model (3.30)-(3.31) and the original form of residual generation, u = | |ˆ> we introduce the following matrix pencil ¸ ˜ = D sL E 0 Hi Hg = (6.71) D˜ sH F G L Ii Ig That means we consider a dynamic system whose inputs are the process input vector x and output vector | and output is dierence between the process output | and its estimate |˙ delivered by the parallel model. Suppose a regular transformation by S1 leads to ¸ ¸ ³ ´ 0 Hg ˜g = ng = ˜ > udqn H S1 Ig Hg and denote S1

¸ ¸ ˜i 0 sL + D E 0 Hi Hg sH1 + D1 E1 H = ˜g F G L Ii Ig × H

where and × denote constant matrices and matrix pencil of appropriate dimensions, respectively, and their forms and values are not of interest. Then, by a suitable regular transformation of the form, we obtain ¸ D sL E 0 Hi Hg gldj(T> L) gldj(S> L)S1 F G L Ii Ig ¸ ˜i 0 D1 sH1 E1 H = gldj(S> L) ˜g gldj(T> L) × H 5 6 ˜11 H ˜i 1 0 Dh sHh 0 E ˜12 H ˜i 2 0 8 =7 (6.72) 0 Df sHf E ˜g × × H where S> T are regular matrices that transform the matrix pencil D1 sH1 into its Kronecker canonical form and the matrix pencil Dh sHh is the composite of ﬁnite, inﬁnite and row parts of the Kronecker form which have full row rank, as stated in Lemma 6.4. Since it is supposed that ¸ D sL udqn =q F i.e. (F> D) is observable, the 0 block in (6.70) disappears. Due to its special form, matrix pencil Df sHf can also be equivalently rewritten into ¸ Dr sL Fr

6.5 UIDO design

149

where Dr sL and Fr are diagonal matrices with blocks in the form 6 5 s : 9 : 9 1 ... £ ¤ : > Frl = 0 · · · 1 9 Drl sLl = 9 : .. .. 8 7 . . 1 s Corresponding to it we denote £

¸ ¤ ˜12 H ˜i 2 Er Hi r E Gr Ii r

In conclusion, we have, after carrying out the above-mentioned transformations, ¸ D sL E 0 Hi Hg (6.73) F G L Ii Ig 5 6 ˜11 H ˜i 1 0 Dh sHh 0 E 9 0 Dr sL Er Hi r 0 : : 9 7 Gr Ii r 0 8 0 Fr ˜g × × H From the linear system theory we know • (Fr > Dr ) is observable; • denoting the state vector of the sub-system (Dr > Er > Fr ) by {r , then there exists a matrix W such that {r = W {. Thus, based on sub-system model ¸ ¸ x x + Hi r i> ur = Fr {r + Gr + Ii r i {˙ r = Dr {r + Er | | we are able to construct a residual generator of the form ¸ μ ¸¶ x x ˙{ + Or ur Fr { ˆr + Er ˆr Gr ˆr = Dr { | | ¸ x u = Fr { ˆr + Gr |

(6.74)

(6.75) (6.76)

whose dynamics is governed by h˙ r = (Dr Or Fr )hr + (Hi r Or Ii r )i> u = Fr hr + Ii r i ˆr . with hr = {r { Naturally, the above-mentioned design scheme for UIDO is realizable only certain conditions are satisﬁed. The theorem given below provides us with a clear answer to this problem.

150

6 Perfect unknown input decoupling

Theorem 6.12 The following statements are equivalent: • There exists a UIDO; • In (6.73), block (Dr > Er > Fr > Gr ) exists and ¸ Hi r 6= 0 Ii r • The following condition holds true ¸ ¸ sL D Hi Hg sL D Hg ? udqn q + p= udqn F Ig F Ii Ig Due to the requirement on the knowledge of Kronecker canonical form and decomposition of matrix pencil, we omit the proof of this theorem and refer the interested reader to the references given at the end of this chapter. Nevertheless, we can see that the existence condition for a UIDO being designed by the matrix pencil approach described above is identical with the one stated in Theorem 6.2. This condition, as we have illustrated, is weaker than the one for UIO. As a summary, the design algorithm for UIDO using the matrix pencil approach is outlined below. Algorithm 6.11 The matrix pencil approach to the design of UIDO Step 1: Decompose the matrix pencil (6.71) into (6.73) by regular transformations; Step 2: Deﬁne UIDO according to (6.75)-(6.76) by choosing Or properly. 6.5.4 A numerical approach to the UIDO design The approach stated below is in fact a summary of the results presented in Subsections 5.7.1 and 6.2.3. Consider system model (3.30)-(3.31). As shown in Subsections 5.7.1 and 6.2.3, residual generator }˙ = J} + Kx + O|> u = y| z} tx

(6.77)

delivers a residual signal u whose dynamics, expressed in the non-recursive form, is governed by ¡ ¢ u(s) = zJv sv h(s) + yv Ki>v L¯i v iv (s) + Kg>v L¯gv gv (s) where

6.5 UIDO design

£ ¤ J = Jr j > Jr

0 0 ··· 0 91 0 ··· 0: : 9 : 9 = 9 ... . . . . . . ... : 5 Rv×(v1) : 9 70 ··· 1 08 0 ··· 0 1 6 F FD : ¤ £ : .. : = 0> yv = yv>0 · · · yv>v . 8

5 6 j1 9 9 9 : j = 7 ... 8 > yv 9 7 jv FDv £ ¤ z = 0 · · · 0 1 > y = yv>v > t = yG 5

5

yv>0 yv>1 .. .

9 9 K = W E OG> O = 9 7

yv>v1 5

yv>1 9 yv>2 9 W =9 . 7 .. yv>v 5 9 9 L¯i v = 9 9 7 5

yv>2 · · · yv>v1 · · · · · · yv>v .. ··· ··· . 0 ···

···

Lni ×ni zjLni ×ni .. . zJv1 jLni ×ni

(6.78)

(6.79)

(6.80) 6 : : : jyv>v 8

6 F 69 : yv>v 9 FD : 9 .. : : 0 :9 . : : .. : 9 : 9 . 8 9 ... : : 9 0 7 FDv2 8 FDv1

(6.81)

5

··· R .. .. . . .. .. . . R · · · zjLni ×ni Lni ×ni R .. .

(6.82)

6 : : : : 8

6 ··· R 9 .. : .. 9 zjLng ×ng . . : ¯ 9 : Lgv = 9 : .. . . . . 7 . . R 8 . zJv1 jLng ×ng · · · zjLng ×ng Lng ×ng 6 6 5 5 i (s)sv g(s)sv : : 9 9 .. .. : : 9 9 . . iv (s) = 9 : > gv (s) = 9 := 7 i (s)s1 8 7 g(s)s1 8 i (s) g(s) Lng ×ng

151

6

5

R .. .

Following Theorem 6.4, under condition ¤ £ ¤ £ udqn Ki>v Kr>v Kg>v A udqn Kr>v Kg>v

152

6 Perfect unknown input decoupling

we are able to solve equations £ ¤ yv Ki>v 6= 0 and yv Kr>v Kg>v = 0

(6.83)

for yv such that residual generator (6.77) becomes a UIDO, i.e. its dynamics fulﬁlls h˙ = Jh + (W Hi OIi )i> u = zh + yIi i in the recursive form or equivalently u(s) = zJv sv h(s) + yv Ki>v L¯i v iv (s) in the non-recursive form. In summary, we have Algorithm 6.12 The UIDO design approach by Ding et al. Step 1: Solve (6.83) for yv ; Step 2: Choose j and set J> K> O> t> W> y> z according to (6.78)-(6.81); Step 3: Construct residual generator according to (6.77). Example 6.7 We continue our study in Example 6.3 and now design a UIDO for the benchmark system EHSA. Remember that we have found out that beginning with v = 3 condition (6.83) is satisﬁed. Below, we design a reduced order UIDO (for v = 3) using the above algorithm. Step 1: Solve (6.83) for yv ¤ £ £ ¤ yv>0 = 6=73 × 1016 2=99 × 1012 > yv>1 = 3=04 × 1011 1=00 ¤ ¤ £ £ yv>2 = 8=79 × 1014 0=29 × 102 > yv>3 = 9=94 × 1017 3=27 × 106 Step 2: j is chosen to be

5

6 6=0 × 106 j = 7 1=1 × 105 8 6=0 × 102

and compute K> O> t> W> y> z> which results in 5 6 0=0005 4=26 × 108 0=4416 3=04 × 1011 1 W = 7 0 1=25 × 1012 0=0005 8=79 × 1014 0=0029 8 0 0 0 9=94 × 1017 3=27 × 106 5 6 5 6 5=96 × 1010 16=959 19=626 O = 7 1=94 × 1011 6=40 × 101 8 > K = 7 0 8 2=83 × 1014 9=3 × 104 0 £ ¤ £ ¤ y = 9=94 × 1017 3=27 × 106 > t = 0> z = 0 0 1 = Step 3: Construct residual generator according to (6.77) using the obtained system matrices.

6.7 An alternative scheme - null matrix approach

153

6.6 Unknown input parity space approach With the discussion in the last subsection as background, the parity space approach introduced in the last chapter can be readily extended to solve the PUIDP. Since the underlying idea and the solution are quite similar to the ones given in the last subsection, below we just give the algorithm for the realization of the unknown input parity space approach without additional discussion. Algorithm 6.13 The unknown input parity space approach Step 1: Solve

£ ¤ yv Ki>v 6= 0 and yv Kr>v Kg>v = 0

for yv ; Step 2: Construct residual generator as follows u(n) = yv (|v (n) Kx>v xv (n)) = Note that the application of this algorithm leads to a residual signal decoupled form g: u(n) = yv (|v (n) Kx>v xv (n)) = yv Ki>v iv (n)=

6.7 An alternative scheme - null matrix approach Recently, Frisk and Nyberg have proposed an alternative scheme to study residual generation problems and in particular to solve PUIDP. Below, we brieﬂy introduce the basic ideas of this scheme. Consider system model (3.29) and rewrite it into 6 5 ¸ ¸ g(s) |(s) J|g (v) J|i (v) J|x (v) 7 i (s) 8 = (6.84) = x(s) 0 0 L x(s) Now, we are able to formulate the residual generation problem in an alternative manner, i.e. ﬁnd a dynamic system U(v) 5 RH4 with | and x as its inputs and residual signal u as its output so that 5 6 ¸ ¸ g(s) |(s) J|g (v) J|i (v) J|x (v) 7 i (s) 8 u(s) = U(s) = U(v) x(s) 0 0 L x(s) ¸ ¸ J|g (v) J|i (v) g(s) = U(v) = (6.85) 0 0 i (s) In particular, if there exists a U(v) 5 RH4 such that

154

6 Perfect unknown input decoupling

¸ J|g (v) J|x (v) U(v) =0 0 L

(6.86)

then we have

¸ ¸ |(s) J|i (v) i (s)= = U(v) 0 x(s) ¸ J|g (v) J|x (v) Note that solving (6.86) is a problem of ﬁnding null matrix of = 0 L In this way, solving PUIDP is transformed into a problem of ﬁnding a null matrix. Nowadays, there exist a number powerful algorithms and software tools that provide us with numerically reliable and computationally e!cient solutions for (6.86). It is worth mentioning that application of the so-called minimal polynomial basis method for solving (6.86) leads to a residual generator of the minimum order.

u(s) = U(s)

6.8 Minimum order residual generator Remember that the minimum order of a parity relation or an observer-based residual generator is given by the minimum observability index plq . How can we design a UIDO or a parity relation based unknown input residual generator of a minimum order? The answer to this question is of practical interest, since a minimum order residual generator implies a minimal on-line computation. In Subsections 6.5.1, 6.4.2 and Section 6.7, we have mentioned that • the algebraic approach by Ge and Fang • the geometric approach and • the minimal polynomial basis method can be used to construct residual generators of a minimum order. Below, we shall introduce two approaches in details. 6.8.1 Minimum order residual generator design by geometric approach In this subsection, we propose a design procedure for constructing minimum order UIFDF based on the results achieved in Subsection 6.4.2. Assume that the existence condition (6.45) for a UIFDF is satisﬁed. Then, ¢ ¡ applying Algorithm 6.8 leads to an observable pair F¯21 > D¯221 as shown in (6.40). Now, instead of constructing a residual generator described by (6.42), we reconsider ¯x ¯ }˙21 = D¯221 }21 + W21 Ex + W21 Omax | = D¯221 }21 + E |¯ = Y2 | = F¯21 }21

(6.87) (6.88)

6.8 Minimum order residual generator

with

155

¸ ¤ £ x ¯ = W21 E W21 Omax > x ¯= = E |

¡ ¢ Suppose that the minimum observability index of the observable pair F¯21 > D¯221 is 2>min = It is known from Chapter 5 that the minimum order residual generator for (6.87)-(6.88) is 2>min and we are able to apply Algorithm 5.1 to design a (minimum order) residual generator with v = 2>min = To show that 2>min is also the minimum order of (reduced order) UIFDF, we call the reader’s attention to the following facts: Given system model (6.31) • any pair (O> Y ) that solves the PUIDP leads to (D OF> Hg > Y F) ¸ ¸ ¸¶ ˜ ˜g1 D11 D˜12 F˜11 F˜12 H > > 0 0 D˜22 0 F˜22 ´ ³ ˜g1 > F˜11 includes the perfect controllable • the subspace spanned by D˜11 > H ¯g1 > F¯1 ) given in Lemma 6.3 subspace (D¯11 > H ³ ´ ˜ 1 > Y˜1 > • by a suitable selection of a pair O μ

³ ´ ˜ 1 F˜11 > H ˜g1 > Y˜1 F˜11 D˜11 O ¸ ¸ ¸¶ μ ˜g1>1 F˜11>11 F˜11>12 D˜11>11 D˜11>12 H > > 0 0 D˜11>22 0 F˜11>22 ´ ³ ˜g1>1 > F˜11>11 is perfect controllable. where D˜11>11 > H • Due to the special form of μ ¸ ¸¶ D˜11>22 [ F˜11>22 [ > 0 D˜22 0 F˜22

(6.89)

with [ denoting some block of no interest, it is evident that the minimum order of the residual generator for the pair (6.89) is ³not larger´ than the minimum order of the residual generator for the pair D˜22 > F˜22 • the pair (6.89) is equivalent to the pair (D¯22 > F¯2 ) given in Theorem 6.9. Based on these facts, the following theorem becomes clear. Theorem 6.13 Given system (6.25)-(6.26) and suppose that the PUIDP is solvable. Then, using Algorithms 6.8 and 5.1, a minimum order UIFDF can be constructed. Algorithm 6.14 The geometric approach based design of minimum order residual generator

156

6 Perfect unknown input decoupling

Step 1: Apply Algorithm 6.8 to system (6.25)-(6.26) and bring the resulted system into form (6.87)-(6.88) Step 2: Find the minimum observability index 2>min and set v = 2>min Step 3: Using Algorithm 5.1 to construct a minimum order residual generator. Example 6.8 We now apply Algorithm 6.14 to design a minimum order UIDO for the benchmark system LIP100. For this purpose, we continue our study in Example 6.4, from which we can ﬁnd out 2>min = 1 and thus set v = 1= It follows the determination of the observer gain that is set to be 1=0 and the other system matrices (parameters). Fig.6.3 gives the response of the residual signal to a simulated fault that occurred at w = 15vhf. 0.5 0.4 0.3

Residual Signal

0.2 0.1 0 Ŧ0.1 Ŧ0.2 Ŧ0.3 Ŧ0.4 Ŧ0.5 0

5

10

15

20

25

30

35

40

45

50

Time [s]

Fig. 6.3 Response of the residual signal generated by a minimum order UIDO

6.8.2 An alternative solution A natural way to approach the reach for a minimum order residual generator is a repeated use of Algorithm 6.13 or 6.12 by increasing v step by step. This implies, however, equation £ ¤ (6.90) yv Ki>v 6= 0 and yv Kr>v Kg>v = 0 should be repeatedly solved, which, for a large v, results in an involved computation and may also lead to some numerical problems, for instance due to high power of D, Kr>v > Kg>v may become ill-conditioned. Below is an approach that oers a solution to this problem.

6.8 Minimum order residual generator

157

Recall that a parity vector can be parametrized by ¯ edvh>v > zv 6= 0> Tedvh>v = gldj(Qplq > · · · > Qpd{ 1 > Qpd{ > · · · > Qv ) yv = zv T Ql FDlr = 0> l = 1> · · · > pd{1 > Qpd{ = Qpd{ +1 = · · · = Qv = Lp×p and further yv Kg>v by

¯ edvh>v K ¯ g>v yv Kg>v = zv T

¯ g>v as deﬁned in (6.14), Dr > Or in (5.41). Note that the elements of with K matrix Dr are either one or zero, hence computation of high power of Dr is not critical. Moreover, for v pd{ we have Dvr = 0= Taking into account these facts, the following algorithm is developed, which can be used to determine the minimum order parity vector that ensures a residual generation decoupled from the unknown inputs. Algorithm 6.15 Calculation of minimum order parity vector Step 1: Transform (F> D) into its observer canonical form and determine the observability indices and matrices Dr > Or ; Step 2: Set initial conditions ¤ ¤ £ £ ˜ g>v = FDr v1 H ¯ g>v = K ¯g ¯g · · · F H ˜ g>v Ig > K v = plq > K ¯ edvh>v = Qvljpdplq > Qvljpdplq FDr plq = 0 Sr = T Step 3: Solve ¯ g>v = 0 zv Sr K If it is solvable, then set yv = zv Sr K1>v with K1>v as deﬁned in (6.13) and end; Step 4: If v = plq + pd{ , no solution and end; Step 5: If v ? pd{ 1, set ¸ ¯ g>v1 0 ¤ £ K v1 ¯ ˜ ¯ ¯ v = v + 1> Kg>v = ˜ g>v Ig > Kg>v = FDr Hg · · · F Hg K ¯ edvh>v1 > Qv )> Qv FDvr = 0 Sr = gldj(T and go to Step 3; Step 6: If v = pd{ 1, set ¸ ¯ g>v1 0 ¤ £ K v1 ¯ ˜ ¯ ¯ v = v + 1> Kg>v = ˜ g>v Ig > Kg>v = FDr Hg · · · F Hg K ¯ edvh>v1 > Lp×p ) Sr = gldj(T and go to Step 3;

158

6 Perfect unknown input decoupling

¯ g>v as a new matrix K ˆ g>v , remove them Step 7: Form the ﬁrst ng columns of K ¯ g>v as new K ¯ g>v , i.e. ¯ g>v and deﬁne the rest of K from K ¯ g>v (ng + 1> ) ¯ g>v = K K ¯ g>v (ng + ¯ g>v and K where denotes the number of the columns of the old K 1> ) the columns from the ng + 1 to the last one. Step 8: Set ¸ ¯ ¤ £ ˜ g>v = FDr pd{ 1 H ¯ g>v = Kg>v1 0 > K ¯g ¯g · · · F H v = v + 1> K ˜ Kg>v Ig and solve ˆ g>v = 0 S1 K for S1 and set

¯ edvh>v1 > Lp×p ) Sr = gldj(S1 T

and go to Step 3. The purpose of Algorithm 6.15 is to solve £ ¤ yv Kr>v Kg>v = 0 for yv with the minimum order. The underlying ideas adopted are • to do it iteratively, • to utilize the facts — for v = pd{ , FDvr = 0 and so for S 6= 0 6 6 5 5 ¯g ¯g FDr l H FDr l H : : 9 9 .. .. S7 8 = 0 =, gldj(S1 > Lp×p ) 7 8=0 . . v¯ v ¯ FDr Hg FDr Hg where

5

6 ¯g FDr l H 9 : .. S1 7 8=0 . v1 ¯ FDr Hg

— given matrices T1 > T2 of appropriate dimensions the solvability of the following two equations ¤ £ S T1 T2 = 0 and S1 S2 T2 = 0> S2 T1 = 0 are identical, moreover S = S1 S2 .

6.9 Notes and references

159

• following Theorem 6.3 we only need to carry out searching up to v = pd{ + plq . In comparison with a direct solution of equation (6.90), using Algorithm 6.15 has the following advantages: • The highest power of Dr is limited to pd{ 1; • The maximally dimensional linear equation to be solved is 5 6 0 ··· 0 Ig 9 . : . ¯g 9 FH Ig . . .. : 9 :=0 zv Sr 9 : .. .. .. 7 . . 0 8 . ¯g Ig ¯g · · · F H FDr pd{ 1 H which is the case for v = pd{ + plq and whose dimension is not larger than p( pd{ + 1) × ng ( pd{ + 1). Note that in the same case a direct solution of (6.90) implies 5 6 0 ··· 0 F Ig 9 . : . 9 FD FHg Ig . . .. : :=0 yv 9 9 : .. .. .. .. 7 . 0 8 . . . FDpd{ +plq FDpd{ +plq 1 Hg · · · FHg Ig whose dimension amounts to p( pd{ + plq + 1) × (q + ng ( pd{ + plq + 1))= Note that the one-to-one mapping between the parity vector and DO design also allows applying Algorithm 6.15 for the design of minimum order UIDO.

6.9 Notes and references Unknown input decoupling was an attractive research topic in the past two decades. In this chapter, we have only introduced some representative methods aiming at demonstrating how to approach the relevant issues around this topic. The existence conditions for the PUIDP, expressed in terms of the rank of transfer matrices, was ﬁrst derived by Ding and Frank [38]. Using matrix pencil technique, Patton and Hou [76] have given a proof of the check condition described by the rank of Rosenbrock system matrix, which, dierent from the proof given in Subsection 6.2.2, requires the knowledge of the matrix pencil technique. The existence conditions expressed by the rank of parity space matrices Kr>v > Kg>v have been studied by Chow and Willsky [23], and subsequently by Wuennenberger [148]. The existence condition (6.11) and the

160

6 Perfect unknown input decoupling

results described by Theorem 6.3 have been lately presented, for instance by Ding et al. [42]. Concerning the solution of the PUIDP, we have introduced dierent methods. Signiﬁcant contributions to the frequency domain approaches have been made by Frank with his co-worker [38, 54] and Viswanadham et al. [142]. The eigenstructure assignment approach presented in Subsection 6.4.1 is a summary of the work by Patton and his research group [119]. Massoumnia [102] has initiated the application of the geometric theory to the FDI system design. However, considering the demand on the knowledge of geometric theory, which seems di!cult for the readers without profound knowledge of the advanced control theory, we have adopted a modiﬁed form for the description of this approach. Most of those results can be, in the dual form, found in the books by Wonham [147] and Kailath [87]. Algorithm 6.6 is given in [12]. UIDO and parity space type residual generator design are the two topics in the ﬁeld of model-based FDI which received much attention in the last two decades. The contributions by Chow and Willsky [23] using the parity space approach, by Ge and Fang [61] (see Subsection 6.5.1) and by Wuennenberg and Frank [149] using the Kronecker canonical form are the pioneering works devoted to these topics, in which, above all, the original ideas have been proposed. Their works have been followed by a great number of studies, e.g. the one on the use of UIO technique made by Hou and Müller [73], the matrix pencil approach developed by Patton and Hou [76], in which matrix pencil decompositions are necessary and thus the use of a matrix pencil decomposition technique proposed by Van Dooren [45] is suggested, as well as the work by Wuennenberg [148], just mention some of them. We would like to point out that in this chapter we have only presented the original and simplest form of the UIO technique, although it is one of widely used approach and is of a number of presentation forms, see for instance [74, 141]. The reason why we did not present more lies in the fact that the application of this approach for the FDI purpose is restricted due to the existence conditions. They are stronger than most of the other approaches described in this chapter. We refer the reader to the survey papers, e.g. [50, 51, 52], and the references given there for more information about this technique. The alternative scheme for residual generation and PUIDP solution by means of null matrix formulation has been recently proposed in [57, 139]. Finally, we would like to mention that only few studies on the design of minimum or low order residual generators have been reported, although such residual generators are of practical interest, due to their favorable on-line computation.

7 Residual generation with enhanced robustness against unknown inputs

It has been early recognized that the restriction on the application of the perfect decoupling technique introduced in the last chapter may be too strong for a realistic dissemination of this technique in practice. Taking a look at the general existence condition for a residual generator perfectly decoupled from unknown inputs, udqn (J|g (s)) ? p it becomes clear that a perfect decoupling is only possible when enough number of sensors are available. This is often not realistic from the economic viewpoint. Furthermore, if model uncertainties are unstructured and disturbances possibly appear in all directions of the measurement subspace, the decoupling approaches introduced in the last chapter will fail. Since the pioneering work by Lou et al., in which the above problems were, for the ﬁrst time, intensively and systematically studied and a solution was provided, much attention has been devoted to this topic. The rapid development of robust control theory in the 80’s and early 90’s gave a decisive impulse for the establishment of a framework, in which approaches and tools

Fig. 7.1 Schematic description of robust residual generation

162

7 Residual generation with enhanced robustness against unknown inputs

to deal with robustness issues in the FDI ﬁeld are available. The major objective of this chapter is to present those advanced robust FDI approaches and the associated tools, which are becoming popular for the robust residual generator design. Dierent from a perfect decoupling, the residual generators studied in this chapter will be designed in the context of a trade-o between the robustness against the disturbances and the sensitivity for the faults. As a result, the generated residual signal will also be aected by the disturbances, as shown in Fig.7.1. Generally speaking, robust FDI problems can be approached in three different manners: • making use of knowledge of the disturbances A typical example is the Kalman ﬁlter approach, in which it is assumed that the unknown input is white noise. ¯ |g (s) which, on the one side, • approximating J|g (s) by a transfer matrix J satisﬁes the existence conditions for a PUID and, on the other side, provides an optimal approximation (in some sense) to the original one It is evident that the design procedure of this scheme would consist of two steps: the ﬁrst one is the approximation and the second one the solution ¯ |g (s) of PUIDP based on J • designing residual generators under a certain performance index A reasonable extension of the PUIDP is, instead of a perfect decoupling, to make a compromise between the robustness against the unknown input and the sensitivity to the faults. This compromise will be expressed in terms of a performance index, under which the residual generator design will then be carried out. In the forthcoming sections, we are going to describe these three types of schemes, concentrate ourselves, however, on the third one, due to its important role both in theoretical study and practical applications.

7.1 Mathematical and control theoretical preliminaries Before we begin with our study on the robustness issues surrounding FDI system design, needed mathematical and control theoretical knowledge, skills and associated tools, including • • • • • •

norms for signals and systems algorithms for norm computation singular value decomposition (SVD) co-inner-outer factorization (CIOF), H4 solutions to model matching problem (MMP) and linear matrix inequality (LMI) technique,

will be introduced in this section. Most of them are standard in linear algebra and robust control theory. The detailed treatment of these topics can be found in the references given at the end of this chapter.

7.1 Mathematical and control theoretical preliminaries

163

7.1.1 Signal norms In this subsection, we shall answer the question: how to measure the size of a signal. Measuring the size of a signal in terms of a certain kind of norm is becoming the most natural thing in the world of control engineering. A norm is a mathematical concept that is originally used to measure the size of functions. Given signals x> y 5 6 5 6 1 x1 9 .. : 9 .. : q x = 7 . 8 5 R > y = 7 . 8 5 Rq xq

q

then a norm must have the following four properties: I. kxk 0 II. kxk = 0 +, x = 0 III. kdxk = |d|kxk> d is a constant IV. kx + yk kxk + kyk. Next, three types of norms, which are mostly used in the control engineering and also for the FDI purpose, are introduced. L1 norm: The L1 norm of a vector-valued signal x(w) or x(n) is deﬁned by ! Ã4 q Z 4 q X X X |xl (w)|gw or kxk1 = |xl (n)| = (7.1) kxk1 = l=1

0

l=1

n=0

L2 norm: The L2 norm of a vector-valued signal x(w) or x(n) is deﬁned by μZ kxk2 =

0

4

Ã4 !1@2 ¶1@2 X W x (w)x(w)gw or kxk2 = x (n)x(n) = W

(7.2)

n=0

The L2 norm is associated with energy. While xW (n)x(n) or xW (n)x(n) is generally interpreted as the instantaneous power, kxk22 stands for the total energy. In practice, the root mean square (RMS), instead of L2 norm, is often used. The RMS measures the average energy of a signal over a (large) time interval (0> W ) and is deﬁned by 3 kxkUPV

1 =C W

ZW

41@2 xW ( )x( )g D

=

(7.3)

0

It follows from the Parseval Theorem that the size computation of a signal can also be carried out in the frequency domain:

164

7 Residual generation with enhanced robustness against unknown inputs

Z

4

xW (w)x(w)gw =

0

1 2

Z

4

xW (m$)x(m$)g$

4

for the continuous time signal and 4 X n=0

1 x (n)x(n) = 2 W

Z

xW (hm )x(hm )g

for the discrete time signal, where x(m$) = F(x(w)) and x(hm ) = F(x(n)) with F denoting the Fourier transformation. L4 norm: The L4 norm of a signal x(w) or x(n) is the least upper bound of its absolute value: kxk4 = max sup |xl (w)| or kxk4 = max sup |xl (n)|= l

l

w

n

The L4 norm is the maximum amplitude of a signal. In the FDI study, we are often interesting in checking whether the peak amplitude of a vector-valued residual is below a given threshold. To this end, we introduce next the so-called peak-norm. Peak norm: The peak norm of x 5 Rq is deﬁned by ¡ ¡ ¢1@2 ¢1@2 kxkshdn = sup xW (w)x(w) or kxkshdn = sup xW (n)x(n) = w

n

Remark 7.1 By introducing the above deﬁnitions we have supposed that the signal under consideration is zero for w ? 0, i.e. it starts at time w = 0. A direct application of the signal norms in the FDI ﬁeld is the residual evaluation, where the size (in the sense of a norm) of the residual signal will be on-line calculated and then compared with the given threshold. Since evaluation over the whole time or frequency domain is usually unrealistic, introducing an evaluation window is a practical modiﬁcation. For our purpose, following deﬁnitions are introduced: kxk1> =

q Z X

w1

l=1

μZ kxk2> =

|xl (w)|gw or kxk1> =

n2 q X X

|xl (n)|

W

$2

n=n1

¶1@2

1 xW (m$)x(m$)g$ 2 $1 = max sup |xl (w)| or kxk4> = max sup |xl (n)| l

(7.4)

l=1 n=n1

Ã n !1@2 ¶1@2 2 X W x (w)x(w)gw or kxk2> = x (n)x(n) (7.5)

Z

kxk2>! = kxk4>

w2

w1

μ

w2

w5

l

n5

(7.6) (7.7)

7.1 Mathematical and control theoretical preliminaries

165

where = (w1 > w2 ) or = (n1 > n2 ) and ! = ($ 1 > $ 2 ) stand for the time and frequency domain evaluation windows. Note that a discrete time signal over a time interval can also be written into a vector form. For instance, in our study on parity space methods, we have used the notation gv (n) 5 6 g(n) 9 g(n + 1) : 9 : gv (n) = 9 : .. 7 8 . g(n + v) to represent the disturbance in the time interval [n> n + v] = Thus, in this sense, we are also able to use vector norms to the calculation of the size of a (discrete time) signal. Corresponding to the above-mentioned three kinds of signal norms, we introduce following vector norms: 1 norm: v X |x(n + l)|= (7.8) kxk1>v = l=0

2 norm: kxk2>v =

Ã v X

!1@2 2

x (n + 1)

=

(7.9)

l=0

4 norm:

kxk4>v =

sup

|x(n)|=

(7.10)

l5[n>n+v]

It is obvious that the computation of a vector norm is much more simple than the one of a signal norm. 7.1.2 System norms In this subsection, we shall answer the question: how to measure the size of a system. Consider a dynamic system |(s) = J(s)x(s). For our purpose, we only consider those LTI systems, which are causal and stable. Causality means J(w) = 0 for w ? 0 or J (n) = 0 for n ? 0 with J(w) or J (n) as impulse response. Mathematically, the causality requires that J(s) is proper, i.e. lim J(s) ? 4=

s$4

A system is called strictly proper if lim J(s) = 0=

s$4

System J(v) or J(}) is called stable if it is analytic in the closed RHP (Re (v) 0) or for |}| 1=

166

7 Residual generation with enhanced robustness against unknown inputs

One way to describe the size of the transfer matrix J(s) is in terms of norms for systems or norms for transfer function matrices. There are two different ways to introduce norms for systems. From the mathematical viewpoint J is an operator that maps the vector-valued input function x to the vectorvalued output function |. The operator norm kJks is deﬁned in terms of the norms of input and output functions as follows: k|ks kJxks = sup = x6=0 kxks x6=0 kxks

kJks = sup

(7.11)

It is thus also known as induced norm. Suppose that the input signal x is not ﬁxed and can be any signal of L2 norm. It turns out k|k2 kJxk2 = sup = sup ¯ (J($)) kxk 2 x6=0 x6=0 kxk2 $5[0>4]

sup

(7.12)

for continuous time systems and ¡ ¢ k|k2 kJxk2 = sup = sup ¯ J(hm ) kxk kxk 2 2 x6=0 x6=0 5[0>2]

sup

¢ ¡ for discrete time systems, where ¯ (J($)) or ¯ J(hm ) denotes the maximum singular value of J(m$) or J(hm ). This induced norm equals to the H4 norm of J(s) deﬁned by H4 norm: ¡ ¢ ¯ (J($)) or kJk4 = sup ¯ J(hm ) = (7.13) kJk4 = sup $5[0>4]

5[0>2]

H4 norm can be interpreted as the ampliﬁcation of a transfer function matrix that maps the input signal with ﬁnite energy but being any kind of signals into the output signal. Remember that in the design form of a residual generator, ˆ x (s) (J|i (s)i (s) + J|g (s)g(s)) u(s) = U(s)P both signals, g(s)> i (s)> are unknown. If their energy level is bounded, then H4 norm can be used to measure their inﬂuence on the residual signal. It is interesting to note that even if the input signal x is not L2 bounded but kxkUPV ? 4> we have sup x6=0

k|kUPV = sup ¯ (J($)) = kJk4 = kxkUP V $

(7.14)

Let J(w) (for continuous time) as well as J(0)> J(1)> · · · (for discrete time) be the impulse response function of system |(s) = J(s)x(s) 5 R, then H1 norm:

7.1 Mathematical and control theoretical preliminaries

Z kJk1 =

4

4

|J( )|g or kJk1 = |J(0)| + |J(1)| + · · · =

167

(7.15)

The H1 norm of J is the L4 /L4 norm induced norm, i.e. k|k4 kJxk4 = sup = kJk1 = kxk 4 x6=0 x6=0 kxk4

(7.16)

sup

Thus, the H1 norm of J can be interpreted as the ampliﬁcation of the maximum value of the input signal. In the FDI study, for |(s) 5 Rp > k|kshdn instead of k|k4 is often used for the purpose of residual evaluation. In this case, peak-to-peak gain: kJkshdn = sup x6=0

k|kshdn kxkshdn

(7.17)

is useful for the threshold computation. A further induced norm is the so-called generalized H2 norm, generalized H2 norm: k|kshdn x6=0 kxk2

kJkj = sup

(7.18)

which is rarely applied in the control theory but provides us with a helpful tool to answer the question: how large does the disturbance (input variable) with bounded energy cause instantaneous power change in the residual signal (output variable)? Another norm for transfer function matrices is H2 norm: μ kJk2 = μ kJk2 =

1 2 1 2

Z

4

¡ ¢ wudfh JW (m$)J(m$) g$

4

Z

0

2

¡ ¢ wudfh JW (hm )J(hm ) g

¶1@2 or

(7.19)

¶1@2 =

(7.20)

H2 norm is not an induced norm, but widely used in the control theory. Given transfer matrix J> when the input is a realization of a unit variance white noise process, then the H2 norm of J equals to the expected RMS value of the output. A well-known application example of the H2 norm is the optimal Kalman ﬁlter, in which the H2 norm of the transfer function matrix from the noise to the estimation error is minimized. Motivated by the study on parity space methods, we introduce next some norms for matrices. Compared with the norms for transfer function matrices, the norms for matrices are computationally much simpler. Let J 5 Rp×q be a matrix with elements Jl>m > l = 1> · · · > p> m = 1> · · · > q, then we have

168

7 Residual generation with enhanced robustness against unknown inputs

matrix norm induced by the 2 norm for vectors, which is also called spectral norm: ³ ´1@2 kJxk2 = ¯ (J) = max l (JW J) (7.21) kJk2 = sup l x6=0 kxk2 Frobenius- or Euclidian norm: 3 41@2 Ã !1@2 q p X q X X 2 W |Jlm | D = l (J J) kJkI = C l=1 m=1

1 norm: kJk1 = max m

for which equation max m

(7.22)

l=1

p X

p X

|Jlm |

(7.23)

kJxk1 x6=0 kxk1

(7.24)

l=1

|Jlm | = sup

l=1

holds, i.e. kJk1 is a matrix norm induced by the 1 norm for vectors. 4 norm: q X kJk4 = max |Jlm | l

m=1

which also equals to the induced norm by the 4 norm for vectors, i.e. max l

q X m=1

kJxk4 = x6=0 kxk4

|Jlm | = sup

(7.25)

7.1.3 Computation of H2 and H4 norms Suppose system J(s) has a minimal state space realization J(s) = G+F(sL D)1 E and D is stable, then • for continuous time systems kJk2 is ﬁnite if and only if G = 0 and kJk2 = wudfh(FS F W ) = wudfh(E W TE)

(7.26)

where S> T are respectively the solution of Lyapunov equations DS + S DW + EE W = 0> TD + DW T + F W F = 0

(7.27)

• for discrete time systems kJk2 = wudfh(FS F W + GGW ) = wudfh(E W TE + GW G)

(7.28)

where S> T are respectively the solution of Lyapunov equations DS DW S + EE W = 0> DW TD T + F W F = 0=

(7.29)

7.1 Mathematical and control theoretical preliminaries

169

Unlike the H2 norm, an iterative procedure is needed for the computation of the H4 norm, where an algorithm of determining whether kJk4 ? will be repeatedly used until inf {kJk4 ? } := kJk4

is found. Below is the so-called Bounded Real Lemma that characterizes the set {kJk4 ? }. Lemma 7.1 Given a continuous time system J(v) = G + F(vL D)1 E 5 RH4 , then kJk4 ? if and only if U := 2 L GW G A 0 and there exists S = S W 0 satisfying the Riccati equation S (D + EU1 GW F) + (D + EU1 GW F)W S + S EU1 E W S + F W (L + GU1 GW )F = 0= Lemma 7.2 Given a discrete time system J(}) = G + F (}L D)1 E 5 RH4 > then kJk4 ? if and only if <[ 0 such that 2 L GW G E W [E A 0 D¯W [ D¯ [ D¯W [J (L + [J)1 [ D¯ + 2 T = 0 ¢1 W ¡ ¢1 W ¡ G F> J = E 2 L GW G E D¯ = D + E 2 L GW G ¡ 2 ¢1 W W T=F L G G F and (L + [J)1 D¯ is stable. We see that the core of the above computation is the solution of Riccati equations which may be, when the system order is very high, computationally consuming. There exists a number of CAD programs for that purpose. In subsection 7.1.7, an LMI based algorithm will be introduced for the H4 norm computation as well as the computation of other above-mentioned norms. 7.1.4 Singular value decomposition The SVD of a matrix J 5 Rq×n is expressed by J = X Y W where X 5 Rq×q > Y 5 Rn×n > X X W = Lq×q > Y Y W = Ln×n

170

7 Residual generation with enhanced robustness against unknown inputs

and for q A n

=

for q n

gldj( 1 > · · · > n ) 0(qn)×n

¸

¤ £ = gldj(1 > · · · > q ) 0q×(nq)

with 1 2 · · · n denoting the singular values of J. The SVD of J 5 Rq×n is of the following two interesting properties: n X ° ° kJkI = °X Y W °I = k kI = l

(7.30)

l=1

kJk4 = kX Y W k4 = k k4 = 1 =

(7.31)

7.1.5 Co-inner-outer factorization Inner-outer factorization (IOF) technique is a powerful tool for solving robustness related control problems. For the FDI purpose, the so-called coinner-outer factorization (CIOF) plays an important role. Roughly speaking, a CIOF of a transfer function matrix J(s) is a decomposition of J(s) into J(s) = Jfr (s)Jfl (s)

(7.32)

where Jfl (s) is called co-inner and satisﬁes Jfl (m$)Jfl (m$) = L for all $ (for continuous time systems) or J (hm )J(hm ) = L for all 5 [0> 2] (for discrete time systems) and Jfr (s) is called co-outer and has as its zeros all the (left) zeros of J(s) in the LHP and on the m$-axis, including at inﬁnity (for continuous time systems), or within |}| 1 (for discrete time systems). IOC is a dual form of CIOF and thus the IOC of J(s) can be expressed in terms of the CIOF of JW (s) = (Jfr (s)Jfl (s))W = Jl (s)Jr (s)= Jl (s)> Jr (s) are respectively called inner and outer of J(s)= In most of textbooks on robust control, study is mainly focused on IOF instead of CIOF. Also, it is generally presented regarding to continuous time systems. Next, we shall introduce the existence conditions for CIOF and the associated algorithms by "translating" the results on IOF into the ones of CIOF based on the duality. We ﬁrst introduce some relevant deﬁnitions. A rational matrix J(s) is called surjective if it has full row rank for almost all s and injective if it has full column rank for almost all s= A co-outer is analytic in C¯+ and has a left inverse analytic in C+ = If there exists J (s) 5 RH4 such that J (s)J(s) = L, then J(s) is called left invertible in RH4 . The following results are well-known in the robust control theory. is surjective and Lemma 7.3 Assume that J(s) 5 LHp×n 4 • in case of a continuous time system: ;$ 5 [0> 4] rank (J(m$)) = p +, J(m$)J (m$) A 0

(7.33)

7.1 Mathematical and control theoretical preliminaries

• in case of a discrete time system: ; 5 [0> 2] ¢ ¡ rank J(hm ) = p +, J(hm )J (hm ) A 0

171

(7.34)

then there exists a CIOF J(s) = Jfr (s)Jfl (s)=

(7.35)

is surjective and ;$ 5 [0> 4] Lemma 7.4 Assume that J(v) 5 LHp×n 4 J(m$)J (m$) A 0=

(7.36)

ˆ (v) that also gives a CIOF ˆ 1 (v)Q Then there exists an LCF J(v) = P ˆ (v) = Jfr (v)Jfl (v) ˆ 1 (v)Q J(v) = P

(7.37)

ˆ 1 (v) as co-outer and Jfl (v) = Q ˆ (v) 5 RH4 as co-inner. with Jfr (v) = P This factorization is unique up to a constant unitary multiple. If J(v) 5 RH4 > then J1 fr (v) 5 RH4 = Furthermore, assume that the realization of J(v) = (D> E> F> G) with D 5 Rq×q is detectable and ;$ 5 [0> 4] ¸ D m$L E rank = q + p= (7.38) F G Then the above LCF can be expressed by ˆ (v) = (D OF> O> TF> T)> Q ˆ (v) = (D OF> E OG> TF> TG) 5 RH4 P W 1@2 W T = (GG ) > O = (\ F + EGW )(GGW )1 (7.39) where \ 0 is the stabilizing solution of the Riccati equation D\ + \ DW + EE W (\ F W + EGW )(GGW )1 (F\ + GE W ) = 0=

(7.40)

is surjective and ; 5 [0> 2] Lemma 7.5 Assume that J(}) 5 LHp×n 4 J(hm )J (hm ) A 0=

(7.41)

ˆ (}) that also gives a CIOF ˆ 1 (})Q Then there exists an LCF J(}) = P ˆ (}) = Jfr (})Jfl (}) ˆ 1 (})Q J(}) = P

(7.42)

ˆ 1 (}) as co-outer and Jfl (}) = Q ˆ (}) 5 RH4 as co-inner. with Jfr (}) = P 1 If J(}) 5 RH4 > then Jfr (}) 5 RH4 = This factorization is unique up to a constant unitary multiple. Furthermore, assume that the realization of J(}) = (D> E> F> G) with D 5 Rq×q is detectable and ; 5 [0> 2] ¸ D hm L E rank = q + p= (7.43) F G

172

7 Residual generation with enhanced robustness against unknown inputs

Then the above LCF can be expressed by ˆ (v) = (D OF> O> UF> U)> Q ˆ (v) = (D OF> E OG> UF> UG) 5 RH4 P ¢1@2 ¡ U = GGW + F[F W > O = (DW [F W + EGW )(GGW + F[F W )1 (7.44) where [ 0 is the stabilizing solution of the Riccati equation ³ ´1 ¡ ¢1 W F[ DWG [ + EGB GB E W = 0 DG [ L + F W GGW ¡ ¢1 GE W = DG = D F W GGW

(7.45)

Note that Lemmas 7.4 and 7.5 establish an important connection between CIOF and LCF, which is useful for our latter study. In Lemmas 7.3 - 7.5, the LCF is achieved on the assumption that the transfer matrix is surjective. Removing this condition, the LCF would be computationally more involved. Below, we introduce a recent result by Oara and Varga for a general CIOF. For the sake of simpliﬁcation, we restrict ourselves to the continuous time systems. be a real rational matrix of rank u. A CIOF Lemma 7.6 Let J(v) 5 RHp×n 4 J(v) = Jfr (v)Jfl (v) with Jfl (v) co-inner and Jfr (v) co-outer, can be computed using the following two-step algorithm: • Column compression by all-pass factors: J is factorized as £ ¤ ˜ J(v) = J(v) 0 Jd (v) ˜ where Jd (v) is square and inner, J(v) 5 RHp×u is injective and has the ¯ same zeros in C+ as J(v)> and its zeros in C include the zeros of J1 d (v)= By this step, Jd (v) is chosen to have the smallest possible Mcmillan degree which is equal to the sum of all right minimal indices of J(v)= The computation of Jd (v) amounts to solving for the stabilizing solution of a ˜ standard Riccati£ equation. J(v) into J(v) = J(v)J d1 (v)> ¤ can be rewritten u×n ˜ is inner. where Jd (v) = Jd1 (v) Jd (v) > Jd1 (v) 5 R ˜ ˜ • Dislocation of zeros by all-pass factors. J(v) is further factorized as J(v) = ¯ r (v) is injective and has ¯ r (v)Jd2 (v), where Jd2 (v) is square, inner and J J no zeros in C+ . By this step, Jd2 (v) is chosen to have the smallest possible ˜ Mcmillan degree which is equal to the number of zeros of J(v) in C+ . The computation of Jd2 (v) is achieved by solving a Lyapunov equation. The CIOF is ﬁnally given by ¯ r (v)> Jfl (v) = Jd2 (v)Jd1 (v)= J(v) = Jfr (v)Jfl (v)> Jfr (v) = J

(7.46)

7.1 Mathematical and control theoretical preliminaries

173

7.1.6 Model matching problem H4 optimization technique is one of the most celebrated frameworks in the control theory, which has been well established between the 80’s and 90’s. The application of H4 optimization technique to the FDI system design is many-sided and covers a wide range of topics like design of robust FDF, fault identiﬁcation, handling of model uncertainties, threshold computation etc. MMP is a standard problem formulation in the H4 framework. Many approaches to the FDI system design can be, as will be shown in the next sections, reformulated into an MMP. The MMP met in the FDI framework is often of the following form: given W1 (v)> W2 (v) 5 RH4 > ﬁnd U(v) 5 RH4 so that kW1 (v) U(v)W2 (v)k4 $ min = (7.47) The following result oers a solution to the MMP in a way that is very helpful for the FDI system design. Lemma 7.7 Given (scalar) transfer functions W1 (v)> W2 (v)> U(v) 5 RH4 and assume that W2 (v) has zeros v = vl > l = 1> · · · > s> in the RHP, then 1@2

¯ kW1 (v) U(v)W2 (v)k4 =

(W )

(7.48)

¯ ) denotes the maximum eigenvalue of matrix W which is formed as where (W follows: • form 5

1 v1 +v 1

9 S1 = 7 · · ·

1 vs +v 1

··· 1 vl +v m

···

1 v1 +v s

5 W1 (v1 )W (v1 )

6

···

1

9 ··· : 8 > S2 = 9 7

1 vs +v s

v1 +v 1

···

W1 (vs )W1 (v1 ) vs +v 1

W1 (vl )W1 (vm ) vl +v m

···

W1 (v1 )W1 (vs ) v1 +v s

··· W1 (vs )W1 (vs ) vs +v s

6 : : 8

(7.49) • set

1@2

W = S1

1@2

S2 S1

=

(7.50)

It follows from Lemma 7.7 that the model matching performance depends on the zeros of W2 (v) in the RHP. Moreover, if W1 (v) = > a constant, then kW1 (v) U(v)W2 (v)k4 = || =

(7.51)

These two facts would be useful for our subsequent study. 7.1.7 Essentials of the LMI technique In the last decade, the LMI technique has become an important formulation and design tool in the control theory, which is not only used for solving standard robust control problems but also for multiobjective optimization and

174

7 Residual generation with enhanced robustness against unknown inputs

handling of model uncertainties. As FDI problems are in their nature a multiobjective trade-o, i.e. enhancing the robustness against the disturbances, model uncertainty and the sensitivity to the faults simultaneously, application of the LMI technique to the FDI system design is currently receiving considerable attention. In the H4 framework, the Bounded Real Lemma that connects the H4 norm computation to an LMI plays a central role. Next, we brieﬂy introduce the "LMI-version" of the Bounded Real Lemma for continuous and discrete systems. 1

Lemma 7.8 Given a stable LTI system J(v) = G + F (vL D) kJ(v)k4 ? if and only if there exists a symmetric \ with 6 5 W D \ + \ D \ E FW 7 EW \ L GW 8 ? 0> \ A 0= F G L

E, then

(7.52) 1

Lemma 7.9 Given a stable LTI system J(}) = G + F (}L D) kJ(})k4 ? if and only if there exists a symmetric [ with 5 6 [ [D [E 0 9 DW [ [ 0 F W : 9 W : 7 E [ 0 L GW 8 ? 0> [ A 0= 0 F G L

E, then

(7.53)

In the LMI framework, the so-called Schur complement is often used for checking the deﬁniteness of a matrix. Given matrix ¸ D11 D12 D= D21 D22 and suppose that D11 is invertible, then D A 0 or D ? 0

(7.54)

= D22 D21 D1 11 D12 A 0 or ? 0

(7.55)

if and only if where is known as the Schur complement of D=

7.2 Kalman ﬁlter based residual generation In this section, we present one of the ﬁrst residual generation schemes, the Kalman ﬁlter based residual generation scheme. Consider a discrete time dynamic system described by

7.2 Kalman ﬁlter based residual generation

{(n + 1) = D{(n) + Ex(n) + Hi i (n) + H (n) |(n) = F{(n) + Gx(n) + Ii i (n) + (n)

175

(7.56) (7.57)

where {(n) 5 Rq > x(n) 5 Rnx > |(n) 5 Rp are the state, input, output vectors of the system, i (n) 5 Rni stands for the fault vector. (n) 5 Rn > (n) 5 Rp represent process and measurement noise vectors. It is evident that for such a system there exists no residual generator decoupled from the unknown inputs (n)> (n). On the other hand, from the well-established stochastic control theory we know that a Kalman ﬁlter delivers residual that is a white Gaussian process if the noise signals (n)> (n) are white Gaussian processes and independent of initial state vector {(0) with H[(n)] = 0> H[(l) W (m)] = lm > 0 H[(n)] = 0> H[(l) W (m)] = lm > A 0 H[{(0)] = { ¯> H[({(0) { ¯)({(0) { ¯)W ] = Sr =

(7.58) (7.59) (7.60)

The Kalman ﬁlter technique makes use of this fact and performs a fault detection in two steps: • residual generation using a Kalman ﬁlter • residual evaluation by doing the so-called Generalized Likelihood Ratio (GLR) test that allows us to detect changes in the residual signal. In Chapter 10, the GLR test will be studied. In this section, we devote our attention to the problem of residual generation using a Kalman ﬁlter. We suppose that the noises (n)> (n) and the initial state vector {(0) possess the properties described by (7.58)-(7.60). A Kalman ﬁlter is, although structured similar to an observer of full order, a time-varying system given by the following recursions: recursive scheme for optimal state estimation: { ˆ (0 | 0) = { ¯ { ˆ(n | n 1) = Dˆ {(n 1 | n 1) + Ex(n 1)> n = 1> 2> · · · { ˆ(n | n) = { ˆ(n | n 1) + O(n) (|(n) F { ˆ(n | n 1) Gx(n)) =

(7.61) (7.62) (7.63)

recursive scheme for Kalman ﬁlter gain: S (0 | 0) = Sr

(7.64)

S (n | n 1) = DS (n 1 | n 1)D + H H ¡ ¢1 O(n) = S (n | n 1)F W + FS (n | n 1)F W S (n | n) = (L O(n)F) S (n | n 1)> n = 1> 2> · · · W

W

(7.65) (7.66) (7.67)

where { ˆ(l | m) denotes the estimation of {(l) given the measurements |(1)> · · · > |(m) and

176

7 Residual generation with enhanced robustness against unknown inputs

¢ ¡ S (n | n) = H [{(n) { ˆ(n | n)][{(n) { ˆ(n | n)]W ¢ ¡ S (n | n 1) = H [{(n) { ˆ(n | n 1)][{(n) { ˆ(n | n 1)]W

(7.68) (7.69)

are the associated estimation error covariance. The signiﬁcant characteristics of Kalman ﬁlter is • the state estimation is optimal in the sense of S (n | n) = H[{(n) { ˆ(n | n)][{(n) { ˆ(n | n)]W = plq • the so-called innovation process, |(n) F { ˆ(n | n 1) Gx(n) is a white Gaussian process with covariance + FS (n | n 1)F W = The underlying idea of applying Kalman ﬁlter to solve FDI problems lies in making use of the second property. Let residual signal u(n) be the innovation process u(n) = |(n) F { ˆ(n | n 1) Gx(n)= Under the normal operating condition, i.e. fault-free, u(n) should be a zero mean white Gaussian process. When a fault occurs, i.e. i (n) 6= 0, u(n) is no longer white, which can be determined, for instance, by means of a GLR test that will be discussed in the next part. In such a way, a successful fault detection is performed. Note that the signal F { ˆ(n | n 1) + Gx(n) is in fact an optimal estimation of the measurement |(n). Remark 7.2 Although given in the recursive form, the Kalman ﬁlter algorithm (7.61)-(7.63) is highly computation consuming. The most involved com¢1 ¡ > which may also cause numerical staputation is + FS (n | n 1)F W bility problem. There are a great number of modiﬁed forms of the Kalman ﬁlter algorithm. The reader is referred to the references given at the end of this chapter. Suppose the process under consideration is stationary, then lim O(n) = O = constant matrix

n$4

which is subject to with

¡ ¢1 O = \ F W F\ F W +

(7.70)

³ ¡ ¢1 ´ \ = L [F W F\ F W + F [> \ = lim S (n | n)

(7.71)

[ = D\ DW + H H W > [ = lim S (n | n 1)=

(7.72)

n$4

n$4

7.2 Kalman ﬁlter based residual generation

177

It holds

³ ¡ ¢1 ´ F [DW + H H W = [ = D L [F W F\ F W +

(7.73)

(7.73) is an algebraic Riccati equation whose solution [ is positive deﬁnite if the pairs (D> H ) and (F> D) are respectively controllable and observable. It thus becomes evident that given system model (7.56)-(7.57) the gain matrix O can be calculated o-line by solving Riccati equation (7.73). The corresponding residual generator is then given by { ˆ(n | n) = Dˆ {(n 1 | n 1) + Ex(n 1) +O (|(n) F { ˆ(n | n 1) Gx(n)) u(n) = |(n) F { ˆ(n | n 1) Gx(n)=

(7.74) (7.75)

Note that we now have in fact an observer of the full-order. Below is an algorithm for the on-line implementation of the Kalman ﬁlter algorithm given by (7.61)-(7.67). Algorithm 7.1 On-line implementation of the Kalman ﬁlter algorithm Step 0: O-line set up of initial conditions: set { ˆ(0 | 0)> S (0 | 0) as given in (7.61) and (7.64) Step 1: Calculate { ˆ(n | n 1)> S (n | n 1)> O(n) according to (7.62), (7.65) and (7.66) Step 2: Calculate { ˆ(n | n)> S (n | n) according to (7.63) and (7.67) Step 3: Increase n and go Step 1. Remark 7.3 The o-line set up (Step 0) is needed only for one time, but Steps 1 - 3 have to be repeated at each time instant. Thus, the on-line implementation, compared with the steady-state Kalman ﬁlter, is computationally very consuming. For the FDI purpose, we can generally assume that the system under consideration is operating in its steady state before a fault occurs. Therefore, the use of the steady-state type residual generator (7.74)-(7.75) is advantageous. In this case, the most involved computation is ﬁnding a solution for Riccati equation (7.73), which, nevertheless, is carried out o-line, and moreover for which there exist a number of numerically reliable methods and CAD programs. Example 7.1 In this example, we design a steady Kalman ﬁlter for the vehicle lateral dynamic system. For our purpose, the linearized, discrete time model is used with ¸ ¸ ¸ 0=6333 0=0672 0=0653 152=7568 1=2493 D= >E = >F = 2=0570 0=6082 0=4039 0 1=0000 ¸ ¸ ¸ ¸ 0 0 0=0653 1 0 56 0=0653 56 > Ii = > H = = G= > Hi = 0 0 0=4039 01 0 0=4039 0

178

7 Residual generation with enhanced robustness against unknown inputs

Using the given technical data, we get ¸ 0=0025 0 = 0=0012> = 0 1=2172h 5 and based on which the observer gain matrix has been computed ¸ 0=0025 0=0086 O= = 0=0122 0=9487

7.3 Approximation of UI-distribution matrix The underlying the idea of the approaches introduced in this section can be simply formulated as follows: given system model |(s) = J|x (s)x(s) + J|i (s)i (s) + J|g (s)g(s) which does not satisfy the existence condition of the PUIDP, (6.5), ﬁnd a ˆ g (s) that approximates J|g (s) in some optimal transfer function matrix J sense and simultaneously ensures that (6.5) is satisﬁed. In a next step, we are then able to design a residual generator which is designed on the basis of the approximated model and delivers an approximated decoupling. 7.3.1 Approximation of matrices Eg , Fg Consider the minimal state space realization of the above model {˙ = D{ + Ex + Hi i + Hg g | = F{ + Gx + Ii i + Ig g

(7.76) (7.77)

for which the existence condition for the PUIDP, (6.9), is not satisﬁed. We suppose that ¸ Hg = ng A p= udqn Ig Patton and Chen have proposed the idea of approximating the unknown input ˆg > Iˆg that ensure the solvability (UI) distribution matrices Hg > Ig by matrices H of the PUIDP. Remember that ng ? p is a su!cient condition for the solution of the PUIDP. Thus, we formulate the approximation problem as follows: given maˆg 5 Rq×ng > Iˆg 5 Rp×ng such that trices Hg 5 Rq×ng > Ig 5 Rp×ng , ﬁnd H ¸ ˆ H Condition I: udqn ˆg = ng ? p (7.78) Ig ° ¸° ˆg ° ° Hg H °= ° Condition II: min ° (7.79) ˆg >Iˆg Ig Iˆg ° H

7.3 Approximation of UI-distribution matrix

179

Condition I, (7.78), ensures that ³ ´ ˆg + Iˆg ? p udqn F(sL D)1 H and the solution of the optimization problem (7.79) delivers an optimal approximation of matrix Hg > Ig in sense of a norm k · k. In the following of this subsection, we consider two matrix norms: the Frobenius and the 2 norm. We shall use SVD as the mathematical tool for the solution of the approximation problem (7.78)-(7.79). We now do an SVD of matrix ¸ Hg Ig which yields

Hg Ig

¸

=

Xg YgW >

gldj( 1 > · · · > ng ) = 0(q+png )×ng

¸

Xg 5 R(q+p)×(q+p) > Xg X W = L(q+p)×(q+p) ng ×ng

Yg 5 R Setting

> Yg Y

W

= Lng ×ng =

¸ ¸ ˆg gldj(1 > · · · > p1 > 0> · · · > 0) H = Xg YW 0(q+png )×ng Iˆg

(7.80) (7.81) (7.82)

(7.83)

results in

ˆg Hg H Ig Iˆg

¸

= Xg

¸ gldj(0> · · · > 0> p > · · · > ng ) Y W= 0(q+png )×ng

Hence, we ﬁnally have ° ° ¸° ¸° ng X ˆg ° ˆg ° ° Hg H ° Hg H ° = ° = p = ° ° > l ° ° Ig Iˆg ° Ig Iˆg °2 I l=p Note that the matrix deﬁned by (7.83) satisﬁes Condition I and furthermore, known from the linear algebra, it does also solve the optimization problem (7.79). Thus, we claim that (7.83) is a solution for the above-mentioned problem. In fact, this result is not surprising, since it is reasonable that the approximation has at least to maintain the dominant part of the matrix being approximated, which is, in our case, given by the ﬁrst p 1 largest singular values. In summary, we have Algorithm 7.2 Optimal approximation of Hg > Ig

180

7 Residual generation with enhanced robustness against unknown inputs

Step 1: Do an SVD according to (7.80)-(7.82); Step 2: Set ¸ ˆg H Iˆg according to (7.83). Once an approximation for Hg > Ig has been found, we can apply the schemes described in the last chapter to solve the decoupling problem ³ ´ ˆ x (s) F(sL D)1 H ˆg + Iˆg = 0= U(s)P 7.3.2 Approximation of matrices Hg>v Remember that the dynamics of a residual generator can be expressed in a non-recursive form (see Subsection 6.5.4) ¡ ¢ (7.84) u(s) = zJv sv h(s) + yv Ki>v L¯i v iv (s) + Kg>v L¯gv gv (s) ¤ £ (7.85) yv = yv>0 yv>1 · · · yv>v 5 Sv £ ¤ z = 0 ··· 0 1 (7.86) and the necessary and su!cient condition for a successful unknown input decoupling is given by ¤ £ ¤ £ (7.87) udqn Kr>v Kg>v ? the row number of Kr>v Kg>v or equivalently udqn (Tedvh>v Kg>v ) ? the row number of Tedvh>v Kg>v where matrix Tedvh>v denotes the base matrix of parity space Sv of order s. Since yv 5 Sv is a necessary condition for the existence of a residual generator, what can be approximated is only matrix Kg>v . It follows from Theorems 5.6 and 5.11 that the rank of matrix Tedvh>v is equal to its row number given by glp(Sv ) = udqn(Tedvh>v ) =

v X

(p pl )> for plq v ? pd{

l=plq

= p × (v pd{ + 1) +

pd{ X1

(p pl )> for v pd{ =

l= plq

This motivates us to deﬁne the following approximation problem: given matrix ˆ g>v such that Kg>v , ﬁnd K

7.3 Approximation of UI-distribution matrix

181

ˆ g>v ) ? n˜g Condition I: udqn(K with n˜g =

v X

(p pl )> for plq v ? pd{

l= plq

= p × (v pd{ + 1) +

pd{ X1

(p pl )> for v pd{

l=plq

ˆ g>v kI as well as inf kKg>v K ˆ g>v k2 = Condition II: min kKg>v K ˆ g>v K

ˆ g>v K

It is evident that Condition I is a su!cient condition such that (7.87) holds. On account of a similar procedure carried out in the last subsection, we give the following algorithm to solve the above-mentioned approximation problem. Algorithm 7.3 Optimal approximation of Kg>v Step 1: Do an SVD on Kg>v Kg>v = Xg Yg W where

¤ £ = gldj(1 > · · · > p(v+1) ) Rp(v+1)×(ng p)(v+1)

(7.88)

Xg 5 R(p(v+1))×(p(v+1)) > Xg X W = Lp(v+1)×p(v+1) Yg 5 Rng (v+1)×ng (v+1) > Yg Y W = Lng (v+1)×ng (v+1) Step 2: Set

£ ¤ ˆ g>v = Xg gldj( 1 > · · · > ˜ > 0> · · · > 0) 0p(v+1)×(ng p)(v+1) Y W = K ng 1

Again, once an approximation has been found, we are able to ﬁnd a vector z solving ˆ g>v = 0= zTedvh>v K The achieved parity vector deﬁned by yv = zTedvh>v can then be used to construct a parity relation based residual generator or, as shown in the last two chapters, to construct a diagnostic observer. We would like to point out that the assumption that the SVD of matrix Kg>v can be written into (7.88) holds, since ng A p, otherwise, as shown in Subsection 6.5.4, the PUIDP becomes trivial and solvable. Also, as shown above, the approximation error is given by X

p(v+1)

ˆ g>v kI = kKg>v K

˜g l=n

ˆ g>v k2 = ˜ = l > kKg>v K ng

182

7 Residual generation with enhanced robustness against unknown inputs

7.3.3 Some remarks The most signiﬁcant advantage of approaching the robust residual generation problem by an approximation of UI distribution matrices lies in its mathematical simplicity and its evident relationship to the PUIDP. In fact, it is a natural extension of the PUIDP. On the other hand, this kind of approach, although already known in the late 80’s, has not received so much attention like the robust residual generation approaches described in the forthcoming sections of this chapter. We now study the approximation errors and their inﬂuence on the residual signals. We ﬁrst consider the approach based on an approximation of matrices Hg > Ig . The residual dynamics is governed by ³ ´ ˆi (s)i (s) + Q ˆg (s)g(s) u(s) = U(s) Q (7.89) where ˆi (s) = Ii + F(sL D + OF)1 (Hi OIi ) Q ˆg (s) = Ig + F(sL D + OF)1 (Hg OIg )= Q Under consideration that ´ ³ ˆg OIˆg ) = 0 U(s) Iˆg + F(sL D + OF)1 (H

(7.90)

(7.89) can be brought into μ ˆi (s)i (s) + U(s) u(s) = U(s)Q

Ig + F(sL D + OF)1 (Hg OIg )

¶ g(s)=

Thus, the maximum (possible) inﬂuence of the unknown input vector g on u can be measured by ¢ ¡ (7.91) kU(s) Ig + F(sL D + OF)1 (Hg OIg ) k4 kg(s)k2 = Note that in (7.91), U(s) is the solution of (7.90) and Ig > Hg are subject to ° ¸° ° Hg ° ° ° ° Ig ° = p = 2 It turns out ¡ ¢ inf kU(s) Ig + F(sL D + OF)1 (Hg OIg ) k4 (7.92) U(s) ¢ ¡ kU(s) Ig + F(sL D + OF)1 (Hg OIg ) k4 = inf U(s) solving (7.90) The situation with the approximation of Kg>v is almost the same, in which we have

7.3 Approximation of UI-distribution matrix

¡ ¢ u(s) = zJv sv h(s) + yv Ki>v L¯i v iv (s) + Kg>v L¯gv gv (s) ¡ ¢ = zJv sv h(s)yv Ki>v L¯i v iv (s) + Kg>v L¯gv gv (s) where yv 5 Sv solves

183

(7.93) (7.94)

ˆ g>v = 0 yv K

ˆ g>v . To simplify the discussion, we set j = 0 which leads and Kg>v = Kg>v K to zJv = 0= This results in that the maximum inﬂuence of the unknown input vector g can be measured by kyv Kg>v L¯gv k4 kgv (s)k2 = It is clear that inf kyv Kg>v L¯gv k4

yv 5Sv

inf

ˆ g>v =0 yv 5Sv >yv K

kyv Kg>v L¯gv k4 =

(7.95)

As a conclusion of (7.92) and (7.95), we claim Claim. A direct optimization in the sense of ¡ ¢ inf kU(s) Ig + F(sL D + OF)1 (Hg OIg ) k4 U(s)

or inf kyv Kg>v L¯gv k4

yv 5Sv

will provide us with a better FDI performance than approximating the robust FDI problems based on an approximation of unknown input matrices Hg > Ig or Kg>v . Remark 7.4 Recall that ¸ ˆ H ˆ g>v ) ? n˜g udqn ˆg ? p =, udqn(K Ig therefore the approach based on the approximation of Kg>v delivers a residual generator with higher robustness. We would like to point out that the above-described approximations are not unique, thus it makes sense to carry out a further optimization ¢ ¡ inf kU(s) Ig + F(sL D + OF)1 (Hg OIg ) k4 U(s) solving (7.90) or inf

ˆ g>v =0 yv 5Sv >yv K

kyv Kg>v L¯gv k4

in order to improve the FDI performance.

184

7 Residual generation with enhanced robustness against unknown inputs

Finally, it is worth mentioning that following the same idea we can also make an approximation of the unknown input transfer matrix in the H4 optimization framework ˆ g (s)k min kJg (s) J

ˆ g (s) J

ˆ g (s) has to satisfy where J ˆ g (s)) ? p= udqn(J However, carrying out such an approximation requires some special mathematical knowledge.

7.4 Robustness, fault sensitivity and performance indices Beginning with this section, we shall study the FDI problems in the context of a trade-o between the robustness against the disturbances and sensitivity to the faults. To this end, we are ﬁrst going to ﬁnd a way to evaluate the robustness and sensitivity and then to deﬁne performance indices that would give a fair evaluation of the trade-o between the robustness and sensitivity. To simplify the notations, in this section we express a residual generator in terms of (7.96) u = Kg (S )g + Ki (S )i where u stands for residual vector which is either u(s) for the residual generators in the recursive form (observer-based residual generators) or uv (n) for the residual generators in the non-recursive form (parity space residual generators). Corresponding to it, we have ˆ x (s)J|g (s)> Ki (S ) = U(s)P ˆ x (s)J|i (s) Kg (S ) = U(s)P

(7.97)

or Kg (S ) = Yv Kg>v > Ki (S ) = Yv Ki>v

(7.98)

where variable S is used to denote design parameters, which are, in case of a residual generator in the recursive form, the post-ﬁlter U(s) and the observer matrix O, and the parity vector Yv for a residual generator in the non-recursive form. 7.4.1 Robustness and sensitivity A natural way to evaluate the robustness of residual generator (7.96) against g is the use of an induced norm, which is formally deﬁned by

7.4 Robustness, fault sensitivity and performance indices

Ug := kKg (S )k = sup g6=0

kKg (S )gk = kgk

185

(7.99)

It is well known that this is a worst-case evaluation of the possible inﬂuence of g on u= Compared with the robustness, evaluation of sensitivity of an FDI system to the faults is not undisputed. The way of using an induced norm like kKi (S )i k ki k i 6=0

Vi>+ = kKi (S )k = sup

(7.100)

is popular and seems even logical. However, when we take a careful look at the interpretation of (7.100), which means a best-case handling of the inﬂuence of i on u, the sense of introducing (7.100) for the sensitivity becomes questionable. A worst-case for the sensitivity evaluation should, in fact, be the minimum inﬂuence of i on u, which can be expressed in terms of Vi> = kKi (S )k = inf

i 6=0

kKi (S )i k = ki k

(7.101)

Note that Vi> is not a norm, since there may exist i 6= 0 such that Ki (S )i = 0. This is also the reason why in some cases the sensitivity deﬁned by (7.101) makes less sense. Both Vi>+ > Vi> have been adopted to measure the sensitivity of the FDI system to the faults, although Vi> was introduced much late than Vi>+ . We would like to remark that both Vi>+ > Vi> are some extreme value of transfer matrix. From the practical viewpoint, it is desired to deﬁne an index that gives a fair evaluation of the inﬂuence of the faults on the residual signal over the whole time or frequency domain and in all directions in the measurement subspace. We shall introduce such an index at the end of this chapter, after having studied the solutions under the standard performance indices. 7.4.2 Performance indices: robustness vs. sensitivity With the aid of the introduced concepts of the robustness and sensitivity we are now able to formulate our wish of designing an FDI system: the FDI system should be as robust as possible to the disturbances and simultaneously as sensitive as possible to the faults. It is a multiobjective optimization problem: given (7.96), ﬁnd S such that Ug $ min and simultaneously Vi $ max = It is well known that solving a multiobjective optimization problem is usually much more involved than solving a single-objective optimization. Driven by this idea, a variety of attempts have been made to reformulate the optimization objective as a compromise between the robustness and sensitivity. A ﬁrst

186

7 Residual generation with enhanced robustness against unknown inputs

kind of these performance indices was introduced by Lou et al., which takes the form MVU = sup(i Vi g Ug )> i > g A 0 (7.102) S

where i > g are some given weighting constants. Wuennenberg and Frank suggested to use the following the performance index MV@U = sup S

Vi Ug

(7.103)

which, due to its intimate connection to the sensitivity theory, is widely accepted. Currently, the index of the form Ug ? and Vi A

(7.104)

becomes more popular, where > are some positive constant. The FDI system design is then formulated as maximizing and minimizing by selecting S . 7.4.3 Relations between the performance indices Next, we are going to demonstrate that the above-introduced three types of indices are equivalent in a certain sense. Suppose that ¶ μ SV@U>rsw = arg sup MV@U S

then it follows from (7.96) that for any constant &> &SV@U>rsw also solves supS MV@U = This means that the optimal solution to the ratio-type optimization is unique up to a constant. To demonstrate the relationship between the optimal performance under indices (7.103) and (7.104), suppose that Srsw solves max and min subject to (7.104) S

S

and yields kKi (Srsw )k = 1 > kKg (Srsw )k = 1 =,

kKi (Srsw )k = = 1= kKg (Srsw )k 1

On the other side, SV@U>rsw ensures that ;& A 0 ° ° ° ° °Ki (SV@U>rsw )° °&Ki (SV@U>rsw )° ° ° ° ° °Kg (SV@U>rsw )° = °&Kg (SV@U>rsw )° = As a result, it is possible to ﬁnd a & such that ° ° ° ° °&Ki (SV@U>rsw )° 1 and °&Kg (SV@U>rsw )° = 1 =

(7.105)

To illustrate the relation between optimizations under (7.103) and (7.102), suppose that SVU>rsw solves

7.5 Optimal selection of parity matrices and vectors

187

sup MVU = sup (i kKi (S )k g kKg (S )k) S

S

and results in i kKi (SVU>rsw )k g kKg (SVU>rsw )k = > kKg (SVU>rsw )k = =,

i kKi (SVU>rsw )k = g + = kKg (SVU>rsw )k

Once again, we are able to ﬁnd a & such that i k&Ki (SVU>rsw )k g + and k&Kg (SVU>rsw )k = k&Kg (SVU>rsw )k =, i k&Ki (SVU>rsw )k g k&Kg (SVU>rsw )k =

(7.106)

(7.105) and (7.106) demonstrate that the optimal solution under ratio-type performance index (7.103) is, up to a constant, equivalent to the ones under indices (7.102) and (7.104). With this fact in mind, in this chapter we mainly consider optimizations under indices (7.103) and (7.104), which are also mostly considered in recent studies.

