Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB® [1st ed. 2016] 3319323237, 9783319323237

Table of contents :
Series Editors’ Foreword
Preface
Contents
Abbreviations
1 Introduction
1.1 Fault Diagnosis Methodologies
1.2 Robust Observer-Based Fault Diagnosis: An Overview
1.3 Outline of the Book
References
2 Detection and Isolation of Actuator Faults
2.1 Introduction
2.2 Problem Formulation
2.3 Actuator FD Scheme
2.4 Actuator FI Scheme
2.5 Simulation Results
2.6 Conclusions
References
3 Detection and Isolation of Sensor Faults
3.1 Introduction
3.2 Problem Formulation
3.3 Sensor FD Scheme
3.4 Sensor FI Scheme
3.5 Simulation Results
3.6 Conclusions
References
4 Robust Estimation of Actuator Faults
4.1 Introduction
4.2 Problem Formulation
4.3 Actuator FE Scheme
4.3.1 Observer Design
4.3.2 Estimation of Actuator Faults
4.4 A Generalization to Sensor FE
4.4.1 Observer Design
4.4.2 Estimation of Sensor Faults
4.5 Simulation Results
4.5.1 Actuator Fault Estimation
4.5.2 Sensor Fault Estimation
4.6 Conclusions
References
5 Robust Estimation of Sensor Faults
5.1 Introduction
5.2 Problem Formulation
5.3 SMO-Based Sensor FE
5.4 AO-Based Sensor FE
5.5 Simulation Results
5.6 Conclusions
References
6 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and AO
6.1 Introduction
6.2 Problem Formulation
6.3 SMOs-Based FE Scheme
6.3.1 Design of Observers
6.3.2 Estimation of Faults
6.4 SMO- and AO-Based FE Scheme
6.4.1 Design of Observers
6.4.2 Estimation of Faults
6.5 Simulation Results
6.6 Conclusions
References
7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO
7.1 Introduction
7.2 Problem Formulation
7.3 Design of Observers
7.4 Estimation of Faults
7.5 Simulation Results
7.6 Conclusions
References
8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems
8.1 Introduction
8.2 Problem Formulation
8.3 Design of Observer
8.4 Simulation Results
8.4.1 Example of FE for Descriptor Systems
8.4.2 Example of FE for Normal Systems
8.5 Conclusion
References
9 Conclusions and Future Work
9.1 Conclusions
9.2 Future Work
References
Appendix ASolving Linear Matrix Inequality (LMI)Problems
Appendix BYALMIP Toolbox: A Short Tutorial
Index

Citation preview

Advances in Industrial Control

Jian Zhang Akshya Kumar Swain Sing Kiong Nguang

Robust ObserverBased Fault Diagnosis for Nonlinear Systems Using MATLAB®

Advances in Industrial Control Series editors Michael J. Grimble, Glasgow, UK Michael A. Johnson, Kidlington, UK

More information about this series at http://www.springer.com/series/1412

Jian Zhang Akshya Kumar Swain Sing Kiong Nguang •

Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB®

123

Jian Zhang Department of Electrical and Computer Engineering University of Auckland Auckland New Zealand

Sing Kiong Nguang Department of Electrical and Computer Engineering University of Auckland Auckland New Zealand

Akshya Kumar Swain Department of Electrical and Computer Engineering University of Auckland Auckland New Zealand

MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA, http://www.mathworks.com ISSN 1430-9491 Advances in Industrial Control ISBN 978-3-319-32323-7 DOI 10.1007/978-3-319-32324-4

ISSN 2193-1577

(electronic)

ISBN 978-3-319-32324-4

(eBook)

Library of Congress Control Number: 2016936959 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology has an impact on all the areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies…, new challenges. Much of this development work resides in the industrial reports, the feasibility study papers, and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. An important area of industrial control is that of fault detection (FD), isolation (FI), and estimation and the related area of fault-tolerant control (FTC) design. In industry it is usual to find different sectors adopting fault analysis methods that suit the characteristics of their particular processes. For example, the process industries often use statistical techniques to monitor performance and knowledge-based methods to perform fault analysis. The industries that have processes involving rotating machinery usually resort to signal analysis and pattern recognition techniques that form the basis of condition-monitoring methods. In the aerospace industries where more accurate models are available, model-based methods that use residual generation techniques are more likely. If the application is safety-critical as in the nuclear industry, then hardware redundancy is used to overcome fault situations. The Series Editors have known triple computer hardware redundancy to be used in some safety-critical marine applications. The method by which FTC systems are designed will use the information that is available on the process. For example, the model-based fault analyses of the processes of the aerospace industries are frequently supported by model-based FTC methods. This range of fault analysis techniques and industrial applications has supplied the Advances in Industrial Control monographs with a continual succession of diverse contributions ever since the series began in 1992. In recent years the entries to these fields have included the following:

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Series Editors’ Foreword

• Fault-tolerant Flight Control and Guidance Systems by Guillaume J.J. Ducard (ISBN 978-1-84882-560-4, 2009) • Fault-tolerant Control Systems by Hassan Noura, Didier Theilliol, JeanChristophe Ponsart and Abbas Chamseddine (ISBN 978-1-84882-652-6, 2009) • Detection and Diagnosis of Stiction in Control Loops edited by Mohieddine Jelali and Biao Huang (ISBN 978-1-84882-774-5, 2010) • Multivariate Statistical Process Control by Zhiqiang Ge and Zhihuan Song (ISBN 987-1-4471-4512-7, 2012) • Model-based Fault Diagnosis Technique (second edition) by Steven X. Ding (ISBN 978-1-4471-4798-5, 2013) • Fault Diagnosis and Fault-Tolerant Control and Guidance for Aerospace Vehicles by Ali Zolghadri, David Henry, Jérôme Cieslak, Denis Efimov and Philippe Goupil (ISBN 978-1-4471-5312-2, 2013) • Data-driven Design of Fault Diagnosis and Fault-tolerant Control Systems by Steven X. Ding (ISBN 978-1-4471-6409-8, 2014) This entry to the Advances in Industrial Control monograph series, Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB® by Jian Zhang, Akshya K. Swain, and Sing K. Nguang continues the theory and development of model-based fault detection techniques for nonlinear systems. After an introductory overview of FDI methods (Chap. 1), the monograph focuses on techniques for a nonlinear system model that is used from Chap. 2 onwards. Each chapter demonstrates the techniques presented using particular mathematical models: a. b. c. d.

a single-link robotic arm problem in Chaps. 3, 4, and 6; a satellite attitude control problem in Chap. 5; an aircraft model in Chap. 2; and a convenient mathematical model in Chaps. 7 and 8.

Interestingly the monograph deals with separate sensor and actuator faults and then the more difficult situation of simultaneous sensor–actuator faults. Each chapter follows a similar structure: introduction, problem formulation, chapter body sections, simulation results, and finally conclusions; this makes it very appropriate for the purposes of teaching or self-study. There is a useful mathematical appendix and all the simulations use MATLAB® for which there is an appendix describing the authors’ MATLAB® toolbox. This monograph will enable industrial control engineers, academic control researchers, and control graduate students to gain insight and simulations experience with a range of fault analysis situations. It is a very welcome addition to the Advances in Industrial Control series. Glasgow, Scotland, UK

Michael J. Grimble Michael A. Johnson Industrial Control Centre

Preface

The detection and isolation of faults in industrial systems is an important problem and has attracted lots of attention from researchers around the globe during the past few decades. The timely detection, diagnosis, and correction of faults in a system are critical in avoiding abnormal event progression and reducing significantly the productivity loss. Research on fault diagnosis and isolation (FDI) continues to progress at a rapid pace and there is an abundance of literature on this topic using various approaches which range from analytical methods to statistical approaches. Many books and research monographs on FDI and related areas have been published in the past. The mathematics behind the theories of FDI is quite rigorous and involved. From our experience in teaching and research, we have felt that a reasonable percentage of people, students, as well as practicing engineers have difficulties in applying the FDI theories to practical problems using the available software packages. This is a serious bottleneck in popularizing the FDI theories among early researchers and practicing engineers. We, therefore, were motivated to prepare this book. The focus of this book is to address the problem of robust fault detection, isolation, and estimation of actuator and sensor faults for Lipschitz nonlinear systems using sliding-mode, adaptive, and descriptor system approaches. The problem of detecting and isolating actuator faults is initially discussed. Sensor faults are treated as actuator faults by using integral observer-based approach and then the problem of sensor fault diagnosis, including detection, isolation, and estimation is studied. The proposed scheme has the ability of successfully diagnosing incipient faults in the presence of system uncertainties. The problem of fault estimation is then studied. We first consider the situation when there is only actuator fault or sensor fault at a particular time. Then simultaneous estimation of actuator and sensor faults are studied. Sliding-mode observers (SMOs), adaptive observers (AOs), unknown-input observers (UIOs), and descriptor system approaches are applied in this book. H1 filtering is integrated into the observers to ensure that the fault estimation error as well as the state estimation error is less than a prescribed performance level. The existence of the

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proposed fault estimators and their stability analysis are carried out in terms of linear matrix inequalities (LMIs). It has been observed that when the Lipschitz constant is unknown or too large, it may fail to find feasible solutions for observers. In order to deal with this situation, adaptation laws are used to generate an additional control input to the nonlinear system. The additional control input can eliminate the effect of Lipschitz constant on the solvability of LMIs. The effectiveness of various methods proposed in this research has been demonstrated using several numerical and practical examples. The simulation results demonstrate that the proposed methods can achieve the prescribed performance requirements. The monograph is intended for use of mainly upper division engineering students and researchers. The level of mathematical rigor is relatively high and we have therefore proved almost everything and given the prerequisite in the appendix to keep this monograph self-contained. MATLAB and Simulink files for each of the examples can be found on the monograph webpage: www.springer.zhang-swain-nguang.robust_fault_diagnosis Auckland, New Zealand November 2015

Jian Zhang Akshya Kumar Swain Sing Kiong Nguang

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fault Diagnosis Methodologies . . . . . . . . . . . . . . . . . 1.2 Robust Observer-Based Fault Diagnosis: An Overview 1.3 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Detection and Isolation of Actuator Faults 2.1 Introduction . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation . . . . . . . . . . . . . 2.3 Actuator FD Scheme . . . . . . . . . . . . . 2.4 Actuator FI Scheme . . . . . . . . . . . . . . 2.5 Simulation Results. . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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3 Detection and Isolation of Sensor Faults . 3.1 Introduction . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . 3.3 Sensor FD Scheme . . . . . . . . . . . . . 3.4 Sensor FI Scheme . . . . . . . . . . . . . . 3.5 Simulation Results. . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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4 Robust Estimation of Actuator Faults . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Problem Formulation . . . . . . . . . . . . 4.3 Actuator FE Scheme . . . . . . . . . . . . 4.3.1 Observer Design . . . . . . . . . . 4.3.2 Estimation of Actuator Faults .

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4.4 A Generalization to Sensor FE . . . 4.4.1 Observer Design . . . . . . . . 4.4.2 Estimation of Sensor Faults 4.5 Simulation Results. . . . . . . . . . . . 4.5.1 Actuator Fault Estimation. . 4.5.2 Sensor Fault Estimation . . . 4.6 Conclusions . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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6 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and AO . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 6.3 SMOs-Based FE Scheme . . . . . . . . . . . . . . . . . . . . 6.3.1 Design of Observers . . . . . . . . . . . . . . . . . . 6.3.2 Estimation of Faults . . . . . . . . . . . . . . . . . . 6.4 SMO- and AO-Based FE Scheme . . . . . . . . . . . . . . 6.4.1 Design of Observers . . . . . . . . . . . . . . . . . . 6.4.2 Estimation of Faults . . . . . . . . . . . . . . . . . . 6.5 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 7.3 Design of Observers . . . . . . . . . . . . . . . . . . . . . . . 7.4 Estimation of Faults. . . . . . . . . . . . . . . . . . . . . . . . 7.5 Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Robust Estimation of Sensor Faults 5.1 Introduction . . . . . . . . . . . . . . 5.2 Problem Formulation . . . . . . . . 5.3 SMO-Based Sensor FE. . . . . . . 5.4 AO-Based Sensor FE . . . . . . . . 5.5 Simulation Results. . . . . . . . . . 5.6 Conclusions . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .

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8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

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8.3 Design of Observer . . . . . . . . . . . . . . . . . . . 8.4 Simulation Results. . . . . . . . . . . . . . . . . . . . 8.4.1 Example of FE for Descriptor Systems 8.4.2 Example of FE for Normal Systems . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusions and Future Work 9.1 Conclusions . . . . . . . . . . 9.2 Future Work . . . . . . . . . . References . . . . . . . . . . . . . . .

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Appendix A: Solving Linear Matrix Inequality (LMI) Problems . . . . . . 203 Appendix B: YALMIP Toolbox: A Short Tutorial . . . . . . . . . . . . . . . . 213 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Abbreviations

AO BMI FD FDI FE FI FTC LMI SMO UIO

Adaptive Observer Bilinear Matrix Inequality Fault Detection Fault Detection and Isolation Fault Estimation Fault Isolation Fault-Tolerant Control Linear Matrix Inequality Sliding-mode Observer Unknown-input Observer

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Chapter 1

Introduction

A fault can generally be defined as an unexpected deviation of at least one characteristic property, called the feature of the system, from the normal condition which tends to degrade the overall performance of a system and leads to undesirable but still tolerable behavior of the system [1]. The increased productivity requirements and stringent performance specifications have led to more demanding operating conditions in many modern engineering systems such as aircraft, automotive vehicles, high-speed railways and power systems. Such conditions increase the possibility of faults which will result in off-specification production, increased operating costs, detrimental environmental impacts and even catastrophic disasters that claim both property and human life [2, 3]. Faults can be of several types which may arise due to different conditions such as malfunctions in actuators and sensors, abnormal parameter variations of the process, and hard failures in equipment due to structural changes, etc. Typical examples of faults are: • Actuator faults, such as damage in the bearings, deficiencies in force and momentum, defects in the gears, aging effects, and stuck faults. Actuators are used to generate the desired inputs to control the process to behave normally. When actuator faults occur, the faulty actuators are no longer able to generate the desired control inputs. • Sensor faults, such as scaling errors, drifts, dead zones, short cuts, and contact failures. Sensors are used to provide measurements that are needed for monitoring the system and computing the desired inputs. When sensor faults occur, the faulty sensors are no longer able to provide accurate measurements which are needed to generate the control inputs. • Abnormal parameter variations in the system. When some components of the plant are faulty, the original process is changed into a different process so that the controller designed for the original process is no longer able to achieve the expected system performance. • Construction defects such as cracks, ruptures, fractures, leaks, and loose parts etc. • External obstacles such as collisions and clogging of outflows. © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_1

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1 Introduction

Fig. 1.1 A faulty system which is subject to actuator faults and sensor faults

Throughout this book, we will focus on the type of faults which can be modelled as additive changes appearing in actuators and/or sensors. A faulty system with actuator and sensor faults is depicted in Fig. 1.1. The growing demands for reliability, availability, safety and maintainability of modern control systems call for the research of fault diagnosis. This research field has attracted consideration worldwide in both theory and application in the past two decades [4–10]. A fault diagnosis system normally performs three major tasks: fault detection (FD), fault isolation (FI), and fault estimation (FE) [11]. FD is the first step of fault diagnosis and is used to make a decision regarding the system’s working conditions, namely whether or not the system is working under normal conditions. After a fault is detected, the next step is FI. In this step, the locations of faults are determined, i.e., the faulty sensors or actuators are identified from amongst several sensors and actuators. Most practical diagnosis systems contain only fault detection and isolation(FDI). However, FDI cannot provide more comprehensive information of the fault, such as the magnitude, location, and nature of the fault. This information can be provided by the step of FE, which is also referred to as fault identification [1]. This procedure can be regarded as an extension to FDI, since accurate fault estimates implies awareness of their occurrences and locations. Moreover, FE plays an important role in fault-tolerant control (FTC), which stabilizes the closed-loop system and guarantees a prescribed performance level after the occurrence of any fault [12, 13].

1.1 Fault Diagnosis Methodologies During the past two decades, there have been significant research activities in the development of new methodologies for fault diagnosis; for example see [2, 14–17]. These methods can be divided into two categories: hardware redundancy approach and analytical redundancy approach. In hardware redundancy approach, multiple sensors, actuators, and components are equipped to measure and/or control particular variables. Then outputs from identical components are compared for consistency. This kind of approach is commonly used in mission and safety-critical systems such as digital fly-by-wire flight systems and nuclear reactors. However, the use of

1.1 Fault Diagnosis Methodologies

3

physical redundancy dramatically increases system size, expense of extra equipment, and maintenance cost [7]. And sometimes it may cause complex problems when incorporated with other redundant devices. Instead of using extra hardware to create redundancy, the analytical redundancy approach uses analytical or mathematical model of the system to to track the changes in the plant dynamics. The main idea behind this method is to generate directional residuals by failure detection filters. Different fault effects can be mapped into different directions or planes in the residual vector space so that FDI can be achieved [6]. The existing analytical redundancy fault diagnosis approaches can be broadly divided into knowledge-based FDI methods, signal-based FDI methods, and model-based FDI methods: 1. Knowledge-based FDI methods The knowledge-based methods of fault diagnosis are essentially developed from the heuristic symptoms. These are obtained either from expert human operators, or from a qualitative model. In some of the knowledge-based methods, the model is built up by expert reasoning [18], fuzzy reasoning [19], and neural networks [20, 21], for mapping the inputs and outputs of the unknown system. In many other knowledge-based methods, measurement data is mapped to a known pattern which includes different normal and abnormal operating conditions directly, so that the system condition can be identified. One of the major advantages of knowledge-based methods is that they do not require an explicit mathematical model of the monitored system. However, they require, in advance, the knowledge of the monitored plants, knowledge such as the training data which contains faults and the corresponding symptoms, under different faulty conditions. 2. Signal-based FDI methods Many signal-based FDI methods have been developed and can be divided into two categories: spectral analysis [22] (time-frequency, time-scale analysis, etc.) and statistical methods [23] (signal classification, pattern recognition, etc.). These methods extract proper signals or symptoms from the system such as spectral power densities, correlation coefficients, and covariances, for the analysis of faults. Although, unlike knowledge-based methods, signal-based techniques do not require a complete analytical model, their efficiency is particularly limited for early fault detection and for the detection of faults which occur during transient operation. 3. Model-based FDI methods With the development of digital computers and system identification techniques, model-based FDI methods have received considerable attention in recent years [7, 24–26]. Unlike physical redundancy methods, where measurements from parallel components are compared to each other, model-based FDI methods utilize analytically computed outputs and compare the values with the sensor measurements to indicate the presence of faults in systems of interest. The resulting differences are called residuals. These signals are supposed to be zero when the system is fault free and nonzero when the system is faulty. Model-based FDI methods comprise two principal steps: residual generation and residual evaluation. The most important and

4

1 Introduction

difficult task in model-based fault diagnosis is the generation of residuals. Corresponding to different residual generation techniques, model-based FDI methods that have been developed in the literature can be divided into three groups [27]: (1) Parity-equation approach [1, 28]: This approach is based on the test of consistency of parity equations using measurements and inputs. Any inconsistency of the parity equations can be used to detect the faults. (2) Parameter-estimation approach [20, 29, 30]: This approach is based on the assumption that the occurrence of any fault will change the values of the physical system parameters such as mass, friction, resistance, etc. The parameters of the actual process can be repeatedly estimated using on-line parameter-estimation methods. A fault can be declared if there exist discrepancies between the true values of the system parameters and their estimated values. (3) Observer-based approach [8, 31–38]: Observer-based approach is the most extensively used method in model-based fault diagnosis. The basic idea behind this approach is to use observers or filters to estimate system outputs from measurements. The output estimation errors are taken as residuals. Compared with other approaches, the observer-based approach can be implemented using only on-line measurements and offers more design freedom, and therefore attracted the most interests during the past decade. In this book, we will focus on model-based fault diagnosis methods. More specifically, observer-based fault diagnosis methods will be the main concern.

1.2 Robust Observer-Based Fault Diagnosis: An Overview Owing to the high dependence of the model-based fault diagnosis methods to the corresponding mathematical models, a major downside of this class of approaches is that they require an accurate mathematical model of the considered system, which is often difficult to obtain in many practical situations. Moreover, the system parameters often vary during the process and the characteristics of disturbances are also unknown. The existence of system uncertainties and disturbances can cause a misleading alarm and therefore make the model-based fault diagnosis system ineffective [39]. Consequently, it is vital to take the issue of robustness into account while designing observer-based fault diagnosis systems. A robust fault diagnosis system should have the ability to be sensitive to fault signals but insensitive to other signals. During the past two decades, a lot of robust observer-based fault diagnosis methods have been developed, and some of which are described as follows: 1. Beard–Jones fault detection filter (BJFDF)-based fault diagnosis. The development of the observer-based FDI began in the early 1970 s when BJFDF was first introduced in [40, 41]. The main idea behind BJFDF is that each directional residual is designed in correspondence to a particular fault or a particular group of faults. This approach was refined in a geometric framework in [42]. The design

1.2 Robust Observer-Based Fault Diagnosis: An Overview

5

problem of BJFDF was later investigated in [43] using a spectral approach and in [44] using eigenstructure assignment. Some methods to improve the robustness of BJFDF have been proposed in [45–47]. 2. Unknown-input observer (UIO)-based fault diagnosis. The design of observers for systems subject to unknown-inputs has attracted considerable attention in the past and different UIOs have been developed. For example, in [48] full-order linear UIOs were designed while in [49] reduced-order linear UIOs were designed; Nonlinear UIOs were designed for bilinear systems in [50] and for Lipschitz nonlinear systems in [51, 52]. If uncertainties are treated as unknown-inputs, UIOs can be readily used for fault diagnosis. In [53–55], fault diagnosis schemes based on UIO were proposed for linear systems with uncertainties. For special classes of nonlinear systems such as bilinear systems and Lipschitz nonlinear systems, UIO-based fault diagnosis were designed in [56, 57]. Linear UIO was extended to a more general class of nonlinear systems using a nonlinear state transformation, and was applied to fault diagnosis for uncertain nonlinear systems in [58]. It is worth noting that most existing UIO-based schemes assume that the fault distribution matrix is known, which is often not the case, and many of them are only devoted to FD or single FI. 3. Adaptive observer (AO)-based fault diagnosis. The AO has the ability that they can estimate not only system states, but also the slowly varying unknown parameters of the observed systems. In [59–61], systems are assumed to be known and faults can be properly parameterized. In [62, 63], AO was designed for systems with unknown parameters. The main disadvantage of AO-based fault diagnosis methods is that the they are normally only suitable for the constant fault case. However, faults are often time-varying and even fast time-varying sometimes. 4. Sliding-mode observer (SMO)-based fault diagnosis. Due to the inherent robustness of sliding-mode algorithms to unknown modeling uncertainties and disturbances, the fault diagnosis methods based on SMOs have been widely studied in recent years. Considerable success has been achieved in many areas; for example, see [8, 39, 64–70]. Early work on applying the SMO for fault diagnosis was shown in [71] where an SMO is designed with the assumption that the states of the system are available. In [66, 67], the authors attempted to design an SMO for systems with uncertainties. When a fault occurs, the sliding motion will be destroyed and the residual will deviate from zero. On the other hand, the SMO proposed in [8], which is similar to that of [72], can maintain the sliding mode even after the presence of faults by selecting an appropriate gain. Therefore, the constant actuator faults and sensor faults can be reconstructed by the so-called equivalent output injection concept under certain conditions. This result was extended to a more general case in [73] where the derivative of the sensor fault is nonzero. However, the requirement of a complicated coordinate transformation and that the system is accurately known and limits its application. The assumption of open-loop stability in [73] was later relaxed in [68] to achieve robust sensor-fault estimation using a linear matrix inequality (LMI) formulation. In [74], a nonlinear diffeomorphism was introduced to explore the system structure and the sensor fault was transformed

6

1 Introduction

into a pseudo-actuator-fault scenario using a filter. An SMO was then designed to estimate the sensor fault based on the filtered system. In [75], the authors proposed a scheme to estimate incipient sensor faults for both open-loop stable and unstable systems. The problem of early detection of incipient faults using SMOs is also studied in [76]. A high-order SMO was designed in [69] to estimate sensor faults. It has been shown in [77, 78] that if certain conditions on fault and uncertainty distribution matrices are met, then the system uncertainties can be perfectly decoupled from faults and the reconstruction of actuator faults can be achieved. If this condition is not satisfied, the approach must settle for minimizing the effect of the uncertainties on the fault estimation. In [68], an SMO-based FDI scheme which can minimize the L2 gain between the uncertainty and the fault reconstruction signal was proposed for a class of linear systems with uncertainties.

1.3 Outline of the Book In this book, we restrict our attention to design novel observers to detect, isolate and estimate both actuator and sensor faults. As most plants are inherently nonlinear and the faults may often amplify the nonlinearities by driving the plants from a relatively linear operating point into a more nonlinear operating region, the study of fault diagnosis for nonlinear systems has become a very active research topic for both theoretical and practical reasons. The presence of nonlinearities in control systems constitutes a major challenge to observer-based fault diagnosis. Due to the importance and practicality of Lipschitz nonlinear systems, we consider this class of nonlinear system throughout this book. Detailed discussions on the backgroud of each problem and literature surveys may be found in the introduction of each chapter. In order to improve the readability of the book and facilitate the partial readings, we write the chapters in a self-sufficient format. The rest of the book is organized as follows: The book starts with an introduction of actuator FDI in Chap. 2. An SMO is proposed based on a constrained Lyapunov equation and the sufficient conditions for the design of such an observer have been derived. The fault signal is estimated by employing the concept of equivalent output error injection. The estimated fault signal provides information about the occurrence of a fault as well as about the size and severity of the fault. Under certain conditions, the actuator fault can be precisely reconstructed. The effectiveness of the proposed scheme is illustrated considering the example of a modified seventh-order aircraft model. Compared with actuator FDI, very little research has been carried out on the sensor fault diagnosis, especially for the incipient sensor fault diagnosis. Therefore, a robust sensor FDI scheme is proposed in Chap. 3. The essential idea behind the proposed scheme is to employ a coordinate transformation such that the sensor faults can be separated from system uncertainties. More specifically, after the coordinate transformation of the original system, one of the subsystems is only subject to sensor faults , but without modeling uncertainties, which only appears in another subsystem.

1.3 Outline of the Book

7

Based on the transformed system and the integral observer-based approach, multiple sensor faults are detected using the classical Luenberger observer and then isolated using a bank of SMOs. FDI can only indicates when and where a fault occurs in the system, but can not provide the magnitude, shape, and duration of the fault. In the following Chaps. 4–8, a more comprehensive topic, fault estimation, is studied. We start to discuss this topic with a comparatively simpler situation where there occur only actuators or sensor faults occurring at a particular time. In Chap. 4, actuator FE is first discussed. A new SMO-based FE scheme which is inspired by the work presented in [78] is developed. Then the results are extended to sensor FE. In Chap. 5, two methods are proposed for sensor FE. In the first method, the sensor fault is estimated using SMOs, while in the second method, it is estimated based on adaptive technique. Chapters 4 and 5 deal separately with the actuator faults and sensor faults. In Chap. 6, the situation when both actuator faults and sensor faults occur in a process simultaneously is studied. By integrating H∞ criteria, the proposed observer is not only capable of estimating the states and faults, but also minimizes the H∞ gain between the estimation error and system uncertainties. Since the integrated H∞ filtering ensures more accurate state estimations, the fault estimation is much more robust against the system uncertainties, compared with the fault estimation obtained from observers without H∞ filtering. The essence behind the methods of Chap. 6 is to employ two cascade coordinate transformations to transform the system matrices to a special structure. However, the use of such transformations may bring complexities to the design of observers. In Chap. 7, we develop a different FE method which consists of an SMO and an UIO for the same faulty system. A descriptor system approach is introduced to investigate the simultaneous estimation of actuator fault and sensor fault in Chap. 8. The descriptor system is constructed by taking sensor faults as auxiliary states. An observer together with adaptive technique is designed for the augmented system to estimate faults. Asymptotic estimates of the original system states and the sensor faults can be directly obtained from the descriptor observer. Meanwhile, the actuator faults are obtained using adaptive observer-based approach. The difference between the method developed in this chapter and those proposed in Chaps. 4–7 is that, this approach does not require any coordinate transformation and there is no need to determine matrices, which satisfy the matching condition. Concluding remarks and suggestions for future research are discussed in Chap. 9.