7.5 Optimal selection of parity matrices and vectors In this section, approaches to optimal selection of parity vectors will be presented. The starting point is the design form of the parity relation based residual generator u(n) = Yv (Kg>v gv (n) + Ki>v iv (n)) > Yv 5 Sv

(7.107)

where u(n) 5 R , denotes the dimension of the parity space of order s, which, following Theorem 5.11, is given by =

v X

(p pl )> for plq v ? pd{

l=plq

= p × (v pd{ + 1) +

pd{ X1

(p pl )> for v pd{ =

l=plq

Our task is to choose Yv under a given performance index. Recall that it holds for Yv Kg>v > Yv Ki>v with Yv 5 Sv that ¯ g>v Yv Kg>v = Y¯v Tedvh>v Kg>v := Y¯v K ¯ i>v Yv Ki>v = Y¯v Tedvh>v Ki>v := Y¯v K

(7.108) (7.109)

with Tedvh>v being the base matrix of parity space and Y¯v 6= 0 an arbitrary matrix with appropriate dimensions. Hence, residual generator (7.107) can be rewritten into

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7 Residual generation with enhanced robustness against unknown inputs

¡ ¢ ¯ i>v iv (n) = ¯ g>v gv (n) + K u(n) = Y¯v K

(7.110)

We suppose that ¢ ¡ ¯ g>v ¯ g>v = the row number of K udqn K i.e. the PUIDP is not solvable, which motivates us to ﬁnd an alternative way to design the parity matrix Yv = 7.5.1 Si>+ /Rg as performance index The idea of using the ratio of the robustness (Ug ) to the sensitivity (Vi ) was initiated by Wuennenberg and Frank in the middle 80’s. We ﬁrst consider the case kKi i k Vi>+ = sup i 6=0 ki k and express the performance index in terms of MV>+@U = max

Yv 5Sv

¯ i>v k Vi>+ kY¯v K = max ¯ ¯ Ug Y¯v kYv Kg>v k

(7.111)

¯ g>v where k·k denotes some induced norm of a matrix. Doing an SVD of K gives ¯ g>v = X Y W > X X W = L× K ¤ £ Y Y W = L× > = gldj( 1 > · · · > ) 0×()

(7.112) (7.113)

where is the column number of Kg>v > i.e. = ng (v + 1 pd{ )= Set Y¯v = Y˜v V 1 X W > V = gldj( 1 > · · · > ) it turns out

(7.114)

° £ ° ° ¤° ° °Y¯v K ¯ g>v ° k = ° °Y˜v L> 0×() ° =

Following the deﬁnitions of 1, 2, and 4 norms for a matrix, we have ° £ ¤° ° °˜ (7.115) °Yv L> 0×() ° = kY˜v k1 ° £ °1 ¤ °˜ ° ¯ (Y˜v ) = kY˜v k2 (7.116) °Yv L> 0×() ° = 2 ° £ ° ¤° °˜ (7.117) °Yv L> 0×() ° = kY˜v k4 = 4

Note that ¯ i>v k = kY˜v V 1 X W K ¯ i>v k kY˜v kkV 1 X W K ¯ i>v k= kY¯v K

7.5 Optimal selection of parity matrices and vectors

189

As a result, the following inequality holds: ; Y˜v 6= 0 ¯ i>v k ¯ i>v k ¯ i>v k kY˜v V 1 X W K kY˜v kkV 1 X W K kY¯v K ¯ i>v k= = = kV 1 X W K ¯ g>v k kY¯v K kY˜v k kY˜v k In other words, we have ¯ i>v k= MV>+@U kV 1 X W K On the other hand, it is evident that setting Y˜ = L gives ¯ i>v k kY¯v K 1 W ¯ ¯ ¯ g>v k = kV X Ki>v k kYv K thus, we ﬁnally have ¯ i>v k kY¯v K ¯ i>v k= = kV 1 X W K MU>+@V = max ¯ ¯ ¯v kYv Kg>v k Y This proves the following theorem. Theorem 7.1 Given system (7.110), then Y¯v = V 1 X W

(7.118)

solves the optimization problems ¯ i>v k1 ¯ i>v k2 kY¯v K kY¯v K > MV@U>2 = max ¯ ¯ MV@U>1 = max ¯ ¯ Y¯v kYv Kg>v k1 Y¯v kYv Kg>v k2 ¯ ¯ kYv Ki>v k4 MV@U>4 = max ¯ ¯ ¯ Yv kYv Kg>v k4

(7.119) (7.120)

which results in ¯ i>v k1 kY¯v K ¯ i>v k1 = kV 1 X W K MV@U>1 = max ¯ ¯ ¯ Yv kYv Kg>v k1 ¯ i>v k2 kYv K 1 W ¯ MV@U>2 = max ¯ g>v k2 = kV X Ki>v k2 Y¯v kYv K ¯ i>v k4 kY¯v K 1 W ¯ MV@U>4 = max ¯ i>v k4 = kV X Kg>v k4 = Y¯v kYv K

(7.121) (7.122) (7.123)

Note that the optimal solution (7.118) solves all above-mentioned three optimization problems. This fact is of great interest for the sensitivity and performance analysis of FDI systems. We now summarize the main results achieved above into an algorithm. Algorithm 7.4 Solution of optimization problem Vi>+ @Ug

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7 Residual generation with enhanced robustness against unknown inputs

¯ g>v ; Step 1: Do an SVD on K Step 2: Set Y¯v according to (7.118). Next, we study the relationship between the optimization problems whose solutions are presented above and the optimal design of parity vectors under the performance index M = max yv

¯ i>v K ¯ W yW yv K i>v ¯ g>v K ¯ W yW yv K

(7.124)

g>v

which was introduced by Wuennenberg and Frank and is now one of the mostly used performance indices. To begin with, we take a look at the solution of optimization problem (7.124). Let the optimal solution be denoted by yv>rsw and rewrite (7.124) into ¡ ¢ W W ¯ g>v K ¯ i>v K ¯ g>i ¯ i>v (yv>rsw )W = 0= K yv>rsw M K ¯ g>v > K ¯ g>v = X Y W > we obtain By an SVD of K ¡ ¢ W ¯ i>v K ¯ i>v (yv>rsw )W = 0= yv>rsw MX W X W K Setting yv>rsw = y¯v V 1 X W yields

¡ ¢ ¯ i>v y¯v V 1 X W K ¯ i>v W = 0= M y¯v (¯ yv )W y¯v V 1 X W K

It is clear that choosing the nominal eigenvector corresponding to the maximal ¯ i>v K ¯ W X V 1 as y¯v , i.e. eigenvalue of matrix V 1 X W K i>v ¡ ¢ W ¯ i>v K ¯ i>v y¯v pd{ L V 1 X W K X V 1 = 0> y¯v (¯ yv )W = 1

(7.125)

¯ i>v K ¯ W X V 1 , with pd{ being the maximal eigenvalue of matrix V 1 X W K i>v gives M = pd{ = (7.126) Theorem 7.2 Given system (7.110), then the optimal solution of (7.124) is given by (7.127) yv>rsw = y¯v V 1 X > with y¯v solving (7.125), and in this case M = max yv

¯ i>v K ¯ W yvW yv K i>v ¯ g>v K ¯ W yvW = pd{ = yv K g>v

Comparing M in (7.124) with MV@U>2 and noting the fact ¢ ¡ ¢ ¡ W ¯ i>v ¯ i>v X V 1 = ¯ V 1 X W K pd{ = ¯ K

(7.128)

7.5 Optimal selection of parity matrices and vectors

we immediately see

2 = M = MV@U>2

191

(7.129)

This reveals the relationship between both the performance indices and veriﬁes that the optimal solution is not unique. In fact, we have ¯ i>v k2 ¯ i>v k2 kY¯v K kyv K = max MV@U>2 = max ¯ ¯ ¯ g>v k2 = ¯v kYv Kg>v k2 yv kyv K Y Nevertheless, both the FDI systems have quite dierent fault detectability, as the discussion in Chapter 12 will show. Bringing (7.125) into the following form ¡ ¢ W ¯ i>v K ¯ i>v X V 1 = 0 +, y¯v pd{ L V 1 X W K ¡ ¢ W W ¯ g>v K ¯ i>v K ¯ g>v ¯ i>v =0 (7.130) yv M K K shows that the optimization problem (7.124) is equivalent to a generalized eigenvalue-eigenvector problem deﬁned by (7.130). The maximal eigenvalue is the optimal value of performance index M, and the corresponding eigenvector is the optimal parity vector. 7.5.2 Si> /Rg as performance index As mentioned in the last section Vi> =

¯ i>v iv (n)k kK kiv (n)k iv (n)6=0 inf

is also a reasonable index to evaluate the fault sensitivity. However, it is not a norm. For this reason, we restrict our attention just to the following case ¯ i>v ) Vi> = (Y¯v K ¯ i>v . It is ¯ i>v ) denotes the minimum singular value of matrix K where (Y¯v K ¯ ¯ worth noting that if Yv Ki>v has full column rank, then ¯ i>v iv (n)k kY¯v K ¯ i>v )= = (Y¯v K kiv (n)k iv (n)6=0 inf

Analogous to the calculation made in the last subsection we have MV>@U = max ¯v Y

¯ i>v ) ¯ i>v ) Vi> (Y¯v K (Y˜v V 1 X W K = max ¯ ¯ = max = ¯v kYv Kg>v k2 Ug Y kY˜v k2 Y˜v

Since ¯ i>v ) ¯ i>v ) (Y˜v V 1 X W K ¯ (Y˜v )(V 1 X W K and ¯ (Y˜v ) = kY˜v k2 , it turns out

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7 Residual generation with enhanced robustness against unknown inputs

Theorem 7.3 Given system (7.110), then Y¯v = V 1 X W

(7.131)

solves the optimization problem MV>@U = max ¯v Y

which results in

¯ i>v ) Vi> (Y¯v K = max ¯ ¯ ¯v kYv Kg>v k2 Ug Y

(7.132)

¡ ¢ ¯ i>v = MV>@U = V 1 X > K

It is very interesting to notice that the optimal solution Y¯v = V 1 X W > recalling the results described in Theorem 7.1, solves both optimization problems Vi>+ @Ug and Vi> @Ug . As mentioned early, the optimization solution is not unique. Setting Yv = yv = y¯v V 1 X W where y¯v is the eigenvector corresponding to the minimum eigenvalue of matrix ¯ i>v K ¯ W X V 1 , i.e. V 1 X W K i>v ¡ ¢ W ¯ i>v K ¯ i>v y¯v V 1 X W K X V 1 plq L = 0> plq 6= 0> y¯v (¯ yv )W = 1 delivers the same performance value, ¡ ¢ ¯ i>v = MV>@U = V 1 X W K On the other side, in this case MV>+@U ? max MV=+@U Y¯v

that is, the solution is not optimal in the sense of MV>+@U . We see that dierent optimal solutions may provide us with quite dierent system performance. Which one is the best one can only be answered in the context of an analysis of FDI system performance, in which relationships between the performance indices and system properties are established and the functions of residual generations and evaluation are integrated considered. This is the central topic in Chapter 12. Let l (·) denote the l-th non-zero singular value of a matrix. Due to ³ ´ ¡ ¢ ¯ i>v ¯ i>v l Y˜v V 1 X W K ¯ (Y˜v ) l V 1 X W K we have

7.5 Optimal selection of parity matrices and vectors

193

³ ´ ¯ i>v l Y˜v V 1 X W K

max ¯v Y

max

¡ ¢ ¯ i>v l Y¯v K ¯ g>v k2 = max ˜v kY¯v K Y ¡ 1 W ¢ ¯ i>v ¯ (Y˜v ) l V X K

˜v Y

¯ (Y˜v )

¯ (Y˜v ) ¡ ¢ ¯ i>v = l V 1 X W K

(7.133) (7.134)

and so the following theorem. Theorem 7.4 Given system (7.110), then Y¯v = V 1 X W

(7.135)

solves the optimization problem MV@U>l

¡ ¢ ¯ i>v l Yv K = max ¯ g>v k2 for all l ¯v kYv K Y

(7.136)

for which we have ¡ ¢ ¯ i>v ¡ ¢ l Yv K ¯ i>v = = l V 1 X W K max ¯ ¯ kYv Kg>v k2 Yv We would like to call reader’s attention that Theorem 7.1 and Theorem 7.3 are indeed two special cases of Theorem 7.4. From the practical viewpoint, performance index MV@U>l gives a fair evaluation of the inﬂuence of the faults on the residual signal over the time interval [n v> n] and in all directions in the measurement subspace. For these reasons, solution (7.135) is called uniﬁed parity space solution. 7.5.3 JV U as performance index The ﬁrst version of the performance index in the MVU form was proposed by Lou et al. We consider in the following a modiﬁcation form expressed in terms of ¡ ¢ ¯ i>v k g kY¯v K ¯ g>v k > i > g A 0 (7.137) MVU = max i kY¯v K Y¯v

where k · k denotes 1, 2, and 4 norms of a matrix. Since MVU is proportional to the size of parity matrix Y¯v , we suppose that kY¯v k = 1, i.e. we are only interested in those nominal solutions. Repeating the same procedure adopted in the previous two subsections allows us to rewrite (7.137) into ¡ ¢ ¢ ¡ ¯ i>v k g kY¯v K ¯ g>v k kY˜v k i kV 1 X W K ¯ i>v k g MVU = max i kY¯v K Y¯v

with Y¯v = Y˜v V 1 X W = Thus, we claim

194

7 Residual generation with enhanced robustness against unknown inputs

Theorem 7.5 Given system (7.110), then the optimal solution

leads to MVU =

V 1 X W Y¯v = kV 1 X W k

(7.138)

¯ i>v k g i kV 1 X W K = kV 1 X W k

(7.139)

Recall that the optimization problems MV>+@U > MV>@U are independent of the size of Y¯v , i.e. for all 6= 0 Y¯v = V 1 X W also solves the optimization problems MV>+@U > MV>@U , and furthermore ¯ i>v k= MV>+@U = kV 1 X W K It leads to Corollary 7.1 MV>+@U > MV>@U > MVU have the identical solution: V 1 X W = Y¯v = kV 1 X W k

(7.140)

Deﬁnition 7.1 Given system (7.110). V 1 X W Y¯v = kV 1 X W k is called nominal uniﬁed solution of parity matrix. From the mathematical viewpoint, Y¯v satisfying (7.140) can be interpreted ¯ i>v > i.e. ¯ g>v and used for weighting K as the inverse of the amplitude of K ¡ ¢ ¯ i>v iv (n) ¯ g>v gv (n) + K u(n) = Y¯v K £ ¤ L 0×() Y W V 1 X W ¯ gv (n)= Ki>v iv (n) + = kV 1 X W k kV 1 X W k From the FDI viewpoint, this solution ensures that the inﬂuence of the faults will be stronger weighted at the places where the inﬂuence of the disturbances is weaker. In this manner, an optimal trade-o between the robustness against the disturbances on the one side and the fault sensitivity on the other side is achieved. We would like to call reader’s attention that this idea will also be adopted in the observer-based residual generator design. Corollary 7.2 It holds MVU =

i MV>+@U g = kV 1 X W k

(7.141)

7.5 Optimal selection of parity matrices and vectors

195

It follows from (7.141) that increasing MU@V>+ simultaneously enhances MUV and vice verse. This also veriﬁes our early statement that the performance indices (7.103) and (7.102) are equivalent. Analogous to the discussion in the last two subsections, it can readily be demonstrated that • the optimal solution Y¯v satisfying (7.140) also solves the optimization problem ¡ ¢ ¯ i>v g ¡ ¡ ¢ ¢ i l V 1 X W K ¯ ¯ ¯ ¯ i l Yv Ki>v g kYv Kg>v k2 = max ¯v >kY¯v k=1 kV 1 X W k Y (7.142) • the optimal parity vector yv = y¯v

V 1 X W > y¯v y¯vW = 1 kV 1 X W k

where y¯v is the eigenvector corresponding to the maximum eigenvalue of ¯ i>v , solves the optimization problem matrix V 1 X W K ¡ ¢ ¯ i>v k2 kyv K ¯ g>v k2 = kyv K max yv >kyv k2 =1

7.5.4 Optimization performance and system order Until now, our study on the parity space relation based residual generation has been carried out for a given v. Since v is a design parameter, the question may arise: How can we choose a suitable v? The fact that the choice of the system order v may have considerable inﬂuence on the optimization performance has been recognized, but only few attention has been devoted to this subject. In this subsection, we will ﬁnd out an answer to this problem, which may, although not complete, build a basis for further investigation. To begin with, we concentrate ourselves on a modiﬁed form of the optimization problem (7.124), for which the following theorem is known. Theorem 7.6 The inequality min

yv 5Sv

min

yv+1 5Sv+1

> > yv Kg>v Kg>v yv > y> yv Ki>v Ki>v v

> > yv+1 Kg>v+1 Kg>v+1 yv+1 > > yv+1 Ki>v+1 Ki>v+1 yv+1

= plq>v A = plq>v+1

(7.143)

holds. The proof of this theorem is strongly related to the study on the minimum order of residual generators in Section 6.8 and much involved, hence it is omitted. We refer the interested reader to a paper by Ding et al. listed at the end of this chapter.

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7 Residual generation with enhanced robustness against unknown inputs

Remark 7.5 It can be shown that the performance index plq>v converges to a limit with v $ 4. Theorem 7.6 reveals that increasing the order of parity space does really improve the system robustness and sensitivity. On the other hand, increasing v means more on-line computation. Thus, a compromise between the system performance and the on-line implementation is desired. To this end, we propose the following algorithm. Algorithm 7.5 Selection of v: Step 0: Set the initial value of the order of parity relation v (note that it should be larger than or equal to plq ) and a tolerance; Step 1: Calculate the base matrix of parity space Tedvh>v and Tedvh>v Ki>v > Tedvh>v Kg>v ; Step 2: Solve the generalized eigenvalue-eigenvector problem; Step 3: If plq>v1 plq>v tolerance, end, otherwise go back to Step 1. We would like to point out that a repeated calculation of Step 1 is not necessary. In fact, once the system model is transformed into the canonical ¯ mr = R> plq m pd{ 1, are solved, we observer form and equations Qm FD can determine the base matrix of the parity space and Tedvh>v Ki>v > Tedvh>v Kg>v for dierent v without solving additional equations (see also Subsection 5.6.2 and Section 6.8.). This fact promises a strong reduction of computation for a (sub-)optimal selection of the order of the parity matrices. Remember that MV@U>2 =

p ¡ ¢ ¯ i>v k2 kY¯v K ¯ g>v k2 = i MV>+@U g max > MVU>2 = max kY¯v K kV 1 X W k2 Y¯v

the following corollary becomes clear. Corollary 7.3 The inequalities max

Yv 5Sv

kYv Ki>v k2 kYv+1 Ki>v+1 k2 A max kYv Kg>v k2 Yv+1 5Sv+1 kYv+1 Kg>v+1 k2

(7.144)

max (i kYv Ki>v k2 g kYv Kg>v k2 ) A

Yv 5Sv

max

Yv+1 5Sv+1

(i kYv+1 Ki>v+1 k2 g kYv+1 Kg>v+1 k2 )

(7.145)

max (kyv Ki>v k2 kyv Kg>v k2 ) A

Yv 5Sv

max

Yv+1 5Sv+1

(kyv+1 Ki>v+1 k2 kyv+1 Kg>v+1 k2 )

(7.146)

hold. In fact, we can expect that the results of Theorem 7.6 as well as Corollary 7.3 are applicable for other performance indices.

7.5 Optimal selection of parity matrices and vectors

197

7.5.5 Summary and some remarks In this section, we have introduced a number of performance indices and, based on them, formulated and solved a variety of optimization problems. Some of them sound similar but have dierent meanings, and the others may be deﬁned from dierent aspects but have identical solutions. It seems that a clear classiﬁcation and a summary of the approaches described in this section would be useful for the reader to get a deep insight into the framework of model-based residual generation schemes. We have deﬁned two types of performance indices ¯ i>v k kYv Ki>v k kY¯v K = max ¯ ¯ ¯v kYv Kg>v k Yv 5Sv kYv Kg>v k Y = max (i kYv Ki>v k g kYv Kg>v k) Yv 5Sv ¡ ¢ ¯ i>v k g kY¯v K ¯ g>v k = max i kY¯v K

Type I : MV@U = max Type II : MVU

(7.147) (7.148)

¯v Y

and each of them can be expressed in four dierent forms, depending on which of the norms, 1, 2, 4 norm, or the minimum singular value is used for the evaluation of the robustness and sensitivity. It is worth noting that the minimum singular value (Yv Ki>v ) is not a norm, but it, together with kYv Kg>v k2 , measures the worst-case from the FDI viewpoint. A further variation of (7.147) and (7.148) is given by the selection of parity matrix Y¯v : it can be a × -dimensional matrix or just a -dimensional row ¯ g>v = Tedvh>v Kg>v . Of vector, where denotes the number of the rows of K ¯ course, Yv can also be a × -dimensional matrix with 1 , the results will remain the same. Considering the fact that the solution of the optimization problem Type I is independent of kY¯v k and the one of Type II is proportional to kY¯v k, we have introduced the concept of nominal optimal solution whose size (norm) is one (kY¯v k = 1). The most signiﬁcant results derived in this section can then be stated as follows: • Given Y¯v 5 R× , then

V 1 X W Y¯v = kV 1 X W k

is the nominal optimal solution for the both types optimization problems, independent of which norm is used; • Given a row vector yv 5 R , then yv = y¯v

V 1 X W kV 1 X W k

is the nominal optimal solution for the both types optimization problems, where y¯v is, depending on the norm used, chosen as follows:

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7 Residual generation with enhanced robustness against unknown inputs

£ ¤£ ¤ 1 norm : y¯v = 1 · · · 1 0 · · · 0 hl 0 · · · 0 (7.149) ¡ 1 W ¢ W 1 W ¯ i>v ) V X K ¯ i>v pd{ L = 0 or 2 norm : y¯v (V X K W W ¯ i>v K ¯ i>v ¯ g>v K ¯ g>v max>v K )=0 (7.150) y¯v (K 5 6 0 9 .. : 9 : £ ¤ 9 .W : : 4 norm : y¯v = 0 · · · 0 1 0 · · · 0 9 (7.151) 9 hl : 9 0 : 7 8 .. . £ ¤ W hl = · · · 0 1 0 · · · = Recall that

° ° ° ° ¯ i>v ° > MV@U>4 = °V 1 X W K ¯ i>v ° MV@U>1 = °V 1 X W K 1 4

and remember the deﬁnitions of the 1 and 4 norms of a matrix, we have y¯v in (7.149) that selects the the largest absolute column sum, assumed to be the l-th column, of V 1 X W and y¯v in (7.151) that selects the largest absolute row sum, assumed to be the m-th row, of V 1 X W = • The optimal value of performance index MU@V is ¡ ¢ ¯ i>v k> MV>@U = V 1 X W K ¯ i>v MV>+@U = kV 1 X W K and MVU is MVU>+

¡ ¢ ¯ i>v g ¯ i>v k g i V 1 X W K i kV 1 X W K > MVU> = = = kV 1 X W k kV 1 X W k2

Either for Y¯v 5 R× or for yv 5 R these results always hold. The last statement is worth a brief discussion. We see that using a parity vector or a parity matrix has no inﬂuence on the optimal value of the performance indices. But these two dierent constructions do have considerably dierent inﬂuences on the system performance. Taking a look at the design form, u(n) = Yv (Kg>v gv (n) + Ki>v iv (n)) > Yv 5 Sv shows the role of Yv evidently: It is a ﬁlter and selector. From the geometric viewpoint, it spans a subspace and thus allows only the signals, whose components lie in this subspace, to have an inﬂuence on the residual u(n). When Yv is selected as a vector, the dimension of the subspace spanned by yv is one, i.e. it selects signals only in one direction. Of course, in this direction the ratio of the robustness to the sensitivity is optimal in a certain sense, but if the strength of the fault in this direction is weak, a fault detection will become very difﬁcult. In contrast, choosing Yv to be matrix ensures that all components of the fault will have inﬂuence on the residual, although in some directions the ratio of the robustness to the sensitivity may be only suboptimal. Following this discussion, it can be concluded that

7.5 Optimal selection of parity matrices and vectors

199

• if we have information about the faults and know they will appear in a certain direction, then using a parity vector may reduce the inﬂuence of model uncertainties and improve the sensitivity to the faults, • in other cases, using a parity matrix is advisable. Another interesting aspect is the computation of optimal solutions. All of ¯ g>v = Tedvh>v Ki>v > derived results rely on the SVD of K ¯ g>v = X Y W > X X W = L× > Y Y W = L× K ¤ £ ¤ £ = gldj(1 > · · · > ) 0×() = V 0×() = Remark 7.6 The assumption made at the beginning of this section, ¢ ¡ ˆ g>v ¯ g>v = the row number of K udqn K does not lead to the loss of generality of our results. In fact, if this condition ¯ g>v results in is not true, then an SVD of K ¯ g>v = X Y W > X X W = L× > Y Y W = L× K ¸ gldj( 1 > · · · > ) = = 0()× Note that there exists a matrix V 5 R× such that V = L× = It is easy to prove that by replacing V 1 with V all results and theorems derived in this section hold true. In case that the 2 norm is used, we have an alternative way to compute the solution, namely by means of the generalized eigenvalue-eigenvector problem, ¡ ¢ W W ¯ g>v K ¯ i>v ¯ g>v ¯ i>v K = 0> Yv YvW = L= K Y¯v K For Y¯v 5 R× , Y¯v consists of all the eigenvectors, while for yv 5 R , yv is the eigenvector corresponding to the maximum eigenvalue. Thanks to the work by Wuennenberg and Frank, the generalized eigenvalue-eigenvector problem as the solution is much popular than the one of using SVD, although many of numerical solutions for the generalized eigenvalue-eigenvector problem are based on the SVD. Finally, we would like to place particular emphasis on the application of the achieved results to the design of observer-based residual generators. We have in Subsection 5.7.1 shown the interconnections between the parity space and observer-based approaches, and revealed the fact that the observer-based residual generator design can equivalently be considered as a selection of a parity vector or matrix. Thus, the results achieved here are applicable for the design of observer-based residual generators. To illustrate it, consider the non-recursive design form of observer-based residual generators given by

200

7 Residual generation with enhanced robustness against unknown inputs

¡ ¢ u(s) = zJv sv h(s) + yv Ki>v L¯i>v iv (s) + Kg>v L¯g>v gv (s) > yv 5 Sv = Let j = 0, i.e. the eigenvalues of matrix J equal zero, we have u(n) = yv (Ki>v iv (n) + Kg>v gv (n)) or in a more general form u(n) = Yv (Ki>v iv (n) + Kg>v gv (n)) = This is just the form, on account of which we have derived our results. Once yv or Yv is determined under a given performance index, we can set the parameter matrices of the residual generator according to Theorem 5.12. Example 7.2 In this example, we brieﬂy demonstrate the computation of the uniﬁed solution of parity space matrix for the benchmark system vehicle lateral ¯ g>v : dynamics. For our purpose, we ﬁrst set v = 2 and compute K 5 6 12=4385 0=6892 3=0551 0 0 1=0000 0 0 0 9 0=1179 0=0810 29=8597 0 0 0 00 0 : 9 := 7 20=6950 11=9554 0 12=4385 0=6892 3=0551 0 0 1=0000 8 373=8685 20=6950 0 0=1179 0=0810 29=8597 0 0 0 ¯ g>v = X Y W , which yields Next, do an SVD on K 5 6 0=4343 0=0223 0=2416 0=8675 9 0=0033 0=0005 0=9637 0=2668 : : X =9 7 0=0592 0=9982 0=0004 0=0039 8 0=8988 0=0550 0=1132 0=4199 5 6 28=7790 0 0 0 9 0 12=5281 0 0 : := V=9 7 0 0 2=1373 0 8 0 0 0 0=0416 Finally, we compute the optimal solution Y¯v = V 1 X W > 5 6 0=0151 0=0001 0=0021 0=0312 9 0=0018 0=0000 0=0797 0=0044 : : Y¯v = 9 7 0=1130 0=4509 0=0002 0=0530 8 20=8489 6=4123 0=0939 10=0922 as well as the nominal uniﬁed solution of parity matrices given by

7.6 H" optimal fault identiﬁcation scheme

5

Y¯v

Y¯v

Y¯v

201

6

0=0007 0=0000 0=0001 0=0015 9 0=0001 0=0000 0=0038 0=0002 : V 1 X W : = =9 kV 1 X W k1 7 0=0054 0=0215 0=0000 0=0025 8 0=9938 0=3057 0=0045 0=4811 5 6 0=0006 0=0000 0=0001 0=0013 9 0=0001 0=0000 0=0033 0=0002 : V 1 X W : = =9 1 W kV X k2 7 0=0047 0=0188 0=0000 0=0022 8 0=8675 0=2668 0=0039 0=4199 5 6 0=0004 0=0000 0=0001 0=0008 9 0=0000 0=0000 0=0021 0=0001 : V 1 X W := = =9 1 W kV X k4 7 0=0030 0=0120 0=0000 0=0014 8 0=5568 0=1712 0=0025 0=2695

7.6 H4 optimal fault identiﬁcation scheme In this section, we brieﬂy discuss about the H4 optimal fault identiﬁcation problem (OFIP), one of the most popular topics studied in the FDI area. The OFIP is formulated as ﬁnding residual generator (5.24) such that (A 0) is minimized under a given (A 0), where ° ° ° ° °U(s)J ¯ g (s)° ? > °L U(s)J ¯ i (s)° ? (7.152) 4 4 ° ° °i (s) U(s)J ¯ i (s)i (s)° ° ° 2 ¯ g (s)° ? = =, subject to °U(s)J 4 ki k2 Considering that it is often unnecessary to reconstruct i (s) over the whole frequency domain, a weighting matrix Z (s) 5 RH4 can be introduced, which deﬁnes the frequency range of interest, and the H4 OFIP (7.152) is then reformulated into ° ° ° ° °U(s)J ¯ g (s)° ? > °Z (s) U(s)J ¯ i (s)° ? = (7.153) 4 4 Although H4 OFIP is a formulation for the purpose of fault identiﬁcation, it has been originally used for the integrated design of robust controller and FDI. From the fault detection viewpoint, H4 OFIP or its modiﬁed form (7.153) can also be interpreted as a reference model-based design scheme, which is formulated as: given reference model uuhi = i> ﬁnd U(s) so that ku uuhi k2 = ku i k2 $ min +, °£ ¤° ¯ i (s) U(s)J ¯ g (s) ° min ° L U(s)J

U(s)5RH4

4

=

One essential reason for the wide application of H4 OFIP solutions is that it is of the simplest MMP form. Maybe for this reason, in the most studies, optimization problem (7.152) or its modiﬁed form (7.153) are considered as

202

7 Residual generation with enhanced robustness against unknown inputs

being solvable and no special attention has been paid to the solution. The following discussion calls for more attention to this topic. To simplify the discussion, we consider continuous time systems and as¯ i (v) has a RHP-zero v0 = It follows from Lemma 7.7 sume that i 5 R and J that for any given weighting factor Z (v) 5 RH4 ° ° ¯ i (v)° = |Z (v0 )| = min °Z (v) U(v)J (7.154) 4 U5RH4

In case that Z (v) = L> we have ¯ i (v)k4 = 1= min kL U(v)J

U5RH4

(7.155)

¯ i (v)k4 = 1> and as a Note that in (7.155) setting U(v) = 0 gives kL U(v)J result, we have u(v) = 0 =, i (v) u(v) = i (v) =,

ki (v) u(v)k2 = 1= ki (v)k2

That means zero is the best estimation for i (v) (although may not be the only one) in the sense of (7.155) and the estimation error equals to i (v)= Consider further that ¯ g (v)k4 = 0 ¯ g (v) = 0 =, kU(v)J U(v) = 0 =, U(v)J then it becomes evident that U(v) = 0 also solves H4 OFIP= Of course, such an estimation with a relative estimation error equal to ki (v) u(v)k2 =1 ki (v)k2 is less useful in practice. Generally speaking, (7.154) reveals that adding a weighting matrix Z (v) does not automatically ensure a good estimation performance. On the other side, it provides us with a useful relation, based on which the weighting matrix can be suitably selected. Equation (7.154) can be understood that Z (v) should ¯ i (v)> i.e. if v0 is a RHP-zero have a RHP-zero structure similar to the one of J ¯ of Ji (v), then the best solution can be achieved as v0 is also a zero of Z (v)= In the Chapter 14, we shall study H4 OFIP in more details.

7.7 H2 /H2 design of residual generators Beginning with this section, we study design schemes for the residual generator (5.24) whose dynamics is governed by ˆ x (s) (J|g (s)g(s) + J|i (s)i (s)) u(s) = U(s)P ¡ ¢ ¯ i (s)i (s) > U(s) 5 RH4 = ¯ g (s)g(s) + J = U(s) J

(7.156)

7.7 H2 @H2 design of residual generators

203

The major dierence between these schemes lies in the performance index, under which the residual generator design is formulated as an optimization problem. The design problem addressed in this section is the so-called H2 @H2 design scheme, which is formulated as follows. Deﬁnition 7.2 (H2 @H2 design) Given system (7.156), ﬁnd a transfer vector U(s) 5 RH4 that solves ° ° ¯ i (s)° °U(s)J ° °2 = sup M2 (U) = sup (7.157) ¯ ° ° U(s)5RH4 U(s)5RH4 U(s)Jg (s) 2

H2 @H2 design has been proposed in 1989 and was the ﬁrst design scheme using the MV@U type performance index for the post-ﬁlter design. It has been inspired by the optimal selection of parity vectors proposed by Wuennenberg. This can also be observed by the solution to (7.157) that is given in the next theorem. Theorem 7.7 Given continuous time system (7.156), then ° ° °U(v)J ¯ i (v)° 1@2 2 ° ° sup M2 (U) = sup ° ¯ ° = max ($ rsw ) U(v)5RH4 U(v)5RH4 U(v)Jg (v)

(7.158)

2

max ($ rsw ) = sup max ($) $

where max ($) is the maximal eigenvalue of the generalized eigenvalue-eigenvector problem ¡ ¢ ¯ i (m$) max ($)J ¯ g (m$)J ¯ g (m$) = 0 ¯ i (m$)J (7.159) ymax (m$) J with ymax (m$) being the corresponding eigenvector. The optimal solution Ursw (v) is given by Ursw (v) = te (v)ymax (v)> te (v) 5 RHp 2

(7.160)

where te (v) represents a band pass ﬁlter at frequency $ rsw , which gives Z 4 1 ¯ g (m$)J ¯ g (m$)Ursw Ursw (m$)J (m$)g$ (7.161) 2 4 ¯ g (m$)J ¯ g (m$)ymax (m$ rsw )= ymax (m$ rsw )J Proof. The original proof given by Ding and Frank consists of three steps Step1: prove that the optimization problem ° ° °U(s)J ¯ i (s)° 2 with constraint M= sup U(s)5RH4 ° ° ¯ g (s)° 0 ? °U(s)J 2

is equivalent to a generalized eigenvalue-eigenvector problem;

(7.162)

204

7 Residual generation with enhanced robustness against unknown inputs

Step 2: ﬁnd the solution for the generalized eigenvalue-eigenvector problem; Step 3: prove sup M2 (U) = M= U(s)5RH4

Here, we only outline the ﬁrst step. The next two steps are straightforward and the reader can refer to the paper by Ding and Frank given at the end of this chapter. Note that (7.162) is, according to the duality theorem, equivalent to ! Ã Ã ! ´ ³ ˜ ($) ˜i ($) U H ˜ ˜ M = sup inf S + S Hi ($) U ($) ¯ 0 U where S (·) is an operator, Z4 ³ ´ ³ ´ ˜ ˜ ˜g ($) U ˜ ($) g$ S Hg ($) U ($) = wudfh H ³

4 Z4

´

˜i ($) U ˜ ($) = S H

³ ´ ˜i ($) U ˜ ($) g$ wudfh H

4

¯gW (m$) H ¯g (m$) > H ˜i ($) = H ¯iW (m$) H ¯i (m$) ˜g ($) = H H W ˜ ($) = U (m$) U (m$) = U It turns out M = inf () 0

with variable satisfying ;$

˜i ($) H ˜g ($) 0 +, H ˜g ($) H ˜i ($) 0= + H

(7.163) can be equivalently written as max ($) with max ($) denoting the maximal eigenvalue of matrix pencil ˜g ($) = ˜i ($) max ($) H H As a result, we ﬁnally have M = inf () = max ($) = 0

t u

(7.163)

7.7 H2 @H2 design of residual generators

205

¯ g (v) is left invertible in RH4 > i.e. ;$ 5 [0> 4] Suppose that J ¯ g (m$) A 0= ¯ g (m$)J J

(7.164)

¯ g (v) It follows from Lemma 7.4 that we are able to do a CIOF of J ¯ g (v) = Jgr (v)Jgl (v) J with a left invertible co-outer Jgr (v) and co-inner Jgl (v)= Let ymax (v)J1 Ursw (v) = te (v)¯ gr (v) with ¡ ¢ ¯ ¯ y¯max (m$) J1 gr (m$)Ji (m$)Ji (m$)Jgr (m$) max ($)L = 0=

(7.165)

In other words, in this case sup U(v)5RH4

¡ ¢ ¯ M2 (U) = max ($ rsw )> 1@2 ¯ J1 max ($ rsw ) = sup gr (m$)Ji (m$) = $

Without proof, we give the analogous result of the above theorem for discrete time systems. The interested reader is referred to the paper by Zhang et al. listed at the end of this chapter. Corollary 7.4 Given discrete time system (7.156), then ° ° °U(})J ¯ i (})° ° °2 = 1@2 M2 (U) = sup sup max ( rsw ) ° ¯ ° (}) U(}) J U(})5RH4 U(})5RH4 g 2

max (rsw ) = sup max ()

where max () is the maximal eigenvalue of the generalized eigenvalue- eigenvector problem ¯ i (hm )J ¯ i (hm ) max ()J ¯ g (hm )J ¯ g (hm )) = 0 smax (hm )(J

(7.166)

with smax (hm ) being the corresponding eigenvector. The optimal solution Ursw (}) is given by Ursw (}) = irsw (})s(}) where irsw (}) is an ideal band pass with the selective frequency at rsw , which satisﬁes Z 0

;t W (}) 5 RH2 > irsw (hm )t(hm ) = 0> 6= rsw 2

irsw (hm$ )t(hm$ )t (hm$ )irsw (hm$ )g = t(hmrsw )t (hmrsw ).

(7.167)

206

7 Residual generation with enhanced robustness against unknown inputs

Although the H2 @H2 design is the ﬁrst approach proposed for the optimal design of observer-based residual generators using the advanced robust control technique, only few study has been devoted to it. In our view, there are two reasons for this situation. The ﬁrst one is that the derivation of the solution, dierent from the standard H2 control problem, is somewhat involved. The second one is that the implementation of the resulting residual generator seems unpractical. The reader may notice that the most signiﬁcant characterization of an H2 @H2 optimal residual generator is its bandpass property. It is this feature that may considerably restrict the application of H2 @H2 optimal residual generator due to the possible loss of fault sensitivity. On the other side, this result is not surprising. Remember the interpretation of the H2 norm as the RMS value of a system output when this system is driven by a zero mean white noise with unit power spectral densities. It is reasonable that an optimal fault detection will be achieved at frequency $ rsw > since at other frequencies the relative inﬂuence of the fault, whose power spectral density is, as assumed to be a white noise, a constant, would be deﬁnitively smaller. Unfortunately, most kinds of faults are deterministic and therefore H2 @H2 design makes less practical sense.

7.8 Relationship between H2 /H2 design and optimal selection of parity vectors The analogous form between the H2 @H2 solution (7.166) and the optimal selection of parity vectors (7.130) motivates our discussion in this section. For our purpose, we consider discrete time model {(n + 1) = D{(n) + Ex(n) + Hg g(n) + Hi i (n) |(n) = F{(n) + Gx(n) + Ig g(n) + Ii i (n)=

(7.168) (7.169)

Suppose that {jg (0)> jg (1)> · · · } is the impulse response of system (7.168)(7.169) to the unknown disturbances g. Apparently, jg (0) = Ig > jg (1) = FHg > · · · > jg (v) = FDv1 Hg > · · · =

(7.170)

We can then express matrix Kg>v in parity space residual generator (5.92) in terms of the impulse response as follows 5 6 ··· 0 jg (0) 0 9 .. : 9 jg (1) jg (0) . . . . : := Kg>v = 9 9 . : .. .. 7 .. . . 0 8 jg (v) · · · jg (1) jg (0) Partition the parity vector yv as

7.8 Relationship between H2 @H2 design and optimal selection of parity vectors

£

yv = yv>0 yv>1 · · · yv>v

207

¤

where the row vector yv>l 5 Rp > l = 0> 1> · · · v. Then, we have £ ¤ yv Kg>v = *(v) *(v 1) · · · *(0) with *(l) =

l X

lo jg (o)> l = yv>vl > l = 0> 1> · · · v=

o=0

Let v go to inﬁnity. It leads to £ ¤ lim yv Kg>v = *(4) · · · *(0)

(7.171)

lo jg (o) = (l) jg (l) = Z 1 (S (})Jg (}))

(7.172)

v$4

and in this case *(l) =

l X o=0

S (}) = Z[(l)]> (l) = {0 > 1 > · · · }

(7.173)

where denotes the convolution. Equation (7.173) means that S (}) is the }-transform of the sequence {0 > 1 > · · · }. According to the Parseval Theorem, we have W W yv = lim yv Kg>v Kg>v

v$4

=

1 2

Z 0

2

4 X

*(l)*W (l)

(7.174)

l=0

S (hm$ )J|g (hm$ )J|g (hm$ )S (hm$ )g$

with J|g (}) = F (}L D)1 Hg + Ig = Similarly, it can be proven that Z 2 1 W W lim yv Ki>v Ki>v yv = S (hm$ )J|i (hm$ )J|i (hm$ )S (hm$ )g$ (7.175) v$4 2 0 1

with J|i (}) = F (}L D) Hi + Ii = On the other side, if given a residual generator ´ ³ ˆx (})x(}) ˆ x (})|(}) Q (7.176) u(}) = U(}) P we can always construct a parity vector, as stated in the next lemma. Lemma 7.10 Given system (7.168)-(7.169) and a residual generator (7.176) with U(}) 5 RH1×p 4 . Then the row vector deﬁned by £ ¤ ¯ F¯ D¯E ¯ F¯ E ¯ G ¯ y = · · · F¯ D¯2 E (7.177) ¯ E> ¯ F> ¯ G) ¯ is the state space realization of U(})P ˆ x (}), belongs to the where (D> parity space Sv (v $ 4).

208

7 Residual generation with enhanced robustness against unknown inputs

Proof. Assume that (Du > Eu > Fu > Gu ) is a state space realization of U(}). Recalling Lemma 3.1, we know that ¸ ¸ ¤ £ D OF 0 ¯ = Gu = ¯ = O > F¯ = Gu F Fu > G >E D¯ = Eu Eu F Du It can be easily obtained that 6 F £ ¤ 9 FD : : ¯ F¯ D¯E ¯ F¯ E ¯G ¯ 9 = lim · · · F¯ D¯2 E 9 FD2 : v$4 8 7 .. . 6 5 F : £ ¤9 9 F(D OF) : = lim · · · Fu Du Eu Fu Eu Gu 9 F(D OF)2 : = v$4 8 7 .. . 5

lim yKr>v

v$4

(7.178)

For a linear discrete time system (n + 1) = (D OF)(n)> (n) = F(n)

(7.179)

with any initial state vector (0) = 0 5 Rq , apparently (0) = F0 > (1) = F(D OF)0 > (2) = F(D OF)2 0 > · · · = and O is selected to ensure the stability of D OF, the Since U(}) 5 RH1×p 4 cascade connection of system (7.179) and U(}) is stable. So lim Z 1 {U(})(})} = 0=

n$4

Note that 6 F0 : £ ¤9 9 F(D OF)0 : lim Z 1 {U(})(})} = lim · · · Fu Du Eu Fu Eu Gu 9 F(D OF)2 0 : v$4 n$4 8 7 .. . 5

we get

6 F : £ ¤9 9 F(D OF) : lim · · · Fu Du Eu Fu Eu Gu 9 F(D OF)2 : 0 = 0 v$4 8 7 .. . 5

for any initial state vector 0 5 Rq . Thus it can be concluded that

7.8 Relationship between H2 @H2 design and optimal selection of parity vectors

lim

v$4

£

209

6

5

F F(D OF) : ¤9 : 9 · · · Fu Du Eu Fu Eu Gu 9 F(D OF)2 : = 0= 8 7 .. .

At last, from (7.178) we obtain lim yKr>v = 0

v$4

i.e. the vector y deﬁned by (7.177) belongs to the parity space Sv (v $ 4). The lemma is thus proven. u t It is of interest to note that vector y is indeed composed of the impulse ¯ ¯ ¯ 1 ¯ ˆ response of ©the residual generator U(}) ª Px (}) = G + F(}L D) E, which 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ is given by G> F E> F DE> F D E> · · · . Based on the above analysis, the following theorem can be obtained. Theorem 7.8 Given system (7.168)-(7.169) and assume that yv>rsw > Mv>rsw and Ursw (})> Mrsw are the optimal solutions of optimization problems Mv>rsw = max Mv = max yv 5Sv

W W yv Ki>v Ki>v yv

yv 5Sv

W yW yv Kg>v Kg>v v

Mrsw = R 2

max 1×p R02

=

U(})5RH4

R 2

0

max

=

W W yv>rsw yv>rsw Ki>v Ki>v W yW yv>rsw Kg>v Kg>v v>rsw

U(})5RH1×p 4

(7.180)

M

ˆ x (hm )J|i (hm )J (hm )P ˆ x (hm )U (hm )g$ U(hm )P |i ˆ x (hm )J|g (hm )J (hm )P ˆ x (hm )U (hm )g$ U(hm )P |g

ˆ x (hm$ )J|i (hm$ )J (hm$ )P ˆ x (hm$ )Ursw Ursw (hm$ )P (hm$ )g$ |i = R02 (hm$ )g$ ˆ x (hm$ )J|g (hm$ )J (hm$ )P ˆ x (hm$ )Ursw Ursw (hm$ )P 0

(7.181)

|g

respectively. Then lim Mv>rsw = Mrsw

v$4

ˆ x (}) S (}) = Ursw (})P

(7.182) (7.183)

where S (}) = Z[(l)]> (l) = {yv$4>rsw>v > yv$4>rsw>v1 > · · · > yv$4>rsw>0 }= (7.184) Proof. Let yv$4>rsw denote the optimal solution of optimization problem (7.180) as v $ 4. Remembering Theorem 7.6 and the associated remark, ˆ x1 (})Q ˆx (}), it follows from (7.173)-(7.175) that for any LCF of J|x (}) = P the post-ﬁlter Ur (}) given by ˆ x1 (}) Ur (}) = S (})P

210

7 Residual generation with enhanced robustness against unknown inputs

where S (}) is deﬁned by (7.184), leads to M |U(})=Ur (}) = lim Mv>rsw = lim max Mv = max max Mv v$4

v$4 yv 5Sv

v

yv 5Sv

max

U(})5RH1×p 4

M=

(7.185) We now demonstrate that M |U(})=Ur (}) = Mrsw =

max

U(})5RH1×p 4

M=

(7.186)

Suppose that (7.186) does not hold. Then, the optimal solution of optimization and dierent from Ur (})> should problem (7.181), denoted by Uf (}) 5 RH1×p 4 lead to M |U(})=Uf (}) = max 1×p M A M |U(})=Ur (}) = (7.187) U(})5RH4

According to Lemma 7.10, we can ﬁnd a parity vector y 5 Sv whose comˆ x (}). ponents are just a re-arrangement of the impulse response of Uf (})P Moreover, because of (7.173)-(7.175), we have Mv |yv =y = M |U(})=Uf (})

(7.188)

As a result, it follows from (7.185), (7.187) and (7.188) that Mv |yv =y A max max Mv v

yv 5Sv

which is an obvious contradiction. Thus we can conclude that Mrsw =

max

U(})5RH1×p 4

M = M |U(})=Ur (}) = lim Mv>rsw v$4

and ˆ x1 (}) := Ursw (}) Ur (}) = S (})P solve optimization problem (7.181). The theorem is thus proven. u t Theorem 7.8 gives a deeper insight into the relationship between the parity space approach and the H2 @H2 design and reveals some very interesting facts when the order of the parity relation v increases: • The optimal performance index Mv>rsw of the parity space approach converges to a limit which is just the optimal performance index Mrsw of the H2 @H2 optimization. • There is a one-to-one relationship between the optimal solutions of optimization problems (7.180) and (7.181) when the order of the parity relation v $ 4. Since Ursw (}) is a band-limited ﬁlter, the frequency response of yv$4>rsw is also band-limited. The above result can be applied in several ways, for instance:

7.9 LMI aided design of FDF

211

• for multi-dimensional systems, the optimal solution of the H2 @H2 design can be approximately computed by at ﬁrst calculating the optimal solution of the parity space approach with a high order v and then doing the ztransform of the optimal parity vector. It is worth noticing that numerical problem may be met for some systems, especially when D is unstable. • In the parity space approach, a high order v will improve the performance index Mv>rsw but, on the other side, increase the on-line computational eort. To determine a suitable trade-o between performance and implementation eort, the optimal performance index Mrsw of the H2 @H2 design can be used as a reference value. • Based on the property that the frequency response of yv$4>rsw is bandlimited, advanced parity space approaches can be developed to achieve both a good performance and a low order parity vector. For instance, inﬁnite impulse response (IIR) ﬁlter and wavelet transform have been introduced, respectively, to design optimized parity vector of low order and good performance. Example 7.3 (A numerical example) Given a discrete time system modelled by (7.168)-(7.169), where ¸ ¸ £ ¤ 1 1=30 2 D= >E = >F = 0 1 0=25 0=25 1 ¸ ¸ 0=4 0=6 > Hi = > G = Ig = Ii = 0= (7.189) Hg = 0=5 0=1 As system (7.189) is stable, matrix O in the LCF can be selected to be ˆ x (}) is an identity matrix. To solve the generalized zero matrix and thus P eigenvalue-eigenvector problem (7.166) to get rsw that achieves max (rsw ) = sup max (), note that max () =

0=0125 + 0=01 cos = 0=41 0=4 cos

Therefore, the optimal performance index of the H2 @H2 design is Mrsw = 2=25 and the selective frequency is rsw = 0. Fig.7.2 demonstrates the change of the optimal performance index Mv>rsw with respect to the order of the parity relation v. From the ﬁgure it can be seen that Mv>rsw increases with the increase of v and, moreover, Mv>rsw converges to Mrsw when v $ 4 . Fig. 7.3 shows the frequency responses of the optimal parity vector yv>rsw when v is chosen as 20> 50> 100 and 200 respectively. We see that the bandwidth of the frequency response of yv>rsw becomes narrower and narrower with the increase of v.

7.9 LMI aided design of FDF Comparing with the methods introduced in the last two sections, the FDF scheme with its ﬁxed structure oers a lower degree of the design freedom.

212

7 Residual generation with enhanced robustness against unknown inputs 2.5

2

Js,opt

Optimal performance index

Jopt

1.5

1

0.5

0 0

50

100

150

The order of the parity relation s

Fig. 7.2 The optimal performance Mv>rsw vs v 50 s=20 s=50 s=100 s=200

45

40

35

25

⏐ν

s,opt

jθ

(e )⏐

30

20

15

10

5

0 Ŧπ

Ŧ3

Ŧ2

Ŧ1

0

θ

1

2

3

π

Fig. 7.3 The frequency response of the optimal parity vector yv>rsw vs v

On the other hand, its observer structure allows a design using the available approaches for the robust observer design. For this reason, the FDF design is receiving much attention currently. In this section, we deal with the optimal design of FDF under dierent performance indices. Recall that for a given system described by (3.30)-(3.31), an FDF delivers a residual whose dynamics with respect to the faults and unknown inputs is described by ˆi (s)i (s) ˆg (s)g(s) + Q u(s) = Q ˆg (s) = F(sL D + OF)1 (Hg OIg ) + Ig Q ˆi (s) = F(sL D + OF)1 (Hi OIi ) + Ii = Q

(7.190) (7.191) (7.192)

7.9 LMI aided design of FDF

213

ˆg (s) is smaller Our main objective is to ﬁnd an observer matrix O such that Q ˆi (s) is as large as possible. We shall than a given bound and simultaneously Q use the H2 and H4 norms as well as the so-called H index to measure the size of these two transfer matrices. To this end, the LMI technique will be used as a mathematical tool for the problem solution. 7.9.1 H2 to H2 trade-o design of FDF We begin with a brief review of the H2 optimization problem described by min kF(sL D + OF)1 (Hg OIg )k2 = O

(7.193)

Remark 7.7 Remember that for a continuous time system its H2 norm exists only if it is strictly proper. For a discrete time system, ¢ ¡ ¢ ¡ kIg + F(sL D + OF)1 (Hg OIg )k2 = wudfh FS F W + wudfh Ig IgW ¢ ¡ = wudfh(Hg OIg )W T(Hg OIg ) + wudfh IgW Ig where S> T are respectively the solutions of two Lyapunov equations. Thus, min kIg + F(sL D + OF)1 (Hg OIg )k2 +, O

min kF(sL D + OF)1 (Hg OIg )k2 = O

For this reason, we only need to consider the optimization problem (7.193). Theorem 7.9 Given system F(sL D + OF)1 (Hg OIg ) and suppose that A1. (F> D) is detectable; A2. Ig has full row rank with Ig IgW = L; ¸ D m$L Hg A3. for a continuous time system or for a discrete time sysF Ig ¸ m D h L Hg tem has full row rank for all $ 5 [0> 4] or 5 [0> 2]> F Ig then the minimum ¡ ¢1@2 min kF(sL D + OF)1 (Hg OIg )k2 = wudfh(F[F W ) or O ¡ ¢1@2 min kF(sL D + OF)1 (Hg OIg )k2 = wudfh(F\ F W ) O

is achieved by O = O2 = [F W + Hg IgW or ¢¡ ¢1 ¡ O = O2 = D\ F W + Hg IgW L + F\ F W where matrix [ 0 solves the Riccati equation

(7.194) (7.195)

214

7 Residual generation with enhanced robustness against unknown inputs

(D Hg IgW F)[ + [(D Hg IgW F)W [F W F[ + Hg HgW Hg IgW Ig HgW = 0 (7.196) +, (D O2 F)[ + [(D O2 F)W + [F > F[ + Hg HgW Hg IgW Ig HgW = 0 (7.197) and matrix \ 0 solves the Riccati equation W

W

(D O2 F) \ (D O2 F) \ + (Hi O2 Ii ) (Hi O2 Ii ) = 0=

(7.198)

This theorem is a dual result of the well-known H2 optimization of the state feedback controller, min k(F GN)(sL D EN)1 Hk2 N

the proof is therefore omitted. Remark 7.8 In A2, if Ig IgW 6= L we are able to ﬁnd an output transformation Y to ensure that Y Ig IgW Y W = L as far as Ig has full row rank. Assumption A3 ensures that no zeros lie on the imaginary axis and at inﬁnity. Recall that the optimal design of FDF diers from the optimal estimation mainly in its additional requirement on the sensitivity to the faults. This requires to add an additional optimization objective to (7.193). Next, we are going to introduce a design scheme, starting from Theorem 7.9, that allows a compromise between the robustness and the fault sensitivity. Set O = O2 + O with O2 given in (7.194) and bring the dynamics of the residual generator (7.190) into h˙ = (D O2 F)h + (Hg O2 Ig )g + (Hi O2 Ii )i + y y = O(| |ˆ)> h = { { ˆ> u = | |ˆ = Fh + Ig g + Ii i with { ˆ denoting the state variable estimation delivered by the FDF. Let Wug (s) denote the dynamic part of the transfer matrix from g(s) to u(s), i.e. Wug (s)g(s) = F(sL D + O2 F)1 ((Hg O2 Ig )g(s) + y(s)) = Since y(s) = O (Fh(s) + Ig g(s)) ¡ ¢ = O F(sL D + O2 F)1 ((Hg O2 Ig )g(s) + y(s)) + Ig g(s) ¢1 ¡ = L + OF(sL D + O2 F)1 ¢ ¡ O F(sL D + O2 F)1 (Hg O2 Ig ) + Ig g(s) we obtain

7.9 LMI aided design of FDF

215

Wug (s)g(s) = F(sL D + O2 F)1 (Hg O2 Ig )g(s) + F(sL D + O2 F)1 ¶ μ ¡ ¢ F(sL D + O2 F)1 1 1 L + OF(sL D + O2 F) (O) (Hg O2 Ig )g(s) + Ig g(s) = F(sL D + O2 F)1 (Hg O2 Ig )g(s) + ¶ μ F(sL D + O2 F)1 = F(sL D + O2 F + OF)1 (O) (Hg O2 Ig )g(s) + Ig g(s) Note that X (s) := F(sL D + O2 F)1 (Hg O2 Ig ) + Ig is co-inner and ¡ ¢W X (s) F(sL D + O2 F)1 (Hg O2 Ig ) 5 RHB 4 , thus we ﬁnally have kWug (s)k22 = kF(sL D + O2 F)1 (Hg O2 Ig )k22 +kF(sL D + (O2 + O)F)1 (O)k22 =

(7.199)

With the aid of (7.199), we are able to formulate our design objective as ﬁnding O such that D (O + O)F is stable kWug (s)k2 ? +,

(7.200) (7.201)

kF(sL D + (O2 + O)F)1 Ok22 ? 2 wudfh(F[F W ) or kF(sL D + (O2 + O)F)1 Ok22 ? 2 wudfh(F\ F W ) kF(sL D + (O2 + O)F)1 (Hi (O2 + O)Ii )k2 $ max =

(7.202)

Following the computing formula for the H2 norm, (7.200)-(7.202) can further be reformulated as: for the continuous time system ¡ ¢ ¯i OIi )W Z (H ¯i OIi ) max wudfh (H (7.203) O

wudfh(OW Z O) ? 2 wudfh(F[F W ) := 1 W

W

(DO2 OF)Z + Z (DO2 OF) + F F = 0> Z A 0 ¯i = Hi O2 Ii DO2 = D O2 F> H and for the discrete time system ¡ ¢ ¯i OIi )W ](H ¯i OIi ) max wudfh (H O

wudfh(OW ]O) ? 2 wudfh(F\ F W ) := 1

(7.204) (7.205)

(7.206) (7.207)

(7.208) (DO2 OF)W ](DO2 OF) ] + F W F = 0> ] A 0= 1 ¯ ¯ S = Setting O = S O> S = Z for the continuous time case and O = S O> 1 ] for the discrete time case leads respectively to ¯ W S O)> ¯ wudfh(OW ]O) = wudfh(O ¯ W S O) ¯ wudfh(OW Z O) = wudfh(O ¢ ¡ ¯i OIi ) ¯i OIi )W Z (H wudfh (H ¢ ¡ W ¯ i ) S (Z H ¯i OI ¯ i) ¯i OI = wudfh (Z H ¢ ¡ ¯i OIi ) ¯i OIi )W ](H wudfh (H ¢ ¡ ¯ i )W S (] H ¯i OI ¯ i) = ¯i OI = wudfh (] H

216

7 Residual generation with enhanced robustness against unknown inputs

Using Schur complement we have that ¯ W S O) ¯ ? 1 kF(sL DO2 + OF)1 Ok2 = wudfh(O ¯ and (DO2 S OF) is stable if and only if • for the continuous time system: there exist T1 > Z such that ¯ + FW F ? 0 ¯ W OF DWO2 Z + Z DO2 F W O ¸ ¯ Z O A 0> wudfh(T1 ) ? 1 W ¯ O T1 • for the discrete time system: there exist T2 > ] such that ¸ ¯ ] ]DO2 OF A0 ¯W ] FW F DWO2 ] F W O ¸ ¯ ] O ¯ W T2 A 0> wudfh(T2 ) ? 1 = O

(7.209) (7.210)

(7.211) (7.212)

In summary, we obtain the following optimization design scheme for FDF. Theorem 7.10 The optimization problem (7.200)-(7.202) is equivalent to • for continuous time systems ³¡ ¢ ¡ ¢´ ¯i OI ¯i OI ¯ i W Z 1 Z H ¯ i max wudfh Z H ¯ Z>O

(7.213)

subject to ¯ + FW F ? 0 ¯ W OF DWO2 Z + Z DO2 F W O ¸ ¯ Z O ¯ W T1 A 0> wudfh(T1 ) ? 1 O • for discrete time systems ³¡ ¢ ¡ ¢´ ¯i OI ¯i OI ¯ i W ] 1 ] H ¯ i max wudfh ] H ¯ ]>O

(7.214) (7.215)

(7.216)

subject to

¸ ¯ ] ]DO2 OF A0 ¯W ] FW F DWO2 ] F W O ¸ ¯ ] O ¯ W T2 A 0> wudfh(T2 ) ? 1 = O

(7.217) (7.218)

On account of the above-achieved results, following algorithm for the H2 to H2 optimal design of FDF is proposed.

7.9 LMI aided design of FDF

217

Algorithm 7.6 H2 to H2 optimization of continuous time FDF Step 1: Solve Riccati equation (7.196) for [ A 0 and further O2 ; ¯ Z; Step 2: Solve optimization problem (7.214)-(7.215) for O> Step 3: Set the optimal solution as follows: ¯ O = [F W + Hg I W + Z 1 O= Algorithm 7.7 H2 to H2 optimization of discrete time FDF Step 1: Solve Riccati equation (7.198) for \ A 0; ¯ ]; Step 2: Solve optimization problem (7.217)-(7.218) for O> Step 3: Set the optimal solution as follows: ¢¡ ¢1 ¡ ¯ + ] 1 O= O = D\ F W + Hg IgW L + F\ F W Remark 7.9 Notice that the cost functions (7.213) and (7.216) are nonlinear regarding to Z or ]= Moreover, due to the constraints (7.215) and (7.218), the ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¯ i and OI ¯ i W ] 1 OI ¯ i are bounded. On account ¯ i W Z 1 OI terms OI of this fact, the cost functions can be replaced by ¢ ¡ W ¯i IiW O ¯iW OI ¯W H ¯i H ¯ i as well as ¯i Z H max wudfh H ¯ Z>O ¢ ¡ W ¯i IiW O ¯iW OI ¯W H ¯i H ¯ i = ¯i ] H max wudfh H ¯ ]>O

Example 7.4 We now apply Algorithm 7.6 to the benchmark system EHSA with model (3.83). We suppose that measurement noises are present in the sensor signals and model them by extending Hg > Ig to 5 6 0 00 9 0=143 0 0 : ¸ 9 : 010 9 : 0 0 : > Ig = = Hg = 9 0 001 7 0 0 08 0 00 Solving Riccati equation (7.196) gives 5 6 8=69 × 1022 1=03 × 1025 9 0=067976 2=06 × 1012 : 9 : ¢ ¡ 25 9 1=538 × 10 8=078 × 1028 : = O2 9 > wudfh F[F W 0=7 × 105 = : 7 6=6889 × 104 2=03 × 106 8 2=03 × 106 2=02 × 1015 ¯ Z with In the next step, optimization problem (7.214)-(7.215) is solved for O> 1 1=7 × 105 >

218

7 Residual generation with enhanced robustness against unknown inputs

5

6 0=1703 4810=7 2=84 × 106 1=47 × 105 1=81 × 105 91=47 × 105 8=0014 × 104 5=08 × 106 : 0=0456 278=21 9 : 5 6 8 4: 9 9=95 × 10 0=010 2=6103 × 10 : Z = 9 1=81 × 10 5=08 × 10 7 0=1703 0=0456 0=010 1=6756 × 106 0=0003 8 4 0=0003 265=89 4810=7 278=21 2=6103 × 10 6 5 21561 145=29 : 9 45507 19=36 : 9 6 5 : ¯ 9 O = 9 8=01 × 10 1=54 × 10 : = 7 0=389 9=77 × 105 8 40=74 1460=2 Finally, the optimal solution is 5

6 0=0177 0=0093 9 0=0048 0=0042 : 9 : ¯ = 9 0=0084 2=97 × 105 : = O = O2 + Z 1 O 9 : 7 3=03 × 105 8 53=96 0=8745 5=6617 7.9.2 On H index Remember the discussion on the fault sensitivity in Subsection 7.4.1, which provides us with reasonable arguments to evaluate the fault sensitivity by means of the so-called Vi> index. In this subsection, we shall address the deﬁnition of Vi> index for a transfer matrix and its computation. This index is called H index and will be, instead of the H4 norm or the H2 norm as required in (7.202), used for the evaluation of the fault sensitivity. Deﬁnition 7.3 Given system |(s) = J(s)x(s)= The strict H index of J(s) is deﬁned by kJ(s)x(s)k2 = (7.219) kJ(s)k = inf x6=0 kx(s)k2 Note that kJ(s)k is not a norm. For instance, if the row number of J(s) is smaller than its column number, then there exists some x(s) 6= 0 so that J(s)x(s) = 0 and therefore kJ(s)k = 0= Consider that an evaluation of those faults, which are structurally undetectable (see Chapter 4), makes no sense. We are only interested in evaluation of the minimum value of kJ(s)x(s)k2 > for kx(s)k2 = 1> which is dierent from zero. Since Z 4 1 kJ(s)x(s)k2 = x (m$)J (m$)J(m$)x(m$)g$ 2 4 we have the nonzero minimum value of kJ(s)x(s)k2 • for surjective J(s)

¡ ¡ ¢ ¢ kJ(s)k = min JW (m$) or min JW (hm ) $

7.9 LMI aided design of FDF

219

• for injective J(s) ¡ ¢ kJ(s)k = min (J(m$)) or min J(hm ) $

where (·) denotes the minimum singular value of a matrix. For the FDI purpose, we introduce the following deﬁnition. Deﬁnition 7.4 Given system |(s) = J(s)x(s)= The H index of J(s) is deﬁned by ¡ ¡ ¢ ¢ kJ(s)k = min JW (m$) or min JW (hm ) $

for surjective J(s) satisfying ;$> J(m$)JW (m$) A 0 or ;> J(hm )JW (hm ) A 0 and

(7.220)

¡ ¢ kJ(s)k = min (J(m$)) or min J(hm ) $

for injective J(s) satisfying ;$> JW (m$)J(m$) A 0 or ;> JW (hm )J(hm ) A 0=

(7.221)

Remark 7.10 Note that if both (7.220) and (7.221) are satisﬁed, then ¡ ¡ ¡ ¢ ¢ ¢ min JW (m$) = min (J(m$)) > min JW (hm ) = min J(hm ) = $

$

Moreover, ;$> J(m$)JW (m$) = J(m$)JW (m$) and ;> J(hm )JW (hm ) = J(hm )JW (hm )= Next, we study the computation of the H index of a transfer matrix as deﬁned in Deﬁnition 7.4. We start with a detailed discussion about continuous time systems and give the "discrete time version" at the end of the discussion. Our major results rely on the following lemma. Lemma 7.11 Let D> E> S> V> U be matrices of compatible dimensions with S> U symmetric, U A 0 and (D> E) stabilizable. Suppose either one of the following assumptions is satisﬁed: A1. D has no eigenvalues on the imaginary axis; A2. S 0 or S 0 and (S> D) has no unobservable modes on the imaginal axis. Then, the following statements are equivalent:

220

7 Residual generation with enhanced robustness against unknown inputs

I. The parahermitian rational matrix ¤ £ (v) = E W (vL DW )1 L

S V VW U

¸

(vL D)1 E L

¸

satisﬁes (m$) A 0 for all 0 $ 4 II. There exists a unique real and symmetric [ such that (D EU1 V W )W [ + [(D EU1 V W ) [EU1 E W [ + S VU1 V W = 0 and that D EU1 V W EU1 E W [ is stable. Lemma 7.11 is a standard result in the robust control theory. Hence, its proof is omitted. Theorem 7.11 Given system J(v) = G + F (vL D)1 E with A1. GW G 2 L A 0 and A2. (F> D) has no unobservable modes on the imaginary axis, then inequality ¡ ¢W ¡ ¢ F(m$L D)1 E + G F(m$L D)1 E + G A 2 L

(7.222)

holds for all $, including at inﬁnity, if and only if there exists a symmetric matrix [ such that D¯W [ + [ D¯ [EU1 E W [ + F W (L GU1 GW )F = 0 with

(7.223)

D¯ = D EU1 GW F> U = GW G 2 L=

Proof. The proof is straightforward. We ﬁrst substitute S = F W F> V = F W G> U = GW G 2 L into (v) given in Lemma 7.11, which gives ¸ ¸ ¤ FW F £ (vL D)1 E FW G (v) = E W (vL DW )1 L GW F G W G 2 L L ¢¡ ¢ ¡ W W 1 W W W 1 F(vL D ) E + G 2 L= = E (vL D ) F + G As a result, (m$) A 0 +,(7.222) holds. Finally, using Lemma 7.11 the theorem is proven. u t With the proof of Theorem 7.11, the following corollary becomes evident. Corollary 7.5 Given system J(v) = G + F (vL D)

1

E with

7.9 LMI aided design of FDF

A1. GGW 2 L A 0 and A2. (D> E) has no uncontrollable modes on the imaginary axis, then inequality ¡ ¢¡ ¢W F(m$L D)1 E + G F(m$L D)1 E + G A 2 L

221

(7.224)

holds for all $, including at inﬁnity, if and only if there exists a symmetric matrix \ such that ¯ + \ D¯W \ F W U1 F\ + E(L GW U1 G)E W = 0 D\ with

(7.225)

D¯ = D EGW U1 F> U = GGW 2 L=

Remember that with the H index deﬁned in Deﬁnition 7.4 we are only interesting in the minimal nonzero singular value of a transfer matrix, which is equivalent to, for given J(v) = F (vL D)1 E + G> J (m$)J(m$) A 0> ;$ if J(v) is injective or

J(m$)J (m$) A 0> ;$

if J(v) is surjective. Note that ;$ ¸ D m$L E I= J (m$)J(m$) A 0 +, udqn =q+n F G ¸ D m$L E II= J(m$)J (m$) A 0 +, udqn =q+p F G

where q> n> p denote the number of the state variables, the inputs and the outputs respectively. Hence, it also ensures that there exists no unobservable mode on the imaginary axis in case I and no uncontrollable mode on the imaginary axis in case II. As a result of Theorem 7.11, Corollary 7.5 and the above discussion, we have 1

Theorem 7.12 Given system J(v) = G + F (vL D)

E that satisﬁes

A1. (a) GW G 2 L A 0> if J(v) is injective, or (b) GGW 2 L A 0> if J(v) is surjective A2. (a) for J(v) being injective ;$ 5 [0> 4] ¸ D m$L E udqn =q+n (7.226) F G or (b) for J(v) being surjective ;$ 5 [0> 4] ¸ D m$L E =q+p udqn F G

(7.227)

222

7 Residual generation with enhanced robustness against unknown inputs

then for a given constant A 0 the following two statements are equivalent: S1. H index satisﬁes

kJ(v)k A

(7.228)

S2. for case (a) there exists a symmetric matrix [ such that [ D¯ + D¯W [ [EU1 E W [ + F W (L GU1 GW )F = 0 D¯ = D EU1 GF> U = G> G 2 L

(7.229)

or for case (b) there exists a symmetric matrix \ such that ¯ + \ D¯W \ F W U1 F\ + E(L GW U1 G)E W = 0 D\ D¯ = D EGW U1 F> U = GGW 2 L=

(7.230)

(7.229) and (7.230) are Riccati equations, which can also be equivalently reformulated as Riccati inequalities. To this end, dierent methods are available. Next, we introduce one approach proposed by Zhang and Ding. Recalling Lemma 7.4 and its dual form for the IOF, we can, under condi1 tion (7.226) or (7.227), factorize J(v) = G + F (vL D) E 5 RH4 into ˆ (v) = Q (v)P 1 (v) ˆ 1 (v)Q J(v) = P ˆ (v)> Q (v) are co-inner and inner respectively and P ˆ 1 (v)> P 1 (v) 5 where Q RH4 = It turns out that kJ(v)k = kP 1 (v)k =

1 kP (v)k4

(7.231)

for J(v) being injective and satisfying (7.226) and ˆ 1 (v)k = kJ(v)k = kP

1 ˆ (v)k4 kP

(7.232)

for J(v) being surjective and satisfying (7.227). As a result, the requirement that kJ(v)k A can be equivalently expressed by kP (v)k4 ?

1 ˆ (v)k4 ? 1 = or kP

(7.233)

The following theorem follows directly from (7.233) and the Bounded Real Lemma, Lemma 7.8. Theorem 7.13 Given J(v) = G + F (vL D)1 E 5 RHp×n and A 0> 4 suppose that I. for J(v) being injective ¸ D m$L E ;$> udqn = q + n> GW G 2 L A 0 F G

7.9 LMI aided design of FDF

223

II. for J(v) being surjective ¸ D m$L E ;$> udqn = q + p> GGW 2 L A 0 F G then kJ(v)k A if and only if for case I there exists [ = [ W such that [D+DW [ +F W F +([E +F W G)( 2 L GW G)1 (E W [ +GW F) A 0 (7.234) and for case II there exists \ = \ W such that \ DW +D\ +EE W +(\ F W +EGW )( 2 L GGW )1 (F\ +GE W ) A 0= (7.235) Proof. We only prove (7.234) for case I. (7.235) for case II is a dual result of (7.234). It follows from Bounded Real Lemma that for kP (s)k4 ? 1 there exists a matrix S A 0 (S ) = (D EI )S + S (D EI )W + EY Y W E W 1 +(EY Y W [I W )( 2 L Y Y W )1 (Y Y W E W I [) ? 0

(7.236)

where, as a dual result of Lemma 7.4, Y = (GW G)1@2 > I = (GW G)1 (E W T + GW F)

(7.237)

with T 0 being the solution of Riccati equation TD + DW T + F W F (TE + F W G)(GW G)1 (E W T + GW F) = 0= Substituting (7.237) into the left side of (7.236) yields (S ) = DS + S DW S I W GW GI S + (E(GW G)1 S I W ) ( ¶1 ) μ 1 W W 1 × (G G) + L (G G) ((GW G)1 E W I S ) 2 = DS + S DW S I W GW GI S + (E(GW G)1 S I W )(GW G) ¡ ¢1 W × GW G 2 L (G G)((GW G)1 E W I S ) = DS + S DW S I W GW GI S + (E S TE S F W G) ¡ ¢1 ¡ W ¢ E E W TS GW FS = × GW G 2 L Let [ = T S 1 . Then we have S 1 (S )S 1 = (T [)D + DW (T [) I W GW GI ¡ ¢1 ¡ W ¢ E [ + GW F = +([E + F W G) GW G 2 L

(7.238)

224

7 Residual generation with enhanced robustness against unknown inputs

Because S is a nonsingular matrix, (7.236) holds if and only if S 1 (S )S 1 ? 0. By noting that (7.237) and (7.238) imply TD+DW TI W GW GI = F W F, (7.236) is equivalent to ¡ ¢1 ¡ W ¢ E [ + GW F A 0= [D + DW [ + F W F + ([E + F W G) 2 L GW G (7.239) The theorem is thus proven. u t Without proof, we introduce the "discrete time" version of Theorem 7.13. Corollary 7.6 Given J(}) = G + F (}L D)1 E 5 RHp×n and A 0> 4 suppose that I. for J(}) being injective ¸ D hm L E ; 5 [0> 2]> udqn =q+n F G

II. for J(}) being surjective ¸ D hm L E ; 5 [0> 2]> udqn =q+p F G

then kJ(})k A if and only if for case I there exists [ = [ W such that GW G + E W [E 2 L A 0 (7.240) D¯W [ D¯ [ D¯W [J (L + [J)1 [ D¯ + 2 T A 0 +, ¡ W ¢1 W W W 2 W ˜ G G + E [E L ˜ +F F A0 D [D [ H (7.241) H ¢ ¡ ¢ ¡ 2 1 1 GW F> J = E 2 L GW G EW D¯ = D + E L GW G ¡ ¢ 1 ˜ = E W [D + GW F T = F W 2 L GW G F> H or for case II there exists \ = \ W such that GGW + F\ F W 2 L A 0 ¯ J (L + \ J)1 \ D¯W + 2 T A 0 +, ¯ D¯W [ D\ (7.242) D\ ¡ ¢1 W W W 2 W I˜ + EE A 0 D\ D \ I˜ GG + F\ F L (7.243) ¡ ¢ ¡ ¢ 1 1 F> J = F W 2 L GGW F D¯ = D + EGW 2 L GGW ¢ ¡ 2 1 T = E L GGW E W > I˜ = F\ DW + GE W = In the next subsections, the H index and its LMI aided computation will be integrated into the FDF design.

7.9 LMI aided design of FDF

225

7.9.3 H2 to H trade-o design of FDF The so-called H2 to H sub-optimal design of FDF is deﬁned as follows: Find O such that L=D OF is stable LL=kF(sL D + OF) LLL=kF(sL D + OF)

1

1

(7.244)

(Hg OIg )k2 ?

(7.245)

(Hi OIi ) + Ii k $ max

(7.246)

i where it is assumed that F(sL D + OF)1 (Hi OIi ) + Ii 5 RHp×n is 4 injective and for continuous time systems

kF(sL D + OF)1 (Hi OIi ) + Ii k ¡ ¢ = min F(m$L D + OF)1 (Hi OIi ) + Ii A 0 $

as well as for discrete time systems kF(sL D + OF)1 (Hi OIi ) + Ii k ¡ ¢ = min F(hm L D + OF)1 (Hi OIi ) + Ii A 0=

Analogous to the derivation given in Subsection 7.9.1, we ﬁrst set O = O2 +O and reformulate the optimization problem as ﬁnding O such that L= D (O + O)F is stable LL= kF(sL D + (O2 + O)F)

1

Ok22

2

(7.247) W

? wudfh(F[F ) = 1 or (7.248)

LL= kF(sL D + (O2 + O)F)1 Ok22 ? 2 wudfh(F\ F W ) = 1 (7.249) ¯i OIi ) + Ii k $ max = (7.250) LLL= kF(sL D + (O2 + O)F)1 (H Recall that conditions I and II can be expressed in terms of the following LMIs: • for continuous time systems there exist T1 > \1 such that ¯ + FW F ? 0 ¯ W OF DWO2 \1 + \1 DO2 F W O ¸ ¯ \1 O A 0> wudfh(T1 ) ? 1 W ¯ O T1 • for the discrete time system: there exist T2 > ]1 such that ¸ ¯ ]1 ]1 DO2 OF A0 ¯ W ]1 F W F DWO2 ]1 F W O ¸ ¯ ]1 O ¯ W T2 A 0> wudfh(T2 ) ? 1 O

(7.251) (7.252)

(7.253) (7.254)

226

7 Residual generation with enhanced robustness against unknown inputs

where ¯ S = \ 1 or S = ] 1 > DO = D O2 F O = S O> 2 1 1 and O2 is the H2 optimal observer gain as given in Theorem 7.9. It follows from Theorem 7.13, Corollary 7.6 and Schur complement that we can reformulate (7.250) as max

"

O>\2 =\2W

2 subject to

(7.255)

˜ W \2 + I W F H GGW 22 L i i ˜i + F W I \2 DO + DW \2 + F W F \2 H O i

# A0

for continuous time systems and max

"

O>]2 =]2W

2 subject to

(7.256)

˜ W ]2 H ˜ W ]2 DO + I W F ˜i 2 L H IiW Ii + H i i i W W W ˜ DO ]2 Hi + F Ii DO ]2 DO ]2 + F W F

# A0

for discrete time systems, where ˜i = Hi (O2 + O) Ii = H ¯i OIi = DO = DO2 OF> H In summary, the H2 to H sub-optimal design of FDF can be formulated as the following optimization problem with nonlinear matrix inequalities (NMI): • for continuous time systems max

¯ 1 >\2 =\ W O>\ 2

2 subject to

(7.257)

¯ + FW F ? 0 ¯ W OF DWO2 \1 + \1 DO2 F W O ¸ ¯ \1 O A 0> wudfh(T1 ) ? 1 W ¯ O T1 " # ˜ W \2 + I W F H GGW 22 L i i ˜i + F W I \2 DO + DW \2 + F W F A 0 \2 H O i • for discrete time systems max

¯ 1 >]2 =] W O>] 2

¯ ]1 ]1 DO2 OF W W ¯W W DO2 ]1 F O ]1 F F

"

2 subject to

(7.258) ¸ A0

¸ ¯ ]1 O ¯ W T2 A 0> wudfh(T2 ) ? 1 O

˜ W ]2 H ˜ W ]2 DO + I W F ˜i 2 L H IiW Ii + H i i i W W W ˜ DO ]2 Hi + F Ii DO ]2 DO ]2 + F W F

# A 0=

7.9 LMI aided design of FDF

227

Remark 7.11 Optimization problems (7.257) and (7.258) have been formulated on the assumption that F(sL D + OF)1 (Hi OIi ) + Ii is injective. In case that it is surjective, using the dual principle we are able to derive the solution. Remark 7.12 (7.257) and (7.258) are optimization problems with NMI constraints. Such optimization problems can be solved in an iterative way. We refer the reader to some literatures on this topic, given at the end of this chapter. 7.9.4 H4 to H trade-o design of FDF The so-called H4 to H optimization of FDF is formulated as ﬁnding O such that L= D OF is stable LL= kF(sL D + OF)1 (Hg OIg ) + Ig k4 ? LLL= kF(sL D + OF)1 (Hi OIi ) + Ii k $ max =

(7.259) (7.260) (7.261)

The basic idea of solving the above optimization problem is, again, to transform it into an optimization problem with matrix inequalities as constraints. Initiated by Hou and Patton in 1997, study on the H4 to H optimization of FDF has received considerable attention. In this subsection, we only introduce an essential formulation of this optimization problem. For further details and results published in the past, we refer the reader to the literature cited at the end of this chapter. To ensure the existence of a nonzero minimum H index, we assume that i is injective and for continuous F(sL D + OF)1 (Hi OIi ) + Ii 5 RHp×n 4 time systems kF(sL D + OF)1 (Hi OIi ) + Ii k ¡ ¢ = min F(m$L D + OF)1 (Hi OIi ) + Ii A 0

(7.262)

$

as well as for discrete time systems kF(sL D + OF)1 (Hi OIi ) + Ii k ¡ ¢ = min F(hm L D + OF)1 (Hi OIi ) + Ii A 0=

(7.263)

Once again, we would like to mention that using the dual principle a solution for the surjective case can also be found. For the sake of simplicity, we only concentrate ourselves on the injective case. It follows from Lemmas 7.8 and 7.9 that requirements (7.259) and (7.260) can be written into a matrix inequality form

228

7 Residual generation with enhanced robustness against unknown inputs

• for continuous time systems: there exists a \ A 0 such that 6 5 (D OF)W \ + \ (D OF) \ (Hg OIg ) F W 7 (Hg OIg )W \ L IgW 8 ? 0 F Ig L

(7.264)

• for discrete time systems: there exists a [ A 0 such that 5 6 [ [ (D OF) [(Hg OIg ) 0 9 (D OF)W [ [ 0 FW : 9 : ? 0= W 7 (Hg OIg ) [ 0 L IgW 8 0 F Ig L

(7.265)

Combining (7.264), (7.265) with the results given in Theorem 7.13, Corollary 7.6 leads to the following reformulation of H4 to H optimization (7.259)(7.261): • for continuous time systems: max

1 subject to

6 (D OF)W \ + \ (D OF) \ (Hg OIg ) F W 7 (Hg OIg )W \ L IgW 8 ? 0 F Ig L 5

O>\ A0>\1 =\1W

(Hi OIi )W \1 + IiW F IiW Ii 21 W \1 (Hi OIi ) + F Ii \1 (D OF) + (D OF)W \1 + F W F

(7.266)

¸ A0

• for discrete time systems: max

O>[>[1 =[1W

1 subject to

(7.267)

6 [ [ (D OF) [(Hg OIg ) 0 9 (D OF)W [ [ 0 FW : :?0 9 7 (Hg OIg )W [ 0 L IgW 8 0 F Ig L 5

¸ 11 12 A0 W 12 22

11 = IiW Ii + (Hi OIi )W [1 (Hi OIi ) 21 L 12 = (Hi OIi )W [1 (D OF) + IiW F 22 = (D OF)W [1 (D OF) [1 + F W F= Again, solving (7.266) and (7.267) deals with an optimization with NMI constraints and requires the application of advanced nonlinear optimization technique.

7.9 LMI aided design of FDF

229

7.9.5 An alternative H4 to H trade-o design of FDF Although the approach proposed in the last subsection provides an elegant LMI solution to the H4 to H trade-o design, it is computationally involved and delivers often only local optimum. We now derive an alternative solution to the same problem with less computation and guaranteeing the global optimum. i is surjective Assume that F(sL D + OF)1 (Hi OIi ) + Ii 5 RHp×n 4 and satisﬁes ¸ D m$L Hi = q + p or ;$> udqn F Ii ¸ D hm L Hi ; 5 [0> 2]> udqn = q + p= F Ii It follows from Lemma 7.4 or Lemma 7.5 that setting (7.268) O0 = (\ F W + Hi IiW )(Ii IiW )1 > Y0 = (Ii IiW )1@2 > A 0 or ¢1@2 ¡ O0 = (DW [F W + Hi IiW )(Ii IiW + F[F W )1 > Y0 = Ii IiW + F[F W (7.269) gives ° ° °F(m$L D + OF)1 (Hi OIi ) + Ii ° = or ° ° °F(hm L D + OF)1 (Hi OIi ) + Ii ° = where \ is the solution of Riccati equation D\ + \ DW + Hi HiW (\ F W + Hi IiW )(Ii IiW )1 (F\ + Ii HiW ) = 0 (7.270) and [ of

³ ´1 ¡ ¢1 DG [ L + F W Ii IiW F[ DWG [ + Hi IiWB Ii B HiW = 0 ¡ ¢1 Ii HiW = DG = D F W Ii IiW

(7.271)

Note that the dynamics of the corresponding FDF is governed by ˆg>0 (s)g(s) ˆi>0 (s)i (s) + Q u(s) = Q

(7.272)

with ˆ 1 (s)Q ˆi>0 (s)> J|g (s) = P ˆ 1 (s)Q ˆg>0 (s) J|g (v) = P i>0 g>0 ¡ ¢ ˆ i>0 (s) = P ˆ g>0 (s) = Y0 L F(sL D + O0 F)1 O0 ˆ x (s) = P P ¡ ¢ ˆi>0 (s) = Y0 F(sL D + O0 F)1 (Hi O0 Ii ) + Ii Q ¡ ¢ ˆg>0 (s) = Y0 F(sL D + O0 F)1 (Hg O0 Ig ) + Ig = Q

(7.273) (7.274) (7.275)

230

7 Residual generation with enhanced robustness against unknown inputs

Next, we would like to demonstrate that O0 > Y0 given in (7.268) or (7.269) solve the optimization problem of ﬁnding O> Y such that L= D OF is stable LL= kY F(sL D + OF)1 (Hg OIg ) + Y Ig k4 ? LLL= kY F(sL D + OF)1 (Hi OIi ) + Y Ii k $ max =

(7.276) (7.277) (7.278)

Remember that the dynamics of an FDF is generally described by ˆi (s)i (s) + Q ˆg (s)g(s) u(s) = Q which can be further written as

³ ´ ˆ 1 (s) Q ˆi>0 (s)i (s) + Q ˆg>0 (s)g(s) = u(s) = Px (s)P i>0

Suppose that O> Y ensure ° ° ° ° ° ° °ˆ ˆ 1 (s)Q ˆg>0 (s)° ° °Qg (s)° = °Px (s)P i>0 4

4

?

and make ° ° ° ° ° ° °ˆ ˆ 1 (s)Q ˆi>0 (s)° ° ° =° °Qi (s)° = °Px (s)P i>0 ° ° 1 ˆ (s)° °Px (s)Pi>0

4

reaching its maximum. Consider the following relations (see also (7.232)) ° ° ° ° ° ° ° °ˆ ° ° ° °ˆ 1 ˆ 1 ˆ °Qg>0 (s)° °P i>0 (s)Px (s)° °Px (s)Pi>0 (s)Qg>0 (s)° 4 4 ° ° ° 4 ° ° ° ° °ˆ 1 1 ˆ ˆ ˆ (s) = P (s) P (s) P (s)P (s) Q (s) Q ° x ° ° i>0 ° i>0 i>0 x i>0

° ° ° °ˆ °Pi>0 (s)Px1 (s)°

4

° ° ° ˆ 1 (s)° ° °Px (s)P i>0

4

and set 1 ° = ° ° ° W 1@2 (L F(m$L D + O0 F)1 O0 ) Px1 (m$)° °(Ii Ii )

or

4

1

° = ° ´1@2 °³ ° m L D + O F)1 O ) P 1 (hm )° ° Ii I W + F[F W (L F(h x 0 0 i ° °

4

we get

° ° ° °ˆ °Qg>0 (s)°

° ° ° ° ° °ˆ ° ˆ 1 (s)Q ˆg>0 (s)° °Px (s)P = (s) Q ° ° ° ? g i>0 4 4 4 ° ° ° ° 1 ° ° ° °ˆ ˆi (s)° = ° = °Q °Qi>0 (s)° ° ° ˆ 1 (s)° ° °Px (s)P i>0 4

This result veriﬁes that O0 > Y0 given in (7.268) or (7.269) do solve the optimization problem (7.276)-(7.278) and thus proves the following theorem.

7.9 LMI aided design of FDF

231

Theorem 7.14 O0 > Y0 given in (7.268) or (7.269) with 0?? ° ¡ ¢° or ° W 1@2 ¯ 1 H ¯g + Ig ° F(m$L D) °(Ii Ii ) ° 4 ° 0?? ° ´1@2 ¡ °³ ¢° m 1 ¯ ¯ ° Ii I W + F[F W F(h L D) Hg + Ig ° i ° °

(7.279) (7.280)

4

¯g = Hg O0 Ig D¯ = D O0 F> H solve the optimization problem (7.276)-(7.278). It is worth noting that for the determination of O0 > Y0 only the solution of one Riccati equation is needed. Moreover, the solution is analytically achievable and ensures a global optimum. Remark 7.13 We would like to mention that O0 > Y0 given in (7.268) or (7.269) also solves sup O>Y

kY F(sL D + OF)1 (Hi OIi ) + Y Ig k kY F(sL D + OF)1 (Hg OIg ) + Y Ig k4

where (A 0) can be any constant. In Section 12.3, we shall prove it. We would like to call reader’s attention that this result also veriﬁes the comparison study in Section 7.4. Algorithm 7.8 H4 to H trade-o design of FDF - an alternative solution Step 1: Compute O0 > Y0 according to (7.268) or (7.269) Step 2: Set satisfying (7.279) or (7.280). Example 7.5 In this example, we design an H4 to H optimal FDF for the benchmark vehicle dynamic system via Algorithm 7.8. To this end, system model (3.76) in Subsection 3.7.4 is slightly modiﬁed, where 0 = 103600 + FY > FY 5 [10000> 0] FY

is rewritten as 0 = 9360 + FY > FY 5 [5000> 5000] = FY

This change is due to the need in the late study. It results in ¸ ¸ 2=9077 0=9762 1=0659 D= >E = 28=4186 3=2546 38=9638 ¸ ¸ 145=3844 1=1890 53=2973 F= >G = 0 1=0000 0 with

232

7 Residual generation with enhanced robustness against unknown inputs

¸ ¸ 0 0 1=0659 1 0 53=2973 > Ii = = Hi = 0 0 38=9638 01 0

The achieved results are ¸ ¸ 0=0200 0=0002 0=0188 0 O0 = > Y0 = = 0=7308 0=0707 0 1 7.9.6 A brief summary and discussion In this section, we have derived three approaches to the optimal design of FDF regarding to three dierent performance indices: • H2 @H2 optimization • H2 to H optimization • H4 to H optimization. Remember that our design objective is indeed a multiobjective optimization, i.e. minimizing the inﬂuence of the disturbances on the residual and simultaneously maximizing the one of the possible faults. The underlying idea adopted here for solving such optimization problems is to reduce the multiobjective optimization problem to a single optimization problem with constraints. To this end, the well-established robust control theory and LMItechniques have been used. As a result, all requirements and constraints given in the form of norms of transfer function matrices are equivalently expressed in terms of matrix inequalities. Although these matrix inequalities look formally linear, part of them is indeed bilinear related to the optimization parameters. Bearing in mind the main objective of our handling, we shall continue our study without detailed dealing with the solution for the formulated optimization problems. At the end of this chapter, however, a number of references are given, where the reader is provided with useful materials like essentials, algorithms and even software solutions for such problems. The reader might notice that the H4 norm has not been taken into account for measuring the inﬂuence of faults . This is mainly due to the di!culty met by handling the inequality ˆi (s)k4 A A 0= kQ A further discussion upon this will be carried out in the subsequent sections.

7.10 The uniﬁed solution In the last sections, dierent norms and indices have been used to describe the inﬂuence of the unknown disturbances and faults on the residual signal. Remember that both the H4 norm and index H are some extreme value of a transfer function matrix. From the practical viewpoint, it is desired to

7.10 The uniﬁed solution

233

deﬁne an index that gives a fair evaluation of the inﬂuence of the faults on the residual signal over the whole frequency domain and in all directions in the measurement subspace. The major objective of this section is to introduce an index for a practical evaluation of the fault sensitivity and, based on it, to achieve an optimal design of the observer-based FD systems. 7.10.1 Hl /H4 index and problem formulation Consider system (7.156). To simplify our discussion, we ﬁrst focus on the continuous time systems. The extension to the discrete time systems is straightforward and will be given at the end of this section. For our purpose, we now introduce a deﬁnition of fault sensitivity and, associated with it, the so-called Hl @H4 performance index. Recall that the singular values of a matrix give a measurement of the "gain" in each direction of the subspace spanned by the matrix. In this context, all singular ¯ i (m$))> $ 5 [0> 4]> together build a natural measurement values l (U(m$)J of the fault sensitivity. They cover ° all°directions ° of° the subspace spanned by ¯ i (m$)= In comparison, °UJ ¯ i ° or °UJ ¯ i ° are only two extreme U(m$)J 4 points in this subspace. It holds ;$ 5 [0> 4]> ° ° ° ° ¯ i (m$)) °UJ °U J ¯ i ° l (U(m$)J ¯i ° (7.281) 4 Associated with it, we introduce ¯ i (m$))> Deﬁnition 7.5 (Hl @H4 design) Given system (7.156) and let l (U(m$)J ¯ i (m$). l = 1> · · · > ni > be the singular values of U(m$)J ¯ i (m$)) l (U(m$)J ° Ml>$ (U) = ° °U(v)J ¯ g (v)°

(7.282)

4

is called Hl @H4 performance index. We would like to call reader’s attention that Hl @H4 index indicates a set of (ni ) functions. It is clear that M4 (U) and M0 (U)> ° ° ¯ i (v)° °U(v)J ¯ i (m$)) inf $ (U(m$)J ° °4 ° M4 (U) = ° ¯ g (v)° > M0 (U) = ¯ g (v)° °U(v)J °U(v)J 4 4 are only two special functions in the set of Ml>$ (U)= Under Hl @H4 performance index, we now formulate the residual gener¯ i (m$))> l = ator design as ﬁnding U(v) 5 RH4 such that for all l (U(m$)J 1> · · · > ni > $ 5 [0> 4]> Ml>$ (U) is maximized, i.e. sup U(v)5RH4

Ml>$ (U) =

sup U(v)5RH4

¯ i (m$)) l (U(m$)J ° ° = °U(v)J ¯ g (v)° 4

(7.283)

234

7 Residual generation with enhanced robustness against unknown inputs

It is worth emphasizing that (7.283) is a multiobjective (ni objectives!) optimization and the solution of (7.283) would also solve ° ° °U(v)J ¯ i (v)° ¯ i (m$)) inf $ (U(m$)J 4 ° ° ° ° and sup (7.284) sup ° ¯ °U(v)J ¯ g (v)° ° U(v)5RH4 U(v)Jg (v) U(v)5RH4 4

4

which have been discussed in the previous sections. 7.10.2 Hl /H4 optimal design of FDF: the standard form Now, we are going to solve (7.283). For our purpose, we ﬁrst assume that ¯ g (m$) A 0= ¯ g (m$)J ;$ 5 [0> 4]> J

(7.285)

This assumption will be removed in the next section. It follows from Lemma ¯ g (v) = P ˆ 1 (v)Q ˆ0 (v)> where P ˆ 1 (v) is ¯ g (v) can be factorized into J 7.4 that J 0 0 ˆ0 (v) a co-inner of J ¯ g (v)= Let U(v) = T(v)P ˆ 0 (v) for some T(v) 5 a co-outer, Q ¯ i (m$))> l = RH4 = It then turns out that ;$ 5 [0> 4] and for all l (U(m$)J 1> · · · > ni > ´ ³ ¯ i (m$) ˆ 0 (m$)J ´ ³ l T(m$)P ¯ i (m$) = ˆ 0 (m$)J l P (7.286) Ml>$ (U) = kT(v)k4 ˆ 0 (v) leads to On the other side, setting U(v) = P ´ ³ ¯ i (m$)) Ml>$ (U) = l P ¯ i (m$) ˆ 0 (m$)J ;$> l (U(m$)J ˆ 0 (v) leads to the i.e. ;$ 5 [0> 4], l = 1> · · · > ni > the postﬁlter U(v) = P maximum Ml>$ (U)= As a result, the following theorem is proven. Theorem 7.15 Given system (7.156) and assume that (7.285) holds, then ¯ i (m$))> l = 1> · · · > ni > ;$ 5 [0> 4] and l (U(m$)J Ã ! ˆ 0 (v) sup Ml>$ (U) = P (7.287) Ursw (v) = arg U(v)5RH4

ˆ 1 (v) is a co-outer of J ¯ g (v)= where P 0 Theorem 7.15 reveals that Ursw (v) leads to a simultaneous optimum of ¯ g (m$)= It also performance index (7.283) in the whole subspace spanned by J covers the special case (7.284). For this reason, Ursw (v) is the uniﬁed solution. It can be shown (see Chapter 12) that the uniﬁed solution delivers not only an optimal solution in the sense of (7.283) or (7.284) but also an optimal trade-o in the sense that given an allowable false alarm rate, the fault detection rate is maximized. This also gives a practical explanation why the uniﬁed

7.10 The uniﬁed solution

235

solution, dierent from the existing optimization methods, solves (7.283) si¯ i (m$))> $ 5 [0> 4]> l = 1> · · · > ni . multaneously for all l (U(m$)J Applying Ursw (v) to (7.156) yields ˆ0 (v)g(v) + P ˆ 0 (v)J ¯ i (v)i (v) u(v) = Q

(7.288)

ˆ 0 (m$) can and in the fault-free case kuk2 = kgk2 = In the above expression, P be considered as a weighting matrix of the inﬂuence of i on u= Remember that ¯ g (v) and the co-outer of a transfer ˆ 0 (v) is the inverse of the co-outer of J P function matrix can be interpreted as the magnitude proﬁle of the transfer function matrix in the frequency domain. In this context, it can be concluded that the optimal solution is achieved by inversing the magnitude proﬁle of ¯ g (v)= As a result, the inﬂuence of g on u becomes uniform in the whole J subspace spanned by the possible disturbances, while the inﬂuence of i on u is ¯ g (m$)> i.e. where J ¯ g (m$) weighted by the inverse of the magnitude proﬁle of J ¯ is strong (weak), Ji (m$) will be weakly (strongly) weighted. Remark 7.14 Remember that in Subsection 7.9.5, we have derived a solution for the optimization problem ¯ i (m$)) inf $ (U(m$)J ° ° M0 (U) = sup °U(v)J ¯ g (v)° U(v)5RH4 4

which is dierent from the one given above. This shows that the solution for this problem is not unique. In Chapter 12, we shall further discuss this problem. Following Lemma 7.4, the results given in Theorem 7.15 can also be presented in the state space form. To this end, suppose that the minimal state space realization of system (7.156) is given by {(w) ˙ = D{(w) + Ex(w) + Hg g(w) + Hi i (w) |(w) = F{(w) + Gx(w) + Ig g(w) + Ii i (w)=

(7.289)

Using an FDF, { ˆ˙ (w) = Dˆ {(w) + Ex(w) + O(|(w) |ˆ(w)) |ˆ(w) = F { ˆ(w) + Gx(w)> u(w) = Y (|(w) |ˆ(w)) for the purpose of residual generation gives ³ ´ ˆ x (v)J|g (v)g(v) + P ˆ x (v)J|i (v)i (v) u(v) = Y P ³ ´ ˆg (v)g(v) + Q ˆi (v)i (v) =Y Q ˆ x (v) = L F(vL D + OF)1 O = P ˆ g (v) = P ˆ i (v) P ˆg (v) = Ig + F(vL D + OF)1 (Hg OIg ) Q ˆi (v) = Ii + F(vL D + OF)1 (Hi OIi ) Q ˆ 1 (v)Q ˆg (v)> Ji (v) = P ˆ 1 (v)Q ˆi (v)= Jg (v) = P g

i

(7.290)

(7.291)

236

7 Residual generation with enhanced robustness against unknown inputs

The following theorem represents a state space version of the optimal solution (7.287) and gives the optimal design for O> Y . Theorem 7.16 Given system (7.289) that is detectable and satisﬁes ;$ 5 [0> 4] ¸ D m$L Hg =q+p (7.292) udqn F Ig and the residual generator (7.290), then Orsw = (Hg IgW + \g F W )(Ig IgW )1 > Yrsw = (Ig IgW )1@2

(7.293)

with \g 0 being the stabilizing solution of the Riccati equation D\g + \g DW + Hg HgW (Hg IgW + \g F W )(Ig IgW )1 (Ig HgW + F\g ) = 0 (7.294) ˆi (m$))> l = 1> · · · > ni > deliver an optimal FDF (7.290) in the sense of ;$> l (Y Q ˆi (m$)) l (Y Q ˆi>rsw (m$)) ° = l (Yrsw Q sup Ml>$ (O> Y ) = sup ° ° ˆg (v)° O>Y O>Y °Y Q °

(7.295)

4

ˆi>rsw (v) = Ii + F(vL D + Orsw F)1 (Hi Orsw Ii )= Q The proof of this theorem follows directly from Lemma 7.4 and Theorem 7.15. Theorem 7.16 provides us not only with a state space expression of optimization problem (7.283) but also with the possibility for a comparison with the existing methods from the computational viewpoint. Remember that most of the LMI aided design methods handle the optimization problems as a multiobjective optimization. As a result, the solutions generally include two Riccati LMIs. In comparison, the uniﬁed solution only requires solving Riccati equation (7.294) and thus demands less computation. Example 7.6 We now design an FDF using the uniﬁed solution for the benchmark system LIP100. Our design purpose is to increase the system robustness against the unknown inputs including measurement noises. Based on model (3.57) with the extended Hg > Ig (to include the measurement noises) £ ¤ £ ¤ Hg = 0 E > Ig = L3×3 0 we get

5

Orsw

6 0=0006 0=0711 0=0013 9 0=0711 8=8764 0=1619 : : =9 7 6=1356 0=1619 0=0030 8 > Yrsw = L3×3 = 84=6187 39=4106 0=7188

7.10 The uniﬁed solution

237

7.10.3 Discrete time version of the uniﬁed solution In this subsection, we shall brieﬂy present the analogous version of Theorems 7.15 and 7.16 for discrete time systems without proof. Theorem 7.17 Given system (7.156) and assume that ¯ (hm ) A 0= ¯ g (hm )J ; 5 [0> 2]> J g

(7.296)

¯ i (hm ))> l = 1> · · · > ni > then ; 5 [0> 2]> and l (U(hm )J Ã ! Ursw (}) = arg

sup

Ml>$ (U)

U(v)5RH4

ˆ 0 (}) =P

(7.297)

¯ g (})= ˆ 1 (}) is a co-outer of J where P 0 Theorem 7.18 Given system {(n + 1) = D{(n) + Ex(n) + Hg g(n) + Hi i (n) |(n) = F{(n) + Gx(n) + Ig g(n) + Ii i (n) that is detectable and satisﬁes ; 5 [0> 2] ¸ D hm L Hg = q + p= udqn F Ig Then residual generator { ˆ(n + 1) = (D Orsw F) { ˆ(n) + (E Orsw G) x(n) + Orsw |(n) u(n) = Yrsw (|(n) F { ˆ(n) Gx(n)) with Orsw = OWg > Yrsw = Zg

(7.298)

delivers residual ´ signal u(n) that is optimum in the sense that ; 5 [0> 2] and ³ ˆi (hm ) l Y Q ´ ³ ˆi (hm ) ´ ³ l Y Q ˆi>rsw (hm ) ° sup ° = Y Q l rsw ˆi (})° O>Y ° ° °Y Q 4

ˆi>rsw (}) = Ii + F (}L D + Orsw F)1 (Hi Orsw Ii ) = Q In (7.298), Zg is the left inverse of a full column rank matrix K satisfying Kg KgW = F[g F W + Ig IgW , and ([g > Og ) is the stabilizing solution to the DTARS (discrete time algebraic Riccati system) ¸ ¸ D[g DW [g + Hg HgW D[g F W + Hg IgW L = 0= (7.299) Og F[g DW + Ig HgW F[g F W + Ig IgW

238

7 Residual generation with enhanced robustness against unknown inputs

7.11 The general form of the uniﬁed solution Due to the complexity, the study in this section will focus on continuous time systems. Recall that the uniﬁed solution proposed in the last section is based on ¯ g (v) is surjective assumption (7.285) or its state space expression (7.292), i.e. J and has no zero on the m$-axis or at inﬁnity. Although it is standard in the robust control theory and often adopted in the observer-based residual generator design, this assumption may considerably restrict the application of design schemes. For instance, the case udqn(Ig ) ? p> which is often met in practice, leads to invalidity of (7.292). Another interesting fact is that for ng ? p a PUID can be achieved, as shown in Chapter 6. It is evident that for ng ? p (7.285) does not hold. It is well-known that a zero $ l on the m$-axis or at inﬁnity means that a disturbance of frequency $l or with inﬁnitively high frequency will be fully blocked. Also, for ng ? p there exists a subspace in the measurement subspace, on which g has no inﬂuence. From the FDI viewpoint, the existence of such a zero or subspace means a "natural" robustness against the unknown disturbances. Moreover, remember that the uniﬁed solution can be interpreted as weighting the inﬂuence of the faults on the residual signal by ¯ g (v). Following it, around zero means of inversing the magnitude proﬁle of J will $ l > say $ 5 ($ l ± $) > the inﬂuence of the faults on the ³residual signal ´ 1 ˆ ˆ be considerably strongly weighted by P0 (m$) (because l P0 (m$) is very small). From this observation we learn that it is possible to make use of the information about the available zeros on the m$-axis or at inﬁnity or the exist¯ g (v) to improve the fault sensitivity considerably while ing null subspace of J keeping the robustness against g= This is the motivation and the basic idea behind our study on extending the uniﬁed solution so that it can be applied to system (7.156) without any restriction. Our extension study consists of two parts: (a) a special factorization of ¯ g (v) is developed, based on it (b) an approximated "inverse" of J ¯ g (v) in the J whole measurement subspace will be derived. 7.11.1 Extended CIOF g ¯ (v) 5 RHp×n , Now, under the assumption that J 4 g ¢ ¡ ¯ g (v) = min {p> ng } udqn J

¯ g (v) will be factorized into J ¯ g (v) = Jgr (v)J4 (v)Jm$ (v)Jgl (v) J

(7.300)

with co-inner Jgl (v)> left invertible Jgr (v), Jm$ (v) having the same zeros on ¯ g (v). This special the m$-axis and J4 (v) having the same zeros at inﬁnity as J

7.11 The general form of the uniﬁed solution

239

¯ g (v) is in fact an extension of the standard CIOF introduced factorization of J at the beginning of this chapter. We present this factorization in the form of an algorithm. Algorithm 7.9 Algorithm for the extended CIOF Step 0: Do a column compression by all-pass factors as described in Lemma ˜ ¯ g (v) = J(v)J 7.6: J d1 (v)= Note that this step is necessary only if ng A p= ˜ ˜ Step 1 : Do a dislocation of zeros for J(v) by all-pass factors: J(v) = ¯ r (v) in C by vl> = ¯ r (v)Jl1 (v). Denote the zeros and poles of J J Step 2: Set v = d + with d satisfying ;vl> Re(vl> ) ? d ? 0

(7.301)

¯ r (d + ) = J ¯ r (v)= ¯ r (v): J1 () = J and substitute v = d + into J ¯ r (v) in Cm$ > C We denote the zeros of J1 () corresponding to the zeros of J and at inﬁnity by l>m$ ,l> and l>4 respectively. It follows from (7.301) that Re(l>m$ ) A 0> Re(l> ) ? 0> l>4 = 4= Also, all poles of J1 () are located in C . Step 3: Do a CIOF of J1 () following Lemma 7.6: J1 () = J1r ()J1l ()= Note that J1l () is inner and J1r () is an outer factor whose zeros belong to C and at inﬁnity= ¯ r (v) equal to Step 4: Substitute = v d into J1l ()> J1r () and set J ¯ 1r (v)= ¯ r (v) = J1 () = J1l (v d) J1r (v d) = JWm$ (v)J J ¯ r (v) in Cm$ > it is evident Remembering that l>m$ is corresponding to a zero of J that Jm$ (v) has as its zeros all the zeros of J(v) on the imaginary axis. Noting ¯ 1r (v) only has zeros that v = d + > d ? 0> it can be further concluded that J ¯ 1r (v) are all located in in C as well as at inﬁnity and the poles of Jm$ (v)> J ¯ 1r (v) in C by vl> = C = Denote the zeros and poles of J ³ ´ f ¯ 1r (v) : J ¯ 1r (v) = J ¯ 1r f and substitute it into J Step 5: Set v = = 1

1

J2 ()> where f is a constant satisfying ;vl> f A

(Im(vl> ))2 + |Re(vl> )| | = |Re(vl> )|

(7.302)

¯ 1r (v) at inﬁnity It is straightforward to prove that after Step 5 the zeros of J and in C are located in C+ and C of the -complex plane respectively. Also ¯ 1r (v) are in C of the -complex plane. the poles of J Step 6: Do a CIOF on J2 () following Lemma 7.6: J2 () = J2r ()J2l ()= Note that J2l () is inner and has as its zeros all the zeros of J2 () in C+ = Since J2 () has no zero in Cm$ and at inﬁnity, J2r () is right invertible in RH4 = ¯ Step 7: Substitute = f+v v into J2l ()> J2r () and set J1r (v) equal to

240

7 Residual generation with enhanced robustness against unknown inputs

μ J2 () = J2r

f+v v

¶

μ J2l

f+v v

¶ = Jr (v)J4 (v)=

¯ g (v) at inﬁnity. Denote It is evident that J4 (v) has as its zeros only zeros of J a zero or a pole of J2r () in C by m> 5 C . It turns out v=

f f(Re(m> ) 1) f Im(m> )m = m> 1 (Re(m> ) 1)2 + (Im(m> ))2

Since f A 0> Re(m> ) ? 0> we have Re(v) ? 0 +, v 5 C . It can thus be concluded that the poles of JW4 (v)> Jr (v) lie in C > and Jr (v) is right invertible in RH4 = As a result, the desired factorization ¯ g (v) = Jgr (v)J4 (v)Jm$ (v)Jgl (v)> Jgl (v) = Jd1 (v)Jl1 (v)= J is achieved. Remark 7.15 We would like to point out that the study on the extended CIOF primarily serves as a mathematical formulation. Below, we shall demonstrate that the information provided by those zeros at the m$-axis can be utilized to improve the fault detection performance. From the numerical viewpoint, there should be more e!cient algorithms to realize such an extended CIOF. 7.11.2 Generalization of the uniﬁed solution We are now in a position to extend and generalize the uniﬁed solution. Recalling the idea behind the uniﬁed solution and our discussion at the beginning of this section, our focus is on approximating the inverse of Jgr (v)J4 (v)Jm$ (v) by a post-ﬁlter in RH4 = To this end, we shall approach the inverse of Jm$ (v)> J4 (v),Jgr (v) separately. Remember that Jm$ (v) has as its zeros all the zeros of J(v) on the m$˜ m$ (v) by Jm$ (v %)> % A 0= Note that all zeros of J ˜ m$ (v) are axis. Deﬁne J 1 ˜ located in C = If % is chosen to be small enough, then we have Jm$ (v) as an approximation of the inverse of Jm$ (v) with ˜ 1 (v) = J1 (v %) 5 RH4 = ˜ 1 (v)Jm$ (v) L> J J m$ m$ m$ ˜ 4 (v) = J4 To approximate the inverse of J4 (v)> we introduce J 1

(7.303) ³

5 C . Thus, choosing small enough yields ¶ μ v 1 1 1 ˜ ˜ 5 RH4 = J4 (v)J4 (v) L> J4 (v) = J4 v + 1

0> whose zeros are

v v+1

´ > A

(7.304)

Recalling that Jgr (v) = JWr (v) is left invertible in RH4 > for p = ng the 1 solution is trivial, i.e. J gr (v) = Jgr (v)= We study the case p A ng = As

7.11 The general form of the uniﬁed solution

241

described in Lemma 7.6, using a row compression by all-pass factors Jgr (v) can be factorized into ¸ Jgr>2 (v) (7.305) 5 RHp×ng Jgr (v) = Jgr>1 (v) 0 with Jgr>1 (v) 5 RHp×p , Jgr>2 (v) 5 RHng ×ng . Both Jgr>1 (v)> Jgr>2 (v) are ¯ g (v) only spans a p×ng -dimensional invertible in RH4 = (7.305) means that J subspace of the p × p-dimensional measurement space. In order to inverse ¯ g (v) ¯ g (v) in the whole measurement space approximately, we now extend J J and g(v) to ¸ £ ¤ ¯ g (v) 0 5 RHp×p > gh (v) = g(v) 5 Rp ¯ g>h (v) = J J 0 which yields no change in the results achieved above. As a result, we have ¸ Jgr>2 (v)J4 (v)Jm$ (v) 0 ¯ ¯ Jg>h (v) = Jgr>1 Jgl (v) 0 0 ¸ ¯ gl (v) = Jgl (v) 0 is co-inner. J 0 L ¸ Jgr>2 (v)J4 (v)Jm$ (v) 0 Since is not invertible, we can introduce 0 0 ¸ Jgr>2 (v)J4 (v)Jm$ (v) 0 0 L with a very small constant to approximate it. Together with (7.303)-(7.305), we now deﬁne the optimal post-ﬁlter as ¸ 1 1 ˜ 1 ˜ (v)J J 4 (v)Jgr>2 (v) 0 m$ J1 Ursw (v) = (7.306) 1 gr>1 (v)= 0 L Ursw (v) satisfying (7.306) is an approximation of the inverse of the magnitude ¯ g (v). In order to understand it well, we apply Ursw (v) to residual proﬁle of J generator ¡ ¢ ¯ i (v)i (v) ¯ g (v)g(v) + J u(v) = U(v) J and study the generated residual signal. It turns out ¡ ¢ ¯ i (v)i (v) = ¯ g (v)g(v) + J u(v) = Ursw (v) J ¸ ¡ ¢ u1 (v) ¯ ¯ = Ursw (v) Jg>h (v)gh (v) + Ji (v)i (v) = u2 (v) 1 ¸ ˜ 1 ˜ (v)J J 4 (v)J4 (v)Jm$ (v)Jgl (v)g(v) + Ji 1 (v)i (v) m$

1 Ji 2 (v)i (v)

(7.307)

242

with

7 Residual generation with enhanced robustness against unknown inputs

¸ ¸ 1 1 ˜ 1 ˜ (v)J Ji 1 (v) J 4 (v)Jgr>2 (v) 0 m$ ¯ J1 = 1 gr>1 Ji (v)= Ji 2 (v) L 0

Note that g has no inﬂuence on u2 (v)= Following the basic idea of the uniﬁed solution, the transfer function of the faults to u2 (v) should be inﬁnitively large weighted. In solution (7.306), this is realized by introducing factor 1 . It is very interesting to notice that in fact u2 (v) corresponds to the solution of the full disturbance decoupling problem, where only this part of the residual vector is generated and used for the FD purpose. This also means that the dimension of the residual vector, p ng > is smaller than the dimension of the measurement p= In against, the uniﬁed solution results in a residual vector with the same dimension like the measurement vector, which allows also to detect those faults, which satisfy Ji 2 (v)i (v) = 0 and thus are undetectable using the full disturbance decoupling schemes. In case p = ng , the solution is reduced to 1 ˜ 1 (v)J ˜ 1 Ursw (v) = J 4 (v)Jgr (v)= m$

(7.308)

Summarizing the results achieved in this and the last sections, the uniﬁed ¯ g (v) solution can be understood as the inverse of the magnitude proﬁle of J and described in the following general form: given system (7.156), ½ the uniﬁed solution is given by (7=306) if p A ng = (7.309) the uniﬁed solution is given by (7=308) if p ng ¯ g (v) has no zero in Cm$ or at inﬁnity, then J ˜ m$ (v) = L or Note that if J ˜ J4 (v) = L= We would like to call reader’s attention to the fact that the uniﬁed solution will be re-studied in Chapter 12 under a more practical aspect. The physical meaning of the uniﬁed solution will be revealed. Example 7.7 We now illustrate the discussion in this subsection by studying the following example. Consider system (7.156) with 6 5 ¸ ¸ u1 (v) i1 (v) g1 (v) 8 7 > g(v) = u(v) = u2 (v) > i (v) = i2 (v) g2 (v) u3 (v) 6 5 5 v+5 6 v v2 +3v+2 v2 +4v+1 0 2 v+1 8 > J ¯ i (v) = 7 2 1 8 J|x (v) = 7 vv2 +3v+2 v +4v+1 0 1 0 1 v+1 " # ¸ (v+1)(v+4) 1 ¯ g1 (v) J 2 +4v+1 2 +4v+1 ¯ ¯ v v Jg (v) = > Jg1 (v) = = v+2 v+5 0 v2 +4v+1 v2 +4v+1 ¯ W (v) has two zeros at 3> 4= Moreover, it is evident that u3 (v) = i2 (w)> i.e. a J g full decoupling is achieved. However, using u3 (w) i1 cannot be detected. In the

7.11 The general form of the uniﬁed solution

243

following, we are going to apply the general optimal solution given in (7.309) to solve the FD problem. It will be shown that using (7.309) we can achieve the similar performance of detecting i2 as using the full decoupling scheme. In addition, i1 can also be well detected and the zero at inﬁnity can be used to enhance the fault sensitivity. We now ﬁrst apply the algorithm given in the last section to achieve the special factorization (7.300) and then compute the ¯ W (v) has only zeros optimal solution according to (7.309). Considering that J g1 f in C and at 4> we start from Step 5. Let f = 4=5 and substitute v = 1 ¯ W (v)> into J g1 " 4(+3=5)(+0=125) 5(1)(0=1) # 2 +16+3=25 2 +16+3=25 ¯ Wg1 (v) = J2 () = = J 2 (1) 2 +16+3=25

2(1)(+1=25) 2 +16+3=25

J2 () has poles at 15=7942> 0=2058 and zeros at 1> 0=5. As the next step, do an IOF of J2 () using Lemma 7.6. It results in J2 () = J2l ()J2r (), where ¸ 0=8321 0=5547 J2l () = 0=5547(v1) 0=8321(v1) > J2r () = "

v+1

v+1

3=0509(2 2=31820=3182) 2=7735(2 +4=35+0=725) 2 +16+3=25 2 +16+3=25 2 3=0509( +2=6364+0=0455) 4=4376(2 +0=1562+0=5312) 2 +16+3=25 2 +16+3=25

#

Note that J2l () has a pole at 1 and a zero at 1 and J2r () has poles at 15=7942> 0=2058 and zeros 1> 0=5. The next step is to transform J2l ()> J2r () back to the v-plane by letting = 4=5+v v . It yields ¸ 0=8321 0=5547 JW4 (v) = 1=2481 1=8721 > Jr (v) = v+2=25 v+2=25 " # 2 2 0=8321(v +4=7037v+3=3333) 0=2465(v +0=875v12=375) v2 +4v+1 v2 +4v+1 0=5547(v2 +5=6667v+5=5) 0=3695(v2 +5=75v+12) v2 +4v+1 v2 +4v+1

where JW4 (v) has a pole at 2=25 and zero at 4, Jr (v) has poles at 3=7321> 0=2679 and zeros at 3> 2=25. Thus, JWg1 (v) = JW4 (v)Jr (v), ¸ ¸ W ¯ g1 (v) J Jr (v)J4 (v) ¯ = = Jg (v) = R R As a result, the optimal post ﬁlter Ursw (v) is given by ¸ 1 ˜ 4 (v)(JWr (v))1 0 J with Ursw (v) = 1 0 L ¸ 0=8321 0=5547 ˜ 1 (v) = 0=2465(1+2=25)v0=5547 J 0=3698(1+2=25)v+0=8321 4 " JW r

=

v+1

v+1

0=8321(v2 +5=75v+12) 1=2481(v2 +5=6667v+5=5) v2 +5=25v+6=75 v2 +5=25v+6=75 2 0=5547(v +0=875v12=375) 1=8721(v2 +4=7037v+3=3333) v2 +5=25v+6=75 v2 +5=25v+6=75

#

244

7 Residual generation with enhanced robustness against unknown inputs

and the small positive numbers > are selected as = 0=01> = 0=01. The optimal performance indexes Ml>$ (Ursw ), l = 1> 2 are shown in Fig.7.4. It can be read from the ﬁgure that M4 (Ursw ) = max Ml>$ (Ursw ) = 99=1( 40gE) l>$

M0 (Ursw ) = min Ml>$ (Ursw ) 0= l>$

We would like to point out that, if $ 0, the value of M4 (Ursw ) will converge to 1 = 100. It is evident that M4 (Ursw ) may become inﬁnitively large as goes to zero. 80

60

Performance index (dB)

40

20

0

Ŧ20

Ŧ40

Ŧ60

Ŧ80

Ŧ100 Ŧ2 10

Ŧ1

0

10

10

1

10

2

10

3

10

Frequency (rad/sec)

Fig. 7.4 Optimal performance index Ml>$ (Ursw ), l = 1> 2=

7.12 Notes and references The topics addressed in this chapter build one of the vital research ﬁelds of the recent years in the area of the model-based fault diagnosis technique. The results presented in Sections 5-10 mark the state of the art of the model and observer-based FDI methods. After working with this chapter, we can identify the major reasons for this development: • the development of these methods are well and similarly motivated They are driven by the increasing needs for enhanced robustness against disturbances and simultaneously by the demands for reliable and fault sensitive residual generation.

7.12 Notes and references

245

• the ideas behind these methods are similar The residual generation problems are formulated in terms of robustness and sensitivity and then solved in the framework of the robust control theory. • each method and design scheme is coupled with a newly developed method in the framework of the robust control theory The duality between the control and estimation problems enables a direct application of advanced control theory and technique to approaching the residual generation problems. Due to this close coupling with the advanced control theory, needed preliminaries of the advanced control theory have ﬁrst been introduced in this chapter. We refer the reader to [49, 46, 161, 160, 134] for the essential knowledge of signal and system norms, the associated norm computation and the H2 @H4 technique. To our knowledge, [14, 130] are two mostly cited literatures in the area of the LMI technique, which contain both the needed essentials and computational skills. The factorization technique plays an important role in our study. We refer the reader to [161] for a textbook styled presentation on this topic and [110, 137] for a deeper study, for which some special mathematical knowledge is required. The proofs of Lemmas 7.1 - 7.5 are given in [161], the proof of Lemma 7.6 in [110] and the one of Lemma 7.7 on the MMP solution in [46]. The LMI version of the Bounded Real Lemma, Lemmas 7.8 - 7.9, is well known, see for instance [14]. The Kalman ﬁlter technique is standard and can be found in almost any standard textbooks of control engineering, see for instance [6, 20]. Patton and Chen [115] initiated the technique of residual generator design via an approximation of unknown input distribution matrices and made the major contributions to it. The study on the comparison of dierent performance indices presented in Section 7.4 gives us a deeper insight into the optimization strategies. To our knowledge, no study has been published on this topic. The optimal selection of parity matrices and vectors addressed in Section 7.5 is mainly due to the work by Ding and co-worker [27, 28, 30] . They extended the ﬁrst results by Chow and Willsky [23, 98] and by Wuennenberg [148] in handling residual generator design via parity space technique and gave a systematic and complete procedure to the residual generator design. Although the H2 @H2 design is the ﬁrst approach proposed in [37, 97] for the optimal design of observer-based residual generators using the advanced robust control technique, only few study has been devoted to it. The interesting result on the relationship between the parity vector and H2 @H2 solution has been recently published by Zhang et al. [155]. Based on it, Ye et al. [150, 151] have developed time-frequency domain approaches for the residual generator design . The core of an observer-based residual generator is an observer or postﬁlter based residual generator. Some works have formulated the design problems in

246

7 Residual generation with enhanced robustness against unknown inputs

the framework of H2 or H4 or mixed H2 /H4 ﬁltering [47, 88, 89, 101, 109], or using the game theory [24]. The H4 @H4 design problem was ﬁrst proposed and solved in [39], lately in [121, 54, 129]. In the literature, few results have been reported on the LMI technique based solution of H4 @H4 design. Most relevant works are focused on the FDF design with H4 robustness against disturbances, see for instance [109, 47, 22] or [89]. Although it has been proposed and addressed in 1993 [43], H @H4 design problem has been extensively studied after the publication of the ﬁrst LMI solution to this problem [75]. The discussion about the H index in Subsection 7.9.2 is based on the work by Zhang and Ding [153] and strongly related to the results in [75, 122, 96]. Roughly speaking, there are three dierent design schemes relating to the H index • LMI technique based solutions, which also build the mainstream of the recent study on observer-based FD, • H4 solution by means of a reformulation of H @H4 design into a standard H4 problem as well as • factorization technique based solutions. In this chapter, we have studied the ﬁrst and the third schemes in the extended details. The second one has been brieﬂy addressed. The most significant contributions to the ﬁrst scheme are [75, 122, 96], while [72, 123] have provided solutions to the second scheme. In [32], the factorization technique has been used for the ﬁrst time to get a complete solution. This work is the basis for the development of the uniﬁed solution. A draft version of the uniﬁed solution has been reported in [29]. Further contributions to this scheme can be found in [82, 97, 152]. The uniﬁed solution plays a remarkable role in the subsequent study. The fact that the uniﬁed solution oers a simultaneous solution to the multiobjective Hl @H4 (including H @H4 and H4 @H4 ) optimization problem with, in comparison with other LMI solutions, considerably less computation is only one advantage of the uniﬁed solution, even though it seems attractive for the theoretical study. It will be demonstrated, in Chapter 12, that the general form of the uniﬁed solution leads to an optimal trade-o between the false alarm rate and fault detection rate and thus meets the primary and practical demands on an FDI system. This is the most important advantage of the uniﬁed solution. We would like to call reader’s attention that the study on the extended CIOF in Subsection 7.11.1 primarily serves as a mathematical formulation. Aided by this formulation, we are able to prove that making use of the information provided by those zeros at the m$-axis will lead to an improvement of the fault detection performance. From the numerical viewpoint, there should be more e!cient algorithms to realize such an extended CIOF.

8 Residual generation with enhanced robustness against model uncertainties

In this chapter, we shall deal with robustness problems met by generating residual signals in uncertain systems. As sketched in Fig.8.1, model uncertainties can be caused by changes in process and in sensor, actuator parameters. These changes will aect the residual signal and complicate the FDI process. The major objective of addressing the robustness issues is to enhance the robustness of the residual generator against model uncertainties and disturbances without signiﬁcant loss of the faults sensitivity.

Fig. 8.1 Schematic description of residual generation in a uncertain dynamic system

Model uncertainties may be present in dierent forms. It makes the handling of FDI in uncertain systems much more complicated than FDI for systems with unknown inputs. Bearing in mind that there exists no systematic way to address FDI problems for uncertain systems, in this chapter we shall focus on the introduction of some basic ideas, design schemes and on handling of representative model uncertainties.

248

8 Residual generation with enhanced robustness against model uncertainties

8.1 Preliminaries The major mathematical tool used for our study is the LMI technique introduced in the last chapter. Next, we shall introduce some additional mathematical preliminaries that are needed for the study on uncertain systems. 8.1.1 LMI aided computation for system bounds The following lemma plays an important role in boundness computation for uncertain systems. Lemma 8.1 Let J> O> H and I (w) be real matrices of appropriate dimensions with I (w) being a matrix function and I (w)W I (w) L. Then (a) for any % A 0, OI (w)H + H W I W (w)OW

1 W OO + %H W H %

(8.1)

(b) for any % A 0> S A 0 satisfying S 1 %H W H A 0, 1 (J + OI (w)H)S (J + OI (w)H)W J(S 1 %H W H)1 JW + OOW = (8.2) % Consider a system ¯ + I¯g g ¯ +H ¯g g> | = F{ {˙ = D{ ¯g = Hg + H> I¯g = Ig + I D¯ = D + D> F¯ = F + F> H

(8.3) (8.4)

with polytopic uncertainty

D H F I

¸ =

o X ol=1

l

¸ o Dl Hl X > l = 1> l 0> l = 1> · · · > o= Fl Il

(8.5)

ol=1

It holds Lemma 8.2 Given system (8.3)-(8.5) and a constant A 0> then k|k2 ? kgk2 if and only if there exists S A 0 so that ;l = 1> · · · > o 6 5 (D + Dl )W S + S (D + Dl ) S (Hg + Hl ) (F + Fl )W 7 (Hg + Hl )W S L (Ig + Il )W 8 ? 0= F + Fl Ig + Il L

(8.6)

8.1 Preliminaries

249

The proof of this lemma can be found in the book by Boyd et al. (see the reference given at the end of this chapter). Along with the lines of this proof, we can ﬁnd an LMI solution for the computation of the H index by extending Theorem 7.13 to the systems with polytopic uncertainties. Without proof, we summarize the results into the following lemma. Lemma 8.3 Given system (8.3)-(8.5) and a constant A 0> suppose that for l = 1> · · · > o ¸ D + Dl m$L Hg + Hl W = q + ng > (Ig + Il ) (Ig + Il ) A ;$> udqn F + Fl Ig + Il then k|k2 A kgk2 if and only if there exists S = S W so that ;l = 1> · · · > o 5 W W 6 (D + Dl ) S + S (D + Dl ) S (Hg + Hl ) (F + Fl ) 7 (Hg + Hl )W S L (Ig + Il )W 8 A 0= F + Fl Ig + Il L

(8.7)

8.1.2 Stability of stochastically uncertain systems Given a stochastically uncertain system Ã ! o X {(n + 1) = D + Dl sl (n) {(n)

(8.8)

l=1

where sl (n)> l = 1> · · · > o> represents a stochastic process with ³£ ¤W £ ¤´ s1 (n) · · · so (n) = gldj( 1 > · · · > o )= E (sl (n)) = 0> E s1 (n) · · · so (n) l > l = 1> · · · o> are known. It is further assumed that s(0)> s(1)> · · · > are independent and {(0) is independent of s(n)= The stability of (8.8) should be understood in the context of statistics. The so-called mean square stability serves for this purpose. Deﬁnition 8.1 Mean square stability: Given system (8.8) and denote ¡ ¢ P (n) = E {(n){W (n) = The system is called mean-square stability if for any {(0) lim P (n) = 0=

n$4

It is straightforward that P (n + 1) = DP (n)DW +

o X l=1

2l Dl P (n)DWl =

250

8 Residual generation with enhanced robustness against model uncertainties

Lemma 8.4 Given system (8.8). It is mean square stable if and only if there exists S A 0 so that DS DW S +

o X

2l Dl S DWl ? 0=

l=1

We refer the reader to the book by Boyd et al. for a comprehensive study on systems with stochastic uncertainties.

8.2 Transforming model uncertainties into unknown inputs As introduced in Chapter 3, systems with norm bounded uncertainties can be described by ¯ + Gx ¯ + I¯g g + Ii i ¯ + Ex ¯ +H ¯g g + Hi i> | = F{ {˙ = D{ ¯ = E + E> F¯ = F + F D¯ = D + D> E ¯ = G + G> H ¯g = Hg + H> I¯g = Ig + I G where

¸ ¸ £ ¤ D E H H = (w) J K M F G I I

(8.9) (8.10) (8.11)

(8.12)

with known H> I> J> K> M which are of appropriate dimensions and unknown (w) which is bounded by (8.13) ¯ () = Applying residual generator ³ ´ ˆ x (s)|(s) Q ˆx (s)x(s) > U(s) 5 RH4 u(s) = U(s) P

(8.14)

to (8.9)-(8.11) yields ¯ + Ex ¯ +H ¯g g + Hi i {˙ = D{ h˙ = (D OF) h + (D OF) { + (E OG) x ¢ ¡ ¯g OI¯g g + (Hi OIi ) i + H ¢ ¡ u(s) = U(s) Fh + F{ + Gx + I¯g g + Ii i =

(8.15) (8.16)

It is evident that system (8.15)-(8.16) is stable if and only if the original system (8.9) is stable and the observer gain O is so chosen that D OF is stable. For this reason, we assume in the following study that for any (w) (8.9) is stable. Note that, due to (8.12), (8.15) and (8.16) can be further written into

8.2 Transforming model uncertainties into unknown inputs

251

h˙ = (D OF) h + (H OI ) * + (Hg OIg ) g + (Hi OIi ) i u(s) = U(s) (Fh + I * + Ig g + Ii i ) 5 6 £ ¤ { * = J K M 7x8= g Let

¸ ¤ ¤ £ £ * g˜ = > Hg˜ = H Hg > Ig˜ = I Ig g

we have ¢ ¡ h˙ = (D OF) h + Hg˜ OIg˜ g˜ + (Hi OIi ) i ³ ´ u(s) = U(s) Fh + Ig˜g˜ + Ii i

(8.17) (8.18)

i.e. the dynamics of the residual generator is now represented by (8.17)-(8.18). In this way, the inﬂuence of the model uncertainty of the norm bounded type ˜ Thanks to its standard is modelled as a part of the unknown input vector g= form, optimal design of (8.17)-(8.18) can be realized using the approaches presented in Chapter 7. Remark 8.1 Note that * is a function of g and i , which can be expressed by ¸ £ ¤ x * = J ({g + {i ) + K M g with ¯ g + Ex ¯ +H ¯g g> {˙ i = D{ ¯ i + Ex ¯ + Hi i= {˙ g = D{ In the fault-free case, 5 6 ¸ £ ¤ {g * g˜ = >* = J K M 7 x 8= g g Thus, the uniﬁed solution can be achieved based on (8.17)-(8.18), even if g˜ depends on i= This way of handling system uncertainties can also be extended to dealing with other types of model uncertainties. It is worth pointing out that modelling the model uncertainty as unknown input vector may lead to a conservative design of the residual generator, since valuable information about the structure of the model uncertainty has not been taken into account.

252

8 Residual generation with enhanced robustness against model uncertainties

8.3 Reference model strategies 8.3.1 Basic idea Among the existing FDI schemes for uncertain systems, the so-called reference model based scheme has received considerable attention. The basic idea behind this scheme is the application of a reference model. In this way, similar to the solution of the H4 OFIP, the original FDI problem can be transformed into a standard design problem kuuhi uk2 °5 6° with respect to (8.9)-(8.11) O>U(s)5RH4 >i>g ° x ° ° ° °7 g 8° ° ° ° i ° 2 min

sup

(8.19)

with uuhi denoting the reference model. (8.19) is an MMP and there exist a number of methods to approach (8.19). The major dierence between those methods lies in the deﬁnition of the reference model. The earliest and most studied strategy is to handle the FDI problems in the form of the H4 OFIP. That means the reference model uuhi is deﬁned as uuhi (s) = i (s) or uuhi (s) = Z (s)i (s)

(8.20)

with a given weighting matrix Z (s) 5 RH4 = This method has been ﬁrst introduced in solving the integrated design of controller and FD unit and lately for the FD purpose, where optimization problem (8.19) is solved in the H4 / framework. As mentioned in Section 7.6 in dealing with the solution of H4 OFIP, the performance of the FDI systems designed based on reference model (8.20) strongly depends on the system structure regarding to the faults and on the selection of the weighting matrix Z (s). Next, we shall present an approach proposed by Zhong et al., which provides us with a more reasonable solution for the FDF design. 8.3.2 A reference model based solution for systems with norm bounded uncertainties The proposed approach consists of a two-step procedure for the design of FDI system: • Find the uniﬁed solution for system (8.9)-(8.11) with (w) = 0= Let Orsw > Yrsw be computed according to (7.293) and (8.21) uuhi (s) = Juuhi i (s)i (s) + Juuhi g (s)g(s) ¡ ¢ 1 Juuhi i (s) = Yrsw F(sL D + Orsw F) (Hi Orsw Ii ) + Ii ¡ ¢ Juuhi g (s) = Yrsw F(sL D + Orsw F)1 (Hg Orsw Ig ) + Ig

8.3 Reference model strategies

253

• Solve optimization problem kuuhi uk2 °5 6° ° ° >i>g ° x ° °7 g 8° ° ° ° i ° 2

min sup O>Y

(8.22)

by means of a standard LMI optimization method. Comparing reference models (8.20) and (8.21) makes it clear that including the inﬂuence of g in the reference model is the distinguishing dierence between (8.21) on the one side and (8.20) on the other side. At the ﬁrst glance, it seems contradictory that g is integrated into the reference model though reducing the inﬂuence of g is desired. On the other hand, we have learnt from the uniﬁed solution that the optimum is achieved by a suitable trade-o between the inﬂuences of the faults and disturbances. Simply reducing the inﬂuence of the disturbances does not automatically lead to an optimal trade-o. Now, we describe the second step of the approach, i.e. the solution of (8.22), in the extended detail. Let {uhi be the state vector of the reference model, i.e. {˙ uhi = Duhi {uhi + Hi>uhi i + Hg>uhi g> uuhi = Fuhi {uhi + Ii>uhi i + Ig>uhi g (8.23) Duhi = D Orsw F> Hi>uhi = Hi Orsw Ii > Hg>uhi = Hg Orsw Ig Fuhi = Yrsw F> Ii>uhi = Yrsw Ii > Ig>uhi = Yrsw Ig = Recalling that the dynamics of residual generator (8.14) with U(v) = Y (i.e. an FDF) can be written as ¸ ¸ ¸ ¸ ¯ {˙ D¯ 0 { E = {+ x h˙ D OF D OF h E OG ¸ ¸ ¯g Hi H g+ i (8.24) + ¯ Hi OIi Hg OI¯g μ ¸ ¶ £ ¤ { F F u=Y + Gx + I¯g g + Ii i (8.25) h it turns out ¢ ¡ {˙ r = (Dr + Dr ) {r + Hr>g¯ + Hr>g¯ g¯ ¢ ¡ uuhi u = (Fr + Fr ) {r + Ir>g¯ + Ir>g¯ g¯

(8.26) (8.27)

with 6 5 6 5 6 {uhi Duhi 0 x 0 £ ¤ 8 > Fr = Fuhi 0 Y F 0 {r = 7 { 8 > g¯ = 7 g 8 > Dr = 7 0 D 0 0 D OF h i 5

254

8 Residual generation with enhanced robustness against model uncertainties

6 0 Hg>uhi Hi>uhi ¤ £ 8 > Ir>g¯ = 0 Ig>uhi Y Ig Ii>uhi Y Ii Hg Hi Hr>g¯ = 7 E 0 Hg OIg Hi OIi 6 6 5 5 0 0 0 0 £ ¤ 8 (w) 0 J 0 D 08 = 7 H Dr = 7 0 H OI 0 D OF 0 £ ¤ £ ¤ Fr = 0 Y F 0 = Y I (w) 0 J 0 6 6 5 5 0 0 0 0 £ ¤ 8 (w) K M 0 E H 08 = 7 H Hr>g¯ = 7 H OI E OG H OI 0 £ ¤ £ ¤ Ir>g¯ = Y G Y I 0 = Y I (w) K M 0 = 5

The following theorem builds the basis for the solution of (8.22). Theorem 8.1 Given system (8.26)-(8.27) and suppose that {r (0) = 0 and W (w)(w) 1= Then

Z4

W

(uuhi u) (uuhi u) gw ?

2

0

Z4 ¯ g¯W ggw

if there exist some % A 0 and S A 0 so that 5 W ¯W J ¯W K ¯ 6 ¯ S Hr>g¯ + %J ¯ SH FrW Dr S + S Dr + %J ¯WJ ¯WK ¯ ¯ 9 H W ¯S + %K 0 : 2 L + %K Ir>Wg¯ r>g :?0 9 7 Fr Ir>g¯ L Y I 8 ¯W S 0 I W Y W %L H where

i £ ¤ £ ¤ W h ¯ = 0 J 0 >K ¯ = K M 0 >H ¯ = 0 H W (H OI )W = J

Proof. Let Y ({) = {Wr S {r > S A 0= It holds W (uuhi u) (uuhi u) 2 g¯W g¯ + Y˙ (w) ? 0 =, Z4 Z4 Z4 W 2 W ¯ ¯ (uuhi u) (uuhi u) gw g ggw + Y˙ (w)gw 0

Z4

=

0

(uuhi u)W (uuhi u) gw 2

0

Z4

=, 0

(8.28)

0

0

Z4

¯ + Y (4) ? 0 g¯W ggw 0

(uuhi u) (uuhi u) gw 2 W

Z4 ¯ ? 0= g¯W ggw 0

(8.29)

8.3 Reference model strategies

255

Since (uuhi u)W (uuhi u) 2 g¯W g¯ + Y˙ (w) = # Ã" ¸! ¸ W £ ¤ £ W W¤ (Fr + Fr ) 0 0 {r ¯ ¡ ¢W Fr + Fr Ir>g¯ + Ir>g¯ {r g 2 0 L g¯ Ir>g¯ + Ir>g¯ #! Ã" ¢ ¡ ¸ W ¤ £ (Dr + Dr ) S + S (Dr + Dr ) S Hr>g¯ + Hr>g¯ {r ¡ ¢W + {Wr g¯W g¯ Hr>g¯ + Hr>g¯ S 0 it turns out # " ¸ £ ¤ (Fr + Fr )W 0 0 ¡ ¢W Fr + Fr Ir>g¯ + Ir>g¯ 0 2L Ir>g¯ + Ir>g¯ " ¢# ¡ (Dr + Dr )W S + S (Dr + Dr ) S Hr>g¯ + Hr>g¯ ¡ ¢W + ?0 Hr>g¯ + Hr>g¯ S 0 Z4 Z4 W 2 ¯ ? 0= g¯W ggw =, (uuhi u) (uuhi u) gw 0

(8.30)

0

Applying the Schur complement we can rewrite (8.30) into 6 5 ¢ ¡ (Dr + Dr )W S + S (Dr + Dr ) S Hr>g¯ + Hr>g¯ (Fr + Fr )W ¡ ¢W ¢W : ¡ 9 7 Hr>g¯ + Hr>g¯ S 2 L Ir>g¯ + Ir>g¯ 8 ? 0 Fr + Fr Ir>g¯ + Ir>g¯ L (8.31) 5 5 W W 6 W W 6 Dr S + S Dr S Hr>g¯ Fr Dr S + S Dr S Hr>g¯ Fr W 2 W 8 W 7 Hr> S L I Hr> S 0 Ir>Wg¯ 8 ? 0= + +, 7 ¯ ¯ g r>g g¯ Fr Ir>g¯ L Fr Ir>g¯ 0 Split the second matrix in the above inequality into 5 6 ¯1 H 9 ¯ : 6 9 H2 : 5 ¯ : 9 H DWr S + S Dr S Hr>g¯ FrW 9 3 : £ ¤ W W 8 9 : 7 Hr> S 0 I = g¯ r>g¯ 9 0 : (w) 0 J 0 K M 0 0 9 0 : Fr Ir>g¯ 0 9 : 7 0 8 Y I 6 4W 35 ˜1 H F E9 H ˜ : F 5 6 E9 2 : ¯1 : F E9 H ˜ H E9 3 : £ ¤F ¯2 8 = S H= ¯ 9 0 : (w) 0 J 0 K M 0 0 F > 7 H +E : F E9 ¯3 F E9 0 : H : F E9 D C7 0 8 Y I

256

8 Residual generation with enhanced robustness against model uncertainties

Then, according to Lemma 8.1, we know that (8.31) holds if there exists a % A 0 so that 5 65 6 ¯1 ¯1 W H H ¯ :9 ¯ : 9 H 9 2 : 9 H2 : 6 5 W ¯ :9 ¯ : 9 H Dr S + S Dr S Hr>g¯ FrW 9 3 : 9 H3 : W 2 W 8 9 : 9 7 Hr> S L I 0 : g¯ r>g¯ + 1@% 9 :9 0 : 9 : : 9 Fr Ir>g¯ L 9 0 :9 0 : 7 0 87 0 8 Y I Y I ¤ £ ¤W £ 0 J 0 K M 0 0 ? 0= +% 0 J 0 K M 0 0 Finally, applying the Schur complement again yields 5 W ¯W J ¯W K ¯ 6 ¯ S Hr>g¯ + %J ¯ Dr S + S Dr + %J SH FrW W W 2 W W ¯ J ¯ K ¯ ¯ I ¯ 9 H ¯S + %K 0 : L + %K r>g r>g 9 : ? 0= 7 Fr Ir>g¯ L Y I 8 ¯W S 0 I W Y W %L H t u

The theorem is thus proven. Remark 8.2 The assumptions

{r (0) = 0 and W (w)(w) L do not lead to the loss of the generality of Theorem 8.1. If {r (0) 6= 0> it can be considered as an additional unknown input. In case that W (w)(w) L> 6= 1> we deﬁne p ¯ = (w)@ (w) £ ¤ £ ¤p ˜0K ˜ M˜ 0 = 0 J 0 K M 0 = 0J As a result, Theorem 8.1 holds. Remark 8.3 If Z4

W

(uuhi u) (uuhi u) gw 0

2

Z4 ¯ g¯W ggw 0

instead of (8.28) is required, condition (8.29) can be released and replaced by 5 W ¯W J ¯W K ¯ 6 ¯ S Hr>g¯ + %J ¯ SH Dr S + S Dr + %J FrW ¯WJ ¯WK ¯ ¯ 9 H W ¯S + %K 0 : 2 L + %K Ir>Wg¯ r>g 9 :0 7 Fr Ir>g¯ L Y I 8 ¯W S 0 I W Y W %L H ¯W J ¯ ? 0= DWr S + S Dr + %J

8.3 Reference model strategies

257

Based on Theorem 8.1, the optimization problem (8.22) can be reformulated as min subject to O>Y

5

¯W J ¯W K ¯ S Hr>g¯ + %J ¯ DWr S + S Dr + %J FrW W W 2 W ¯ J ¯ K ¯ ¯ IW¯ 9 H ¯S + %K L + %K r>g r>g 9 7 Fr Ir>g¯ L ¯W S 0 I W Y W H

(8.32) ¯ 6 SH 0 : :?0 Y I 8 %L

(8.33)

for some S A 0> % A 0= For our purpose of solving (8.32), let 6 5 S11 S12 0 1 \= S = 7 S21 S22 0 8 > O = S33 0 0 S33 Then (8.33) becomes an LMI regarding to S A 0> Y> \ and % A 0> as described by Q W = Q = [Qlm ]7×7 ? 0 where ¸ ¸ ¸ ¸ S11 S12 S11 S12 Duhi 0 0 0 + + S21 S22 S21 S22 0 D 0 %JW J ¸ ¸ ¸ 0 0 S11 S12 + K = 0> Q13 = %JW E S21 S22 ¸ ¸ ¸ ¸ ¸ Hg>uhi 0 Hi>uhi S11 S12 S11 S12 + M> Q = = 15 S21 S22 Hg S21 S22 Hi %JW ¸ ¸ W ¸ Fuhi 0 S11 S12 > Q17 = = H S21 S22 0

Q11 = Q12 Q14 Q16 Q22 Q25 Q33 Q44

Duhi 0 0 D

¸W

= DW S33 + S33 D F W \ W \ F> Q23 = 0> Q24 = S33 Hg \ Ig = S33 Hi \ Ii > Q26 = F W Y W > Q27 = S33 H \ I = 2 L + %K W K> Q34 = %K W M> Q35 = 0> Q36 = 0> Q37 = 0 W = 2 L + %M W M> Q45 = 0> Q46 = Ig>uhi IgW Y W > Q47 = 0> Q55 = 2 L

W IiW Y W > Q57 = 0> Q66 = L> Q67 = Y I> Q77 = %L= Q56 = Ii>uhi

As a result, we have an LMI solution that is summarized in the following algorithm. Algorithm 8.1 LMI solution of optimization problem (8.32) Step 0: Form matrix Q = [Qlm ]7×7 Step 1: Given A 0> ﬁnd S A 0> \> Y and A 0 so that Q ? 0=

258

8 Residual generation with enhanced robustness against model uncertainties

Step 2: Decrease and repeat Step 1 until the tolerant value is reached 1 \= Step 3: O = S33 Since for (w) = 0 the inﬂuence of x(w) on u(w) is nearly zero, neither in reference model (8.20) nor in (8.21) x(w) is included. However, we see that the system input x(w) does aect the dynamics of the residual generator= It is thus reasonable to include x(w) as a disturbance into the FDI system design. On the other side, it should be kept in mind that x(w)> dierent from g(w)> is online available. In order to improve the FDI system performance, knowledge of x(w) should be integrated into FDI system design and operation. This can be done, for instance, in form of an adaptive threshold or the so-called threshold selector, as will be shown in the Chapter 9. Remark 8.4 The above-presented results have been derived for continuous time systems. Analogous results for discrete time systems can be achieved in a similar way. For this purpose, inequality (8.2) in Lemma 8.1 is helpful. It would be a good exercise for the interested reader. Example 8.1 In this example, we design an FDF for the benchmark system LIP100 by taking into account the model uncertainty. In Subsection 3.7.2, the model uncertainty is well described, which is mainly caused by the linearization error. For our design purpose, the uniﬁed solution is used for the construction of the reference model and Algorithm 8.1 is applied to compute the observer gain O and post-ﬁlter Y= Remember that the open loop of the inverted pendulum system is not asymptotically stable. Thus, dierent from our previous study on this benchmark, the closed loop model of LIP100 builds the basis for our design. For the sake of simplicity, we assume that a state feedback controller is used, which places the closed loop poles at 3=1> 3=2> 3=3> 3=4. We are in a position to design the FDF. • Design of the reference model: The reference model is so designed that it is robust against unknown input and measurement noises. It results in 5 6 6 5 1=1338 0=1718 0=3728 100 9 0=1718 0=0260 0=0565 : : 8 7 Orsw = 9 7 6=5071 0=0565 0=1226 8 > Yrsw = 0 1 0 001 84=1928 0=0167 0=0362 • Determination of O> Y via Algorithm 8.1: We get 5 6 0=0040 0=0076 0=0040 9 0=0378 0=1211 0=0559 : : O = 1=0 × 104 9 7 0=1179 0=2152 1=5578 8 0=4436 1=3666 0=6366 5 6 0=6741 0=2586 0=4970 Y = 7 0=2586 1=1533 0=4906 8 0=4970 0=4906 0=7928 with = 2=8216=

8.4 Residual generation for systems with polytopic uncertainties

259

8.4 Residual generation for systems with polytopic uncertainties In this section, we address residual generation for systems with polytopic uncertainties. As described in Chapter 3, those systems are modelled by ¯ + Gx ¯ + I¯g g ¯ + Ex ¯ +H ¯g g> | = F{ {˙ = D{ ¯ = E + E> F¯ = F + F D¯ = D + D> E ¯ = G + G> H ¯g = Hg + H> I¯g = Ig + I G with

D E H F G I

¸ =

o X

l

ol=1

(8.34)

¸ o Dl El Hl X > l = 1> l 0> l = 1> · · · > o= Fl Gl Il ol=1

(8.35) Two approaches will be presented. The ﬁrst one is based on the reference model scheme, while the second one is an extension of the LMI based H to H4 design scheme. 8.4.1 The reference model scheme based scheme For our purpose, we apply again reference model (8.23) and formulate the residual generator design as ﬁnding O> Y such that A 0 is minimized, where is given in the context of Z4

W

(uuhi u) (uuhi u) gw ? 0

2

Z4 ¯ g¯W ggw

(8.36)

0

and uuhi u is governed by

uuhi

¢ ¡ {˙ r = (Dr + Dr ) {r + Hr>g¯ + Hr>g¯ g¯ ¢ ¡ u = (Fr + Fr ) {r + Ir>g¯ + Ir>g¯ g¯

¯ Dr > H ¯> Fr > I ¯ deﬁned in (8.26)-(8.27) and with {r > g> r>g r>g 6 5 o o 0 0 0 X X £ ¤ Dl 0 8 > Fr = Dr = l D¯l > D¯l = 7 0 l F¯l > F¯l = 0 Y Fl 0 0 Dl OFl 0 ol=1 ol=1 6 5 o 0 0 0 X ¯l = 7 ¯l > H El Hl 08 Hr>g¯ = lH El OGl Hl OIl 0 ol=1 Ir>g¯ =

o X ol=1

£ ¤ l I¯l > I¯l = Y Gl Y Il 0 =

260

8 Residual generation with enhanced robustness against model uncertainties

It follows from Lemma 8.2 that for given A 0 (8.36) holds if and only if there exists S A 0 so that ;l = 1> · · · > o 5¡ ¢W ¢ ¡ ¢ ¡ ¢W 6 ¡ ¯l Dr + D¯l S + S Dr + D¯l S Hr>g¯ + H Fr + F¯l ¡ ¢ ¢W : ¡ 9 ¯l W S 7 Hr>g¯ + H L Ir>g¯ + I¯l 8 ? 0= (8.37) Fr + F¯l Ir>g¯ + I¯l L Setting

5

6 S11 S12 0 1 S = 7 S21 S22 0 8 A 0> O = S33 \ 0 0 S33

(8.38)

(8.37) +, Ql = QlW = [Qmn ]7×7 ? 0> l = 1> · · · > o

(8.39)

yields with ¸W ¸ ¸ ¸ S11 S12 S11 S12 Duhi Duhi 0 0 + S21 S22 S21 S22 0 D + Dl 0 D + Dl ¸ ¸ ¸ 0 0 S11 S12 Q12 = > Q = 13 E + El DWl S33 FlW \ W S21 S22 ¸ ¸ ¸ ¸ Hg>uhi Hi>uhi S11 S12 S11 S12 Q14 = > Q15 = S21 S22 Hg + Hl S21 S22 Hi

Q11 =

Q16 =

¸ W Fuhi > Q22 = DW S33 F W \ W + S33 D \ F FlW Y W

Q23 = S33 El \ Gl > Q24 = S33 (Hg + Hl ) \ (Ig + Il ) Q25 = S33 Hi \ Ii > Q26 = F W Y W > Q33 = L> Q34 = 0> Q35 = 0 W Q36 = GlW Y W > Q44 = L> Q45 = 0> Q46 = Ig>uhi (Ig + Il )W Y W W Q55 = L> Q56 = Ii>uhi IiW Y W > Q66 = L=

Based on this result, the optimal design of residual generators for systems with polytopic uncertainties can be achieved using the following algorithm. Algorithm 8.2 LMI solution of (8.36) Step 0: Form matrix Ql = [Qnm ]7×7 > l = 1> · · · > o Step 1: Given A 0> ﬁnd S A 0 satisfying (8.38)> \> Y so that Ql ? 0= Step 2: Decrease and repeat Step 1 until the tolerant value is reached. Step 3: Set O according to (8.38).