References 1. Gertler JJ (1998) Fault detection and diagnosis in engineering systems. Marcel Dekker, New York 2. Venkatasubramanian V, Rengaswamy R, Yin K, Kavuri SN (2003) A review of process fault detection and diagnosis part 1: quantitative model-based methods. Comput Chem Eng 27:293– 311

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3. Chen, W (2007) Model based fault diagnosis in complex control systems-robust and adaptive approaches. PhD thesis, Simon Fraser University 4. Patton RJ, Frank PM, Clark RN (1989) Fault diagnosis in dynamic systems: theory and applications. Prentice-Hall, Upper Saddle River 5. Frank PM (1990) Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy-a survey and some new results. Automatica 26:459–474 6. Frank PM (1996) Analytical and qualitative model-based fault diagnosis-a survey and some new results. Eur J Control 2:6–28 7. Chen J, Patton RJ (1999) Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, Boston 8. Edwards C, Spurgeon SK, Patton RJ (2000) Sliding mode observers for fault detection and isolation. Automatica 36:541–553 9. Nguang SK, Shi P, Ding S (2006) Delay-dependent fault estimation for uncertain time-delay nonlinear systems: an LMI approach. Int J Robust Nonlinear Control 16:913–933 10. Nguang SK, Shi P, Ding S (2007) Fault detection for uncertain fuzzy systems: an LMI approach. IEEE Trans Fuzzy Syst 15:1251–1262 11. Willsky AS (1976) A survey of design methods for failure detection in dynamic systems. Automatica 12:601–611 12. Tan CP, Edwards C (2007) Sensor and/or actuator fault reconstruction plays a key role in the ftc design. Asian J Control 9(3):340–344 13. Edwards C, Tan CP (2006) Sensor fault tolerant control using sliding mode observers. Control Eng Pract 14(8):897–908 14. Venkatasubramanian V, Rengaswamy R, Yin K, Kavuri SN (2003) A review of process fault detection and diagnosis part II: qualitative models and search strategies. Comput Chem Eng 27:313–326 15. Venkatasubramanian V, Rengaswamy R, Yin K, Kavuri SN (2003) A review of process fault detection and diagnosis part III: process history based methods. Comput Chem Eng 27:327–346 16. Isermann R (2005) Model-based fault detection and diagnosis-status and applications. Annu Rev Control 29:71–85 17. Angeli C, Chatzinikolaou A (2004) On-line fault detection techniques for technical systems: a survey. Int J Comput Sci Appl 1:12–30 18. Allen SM, Cagavan AK (1990) An expert system approach to global fault detection and isolation design. Charles river analytics inc., Cambridge 19. Patton RJ, Frank PM, Clark RN (2000) Issues of fault diagnosis for dynamic systems. Springer, London 20. Liu XQ, Zhang HY, Liu J, Yang J (2000) Fault detection and diagnosis of permanent-magnet dc motor based on parameter estimation and neural network. IEEE Trans Ind Electron 47:1021– 1030 21. Lu PJ, Zhang MC, Hsu TC (2001) An evaluation of engine faults diagnostics using artificial neural networks. J Eng Gas Turbine Power 123:340–347 22. Garcia-Perez A, Romero-Troncoso RDJ, Cabal-Yepez E, Osornio-Rios RA (2011) The application of high-resolution spectral analysis for identifying multiple combined faults in induction motors. IEEE Trans Ind Electron 58:2002–2010 23. Carrillo MG, Kinnaert M (2010) Sensor fault detection and isolation in three-phase systems using a signal-based approach. IET Control Theory Appl 4:1838–1848 24. Simani S, Fantuzzi C, Patton R (2003) Model-based fault diagnosis in dynamic systems using identification techniques. Springer, London 25. Floquet T, Barbot JP, Perruquetti W, Djemai M (2004) On the robust fault detection via sliding mode disturbance observer. Int J Control 77:622–629 26. Shields DN (2005) Observer-based residual generation for fault diagnosis for non-affine nonlinear polynomial systems. Int J Control 78:363–384 27. Isermann R (2006) Fault diagnosis of technical process-applications. Springer, Heidelberg 28. Ding X, Guo L, Jeinsch T (1999) A characterization of parity space and its application to robust fault detection. IEEE Trans Autom Control 44:337–343

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29. Balle P, Isermann, R (1998) Fault detection and isolation for nonlinear processes based on local linear fuzzy models and parameter estimation. In: Proceedings of American Control Conference, pp 1605–1609 30. Jiang T, Khorasani K, Tafazoli S (2008) Parameter estimation-based fault detection, isolation and recovery for nonlinear satellite models. IEEE Trans Control Syst Technol 16:799–808 31. Zhang K, Jiang B, Shi P (2012) Observer-based fault estimation and accomodation for dynamic systems. Springer, Heidelberg 32. Hammouri H, Kinnaert M, El-Yaagoubi EH (1999) Observer-based approach to fault detection and isolation for nonlinear systems. IEEE Trans Autom Control 44:1879–1884 33. Jiang B, Chowdhury F (2004) Observer-based fault diagnosis for a class of nonlinear systems. Proc Am Control Conf 6:5671–5675 34. Commault C, Dion J-M, Sename O, Motyeian R (2002) Observer-based fault detection and isolation for structured systems. IEEE Trans Autom Control 47:2074–2079 35. Duan GR, Patton RJ (2001) Robust fault detection using luenberger-type unknown input observers-a parametric approach. Int J Syst Sci 32:533–540 36. Brumback B (1987) A chi-square test for fault-detection in kalman filters. IEEE Trans Autom Control 32:552–554 37. Chen J, Patton RJ, Zhang HY (1996) Design of unknown input observers and robust fault detection filters. Int J Control 63:85–105 38. Zhang J, Shi P, Qiu J, Nguang SK (2015) A novel observer-based output feedback controller design for discrete-time fuzzy systems. IEEE Trans Fuzzy Syst 23:223–229 39. Edwards C, Spurgeon SK (1998) Sliding mode contol: theory and applications. Taylor and Francis, Abingdon 40. Beard RV (1971) Failure accommodation in linear systems through self-reorganization. PhD thesis, MIT 41. Jones HL (1973) Failure detection in linear systems. PhD thesis, MIT 42. Massoumnia M (1986) Geometric approach to the synthesis of failure detection filters. IEEE Trans Autom Control 31:839–846 43. White JE, Speyer J (1987) Detection filter design: spectral theory and algorithms. IEEE Trans Autom Control 32:593–603 44. Park J, Rizzoni G (1993) An eigenstructure assignment algorithm for the design of fault detection filters. IEEE Trans Autom Control 39:1521–1524 45. Douglas R (1993) Robust fault detection filter design. PhD thesis, University of Texas 46. Chung W (1997) Game theoretic and decentralized estimation for fault detection. PhD thesis, University of California, Los Angeles 47. Zhang M, Ding SX, Lam J, Wang H (2003) An LMI approach to design robust fault detection filter for uncertain LTI systems. Automatica 39:543–550 48. Darouach M, Zasadzinski M, Xu SJ (1994) Full-order observers for linear systems with unknown inputs. IEEE Trans Autom Control 39:607–609 49. Hou M, Muller PC (1992) Design of observers for linear systems with unknown inputs. IEEE Trans Autom Control 37:871–874 50. Zasadzinski M, Magarotto E, Darouach M (2000) A disturbance accommodating estimator for bilinear systems. Proceedings of the CDC. Australia, Sydney, pp 796–801 51. Pertew AM, Marquez HJ, Zhao Q (2005) Design of unknown input observers for Lipschitz nonlinear systems. Proceedings of American Control Conference. Portland, Oregon, USA, pp 4198–4203 52. Chen W, Saif M (2006) Unknown input observer design for a class of nonlinear systems: an LMI approach. Proceedings of American Control Conference. Minneapolis, Minnesota, pp 834–838 53. Hou M, Muller PC (1994) Fault detection and isolation observers. Int J Control 60:827–846 54. Chen J, Patton RJ, Zhang H (1996) Design of unknown input observer and robust fault detection filters. Int J Control 63:85–105 55. Guang-Ren D, Patton RJ (2001) Fault detection using Luenberger-type unknown input observers-a parametric approach. Int J Syst Sci 32:533–540

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56. Yang H, Saif M (1997) State observation, failure detection and isolation (fdi) in bilinear systems. Int J Control 67:901–920 57. Chen W (2006) Fault detection and isolation based on novel unknown input observer design. In: Proceedings of American Control Conference, pp 5129–5134 58. Yang H, Saif M (1996) Monitoring and diagnostics of a class of nonlinear systems using a nonlinear unknown input observer. In: Proceedings of IEEE CCA, pp 1006–1011 59. Xu A, Zhang Q (2004) Nonlinear system fault diagnosis based on adaptive estimation. Automatica 40:1181–1193 60. Zhang X, Parisini T, Polycarpou MM (2005) Sensor bias fault isolation in a class of nonlinear systems. IEEE Trans Autom Control 50:370–376 61. Zhang K, Jiang B, Cocquempot V (2008) Adaptive observer-based fast fault estimation. Int J Control Autom Syst 6:320–326 62. Shafai B, Pi CT, Bas O, Nork S, Linder SP (2001) A general purpose observer architecture with application to failure detection and isolation. Proceedings of American Control Conference. Arlington, VA, pp 1133–1138 63. Jiang B, Staroswiecki M, Cocquempot V (2004) Fault diagnosis based on adaptive observer for a class of nonlinear systems with unknown parameters. Int J Control 77:415–426 64. Utkin VI (1978) Sliding modes and their application in variable structure systems. Mir, Moscow 65. Utkin VI (1992) Sliding modes in control optimization. Springer, Berlin 66. Yang H, Saif M (1995) Fault detection in a class of nonlinear systems via adaptive sliding mode observer. In: Proceedings of the IEEE International Conference on System, Man and Cybernetics 67. Hermans F, Zarrop M (1996) Sliding mode observers for robust sensor monitoring. In: Proceedings of the 13th IFAC World Congress, pp 211–216 68. Tan CP, Edwards C (2003) Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int J robust nonlinear control 13:443–463 69. Fridman L, Levant A, Davila J (2007) Observation of linear systems with unknown inputs via high-order sliding-modes. Int J Syst Sci 38:773–791 70. Alwi H, Edwards C, Tan CP (2011) Fault detection and fault-tolerant control using sliding modes. Springer, London 71. Sreedhar R, Fernandez B, Masada G (1993) Robust fault detection in nonlinear systems using sliding mode observers. In: Proceedings of IEEE Conference on Control Application 72. Walcott BL, Zak SH (1987) State observation of nonlinear uncertain dynamical systems. IEEE Trans Autom Control 32:166–170 73. Tan CP, Edwards C (2002) Sliding mode observers for detection and reconstruction of sensor faults. Automatica 38:1815–1821 74. Yan XG, Edwards C (2007) Sensor fault detection and isolation for nonlinear systems based on a sliding mode observer. Int J Adapt Control Signal Process 21:657–673 75. Alwi H, Edwards C, Tan CP (2009) Sliding mode estimation schemes for incipient sensor faults. Automatica 45:1679–1685 76. Chen W, Chowdhury FN (2010) A synthesized design of sliding-mode and Luenberger observers for early detection of incipient faults. Int J Adapt Control Signal Process 24:1021– 1035 77. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 78. Yan XG, Edwards C (2008) Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems. Int J Syst Sci 39(4):349–359

Chapter 2

Detection and Isolation of Actuator Faults

This chapter presents an SMO-based actuator FDI approach for uncertain Lipschitz nonlinear systems. It is assumed in this chapter that only actuator faults occur in the system.

2.1 Introduction The research on FDI has received considerable attention during the past two decades due to increasing demand for safety and reliability of an automatic control system. Fruitful results can be found in [1–6] and the references therein. Due to the robustness to uncertainties such as system deviation, disturbances, and unknown nonlinearities, sliding-mode control (SMC) has been recognized as a promising robust control approach to confront uncertain systems. Since early work of applying SMOs for FDI in [7], the SMOs-based FDI methods have been developed extensively [8–13]. However, almost all these approaches mainly focus on relatively large-sized faults. The research on the detection and isolation of incipient faults has been less studied and still remains a challenge to model-based FDI techniques, because they are almost unnoticeable during their initial stage and their effects to residuals are most likely to be concealed by system uncertainties. Moreover, the robustness of the SMO to uncertainties makes it also robust to these type of faults. Inspired by the work presented in [14], we develop a novel method to detect and isolate incipient actuator faults for uncertain Lipschitz nonlinear systems. The proposed method essentially transforms the original system into two subsystems (subsystem-1 and 2), where subsystem-1 includes the effects of system uncertainties and actuator faults while subsystem-2 only has actuator faults. For the purpose of fault detection, we design an SMO for subsystem-1 and a traditional Luenberger observer for subsystem-2. Taking the output estimation error of subsystem-2 as the residual and comparing it with a predefined threshold, the occurrence of an actuator fault can then be detected if the residual goes over the threshold. © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_2

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2 Detection and Isolation of Actuator Faults

After a fault is detected, the next step is to determine the location of the fault, which is the purpose of FI. In principle, the use of one single observer may permit the isolation of faults if their effect has independent projections onto the residual space. However, if the system has significant nonlinearities, it is difficult to assure this independence. Therefore, a bank of observers are needed to isolate faults if they occur on different actuators at the same time. There are two schemes for FI. The first one is called dedicated observer scheme [15]. In this scheme, N observers are designed to generate N residuals and the ith residual is expected to be only sensitive to the ith fault, but is insensitive to others. The other scheme is called generalized observer scheme [16], where N observers are also designed to produce N residuals. However, the difference is that the ith residual is sensitive to all possible faults except the ith one. In this chapter, the actuator FI is carried out using the modified dedicated observer scheme to subsystem-2. Multiple SMOs, one for each possible actuator fault, are used to generate the estimated output vector. The estimated output vector is then compared with the actual output vector in order to determine which actuator is affected by the fault. The remaining sections of this chapter are organized as follows: Sect. 2.2 briefly describes the mathematical preliminaries required for designing observers. Section 2.3 designs observers for FD and derives the sufficient condition for the stability of the proposed observers based on Lyapunov approach. The actuator FI scheme is presented in Sect. 2.4. Simulation results are shown in Sect. 2.5 with conclusions in Sect. 2.6.

2.2 Problem Formulation Consider a nonlinear system described by 

x˙ (t) = Ax(t) + f (x, t) + Bu(t) + D f a (t) + EΔψ(t), y(t) = Cx(t),

(2.1)

where x ∈ R n , u ∈ R m , and y ∈ R p denote respectively the vector of state variables, inputs and outputs. f a ∈ R h is the vector of unknown actuator faults. Δψ ∈ R r models the lumped uncertainties and disturbances experienced by the system and f (x, t) represents the known nonlinear continuous term. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R n×h , and E ∈ R n×r are known constant matrices with C and E both being of full rank. Note that a nonlinear system of the form x˙ (t) = Ω(x, u, t) can be expressed as x˙ (t) = Ax(t) + f (x, t) if Ω(x, u, t) is continuously differentiable with respect to x. Remark 2.1 It is assumed in the book that the fault distribution matrix D is known. This is not a restrictive assumption since much work has been done on estimating the fault distribution matrix when it is not fully known. Some examples can be seen in [17, 18]. In this chapter, we assume that the actuator faults could occur in each input channel, and therefore we have D = B and f a ∈ R m .

2.2 Problem Formulation

13

Before starting the main results of this chapter, we make the following assumptions on System (2.1). Assumption 2.1 rank(CE) = rank(E). Assumption 2.2 For every complex number s with nonnegative real part 

sI − A E rank C 0

 = n + rank(E).

(2.2)

Assumption 2.3 The nonlinear continuous term f (x, t) is assumed to be known and Lipschitz about the state x uniformly, i.e.,  f (x, t) − f (ˆx , t) ≤ L f x − xˆ , ∀x, xˆ ∈ R n

(2.3)

where L f is the known Lipschitz constant. Assumption 2.4 The actuator fault vector f a and uncertainty vector Δψ satisfies the following constraint:  f a  ≤ ρa and Δψ ≤ ξ,

(2.4)

where ρa and ξ are two known positive constants. Lemma 2.1 Under Assumption 2.1, there exist state and output transformations 

   z1 w1 z = Tx = , w = Sy = z2 w2

(2.5)

such that in the new coordinate, the system matrices become, TAT −1 =



       B1 E1 A1 A2 C1 0 , SCT −1 = , TB = , TE = A3 A4 B2 0 0 C4

(2.6)

where T ∈ R n×n , S ∈ R p×p , z1 ∈ R r , z2 ∈ R n−r , w1 ∈ R r , w2 ∈ R p−r , A1 ∈ R r×r , A2 ∈ R r×(n−r) , A3 ∈ R (n−r)×r , A4 ∈ R (n−r)×(n−r) , B1 ∈ R r×m , E1 ∈ R r×r , C1 ∈ R r×r , C4 ∈ R (p−r)×(n−r) and D2 ∈ R (p−r)×q . E1 and C1 are invertible. Partition T and S as,     S T1 , S= 1 (2.7) T= T2 S2 where T1 ∈ R r×n , T2 ∈ R (n−r)×n , S1 ∈ R r×p , and S2 ∈ R (p−r)×p . After introducing the state and output transformations (2.5), System (2.1) is expressed as, z˙ = TAT −1 z + T f (T −1 z, t) + TB(u + f a ) + TEΔ y = CT −1 z.

(2.8)

14

2 Detection and Isolation of Actuator Faults

Using the relations in (2.6), System (2.8) is converted into two subsystems as  

z˙1 = A1 z1 + A2 z2 + f 1 (T −1 z, t) + B1 (u + f a ) + E1 Δψ w1 = C1 z1 z˙2 = A3 z1 + A4 z2 + f 2 (T −1 z, t) + B2 (u + f a ) w2 = C4 z2 ,

(2.9) (2.10)

where f 1 (T −1 z, t) = T1 f (T −1 z, t) and f 2 (T −1 z, t) = T2 f (T −1 z, t). Lemma 2.2 The pair (A4 , C4 ) is detectable if and only if Assumption 2.2 holds. 

Proof See [19, 20].

It follows from Lemma 2.2 that there exists a matrix L ∈ R (n−r)×(p−r) such that A4 − LC4 is stable, and thus for any Q2 > 0, the Lyapunov equation, (A4 − LC4 )T P2 + P2 (A4 − LC4 ) = −Q2 ,

(2.11)

has a unique solution P2 > 0 [21]. Remark 2.2 It is seen from Lemma 2.1 that the satisfaction of Assumption 2.1 ensures the existence of coordinate transformations T and S, such that in the new coordinate, the subsystem-1, formulated in (2.9), is prone to both actuator faults and system uncertainties, while the subsystem-2, formulated in (2.10), is only prone to actuator faults but free from system uncertainties. It follows from Assumption 2.2 that the pair (A4 , C4 ) is detectable, which provides the necessary condition for the existence of an observer for system (2.10). Assumption 2.3 states that the nonlinear systems considered is Lipschitz. Many practical systems satisfy the Lipschtiz condition, at least locally. For example, trigonometric nonlinearities occurring in robotic applications and the nonlinearities which are square or cubic in nature, can be assumed to be Lipschitz.

2.3 Actuator FD Scheme FD is the first step of fault diagnosis to determine whether a fault has occurred or not. The decision on the occurrence of a fault can be made if a significant residual change is observed. If we design SMOs directly for the original system, the effect of actuator faults, especially the ones with small magnitudes, on state estimation errors could be attenuated or even eliminated by the variable structure term [14]. The detection of faults therefore becomes difficult. Observing subsystem-2 in (2.10), one can find out that the state z2 is neither subject to system uncertainties nor faults before the occurrence of any actuator fault. If we can design an observer for this particular subsystem and take the output estimation error w2 − wˆ 2 (wˆ 2 is the estimation of w2 ) as the residual, then the problem caused by designing conventional SMOs for the

2.3 Actuator FD Scheme

15

original system can be solved. This intuition inspires the proposed FD scheme in this section. For Subsystem (2.9), we construct the following SMO: 

zˆ˙1 = A1 zˆ1 + A2 zˆ2 + f 1 (T −1 zˆ , t) + B1 u + (A1 − As1 )C1−1 (w1 − wˆ 1 ) + ν1 (2.12) wˆ 1 = C1 zˆ1 ,

where As1 ∈ R r×r is a stable matrix which needs to be determined and zˆ is defined as zˆ := col(C1−1 w1 , zˆ2 ). It is worth noting that zˆ does not represent the state estimate vector col(ˆz1 , zˆ2 ). The discontinuous output error injection term ν1 , that is used to eliminate the effects of uncertainties, is defined by  ν1 =

P (C −1 w −ˆz )

k1 P1 (C1−1 w1 −ˆz1 )

if C1−1 w1 − zˆ1 = 0

0

otherwise,

1

1

1

1

(2.13)

where k1 = E1 ξ + η1 . The parameter η1 is a positive scalar which is to be determined for System (2.15) to be driven to the predefined sliding surface (2.32). P1 ∈ R r×r > 0 is the Lyapunov matrix of As1 . It is worth noting that state z1 can be obtained by the measured output y as z1 = C1−1 w1 = C1−1 S1 y. For Subsystem (2.10), a Luenberger observer with the following form is designed: 

zˆ˙2 = A4 zˆ2 + A3 C1−1 w1 + f 2 (T −1 zˆ , t) + B2 u + L(w2 − wˆ 2 ) wˆ 2 = C4 zˆ2 ,

(2.14)

where L ∈ R (n−r)×(p−r) is the gain of a traditional Luenberger observer. If the state estimation errors are defined as e1 = z1 − zˆ1 and e2 = z2 − zˆ2 , then the state estimation error dynamics, before the occurrence of actuator faults, can be obtained as e˙ 1 = z˙1 − z˙ˆ1 = A1 z1 + A2 z2 + f 1 (T −1 z, t) + B1 u + E1 Δψ −A1 zˆ1 − A2 zˆ2 − f 1 (T −1 zˆ , t) − B1 u − (A1 − As1 )C1−1 (w1 − wˆ 1 ) − ν1   = As1 e1 + A2 e2 + f 1 (T −1 z, t) − f 1 (T −1 zˆ , t) + E1 Δψ − ν1 = As1 e1 + A2 e2 + Δf 1 + E1 Δψ − ν1 e˙ 2 = z˙2 − zˆ˙2

(2.15)

= A3 z1 + A4 z2 + f 2 (T −1 z, t) − A4 zˆ2 − A3 C1−1 w1 − f 2 (T −1 zˆ , t) − L(w2 − wˆ 2 )   = (A − LC4 )e2 + f 2 (T −1 z, t) − f 2 (T −1 zˆ , t) = (A − LC4 )e2 + Δf 2 , (2.16) where Δf 1 = f 1 (T −1 z, t) − f 1 (T −1 zˆ , t) and Δf 2 = f 2 (T −1 z, t) − f 2 (T −1 zˆ , t).

16

2 Detection and Isolation of Actuator Faults

We now present Theorem 2.1 which establishes the sufficient condition for the stability of the above error dynamics (2.15) and (2.16). Theorem 2.1 Given System (2.1) with Assumptions 2.1–2.4. When the system is free of actuator faults, the error dynamics (2.15) and (2.16) are asymptotically stable, if there exist matrices As1 < 0, L, P1 = P1T > 0 and P2 = P2T > 0, and positive scalars α1 and α2 such that 

Π1 + α11 P1 P1 Λ := AT2 P1 Π2 +

P1 A2 1 P P + aIn−r α2 2 2

 0 [22]. Taking X = P1T e1 and Y = Δf 1 , we can obtain 2eT1 P1 Δf 1 ≤

1 T e P1 P1T e1 + α1 Δf 1T Δf 1 . α1 1

(2.18)

Note that zˆ := [(C1−1 w1 )T , (ˆz2 )T ]T . Then, before the occurrence of actuator faults we have,   0 (2.19) z − zˆ = e2 Therefore T −1 z − T −1 zˆ  = T −1 e2  and Δf 1  ≤ L f1 T −1 e2  Δf 2  ≤ L f2 T −1 e2 , where L f1 = T1 L f and L f2 = T2 L f .

(2.20)

2.3 Actuator FD Scheme

17

Then we have 1 V˙ 1 ≤ eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A2 e2 + eT1 P1 P1T e1 + α1 Δf 1T Δf 1 α1 + 2eT1 P1 E1 Δψ − 2eT1 P1 ν1 ≤ eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A2 e2 +

1 T e P1 P1T e1 + α1 L f21 T −1 2 e2 2 α1 1

+ 2eT1 P1 E1 Δψ − 2eT1 P1 ν1 .

(2.21)

From the definition of ν1 in (2.13), it can be shown that eT1 P1 ν1 = k1 eT1 P1

k1 P1 e1 2 P1 e1 = = k1 P1 e1 . P1 e1  P1 e1 

Now 2eT1 P1 E1 Δψ ≤ 2E1 ΔψP1 e1  ≤ 2E1 ξ P1 e1 .

(2.22)

Since k1 = E1 ξ + η1 , then 2eT1 P1 E1 Δψ − 2k1 P1 e1  ≤ 2E1 ξ P1 e1  − 2(E1 ξ + η1 )P1 e1  = −2η1 P1 e1 .

(2.23)

Therefore (2.21) can further be simplified as 1 V˙ 1 ≤ eT1 Π1 e1 + 2eT1 P1 A2 e2 + eT1 P1 P1 e1 + α1 L f21 T −1 2 e2 2 − 2η1 P1 e1  α1  1 ≤ eT1 Π1 + P1 P1 e1 + 2eT1 P1 A2 e2 + α1 L f21 T −1 2 e2 2 . (2.24) α1 Similarly, the time derivative of V2 = eT2 P2 e2 along the trajectory of System (2.15) can be computed as V˙ 2 = eT2 P2 e˙ 2 + e˙ T2 P2 e2   = eT2 P2 (A4 − LC4 ) + (A4 − LC4 )T P2 e2 + 2eT2 P2 Δf 2  1 T ≤ e2 Π2 + P2 P2 e2 + α2 L f22 T −1 2 e2 2 . α2

(2.25)

18

2 Detection and Isolation of Actuator Faults

Combining (2.24) and (2.25) yields V˙ = V˙ 1 + V˙ 2   1 1 ≤ eT1 Π1 + P1 P1 e1 + eT2 Π2 + P2 P2 + aIn−r e2 + 2eT1 P1 A2 e2 α1 α2  T   e e (2.26) = 1 Λ 1 e2 e2 If there exist matrices As1 < 0, L, P1 = P1T > 0 and P2 = P2T > 0, and positive ˙ scalars α1 and  α2 such that Inequality (2.17) is satisfied, then V < 0 for any e = 0, e1 . This implies that the error dynamics (2.15) and (2.16) are asympwhere e = e2 totically stable. This completes the proof.  Remark 2.3 Using the Schur complement result, the Inequality (2.17) can be transformed into the following LMI feasibility problem: there exist matrices X, Y , P1 > 0, P2 > 0 and positive scalars α1 , α2 such that ⎡

⎤ X + X T P1 P1 A2 0 ⎢ P1 −α1 I 0 0 ⎥ ⎢ T ⎥ 0. This completes the proof. 

20

2 Detection and Isolation of Actuator Faults

Let the actuator fault occurs at time instant t f . Then the error dynamics (2.15) and (2.16) become   e˙ 1 = As1 e1 + A2 e2 + f 1 (T −1 z, t) − f 1 (T −1 zˆ , t) + E1 Δψ + B1 f a − ν1   e˙ 2 = (A4 − LC4 )e2 + f 2 (T −1 z, t) − f 2 (T −1 zˆ , t) + B2 f a .

(2.37) (2.38)

Observing (2.38), one can find out that e2 is only affected by actuator faults f a , but not subject to system uncertainties Δψ as well as the error injection term ν1 . From (2.31), it can be seen that the bound of the norm of e2 (t) depends on the bound of the −1 unknown initial condition e2 (0). Since e2 (0) is multiplied by e(c0 L f2 T −a0 )t , its effect will decrease exponentially and e2 will approach to zero if there is no actuator fault. Otherwise it will deviate from zero. Therefore ew2  = C4 e2  provides a good choice, as the residual, to detect the occurrence of actuator faults. The actuator FD scheme can be devised as follows: Actuator FD scheme: Actuator faults can be detected if the residual ew2  exceeds a predefined threshold ς . Otherwise the system is healthy within the considered time. The detection time td (td ≥ t f ) is defined as the first time instant such that ew2  is observed greater than ς . Remark 2.4 It follows from Lemma 2.3 that e2 will approach to zero when System (2.1) is healthy. This implies that a small threshold ς can be selected. The value of ς does not significantly affect the performance of the proposed FD scheme.

2.4 Actuator FI Scheme After detecting the occurrence of actuator faults, the next objective is to determine their locations, if the system suffers from multiple faults simultaneously. Denote T T f a as f a = [ f a1 , f a2 , . . . , f am T ]T . If we can decide whether or not f ai = 0, i = 1, 2, . . . , m, then the actuator fault isolation can be achieved according to the known fault distribution matrix B. In order to do this, we adopt the modified dedicated observer scheme. More specifically, for each possible f ai = 0, i = 1, 2, . . . , m, we design two SMOs (one is designed for subsystem-1 and the other is designed for subsystem-2) and a total number of 2m SMOs are designed. The observer that is designed for f ai is required to satisfy the following constraint: the obtained residual is only sensitive to f ai , but is insensitive to all other faults. For the ith actuator fault f ai , i = 1, 2, . . . , m, we design the following SMO for Subsystem (2.9): 

z˙ˆ1i = A1 zˆ1i + A2 zˆ1i + f 1 (T −1 zˆ i , t) + B1 u + (A1 − As1 )C1−1 (w1i − wˆ 1i ) + ν1i wˆ 1i = C1 zˆ1i , (2.39)

2.4 Actuator FI Scheme

21

where zˆ i and wˆ i denote respectively the estimated state and output obtained by this isolation estimator. zˆ i is defined as zˆ i := col(C1−1 w1 , zˆ2i ). The output error injection term ν1i is defined as ν1i

=

 P (C −1 w −ˆzi ) (E1 ξ + B1 ρa + η1 ) P1 (C1−1 w1 −ˆz1i ) 1

1

1

0

1

if C1−1 w1 − zˆ1i = 0

(2.40)

otherwise,

where P1 ∈ R r×r is a symmetric positive definite matrix which is to be determined and η1 is a positive scalar defined by (2.33). For Subsystem (2.10), an SMO is designed instead of a normal Luenberger observer that has been used in Sect. 2.3 for FD. The proposed SMO has the following form: 

zˆ˙2i = A4 zˆ2i + A3 C1−1 w1i + f 2 (T −1 zˆ i , t) + B2 u + L(w2i − wˆ 2i ) + B¯ 2i ν2i wˆ 2i = C4 zˆ2i ,

(2.41)

where L is the observer gain to be determined. Partition B¯ 2 into B¯ 2 = [B21 , . . . , B2m ]. Then B2i represents the ith column of B2 and B¯ 2i denotes the rest of the columns. The output error injection term ν2i is defined by ν2i

=

 F¯ i (wi −wˆ i ) (ρa + η3 )  F¯ i (w2i −wˆ 2i ) 2

0

2

if w2i − wˆ 2i = 0

(2.42)

otherwise,

where η3 is a positive scalar and F ∈ R m×(p−r) is a matrix to be determined. F i represents the ith row of F and F¯ i consists of all other rows. If the state estimation errors obtained from the SMOs, which are designed for the f ai , are defined as ei1 = z1i − zˆ1i and ei2 = z2i − zˆ2i , then the error dynamics after the occurrence of actuator faults can be obtained as   e˙ i1 = As1 ei1 + A2 ei2 + f 1 (T −1 z, t) − f 1 (T −1 zˆ i , t) + B1 f a + E1 Δψ − ν1i (2.43)   e˙ i2 = (A4 − LC4 )ei2 + f 2 (T −1 z, t) − f 2 (T −1 zˆ i , t) + B2 f ai − B¯ 2i ν2i   i = (A4 − LC4 )ei2 + f 2 (T −1 z, t) − f 2 (T −1 zˆ i , t) + B2i f ai + B¯ 2i ( f¯a − ν2i ), (2.44) where f¯a represents the vector of actuator faults excluding f ai . The sufficient conditions for the stability of the above error dynamics are presented in the following result: i

Theorem 2.3 Given System (2.1) with Assumptions 2.1–2.4. If there exist matrices As1 < 0, L, P1 = P1T > 0, P2 = P2T > 0, and F, and positive scalars α1 and α2 such that

22

2 Detection and Isolation of Actuator Faults

B2T P2 = FC4  Π1 + α11 P1 P1 AT2 P1

P1 A2 Π2 +

1 P P α2 2 2

(2.45)

 + aIn−r

0, (2.27) and (2.52) Theorem 2.3 presents the sufficient conditions for the stochastic stability of the dynamics (2.43) and (2.44) when f ai = 0 and also forms the intuitive principle of the FI scheme as follows: the decision on which actuator is faulty is equivalent to the problem of determining which f ai is not equal to zero. After the system is detected to be faulty at some time instant td , a bank of 2m SMOs are designed according to all possible faulty models. More specifically, for each f ai , i = 1, 2, . . . , m, two observers given by (2.39) and (2.41) are designed to estimate states and outputs. It can be seen from the proof of Theorem 2.3 that the output error injection term ν2i can j attenuate the effect of f a , j ∈ {1, 2, . . . , m}\{i} to the residual, but can not eliminate i the effect of f a . Therefore, if f ai = 0, the state estimation error ei2 obtained by the observers which are designated for f ai will converge to zero. Otherwise ei2 will go beyond a predefined threshold for some finite time ti > td if f ai = 0. Based on this analysis, we can choose eiw2  = C4 ei2  as the residual and compare it with the corresponding threshold ςi , then the location of actuator faults can be concluded. The selection of the isolation threshold ςi is similar to the selection of the detection threshold ς . Since the residual eiw2  obtained from the observer which is designed

24

2 Detection and Isolation of Actuator Faults

Fig. 2.1 An example of actuator FI using a bank of SMOs

to isolate f ai , is close to zero if f ai = 0, a small value of ςi can be chosen. The FI scheme can be summarized as follows: Actuator FI scheme: Multiple actuator faults can be isolated by comparing the residual eiw2 , (i = 1, 2, . . . , m) with a predefined threshold ςi . It can be concluded that f ai = 0 if eiw2  goes over the threshold for some finite time ti > td . Otherwise f ai = 0 if eiw2  is always below the threshold ςi during the time studied. Considering the structure of B, the decision on which actuator is faulty can then be made. To further illustrate this scheme, we consider an example of a system with two actuator faults. As shown in Fig. 2.1, f a1 and f a3 can be isolated at time t1 and t3 respectively.

2.5 Simulation Results In this section, the effectiveness of the proposed schemes in detecting and isolating actuator faults has been demonstrated by an example of a modified seventh-order aircraft model used in [26], in which the states are defined as

2.5 Simulation Results

25

x1 = φ − bank angle(rad) x2 = r − yaw rate(rad/s) x3 = p − roll rate(rad/s) x4 = δ − sideslip angle(rad) x5 = x7 − washout f ilter state x6 = δr − rudder de f lection(rad) x7 = δa − aileon de f lection(rad) The inputs are u1 = δrc − rudder command(rad) u2 = δac − aileon command(rad) and outputs are y1 = ra − roll acceleration(rad/s) y2 = pa − yaw acceleration(rad/s) y3 = φ − bank angle(rad) y4 = x7 − washout f ilter state The system is in the form of (2.1) with ⎡

⎤

⎡ 0 ⎥ ⎢ ⎢ 0 −0.154 −0.04 1.54 0 −0.744 −0.032 ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎢ 0 0.249 −1 −5.2 0 0.337 −1.12 ⎥ ⎢ 0 ⎥ ⎢ ⎢ ⎥ ⎢ 0.02 0 ⎥, B = ⎢ A = ⎢ 0.0386 −0.996 0 −2.117 0 ⎢ 0 ⎥ ⎢ ⎢ 0 ⎥ ⎢ 0 0.5 0 0 −4 0 0 ⎢ ⎥ ⎢ ⎣ 20 ⎥ ⎢ 0 0 0 0 −20.000 0 ⎦ ⎣ 0 0 0 0 0 0 0 0 −25 ⎡ ⎤ 0 −0.154 −0.04 1.54 0 −0.744 −0.032 ⎢ 0 0.249 −1 −5.2 0 0.337 −1.12 ⎥ ⎢ ⎥ C=⎢ ⎥ ⎣1 0 0 0 0 0 0 ⎦ 0

0 

0

0

E= 1100100

1

0 T

0

0

1

0

0 

0

0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ 25

0

, f (x, t) = sin x3 sin x3 0 0 sin x3 0 0

T

, Δψ = 2 sin t.

Notice that in [26], the original model is linear and there is no system uncertainty. In our simulation, the terms associated with the nonlinearity and system uncertainty are added to show the effectiveness of the proposed actuator FDI method for uncertain Lipschitz nonlinear systems.

26

2 Detection and Isolation of Actuator Faults

The actuator fault f a = col( f a1 , f a2 ) is applied to the system and defined as  f a1 =  f a2 =

0 , t ≤ 15 s 0.05 exp(0.01t) , t ≥ 15 s 0 , t ≤ 20 s 0.07 exp(0.03t) , t > 20 s

The nonsingular transformation matrices T and S are selected as ⎤ 0.8440 0.1560 0.0405 −1.5598 0 0.7535 0.0324 ⎢ −1.0000 1.0000 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 1.0000 0 0 0 0 ⎥ ⎥ ⎢ 0 0 0 1.0000 0 0 0 ⎥ T =⎢ ⎥ ⎢ ⎢ −1.0000 0 0 0 1.0000 0 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 0 1.0000 0 ⎦ 0 0 0 0 0 0 1.0000 ⎡ ⎤ 1.0000 0 −0.8333 0 ⎢ −1.4359 1.0000 −0.4701 0 ⎥ ⎥ S=⎢ ⎣ 1.0128 0 0.1560 0 ⎦ 1.0128 0 −0.8440 1.0000 ⎡

The system matrices under the new coordinate become ⎤ 1.4794 1.3088 0.7373 5.6393 0 −16.3183 −0.9083 ⎢ −0.1540 −0.1300 −1.0338 1.2998 0 −0.6280 −0.0270 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0.1494 −1.1281 ⎥ ⎢ 0.2490 0.2102 −1.0101 −4.8116 ⎥ ⎢ 0 0.7414 0.0310 ⎥ TAT −1 = ⎢ ⎥ ⎢ −0.9574 −0.8466 0.0388 −3.6104 ⎢ −3.5000 1.0460 −0.8583 −5.4593 −4.0000 2.6372 0.1134 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 −20.0000 0 0 ⎦ ⎣ 0 0 0 0 0 0 −25.0000 ⎤ ⎡ −0.9873 0.0000 −0.0000 0.0000 0 −0.0001 −0.0000 ⎢ 0.0000 0.4701 −0.9426 −7.4112 0 1.4053 −1.0741 ⎥ ⎥ SCT −1 = ⎢ ⎣ 0.0000 −0.1560 −0.0405 1.5598 0 −0.7535 −0.0324 ⎦ 0.0000 −0.1560 −0.0405 1.5598 1.0000 −0.7535 −0.0324 ⎡ ⎤ ⎤ ⎡ 1 15.0700 0.8100 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ 0 0 ⎢ ⎥ ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎥ 0 0 ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ 0 , TE = TB = ⎢ 0 0 ⎢ ⎥ ⎥ ⎢ ⎢0⎥ ⎥ ⎢ 0 0 ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎦ ⎣ 20.0000 0 ⎣0⎦ 0 25.0000 0 ⎡

2.5 Simulation Results

27

Imposing the stability constraint to the transformed system and formulating the problem in an LMI framework gives the values of the parameters of the proposed observers. The simulation results are obtained by running successively the files chapter2_lmi.m, chapter2.mdl, and chapter2_plot.m. The solutions of various LMIs are obtained by running the M-file chapter2_lmi.m which are given below. File Chapter2_lmi.m

28

2 Detection and Isolation of Actuator Faults

2.5 Simulation Results

Parameters are obtained as P1 = 0.0048, As1 = −25.8618, α1 = 0.0021, γ = 2.0190 × 10−11 ⎤ ⎡ 0.3072 0.0109 −0.0689 −0.2578 0.0144 0.0058 ⎢ 0.0109 0.2432 0.0527 0.0969 0.0506 0.0342 ⎥ ⎥ ⎢ ⎢ −0.0689 0.0527 0.4599 0.1648 −0.0253 0.0254 ⎥ ⎥ ⎢ P0 = ⎢ ⎥ ⎢ −0.2578 0.0969 0.1648 0.4662 −0.0008 0.0002 ⎥ ⎥ ⎢ ⎣ 0.0144 0.0506 −0.0253 −0.0008 0.1079 0.0545 ⎦ 0.0058 0.0342 0.0254 0.0002 0.0545 0.0372 ⎤ ⎡ 2.3497 6.3662 1.6347 ⎢ −3.1985 −13.8943 −1.7068 ⎥ ⎥ ⎢ ⎢ −3.8256 −11.5108 −1.5464 ⎥ ⎥ ⎢ L=⎢ ⎥ ⎢ 3.6515 10.8768 −1.9943 ⎥ ⎥ ⎢ ⎣ 58.2398 282.3094 −3.8605 ⎦ −57.1666 −394.4680 8.0411  −0.8797 −4.4882 −0.0156 F= −0.7677 −3.2451 0.0039 

29

30

2 Detection and Isolation of Actuator Faults

It is worth noting that the parameters obtained from LMI may differ from that shown here. This is expected because these are obtained by solving LMIs which does not give unique solutions. Note that although these parameters are computed for fault isolation, they can also be applied for fault detectors (2.12) and (2.14). The above obtained parameters are used to simulate the system described in Simulink model chapter2.mdl and the figures are plotted by running the file chapter2_plot.m. In the simulation, we have selected the initial state as x(0) = [0, 0, 0, 0, 0, 0, 0]T and xˆ (0) = [−0.1, 0, 0.2, −0.1, −0.1, −0.1, −0.1]T . In the Simulink model, a parameter “δ” has been added to the denominator of (2.40) and (2.42) to reduce the chattering effect. We have selected δ = 0.01 in the simulation. The detectability of the proposed scheme is shown in Fig. 2.2. It shows that the proposed method could successfully detect the occurrence of a fault at around t = 10.05 s (the fault occurs at 10 s) with the corresponding threshold being chosen as 0.05. After the detection of faults, the next stage is to determine which actuator, amongst the various actuators, is faulty. This is carried out using two isolation observers. Results of the simulation are shown in Figs. 2.3 and 2.4. The residual generated by the first isolation observer, which is designed for f a1 , is compared with the threshold that is set to 0.02 in Fig. 2.3. From the figure it is observed that the residual exceeds the threshold at about 10.1 s, which implies that f a1 can be found to be nonzero at approximately 10.1 s. Figure 2.4 shows the simulation result when the isolation observer designed for f a2 is used. It is seen from the figure that the residual obtained by the this observer exceeds the threshold (0.06) at approximately 20.1 s, which denotes that f a2 can be found to be nonzero at about 20.1 s. 0.7 residual ||e w2 || threshold for fault detection

0.6

Magnitude

0.5 0.4 0.3 0.2 fault detected

0.1 0

0

5

10

15

Time (s)

Fig. 2.2 Detection of the occurrence of actuator faults

20

25

30

2.6 Conclusions

31 0.8 residual ||e1w2||

0.7

1

threshold for isolating fa

Magnitude

0.6 0.5 0.4 0.3 0.2

1

fa ≠ 0 can be concluded

0.1 0

0

5

10

15

20

25

30

Time (s)

Fig. 2.3 Isolation of f a1 0.7 2

residual ||ew2||

0.6

2

threshold for isolating fa

Magnitude

0.5 0.4 0.3 0.2

2

fa ≠ 0 can be concluded

0.1 0

0

5

10

15

20

25

30

Time (s)

Fig. 2.4 Isolation of f a2

2.6 Conclusions In this chapter, we propose a new scheme to robustly detect and isolate incipient actuator faults for uncertain Lipschitz nonlinear systems. The proposed FDI scheme essentially transforms the original system into two subsystems where subsystem-1 includes both actuator faults and system uncertainties while subsystem-2 has actuator faults but without uncertainties. Actuator faults can be detected by applying a Luenberger observer for subsystem-2, and isolated using a bank of SMOs for both subsystems based on the modified dedicated observer scheme. The most distinct feature of the proposed FDI scheme is that, by imposing a coordinate transformation to the original system, the effects of system uncertainties to the residual of subsystem-

32

2 Detection and Isolation of Actuator Faults

2 are completely decoupled, which makes the scheme sensitive to incipient faults while still robust to modelling uncertainty. Thus, early detection can be achieved and a false alarm caused by modeling uncertainties can be totally avoided. The sufficient conditions of stability of the proposed observers have been studied and represented in the form of LMI. Its effectiveness has been demonstrated considering the example of a modified aircraft model.

References 1. Yeu TK, Kim HS, Kawaji S (2005) Fault detection, isolation and reconstruction for descriptor systems. Asian J Control 7:356–367 2. Ding SX, Zhang P, Naik A, Ding EL, Huang B (2009) Subspace method aided data-driven design of fault detection and isolation systems. J Process Control 19:1496–1510 3. Zhang P, Zou J (2012) Observer-based fault diagnosis and self-restore control for systems with measurement delays. Asian J Control 14:1717–1723 4. Wang Z, Shen Y, Zhang X (2014) Actuator fault estimation for a class of nonlinear descriptor systems. Int J Syst Sci 45:487–496 5. Zhang J, Swain AK, Nguang SK (2012) Detection and isolation of incipient sensor faults for a class of uncertain nonlinear systems. IET Control Theory Appl 6:1870–1880 6. Zhang J, Swain AK, Nguang SK (2013) Robust sensor fault estimation scheme for satellite attitude control systems. J Frankl Inst 350:2581–2604 7. Sreedhar R, Fernandez B, Masada G (1993) Robust fault detection in nonlinear systems using sliding mode observers. In: Proceedings of IEEE conference on control application 8. Edwards C, Spurgeon SK, Patton RJ (2000) Sliding mode observers for fault detection and isolation. Automatica 36:541–553 9. Tan CP, Edwards C (2002) Sliding mode observers for detection and reconstruction of sensor faults. Automatica 38:1815–1821 10. Tan CP, Edwards C (2003) Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int J Robust Nonlinear Control 13:443–463 11. Tan CP, Edwards C (2007) Sensor and/or actuator fault reconstruction plays a key role in the ftc design. Asian J Control 9(3):340–344 12. Yan XG, Edwards C (2007) Sensor fault detection and isolation for nonlinear systems based on a sliding mode observer. Int J Adapt Control Signal Process 21:657–673 13. Raoufi R, Marquez HJ (2010) Simultaneous sensor and actuator fault reconstruction and diagnosis using generalized sliding mode observers. In: Proceedings of American control conference, pp 7016–7021 14. Chen W, Chowdhury FN (2010) A synthesized design of sliding-mode and Luenberger observers for early detection of incipient faults. Int J Adapt Control Signal Process 24:1021– 1035 15. Clark R (1978) Instrument fault detection. IEEE Trans Aerosp Electron Syst 14:456–465 16. Frank PM (1990) Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy-a survey and some new results. Automatica 26:459–474 17. Chen J, Patton RJ (1991) Optimal selection of unkown input distribution matrix in the design of robust observers for fault diagnosis. Automatica 29:837–841 18. Gertler J (1995) Optimal residual decoupling for robust fault diagnosis. Int J Control 61:395– 421 19. Corless M, Tu J (1998) State and input estimation for a class of uncertain systems. Automatica 34:757–764 20. Hui S, Zak SH (2005) Observer design for system with unknown inputs. Int J Appl Math Comput Sci 15(4):431–446

References

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21. Yan XG, Edwards C (2008) Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems. Int J Syst Sci 39(4):349–359 22. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 23. Ioannou PA, Sun J (1996) Robust adaptive control. Prentice Hall, Englewood Cliffs 24. Utkin VI (1992) Sliding modes in control optimization. Springer, Berlin 25. Zhang K, Jiang B, Shi P (2009) Fast fault estimation and accommodation for dynamical systems. IET Control Theory Appl 3:189–199 26. Tan CP, Edwards C (2000) An LMI approach for designing sliding mode observers. In: Proceedings of 39th IEEE conference on decision and control, Sydney, Australia

Chapter 3

Detection and Isolation of Sensor Faults

In this chapter, the actuator fault detection and isolation (FDI) scheme proposed in Chap. 2 is extended to sensor FDI.

3.1 Introduction With the development of modern technology, autonomous systems are more and more dependent on sensors which often carry the most important information in automated/feedback control systems. Faults occurring in sensors may lead to poor regulation or tracking performance, or even affect the stability of the control system. Therefore, the study of sensor FDI is becoming increasingly important. However, compared with the study of actuator FDI using SMOs, the research on sensor FDI is less studied in this realm. In this chapter, we will extend the method proposed in Chap. 2 to sensor FDI. By applying coordinate transformations, we first transform the the original system into two subsystems (subsystem-1 and 2) where subsystem-1 includes the effects of system uncertainties but is free from sensor faults and subsystem-2 has sensor faults but without any uncertainties. Then, we treat sensor faults in subsystem-2 as actuator faults using integral observer-based approach [1]. For the purpose of fault detection (FD), an SMO is designed for subsystem-1, while a traditional Luenberger observer is designed for subsystem-2. The occurrence of a sensor fault is detected if the output estimation error of subsystem-2 exceeds the predefined threshold. Furthermore, based on the modified dedicated observer scheme, we develop a scheme which consists of a bank of SMOs to isolate multiple sensor faults. The chapter is organized as follows: Sect. 3.2 briefly describes the mathematical preliminaries required for designing observers. Section 3.3 proposes a sensor FD scheme and derives the stability condition of the proposed observers based on Lyapunov approach. The sensor fault isolation (FI) scheme is presented in Sect. 3.4. The results of simulation are shown in Sect. 3.5 with conclusions in Sect. 3.6. © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_3

35

36

3 Detection and Isolation of Sensor Faults

3.2 Problem Formulation It is assumed only sensor faults occur in the system. In this case, the considered system has the following form: 

x˙ (t) = Ax(t) + f (x, t) + Bu(t) + EΔψ(t) y(t) = Cx(t) + Dfs (t),

(3.1)

where x ∈ R n , u ∈ R m , and y ∈ R p denote respectively the vector of state variables, inputs and outputs. fs ∈ R q represents the vector of unknown sensor faults. Δψ ∈ R r stands for the system uncertainties and f (x, t) represents the known nonlinear continuous term. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R p×q , and E ∈ R n×r (p ≥ q + r) are known constant matrices, and C, D, and E have full rank. Prior to presenting the main results of this chapter, we make the following assumptions on System (3.1). Assumption 3.1 rank(CE) = rank(E). Assumption 3.2 For every complex number s with nonnegative real part: 

sI − A E rank C 0

 = n + rank(E).

(3.2)

Assumption 3.3 The nonlinear continuous term f (x, t) is assumed to be known and Lipschitz about the state x uniformly, i.e., f (x, t) − f (ˆx , t) ≤ Lf , ∀ x, xˆ ∈ R n ,

(3.3)

where Lf is the known Lipschitz constant. Assumption 3.4 The sensor fault vector fs and uncertainty vector Δψ satisfy the following constraints: fs  ≤ ρs and Δψ ≤ ξ,

(3.4)

where ρs and ξ are two known positive constants. Lemma 3.1 Under Assumption 3.1, there exist state and output transformations 

   z1 w1 z = Tx = , and w = Sy = z2 w2

3.2 Problem Formulation

37

respectively, such that in the new coordinate, system matrices become TAT

−1

SCT −1

     B1 E1 A1 A2 , TB = , TE = = , A3 A4 B2 0     0 C1 0 , SD = = 0 C4 D2 

(3.5)



   T1 S1 n×n where T = ∈R ,S = ∈ R p×p , T1 ∈ R r×n , S1 ∈ R r×p , z1 ∈ R r , T2 S2 w1 ∈ R r , A1 ∈ R r×r , A4 ∈ R (n−r)×(n−r) , B1 ∈ R r×m , E1 ∈ R r×r , C1 ∈ R r×r , C4 ∈ R (p−r)×(n−r) and D2 ∈ R (p−r)×q . C1 is invertible. After introducing the transformations T and S, System (3.1) is converted into the following two subsystems:  

z˙1 = A1 z1 + A2 z2 + f1 (T −1 z, t) + B1 u + E1 Δψ w1 = C1 z1

(3.6)

z˙2 = A3 z1 + A4 z2 + f2 (T −1 z, t) + B2 u w2 = C4 z2 + D2 fs ,

(3.7)

where f1 (T −1 z, t) = T1 f (T −1 z, t) and f2 (T −1 z, t) = T2 f (T −1 z, t). Lemma 3.2 The pair (A4 , C4 ) is detectable if and only if Assumption 3.2 holds. 

Proof See [2, 3].

 t In order to apply the method developed in Chap. 2, we define a new state z3 = 0 w2 (τ )dτ so that z˙3 (t) = C4 z2 + D2 fs .

(3.8)

Equations (3.7) and (3.8) can be combined to form an augmented system of order n + p − 2r as 

z˙2 z˙3



 =

A4 0 C4 0



         A3 f (T −1 z, t) z2 0 B2 + f z1 + 2 u+ + z3 0 0 0 D2 s

w3 = z3

(3.9)

System (3.9) can then be rewritten in a more compact form as 

z˙0 = A0 z0 + A¯ 3 z1 + f¯2 (T −1 z, t) + B0 u + D0 fs w3 = C0 z0 ,

(3.10)

38

3 Detection and Isolation of Sensor Faults

    z2 A4 0 ∈ R (n+p−2r)×(n+p−2r) , ∈ R n+p−2r , w3 ∈ R p−r , A0 = where z0 = z3 C4 0       0 ¯A3 = A3 ∈ R (n+p−2r)×r , B0 = B2 ∈ R (n+p−2r)×m , D0 = ∈ 0 0 D2   −1   f (T z, t) R (n+p−2r)×q , C0 = 0 Ip−r ∈ R (p−r)×(n+p−2r) , f¯2 (T −1 z, t) = 2 . 0 Note that (3.10) is in the form of (2.10) and represents a system with an actuator fault vector fs . Accordingly, System (3.6) can be rewritten as 

z˙1 = A1 z1 + A¯ 2 z0 + f1 (T −1 z, t) + B1 u + E1 Δψ w1 = C1 z1 ,

(3.11)

where A¯ 2 = [ A2 0r×(p−r) ]. Lemma 3.3 The pair (A0 , C0 ) is observable if Assumption 3.2 holds. Proof From the Popov–Belevitch–Hautus (PBH) test, the pair (A0 , C0 ) is observable if and only if 

sI − A0 rank C0



⎡

⎤ sI − A4 0 = rank ⎣ −C4 sI ⎦ = n + p − 2r, 0 I

for all s ∈ C . If s = 0, it is obvious that ⎡ ⎤   sI − A4 0 −A4 + p − r. rank ⎣ −C4 sI ⎦ = rank −C4 0 I

(3.12)

(3.13)

If Assumption 3.2 holds, it follows that (A4 , C4 ) is observable and thus 

sI − A4 rank −C4

 = n − r for all s ∈ C .

(3.14)

It follows that the rank test (3.12) holds when s = 0. Moreover, since (A4 , C4 ) is observable, if s = 0, ⎡

⎤   sI − A4 0   ⎣ −C4 sI ⎦ a1 = 0 ⇒ a1 = 0 a2 a2 0 I ⎡

(3.15)

⎤ sI − A4 0 It means that the columns of ⎣ −C4 sI ⎦ are linearly independent and its rank 0 I is n + p − 2r. This completes the proof. 

3.2 Problem Formulation

39

It follows from Lemma 3.3 that there exists a matrix L0 ∈ R (n+p−2r)×(p−r) such that A0 − L0 C0 is stable, and thus for any Q0 > 0, the Lyapunov equation (A0 − L0 C0 )T P0 + P0 (A0 − L0 C0 ) = −Q0 ,

(3.16)

has a unique solution P0 > 0 [4]. Remark 3.1 Assumption 3.1 ensures the existence of coordinate transformations T and S, such that in the new coordinate sensor faults can be completely decoupled from uncertainties. After the transformation, subsystem-1, which is formulated in (3.11), is free from sensor faults but subject to system uncertainties; and subsystem-2, which is formulated in (3.10), is prone to sensor faults but free from system uncertainties.

3.3 Sensor FD Scheme For Subsystem (3.11), we design an SMO which has the form as 

zˆ˙1 = A1 zˆ1 + A¯ 2 zˆ0 + f1 (T −1 zˆ , t) + B1 u + (A1 − As1 )C1−1 (w1 − wˆ 1 ) + ν1 (3.17) wˆ 1 = C1 zˆ1 ,

where As1 ∈ R r×r is a stable matrix which needs to be determined. zˆ is defined as zˆ := col(C1−1 w1 , zˆ2 ). The discontinuous output error injection term ν1 is defined by ν¯ 1 =

P (C −1 w −ˆz )

k1 P1 (C1−1 w1 −ˆz1 )

if C1−1 w1 − zˆ1 = 0

0

otherwise,

1

1

1

1

(3.18)

where k1 = E1 ξ + η1 and η1 is a positive scalar which needs to be determined. P1 ∈ R r×r is a symmetric positive definite matrix that will be defined later. For Subsystem (3.10), we design the following Luenberger observer: 

z˙ˆ0 = A0 zˆ0 + A¯ 3 C1−1 w1 + f¯2 (T −1 zˆ , t) + B0 u + L0 (w3 − wˆ 3 ) wˆ 3 = C0 zˆ0 ,

(3.19)

where L0 ∈ R (n+p−2r)×(p−r) is the gain of the Luenberger observer. If the state estimation errors are defined as e1 = z1 − zˆ1 and e0 = z0 − zˆ0 , then the error dynamics before the occurrence of sensor faults can be obtained as

 e˙ 1 = As1 e1 + A¯ 2 e0 + f1 (T −1 z, t) − f1 (T −1 zˆ , t) + E1 Δψ − ν1

 e˙ 0 = (A0 − L0 C0 )e0 + f¯2 (T −1 z, t) − f¯2 (T −1 zˆ , t) .

(3.20) (3.21)

40

3 Detection and Isolation of Sensor Faults

We now present Theorem 3.1 which establishes sufficient conditions for the existence of the proposed observers (3.17)–(3.19) and outlines a constructive design procedure. Theorem 3.1 Given System (3.1) with Assumptions 3.1–3.4. When the the system is free of sensor faults, the error dynamics (3.20) and (3.21) are asymptotically stable if there exist matrices As1 < 0, L0 , P1 = P1T > 0 and P0 = P0T > 0, and positive scalars α1 and α0 such that   Π1 + α11 P1 P1 P1 A¯ 2 0 [5], then 1 V˙ 1 ≤ eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A¯ 2 e0 + 2eT1 P1 E1 Δψ + eT1 P1 P1T e1 α1 T 

−1 −1 −1 −1 + α1 f1 (T z, t) − f1 (T zˆ , t) f1 (T z, t) − f1 (T zˆ , t) − 2eT1 P1 ν1 . (3.23) Note that zˆ := [(C1−1 w1 )T , (ˆz0 )T ]T , then before the occurrence of sensor faults we have   0 (3.24) z − zˆ = e2 Therefore T −1 z − T −1 zˆ  = T −1 e2  ≤ T −1 e0  and f1 (T −1 z, t) − f1 (T −1 zˆ , t) ≤ Lf1 T −1 e0  f2 (T −1 z, t) − f2 (T −1 zˆ , t) ≤ Lf2 T −1 e0 ,

(3.25)

where Lf1 = T1 Lf and Lf2 = T2 Lf . Moreover, from the definition of ν1 , it can be obtained that eT1 P1 ν1 = k1 P1 e1 .

(3.26)

3.3 Sensor FD Scheme

41

Then Eq. (3.23) can be simplified as 1 V˙ 1 ≤ eT1 Π1 e1 + 2eT1 P1 A¯ 2 e0 + eT1 P1 P1 e1 + α1 Lf21 T −1 2 e0 2 α1 + 2E1 ξ P1 e1  − 2(E1 ξ + η1 )P1 e1  1 ≤ eT1 Π1 e1 + 2eT1 P1 A¯ 2 e0 + eT1 P1 P1 e1 + α1 Lf21 T −1 2 e0 2 . α1

(3.27)

Similarly, the derivative of V0 can be obtained as 

V˙ 0 = eT0 Π2 e0 + 2eT0 P0 f¯2 (T −1 z, t) − f¯2 (T −1 zˆ , t) 1 ≤ eT0 Π2 e0 + eT0 P0 P0 e0 + α0 Lf22 T −1 2 e0 2 . α0

(3.28)

Combining (3.27) and (3.28) yields  T   e e V˙ = V˙ 1 + V˙ 0 ≤ 1 Λ 1 e2 e2

(3.29)

If there exist matrices As1 < 0, L0 , P1 = P1T > 0 and P0 = P0T > 0, and positive scalars α1 and α0 such that Inequality (3.22) is satisfied, then V˙ < 0 for any e e = 0, where e = 1 . This implies that the error dynamics (3.20) and (3.21) are e2 asymptotically stable. This completes the proof.  Remark 3.2 The problem of finding matrices to satisfy Inequality (3.22) can be transformed into the following LMI feasibility problem using Schur complement: there exist matrices X, Y0 , P1 = P1T > 0, P0 = P0T > 0, and positive scalars α1 and α0 such that ⎤ P1 A¯ 2 0 X + X T P1 ⎢ P1 −α1 I 0 0 ⎥ ⎥ 0. This completes the proof.  After the occurrence of any sensor fault at time instant tf , the state estimation error dynamics (3.20) and (3.21) become

 e˙ 1 = As1 e1 + A¯ 2 e0 + f1 (T −1 z, t) − f1 (T −1 zˆ , t) + E1 Δψ − ν1

 e˙ 0 = (A0 − L0 C0 )e0 + f¯2 (T −1 z, t) − f¯2 (T −1 zˆ , t) + D0 fs .