8.4 Residual generation for systems with polytopic uncertainties

261

Example 8.2 In our previous examples concerning the benchmark system EHSA, we have learned that the linearization model works only in a neighborhood around the linearization point. In Subsection 3.7.5, the linearization errors are modelled into the polytopic type uncertainty. In this example, we design an FDF for the benchmark system EHSA under consideration of the polytopic type uncertainty. The design procedure consists of • Design of a reference model: For this purpose, the uniﬁed solution is applied to the linearization model with 5 6 0 00 9 0=143 0 0 : ¸ 9 : 010 9 : 0 0 : > Ig = = Hg = 9 0 001 7 0 0 08 0 00 The resulted observer gain and post-ﬁlter (of the reference model) are 5 6 8=69 × 1022 1=03 × 1025 9 0=067976 2=06 × 1012 : 9 : 25 28 : Orsw == 9 9 1=538 × 10 4 8=078 × 10 6 : > Yrsw = L= 7 6=6889 × 10 2=03 × 10 8 2=03 × 106 2=02 × 1015 • Determination of O> Y via Algorithm 8.2: We get 5 6 0=1111 0=0004 9 0=5250 0=9723 : ¸ 9 : 1=0349 0=0019 209=96 0=5119 : Y = >\ = 9 9 : 0=0019 0=3689 7 516=69 0=0007 8 0=0005 541=3 6 5 1=734 6=04 × 105 9 72=201 0=105 : : 9 : > = 1000= 105=46 0=0008 O=9 : 9 7 3=96 × 1011 5=65 × 105 8 0=2949 199=96 In our simulation study, we ﬁrst compare the residual signals generated respectively by FDF with and without considering the polytopic uncertainty. Fig.8.2 veriﬁes a signiﬁcant performance improvement, as the residual generator is designed by taking into account the polytopic uncertainty. To demonstrate the application of the FDF designed above, an actuator fault, which results in a piston runaway, is generated at w = 3 v= Fig.8.3 shows a successful fault detection. Using the analog version of Lemma 8.2 for discrete time systems, it is easy to ﬁnd an LMI solution of a reference model based design of the discrete time residual generator given by

262

8 Residual generation with enhanced robustness against model uncertainties Ŧ5

14

x 10

Without polytopic uncertainties With polytopic uncertainties

12 10

Residual [m]

8 6 4 2 0 Ŧ2 Ŧ4 Ŧ6 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time [s]

Fig. 8.2 Residual signal regarding to {s > generated by FDF with and without considering polytopic uncertainty

Ŧ4

1

x 10

0

Residual [m]

Ŧ1 Ŧ2 Ŧ3 Ŧ4 Ŧ5 Ŧ6 Ŧ7 0

1

2

3

4

5

Time [s]

Fig. 8.3 Response of residual signal regrading to {s to an actuator fault

{ ˆ(n + 1) = Dˆ {(n) + Ex(n) + O (|(n) |ˆ(n)) |ˆ(n) = F { ˆ(n) + Gx(n)> u(n) = Y (|(n) |ˆ(n)) =

(8.40) (8.41)

To this end, the uniﬁed solution described in Theorem 7.18 will be used as reference model and the design problem is formulated as ﬁnding O> Y such that A 0 is minimized, where is given by 4 X n=0

W

(uuhi (n) u(n)) (uuhi (n) u(n)) ?

2

4 X

¯ g¯W (n)g(n)=

(8.42)

n=0

Without derivation, we present below the LMI solution of this optimization problem:

8.4 Residual generation for systems with polytopic uncertainties

min subject to

263

(8.43)

Y>S A0>\

Ql = QlW = [Qmn ]8×8 ? 0> l = 1> · · · > o where Q11 =

¸ ¸ ¸ Duhi S11 S12 S11 S12 0 > Q12 = 0> Q13 = S21 S22 S21 S22 0 D + Dl

¸ ¸ ¸ ¸ 0 Hg>uhi S11 S12 S11 S12 > Q16 = = 0> Q15 = E + El S21 S22 S21 S22 Hg + Hl ¸ ¸ ¤ £ Hi>uhi S11 S12 > Q18 = 0> Q22 = S33 > Q23 = 0 S33 Dl \ Fl = S21 S22 Hi

Q14 Q17

Q24 = S33 D \ F> Q25 = S33 El \ Gl > Q26 = S33 (Hg + Hl ) \ (Ig + Il ) Q27 = S33 Hi \ Ii > Q28 = 0> Q33 = Q11 > Q34 = 0> Q35 = Q36 = Q37 = 0 W ¸ Fuhi Q38 = > Q44 = Q22 > Q45 = Q46 = Q47 = 0> Q48 = F W Y W FlW Y W Q55 = L> Q56 = Q57 = 0> Q58 = GlW Y W > Q66 = Q55 > Q67 = 0 W W Q68 = Ig>uhi (Ig + Il )W Y W > Q77 = Q55 > Q78 = Ii>uhi IiW Y W > Q88 = L=

8.4.2 H_ to H4 design formulation Denote the dynamics of residual generator (8.24)-(8.25) by {˙ u = Du {u + Eu x + Hu>g g + Hu>i i> u = Fu {u + Gu x + Iu>g g + Iu>i i ¸ X ¸ o o X Dl D 0 0 + := Du>0 + l l Du>l Du = 0 D OF Dl OFl 0 Eu =

¸

ol=1

E + 0

o X

l

ol=1

ol=1

¸

El := Eu>0 + El OGl

o X

l Eu>l

ol=1

o o X £ £ ¤ X ¤ Fu = 0 Y F + l Y Fl 0 := Fu>0 + l Fu>l ol=1

Gu =

o X

l Y Gl :=

ol=1

ol=1

Hu>g =

o X

l Gu>l > Iu>g = Y Ig +

ol=1 o X ol=1

l Y Il := Iu>0 +

o X ol=1

¸ X ¸ o o X Hl Hg + := Hu>0 + l l Hu>l Hg OIg Hl OIl ol=1 ol=1 ¸ Hi > Iu>i = Y Ii = Hu>i = Hi OIi

l Iu>l

264

8 Residual generation with enhanced robustness against model uncertainties

Along with the idea of the H_ to H4 design of residual generators presented in Subsection 7.9.4, we formulate the residual generator design as ﬁnding O> Y such that ° £ ¤ £ ¤° °Fu (sL Du )1 Eu Hu>g + Gu Iu>g ° ? (8.44) 4 kFu (sL Du )1 Hu>i + Iu>i k $ max =

(8.45)

It follows from Lemmas 8.2-8.3 that (8.44)-(8.45) can be equivalently written into max

O>Y>S A0>T=TW

1 subject to

(8.46)

6 Q S (Eu>0 + Eu>l ) S (Hu>0 + Hu>l ) (Fu>0 + Fu>l )W : 9 (Eu>0 + Eu>l )W S L 0 GuW :?0 9 8 7 (H + H )W S W 0 L Iu>g u>0 u>l Fu>0 + Fu>l Gu Iu>g L (8.47) 5

W

Q = (Du>0 + Du>l ) S + S (Du>0 + Du>l ) > l = 1> · · · > o W W 6 (Du>0 + Du>l ) T + T (Du>0 + Du>l ) THu>i (Fu>0 + Fu>l ) W W 7 8 A 0> l = 1> · · · > o= Hu>i T 1 L Iu>i Fu>0 + Fu>l Iu>i 1 L (8.48) 5

Now, let

S =

S1 0 0 S2

¸

A 0> T = TW =

¸ T1 0 = 0 T2

Then, (8.46)-(8.48) become max

W O>Y>S1 A0>S2 A0>T1 =TW 1 >T2 =T2

1 subject to

(8.49)

6 W Q21 S1 (E + El ) S1 (Hg + Hl ) FlW Y W Q11 W W 9 : Q21 Q22 S2 (El OGl ) Q24 F Y 9 : W W 9 (E + El )W S1 (El OGl )W S2 : L 0 Gl Y 9 : W W W 8 7 (Hg + Hl )W S1 Q24 0 L (Ig + Il ) Y Y Fl YF Y Gl Y (Ig + Il ) L 5

? 0> l = 1> · · · > o

(8.50)

W

Q11 = (D + Dl ) S1 + S1 (D + Dl ) > Q21 = S2 (Dl OFl ) W

Q22 = S2 (D OF) + (D OF) S2 > Q24 = S2 (Hg + Hl OIg OIl ) 5 6 W P21 T1 Hi FlW Y W P11 9 P21 P22 T2 (Hi OIi ) F W Y W : 9 : A 0> l = 1> · · · > o (8.51) 7 T1 H W T2 (Hi OIi )W 1 L IiW Y W 8 i Y Fl YF Y Ii 1 L W

P11 = (D + Dl ) T1 + T1 (D + Dl ) > P21 = T2 (Dl OFl )

8.5 Residual generation for stochastically uncertain systems

265

W

P22 = T2 (D OF) + (D OF) T2 = As mentioned in our study on the H_ to H4 design, (8.49)-(8.51) are an optimization problem with NMI constraints due to the terms S2 O> T2 O> which can be approached by advanced nonlinear optimization technique. A (very) conservative solution could be achieved by setting T2 = S2 > O = S21 \= In this case, (8.50)-(8.51) becomes LMIs regarding to S1 A 0> S2 A 0> T1 > \> Y=

8.5 Residual generation for stochastically uncertain systems In this section, we deal with residual generation for stochastically uncertain systems, which, as introduced in Chapter 3, are described by ¯ ¯ ¯g g(n) + Hi i (n) {(n + 1) = D{(n) + Ex(n) +H ¯ ¯ |(n) = F{(n) + Gx(n) + I¯g g(n) + Ii i (n)

(8.52) (8.53)

where ¯ = E + E> F¯ = F + F D¯ = D + D> E ¯ = G + G> H ¯g = Hg + H> I¯g = Ig + I G D> E> F> G> H and I represent model uncertainties satisfying

D E H F G I

¸

¸ ¶ o μ X Dl El Hl = s (n) Fl Gl Il l

(8.54)

l=1

with known matrices Dl > El > Fl > ¤Gl > Hl > Il > l = 1> · · · o> of appropriate dimen£ sions. sW (n) = s1 (n) · · · so (n) represents model uncertainties and is expressed as a stochastic process with ¢ ¡ s¯(n) = E (s(n)) = 0> E s(n)sW (n) = gldj( 1 > · · · > o ) where l > l = 1> · · · o> are known. It is further assumed that s(0)> s(1)> · · · > are independent and {(0)> x(n)> g(n)> i (n) are independent of s(n)= For the purpose of residual generation, the standard residual generator { ˆ(n + 1) = Dˆ {(n) + Ex(n) + O (|(n) |ˆ(n)) |ˆ(n) = F { ˆ(n) + Gx(n)> u(n) = Y (|(n) |ˆ(n)) is considered in the following study.

(8.55) (8.56)

266

8 Residual generation with enhanced robustness against model uncertainties

8.5.1 System dynamics and statistical properties For our purpose, the dynamics and the statistical properties of residual generator (8.55)-(8.56) will ﬁrst be studied. Introducing the following notations, ¸ ¸ ¸ {(n) D 0 Dl 0 > Du>0 = > Du>l = {u (n) = {(n) { ˆ(n) 0 D OF Dl OFl 0 ¸ ¸ o X E El > Eu>l = Du>l sl (n)> Eu>0 = Du = Du>0 + 0 El OGl l=1

Eu = Eu>0 +

o X

£ ¤ £ ¤ Eu>l sl (n)> Fu>0 = 0 Y F > Fu>l = Y Fl 0

l=1

Fu = Fu>0 +

o X l=1

Hu>l =

Fu>l sl (n)> Gu>l = Y Gl > Gu =

o X

Gu>l sl (n)> Hu>0

l=1

Hg = Hg OIg

¸

¸ o X Hl > Hu = Hu>0 + Hu>l sl (n)> Iu>0 = Y Ig Hl OIl l=1

Iu>l = Y Il > Iu = Iu>0 +

o X

Iu>l sl (n)> Hu>i =

l=1

¸ Hi > Iu>i = Y Ii Hi OIi

we have {u (n + 1) = Du {u (n) + Eu x(n) + Hu g(n) + Hu>i i (n) u(n) = Fu {u (n) + Gu x(n) + Iu g(n) + Iu>i i (n)=

(8.57) (8.58)

Note that the overall system (the plant + the residual generator) is mean square stable if and only if the plant is mean square stable, since the observer gain O has no inﬂuence on system (8.52). In the following of this section, the mean square stability of the plant is assumed. Remember that s(n) is independent of g(n)> x(n)> {(n)> h(n) and s¯(n) = 0= Thus, the mean of u(n) can be expressed by ¯u (n) + Eu>0 x(n) + Hu>0 g(n) + Hu>i i (n) { ¯u (n + 1) = Du>0 { u¯(n) = Fu>0 { ¯u (n) + Iu>0 g(n) + Iu>i i (n) which is equivalent to h¯(n + 1) = (D OF)¯ h(n) + (Hg OIg )g(n) + (Hi OIi )i (n) (8.59) u¯(n) = Y (F h¯(n) + Ii g(n) + Ii i (n)) = (8.60) 8.5.2 Basic idea and problem formulation Note that the mean of the residual signal given by (8.59)-(8.60) is exactly presented in a form, to which the uniﬁed solution can be used. Bearing in mind

8.5 Residual generation for stochastically uncertain systems

267

the stochastic property of the model uncertainty, we introduce the following performance index M = E(u(n) uuhi (n))W (u(n) uuhi (n))

(8.61)

which will be minimized by selecting O and Y= In (8.61), uuhi (n) stands for the reference model given by

Duhi

{uhi (n + 1) = Duhi {uhi (n) + Hi>uhi i (n) + Hg>uhi g(n) uuhi (n) = Fuhi {uhi (n) + Ii>uhi i (n) + Ig>uhi g(n) = D Orsw F> Hi>uhi = Hi Orsw Ii > Hg>uhi = Hg Orsw Ig Fuhi = Yrsw F> Ii>uhi = Yrsw Ii > Ig>uhi = Yrsw Ig =

with Orsw > Yrsw chosen using the uniﬁed solution described in Theorem 7.18. It is evident that M is a standard evaluation of the dierence between the residual signal u and the reference model uuhi in the statistic context. Since E(u(n) uuhi (n))W (u(n) uuhi (n)) = u(n) uuhi (n))W (¯ u(n) uuhi (n)) E(u(n) u¯(n))W (u(n) u¯(n)) + (¯ we formulate the design problem as ﬁnding O> Y such that for some given A0 u(n) uuhi (n)) $ min (¯ u(n) uuhi (n))W (¯ subject to

2u (n)

(8.62)

= E(u(n) u¯(n)) (u(n) u¯(n)) is bounded= W

Next, we shall derive an LMI solution for (8.62). 8.5.3 An LMI solution For our purpose of solving (8.62), we are ﬁrst going to ﬁnd LMI conditions for u(n) uuhi (n)) ? 21 (¯ u(n) uuhi (n))W (¯

n1 X

¡ W ¢ g (m)g(m) + i W (m)i (m) (8.63)

m=0

+22 2u (n) ? 21

n1 X

¡ W ¢ g (n)g(n) + i W (n)i (n)

¡ W ¢ g (m)g(m) + i W (m)i (m) + xW (m)x(m)

(8.64)

m=0

¡ ¢ + 22 gW (n)g(n) + i W (n)i (n) + xW (n)x(n) for some 1 A 0> 2 A 0> 1 A 0> 2 A 0= We start with problem (8.63). Introducing notions

268

8 Residual generation with enhanced robustness against model uncertainties

¸ ¸ ¸ g(n) h¯(n) D OF 0 ¯ > g(n) = > D = (n) = {uhi (n) 0 Duhi i (n) ¸ ¤ £ Hg OIg Hi OIi F = Y F Fuhi > H>g¯ = Hg>uhi Hi>uhi ¤ £ I>g¯ = Y Ig Ig>uhi Y Ii Ii>uhi

yields ¯ (n + 1) = D (n) + H>g¯g(n) ¯ u¯(n) uuhi (n) = F (n) + I>g¯g(n)=

(8.65) (8.66)

The following theorem provides an LMI condition for (8.63). Theorem 8.2 Given system (8.65)-(8.66), the constants 1 A 0> 2 A 0 and suppose that (0) = 0, then ;n u(n) uuhi (n)) ? 21 (¯ u(n) uuhi (n))W (¯

n1 X

¡ W ¢ g (m)g(m) + i W (m)i (m)

m=0

+22

¡ W ¢ g (n)g(n) + i W (n)i (n)

if the following three LMI’s hold for some S A 0 6 5 S S D S H>g¯ 7 DW S S 0 8A0 W S 0 L H> g¯ ¸ S FW 0 F 21 L ¸ L I>Wg¯ 0= I>g¯ 22 L

(8.67)

(8.68) (8.69)

Proof. Let Y (m) = W (m)S (m)> S A 0> m = 1> · · · = It is evident that ¯ Y (m + 1) Y (m) ? g¯W (m)g(m) ensures W

(n)S (n) ?

n1 X

¯ g¯W (m)g(m)=

(8.70)

(8.71)

m=0

(8.70) is equivalent with ¸ W ¸ ¤ £ D S 0 D H ? 0= S W >g¯ H> 0 L g¯ If (8.71) holds, then

(8.72)

8.5 Residual generation for stochastically uncertain systems

FW F 21 S =, W (n)FW F (n) ? 21

n1 X

¯ g¯W (m)g(m)=

269

(8.73)

m=0

Applying the Schur complement yields 6 5 6 5 1 S S D S H>g¯ S D H>g¯ 7 DW S 0 8 A 0 +, 7 DW S S 0 8A0 W W L H>g¯ 0 L H>g¯S 0 ¸ W S F 0= FW F 21 S +, F 21 L Since ¯ (¯ u(n) uuhi (n))W (¯ u(n) uuhi (n)) W (n)FW F (n) + g¯W (n)I>Wg¯I>g¯g(n) ¸ L I>Wg¯ ¯ ¯ 2 g¯W (n)g(n) 0 =, g¯W (n)I>Wg¯I>g¯g(n) I>Wg¯I>g¯ 22 L +, 2 I>g¯ 22 L the theorem is proven. u t The solution of (8.64) is somewhat involved. We start with some preliminary work. Deﬁne ¢ ¡ (8.74) Y (n) = E {Wu (n)S¯ {u (n) ¯ for some S A 0= We know from the basic statistics that ¤ ¡ W ¢ £ ¯u (n)S¯ { ¯u (n) > h{u (n) = {u (n) { ¯u (n) Y (n) = E hW{u (n)S¯ h{u (n) + E { and moreover £ ¤ ¡ ¢ £ ¤ E hW{u (n)S¯ h{u (n) = wudfh S¯ H{u (n) > H{u (n) = E h{u (n)hW{u (n) = Hence 5

5 6W 6 { ¯u (n) { ¯u (n) 9 x(n) : 9 x(n) : 9 : : Y (n + 1) = wudfh (PD H{u (n)) + 9 7 g(n) 8 P1 7 g(n) 8 i (n) i (n)

(8.75)

where o X ¡ ¢ 2l DWu>l S Du>l H{u (n) = E h{u (n)hW{u (n) > PD = DWu>0 S¯ Du>0 +

5

l=1

6 5 W 6 DWu>0 Du>l o W : W : 9 Eu>0 9 ¤ X ¤ £ £ 2 9 Eu>l : ¯ : S¯ Du>0 Eu>0 Hu>0 Hu>i + P1 = 9 W l W 7 Hu>0 8 7 Hu>l 8 S Du>l Eu>l Hu>l 0 = l=1 W Hu>i 0 (8.76)

270

8 Residual generation with enhanced robustness against model uncertainties

Suppose

5

S¯ 90 P1 ? 9 70 0

0 L 0 0

0 0 L 0

6 0 0: := 08 L

(8.77)

Note that (8.77) also implies PD ? S¯ then we have ¤ £ ¯Wu (n)S¯ { ¯u (n) Y (n + 1) ? E hW{u (n)S¯ h{u (n) + { +gW (n)g(n) + xW (n)x(n) + i W (n)i (n) W

W

(8.78)

W

= Y (n) + g (n)g(n) + x (n)x(n) + i (n)i (n) which leads to Y (n) ?

n1 X

£ W ¤ g (m)g(m) + xW (m)x(m) + i W (m)i (m) =,

(8.79)

m=0 n1 X£ ¢ ¤ ¡ gW (m)g(m) + xW (m)x(m) + i W (m)i (m) = ¯Wu (n)S { ¯u (n) ? wudfh S¯ H{u (n) + { m=0

We now consider 2u (n) and write it into 6 6W 5 x(n) x(n) 2u (n) = wudfh (PF H{u (n)) + { ¯Wu (n)P2 { ¯u (n) + 7 g(n) 8 P3 7 g(n) 8 i (n) i (n) 5

W Fu>r Fu>r

PF =

+

o X

W 2l Fu>l Fu>l > P2

l=1

=

o X

W 2l Fu>l Fu>l PF

6 W Gu>l £ ¤ W 8 Gu>l Iu>l 0 = 2l 7 Iu>l P3 = l=1 0 o X

5

(8.80)

l=1

(8.81)

As a result, if ¡ ¡ ¢ ¢ ¯Wu (n)P2 { ¯u (n) 21 wudfh S¯ H{u (n) + { ¯Wu (n)S¯ { ¯u (n) wudfh (PF H{u (n)) + { (8.82) P3 22 then it holds 2u (n) ? 21

n1 X m=0

+ 22

£ W ¤ g (m)g(m) + xW (m)x(m) + i W (m)i (m)

¡ W ¢ g (n)g(n) + i W (n)i (n) + xW (n)x(n) =

(8.83)

8.5 Residual generation for stochastically uncertain systems

271

It is evident that (8.82) holds, when PF 21 S¯ =

(8.84)

In summary, we have proven the following theorem. Theorem 8.3 Given system (8.57)-(8.58) and constants 1 A 0> 2 A 0= Then (8.64) holds if there exists S¯ A 0 so that 5 6 S¯ 0 0 0 9 0 L 0 0: : P1 ? 9 (8.85) 7 0 0 L 08 0 00L 2¯ (8.86) PF 1 S P3 22

(8.87)

where P1 > PF > P3 are respectively deﬁned in (8.76),(8.80), (8.81). Remark 8.5 It follows from Lemma 8.4 that the LMI (8.77) ensures the stability of the overall system. Starting from Theorems 8.2 and 8.3, we are now in a position to describe optimization problem (8.62) more precisely. The design objective is to solve the optimization problem ¡ ¢ (8.88) min z1 21 + z2 22 O>Y

subject to (8.67)-(8.69) and (8.85)-(8.87) for given constants 1 A 0> 2 A 0= In this formulation, z1 > z2 are two weighting factors whose values depend on the bounds of the L2 norm and L4 of x> g> i= Let S matrix given in (8.67)-(8.69) be ¸ S1 0 A0 S = 0 S2 and set S¯ matrix given in (8.85)-(8.87) equal to ¸ S3 0 A 0= S¯ = 0 S1 Moreover, deﬁne O = S11 \= As a result, (8.67)-(8.69) can be respectively rewritten into

(8.89)

272

8 Residual generation with enhanced robustness against model uncertainties

5

6 0 S1 D \ F 0 S1 Hg \ Ig S1 Hi \ Ii S1 9 0 S2 0 S2 Duhi S2 Hg>uhi S2 Hi>uhi : 9 : 9 (S D \ F)W : 0 S1 0 0 0 1 9 : 9 : W 0 S2 0 0 0 Duhi S2 9 : 9 : W W S2 0 0 L 0 7 (S1 Hg \ Ig ) Hg>uhi 8 W (S1 Hi \ Ii )W Hi>uhi S2 0 0 0 L (8.90) A0

6 0 FW Y W S1 W 8 7 0 S2 Fuhi 0 2 Y F Fuhi 1 L 5

5

(8.91)

6 W L 0 IgW Y W Ig>uhi W 7 8 0= 0 L IiW Y W Ii>uhi 2 Y Ig Ig>uhi Y Ii Ii>uhi 1 L

(8.92)

As to (8.85)-(8.87), a reformulation is needed. To this end, rewrite P1 > PF and P3 into 5 6 65 S¯ 0 · · · 0 Q0 2¯ : :9 ¤9 £ 9 0 1 S · · · 0 : 9 Q1 : P1 = Q0W Q1W · · · QoW 9 . . : : 9 . . 7 .. . . . . . .. 8 7 .. 8 0 · · · 0 2o S¯ Qo ¤ £ £ ¤ Q0 = Du>0 Eu>0 Hu>0 Hu>i > Ql = Du>l Eu>l Hu>l 0 > l = 1> · · · > o 5 65 6 L 0 ··· 0 Fu>0 2 :9 : ¤9 £ W 9 0 1 L · · · 0 : 9 Fu>1 : W W Fu>1 · · · Fu>o PF = Fu>0 9 .. . . . . .. : 9 .. : 7 . . . . 87 . 8 0 · · · 0 2o L Fu>o 6 5 2 65 1 L · · · 0 Wu>1 ¤9 £ W :9 . : W · · · W Fu>o P3 = Wu>1 7 0 . . . 0 8 7 .. 8 Wu>o 0 · · · 2o L £ ¤ Wu>l = Gu>l Iu>l 0 > l = 1> · · · > o= Then applying the Schur complement yields 5 Q1W S˜ Q0W 9 Q0 S¯ 1 0 9 ¡ 2 ¢1 9 ¯ Q 0 9 1S (8.85) +, 9 1 .. .. 9 .. . . 7 . Qo

0

···

··· ··· ··· .. .

QoW 0 0 .. .

¢1 ¡ 0 2o S¯

6 : : : :?0 : : 8

8.5 Residual generation for stochastically uncertain systems

5

273

6

S˜ Q0W S¯ Q1W S¯ · · · QoW S¯ 5 6 S¯ 0 0 0 9 S¯ Q0 S¯ : 0 0 9 : 9 0 L 0 0: 2 ¯ 9¯ : 0 : +, 9 S Q1 0 1 S · · · : ? 0> S˜ = 9 7 0 0 L 08 9 . : . . . . .. .. .. .. 7 .. 8 0 00L 2 ¯ ¯ S Qo 0 ··· 0 o S 6 5 2¯ W W W · · · Fu>0o 1 S Fu>0 Fu>1 9 Fu>0 L 0 ··· 0 : : 9 : 9 Fu>1 0 2 L · · · 0 1 (8.86) +, 9 :0 : 9 .. . .. . . . . . 7 . . . . 8 . Fu>o 0 ··· 0 2 o L 6 5 2 W W 2 L Wu>1 · · · Wu>u 9 Wu>1 2 0 : 1 L ··· : 9 (8.87) +, 9 . .. : 0= . . .. .. 7 .. . 8 Wu>o ··· 0 2 o L

(8.93)

(8.94)

(8.95)

Note that

¸ S3 D 0 S3 E S3 Hg S3 Hi ¯ S Q0 = 0 S1 D \ F 0 S1 Hg \ Ig S1 Hi \ Ii ¸ S3 Dl 0 S3 El S3 Hl 0 ¯ S Ql = 0 S1 Dl \ Fl S1 El \ Gl S1 Hl \ Il 0 £ ¤ £ ¤ Fu>0 = 0 Y F > Fu>l = Y Fl 0 > l = 1> · · · > o £ ¤ Wu>l = Y Gl Y Il 0 > l = 1> · · · > o=

Thus, (8.90)-(8.95) are LMIs regarding to S1 > S2 > S3 > \> Y= It allows us to use the following algorithm to solve (8.88). Algorithm 8.3 LMI aided FDI design for stochastically uncertain systems Step 0: Set 1 > 2 > 1 > 2 and z1 > z2 Step 1: Find S1 A 0> S2 A 0> S3 A 0> \> Y so that (8.90)-(8.95) are satisﬁed Step 2: Decrease ¢ ¡ z1 21 + z2 22 and repeat Step 1 until the tolerant value is reached Step 3: set O = S11 \= Remark 8.6 The solution may become conservative due to deﬁnition (8.89). Using an iterative algorithm, this problem can be solved. Example 8.3 In this example, we continue our study on the benchmark vehicle dynamic system (see Subsection 3.7.4). Our purpose is to design an FDF 0 . To via Algorithm 8.3, which takes into account the stochastic change in FY this end, the discrete time system model (3.77) with a slight modiﬁcation

274

8 Residual generation with enhanced robustness against model uncertainties 0 FY = 9360 + FY > FY 5 [5000> 5000]

is adopted. D> E are respectively ¸ ¸ 0=0388 0=0024 0=0108 4 4 > E = 10 × = D = 10 × 0=1208 0=0201 0=3952 We assume that only yaw rate measurement is available for the fault detection purpose. Our design procedure is as follows: • Design of the reference model: ¸ 0=2000 > Yrsw = 5=7014= Orsw = 1=4495 • Under the setting z1 = z2 = 1> we get ¸ ¸ 31=2660 0=6697 0=9343 0=1390 > S2 = Y = 5=7014> S1 = 0=6697 0=0419 0=1390 0=0483 ¸ ¸ ¸ 1=3667 0=0550 2=1358 0=0796 S3 = >\ = >O = 0=0550 0=0237 0=0312 0=5288 1 = 50> 2 = 7=1125 × 107 = 8.5.4 An alternative approach In the above presented approach, the optimization objective is described by (8.62). Alternatively, we can also deﬁne n n X X ¡ W ¢ g (m)g(m) + i W (m)i (m) (8.96) (¯ u(m) uuhi (m))W (¯ u(m) uuhi (m)) ? 2 m=0

m=0

subject to n X m=0

2u (m) ? 2

n X ¡ W ¢ g (m)g(m) + i W (m)i (m) + xW (m)x(m)

(8.97)

m=0

as a cost function and formulate the design problem as ﬁnding O> Y so that 2 is minimized for a given constant 2 . Its solution can be easily derived along with the lines given above and the standard solution for H4 norm computation (Bounded Real Lemma). Next, we sketch the basic steps of the solution and give the design algorithm. We assume that (0) = 0> h{u (0) = 0= It follows from Lemma 7.9 that (8.96) holds if and only if there exists a S A 0 so that 6 5 S S D S H>g¯ 0 9 DW S S 0 FW : : 9 W 7 H ¯S 0 L I W ¯ 8 ? 0= >g

0

>g

F

I>g¯ L

8.5 Residual generation for stochastically uncertain systems

Setting

275

¸ S1 0 S = A 0> O = S11 \ 0 S2

leads to 5 0 Q13 0 Q15 Q16 0 S1 9 0 0 S D S H S H 0 S 2 2 2 g>uhi 2 i>uhi 9 W W W 9 Q13 0 S 0 0 0 F Y 1 9 W 9 0 DWuhi S2 0 S2 0 0 F uhi 9 W W W 9 Q15 Hg>uhi S2 0 0 L 0 Q75 9 W W W 7 Q16 Hi>uhi S2 0 0 0 L Q76 0 0 Y F Fuhi Q75 Q76 L

6 : : : : :?0 : : : 8

(8.98)

Q13 = S1 D \ F> Q15 = S1 Hg \ Ig > Q16 = S1 Hi \ Ii Q75 = Y Ig Ig>uhi > Q76 = Y Ii Ii>uhi = To ﬁnd a su!cient LMI condition for (8.97), we introduce £ ¤ Y (m) = E hW{u (m)S¯ h{u (m) and consider

¡ ¢ 2u (m) 2 gW (m)g(m) + i W (m)i (m) + xW (m)x(m) + Y (m + 1) Y (m) ? 0

which ensures that (8.97) holds. Remember that £ ¤ ¡ ¢ 2u (m) = E uW (m)u(m) E u¯(m)W u¯(m) ¤ ¡ W ¢ £ ¯u (m)S¯ { ¯u (m) = Y (m) = E {Wu (m)S¯ {u (m) E { It turns out £ ¤ £ ¤ £ ¤ E uW (m)u(m) + E {Wu (m + 1)S¯ {u (m + 1) E {Wu (m)S¯ {u (m) ¡ ¢ 2 gW (m)g(m) + i W (m)i (m) + xW (m)x(m) ¢ ¡ W ¢ ¡ W ¢¢ ¡ ¡ ¯u (m + 1)S¯ { ¯u (m + 1) E { ¯u (m)S¯ { ¯u (m) ? 0 E u¯(m)W u¯(m) + E { which is equivalent to 5 W 6 5 6 Du>0 S¯ 0 0 0 W : 2 9 Eu>0 : ¤ 9 £ 9 W : S¯ Du>0 Eu>0 Hu>0 Hu>i 9 0 L 02 0 : 7 Hu>0 8 70 0 L 0 8 W 0 0 0 2L Hu>i 5 W 6 W Du>l Fu>l ¸ ¸ o W W : X ¯ 9 Du>l Eu>l Hu>l 0 2 9 Eu>l Gu>l : S 0 ?0 l 7 W + W 8 0 L Fu>l Gu>l Iu>l 0 Hu>l Iu>l l=1 0 0 5 5 W 6 6 Du>0 S¯ 0 0 0 W : 2 9 Eu>0 : £ ¤ 9 9 W : S¯ Du>0 Eu>0 Hu>0 Hu>i 9 0 L 02 0 : ? 0= 70 0 L 0 8 7 Hu>0 8 W 0 0 0 2L Hu>i

(8.99)

(8.100)

276

8 Residual generation with enhanced robustness against model uncertainties

It is evident that (8.99) implies (8.100). Now, let ¸ S3 0 A0 S¯ = 0 S1 and apply Schur complement to (8.99). We have for (8.99) 5 ˜ S Q0W S¯ 9 S¯ Q0 S¯ 9 9 Sˆ Q ˜1 0 9 9 . .. 7 .. . ˆ ˜ S Qo 0

˜ W Sˆ · · · Q ˜ W Sˆ 6 Q 1 o 0 0 : : 2 ¯ 1 S · · · 0 : :A0 .. : .. .. . . . 8 ¯ · · · 0 2 o S

(8.101)

where 6 S¯ 0 0 0 9 0 2L 0 0 : : =9 7 0 0 2L 0 8 0 0 0 2L ¸ S3 D 0 S3 E S3 Hg S3 Hi = 0 S1 D \ F 0 S1 Hg \ Ig S1 Hi \ Ii 6 5 S3 Dl 0 S3 El S3 Hl 0 = 7 0 S1 Dl \ Fl S1 El \ Gl S1 Hl \ Il 0 8 > l = 1> · · · > o= 0 Y Gl Y Il 0 Y Fl 5

S˜

S¯ Q0 ˜l Sˆ Q

In summary, we have the following algorithm. Algorithm 8.4 An alternative approach to LMI aided FDI design for stochastically uncertain systems Step 0: Set A 0 and A 0 Step 1: Find S1 A 0> S2 A 0> S3 A 0> \> Y so that (8.98) and (8.101) are satisﬁed Step 2: Decrease A 0 and repeat Step 1 until the pre-deﬁned tolerant value is reached Step 3: Set O = S11 \=

8.6 Notes and references In this chapter, we have focused our study on the application of the LMI technique to dealing with the robustness issues surrounding the design of residual generators for systems with model uncertainties. Although dierent types of model uncertainties have been handled, the underlying ideas of the presented methods are similar. The core of these methods is the application of a reference model. In this way, similar to the solution of the H4 OFIP,

8.6 Notes and references

277

the original residual generation problem is transformed into a, more or less, standard MMP problem. A key and also critical point surrounding the reference model based residual generation strategy is the selection of the reference model. Among the dierent selection schemes, handling the residual generation in the H4 OFIP framework is the most popular one, where the faults themselves or the weighted faults are deﬁned as the reference model. This method has been ﬁrst introduced in solving the integrated design of controller and FD unit [105, 136] and lately for the residual generation purpose [22, 56, 100, 123], where the optimization problem can also be solved in the H4 / framework [161]. Significantly dierent from it, disturbances are integrated into the reference model used in our study in this chapter. The basic idea behind such a reference model is the trade-o between the robustness and fault detectability. This idea has been ﬁrst proposed by Zhong et al. [159], where the uniﬁed solution is, due to its optimal trade-o, adopted as reference model. The methods presented in this chapter are the results of the application of this idea to the systems with dierent kinds of model uncertainties, where the LMI technique as the tool for the solution plays a central role. We refer the reader again to [14, 130] for the needed knowledge of the LMI technique. A comprehensive discussion on Lemma 8.1 can be found in [145]. We would like to call reader’s attention to the systematical and comprehensive study on the interpretation of the uniﬁed solution in Chapter 12. It will be demonstrated that the uniﬁed solution provides us with a reference model that is optimum in the sense of a trade-o between the false alarm rate and the fault detectability. Another way of handling residual generation problems for uncertain systems is to extend the H @H4 or H4 @H4 solutions. For instance, [126] proposed to solve H4 and H H4 problems in the H4 / framework. [72] developed a two-step scheme, in which H @H4 design of the residual generator is ﬁrst transformed and solved by means of the LMI technique and then in the second step the fault sensitivity performance is addressed with the aid of -synthesis. Comparing with the study in the previous two chapters, the reader may notice that the results presented in this chapter are considerably limited. This is also the state of the art in the model-based FDI technique. If we say, the results in Chapters 6 and 7 mark the state of the art of yesterdays’ and today’s FDI technique respectively, then it can be concluded that the study on the model-based FDI for uncertain systems would be a major topic in the ﬁeld of the model-based FDI technique in the coming years.

Part III

Residual evaluation and threshold computation

9 Norm based residual evaluation and threshold computation

In this and the next two chapters, we shall study residual evaluation and threshold computation problems. The study in the last part has clearly shown that the residual signal is generally corrupted with disturbances and uncertainties caused by parameter changes. To achieve a successful fault detection based on the available residual signal, further eorts are needed. A widely accepted way is to generate such a feature of the residual signal, by which we are able to distinguish the faults from the disturbances and uncertainties. Residual evaluation and threshold setting serve for this purpose. A decision on the possible occurrence of a fault will then be made by means of a simple comparison between the residual feature and the threshold, as shown in 9.1.

Fig. 9.1 Schematic description of residual evaluation and threshold generation

282

9 Norm based residual evaluation and threshold computation

Depending on the type of the system under consideration, there exist two residual evaluation strategies. The statistic testing is one of them, which is well established in the framework of statistical methods. Another one is the so-called norm based residual evaluation. Besides the less on-line calculation, the norm based residual evaluation allows a systematic threshold computation using the well-established robust control theory. In this chapter, we shall focus on the norm based residual evaluation and the associated threshold computation, as sketched in Fig.9.1. The statistic testing methods and the integration of the norm based and statistic methods will be addressed in the next two chapters.

9.1 Preliminaries The concepts with the signal and system norms introduced in Sections 7.1 and 8.1 are essential for our study in this chapter. Remember that in Section 7.1 we have introduced the so-called peak-topeak gain and the generalized H2 norm. Both of them are the induced system norm and useful for our study in this chapter. Below, we present the known results on the LMI aided computation of these two norms, published by Scherer et al. in their celebrated paper entitled multiobjective output-feedback control via LMI optimization. Lemma 9.1 Given system G : {˙ = D{ + Hg g> | = F{> {(0) = 0= Then for a given constant A 0 kGkj ? +, k|kshdn ? kgk2 if and only if there exists a S A 0 so that W ¸ ¸ D S + S D S Hg S FW ? 0> A 0= HgW S L F 2L Lemma 9.2 Given system G : {˙ = D{ + Hg g> | = F{ + Ig g> {(0) = 0 where g is bounded by

;w> gW (w)g(w) 1=

Then for a given constant A 0 kGkshdn ? +, k|kshdn ? kgkshdn if there exist A 0> A 0 and S A 0 so that

9.1 Preliminaries

283

¸

DW S + S D + S S Hg ?0 HgW S L 5 6 S 0 FW 7 0 ( ) L IgW 8 A 0= F Ig L

The following two lemmas are the extension of Lemmas 9.1 and 9.2 to the systems with polytopic uncertainties. For their proof, the way of handling polytopic uncertainty described in the book by Boyd et al. can be adopted. Lemma 9.3 Given system G : {˙ = (D + D) { + (Hg + H) g> | = (F + F) {> {(0) = 0 ¸ X ¸ o D H Dl Hl = l F 0 Fl 0 ol=1

o X

l = 1> l 0> l = 1> · · · > o=

l=1

Then for a given constant A 0 kGkj ? +, k|kshdn ? kgk2 if and only if there exists a S A 0 so that ;l = 1> · · · > o> " # W (D + Dl ) S + S (D + Dl ) S (Hg + Hl ) ?0 W (Hg + Hl ) S L ¸ S (F + Fl )W A 0= F + Fl 2L Lemma 9.4 Given system G : {˙ = (D + D) { + (Hg + H) g> | = (F + F) { + (Ig + I ) g> {(0) = 0 ¸ X ¸ o D H Dl Hl = l F I Fl Il ol=1

o X

l = 1> l 0> l = 1> · · · > o=

l=1

Then for a given constant A 0 kGkshdn ? +, k|kshdn ? kgkshdn if there exist A 0> A 0 and S A 0 so that ;l = 1> · · · > o>

284

9 Norm based residual evaluation and threshold computation

"

# W (D + Dl ) S + S (D + Dl ) + S S (Hg + Hl ) ?0 (Hg + Hl )W S L 5 W 6 S 0 (F + Fl ) W 7 0 ( ) L (Ig + Il ) 8 A 0= F + Fl Ig + Il L

9.2 Basic concepts In practice, the so-called limit monitoring and trend analysis are, due to their simplicity, widely used for the purpose of fault detection. For a given signal |, the primary form of limit monitoring is | ? |min or | A |max =, alarm, a fault is detected |min | |max =, no alarm, fault-free where |min > |max denote the minimum and maximum values of | in the faultfree case. They are also called threshold. The trend analysis of a signal | can be in fact interpreted as limit monitoring of |> ˙ and thus formulated as |˙ ? |˙min or |˙ A |˙max =, alarm, a fault is detected |˙min |˙ |˙max =, no alarm, fault-free. Also widely accepted in practice is the root-mean-square (RMS) (see also Section 7.1), denoted by k·kUPV > that measures the average energy of a signal over a time interval (0> W )= The fault detection problem is then described by: k|kUPV ? k|kUPV>min or k|kUPV A k|kUP V>max =, alarm, a fault is detected k|kUPV>min k|kUPV k|kUPV>max =, no alarm, fault-free with k|kUP V>min > k|kUPV>max as minimum and maximum values of k|kUPV = In order to overcome the di!culty with noises, the average value of a signal over a time interval [w> w + W ] > instead of its maximum/minimum value or RMS, is often used for the purpose of fault detection. In this case, the limit monitoring can be formulated as: 1 |¯(w) = W |¯min

w+W Z

u¯( )g ? |¯min w

1 or |¯(w) = W

w+W Z

u¯( )g A |¯max w

=, alarm, a fault is detected |¯(w) |¯max =, no alarm, fault-free

9.3 Some standard evaluation functions

285

where |¯min > |¯max represent the minium and maximum value of |¯(w) respectively. In summary, it is the state of the art in practice that for the purpose of fault detection an evaluation function is ﬁrst deﬁned, which gives some mathematical feature of the signal, and, based on it, a threshold is established. The last step is then the decision making. In the subsequent sections, we shall study these issues in a more generalized form.

9.3 Some standard evaluation functions Consider a dynamic process. Driven by the process input signal x> the value or average value or the energy of the process output | may become very large. In order to achieve an e!cient and highly reliable FDI, it is reasonable to analyze the system performance on account of a residual signal instead of |= In our subsequent study in this chapter, we assume that for the FDI purpose a residual vector, u 5 Rnu > is available. Next, we describe some standard evaluation functions which are in fact a generalization of the above-mentioned evaluation functions of |. Peak value: The peak value of residual signal u is deﬁned and denoted by, for continuous time u(w) Mshdn = kukshdn := sup ku(w)k > ku(w)k =

Ãn u X

w0

!1@2 ul2 (w)

(9.1)

l=1

and for discrete time u(n) Mshdn = kukshdn := sup ku(n)k > ku(n)k =

Ãn u X

n0

!1@2 ul2 (n)

=

(9.2)

l=1

The peak value of u is exactly the peak norm of u, as introduced in Section 7.1. Using the peak value of u, the limit monitoring problem can be reformulated as Mshdn A Mwk>shdn =, alarm, a fault is detected Mshdn Mwk>shdn =, no alarm, fault-free where Mwk>shdn is the so-called threshold deﬁned by Mwk>shdn =

sup ku(w)kshdn or Mwk>shdn =

fault-free

sup ku(n)kshdn =

(9.3)

fault-free

Also, we can use the peak value of u˙ or u(n) = u(n + 1) u(n) to reformulate the trend analysis. Let

286

9 Norm based residual evaluation and threshold computation

Mwuhqg = kuk ˙ shdn = sup ku(w)k ˙ for the continuous time case

(9.4)

Mwuhqg = ku(n)kshdn = sup ku(n)k for the discrete time case

(9.5)

w0

n0

Mwk>wuhqg =

sup ku(w)k ˙ shdn or Mwk>shdn =

fault-free

sup ku(n)kshdn

(9.6)

fault-free

then Mwuhqg A Mwk>wuhqg =, alarm, a fault is detected Mwuhqg Mwk>wuhqg =, no alarm, fault-free. ˆ˙ Often, for the practical implementation u˙ is replaced by u> ˆ˙ u(s) =

s u(s) s + 1

(9.7)

with 0 ? ¿ 1 or u(n) by u(n) = u(n) u(n 1)=

(9.8)

As for the average value evaluation, we deﬁne for the continuous time case

Mdyhudjh = ku(w)kdyhudjh

1 = sup k¯ u(w)kshdn > u¯(w) = W w0

w+W Z

u( )g

(9.9)

w

and for the discrete time case Mdyhudjh = ku(n)kdyhudjh = sup k¯ u(n)kshdn > u¯(n) = n0

Q 1 X u(n + m) Q m=1

(9.10)

and moreover, Mwk>dyhudjh =

sup ku(w)kdyhudjh or

fault-free

sup ku(n)kdyhudjh =

(9.11)

fault-free

As a result, the decision logic for detecting a fault is Mdyhudjh A Mwk>dyhudjh =, alarm, a fault is detected Mdyhudjh Mwk>dyhudjh =, no alarm, fault-free. The following modiﬁed form of average value u¯ given in (9.9) or (9.10) is often adopted u¯˙ (w) = u¯(w) + u(w) u¯(n + 1) = (1 )¯ u(n) + u(n)

(9.12) (9.13)

9.4 Basic ideas of threshold setting and problem formulation

287

where 0 ? ?? 1 and 0 ¿ ? 1= RMS value: As introduced in Section 7.1, the RMS value of u is deﬁned by, for the continuous time case, 3 MUPV = ku(w)kUPV = C

1 W

41@2

w+W Z

2

ku( )k g D

(9.14)

w

and for the discrete time case, 3

MUPV = ku(n)kUPV

41@2 Q X 1 2 =C ku(n + m)k D = Q m=1

(9.15)

MUPV measures the average energy of u over time interval (w> w + W ) as well as (n> n + Q )= Remember that the RMS of a signal is related to the L2 norm of this signal. In fact, it holds 2

ku(w)kUPV as well as 2

ku(n)kUPV

1 2 ku(w)k2 W

(9.16)

1 2 ku(n)k2 = Q

(9.17)

Let Mwk>UPV =

sup kukUPV

fault-free

be the threshold, then the detection logic becomes MUPV A Mwk>UPV =, alarm, a fault is detected MUPV Mwk>UPV =, no alarm, fault-free.

9.4 Basic ideas of threshold setting and problem formulation From the engineering viewpoint, the determination of a threshold is to ﬁnd out the tolerant limit for disturbances and model uncertainties under faultfree operation conditions. There are a number of factors that can signiﬁcantly inﬂuence this procedure. Among them are • the dynamics of the residual generator • the way of evaluating the unknown inputs (disturbances) and model uncertainties as well as • the bounds of the unknown inputs and model uncertainties. Next, we shall brieﬂy address these issues.

288

9 Norm based residual evaluation and threshold computation

9.4.1 Dynamics of the residual generator We assume that the system model is given by (8.9)-(8.11), where the model uncertainties are either the norm bounded type (8.12) or the polytopic type (8.35). Remark 9.1 The model uncertainty of the stochastic type given in (8.54) will be handled in a separate chapter. Applying residual generator (8.14) to this process model yields ¯ + Ex ¯ +H ¯g g + Hi i {˙ = D{ h˙ = (D OF) h + (D OF) { + (E OG) x ¢ ¡ ¯g OI¯g g + (Hi OIi ) i + H ¢ ¡ u(s) = U(s) Fh + F{ + Gx + I¯g g + Ii i =

(9.18) (9.19) (9.20)

Note that the modiﬁed forms (9.7) or (9.8) of the trend analysis or (9.12) as well as (9.13) of the average value analysis can be handled as a ﬁltering of the residual signal and thus included in the post-ﬁlter U(s). Hence, without loss of generality, we use below (9.18)-(9.20) to represent all the three possible forms of the residual signal under consideration. Let’s denote the minimal state space realization and the state vector of U(s) by (Ds > Es > Fs > Gs ) and {s respectively with subscript s standing for post-ﬁlter. For our purpose, write (9.18)-(9.20) into the following compact form {˙ u = (Du + Du ) {u + (Hg>u + Hu ) gu + Hu>i i u = (Fu + Fu ) {u + (Ig>u + Iu ) gu + Iu>i i where

(9.21) (9.22)

6 5 5 6 6 { D 0 0 D 00 {u = 7 h 8 > Du = 7 0 D OF 0 8 > Du = 7 D OF 0 0 8 {s 0 Es F Ds Es F 0 0 6 6 5 5 ¸ E Hg E H x > Hu>g = 7 0 Hg OIg 8 > Hu = 7 E OG H OI 8 gu = g Es G 0 Es Ig Es I 6 5 Hi ¤ £ £ ¤ Hu>i = 7 Hi OIi 8 > Fu = 0 Gs F Fs > Fu = Gs F 0 0 Es Ii ¤ £ £ ¤ Iu>g = 0 Gs Ig > Iu = Gs G Gs I > Iu>i = Gs Ii i= 5

In case of the norm bounded model uncertainty 6 6 5 5 H H £ ¤ £ ¤ Du = 7 H OI 8 (w) J 0 0 > Hu = 7 H OI 8 (w) K M Es I Es I £ ¤ £ ¤ Fu = Gs I (w) J 0 0 > Iu = Gs I (w) K M > W (w)(w) L

9.4 Basic ideas of threshold setting and problem formulation

289

while for the polytopic uncertainty 6 Dl 00 Du = l Du>l > Du>l = 7 Dl OFl 0 0 8 Es Fl 0 0 ol=1 6 5 o El Hl X Hu = l Hu>l > Hu>l = 7 El OGl Hl OIl 8 Es Gl Es Il ol=1 5

o X

Fu =

o X

¤ £ l Fu>l > Fu>l = 0 0 Gs Fl

ol=1

Iu =

o X

¤ £ l Iu>l > Iu>l = Gs Gl Gs Il =

ol=1

9.4.2 Deﬁnitions of thresholds and problem formulation Recall that the threshold is understood as the tolerant limit for the unknown inputs and model uncertainties during the fault-free system operation. Under this consideration, the threshold can be generally deﬁned by Mwk =

sup M

i =0>g>

with denoting the model uncertainties and M the feature of the residual signal like Mshdn > Mwuhqg > MUPV deﬁned in the last subsection. Also, the way of evaluating the unknown inputs plays an important role by the determination of thresholds. Typically, the energy level and the maximum value of unknown inputs are adopted in practice for this purpose. In this context, we introduce below dierent kinds of thresholds to cover these possible practical cases. Deﬁnition 9.1 Suppose that gu is bounded by and in the sense of kgu kshdn kgkshdn + kxkshdn g>4 + x>4 =

(9.23)

Then the threshold Mwk>shdn>shdn is deﬁned by Mwk>shdn>shdn =

sup kgu kshdn g>4 + x>4 i =0>¯ ()

Mshdn

(9.24)

Mshdn

(9.25)

for the norm bounded uncertainty or Mwk>shdn>shdn =

for the polytopic uncertainty.

sup kgu kshdn g>4 + x>4 i =0> l >l=1··· >o

290

9 Norm based residual evaluation and threshold computation

Mwk>shdn>shdn measures the maximum (instantaneous) change in u caused by the instantaneous (bounded) changes of > gu . Note that Mwk>shdn>shdn can be reached even if the energy level of signal gu may be very low but its size at some time instance is very large. Deﬁnition 9.2 Suppose that gu is bounded by and in the sense of kgu k2 g>2 + x>2 and kgu kshdn g>4 + x>4 =

(9.26)

Then the threshold Mwk>shdn>2 is deﬁned by Mwk>shdn>2 =

sup kgu k2 g>2 +x>2 kgu kshdn g>4 +x>4 i =0>¯ ()

Mshdn

(9.27)

Mshdn

(9.28)

for the norm bounded uncertainty or Mwk>shdn>2 =

sup kgu k2 g>2 +x>2 kgu kshdn g>4 +x>4 i =0> l >l=1··· >o

for the polytopic uncertainty. Although Mwk>shdn>2 also measures the maximum change in u> but dierent from Mwk>shdn>shdn > Mwk>shdn>2 does it with respect to the bounded energy in gu . Deﬁnition 9.3 Suppose that gu is bounded by and in the sense of kgu k2 g>2 + x>2 = Then the threshold Mwk>UPV>2 is deﬁned by Mwk>UPV>2 =

sup kgu kUPV g>2 + x>2 i =0>¯ ()

MUPV

(9.29)

MUPV

(9.30)

for the norm bounded uncertainty or Mwk>UPV>2 =

sup kgu kUPV g>2 + x>2 i =0> l >l=1··· >o

for the polytopic uncertainty. Mwk>UPV>2 measures the maximum change in the (average) energy level of u in response to the model uncertainty and unknown inputs which are of certain energy level. In practice, aiming at an early fault detection on the one side and a low false alarm rate on the other side, g>4 is often set low and Mwk>shdn>shdn is used to activate the computation of Mwk>shdn>2 or Mwk>UPV>2 = While Mwk>shdn>2 is generally set higher than Mwk>shdn>shdn , due to the assumption on the energy level of gu > Mwk>UPV>2 requires an observation of the residual signals over a (long) time window= This scheme is used to reduce the false alarm rate.

9.5 Computation of Mwk>UP V>2

291

Remark 9.2 Although the input signal x is treated as a "unknown input", the available information about it will be used to realize the so-called adaptive threshold, which will then recover the performance. From the mathematical and system theoretical viewpoint, the abovedeﬁned thresholds can be understood as induced norms or ”system gains”. In this context, we are able to formulate the threshold computation as an optimization problem: • Computation of Mwk>shdn>shdn Mwk>shdn>shdn = min ( g>4 + x>4 ) with subject to (9.31) ;gu satisfying (9=23) > either norm bounded or polytopic sup ku(w)k sup kgu (w)k or sup ku(n)k sup kgu (n)k = w0

w0

n0

n0

• Computation of Mwk>shdn>2 Mwk>shdn>2 = min 1 ( g>2 + x>2 ) + 2 ( g>4 + x>4 ) (9.32) with 1 subject to ;gu satisfying (9=26) > either norm bounded or polytopic sup ku(w)k 1 kgu (w)k2 or sup ku(n)k 1 kgu (n)k2 = w0

n0

Remark 9.3 The term 2 ( g>4 + x>4 ) in (9.32) is due to the existence of Ig>u + Iu , by which gu will act on u instantaneously. In the section dealing with the computation of Mwk>shdn>2 > we shall explain it in more detail. • Computation of Mwk>UPV>2 Mwk>UPV>2 = min ( g>2 + x>2 ) with subject to (9.33) ;W> gu satisfying (9=26) > either norm bounded or polytopic ku(w)k2 kgu (w)k2 or ku(n)k2 kgu (n)k2 = Using the LMI technique, we shall derive algorithms for solving these problems. This is the major objective of the rest of the sections in this chapter.

9.5 Computation of Jwk>UPV>2 In this section, we address the computation of Mwk>UP V>2 for the systems with both the norm bounded and polytopic model uncertainty.

292

9 Norm based residual evaluation and threshold computation

9.5.1 Computation of Jwk>UP V>2 for the systems with the norm bounded uncertainty For our purpose, we ﬁrst give a theorem, which builds the basis for the computation of Mwk>UPV>2 = Theorem 9.1 Given system (9.21)-(9.22) with the norm bounded uncertainty and A 0> and suppose that {u (0) = 0> W (w)(w) L, then ku(w)k2 ? kgu (w)k2 if there exist % A 0> S A 0 so that 5 W 6 ¯W J ¯W K ¯ ¯ S Hu>g + %J ¯ FuW S H Du S + S Du + %J W ¯WK ¯ IW 9 2 L + %K 0 : Hu>g g>u 9 :?0 7 Ig>u L Gs I 8 Fu ¯W S 0 I W GsW %L H where

6 H ¯ = J 0 0 >K ¯ = K M >H ¯ = 7 H OI 8 = J Es I £

¤

£

¤

(9.34)

5

(9.35)

The proof of this theorem is similar with the one of Theorem 8.1 and follows directly from the Bounded Real Lemma and Lemma 8.1. The "discrete time version" of Theorem 9.1 is given in the following theorem. Theorem 9.2 Given system {u (n + 1) = (Du + Du ) {u (n) + (Hg>u + Hu ) gu (n) u(n) = (Fu + Fu ) {u (n) + (Ig>u + Iu ) gu (n) with the norm bounded uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and A 0> and suppose that {u (0) = 0> W (n)(n) L, then ku(n)k2 ? kgu (n)k2 if there exist A 0> S A 0 so that 6 5 ¯ S Hu SH S 0 S Du 9 0 Iu Gs I : L Fu : 9 W W ¯ ¯W J ¯S ¯ 9 Du S FuW J J K 0 : :?0 9 W W ¯WJ ¯ K ¯WK ¯ 2L 0 8 7 Hg>u S Ig>u K ¯ W S I W GsW 0 0 L H ¯ J> ¯ K ¯ as deﬁned in (9.35). with H>

(9.36)

(9.37)

9.5 Computation of Mwk>UP V>2

293

Proof. Due to the similarity to Theorem 8.1, we only brieﬂy sketch the proof. It is evident that (9.36) holds if # " ¸ ¸ W W Du + Du Hg>u + Hu S 0 (Du + Du ) (Fu + Fu ) 0 L Fu + Fu Ig>u + Iu (Hg>u + Hu )W (Ig>u + Iu )W ¸ S 0 ? 0= 0 2L Recall that

Du Hu Fu Iu

¸

5

6 H 9 H OI : £ ¤ : =9 7 Es I 8 (n) J 0 0 K M = Gs I

It follows from Lemma 8.1 that the above inequality holds, provided that for some % A 0

DWu W Hg>u

3 5 65 6W 41 H H ¸ ¸ 1 9 H OI : 9 H OI : F FuW E S 0 Du Hu E 9 F : 9 : % W C 0 L 7 Es I 8 7 Es I 8 D Ig>u Fu Iu Gs I Gs I ¸ ¤ ¤W £ 1£ S 0 J00KM + J00KM ? 0= 0 2L % ¸

Now, applying the Schur complement yields 5 5 6 65 6W H H ¸1 ¸ 9 9 H OI : 9 H OI : : Du Hu 9%9 : :9 : S 0 9 7 Es I 8 7 Es I 8 : 0 L Fu Iu 9 :?0 9 : I Gs I G s 9 ¸ 1 W ¸: W 1 ¯W ¯ W 7 8 ¯ ¯ Du Fu %J J S %J K W W 1 ¯W ¯ 1 ¯W ¯ 2 Hg>u Ig>u %K J %K K L 5 ¯ 6 S Hu SH S 0 S Du 9 0 L Fu Iu Gs I : : 9 W 1 ¯W ¯ 1 ¯W ¯ W 9 0 : +, 9 Du S Fu % J J S %J K : ? 0= 1 ¯W ¯ 1 ¯W ¯ W W 2 8 7 Hg>u S Ig>u L 0 K J K K % % 1 W W W ¯ 0 0 %L H S I Gs Finally, setting =

1 %

completes the proof. u t

With the aid of Theorems 9.1 and 9.2 as well as the relation between the L2 norm and the RMS, (9.16) or (9.17), we have Algorithm 9.1 Computation of Mwk>UPV>2 for the systems with the norm bounded uncertainty

294

9 Norm based residual evaluation and threshold computation

s s ¯ K ¯ in (9.35) by J@ ¯ > K@ ¯ Step 0: Substitute J> Step1: Solve optimization problem min subject to (9=34) or (9=37) for % A 0> S A 0 and set = min Step 2: Set Mwk>UPV>2 =

( g>2 + x>2 ) ( g>2 + x>2 ) s s or Mwk>UP V>2 = = W Q

(9.38)

Example 9.1 In this example, we illustrate the application of the above algorithm to the threshold computation via the benchmark system DC motor DR300. In order to demonstrate that the proposed approach is also applicable for systems modelled in terms of transfer functions, our study is based on the input-output description of the DC motor DR300 given in Section 3.7.1. We assume that the gain of the nominal model is uncertain with J|x (v) =

e0 + v3 + d2 v2 + d1 v + d0

s s where 5 [ > ], = 10000, and moreover the measurement | is corrupted with a noise, |(v) = J|x (v)x(v) + 0=01g> kgk2 g>2 = 1=8= We now apply the residual generator developed in Example 5.9 to this system. It leads to ¸ ¸ £ ¤ x £ ¤ x h˙ = Jh + K 0=01O > u = Z h + 0 0=01 g g with

5

6 5 6 1 £ ¤ ¯ K ¯ = 7 0 8 (w) 1 0 = K = 7 0 8 =, H 0 0

By solving optimization problem min subject to (9=34) or (9=37) we get = 0=27= Under the assumption that x>2 = 2=1 and the evaluation time window W = 10v, the threshold is ﬁnally set to be Mwk>2>2 =

( g>2 + x>2 ) s = 0=33= 10

To verify the design result, simulations with dierent faults are made. Fig.9.2 and Fig.9.3 show the threshold and the responses of the evaluated residual signal to an actuator fault and a sensor fault.

9.5 Computation of Mwk>UP V>2

295

3 ||r||RMS Jth,RMS,2

2.5

||r||RMS

2

1.5

1

0.5

0

15

20

25

30 Time [s]

35

40

45

50

Fig. 9.2 Threshold and the evaluated residual signal: = 3100> iD = 0=05Y> occurred at w = 25v 1.6 ||r||RMS Jth,RMS,2

1.4 1.2

||r||RMS

1 0.8 0.6 0.4 0.2 0

15

20

25

30 Time [s]

35

40

45

50

Fig. 9.3 Threshold and the evaluated residual signal: = 3100> iV1 = 30=25Y> occurred at w = 25v

9.5.2 Computation of Jwk>UP V>2 for the systems with the polytopic uncertainty Now, we consider system (9.21)-(9.22) with the polytopic uncertainty. The following two theorems follow directly from Lemma 8.2 and its "discrete time version". Theorem 9.3 Given system (9.21)-(9.22) with the polytopic uncertainty and A 0> and suppose that {u (0) = 0> then ku(w)k2 ? kgu (w)k2

(9.39)

296

9 Norm based residual evaluation and threshold computation

if there exists S A 0 so that ;l = 1> · · · > o> 5 W W 6 (Du + Du>l ) S + S (Du + Du>l ) S (Hu>g + Hu>l ) (Fu + Fu>l ) W W 7 (Hu>g + Hu>l ) S L (Iu>g + Iu>l ) 8 ? 0= Fu + Fu>l Iu>g + Iu>l L (9.40) Theorem 9.4 Given system {u (n + 1) = (Du + Du ) {u (n) + (Hg>u + Hu ) gu (n) u(n) = (Fu + Fu ) {u (n) + (Ig>u + Iu ) gu (n) with the polytopic uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and A 0> and suppose that {u (n) = 0> then ku(n)k2 ? kgu (n)k2 if there exists a S A 0 so that ;l = 1> · · · > o> 5 0 S S (Du + Du>l ) S (Hu>g + Hu>l ) W 9 (Du + Du>l )W S S 0 (Fu + Fu>l ) 9 7 (Hu>g + Hu>l )W S 0 L (Iu>g + Iu>l )W 0 Fu + Fu>l Iu>g + Iu>l L

(9.41)

6 : : ? 0= (9.42) 8

Based on Theorems 9.3 and 9.4, we have Algorithm 9.2 Computation of Mwk>UPV>2 for the systems with the polytopic uncertainty Step1: Solve optimization problem min subject to (9=40) or (9=42) for S A 0 and set = arg (min ) Step 2: Set Mwk>UPV>2 =

( g>2 + x>2 ) ( g>2 + x>2 ) s s or Mwk>UPV>2 = = W Q

Example 9.2 We continue our study in Example 8.2, in which an FDF is designed for the benchmark system EHSA with polytopic model uncertainty. Our objective is now to compute the corresponding Mwk>UPV>2 via Algorithm 8.2. We assume that g>2 is bounded by 2 and the evaluation window is 5v= The computation of Step 1 gives = 0=98= Following it, we have

9.6 Computation of Mwk>shdn>shdn

Mwk>UPV>2 =

297

0=98 (2 + x>2 ) s = 5

In our simulation, x>2 is on-line estimated (see Section 9.8). In Fig.9.4, both the RMS value of the residual signal and the corresponding threshold are shown, where a fault in s sensor occurred at w = 3v= 14 12

||r||

RMS

Jth,RMS,2

Residual [Ŧ]

10 8 6 4 2 0 0

1

2

3

4

5

Time [s]

Fig. 9.4 Residual response and threshold

9.6 Computation of Jwk>shdn>shdn 9.6.1 Computation of Jwk>shdn>shdn for the systems with the norm bounded uncertainty We start with the (su!cient) condition for ku(w)kshdn ? kgu (w)kshdn under the assumption that i = 0> {u (0) = 0 and a given A 0= Theorem 9.5 Given system (9.21)-(9.22) with the norm bounded uncertainty and A 0> suppose that {u (0) = 0> kgu (w)kshdn 1> W (w)(w) L= Then ku(w)kshdn ? if there exists A 0> A 0> %1 A 0> %2 A 0> S A 0 so that

298

9 Norm based residual evaluation and threshold computation

6 ¯W J ¯ S Hg>u + %1 J ¯W K ¯ SH ¯ S Du + DWu S + S + %1 J W ¯WJ ¯ ¯WK ¯ 7 S + %1 K L + %1 K 0 8 ? 0 (9.43) Hg>u W ¯ S 0 %1 L H 6 L Fu Ig>u 1@2 Gs I ¯W J ¯ ¯W K ¯ : S %2 J FuW %2 J 0 : 0 (9.44) W W ¯ J ¯ ( )L %2 K ¯WK ¯ 8 0 %2 K Ig>u 5

5 9 9 7

1@2 (Gs I )W

0

0

%2 L

¯ J> ¯ K ¯ are given in (9.35). where H> The proof of this theorem can be achieved along with the lines in the proof of Lemma 9.2 provided by Scherer et al., together with the application of Lemma 8.1, see also the proof of the next theorem. Theorem 9.6 Given system {u (n + 1) = (Du + Du ) {u (n) + (Hu>g + Hu ) gu (n) u(n) = (Fu + Fu ) {u (n) + (Iu>g + Iu ) gu (n) with the norm bounded uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and A 0> and suppose that {u (0) = 0> W (n)(n) L> gWu (n)gu (n) 1 then ku(n)kshdn ? if there exist A 0> A 0> %1 A 0> %2 A 0> S A 0 so that 6 5 ¯ S Hu>g SH S S Du ¯W J ¯ %1 J ¯W K ¯ 9 DWu S (1 )S %1 J 0 : : A 0 (9.45) 9 W W ¯ W ¯ ¯ ¯ 7 Hu>g S %1 K J L %1 K K 0 8 ¯W S 0 0 %1 L H 5 6 1@2 L Fu Iu>g Gs I ¯W J ¯ ¯W K ¯ 9 : S %2 J FuW %2 J 0 9 : 0 (9.46) W W ¯ ¯ ¯WK ¯ 7 8 0 %2 K J ( )L %2 K Iu>g 1@2 (Gs I )W

0

0

%2 L

¯ J> ¯ K ¯ are given in (9.35). where H> Proof. Let Y ({u (n)) = {Wu (n)S {u (n) for some S A 0 and assume that Y ({u (n)) ?

for 0 ? ? 1> A 0= Note that Y ({(n)) satisfying

(9.47)

9.6 Computation of Mwk>shdn>shdn

Y ({u (n + 1)) + ( 1)Y ({u (n)) ? > Y ({(0)) = 0

299

(9.48)

is bounded by the solution of dierence equation Y ({u (n + 1)) = (1 )Y ({u (n)) + i.e. Y ({u (n)) ?

=

On the other side, matrix inequality " # ¤ £ (Du + Du )W S (Du + Du ) (Hg>u + Hu ) (Hg>u + Hu )W ¸ ¸ S 0 00 +(1 ) ? 0 0 0L

(9.49)

ensures that ;gW (n)g(n) Y ({u (n + 1)) + ( 1)Y ({u (n)) ? gW (n)g(n) =, Y ({u (n)) ?

=

Thus, (9.47) holds if (9.49) is satisﬁed. Note that ;gu > (n)> bounded by kgu kshdn 1 and W (n)(n) L respectively, ¡ ¢ uW (n)u(n) gWu (n)gu (n) + Y ({u (n) gWu (n)gu (n) W

=, u (n)u(n) ?