(3.35) (3.36)

Observing (3.36), one can find out that e0 is only affected by sensor faults fs , but not subject to system uncertainties Δψ  or the error injection term ν1 . Note that the 0 . Thus, the sensor fault fs affects the last sensor fault distribution matrix D0 = D2 p − r components of e0 , namely ez3 , directly (ez3 = z3 − zˆ3 ). More specifically, if there occurs a fault, ez3 will definitely change. The situation that the sensor fault only affects the first n-r components of e0 does not exist. Since e0 will approach to zero only if there is no sensor fault, then after the occurrence of any sensor fault, ez3 will deviate from zero. Based on this analysis, we can choose ew3  = C0 e0  = ez3  as the residual to detect the occurrence of sensor faults. The sensor FD scheme can be devised as follows:

3.3 Sensor FD Scheme

43

Sensor FD scheme: Sensor faults can be detected if the residual ew3  exceeds a predefined threshold ς . Otherwise the system is healthy within the considered time. The detection time td (td ≥ tf ) is defined as the first time instant such that ew3  is observed greater than ς .

3.4 Sensor FI Scheme Assume that multiple sensor faults occur in the system simultaneously. Denote the T T qT vector of sensor faults as fs = [fs1 , fs2 , . . . , fs ]T . The objective of this section is to design a scheme that is capable of determining whether or not fsi = 0, i = 1, 2, . . . , q. Then, according to the known fault distribution matrix D, the location of sensor faults can be determined. For each possible fsi = 0, i = 1, 2, . . . , q, we design two SMOs (one is designed for subsystem-1 and the other is designed for subsystem-2). Therefore for q faulty models, a total number of 2q SMOs are designed. The observers should meet the following requirement by design: the residuals obtained from the observers that are designed for fsi should be only sensitive to fsi , but insensitive to all other faults. To isolate the ith sensor fault fsi , we design the following SMO for Subsystem (3.11): 

zˆ˙1i = A1 zˆ1i + A˜ 2 zˆ0i + f1 (T −1 zˆ i , t) + B1 u + (A1 − As1 )C1−1 (w1i − wˆ 1i ) + ν1i (3.37) wˆ 1i = C1 zˆ1i ,

where zˆ i and wˆ i denote, respectively, the estimated state and output obtained by this isolation estimator. zˆ i is defined as zˆ i := col(C1−1 w1 , [ In−r 0 ]ˆz0i ). The output error injection term ν1i is defined as ν1i

=

P (C −1 w −ˆzi )

(E1 ξ + η1 ) P1 (C1−1 w1 −ˆz1i )

if C1−1 w1 − zˆ1i = 0

0

otherwise,

1

1

1

1

(3.38)

where P1 ∈ R r×r is a symmetric positive definite matrix to be determined and η1 is a positive scalar defined in (3.32). For Subsystem (3.10), the proposed SMO has the following form: 

¯ 0i ν2i z˙ˆ0i = A0 zˆ0i + A¯ 3 C1−1 w1i + f¯2 (T −1 zˆ i , t) + B0 u + L0 (w3i − wˆ 3i ) + D i i wˆ 3 = C0 zˆ0 ,

(3.39)

where L0 is the observer gain which is to be determined. Partition D0 into D0 = q ¯ 0i denotes the rest of the [D01 , . . . , D0 ]. D0i represents the ith column of D0 and D columns. The discontinuous output error injection term ν2i is defined by

44

3 Detection and Isolation of Sensor Faults

⎧ ⎨(ρ + η ) F¯ 0i eiw3 if e = 0 s 3 F¯ i ei  w3 i 0 w3 ν2 = ⎩0 otherwise,

(3.40)

where eiw3 = w3i − wˆ 3i , η3 is a positive scalar and F0 ∈ R q×(p−r) is a matrix to be determined. F0i is the ith row of F0 and F¯ 0i consists of all other rows. Define the state estimation errors obtained by the SMOs that are designed for possible fsi = 0 as ei1 = z1i − zˆ1i and ei0 = z0i − zˆ0i . Then the error dynamics after the occurrence of sensor faults can be obtained as

 (3.41) e˙ i1 = As1 ei1 + A¯ 2 ei0 + f1 (T −1 z, t) − f1 (T −1 zˆ i , t) + E1 Δψ − ν1i

 i i −1 −1 i i i ¯ 0 ν2 e˙ 0 = (A0 − L0 C0 )e0 + f¯2 (T z, t) − f¯2 (T zˆ , t) + D0 fs − D

 ¯ 0i (f¯s i − ν2i ), (3.42) = (A0 − L0 C0 )ei0 + f¯2 (T −1 z, t) − f¯2 (T −1 zˆ i , t) + D0i fsi + D i where f¯s represents the vector of all other sensor faults except fsi .

Theorem 3.3 Given System (3.1) with Assumptions 3.1–3.4. If there exist matrices As1 < 0, L0 , P1 = P1T > 0, P0 = P0T > 0 and F0 , and positive scalars α1 and α0 such that D0T P0 = F0 C0  Π1 + α11 P1 P1 A¯ T2 P1 Π0 +

¯

P1 A2 1 P P + α0 0 0

(3.43)

 aIn+p−2r

0, P0 = P0T > 0 and F0 , and positive scalars α1 and α0 such that Inequality (3.44) is satisfied, then we have V˙ i = V˙ 1i + V˙ 0i < 0. This concludes that ei tends to zero exponentially if fsi = 0 even after the occurj rence of sensor faults fs = 0, j ∈ {1, 2, . . . , q}\{i}. On the other hand, if fsi = 0, ¯ 0i (f¯s i − ν2i ), because D0 is of full the term D0i fsi in (3.42) can not be attenuated by D column rank. Therefore, we can conclude that limt→∞ ei0 = 0 if fsi = 0. This completes the proof.  Remark 3.3 For a special case when P0 has the diagonal structure as  P0 =

P01 0 0 P02

 (3.48)

where P01 ∈ R (n−r)×(n−r) and P02 ∈ R (p−r)×(p−r) are symmetric positive definite matrices. Then the inequality (3.44) can be transformed into the following LMI feasibility problem: there exist matrices X, Y02 , P1 > 0, P01 > 0, P02 > 0 and positive scalars α0 , α1 such that ⎡

⎤ X + X T P1 P1 A2 0 0 0 ⎢ P1 −α1 I 0 0 0 0 ⎥ ⎢ T ⎥ T T ⎢ A P1 0 A4 P01 + P01 A4 + aI C4 P02 P01 0 ⎥ ⎢ 2 ⎥ < 0, T ⎢ 0 −Y02 − Y02 + aI 0 P02 ⎥ 0 P02 C4 ⎢ ⎥ ⎣ 0 0 P01 0 −α0 I 0 ⎦ 0 −α0 I 0 0 0 P02 (3.49)  0 where X = , where L02 ∈ R (p−r)×(p−r) . Y02 = P02 L02 and L0 = L02 Substituting the diagonal block matrix P0 into (3.44) yields F0 = D2T P02 . 

P1 As1 ,

46

3 Detection and Isolation of Sensor Faults

Remark 3.4 For a more general case of P0 , the problem of finding matrices P0 , F0 to simultaneously satisfy both (3.44) and (3.43) can be transformed into the following LMI optimization problem: minimize γ subject to P1 > 0, P0 > 0, (3.30) and   −γ In+p−2r (D0T P0 − F0 C0 )T < 0, D0T P0 − F0 C0 −γ Iq

(3.50)

where X = P1 As1 and Y0 = P0 L0 . Theorem 3.3 characterizes the property of the proposed SMOs and also forms the intuitive principle of the FI scheme as follows: According to all possible faulty models, a bank of 2q SMOs are activated when a fault is detected at some time instant td . More specifically, for each fsi , i = 1, 2, . . . , q, two observers given by (3.37) and (3.39), are designed to estimate the states and the outputs. If fsi = 0, the state estimation error ei0 obtained by the observers which are designated for fsi will converge to zero. Otherwise, the state estimation error ei0 will go beyond a predefined threshold for some finite time ti > td if fsi = 0. Based on this analysis, we can choose eiw3  = C0 ei0  as the residual and summarize the FI scheme as follows: Sensor FI scheme: Multiple sensor faults can be isolated by comparing the residual eiw3 , (i = 1, 2, . . . , q) with a predefined threshold ςi . If eiw3  goes over the threshold for some finite time ti > td , then it is concluded that fsi = 0. Otherwise if eiw3  is always below the threshold ςi during the time studied, then fsi = 0. Considering the structure of D0 , the decision on which sensor is faulty can be made. To further illustrate the above scheme, an example of a system with three possible faulty models is given: 1. All eiw3 , i = 1, 2, . . . , q are below the corresponding threshold ςi during the experiment time, in which case it implies that all sensors are healthy; 2. There exist some finite time t1 > td and t3 > td , such that e1w3  > ς1 and e3w3  > ς3 respectively, whereas others do not, in which case it implies that fs1 and fs3 do not equal to zero while others do; 3. All eiw3  > ςi , i = 1, 2, . . . , q after some finite time, in which case it implies that all fsi do not equal to zero.

3.5 Simulation Results In this section, the effectiveness of the proposed scheme in detecting and isolating sensor faults has been demonstrated considering an example of a single-link robotic arm with a revolute elastic joint.

3.5 Simulation Results

47

The dynamics is described by Jl q¨ 1 + Fl q˙ 1 + k(q1 − q2 ) + mglsinq1 = 0 Jm q¨ 2 + Fm q˙ 2 − k(q1 − q2 ) = u

(3.51)

where q1 and q2 denote the link position and the rotor position, respectively; u is the torque delivered by the motor; m is the link mass, l is the center of mass, Jm is the link inertia, Jl is the motor rotor inertia, Fm is the viscous friction coefficient, Fl is the viscous friction coefficient, k is the elastic constant, and g is the gravity constant. In the simulation, the values of these parameters are chosen as m = 4, l = 0.5, Jm = 1, Jl = 2, Fm = 1, Fl = 0.5, k = 2, and g = 9.8 (all in SI units). Choosing x1 = q1 , x2 = q˙ 1 , x3 = q2 , x4 = q˙ 2 and assuming that the link position, the link velocity and the rotor position can be measured, the dynamics (3.51) can be represented in the following state-space form as ⎡

⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ 0 1 0 0 0 0 x˙ 1 x1 −Fl k −mgl ⎢ x˙ 2 ⎥ ⎢ −k ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 0 x sinx 2 1 ⎢ ⎥ = ⎢ Jl Jl Jl ⎥ ⎢ ⎥ ⎢ Jl ⎥+⎢ 0 ⎣ x˙ 3 ⎦ ⎣ 0 0 0 1 ⎦ ⎣ x3 ⎦ + ⎣ ⎦ ⎣0 0 1 k −k −Fm x˙ 4 x 0 0 4 Jm Jm Jm Jm ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x 10 1 0 0 0 ⎢ 1⎥ x 2⎥ ⎣ ⎦ y = ⎣0 0 1 0⎦⎢ ⎣ x3 ⎦ + 2 0 fs . 01 0001 x4 The nonlinear term f (x, t) =

−mgl sinx1 Jl

⎤

⎡ ⎤ 1 ⎥ ⎢1⎥ ⎥ u + ⎢ ⎥ Δψ, ⎦ ⎣0⎦ 0

(3.52)

mgl . Note that Jl   fs1 and sensor faults fs = 2 have fs

has a Lipschitz constant of

the terms associated with system uncertainties Δψ

been added in the system to demonstrate the effectiveness of the proposed sensor FDI scheme. In the simulation, Δψ is assumed to be −0.045 mgl sin(t). Jl Two transformations z = Tx and w = Sy with ⎡

⎤ 1.0000 0 −0.5000 0 ⎢ −1.0000 1.0000 0 0 ⎥ ⎥, T =⎢ ⎣ 0 0 1.0000 0 ⎦ 0 0 0 1.0000 ⎡ ⎤ 1.0000 −0.5000 0 1.0000 0 ⎦, S=⎣ 0 0 0 1.0000

48

3 Detection and Isolation of Sensor Faults

are introduced such that under the new coordinate, the system matrices become ⎤ ⎡ ⎤ 1 0.5 −0.5 1 1000 ⎥ ⎢ −2.25 −1.25 −0.125 0 ⎥ , SCT −1 = ⎣ 0 0 1 0 ⎦ , TAT −1 = ⎢ ⎣ 0 0 0 1 ⎦ 0001 0 −1 −1 2 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 1 00 ⎢0⎥ ⎢0⎥ ⎥ ⎢ ⎥ ⎣ ⎦ TB = ⎢ ⎣ 0 ⎦ , TE = ⎣ 0 ⎦ , SD = 2 0 . 01 1 0 ⎡

The simulation results are obtained by running successively the files chapter3_lmi.m, chapter3.mdl, and chapter3_plot.m. For the transformed system, we are now in the position to compute the values of the parameters of the proposed isolators. This is equivalent to the problem of solving the LMI (3.49). The following M-file chapter3_lmi.m shows how this can be done. File chapter3_lmi.m

3.5 Simulation Results

49

50

3 Detection and Isolation of Sensor Faults

3.5 Simulation Results

51

Parameters are obtained as P1 = 0.2393, As1 = −2.8737 ⎡ ⎤   0.5676 0.0279 −0.0327 0.1816 0.0394 ⎣ ⎦ 0.0279 0.6891 0.3339 , P02 = P01 = , 0.0394 0.2298 −0.0327 0.3339 0.7101 ⎤ ⎡ 0 0 ⎥ ⎢   0 0 ⎥ ⎢ 0.3632 0.0787 ⎥ ⎢ 0 0 L0 = ⎢ ⎥ , F0 = 0.0394 0.2298 . ⎣ 4.5704 −0.7920 ⎦ −0.7057 3.6529 Note that although these parameters are computed for fault isolation, they can also be applied for fault detectors (3.17) and (3.19). The above obtained parameters are used to simulate the system described in simulink program chapter3.mdl and the figures are plotted running the file chapter3_plot.m. The initial states have been selected as x(0) = [0, 0, 0, 0]T , zˆ1 (0) = −0.1 and zˆ0 (0) = [0, 0.2, −0.1, −0.1, −0.1]T . In the simulink model, a parameter “δ” has been added to the denominator

52

3 Detection and Isolation of Sensor Faults

of (3.38) and (3.40) to reduce the chattering effect. We have selected δ = 0.006 for ν1 and δ = 0.01 for ν2 in the simulation. Case-1 In this case the sensor faults are given as  fs1 =

0 , t ≤ 18 s 0.05 exp(0.01t), t ≥ 18 s

fs2 = 0, ∀t To detect the occurrence of faults, we choose the norm of the output estimation error ew3 , i.e., ew3  as the residual. It is seen from Fig. 3.1 that ew3  initially approaches to zero and then deviates from zero. The threshold is selected to be 0.02 in this simulation. This selection is reasonable because the FD threshold is dependent on the magnitude of the fault, which is assumed to be small in size. From Fig. 3.1, a significant change of the residual can be observed at about 18 s. The residual exceeds the threshold 0.02 at around t = 18.25 s (the fault occurs at 18 s), which implies that at least one of the sensors is faulty. After the detection of a fault, the next stage is to determine which sensor amongst the various sensors is faulty. For this specific example, the problem is to determine whether or not fsi = 0, i = 1, 2. Results of the simulation are shown in Figs. 3.2 and 3.3. The residual generated by the isolation observer, which is designed for fs1 , is compared with the threshold that is set to 0.02 in Fig. 3.2. From the figure it is observed that the residual exceeds the threshold at about 18.23 s, which implies that fs1 can be concluded to be nonzero after approximately 18.23 s. Further, from Fig. 3.3 it is seen that the residual generated by the isolation observer, which is designed for

0.16 residual ||e w3|| threshold for fault detection

0.14

Magnitude

0.12 0.1 0.08 0.06 0.04

fault detected

0.02 0

0

5

10

15

Time (s) Fig. 3.1 Detection of the occurrence of sensor faults

20

25

30

3.5 Simulation Results

53

0.16

residual ||e1 || w3

0.14

threshold for isolating f1 s

Residual

0.12 0.1 0.08 0.06 f1≠0 can be concluded

0.04

s

0.02 0

0

5

10

15

20

25

30

Time (s) Fig. 3.2 Isolation of fs1 0.16 residual ||e1

w3

0.14

||

threshold for isolating f 1 s

Magnitude

0.12 0.1 0.08 0.06 1

0.04

fs ≠ 0 can be concluded

0.02 0

0

5

10

15

20

25

30

Time (s) Fig. 3.3 Isolation of fs2

fs2 , always remains at zero. Thus the possibility of fs2 = 0 can be excluded. It can therefore be concluded that the first and second sensors are faulty after some time instant, while the third sensor is healthy according to the fault distribution D.

54

3 Detection and Isolation of Sensor Faults

Case-2 In this case the sensor faults are given as  fs1 =  fs2 =

0 , t ≤ 18 s 0.05 exp(0.01t), t ≥ 18 s 0 , t ≤ 25 s 0.07 exp(0.03t), t ≥ 25 s

The detectability of the proposed scheme is shown in Fig. 3.4. Significant changes of the residual can be observed at about 18–25 s. The residual first exceeds the threshold 0.02 at around t = 18.25 s (the fault occurs at 18 s), which implies that an incipient sensor fault has occurred at around t = 18.25 s. The simulation result obtained by the isolation observer, designed for fs1 , is shown in Fig. 3.5. It is observed from the figure that the residual exceeds the threshold 0.02 at about 18.25 s, which implies that fs1 = 0 after about 18.25 s. Figure 3.6 shows the residual obtained by the isolation observer designed for fs2 . The residual exceeds the threshold (0.02) at approximately 25.17 s, which denotes that fs2 can be found to be nonzero at about 25.17 s. Thus, it can be concluded that sensor faults appear in the first and second sensors after about 18.25 s and in the third sensor after about 25.17 s. 0.16 residual ||e w3 || threshold for fault detection

0.14

Magnitude

0.12 0.1 0.08 0.06 0.04

fault detected

0.02 0

0

5

10

15

Time (s) Fig. 3.4 Detection of the occurrence of sensor faults

20

25

30

3.5 Simulation Results

55

0.16 1

residual ||e w3 ||

0.14

1

threshold for isolating f s

Magnitude

0.12 0.1 0.08 0.06 f1≠0 can be determined

0.04

s

0.02 0

0

5

10

15

20

25

30

Time (s) Fig. 3.5 Isolation of fs1 0.16 2

residual ||e w3 || 2 threshold for isolating f

0.14

s

Magnitude

0.12 0.1 0.08 0.06 2 s

f ≠0 can be determined

0.04 0.02 0

0

5

10

15

Time (s) Fig. 3.6 Isolation of fs2

20

25

30

56

3 Detection and Isolation of Sensor Faults

3.6 Conclusions In this chapter, a new sensor FDI scheme is developed based on the results of Chap. 2. The proposed FDI scheme essentially transforms the original system into two subsystems where subsystem-1 includes system uncertainties, but is free from sensor faults and subsystem-2 has sensor faults but without uncertainties. Using the integral observer-based approach, sensor faults in subsystem-2 are transformed into actuator faults and detected by designing a Luenberger observer for this subsystem. After being detected, multiple transformed sensor faults are then isolated based on the modified dedicated observer scheme using a bank of SMOs. The sufficient condition of stability of the proposed FDI scheme has been studied and represented in the form of LMI. Its effectiveness has been demonstrated considering the example of a single-link robotic arm with a revolute elastic joint. Simulation results confirm that the proposed method can effectively detect and isolate incipient sensor faults in the presence of system uncertainties.

References 1. Jiang GP, Zheng WX, Tang WK, Chen GR (2006) Integral-observer-based chaos synchronization. IEEE Trans Circuits Syst 53:110–114 2. Corless M, Tu J (1998) State and input estimation for a class of uncertain systems. Automatica 34:757–764 3. Hui S, Zak SH (2005) Observer design for system with unknown inputs. Int J Appl Math Comput Sci 15(4):431–446 4. Yan XG, Edwards C (2008) Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems. Int J Syst Sci 39(4):349–359 5. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 6. Utkin VI (1992) Sliding Modes in Control Optimization. Springer, Berlin

Chapter 4

Robust Estimation of Actuator Faults

We have discussed the issue of actuator/sensor fault detection and isolation (FDI) in the last two chapters. However, FDI can only indicate when and where a fault occurs in the system, but cannot provide more comprehensive information of the fault such as magnitude, shape, and duration. In this chapter, we will move on to the topic on fault estimation (FE) to further explore the nature of a fault.

4.1 Introduction FE is different from the majority of FDI in the sense that it not only detects and isolates the fault, but also provides details of the fault, such as the location, size, and duration. Thus, it is especially useful for incipient faults and slow drifts, which are very difficult to detect. Also FE is vital in fault-tolerant control (FTC) systems. During the last two decades, the research on sliding-mode observers (SMO)-based FE has received considerable attention. Several results have been reported on this topic [1–7]. In this chapter, a new SMO-based FE scheme which is inspired by the work presented in [8] is proposed for uncertain Lipschitz nonlinear systems. Actuator FE is discussed first and then the results are extended to sensor FE. For the case when there are only actuator faults, we initially introduce a linear coordinate transformation to transform the original state vector into two parts such that actuator faults only appear in the dynamics of the second state vector. By analyzing the second state error dynamics during the sliding motion, actuator faults are estimated by employing the concept of equivalent output error injection. In this chapter, we assume that the system uncertainties are unstructured. While in many papers such as [3, 8, 9], they are assumed to be structured. Therefore, system uncertainties cannot be completely decoupled from faults. However, their effects on the estimation errors of states and faults can be minimized by integrating a prescribed H∞ uncertainty attenuation level into the observer. The sufficient conditions for the stability of the proposed observer © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_4

57

58

4 Robust Estimation of Actuator Faults

with a prescribed H∞ performance are derived using Lyapunov stability theory, and finally formulated as an LMI optimization problem, such that the observer parameters can be obtained. It should be noted that the problem of solving the matching condition is converted into an LMI minimization problem in [10]. However, this method can only provide an approximate solution. In this chapter, we propose a new way to solve this problem. For the case when there are only sensor faults, we use the integral observer to transform the sensor faults into the form of actuator faults. Then the proposed actuator FE method can be extended to estimate sensor faults directly. The remainder of this chapter is organized as follows: Sect. 4.2 briefly describes the mathematical preliminaries required for developing the FE method. Section 4.3 describes the design procedure of the proposed SMO and derives the sufficient condition for its existence. The proposed actuator FE method is further extended to sensor FE in Sect. 4.4. The results of simulation are shown in Sect. 4.5 considering the example of single-link flexible joint robot arm system with conclusions in Sect. 4.6.

4.2 Problem Formulation Consider the following nonlinear system: 

x(t) ˙ = Ax(t) + f (x, t) + Bu(t) + D f a (t) + Δψ(t), y(t) = C x(t)

(4.1)

where x ∈ R n , u ∈ R m , and y ∈ R p denote, respectively, the vector of state variables, inputs, and outputs. f a ∈ R h represents the vector of unknown actuator faults. f a = 0 represents an actuator fault-free condition. Δψ models lumped uncertainties and disturbances experienced by the system and f (x, t) is the nonlinear term and assumed to be known. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , and D ∈ R n×h (h ≤ p < n) are known constant matrices with C and D both being of full rank. Without loss of generality, it is assumed that the output matrix C has the form C = [ 0 I p ]. Remark 4.1 It is worth noting that the system uncertainty under consideration is unstructured, which is more general than the type of structured uncertainty that has been considered for fault diagnosis of Lipschitz nonlinear systems in literature [3, 5, 8, 11]. In the case of structured system uncertainty, certain rank conditions of the uncertainty distribution matrix are assumed to be satisfied such that the fault can be completely decoupled from the uncertainty. Remark 4.2 If C does not have theassumed  structure, it is possible to find a nonsinNcT gular transformation matrix Tc = , where Nc ∈ R n×(n− p) and the columns C span the null space of C [12], such that in the new coordinate, C Tc−1 = [ 0 I p ].

4.2 Problem Formulation

59

Before starting the main results of this chapter, we make the following assumptions on System (4.1). Assumption 4.1 The matrix pair (A, C) is detectable. It follows from Assumption 4.1 that there exists a matrix L ∈ R n× p such that A − LC is stable, and thus for any Q > 0, the Lyapunov equation (A − LC)T P + P(A − LC) = −Q has an unique solution P > 0 [8]. Assume that P ∈ R n×n , Q ∈ R n×n are in the following form:     Q1 Q2 P1 P2 , Q= . P= P2T P3 Q 2T Q 3

(4.2)

(4.3)

It is obvious that P1 ∈ R (n− p)×(n− p) > 0, P3 ∈ R p× p > 0, Q 1 ∈ R (n− p)×(n− p) > 0, and Q 3 ∈ R p× p > 0 if P > 0 and Q > 0. Assumption 4.2 There exists an arbitrary matrix F ∈ R h× p such that D T P = FC.

(4.4)

Assumption 4.3 The known nonlinear function f (x, t) is Lipschitz with respect to the state x, i.e., ∀x, xˆ ∈ X , ˆ  f (x, t) − f (x, ˆ t) ≤ L f x − x,

(4.5)

where L f is the known Lipschitz constant. Assumption 4.4 The actuator fault vector f a and uncertainty vector Δψ satisfies the following constraint:  f a  ≤ ρa and Δψ ≤ ξ,

(4.6)

where ρa and ξ are two known positive constants. Assume that the pair (A, D) has the following structure:  A=

   D1 A1 A2 , D= , A3 A4 D2

(4.7)

where A1 ∈ R (n− p)×(n− p) , A4 ∈ R p× p , D1 ∈ R (n− p)×h , and D2 ∈ R p×h . Then the following conclusions can be obtained: Lemma 4.1 If P and Q have been partitioned as in (4.3), then 1. D1 + P1−1 P2 D2 = 0 if (4.4) is satisfied; 2. The matrix A1 + P1−1 P2 A3 is stable if (4.2) is satisfied.

60

4 Robust Estimation of Actuator Faults

Proof 1. From the matrix partitions, it follows that D P= T



D1T

D2T





P1 P2 P2T P3



  = D1T P1 + D2T P2T D1T P2 + D2T P3   = (P1 (D1 + P1−1 P2 D2 ))T D1T P2 + D2T P3   FC = 0 F

(4.8) (4.9)

By comparing (4.8) and (4.9), conclusion-1 can be obtained. 2. Applying block matrix multiplication to (4.2) yields A1T P1 + P1 A1 + A3T P2T + P2 A3 = −Q 1 .

(4.10)

(A1 + P1−1 P2 A3 )T P1 + P1 (A1 + P1−1 P2 A3 ) = −Q 1 .

(4.11)

This implies that

Therefore, conclusion-2 can be obtained from the fact that Q 1 and P1 > 0.

>

0 

4.3 Actuator FE Scheme In this section, an SMO is designed to simultaneously estimate actuator faults and system states for System (4.1).

4.3.1 Observer Design The design of SMO begins by introducing a linear change of coordinates z = T x [8, 13] so as to impose a specific structure on the fault distribution matrix D. The transformation matrix T has the following form:  T :=

 In− p P1−1 P2 . 0 Ip

(4.12)

Based on the conclusion-1 of Lemma 4.1, System (4.1) can be transformed into the following system: 

z˙ (t) = A z z(t) + T f (T −1 z, t) + Bz u(t) + T Δψ(t) + Dz f a (t) y(t) = C z z(t)

(4.13)

4.3 Actuator FE Scheme

61

where    A¯ 2 A1 + P1−1 P2 A3 A2 − A1 P1−1 P2 + P1−1 P2 (A4 − A3 P1−1 P2 ) = , A¯ 4 A4 − A3 P1−1 P2 A3     B¯ 1 B1 + P1−1 P2 B2 Bz = ¯ = , B2 B2       0 D¯ D1 + P1−1 P2 D2 Dz = ¯ 1 = = , D2 D2 D2   Cz = 0 I p . 

Az =

A¯ 1 A¯ 3

System (4.13) can further be written as ⎧ ⎨ z˙ 1 = A¯ 1 z 1 + z˙ = A3 z 1 + ⎩ 2 y = z2

A¯ 2 z 2 + T1 f (T −1 z, t) + B¯ 1 u(t) + T1 Δψ(t) A¯ 4 z 2 + T2 f (T −1 z, t) + B¯ 2 u(t) + D2 f a + T2 Δψ(t)

(4.14)

where z = col(z 1 , z 2 ) with z 1 ∈ R n− p and z 2 ∈ R p , T1 = [ In− p P1−1 P2 ], T2 = [ 0 I p ]. For the transformed System (4.14), the following SMO is designed: ⎧ ⎨ z˙ˆ 1 = A¯ 1 zˆ 1 + z˙ˆ = A3 zˆ 1 + ⎩ 2 yˆ = zˆ 2

A¯ 2 y + T1 f (T −1 zˆ , t) + B¯ 1 u(t) A¯ 4 zˆ 2 + T2 f (T −1 zˆ , t) + B¯ 2 u(t) + ( A¯ 4 − A0 )(y − yˆ ) + ν1 (4.15)

where zˆ 1 and zˆ 2 denote the estimates of z 1 and z 2 , respectively; yˆ denotes the estimate of y; zˆ := col(ˆz 1 , y). Note that zˆ does not represent the state estimate col(ˆz 1 , zˆ 2 ). A0 ∈ R p× p is a stable design matrix and plays the role of Luenberger observer gain. The discontinuous vector ν1 is defined by

ν1 =

yˆ ) k PP00 (y− if y − yˆ = 0 (y− yˆ )

0

otherwise

(4.16)

where P0 ∈ R p× p is a symmetric positive definite matrix and k = D2 ρa + η1 . The constant η1 is a positive scalar which is to be determined later. Define the state estimation errors as e1 = z 1 − zˆ 1 and e2 = z 2 − zˆ 2 . Then the error dynamics after the occurrence of actuator faults are expressed as  (4.17) e˙1 = A¯ 1 e1 + T1 f (T −1 z, t) − f (T −1 zˆ , t) + T1 ψ,  −1 −1 e˙2 = A0 e2 + A3 e1 + T2 f (T z, t) − f (T zˆ , t) + D2 f a + T2 Δψ − ν1 . (4.18)

62

4 Robust Estimation of Actuator Faults

The following theorem provides the existence condition of the proposed observer in the form of (4.15) with the prescribed H∞ performance index which is defined by H ∞ :=

eL2 √ ≤ μ, Δψ L2 ΔψL2 =0 sup

(4.19)

where e := col(e1 , e2 ) and μ is a positive scalar. Theorem 4.1 Consider System (4.1) with Assumptions 4.1–4.4. Given a positive scalar μ, if there exist matrices P1 = P1T > 0, P2 , P0 = P0T > 0 and A0 , and positive scalars α1 , α0 such that P1 D1 + P2 D2 = 0, ⎡ ⎤ Π1 + In− p A3T P0 P1 P2 ⎢ P0 A3 Π2 + I p 0 P0 ⎥ ⎥ < 0, Λ := ⎢ ⎣ P1 0 −μIn− p 0 ⎦ P0 0 −μI p P2T

(4.20) (4.21)

where Π1 = A1T P1 + P1 A1 + A3T P2T + P2 A3 + α11 (P1 P1 + P2 P2T )+(α1 +α0 )L f2 In− p and Π2 = A0T P0 +P0 A0 + α10 P0 P0 , then the observer error dynamics is asymptotically √ stable with an H∞ disturbance attenuation level μ > 0 subject to H ∞ ≤ √ μ. Proof It follows from Lemma 4.1 that if Assumption 4.2 is satisfied, then D1 + P1−1 P2 D2 = 0,

(4.22)

which is equivalent to (4.20). Consider the Lyapunov function as V = e T Pz e,

(4.23)

where Pz = (T T )−1 P T −1 . It is worth noting that Pz in the new coordinate has the following quadratic form:  Pz =

 P1 0 , 0 P0

(4.24)

where P0 = −P2T P1−T P2 + P3 . Therefore, the Lyapunov function can be rewritten as V = V1 (e1 ) + V2 (e2 ) = e1T P1 e1 + e2T P0 e2 .

(4.25)

4.3 Actuator FE Scheme

63

The time derivative of V1 along the trajectories of System (4.17) can be shown to be equal to V˙1 = e˙1T P1 e1 + e1T P1 e˙1  = e1T ( A¯ 1T P1 + P1 A¯ 1 )e1 + 2e1T P1 T1 f (T −1 z, t) − f (T −1 zˆ , t) + 2e1T P1 T1 Δψ. (4.26) Note that zˆ := col(ˆz 1 , y). It is easy to see that       −1  T z − T −1 zˆ  = T −1 e1  = e1  ,  0     f (T −1 z, t) − f (T −1 zˆ , t) ≤ L f e1  . Since the inequality 2X T Y ≤ then

1 α

X T X + αY T Y holds for any scalar α > 0 [3],

1 V˙1 ≤ e1T ( A¯ 1T P1 + P1 A¯ 1 )e1 + e1T P1 T1 T1T P1T e1 + α1  f (T −1 z, t) − f (T −1 zˆ , t)2 α1 + 2e1T P1 T1 Δψ   1 T T T T 2 ¯ ¯ P1 T1 T1 P1 + α1 L f In− p e1 + 2e1T P1 T1 Δψ. ≤ e1 A1 P1 + P1 A1 + α1 (4.27) Similarly, from the definition of ν1 in (4.16) and the bound on f a , we can obtain the time derivative of V2 as  V˙2 = e2T (A0T P0 + P0 A0 )e2 + 2e2T P0 A3 e1 + 2e2T P0 T2 f (T −1 z, t) − f (T −1 zˆ , t) + 2e2T P0 D2 f a + 2e2T P0 T2 Δψ − 2e2T P0 ν1

 ≤ e2T (A0T P0 + P0 A0 )e2 + 2e2T P0 A3 e1 + 2e2T P0 T2 f (T −1 z, t) − f (T −1 zˆ , t) + 2e2T P0 T2 Δψ   1 P0 T2 T2T P0 e2 + 2e2T P0 A3 e1 + α0 L f2 e1 2 ≤ e2T A0T P0 + P0 A0 + α0 + 2e2T P0 T2 Δψ.

(4.28)

From the structure of A¯ 1 , T1 and T2 , we can obtain V˙ as V˙ = V˙1 + V˙2   1 ≤ e1T A¯ 1T P1 + P1 A¯ 1 + P1 T1 T1T P1T + (α1 + α0 )L f2 In− p e1 α1   1 + e2T A0T P0 + P0 A0 + P0 T2 T2T P0 e2 + 2e2T P0 A3 e1 + 2e1T P1 T1 Δψ + 2e2T P0 T2 Δψ α0

64

4 Robust Estimation of Actuator Faults

= e1T Π1 e1 + e2T Π2 e2 + 2e2T P0 A3 e1 + 2e1T P1 T1 Δψ + 2e2T P0 T2 Δψ.

(4.29)

When Δψ = 0, it follows that    T  Π1 A3T P0 e1 e . V˙ ≤ 1 e2 P0 A3 Π2 e2   

(4.30)

W

If there exists a feasible solution to (4.21), then we can conclude that W < 0, thus V˙ < 0. This implies that e → 0 as t → ∞. Therefore, the error dynamics is asymptotically stable when Δψ = 0. When Δψ = 0, to attain robustness to the disturbance on the estimation error in the L2 sense, we impose the following performance index:  J=

∞



e2 − μΔψ2 dt.

(4.31)

0

Then 

∞

J= =

0 ∞

 

e2 − μΔψ2 + V˙ dt −



∞

V˙ dt

0

e2 − μΔψ2 + V˙ dt − V (∞) + V (0).

0

Since V (∞) > 0, under zero initial conditions, we have 

∞

 2 e − μΔψ2 + V˙ dt 0 ⎛⎡ ⎤T ⎡ ⎤⎞  ∞ e1 e1 ⎜⎣ ⎟ = ⎝ e2 ⎦ Λ ⎣ e2 ⎦⎠ dt. 0 Δψ Δψ

J≤

(4.32)

If (4.21) holds, then J < 0, namely !

∞ 0

√ e T edt ≤ μ

!

∞

Δψ T Δψdt, ∀ t > 0,

(4.33)

0

√ which means eL2 ≤ μΔψL2 . Therefore the H∞ performance has been established. This completes the proof.  Remark 4.3 It is noticed in Theorem 4.1 that the derived sufficient conditions for the existence of the proposed observer include a linear matrix equality (4.20), which is difficult to directly solve by MATLAB toolbox. The solvability of (4.20) can be

4.3 Actuator FE Scheme

65

converted into the problem of finding the minimum of a positive scalar γ satisfying the following inequality constraint: 

γ In− p P1 D1 + P2 D2 (P1 D1 + P2 D2 )T γ Ih

 > 0.

However, this method can only make P1 D1 approximate to −P2 D2 . In the following theorem, we propose a new method to solve this problem. Theorem 4.2 Consider System (4.1) with Assumptions 4.1–4.4. Given a positive scalar μ, if there exist matrices P0 = P0T > 0, Y and Z , and positive scalars α1 , α0 such that the following LMI feasibility problem has a solution: ⎡

⎤ Π3 A3T P0 Z G 1 Z G 2 Z G 1 Z G2 0 ⎢ P0 A3 Π4 0 P0 0 0 P0 ⎥ ⎢ T T ⎥ ⎢G Z 0 0 0 ⎥ ⎢ 1T T 0 −μIn− p 0 ⎥ ⎢G Z 0 −μI p 0 0 0 ⎥ ⎢ 2T T P0 ⎥ < 0, ⎢G Z ⎥ 0 0 0 −α I 0 0 1 n− p ⎢ 1T T ⎥ ⎣ G2 Z 0 0 0 0 −α1 I p 0 ⎦ 0 0 0 0 −α0 I p 0 P0

(4.34)

where Π3 = A1T G 1T Z T + Z G 1 A1 + A3T G 2T Z T + Z G 2 A3 + (α1 + α0 )L f2 In− p + In− p ,       In− p 0 T + + , and G 2 = In − D D Π4 = Y + Y + I p , G 1 = I n − D D with 0 Ip D + = (D T D)−1 D T , then the observer error dynamics is asymptotically stable with √ √ an H∞ disturbance attenuation level μ > 0 subject to H ∞ ≤ μ. Proof Equation (4.20) can be rewritten as [ P1 P2 ]D = 0.

(4.35)



 D Since rank = rank(D) is straightforward, then the general solution of 0 (4.35) exists and can be given by  [ P1 P2 ] = Z In − D D + ,

(4.36)

where Z ∈ R (n− p)×n is a design matrix and D + is the generalized inverse of D, i.e., D + = (D T D)−1 D T . Such a D + always  since D is of full column rank.  exists   0   I n− p + + and P2 = Z In − D D into Substituting P1 = Z In − D D 0 Ip (4.21), and letting Y = P0 A0 , and then using Schur complement, the LMI (4.34) can be obtained immediately. This completes the proof. 

66

4 Robust Estimation of Actuator Faults

After solving the LMI (4.34), we can compute the observer gain A0 from A0 = P0−1 Y . Substituting C = C z T , D = T −1 Dz , and P = T T Pz T (Pz is a diagonal matrix as shown in (4.24)) into the matching condition (4.4) yields DzT Pz = FC z . From the structure of Dz , Pz , and C z , we can obtain that F = D2T P0 . The obtained P and F can ensure the simultaneous satisfaction of Lyapunov equation (4.2) and the matching condition (4.4). Remark 4.4 The effect of the system uncertainty Δψ on the state estimation error is decided by the value of μ. The accuracy of fault estimation increases as the value of μ becomes smaller and smaller. Therefore the robustness of the observer can be enhanced by minimizing the value of μ. The minimum μ can be obtained by solving the following LMI optimization problem: minimize μ subject to P0 > 0 and (4.34). Theorem 4.2 has outlined a constructive design procedure for the proposed observer. The objective is now to design the gain of discontinuous input ν1 (t) such that the error systems can be driven to the sliding surface (4.37) in finite time and a sliding motion can be maintained on it thereafter. For error dynamics (4.17) and (4.18), we define the sliding-mode surface as S = {(e1 , e2 )|e2 = 0}.

(4.37)

Theorem 4.3 Given System (4.1) with Assumptions 4.1–4.4 and the proposed observers (4.15). Then the error dynamics (4.18) can be driven to the sliding surface given by (4.37) in finite time and remain on it if the LMI optimization problem formulated in (4.34) is solvable and the gain from (4.16) satisfies k = D2 ρa + η1 ≥ D2 ρa + A3 ε + L f ε + ξ + η2 .

(4.38)

where η1 ≥ A3 ε + L f ε + ξ + η2 , ε is the upper bound of e and η2 is a positive scalar. Proof Consider a Lyapunov candidate function as Vs (e2 ) = e2T P0 e2 . Then the derivative of Vs along the trajectory of the error system (4.18) becomes  V˙s = e2T (A0T P0 + P0 A0 )e2 + 2e2T P0 A3 e1 + 2e2T P0 T2 f (T −1 z, t) − f (T −1 zˆ , t) + 2e2T P0 D2 f a + 2e2T P0 T2 Δψ − 2e2T P0 ν.

(4.39)

Since A0 is a stable matrix by design, therefore A0T P0 + P0 A0 < 0. It follows from Theorem 4.1 that e ≤ ε. Then (4.39) becomes  V˙s ≤ 2e2T P0 A3 e1 + 2e2T P0 T2 f (T −1 z, t) − f (T −1 zˆ , t) + 2e2T P0 D2 f a + 2e2T P0 T2 Δψ − 2kP0 e2 

4.3 Actuator FE Scheme

67

 ≤ 2P0 e2  A3 e1  + T2  f (T −1 z, t) − f (T −1 zˆ , t) + D2  f a  +T2 Δψ − k) ≤ 2P0 e2 (A3 e1  + L f e1  + D2 ρa + ξ − k) = 2P0 e2 (A3 ε + L f ε + ξ − η1 ).

(4.40)

If the condition (4.38) holds, then " V˙s ≤ −2η2 P0 e2  ≤ −2η2 λmin (P0 )Vs1/2 ,

(4.41)

where λmin (P0 ) is the smallest eigenvalue of P0 . This shows that the reachability condition is satisfied. As a consequence, an ideal sliding motion will take place on the surface S in finite time. This completes the proof. 

4.3.2 Estimation of Actuator Faults Given the proposed observer in the form of (4.15), the objective now is to estimate actuator faults using the so-called equivalent output injection [1]. Assuming that a sliding motion has been achieved, then e2 = 0 and e˙2 = 0 and (4.18) becomes  0˙ = 0 + A3 e1 + T2 f (T −1 z, t) − f (T −1 zˆ , t) + D2 f a + T2 Δψ − ν1eq , (4.42) where ν1eq is the equivalent output error injection signal which is required to maintain a sliding motion. It can be approximated to any degree of accuracy by replacing (4.16) with P0 (y − yˆ ) , (4.43) ν1 = k P0 (y − yˆ ) + δ where δ is a small positive scalar to reduce the chattering effect. It is worth noting that the analysis of fault estimation in this section is based on the assumption that e2 = e˙2 = 0 which represents perfect sliding. However, with the above approximation, the error dynamics does not slide on the surface S perfectly, but within a small boundary layer around it [14]. Define fˆa = D2+ ν1eq ,

(4.44)

where D2+ is the left pseudoinverse of D2 . Then, (4.42) can be rewritten as  f a − fˆa = −D2+ A3 e1 − D2+ T2 f (T −1 z, t) − f (T −1 zˆ , t) − D2+ T2 Δψ.

(4.45)

68

4 Robust Estimation of Actuator Faults

Computing the L2 norm of (4.45) yields   f a − fˆa L2 = D2+ A3 e1 + D2+ T2 f (T −1 z, t) − f (T −1 zˆ , t) + D2+ T2 ΔψL2 ≤ σmax (D2+ A3 )e1 L2 + L f σmax (D2+ )e1 L2 + σmax (D2+ )ΔψL2

≤ σmax (D2+ A3 )eL2 + L f σmax (D2+ )eL2 + σmax (D2+ )ΔψL2 . (4.46)

Since eL2 ≤

√

μΔψL2 , we obtain

√ √  f a − fˆa L2 ≤ μσmax (D2+ A3 ) + μL f σmax (D2+ ) + σmax (D2+ ) ΔψL2 √ = ( μβ1 + β2 )ΔψL2 , (4.47) where β1 = σmax (D2+ A3 ) + L f σmax (D2+ ) and β2 = σmax (D2+ ). Thus for a small √ ( μβ1 + β2 )ΔψL2 , the actuator fault can be approximated as D2+ P0 (y − yˆ ) fˆa = k . P0 (y − yˆ ) + δ

(4.48)

Remark 4.5 The estimated fault signal in (4.48) only depends on known system information such as the measured output y, estimated output yˆ , and P0 which can be obtained off-line. Therefore, the proposed FE scheme can be implemented online. Remark 4.6 From (4.47), it is observed that the size of the FE error is directly √ related to the system uncertainty by the term μβ1 + β2 . As a result, the precise reconstruction of actuator fault is not possible due to the presence of Δψ. However, the developed SMO can still preserve the shape of the fault signal effectively if μ is made sufficiently small. By minimizing its value, the robustness of the proposed observer against the unknown signals can be enhanced and the desired accuracy in the estimation of faults can be obtained.

4.4 A Generalization to Sensor FE In this section, the actuator FE method proposed in Sect. 4.3 will be extended to sensor FE in the presence of system uncertainties. In this case, the system under consideration has the following form: 

x(t) ˙ = Ax(t) + f (x, t) + Bu(t) + Δψ(t) y(t) = C x(t) + E f s (t)

(4.49)

where E ∈ R p×q is a known constant matrix with full column rank indicating which of the sensors are faulty. f s ∈ R q is the vector of sensor faults and satisfies  f s  ≤ ρs , where ρs is a known real constant.

4.4 A Generalization to Sensor FE

69

In order to apply the method proposed in Sect. 4.3, we need to transform#the sensor t faults into the form of actuator faults. To do so, we define a new state ϕ = 0 y(τ )dτ so that ϕ˙ = C x + E f s . Then the following augmented system with the new state ϕ and output w can be obtained: 

˜ + E˜ f s + Δψ ˜ ˜ t) + Bu x˜˙ = A˜ x˜ + f˜([In 0]x, w = C˜ x˜

(4.50)

      x A0 B ∈ R n+ p , w ∈ R p , A˜ = ∈ R (n+ p)×(n+ p) , B˜ = ∈ ϕ C0 0     0 R (n+ p)×m , E˜ = ˜ t) = ∈ R (n+ p)×q , C˜ = 0 I p ∈ R p×(n+ p) , f˜([In 0]x, E     ˜ t) f ([In 0]x, ˜ = Δψ ∈ R n+ p . ∈ R n+ p , Δψ 0 0 where x˜ =

˜ C) ˜ is detectable if Assumption 4.1 is satisfied. Lemma 4.2 The pair ( A, Proof ∀ R(s) ≥ 0, ⎡ ⎤  s In − A 0 s In+ p − A˜ = rank ⎣ −C s I p ⎦ rank C˜ 0 Ip   s In − A + p. = rank C 

If Assumption 4.1 is satisfied, then we have 

 s In+ p − A˜ rank = n + p, ∀R(s) ≥ 0, C˜ ˜ C) ˜ is detectable. which means that the pair ( A, This completes the proof. It follows from Lemma 4.2 that there exists a matrix L˜ ∈ R A˜ − L˜ C˜ is stable, and for any Q˜ > 0,

 (n+ p)× p

˜ A˜ − L˜ C) ˜ = − Q˜ ˜ T P˜ + P( ( A˜ − L˜ C)

such that

(4.51)

has solution P˜ > 0. Assume that P˜ is in the following form: P˜ =   an unique P˜1 P˜2 with P˜1 = P˜1T ∈ R n×n > 0 and P˜3 = P˜3T ∈ R p× p > 0. Applying block P˜2T P˜3 multiplication to (4.51), it follows that A + P˜1−1 P˜2 C is stable. Assumption 4.5 There exists an arbitrary matrix F˜ ∈ R q× p such that E˜ T P˜ = F˜ C˜

(4.52)

70

4 Robust Estimation of Actuator Faults

4.4.1 Observer Design From Assumption 4.5, it follows that P˜2 E = 0. Based on this observation, we can transform System (4.50) into the following system  by employing a nonsingular state  −1 ˜ ˜ P P I transformation z˜ = T˜ x˜ with T˜ = n 1 2 : 0 Ip 

˜ + E˜ z f s z˙˜ = A˜ z z˜ + T˜ f˜([In 0]T˜ −1 z˜ , t) + B˜ z u + T˜ Δψ ˜ w = C z z˜

(4.53)

where    A + P˜1−1 P˜2 C −A P˜1−1 P˜2 − P˜1−1 P˜2 C P˜1−1 P˜2 A˜ 1 A˜ 2 = , C −C P˜1−1 P˜2 A˜ 3 A˜ 4   B = , 0  −1    0 P˜1 P˜2 E = = , E E   = 0 Ip .

A˜ z = B˜ z E˜ z C˜ z



System (4.53) can be rewritten as ⎧ ⎨ z˙˜ 1 = A˜ 1 z˜ 1 + A˜ 2 z˜ 2 + f ([In 0]T˜ −1 z˜ , t) + Bu + Δψ z˙˜ = A˜ 3 z˜ 1 + A˜ 4 z˜ 2 + E f s ⎩ 2 w = z˜ 2

(4.54)

where z˜ = col(˜z 1 , z˜ 2 ) with z˜ 1 ∈ R n and z˜ 2 ∈ R p . For System (4.54), we develop the following observer: ⎧ ˙ ⎪ ⎨ zˆ˜ 1 = A˜ 1 zˆ˜ 1 + ˙ˆ z˜ 2 = A˜ 3 zˆ˜ 1 + ⎪ ⎩ wˆ = zˆ˜ 2

A˜ 2 w + f ([In 0]T˜ −1 zˆ˜ , t) + Bu A˜ 4 zˆ˜ 2 + ( A˜ 4 − A˜ 0 )(w − w) ˆ + ν2

(4.55)

where zˆ˜ 1 and zˆ˜ 2 denote the estimates of z˜ 1 and z˜ 2 , respectively; wˆ denotes the estimate of w; zˆ˜ := col(zˆ˜ 1 , w); A˜ 0 ∈ R p× p is a stable design matrix; and ν2 is defined by ν2 =

˜ w) ˆ if w − wˆ = 0 k˜  PP˜0 (w− (w−w) ˆ 0

0

otherwise

(4.56)

where k˜ = Eρs + η˜ 1 and η˜ 1 is a positive scalar which needs to be determined.

4.4 A Generalization to Sensor FE

71

Define the state estimation errors in the new coordinate as e˜1 = z˜ 1 − zˆ˜ 1 and e˜2 = z˜ 2 − zˆ˜ 2 , and then the error dynamics can be obtained from % & e˙˜1 = A˜ 1 e˜1 + f ([In 0]T˜ −1 z˜ , t) − f ([In 0]T˜ −1 zˆ˜ , t) + Δψ, e˙˜2 = A˜ 0 e˜2 + A˜ 3 e˜1 + E f s − ν2 .

(4.57)

We now present the following theorem which provides sufficient conditions for ˜ L2 ≤ the √ existence of the proposed observer with the prescribed H∞ performance e ϑΔψL2 , where e˜ := col(e˜1 , e˜2 ). Theorem 4.4 Given System (4.49) with Assumptions 4.1, 4.3–4.5. If there exist matrices P˜1 = P˜1T > 0, P˜0 = P˜0T > 0, Z˜ , Y˜ and positive scalars ϑ, κ such that the following LMI optimization problem has a solution: Minimize ϑ subject to P˜1 > 0, P˜0 > 0 and ⎡ ⎤ Ξ1 C T P˜0 P˜1 P˜1 ⎢ P˜0 C Ξ2 0 0 ⎥ ⎢ ⎥ < 0, ⎣ P˜1 0 −ϑ In 0 ⎦ P˜1 0 0 −κ In

(4.58)

where Ξ1 = A T P˜1 + P˜1 A + C T (I p − E E + )T Z˜ T + Z˜ (I p − E E + )C + (κL f2 + 1)In and Ξ2 = Y˜ + Y˜ T + I p with E + = (E T E)−1 E T , then the observer error √dynamics is asymptotically stable with the prescribed H∞ performance e ˜ L2 ≤ ϑΔψL2 . Furthermore, F˜ can be obtained by F˜ = E T P˜0 . Proof Consider the Lyapunov function as V˜ = e˜ T P˜z e, ˜ where P˜z = (T˜ −1 )T P˜ T˜ −1 = The derivative of V˜ is



(4.59)

 P˜1 0 with P˜0 = − P˜2T P˜1−1 P˜2 + P˜3 . 0 P˜0

% & V˙˜ = e˜1T ( P˜1 A˜ 1 + A˜ 1T P˜1 )e˜1 + 2e˜1T P˜1 f ([In 0]T˜ −1 z˜ , t) − f ([In 0]T˜ −1 zˆ˜ , t) 2e˜1T P˜1 Δψ + e˜2T ( P˜0 A˜ 0 + A˜ 0T P˜0 )e˜2 + 2e˜2T P˜0 A˜ 3 e˜1 + 2e˜2T P˜0 E f s − 2e˜2T P˜0 ν2   1 ≤ e˜1T P˜1 A + A T P˜1 + P˜2 C + C T P˜2T + P˜1 P˜1 + κ L f2 In e˜1 + e˜2T ( P˜0 A˜ 0 + A˜ 0T P˜0 )e˜2 κ T ˜ T ˜ + 2e˜2 P0 C e˜1 + 2e˜1 P1 Δψ (4.60)

+

72

4 Robust Estimation of Actuator Faults

Define



∞

J=

 2 e ˜ − ϑΔψ2 dt.

(4.61)

0

Under zero initial conditions, it follows from (4.60) that  ∞% & e ˜ 2 − ϑΔψ2 + V˙˜ dt J≤ 0 ⎛  = 0

∞

⎞

⎤⎡ ⎤T ⎡ ⎤⎟ ⎜⎡ C T P˜0 Ξ3 + κ1 P˜1 P˜1 P˜1 e˜1 ⎟ ⎜ e˜1 ⎜⎣ ⎟ P˜0 A˜ 0 + A˜ 0T P˜0 + I p 0 ⎦ ⎣ e˜2 ⎦⎟ dt, P˜0 C ⎜ e˜2 ⎦ ⎣ ⎜ ⎟ Δψ ⎠ ⎝ Δψ P˜1 0 −ϑ In    M

(4.62) where Ξ3 = A T P˜1 + P˜1 A + C T P˜2T + P˜2 C + (κL f2 + 1)In .   E = rank(E), the general solution of P˜2 E = 0 can be expressed as Since rank 0 P˜2 = Z˜ (I p − E E + ),

(4.63)

where Z˜ ∈ R n× p is a design matrix to be determined and E + is the generalized inverse of E. A sufficient condition for J < 0 is M < 0. Letting Y˜ = P˜0 A˜ 0 and substituting (4.63) into M < 0 gives ⎡ ⎤ Ξ1 + κ1 P˜1 P˜1 C T P˜0 P˜1 ⎣ (4.64) Ξ2 0 ⎦ < 0. P˜0 C P˜1 0 −ϑ In Using Schur complement, the inequality (4.58) can be obtained. This completes the proof.



Define the sliding-mode surface as S = {(e˜1 , e˜2 )|e˜2 = 0}.

(4.65)

Theorem 4.5 Given System (4.49) with Assumptions 4.1, 4.3–4.5, and the proposed observer (4.55). Then the error dynamics (4.57) can be driven to the sliding surface defined by (4.65) in finite time and remain on it if the LMI optimization problem formulated in (4.58) is solvable and the gain k˜ in (4.56) satisfies k˜ = Eρs + η˜ 1 ≥ Eρs + C˜ε + η˜ 2 ,

(4.66)

˜ and η˜ 2 is a positive scalar. where η˜ 1 ≥ C˜ε + η˜ 2 , ε˜ is the upper bound of e,

4.4 A Generalization to Sensor FE

73

Proof Consider a Lyapunov candidate function as V˜s (e˜2 ) = e˜2T P˜0 e˜2 . Then the derivative of V˜s along the trajectory of the error system (4.57) becomes V˙˜s = e˜2T ( A˜ 0T P˜0 + P˜0 A˜ 0 ) + 2e˜2T P˜0 A˜ 3 e˜1 + 2e˜2T P˜0 E f s − 2e˜2T P˜0 ν2 .

(4.67)

Since A˜ 0 is a stable matrix by design, therefore A˜ 0T P˜0 + P˜0 A˜ 0 < 0 and (4.67) becomes V˙˜s ≤ 2e˜2T P˜0 A˜ 3 e˜1 + 2e˜2T P˜0 E f s − 2e˜2T P˜0 ν2 ˜ ≤ 2 P˜0 e˜2 ( A˜ 3 e˜1  + Eρs − k).

(4.68)

If the condition (4.66) holds, then ' V˙˜s ≤ 2η˜ 2  P˜0 e˜2  ≤ −2η˜ 2 λmin ( P˜0 )V˜s1/2 ,

(4.69)

where λmin ( P˜0 ) is the smallest eigenvalue of P˜0 . This shows that the reachability condition is satisfied. As a consequence, an ideal sliding motion will take place on the surface S in finite time. This completes the proof. 

4.4.2 Estimation of Sensor Faults It can be verified that for a small approximated as

√ ϑσmax (E + C)ΔψL2 , the sensor fault can be

ˆ E + P˜0 (w − w) , fˆs = k˜  P˜0 (w − w) ˆ + δ˜

(4.70)

where δ˜ is a small positive scalar.