(9.50)

2

if (9.47) holds. Moreover, (9.50) can be expressed in terms of matrix inequality " # ¸ ¤ S 0 (Fu + Fu )W £ 1 F + F I + I = (9.51) u u g>u u W 0 ( ) L (Ig>u + Iu ) According to Lemma 8.1, we know that for 1 A 0> 2 A 0 W ¸ W¸ ¯ ¢ £ ¤ £ ¤ 1 J Du ¡ 1 W 1 ¯K ¯ ¯ ¯ Du Hg>u + J S 1H H W W ¯ Hg>u K 1 ¸ (1 )S 0 ? 0 L W ¸³ W¸ ´ 1 £ ¯ ¤ £ ¤ 1 J W 1 Fu ¯K ¯ Fu Ig>u + J L 2 Gs I (Gs I ) W W ¯ Ig>u K 2 ¸ S 0 0 ( ) L are su!cient for (9.49) and (9.51) respectively. Applying the Schur complement we have

300

9 Norm based residual evaluation and threshold computation

5

6 ¯H ¯W S 1 1 H Du Hg>u ¯W J ¯ 1J ¯W K ¯ 8 A 0 +, 7 DWu (1 )S 1 J 1 1 1 1 W ¯WJ ¯ ¯WK ¯ K Hg>u L K 1 1 6 5 ¯ S Hg>u SH S S Du 9 DW S (1 )S 1 J ¯W J ¯ 1J ¯W K ¯ 0 : : 9 u 1 1 A0 9 HW S 1 ¯W ¯ 1 ¯W ¯ K J L K K 0 : 8 7 g>u 1 1 1 ¯W S 0 0 H 1 L 5 6 L Fu Ig>u 1@2 Gs I ¯W J ¯ ¯W K ¯ 9 : S 1 J FuW 1 J 0 9 : 2 2 9 : 0= 1 ¯W ¯ 1 ¯W ¯ W K J ( )L K K Ig>u 0 7 8 2 2 W 1 1@2 0 0 (Gs I ) L 2

The theorem is ﬁnally proven by setting %1 =

1 1 > %2

=

1 2 =

t u

Algorithm 9.3 Computation of Mwk>shdn>shdn for the systems with the norm bounded uncertainty s s ¯ ¯ K ¯ in (9.35) by J@ ¯ > K@ Step 0: Substitute J> Step1: Solve optimization problem min subject to (9=43) (9=44) or (9=45) (9=46) for A 0> A 0> %1 A 0> %2 A 0> S A 0 and set = min Step 2: Set Mwk>shdn>shdn = ( g>4 + x>4 ) =

(9.52)

Example 9.3 In this example, we study the residual evaluation and threshold setting problems via the benchmark system LIP100. The same model like the one used in Example 8.1 is adopted. Further, we suppose the use of a residual generator designed by the uniﬁed solution with 5 6 6 5 1=1338 0=1718 0=3728 100 9 0=1718 0=0260 0=0565 : : 8 7 Orsw = 9 7 6=5071 0=0565 0=1226 8 > Yrsw = 0 1 0 = 001 84=1928 0=0167 0=0362 The application of Algorithm 9.3 leads to = 4=6762= On the assumption that g>4 = 0=0096> x>4 = 0=1 with an evaluation window of 10v, we have ﬁnally Mwk>shdn>shdn = 0=5125= In Fig.9.5, we show a simulation of the evaluated residual signals in comparison with the threshold set above, where a fault in the velocity sensor of the cart occurred at w = 6v=

9.6 Computation of Mwk>shdn>shdn

301

Evaluated Residual/Threshold

1.5

1

0.5

0 0

1

2

3

4

5

6

7

8

9

10

Time [s]

Fig. 9.5 Residual evaluation and comparison with the threshold by a sensor fault

9.6.2 Computation of Jwk>shdn>shdn for the systems with the polytopic uncertainty Consider system (9.21)-(9.22) with the polytopic uncertainty. Following Lemma 9.4 and its "discrete time version", we have Theorem 9.7 Given system (9.21)-(9.22) with the polytopic uncertainty and A 0> and suppose that {u (0) = 0> then ku(w)kshdn ? kgu (w)kshdn if there exist A 0> A 0 and S A 0 so that ;l = 1> · · · > o> " # (Du + Du>l )W S + S (Du + Du>l ) + S S (Hu>g + Hu>l ) ?0 (Hu>g + Hu>l )W S L 5 W 6 S 0 (Fu + Fu>l ) 7 0 ( ) L (Iu>g + Iu>l )W 8 0= Fu + Fu>l Iu>g + Iu>l L

(9.53)

(9.54)

Theorem 9.8 Given system {u (n + 1) = (Du + Du ) {u (n) + (Hu>g + Hu ) gu (n) u(n) = (Fu + Fu ) {u (n) + (Iu>g + Iu ) gu (n) with the polytopic uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and A 0> and suppose that {u (n) = 0> then ku(n)kshdn ? kgu (n)kshdn

302

9 Norm based residual evaluation and threshold computation

if there exist A 0> A 0 and S A 0 so that ;l = 1> · · · > o> 6 5 S S (Du + Du>l ) S (Hu>g + Hu>l ) W 8A0 7 (Du + Du>l ) S (1 )S 0 W (Hu>g + Hu>l ) S 0 L 6 5 S 0 (Fu + Fu>l )W 7 0 ( ) L (Iu>g + Iu>l )W 8 0= Fu + Fu>l Iu>g + Iu>l L

(9.55)

(9.56)

Algorithm 9.4 Computation of Mwk>shdn>shdn for the systems with the polytopic uncertainty Step1: Solve optimization problem min subject to (9=53) (9=54) or (9=55) (9=56) for A 0> A 0 and S A 0 and set = min Step 2: Set Mwk>shdn>shdn = ( g>4 + x>4 ) =

9.7 Computation of Jwk>shdn>2 9.7.1 Computation of Jwk>shdn>2 for the systems with the norm bounded uncertainty Consider system (9.21)-(9.22) with the norm bounded uncertainty. It is evident that gu (w) acts directly on u(w) via the crossing matrix Iu>g + Iu = The maximum change in u(w) caused by gu (w) via Iu>g + Iu is given by 2 ( g>4 + x>4 ) with sup ¯ ((w))

¡ ¢ ¡ ¢ ¯ W Iu>g + Gs I (w)K ¯ 2 L Iu>g + Gs I (w)K

¯ is given in (9.35). Using Lemma 8.1, can be determined by solving where K 2 2 = min 2 subject to 3 A0 6 5 L Iu>g Gs I W ¯WK ¯ 0 8 0= 7 Iu>g 2 L 3 K W W 0 3 L I Gs

(9.57)

Write u into two parts, u = u1 + u2 > u1 = (Fu + Fu ) {u > u2 = (Ig>u + Iu ) gu = Using Lemmas 9.1 and 8.1, we are able to compute the bound of the inﬂuence of gu on u1 > as stated in the following theorem.

9.7 Computation of Mwk>shdn>2

303

Theorem 9.9 Given system {˙ u = (Du + Du ) {u + (Hg>u + Hu ) gu u1 = (Fu + Fu ) {u with the norm bounded uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and 1 A 0> and suppose that {u (0) = 0> W (w)(w) L> then ku1 (w)kshdn ? 1 kgu (w)k2 if there exist 1 A 0> 2 A 0> S A 0 so that 6 5 W ¯W J ¯ S Hu>g + 1 J ¯W K ¯ SH ¯ Du S + S Du + 1 J W ¯WJ ¯ ¯WK ¯ 21 L 7 Hu>g S + 1 K 0 8?0 1K W ¯ S 0 1 L H 5 6 L Fu Gs I ¯W J ¯ 0 80 7 FuW S + 2 J W W 0 2 L I Gs

(9.58)

(9.59)

¯ J> ¯ K ¯ are given in (9.35). where H> The proof of this theorem is similar to the one of the next theorem. Theorem 9.10 Given system {u (n + 1) = (Du + Du ) {u (n) + (Hu>g + Hu ) gu (n) u1 (n) = (Fu + Fu ) {u (n) with the norm bounded uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and 1 A 0> and suppose that {u (0) = 0> W (n)(n) L then ku1 (n)kshdn ? 1 kgu (n)k2 if there exist 1 A 0> 2 A 0> S A 0 so that 5 6 ¯ S S Du S Hu>g SH ¯W J ¯S ¯W K ¯ 9 DWu S 1 J 1 J 0 : 9 W : W W ¯ J ¯ 1 K ¯ K ¯ 21 L 0 8 ? 0 7 Hu>g S 1 K ¯W S 0 0 1 L H 6 5 Gs I L Fu ¯W J ¯ 0 80 7 FuW S + 2 J 0 2 L I W GsW ¯ J> ¯ K ¯ are given in (9.35). where H>

(9.60)

(9.61)

304

9 Norm based residual evaluation and threshold computation

Proof. Let Y ({u (n)) = {Wu (n)S {u (n) for some S A 0= Considering that Y ({u (n + 1)) Y ({u (n)) ? 21 kgu (n)k2

(9.62)

yields Y ({u (n)) ? 21

n1 X

kgu (m)k2

m=0

we have u1 (n) ? 21

n1 X

2

kgu (m)k

m=0

provided that

(Fu + Fu )W (Fu + Fu ) S=

We now express (9.62) in terms of matrix inequality: " # ¸ W ¤ £ (Du + Du ) S 0 ? 0= W S Du + Du Hu>g + Hu 0 21 L (Hu>g + Hu )

(9.63)

(9.64)

Using Lemma 8.1 leads to a su!cient condition for (9.64) as well as (9.63), respectively 5 6 ¯ S Hu>g SH S S Du ¯W J ¯S ¯W K ¯ 9 DWu S 1 J 1 J 0 : : 9 W W W ¯ J ¯ 1 K ¯ K ¯ 21 L 0 8 ? 0 7 Hu>g S 1 K ¯W S 0 0 1 L H 6 5 Gs I L Fu ¯W J ¯ 0 80 7 FuW S + 2 J 0 2 L I W GsW for some 1 A 0> 2 A 0=

t u

Algorithm 9.5 Computation of Mwk>shdn>2 for the systems with the norm bounded uncertainty s s ¯ ¯ K ¯ in (9.35) by J@ ¯ > K@ Step 0: Substitute J> Step 1: Solve optimization problem (9.57) for 2 Step 2: Solve optimization problem min 1 subject to (9=58) (9=59) or (9=60) (9=61) for 1 A 0> 2 A 0> S A 0 and set 1 = min 1 Step 3: Set Mwk>shdn>2 = 1 ( g>2 + x>2 ) + 2 ( g>4 + x>4 ) =

(9.65)

9.7 Computation of Mwk>shdn>2

305

Example 9.4 Under the exactly same conditions with the ones of Example 9.3, we now determine Mwk>shdn>2 for the benchmark system LIP100 with the norm bounded uncertainty. Under the application of Algorithm 9.5 we get 1 = 3=3197> 2 = 9=0651 × 1011 and further, on the assumption that x>4 = 0=1000> g>4 = 0=0099> x>2 = 0=3162> g>2 = 0=0163 Mwk>shdn>2 = 3=3197(0=3162 + 0=0163) +9=0651 × 1011 (0=100 + 0=0099) = 1=1040= In Fig.9.6, we see simulation of the evaluated residual signals in comparison

Evaluated Residual / Threshold

1.5

1

0.5

0 0

1

2

3

4

5

6

7

8

9

10

Time [s]

Fig. 9.6 Residual evaluation and comparison with the threshold by a sensor fault

with the threshold set above, where a fault in the velocity sensor of the cart occurred at w = 6v= 9.7.2 Computation of Jwk>shdn>2 for the systems with the polytopic uncertainty We now study the computation of Mwk>shdn>2 for system (9.21)-(9.22) with the polytopic uncertainty. In order to evaluate the inﬂuence of gu (w) on u(w) via the crossing matrix Iu>g + Iu > we propose to solve the following optimization problem: ﬁnding 2 such that ;l = 1> · · · > o> W

(Iu>g + Iu>l ) (Iu>g + Iu>l ) 2 L=

(9.66)

306

9 Norm based residual evaluation and threshold computation

For the evaluation of the inﬂuence of L2 norm of gu on u1 we have the following two theorems which are a straightforward extension of Lemma 9.3 and its "discrete time version". Theorem 9.11 Given system {˙ u = (Du + Du ) {u + (Hg>u + Hu ) gu u1 = (Fu + Fu ) {u with the polytopic uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and 1 A 0> and suppose that {u (0) = 0> then ku1 (w)kshdn ? 1 kgu (w)k2 if there exist S A 0 so that ;l = 1> · · · > o> # " (Du + Du>l )W S + S (Du + Du>l ) S (Hu>g + Hu>l ) ?0 (Hu>g + Hu>l )W S L ¸ W S (Fu + Fu>l ) 0= Fu + Fu>l L

(9.67) (9.68)

Theorem 9.12 Given system {u (n + 1) = (Du + Du ) {u (n) + (Hu>g + Hu ) gu (n) u1 (n) = (Fu + Fu ) {u (n) with the polytopic uncertainty, where all system matrices are identical with the ones used in (9.21)-(9.22), and 1 A 0> and suppose that {u (n) = 0> then ku1 (n)kshdn ? 1 kgu (n)k2 if there exists S A 0 so that ;l = 1> · · · > o> 6 5 S S (Du + Du>l ) S (Hu>g + Hu>l ) W 8?0 7 (Du + Du>l ) S S 0 W 2 (Hu>g + Hu>l ) S 0 1 L ¸ L Fu + Fu>l 0= W (Fu + Fu>l ) S

(9.69)

(9.70)

Algorithm 9.6 Computation of Mwk>shdn>2 for the systems with the polytopic uncertainty Step1: Solve optimization problem (9.66) 2 Step 2: Solve optimization problem min 1 subject to (9=67) (9=68) or (9=69) (9=70) for S A 0 and set 1 = min 1

9.8 Threshold generator

307

14

||r||

RMS

12

J

th,RMS,2

Residual [Ŧ]

10 8 6 4 2 0 0

1

2

3

4

5

Time [s] Fig. 9.7 Evaluated residual signal and threshold

Step 3: Set Mwk>shdn>2 = 1 ( g>2 + x>2 ) + 2 ( g>4 + x>4 ) = Example 9.5 Similar to Example 9.2, we now compute Mwk>shdn>2 for the benchmark system EHSA via Algorithm 9.6: Step 1: 2 = 1=5168 Step 2: 1 = 1=0006 Step 3: Mwk>shdn>2 = 3=5095 + 1=0006 x>2 + 1=5083 x>4 for g>4 = 1 Again, by the simulation the on-line estimation of x>2 and x>4 is used. In Fig.9.7, both the RMS value of the residual signal and the corresponding threshold are shown, where a fault in s sensor occurred at w = 3v=

9.8 Threshold generator The thresholds derived in the last sections have in common that they are constant and a function of a bound on the input vector x. Since x is generally on-line available during process operation, substituting the bound on x by an on-line computation would considerably reduce the threshold size and thus increase the fault detection sensitivity. Those thresholds which are driven by the system input signals, as shown in Fig.9.1, are known as adaptive thresholds or threshold selectors. Analog to the concept of residual evaluator, we call them threshold generator. While the bound on the peak of x>4 can be easily replace by the instantaneous value

308

9 Norm based residual evaluation and threshold computation

kx(w)k =

q q xW (w)x(w) or kx(n)k = xW (n)x(n)

x>2 will be approximated by kx(w)k2>W

3 w+W 41@2 3 41@2 Z Q X 2 2 =C ku( )k g D or kx(n)k2>Q = C ku(n + m)k D m=1

w

in an iterative way or with a weighting, e.g. 2

2

2

kx(n)k2>m+1 = kx(n)k2>m + ku(n + m + 1)k or kx(n)k22>m+1 = kx(n)k22>m + ku(n + m + 1)k2 with 0 ? 1= The three kinds of constant thresholds introduced in the last sections, Mwk>UPV>2 > Mwk>shdn>shdn and Mwk>shdn>2 given by (9.38), (9.52) and (9.65) respectively, will be replaced by the threshold generators g>2 j (w) = s + kx(w)kUPV or Mwk>UPV>2 W g>2 j (n) = s + kx(n)kUPV Mwk>UPV>2 Q j Mwk>shdn>shdn (w) = g>4 + kx(w)k or j (w) Mwk>shdn>2

(9.71)

(9.72)

j (n) = g>4 + kx(n)k Mwk>shdn>shdn = 1 g>2 + 2 g>4 + 1 kx(w)k2>W + 2 kx(w)k

j (n) Mwk>shdn>2

=

1 g>2

+

2 g>4

+

1

kx(n)k2>Q +

2

or

(9.73)

kx(n)k

where the superscript j stands for generator. It is interesting to notice that the threshold generators consist of two parts: a constant part and a time varying part. This time varying part depends on the instantaneous energy change in the input signals. In other words, under dierent operating conditions, expressed in terms of the input signals, the threshold will be dierent. In this context, the threshold generator is an adaptive threshold. Since j j j Mwk>UPV>2 > Mwk>shdn>shdn Mwk>shdn>shdn > Mwk>shdn>2 Mwk>shdn>2 Mwk>UPV>2

it is clear that substituting the thresholds by the corresponding threshold generators will enhance the fault detection sensitivity. Example 9.6 In this example, we replace the constant threshold computed in Example 9.1 by a threshold generator and repeat the simulation. It follows from (9.71)that j (w) = 0=15 + 0=27 kx(w)kUPV Mwk>UPV>2

9.8 Threshold generator

309

0.5 ||r||RMS

0.45

Jgth,RMS,2

u

0.4 0.35

||r||RMS

0.3 0.25 0.2 0.15 0.1 0.05 0

15

20

25

30 Time [s]

35

40

45

50

Fig. 9.8 Threshold generator and the evaluated residual signal: = 3100> iD = 0=006Y> occurred at w = 25v ||r||RMS

Jgth,RMS,2

u

0.5

||r||RMS

0.4

0.3

0.2

0.1

0

15

20

25

30 Time [s]

35

40

45

50

Fig. 9.9 Threshold generator and the evaluated residual signal: = 3100> iV1 = 30=125Y> occurred at w = 25v

where kx(w)kUP V will be on-line computed. Fig.9.4 and Fig.9.5 show the threshold generator and the responses of the evaluated residual signal to an actuator fault and a sensor fault. Comparing Fig.9.8 and Fig.9.9 with Fig.9.2 and Fig.9.3, we clearly see that the threshold generator scheme delivers higher fault detectability, also in case of a large size input signal.

310

9 Norm based residual evaluation and threshold computation

9.9 Notes and references Although the norm based residual evaluation was initiated by Emami-naeini et al. [48] almost twenty years ago, only few research results on this topic have been published, see for instance [39, 54, 77, 85, 123]. On the other side, in practice limit monitoring and trend analysis schemes are very popular, where the determination of thresholds plays a central role. It is the state of the art in practice that thresholds are generally determined based on experiences or by means of real tests and simulation. The results and algorithms presented in this chapter are a considerable extension of the results reported in [34]. They have been achieved in the norm based framework and therefore may lead to a conservative threshold setting. Even though, they provide the system designer with a reliable and reasonable estimate of the value range of the thresholds. It can save a great number of real tests and therefore are valuable both from the technical and economic viewpoint. The major tools used for our study in this chapter is the robust control theory and LMI technique. We refer the reader to [14, 130] as well as [160] for the needed knowledge and computation skills in this area. The proofs of Lemmas 9.1, 9.2 on the generalized H2 norm and peak-topeak gain can be found in [130] and as well as in [14]. The extension of these results to the systems with polytopic model uncertainties, as given in Lemmas 9.3 and 9.4, is schematically described in [14]. A major conclusion of this chapter is that for dierent application purposes dierent residual evaluation functions and correspondingly dierent induced norms should be used. This conclusion also reveals the deﬁcit in the current research. The eorts for achieving an optimization without considering the evaluation function and the associated threshold computation can result in poor FDI performance. Research on the optimization schemes under performance indices dierent from the H4 or H2 norm is urgently demanded in order to ﬁll in the gap between the theoretical study and practical applications.

10 Statistical methods based residual evaluation and threshold setting

10.1 Introduction The objective of this chapter is to present some basic statistical methods which are typically used for residual evaluation, threshold setting and decision making. In working with this chapter, the reader will observe that the way of problem handling and the mathematical tools used for the problem solution are signiﬁcantly dierent from those presented in the previous chapters. We shall ﬁrst introduce some elementary statistical testing methods and the basic ideas behind them. Although no dynamic process is taken into account, those methods and ideas build the basis for the study in the sequent sections. A further section is devoted to the criteria for the selection of thresholds. In the last section, we shall brieﬂy deal with residual evaluation problems for stochastic dynamic processes, as sketched in Fig.10.1.

Fig. 10.1 Schematic description of statistic testing based residual evaluation and decision making

312

10 Statistical methods based residual evaluation and threshold setting

10.2 Elementary statistical methods In this section, a number of elementary statistical methods will be introduced. 10.2.1 Basic hypothesis test The problem under consideration is formulated as follows: Given a model | =+5R with 5 N (0> 2 ) (i.e. normally distributed with zero mean and variance 2 )> = 0 or || A 0> a number of samples of |> |1 > · · · |Q > and a constant A 0 (the so-called signiﬁcance level), ﬁnd a threshold Mwk such that prob {|¯ | | A Mwk | = 0} ? > |¯ =

Q 1 X |l Q l=1

(10.1)

| | A Mwk under where prob{|¯ | | A Mwk | = 0} denotes the probability that |¯ condition = 0= It is well-known that the probability prob{|¯ | | A Mwk | = 0} is the false alarm rate if the following decision rule is adopted: |¯ | | Mwk : = 0 (K0 > null hypothesis) |¯ | | A Mwk : 6= 0 (K1 > alternative hypothesis).

(10.2) (10.3)

From the viewpoint of fault detection, the above mathematical problem is the answer to the fault detection problem: Given system model, how can we select the threshold towards a reliable detection of the change (fault) in based on the samples of the output |? In the problem formulation and the way of approaching the solution we can observe some key steps: • the objective is formulated in the statistical context: the probability of a false decision, i.e. prob{|¯ | | A Mwk | = 0} > should be smaller than the given signiﬁcance level Q P |l > • an estimation of the mean value of | based on the samples> |¯ = Q1 l=1

is included in the testing process • the decision is made based on two hypotheses: K0 > the null hypothesis, means no change in > while K1 > the alternative hypothesis, means a change of = Throughout this chapter, these three key steps to the problem solutions will play an important role. The solutions of the above-formulated problem are summarized into two algorithms, depending on whether is known. For details, the interested reader is referred, for instance, to the textbook by Lapin. Algorithm 10.1 Computing Mwk if is known

10.2 Elementary statistical methods

313

Step 1: Determine the critical normal deviate }@2 using the table of critical normal deviate values or the table of standard normal distribution, i.e. ª © (10.4) prob } A }@2 = @2 Step 2: Set Mwk

Mwk = }@2 s Q since |¯ is normally distributed with

(10.5)

2 = Q

H (¯ | ) = 0> ydu (¯ |) =

(10.6)

Algorithm 10.2 Computing Mwk if is unknown Step 1: Determine w@2 using the table of w distribution with degree of freedom equal Q 1, i.e. ª © (10.7) prob w A w@2 = @2 Step 2: Set Mwk

Q P

Mwk

v = w@2 s > v2 = Q

where w=

(|l |¯)

l=1

Q 1

2

(10.8)

|¯ s v@ Q

satisﬁes Student distribution with the degree of freedom equal to Q 1= Remark 10.1 The idea behind the above algorithm is an estimation of the variance by v= It is clear that for the purpose of change detection, following on-line computation (evaluation of the samples of |) is needed: In case that is known it is Q 1 X |l |¯ = Q l=1 otherwise |¯ and

v=

v u Q uP u (| |¯)2 t l=1 l Q 1

=

314

10 Statistical methods based residual evaluation and threshold setting

10.2.2 Likelihood ratio and generalized likelihood ratio Likelihood ratio (LR) methods are very popular in the framework of change detection. In this subsection, we brieﬂy introduce two basic versions of these methods. We refer the interested reader to the excellent monograph by Basseville and Nikiforov for details on the topics introduced in this and the next subsections. Given the system model ½ 0 = 0> K0 (no change) 2 | = + > 5 N (0> )> = 1 > K1 (change but constant) the log likelihood ratio for data |l is deﬁned by i (|l )2 s (|l ) 1 1 h 2 2 = 2 (|l 0 ) (|l 1 ) > s (|l ) = s v(|l ) = ln 1 h 22 s0 (|l ) 2 2 (10.9) where s (|l ) is the probability density of | for | = |l = The basic idea of the LR methods can be clearly seen from the decision rule ½ ? 0> K0 ( = 0) is accepted = v(|l ) = A 0> K1 ( = 1 ) is accepted Note that v(|l ) A 0 means s1 (|l ) A s0 (|l )> i.e. given |l the probability of = 1 is higher than the one of = 0 = Thus, it is reasonable to make a decision in favour of K1 = In case that Q samples of |> |l > l = 1> · · · > Q , are available, the (log) LR is deﬁned by Q i s1 (|l ) 1 Xh 2 2 = 2 (|l 0 ) (|l 1 ) s0 (|l ) 2 l=1 l=1 l=1 ¶ Q μ 1 + 0 1 0 X = (10.10) | = l 2 l=1 2

V1Q =

Q X

vl =

Q X

ln

We distinguish two dierent cases: 1 is known and 1 is unknown. Detection when 1 (A 0) is known and 0 = 0 Note that ¶ X ¶ Q μ Q μ Q X 1 X 1 + 0 1 1 Q = A 0 +, |l |l |l A V1 A 0 +, 2 2 Q l=1 2 l=1 l=1 (10.11) and moreover ¶ μ Q 2 1 X = |l 5 N 0> Q l=1 Q Thus, given allowed false alarm rate > the following algorithm can be used to compute the threshold.

10.2 Elementary statistical methods

315

Algorithm 10.3 Computing Mwk if 1 is known Step 1: Determine } i.e.

1 2

using the table of standard normal distribution, prob {} A }> } =

Step 2: Set Mwk

Mwk = } s = Q

(10.12)

It is very interesting to see the interpretation of condition (10.11). Recall Q P that Q1 |l gives in fact an estimate of the mean value of | based on the l=1

available samples. (10.11) tells us: if the estimate of the mean value is larger than 21 , then a change is detected. This is exactly what we would instinctively do in such a situation. Detection when 1 is unknown and 0 = 0 In practice, it is the general case that 1 is unknown. For the purpose of detecting change in with unknown 1 > the so-called generalized likelihood ratio (GLR) method was developed, where 1 is replaced by its maximum likelihood estimate. The maximum likelihood estimate of 1 is an estimate achieved under the cost function that the LR is maximized. Thus, the maximum LR as well as the maximum likelihood estimate of 1 are the solution of the following optimization problem Q i 1 Xh 2 2 (| ) | l 1 l 1 2 2 l=1 5 Ã 6 !2 Q X 1 1 = max 2 7 |l Q (1 |¯)2 8 =, 1 2 Q l=1

max V1Q = max 1

ˆ1 =

arg max V1Q 1

+, max V1Q 1

Q 1 X 1 = |¯ = |l > max V1Q = 2 1 Q l=1 2 Q

1 2 (¯ |) = = 2 2 @Q

(10.13) Ã Q !2 X |l l=1

(10.14)

It is of practical interest to notice that • the maximum likelihood estimate of 1 is the estimate of the mean value, Q P |¯ = Q1 |l l=1

2

• the maximum LR, 221@Q (¯ | ) > is always larger than zero and thus • a suitable threshold should be established to avoid high false alarm rate. Note that the distribution of 21@Q (¯ | )2 is "2 (1). Therefore, given allowed false alarm rate > the following algorithm can be used to compute the threshold.

316

10 Statistical methods based residual evaluation and threshold setting

Algorithm 10.4 Computing Mwk if 1 is unknown Step 1: Determine " using the table of "2 distribution with 1 degree of freedom, i.e. prob {" A " } = Step 2: Set Mwk Mwk = " @2=

(10.15)

For both the cases, the decision rule is ½ ? Mwk > K0 ( = 0) is accepted V1Q = A Mwk > K1 ( 6= 0) is accepted which ensures that false alarm rate is not larger than = It has been theoretically proven that the LR based change detection leads to a minimization of the missed detection rate for a given false alarm rate. For the implementation of the above-described LR methods, computation of V1Q is needed, which is Q 1 X |l |¯ = Q l=1 in case that 1 is known and otherwise Ã Q !2 X 1 |l = 2 2 Q l=1 10.2.3 Vector-valued GLR In this subsection, the generalized likelihood ratio (GLR) test will be presented in the vector form. Given the system model ½ 0 > no change | = + > 5 N (0> )> = 1 > change where |> > 5 Rq and the probability density of Gaussian vector | is deﬁned by W 1 1 1 (10.16) h 2 (|) (|) = s> (|) = p q (2) det( ) Hence, the LR for given vector | satisﬁes i 1h s (|) = v(|) = ln 1 (| 0 )W 1 (| 0 ) (| 1 )W 1 (| 1 ) = s0 (|) 2 Under the assumption that 0 = 0 and Q (vector-valued) samples of |> |n > n = 1> · · · > Q> are available, the maximum likelihood estimate of 1 and the maximum LR are given by

10.2 Elementary statistical methods

317

# "Q Q X X 1 W max V1Q = max |nW 1 |n (|n 1 ) 1 (|n 1 ) = 1 1 2 n=1 n=1 Ã !# "Q Q Q X X X 1 W 1 W 1 1 W 1 W 1 |n |n |n |n Q 1 1 21 |n max 1 2 Q l=1 n=1

= max 1

n=1

Q i 1 X 1 h W 1 | 1 )W 1 (¯ | 1 ) > |¯ = |n Q |¯ |¯ Q (¯ 2 Q l=1

=, ˆ 1 = arg max V1Q = |¯ =, max V1Q = 1

1

Q W 1 |¯ |¯= 2

(10.17)

Once again, we can see that also in the vector-valued case, the maximum Q P likelihood estimate of 1 is |¯ = Q1 |n = Since |¯ is a q-dimensional vector l=1

with |¯ 5 N (0> @Q ) Q |¯W 1 |¯ is distributed as a "2 (q)= As a result, the following algorithm can be used for computing the threshold if the decision rule is deﬁned as ½ ? Mwk > K0 ( = 0) is accepted Q = V1 = A Mwk > K1 ( 6= 0) is accepted Algorithm 10.5 Computing Mwk if vector 1 is unknown Step 1: Determine " using the table of "2 distribution with q degrees of freedom, i.e. prob {" A " } = Step 2: Set Mwk Mwk = " @2=

(10.18)

10.2.4 Detection of change in variance Given the system model | = + 5 R> 5 N (0> 20 ) a number of samples of |> |1 > · · · |Q and a constant A 0 (the signiﬁcance level), ﬁnd a statistic and a threshold Mwk such that the change in variance (assume that 2 A 20 ) can be detected with a false alarm rate smaller than = We present two testing methods for our purpose. Testing with the "2 statistic given by Lapin The statistic

318

10 Statistical methods based residual evaluation and threshold setting Q P

"Q 1

(Q 1)v2 2 := >v = 20

(|l |¯)2

l=1

Q 1

(10.19)

has the standard "2 sampling distribution with the degree of freedom equal to Q 1= Thus, given > the threshold is determined by (using the standard "2 distribution table) ª © (10.20) Mwk = "2 > prob "2 A "2 = = The decision rule is

½

"Q 1

=

? Mwk > K0 (2 20 ) is accepted = A Mwk > K1 (2 A 20 ) is accepted

Testing using GLR given by Basseville and Nikiforov For this purpose, ﬁrst consider LR which is described by V1Q =

Q X

vl =

l=1

Q X

ln

l=1

Q Q 0 s1 (|l ) 1 X 2 1 X 2 = Q ln + 2 |l 2 | = s0 (|l ) 1 2 0 l=1 2 1 l=1 l

Thus, solving the optimization problem Ã ! Q Q X X 1 1 0 max V1Q = max Q ln + |2 2 | 2 =, 1 1 1 2 20 l=1 l 21 l=1 l Q 1 X 2 | 1 Q l=1 l !# " Ã Q Q Q 1 X 2 1 X 2 = ln 0 |l | + 2 1 + ln 2 Q l=1 2 0 l=1 l

ˆ 21 = arg max V1Q = V1Q

(10.21)

gives the GLR. 10.2.5 Aspects of on-line realization The above-presented detection methods can be realized on-line in dierent ways. On-line implementation with a ﬁxed sample size N In this case, the decision rule, for instance for the GLR test, is deﬁned by (n+1)Q VnQ+1 (n+1)Q

VnQ+1

½ = =

? Mwk > K0 ( = 0) is accepted 6 0) is accepted A Mwk > K1 ( =

Q X l=1

(n+1)Q

vnQ+l =

X

l=nQ +1

ln

s1 (|l ) = s0 (|l )

10.2 Elementary statistical methods

319

The observation will be stopped after the ﬁrst sample of size Q for which the decision is made in favor of K1 ( 6= 0). Note that in this case the maximal (possible) delay is Q × Wv > where Wv is the sampling time. On-line implementation in a recursive manner In practice, for the reason of achieving a su!ciently large sample size and continuously computing the LR, GLR is often realized in a recursive form. For this purpose, we deﬁne Ã n ! Ã n !W X X 1 W 1 |>n |>n |l 1 |l = 2n l=1 l=1

1 Vn = 2n |>n =

n X

|l > n = 1> · · · >

n=l

and write V n+1 into Ãn+1 ! Ãn+1 !W X X 1 n+1 1 = |l |l V 2(n + 1) l=1 l=1 ¸ ¢ ¡ W 1 1 1 1 W W = |>n 2 |>n |n+1 + |n+1 1 |>n + 1 |n+1 2 (n + 1) (n + 1) μ ¶W n 1 1 n V + |>n+1 |n+1 = 1 |n+1 = (n + 1) (n + 1) 2 Based on it, the following recursive calculation is introduced: ¶W μ 1 V n+1 = V n + (1 ) |>n+1 |n+1 1 |n+1 > V 0 = 0 (10.22) 2 |>n+1 = |>n + |n+1 where 0 ? ? 1 and acts as a forgetting factor. In order to avoid |>n being too large, |¯n := n1 |>n can be replaced by |n + (1 )|n+1 > |¯0 = 0= |¯n+1 = ¯ As a result, (10.22) can then be written into ¶W μ (1 ) |n+1 1 |n+1 V n+1 = V n + |¯n+1 2

(10.23)

and, in case that 1 is very small, furthermore W 1 |n+1 = V n+1 = V n + |¯n+1

Setting a counter

(10.24)

320

10 Statistical methods based residual evaluation and threshold setting

An eective way to make the decision making procedure to be robust against strong noises is to set a counter. Let L{V n AMwk } be an indicator that the GLR is larger than the threshold, i.e. ½ 1> V n A Mwk = L{V n AMwk } = 0> V n ? Mwk Then the stopping rule is set to be ( wd = min n :

Q X

) L{V nl AMwk } A

l=0

where is a threshold for the number of the crossings of threshold Mwk =

10.3 Criteria for threshold computation In the last section, the threshold is determined in such a way that the admissible false alarm rate will not be exceeded. In this section, we ﬁrst study this criterion from the theoretical viewpoint and then present a number of dierent criteria for the threshold computation given by Mcdonough and Whalen. 10.3.1 The Neyman-Pearson criterion Let us introduce notations Si = prob(G1 | K0 )> Sp = prob(G0 | K1 ) for the probability that decision for K1 is made (G1 ) in case of no change (K0 ) and the probability that decision for K0 is made (G0 ) as the change is present (K1 )> respectively, i.e. false alarm rate = Si missed detection rate = Sp = The scheme adopted in the last section for the threshold computation can then be formulated as: Given an admissible false alarm rate Si>d > the threshold should be selected such that Si = prob(G1 | K0 ) Si>d =

(10.25)

It is also desired that the missed detection rate is minimized under the condition that (10.25) is satisﬁed. It leads to the following optimization problem:

10.3 Criteria for threshold computation

321

min Sp = min prob(G0 | K1 ) subject to Si = prob(G1 | K0 ) Si>d = (10.26) Note that Sp = prob(G0 | K1 ) = 1 prob(G1 | K1 ) and Sg :=prob(G1 | K1 ) means the detection rate, thus optimization problem (10.26) can be equivalently reformulated as: max Sg = max prob(G1 | K1 ) subject to Si = prob(G1 | K0 ) Si>d = (10.27) Optimization problem (10.27) is called Neyman-Pearson criterion. On the assumptions that • the conditional densities s0 (|) := s (K0 | |) > s1 (|) := s (K1 | |) are known • there exist no unknown parameters in s0 (|)> s1 (|) the so-called Neyman-Pearson Lemma provides a solution to the optimization problem (10.27), which can be roughly stated as follows: Given s0 (|) (6= 0)> s1 (|) and Si>d • if s1 (|)@s0 (|) ? Mwk > choose K0 • if s1 (|)@s0 (|) A Mwk > choose K1 • Mwk is determined by prob(s1 (|)@s0 (|) A Mwk | K0 ) = Si>d =

(10.28)

Following Neyman-Pearson Lemma, it becomes clear that the LR method introduced in the last section for the case of both 0 > 1 being known ensures a maximum fault detection rate. Moreover, the GLR provides us with a suboptimal solution, since s1 (|) contains a unknown parameter (1 is unknown and estimated). 10.3.2 Maximum a posteriori probability (MAP) criterion Consider again the system model ½ | = + > 5 N (0> )> =

0 > no change = 1 > change

Assume that a posteriori probability is available, i.e. S0 = prob ( = 0 ) > S1 = prob ( = 1 ) are known then it turns out

322

10 Statistical methods based residual evaluation and threshold setting

s1 (|) = s (K1 | |) =

s| (| | K1 )S1 s| (| | K0 )S0 > s0 (|) = s (K0 | |) = = s| (|) s| (|)

Now consider the (log) LR v(|) = ln

s| (| | K1 )S1 S1 s| (| | K1 ) s1 (|) = ln + ln = = ln s0 (|) s| (| | K0 )S0 s| (| | K0 ) S0

The MAP criterion results in a decision in favour of K1 if v(|) = ln ss10 (|) (|) A 0> otherwise K0 = Thus, following the MAP criterion, the threshold is computed by solving S1 s| (Mwk | K1 ) + ln = 0= (10.29) ln s| (Mwk | K0 ) S0 For instance, for the above-given system model we have ¶ μ 1 0 1 + 0 s| (Mwk | K1 ) S1 S1 + ln + ln Mwk = =, ln 2 s| (Mwk | K0 ) S0 2 S0 ¶ μ s| (Mwk | K1 ) S0 S1 1 0 1 + 0 ln + ln = ln Mwk = 0 +, =, 2 s| (Mwk | K0 ) S0 2 S1 2 S0 1 + 0 = Mwk = ln + 1 0 S1 2 To determine the false alarm rate, the probability prob(| A Mwk | K0 ) will be calculated. 10.3.3 Bayes’ criterion Bayes’ criterion is a general criterion which allows us to make a decision among a number of hypotheses. For the sake of simplicity, we only consider the case with two hypotheses, K0 and K1 = The basic idea of the Bayes’ criterion consists in the introduction of a cost function of the form M = F00 S (G0 | K0 )S0 + F10 S (G1 | K0 )S0 + F01 S (G0 | K1 )S1 +F11 S (G1 | K1 )S1 (10.30) where S0 =prob(K0 ) > S1 =prob(K1 ) and are assumed to be known, Flm > l> m = 0> 1, is the "cost" for choosing decision Gl when Km is true. Thus, it is reasonable to assume that ;l> m Flm>l6=m A Fll = The decision rule is then derived based on the minimization of the cost function M= For this purpose, (10.30) is re-written into

10.3 Criteria for threshold computation

323

M = F00 (1 S (G1 | K0 )) S0 + F01 (1 S (G1 | K1 )) S1 + F10 S (G1 | K0 )S0 + F11 S (G1 | K1 )S1 Z = F00 S0 + F01 S1 + [S0 (F10 F00 ) s0 (|) + S1 (F11 F01 )s1 (|)] g| where s0 (|)> s1 (|) stand for the densities of K0 > K1 = It turns out that S0 (F10 F00 ) s0 (|) + S1 (F11 F01 )s1 (|) ? 0 =, S0 (F10 F00 ) s1 (|) A s0 (|) S1 (F01 F11 ) will reduce the cost function. As a result, the threshold is deﬁned Mwk = ln

S0 F10 F00 + ln S1 F01 F11

and the decision rule is ( F10 F00 0 s1 (|) A Mwk = ln S S1 + ln F01 F11 > decision for K1 = v(|) = ln S0 F10 F00 s0 (|) ? Mwk = ln S1 + ln F01 F11 > decision for K0 10.3.4 Some remarks It is evident that the main dierence among the above-introduced methods consists in the fact that using Neyman-Pearson strategy the prior probabilities of K0 > K1 are not needed, while Bayes’ criterion and MAP criterion are based on them. It is remarkable that all three methods lead to the computation of (log) LR. Neyman-Pearson scheme is mostly suitable for the solution of the fault detection problem formulated as: Given an admissible false alarm rate, ﬁnd a threshold and a decision rule such that the missed detection rate is minimized, although the GLR may only give a sub-optimal solution. On the other side, the Neyman-Pearson scheme is a traditional statistical method whose core is performing hypotheses tests towards decisions consistent with sample evidence. In against, Bayes’ and MAP schemes allow to make a decision even if the usual sample data are not available. In particular, Bayes’ criterion takes into account the possible "costs" for a decision. This will make the whole decision procedure more reasonable. It should be pointed out that the Bayes’ scheme can also be extended to the case where the probabilities of K0 > K1 are not available. In this case, the worst case due to the unknown S0 > S1 = 1 S0 should be taken into the optimization procedure. For instance, instead of minimizing M in (10.30) a so-called minmax optimization problem is solved: max

max

min

S0 >S1 =1S0 Flm

min M =

S0 >S1 =1S0 Flm

¸ F00 S (G0 | K0 )S0 + F10 S (G1 | K0 )S0 + = F01 S (G0 | K1 )S1 + F11 S (G1 | K1 )S1

324

10 Statistical methods based residual evaluation and threshold setting

10.4 Application of GLR testing methods The methods presented in this section are in fact the application and extension of the above introduced methods to the solution of fault detection problems met in linear dynamic systems. 10.4.1 Kalman ﬁlter based fault detection Consider an LTI system given by {(n + 1) = D{(n) + Ex(n) + Hi i (n) + (n) |(n) = F{(n) + Gx(n) + Ii i (n) + (n)

(10.31) (10.32)

where (n) 5 N (0> )> (n) 5 N (0> ) are independent white noises. Using a steady Kalman ﬁlter introduced in Section 7.2 an innovation process u(n) = |(n) F { ˆ(n) Gx(n)

(10.33)

is created with white Gaussian process u(n) 5 N (0> u )> u = + F\ F W > when i (n) = 0= We are interested in the problem of detecting those faults whose energy level is higher than a tolerant limit Oi > i.e. v u ½ v u 1 X Oi > Kr (fault-free) (10.34) ki (n)kv = t i W (n l)i (n l) = A Oi > K1 (fault) v + 1 l=0 by using u(n) as the residual signal and on the assumption that u(n l)> l = 0> · · · > v> are available for the detection purpose. Next, we apply the GLR scheme to solve this problem. Write the available residual data into a vector 5 6 u(n v) 9 u(n v + 1) : 9 : unv>n = 9 := .. 7 8 . u(n) It turns out unv>n = unv>n>0 + unv>n>i

(10.35)

where unv>n>0 represents the fault-free and stochastic part of the residual signal ˜u )> ˜u = gldj ( u > · · · > u ) unv>n>0 5 N (0> and unv>n>i is described by unv>n>i = Dv h(n v) + Pi>v inv>n

10.4 Application of GLR testing methods

325

with 6 5 6 5 i (n v) Ii F : 9 9 .. ¯ 9 FD : ¯i : 9 9 FH . : : > Dv = 9 9 =9 = > P : 9 . i>v : 9 9 .. .. 7 .. 8 8 7 7 . . v ¯ FD ¯i F D¯v1 H i (n) 5

inv>n

6 0 : .. .. : . . : : .. .. . 0 8 . ¯i Ii · · · FH

h(n) denoting the mean of the state estimate delivered by the Kalman ﬁlter (see Section 7.2), i.e. ¯i = Hi OIi ¯ ¯i i (n)> D¯ = D OF> H h(n + 1) = Dh(n) +H

(10.36)

and O the observer gain given by (7.70). We assume that h(n) = 0 before the fault occurs. For our purpose, the GLR for the given model (10.35) is computed as follows sup 2Vnv>n = 2 ln

ki (n)kv AOi

sup ki (n)kv Oi

+

sup ki (n)kv Oi

sup ki (n)kv AOi

skinv>n kAOi (unv>n ) skinv>n kOi (unv>n )

=

i h (unv>n u¯nv>n )W (unv>n u¯nv>n )

(10.37)

i h W (unv>n u¯nv>n ) (unv>n u¯nv>n )

(10.38)

whose solution can be approached by solving optimization problems (10.37) and (10.38) separately. To this end, we ﬁrst assume that h(n v) is small enough so that (10.39) u¯nv>n = unv>n>i Pi>v inv>n = Note that 1 iW inv>n =, v + 1 nv>n W ¯ 2i inv>n (v + 1) O2i := O ki (n)k2v O2i +, inv>n 2

ki (n)kv =

we have for ki (n)kv Oi iˆnv>n>0 = arg

h

sup ki (n)kv Oi

h

= arg

inf

1 W 2 v+1 inv>n inv>n Oi

and for ki (n)kv A Oi

(10.40)

i (unv>n u¯nv>n )W (unv>n u¯nv>n ) i

(unv>n Pi>v inv>n )W (unv>n Pi>v inv>n )

326

10 Statistical methods based residual evaluation and threshold setting

iˆnv>n>1 = arg = arg

inf

h sup ki (n)kv AOi

1 W 2 v+1 inv>n inv>n AOi

i (unv>n u¯nv>n )W (unv>n u¯nv>n )

i h W (unv>n Pi>v inv>n ) (unv>n Pi>v inv>n ) =

On the assumption that Pi>v is right invertible, i.e. of full row rank, we have ¡ ¢ W W 1 Pi>v Pi>v unv>n iˆnv>n>0 = Pi>v ³ ´1 W W ¯2 , if unv>n unv>n O Pi>v Pi>v i ¡ ¢ W W 1 Pi>v Pi>v unv>n r iˆnv>n>0 = Pi>v

¯i O ³ ´1 W W unv>n unv>n Pi>v Pi>v

³ ´1 W W ¯ 2 > and if unv>n unv>n A O Pi>v Pi>v i ¡ ¢ W W 1 Pi>v Pi>v unv>n iˆnv>n>1 = Pi>v ³ ´1 W W ¯2 , if unv>n unv>n A O Pi>v Pi>v i ¡ ¢ W W 1 Pi>v Pi>v unv>n r iˆnv>n>1 = Pi>v

¯i + % O ³ ´1 W W unv>n unv>n Pi>v Pi>v

³ ´1 W W ¯ 2 > where % A 0 is an arbitrarily small conif unv>n unv>n O Pi>v Pi>v i stant. It turns out ; Ã !2 A ¯ A +% O i W ¯2 A unv>n > O 1 t W A unv>n i 1 ? W unv>n (Pi>v Pi>v ) unv>n Ã !2 2Vnv>n = A A ¯i O A W ¯2 A unv>n > A O 1 t W = unv>n i 1 W unv>n (Pi>v Pi>v ) unv>n (10.41) where ¡ ¢ W W 1 Pi>v Pi>v = unv>n unv>n = As a result, the decision rule follows (10.41) directly and is described by ¡ ¢ W W 1 ¯ 2i : K0 Pi>v Pi>v unv>n O (10.42) unv>n ¡ ¢ W W 1 ¯ 2i : K1 = Pi>v Pi>v unv>n unv>n A O (10.43) ³ ´1 W W unv>n builds the evaluation function (testing staPi>v Pi>v Thus, unv>n tistic) for our fault detection purpose.

10.4 Application of GLR testing methods

327

Next, we study the following two problems: (a) given residual evaluation ³ ´1 W W ¯ 2 > ﬁnd the false function unv>n unv>n and threshold Mwk = O Pi>v Pi>v i alarm rate deﬁned by ³ ´ ¡ ¢ W W 1 Pi>v Pi>v = prob unv>n unv>n A Mwk | ki (n)kv Oi (10.44) ³ ´1 W W unv>n and allowPi>v Pi>v (b) given residual evaluation function unv>n able false alarm rate > ﬁnd the threshold. To solve these two problems, let ˜u1@2 unv>n = u˜nv>n = It holds ¡ ¢ W W 1 W Pi>v Pi>v unv>n u˜nv>n u˜nv>n 1 unv>n ³ ´ W ˜ 1@2 ˜u1@2 Pi>v Pi>v = min u

(10.45)

³ ´ ˜u1@2 Pi>v P W ˜ 1@2 denoting the minimum eigenvalue of mawith min i>v u 1@2

1@2

W ˜ ˜u trix Pi>v Pi>v . Thus, we can estimate the false alarm rate by u means of ¢ ¡ W (10.46) prob u˜nv>n u˜nv>n A Mwk | ki (n)kv Oi = W Considering that in the fault-free case u˜nv>n u˜nv>n is noncentrally "2 distributed with noncentrality parameter ³ ´ W W ˜ 1@2 W ˜ 1@2 ¯ 2i max ˜u1@2 Pi>v Pi>v ˜u1@2 Pi>v Pi>v inv>n O inv>n u u

and the degrees of the freedom equals to the dimension of u˜nv>n > the probability in (10.46) can be computed using the noncentral "2 distribution. To solve the second problem, we can directly use the following relation ³ ´ 2 unv>n )> ¯ ) A Mwk = prob "2 (dim(˜ (10.47) ³ ´ W ˜ 1@2 ¯ 2 = O ¯ 2i max ˜u1@2 Pi>v Pi>v u unv>n )> 2 stand for the for the determination of Mwk by given > where dim(¯ degrees of the freedom and the (maximum) non-centrality parameter of the non-central "2 distribution respectively. Algorithm 10.6 Threshold computation ³ ´ ˜u1@2 Pi>v P W ¯ 2 max ˜ 1@2 Step 1: Computation of O i i>v u Step 2: Determination of Mwk according to (10.47). Algorithm 10.7 Computation of false alarm rate

328

10 Statistical methods based residual evaluation and threshold setting

³ ´ ¯ 2 and O ¯ 2 max ˜u1@2 Pi>v P W ˜ 1@2 Step 1: Computation of O i i i>v u ³ ´ 2 ¯2 = Step 2: Computation of prob "2 (dim(˜ unv>n )> ¯ ) A O i Algorithm 10.8 on-line realization Step 1: Computation of evaluation function ¡ ¢ W W 1 Pi>v Pi>v unv>n unv>n ³ ´1 W W Step 2: Comparison between unv>n unv>n and threshold Mwk = Pi>v Pi>v Remember that the above solution is achieved on the assumption of (10.39). We now remove this assumption. Let 5 6 5 6 h(n v) 0 F Ii 9 i (n v) : 9 : 9 : .. .. ¯i 9 F D¯ : 9 : .. . . FH ¯ 9 : 9 : ¯ . inv>n = 9 : : > Pi>v = 9 . . . . .. .. 0 8 .. 9 : 7 .. .. 7 8 . ¯i Ii ¯i · · · F H F D¯v FDv1 H i (n) and rewrite u¯nv>n into ¯ i>v i¯nv>n = u¯nv>n = unv>n>i = P Since h(n v) is driven by the fault, we replace our original problem formulation (10.34) by r ½ 1 W Ou¯> Kr (fault-free) u¯nv>n u¯nv>n = (10.48) k¯ unv>n k = A Ou¯> K1 (fault) v+1 where Ou¯ is a constant determined by ° ¡ ¢1 ° ° ¯i ° H Ou¯ = °F }L D¯ °

4

Oi =

That means, we now deﬁne the fault detection problem in terms of the inﬂuence of the fault on the mean of the residual signal. Applying the GLR to the given model (10.35) yields sup 2Vnv>n = 2 ln h +

k¯ unv>n kOu¯

=

sk¯unv>n kOu¯ (unv>n ) W

i (10.49)

i h (unv>n u¯nv>n )W (unv>n u¯nv>n ) =

(10.50)

k¯ unv>n kOu¯

k¯ unv>n kAOu¯

sup

sk¯unv>n kAOu¯ (unv>n )

(unv>n u¯nv>n ) (unv>n u¯nv>n )

sup sup

k¯ unv>n kAOu¯

10.4 Application of GLR testing methods

329

Along with the way of solving (10.37) and (10.38), we can ﬁnd out that for k¯ unv>n k Ou¯ i h sup (unv>n u¯nv>n )W (unv>n u¯nv>n ) uˆ ¯nv>n>0 = arg k¯ unv>n kOu¯

= unv>n W if unv>n unv>n O2u¯ and

uˆ ¯nv>n>0 = arg

sup k¯ unv>n kOu¯

i h (unv>n u¯nv>n )W (unv>n u¯nv>n )

Ou¯ = unv>n q W unv>n unv>n W if unv>n unv>n A O2u¯> as well as for k¯ unv>n k A Ou¯ i h W sup (unv>n u¯nv>n ) (unv>n u¯nv>n ) uˆ ¯nv>n>1 = arg k¯ unv>n kAOu¯

= unv>n W if unv>n unv>n A O2u¯ and

uˆ ¯nv>n>0 = arg

sup k¯ unv>n kAOu¯

i h (unv>n u¯nv>n )W (unv>n u¯nv>n )

Ou¯ + % = unv>n q W unv>n unv>n W if unv>n unv>n O2u¯> where % A 0 is a arbitrarily small constant. It leads to ; μ ¶2 A Ou¯+% W W A s u 1 unv>n > unv>n unv>n O2u¯ ? nv>n W unv>n unv>n μ ¶2 = 2Vnv>n = A A O W W 2 u ¯ = unv>n 1 s W unv>n > unv>n unv>n A Ou¯ unv>n unv>n

(10.51) Thus, the decision rule can be deﬁned as W unv>n O2u¯ : K0 unv>n W unv>n unv>n

A

O2u¯

: K1

(10.52) (10.53)

W with unv>n unv>n as the testing statistic. Similar to the study in the ﬁrst part of this section, we can introduce

˜u1@2 unv>n u˜nv>n = and, based on it, estimate the false alarm rate by applying decision rule (10.52) and (10.53) or determine the threshold for a given allowable false alarm rate > as described in the following two algorithms.

330

10 Statistical methods based residual evaluation and threshold setting

Algorithm 10.9 Computation of false alarm rate Step 1: Compute

³ ´ ˜ > ¯ 2u = O2u = min

Step 2: Compute

´ ³ 2 unv>n )> ¯ u ) A O2u¯ = prob "2 (dim(˜

(10.54)

Algorithm 10.10 Threshold computation 2 Step 1: Compute ¯ u Step 2: Solve

³ ´ 2 prob "2 (dim(˜ unv>n )> ¯ u ) A Mwk =

(10.55)

for Mwk = 10.4.2 Parity space based fault detection Applying the parity space method to system (10.31) and (10.32) yields 3 5 65 64 G 0 x(n v) 5 6 E |(n v) 9 :9 :F .. .. .. E9 :9 :F . . . : 9 .. 9 FE :9 :F(10.56) unv>n = Y E 7 8 . E 9 : 9 :F .. . .. .. .. C 7 87 8D . . 0 . |(n) FDv1 E · · · FE G x(n) = Pi inv>n + nv>n with

5 9 9 Pi>v = Y 9 9 7

nv>n

Ii

0 .. .. . .

(10.57) 6

: : : : .. .. . 0 8 . FDv1 Hi · · · FHi Ii 6 5 5 0 y(n v) : 9 9 .. .. : 9 F 9 . . : 9 9 =9 :+9 . .. . .. ... 8 7 .. 7 . FDv1 · · · F y(n)) FHi .. .

65

6 (n v) :9 : .. :9 : . :9 : 5 N (0> )= :9 : .. 87 8 . 0 (n)

Again, we are interested in detecting those faults whose energy level is higher than a tolerant limit Oi > i.e. v u ½ v u 1 X Oi > Kr (fault-free) ki (n)kv = t = i W (n l)i (n l) = A Oi > K1 (fault) v + 1 l=0 (10.58)

10.4 Application of GLR testing methods

331

Comparing (10.57) with (10.35) makes it clear that we are able to use the same method to solve the above-deﬁned fault detection problem. Thus, without a detailed derivation, we give the major results in the following two algorithms. Algorithm 10.11 Threshold computation ³ ´ 2 ¯ 2 max 1@2 Pi>v P W 1@2 Step 1: Compute ¯ = O i i>v Step 2: Solve ³ ´ 2 prob "2 (dim(˜ unv>n )> ¯ ) A Mwk = for Mwk . Algorithm 10.12 Computation of false alarm rate ³ ´ ¯ 2 and ¯ 2 = O ¯ 2 max 1@2 Pi>v P W 1@2 Step 1: Compute O i i i>v ³ ´ 2 ¯2 = Step 2: Compute the false alarm rate prob "2 (dim(˜ unv>n )> ¯ ) A O i ¯ 2 are identical with the In the above two algorithms, the parameters > O i ones given in (10.45) and (10.40), and vector u˜nv>n is given by u˜nv>n = 1@2 unv>n = Remark 10.2 The above results have been achieved on the assumption that Pi>v is right invertible. In case that it does not hold, we can use the method described in Section 7.3 to replace Pi>v by its approximation which is then invertible. Example 10.1 To illustrate the application of Algorithm 10.6, we consider three tank system DTS200 given in Subsection 3.7.3. In order to get more insight into the system design and threshold computation, we design two different Kalman ﬁlters, based on model (3.66). The ﬁrst one is a Kalman ﬁlter driven only by one sensor (the level sensor of tank 1). Such a residual generator is often integrated into a bank of residual generators for the isolation purpose, see Section 13.5.1. Under the assumption that = 0=1L3 > = 0=1 the observer gain is given by 5

6 0=3816 O = 7 0=0452 8 = 0=1107 Suppose that Oi = 0=05 and the length of the evaluation window v = 15, then we get W ¯ 2i pd{ ( u1@2 Pi>v Pi>v u1@2 ) = 0=3787= O

332

10 Statistical methods based residual evaluation and threshold setting 100

testing statistic

80

60

40

25.6 20

0

0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 10.2 Testing statistic and the threshold: one sensor case

In the next step, Mwk is determined according to (10.47). Setting = 0=05 results in Mwk = 25=6222= Fig.10.2 provides us with a simulation result of the testing statistic and threshold by a oset fault (5fp) in sensor 1 at w = 12vhf= The second Kalman ﬁlter is designed using all three sensor signals. For = = 0=1L3 > we get 5 6 0=3813 0=0000 0=0007 O = 7 0=0000 0=3804 0=0007 8 = 0=0007 0=0007 0=3806 On the assumption Oi = 0=1 and length of evaluation window v = 15, it turns out W ¯ 2i pd{ ( u1@2 Pi>v Pi>v u1@2 ) = 1=5213= O On the demand of = 0=05> we have Mwk = 27=4760= Fig.10.3 gives a simulation result of the testing statistic and threshold by the same oset sensor fault (i = 5fp> w = 12vhf) like the last simulation= Comparing both the simulation results, we can evidently see that fault detectability is enhanced with more measurements.

10.5 Notes and references

333

100

testing statistic

80

60

40

27.47 20

0 0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 10.3 Testing statistic and the threshold: three sensors case

10.5 Notes and references In this chapter, essentials of statistic methods for the residual evaluation and decision making have been brieﬂy reviewed. In Section 10.2, basic ideas, important concepts and basic statistic testing tools have been introduced. For the basic knowledge and elementary methods of probability and statistics, we have mainly referred the book by Lapin [92]. By the introduction of the LR and GLR technique, the monograph by Basseville and Nikiforov [10] serves as a major reference. The discussion in Section 10.3 about criteria for threshold computation is intended for providing the reader with deeper background information about the LR method and other useful alternative schemes. It is mainly based on [103]. From the FDI viewpoint, Section 10.4 builds the main focus of this chapter. Along with the ideas presented in [10] and equipped with the skill of applying the GLR technique to solve change detection problems learned from [10], we have introduced two methods for detecting faults in stochastic systems. They serve as a bridge between the model-based FDI methods presented in the previous chapters and the statistical methods, and build the basis for an extended study in the forthcoming chapter. We would like to emphasize that the statistical methods introduced in this chapter is only a small part of the statistical methods based FDI framework. For more detailed and comprehensive study, we refer the reader to the excellent monographs by Basseville and Nikiforov [10] and by Gustafsson [69] as well as the frequently cited book [93]. There also are a great number of excellent papers, for instance [7, 8, 9, 91, 157].

11 Integration of norm based and statistical methods

In this chapter, we study the integration of norm based and statistical methods to address FDI in systems with both deterministic disturbances and stochastic uncertainties. Three schemes with dierent solution strategies and supported by dierent tools will be presented. The ﬁrst scheme deals with FDI in systems with deterministic disturbances and stochastic noises, while the second and third ones address systems with stochastically varying parameters.

11.1 Residual evaluation in stochastic systems with deterministic disturbances As sketched in Fig.11.1, in this section we continue our study started in the last chapter.

Fig. 11.1 FDI in systems with deterministic disturbances and stochastic noises

336

11 Integration of norm based and statistical methods

We consider systems modelled by {(n + 1) = D{(n) + Ex(n) + Hg g(n) + Hi i (n) + (n) |(n) = F{(n) + Gx(n) + Ig g(n) + Ii i (n) + (n)=

(11.1) (11.2)

The terms Hg g(n)> Ig g(n) represent the inﬂuence of some deterministic unknown inputs with known distribution matrices Hg > Ig and vector of unknown inputs g(n) 5 Rng > which is bounded by v u n u X ;n t gW (l)g(l) g l=nv

with v denoting the length of the evaluation window. (n) 5 Rq > y(n) 5 Rp are assumed to be discrete time, zero-mean, white noise and satisfy ¸ ¶ μ ¸ ¤ T V (l) £ W W (m) y (m) = = E V W U lm y(l) Further, (n)> y(n) are assumed to be statistically independent of the input vector x(n)= Our objective is to detect the fault vector i (n) 5 Rni if it diers from zero. 11.1.1 Residual generation For the residual generation purpose, we use, without loss of generality, an FDF { ˆ(n + 1) = (D OF) { ˆ(n) + (E OG) x(n) + O|(n) u(n) = Y (|(n) F { ˆ(n) Gx(n)) 5 Rpu which yields ¯i i (n) + ¯(n) ¯ ¯g g(n) + H h(n + 1) = Dh(n) +H u(n) = Y (Fh(n) + Ig g(n) + Ii i (n) + y(n))

(11.3) (11.4)

with h(n) = {(n) { ˆ(n)> ¯(n) = (n) Oy(n) ¯i = Hi OIi = ¯ ¯ D = D OF> Hg = Hg OIg > H The observer matrix O and post-ﬁlter Y can be selected e.g. using the uniﬁed solution or Kalman-ﬁlter scheme. In the steady state, the means of h(n)> u(n)> h¯(n) = E (h(n)) > u¯(n) = E (u(n)) > satisfy

11.1 Residual evaluation in stochastic systems with deterministic disturbances

¯h(n) + H ¯g g(n) + H ¯i i (n) h¯(n + 1) = D¯ u¯(n) = Y (F h¯(n) + Ig g(n) + Ii i (n)) =

337

(11.5) (11.6)

For our purpose, we write u¯(n) into u¯(n) = ug (n) + ui (n) with

³ ¡ ´ ¢1 ¯g + Ig g(}) H ug (}) = Y F }L D¯ ³ ¡ ´ ¢1 ¯i + Ii i (})= H ui (}) = Y F }L D¯

Note that in the fault-free case the mean of the residual signal is bounded by v u n ° ³ ¡ ´° u X ¢1 ° ¯g + Ig ° kug knv>n = t ugW (l)ug (l) °Y F }L D¯ H ° g := ug 4

l=nv

(11.7) for all n= The covariance matrix of u(n) is given by ¡ ¢ W E (u(n) u¯(n)) (u(n) u¯(n)) = Y FS F W + U Y W

(11.8)

where S A 0 solves ¯ D¯W S + = 0 DS ¸ ¸ £ ¤ T V L = T OV W VOW + OUOW = = L O VW U OW

11.1.2 Problem formulation Along with the lines in Section 10.4, we formulate two problems for our study. Problem 1: Given {u(l)> l = n v> · · · > n} > ﬁnd a residual evaluation function (testing statistic), kukh > a threshold Mwk and compute the false alarm rate deﬁned by = prob {ku(n)kh A Mwk | i = 0} = (11.9) Problem 2: Given {u(l)> l = n v> · · · > n} > an allowable false alarm rate and the residual evaluation function (testing statistic), kukh > as deﬁned in Problem 1, ﬁnd a threshold Mwk such that prob {ku(n)kh A Mwk | i = 0} = Both of these two problems are of strong practical interests.

(11.10)

338

11 Integration of norm based and statistical methods

11.1.3 GLR solutions Below, we shall use the GLR method to solve the above two problems. For this purpose, the GLR for given u(l)> l = n v> · · · > n> is computed. It results in

n 2Vnv

n X

=2

n Q

sup vl = 2 log

kug knv>n ug >i 6=0 l=nv

l=nv

n Q

sup

kug knv>n ug >i =0 l=nv

"

=

"

+

sup kug knv>n ug >i =0

sup kug knv>n ug >i 6=0

n X

si 6=0 (u(l)) si =0 (u(l)) #

W

(ug (l)) ug (l)

l=nv

n X

# W

(ug>i (l)) ug>i (l)

(11.11)

l=nv

where 1 12 (ug>i (l))W ug>i (l) si 6=0 (u(l)) = p pu h (2) 1 12 (ug (l))W ug>i (l) si =0 (u(l)) = p p h (2) u ug (l) = u(l) ug (l)> ug>i (l) = u(l) ug (l) ui (l)= Introduce the notations ug>nv>n = unv>n ug>nv>n > ug>i>nv>n = unv>n ug>nv>n ui>nv>n 5 5 5 6 6 6 u(n v) ug (n v) ui (n v) 9 : 9 9 : : .. .. .. 9 : 9 9 : : . . . : > ug>nv>n = 9 : > ui>nv>n = 9 : unv>n = 9 9 : 9 9 : : .. .. .. 7 8 7 7 8 8 . . . ug (n) ui (n) u(n) then we have n = 2Vnv

+

sup kug knv>n ug i =0

h i W (ug>nv>n ) ug>nv>n h

sup kug knv>n ug >i 6=0

i (ug>i>nv>n )W ug>i>nv>n = (11.12)

Moreover, the boundedness of ug (n) gives W ug>nv>n ug>nv>n =

n X l=nv

ugW (l)ug (l) 2ug =

11.1 Residual evaluation in stochastic systems with deterministic disturbances

339

Next, we compute the LR estimate for ug>nv>n : h i W sup uˆg>nv>n>0 = arg (ug>nv>n ) ug>nv>n W ug>nv>n ug>nv>n 2u g inv>n =0

uˆg>nv>n>0

W unv>n 2ug = unv>n > if unv>n h i = arg sup (ug>nv>n )W ug>nv>n

uˆg>nv>n>1

ug W > if unv>n unv>n A 2ug = unv>n q W unv>n unv>n h i = arg sup (ug>i>nv>n )W ug>i>nv>n = 0

W ug>nv>n ug>nv>n 2u g inv>n =0

W ug>nv>n ug>nv>n 2u g inv>n 6=0

as well as for ui>nv>n uˆi>nv>n>1 = arg

sup W ug>nv>n ug>nv>n 2u g inv>n 6=0

h i (ug>i>nv>n )W ug>i>nv>n = unv>n =

As a result, we have ; W 0 for unv>n unv>n 2ug ? μ ¶2 n = 2Vnv = ug W W = unv>n unv>n 1 s W for unv>n unv>n A 2ug unv>n unv>n

(11.13) Recall that in the context of the GLR scheme a decision for a fault will be n made if Vnv A 0= Thus, it follows from (11.13) that the probability of a false alarm (the false alarm rate) equals to ¢ ¡ W ¢ ¡ n A 0 | inv>n = 0 = prob unv>n unv>n A 2ug | inv>n = 0 = = prob Vnv (11.14) W In this way, unv>n unv>n deﬁnes a residual evaluation function (testing statistic) and 2ug the threshold. For the computation of the false alarm rate> we W unv>n = Remember that in need a further study on the testing statistic unv>n the fault-free case 5 6 5 65 6 Y y(n v) 0 ¯(n v) 9 : 9 :9 : .. .. .. 9 : 9 YF :9 : . . . :+9 :9 := ug>nv>n = 9 9 : 9 : 9 : .. .. . .. .. .. 7 8 7 8 . . 87 . . Y F D¯v1 · · · Y F 0 Y y(n) ¯(n)

340

5 9 9 9 9 7

11 Integration of norm based and statistical methods

0 Y

0 .. . Y F Y FO .. .. . . v1 v1 Y F D¯ Y F D¯ O · · ·

with

5 6 6 (n v) 9 y(n v) : 9 : :9 : .. .. :9 : . . ˜ :9 : 5 N (0> ) : 9 : . .. .. : . 0 89 9 : Y F Y FO 0 Y 7 (n) 8 y(n)

μ ˜ = S˜ gldj 5 9 9 ˜ S =9 9 7

0 Y

¸ ¸¶ T V T V > · · · > S˜ W VW U VW U 0 .. .

.. . Y F Y FO .. .. .. . . 0 . Y F D¯v1 Y F D¯v1 O · · · Y F Y FO 0 Y

(11.15) 6 : : := : 8

W ˜ 1 unv>n is noncentrally "2 distributed with the Thus, for i = 0> unv>n degree of freedom equal to pu (v + 1) and the noncentrality parameter W ˜ 1 ug>nv>n = Consider that ug>nv>n ¢ ¡ W unv>n A 2ug | inv>n = 0 = prob unv>n ´ ³ ´ ³ W ˜ 2 ˜ 1 unv>n A 1 prob unv>n max ug | inv>n = 0 μ ¶ ³ ´ 1 2 W 1 ˜ ˜ prob max unv>n unv>n A max ug | inv>n = 0 g μ ¶ ³ ´ 1 2 W 1 ˜ ˜ = 1 prob max unv>n unv>n max ug | inv>n = 0 = g

Hence, is bounded by ³ ³ ³ ´ ´ ³ ´ ´ 1 ˜ 2 ˜ 2 1 prob "2 pu (v + 1)> 1 min ug max ug

(11.16)

³ ³ ´ ´ 2 ˜ 2 where "2 pu (v + 1)> 1 min ug denotes " distribution with the degree of ³ ´ ˜ 2 the freedom equal to pu (v+1) and the noncentrality parameter 1 min ug = As a result, Problem 1 is solved. The reader may notice that in (11.16) ³ ´ ³ ´ ˜ 2u threshold 1 ˜ 2 noncentrality parameter 1 max ug = min g It will cause a high false alarm rate. To overcome this di!culty, we can simply W substitute unv>n unv>n by W ˜ 1 unv>n unv>n

11.1 Residual evaluation in stochastic systems with deterministic disturbances

341

as the testing statistic. Although additional on-line computation is now needed, we have μ ¶ ³ ´ W ˜ 1 2u | inv>n = 0 ˜ 1 unv>n A max prob max unv>n g g ¢ ¢ ¡ 2¡ (11.17) = 1 prob " pu (v + 1)> 2ug 2ug which ensures that the noncentrality parameter is equal to the threshold. We summarize the major result in the following algorithm. Algorithm 11.1 Computation of for given statistic and threshold Step 1: Compute 2ug according to (11.7) ˜ according to (11.15) Step 2: Form ¢ ¢ ¡ ¡ Step 3: Compute prob "2 pu (v + 1)> 2ug 2ug and ﬁnally the bound of using (11.17). Algorithm 11.2 on-line realization Step 1: Compute testing statistic W ˜ 1 unv>n unv>n

Step 2: Comparison between the testing statistic threshold Mwk = 2ug = W ˜ 1 unv>n and Now, we solve Problem 2 for given testing statistic unv>n an allowable false alarm rate = It follows from (11.17) that the threshold Mwk can be determined by solving ³ ³ ³ ´ ´ ´ ˜ 2 = 1 prob "2 pu (v + 1)> 1 (11.18) min ug Mwk =

It leads to the following algorithm. Algorithm 11.3 Threshold computation Step 1: Compute 2ug Step 2: Determine Mwk according to (11.18). 11.1.4 Discussion and example W ˜ unv>n We see that the residual evaluation function is the statistic unv>n which measures the weighted energy level of the residual signal over the time interval (n v> n)= It is interesting to note that this function is similar to W unv>n unv>n > the one used in the norm based residual evaluation methods. On the other hand, dierent from the well-established norm based methods, where the threshold is set to be 2ug > the threshold proposed here is determined in the statistical context as deﬁned by (11.18).

342

11 Integration of norm based and statistical methods

The achieved results evidently reveal that, both in the norm based methods and the approach presented in this section, the boundedness of ug and the covariance of the residual signal given in (11.8) play an important role in threshold determination, as we can see from (11.18). This is a convincing argument for a system designer to make use of the degree of the design ³ freedom ´ 1 ˜ 2u oered by the observer to achieve an optimal trade-o between min g and Mwk . Example 11.1 We continue our study in Example 10.1, where a fault detection system is designed for the three tank system benchmark. Now, in addition to the noises, oset in the sensors is taken into account and modelled as unknown inputs by Ig g> Ig = L and g 5 R3 = It is assumed that g is bounded by g = 0=05= Our design objective is to determine the threshold Mwk using Algorithm 11.3. For the residual generation purpose, we use the same two Kalman ﬁlters designed in Example 10.1, i.e. (a) a Kalman ﬁlter driven by the level sensor of tank 1 (b) a Kalman ﬁlter driven by all three sensors. Under the same assumptions with = 0=05> we have Case (a) with one sensor: Mwk = 26=3349 Case (b) with three sensors: Mwk = 68=0159= Fig.11.2 and Fig.11.3 show the simulation results of the testing statistic and 100 90

testing statistic

80 70 60 50 40 30 26.33 20 10 0

0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 11.2 Testing statistic and the threshold: one sensor case

11.2 Residual evaluation scheme for stochastically uncertain systems

343

200 180

testing statistic

160 140 120 100 80 68.01 60 40 20 0

0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 11.3 Testing statistic and the threshold: three sensors case

threshold by an oset fault (5fp) in sensor 1 at w = 12vhf> with respect to the designed FD systems.

11.2 Residual evaluation scheme for stochastically uncertain systems In Section 8.5, we have studied the residual generation problems for stochastically uncertain systems. The objective of this section is to address the residual evaluation problems, as sketched in Fig.11.4. 11.2.1 Problem formulation As studied in Section 8.5, we consider system model ¯ ¯ ¯g g(n) + Hi i (n) {(n + 1) = D{(n) + Ex(n) +H ¯ ¯ |(n) = F{(n) + Gx(n) + I¯g g(n) + Ii i (n)

(11.19) (11.20)

where ¯ = E + E> F¯ = F + F D¯ = D + D> E ¯ = G + G> H ¯g = Hg + H> I¯g = Ig + I= G D> E> F> G> H and I represent model uncertainties satisfying

344

11 Integration of norm based and statistical methods

Fig. 11.4 FDI in systems with deterministic disturbances and stochastic uncertainties

D E H F G I

¸ =

¸ ¶ o μ X Dl El Hl sl (n) Fl Gl Il

(11.21)

l=1

with known matrices Dl > El > Fl > ¤Gl > Hl > Il > l = 1> · · · o> of appropriate dimen£ sions. sW (n) = s1 (n) · · · so (n) represents model uncertainties and is expressed as a stochastic process with ¢ ¡ s¯(n) = E (s(n)) = 0> E s(n)sW (n) = gldj( 1 > · · · > o ) where l > l = 1> · · · o> are known. It is further assumed that s(0)> s(1)> · · · > are independent and {(0)> x(n)> g(n)> i (n) are independent of s(n)= For the purpose of residual generation, an FDF { ˆ(n + 1) = Dˆ {(n) + Ex(n) + O (|(n) |ˆ(n)) |ˆ(n) = F { ˆ(n) + Gx(n)> u(n) = Y (|(n) |ˆ(n))

(11.22) (11.23)

is used. The dynamics of the above residual generator is governed by {u (n + 1) = Du {u (n) + Eu x(n) + Hu g(n) + Hu>i i (n) u(n) = Fu {u (n) + Gu x(n) + Iu g(n) + Iu>i i (n) ¸ ¸ {(n) {(n) = {u (n) = h(n) {(n) { ˆ(n)

(11.24) (11.25)

and the mean of u(n) is h¯(n + 1) = (D OF)¯ h(n) + (Hg OIg )g(n) + (Hi OIi )i (n) (11.26) u¯(n) = Y (F h¯(n) + Ig g(n) + Ii i (n)) = (11.27)

11.2 Residual evaluation scheme for stochastically uncertain systems

345

The matrices Du > Eu > Fu > Gu > Hu > Iu > Hu>i and Iu>i in (11.24)-(11.25) are described in Section 8.5. We assume that the system is mean square stable. In the remainder of this section, the standard variance of u(n) is denoted by £ ¤ £ ¤ u (n) = E (u(n) u¯(n))W (u(n) u¯(n)) = E hWu (n)hu (n) with hu (n) = u(n) u¯(n)= It is the objective of our study in this section that a residual evaluation strategy will be developed and integrated into a procedure of designing an observer-based FDI system. This residual evaluation strategy should take into account a prior knowledge of the model uncertainties and combine the statistic testing and norm based residual evaluation schemes. Note that the residual signal considered in the last section is assumed to be a normal distributed. Dierently, we have no knowledge of the distribution of the residual signal addressed in this section. The problems to be addressed in the next subsections are • selection of a residual evaluation function and • threshold determination for the given residual evaluation function and an allowable false alarm rate = 11.2.2 Solution and design algorithms A simplest way to evaluate the residual signal is to compute its size at each time instant and compare it with a threshold. Considering that u(n) is a stochastic process whose distribution is unknown, it is reasonable to set the threshold equal to r r sup u¯W (n)¯ u(n) + sup u (n) (11.28) Mwk = g>i =0

g>x

and deﬁne the decision logic as q M = uW (n)u(n) A Mwk =, fault q M = uW (n)u(n) Mwk =, fault-free

(11.29) (11.30)

where (A 1) is some constant used to reduce the false alarm rate. In (11.28), the ﬁrst term represents the bound on the mean value of the residual signal in the fault-free case, while the second term, considering the stochastic character of u(n)> is used to express the expected derivation of u(n) from its mean value. It is evident that the above decision logic with threshold (11.28) may result in a high false alarm rate if the standard variance of u(n) is large. For this reason, we propose the following residual evaluation function

346

11 Integration of norm based and statistical methods

v !W Ã ! uÃ Q Q u 1 X 1 X t M= u(n l) u(n l) Q l=1 Q l=1 for some Q= In fact

(11.31)

Q 1 X u(n l) Q l=1

is the average of the residual signal over the time interval (n Q> n), which is inﬂuenced by both the additive and multiplicative faults. The following theorem reveals an important statistical property of evaluation function (11.31). Theorem 11.1 Given system model and ¢ £ suppose ¤that the ¡ (11.24)-(11.25) system is mean square stable, i.e. E {Wu (n){u (n) and E hW{ (n)h{ (n) with ¯u (n) h{ (n) = {u (n) { are bounded. Then, Ã E

!W Ã ! Q Q 1 X 1 X hu (n l) hu (n l) Q l=1 Q l=1 Q

where A 0 is some constant. Proof. Note that for l A 0 h i ¢W £ ¤ ¡ E hWu (n)hu (n l) = E hW{ (n l) Fu>0 Dlu>0 Fu>0 h{ (n l) 3 5 65 6W 4 {u (n l) {u (n l) F E +wudfh CE 7 x(n l) 8 7 x(n l) 8 Tl D g(n l) g(n l) 5 W 6 Du>m ¡ o X ¢W £ ¤ W 8 Fu>m Gu>m Iu>m = Fu>0 Dl1 Tl = 2m 7 Eu>m u>0 W m=1 Hu>m It leads to 5Ã !W Ã Q !6 Q Q X X X £ ¤ E7 hu (n l) hu (n l) 8 = E hWu (n l)hu (n l) l=1

+

l=1

l=1

m1 Q X X ¡ £ W ¤ £ ¤¢ E hu (n l)hu (n m) + E hWu (n m)hu (n l) m=2 l=1

=

Q X l=1

Q ¡ ¢ £ ¤ X m + mW E hWu (n l)hu (n l) + m=2

11.2 Residual evaluation scheme for stochastically uncertain systems

347

with 5 m = E 7hW{ (n m)

Ãm1 X

6

!W

Fu>0 h{ (n m)8

Fu> 0 Dlu>0

l=1

3 5 4 65 6W {u (n m) {u (n m) E F +wudfh CE 7 x(n m) 8 7 x(n m) 8 Tm D = g(n m) g(n m) Recall that (L Du>0 )

m1 X

Dlu>0 = Du>0 (L Dm1 u>0 )

l=1

and moreover, considering that the size of all eigenvalues of Du>0 is smaller than one, we also have ;m m1 X

Dlu>0 = (L Du>0 )1 Du>0 (L Dm1 u>0 )

l=1

is bounded by lim

m$4

m1 X

Dlu>0 = (L Du>0 )1 Du>0 =

l=1

It turns out m + mW = 4 3 5 65 6W {u (n m) {u (n m) ¤ £ W F E E h{ (n m)m h{ (n m) + wudfh CE 7 x(n m) 8 7 x(n m) 8 m D g(n m) g(n m) m =

W mW Fu>0 Fu>0

+

W Fu>0 Fu>0 m > m

m1 X ¡ ¢ Tl + TWl =

o X

5

l=1

6 Du>l £ ¤ W 8 W W m Fu>0 Fu>l Gu>l Iu>l Tl = 2l 7 Eu>l W l=1 l=1 Hu>l

m1 X

m = (L Du>0 )1 Du>0 (L Dm1 u>0 )= Note ;m m > m are bounded, i.e. <% > % so that ;m max (m ) % > max (m ) % we have

348

11 Integration of norm based and statistical methods

5Ã !W Ã Q !6 Q Q X X X E7 hu (n l) hu (n l) 8 u (n l)+ l=1

l=1

l=1

6W 5 6 Q {u (n m) {u (n m) X £ W ¤ E h{ (n m)h{ (n m) + % E 7 x(n m) 8 7 x(n m) 8 Q % m=2 m=2 g(n m) g(n m) £ W ¤ £ W ¤ = max u (n) + % max E h{ (n)h{ (n) + % max E {u (n){u (n) g>x g>x g>x ¡ ¢ (11.32) +% max xW (n)x(n) + gW (n)g(n) 5

Q X

g>x

¤ £ ¤ £ where, due to the boundness of E hW{ (n)h{ (n) and E {Wu (n){u (n) > is a constant and independent of Q= It results in ﬁnally Ã

!W Ã ! Q Q 1 X 1 X E hu (n l) hu (n l) = Q l=1 Q l=1 5Ã !W Ã Q !6 Q X 1 7 X E hu (n l) hu (n l) 8 = Q2 Q l=1 l=1 The theorem has thus been proven. u t Note that v !W Ã ! v u Ã u Q Q Q u u1 X X X 1 tE 1 u(n l) E u(n l) t (¯ uW (n l)¯ u(n l)) Q l=1 Q l=1 Q l=1 r max u¯W (n)¯ u(n) := u¯max = n>g>i =0

We have, following Theorem 11.1, Ã

!W Ã ! Q Q 1 X 1 X EM = E u(n l) E u(n l) + Q l=1 Q l=1 5Ã !W Ã !6 Q Q X X 1 1 hu (n l) hu (n l) 8 2u¯max + = E7 Q l=1 Q l=1 Q 2

(11.33) and Theorem 11.1 reveal that for Q $ 4 5Ã !W Ã !6 Q Q X X 1 1 E7 hu (n l) hu (n l) 8 $ 0 Q l=1 Q l=1 EM 2 2u¯max

(11.33)

(11.34) (11.35)

11.2 Residual evaluation scheme for stochastically uncertain systems

349

i.e. M will deliver a good estimate for the mean value of the residual signal. Motivated and guided by the above discussion, we propose, corresponding to evaluation function (11.31), the following general form for setting the threshold: q Mwk = 2u¯max + (Q ) u>max (n)> u>max (n) = sup u (n) (11.36) g>x

where (Q ) is a constant for a given Q . In this way, the problem of determining the threshold is reduced to ﬁnd (Q )= Next, we approach this problem for a given allowable false alarm rate = To this end, we ﬁrst introduce the well-known Tchebyche Inequality, which says: for a given random number { ¯)2 > it holds and a constant A 0 satisfying 2 E ({ { prob (|{ { ¯| )

E ({ { ¯)2 = 2

(11.37)

Recall that the false alarm rate is deﬁned by prob (M A Mwk | i = 0) and moreover prob (M A Mwk | i = 0) = prob (M E (M) A Mwk E (M) | i = 0) prob (M E (M) A Mwk | i = 0) prob (|M E (M)| Mwk | i = 0) = If follows from the Tchebyche Inequality that setting q Mwk = 2u¯max + (Q ) u>max which satisﬁes 2 2u¯max + Q EM 2 E (M EM) 2 2 Mwk Mwk 2u¯max + (Q ) u>max

=, (Q )

(1 ) 2u¯max + u>max

Q

(11.38)

ensures prob (M A Mwk | i = 0) = From (11.38) it can be seen that a lower allowable false alarm rate requires a larger (Q )= To complete our design procedure, it remains to ﬁnd 2u¯max and u>max as £ W ¤ £ W ¤ well as E h{ (n)h{ (n) and E {u (n){u (n) which are needed for the computation of threshold (11.36) as well as in (11.32). Using the LMI technique introduced in Chapter 9, we obtain the following results.

350

11 Integration of norm based and statistical methods

Theorem 11.2 Given system model (11.26)-(11.27), 1 A 0> and assume that q kg(n)k2 g>2 > ;n gW (n)g(n) g>4 = Then, ;n

u(n) 1 2g>2 + 2 2g>4 := 2u¯max u¯W (n)¯

(11.39)

if the following two LMI’s hold for some S A 0 6 5 S S (D OF) S (Hg OIg ) 8A0 7 (D OF)W S S 0 W (Hg OIg ) S 0 L ¸ W W S F Y 0 Y F 1L 1@2

where 2

(11.40)

(11.41)

denotes the maximum singular value of matrix Y Ig =

Proof. The proof of this theorem is straightforward. Indeed, it follows from (11.27) that q q q W W u¯W (n)¯ u(n) (Y F h¯(n)) Y F h¯(n) + (Y Ig g(n)) Y I g(n)= Thus, according to the discrete time version of Lemma 9.1 the ﬁrst term is bounded by q 1@2 (Y F h¯(n))W Y F h¯(n) 1 g>2 and the second term by q 1@2 (Y I g(n))W Y I g(n) max (Y Ig ) g>4 = 2 g>4 =

t u

Theorem 11.3 Given system model (11.24)-(11.25) and 1 A 0> 2 A 0, and assume that q kg(n)k2 g>2 > ;n gW (n)g(n) g>4 = Then, ;n £ ¤ u (n) = E (u(n) u¯(n))W (u(n) u¯(n)) 1

n1 X

¡ W ¡ ¢ ¢ g (m)g(m) + xW (m)x(m) + 2 gW (n)g(n) + xW (n)x(n)

(11.42)

m=0

1 2g>2 + 2 2g>4 + 1

n1 X

¡ W ¢ x (m)x(m) + 2 xW (n)x(n) := u>max (x) (11.43)

m=0

if the following matrix inequalities hold for some S A 0 so that

11.2 Residual evaluation scheme for stochastically uncertain systems

351

6

5

S 00 P1 ? 7 0 L 0 8 > PF 1 S> PG 2 L (11.44) 0 0L 5 W 6 5 W 6 Du>l Du>0 o X ¤ ¤ £ £ W 8 W 8 S Du>0 Eu>0 Hu>0 + S Du>l Eu>l Hu>l 2l 7 Eu>l P1 = 7 Eu>0 W W l=1 Hu>0 Hu>l (11.45) W PF = Fu>r Fu>r +

PG =

o X

2l

l=1

o X

W 2l Fu>l Fu>l

(11.46)

l=1

¸ W ¤ £ Gu>l Gu>l Iu>l = W Iu>l

(11.47)

The proof of this theorem is identical with the one of Theorem 8.3 and is thus omitted here. Theorem 11.4 Given system model (11.24)-(11.25) and 1 A 0> 2 A 0, and assume that max kg(n)k2 g>2 then ;n and for = 1 2 (1 + 1 ) > n1 X¡ ¤ ¢ £ gW (m)g(m) + xW (m)x(m) E hW{ (n)h{ (n) ?

(11.48)

m=0

2g>2 +

n1 X

¡ W ¢ x (m)x(m) := {>max (x)

(11.49)

m=0

if the following matrix inequalities hold for some S A 0 W ¸ ˜u>0 ¤ £ Du>0 S Du>0 S DWu>0 S E ˜u>0 = Eu>0 Hu>0 0> E W ˜ 1 L Eu>0 S Du>0 6 5 S 00 P1 ? 7 0 L 0 8 0 0L S 2L

(11.50)

(11.51) (11.52)

where P1 is given in (11.45). Proof. By proving Theorem 8.3, it has been shown that for given S A 0 wudfh (S H{ (n)) +

{ ¯Wu (n)S { ¯u (n)

?

n1 X m=0

if (11.51) is true. Since

¡ W ¢ g (m)g(m) + xW (m)x(m)

352

11 Integration of norm based and statistical methods

{ ¯Wu (m)S { ¯u (m) 1

n1 X

¡ W ¢ g (m)g(m) + xW (m)x(m) +,

m=0

1

n1 X

¡ W ¢ g (m)g(m) + xW (m)x(m) { ¯Wu (m + 1)S { ¯u (m + 1) + { ¯Wu (m)S { ¯u (m) 0

m=0

+, (11.50) holds it turns out n1 X¡ £ ¤ ¢ gW (m)g(m) + xW (m)x(m) 2 E hW{ (n)h{ (n) ? (1 + 1 ) m=0

when S 2 L= t u

The theorem is thus proven.

Theorem 11.5 Given system model (11.24)-(11.25) and A 0, and assume that max kg(n)k2 g>2 then ;n and for % = 1 n1 X¡ £ W ¤ ¢ gW (m)g(m) + xW (m)x(m) E {u (n){u (n) ? %

(11.53)

m=0

% 2g>2 + %

n1 X

¡

¢ xW (m)x(m) := 2{u (x)

(11.54)

m=0

if there exists S A 0 so that

6 S 00 P1 ? 7 0 L 0 8 0 0L 5

S L

(11.55) (11.56)

where P1 is given in (11.45). Proof. In the proof of Theorem 8.3, it has been shown that for given S A 0 X¡ ¤ n1 ¢ £ gW (m)g(m) + xW (m)x(m) E {Wu (n)S {u (n) ? m=0

if (11.55) holds. As a result, for given A 0> S L leads to X¡ ¤ n1 ¢ £ gW (m)g(m) + xW (m)x(m) = E {Wu (n){u (n) ? m=0

The theorem is thus proven.

t u

11.2 Residual evaluation scheme for stochastically uncertain systems

353

We would like to call¡ reader’s attention to the fact that the bounds of ¢ { (n)> u (n) as well as E {Wu (n){u (n) are respectively a function of the input signal x(l)> l 5 [n Q> n)= As a result, the threshold deﬁned by (11.36) is an adaptive threshold or a threshold generator driven by x(n). Based on the above theorems, we are now able to present the following algorithm for the threshold computation by a given false alarm rate and evaluation window [n Q> n)= Algorithm 11.4 Threshold computation Step 1: Compute deﬁned by (11.32) using the results given in Theorems 11.3-11.5 Step 2: determine (Q ) as deﬁned by(11.38) Step 3: Set Mwk according to (11.36). Remark 11.1 We would like to emphasize that increasing Q may signiﬁcantly decrease the threshold and thus enhance the fault detectability. But, this is achieved at the cost of an early fault detection. Example 11.2 Consider the benchmark system lateral dynamic system introduced in Subsection 3.7.4. For our purpose, the discrete time model (3.77) is used. It is well-known that among the parameters in model (3.74) the front 0 and the rear cornering stiness FK may vary over cornering stiness FY a large range, depending on the road condition and driving maneuvers. Following (3.79) and (3.80), we have ¸ ¸ 0=0388 0=0024 0=0108 > E1 = 104 D1 = 104 0=1208 0=0201 0=3952 ¸ ¸ 1=475 0=0002 0=5405 F1 = 103 > G1 = 103 = 0 0 0 Note that D1 and E1 have been calculated after a discretization. For our study purpose, we also make the following assumptions on the bounds of dierent variables and signals: (a) the variance of F>y is limited to 5000 (b) the upper bound of L2 norm and L4 norm of the disturbance g and the control input x are respectively g>2 = 0=046> g>4 = 0=085 and x>2 = 0=23> x>4 = 0=045 (c) the uniﬁed solution has been applied for the construction of the residual generator with ¸ ¸ 0=0010 0=0038 0=0125 0 > Yrsw = = Orsw = 0=0015 0=0810 0 1 Now, we are in a position to apply Algorithm 11.4 to set the threshold. For the computation of in Step 1, we get

354

11 Integration of norm based and statistical methods

5

m=4

6 0=3993 0=2542 0 0 9 7=4014 0=3089 : 0 0 : = (L Du0 )1 Du0 = 9 7 0 0 0=9384 0=2771 8 0 0 9=2621 0=1516 5 W 6 Du>0 £ ¤ W W 8 W Du>0 Eu>0 0 (m=4) Fu0 T(m=4) = 2FY 7 Eu>0 0 5 6 0=0994 0=0008 0 0 0=0365 0 9 0=0127 0=0001 0 0 0=0046 0 : 9 : 9 0 0 00 0 0: 5 9 : = 10 9 0 0 00 0 0: 9 : 7 0=1715 0=0014 0 0 0=0629 0 8 0 0 00 0 0 5

(m=4)

6 0=1989 0=0135 0 0 0=1351 0 9 0=0135 0=0002 0 0 0=0032 0 : 9 : 9 0 0 00 0 0: 5 9 : = 10 9 0 0 00 0 0: 9 : 7 0=1351 0=0032 0 0 0=1258 0 8 0 0 00 0 0 5 6 00 0 0 90 0 0 : 0 : !(m=4) = 9 7 0 0 5=6969 8=3274 8 0 0 8=3274 0=2884

as well as pd{ (m ) = 2=5 × 106 = % > pd{ (!m ) = 11=55 = %! £ ¤ u>pd{ = 3=3579 × 104 > max H {Wu (n){u (n) = 0=7512 which gives = 15=7= To compute (Q ) required in Step 2, it is further calculated that 2u¯(n) = 0=0072 which, for an allowable FAR = 5% and Q = 30, leads to (Q ) = 214=0329= Finally, Mwk is set constructed according to (11.36). Considering that Mwk depends on the input signal x(n)> Mwk will be on-line computed. Fig.11.5 and Fig.11.6 are the simulation results of the evaluated residual signals, in comparison with the threshold that is "on-line" calculated.

11.2 Residual evaluation scheme for stochastically uncertain systems

355

1 fault

0.8 0.6 0.4 0.2

evaluated residual/ threshold

0

0

5

10

15 Time [s]

20

25

30

1 0.8 0.6 0.4

residual threshold

0.2 0

0

5

10

15 Time[s]

20

25

30

Fig. 11.5 The evaluated residual signal, threshold and a fault in yaw rate sensor

1 fault

0.8 0.6 0.4 0.2

evaluated residual/ threshold

0

0

5

10

15 Time [s]

20

25

30

1 0.8 0.6 0.4

residual threshold

0.2 0

0

5

10

15 Time[s]

20

25

30

Fig. 11.6 The evaluated residual signal, theshold and a fault in the steering sensor

From the above example we see that the threshold setting scheme may be conservative. Also, the design is involved, in which a number of LMIs should be solved for achieving the needed bound estimations of dierent variables. It also leads to the conservative threshold setting.

356

11 Integration of norm based and statistical methods

11.3 Probabilistic robustness technique aided threshold computation In this section, we introduce a new strategy for the computation of thresholds and false alarm rate. This study is motivated by the observation that a new research line has recently, parallel to the well-developed robust control theory, emerged, where the robust control problems are solved in a probabilistic framework. Comparing with the method introduced in the last section, the implementation of this technique demands less involved computation and the threshold setting is less conservative. It opens a new and eective way to solve FD problems and builds an additional link between the traditional statistic testing and the norm based residual evaluation methods. 11.3.1 Problem formulation Consider the system model ¯ + Gx ¯ + I¯g g ¯ + Ex ¯ +H ¯g g> | = F{ {˙ = D{ ¯ = E + E> F¯ = F + F D¯ = D + D> E ¯ ¯g = Hg + Hg > I¯g = Ig + Ig G = G + G> H

(11.57)

where D> E> F> G> Hg ,Ig represent model uncertainties satisfying ¸ ¸ £ ¤ D E Hg H JKM = I F G Ig with known matrices H> I> J> K> M of appropriate dimensions. Dierent to the similar model form introduced in Chapter 3, represents a norm-bounded uncertainty and is expressed in terms of a random matrix with a known prob¯ ( ) } > where ability distribution i over its support set > := { : ¯ (·) denotes the maximal singular value of a matrix. As described in Chapter 3, we model the inﬂuence of these faults by ¯ + EI )x + H ¯g g + Hi i {˙ = (D¯ + DI ){ + (E ¯ ¯ | = (F + FI ){ + (G + GI )x + I¯g g + Ii i

(11.58) (11.59)

where DI > EI ,FI > GI and Hi i> Ii i represent multiplicative and additive faults in the plant, actuators and sensors, respectively. It is assumed that i 5 Uni is a unknown vector, Hi > Ii are known matrices with appropriate dimensions and DI > EI ,FI > GI are unknown. To simplify the notation, we shall use ¸ DI EI I = FI GI to denote the set of multiplicative faults so that I = 0> i = 0 indicate the fault-free case and otherwise there exists at least one fault.

11.3 Probabilistic robustness technique aided threshold computation

357

For the residual generation purpose, an observer-based fault detection system of the following form { ˆ˙ = Dˆ { + Ex + O(| |ˆ)> |ˆ = F { ˆ + Gx> u(v) = U(v)(|(v) |ˆ(v))

(11.60)

is applied, where U(v) 5 RH4 denotes the post-ﬁlter. The dynamics of the above residual generator is governed by u(v) = U1 (v)*x>g (v) + U2 (v)g(v) ¯ ¯ +H ¯g g> *x>g = (J{ + Kx + Mg) {˙ = D{ + Ex with

(11.61) (11.62)

³ ´ 1 U1 (v) = U(v) I + F (vL D + OF) (H OI ) 1

: = Gu1 + Fu (vL Du ) Eu1 ³ ´ U2 (v) = U(v) Ig + F (vL D + OF)1 (Hg OIg ) : = Gu2 + Fu (vL Du )1 Eu2 = For the purpose of residual evaluation, the L2 norm of u(w) is used: μZ M = kuk2 = We know that

0

4

¶1@2 u> (w)u(w)gw =

(

° ° ° ° M sup kuk2 : kgk2 g > °*x>g °2 sup °*x>g °2 ° ° kU1 k4 sup °*x>g °2 + kU2 k4 g

) (11.63)

g>x

(11.64)

g>x

where sup kgk2 = g = We would like to emphasize: • the system model is assumed to be stable and (D> F) is detectable • due to the existence of model uncertainty the residual signal is also a function of the system input x. Since x and its norms are, dierent from the unknown inputs, known during the on-line implementation, this information should be used to improve the performance of the FD system. As a result, the threshold is driven by x. • although the following study can be carried out on the basis of (8.24)(8.25), we adopt (11.61)-(11.62) for our study, because it will lead to a considerable simpliﬁcation of the problem handling.

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11 Integration of norm based and statistical methods

We now begin with the formulation of the problems to be solved in this section. Let the false alarm rate I DU be deﬁned as I DU = prob {kuk2 A Mwk | I = 0> i = 0} =

(11.65)

Our ﬁrst problem to be addressed in this section is to estimate I DU by a given threshold Mwk = It follows from (11.64) that prob {M Mwk | I = 0> i = 0}

(

° ° prob kU1 k4 sup °*x>g °2 + kU2 k4 g Mwk

) =,

g>x

I DU = 1 prob {M Mwk | I = 0> i = 0} 1 ( ) ° ° = prob kU1 k4 sup °*x>g °2 + kU2 k4 g Mwk (

g>x

° ° Mwk kU2 k4 g = prob sup °*x>g °2 kU1 k4 g>x

(11.66) (11.67)

) =

Thus, the problem of estimating I DU is reduced to ﬁnd an estimate for when Mwk is given. On the other side, following the deﬁnition of I DU and (11.66), we have, for a given allowable I DUd > 1 I DUd , 1 I DUd , prob {M A Mwk | I = 0> i = 0} I DUd = As a result, the second problem of ﬁnding Mwk so that the real I DU is smaller than I DUd can be formulated as ﬁnding Mwk so that the following inequality holds 1 I DUd = (11.68) 11.3.2 Outline of the basic idea It is clear that the core of the problems to be solved is the computation of the probability that an inequality related to the random matrix holds= We propose to solve those two probabilistic problems formulated in the last subsection using the procedure given below: • Denote inequality ° ° sup °*x>g °2 (Mwk kU2 k4 g ) @ kU1 k4 g>x

by j( ) > where is independent of = Find an algorithm to compute the norms used in j( ) on the assumption that is given

11.3 Probabilistic robustness technique aided threshold computation

359

• Generate Q matrix samples of > 1 > · · · > Q on the assumption that the random matrix is uniformly distributed in the spectral norm ball • Generate Q samples j( 1 )> · · · > j( Q ) using the algorithms developed in the ﬁrst step • Construct an indicator function of the form ½ ¡ ¢ 1> if j( l ) L l = l = 1> · · · > Q 0> otherwise • An estimation for prob{j( ) } is ﬁnally given by

1 Q

Q ¡ ¢ P L l = l=1

In the following of this section, we are going to realize this idea step by step. 11.3.3 LMIs needed for the solutions We now consider inequality ° ° sup °*x>g °2 (Mwk kU2 k4 g ) @ kU1 k4 = g>x

It follows from (11.62) that ° ° sup °*x>g °2 = k x k4 kxk2 + k g k4 g g>x

where k x k4 x k g k4 g

= sup {k*x k2 : kxk2 = 1} ¯ x + Ex> ¯ *x = (J{x + Kx) : {˙ x = D{ = sup {k*g k2 : kgk2 = 1} ¯ g+H ¯g g> *g = (J{g + Mg) = : {˙ g = D{

Hence, that the above inequality can be further written into k x k4 kxk2 + k g k4 g (Mwk kU2 k4 g ) @ kU1 k4 =

(11.69)

Note that the term on the right side of (11.69) is independent of the model uncertainty regarding to = We denote it by := (Mwk kU2 k4 g ) @ kU1 k4

(11.70)

where kU1 k4 > kU2 k4 can be computed, according to Lemma 7.8 by solving the following LMI problem: kUl k4 = min l := l min , l = 1> 2 5 W 6 Du S + S Du S Eul FuW W W 8 Eul S l L Gul subject to 7 ? 0> S A 0= Fu Gul l L

(11.71)

360

11 Integration of norm based and statistical methods

Dierently, both k x k4 and k g k4 depend on and will be computed, again using Lemma 7.8, as follows: 5 7

5 7

k x k4 = min 1 := 1 min ( ) subject to W

W

6

W

6

(11.72)

(D + H J) S + S (D + H J) S (E + H K) ( J) W W (E + H K) S 1 L ( K) 8 ? 0> S A 0 J K 1 L k g k4 = min 2 := 2 min ( ) subject to W

(11.73)

(D + H J) S + S (D + H J) S (Hg + H M) ( J) W W (Hg + H M) S 2 L ( M) 8 ? 0> S A 0= J M 2 L

Hence, inequality (11.69) can be rewritten into 1 min ( ) kxk2 + 2 min ( ) g := j( ) =

(11.74)

11.3.4 Problem solutions in the probabilistic framework Assume that the probability distribution i of over its support set is given. Using (11.71)-(11.74), our original problems can be reformulated as: • Problem 1: estimate for a given Mwk = prob {j( ) } =

(11.75)

• Problem 2: for a given admissible I DUd > determine Mwk such that I DU = prob(M A Mwk | I = 0> i = 0) I DUd =

(11.76)

The core of these two problems is the computation of the probability that some LMIs are solvable. For this purpose, the so-called probabilistic robustness technique can be used in the form of the procedure described in Subsection 11.3.2. In the framework of the probabilistic robustness technique, the so-called randomized algorithms provide an eective method to generate samples of a matrix uniformly distributed in the spectral norm ball. Assume that j( ) is a Lebesgue measurable function of and Q matrix samples of > 1 > · · · > Q , are generated. Then an empirical estimation of the probability of j( ) is given by q* ˆ= Q where q* is the number of the samples for which j( l ) holds. It is well known that for any 5 (0> 1) and 5 (0> 1), if Q

log 2 22

(11.77)

11.3 Probabilistic robustness technique aided threshold computation

361

then it holds prob {|ˆ | } 1 > = prob {j( ) } =

(11.78)

In fact, describes the accuracy of the estimate and the conﬁdence. In the following, we focus our attention on the application of the probabilistic robustness technique for solving the above-deﬁned Problem 1 and Problem 2. We refer the interested reader to the references given at the end of this chapter for the details of the randomized algorithms. To solve Problem 1, we propose the following algorithm. Algorithm 11.5 Solution of Problem 1 Step 1: Generation of Q matrix samples 1 > · · · > Q using the available randomized algorithms, where Q is chosen to satisfy (11.77) for given and Step 2: Construction of indicator functions for given Mwk : for l = 1> · · · > Q ½ ¡ ¢ 1> if 1 min ( l ) kxk2 + 2 min ( l ) g L2 l = (11.79) 0> otherwise Step 3: Computation of the empirical probability ˆQ =

Q 1 X ¡ l¢ L2 = Q l=1

(11.80)

As a result, we have an estimation for I DU, denoted by I DUh > ˆQ = I DUh = 1

(11.81)

According to (11.78), we know that Q | } 1 prob {|I DUh I DU| } = prob {|ˆ +, prob {|I DUh I DU| A } =

(11.82)

(11.82) gives the conﬁdence of I DUh as an estimate of I DU= For the solution of Problem 2, we propose the following algorithm. Algorithm 11.6 Solution of Problem 2 Step 1: Generation of Q matrix samples 1 > · · · > Q using the available randomized algorithms, where Q is again chosen to satisfy (11.77) for given and Step 2: Computation of l = 1 min ( l ) kxk2 + 2 min ( l ) g > l = 1> · · · > Q

362

11 Integration of norm based and statistical methods

Step 3: Construction of indicator functions ½ ¡ ¢ 1> if m l > l> m 5 {1> · · · > Q } L2l m = 0> otherwise Step 4: Computation of ˆlQ =

Q 1 X l ¡ m¢ L Q m=1 2

(11.83)

Step 5: Determination of threshold for given I DUd and Mwk = 1 min n + 2 min g > n = 1 min ( n ) kxk2 + 2 min ( n ) g (11.84) © l ª ˆQ | ˆlQ 1 I DUd + = n = arg min l5(1>··· >Q )

We now check the real false alarm rate I DU = prob {M A Mwk | I = 0> i = 0} with Mwk satisfying (11.84). Since ¯ n¯ o ¯ n ¯ Pr ¯ ˆQ 1 + I DU¯ 1 we ﬁnally have that setting Mwk according to (11.84) ensures prob {I DUd ˜ I DU I DUd } 1 where ˜ is some constant larger than zero. Thus, the requirement that I DU I DUd is satisﬁed with a probability not smaller than 1 = 11.3.5 An application example We now study an application example and illustrate the results achieved in this section. The system under consideration is again the benchmark system vehicle lateral dynamics introduced in Subsection 3.7.4. In our study, it is assumed that only yaw rate sensor is used. The purpose of our study is to • estimate the FAR of an observer-based FD system designed based on (3.76), where the false alarms are caused by the stochastic model uncer0 and FK ; tainty due to FY • compute the threshold for the observer-based FD system under a given FAR. The following assumptions are made: 0 r • FY = FY + FY > FY 5 [10000> 0] is a random number with uniform distribution; 0 • FK = nFY > n = 1=7278=

11.3 Probabilistic robustness technique aided threshold computation

363

For our study, (3.74)-(3.75) are rewritten with x = O > | = u and 6 5 r0 " Fr # (1+n)F r (noK oY )FY Y pyuhiY 1 2 py uhi 8 > E = pyuhi D = 7 (no o )F r0 r 2 2 r o F Y Y (o +no )F K Y Y Y L} yKuhi Y L} L} " " 1+n noK oY # # 1 pyuhi py2 pyuhi uhi D = no o FY > E = FY oY o2 +no2 K Y YL} yuhiK L} L} F = 0> G = 0> Hg = 0> Ig = 0= For the purpose of residual generation, the following observer is used 6 r r0 # " # 5 (1+n)FY r (noK oY )FY ¸ " FY ¸ ˆ g pyuhi 1 ˆ 2 o1 py py uhi uhi 8 7 gw (u uˆ) + = x + r r0 2 r oY FY gˆ u (noK oY )FY (oY +no2K )FY o u ˆ 2 gw L } L} L} yuhi residual signal u = u uˆ with o1 = 10=4472> o2 = 31=7269= In our study, dierent driving maneuvers have been simulated by CARSIM, a standard program for the simulation of vehicle dynamics. Here, results are presented, which have been achieved using the input data generated by CARSIM during the so-called Double Lane Changing (DLC), a standard driving maneuver for simulation. The sample size N: It follows from (11.77) that for given = 0=02> = 0=01> log 2 = 10000= Q 22 Q = 10000 has been used in our study. Table 11.1 Estimation of FAR for a given threshold The given threshold Mwk Estimation of I DU (%)

0.078 8.43

0.080 6.22

0.082 3.90

0.084 1.38

Table 11.2 Computation of threshold for a given I DUd The given I DUd (%) The achieved threshold Mwk

10 0.0766

5 0.0811

2 0.0836

In Table 11.1, FAR estimations with respect to dierent threshold values, achieved using Algorithm 11.5, are listed, while in Table 11.2 the result of the threshold computation for dierent I DUd > using Algorithm 11.6, is included.

364

11 Integration of norm based and statistical methods

11.3.6 Concluding remarks It is well-known that the sample size Q plays an important role in estimating empirical probability. Increasing Q will improve the estimation performance but also lead to higher computation costs. For this reason, eective algorithms should be used for solving the above-mentioned problems. The reader is referred to the references given at the end of this chapter for such algorithms. Also for the same reason, our study has been carried out on the basis of (11.61) instead of the original form, in order to avoid norm computations for systems of the 2q-th order (the order of the system + the order of the observer). The solution of Problem 1 is useful for the analysis of observer-based FD systems. It is shown that for a given constant threshold the FAR will be a function of system input signals. To ensure a constant FAR, the adaptive threshold should be adopted, as demonstrated by the solution of Problem 2. The solution of Problem 2 provides a useful tool to set a suitable threshold and to integrate it into the design of observer-based FD systems. The introduction of the adaptive threshold ensures that the requirement on the FAR will be satisﬁed for all possible operation states of the process under consideration. The basic idea behind our study in this section is the application of the probabilistic robustness theory for computing the thresholds and FAR. Different from the known norm based residual evaluation methods, in which the threshold computation is based on the worst-case handling of model uncertainty and unknown inputs, our study leads to the problem solutions in the probabilistic framework and may build a bridge between the well-established statistic testing methods and the norm based evaluation methods. Although the study carried out in this section aims at solving the fault detection problem, it is expected that the achieved results can also be extended to solving the fault isolation problem if statistical knowledge of faults, for instance their probability distribution, is available. The methods presented here can also be extended to the cases, where L4 norm or the modiﬁed forms of L2 , L4 norms like RMS or peak value are used as residual evaluation function.

11.4 Notes and references In this chapter, we have introduced three dierent schemes for the purpose of residual evaluation and threshold setting. But, they have one in common: the integration of the norm based methods and statistical methods builds the core of these schemes. Section 11.1 is in fact an extension of the discussion in Section 10.4. Some of the results have been provisionally published in [35]. [70] also addresses fault detection in systems with both stochastic and deterministic unknown inputs, where the inﬂuence of the deterministic unknown inputs is, however, decoupled from the residual signal by means of solving the PUIDP (see Chapter 6).

11.4 Notes and references

365

In our view, the most important message of this section is that the integration of the norm based and statistic based methods may help us to improve the performance of FDI systems. As mentioned in the previous chapter, the reader is referred to [8, 10] for the needed knowledge of the GLR technique. Motivated by the observation in practice that a priori knowledge of the model uncertainties is generally limited, the study in Section 11.2 has been devoted to those systems, which are corrupted with stochastically uncertain changes in the model parameters. Dierent from the study in Section 11.1, we have only information about mean values and variances instead of the distribution of the stochastically uncertain variables. To handle such systems and to solve the associated FDI problems, the LMI technique based analysis and synthesis methods for systems with multiplicative stochastic noises are adopted as tool, which has been introduced in Chapter 8. For more details and the needed mathematical skills, we refer the reader again to [14, 130]. We would like to call reader’s attention to the fact that the key to link the norm based methods and statistical handling, as done in this section, is the Tchebyche Inequality [112]. This idea is ﬁrst proposed by Li et al. [94]. The probabilistic robustness technique is a new research line that has recently, parallel to the well-developed robust control theory, emerged [17, 18, 16]. This technique allows to solve robust control problems in a probabilistic framework and opens a new and eective way to solve fault detection problems. The design scheme presented in Section 11.3 is the preliminary result achieved by the ﬁrst application of the probabilistic robustness technique to addressing the fault detection problems. A draft version of this scheme has been reported in [33]. An advanced study with an application can be found in [125, 154]. In our view, the probabilistic robustness technique is an alternative way to build a bridge between the well-established statistic testing schemes and the norm based methods.

Part IV

Fault detection, isolation and identiﬁcation schemes

12 Integrated design of fault detection systems

The objective of this chapter is the design of model-based fault detection systems. In the literature related to the observer-based FDI technique, this task is often (mis)understood as the design of a residual generator. In the last part, we have studied the residual evaluation issues and learned the important role of the residual evaluation unit in an FDI system. A three-step design procedure with • construction of a residual generator under a given performance index • deﬁnition of a suitable residual evaluation function and, based on it, • determination of a threshold seems a logical consequence of our study in the last two parts towards the design of the model-based FDI system. On the other side, it is of considerable practical interests to know if an integrated design of the fault detection system, i.e. the design of the residual generator, evaluator and threshold in an integrated manner instead of separate handling of these units, will lead to an improvement of the FD system performance. This question is well motivated by the observation that the residual evaluation function and threshold computation have not been taken into account by the development of the optimal residual generation methods, as introduced in Part II. A residual generator optimized under some performance index does not automatically result in an optimal fault detection system. Now, a critical question may arise: what is the criterion for an optimal fault detection system? Having studied the last two parts, the reader should have gained the impression that many model-based FDI problems have been handled in the context of the advanced control theory and an optimum FD system is understood in the context of robustness vs. sensitivity. In practice, essential requirements on a fault diagnosis system are generally expressed in terms of a lower false alarm rate and a higher fault detection rate, and an optimal trade-o between them is of primary interest in designing an

370

12 Integrated design of fault detection systems

FD system. In this context, a separate study on residual generation, evaluation and threshold computation makes less sense. To achieve a successful design of a model-based FD system, an integrated handling of residual generator, evaluator and threshold is needed. False alarms are caused by disturbances and model uncertainties. In order to reduce them, thresholds are introduced, which result, in turn, in a reduction of fault detection rate. The core of designing a fault detection system is to ﬁnd out a suitable trade-o between the false alarm rate and the fault detection rate. In fact, the concepts robustness and sensitivity are the "translation" of false alarm rate and fault detection rate into the language of control theory. Unfortunately, their application is mostly restricted to the residual generator design. In this chapter, we shall study the integrated design of fault detection systems, as sketched in Fig.12.1, and introduce two design strategies. A further focus of our study is to re-view some residual generation methods introduced in Part II in the context of the trade-o between the false alarm rate and the fault detection rate.

Fig. 12.1 Integrated design of fault detection systems

12.1 FAR and FDR As introduced in the last two chapters, false alarm rate (FAR) and fault detection rate (FDR) are two concepts that are originally deﬁned in the statistic framework. Suppose that u is a residual vector that is a stochastic process corrupted with the unknown input vector g and the fault vector i= We denote the residual evaluation function, also called testing statistic, by M = kukh and the corresponding threshold by Mwk > and suppose that the fault detection decision

12.1 FAR and FDR

371

logic is M Mwk =, fault-free M A Mwk =, faulty.

(12.1) (12.2)

Deﬁnition 12.1 The probability I DU I DU = prob (M A Mwk | i = 0)

(12.3)

is called false alarm rate in the statistical framework. Deﬁnition 12.2 The probability I GU I GU = prob (M A Mwk | i 6= 0)

(12.4)

is called fault detection rate in the statistical framework. Deﬁnition 12.3 The probability 1 I GU = prob (M Mwk | i 6= 0)

(12.5)

is called missed detection rate (P GU) in the statistical framework. For FD systems with deterministic residual signals, it is obvious that new deﬁnitions are needed. Ding et al. have ﬁrst introduced the concepts FAR and MDR in the context of a norm based residual evaluation. Below, we shall concentrate ourselves on the establishment of a norm based framework, which will help us to evaluate the performance of a model-based FD system in the context of the trade-o between the FAR and FDR. To simplify the presentation, we ﬁrst introduce the following notations. We denote the residual generator by Gu and suppose that Gu generates a residual vector u which is driven by the unknown input vector g> the fault vector i and aected by the model uncertainty = We assume that g is bounded and express it by kgk g where k·k stands for some signal norm. We denote the norm based evaluation of u by M = kukh threshold by Mwk > and suppose that the decision logic (12.1)-(12.2) is adopted for the detection purpose. The objective of introducing the concept FAR is to characterize the FD system performance in terms of the intensity of false alarms during system operation. A false alarm is created if M A Mwk for i = 0=

(12.6)

372

12 Integrated design of fault detection systems

Deﬁnition 12.4 Given residual generator Gu and Mwk > the set I D (Gu > Mwk ) deﬁned by

I D (Gu > Mwk ) = {g | (12.6) is satisﬁed} (12.7) is called set of disturbances that cause false alarms (SDFA). The size of SDFA indicates the number of the possible false alarms and thus builds a direct measurement of the FD system performance regarding to the intensity of false alarms. On the other side, it is very di!cult to express FAR in terms of the size of SDFA and moreover the determination of the size of SDFA depends on Mwk = For this reason, we introduce the following simpliﬁed deﬁnition for FAR. Consider that in the fault-free case ;g> > kukh kgk g with denoting the induced norm deﬁned by =

sup i =0>>kgkg

kukh =

Thus, a threshold equal to g will guarantee a zero FAR. It motivates us to introduce Deﬁnition 12.5 I DU given by I DU = 1

Mwk g

(12.8)

is called FAR in the norm based framework. Note that for Mwk = 0 I DU = 1> I D (Gu > 0) = max I D (Gu > Mwk ) Mwk

(12.9)

i.e. ;Mwk A 0> I D (Gu > Mwk ) I D (Gu > 0) and for Mwk = g I DU = 0> I D (Gu > g ) = min I D (Gu > Mwk ) Mwk

(12.10)

i.e. ;Mwk ? g > I D (Gu > g ) I D (Gu > Mwk )= The introduction of the concept FDR is intended to evaluate the FD system performance from the viewpoint of fault detectability, which is understood as the set of all detectable fault. Recall that a fault is detected if M A Mwk for i 6= 0= We have

(12.11)

12.2 Maximization of fault detectability by a given FAR

373

Deﬁnition 12.6 Given residual generator Gu and Mwk > the set GH (Gu > Mwk ) deﬁned by

GH (Gu > Mwk ) = {i | (12.11) is satisﬁed} (12.12) is called set of detectable faults (SDF). The size of SDF is a direct measurement of the FD system performance regarding to the fault detectability. Similar to the case with the FAR, we now introduce the simpliﬁed deﬁnition of FDR in the norm based framework. For our purpose, we call reader’s attention to the fact that on the assumption g = 0> = 0> ;i 6= 0> kukh ki k where =

inf

i 6=0>g=0>=0

kukh =

Suppose that those i vectors whose size is larger than i>min are deﬁned as faults to be detected. Then, setting threshold equal to i>min will give a 100% fault detection. Bearing this in mind, we have Deﬁnition 12.7 I GU given by I GU =

i>min Mwk

(12.13)

is called FDR in the norm based framework. Note that on the assumption g = 0> = 0 we have for Mwk = i>min

GH (Gu > i>min ) =

max

Mwk i>min

GH (Gu > Mwk )=

According to this deﬁnition, given an I GU> the threshold should be set as Mwk =

i>min = I GU

(12.14)

12.2 Maximization of fault detectability by a given FAR In this section, we present an approach in the norm based framework, which will lead to a trade-o between the FDR and FAR, expressed in terms of maximizing the number of detectable faults by a given FAR. Our focus is not only on the derivation of this approach but also on an evaluation of the existing optimal residual generation approaches in the context of the trade-o strategy.

374

12 Integrated design of fault detection systems

12.2.1 Problem formulation For the sake of simplicity, we consider system model |(s) = J|x (s)x(s) + J|g (s)g(s) + J|i (s)i (s)

(12.15)

and apply residual generator ´ ³ ˆx (s)x(s) > U(s) 5 RH4 ˆ x (s)|(s) Q u(s) = U(s) P

(12.16)

for the residual generation purpose as well as use L2 norm as the residual evaluation function. Recall that according to detection logic (12.1)-(12.2) a fault can be detected if and only if (12.17) kuk2 A Mwk +, ¯ g (s)g(s) + J ¯ i (s)i (s))k2 A Mwk - detection condition kU(s)(J and in the fault-free case if ¯ g (s)g(s)k2 A Mwk - false alarm condition kuk2 = kU(s)J

(12.18)

then a false alarm will be released, where ¯ g (s) = P ˆ x (s)J|g (s)> J ¯ i (s) = P ˆ x (s)J|g (s) J and are assumed to be stable. Suppose that the allowable I DU is now given. It follows from the false alarm condition (12.18) and the deﬁnition of FAR that the threshold should be set as ¯ g (s)k4 g = (12.19) Mwk = (1 I DU) kU(s)J For our design purpose, we formulate our trade-o design problem as follows: Problem of Maximizing SDF under a given FAR (PMax-SDF): Given I DU and Mwk setting according to (12.19), ﬁnd Ursw>GH (s) 5 RH4 so that ;U(s) 5 RH4 > GH (U> Mwk ) GH (Ursw>GH > Mwk )= (12.20) (12.20) means that GH (Ursw>GH > Mwk > g) is the maximum SDF and thus should give the maximum FDR. At the end of the next subsection, we shall prove that maximizing SDF is equivalent to maximizing FDR. 12.2.2 Essential form of the solution In this subsection, we shall outline the basic idea and present a solution for PMax-SDF on the following assumptions:

12.2 Maximization of fault detectability by a given FAR

S1:

¡ ¢ ¯ g (s) = ng = p udqn (Jg (s)) = udqn J

375

(12.21)

¯ g (s) = Jgr (s)Jgl (s)> Jgr (s)> is left invertible in RH4 > S2: The co-outer of J i.e. ¯ g (m$) A 0 for continuous time systems ¯ g (m$)J ;$ 5 [0> 4]> J ¯ g (hm ) A 0 for discrete time systems. ¯ g (hm )J ; 5 [0> 2]> J

(12.22) (12.23)

We now start to solve the design problem PMax-SDF. Let U(s) = T(s)J1 gr (s) with T(s) 5 RH4 standing for an arbitrarily selectable matrix of an appropriate dimension. It yields ¯ M Mwk = kT(s)J1 gr (s)Ji (s)i (s) + T(s)Jgl (s)g(s)k2 (1 I DU) kT(s)Jgl (s)k4 g = Considering that

kT(s)Jgl (s)k4 = k (T(s)Jgl (s)) k4 = kT(s)k4 ¯ i (s)i (s) + T(s)Jgl (s)g(s)k2 kT(s)J1 (s)J gr

¯ kT(s)k4 kJ1 gr (s)Ji (s)i (s) + Jgl (s)g(s)k2 it turns out ;T(s) 6= 0 5 RH4 ¢ ¡ ¯ i (s)i (s) + Jgl (s)g(s)k2 (1 I DU) g M Mwk kT(s)k4 k J1 (s)J gr

which means ¯ kJ1 gr (s)Ji (s)i (s) + Jgl (s)g(s)k2 (1 I DU) g A 0

(12.24)

is a necessary condition under which fault i (s) becomes detectable. We have proven the following theorem. Theorem 12.1 Given system (12.15), residual generator (12.16), I DU and threshold setting (12.19), a fault i (s) can then be detected only if (12.24) holds. Note that setting T(s) = L and therefore U(s) = J1 gr (s) leads to ¯ M Mwk = kJ1 gr (s)Ji (s)i (s) + Jgl (s)g(s)k2 (1 I DU) g = This means that (12.24) is also a su!cient condition for i (s) to be detectable, provided that U(s) is set to be J1 gr (s). This result provides us with the proof of the solution for PMax-SDF which is summarized into the following theorem.

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12 Integrated design of fault detection systems

Theorem 12.2 Given system (12.15), residual generator (12.16), I DU and threshold setting (12.19), then Ursw>GH (s) = J1 gr (s)

(12.25)

is the optimal solution of PMax-SDF (12.20). The following corollary follows from Lemmas 7.4 and 7.5. Corollary 12.1 Given system (12.15), residual generator (12.16), I DU and threshold setting (12.19), then ˆ g (s) Ursw>GH (s) = P

(12.26)

solves PMax-SDF (12.20). Theorem 12.2 and Corollary 12.1 reveal that the uniﬁed solution given in Theorem 7.10 for continuous time systems and Theorem 7.18 for discrete time systems solves the PMax-SDF. That also explains why the uniﬁed solution does deliver the highest fault sensitivity in the sense of Hl @H4 index. We refer the reader to Section 7.10 for the detailed discussion and description of the uniﬁed solution. Below is a summary of some important properties: • Theorems 7.11 and 7.18 provide us with the state space form of the solution (12.26). • The solution (12.26) ensures that ¯ i (m$)) ¯ i (m$)) min (U(m$)J min (Ursw>GH (m$)J ° ° ° ° = sup as well as ¯ g (v)° °Ursw>GH J °U(v)J ¯ g (v)° U(v)5RH 4 4 4 ¯ i (hm )) ¯ i (hm )) min (Ursw>GH (hm )J min (U(hm )J ° ° ° ° = sup ¯ g (})° ¯ g (})° °Ursw>GH (})J °U(})J U(})5RH4 4

4

thus, it also results in maximizing the FDR ¯ i (m$)) i>min min (Ursw>GH (m$)J i>min = as well as Mwk (1 I DU) g ¯ i (hm )) i>min i>min min (Ursw>GH (hm )J I GU = = = Mwk (1 I DU) g I GU =

• The threshold is given by Mwk = (1 I DU) g =

(12.27)

12.2.3 A general solution In this subsection, we remove Assumptions S1-S2 and present a general solution.

12.2 Maximization of fault detectability by a given FAR

377

Having learned that the uniﬁed solution given in Theorem 7.10 solves the PMax-SDF on the Assumptions S1-S2, it is reasonable to apply the general form of the uniﬁed solution given in Section 7.11 to deal with our problem. Similar to Section 7.11, due to the complexity we only consider continuous time systems in the following study. ¯ g (v) can be factorized, by means As described in Section 7.11, any given J of an extended CIOF (see Algorithm 7.9), into ¸ ¸ Jgl (v) 0 ¯ g (v) = Jgr>1 Jgr>2 (v)J4 (v)Jm$ (v) 0 (12.28) J 0 0 0 L where Jgr>1 (v)> Jgr>2 (v) are invertible in RH4 and Jgl (v) 5 RH4 is coinner. Note that the zero-blocks in the above transfer matrices exist only if Assumption S1 is not satisﬁed, i.e. ¡ ¢ ¯ g (s) = ng ? p= udqn (Jg (s)) = udqn J As a result, the generalized uniﬁed solution is given by ¸ 1 1 ˜ 1 ˜ (v)J J 4 (v)Jgr>2 (v) 0 m$ J1 Ursw (v) = 1 gr>1 (v) 0 L

(12.29)

˜ 1 ˜ 1 (v)> J where A 0 is some constant that can be enough small and J 4 (v) m$ satisfy (7.303)-(7.304). We now check if the generalized uniﬁed solution (12.29) solves the PMax-SDF. Recall that applying (12.29) to (12.16) yields ¡ ¢ ¯ i (v)i (v) = ¯ g (v)g(v) + J (12.30) u(v) = Ursw (v) J ¸ ¸ 1 ˜ 1 ˜ (v)J u1 (v) J 4 (v)J4 (v)Jm$ (v)Jgl (v)g(v) + Ji 1 (v)i (v) m$ = 1 u2 (v) Ji 2 (v)i (v) with

¸ ¸ 1 1 ˜ 1 ˜ (v)J Ji 1 (v) J 4 (v)Jgr>2 (v) 0 m$ ¯ J1 = 1 gr>1 Ji (v)= Ji 2 (v) L 0

It turns out

° ° ° ˜ 1 ° ˜ 1 M Mwk = °J m$ (v)J4 (v)J4 (v)Jm$ (v)Jgl (v)g(v) + Ji 1 (v)i (v)° 2 ° ° 1 ° ˜ 1 ° ˜ 1 + kJi 2 (v)i (v)k2 (1 I DU) °Jm$ (v)J 4 (v)J4 (v)Jm$ (v)Jgl (v)° g 4 ° ° ° ˜ 1 ° 1 ˜ 4 (v)J4 (v)Jm$ (v)Jgl (v)g(v) + Ji 1 (v)i (v)° °Jm$ (v)J 2

1 + kJi 2 (v)i (v)k2 (1 I DU) g = Thus, any fault i (v) that causes

378

12 Integrated design of fault detection systems

° ° 1 ° ° ˜ 1 ˜ 1 (v)J (v)J (v)J (v)g(v) + J (v)i (v) ° + kJi 2 (v)i (v)k2 °Jm$ (v)J 4 m$ gl i1 4 2 A (1 I DU) g (12.31) will be detected. On the other side, for any U(v) = T(v)Ursw (v)> T(v)(6= 0) 5 RH4 , it holds ° ! Ã° ° ° ˜ 1 ˜ 1 °Jm$ (v)J 4 (v)J4 (v)Jm$ (v)Jgl (v)g(v) + Ji 1 (v)i (v)° M Mwk kTk4 = 2 + 1 kJi 2 (v)i (v)k2 (1 I DU) g (12.32) Summarizing (12.31)-(12.32) gives a proof of the following theorem. Theorem 12.3 Given system (12.15), residual generator (12.16), I DU and threshold setting (12.19), then ¸ 1 1 ˜ 1 ˜ (v)J J 4 (v)Jgr>2 (v) 0 m$ J1 (12.33) Ursw>GH (v) = 1 gr>1 (v) 0 L solves the PMax-SDF. It is very interesting to note that any fault i (v) satisfying Ji 2 (v)i (v) 6= 0> kJi 2 (v)i (v)k2 A (1 I DU) g can be detected. Since can be set small enough, we can claim that any fault i (v) with Ji 2 (v)i (v) 6= 0 can be detected. Recall that Ji 2 (v) is spanned by the null space of Jg (v)> i.e. B Ji 2 (v) = JB g (v)Ji (v)> Jg (v)Jg (v) = 0=

(12.34)

Thus, any fault, which can be decoupled from the unknown input vector g(v) in the measurement subspace, can be detected. Below is the algorithm for the optimal design of FD systems in the context of maximizing the fault detectability by a given I DU. Algorithm 12.1 Optimal design of FD systems by given I DU Step Step Step Step

1: 2: 3: 4:

¯ g (v) into (12.28) Bring J ˜ 1 (v)> J ˜ 1 Find J 4 (v) according to (7.303) and (7.304) m$ Set Ursw>GH (v) according to (12.29) Set threshold Mwk according to (12.27).

12.2.4 Interconnections and comparison In this subsection, we study the relationships between the solution (12.26), i.e. the uniﬁed solution, and the design approaches presented in Chapters 6 and 7.

12.2 Maximization of fault detectability by a given FAR

379

Relationship to the PUIDP In Chapter 6, we have studied the PUIDP and learned that under condition ¤ £ (12.35) udqn Jg (s) Ji (s) A udqn(Jg (s)) = ng there exists a residual generator U(s) so that ¢ ¡ ¯ i (s)i (s) = U(s)J ¯ i (s)i (s)= ¯ g (s)g(s) + J u(s) = U(s) J As a result, the threshold will be set equal to zero. We denote the set of all detectable faults using the solution to the PUIDP by © ª ¯ i (s)i (s) 6= 0 =

GH (U> 0) = i | U(s)J It follows from (12.34) and the associated discussion that ;i 5 GH (U> 0) we also have i 5 GH (Ursw>GH > (1 I DU) g )= On the other side, it is evident that for any i satisfying Ji 2 (v)i (v) = 0 ° ° ° ˜ 1 1 ˜ 4 (v)J4 (v)Jm$ (v)Jgl (v)g(v) + Ji 1 (v)i (v)° °Jm$ (v)J ° A (1 I DU) g 2

it holds @ GH (U> 0)= i 5 GH (Ursw>GH > (1 I DU) g ) but i 5 In this way, we have proven the following theorem, which demonstrates that the solution (12.26) provides us with a better fault detectability in comparison with the PUID scheme. Theorem 12.4 Given system (12.15), residual generator (12.16) and assume that (12.35) holds, then

GH (U> 0) GH (Ursw>GH > (1 I DU) g )=

(12.36)

Relationship to H2 @H2 optimal design scheme For the sake of simplicity, we only consider continuous time systems and assume (a) I DU is set to be zero (b) Assumptions S1-S2, (12.21)-(12.22), are satisﬁed. In Section 7.7, the solution of optimization problem is given by ° ° ° ° ¯ i (v)° °te (v)ymax (v)J1 (v)J °t(v)J ¯ i (v)° gr 2 2 ° ° = sup = 1@2 max ($ rsw ) ° ¯ ° kt (v)y (v)k (v) t(v) J rsw max t(v)5RH4 g 2 2 ¡ 1 ¢ ¯ ¯ max ($)) = 0 ymax (m$)(J1 gr (m$)Ji (m$)Ji (m$) Jgr (m$) max ($ rsw ) = max max ($)> $ rsw = arg max max ($)= $

$

380

12 Integrated design of fault detection systems

Here, te (v) represents a band pass ﬁlter at frequency $ rsw , which gives μ Ã =

1 2

1 2

Z

4

4

Z

¶1@2 te (m$)ymax (m$)($)ymax (m$)te (m$)g$

$ rsw +

$ rsw

!1@2 te (m$)ymax (m$)($)ymax (m$)te (m$)g$

(ymax (m$ rsw )($ rsw )ymax (m$ rsw ))1@2 ¡ ¢ ¯ i (m$)J ¯ i (m$) J1 (m$) = ($) = J1 (m$)J gr

gr

Since (m$)te (m$) = te (m$)ymax (m$)ymax

ymax (m$ rsw )($ rsw )ymax (m$ rsw ) max ($ rsw )

the threshold Mwk>2 should be set as s ymax (m$ rsw )($ rsw )ymax (m$ rsw ) g Mwk>2 = max ($ rsw ) with = 2= Denote the SDF achieved by using H2 @H2 optimal scheme with ª © ¯

GH (trsw > Mwk>2 ) = i | ktrsw (v)(Jgl (v)g(v) + J1 gr (v)Ji (v)i (v))k2 A Mwk>2 trsw (v) = te (v)ymax (v)= (12.37) Considering

s =

¯ ktrsw (v)(Jgl (v)g(v) + J1 gr (v)Ji (v)i (v))k2 ¯ i (v)i (v)k2 ktrsw (v)k kJgl (v)g(v) + J1 (v)J 4

ymax (m$ rsw )($ rsw )ymax (m$ rsw )

max ($ rsw )

gr

¯ kJgl (v)g(v) + J1 gr (v)Ji (v)i (v)k2

we know ¯ ktrsw (v)(Jgl (v)g(v) + J1 gr (v)Ji (v)i (v))k2 A Mwk>2 only if s

ymax (m$ rsw )($ rsw )ymax (m$ rsw ) ¯ kJgl (v)g(v) + J1 gr (v)Ji (v)i (v)k2 max ($ rsw ) s ymax (m$ rsw )($ rsw )ymax (m$ rsw ) g =, A max ($ rsw )

¯ kJgl (v)g(v) + J1 gr (v)Ji (v)i (v)k2 A g =

(12.38)

12.2 Maximization of fault detectability by a given FAR

381

Recall that the last inequality is exactly the fault detection condition if the uniﬁed solution is used, i.e. for any i i 5 GH (trsw > Mwk>2 ) it also holds i 5 GH (Ursw>GH > g )= On the other side, since trsw (v) is a vector-valued post-ﬁlter with a strongly limited band, there do exist faults which belong to GH (Ursw>GH > g ) but lead to ¯ trsw (v)J1 gr (v)Ji (v)i (v) = 0 or 1 ¯ i (m$ rsw )i (m$ rsw ) 0 trsw (m$ rsw )Jgr (m$ rsw )J and thus i (v) 5 @ GH (trsw > Mwk>2 )= This proves that the solution (12.26) provides us with a better fault detectability than the H2 @H2 scheme, as summarized in the following theorem. Theorem 12.5 Given system (12.15) and residual generator (12.16), then

GH (trsw > Mwk>2 ) GH (Ursw>GH > g )=

(12.39)

Relationship to H4 @H4 and H @H4 optimal schemes Recall that H4 @H4 and H @H4 are respectively formulated as ° ° ° ° ¯ i (s)° °U(s)J ¯ i (s)° °U(s)J 4 ° ° ° and ° = sup (12.40) sup ¯ ¯ ° ° ° ° U(s)5RH4 U(s)Jg (s) U(s)5RH4 U(s)Jg (s) 4

4

Since in both formulations, the inﬂuence of g is evaluated by L2 norm, the threshold setting should follow (12.19). It is clear that both GH (U4@4 > (1 I DU) g ) and GH (U@4 > (1 I DU) g )> i.e. the sets of detectable faults that are delivered by an H4 @H4 and an H @H4 optimal residual generator, should belong to GH (Ursw>GH > g )= Without proof, we provide the following theorem. Theorem 12.6 Given system (12.15) and residual generator (12.16), then

GH (U4@4 > (1 I DU) g ) GH (Ursw>GH > (1 I DU) g ) (12.41)

GH (U@4 > (1 I DU) g ) GH (Ursw>GH > (1 I DU) g )= (12.42) At the end of this subsection, we would like to evaluate the solutions to H4 OFIP scheme and compare dierent reference model-based FD schemes introduced in Chapter 8 in the context of the trade-o between the FAR and FDR.

382

12 Integrated design of fault detection systems

Consider H4 OFIP scheme given in Section 7.6 and the reference model based design scheme with reference model uuhi = i= They can be uniﬁedly formulated as ﬁnding U(v) so that ku i k2 = ku uuhi k2 $ min +, °£ ¤° ¯ i (v) U(v)J ¯ g (v) ° = min ° L U(v)J 4

(12.43)

U(v)5RH4

In this context, we rewrite detection condition into ¯ i (v)i (v))k2 A Mwk>uhi +, kuuhi + (u uuhi ) k2 A Mwk>uhi = ¯ g (v)g(v) + J kU(v)(J Since kuuhi + (u uuhi ) k2 kuuhi k2 +

min

U(v)5RH4

ku uuhi k2

for a good optimization with a (very) small minU(v)5RH4 ku uuhi k2 > the detection condition, under a given I DU, can be approximately expressed by ¯ g (v)k4 (12.44) kuuhi k2 A Mwk>uhi = (1 I DU) g kUrsw>uhi (v)J °£ ¤° ¯ ¯ ° ° L U(v)Ji (v) U(v)Jg (v) = Ursw>uhi (v) = arg min U(v)5RH4

For our comparison purpose, we now apply the uniﬁed solution as the reference model, ˆg (v)g(v) + P ˆ g (v)J ¯ i (v)i (v) uuhi>VGI (v) = Q and rewrite detection condition into ¯ g (v)g(v) + J ¯ i (v)i (v))k2 A Mwk>uhi>VGI kU(v)(J +, kuuhi>VGI + (u uuhi>VGI ) k2 A Mwk>uhi>VGI = Under the same I DU, the detection condition, by a good optimization, is approximately given by kuuhi>VGI k2 A Mwk>uhi>VGI = (1 I DU) g =

(12.45)

Following Corollary 12.1, the residual signal generated by means of H4 OFIP solutions or the reference model based design scheme with reference model uuhi = i , would lead, under the same I DU, to a poorer fault detectability than the residual signal generated by the reference model based design scheme with reference model uuhi>VGI = In summary, our above discussion evidently demonstrates that the optimal solution (12.33) delivers the best fault detectability for a given I DU= 12.2.5 Examples The following two examples are used to illustrate our discussion in the last subsection.

12.2 Maximization of fault detectability by a given FAR

Example 12.1 Given system model ¸ 1 v1 2 +1=5v+0=5 v g(s) + v+1 |(v) = v1 0 v+0=5

1 v2 +2v+1 1 v+1

383

¸ i (v)=

and assume that I DU is required to be 0. It is easy to prove that ¤ £ 1 U(v) = 1 v+1 delivers a residual signal decoupled from g(v)> i.e.

¸ ¤ £ 1 i (v) 1 0 i (v) = i1 (v)> i (v) = 1 = u(v) = U(v)|(v) = v+1 i2 (v) v+1

The corresponding SDF is

GH (U> 0) = {i | i1 (v) 6= 0} = In comparison, applying Algorithm 12.1 yields ¸ v+0=5 ¸ 0 1 0 Ursw>GH (v) = v+1 1 1 1 v+1 0 which leads to ¸ v1 ¸ 0 u1 (v) v+1 g(v) + = 1 1 v+1 u2 (v) 0

v+0=5 v2 +2v+1

¸

0

(12.46) ¸ i1 (v) = i2 (v)

Thus, for a enough small 1 >

GH (Ursw>GH > g ) = ° ¾ ½ ° °v 1 ° v + 0=5 ° ° g(v) + 2 i2 (v)° A g {i | i1 (v) 6= 0} ^ i | ° v+1 v + 2v + 1 ° 2 ¾ ½ ° °v 1 ° v + 0=5 ° ° = GH (U> 0) ^ i | ° g(v) + 2 i2 (v)° A g = v+1 v + 2v + 1 2 This result veriﬁes the result in Theorem 12.4. Example 12.2 In this example, we concentrate ourselves on the detection of i2 (v) given in the above example, i.e. we consider system model |2 (v) =

1 v1 g(v) + i (v)= v + 0=5 v+1

(12.47)

We ﬁrst design an H2 @H2 optimal residual generator. To this end, we compute max ($) and $ rsw > 1 m$ 1 m$ 1 1 max ($) = 0 +, 1 + m$ 1 m$ 0=5 + m$ 0=5 m$ s 0=25 + $ 2 0=5= max ($) = 2 =, $ rsw = 2 (1 + $ )

384

12 Integrated design of fault detection systems

In the next step, we design a band pass around $ rsw > te>2 (v) =

v2

1 + 0=001v + 0=7

which delivers a (sub-)optimum ° ° ° 1 ° °te (v) v+1 ° °2 = 1@2 max ° max ($ rsw ) ° v1 ° te (v) °te (v) v+0=5 ° 2

(12.48)

° ° ° 1 ° °te>2 (v) v+1 ° ° °2 = ° v1 ° °te>2 (v) v+0=5 ° 2

Note (12.47) is a single output system. Hence, with the above post-ﬁlter also an H4 @H4 optimum is reached, i.e. ° ° ° ° ° ° 1 ° 1 ° °te (v) v+1 ° °te (v) v+1 ° °2 = max ° °4 = 1@2 max ° max ($ rsw ) = te (v) °t (v) v1 ° te (v) °t (v) v1 ° ° e v+0=5 ° ° e v+0=5 ° 2

4

Next, for our comparison purpose, we design an H @H4 optimal residual generator. Remember that the H @H4 optimal design is not unique (see also the next section), we have decided to use the one which is dierent from the uniﬁed solution given in (12.46), as shown below, U@4 (v) =

v+1 > = 0=005 v + 1

(12.49)

and yields ° ° ° ° ° v+1 1 ° ° 1 ° s ° v+1 v+1 ° °U(v) v+1 ° 0=25 + $ 2 ° ° ° ° = max = ° v+1 v1 ° max ° ° $ v1 1 + $2 U(v) ° v+1 v+0=5 ° °U(v) v+0=5 ° 4

4

In comparison, the optimal solution proposed in this section is given by Ursw>GH (v) = which delivers

v + 0=5 v+1

(12.50)

° ° ° v+0=5 1 ° s ° v+1 v+1 ° 0=25 + $ 2 ° ° = max = ° v+0=5 v1 ° $ 1 + $2 ° v+1 v+0=5 ° 4

Below, we compare the performance of residual generators (12.48), (12.49) and (12.50) by means of two simulation cases: Case I : g(w) is a white noise> i (w) = 10(w 20) sin (3w) Case II : g(w) is a white noise> i (w) = 5(w 20) sin (0=5w) =

12.2 Maximization of fault detectability by a given FAR

385

5 4 3

Residual [Ŧ]

2 1 0 Ŧ1 Ŧ2 Ŧ3 Ŧ4 Ŧ5

0

5

10

15 Time [s]

20

25

30

Fig. 12.2 Response of the residual signal generated by an H2 @H2 optimal residual generator (Case I) 1000 800 600

Residual [Ŧ]

400 200 0 Ŧ200 Ŧ400 Ŧ600 Ŧ800

0

5

10

15 Time [s]

20

25

30

Fig. 12.3 Response of the residual signal generated by an H3 @H" optimal residual generator (Case I)

Fig.12.2-Fig.12.4 show the simulation results using the three residual generators, the H2 @H2 optimal residual generator (12.48) (te>2 (v))> H @H4 optimal residual generator (12.49) ( U@4 (v)) and the residual generator Ursw>GH (v) (12.50) designed using the approach proposed in this section, for Case I and Fig.12.5-Fig.12.7 for Case II. We can see that residual generator Ursw>GH (v) is sensitive to the faults in both cases, while te>2 (v) delivers a good FD performance only in Case II and the FD performance of U@4 (v) is poor in both cases. These results conﬁrm the theoretical results achieved in this section and demonstrate. In Section 12.4, we shall brieﬂy study the application of the trade-o approach proposed in this section to the stochastic systems.

386

12 Integrated design of fault detection systems 5 4 3

Residual [Ŧ]

2 1 0 Ŧ1 Ŧ2 Ŧ3 Ŧ4 Ŧ5

0

5

10

15 Time [s]

20

25

30

Fig. 12.4 Response of the residual signal generated by the residual generator designed using the uniﬁed solution (Case I) 20

15

Residual [Ŧ]

10

5

0

Ŧ5

Ŧ10

Ŧ15

0

5

10

15 Time [s]

20

25

30

Fig. 12.5 Response of the residual signal generated by an H2 @H2 optimal residual generator (Case II)

12.3 Minimizing false alarm number by a given FDR The last section has shown that the uniﬁed solution provides an optimal tradeo in the sense that by a given allowable FAR, the fault detectability is maximized. From the practical viewpoint, it is of considerable interest to approach the dual form of the above trade-o, i.e. by a given FDR, how to achieve a minimization of false alarm number. This is the objective of this section. Beside of problem formulation and solution, we shall, in this section, address the interpretation of the developed trade-o scheme and the comparison of the achieved solution with the existing ones.

12.3 Minimizing false alarm number by a given FDR

387

50 40 30

Residual [Ŧ]

20 10 0 Ŧ10 Ŧ20 Ŧ30 Ŧ40 Ŧ50

0

5

10

15 Time [s]

20

25

30

Fig. 12.6 Response of the residual signal generated by an H3 @H" optimal residual generator (Case II) 5 4 3

Residual [Ŧ]

2 1 0 Ŧ1 Ŧ2 Ŧ3 Ŧ4 Ŧ5

0

5

10

15 Time [s]

20

25

30

Fig. 12.7 Response of the residual signal generated by the residual generator designed using the uniﬁed solution (Case II)

12.3.1 Problem formulation Again, we consider system model (12.15) and residual generator (12.16). The fault detection condition and false alarm condition are respectively given in (12.17) and (12.18). We formulate our problem as Problem of Minimizing SDFA under a given FDR (PMin-SDFA): Given I GU in the context of Deﬁnition 12.7 and Mwk setting according to (12.14), ﬁnd Ursw>I D (v) 5 RH4 so that ;U(v) 5 RH4 > I D (Ursw>I D > Mwk ) I D (U> Mwk )=

(12.51)

388

12 Integrated design of fault detection systems

It is evident that applying Ursw>I D (v) would ensure the least number of false alarms. 12.3.2 Essential form of the solution In the subsequent discussion, it is ﬁrst assumed that ¯ i (m$) A 0 for continuous time systems ¯ i (m$)J ;$ 5 [0> 4]> J ¯ g (hm ) A 0 for discrete time systems. ¯ g (hm )J ; 5 [0> 2]> J

(12.52) (12.53)

and p = ni =

(12.54)

¯ i (s) = Ji r (s)Ji l (s)> Ji r (s)> is left invertible in As a result, the co-outer of J RH ° 4 . °Note that, assumptions (12.52)/(12.53) and (12.54) also ensure that ¯ i (s)° A 0= These two assumptions will be removed in the next subsection. °J Theorem 12.7 Given system model (12.15), residual generator (12.16) and ¯ i (v) 5 RH4 satisﬁes (12.52)/(12.53) and (12.54) and I GU, assume that J ¯ g (v) 5 RH4 , then J Ursw>I D (s) = J1 i r (s) 5 RH4

(12.55)

is the solution of PMin-SDFA (12.51). ¯ i (s) = Ji r (s)Ji l (s)> where J1 (s) 5 RH4 > Ji l (s) 5 Proof. Do an LCF of J ir RH4 and Ji l (s) is a co-inner. Assume that U(s) = T(s)J1 i r (s)> T(s) 5 RH4 = Then, false alarm condition (12.18) can be rewritten into ° ° i>min ° ¯ g (s)g(s)° kT(s)Ji l (s)k A 0= (s) J ° °T(s)J1 ir I GU 2

(12.56)

Note that kT(s)Ji l (s)k kT(s)k kJi l (s)k4 = kT(s)k ° ° ° ° ° ° ° ° 1 ¯ ¯ °T(s)J1 i r (s)Jg (s)g(s)° kT(s)k °Ji r (s)Jg (s)g(s)° = 2

It turns out

As a result,

¶ μ° ° i>min ° ° 1 ¯ kT(s)k °Ji r (s)Jg (s)g(s)° I GU 2 ° ° i>min ° ° ¯ kT(s)Ji l (s)k = °T(s)J1 i r (s)Jg (s)g(s)° I GU 2

2

12.3 Minimizing false alarm number by a given FDR

° ° i>min ° ¯ g (s)g(s)° A0 kT(s)k A 0> °J1 (s) J ° ir I GU 2

389

(12.57)

lead to (12.56). In other words, (12.57) is su!cient for a false alarm. Hence any g satisfying (12.57) will result in ° ° ° ° ¯ g (s)g(s)° i>min °U(s)J ¯ i (s)° = °U(s)J 2 I GU ° ° i>min ° ° ¯ kTJi l (s)k A 0= °T(s)J1 i r (s)Jg (s)g(s)° I GU 2 Considering that (12.57) can be achieved by setting U(s) = J1 i r (s)> we ﬁnally have ;T(s) 5 RH4 with kT(s)k A 0> ³ ´ ³ ´ 1

I D J1 i r > Mwk I D TJi r > Mwk which is equivalent to (12.51). The theorem is proven.

t u

Theorem 12.7 provides us with an approach, by which we can achieve an optimal trade-o in the sense of minimizing the FAR under a given FDR in the context of norm based residual evaluation. It is interesting to notice that the role of post-ﬁlter Ursw>I D (s) is in fact to inverse the magnitude proﬁle of ¯ i (s)= As a result, we have J ° ° ° ° ¯ i (s)° = °Ursw>I D (s)J ¯ i (s)° = 1= °Ursw>I D (s)J (12.58) 4 Moreover, the residual dynamics is governed by ¯ u(s) = J1 i r (s)Jg (s)g(s) + Ji l (s)i (s) and the threshold Mwk should be set, according to (12.14), as Mwk =

° i>min ° ¯ i (s)° = i>min = °Ursw>I D (s)J I GU I GU

(12.59)

Note that in case of weak disturbances, Ursw>I D (s) also delivers an estimation of the size of the fault (i.e. the energy of the fault), as kuk2 kJi l (s)i (s)k2 = ki k2 =

(12.60)

We would like to mention that the application of the well-established factorization technique to the problem solution is very helpful for getting a deep insight into the optimization problem. Dierent from the LMI solutions, the ¯ i (s) is interpretation of (12.55) as the inverse of the magnitude proﬁle of J evident. Based on this knowledge, the achieved solution will be used for the comparison study in the next section. From the computational viewpoint, solution (12.55) is an analytical one and the major computation is to solve a Riccati equation for the computation of Ji r (s).