4.5 Simulation Results In this section, the effectiveness of the proposed fault estimation schemes is illustrated by considering a single-link flexible joint robot arm system [10]. The dynamical model is described by

74

4 Robust Estimation of Actuator Faults

θ˙m = ωm , k Bv Kτ (θl − θm ) − ωm + u, ω˙ m = Jm Jm Jm θ˙l = ωl , k mgh sinθl , ω˙ l = − (θl − θm ) − Jl Jl

(4.71)

where θm and ωm are the motor position and velocity, respectively, θl and ωl are the link position and velocity, Jm is the inertia of the DC motor, Jl is the inertia of the link, 2h is the length of the link, m represents its mass, Bv is the viscous friction, k is the torsional spring constant, and K τ is the amplifier gain. Defining x = col(x1 , x2 , x3 , x4 ) := col(θm , ωm , θl , ωl ) as the state vector and assuming that the motor position, motor velocity, and link position are measured, then the flexible joint robot arm system can be described in the form of (4.1) with ⎤ ⎤ ⎡ ⎤ ⎡ 0 1 0 0 0 0.1(sin0.2t)2 ⎥ ⎢ −48.6 −1.25 48.6 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎥ , B = ⎢ 21.6 ⎥ , Δψ = ⎢ A=⎢ 2 ⎦, ⎣ 0 ⎦ ⎣ ⎦ ⎣ 0.2(sint) 0 0 10 0 0 1.95 0 −1.95 0 0 ⎡ ⎤ ⎡ ⎤ 0 1000 ⎢ ⎥ 0 ⎥. C = ⎣ 0 1 0 0 ⎦ , f (x, t) = ⎢ ⎣ ⎦ 0 0010 −0.333 sin(x3 ) ⎡

For simulation purpose, we set the control input u(t) as sin(π t) and the system ⎡ ⎤ 0.01(sin0.2t)2 ⎢ ⎥ 0 ⎥ uncertainty Δψ(t) as ⎢ ⎣ 0.02(sint)2 ⎦. 0

4.5.1 Actuator Fault Estimation Suppose that a fault occurs in the input channel, namely, the fault distribution matrix D is equal to the input matrix B. Consider the fault f a as ⎧ ⎨0 f a = 0.03(t − 3) ⎩ 0.12

t ≤ 3s 3 < t < 7s t ≥ 7s

Notice that C does not have the assumed structure. A nonsingular transformation matrix

4.5 Simulation Results

75

⎡

⎤ −0.1 −0.1 −0.1 0.1 ⎢ 1 0 0 0 ⎥ ⎥ Tc = ⎢ ⎣ 0 1 0 0 ⎦ 0 0 1 0 is therefore introduced to obtain C Tc−1 = [0 I3 ]. It follows that ⎡

⎤ −10 4.0550 −0.9750 −6.0550 ⎢ 0 ⎥ 0 1 0 ⎥ A=⎢ ⎣ 0 −48.6000 −1.2500 48.6000 ⎦ , 100 10 10 10 ⎡ ⎤ −2.1600 ⎢ ⎥ 0 ⎥ B=⎢ ⎣ 21.6000 ⎦ , 0 ⎤ ⎡ −0.033 sin x1 ⎥ ⎢ 0 ⎥. f (x) = ⎢ ⎦ ⎣ 0 0 Note that the transformed system is globally Lipschitz with Lipschitz constant L f = 0.033. Imposing the stability constraint to the system and formulating the problem in an LMI framework gives the following solutions: α = 0.4711, μ = 0.0002, P1 = 0.0070,   P2 = 0 0.0007 −0.0088 , ⎡ ⎤ 0.4672 0 0 P0 = ⎣ 0 0.4672 0 ⎦ , 0 0 0.0028 ⎡ ⎤ −1.5702 0 0 ⎦. 0 −1.5702 0 A0 = ⎣ 0 0 −313.2561 F can be computed as F = [ 0 10.0916 0 ]. It can be verified that both conditions (4.4) and (4.21) are satisfied. The simulation results are obtained by running successively the files chapter4_actuatorlmi.m, chapter4_fa.mdl, and chapter4 _fa_plot.m. The LMI was solved using the source file chapter4_actuatorlmi.m.

76

4 Robust Estimation of Actuator Faults

File chapter4_actuatorlmi.m

4.5 Simulation Results

77

78

4 Robust Estimation of Actuator Faults

The simulation is carried out in the Simulink program chapter4_fa.mdl and the figures are plotted by running the file chapter4_fa_plot.m. In the simulation, the initial states have been selected as x(0) = [0, 0, 0, 0]T and zˆ (0) = [0.3, 0.1, −1, 0.1]T , and k and δ in (4.43) have been selected as 5.5 and 0.01, respectively. Figure 4.1 shows the trajectories of the true states and their estimates. It can be seen from this figure that the proposed SMO can estimate the states very accurately, even after the occurrence of an actuator fault. Figure 4.2 is concerned with the actuator fault estimation. It shows that the proposed SMO can reject the effects of system uncertainties successfully and accurately estimate the actuator fault.

4.5 Simulation Results

79

x1

5 0 −5

0

2

4

6

8

10

0

2

4

6

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10

0

2

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6

8

10

0

2

4

6

8

10

x

2

5 0 −5

x3

5 0 −5

x4

5 0 −5

Time (s)

Fig. 4.1 System states x (solid line) and their estimates xˆ (dashed line) 0.3 Actual f a Estimated f a

0.25

Magnitude

0.2 0.15 0.1 0.05 0 −0.05

0

2

4

6

8

10

Time (s)

Fig. 4.2 Actuator fault f a and its estimation fˆa

4.5.2 Sensor Fault Estimation To verify the efficiency of the proposed sensor FE scheme, we assume that only the first sensor is prone to fault, while other sensors are fault free. In this case, the sensor fault distribution matrix E is E = [ 1 0 0 ]T . For a good estimation performance and a minimal λ solution to the convex optimization problem defined in Theorem 4.4, we apply one of synthesis methods placing the eigenvalues of the error dynamics (4.57) in a specified region D = λ ∈ C | |λ| < d1 , Re(λ) < −d2 . The constraints to locate the eigenvalues of the error system are

80

4 Robust Estimation of Actuator Faults

(

−d1 P˜z P˜z A˜ s A˜ sT P˜z −d1 P˜z

) < 0,

A˜ sT P˜z + P˜z A˜ s + 2d2 P˜z < 0,  A˜ 1 0 . A˜ 3 A˜ 0 Equations (4.72) and (4.73) can be expressed in terms of LMI as ⎡ ⎤ ˜ −d1 P˜1 0 P˜1 A + Z˜ GC 0 ⎢ ⎥ ⎢ 0 −d1 P˜0 Y˜ ⎥ P˜0 C ⎢ ⎥ < 0, ⎢ A T P˜ + C T G˜ T Z˜ T C T P˜ ⎥ ˜ −d 0 P 1 0 1 1 ⎣ ⎦

where A˜ s =



(4.72) (4.73)



Y˜ T

−d1 P˜0  ˜ + C T G˜ T Z˜ T + 2d2 P˜1 P˜1 A + A T P˜1 + Z˜ GC C T P˜0 < 0, Y˜ + Y˜ T + 2d2 P˜0 P˜0 C 0

(4.74)

0

(4.75)

where G˜ = I p − E E + . Therefore, the optimization problem defined in Theorem 4.4 becomes finding P˜1 , P˜0 , Z˜ , Y˜ , κ and the minimal ϑ subject to P˜1 > 0, P˜0 > 0, (4.58), (4.74), and (4.75). In this simulation, we set d1 = 10 and d2 = 1. After solving this optimization problem, we get κ = 1.9392, ϑ = 0.3896, ⎤ ⎡ 4.3768 0.6357 0.0250 −0.0203 ⎢ 0.6357 0.1897 −0.1187 0.1184 ⎥ ⎥ ⎢ P˜1 = ⎢ ⎥, ⎣ 0.0250 −0.1187 3.1299 −3.0624 ⎦ −0.0203 0.1184 −3.0624 3.1307 ⎤ ⎡ 0 −3.6236 −31.3892 ⎢ 0 −1.7515 −7.8897 ⎥ ⎥ ⎢ P˜2 = ⎢ ⎥, ⎣ 0 1.0108 −30.8199 ⎦ 0 −0.9374 29.0267 ⎡ ⎤ 0.0876 −0.0018 0 P˜0 = ⎣ −0.0018 0.0954 0.0016 ⎦ , 0 0.0016 0.7106 ⎡ ⎤ −6.0003 0.6508 −0.0017 ⎢ ⎥ A˜ 0 = ⎣ 0.3085 −7.2920 0.0026 ⎦ , −0.0006 0.0026 −6.2805  F˜ = 0.0876 −0.0018 0 . 

4.5 Simulation Results

81

It can be verified that the eigenvalues of A˜ s are {−7.4322, −6.2805, −5.8601, −9.5516±2.0065i, −7.0288±7.0285i}. The process of computing these parameters is shown in the following program chapter4_sensorlmi.m. File chapter4_sensorlmi.m

82

4 Robust Estimation of Actuator Faults

4.5 Simulation Results

83

It is assumed that there is an abrupt sensor fault which is represented as ⎧ ⎨0 f s = 0.1 ⎩ 0

t ≤ 3s 3 s < t < 7 s, t ≥ 7s

The simulation is carried out in the Simulink program chapter4_fs.mdl and the figures are plotted running the program chapter4_fs_plot.m. In the simulation, the initial states have been selected as x(0) = [0, 0, 0, 0]T and zˆ˜ (0) = [0.3, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1]T , and k˜ and δ˜ in (4.70) have been selected as 400 and 0.08, respectively. The estimation of system states and sensor fault is shown in Figs. 4.3 and 4.4. It should be noticed that we use an integrator # t to transform the sensor fault into the form of an actuator fault and consider h = 0 y(τ )dτ as the new output. The term #t 0 y(τ )dτ actually acts as a low-pass filter and therefore can attenuate the effect of measurement noise. In order to demonstrate this property, we add noise v(t) to the measurement y(t). v(t) is assumed to satisfy ⎡

⎤ 0.0001 0 0 E {v(t)} = 0, Cov{v(t)} = ⎣ 0 0.0001 0 ⎦ , 0 0 0.0001 where E {·} and Cov{·} denote the expectation and covariance of a signal, respectively. The sensor fault estimation is shown in Fig. 4.5. It can be seen that despite the presence of measurement noises, the proposed method could still track the sensor fault f s .

x1

2 0 −2 0

2

4

6

8

10

0

2

4

6

8

10

−5 0

2

4

6

8

10

2

4

6

8

10

x2

5 0 −5

x

3

5 0

x4

0.5 0 −0.5

0

Time (s)

Fig. 4.3 System states x (solid line) and their estimates xˆ (dashed line)

84

4 Robust Estimation of Actuator Faults 0.2 Actual f

s

Estimated f s

0.15

Magnitude

0.1

0.05

0

−0.05

−0.1

0

2

4

6

8

10

Time (s)

Fig. 4.4 Sensor fault f s and its estimation fˆs 0.2 Actual f s Estimated fs 0.15

Magnitude

0.1

0.05

0

−0.05

−0.1

0

2

4

6

8

10

Time (s)

Fig. 4.5 Sensor fault f s and its estimation fˆs (with measurement noises)

4.6 Conclusions A new robust SMO-based actuator/sensor FE scheme with H∞ measure has been presented in this chapter. The unstructured uncertainty under consideration is more general than the structured uncertainty, which implies that the proposed method is applicable for a wider class of systems. The stability conditions of the proposed SMO

4.6 Conclusions

85

have been derived using Lyapunov stability theory and expressed as a LMI optimization problem. This results in calculating the design parameters of the observer and the maximum uncertainty attenuation level that the observer can tackle. The effectiveness of the proposed method has been demonstrated considering the example of a single-link flexible joint robot system and has been found to be satisfactory.

References 1. Edwards C, Spurgeon SK, Patton RJ (2000) Sliding mode observers for fault detection and isolation. Automatica 36:541–553 2. Tan CP, Edwards C (2002) Sliding mode observers for detection and reconstruction of sensor faults. Automatica 38:1815–1821 3. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 4. Gao ZW, Ding SX (2008) Fault reconstruction for Lipschitz nonlinear descriptor systems via linear matrix inequality approach. Circuits Syst Signal Process 27:295–308 5. Raoufi R, Marquez HJ, Zinober ASI (2010) H∞ sliding mode observer for uncertain nonlinear Lipschitz systems with fault estimation synthesis. Int J Robust Nonlinear Control 20:1785– 1801 6. Jiang B, Staroswiecki M, Cocquempot V (2004) Fault estimation in nonlinear uncertain systems using robust/sliding-mode observers. IEE Proc Control Theory Appl 151(1):29–37 7. Edgar Thomas F, Ivn Castillo, Fernnndez Benito R (2012) Robust model-based fault detection and isolation for nonlinear processes using sliding modes. Int J Robust Nonlinear Control 22:89–104 8. Yan XG, Edwards C (2008) Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems. Int J Syst Sci 39(4):349–359 9. Wang Z, Shen Y, Zhang X (2014) Actuator fault estimation for a class of nonlinear descriptor systems. Int J Syst Sci 45:487–496 10. Zhang K, Jiang B, Shi P (2009) Fast fault estimation and accommodation for dynamical systems. IET Control Theory Appl 3:189–199 11. Chen W, Chowdhury FN (2010) A synthesized design of sliding-mode and Luenberger observers for early detection of incipient faults. Int J Adapt Control Signal Process 24:1021– 1035 12. Alwi H, Edwards C, Tan CP (2011) Fault detection and fault-tolerant control using sliding modes. Springer, Heidelberg 13. Zhang J, Swain AK, Nguang SK (2011) Reconstruction of actuator fault for a class of nonlinear systems using sliding mode observer. In: Proceedings of American control conference, pp 1370–1375 14. Edwards C, Spurgeon SK (1998) Sliding mode contol: theory and applications. Taylor and Francis, Abingdon

Chapter 5

Robust Estimation of Sensor Faults

5.1 Introduction During the last two decades, considerable research results have been reported on sensor fault estimation (FE), see [1–3] and the references therein. In [1], an online estimation approach based on adaptive observer technique was adopted to reconstruct the sensor fault with an incipient time profile. A descriptor system approach was introduced to investigate sensor fault diagnosis for nonlinear systems in [2], which is applicable for sensor faults of any forms. In [3], a sensor fault-tolerant control (FTC) scheme was presented for a crane system based on the estimated faults. In recent years, the sliding-mode observers (SMO)-based sensor FE methods have been extensively studied due to its inherent robustness to model uncertainties and disturbances. For example, [4] introduced two methods to estimate sensor faults for systems without uncertainties. In both methods, two SMOs were used in cascade. The second approach of [4] was later improved to achieve robust sensor FE in [5] using an LMI formulation. The open-loop stability required in [4] was no longer a necessary condition. The upper bound on the effect of the uncertainty on the reconstructed fault signals is minimized using H∞ concepts. However, for open-loop unstable systems with certain classes of faults, the method proposed in [5] may not be applicable. This restriction was addressed in [6], where a new method for sensor fault reconstruction was proposed, which is applicable for both open-loop stable and unstable systems. A high-order SMO was designed to reconstruct sensor faults in [7], where both disturbances and uncertainties were considered. For descriptor systems with actuator faults and sensor faults, the FE was carried out using SMOs with feedforward signals in [8]. However, those results are either only applicable for linear systems, or can only minimize the effects of system uncertainties on the fault estimates, which motivates us for this study. In this chapter, we propose two sensor FE schemes for uncertain Lipschitz nonlinear systems. The proposed schemes essentially transform the original system into two subsystems (subsystems 1 and 2), where subsystem-1 includes the effects of © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_5

87

88

5 Robust Estimation of Sensor Faults

system uncertainties but is free from sensor faults and subsystem-2 has sensor faults but without any uncertainties. Therefore, the FE can be carried out independently and is not affected in the presence of uncertainties. This offers a significant advantage over some of the existing methods. For example in [2, 3, 6, 9], the effects of uncertainties on the estimation of faults can only be minimized. And in [10], certain geometric conditions should be satisfied such that the fault can be decoupled from the uncertainty, which brings in an extra difficulty. Sensor faults in subsystem-2 are treated as actuator faults using an integral observer-based approach. In the first scheme, we design an SMO for each subsystem. The one which is designed for subsystem-1 is used to eliminate the effects of system uncertainties in this system, while the other one is used to estimate the sensor faults which is present in subsystem-2 using the concept of equivalent output injection. In the second scheme, we still design an SMO for subsystem-1 to eliminate the effects of uncertainties on the state estimation, and an adaptive observer (AO) [11, 12] rather than an SMO for subsystem-2 to estimate sensor faults. The proposed method extends the results of [11, 12] in which actuator FE of linear systems was studied to sensor FE for uncertain Lipschitz nonlinear systems. Moreover, most of the existing FE methods such as those reported in [2, 10, 13–15] assume that the value of the Lipschitz constant L f is known and they incorporate this knowledge to obtain feasible LMI solutions. However, determining the value of L f of a nonlinear system is often difficult. Further, if the value of L f exceeds the admissible value, those methods often fail to find feasible LMI solutions. In this chapter, the value of L f is assumed to be unknown and adaptation laws are integrated into the proposed schemes such that L f does not appear in the LMI formulation and the feasibility of LMIs is not dependent on the knowledge of this parameter. Therefore, the proposed methods offer distinct advantages over some of the existing methods. The rest of the chapter is structured as follows: Sect. 5.2 briefly describes the mathematical preliminaries required for designing the observers. Section 5.3 proposes a scheme using an SMO to estimate sensor faults and derives the stability condition based on Lyapunov approach. In Sect. 5.4, another scheme based on AO is proposed to estimate the faults. The effectiveness of the developed schemes is illustrated via the simulation in Sect. 5.5 considering an example of a satellite control system with conclusions in Sect. 5.6.

5.2 Problem Formulation Consider a nonlinear system described by 

x(t) ˙ = Ax(t) + W f (x, t) + Bu(t) + EΔψ(t) y(t) = C x(t) + D f s (t)

(5.1)

where x ∈ R n , u ∈ R m , and y ∈ R p denote, respectively, the vector of state variables, inputs, and outputs. f s ∈ R q is an immeasurable vector which is considered

5.2 Problem Formulation

89

as an additive bias resulting from sensor faults. The unknown term Δψ models the lumped uncertainties and disturbances experienced by the system. The nonlinear continuous term f (x, t) ∈ R j is assumed to be known. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R p×q , E ∈ R n×r ( p ≥ q + r ), and W ∈ R n× j are known constant matrices with D and E being of full column rank. Before starting the main results of this chapter, the following assumptions are made. Assumption 5.1 rank(C E) = rank(E). Assumption 5.2 For every complex number s with nonnegative real part, 

s In − A E rank C 0

 = n + rank(E).

(5.2)

Assumption 5.3 The nonlinear term f (x, t) is assumed to be known and Lipschitz about x uniformly, i.e., ∀ x, xˆ ∈ R n , ˆ  f (x, t) − f (x, ˆ t) ≤ L f x − x,

(5.3)

where L f is the Lipschitz constant and assumed to be unknown in this chapter. Assumption 5.4 The sensor fault vector f s and its derivative f˙s , and the uncertainty vector Δψ satisfy the following norm bounded constraints:  f s  ≤ ρs ,  f˙s  ≤ ρss , and Δψ ≤ ξ.

(5.4)

Lemma 5.1 Under Assumption 5.1, there exists linear transformation of coordinates    w1 z1 , w = Sy = z = Tx = z2 w2 

(5.5)

such that T AT

−1

SC T −1 

     B1 E1 A1 A2 , , TB = , TE = = A3 A4 B2 0       0 W1 C1 0 , SD = , TW = , = W2 0 C4 D2 

(5.6)

   T1 S1 ∈ R n×n , S = ∈ R p× p , T1 ∈ R r ×n , S1 ∈ R r × p , z 1 ∈ R r , T2 S2 w1 ∈ R r , A1 ∈ R r ×r , A4 ∈ R (n−r )×(n−r ) , B1 ∈ R r ×m , E 1 ∈ R r ×r , C1 ∈ R r ×r , C4 ∈ R ( p−r )×(n−r ) , D2 ∈ R ( p−r )×q and W1 ∈ R r × j . C1 is invertible. where T =

90

5 Robust Estimation of Sensor Faults

Therefore, in the new coordinate system, System (5.1) is converted into the following two Subsystems (5.7) and (5.8):  

z˙ 1 = A1 z 1 + A2 z 2 + W1 f (T −1 z, t) + B1 u + E 1 Δψ w1 = C1 z 1

(5.7)

z˙ 2 = A3 z 1 + A4 z 2 + W2 f (T −1 z, t) + B2 u w2 = C4 z 2 + D2 f s

(5.8)

t For Subsystem (5.8), we define a new state z 3 = 0 w2 (τ )dτ so that z˙3 (t) = C4 z 2 + D2 f s . An augmented system with the new state z 3 is therefore given as 

z˙ 2 z˙ 3





A4 0 = C4 0



         z2 0 B2 A3 W2 f (T −1 z, t) z1 + u+ + + f , z3 0 0 D2 s 0

w3 = z 3 .

(5.9)

This system can further be rewritten in a more compact form as 

z˙ 0 = A0 z 0 + A¯ 3 z 1 + W¯ 2 f (T −1 z, t) + B0 u + D0 f s w3 = C0 z 0 

A4 0 C4 0

(5.10)



∈ R (n+ p−2r )×(n+ p−2r ) , A¯ 3 = where z 0 ∈ R , w3 ∈ R , A0 =       A3 B2 0 ∈ R (n+ p−2r )×r , B0 = ∈ R (n+ p−2r )×m , D0 = ∈ R (n+ p−2r )×q , 0 0 D2     W2 C0 = 0 I p−r ∈ R ( p−r )×(n+ p−2r ) , W¯ 2 = . 0 Accordingly, Subsystem (5.7) is rewritten as n+ p−2r



p−r

z˙ 1 = A1 z 1 + A¯ 2 z 0 + W1 f (T −1 z, t) + B1 u + E 1 Δψ w1 = C1 z 1

(5.11)

  where A¯ 2 = A2 0r ×( p−r ) . Lemma 5.2 The pair (A0 , C0 ) is observable if Assumption 5.2 holds. Remark 5.1 Assumption 5.1 ensures the existence of coordinate transformations T and S, such that in the new coordinate the sensor faults can be completely decoupled from uncertainties. The satisfaction of this assumption makes the accurate FE (i.e., fault reconstruction) possible, which is the most distinct feature of the proposed schemes. Assumption 5.2 implies that there exists an asymptotic estimator for System (5.10). In Assumption 5.4, the sensor fault f s is assumed to be nonzero and differentiable after its occurrence. This assumption is quite general either for constant faults or time-varying faults at limited rates [15–17].

5.3 SMO-Based Sensor FE

91

5.3 SMO-Based Sensor FE In this section, we propose a scheme which consists of two SMOs to estimate sensor faults. One of the SMOs is designed for Subsystem (5.11) to eliminate the effects of system uncertainties on the state estimation, while the other one is designed for Subsystem (5.10) to estimate sensor faults. For Subsystem (5.11), the proposed SMO has the following form: ⎧ ⎨ z˙ˆ 1 = A1 zˆ 1 + A¯ 2 zˆ 0 + W1 f (T −1 zˆ , t) + B1 u + 21 kˆ1 P1 C1−1 (w1 − wˆ 1 ) + (A1 − As1 )C1−1 (w1 − wˆ 1 ) + ν1 ⎩ wˆ 1 = C1 zˆ 1

(5.12)

where As1 ∈ R r ×r is a stable matrix which needs to be determined, P1 is the Lyapunov matrix of As1 , zˆ := [(C1−1 S1 y)T , ([In−r 0]ˆz 0 )T ]T . The discontinuous output error injection term ν1 , that is being used to eliminate the effects of uncertainties, is defined by  ν1 =

P (C −1 w −ˆz )

(E 1 ξ + η1 ) P1 (C1−1 w1 −ˆz1 ) if C1−1 w1 − zˆ 1 = 0 1

0

1

1

1

(5.13)

otherwise

where η1 is a positive scalar to be determined. It is worth noting that the state z 1 can be obtained by the measured output y as z 1 = C1−1 S1 y. kˆ1 satisfies the following adaptation law: k˙ˆ1 = lk1 P1 (C1−1 w1 − zˆ 1 )2 .

(5.14)

where lk1 is a positive constant. For Subsystem (5.10), the proposed SMO has the following form: ⎧ ⎨ z˙ˆ 0 = A0 zˆ 0 + A¯ 3 C1−1 w1 + W¯ 2 f (T −1 zˆ , t) + B0 u + L 0 (w3 − wˆ 3 ) + 21 kˆ2 W¯ 2 H0 (w3 − wˆ 3 ) + D0 ν2 ⎩ wˆ 3 = C0 zˆ 0

(5.15)

 L 01 ∈ R (n+ p−2r )×( p−r ) with L 01 ∈ R (n−r )×( p−r ) , L 02 H0 ∈ R j×( p−r ) and P0 = P0T > 0 ∈ R (n+ p−2r )×(n+ p−2r ) . The discontinuous output error injection term ν2 is defined by 

where the observer gain L 0 =

 ν2 =

ˆ 3) 3 −w (ρs + η2 ) FF00 (w if w3 − wˆ 3 = 0 (w3 −wˆ 3 ) 0 otherwise

where F0 ∈ R q×( p−r ) is a matrix to be determined and η2 is a positive scalar.

(5.16)

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5 Robust Estimation of Sensor Faults

kˆ2 satisfies the following adaptation law: k˙ˆ2 = lk2 H0 (w3 − wˆ 3 )2 ,

(5.17)

where lk2 is a positive constant. If the state estimation errors are defined as e1 = z 1 − zˆ 1 and e0 = z 0 − zˆ 0 , then the state estimation error dynamics after the occurrence of sensor faults can be obtained from

1 e˙1 = As1 e1 + A¯ 2 e0 + W1 f (T −1 z, t) − f (T −1 zˆ , t) − kˆ1 P1 e1 2 + E 1 Δψ − ν1 , (5.18)

1 e˙0 = (A0 − L 0 C0 )e0 + W¯ 2 f (T −1 z, t) − f (T −1 zˆ , t) − kˆ2 W¯ 2 H0 C0 e0 2 + D0 ( f s − ν2 ). (5.19) We now present the following theorem which provides sufficient conditions for the existence of the proposed SMOs. Theorem 5.1 Given System (5.1) with Assumptions 5.1, 5.2, 5.3, and 5.4. If there exist matrices As1 , L 0 , F0 , H0 , P1 = P1T > 0, and P0 = P0T > 0 such that D0T P0 = F0 C0 , W¯ 2T P0 = H0 C0 ,   Π1 P1 A¯ 2 < 0, A¯ 2T P1 Π2 + 2In+ p−2r

(5.20) (5.21) (5.22)

where Π1 = As1 T P1 + P1 As1 and Π2 = (A0 − L 0 C0 )T P0 + P0 (A0 − L 0 C0 ), then the error dynamics (5.18) and (5.19) are asymptotically stable. Proof Consider the Lyapunov function as V = V1 (e1 ) + V2 (e0 ) + V3 (ek1 ) + V4 (ek2 ),

(5.23)

where V1 (e1 ) = e1T P1 e1 , V2 (e0 ) = e0T P0 e0 , V3 (ek1 ) = lk−1 ek21 /2, V4 (ek2 ) = lk−1 ek22 /2, 1 2 ek1 = k1 − kˆ1 , and ek2 = k2 − kˆ2 . k1 and k2 are two constants which can be determined from (5.32). The time derivative of V1 along the trajectory of System (5.18) can be shown to be V˙1 = e1T (As1 T P1 + P1 As1 )e1 + 2e1T P1 A¯ 2 e0 + 2e1T P1 E 1 Δψ

+ 2e1T P1 W1 f (T −1 z, t) − f (T −1 zˆ , t) − 2e1T P1 ν1 − kˆ1 P1 e1 2 .

5.3 SMO-Based Sensor FE

93

Note that since zˆ := [(C1−1 S1 y)T , ([In−r 0]ˆz 0 )T ]T , it is easy to see that  f (T −1 z, t) − f (T −1 zˆ , t) ≤ L f T −1 e0 . (5.24) Since the inequality 2X T Y ≤ [10], then we have

1 α

X T X + αY T Y holds true for any scalar α > 0

1 V˙1 ≤ e1T (As1 T P1 + P1 As1 )e1 + 2e1T P1 A¯ 2 e0 + 2e1T P1 E 1 Δψ + e1T P1 W1 W1T P1T e1 α1

T

−1 −1 −1 −1 + α1 f (T z, t) − f (T zˆ , t) f (T z, t) − f (T zˆ , t) − 2e1T P1 ν1 − kˆ1 P1 e1 2 1 ≤ e1T Π1 e1 + 2e1T P1 A¯ 2 e0 + W1 2 P1 e1 2 + α1 L f2 T −1 2 e0 2 α1 − kˆ1 P1 e1 2 + 2e1T P1 E 1 Δψ − 2e1T P1 ν1 .

(5.25)

Using (5.13), the last two terms of (5.25) can be expressed as 2e1T P1 E 1 Δψ − 2e1T P1 ν1 ≤ −2η1 P1 e1  ≤ 0.

(5.26)

Letting α1 = 1/L f2 T −1 2 and substituting (5.26) into (5.25) yields V˙1 ≤ e1T Π1 e1 + 2e1T P1 A¯ 2 e0 + e0 2 + (L f21 T −1 2 W1 2 − kˆ1 )P1 e1 2 . (5.27) Similarly, it follows from (5.20) and (5.21) that the derivative of V2 can be obtained as

T 1 V˙2 ≤ e0T Π2 e0 + e0T P0 W¯ 2 W¯ 2T P0 e0 + α0 f (T −1 z, t) − f (T −1 zˆ , t) α0

−1 × f (T z, t) − f (T −1 zˆ , t) − kˆ2 H0 C0 e0 2 + 2e0T P0 D0 ( f s − ν2 ) 1 ≤ e0T Π2 e0 + H0 C0 e0 2 + α0 L f2 T −1 2 e0 2 α0 ˆ − k2 H0 C0 e0 2 + 2e0T P0 D0 ( f s − ν2 )   1 − kˆ2 H0 C0 e0 2 + α0 L f2 T −1 2 e0 2 . (5.28) ≤ e0T Π2 e0 + α0 Let α0 = 1/L f2 T −1 2 . Then it follows that V˙2 ≤ e0T Π2 e0 + (L f2 T −1 2 − kˆ2 )H0 C0 e0 2 + e0 2 .

(5.29)

94

5 Robust Estimation of Sensor Faults

Moreover, the time derivatives of V3 and V4 are ek1 e˙k1 = −ek1 P1 e1 2 , V˙3 = lk−1 1 ek2 e˙k2 = −ek2 H0 C0 e0 2 . V˙4 = lk−1 2

(5.30)

k1 = L f2 T −1 2 W1 2 and k2 = L f2 T −1 2 .