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12 Integrated design of fault detection systems

12.3.3 The state space form Following Lemmas 7.4 and 7.5, the results given in Theorem 12.7 can also be expressed in the state space form. To this end, suppose that the minimal state space realization of system (12.15) is given by {(w) ˙ = D{(w) + Ex(w) + Hg g(w) + Hi i (w) |(w) = F{(w) + Gx(w) + Ig g(w) + Ii i (w)

(12.61) (12.62)

where { 5 Rq > D> E> F> G> Hg > Hi > Ig > Ii are known constant matrices of compatible dimensions. For the purpose of residual generation, FDF of the form { ˆ˙ (w) = Dˆ {(w) + Ex(w) + O(|(w) |ˆ(w)) |ˆ(w) = F { ˆ(w) + Gx(w)> u(w) = Y (|(w) |ˆ(w))

(12.63) (12.64)

can be used, which also represents a state space realization of residual generator (12.16) with a constant post-ﬁlter Y . In (12.63)-(12.64), O and Y are constant matrices and can be arbitrarily selected. The dynamics of (12.63)(12.64) can be equivalently written as ´ ³ ˆ x (s)Ji (s)i (s) ˆ x (s)Jg (s)g(s) + P u(s) = Y P ³ ´ ˆg (s)g(s) + Q ˆi (s)i (s) =Y Q ˆ x (s) = L F(sL D + OF)1 O = P ˆ g (s) = P ˆ i (s) P ˆg (s) = Ig + F(sL D + OF)1 (Hg OIg ) Q ˆi (s) = Ii + F(sL D + OF)1 (Hi OIi ) Q ˆ 1 (s)Q ˆg (s)> Ji (s) = P ˆ 1 (s)Q ˆi (s)= Jg (s) = P g

i

The following theorem represents a state space version of optimal solution (12.55) and so that gives the optimal design for O> Y . Theorem 12.8 Given system (12.61)-(12.62) that is detectable and satisﬁes, for continuous time systems, ¸ D m$L Hi =q+p ;$ 5 [0> 4]> udqn F Ii and for discrete time systems

D hm L Hi ; 5 [0> 2]> udqn F Ii

¸ =q+p

and residual generator (12.63)-(12.64), then for continuous time systems Orsw>I D = (Hi IiW + \i F W )(Ii IiW )1 > Yrsw>I D = (Ii IiW )1@2

(12.65)

12.3 Minimizing false alarm number by a given FDR

391

with \i 0 being the stabilizing solution of the Riccati equation D\i + \i DW + Hi HiW (Hi IiW + \i F W )(Ii IiW )1 (Ii HiW + F\i ) = 0 and for discrete time systems Orsw = OWi > Yrsw = Zi

(12.66)

with Zi being the left inverse of a full column rank matrix Ki satisfying Ki KiW = F[i F W +Ii IiW , and ([i > Oi ) the stabilizing solution to the DTARS (discrete time algebraic Riccati system) ¸ ¸ D[i DW [i + Hi HiW D[i F W + Hi IiW L =0 Oi F[i DW + Ii HiW F[i F W + Ii IiW deliver an optimal FDF in the sense of (12.51). The proof of this theorem follows directly from Lemmas 7.4 and 7.5 and thus omitted. 12.3.4 The extended form We are now going to remove assumption (12.52) or (12.53) and assumption (12.54), and extend the proposed approach so that it can be applied for any system described by (12.15). This extension is of practical interest and will enhance the applicability of the proposed approach considerably. For instance, after this extension the approach can also be applied to the detection of actuator faults, which would be otherwise impossible due to the fact that Ji (s) would have zeros at inﬁnity. For the sake of simplicity, we only consider continuous time systems. We ﬁrst release ° and consider system (12.15) with p A ni = Note ° (12.54) ¯ i (v)° 6= 0> which is equivalent to that in this case °J ¯ i (m$)J ¯ i (m$) A 0 ;$ 5 [0> 4]> J

(12.67)

ni ×p ¯ W (v) 5 RH4 Since J and, due to (12.67), ;$ 5 [0> 4]> i

¡ W ¢ ¯ i (m$) = ni rank J

(12.68)

¯ W (v) can it follows from the discussion on the CIOF in Subsection 7.1.5 that J i be factorized into ¯ i (v) = Ji r (v)Ji l (v) ¯ Wi (v) = Jl (v)Jr (v) +, J J

(12.69)

where Ji l (v) is co-inner and Ji r (v) is left invertible in RH4 = As a result, we have the following theorem.

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12 Integrated design of fault detection systems

i ¯ i (v) 5 RHp×n ¯ g (v) 5 Theorem 12.9 Given J > p A ni > satisfying (12.67), J 4 RH4 , then (12.70) Ursw>I D (v) = J i r (v) 5 RH4

ensures that ;U(v)(6= 0) 5 RH4

I D (Ursw>I D > Mwk ) I D (U> Mwk )= ¯ where J i r (v) is the left inverse of Ji r (v) and Ji r (v) the co-outer of Ji (v) as given in (12.69). The proof of this theorem is similar to the one of Theorem 12.7 and thus omitted. We now remove assumptions ° (12.52) and (12.54) and extend the solution. ° ¯ i (v)° = 0> i.e. there exists a class of faults which Note that in this case °J are, independent of their size, structurally not detectable (see Chapter 4). ¯ i (v)> or for They¡ can be,¢ for ni A p> vectors in the right null subspace of J ¯ ¯ rank Ji (m$) ? p> those vectors corresponding to the zeros Ji (m$) in Cm$ or at inﬁnity. The basic idea behind the extension study is to exclude these faults and consider only the structurally detectable faults. For this purpose, ¯ i (v) introduced in Chapter 7 can be used, which is an extended CIOF of J described by ¯ i (v) = Ji r (v)J4 (v)Jm$ (v)Ji l (v) (12.71) J where Ji l (v) is co-inner, Ji r (v) has a left inverse in RH4 , Jm$ (v) has the same zeros on the imaginary axis and J4 (v) the same zeros at inﬁnity as ¯ i (v)= Considering that kJ4 (v)Jm$ (v)Ji l (v)k = 0, it is reasonable to deﬁne J i (v) =

J4 (v)Jm$ (v) Ji l (v)i (v) =, kJ4 (v)Jm$ (v)k4

ki (v)k2 kJi l (v)i (v)k2 ki (v)k2

(12.72) (12.73)

and reformulate the fault detection problem as ﬁnding U(v) such that the residual generator ¢ ¡ ¯ i r (v)i (v) ¯ g (v)g(v) + J (12.74) u(v) = U(v) J ¯ Ji r (v) = Ji r (v) kJ4 (v)Jm$ (v)k4 (12.75) is optimal in the sense of minimizing FAR under a given FDR, as formulated at the beginning of this section. This problem can then be solved using Theorem 12.7 and the optimal solution is given by ¯ 1 (v)= Ursw>I D (v) = J ir We summarize the introduced approach into the following algorithm. Algorithm 12.2 Optimal design of FD systems by given I GU and i>min

12.3 Minimizing false alarm number by a given FDR

393

¯ i (v) into (12.74) using the extended CIOF algorithm 7.9 Step 1: Bring J ¯ i r (v) according to (12.75) Step 2: Compute J Step 3: Set Ursw>I D (v) as ¯ 1 (v) Ursw>I D (v) = J ir Step 4: Set threshold Mwk according to (12.59). 12.3.5 Interpretation of the solutions and discussion In this subsection, we are going to study the proposed approach from the mathematical viewpoint and compare it with some existing results. To this end, we ﬁrst demonstrate that solution (12.55) also solves the optimization problem ° ° ¯ i (s)° °U(s)J ° ° = M0 = sup (12.76) sup ¯ ° ° U(s)5RH4 U(s)5RH4 U(s)Jg (s) 4

¯ i (s)> J ¯ g (s) and Ji r (s) are the ones given in Theorem 12.10 Assume that J Theorem 12.7, then ° ° ¯ i (s)° °U(s)J 1 ° ° ° = ° sup ° ¯ ° ° 1 ¯ g (s)° U(s)5RH4 U(s)Jg (s) 4 ° °Ji r (s)J 4 ° ° °U(s)J ¯ i (s)° ° ° = J1 Ursw (s) = arg sup i r (s)= ¯ ° ° U(s) J (s) U(s)5RH4 g 4 Proof. Let U(s) = T(s)J1 i r (s) 5 RH4 = It leads to ° ° ° ° (s)J (s)J (s) °T(s)J1 ° i r i l ir kT(s)k ° ° = ° M0 = ° ° ° ° 1 ¯ g (s)° ¯ g (s)° (s) J ° °T(s)Ji r (s)J °T(s)J1 ir 4

Due to the relation ° ° ° ¯ g (s)° (s) J ° °T(s)J1 ir

4

4

° ° ° ¯ g (s)° kT(s)k °J1 (s) J ° ir

4

we get 1 ° M0 ° ° 1 ¯ g (s)° ° °Ji r (s)J

4

and the equality holds when T(s) = L. Thus, U(s) = Ursw (s) = J1 i r (s) is the optimal solution to the optimization problem (12.76) and the theorem is proven. t u

394

12 Integrated design of fault detection systems

Theorem 12.10 reveals that the optimization problem (12.76) can be solved analytically and the major involved computation is solving a Riccati equation for achieving Ji r (s)= Remember that the uniﬁed solution U(s) = J1 gr (s)> under certain conditions, also solves (12.76). It is thus of interest to check the equivalence between these two solutions. ¯ i (s)> J ¯ g (s) can be factorized into Theorem 12.11 Assume that J ¯ g (s) = Jgr (s)Jgl (s) ¯ i (s) = Ji r (s)Ji l (s)> J J 1 with J1 i r (s)> Jgr (s) 5 RH4 , Ji l (s) and Jgl (s) co-inner. Then ° ° ° 1 ° 1 ° ¯ i (s)° °J (s)J ° °Ji r (s)J ¯ i (s)° gr ° = ° 1 ° ° = ° ° 1 ¯ ° Jgr (s)Jg (s)°4 ¯ g (s)° °Ji r (s)J

(12.77)

4

Proof. The left side of (12.77) equals to ° ° ° 1 ¯ i (s)° ° °Ji r (s)J 1 ° = ° ° ° ° ° 1 ° 1 ¯ g (s)° ¯ g (s)° ° °Ji r (s)J °Ji r (s)J 4

4

1

° =° ° ° 1 °Ji r (s)Jgr (s)Jgl (s)° Note that

1 ° ° ° ° 1 °Ji r (s)Jgr (s)°

1

4

° = =° ° ° 1 °Ji r (s)Jgr (s)° 4

° ° ° = °J1 gr (s)Ji r (s) =

4

On the right side of (12.77), we have ° 1 ° ¯ i (s)° °J (s)J ° ° ° ° 1 gr ¯ ° ° ° ° 1 ° = °J1 gr Ji = Jgr Ji r = ¯ g (s)° °J (s)J gr

4

t u

The theorem is thus proven.

1 Although solutions J1 i r (s) and Jgr (s) are equivalent in the sense of optimizing (12.76), they deliver dierent results for the following general optimization problem Hl @H4 ,

sup

Ml>$ (U) =

U(s)5RH4

sup U(s)5RH4

sup U(s)5RH4

Ml> (U) =

sup U(s)5RH4

as described in Theorem 12.12.

¯ i (m$)) l (U(m$)J ° ° as well as (12.78) °U(s)J ¯ g (s)° 4 ¯ i (hm )) l (U(hm )J ° ° (12.79) ¯ g (s)° °U(s)J 4

12.3 Minimizing false alarm number by a given FDR

395

¯ i (s)> J ¯ g (s) satisfy the assumptions given Theorem 12.12 : Assume that J in Theorem 12.11, then the following relations hold ³ ´ ³ ´ ³ ´ 1 ° = = Ml>$ J1 = M0 J1 =° M4 J1 ir ir ir ° 1 ¯ g (s)° ° °Ji r (s)J 4 ° ° ¡ ¢ 1 °U(s)J ¯ i (s)° ¯ i (m$) ¡ 1 ¢ ¡ ¢ J (m$) J l gr 1 ° ° Ml>$ Jgr = ° 1 ° sup M0 Jgr = ° ¯ ° ¯ g (s)° °J (s)J U(s)5RH4 U(s)Jg (s) 4 gr ° °4 ¯ i (s)° °U(s)J ¯ i (m$)) ¡ 1 ¢ l (U(m$)J ° ° ° °4 sup = sup M4 Jgr = ¯ g (s)° ¯ g (s)° °U(s)J °U(s)J U(s)5RH4 U(s)5RH 4 4 4 (12.80) as well as

³ ´ ³ ´ ³ ´ 1 ° = = Ml> J1 = M0 J1 =° M4 J1 ir ir ir ° 1 ¯ g (s)° ° °Ji r (s)J 4 ° ° ¡ ¢ 1 m ¯ °U(s)J ¯ i (s)° ¡ 1 ¢ ¡ ¢ J (h )Ji (hm ) l gr 1 ° ° Ml>$ Jgr = ° 1 ° sup M0 Jgr = ° ¯ ° °J (s)J ¯ g (s)° U(s)5RH4 U(s)Jg (s) 4 gr ° °4 ¯ m ¯ m ° ° ¡ ¢ U(s) J (s) l (U(h )Ji (h )) i 1 ° ° ° °4 J = sup = sup M 4 gr °U(s)J ¯ g (s)° ° ¯ ° U(s)5RH4 U(s)5RH4 U(s)Jg (s) 4

4

(12.81) where sup U(s)5RH4

M4 (U) =

sup U(s)5RH4

° ° ¯ i (s)° °U(s)J ° °4 = ¯ g (s)° °U(s)J 4

Proof. We only prove the continuous time case. Noting that ´ ³ ¯ i (m$) = 1 (m$) J l J1 ir for any $ and l, it turns out ;$> l ³ ´ 1 ° Ml>$ J1 =° ir ° 1 ¯ g (s)° ° °Ji r (s)J 4 ³ ´ ³ ´ 1 1 1 ° M4 Ji r = sup Ml>$ Ji r = ° ° 1 ¯ g (s)° l>$ ° °Ji r (s)J 4 ³ ´ ³ ´ 1 1 1 ° = M0 Ji r = inf Ml>$ Ji r = ° ° 1 l>$ ¯ g (s)° ° °Ji r (s)J 4

It follows from Theorem 12.11 that ³ ´ ¡ ¢ M0 J1 = M0 J1 ir gr =

sup U(s)5RH4

° ° ¯ i (s)° °U(s)J ° ° °U(s)J ¯ g (s)° = 4

(12.82)

396

12 Integrated design of fault detection systems

On the other side, it holds that ¡ ¡ ¢ ¢ ¯ ¯ l J1 J1 gr (m$)Ji (m$) = inf gr (m$)Ji (m$) l ¡ ¡ 1 ¢ ¢ ¡ ¢ ¯ ¯ ¯ l J1 ¯ J1 gr (m$)Ji (m$) sup l Jgr (m$)Ji (m$) = gr (m$)Ji (m$) = l

As a result, ;$> l ¡ ¢ ¡ 1 ¢ ¡ 1 ¢ ¡ 1 ¢ ¡ 1 ¢ M0 J1 gr = inf Ml>$ Jgr Ml>$ Jgr sup Ml>$ Jgr = M4 Jgr = l>$

The theorem is thus proven.

l>$

t u

From the FDI viewpoint, the result in Theorem 12.12 can be interpreted as the fact that the FD system designed by the trade-o strategy developed in this paper is less robust in comparison with the FD system designed by using the uniﬁed solution. On the other side, as mentioned in the former subsection, the new trade-o strategy delivers a better estimation of the size of the possible faults. In this context, we would like to emphasize that the decision for a certain optimization approach should be made based on the design objective not on the mathematical optimization performance index. 12.3.6 An example In this subsection, an example is given to illustrate the results achieved in the last two sections. Consider the FD problem of a system in the form of (12.61)-(12.62) with matrices 5 6 5 6 3 0=5 0=8 1 3 ¸ ¸ 9 1 4 0 1 : 92: : > E = 9 : > F = 1 0=25 1 0 > G = 0=5 D=9 7 2 3 1 0=5 8 718 0 1 0 1 0=3 0 1 2 0 1 5 6 5 6 0=5 1 1 1 0 ¸ ¸ 9 0=8 0=5 0 : 9 : : > Hi = 9 0=5 1 : > Ig = 0=5 1 0 > Ii = 1 0 = Hg = 9 7 0 1 1 8 7 0=2 1 8 1 01 0 1=5 0=2 0 0=5 1 0 From Theorem 12.8, we get the optimal gain matrix O1 > Y1 5 6 0=9735 0=1323 ¸ 9 0=5639 0=5791 : 1 0 9 : = O1 = 7 >Y = 0=2118 0=6198 8 1 0 0=6667 0=4837 0=4439

(12.83)

The uniﬁed solution that solves (12.78), (12.82) and (12.76) simultaneously is

12.3 Minimizing false alarm number by a given FDR

5

397

6

1=0072 1=0029 ¸ 9 0=6343 0=2393 : 0=9333 0=1333 9 : O2 = 7 >Y = = 1=1660 0=7751 8 2 0=1333 0=7333 0=0563 0=3878

(12.84)

The optimal performance indexes, as obtained by solving (12.78), (12.82) and (12.76) are shown in Fig.12.8. It can be seen that, ;$, 2.5

Performance index

2

1.5

1

0.5

0 Ŧ2 10

Ŧ1

10

0

10

1

10

2

10

3

10

Frequency (rad/sec)

Fig. 12.8 Performance index M" (O1 > Y1 ) = M0 (O1 > Y1 ) = M1>$ (O1 > Y1 ) = M2>$ (O1 > Y1 ) 0=1769 (dashed line), performance index M1>$ (O2 > Y2 ) (solid line), and performance index M2>$ (O2 > Y2 ) (dotted line)

2=1533 = M4 (O2 > Y2 ) = M1>$=0 (O2 > Y2 ) M1>$ (O2 > Y2 ) M2>$ (O2 > Y2 ) M2>$=1=7870 (O2 > Y2 ) = M0 (O2 > Y2 ) = M0 (O1 > Y1 ) = M1>$ (O1 > Y1 ) = M2>$ (O1 > Y1 ) = M4 (O1 > Y1 ) = 0=1769= These results verify Theorems 12.10-12.12. In the simulation study, the simulation time is set to be 2000 seconds and the control input is a step signal (step time at 0) of amplitude 5. The unknown disturbances are, respectively, a continuous signal taking value randomly from a uniform distribution between [0=1> 0=1], a sine wave 0=1 sin(0=1w), and a chirp signal with amplitude 0=1 and frequency varying linearly from 0=02 Hz to 0=06 Hz. Fault 1 appears at the 1200-th second as a step function of amplitude 0=75. Fault 2 appears at the 1000-th second as a step function of amplitude 0=4. The fault energy is ki k2 = 24=71. The residual signals are shown in Fig.12.9, where u1 denotes the residual vector generated with O1 > Y1 and u2 that by O2 > Y2 . As ku1 k2 = 25=45> ku2 k2 = 21=46, the residual vector obtained by O1 > Y1 gives a better estimation of the energy level of the fault signal. On the other side, we see from the second ﬁgure that the residual vector got by O2 > Y2 shows

398

12 Integrated design of fault detection systems

a better fault/disturbance ratio in the sense of (12.78), (12.82) and (12.76). This demonstrates the results in Theorem 12.12. 1.5

r1

1 0.5 0 Ŧ0.5 Ŧ1 500

1000

1500

2000

1500

2000

1.5

r2

1 0.5 0 Ŧ0.5 Ŧ1 500

1000

Time [s] Fig. 12.9 Residual signals

12.4 On the application to stochastic systems In the last two sections, two trade-o strategies and the associated design methods have been developed in the norm based evaluation framework. It is of practical interests to know if they are still valid for stochastic systems and in the statistic testing framework. In this section, we shall brieﬂy discuss the related problems. 12.4.1 Application to maximizing FDR by a given FAR In Subsection 11.1.3, we have introduced a GLR solution to the residual evaluation and threshold computation for stochastic systems modelled by (11.1)(11.2). The core of this approach is the computation of the FAR in the sense of Deﬁnition 12.1, which is given by (see (11.17)) ¢ ¢ ¡ ¡ (12.85) 1 prob "2 pu (v + 1)> 2ug 2ug = Equation (12.85) can be equivalently written as ¢ ¡ 1 prob "2 (pu (v + 1)> 1) 1 when the residual evaluation function is re-deﬁned by

(12.86)

12.5 Notes and references W ˜ 1 unv>n unv>n

2ug

399

=

Now, if we set the residual generator according to Corollary 12.1, then we have ;U(s) 5 RH4

GH (U> Mwk ) GH (Ursw>GH > Mwk ) As a result, Ursw>GH delivers the maximal probability Ã ! W ˜ 1 unv>n unv>n prob A 1 | inv>n 6= 0 2ug

(12.87)

while keeping the same FAR as given by (12.86). Remember that the probability given in (12.87) is exactly the FDR given in Deﬁnition 12.2. In this context, we claim that the solution presented in this section, namely the uniﬁed solution, also solves the FD systems design problem for stochastic systems (11.1)-(11.2), which is formulated as: given FAR (in the sense of Deﬁnition 12.1) ﬁnd the residual generator, O and Y> so that the FDR (in the sense of Deﬁnition 12.2) is maximized. 12.4.2 Application to minimizing FAR by a given FDR The trade-o strategy proposed in Section 12.3 requires a threshold setting according to (12.59), which also ﬁts the FDR in the sense of Deﬁnition 12.2, ³ ´ W ˜ 1 unv>n A Mwk | inv>n 6= 0 = I GU = prob unv>n For the computation of the associated FAR as deﬁned in Deﬁnition 12.1, we can again use the estimation ¢ ¢ ¡ ¡ (12.88) 1 prob "2 pu (v + 1)> 2ug Mwk = Remember that the optimal residual generator Ursw>I D ensures that ;U(v) 5 RH4 > I D (Ursw>I D > Mwk ) I D (U> Mwk )= ¢ ¢ ¡ ¡ It results in a maximum probability prob "2 pu (v + 1)> 2ug Mwk > which in turn means Ursw>I D oers the minimum bound for among all possible residual generators. In other words, Ursw>I D delivers a minimum I DU by a given I GU.

12.5 Notes and references Although this chapter is less extensive in comparison with the other chapters, it is, in certain sense, the soul of this book. Dierent from the current way

400

12 Integrated design of fault detection systems

of solving the FDI problems in the context of robustness and sensitivity, as introduced in the previous chapters, the model-based FDI problems have been re-viewed in the context of FAR vs. FDR. Inspired by the interpretation of the concepts FAR and FDR in the statistical framework, we have • introduced the concepts of FAR and FDR in the norm based context, • deﬁned SDF and SDFA and, based on them, • formulated two trade-o problems: maximizing fault detectability by a given (allowable) FAR (PMax-SDF) and minimizing false alarm number by a given FDR (PMin-SDFA). In this way, we have established a norm based framework for the analysis and design of observer-based FDI systems. It is important to notice that in this framework the four essential components of an observer-based FD system, the residual generator, residual evaluation function, the threshold and the decision logic, are taken into account by the problem formulations. This requires and also allows us to deal with the FDI system in an integrated manner. The integrated design distinguishes the design procedure proposed in this chapter signiﬁcantly from the current strategies, where residual generation and evaluation are separately addressed. It has been demonstrated that the uniﬁed solution introduced in Chapter 7 also solves PMax-SDF, while the solution with inversing the magnitude proﬁle of the fault transfer function matrix is the one for PMin-SDFA. In the established norm based framework, a comparison study has further been undertaken. The results have veriﬁed, from the aspect of the trade-o FAR vs. FDR, that • the uniﬁed solution leads to the maximum fault detectability under a given FAR and • the ratio between the inﬂuences of the fault and the disturbances is the decisive factor for achieving the optimum performance and thus the inﬂuence of the disturbance should be integrated into the reference model by designing a reference model based FD system. One question may arise: why have we undertaken a so extensive study on the PUIDP in Chapter 6 and on the robustness issues in Chapter 7? To answer this question, we would like to call reader’s attention to the result that the solution of the PUIDP is implicitly integrated into the general form of the uniﬁed solution (12.29). In fact, the solution of the PUIDP gives a factorization in the form of (7.305), which leads then to (12.29). Also, it should be pointed out that in the established norm framework, we have only addressed the FDI design problems under the assumption that the residual signals are evaluated in terms of the L2 norm. As outlined in Chapter 9, in practice also other kinds of signal norms are used for the purpose of residual evaluation. To study the FDI system design under these norms, the methods and tools introduced in Chapter 7 are very helpful. As additional future work we would like to mention

12.5 Notes and references

401

that an "LMI version" of the uniﬁed solution would help us to transfer the results achieved in this chapter to solving FDI problems met in dealing with other types of systems. In Section 12.4, we have brieﬂy discussed the possible application of the proposed approaches to the stochastic systems. It would be also a promising topic for the future investigation. A useful tool to deal with such problems e!ciently is the optimal selection of parity matrices presented in Section 7.5, which builds a link to the GLR technique. A part of the results in this chapter has been provisionally reported in [31].

13 Fault isolation schemes

Fault isolation is one of the central tasks of a fault diagnosis system, a task that can become, by many practical applications, a real challenge for the system designer. Generally speaking, fault isolation is a signal processing process aiming at gaining information about the location of the faults occurred in the process under consideration. Evidently, the complexity of such a signal processing process strongly depends on • • • •

the the the the

number of the possible faults, possible distribution of the faults in the process under consideration, characteristic features of each fault and available information about the possible faults.

Correspondingly, the fault isolation problems will be solved step by step at dierent stages of a model-based fault diagnosis system. Depending on the number of the faults, their distribution and the fault isolation logic adopted in the decision unit, the residual generator should be so designed that the generated residual vector delivers the ﬁrst clustering of the faults, which, in accordance with the fault isolation logic, divides the faults into a number of sets. At the residual evaluation stage, the characteristic features of the faults are then analyzed by using signal processing techniques based on the available information of the faults. As results, a further classiﬁcation of the faults is achieved, and on its basis a decision about the location of the occurred faults is ﬁnally made. If the number of the faults is limited and their distribution is well structured, a fault isolation may become possible without a complex residual evaluation. The main objective of this chapter is to present a number of widely used approaches for the purpose of fault isolation. Our focus is on the residual generation, as shown in Fig.13.1. We will ﬁrst describe the basic principle, and then show the limitation of the fault isolation schemes which only rely on residual generators and without considering the characteristic features of the faults and thus without the application of special signal processing techniques

404

13 Fault isolation schemes

for the residual evaluation, and ﬁnally present and compare dierent observerbased fault isolation approaches.

Fig. 13.1 Description of the fault isolation schemes addressed in Chapter 13

13.1 Essentials In this section, we ﬁrst study the so-called perfect fault isolation (PFIs) problem formulated as: given system model |(s) = J|x (s)x(s) + J|i (s)i (s)

(13.1)

with the fault vector i (s) 5 Rni , ﬁnd a (linear) residual generator such that each component of the residual vector u(s) 5 Rni corresponds to a fault deﬁned by a component of the fault vector i (s). We do this for two reasons: by solving the PFIs problem • the role and, above all, the limitation of a residual generator for the purpose of fault isolation can be readily demonstrated and • the reader can get a deep insight into the underlying idea and basic principle of designing a residual generator for the purpose of fault isolation. On this basis, we will then present some approaches to the solution of the PFIs problem. 13.1.1 Existence conditions for a perfect fault isolation In order to study the existence conditions for a PFIs, we consider again the general form of the dynamics of the residual generator derived in Chapter 5

13.1 Essentials

5 9 u(s) = 7

6

5

405

6

u1 (s) i1 (s) 9 .. : .. : = U(s)P ˆ ˆ x (s)J|i (s)i (s) = U(s)Px (s)J|i (s) 7 . 8 . 8= uni (s) ini (s)

The requirement on a PFIs can then be mathematically formulated as: ﬁnd U(s) such that ˆ x (s)J|i (s) = gldj(w1 (s)> · · · > wn (s)) U(s)P i which gives

(13.2)

5

6 5 6 u1 (s) w1 (s)i1 (s) 9 .. : 9 : .. 7 . 8=7 8 . uni (s) wni (s)ini (s)

(13.3)

where wl (s)> l = 1> · · · > ni > are some RH4 transfer functions. Thus, the PFIs problem is in fact a problem of solving equation (13.2) which is a dual form the well-known decoupling control problem. In the following of this subsection, we will restrict our attention to the existence conditions of (13.2) whose solution will be handled in the next section. It is evident that (13.2) is solvable if and only if ³ ´ ˆ x (s)J|i (s) = ni = udqn P ˆ x (s) 5 Rp×p has a full-rank equal to p, the following theorem Recall that P becomes evident. Theorem 13.1 The PFIs problem is solvable if and only if udqn (J|i (s)) = ni =

(13.4)

Remember that in Section 4.3, we have studied structural fault isolability. We have learned from Corollary 4.2 that additive faults are structurally isolable if and only if the rank of the corresponding fault transfer matrix is equal to the number of the faults. The result in Theorem 13.1 is identical with the one stated in Corollary 4.2. Hence, we can claim that the PFIs is solvable if and only if the faults are structurally isolable. ˆ x (s) 5 Rp×p > J|i (s) 5 Rp×ni and Since P ´ n ³ ´ o ³ ˆ x (s) > udqn (J|i (s)) = min{p> ni } ˆ x (s)J|i (s) min udqn P udqn P we have Corollary 13.1 The PFIs problem is solvable only if p ni =

406

13 Fault isolation schemes

Theorem 13.1 and Corollary 13.1 not only give some necessary and su!cient conditions for the solution of the PFIs problem but also reveal a physical law that is of importance for our further study on the fault isolation problem: Suppose that the FDI system under consideration only consists of a residual generator and furthermore no assumptions on the faults are made, then a successful fault isolation can only be achieved if the number of the faults to be isolated is not larger than the number of the sensors used (i.e. the dimension of the output signals). In other words, we are only able to isolate as many faults as the sensors used. Surely, this is a hard limitation on the application of the model-based FDI systems. Nevertheless, this strict condition is a result of the hard assumptions we made. Removing them, for instance, by introducing a residual evaluation unit which processes the residual signals by taking into account possible knowledge of faults, or assuming that a simultaneous occurrence of faults is impossible, it is possible to achieve a fault isolation, even if the conditions given in Theorem 13.1 or Corollary 13.1 are not satisﬁed. Let the system model be given in the state space representation J|x (s) = (D> E> F> G)> J|i (s) = (D> Hi > F> Ii ) with D 5 Rq×q > E 5 Rp×nx > F 5 Rp×q > G 5 Rp×nx > Hi 5 Rq×ni and Ii 5 Rp×ni = Then, Theorem 13.1 is equivalent with Theorem 13.2 The PFIs problem is solvable if and only if ¸ sL D Hi = q + ni = udqn F Ii Theorem 13.2 provides us with a PFIs check condition via the Rosenbrock system matrix, whose proof can be found in Corollary 4.3. 13.1.2 PFIs and unknown input decoupling Taking, for instance, a look at the ﬁrst row of (13.3), we can immediately recognize that the residual u1 (s) is decoupled from the faults i2 (s)> · · · > ini (s) and therefore only depends on i1 (s). In general, the l-th residual signal, ul (s), is sensitive to il (s) and totally decoupled from the other faults, i1 (s)> · · · > il1 (s)> il+1 (s)> · · · > ini (s). Recall the problem formulation of the so-called fault detection with unknown input decoupling, the selection of the l-th row of the transfer function matrix U(s) is in fact equivalent to the design of a residual generator with unknown input decoupling. In this sense, the residual generator described by (13.3) can be considered as a bank of dynamic systems, and each of them is a residual generator with perfect unknown input decoupling. In other words, we can handle the residual isolation problem as a special problem of designing residual generators with perfect unknown input decoupling formulated as following: Given

13.1 Essentials

407

¯ li (s)i¯l (s)> l = 1> · · · > ni |(s) = J|x (s)x(s) + jli (s)il (s) + J 6 5 i1 (s) : 9 .. : 9 . £ ¤ : 9 ¯ il (s) = 9 il1 (s) : > J|i (s) = j1i (s) · · · jni i (s) : 9 7 il+1 (s) 8 ini (s) £ ¤ ¯ Jli (s) = j1i (s) · · · jl1i (s) jl+1i (s) jni i (s) ﬁnd Ul (s) 5 RH4 > l = 1> · · · > ni such that ¡ ¢ ˆ x (s) jli (s)il (s) + J ¯ li (s)i¯l (s) ul (s) = Ul (s)P ˆ x (s)jli (s)il (s)> l = 1> · · · > ni = = Ul (s)P This fact allows us to apply the well-developed approaches to the design of residual generators with perfect unknown input decoupling introduced in Chapter 6 to the fault isolation. It is indeed also the mostly used way to solve the PFIs problem. The idea of reducing the fault isolation problem to the design of residual generators with perfect unknown input decoupling is often adopted to handle the case where a PFIs is not realizable, which is in fact mostly met in practice. Assume that ni A p. Taking the fact in mind that for a system with p outputs and p 1 unknown inputs there exists a residual generator with a perfect unknown input decoupling, let’s deﬁne a group of subsets, each of which contains ni (p1) faults. For a system with p outputs and ni faults, there exist ¶ μ ¶ μ ni ! ni ni := n = = p1 ni p + 1 (p 1)!(ni p + 1)! such subsets. Now, we design n residual generators, each of them is perfectly decoupled from p1 faults. According to the relationships between the residual signals and the faults, a logical table is then established, by which a decision on the location of a fault is made. Of course, in this case a fault isolation generally means locating the subset, to which the fault belongs, instead of indicating exactly which fault occurred. To demonstrate how this scheme works, we take a look at the following example. Example 13.1 Suppose that £

Hi = hi 1

J|i (s) = F(sL D)1 Hi + Ii > F 5 R3×q ¤ ¤ £ hi 2 hi 2 hi 4 hi 5 5 Rq×5 > Ii = ii 1 ii 2 ii 2 ii 4 ii 5 5 Rp×5

i.e. the system has three outputs, and ﬁve possible faults have to be detected and isolated. Since p = 3 ? 5 = ni , a PFIs is not realizable. To the end of fault isolation, we now use the fault isolation scheme described above. Firstly, we deﬁne

408

13 Fault isolation schemes

μ ¶ 5 = 10 n= 2 fault subsets: V1 = {i1 > i2 > i3 }> V2 = {i1 > i2 > i4 }> V3 = {i1 > i3 > i4 }> V4 = {i2 > i3 > i4 } V5 = {i1 > i2 > i5 }> V6 = {i1 > i3 > i5 }> V7 = {i2 > i3 > i5 }> V8 = {i1 > i4 > i5 } V9 = {i2 > i4 > i5 }> V10 = {i3 > i4 > i5 }= Then, correspondingly we can design ten residual generators with perfect unknown input decoupling on the basis of the following ten unknown input models ¡ £ ¤ £ ¤¢ ¡ £ ¤ £ ¤¢ D> hi 4 hi 5 > F> ii 4 ii 5 > D> hi 3 hi 5 > F> ii 3 ii 5 ¤ £ ¤¢ ¡ £ ¤ £ ¤¢ ¡ £ D> hi 2 hi 5 > F> ii 2 ii 5 > D> hi 1 hi 5 > F> ii 1 ii 5 ¤ £ ¤¢ ¡ £ ¤ £ ¤¢ ¡ £ D> hi 3 hi 4 > F> ii 3 ii 4 > D> hi 2 hi 4 > F> ii 2 ii 4 ¤ £ ¤¢ ¡ £ ¤ £ ¤¢ ¡ £ D> hi 1 hi 4 > F> ii 1 ii 4 > D> hi 2 hi 3 > F> ii 2 ii 3 ¤ £ ¤¢ ¡ £ ¤ £ ¤¢ ¡ £ D> hi 1 hi 3 > F> ii 1 ii 3 > D> hi 1 hi 2 > F> ii 1 ii 2 = As a result, ten residual signals are delivered, u1 (s) = I1 (i1 (s)> i2 (s)> i3 (s))> u2 (s) = I2 (i1 (s)> i2 (s)> i4 (s)) u3 (s) = I3 (i1 (s)> i3 (s)> i4 (s))> u4 (s) = I4 (i2 (s)> i3 (s)> i4 (s)) u5 (s) = I5 (i1 (s)> i2 (s)> i5 (s))> u6 (s) = I6 (i1 (s)> i3 (s)> i5 (s)) u7 (s) = I7 (i2 (s)> i3 (s)> i5 (s))> u8 (s) = I8 (i1 (s)> i4 (s)> i5 (s)) u9 (s) = I9 (i2 (s)> i4 (s)> i5 (s))> u10 (s) = I10 (i3 (s)> i4 (s)> i5 (s)) with ul (s) = Il (il1 (s)> il2 (s)> il3 (s)) denoting the l-th residual as a function of faults il1 (s)> il2 (s) and il3 (s). Finally, a logic table can be established. Surely, using the logic ul (w) 6= 0 indicates that a fault is from Vl > l = 1> · · · > 10 we are able to locate to which sub-set a fault belongs. The fact that a fault may inﬂuence more than one residual, however, allows to get more information about the location of faults. To this end, the following table is helpful. In Table 13.1, "1" in the l-th column and the m-th row indicates that residual ul is a function of im and "0" means that ul is decoupled from im . Following this table, it becomes clear that not every type of faults is locatable. For instance, if any three faults simultaneously occur, then all of the residual signals will dier from zero. That means we cannot, for instance, distinguish the situation i1 6= 0> i2 6= 0> i3 6= 0 from the one i2 6= 0> i3 6= 0> i4 6= 0. Nevertheless, under the assumption that no faults occur simultaneously the ﬁve faults can be isolated. Indeed, in this case, using residual generators u1 > u2 > u3 > u4 and u5 , instead of all ten residual signals, a PFIs can be achieved. Remark 13.1 The above discussion shows that the strong existence condition for a PFIs may become weaker when we have additional information about faults.

13.1 Essentials

409

Table 13.1 Logic table for fault isolation

i1 i2 i3 i4 i5

u1 1 1 1 0 0

u2 1 1 0 1 0

u3 1 0 1 1 0

u4 0 1 1 1 0

u5 1 1 0 0 1

u6 1 0 1 0 1

u7 0 1 1 0 1

u8 1 0 0 1 1

u9 0 1 0 1 1

u10 0 0 1 1 1

13.1.3 PFIs with unknown input decoupling (PFIUID) We now extend our discussion to the process with a unknown input vector, |(s) = J|x (s)x(s) + J|i (s)i (s) + J|g (s)g(s) and study the problem, under which condition there exists a post-ﬁlter U(s) 5 RH4 such that ˆ x (s)J|i (s) = gldj(w1 (s)> · · · > wn (s)) ˆ x (s)J|g (s) = 0> U(s)P U(s)P i

(13.5)

which implies 6 6 5 w1 (s)i1 (s) u1 (s) 9 : : .. ˆ x (s)J|i (s)i (s) = 9 u(s) = 7 ... 8 = U(s)P 7 8= . uni (s) wni (s)ini (s) 5

Below is a theorem which describes a necessary and su!cient condition for the solution of (13.5). Theorem 13.3 (13.5) is solvable if and only if ¤ £ udqn J|i (s) J|g (s) = udqn (J|i (s)) + udqn (J|g (s)) = ni + udqn (J|g (s)) =

(13.6) (13.7)

Proof. Su!ciency: From Algebraic Theory we know that (13.6) implies there exists a U1 (s) 5 RH4 with udqn (U1 (s)) = p such that

Let

¸ ¯ i (s) 0 £ ¤ J ˆ U1 (s)Px (s) J|i (s) Jg (s) = ¯ g (s) 0 J ¢ ¡ ¢ ¡ ¯ g (s) = udqn (J|g (s)) = ¯ |i (s) = ni > udqn J udqn J

410

13 Fault isolation schemes

£ ¤ U(s) = U2 (s) 0 U1 (s) with ¯ i (s) = gldj(w1 (s)> · · · > wn (s)) U2 (s)J i then we have ˆ x (s)J|i (s) = gldj(w1 (s)> · · · > wn (s))> U(s)P ˆ x (s)J|g (s) = 0= U(s)P i Necessity: Assume that (13.6) is not true. Then for all U(s) ensuring ˆ x (s)J|g (s) = 0 U(s)P we have

´ ³ ˆ x (s)J|i (s) ? udqn (J|i (s)) udqn U(s)P

(13.6) is hence a necessary condition for the solvability of (13.5).

t u

It follows from Theorem 13.3 that the solvability of the PFIUID problem depends on the rank of J|g (s) and the number of measurable outputs. In fact, such a problem is solvable if and only if i (s) and g(s) have totally decoupled eects on the measurement |(s). From the practical viewpoint, this is surely a unrealistic requirement on the structure of the system under consideration. On the other side, however, it reveals an intimate relationship between the problem of fault isolation and unknown input decoupling. In the forthcoming sections, we are going to present a number of approaches to the PFIs problem deﬁned in the last subsection, most of which have been developed following the decoupling principle. Without loss of generality, we consider only the situation without unknown inputs.

13.2 A frequency domain approach The approach presented below is in fact an extension of the so-called frequency domain approach described in Section 6.3 for the purpose of unknown input decoupling. The problem to be solved is now formulated as: Given transfer ˆ x (s)> J|i (s) and suppose udqn (J|i (s)) = ni , ﬁnd such function matrices P a post ﬁlter U(s) 5 RH4 that ensures ˆ x (s)J|i (s) = gldj(w1 (s)> · · · > wn (s)) 5 RH4 U(s)P i ˆ x (s)J|i (s) leads to Recall that applying Algorithm 6.3 to U1 (s)P 6 wˆ11 (s) wˆ12 (s) · · · w1ni (s) 9 0 wˆ22 (s) · · · w2ni (s) : : ˆ x (s)J|i (s) = 9 U1 (s)P : 5 RH4 9 .. .. .. .. 8 7 . . . . ˆ 0 · · · 0 wni ni (s) 5

(13.8)

13.2 A frequency domain approach

411

and wˆll (s) 6= 0> l = 1> · · · > ni , provided that udqn (J|i (s)) = ni = Setting

3 U(s) = C

ni Y

4 Wl D U1 (s)

l=1

with

5

1 0 ··· 0

9 90 1 9 9. 9 .. 9 9 90 ··· Wl (s) = 9 9 .. 90 . 9 9. . 9 .. .. 9 70 ···

··· .. . ··· .. . .. .

(s) wwˆ1l(s) ll

(s) 0 wwˆ2l(s) ll .. .. . . (s) 1 wl1l wˆ (s) ll

0 .. .

1 .. .

··· ··· 0 ··· ··· ···

··· ···

0 ··· 0

6

: 0 ··· 0: : .. .. .. : . . .: : : 0 ··· 0: : : 0 ··· 0: : . . . . .. : . . .: : ··· 1 08 ··· 0 1

gives 3 4 ni Y ˆ x (s)J|i (s) = C Wl (s)D U1 (s)P ˆ x (s)J|i (s) U(s)P 5

l=1

4 wˆ11 (s) wˆ12 (s) ni 9 0 wˆ22 (s) Y 9 = C Wl (s)D 9 . .. 7 .. . l=1 0 ··· 3

6 · · · w1ni (s) · · · w2ni (s) : : : .. .. 8 . . 0 wˆni ni (s)

= gldj(w1 (s)> · · · > wni (s))> wl (s) = wˆll (s) We now summarize these results into an algorithm for solving the PFIs. Algorithm 13.1 Frequency domain approach to PFIs Step 1: Application of Algorithm 6.3 to solve (13.8) for U1 (s); Step 2: Set U(s) 4 3 ni Y U(s) = C Wl (s)D U1 (s)= l=1

Remark 13.2 Since the essential calculations in Algorithm 13.1 consist of addition and multiplication operations of transfer functions, the order of the resulted post ﬁlter may become higher than q, the order of the system.

412

13 Fault isolation schemes

13.3 Fault isolation ﬁlter design In this section, three approaches to the design of fault detection ﬁlters for the purpose of a PFIs will be presented. For the sake of simplicity, we only deal with continuous time systems. As known, under the assumption that the system model is given by {(w) ˙ = D{(w) + Ex(w) + Hi i (w)> |(w) = F{(w)

(13.9)

we can construct an FDF of the form { ˆ˙ (w) = Dˆ {(w) + Ex(w) + O(|(w) |ˆ(w))> |ˆ(w) = F { ˆ(w) u(w) = Y (|(w) |ˆ(w))

(13.10) (13.11)

whose dynamics is governed by ˆ(w)> u(w) = Y Fh(w)= h(w) ˙ = (D OF)h(w) + Hi i (w)> h(w) = {(w) { Remembering our design purpose and the PFIs condition, it is in the following assumed that glp(|) = udqn(F) = p ni = udqn(Hi ) = glp(i )> Y 5 Rni ×p = We formulate the design problem as follows: Given system model (13.9) and fault detection ﬁlter (13.10)-(13.11), ﬁnd O and Y such that the fault detection ﬁlter is stable and Y F(sL D + OF)1 Hi is diagonal. Deﬁnition 13.1 A fault detection ﬁlter solving the above-deﬁned problem is called fault isolation ﬁlter. 13.3.1 A design approach based on the duality to decoupling control The basic idea of the approach by Liu and Si for the problem solution is based on the so-called dual principle, namely the duality between the PFIs and the state feedback decoupling which is formulated as: Given the system model ¯ ¯ ¯ {(w) ˙ = D{(w) + Ex(w)> |(w) = F{(w) and a control law x(w) = N{(w) + I¯ z(w) ﬁnd N and I¯ such that the transfer function matrix ¯ I¯ ¯ ¯ 1 E F(sL D¯ EN) is diagonal. Let

13.3 Fault isolation ﬁlter design

413

¯ F W = E> ¯ HiW = F> ¯ OW = N> Y W = I¯ DW = D> we obtain ¡ ¢W ¯ ¯ 1 E ¯ I¯ = Y F(sL D + OF)1 Hi = F(sL D¯ EN) The duality thus becomes evident. Considering that the state feedback decoupling is a standard control problem whose solution can be found in most of textbooks of modern control theory, we shall below present the results without a detailed proof. To begin with, a so-called fault isolability matrix is introduced. Write ¤ £ Hi = hi 1 · · · hi ni and let l = min{m : FDm1 hi l 6= 0> m = 1> 2> · · · }> l = 1> · · · > ni which are also called fault detectability indices. Then, the fault isolability matrix is deﬁned as h i Ilvr = FD1 1 hi 1 · · · FDni 1 hi ni With the deﬁnition of Ilvr we are now able to state a necessary and su!cient condition for the solvability of the design problem. Lemma 13.1 The transfer function matrix Y F(sL D + OF)1 Hi can be diagnosed if and only if Ilvr is left invertible. The following theorem is the core of the approach, which provides a means of designing a fault isolation ﬁlter (13.10)-(13.11). Theorem 13.4 Suppose Ilvr be left invertible. Setting ¤ ¢ + ¡£ + + ]1 (L Ilvr Ilvr ) (13.12) O = D1 hi 1 · · · D ni hi ni Hi Ilvr + + + ]2 (L Ilvr Ilvr ) Y = Z Ilvr

(13.13)

gives a diagonal transfer function matrix Y F(sL D + OF)1 Hi , where ]1 and ]2 are arbitrary matrices with compatible dimensions, Z is any regular + diagonal matrix, Ilvr is the Moore-Penrose generalized inverse of Ilvr with ¢ W ¡ W + = Ilvr Ilvr Ilvr Ilvr and is a diagonal matrix with its entries l > l = 1> · · · > ni > assignable. It is worth to point out that setting O and Y according to Theorem 13.4 only ensures a diagonal transfer function matrix Y F(sL D + OF)1 Hi but not the system stability. Hence, before formulas (13.12)-(13.13) are applied for solving the PFIs problem formulated above, further conditions should be fulﬁlled.

414

13 Fault isolation schemes

Lemma 13.2 Let Ilvr be left invertible and O be given by (13.12). Then, the characteristic polynomial of the matrix (D OF) is of the form () = (1 1 ) · · · (p p ) 1 () = 0 ()1 () where 1 () is the invariant polynomial with the degree equal to q is uniquely determined once the matrices D> F> Hi are given.

(13.14) p P l=1

l and

The following theorem describes a condition, under which a stable PFIs is ensured. Theorem 13.5 Let Ilvr be left invertible. Then, fault detection ﬁlter (13.10)(13.11) with O being given by (13.12) is stable and ensures a fault isolation if and only if l > l = 1> · · · > p>is one and the invariant polynomial 1 () in (13.14) is Hurwitz. Remark 13.3 Following the deﬁnition of matrix Ilvr the conditions that l = 1 and Ilvr is left invertible imply udqn(FHi ) = ni = Furthermore, the following two statements are equivalent: • the invariant polynomial 1 () in (13.14) is Hurwitz • (D> Hi > F) has no transmission zeros in the RHP. We have thus the following corollary: Corollary 13.2 : Fault detection ﬁlter (13.10)-(13.11) with O being given by (13.12) is stable and ensures a fault isolation if and only if • udqn(FHi ) = ni •

L D Hi udqn F 0

(13.15)

¸ = q + ni for all 5 C+ .

(13.16)

It is very interesting to notice the similarity of the existence conditions for a fault isolation ﬁlter, (13.15)-(13.16), with the ones for a UIO stated in Corollary 6.6. This fact may reveal some useful aspects for the design of fault isolation ﬁlter. Recall that the underlined idea of a UIO is to reconstruct the unknown input vector. Following this idea, it should also be possible to reconstruct the fault vector when conditions (13.15)-(13.16) are satisﬁed. The discussion in Section 6.5.2 shows for this purpose we can use system ˆ(w)) { ˆ˙ (w) = Dˆ {(w) + Ex(w) + Hi iˆ(w) + O(|(w) F { ˆ ˙ FDˆ {(w) Fx(w)) = i (w) = (FHi ) (|(w)

(13.17) (13.18)

Note that the implementation of the above system requires the knowledge of |(w)= ˙

13.3 Fault isolation ﬁlter design

415

Theorem 13.6 Given system model (13.9) and suppose that |(w) ˙ is measurable and the system satisﬁes (13.15)-(13.16), then system (13.17)-(13.18) delivers an estimation for the fault vector. Surely, the application of this result is limited due to the practical di!culty of getting |. ˙ Nevertheless, it reveals the real idea behind the approach presented here lies in the reconstruction of the faults. We know that the existence conditions for such kind of systems are stronger than, for instance, the fault isolation systems designed by the frequency domain approach, where the existence condition is udqn (J|i (s)) = ni which is obviously weaker than (13.15)-(13.16). 13.3.2 The geometric approach In this subsection, an algorithm of using the geometric approach to the fault isolation ﬁlter design will be presented, whose existence conditions are less strict than the ones given in Theorem 13.5 and the order is equal to the sum of l > l = 1> · · · > ni = We shall also brieﬂy discuss the relationship between the order and the number of the invariant zeros of the system under consideration. Without loss of generality, it is assumed that ni = p= We begin with the existence conditions of such kind of fault isolation ﬁlters. Theorem 13.7 : Suppose for system (13.9) the matrix Ilvr is left invertible and the transmission zeros of (D> Hi > F) lie in the LHP, then there exist matrices O and Y such that fault detection ﬁlter (13.10)-(13.11) ensures a PFIs. In the following we present an algorithm that serves, on the one side, as a proof sketch for Theorem 13.7 and, on the other side, as a design algorithm for the fault isolation ﬁlters. The theoretical background of this algorithm is the so-called geometric approach that has been introduced and handled in Chapter 6. The interested reader is referred to the literatures given there. Algorithm 13.2 Design of fault isolation ﬁlters using the geometric approach Step 1: Determine Or that makes (D Or F> Hi > F) maximally uncontrollable by using the known geometric approach, for instance Algorithm 6.6, and transform (D Or F> Hi > F) into ¸ ¸ ¯i £ ¤ D¯1 O1 F¯ D¯3 H > H > F F¯ R D Or F i ¯ R D2 R by a state transformation Wr and an output transformation Yr , where (D¯1 ¯ is perfectly controllable, O1 is arbitrary and the eigenvalues ¯ H ¯i > F) O1 F> 1 of D¯2 are zeros of transfer function matrix Ji | (s) = F (sL D) Hi

416

13 Fault isolation schemes

Step 2: Set

h i 1 I¯lvr = F¯ D¯11 1 h¯1 · · · F¯ D¯1ni h¯ni

where

¤ £ ¯i = h¯1 · · · h¯ni > l = min{m : F¯ D¯m1 h¯m 6= 0> m = 1> 2> · · · }> l = 1> · · · ni H 1

Step 3: Set

³h O1 =

with P=

"

sP 1 1 n=0

i ´ n 1 1 D¯11 h¯1 · · · D¯1 i h¯ni P I¯lvr > Y1 = Z I¯lvr

n1 D¯n1 h¯1 · · ·

sni 1

P

n=0

# nni D¯n1 h¯ni

> Z = gldj(1 > · · · > ni )=

(13.19) The coe!cients nl > n = 1> · · · > l 1> l = 1> · · · > ni , are arbitrarily selectable but should ensure the roots of polynomials sX l 1

ln vn = 0> l = 1> · · · > ni

n=0

lie in the LHP Step 4: Construct the residual generator: ¯ r + O1 Yr )|(w) ¯ ¯1 x(w) + (O }(w) ˙ = (D¯1 O1 F)}(w) +E ¡ ¢ ¯ u(w) = Y1 Yr |(w) F}(w) where

¸ ¸ £ ¤ L L > F¯ = Yr FWr D¯1 = L 0 Wr (D Or F)Wr1 0 0 £ ¤ £ ¤ ¯ ¯ Or = L 0 Wr Or > E1 = L 0 Wr E=

The dynamics of the fault isolation ﬁlter designed using the above algorithm is governed by ¯ ¯i i (w) + D¯3 { %˙ (s) = (D¯1 O1 F)%(w) ¯2 (w) +H ¯ ¯ ˙{ ¯2 (w) = D2 { ¯2 (w)> u(w) = Y1 F%(w)= ¯2 (w) on the residual vector u(w) will vanish Since D¯2 is stable, the inﬂuence of { as w approaching inﬁnity, and thus it is reasonable just to consider the part ¯ ¯ 1 H ¯i i (s) u(s) = Y1 F(sL D¯1 + O1 F) which, as will be shown latter, takes the form l ul (s) = > l = 1> · · · > ni = lr + l1 s + · · · ll sl 1 + sl To explain and to get a deep insight into the algorithm we further make following remarks.

13.3 Fault isolation ﬁlter design

417

Remark 13.4 The deﬁnition of matrix P ensures that condition udqn (FHi ) = ni can be replaced by udqn (Ilvr ) = ni i.e. l can be larger than one. Indeed, the dual form of this result is known in the decoupling control theory. We shall also give a proof in the next subsection. Remark 13.5 It is a well known result of the decoupling control theory that the order of the transfer function matrix between the inputs and outputs after the decoupling is equal to the dierence of the order of the system under con¯i > F) ¯ sideration and the number of its invariant zeros. Since the triple (D¯1 > H is perfect controllable, i.e. it has no transmission zeros, it becomes evident that i ¡ ¢ X l = dim D¯1 =

n

l=1

13.3.3 A generalized design approach The approaches presented in the last two subsections are only applicable for the purpose of component fault isolation. In order to handle the more general case, namely isolation of both component and sensor faults, an approach is proposed by Ding et al., which is established on the duality of the fault isolation problem to the well-established decoupling control theory. In this subsection, we brieﬂy introduce this approach with an emphasis on its derivation of the solution, which, we hope, may give the reader a deep insight into the fault isolation technique. The fault model considered here is given by J|i (s) = Ii + F(sL D)Hi

(13.20)

with (D> Hi > F> Ii ) as a minimal state space realization of J|i (s)= Without loss of generality and for the sake of simplicity it is assumed ng = p= To begin with, the concepts of fault isolatability matrix and fault detectability indices introduced in the last subsections are extended to include the case where Ii 6= 0= Denote ¤ ¤ £ £ Hi = hi 1 · · · hi ni > Ii = I1 · · · Ini then the fault detectability indices are deﬁned by ½ 0> Il 6= 0> l = 1> · · · > ni l = min{m : FDm1 hi l 6= 0> m = 1> 2> · · · }> Il = 0> l = 1> · · · > ni and the fault isolatability matrix by

418

13 Fault isolation schemes

¤ £ Ilvr = Ilvr>1 · · · Ilvr>ni ½ Il > Il 6= 0> l = 1> · · · > ni Ilvr>l = = FDl 1 hi l > Il = 0> l = 1> · · · > ni In order to simplify the notation, we assume, without loss of generality, that Il 6= 0> l = 1> · · · > t 1> and Il = 0> l = t> · · · > ni thus, Ilvr can be written as h i Ilvr = I1 · · · It1 FDt 1 hi t · · · FDni 1 hi ni =

(13.21)

Under the assumption that Ilvr is invertible, we now introduce matrices J> O> W> Y and Z : 6 5 6 5 0 · · · 0 l0 0 Jt 91 0 l1 : : : 9 9 .. J=7 : > l = t> · · · > ni (13.22) 8 > Jl = 9 . . .. . 8 7 . . 0 Jni 0 1 ll 1 ¤ ¢ 1 ¡£ t (13.23) O = hi 1 · · · hi t1 D hi t · · · D ni hi ni P Ilvr " # 1 t 1 ni P P P = 0 ··· 0 (13.24) tn Dn hi t · · · ni n Dn hi ni n=0

n=0

ni h i X W = hi t · · · Dt 1 hi t · · · hi ni · · · Dni 1 hi ni 5 Rq× > = l l=t

(13.25) Y = gldj(1 > · · ·

1 > ni )Ilvr

Z = Y FW=

(13.26) (13.27)

In the following, we shall study the properties of J> O> W> Y and Z and how to make use of these properties to construct a fault isolation ﬁlter. To this end, we ﬁrst prove that (13.28) Z2 W = Y2 F holds, where W satisﬁes

W W = L×

and will be speciﬁed below, and 6 6 5 5 0 t 0 1 0 0 ¸ Y1 : 1 9 .. : I 1 > Y = 9 .. .. .. > Y1 = 7 Y = 8 Ilvr . . . 8 lvr 2 7 . Y2 0 0 ni 0 0 t1 0 ¸ Z1 with Z1 = Y1 FW> Z2 = Y2 FW= Z = Z2

(13.29)

(13.30)

13.3 Fault isolation ﬁlter design

419

It is straightforward that h i Z = Y FW = Y Fhi t · · · FDt 1 hi t · · · Fhi ni · · · FDni 1 hi ni 6 5 0(t1)×(t1) 9 j¯t 0 · · · 0 : : 9 9 . . .. : 9 = gldj(1 > · · · > ni ) 9 0 j¯t+1 . . : : : 9 . . 7 .. . . . . . 0 8 0 · · · 0 j¯ni with row vectors j¯l > l = t> · · · > ni >whose entries are zero but the last one which equals one. That implies 65 6 5 t 0 j¯t 0 : 9 .. : 9 .. (13.31) Z1 = 0> Z2 = 7 87 8 . . 0

ni

0

j¯ni

udqn (Z2 ) = ni t + 1 = the row number of Z2 = Now, we solve Y2 FW1 = 0 for W1 with

£

(13.32)

¤ ¤ £ W W1 5 Rq×q > udqn W W1 = q=

Note that due to (13.31) such a W1 does exist. Let ¸ W 5 Rq×q W1 with

It turns out

¸ ¸ ¤ £ ¤ W W £ W W1 = W W1 = Lq×q = W1 W1

(13.33)

¸ ¸ ¤ W £ ¤ W Y2 F W W1 = Y2 F = Z2 0 = Z2 W W1 W1 £

which proves (13.28). Next, we shall prove ¸ ¸ ¤ £ W J J2 W W = (D OF) > J1 = W1 (D OF)W1 1 0 J1 W1 ¸ ¸ W W (Hi OIi ) (H OI ) = = i i 0 W1 For our purpose, we check (D OF)W= Since

(13.34) (13.35)

420

13 Fault isolation schemes

¤ £ DW = Dhi t · · · Dt hi t · · · Dhi ni · · · D ni hi ni i h OFW = OFhi t · · · OFDt 1 hi t · · · OFhi ni · · · OFDni 1 hi ni and furthermore we have FDn hi l = 0> n = 0> · · · > l 2> l = t> · · · > ni X

l 1

OFDl 1 hi l = Dl hi l

ln Dn hi l > l = t> · · · > ni

n=0

where the ﬁrst equation is due to the deﬁnition of fault detectability indices and the second one the deﬁnition of matrix O> it holds # " n 1 t 1 i P P n n (D OF)W = Dhi t · · · tn D hi t · · · Dhi ni · · · ni n D hi ni = n=0

n=0

(13.36) It is of interest to notice that all columns of matrix (DOF)W can be expressed in terms of a linear combination of the columns of matrix W , i.e. Im ((D OF)W ) Im(W )=

(13.37)

Hence, it is obvious that W1 (D OF)W = 0= Let w l be the l-th row of matrix W > then it follows from (13.36) that for l = sm1 + 1> m = t> · · · > ni > t1 = 0 £ ¤ w l (D OF)W = 0 · · · 0 m0 0 · · · 0

with m0 at the m1 = ( m > m = t> · · · > ni

m1 P n=t

n )-th entry and otherwise for m1 + 1 ? l

£ ¤ w l (D OF)W = 0 · · · 0 1 0 · · · 0 mo 0 · · · 0 o = l sm1 1

with "1" at the (l 1)-th entry and m0 at the m -th entry. As a result, we obtain W (D OF)W = J= Thus, the proof of (13.34) is completed. The proof of (13.35) is evident by noting the fact that ¸ ¤ ¢ L(t1)×(t1) ¡£ OIi = hi 1 · · · hi t1 Dt hi t · · · D ni hi ni P 0 ¤ £ ¤ £ = hi 1 · · · hi t1 0 · · · 0 =, Hi OIi = 0 · · · 0 hi t · · · hi ni =, Im(Hi OIi ) Im(W )

13.3 Fault isolation ﬁlter design

421

which leads to W1 (Hi OIi ) = 0= Now, we are in a position to construct a fault isolation ﬁlter based on matrices J> O> W > Y2 > Z2 > respectively deﬁned by (13.22), (13.23), (13.33), (13.29) and (13.30).Note that during the above study no assumption is made on lm> l = t> · · · > ni > m = 0> · · · l 1, they can be so selected that the dynamic system of the form ¸ ¸ ¸ ¸ ¸ W W }1 (w) J J2 }˙1 (w) Ex(w) + O|(w) (13.38) = + }˙2 (w) }2 (w) 0 J1 W1 W1 is stable. Moreover, we deﬁne ¸ u1 (w) = Z }1 (w) Y Gx(w) + Y |(w) Y FW1 }2 (w)= u(w) = u2 (w)

(13.39)

Now, we check if (13.38)-(13.39) build a fault isolation ﬁlter. To this end, we take a look at the dynamics of the residual generator. Introducing variables %(w) = W {(w) }1 (w)> (w) = W1 {(w) }2 (w) and noting that ¤ £ ¤ £ Y F W W1 = Z Y FW1 =, Y F{(w) = Z (%(w) + }1 (w)) + Y FW1 ((w) + }2 (w)) yield ˙ = J1 (w) %˙ (w) = J%(w) + W (Hi OIi )i (w) + J2 (w)> (w) u(w) = Z %(w) + Y Ii i (w) + Y FW1 (w)=

(13.40) (13.41)

We now calculate W (Hi OIi ) and Y Ii i (s)= Since ¤ £ (Hi OIi ) = 0 · · · 0 hi t · · · hi ni we obtain 5

W (Hi OIi ) = W

£

0 · · · 0 hi t · · · hi ni

¤

0 h¯t

9 90 =9 9. 7 .. 0

0 .. . 0

··· . h¯t+1 . . .. .. . . ··· 0 0

6 0 .. : . : : : 0 8 h¯ni

where all entries of column vector h¯l > l = t> · · · > ni are zero but the ﬁrst one which equals one. and ¸ L(t1)×(t1) = Y Ii = gldj(1 > · · · > ni ) 0 (ni t)×(ni t)

422

13 Fault isolation schemes

Under the assumption that J1 is stable and thus its inﬂuence on u(s) will vanish as w approaching inﬁnity, we ﬁnally have u(s) = Z (sL J)1 W (Hi OIi )i (s) + Y Ii i (s) 5 6 1 i1 (s) 9 : .. 9 : . 9 : 9 : i (s) t1 t1 9 : t 9 it (s) : 9 : t 1 S := n =9 9 s t n=0 t1n s : 9 : .. 9 : 9 : . 9 : n i 9 : i (s) ni 1 7 nS 8 i s

ni

n=0

(13.42)

ni 1n sn

It is hence evident that the residual generator (13.38)-(13.39) does deliver a PFIs. From (13.42) we see two interesting facts: • The residual generator can be divided into two independent parts: a static subsystem that delivers an isolation of faults, i1 > · · · > it1 > and a dynamic subsystem used for isolating faults it > · · · > ini ; • Setting l = 1> l = 1> · · · > t 1> gives 5 6 5 6 u1 (s) i1 (s) 9 : 9 : .. .. 7 8=7 8 . . ut1 (s)

it1 (s)

thus these faults can be identiﬁed. This fact can also be interpreted as: The approach described above is applicable for sensor fault identiﬁcation. In order to get some insight into the construction of the residual generator (13.38)-(13.39) we shall brieﬂy discuss its structural properties from the viewpoint of control theory. We know that the observability is not aected by the output feedback, thus the subsystem (Y FW1 > J1 ) should be observable. On the other side, the modes of J1 are not controllable by (Hi OIi )= Indeed, the eigenvalues of J1 are transmission zeros of the transfer function matrix Ji (s)= To demonstrate this claim, we consider the Smith form of system matrix ¸ sL D Hi = S (s) = F Ii Since the output feedback O|(s) and changes of state space and output bases do not modify the Smith form, we ﬁnd

13.3 Fault isolation ﬁlter design

5

423

6

¸ sL J J2 W (Hi OIi ) sL D + OF Hi OIi 7 8 0 0 sL J1 S (s) YF Ii Ii Z Z FW1 5 6 sL J W (Hi OIi ) 0 8= 0 Ii 7 Z 0 0 sL J1

It follows from (13.42) that ¸ ¸ sL J W (Hi OIi ) L = Z (sL J)1 W (Hi OIi ) + Y Ii Z Ii Recall that the transfer function matrix Z (sL J)1 W (Hi OIi ) + Y Ii has no zeros, we ﬁnally have ¸ L 0 S (s) = 0 sL J1 It thus becomes evident that the transmission zeros of S (s) are identical with the eigenvalues of sL J1 which are knowingly invariant to the output feedback. Remark 13.6 It is worth noting that if only sensor faults are under consideration, i.e. udqn (Ii ) = ni > Hi = 0 then O = 0= That means a fault isolation is only possible based on the system model instead of an observer. The physical interpretation of this fact is evident. Since all sensors are corrupted with faults, none of them should be used for the PFIs purpose. On the other hand, in Section 14.1 we shall present an algorithm that allows a perfect sensor fault identiﬁcation, which also solves the sensor fault isolation problem. In summary, we have the following theorem and the algorithm for the fault isolation ﬁlter design. Theorem 13.8 Suppose that for system (13.20) the matrix Ilvr is left invertible and the transmission zeros of (D> Hi > F> Ii ) lie in the LHP, then system (13.38) and (13.39) provides a PFIs. Algorithm 13.3 Design of fault isolation ﬁlters Step 1: Set Ilvr according to (13.21) and J> O> W according to (13.22)-(13.25) Step 2: Set Y> Z according to (13.26)-(13.27) Step 3: Solve (13.32) and (13.33) for W1 > W > W1 Step 4: Compute J1 > J2 according to (13.34) Step 5: Construct residual generator according to (13.38) and (13.39).

424

13 Fault isolation schemes

Example 13.2 In this and the next examples, we are going to demonstrate the application of Algorithm 13.3 to the solution of fault isolation problem. We ﬁrst consider the benchmark system LIP100 given in Subsection 3.7.2. Our intention is to isolate those three faults: position sensor fault, angular sensor fault as well as the actuator fault. Checking the transmission zeros of the corresponding fault transfer matrix reveals that this fault transfer matrix has two RHP zeros (including the origin) v1 = 4=3163> v2 = 0= Thus, it follows from Theorem 13.8 that FDF (13.38) and (13.39) cannot guarantee a stable perfect fault isolation. In order to verify it, Algorithm 13.3 is applied to the LIP100 model. Below is the design result: Step 1: Computation of Ilvr > J> O> W 5 6 6 0 0 1=9500 1=0000 0 0 9 : 8 > O = 9 0 0 13=7429 : 0 Ilvr = 7 0 1=0000 7 0 0 0=1977 8 0 0 6=1343 0 0 0=6960 J = 2=0000> W = E 5

Step 2:Setting

5

6 10 0 8 0 Y = 70 1 0 0 0=1630

Step 3: Solution of (13.32) and (13.33) for W1 > W 5 6 0 1=0000 0 9 1=0000 0 £ ¤ 0 : : W1 = 9 7 0 0=0000 0 8 > W = 0=0000 0 0=1630 0 0 0 1=0000 Step 4: Computation of J1 > J2 5 6 2=0000 0=0210 0=0000 0=0013 ¸ 9 J J2 0 0 0=0000 1=0000 : : =9 7 0=0000 8 0 0=0000 0 0 J1 0=0000 19=7051 0=0000 0=2489 which has four eigenvalues: 2=000> 4=5652> 4=3163> 0= This result veriﬁes our conclusion.

13.3 Fault isolation ﬁlter design

425

Example 13.3 We now apply Algorithm 13.3 to the benchmark lateral vehicle dynamic system given in Subsection 3.7.4. We are interested in isolating the fault in the yaw rate sensor and the presentation of the road bank angle. In our previous study, the road bank angle has been treated as disturbance. Often, it is desired to isolate and indicate the occurrence of the road bank angle. This motivates our example. Checking the transmission zeros of the corresponding fault transfer matrix gives two LHP zeros. Thus, it follows from Theorem 13.8 that FDF (13.38) and (13.39) would delivers a perfect fault isolation. Below are the design procedure and the associated results: Step 1: Set Ilvr ,J> O> W ¸ ¸ 0=0069 0 0 152=7568 >O = Ilvr = 0=1955 0 1=0000 0 ¸ 1 W = > J = 2=000 0 Step 2: Set Y> Z

¸ ¸ 0 1=0000 0 Y = >Z = 0=0065 0 1

Step 3: Find W1 > W > W1 ¸ £ ¤ £ ¤ 0=0082 > W = 1=0000 0=0082 > W1 = 0 1=0000 W1 = 1=0000 Step 4: Computation of J1 > J2 ¸ ¸ J J2 2=0000 0=9740 = 0=0000 3=1754 0 J1 Step 5: Construct residual generator according to (13.38) and (13.39). We now extend Algorithm 13.3 aiming at removing the requirement on the transmission zeros of (D> Hi > F> Ii )= Let us ﬁrst express the dynamics of the residual generator (13.40)-(13.41) in the transfer matrix form ¢ ¡ (13.43) u(s) = Z (sL J)1 W (Hi OIi ) + Y Ii i (s) ¢ ¡ 1 1 + Z (sL J) J2 + Y FW1 (sL J1 ) (0)= Considering that there exists a diagonal U(s) 5 RH4 so that ¢ ¡ U(s) Z (sL J)1 J2 + Y FW1 (sL J1 )1 5 RH4 ¢ ¡ and U(s) Z (sL J)1 W (Hi OIi ) + Y Ii remains diagonal, U(s)u(s) builds a residual generator which results in a PFIs, i.e. satisﬁes (13.5), and is stable, independent of the placement of the transmission zeros of (D> Hi > F> Ii ) in the complex plane. In this way, the requirement on the transmission zeros of (D> Hi > F> Ii ) can be removed. Note that this extension can considerably increase the order of the residual generator.

426

13 Fault isolation schemes

Example 13.4 In this example, we re-study the fault isolation problem for the benchmark system LIP100 by applying the above-described extension. Remember that in Example 13.2, we have found out that due to the transmission zeros equal to 4=3163 and 0 we can achieve a fault isolation using Algorithm 13.3 but the resulted fault isolation ﬁlter is unstable. We now multiply U(v) =

v(v 4=3163) L (v + 1)(v + 3)

to the residual vector described in Example 13.2. It results in 5 9 U(v)u(v) = 7

6 v(v4=3163) (v+1)(v+3) i1 (v) : v(v4=3163) 8+ (v+1)(v+3) i2 (v) v(v4=3163)(v4=4924) i3 (v) (v+1)(v+2)(v+3)

¢ v(v 4=3163) ¡ Z (vL J)1 J2 + Y FW1 (vL J1 )1 (0)= (v + 1)(v + 3) Note that the unstable poles in the second transfer matrix are canceled by the zeros of U(v) so that the second term in the residual signals will vanish as w approaching inﬁnity. In this way, a stable fault isolation is achieved. It is worth mentioning that the on-line implementation of post-ﬁlter U(v) should carried out in the observer form.

13.4 An algebraic approach to fault isolation In the last section, we have introduced dierent approaches whose application to fault isolation is more or less restricted. For the frequency domain approach it is due to the possible higher order of the resulted residual generators, while for the approaches by Liu and Si as well as the generalized design approach the applicability depends on the structure of the system model. In this section, we shall present an approach, which is in fact an extension of the UIDO design approach presented in Subsection 6.5.4. Thus, this approach can be used for both the parity relation based and the observer-based residual generator design. Consider system model |(s) = J|x (s)x(s) + J|i (s)i (s)

(13.44)

with (D> E> F> G) and (D> Hi > F> Ii ) as minimal realization of transfer function matrices J|x (s) and J|i (s) respectively. Recall that a residual generator can be constructed either in a recursive form like }(w) ˙ = J}(w) + Kx(w) + O|(w)> }(w) 5 Rv u(s) = y|(w) z}(w) yGx(w)

13.4 An algebraic approach to fault isolation

427

with J> K> O> y and z satisfying the Luenberger conditions, or in a nonrecursive form like u(s) = yv (|v (s) Kr>v xv (s)) with yv denoting the parity vector. For both of them, the system dynamics related to the fault vector can be described in a uniﬁed form u(s) = zJv sv h(s) + yv Ki>v L¯i v iv (s)

(13.45)

where h(s) is a vector which, in fault-free case, will be zero as time approaching to inﬁnity, 6 5 5 6 0 · · · · · · 0 j1 0 ··· 0 Ii 5 6 9 1 0 · · · 0 j2 : j1 9 : 9 . : . 9 FHi 9 . . .. .. : Ii . . .. : 9 .. : 9 : : 9 . . . : > j = 7 . 8 > Ki>v = 9 J = 90 1 : .. .. .. : 9 .. . . . . 7 . 0 8 . . jv 7 . . . 0 jv1 8 FDv1 Hi · · · FHi Ii 0 · · · 0 1 jv 5 6 6 5 0 ··· 0 Lni ×ni i (s)sv 9 .. : .. .. 9 i (s)sv+1 : 9 zjLni ×ni . . . : : 9 ¯ 9 : > iv (s) = 9 Li v = 9 : .. : .. . . 8 7 .. .. . 7 0 8 . i (s) zJv1 jLni ×ni · · · zjLni ×ni Lni ×ni ¤ £ ¤ £ z = 0 · · · 0 1 > yv = yv>0 · · · yv>v and j = 0 in case of the parity space approach is used (the non-recursive form) as well as 6 5 yv>0 9 yv>1 : : 9 O = 9 . : jyv>v , y = yv>v 7 .. 8 yv>v1 which describes the relationship between these two forms. Starting from (13.45) we now derive an approach to the design of residual generators for the purpose of fault isolation. We ﬁrst introduce following notations: 5 6 6 5 0 ··· 0 Ii l il (s)sv 9 . : 9 il (s)sv+1 : 9 Fhi l Ii l . . . .. : : 9 l l ¯ ¯ 9 : Ki>v = 9 : .. : > l = 1> · · · > ni > iv (s) = 9 .. . . 8 7 .. .. 0 8 . 7 . il (s) FDv1 hi l · · · Fhi l Ii l ¤ ¤ £ £ Hil = hi 1 · · · hi l1 hi l+1 · · · hi ni > Iil = Ii 1 · · · Ii l1 Ii l+1 · · · Ii ni

428

13 Fault isolation schemes

6 i1 (s) 6 5 l : 9 .. R i (s)sv : 9 . : 9 .. : 9 i l (s)sv+1 : : 9 . : : : > i l (s) = 9 il1 (s) : > ivl (s) = 9 : 9 .. : 9 il+1 (s) : 8 7 . : 9 R8 : 9 .. l i l (s) 8 7 . I 5

5 l Ki>v

9 9 =9 9 7

R ··· . FHil Iil . . .. .. .. . . . v1 l FD Hi · · · FHil Iil

5 Lil v

9 9 =9 9 7

6

i

L(ni 1)×(ni 1) zjL(ni 1)×(ni 1) .. . v1

zJ

ini (s) R .. . ..

··· .. . .. .

6

R .. .

. R jL(ni 1)×(ni 1) · · · zjL(ni 1)×(ni 1) L(ni 1)×(ni 1) 5 6 1 0 ··· 0 9 . . . . .. : 9 zj . . . : l ¯ 9 : Li v = 9 : .. .. . . 7 . . R8 . zJv1 j · · · zj 1

then (13.45) can equivalently be written as ¡ l l l ¢ l ¯l ¯l ¯ i>v Li v iv (s) + K u(s) = zJv sv h(s) + yv Ki>v Li v iv (s) =

: : : : 8

(13.46)

Remember the claim that a perfect unknown input decoupling is achievable if the number of the outputs is larger than the number of the unknown inputs. Taking i l (s) as a unknown input vector, we are able to ﬁnd ni parity vectors, yvl l > l = 1> · · · > ni > such that l = 0> l = 1> · · · > ni = yvl l Ki>v l

Note that the order of the parity vectors must not be identical, hence we denote them separately by vl . Corresponding to the ni parity vectors, we obtain ni residual generators ¡ l l l ¢ l ¯l ¯l ¯ i>v Li v iv (s) + K ul (s) = zl Jvl l svl hl (s) + yvl Ki>v Li v iv (s) ¯ l L¯l i¯l (s)= = zl Jvl svl hl (s) + yl K l

v

i>v i v v

Since hl (s) is independent of i l (s)> as a result, we claim that ul (s) only depends on il (s). That means the residual generator bank, ul (s)> l = 1> · · · > ni , delivers a perfect fault isolation. Notice that during the above derivation no assumption on the structure of the system under consideration has been made. The following theorem is thus proven. Theorem 13.9 Given system model (13.44) with p ni > then there exist vl > l = 1> · · · > ni > and ni residual generators ul (s) with dimension vl > l = 1> · · · > ni > such that each residual generator is only inﬂuenced by one fault. And these residual generators can either be in a recursive form like

13.4 An algebraic approach to fault isolation

}˙ l (w) = Jl } l (w) + K l x(w) + Ol |(w)> } l (w) 5 Rvl ul (w) = y l |(w) zl }(w) y l Gx(w)

429

(13.47) (13.48)

or in a non-recursive form like ul (s) = yvl l (|vl (s) Kr>vl xvl (s)) =

(13.49)

Dynamic systems (13.47)-(13.48) are in fact a bank of residual generators. On the other hand, they can also be written in a compact form }(w) ˙ = J}(w) + Kx(w) + O|(w) u(w) = Y |(w) Z }(w) Y Gx(w)

(13.50) (13.51)

with 6 5 1 6 5 6 5 1 6 u J1 K 0 }1 9 .. : 9 : 9 9 .. : . : . .. } = 7 . 8>u = 7 . 8J = 7 8 > K = 7 .. 8 0 Jni } ni uni K ni 5 6 5 6 5 1 6 y1 z1 O R 9 .. : 9 : 9 .. : . . O = 7 . 8>Y = 7 . 8>Z = 7 8= . 5

Oni

yni

R

zni

Recall that in Section 6.8 we have introduced an algorithm of designing minimum order residual generators, and using it we are able to construct residual generators ul (s)> l = 1> · · · > ni > whose order is minimum. As a result, residual generator (13.50)-(13.51) is also of the minimum order, which, as shown in the last section, may be smaller than q, the order of the system under consideration. We now summary the main results achieved above into an algorithm. Algorithm 13.4 An algebraic approach to designing fault isolation systems l and solve Step 1: Form matrix Ki>v l

£ l ¤ £ ¤ ¯l K yvl l Ki>v i>v = 0 > l = 1> · · · > ni l for yvl l > e.g. using Algorithm 6.15 (Calculation of minimum order parity vectors). Note that 6= 0 is some constant vector and yvl l should be a parity vector. Step 2: Construct residual generator in the non-recursive form ul (s) = yvl l (|vl (s) Kr>vl xvl (s)) or

430

13 Fault isolation schemes

Step 2: Set vector j l ensuring the stability of Jl and 6 5 0 · · · · · · 0 j1l 5 l6 9 1 0 · · · 0 j2l : j1 : 9 : 9 . . .. . . : l 9 . : 9 . . . 0 1 Jl = 9 : > j = 7 .. 8 : 9 . .. .. jvl 7 .. . . 0 j l 8 v1 0 · · · 0 1 jvl 65 5 l l l l · · · yv>v1 yv>v yv>1 yv>2 FE l l : 9 9 yv>2 · · · · · · y 0 FDE v>v :9 9 Kl = 9 . .. .. .. : 9 7 .. . . . 87 l 0 yv>v 5 l yv>0 l 9 yv>1 9 Ol = 9 . 7 ..

··· 6

···

0

let

6 : : : 8

FDv1 E

: £ ¤ : : j l yvl l >vl > y l = yvl l >vl > zl = 0 · · · 0 1 8

l yv>v1

Step 3: Construct residual generators according to }˙ l = Jl } l + K l x + Ol |> ul = y l | zl } y l Gx= Example 13.5 In this example, we apply Algorithm 13.4 to design fault isolation ﬁlters for the benchmark example EHSA. Considering that there are three faults to be isolated with 5 6 3=3973 × 104 0 0 9 ¸ 0 0 0: 9 : 010 9 : 0 0 0 : > Ii = Hi = 9 001 7 0 0 08 0 00 and only two sensors are available, we divide the faults into three groups, (G1) i1 and i2 (G2) i1 and i3 (G3) i2 and i3 > and assume that only two faults can simultaneously occur. On this assumption, we are then able to design a bank of fault isolation ﬁlters that delivers a perfect fault isolation. Below, we demonstrate the design of two fault isolation ﬁlters for the third group. In this case, we set Hi = 0> Ii = L2×2 and design two fault isolation ﬁlters: fault isolation ﬁlter 1 should be sensitive to i3 and decoupled from i2 and fault isolation ﬁlter 2 sensitive to i2 and decoupled from i3 = It follows from Algorithm 13.4 that Step 1:

¤ £ yv1 = 0 1 0 0 0 0 0 0 0 5=8598 × 103 £ ¤ yv2 = 0 0 1 0 3=797 × 103 0 =

13.5 Fault isolation using a bank of residual generators

431

Step 2: for the ﬁrst fault isolation ﬁlter v = 4 5 5 6 6 0 0 0 0 2=4 × 105 1=4074 × 103 9 1 0 0 0 5=0 × 104 : 9 2=9299 × 102 : 9 : : J1 = 9 7 0 1 0 0 3=50 × 103 8 > O1 = 7 20=509 8 0 0 1 0 1=0 × 102 0=5860 £ ¤ K1 = 0> z1 = 0 0 0 1 > y1 = 5=8598 × 103 for the second fault isolation v = 2 ¸ ¸ 0 1=2 × 103 4=5564 > K2 = 0 > O2 = J2 = 0=7342 1 70 £ ¤ y2 = 3=797 × 103 > z1 = 0 1 Step 3: construct fault isolation ﬁlters }˙1 (w) = J1 }1 (w) + O1 |1 (w)> u1 (w) = y1 |1 (w) z1 }1 (w) }˙2 (w) = J2 }2 (w) + O2 |1 (w)> u2 (w) = y2 |1 (w) z2 }2 (w) We would like to call reader’s attention to the special form of yv1 > yv2 = It can be so interpreted that each residual signal is only driven by one sensor. This is in fact the so-called DOS isolation scheme, which will be handled in the forthcoming subsection.

13.5 Fault isolation using a bank of residual generators As mentioned at the beginning of this chapter, a fault isolation problem can be in fact equivalently reformulated as a decoupling problem described by 5 6 6 5 u1 (s) w1 (s)i1 (s) 9 .. : : .. ˆ x (s)J|i (s)i (s) = 9 7 . 8 = U(v)P 8 7 . uni (s) wni (s)ini (s) ˆ x (s) with wˆl (v)> l = 1> · · · > ni > then the fault Denoting the rows of U(v)P isolation problem can be interpreted as a search for a bank of residual generators, wˆl (v)> l = 1> · · · > ni . In the early development stage of FDI technique, the strategy of using a bank of fault detection ﬁlters was considered as a special concept for solving the fault isolation problem. The so-called dedicated observer scheme (DOS) and the generalized observer scheme (GOS), developed in the end of the 70’s and at the beginning of the 80’s by Clark and Frank with his co-worker respectively, are two most known fault isolation approaches of using a bank of residual generators.

432

13 Fault isolation schemes

13.5.1 The dedicated observer scheme (DOS) The DOS was proposed by Clark originally for the purpose of sensor fault isolation. The idea behind the DOS is very simple. Under the assumption that ni sensor faults have to be detected and isolated, ni residual generators are then constructed, and each of them is driven by only one output, i.e. 5 6 6 5 I1 (x(s)> |1 (s)) u1 (s) 9 .. : 9 : .. (13.52) 7 . 8=7 8 ¡ . ¢ uni (s) Ini x(s)> |ni (s) where Il (x(s)> |l (s)) > l = 1> · · · > ni , stands for a function of the inputs and the l-th output |l (w). It is evident that the l-th residual, ul (w)> will only be inﬂuenced by the i-th sensor fault il , it thus ensures a sensor fault isolation. Below we brieﬂy show the application of DOS concept for the sensor fault isolation. Suppose the system model takes the form 6 6 5 5 J1 (s)x(s) + i1 (s) |1 (s) : : 9 9 .. |(s) = 7 ... 8 = J|x (s)x(s) + i (s) = 7 8 (13.53) . |p (s)

Jp (s)x(s) + ip (s)

with i (s) standing for the sensor fault vector. We now construct p residual generators as follows ³ ´ 6 5 5 6 ˆ x1 (s)|1 (s) Q ˆx1 (s)x(s) (s) P U 1 u1 (s) : 9 9 : : 9 .. u(s) = 7 ... 8 = 9 (13.54) : . 7 ³ ´8 up (s) ˆ xp (s)|p (s) Q ˆxp (s)x(s) Up (s) P ˆ xl (s)> Q ˆxl (s) denote a left coprime pair of transfer function Jl (s)> l = where P 1> · · · > p, Ul (s)> l = 1> · · · p> a parametrization vector. It leads to ˆ xl (s)il (s)> l = 1> · · · > p ul (s) = Ul (s)P which clearly means a perfect sensor fault isolation. Remark 13.7 The original form of DOS approach was presented in the statespace. Under the assumption 6 5 6 5 G1 F1 : 9 : 9 J|x (s) = (D> E> F> G)> F = 7 ... 8 > G = 7 ... 8 Fp the l-th residual generator is constructed as follows

Gp

13.5 Fault isolation using a bank of residual generators

433

ˆ(w) Gl (w)x(w)) { ˆ˙ (w) = Dˆ {(w) + Ex(w) + Ol (|l (w) Fl { ul (w) = |l (w) Fl { ˆ(w) Gl x(w) and whose dynamics is governed by h(w) ˙ = (D Ol Fl )h(w) Ol il (s)> ul (s) = Fl h(w) + il (w)= It is evident that the residual signal ul only depends on the l-th fault il = Remember the claim that every kind of residual generators can be presented in the form ´ ³ ˆx (s)x(s) = ˆ x (s)|(s) Q (13.55) u(s) = U(s) P As will be shown below, the bank of residual generators (13.54) is in fact a multi-dimensional residual generator. Thus, all results achieved in the last chapters are also available for a bank of residual generators. In order to show it, we ﬁrst rewrite (13.54) into ³ ´ 6 5 6 5 ˆ x1 (s)|1 (s) Q ˆx1 (s)x(s) U1 (s) P u1 (s) 9 : : 9 .. : 9 .. := 7 . 8=9 . 7 ³ ´8 up (s) ˆ xp (s)|p (s) Q ˆxp (s)x(s) Up (s) P 5 65 6 4 6 35 6 5 ˆ x1 (s) ˆx1 (s) U1 (s) |1 (s) 0 0 P Q 9 : 9 .. : 9 : F : E9 .. .. .. 7 87 . 8 7 8 x(s)D = 8 C7 . . . ˆ xp (s) ˆxp (s) 0 Up (s) |p (s) 0 P Q Introducing notations: ˆxl (s) = (D Ol Fl > E Ol Fl > Fl > Gl ) ˆ xl (s) = (D Ol Fl > Ol > Fl > 1)> Q P and hl for a vector whose the l-th entry is one and the others are zero, then we have ˆ xl (s) = hl (L F(sL D + Ol hl F)1 Ol ) P ˆxl (s) = hl (G + F(sL D + Ol hl F)1 (E Ol hl F))= Q Note that £ ¤ ¯ xl (s) ˆ xl (s) 0 · · · 0 = hl (L F(sL D + Ol hl F)1 Ol hl ) := P 0 ··· 0 P so setting ¯ l = Ol hl O gives

434

13 Fault isolation schemes

5

6 4 6 5 6 35 65 6 5 ˆx1 (s) ¯ x1 (s) u1 (s) 0 U1 (s) |1 (s) Q P 9 .. : 9 : F : E9 : 9 .. : 9 .. .. .. 7 . 8=7 8 x(s)D 8 C7 87 . 8 7 . . . ¯ xp (s) ˆxp (s) 0 Up (s) up (s) P |p (s) Q with ¡ ¢ ¯ l F 1 O ¯l) ¯ xl (s) = hl L F(sL D + O P ¡ ¢1 ¯ lF ¯ l F))= ˆxl (s) = hl G + F(sL D + O (E O Q Recalling the coprime factorization of transfer matrices, we are able to rewrite ˆxl (s) into ¯ xl (s) and Q P ˆ r (s)> Q ˆxl (s) = hl Tlr (s)Q ˆr (s) ¯ xl (s) = hl Tlr (s)P P where ˆ r (s) = L F(sL D + Or F)1 Or P ˆr (s) = G + F(sL D + Or F)1 (E Or G) Q ¯ l F)1 (Or O ¯ l) Tlr (s) = L + F(sL D + O with Or denoting some matrix ensuring the stability of matrix D Or F. This leads ﬁnally to 5 6 5 65 6 u1 (s) U1 (s) h1 T1r (s) ³ 0 ´ 9 .. : 9 :9 : ˆ .. .. ˆr (s)x(s) 7 . 8=7 87 8 Pr (s)|(s) Q . . up (s)

5 9 =7

Thus, setting

0

Up (s)

6

hp Tpr (s)

U1 (s)h1 T1r (s) ³ ´ : ˆ .. ˆr (s)x(s) = P (s)|(s) Q 8 r . Up (s)hp Tpr (s) 6 U1 (s)h1 T1r (s) : 9 .. U(s) = 7 8 . Up (s)hp Tpr (s) 5

we obtain the general form of residual generators (13.55). This demonstrates that a bank of residual generators is in fact a special form of (13.55). The DOS has also been extended to the solution of actuator fault isolation problem, where the system inputs are assumed to be xl (w) = xrl (w) + il (w)> l = 1> · · · > ni with il being the l-th fault in the l-th input. Analogues to the (13.52) we are able to use a bank of residual generators

13.5 Fault isolation using a bank of residual generators

5 9 7

6

5

435

6

I1 (x1 (s)> |(s)) u1 (s) : .. .. : = 9 8 . . 8 7 ¡ ¢ uni (s) Ini xni (s)> |(s)

for the purpose of actuator fault isolation. Note that since |(s) may depend on every input signal xl (s) the l-th residual regenerator Il should be so designed that it is decoupled from xm (s)> m 6= l. This, as shown in the last section, becomes possible when the number of the outputs is at least equals the number of the inputs (and so the number of the faults). The advantage of the DOS is its clear structure and working principle. In against, the application ﬁelds are generally limited to the sensor fault isolation. Remark 13.8 In literature, the reader may ﬁnd the statement that less robustness is an essential disadvantage of the DOS. It is argued as follows: For the construction of each residual generators only one output signal is used. In this case, as known, the design freedom is restricted. Remember, however, on the assumption that sensor fault may occur in every sensor (and so every output) we can only isolate sensor faults and have in fact no design freedom for the purpose of robustness. Thus, the above claim is, although it seems reasonable, not correct. Of course, in case that only a part of sensors may fail, for instance, |1 > · · · > |ni > and the rest, |ni +1 > · · · |p > will fault-freely work, we can modify the DOS as follows ¡ ¢ 6 6 5 5 I1 x(s)> |1 (s)> |ni +1 (s)> · · · |p (s) u1 (s) : 9 .. : 9 .. 8= 7 . 8=7 . ¡ ¢ uni (s) Ini x(s)> |ni (s)> |ni +1 (s)> · · · |p (s) It becomes evident that the degree of freedom provided by |ni +1 (s)> · · · |p (s) can be utilized for the purpose of enhancing the robustness of the FDI system. 13.5.2 The generalized observer scheme (GOS) The GOS was proposed by Frank and his co-worker. The working principle of the GOS is dierent from the one of the DOS. Assume again there exist ni p faults to be isolated. The ﬁrst step to a fault isolation using the GOS consists in the generation of a bank of residual signals that fulﬁll the relation: 6 6 5 5 T1 (i2 (s)> · · · > ini (s)) u1 (s) : 9 .. : 9 .. (13.56) 8= 7 . 8=7 . ¡ ¢ uni (s) Tni i1 (s)> · · · > ini 1 (s) A unique fault isolation then follows the logic if all ul 6= 0 except u1 then i1 6= 0 ··· if all ul = 6 0 except uni then ini 6= 0=

(13.57) (13.58)

436

13 Fault isolation schemes

Although this concept seems quite dierent from the logic usually used, namely, 5 6 6 5 T1 (i1 (s)) u1 (s) 9 .. : 9 : .. 7 . 8=7 8 ¡. ¢ uni (s) Tni ini (s) with if ul 6= 0 then il 6= 0> l = 1> · · · > ni

(13.59)

the existence conditions for a PFIs remains same. To explain this, we only need to notice the fact that the logic (13.57)-(13.58) is indeed complementary to the logic (13.59), i.e. if a fault is localizable according to (13.57)-(13.58), then we are also able to locate the fault uniquely using (13.59). To compare with the DOS, we demonstrate the application of the GOS to the sensor fault isolation. To this end, we consider again process model (13.53) with ni = p sensor faults to be detected and isolated. Under the use of notations 6 6 6 5 5 5 J1 (s) |1 (s) i1 (s) : : : 9 9 9 .. .. .. : : : 9 9 9 . . . : : : 9 9 9 9 Jl1 (s) : 9 |l1 (s) : 9 il1 (s) : ¯ l (s) = 9 : > |¯l (s) = 9 : , i¯l (s) = 9 : J 9 Jl+1 (s) : 9 |l+1 (s) : 9 il+1 (s) : : : : 9 9 9 : : : 9 9 9 .. .. .. 8 8 8 7 7 7 . . . Jp (s)

|p (s)

ip (s)

the system model can be rewritten into ¯ l (s)x + i¯l (s)> l = 1> · · · > p |¯l (s) = J on account of which, the GOS residual generators are constructed in the following form ¡ ¢ ¯l (s)x(s) > l = 1> · · · > p ¯ l (s)¯ |l (s) Q (13.60) ul (s) = Ul (s) P whose dynamics is governed by ¯ l (s)i¯l (s)> l = 1> · · · > p ul (s) = Ul (s)P

(13.61)

¯l (s) are a left coprime pair of transfer matrix J ¯ l (s). It is ¯ l (s)> Q where P evident that (13.61) fulﬁlls (13.56). The original version of the GOS was presented in the state space form described by ¯ l (w)x(w)) |l (w) F¯l { ˆ(w) G { ˆ˙ (w) = Dˆ {(w) + Ex(w) + Ol (¯ ¯ ¯ ul (w) = |¯l (w) Fl { ˆ(w) Gl x(w) whose dynamics is then given by

13.5 Fault isolation using a bank of residual generators

437

h(w) ˙ = (D Ol F¯l )h(w) Ol i¯l (w)> ul (w) = F¯l h(w) + i¯l (w) where

6 6 5 F1 G1 9 .. : 9 .. : 6 5 9 . : 9 . : F1 : : 9 9 : 9 9 Gl1 : Fl1 : ¯ 9 .. : ¯ : 9 9 Jx (s) = (D> E> F> G)> F = 7 . 8 > Fl = 9 : > Gl = 9 Gl+1 : = : 9 Fl+1 : 9 Fp 9 . : 9 . : 7 .. 8 7 .. 8 Fp Gp 5

Analogues to the discussion about the DOS, we demonstrate next that (13.60) is equivalent to (13.55), the general form of residual generators. ¯l (s) by ¯ l (s)> Q Denote P ¯l (s) = (D Ol F¯l > E Ol F¯l > F¯l > G ¯ l) ¯ l (s) = (D Ol F¯l > Ol > F¯l > L)> Q P and introduce a matrix L¯l ¤ £ L¯l = h1 · · · hl1 0 hl+1 · · · hp with hl being a vector whose the l-th entry is one and all the others are zero, then (13.60) can be rewritten as ´ ³ ¡ ¢ ¯l (s)x(s) = Ul (s)L¯l P ˜l (s)x(s) ˜ l (s)|(s) Q ¯ l (s)L¯l |(s) Q ul (s) = Ul (s) P where ˜ l F)1 O ˜ l (s) = L F(sL D + O ˜l P 1 ˜ ˜ l G) ˜ Ql (s) = G + F(sL D + Ol F) (E O ˜ l = Ol L¯l . By introducing with O ˆ r (s) = L F(sL D + Or F)1 Or P ˆr (s) = G + F(sL D + Or F)1 (E Or G) Q ˜ l F)1 (Or O ˜ l) Tlr (s) = L + F(sL D + O it turns out

and ﬁnally

³ ´ ˆ r (s)|(s) Q ˆr (s)x(s) ul (s) = Ul (s)L¯l Tlr (s) P

6 u1 (s) ³ ´ : 9 ˆ r (s)|(s) Q ˆr (s)x(s) u(s) = 7 ... 8 = U(s) P 5

5

up (s)

6 U1 (s)L¯1 T1r (s) ³ ´ 9 : ˆ .. ˆr (s)x(s) = =7 8 Pr (s)|(s) Q . Up (s)L¯p Tpr (s) Thus, it is demonstrated that the GOS is also a special form of (13.60).

438

13 Fault isolation schemes

Remark 13.9 We would like to emphasize that both the DOS and GOS have the same degree of freedom for the purpose of fault isolation or robustness enhancement. Also, using a bank of residual generators, either the DOS or the GOS, we do not achieve more degree of design freedom than a multidimensional residual generator. Example 13.6 In this example, the application of the DOS is illustrated via the benchmark lateral dynamic system. The design objective is to isolate the sensor faults, which will be done based on the model (3.76). To this end, two observers are constructed, and each of them is driven by one sensor: Residual generator I: { ˆ˙ (w) = Dˆ {(w) + Ex(w) + O1 u1 (w) u1 (w) = |1 (w) F1 { ˆ(w) G1 x(w) where |1 (w) is the lateral acceleration sensor signal and ¸ 0=0501 O1 = 0=1039 Residual generator II: { ˆ˙ (w) = Dˆ {(w) + Ex(w) + O2 u2 (w) u2 (w) = |2 (w) F2 { ˆ(w) where |2 (w) is the yaw rate sensor signal and ¸ 0=4873 = O2 = 7=5252 Remark 13.10 We would like to mention that in the case of isolating two faults the DOS and the GOS are identical, as we can see from the above example.

13.6 Notes and references During last years, discussion and studies on the PFIs have been carried out from dierent viewpoints and using dierent mathematical and control theoretical tools. Generally speaking, the existing schemes can be divided into three categories: • solving the PFIs using the unknown input decoupling strategy • formulating the PFIs as a dual problem of designing a decoupling controller and then solving it in this context • handling the PFIs by means of a bank of residual generators.

13.6 Notes and references

439

All these three schemes have been addressed in this chapter. Attention has also been paid to the study on the relationships between these schemes. The discussion about the existence conditions of a PFIs is an extension of the results by Ding and Frank [38]. Using the matrix pencil approach Patton and Hou [118] have published very interesting results on the fault isolation problem viewed from the viewpoint of unknown input decoupling. Considering that the main results of their work on the existence conditions of a PFIs is similar with the ones given in Theorem 13.2, it is not included in this chapter. Since the topic PFIs is one of the central problems of observer-based FDI, much attention has been paid to it during the last two decades, and as a result, a great number of approaches have been reported during this time, see for instance the survey papers by Frank, Gertler and Patton [50, 51, 62, 120]. In this chapter, we have consciously only introduced those approaches, which are representative for introducing the basic ideas and major schemes for achieving a PFIs. The frequency domain approach developed by Ding [36] gives a general solution for the fault isolation problems, while the approach introduced in Subsection 13.3.1, which was proposed by Liu and Si [95], and the geometric method as well as the general design solution described in Subsection 13.3.3 provide solutions in the state space form. The solutions using a bank of residual generators, the DOS and GOS, were respectively derived by Clark [25] and Frank [50]. It is worth to mention that Alcorta García and Frank [2] reported a novel approach to the fault isolation system design. The main contribution of this approach is the construction of a bank residual generators which have a common dynamic part. As a result, the order of the whole fault detection system may become low. This approach has an intimate relationship to the approaches proposed by Liu and Si [95] and the general design solution given in Subsection 13.3.3. In conclusion, we would like to make the following notes: • In practice, it is not realistic to expect achieving a perfect fault isolation just using a residual generator, because of the strict conditions on the structure of the system under monitoring. In most of cases, a stage of residual evaluation and a decision unit are needed. However, the approaches introduced here provide us with the possibility for clustering faults into some groups, which may considerably simplify the decision on a fault isolation. • The concepts like structured residuals or ﬁxed direction residuals have not been included in this chapter. We refer the interested reader to [13, 66, 62, 64, 65, 120, 135] for excellent references on this topic. • As mentioned in Chapter 4, the structural fault isolatability is a concept that is independent of the FDI system used. In this chapter, we have illustrated how to design an FDI system to achieve a PFIs if the system is structurally fault isolable. The realization of a PFIs is decided by the structure of the system under monitoring and by the available information about the faults. The more information we have, the more faults become

440

13 Fault isolation schemes

isolable. In the worst case, i.e. in case that we have no information about faults, the number of the isolable faults is given by the number of the measurements (sensors), as required by the structural fault isolability. • The major focus of this chapter is on the PFIs without taking into account the unknown inputs, model uncertainties and without addressing the residual evaluation problems. Solving these problems is also a part of a fault isolation process. On the other hand, if the faults are structurally isolable, then we are able to accomplish fault isolation in a two-step procedure: (a) ﬁrst achieve a fault isolation (b) then detect each (isolated) fault by taking into account the inﬂuence of the unknown inputs and model uncertainties. In this way, after designing a fault isolation ﬁlter for a PFIs, the remaining problems in the second step are the standard fault detection problems, to which the schemes and methods introduced in the previous chapters can be applied.

14 On fault identiﬁcation

In the fault diagnosis framework, fault identiﬁcation is often considered as the ultimate design objective. In fact, a successful fault identiﬁcation also indicates a successful fault detection and isolation implicitly. This is a reasonable motivation for the intensive research in the ﬁeld of model-based fault identiﬁcation. Roughly speaking, there are four types of model-based fault identiﬁcation strategies: • the parameter identiﬁcation technique based fault identiﬁcation, where the faults are modelled as system parameters that are then identiﬁed by means of the well-established parameter identiﬁcation technique, • the extended observer schemes, in which the faults are addressed as state variables and an extended observer is constructed for the estimation of both state variables and the faults, • the adaptive observer scheme, which can be considered, in some sense, as a combination of the above two schemes, and • the observer-based fault identiﬁcation ﬁlter (FIF) scheme. The ﬁrst strategy is generally applied for the identiﬁcation of multiplicative faults, in order to ﬁt the standard model form of the parameter identiﬁcation technique, while the second and the fourth ones are dedicated to the additive faults. A major dierence between these four strategies lies in the demand on a priori knowledge of the faults to be identiﬁed. In the framework of the ﬁrst three strategies, a successful and reliable fault identiﬁcation is based on certain assumptions on the faults, for instance they are quasi constant or vary slowly or they are generated by a dynamic system. In against, no assumption on the faults is needed by applying the fault identiﬁcation ﬁlter scheme. In this chapter, we concentrate ourselves on the last fault identiﬁcation scheme, which is schematically sketched in Fig.14.1. A major reason for this focus is, on the one hand, the close relationship of the fault identiﬁcation ﬁlter scheme to the FDI schemes introduced in the former chapters and, on the other hand,

442

14 On fault identiﬁcation

the fact that few systematic studies have been reported on this topic, while numerous monographs and signiﬁcant papers are available for the ﬁrst three fault identiﬁcation schemes. The reader who is interested in these three fault identiﬁcation schemes is referred to the representative literature given at the end of this chapter.

Fig. 14.1 Observer based fault identiﬁcation ﬁlter scheme

14.1 Fault identiﬁcation ﬁlter and perfect fault identiﬁcation In order to present the underlying ideas and the core of the fault identiﬁcation ﬁlter (FIF) scheme clearly, we ﬁrst consider LTI systems described by |(s) = J|x (s)x(s) + J|i (s)i (s) J|x (s) = (D> E> F> G) > J|i (s) = (D> Hi > F> Ii )

(14.1) (14.2)

without considering the inﬂuence of the unknown input. An FIF is an LTI system that is driven by x and | and its output is an estimation of i= To ensure that the estimate for i is independent of x and the initial condition of the state variables, a residual generator is the best candidate for an FIF. Applying residual generator ´ ³ ˆx (s)x(s) ˆ x (s)|(s) Q (14.3) u(s) = U(s) P to (14.1)-(14.2) gives ¯ i (s)i (s) := iˆ(s)> J ¯ i (s) = P ˆ x (s)J|i (s)= u(s) = U(s)J

(14.4)

14.1 Fault identiﬁcation ﬁlter and perfect fault identiﬁcation

443

iˆ(s) 5 Rni is called an estimate of fault vector i and (14.3) is called FIF. See Section 5.2 for a detailed description of residual generator (14.3). The primary interest of designing an FIF is to ﬁnd a fault estimate that is as close as possible to the fault vector. The ideal case is the so-called perfect fault identiﬁcation. Deﬁnition 14.1 Given system (14.1)-(14.2) and FIF (14.3). A perfect fault identiﬁcation (PFI) is the case that iˆ(s) = i (s)=

(14.5)

Next, we study the existence conditions to achieve a PFI. It follows from (14.4) that (14.5) holds if and only if ˆi (s) = L> J|i (s) = P ˆ x1 (s)Q ˆi (s) ¯ i (s) = L +, U(s)Q U(s)J which is equivalent to the statement that J|i (s) is left invertible in RH4 = The following Theorem is a reformulation of the above result. Theorem 14.1 Given system (14.1)-(14.2) and FIF (14.3). Then the following statements are equivalent S1: the PFI is achievable S2: J|i (s) is left invertible in RH4 S3: the rank of J|i (s) is equal the column number of J|i (s) and J|i (s) has no transmission zero in C¯+ for the continuous time systems and C¯1 for the discrete time systems. The proof of this theorem is obvious and is thus omitted. If J|i (s) is given in the state space presentation with J|i = (D> Hi > F> Ii ) > then the statement S3 in Theorem 14.1 can be equivalently reformulated as Corollary 14.1 Given system (14.1)-(14.2) and FIF (14.3), then the PFI is achievable if and only if for continuous time systems ¸ D L Hi = q + ni (14.6) ; 5 C¯+ > udqn F Ii and for discrete time systems ¸ D L Hi ¯ = q + ni = ; 5 C1 > udqn F Ii

(14.7)

Suppose that the existence condition given in Corollary 14.1 is satisﬁed. Then, the following algorithm can be used for the FIF design. Algorithm 14.1 FIF design for a PFI

444

14 On fault identiﬁcation

Step 1: Select O such that { ˆ˙ = Dˆ { + Ex + O(| |ˆ)> |ˆ = F { ˆ + Gx is stable Step 2: Solve Ii Ii = L for Ii and set Ã ³ ´1 ! F sL D + OF OI I F + H I F L I i i i i i Ii U(s) = (Hi OIi ) (14.8) Step 3: Construct FIF iˆ(s) = U(s) (|(s) |ˆ(s)) =

(14.9)

ˆi (s)= Remark 14.1 U(s) given in (14.8) is the (left) inverse of Q We would like to point out that Algorithm 14.1 is generally used for the identiﬁcation of sensor faults due to the requirement udqn (Ii ) = ni = It is very interesting to note that in this case Algorithm 14.1 can also be used for the purpose of (sensor) fault isolation, while Algorithm 13.3 for the fault isolation ﬁlter design fails, see Remark 13.6. Example 14.1 We now design an FIF to identify the sensor faults in the benchmark vehicle dynamic system. For our purpose, we add a post-ﬁlter to the residual signal generated by an FDF with ¸ 0=0133 0=0001 O= 0 1=0004 which is selected based on model (3.76). This post-ﬁlter is given by " 2 # v +4=2243v+31=3489 1=1802v+145=4182 2 2 v +6=1623v+37=2062 ˆ 1 (v) = v +6=1623v+37=2062 U(v) = Q 6 2 i

1=55×10 v+0=3788 v +7=1627v+40=1169 v2 +6=1623v+37=2062 v2 +6=1623v+37=2062

ˆi (v) = (D OF> Hi > F> Ii )= Q Assume that J|i (s) satisﬁes the conditions given in Corollary 14.1. It follows from Lemmas 7.4 and 7.5 that there exist an LCF and a CIOF of J|i (s) so that ˆ 1 (s)Q ˆ (s) = Jfr (s)Jfl (s)> P ˆ 1 (s) = Jfr (s)> Q ˆ (s) = Jfl (s)= J|i (s) = P Since J|i (s) has no RHP zero, Jfl (s) is a regular constant matrix. Without loss of generality, assume Jfl (s) = L> then we have ˆ (s)J|i (s) = L= P This proves the following theorem.

14.2 FIF design with additional information

445

Theorem 14.2 Given system (14.1)-(14.2) that satisﬁes (14.6) or (14.7). Then the FIF { ˆ˙ = Dˆ { + Ex + O (| F { ˆ Gx) > iˆ = Y (| F { ˆ Gx) with Y = (Ii IiW )1@2 > O = (\ F W + Hi IiW )(Ii IiW )1 for continuous time systems or ¡ ¢1@2 Y = Ii IiW + F[F W > O = (DW [F W + Hi IiW )(Ii IiW + F[F W )1 for discrete time systems gives lim iˆ(w) = i (w) or lim iˆ(n) = i (n)

w$4

n$4

where \ 0> [ 0 respectively solve D\ + \ DW + Hi HiW (\ F W + Hi IiW )(Ii IiW )1 (F\ + Ii HiW ) = 0 ³ ´1 ¡ ¢1 F[ DWG [ + Hi IiWB Ii B GB HiW = 0 DG [ L + F W Ii IiW ¡ ¢1 Ii HiW = DG = D F W Ii IiW Recall our study on the structural fault identiﬁability in Section 4.4, it can be concluded that the PFI is achievable if and only if the system is structurally fault identiﬁable. We can further conclude that, referred to the existence condition given in Theorem 13.1 for a successful fault isolation, the PFI is achievable if and only if • the faults are isolable and ˆi (s) is a minimal phase system. • Q We have learned in Chapter 13 how di!cult it is to achieve a fault isolation. ˆi (s) should not have any zero in the The PFI requires in addition that Q RHP including zeros at inﬁnity for continuous time systems. It is a very hard condition which can often not be satisﬁed in practice. For instance, we are not able to identify process component faults, because in this case Ii = 0> which means J|i (s) will have zeros at inﬁnity. In other words, we can claim that the PFI is achievable if only sensor faults are under consideration. Bearing it in mind, we shall present various schemes in the next sections, for which the hard existence conditions given in Theorem 14.1 can be released.

14.2 FIF design with additional information A natural way to release the hard existence conditions is to increase the sensor number to gain additional information. On the other side, this solution means

446

14 On fault identiﬁcation

more cost. In practice, the utilization of the ﬁrst derivative of | is widely adopted as a compromise solution for additional information but without additional sensors. In our following study, we assume that udqn (Ii ) = ? ni

(14.10)

|(w)> |(w)> ˙ x(w)> x(w) ˙ are available and the system model is given in the state space representation. For the sake of simplicity, we only study FIF design for continuous time systems. We ﬁrst check how far the additional information |(w) ˙ can help us to release the hard conditions given in Theorem 14.1. Since for |(w) = 0 L (|(w)) ˙ = v|(v) =, J|i ˙ (v) = vJ|i (v) it becomes clear that J|i ˙ (v) has all the ﬁnite transmission zeros of J|i (v)= Comparing with the existence condition given in Corollary 14.1, it can be concluded that using |(w) ˙ only helps us to remove the zeros at inﬁnity. On account of this result, we concentrate ourselves below on the the zeros at inﬁnity. We ﬁrst write |(w) ˙ into ˙ + Ii i˙(w)= |(w) ˙ = FD{(w) + FEx(w) + FHi i (w) + Gx(w)

(14.11)

Note that the term Ii i˙(w) means additional faults on the one side and does not lead to removing the transmission zeros at inﬁnity on the other side. To avoid i˙(w)> let S solve S Ii = 0> udqn (S FHi ) = max ni = Denote

¸ ¸ £ ¤ |(w) x(w) > xh (w) = > Eh = E 0 |h (w) = S |(w) ˙ x(w) ˙ ¸ ¸ ¸ F G 0 Ii > Gh = > Ii>h = Fh = S FD S FE S G S FH

we have an extended system model {(w) ˙ = D{(w)+Eh xh (w)+Hi i (w)> |h (w) = Fh {(w)+Gh xh (w)+Ii>h i (w)= (14.12) In (14.12), the number of the transmission zeros at inﬁnite, q}>4 > is determined by q}>4 = ni udqn (Ii>h ) = Considering that

¸ L J|i (v) J|h i (v) = vS

and thus has all the ﬁnite transmission zeros of J|i (v)> the following theorem is proven.

14.2 FIF design with additional information

447

Theorem 14.3 Given system (14.12) and assume that udqn (J|i (v)) = ni = Then, the PFI is achievable if and only if ¸ Ii = ni C1 : udqn (14.13) FHi ¸ D L Hi C2 : ;> 0 Re () > || ? 4> udqn = q + ni = (14.14) F Ii This theorem reveals the role and limitation of the additional information |(w)= ˙ For the realization of the idea, we can use the following algorithm. Algorithm 14.2 FIF design for a PFI under utilization of |(w) ˙ Step 0: Check the existence conditions given in Theorem 14.3. If they are satisﬁed, go to the next step, otherwise stop Step 1: Solve S Ii = 0> udqn (S FHi ) = ni Step 2: Apply Algorithm 14.1 to the design of an FIF for system (14.12). It is interesting to note that for Ii = 0> the existence conditions given in Theorem 14.3 are identical with the ones of Corollary 6.6, which deals with the design of UIO. This motivates us to construct an FIF using the UIO scheme. Without proof, we present an algorithm for this purpose. The interested reader is referred to the discussion in Subsection 6.5.2. Algorithm 14.3 FIF design for a PFI using the UIO scheme Step 1: Solve P

Ii FHi

¸

= Lni ×ni > P =

¸ P11 0 > P11 5 R×p 0 P22

for P Step 2: Set

W = L Hg>2 F> Hg>2 = Hg

0 P22

¸

Step 3: Select O so that ¸ ¶ μ P11 + O F is stable W D Hg 0 Step 4: Construct an observer ´ ³ ´ ³ ´ ³³ ´ ˜ | ˜ ˜ ˜ }(w) ˙ = W D OF }(w) + W E OG x(w) + W D OF Hg>2 + O ¸ ˜ = Hg P11 + O O 0

448

14 On fault identiﬁcation

Step 5: Set iˆ(w) =

¸ P11 ((L Hg>2 F) |(w) F}(w) Gx(w)) = ˙ FD}(w) FDHg>2 |(w) FEx(w)) P22 (|(w)

One question may arise: can we use the higher order derivatives of |(w) as additional information to achieve a PFI which is otherwise not achievable based on |(w)> |(w)? ˙ The following theorem gives a clear answer to this question. Theorem 14.4 Given system (14.12) and assume that udqn (J|i (v)) = ni and | (l) (w)> l = 1> · · · > q> are available for the FIF construction. Then, the PFI is achievable if and only if 6 5 Ii 9 FHi : : 9 : 9 (14.15) C1 : udqn 9 FDHi : = ni : 9 .. 8 7 . FDq1 Hi ¸ D L Hi = q + ni = (14.16) C2 : ;> 0 Re () > || ? 4> udqn F Ii Proof. The proof of this theorem is similar to the one of Theorem 14.3. Two facts are needed to be noticed: | (l) (w) = FDl {(w) + FDl1 Hi i (w) +

l1 X

FDm1 Hi i (lm) (w) + Ii i (l) (w)

m=1

5 9 9 9 7

|(v) v|(v) .. . vq |(v)

6

5

: 9 : 9 :=9 8 7

L vL .. .

6 : : : (J|x (v)x(v) + J|i (v)i (v)) = 8

(14.17)

(14.18)

vq L

From (14.17) we know that the term FDl1 Hi i (w) can contribute to removing the zeros at inﬁnite, while (14.18) tells us that all the ﬁnite transmission zeros of J|i (v) cannot be removed. These prove the theorem. t u

14.3 On the optimal fault identiﬁcation problem The results in the previous section make it clear that a PFI is only achievable under strict conditions. This fact motivates the search for an alternative solution. The H4 OFIP introduced in Section 7.6 has been considered as such a solution. In this section, we present a key result in the H4 OFIP framework,

14.3 On the optimal fault identiﬁcation problem

449

which extends the results given in Section 7.6. In the following study, we only consider continuous time systems. We assume that A1: ¯ i (v)) = ni udqn(J i ¯ i (v) 5 RHp×n A2: J has at least one zero in the RHP including the ze4 ¯ i (v) is non-minimum phase in a ros on the m$-axis and at inﬁnity, i.e. J generalized sense, in order to avoid the trivial instance of the problem.

On these two assumptions, we study the following optimization problem ° ° ¯ i (v)° = (14.19) min °L U(v)J 4 U(v)5RH4

Note that the optimization problem (7.155) with p = ni = 1 is a special case of (14.19). i ¯ i (v) 5 RHp×n Theorem 14.5 Given J which is non-minimum phase (hav4 ing zeros in RHP and at inﬁnity), then we have

¯ i (v)k4 = 1= min kL U(v)J U

(14.20)

¯ i (v) = Jfr (v)Jfl (v) Proof. We begin with a co-inner-outer factorization of J ¯ i (v) respectively. with Jfr (v) and Jfl (v) denoting co-outer and co-inner of J It results in ¯ i (v)k4 = kL U(v)Jfr (v)Jfl (v)k4 kL U(v)J which further leads to ¯ i (v)k4 = min kX (v) U(v)Jfr (v)k4 min kL U(v)J U

U

(14.21)

with X (v) = Jfl (v)= Note that min kX (v) U(v)Jfr (v)k4 kX (v)k4 = 1= U

(14.22)

On the other hand, min kX (v) U(v)Jfr (v)k4 U

min kX (v) T(v)k4 kX kK

T5UK4

where X and kX kK represent the Henkel operator of X (v) and its Henkel norm, and the last inequality can be found in Francis’s book. Since Jfl (v) 5 RH4 , X (v) = Jfl (v) is an anti-stable transfer function matrix. Thus, denoting the minimal space realization of X (v) by (DX > EX > FX > GX ), which gives X = (DX > EX > FX > 0), we have, following,

450

14 On fault identiﬁcation

kX kK = (pd{ )1@2

(14.23)

where pd{ is the maximal eigenvalue of matrix S T with S and T solving > > > DX S + S D> X = EX EX > DX T + TDX = FX FX

Moreover, it holds for X (v) = Jfl (v)> S T = L=

(14.24)

Therefore, kX kK = 1 and so min kX (v) U(v)Jfr (v)k4 1= U

(14.25)

Summarizing (14.22)-(14.25) ﬁnally yields ¯ i (v)k4 = min kX (v) U(v)Jfr (v)k4 = kX kK = 1= min kL U(v)J U

U

t u Remark 14.2 (14.23) and (14.24) are known in the H4 optimization framework, see the literature given at the end of this chapter. Once again, we would like to call reader’s attention to (14.20) that means ¯ i (v)k4 = U(v) = 0 = arg min kL U(v)J U

The real reason for this more or less surprising result seems to be the fact that a satisfactory fault identiﬁcation over the whole frequency domain is not achievable, provided that the transfer function matrix from i to | is non-minimum phase. If this interpretation is true, then introducing a suitable weighting matrix Z (v) which is used to limit the frequency interval interested for the fault identiﬁcation purpose, could improve the performance. The study in the following sections will demonstrate it and show three dierent ways to the alternative problem solutions.

14.4 Study on the role of the weighting matrix In this section, we consider residual generators of the form ¯ i (v) u(v) = U(v)J and study the generalized optimal fault identiﬁcation problem (GOFIP) deﬁned by ° ° ¯ i (v)° (14.26) min °Z (v) U(v)J 4 U(v)5RH4

14.4 Study on the role of the weighting matrix

451

where Z (v) 5 RH4 is a weighting matrix. Our study focus is on the role of Z (v)= Again, the two assumptions A1 and A2 mentioned in the last section are assumed to hold. Considering that a fault isolation is necessary for a fault identiﬁcation, for our purpose and also for the sake of simplicity, we ﬁrst reformulate the GOFIP (14.26). ˜ Let us choose a U(v) 5 RH4 such that ˜ J ¯ i (v) = gldj(j1 (v)> · · · > jn (v)) U(v) i

(14.27)

¡ ¢ and introduce T(v) = gldj t1 (v)> · · · > tni (v) 5 RHn4i ×ni > which leads to 6 t1 (v)j1 (v) : .. ¯ i (v) = 9 ˜ U(v)J jl (v) 5 RH4 > l = 1> · · · > ni 8 > U(v) = T(v)U(v)> 7 . tni (v)jni (v) 5

then we have 5

6 5 6 u1 (v) t1 (v)j1 (v)i1 (v) 9 : 9 : .. u(v) = 7 ... 8 = 7 8= . uni (v) tni (v)jni (v)ini (v)

(14.28)

˜ Note that the selection of U(v) is a fault isolation problem, which is also the ﬁrst step to a successful fault identiﬁcation. The next step is the solution of the modiﬁed GOFIP: given weighting factors zl (v)> jl (v) 5 RH4 > ﬁnd tl (v) 5 RH4 such that kzl (v)il (v) tl (v)jl (v)il (v)k2 = kzl (v) tl (v)jl (v)k4 > l = 1> · · · > ni kil (v)k2 il 6=0 (14.29) is minimized. Before we begin with solving the GOFIP (14.29), we would like to remind the reader of Lemma 7.7, which tells us, on the assumption that jl (v) has a single RHP zero v0 > sup

min kzl (v) tl (v)jl (v)k4 = |zl (v0 )| =

tl 5RH4

(14.30)

Equation (14.30) reveals that zl (v) should structurally have all RHP zeros with the associated structure of jl (v)> in order to ensure that min kzl (v) tl (v)jl (v)k4 = 0=

tl 5RH4

In the following of this section, we focus our attention on the GOFIP (14.29), which is the standard scalar-valued model-matching problem. On the assumption that jl (m$) 6= 0 for all 0 $ 4

452

14 On fault identiﬁcation

and jl1 (m$) 5 @ RH4 to avoid the trivial instance, we have a standard algorithm (see the book by Francis given at the end of this chapter) to compute optimal tl (v) and the value of min kzl (v) tl (v)jl (v)k4 =

tl 5RH4

Algorithm 14.4 Solution of GOFIP (14.29) Step 1: Do an inner-outer factorization jl (v) = jll (v)jlr (v) where jll (v) is inner and jlr (v) outer Step 2: Deﬁne s(v) = (jll (v))1 zl (v) := x(v)zl (v) and ﬁnd a state-space minimal realization s(v) = (Dl > el > fl > 0) + (a function in RH4 ) where (Dl > el > fl > 0) is strictly proper and analytic in Re v 0 with Dl antistable Step 3: Solve the equations Dl Of + Of DWl = el eWl > DWl Or + Or Dl = fWl fl

(14.31)

for Of and Or Step 4: Find the maximum eigenvalue 2 of Of Or and a corresponding eigenvector & Step 5: Deﬁne (v) = (Dl > &> fl > 0)> *(v) = (DWl > 1 Or &> eWl > 0)> "(v) = s(v)(v)@*(v) Step 6: Set tl (v) = (jlr (v))1 "(v)

(14.32)

tl (v) given in (14.32) is the solution of GOFIP (14.29), i.e. tl (v) = (jlr (v))1 "(v) = arg min kzl (v) tl (v)jl (v)k4 = tl 5RH4

Moreover, min kzl (v) tl (v)jl (v)k4 =

tl 5RH4

and zl (v) tl (v)jl (v) is all-pass, i.e. ;$ |zl (m$) tl (m$)jl (m$)| = = Using the above algorithm we now prove the following theorem. First, for the sake of simplicity, it assumed that

14.4 Study on the role of the weighting matrix

453

x(v) = (jll (v))1 := (Dx > ex > fx > gx ) and Dx has only real and dierent eigenvalues. Thus, without loss of generality, we further suppose that ¢ ¡ Dx = gldj(1 > · · · ) =, hDx w = gldj h1 w > · · · > h w = This is achievable by a regular state transformation. Theorem 14.6 Given a weighting function zl (v) = (Dz > ez > fz ) and jl (v) 5 RH4 > then • min kzl (v) tl (v)jl (v)k4

tl 5UK4

34 41@2 Z C (fz hDz w ez )2 gwD 0

3

41@2

Z4

1 = C 2

(14.33)

zl (m$)zl (m$)g$ D

(14.34)

4

• Z4 min kzl (v) tl (v)jl (v)k4

tl 5UK4

¯ ¯ ¯fz hDz w ez ¯ gw = kzl k 1

(14.35)

0

where > are some positive constants. Proof. Remember that jll (v) is an inner factor, so we have x(v) = (jll (v))1 = jl (v)= Thus, the unstable projection of s(v)> (Dl > el > fl > 0)> can be described by (Dl > el > fl > 0) = (Dx > [ez > fx > 0) where [ is the solution of the equation Dx [ [Dz = ex fz =

(14.36)

Solving (14.31) and (14.36), respectively, gives Z4

Z4 Dx w

[=

h

Dz w

ex fz h

hDl w el eWl (hDl w )W gw=

gw> Of =

0

0

Replacing el by [ez leads to Z4 hDl w el eWl (hDl w )W gw =

Of = 0

Z4 0

34 43 4 4W Z Z hDx w C hDx w ex fz hDz w ez gwD C hDx w ex fz hDz w ez gwD (hDx w )W gw= 0

0

454

14 On fault identiﬁcation

Let f be the maximum eigenvalue of Of . Since 4W 3 4 3 Z4 Z4 Z4 f = ChDx w hDx w ex fz hDz w ez gwD ChDx w hDx w ex fz hDz w ez gwD gw 0

0

0

and

6 h1 w ex1 : 9 .. hDx w ex = 7 8 . 5

h w ex we have Z4

3 ChDx w

f =

4W 3

Z4

0

hDx w ex fz hDz w ez gwD ChDx w 0

4

Z4

hDx w ex fz hDz w ez gwD gw 0

(14.37) 34 42 Z4 X Z ¡ l w ¢2 h exl C hl w fz hDz w ez gwD gw = Z4 X 0

0

l=1

(14.38)

0

3 4 Z4 Z4 Z4 ¡ l w ¢2 w 2 D w 2 l z C h exl (h ) gw (fz h ez ) gwD gw = 1 (fz hDz w ez )2 gw=

l=1

0

0

0

(14.39) Note that fz hDz w ez is the impulse response function of the weighting factor zl (v), hence we also have Z4 f 1

zl (m$)zl (m$)g$=

4

Denote the maximum eigenvalue of matrix Or by r and recall that p f r = It then turns out min kzl (v) tl (v)jl (v)k4

tl 5UK4

v u Z4 u u = tr 1 zl (m$)zl (m$)g$ 4

41@2 3 41@2 34 Z Z4 1 = C zl (m$)zl (m$)g$ D = C (fz hDz w ez )2 gwD 2 0

4

This proves (14.33). The proof of (14.35) is evident. It follows from

14.4 Study on the role of the weighting matrix

f =

Z4 X 0

455

34 42 Z ¡ l w ¢2 w D w h exl C h l fz h z ez gwD gw

l=1

0

34 34 42 42 Z4 X Z Z ¯ w ¯ ¯ ¯ ¡ l w ¢2 h exl C ¯h l fz hDz w ez ¯ gwD gw 1 C ¯fz hDz w ez ¯ gwD = l=1

0

0

Notice that

0

34 4 Z ¯ ¯ C ¯fz hDz w ez ¯ gwD = kzl k

1

0

we ﬁnally have min kzl (v) tl (v)jl (v)k4 = kzl k1 =

tl 5RH4

t u From this theorem we evidently see that the performance of a fault identiﬁcation depends on the selection of the weighting factor zl (v)= A suitable selection of zl (v) may strongly improve the estimation accuracy. Although the estimation errors bounds given by (14.33) and (14.35) are conservative, they may help us to have a better understanding regarding to selecting a weighting factor. To demonstrate this, let us observe the following case. Suppose that we would like to recover (identify) a fault over a given frequency interval ($ 1 > $ 2 ). In order to describe this requirement, we introduce a bandpass as weighting factor zl (v), which has the following frequency domain behavior ½ 1> $ 5 ($ 1 > $ 2 ) 2 = |zl (m$)| = ' 0> $ 5 @ ($ 1 > $ 2 ) It follows from (14.33) that in this case 3 min kzl (v) tl (v)jl (v)k4

tl 5RH4

μ '

1 C 2

41@2

Z4

zl (m$)zl (m$)g$ D 4

¶1@2 1 |$ 2 $ 1 | = 2

In extreme case, we even have lim

min kzl (v) tl (v)jl (v)k4 $ 0

$ 2 $$ 1 tl 5RH4

That means we are able to achieve the desired estimation accuracy if the frequency interval is very narrow. In a similar way, we can also design zl (v) in the time domain based on (14.35).

456

14 On fault identiﬁcation

14.5 Approaches to the design of FIF In this section, we introduce three design approaches for the FIF design. We start with the second step of designing an FIF with the following model 6 5 6 5 t1 (v)j1 (v)i1 (v) + t1 (v)jg>1 (v)g(v) u1 (v) : 9 : 9 .. u(v) = 7 ... 8 = 7 8 . uni (v) tni (v)jni (v)ini (v) + tni (v)jg>ni (v)g(v) 6 5 jg>1 (v) : .. 1×n ˜ J ¯ g (v) = 9 U(v) 8 > jg>l (v) 5 RH4 g 7 . jg>ni (v) and try to solve the problem described as: given jl (v)> jg>l (v) 5 RH4 and a constant A 0> ﬁnd a reasonable zl (v) as well as tl (v) 5 RH4 that is the solution of the optimization problem min

tl (v)5RH4

kzl (v) tl (v)jl (v)k4 > ktl (v)jg>l (v)k4 ? > l = 1> · · · > ni = (14.40)

We would like to emphasize that the selection of zl (v) is also a part of our design procedures. 14.5.1 A general fault identiﬁcation scheme The underlying idea of this approach is to ensure that zl (v) have all RHP zeros with the associated structure of jl (v)= To this end, we propose a two-step design procedure. Algorithm 14.5 Two-step solution of design problem solution (14.40) Step 1: Selection of weighting matrix zl (v) • do an extended CIOF using Algorithm 7.9 jl (v) = jr (v)˜ jl (v)

(14.41)

with invertible jr (v)> j˜l (v) having as its zeros all the zeros of jl (v) in the RHP including on the m$-axis and at inﬁnity. Note that jl (v) is a SISO system and thus the computation may become very simple • set (14.42) zl (v) = j˜l (v) Step 2: Solution of the optimization problem min

tl (v)5RH4

k1 tl (v)jr (v)k4 > ktl (v)jg>l (v)k4 ? =

(14.43)

It is evident that solution of (14.43) delivers an estimate for i¯l (v) = j˜l (v)il (v)= We would like remark that the above algorithm is applicable, independent of the placement of the zeros of jl (v) in the complex plane.

14.5 Approaches to the design of FIF

457

14.5.2 An alternative fault detection scheme The basic idea of the design scheme proposed below is the application of possible frequency or time domain information of the faults to improve the identiﬁcation performance, based on Theorem 14.6. Recall that the algorithm given in the last section is developed on the assumption that jl (m$) 6= 0 for all 0 $ 4

(14.44)

which however may be too hard to be satisﬁed in practice. For instance, if the system is strictly proper then we have a zero at inﬁnity, i.e. jl (m4) = 0= In the following, we are going to propose a scheme to overcome this di!culty. We know that if jl (v) has m$-axis zeros, say $ 1 > · · · > $ n > then faults with frequencies $1 > · · · > $ n do not have any inﬂuence on the system output and thus on the residual signals. In other words, a detection and further identiﬁcation of these faults are impossible. For this reason, we propose the following algorithm for the system design. Algorithm 14.6 The alternative FIF design Step 1: Do an extended co-inner-outer factorization of jl (v)> l = 1> · · · > ni > jl (v)jm$ (v) jl (v) = jr (v)¯ with invertible jr (v)>inner j¯l (v) and jm$ (v) having as its zeros all the zeros of jl (v) on the m$-axis and at inﬁnity Step 2: Select z ¯l (v) according to the frequency domain or the time domain requirements Step 3: Solve optimization problem ° ° ¯l (v) t¯l (v)¯ jl (v)k4 > °t¯l (v)jr1 (v)jg>l (v)°4 ? (14.45) min kz t¯l 5RH4

using the known H4 optimization technique. Step 4: Set tl (v) = t¯l>rsw (v)jr1 (v) as the solution, where t¯l>rsw (v) solves the optimization problem (14.45). Dierent from the design scheme introduced in the previous subsection, the generated residual signal ul (v) delivers an estimate for i¯(v) = jm$ (v)il (v) where jm$ (v) is only a part of j˜l (v) given in (14.41). On the other side, this design procedure requires frequency or time domain information about the possible fault il (v), which is necessary for the selection of z ¯l (v) in Step 3.

458

14 On fault identiﬁcation

14.5.3 Identiﬁcation of the size of a fault In practice, identiﬁcation of the size of a fault, expressed in terms of the energy level (L2 norm) or the average energy level (RMS), is often of primary interest. Recall that in Subsection 7.9.5 as well as in Section 12.3, we have introduced a method that provides us with an alternative solution to the H to H4 design problem, which can also be used to estimate the L2 norm of a fault. Based on this result, we propose the following algorithm for the identiﬁcation of the energy level (L2 norm) or the average energy level of a fault. Algorithm 14.7 Identiﬁcation of the size of a fault Step 1: Do an extended co-inner-outer factorization of jl (v)> l = 1> · · · > ni > jl (v)jm$ (v) jl (v) = jr (v)¯ with invertible jr (v)>inner j¯l (v) and jm$ (v) having as its zeros all the zeros of jl (v) on the m$-axis and at inﬁnity Step 2: Solve optimization problem min subject to ° ° 2 (1 k¯ tl (v)k4 ) ? > °t¯l (v)jr1 (v)jg>l (v)°4 ? t¯l 5RH4

using the known H4 optimization technique. Step 3: Set tl (v) = t¯l>rsw (v)jr1 (v)

(14.46) (14.47)

(14.48)

as the solution, where t¯l>rsw (v) solves the optimization problem (14.46). Remember that integrating tl (v) given in (14.48) into the residual generator yields jl (v)il (v) + t¯l>rsw (v)jr1 (v)tl (v)jg>l (v)g(v)= ul (v) = t¯l>rsw (v)˜ As a result, in case of a weak disturbance g, we have tl>rsw (v)˜ jl (v)il (v)k2 = kil k2 as well as kul (v)k2 k¯ kul (v)kUPV k¯ tl>rsw (v)˜ jl (v)il (v)kUPV = kil kUP V =

(14.49) (14.50)

Example 14.2 Remember that in Example 13.4 we have achieved a perfect fault isolation. Now, based on that result, i.e. 6 v(v4=3163) 6 5 5 u1 (v) (v+1)(v+3) i1 (v) 9 : v(v4=3163) u(v) = 7 u2 (v) 8 = 7 8 (v+1)(v+3) i2 (v) v(v4=3163)(v4=4924) u3 (v) i3 (v) (v+1)(v+2)(v+3)

we are going to identify the faults. To simplify the computation and clearly describe the problem, we only consider the identiﬁcation of ﬁrst fault and

14.5 Approaches to the design of FIF

459

assume that the disturbance on the corresponding sensor is very weak. For our purpose, we ﬁrst apply Algorithm 14.5. In the ﬁrst step, we get v(v 4=3163) v + 4=3163 > j1>0 (v) = = (v + 1)(v + 4=3163) (v + 3)

j1 (v)> je1 (v) = j1 (v) = j1>0 (v)e Set

z1 (v) = je1 (v) we have, after the second step, t1 (v) =

v+3 = v + 4=3463

Thus, as a result, iˆ(v) = t1 (v)u1 (v) which delivers an exact estimate for i¯(v) = je1 (v)i (v)= We now illustrate the application of Algorithm 14.6. The ﬁrst step computation yields jl (v)jm$ (v)> j¯1 (v) = j1 (v) = jr (v)¯

v 4=3163 v v + 4=3163 > jm$ = > j0 (v) = = v + 4=3163 v+1 v+3

Now we select the weighting factor z ¯1 (v)= Considering that j1 (v) has a zero equal to 0, in order to avoid the frequency range around 0, we introduce a band pass 10(v + 1) z ¯1 (v) = v + 10 which is very small around $ = 0 and becomes larger as $ increasing and then approaches to a constant for $ A 10. Next, solving (14.45) gives t¯l>rsw = and ﬁnally t1 (v) =

0=5373v2 + 1=0681v + 0=5308 v2 + 1=9878v + 0=9878

(v + 3)(0=5373v2 + 1=0681v + 0=5308) = (v + 4=3463)(v2 + 1=9878v + 0=9878)

The estimate for i (v) is delivered by (v + 3)(0=5373v2 + 1=0681v + 0=5308) u1 (v)= iˆ(v) = (v + 4=3463)(v2 + 1=9878v + 0=9878) We would like to emphasize that iˆ(v) only gives an estimate of i (v) in the frequency range bounded by z ¯1 (v)= Finally, we consider Algorithm 14.7. The ﬁrst step is identical with the one in Algorithm 14.6, i.e.

460

14 On fault identiﬁcation

j1 (v) = jr (v)¯ jl (v)jm$ (v)= In case that the disturbance is not taken into account, we can set, in the second step, t¯l (v) = 1 and ﬁnally t1 (v) = j01 (v) = and iˆ(v) =

v+3 v + 4=3163

v+3 u1 (v)= v + 4=3163

14.6 Notes and references Two topics, the PFI and H4 OFIP, have been treated in this chapter. In our study, no assumption has been made on the faults to be identiﬁed. This is a major dierence between the approaches described here and the other fault identiﬁcation strategies mentioned at the beginning of this chapter. Study on the PFI is strongly related to the topic structural fault identiﬁability introduced in Chapter 4. Since no assumption on the faults is made, this problem is equivalent to the invertibility of a transfer matrix. Our discussion in Sections 14.1 and 14.2 relies on this idea and the main results can be found in [36, 38]. Hou and Patton have investigated this problem in a different way and by means of the matrix pencil technique [76]. Recalling our discussion about the underlying idea of a UIO in Subsection 6.5.2, an intimate relationship between the FIF and UIO can be recognized. In fact, the ﬁrst FIF towards a PFI has been proposed by Park and Stein using the UIO scheme [113]. As mentioned in Section 7.6, the H4 OFIP is one of the popular topics in the FDI research area [107, 108, 124]. Moreover, it is also often adopted in the integrated design of robust controller and FDI, as proposed in [105, 106, 138]. Extension of the H4 OFI strategy to other types of dynamic systems like time delay systems, nonlinear systems has been recently reported. Theorem 14.5 reveals that solving the H4 OFIP in its original form, (14.19), makes less sense, as far as the fault matrix is non-minimum phase. In this case, integrating a weighting matrix into the system design, as formulated in the GOFIP (14.26), allows reasonable and realistic solutions. Unfortunately, there are few publications devoted to this topic. Our study in Section 14.4 is dedicated to the weighting factors. Based on it, we have developed in Section 14.5 two approaches, which provide us with useful solutions both for the design of the residual generator and the selection of the weighting factors. The last design approach introduced in this section solves the fault identiﬁcation problem in an extended sense. Instead of identifying the faults in the form of a time or

14.6 Notes and references

461

frequency domain function, the energy level of the fault is identiﬁed. This work has been originally motivated and driven by some real application cases. The proof of Theorems 14.5 and 14.6 is based on the known results in [49, 68, 78] and Algorithm 14.4 can be found in [49]. As mentioned at the beginning of this chapter, model-based fault identiﬁcation is a vital research area. We have, with the FIF schemes presented in this chapter, only touched a sub-area. In comparison, the parameter identiﬁcation technique based fault identiﬁcation builds, parallel to the observer-based strategy, one of the mainstreams in this research area. The core of this technique consists in the application of the well-established parameter identiﬁcation technique to the identiﬁcation of the faults that are modelled as system parameters. This technique is especially e!cient in dealing with multiplicative faults. On the other hand, it requires intensive on-line computation and is generally applicable for those faults, which are constants or change slowly. We refer the interested reader to [67, 79, 80, 81, 133] for a comprehensive study on this technique. Recently, the application of the extended observer schemes to fault identiﬁcation has received increasing attention. The underlying idea of the extended observer schemes lies in addressing the faults as the extended state variables, which are then reconstructed by an (extended) observer. The well-known PI observer is a special kind of such observers [15, 127]. The extended observer technique is strongly related to the UIO scheme. In this context, the extended observer is also called simultaneous state and disturbance estimator [132]. Often, such observers/estimators are designed based on certain assumption on the faults, for instance the boundedness on the derivative. We refer the reader to [60, 59, 71] for some recent publications on this topic. Application of advanced adaptive observer technique to fault identiﬁcation has been initiated in the 90’s [41, 40, 143, 144]. In certain sense, it can be considered as a combination of the observer-based and parameter identiﬁcation based schemes. Recent research activity in this ﬁeld is focused on the application to uncertain systems, nonlinear systems and time varying systems [83, 84, 156, 158].

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Index

analytical redundancy, 6 output observer based generation, 74 parity relation based generation, 106 benchmarks DC motor, 31, 113, 294, 308 electrohydraulic servoactuator, 46, 78, 81, 94, 119, 125, 152, 217, 261, 296, 306, 428 inverted pendulum , 34, 94, 97, 102, 118, 139, 142, 156, 236, 258, 300, 304, 422, 424, 457 three tank system, 38, 55, 64, 67, 331, 342 vehicle lateral dynamic system, 42, 146, 177, 200, 231, 273, 353, 423, 436, 442 Bezout identity, 24 Bounded Real Lemma, 169, 174

in the statistical framework, 357, 369 fault detectability in the norm based framework, 370 fault detectability indices, 411 fault detection, 4 fault detection ﬁlter (FDF), 80 fault detection rate (FDR) in the norm based framework, 371 in the statistical framework, 369 fault identiﬁcation, 4 fault identiﬁcation ﬁlter (FIF), 441 fault isolability matrix, 411 fault isolation, 4 fault transfer matrix, 54 generalized likelihood ratio (GLR), 315 generalized optimal fault identiﬁcation problem (GOFIP), 448 hardware redundancy, 4

co-inner-outer factorization (CIOF), 170 extended CIOF, 239 coprime factorization left coprime factorization (LCF), 23 right coprime factorization (RCF), 23 design form of residual generators, 79 diagnostic observer (DO), 81 excitation subspace, 56 false alarm rate (FAR) in the norm based framework, 370

implementation form of residual generators, 79 inner-outer factorization (IOF), 170 Kalman ﬁlter scheme, 175, 324 likelihood ratio (LR), 314 Luenberger equations, 81 a numerical solution, 93 characterization of the solution, 89 Luenberger type ouput observer, 81 maximum likelihood estimate, 315

472

Index

mean square stability, 249 minimum order of residual generators, 87 missed detection rate (MDR) in the statistical framework, 369 model matching problem (MMP), 173 model uncertainties norm bounded type, 26 polytopic type, 26 stochastically uncertain type, 27 modelling of faults, 27 actuator faults, 28 additive faults, 28 faults in feedback control systems, 30 multiplicative faults, 28 process or component faults, 28 sensor faults, 28 norm based residual evaluation limit monitoring, 284 trend analysis, 284 norm of vectors 1 norm, 165 2 norm, 165 " norm, 165 norms of matrices 1 norm, 168 " norm, 168 Frobenius- or Euclidian norm, 168 spectral norm, 168 optimal design of residual generators H3 @H" design, 391 H2 optimization problem, 213 H2 to H3 design, 225 H2 to H2 design, 216 H2 @H2 design, 203 H" optimal fault identiﬁcation , 201 H" to H3 design, 227 H" to H3 design - an alternative solution, 229 Hl /H" design, see also the uniﬁed solution, 233 optimal fault identiﬁcation H" optimal fault identiﬁcation problem (H" OFIP), 447 output observer, 74 parameter identiﬁcation methods, 9

parameterization of residual generators, 77 parity space approach characterization of residual generator, 101 construction of residual generator, 99 minimum order of residual generator, 101 perfect fault identiﬁcation (PFI) existence condition, 441 solutions, 441 perfect fault isolation (PFIs) deﬁnition, 403 existence conditions, 403 fault isolation ﬁlter, 410 frequency domain solution, 408 perfect unknown input decoupling problem (PUIDP), 116 performance indices MV3U index, 186 MV@U index, 186 Vi>+ index, 185 Vi>3 index, 185 H3 index, 218 plausibility test, 6 post-ﬁlter, 77 residual evaluation, 8 residual evaluation functions average value, 286 peak value, 285 RMS value, 287 residual generation, 7 residual generator, 7 residual generator bank dedicated observer scheme (DOS), 430 generalized observer scheme (GOS), 433 residual signal observer-based, 75 parity relation based, 99 sensor fault identiﬁcation, 420, 442 sensor fault isolation, 430, 442 set of detectable faults (SDF), 371 set of disturbances that cause false alarms (SDFA), 370 signal norms

Index L1 norm, 163 L2 norm, 163 L" norm, 164 root mean square (RMS), 163, 287 peak norm , 164 signal processing based fault diagnosis, 5 simultaneous state and disturbance estimator, 459 soft- or virtual sensor, 74 software redundancy, 6 structual fault detectability deﬁnition, 52 existence conditions, 52 structural fault identiﬁability deﬁnition, 65 structural fault isolability check conditions, 58 deﬁnition, 57 system norm H1 norm, 166 H2 norm, 167

473

H" norm, 166 generalized H2 norm, 167 peak-to-peak gain, 167 induced norm, 166 Tchebyche inequality, 349 the uniﬁed solution discrete time version, 237 general form, 242, 376 standard form, 234, 374 thresholds Mwk>UP V>2 , 290 Mwk>shdn>2 , 290 Mwk>shdn>shdn , 289 uniﬁed solution of parity matrix, 194 unknown input diagnostic observer (UIDO), 139 unkown input fault detection ﬁlter (UIFDF), 128 unkown input observer (UIO), 142 weighting matrix, 448