(5.32)

(5.31)

Set

From (5.27), (5.29), (5.30), (5.31), and (5.32), the time derivative of V can then be obtained as V˙ = V˙1 + V˙2 + V˙3 + V˙4 ≤ e1T Π1 e1 + 2e1T P1 A¯ 2 e0 + e0T Π2 e0 + 2e0 2    T  P1 A¯ 2 Π1 e1 e . = 1 e0 e0 A¯ 2T P1 Π2 + 2In+ p−2r

(5.33)

It follows from (5.22) that V˙ < 0, which implies that the observer error dynamics (5.18) and (5.19) are asymptotically stable. This completes the proof.  Remark 5.2 In practice, it is often difficult to know precisely the value of the Lipschitz constant in (5.3). This parameter plays an important role in the design of observers for nonlinear systems. It is worth noting that in [2, 9, 10, 14, 15] the Lipschitz constant L f is assumed to be known and is one of the parameters in the LMI formulation. This can be considered as one of the limitations of those methods. We have observed that it often fails to solve LMI if the value of L f is too large. In order to find a feasible solution of LMI, the value of L f may be reduced by introducing coordinate transformations for certain structures of the Lipshcitz function [18, 19]. However, this may bring in an extra difficulty in the design of observers. In contrast to some of the existing methods, the Lipschitz constant L f in this chapter is assumed to be unknown and adaptive SMOs are proposed to deal with this situation. Specifically, the Lipschitz constants L f1 and L f are injected into the constants k1 and k2 which can be adjusted by the adaptation laws (5.14) and (5.17). Note that the asymptotic estimation of states can be guaranteed even if the estimates of k1 and k2 do not approach to their actual values. This is a distinctive feature of the proposed method. Remark 5.3 The problem of finding matrices P1 = P1T > 0, P0 = P0T > 0, L, As1 , F0 , and H0 to simultaneously satisfy the Inequality (5.22) and Equalities (5.20) and (5.21) can be transformed into the following LMI optimization problem:

5.3 SMO-Based Sensor FE

95

minimize γ1 + γ2 subject to P1 > 0, P0 > 0 and   X + XT P1 A¯ 2 < 0, A¯ 2T P1 P0 A0 + A0T P0 − Y0 C0 − C0T Y0T + 2In+ p−2r   D0T P0 − F0 C0 γ1 I q > 0, T T (D0 P0 − F0 C0 ) γ1 In+ p−2r   W¯ 2T P0 − H0 C0 γ2 I j > 0, (W¯ 2T P0 − H0 C0 )T γ2 In+ p−2r

(5.34) (5.35) (5.36)

where X = P1 As1 and Y0 = P0 L 0 . Theorem 5.1 has shown that the error dynamics (5.18) and (5.19) are asymptotically stable. The objective now is to choose constants η1 and η2 such that the error dynamics can be driven to the sliding surface (5.37) in finite time and a sliding motion can be maintained on it thereafter. For error dynamics (5.18) and (5.19), we define the sliding-mode surface as S = {(e1 , e0 )|e1 = 0, C0 e0 = 0}.

(5.37)

Theorem 5.2 Given System (5.1) with Assumptions 5.1, 5.2, 5.3, and 5.4 and the proposed observers (5.12) and (5.15). Then the error dynamics (5.18) and (5.19) can be driven to the sliding surface given by (5.37) in finite time and remain on it if LMIs (5.34)–(5.36) are solvable and the gains η1 and η2 satisfy

η1 ≥  A¯ 2  + L f T −1 W1  e0  + η3 , η2 ≥

L f T

(5.38)

−1

W2 e0  + η4 , D0 

(5.39)

where η3 and η4 are the two positive scalars. Proof Consider the Lyapunov candidate functions V1 = e1T P1 e1 and V2 = e0T P0 e0 , then V˙1 ≤ e1T (P1 As1 + As1 T P1 )e1 + 2e1T P1 A¯ 2 e0 + W1 f (T −1 z, t) − W1 f (T −1 zˆ , t) +E 1 Δψ − ν1 ) .

(5.40)

Since by design As1 is a stable matrix which implies that P1 As1 + As1 T P1 < 0, therefore V˙1 ≤ 2e1T P1 ( A¯ 2 e0 + W1 f (T −1 z, t) − W1 f 1 (T −1 zˆ , t) + E 1 Δψ − ν1 )

≤ 2P1 e1   A¯ 2 e0  + L f T −1 W1 e0  − η1 . It follows from (5.38) that  1 V˙1 ≤ −2η3 P1 e1  ≤ −2η3 λmin (P1 )V12 .

(5.41)

96

5 Robust Estimation of Sensor Faults

Similarly, if (5.39) is satisfied,we have  1 V˙2 ≤ −2η4 P0 e0  ≤ −2η4 λmin (P0 )V22 .

(5.42)

This shows that the reachability condition [20] is satisfied and a sliding motion is achieved and maintained after some finite time ts > 0. This completes the proof.  After reaching the sliding surface, the sliding motion will be maintained thereafter, i.e., C0 e0 = 0 and C0 e˙0 = 0. It follows from (5.19) that

0 = C0 A0 e0 + C0 W¯ 2 f (z, t) − f (ˆz , t) + D2 ( f s − ν2eq ),

(5.43)

where ν2eq is the equivalent output error injection signal representing the average behavior of the discontinuous function ν2 . Since limt→∞ e0 = 0 according to Theorem 5.1, f (z, t) − f (ˆz , t) will also tend to zero. This implies (from (5.43)) that f s → ν2eq

as

t →∞

(5.44)

The equivalent output error injection signal ν2eq can be approximated as ν2eq = (ρs + η2 )

F0 ew3 , F0 ew3  + δ

(5.45)

where δ is a small positive scalar to reduce the chattering effect. It can be shown that ν2eq can be approximated to any degree of accuracy by (5.45) [20]. Therefore, the sensor fault can be approximated as fˆs ≈ (ρs + η2 )

F0 ew3 . F0 ew3  + δ

(5.46)

5.4 AO-Based Sensor FE In Sect. 5.3, a sensor FE scheme using two SMOs is proposed. One of the SMOs is used to eliminate the effects of system uncertainties on the state estimation, while the other one is used to estimate sensor faults. In this section, an alternative method based on using an AO technique [11, 12, 15] is proposed. More specifically, the proposed scheme combines an SMO with an AO, where the SMO is used for Subsystem (5.11) to remove the effects of system uncertainties on the state estimation and the AO is used to directly estimate sensor faults which exist in Subsystem (5.10).

5.4 AO-Based Sensor FE

97

For Subsystem (5.11), the proposed SMO has the similar form as that is used in Sect. 5.3, i.e., ⎧ ⎨ z˙ˆ 1 = A1 zˆ 1 + A¯ 2 zˆ 0 + W1 f (T −1 zˆ , t) + B1 u + 21 kˆ3 P1 C1−1 (w1 − wˆ 1 ) + (A1 − As1 )C1−1 (w1 − wˆ 1 ) + ν ⎩ wˆ 1 = C1 zˆ 1

(5.47)

where As1 ∈ R r ×r is a stable matrix which needs to be determined, P1 is the Lyapunov matrix of As1 , zˆ := [(C1−1 S1 y)T , ([In−r 0]ˆz 0 )T ]T , the discontinuous output error injection term ν is defined by ν=

 C −1 S y−ˆz (E 1 ξ + η5 ) C1−1 S1 y−ˆz1  if e1 = 0 1

1

1

0

(5.48)

otherwise

where η5 is a positive scalar to be determined, and kˆ3 satisfies the following adaptation law: k˙ˆ3 = lk3 P1 (C1−1 w1 − zˆ 1 )2

(5.49)

where lk3 is a positive constant. For Subsystem (5.10), the proposed AO has the following form: ⎧ −1 −1 ⎪ ⎨ z˙ˆ 0 = A0 zˆ 0 + A¯ 3 C1 w1 + W¯ 2 f (T zˆ , t) + B0 u + L 0 (w3 − wˆ 3 ) 1ˆ ¯ + 2 k4 W2 H0 (w3 − wˆ 3 ) + D0 fˆs ⎪ ⎩ wˆ = C zˆ 3 0 0

(5.50)

 L 01 ∈ R (n+ p−2r )×( p−r ) with L 01 ∈ R (n−r )×( p−r ) , where the observer gain L 0 = L 02 H0 ∈ R j×( p−r ) , P0 = P0T > 0 ∈ R (n+ p−2r )×(n+ p−2r ) , kˆ4 satisfies the following adaptation law: 

k˙ˆ4 = lk4 H0 (w3 − wˆ 3 )2 ,

(5.51)

where lk4 is a positive constant. fˆs represents the estimated sensor fault and its dynamics is defined as f˙ˆs = Γ F0 ew3 − εΓ fˆs ,

(5.52)

where Γ ∈ R q×q is a symmetric positive definite matrix representing the learning rate, F0 ∈ R q×( p−r ) is a matrix to be determined, and ε is a positive scalar. Denote e f = f s − fˆs . Then after the occurrence of the faults, the dynamics of the state estimation error are obtained from

98

5 Robust Estimation of Sensor Faults



1 e˙1 = As1 e1 + A¯ 2 e0 + W1 f (T −1 z, t) − f (T −1 zˆ , t) − kˆ3 P1 e1 + E 1 Δψ − ν, 2 (5.53)

1 e˙0 = (A0 − L 0 C0 )e0 + W¯ 2 f (T −1 z, t) − f (T −1 zˆ , t) − kˆ4 W¯ 2 H0 C0 e0 + D0 e f . 2 (5.54) Before presenting the main result of this section, we define the following region:   λmin (P1 ) ew1 2 + Ω1 = (ew1 , ew3 , fˆs ) C1 2  μ4 −1 2 , ≤ λmin (Γ )ρs + μ7   λmin (P1 ) Ω2 = (ew1 , ew3 , fˆs ) ew1 2 + C1 2  μ4 > λmin (Γ −1 )ρs2 + , μ7

λmin (P0 ) λmin (Γ −1 ) ˆ 2 2  fs  e  + w 3 C0 2 2

λmin (P0 ) λmin (Γ −1 ) ˆ 2 2  fs  e  + w 3 C0 2 2

μ1 = λmin (−As1 T P1 − P1 As1 ) μ2 = λmin (−(A0 − L 0 C0 )T P0 − P0 (A0 − L 0 C0 ) − 2In+ p−2r ) > 0 μ3 = λmin (ε I − G) > 0 2 μ4 = ρss λmax (Γ −1 G −1 Γ −1 ) + ερs2

μ5 = min(μ1 , μ2 , μ3 ) μ6 = max(λmax (P1 ), λmax (P0 ), λmax (Γ −1 )) μ7 = μ5 /μ6 , where G ∈ R q×q is a symmetric positive definite matrix. Theorem 5.3 Given System (5.1) with Assumptions 5.1, 5.2, 5.3, and 5.4. If there exist matrices As1 , L 0 , F0 , H0 , P1 = P1T > 0, and P0 = P0T > 0 such that D0T P0 = F0 C0 , W¯ 2T P0 = H0 C0 ,   −Q 1 P1 A¯ 2 < 0, A¯ 2T P1 −Q 0 + 2In+ p−2r

(5.55) (5.56) (5.57)

where −Q 1 = As1 T P1 + P1 As1 , −Q 0 = (A0 − L 0 C0 )T P0 + P0 (A0 − L 0 C0 ), then for a given matrix Γ and a positive scalar ε, the error dynamics (5.53)–(5.54) are uniformly bounded and (ew1 , ew3 , fˆs ) converges to Ω1 at a rate greater than e−μ7 t .

5.4 AO-Based Sensor FE

99

Proof Consider the Lyapunov function as ek23 /2 + lk−1 ek24 /2 + e Tf Γ −1 e f , V = e1T P1 e1 + e0T P0 e0 + lk−1 3 4

(5.58)

where ek3 = k3 − kˆ3 and ek4 = k4 − kˆ4 . k3 and k4 are two constants which are defined as k3 = L f2 T −1 2 W1 2 and k4 = L f2 T −1 2 , respectively. The time derivative of V can be shown to be V˙ ≤ −e1T Q 1 e1 + 2e1T P1 A¯ 2 e0 + e0 2 + (L f2 T −1 2 W1 2 − kˆ3 )P1 e1 2 − e0T Q 0 e0 + e0 2 + (L f2 T −1 2 − kˆ4 )H0 C0 e0 2 − ek3 P1 e1 2 − ek4 H0 C0 e0 2 + 2e Tf Γ −1 f˙s + 2εe Tf f s − 2εe Tf e f ≤ −e1T Q 1 e1 + e0T (−Q 0 + 2I )e0 + 2e1T P1 A¯ 2 e0 + 2e Tf Γ −1 f˙s + 2εe Tf f s − 2εe Tf e f .

(5.59)

Since 2X T Y ≤ α1 X T G X + αY T G −1 Y holds for any scalar α > 0 and symmetric positive definite matrix G [21], therefore 2e Tf Γ −1 f˙s ≤ e Tf Ge f + f˙sT Γ −1 G −1 Γ −1 f˙s ≤ e Tf Ge f + ρss λmax (Γ −1 G −1 Γ −1 )

(5.60)

2εe Tf f s ≤ εe f 2 + ερs2 .

(5.61)

Moreover,

Substituting (5.60) and (5.61) into (5.59) gives V˙ ≤ −e1T Q 1 e1 − e0T (Q 0 − 2I )e0 + e Tf (G − ε I )e f + ρss λmax (Γ −1 G −1 Γ −1 ) + ερs2 ≤ −μ1 e1 2 − μ2 e0 2 − μ3 e f 2 + μ4 ≤ −μ5 (e1 2 + e0 2 + e f 2 ) + μ4 .

(5.62)

Further from (5.58) V ≤ λmax (P1 )e1 2 + λmax (P0 )e0 2 + λmax (Γ −1 )e f 2 ≤ max(λmax (P1 ), λmax (P0 ), λmax (Γ −1 ))(e1 2 + e0 2 + e f 2 ) = μ6 (e1 2 + e0 2 + e f 2 ).

(5.63)

Then V˙ ≤ −μ7 V + μ4 .

(5.64)

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5 Robust Estimation of Sensor Faults

For any real constant p and q ∈ R, we have ( p − q)2 ≥

p2 − q 2. 2

(5.65)

Therefore V ≥ λmin (P1 )e1 2 + λmin (P0 )e0 2 + λmin (Γ −1 )e f 2   2 ˆ  f (P )  λmin (P1 ) λ min 0 s ≥ ew1 2 + ew3 2 + λmin (Γ −1 ) − ρs2 . C1 2 C0 2 2

(5.66)

If (ew1 , ew3 , fˆs ) ∈ Ω2 , then V > μ4 /μ7 and consequently V˙ < 0. Therefore it can be concluded that (ew1 , ew3 , fˆs ) is uniformly bounded and converges to Ω1 exponentially at a rate greater than e−μ7 t . This completes the proof.  For error dynamics (5.53) and (5.54), we define the sliding-mode surface as S = {(e1 )|e1 = 0}.

(5.67)

Theorem 5.4 Given System (5.1) with Assumptions 5.1, 5.2, 5.3, and 5.4 and the proposed observers (5.47) and (5.50). Then the error dynamics (5.53) and (5.54) can be driven to the sliding surface (5.67) in finite time and remain on it if the problem of Theorem 5.3 is solvable and the gain η5 satisfies η5 ≥ ( A¯ 2  + L f T −1 W1 )e0  + η6 ,

(5.68)

where η6 is a positive scalar. Proof The proof is similar to the proof of Theorem 5.2 and thus omitted here.



5.5 Simulation Results The effectiveness of the proposed schemes in estimating the sensor faults has been demonstrated by considering an example of a satellite attitude control system model [14, 22, 23], whose dynamics is described as J ω˙ + ω× J ω = 3ω02 ζ × J ζ + u + Td ,

(5.69)

where J = diag{J1 , J2 , J3 } are the moments of inertia of the satellite along principal axes; ω = [ωx ω y ωz ]T is the angular velocity of body-fixed reference frame of satellite with respect to an inertial reference frame expressed in the body-fixed reference frame; ω0 is the constant orbital rate; u = [u 1 u 2 u 3 ]T is the control torque

5.5 Simulation Results

101

vector; Td = [T1d Td2 Td3 ]T is the external disturbance torque; the nonlinear term ζ = [− sin θ sin ϕ cos θ cos ϕ cos θ ]T ; Euler angles ϕ, θ , and ψ are roll, pitch, and yaw attitude angles along principal axes, respectively; and the skew symmetric matrix is given by ⎡ ⎤ 0 −wz w y ω× = ⎣ wz 0 −wx ⎦ . −w y wx 0 For small attitude angles, the dynamics (5.69) becomes ⎧ ⎨ J1 ω˙ x − (J2 − J3 )ω y ωz + 3ω02 (J2 − J3 )ϕ = u 1 + Td1 J2 ω˙ y − (J3 − J1 )ωz ωx + 3ω02 (J1 − J3 )θ = u 2 + Td2 ⎩ J3 ω˙ z − (J1 − J2 )ωx ω y = u 3 + Td3

(5.70)

The kinematic differential equation of an orbiting rigid body can be described as ⎤ ⎡ ⎡ ⎤⎡ ⎤ ⎤ ϕ˙ cos θ sin ϕ sin θ cos ϕ sin θ sin ψ ωx ⎣ θ˙ ⎦ = 1 ⎣ 0 cos ϕ cos θ − sin ϕ cos θ ⎦ ⎣ ω y ⎦ + ω0 ⎣ cos θ cos ψ ⎦ . cos θ cos θ sin θ sin ψ ωz 0 sin ϕ cos ϕ ψ˙ (5.71) ⎡

This nonlinear system can be linearized as ⎧ ⎨ ϕ˙ = ωx + ω0 ψ θ˙ = ω y + ω0 ⎩ ˙ ψ = ωz − ω0 ϕ

(5.72)

Choosing the state variable x as x = [ϕ θ ψ ωx ω y ωz ]T , the satellite attitude control system, when subjected to the system uncertainties Δψ and sensor faults f s , can be put into the form of (5.1) with ⎤ ⎡ ⎤ 0 0 0 0 0 ω0 1 0 0 ⎢ ⎢ 0 0 0 ⎥ 0 0 0 0 1 0⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ 0 ⎥ 0 0 ⎥ 0 0 0 0 1 −ω 0 ⎥, ⎢ ⎥ A=⎢ , B = ⎢ −3ω2 J −1 (J2 − J3 ) ⎢ J −1 0 0 ⎥ 0 0 0 0 0⎥ 0 1 ⎥ ⎢ ⎢ 1 ⎥ −1 −1 ⎣ ⎣ 0 J 0 ⎦ 0 −3ω02 J2 (J1 − J3 ) 0 0 0 0 ⎦ 2 0 0 0 000 0 0 J3−1 ⎡ ⎤ ⎡ ⎤ 0 0 0 0 0 0 ⎢1 ⎥ ⎢ 0 0 ⎥ 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎢0 ⎥ ⎢ 0 0 ⎥ 0 0 0 ⎥ ⎢ ⎥ W = 0.0001 × ⎢ , D = ⎢ 0 J −1 (J 2 − J 3) ⎥ ⎢ 0 0 ⎥, 0 0 1 ⎢ ⎥ ⎢ ⎥ −1 ⎣0 ⎦ ⎣ 0.4 0 ⎦ 0 J2 (J 3 − J 1) 0 0 0.6 0 0 0 J3−1 (J 1 − J 2) ⎡

102

5 Robust Estimation of Sensor Faults ⎡

0 0 0

⎢ ⎢ ⎢ E =⎢ ⎢ J −1 ⎢ 1 ⎣ 0 0

⎤ 0 0 ⎤ ⎡ 0.2 0 ⎥ w0 ⎥ ⎢ ⎥ 0 0.3 ⎥ x5 x6 ⎥ ⎥ , C = I6 . , f (x, t) = 10000 × ⎢ ⎥ ⎣ 0 0 ⎥ x6 x4 ⎦ −1 x4 x5 J2 0 ⎦ 0 J3−1

Note that the satellite is assumed to be running in a small angle maneuver. Therefore the nonlinear function f (x, t) is locally Lipschitz nonlinear, which satisfies the condition (5.3). Δψ represents the lumped system uncertainties which include the external disturbance torques, modeling uncertainties, and noises, and is ¯ = 10−4 . The inertia matrix is assumed to be bounded with a known constant Δψ assumed to be J = diag{18.73, 20.77, 23.63} kg · m2 . The orbital angular velocity ω0 = 0.0015 rad/s. A classical PD controller is designed such that the closed-loop satellite attitude control system is stable. Figure 5.1 shows the control torque in the fault-free condition. The attitude angles and angular velocities are measured by star sensors and gyros, respectively, which are unavoidably susceptible to possible faults in the severe space environment. In this simulation, gyros are assumed to be prone to potential faults. The mathematical model of a gyro is expressed as that in [24]: ω˜ = ω + β + n,

(5.73)

where ω˜ is the measured angular velocity, ω is the actual angular velocity, β is the gyro drift, and n is the zero-mean Gaussian white noise. The drift β can be 0.4

Control torques (Nm)

0.2 0 −0.2 −0.4 −0.6 −0.8 −1

u1 u

−1.2 −1.4

u

0

50

100

Time (s) Fig. 5.1 The control torques in the fault-free condition

150

2 3

200

5.5 Simulation Results

103

regarded as the net effect of several systematic error sources such as scale factor errors, non-orthogonality, and misalignment. Since a gyro fault will essentially result in an increase of the drift, we take this additive bias term as the sensor fault. In this simulation, it is assumed that only the second and third gyros are prone to faults, while other sensors are fault free. We define the sensor fault vector as f s = [ f sT1 , f sT2 ]T with ⎧ ⎪ , t ≤ 65 s ⎨ 0 rad/s f s1 = 0.05 rad/s , 65 s < t < 140 s ⎪ ⎩ 0 rad/s , t ≥ 140 s ⎧ ⎪ , t ≤ 75 s ⎨ 0 rad/s f s2 = 0.001(t − 75) rad/s , 75 s < t ≥ 150 s ⎪ ⎩ 0 rad/s , t ≥ 150 s It is worth noting that f s1 is a pulse-wise signal and f s2 is a ramp-wise signal. Most common faults in the satellite attitude estimation system such as electronic short circuit, device saturation, data losses in the on-board computer or in the GPS, and gradual creation of bias in gyro measurement can be modeled as these two types of signals [25, 26]. Moreover, it should be emphasized that f s2 is an incipient fault and this type of fault is almost unnoticeable during its initial stage and the effects to residuals are most likely to be concealed by system uncertainties. An incipient fault in a single subsystem may have big impact on the whole closed-loop satellite system and cause potentially catastrophic damage. Hence, it is necessary to detect and isolate incipient faults as early as possible to maintain the reliability of the system. Introduce two nonsingular transformation matrices T and S with ⎤ 2.1962 0.2444 0.6411 0.4000 0.0000 0.0000 ⎢ −0.5935 0.3444 0.6923 0.1000 −0.0000 −0.0000 ⎥ ⎥ ⎢ ⎢ 1.6374 0.2481 0.7141 0.3000 −0.0000 −0.0000 ⎥ ⎥ ⎢ T =⎢ ⎥ ⎢ 0.2877 0.1632 0.2497 0.0000 −0.6777 −1.7701 ⎥ ⎥ ⎢ ⎣ 1.6303 −0.5390 −0.4255 0.0000 2.2390 3.0161 ⎦ −0.9838 1.8366 1.9040 0.0000 −7.6293 −13.4976 ⎡ ⎤ 0.2000 0.5000 1.0000 0.3000 0 0 ⎢ −0.3000 0.6000 0.1000 0.6400 ⎥ 0 0 ⎢ ⎥ ⎢ 1.0000 0.2000 0.4700 0.2300 ⎥ 0 0 ⎢ ⎥ S=⎢ ⎥ ⎢ 1.4372 −0.0722 −0.0451 −0.0000 0.3000 0.3200 ⎥ ⎢ ⎥ ⎣ 8.3172 0.1204 0.0282 −0.0000 −0.5000 −0.2000 ⎦ −2.7387 −0.0481 −0.0987 0.0000 0.2000 0.7000 ⎡

104

5 Robust Estimation of Sensor Faults

such that in the new coordinate the system matrices become ⎤ 12.9325 −2.4458 −9.1072 33.7155 −20.5905 −9.0701 ⎢ −3.1881 1.1399 1.8924 −10.6851 6.3678 2.7729 ⎥ ⎥ ⎢ ⎥ ⎢ 9.6058 −1.7747 −6.7583 24.6382 −15.1766 −6.6753 ⎥ ⎢ T AT −1 = ⎢ ⎥ ⎢ 1.8893 −0.1354 −1.5149 4.1943 −2.4929 −1.1256 ⎥ ⎥ ⎢ ⎣ 8.5201 −2.8346 −4.9811 26.2280 −16.4049 −7.0738 ⎦ −2.6638 3.9189 −1.0340 −17.7328 12.1368 4.8965 ⎡ ⎤ 0.9391 1.2724 −0.6762 −0.0000 0.0000 0.0000 ⎢ 9.5983 3.4392 −11.8108 −0.0000 0.0000 0.0000 ⎥ ⎢ ⎥ ⎢ 0.5862 0.2308 −0.0918 −0.0000 0.0000 0.0000 ⎥ ⎢ ⎥ −1 SC T = ⎢ ⎥ ⎢ 0.0000 0.0000 −0.0000 −2.5252 1.7486 0.6982 ⎥ ⎢ ⎥ ⎣ 0.0000 0.0000 −0.0000 −21.4836 12.2554 5.5708 ⎦ −0.0000 −0.0000 0.0000 5.2212 −3.5207 −1.5233 ⎡ ⎡ ⎤ ⎤ 0.0214 0.0489 0.1923 0.0214 0.0000 0.0000 ⎢ 0.0053 0.0689 0.2077 ⎥ ⎢ 0.0053 −0.0000 −0.0000 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ 0.0160 0.0496 0.2142 ⎥ ⎢ 0.0160 −0.0000 −0.0000 ⎥ TB =⎢ ⎥, T E = ⎢ ⎥ ⎢ 0.0000 0.0000 0.0000 ⎥ ⎢ 0.0000 −0.0326 −0.0749 ⎥ ⎢ ⎢ ⎥ ⎥ ⎣ 0.0000 0.0000 0.0000 ⎦ ⎣ 0.0000 0.1078 0.1276 ⎦ 0.0000 −0.3673 −0.5712 0.0000 −0.0000 −0.0000 ⎡ ⎤ 0 0 ⎢ ⎥ 0 0 ⎢ ⎥ ⎢ ⎥ 0 0 ⎢ ⎥ SD = ⎢ ⎥. ⎢ 0.1200 0.1920 ⎥ ⎢ ⎥ ⎣ −0.2000 −0.1200 ⎦ 0.0800 0.4200 ⎡

Imposing the stability conditions in Theorem 5.1 to the transformed system and formulating the problem in an LMI framework gives the following values for the design parameters of the developed SMO and AO: γ1 = 2.6632e − 9, γ2 = 0.0021, ⎡ ⎤ −216.1744 −152.9506 −197.5088 As1 = 1e3 × ⎣ −182.2166 −176.8936 −185.8210 ⎦ , −241.6414 −186.9670 −228.0913 ⎡ ⎤ 15.8001 5.7527 −18.3682 P1 = ⎣ 5.7527 3.1114 −7.5161 ⎦ , −18.3682 −7.5161 22.1306

5.5 Simulation Results

105

⎤ 90.4332 16.4998 176.7723 83.2541 −57.0233 174.3784 38.4622 297.9475 140.3236 −96.1121 ⎥ ⎥ 38.4622 16.2827 54.5498 25.6912 −17.5967 ⎥ ⎥, 297.9475 54.5498 583.3375 249.0196 −175.3884 ⎥ ⎥ 140.3236 25.6912 249.0196 160.6261 −80.2027 ⎦ −96.1121 −17.5967 −175.3884 −80.2027 97.6945 ⎤ ⎡ −3.4061 −19.1455 4.7231 ⎢ −0.9839 25.7624 −6.7388 ⎥ ⎥ ⎢ ⎢ 2.6196 −3.0986 0.8268 ⎥ ⎥ ⎢ L0 = ⎢ ⎥, ⎢ 1.0907 −5.3703 1.5499 ⎥ ⎥ ⎢ ⎣ 0.4276 −2.7923 0.9179 ⎦ −0.1750 1.6784 0.0280   6.1655 −8.6591 2.8095 F0 = , 8.4553 −5.1485 16.9815 ⎤ ⎡ −0.0032 −0.0015 0.0010 ⎢ −0.0000 0.0000 −0.0000 ⎥ ⎥ ⎢ H0 = ⎢ ⎥. ⎣ 0.0031 0.0015 −0.0010 ⎦ 0.0013 0.0006 −0.0004 The MATLAB code for computing these parameters is given in the following program chapter5_lmi.m. ⎡

100.5731 ⎢ 90.4332 ⎢ ⎢ 16.4998 P0 = ⎢ ⎢ 176.7723 ⎢ ⎣ 83.2541 −57.0233

File chapter5_lmi.m

106

5 Robust Estimation of Sensor Faults

5.5 Simulation Results

107

108

5 Robust Estimation of Sensor Faults

For AO, we further select  Γ =

 40 0 , ε = 0.01 0 40

(5.74)

The system is simulated in the program chapter5.mdl and the plots are obtained by running the file chapter5_plot.m. In the simulation, we have selected the initial states as x(0) = [0.0112, −0.0745, 0.0235, 0.0316, 0.0424, 0.056]T , zˆ 1 (0) = [0, 0, 0]T , zˆ 0 (0) = [0, 0, 0, 0, 0, 0]T , fˆs1 (0) = 0, fˆs2 (0) = 0, kˆ1 (0) = 0, kˆ2 (0) = 0, kˆ3 (0) = 0 and kˆ4 (0) = 0, and δ = 0.018. Figures 5.2, 5.3, 5.4, 5.5, 5.6, and 5.7 illustrate the trajectories of the actual states and their estimates provided respectively by SMObased method and AO-based method. It can be seen from the figures that both schemes can estimate states accurately, before and after the occurrence of sensor faults. When sensor fault occurs at 100 and 200s, overshoots are observed in the trajectories of state estimates. The results of sensor FE are illustrated in Figs. 5.8 and 5.9. The

5.5 Simulation Results

109

Roll angle (rad)

0.06 Actual x1

0.04

Estimation by SMO

0.02 0 −0.02

0

50

100

150

200

Roll angle (rad)

0.06 Actual x1

0.04

Estimation by AO

0.02 0 −0.02

0

50

100

150

200

Time (s)

Pitch angle (rad)

Pitch angle (rad)

Fig. 5.2 ϕ and its estimates by SMO-based method and AO-based method

0.05 0 −0.05 Actual x2

−0.1

Estimation by SMO

−0.15 0

50

100

150

200

0.05 0 −0.05 Actual x2

−0.1

Estimation by AO

−0.15 0

50

100

150

200

Time (s) Fig. 5.3 θ and its estimates by SMO-based method and AO-based method

performance of both methods on estimating sensor faults is found to be satisfactory. The AO-based method has a longer oscillation but less bias. For the objective of comparison, the descriptor system approach proposed in [2] and the robust nonlinear observer proposed in [27] are carried out for this satellite attitude control system. Unfortunately, it shows that for this particular system, there does not exist any solution to the LMIs as proposed in these two papers. Therefore,

5 Robust Estimation of Sensor Faults

Yaw angle (rad)

Yaw angle (rad)

110 0.15

Actual x3

0.1

Estimation by SMO

0.05 0 −0.05 0

50

100

150

200

0.15 Actual x3

0.1

Estimation by AO

0.05 0 −0.05 0

50

100

150

200

Time (s) Fig. 5.4 ψ and its estimates by SMO-based method and AO-based method

ω x(rad/s)

0.06 Actual x4

0.04

Estimation by SMO

0.02 0

−0.02 0

50

100

150

200

ω x(rad/s)

0.15 Actual x4

0.1

Estimation by AO

0.05 0

−0.05 0

50

100

150

200

Time (s) Fig. 5.5 ωx and its estimates by SMO-based method and AO-based method

the sensor FE schemes developed in [2, 27] are no longer applicable for the considered system. However, our methods work well, which shows the significance of our approach.

5.5 Simulation Results

111

0.05

ω y (rad/s)

Actual x 5 Estimation by SMO

0

−0.05 0

50

100

150

200

0.05

ω y (rad/s)

Actual x 5 Estimation by AO

0

−0.05 0

50

100

150

200

Time (s) Fig. 5.6 ω y and its estimates by SMO-based method and AO-based method

ωz(rad/s)

0.15 Actual x 6 Estimation by SMO

0.1 0.05 0 −0.05

0

50

100

150

200

ωz(rad/s)

0.15 Actual x 6 Estimation by AO

0.1 0.05 0 −0.05

0

50

100

150

Time (s) Fig. 5.7 ωz and its estimates by SMO-based method and AO-based method

200

112

5 Robust Estimation of Sensor Faults 0.3 Actual f s1 Estimation by SMO Estimation by AO

0.25

Magnitude (rad/s)

0.2 0.15 0.1 0.05 0 −0.05 −0.1

0

50

100

150

200

Fig. 5.8 Sensor fault f s1 and its estimated value fˆs1 0.3 Actual f s2 Estimation by SMO Estimation by AO

Magnitude (rad/s)

0.25 0.2 0.15 0.1 0.05 0 −0.05

0

50

100

Fig. 5.9 Sensor fault f s2 and its estimated value fˆs2

150

200

5.6 Conclusions

113

5.6 Conclusions In this chapter, we propose two schemes to estimate sensor faults for uncertain Lipschitz nonlinear systems. The proposed schemes essentially transform the original system into two subsystems where subsystem-1 includes system uncertainties but is free from sensor faults and subsystem-2 has sensor faults but without uncertainties. Using the integral observer-based approach, sensor faults in subsystem-2 are transformed into actuator faults. In the first scheme, two SMOs are designed. One of which is used to eliminate the effect of system uncertainties, while the other one is used to estimate sensor faults. In the second scheme, we design an SMO for subsystem-1. The difference is that an AO is designed for subsystem-2 to estimate sensor faults instead of using an SMO in the first scheme. Adaptation laws are integrated into both schemes to deal with the situation when the Lipschitz constant is unknown or too large. An example of a satellite attitude control system has been used to demonstrate the effectiveness of the proposed sensor FE schemes. Simulation results confirm that the both methods can accurately estimate sensor faults, including incipient sensor faults, in the presence of system uncertainties. It is worth noting that the sensor faults are taken as the form of actuator faults using an integral observer; therefore, the main results developed in this chapter can be directly deduced for actuator FE.

References 1. Bonivento C, Isidoria A, Marconia L, Paolia A (2004) Implicit fault-tolerant control: application to induction motors. Automatica 40(3):355–371 2. Gao ZW, Ding SX (2007) Sensor fault reconstruction and sensor compensation for a class of nonlinear state-space systems via a descriptor system approach. IET Control Theory Appl 1:578–585 3. Tan CP, Edwards C (2007) A robust sensor fault tolerant control scheme implemented on a crane. Asian J Control 9(3):340–344 4. Tan CP, Edwards C (2002) Sliding mode observers for detection and reconstruction of sensor faults. Automatica 38:1815–1821 5. Tan CP, Edwards C (2003) Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int J Robust Nonlinear Control 13:443–463 6. Alwi H, Edwards C, Tan CP (2009) Sliding mode estimation schemes for incipient sensor faults. Automatica 45:1679–1685 7. Fridman L, Levant A, Davila J (2007) Observation of linear systems with unknown inputs via high-order sliding-modes. Int J Syst Sci 38:773–791 8. Yeu TK, Kim HS, Kawaji S (2005) Fault detection, isolation and reconstruction for descriptor systems. Asian J Control 7(4):356–367 9. Raoufi R, Marquez HJ, Zinober ASI (2010) H∞ sliding mode observer for uncertain nonlinear Lipschitz systems with fault estimation synthesis. Int J Robust Nonlinear Control 20:1785– 1801 10. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 11. Zhang K, Jiang B, Cocquempot V (2008) Adaptive observer-based fast fault estimation. Int J Control Autom Syst 6:320–326

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12. Zhang K, Jiang B, Shi P (2009) Fast fault estimation and accommodation for dynamical systems. IET Control Theory Appl 3:189–199 13. Raoufi R, Marquez HJ (2010) Simultaneous sensor and actuator fault reconstruction and diagnosis using generalized sliding mode observers. In: Proceedings of American control conference, pp 7016–7021 14. Gao C, Zhao Q, Duan G (2013) Robust actuator fault diagnosis scheme for satellite attitude control systems. J Frankl Inst, 2560–2580. http://dx.doi.org/10.1016/j.jfranklin.2013.02.021 15. Jiang B, Staroswiecki M, Cocquempot V (2006) Fault accommodation for nonlinear dynamic systems. IEEE Trans Autom Control 51:1578–1583 16. Chen J, Patton RJ (1999) Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers, Boston 17. Wang H, Daley S (1996) Actuator fault diagnosis: an adaptive observer-based technique. IEEE Trans Autom Control 41(7):1073–1078 18. Raghavan R (1998) Observer for a class of nonlinear systems. IEEE Trans Autom Control 43:397–401 19. Zhu F, Han Z (2002) A note on observers for lipschitz nonlinear systems. IEEE Trans Autom Control 47:1751–1754 20. Utkin VI (1992) Sliding modes in control optimization. Springer, Berlin 21. Jiang B, Wang JL, Soh YC (2002) An adaptive technique for robust diagnosis of faults with independent effect on system outputs. Int J Control 75:792–802 ˇ 22. Nguyen HQ, Celikovsky S (2012) Fault diagnosis scheme for nonlinear stochastic systems with time-varying fault: application to the rigid spacecraft control. Cybern Phys 1:179–187 23. Jia QX, Zhang YC, Chen W, Shen Y (2012) A novel fault reconstruction approach to satellite attitude control system via learning unknown input observer and H∞ techniques. In: Proceedings of the American control conference, 5160–5162 24. Bae J, Kim Y (2010) Attitude estimation for satellite fault tolerant system using federated unscented Kalman filter. Int J Aeronaut Space 11:80–86 25. Chen RH, Ng HK, Speyer JL, Guntur LS (2006) Health monitoring of a satellite system. J Guid Control Dyn 29:593–605 26. Ahmadi B, Namvar M (2012) Robust detection and isolation of failures in satellite attitude sensors and gyro. Robotica 30:1157–1166 27. Khosrowjerdi MJ (2011) Robust sensor fault reconstruction for Lipschitz nonlinear systems. Math Probl Eng. doi:10.1155/2011/146038

Chapter 6

Simultaneous Estimation of Actuator and Sensor Faults Using SMO and AO

6.1 Introduction Among various types of FE methods that have been developed, SMO-based FE has been proven to be an effective way to estimate/reconstruct faults for many systems [1–8]. These aforementioned researches deal separately with actuator or sensor faults. However, in many practical systems, actuators and sensors are simultaneously prone to faults. Misinterpretation of actuator and sensor faults may cause a high rate of false alarm and unnecessary maintenance. Therefore, it would be desirable to consider actuator and sensor faults under a unified framework. In [9], a new descriptor fuzzy SMO approach was presented to obtain the simultaneous estimates of system states and faults. In [10], the reconstruction of actuator and sensor faults was carried out by means of a sliding-mode high-order differentiator. By taking the output disturbance as an auxiliary state vector, an augmented descriptor was constructed and a robust H∞ SMO was developed to estimate faults as well as states in [11]. In this chapter, two new schemes are developed for uncertain Lipschitz nonlinear systems that are subject to actuator and sensor faults simultaneously. The proposed schemes essentially transform the original system into two subsystems (subsystems 1 and 2), where subsystem-1 includes the effects of actuator faults but is free from sensor faults and subsystem-2 only has sensor faults. Using an integral observerbased approach [12, 13], sensor faults in subsystem-2 are transformed such that they appear as actuator faults. The augmented subsystem-2 is further transformed by a linear coordinate transformation such that a specific structure can be imposed to the sensor fault distribution matrix. In the first scheme, it is assumed that the fault distribution matrix satisfies the matching condition. Then two SMOs are designed to estimate actuator and sensor faults, respectively. However, this assumption is restrictive and sometimes it is difficult to find such matrices to satisfy both the Lyapunov equation and matching condition. To overcome the restriction imposed by the matching condition, many attempts have been made using high-order SMOs © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_6

115

116

6 Simultaneous Estimation of Actuator and Sensor…

[14–16]. In the second scheme, we design an adaptive observer (AO) to estimate sensor faults instead of an SMO, to deal with the situation when the matching condition is not satisfied. For both schemes, the sufficient conditions for the existence of the proposed observers with H∞ performance are derived and expressed as an LMI optimization problem such that the upper bounds of the state and fault estimation errors can be minimized. The rest of the chapter is organized as follows: Sect. 6.2 briefly describes the mathematical preliminaries required for designing observers. Section 6.3 proposes the first scheme based on the matching condition. The sufficient conditions of the stability of the proposed observers are derived and expressed in LMIs. If the matching condition is not satisfied, the second scheme is proposed in Sect. 6.4. The results of simulation are shown in Sect. 6.5 with conclusions in Sect. 6.6.

6.2 Problem Formulation Consider a nonlinear system described by 

x˙ (t) = Ax(t) + f (x, t) + B(u(t) + fa (t)) + EΔψ(t) y(t) = Cx(t) + Dfs (t)

(6.1)

where x ∈ R n , u ∈ R m , and y ∈ R p denote, respectively, the vector of states, inputs, and outputs. fa ∈ R m represents the vector of actuator faults and fs ∈ R q represents the vector of sensor faults. Δψ represents the unknown bounded uncertainties and the nonlinear continuous term f (x, t) ∈ R n is assumed to be known. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R p×q , and E ∈ R n×r (p ≥ q) are known constant matrices with B and D being of full rank. Before starting the main results of this chapter, the following assumptions are made: Assumption 6.1 rank(CB) = rank(B). Assumption 6.2 For every complex number s with nonnegative real part,  rank

 sIn − A B = n + rank(B). C 0

(6.2)

Assumption 6.3 The nonlinear function f (x, t) is Lipschitz about x uniformly, that is, ∀x, xˆ ∈ R n , (6.3) f (x, t) − f (ˆx , t) ≤ Lf x − xˆ , where Lf is the Lipschitz constant. Assumption 6.4 The actuator fault fa and sensor fault fs are norm bounded, i.e., fa  ≤ ρa , fs  ≤ ρs . fs is differentiable after its occurrence (the continuity at the fault occurrence time is not required), and f˙s belongs to L2 [0, ∞).

6.2 Problem Formulation

117

Lemma 6.1 Under Assumption 6.1, there exist state and output transformations 

h h= 1 h2





     x1 w1 y =T , w= =S 1 , x2 w2 y2

(6.4)

such that in the new coordinate, the system matrices become      B1 E1 A1 A2 , TB = , , TE = A3 A4 0 E2     0 C1 0 , SD = , = 0 C4 D2

TAT −1 = SCT −1





   T1 S1 ∈ R n×n , S = ∈ R p×p , T1 ∈ R m×n , S1 ∈ R m×p , h1 ∈ R m , T2 S2 w1 ∈ R m , A1 ∈ R m×m , A4 ∈ R (n−m)×(n−m) , B1 ∈ R m×m , E1 ∈ R m×r , C1 ∈ R m×m , C4 ∈ R (p−m)×(n−m) and D2 ∈ R (p−m)×q . B1 and C1 are invertible. Therefore, in the new coordinate, System (6.1) is converted into the following two subsystems:

where T =



h˙ 1 = A1 h1 + A2 h2 + T1 f (T −1 h, t) + B1 (u + fa ) + E1 Δψ w1 = C1 h1  h˙ 2 = A3 h1 + A4 h2 + T2 f (T −1 h, t) + E2 Δψ w2 = C4 h2 + D2 fs

(6.5) (6.6)

t For Subsystem (6.6), define a new state h3 = 0 w2 (τ )dτ so that h˙3 = C4 h2 +D2 fs . Then an augmented system with the new state h3 is given as 

h˙ 0 = A0 h0 + A˜ 3 h1 + T¯ 2 f (T −1 h, t) + D0 fs + E0 Δψ w3 = C0 h0

(6.7)

    h2 A4 0 ∈ R (n+p−2m)×(n+p−2m) , ∈ R n+p−2m , w3 ∈ R p−m , A0 = where h0 = h C4 0   3     0 E2 (n+p−2m)×q ˜A3 = A3 ∈ R (n+p−2m)×m , D0 = ∈ R ∈ , E0 = 0 0 D2     T2 ∈ R (n+p−2m)×n . R (n+p−2m)×r , C0 = 0 Ip−m ∈ R (p−m)×(n+p−2m) and T¯ 2 = 0 Lemma 6.2 The pair (A0 , C0 ) is observable if Assumption 6.2 holds. Lemma 6.2 indicates that there exists a matrix L0 ∈ R (n+p−2m)×(p−m) such that A0 − L0 C0 is stable, and thus for any Q0 ∈ R (n+p−2m)×(n+p−2m) > 0, the Lyapunov equation (6.8) (A0 − L0 C0 )T P0 + P0 (A0 − L0 C0 ) = −Q0 , has unique solution P0 ∈ R (n+p−2m)×(n+p−2m) > 0.

118

6 Simultaneous Estimation of Actuator and Sensor…

Partition P0 and Q0 as  P0 =

   P01 P02 Q01 Q02 , Q . = 0 T T P02 P03 Q02 Q03

(6.9)

It follows from P0 > 0 and Q0 > 0 that P01 ∈ R (n−m)×(n−m) > 0, P03 ∈ > 0, Q01 ∈ R (n−m)×(n−m) > 0, and Q03 ∈ R (p−m)×(p−m) > 0. If P0 R and Q0 have the structure as shown in (6.9), then the following conclusion is obvious: (p−m)×(p−m)

−1 Lemma 6.3 The matrix A4 + P01 P02 C4 is stable, if Lyapunov equation (6.8) is satisfied.

Proof According to the structure of A0 , C0 , P0 , and Q0 , it is easy to see that the first n − m columns of A0 − L0 C0 are independent of L0 . After the block multiplication to (6.8), the following equation can be obtained: T + P01 A3 + P02 C4 = −Q01 . AT4 P01 + C4T P02

This equation can be rewritten as −1 −1 P02 C4 )T P01 + P01 (A4 + P01 P02 C4 ) = −Q01 . (A4 + P01 −1 Since P01 > 0 and Q01 > 0, it follows that A4 + P01 P02 C4 is stable from the Lyapunov theory. This completes the proof. 

Remark 6.1 Assumptions 6.1 and 6.2 are necessary and sufficient conditions for the design of a stable sliding-mode when the system has matched uncertainties or unknown inputs [17]. It is seen from Lemma 6.1 that Assumption 6.1 ensures the existence of coordinate transformations T and S, such that in the new coordinate actuator and sensor faults can be completely separated. It follows from Assumption 6.2 that the pair (A0 , C0 ) is observable, which provides the necessary condition for the existence of an observer for System (6.7). Assumption 6.3 states that the considered nonlinear system is Lipschitz. Assumption 6.4 requires the boundedness of actuator and sensor faults, which is a general assumption in observer-based fault diagnosis [2, 7, 18].

6.3 SMOs-Based FE Scheme In this section, we will introduce the first FE scheme which consists of two SMOs. One SMO is designed to estimate actuator faults while the other one is designed to estimate sensor faults.

6.3 SMOs-Based FE Scheme

119

6.3.1 Design of Observers By taking the state transformation of z := T0 h0 = [z1T z2T ]T with  −1 P02 In−m P01 , T0 = 0 Ip−m 

(6.10)

where z1 ∈ R n−m and z2 ∈ R p−m , Subsystem (6.7) can be transformed into 

z˙ = Az z + T0 A˜ 3 h1 + T0 T¯ 2 f (T −1 , t) + Dz fs + Ez Δψ w3 = Cz z

(6.11)

where  −1 −1 −1 −1 P02 C4 −A4 P01 P02 − P01 P02 C4 P01 P02 A4 + P01 , −1 C4 −C4 P01 P02   −1     E2 P01 P02 D2 Dz = , Ez = (6.12) , Cz = 0 Ip−m . 0 D2 

Az =

A¯ 1 A¯ 3

A¯ 2 A¯ 4





=

Lemma 6.4 If there exists an arbitrary matrix F0 ∈ R q×(p−m) such that D0T P0 = F0 C0 ,

(6.13)

P02 D2 = 0, F0 = D2T P¯ 03 .

(6.14)

then we have

(6.15)

Proof The Lyapunov matrix P0 in the new coordinate can be proved to have the quadratic form as Pz =

(T0T )−1 P0 T0 −1

 P01 0 , = 0 P¯ 03 

(6.16)

T −T where P¯ 03 = −P02 P01 P02 + P03 . Substituting P0 = T0T Pz T0 , C0 = Cz T0 and −1 D0 = T0 Dz into the matching condition (6.13) yields

DzT Pz = F0 Cz .

(6.17)

From the structure of Dz , Cz , and Pz , the conclusions P02 D2 = 0 and F0 = D2T P¯ 03 can be obtained. This completes the proof. 

120

6 Simultaneous Estimation of Actuator and Sensor…

 It follows from Lemma 6.4 that in the new coordinate system z, Dz =

 0 . D2

Therefore, Systems (6.5) and (6.11) can be rewritten respectively as 

−1 h˙ 1 = A1 h1 + A2 z1 − A2 P01 P02 w3 + T1 f (T −1 h, t) + B1 (u + fa ) + E1 Δψ w1 = C1 h1

⎧ ⎨ z˙1 = A¯ 1 z1 + A¯ 2 z2 + A3 h1 + T2 f (T −1 h, t) + E2 Δψ z˙ = A¯ 3 z1 + A¯ 4 z2 + D2 fs ⎩ 2 w3 = z2

(6.18) (6.19)

For Systems (6.18) and (6.19), we construct the following two SMOs, respectively: ⎧ ˙ −1 ⎪ ˆ t) + B1 (u + ν1 ) ⎨ hˆ 1 = A1 hˆ 1 + A2 zˆ1 − A2 P01 P02 w3 + T1 f (T −1 h, −1 s + (A1 − A1 )C1 (w1 − wˆ 1 ) ⎪ ⎩ wˆ 1 = C1 hˆ 1 ⎧ ˆ t) ⎨ z˙ˆ1 = A¯ 1 zˆ1 + A¯ 2 w3 + A3 C1−1 w1 + T2 f (T −1 h, zˆ˙ = A¯ 3 zˆ1 + A¯ 4 zˆ2 + (A¯ 4 − L)(w3 − wˆ 3 ) + D2 ν2 ⎩ 2 wˆ 3 = zˆ2

(6.20)

(6.21)

where hˆ 1 , zˆ1 , zˆ2 , wˆ 1 , and wˆ 3 denote, respectively, the estimated h1 , z1 , z2 , w1 , and w3 . As1 ∈ R m×m is a stable matrix and L ∈ R (p−m)×(p−m) is the observer gain. It is worth −1 P02 w3 ) and does not represent noting that hˆ is defined as hˆ := col(C1−1 S1 y, zˆ1 − P01 ˆ ˆ the state estimate vector col(h1 , h2 ). The discontinuous output error injection terms ν1 and ν2 are defined by ⎧ ⎨(ρ + η ) B1T P1 (C1−1 S1 y−hˆ 1 ) if C −1 S y − hˆ = 0 a 1 1 1 1 B1T P1 (C1−1 S1 y−hˆ 1 ) ν1 = ⎩0 otherwise T ¯ D P (w −wˆ ) (ρs + η2 ) D2T P¯ 03 (w3 −wˆ 3 ) if w3 − wˆ 3 = 0 3 3 2 03 ν2 = 0 otherwise

(6.22)

(6.23)

where P1 ∈ R m×m is the symmetric definite Lyapunov matrix for As1 , and η1 and η2 are two positive scalars which are to be determined. If the state estimation errors are defined as e1 = h1 − hˆ 1 , e2 = z1 − zˆ1 , and e3 = z2 − zˆ2 , then their dynamics after the occurrence of faults can be obtained as

6.3 SMOs-Based FE Scheme

121

 ˆ t) + B1 (fa − ν1 ) + E1 Δψ, e˙ 1 = As1 e1 + A2 e2 + T1 f (T −1 h, t) − f (T −1 h,

 ˆ t) + E2 Δψ, e˙ 2 = A¯ 1 e2 + T2 f (T −1 h, t) − f (T −1 h, e˙ 3 = A¯ 3 e2 + Le3 + D2 (fs − ν2 ).

(6.24) (6.25) (6.26)

Define the controlled estimation error r as ⎡

⎤ e1 r = He = H ⎣ e2 ⎦ , e3

(6.27)

where H is a prespecified weight matrix and assumed to have full rank: ⎡

⎤ H1 0 0 H := ⎣ 0 H2 0 ⎦ . 0 0 H3

(6.28)

Consider the following worst-case performance measure: H∞ :=

sup

ΔψL2 =0

r2L2 Δψ2L2

.

(6.29)

We now present Theorem 6.1 which establishes the sufficient conditions for the existence of the proposed SMOs in the form of (6.20) and (6.21) with a prescribed H∞ performance H∞ ≤ μ, where μ is a small positive constant. In other words, the H∞ gain of the transfer function from the system uncertainty Δψ to the estimation √ error r is bounded by μ. Theorem 6.1 Consider System (6.1) with Assumptions 6.1–6.4. Given a positive T T > 0, P¯ 03 = P¯ 03 >0 scalar μ, if there exist matrices As1 , L, P1 = P1T > 0, P01 = P01 and P02 , and positive scalars α1 and α2 such that P02 D2 = 0, ⎤ ⎡ P1 A2 0 P1 E1 Π1 + H1T H1 ⎢ AT2 P1 Π2 + H2T H2 C4T P¯ 03 P01 E2 ⎥ ⎥ < 0, Λ := ⎢ ⎣ 0 P¯ 03 C4 Π3 + H3T H3 0 ⎦ T2T P01 0 −μIr T1T P1

(6.30)

(6.31)

T where Π1 = As1 T P1 + P1 As1 + α11 P1 T1 T1T P1 , Π2 = AT4 P01 + P01 A4 + C4T P02 + 1 T 2 −1 2 T¯ ¯ P02 C4 + α2 P01 T2 T2 P01 + (α1 + α2 )Lf T  In−m and Π3 = P03 L + L P03 , then the estimation error dynamics are asymptotically stable with the prescribed H∞ √ tracking performance rL2 ≤ μΔψL2 .

122

6 Simultaneous Estimation of Actuator and Sensor…

Proof It follows from Lemma 6.4 that if there exists a matrix F0 such that D0T P0 = F0 C0 , then we have P02 D2 = 0. Based on the quadratic form of Pz , we consider the Lyapunov function as V = V1 + V2 + V3 ,

(6.32)

where V1 = eT1 P1 e1 , V2 = eT2 P01 e2 , and V3 = eT3 P¯ 03 e3 . The time derivative of V1 along the trajectories of error dynamics (6.24) can be shown to be V˙ 1 = eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A2 e2 + 2eT1 P1 E1 Δψ

 ˆ t) + 2eT1 P1 B1 (fa − ν1 ). + 2eT1 P1 T1 f (T −1 h, t) − f (T −1 h, −1 From the fact that hˆ := col(C1−1 S1 y, zˆ1 − P01 P02 w3 ), it follows that

   −1 ˆ t) f (T h, t) − f (T −1 h,  ≤ Lf T −1 e2 .

(6.33)

Since for any scalar α > 0, the inequality 2X T Y ≤ α1 X T X + αY T Y holds [6], then 1 V˙ 1 ≤ eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A2 e2 + 2eT1 P1 E1 Δψ + T1T P1 e1 2 α1 ˆ t)2 + 2eT P1 B1 (fa − ν1 ) + α1 f (T −1 h, t) − f (T −1 h, 1

1 ≤ eT1 (As1 T P1 + P1 As1 + P1 T1 T1T P1 )e1 + 2eT1 P1 A2 e2 + 2eT1 P1 E1 Δψ α1 + α1 Lf2 T −1 2 e2 2 + 2eT1 P1 B1 (fa − ν1 ). (6.34) From (6.22), it is easy to show that eT1 P1 B1 (fa − ν1 ) = eT1 P1 B1 fa − (ρa + η1 )

B1T P1 e1 2 B1T P1 e1 

≤ −η1 B1T P1 e1  < 0.

(6.35)

Therefore   1 V˙ 1 ≤ eT1 As1 T P1 + P1 As1 + P1 T1 T1T P1 e1 + 2eT1 P1 A2 e2 α1 + 2eT1 P1 E1 Δψ + α1 Lf2 T −1 2 e2 2 .

(6.36)

Similarly, the derivatives of V2 and V3 with respect to time can be obtained as



 ˆ t) + 2eT P01 E2 Δψ V˙ 2 = eT2 A¯ T1 P01 + P01 A¯ 1 e2 + 2eT2 P01 T2 f (T −1 h, t) − f (T −1 h, 2

6.3 SMOs-Based FE Scheme

123

  1 T ≤ eT2 AT4 P01 + P01 A4 + P02 C4 + C4T P02 + P01 T2 T2T P01 + α2 Lf2 T −1 2 In−m e2 α2 + 2eT2 P01 E2 Δψ,

(6.37)

V˙ 3 = eT3 (P¯ 03 L + L T P¯ 03 )e3 + 2eT3 P¯ 03 C4 e2 + 2eT3 P¯ 03 D2 (fs − ν2 ) ≤ eT3 (P¯ 03 L + L T P¯ 03 )e3 + 2eT3 P¯ 03 C4 e2 − 2l2 D2T P¯ 03 e3  ≤ eT3 (P¯ 03 L + L T P¯ 03 )e3 + 2eT3 P¯ 03 C4 e2 .

(6.38)

Therefore, the derivative of V can be obtained from V˙ = V˙ 1 + V˙ 2 + V˙ 3 ⎡ ⎤T ⎡ ⎤⎡ ⎤ Π1 P1 A2 0 e1 e1 ≤ ⎣ e2 ⎦ ⎣ AT2 P1 Π2 C4T P¯ 03 ⎦ ⎣ e2 ⎦ + 2eT1 P1 E1 Δψ + 2eT2 P01 E2 Δψ. e3 e3 0 P¯ 03 C4 Π3    W

(6.39) If there exists a feasible solution to (6.31), then it is straightforward that W < 0. Thus V˙ (t) < 0 when Δψ = 0. This implies that e → 0 as t → ∞. Therefore the error dynamics are asymptotically stable when Δψ = 0. When Δψ = 0, to attain the robustness of the proposed observer to the disturbances Δψ in L2 sense, we define V0 = V˙ + r T r − μΔψ T Δψ.

(6.40)

If Condition (6.31) is satisfied, it follows that ⎡

⎤T ⎡ ⎤ e1 e1 ⎢ e2 ⎥ ⎢ e2 ⎥ ⎢ ⎥ ⎥ V0 ≤ ⎢ ⎣ e3 ⎦ Λ ⎣ e3 ⎦ < 0. Δψ Δψ

(6.41)

Then under zero initial conditions, we obtain the following: 

∞



 r2 − μΔψ2 dt =

 

0

0

= 

∞

 2  r − μΔψ2 + V˙ dt −



∞

V˙ dt

0

 2  r − μΔψ2 + V˙ dt − V (∞) + V (0)

0 T

≤ 0

which implies that

∞

V0 dt < 0,

(6.42)

124

6 Simultaneous Estimation of Actuator and Sensor…



T



T

(r r)dt ≤ μ T

0

(Δψ T Δψ)dt,

(6.43)

0

namely, rL2 ≤

√

μΔψL2 .

(6.44) 

This completes the proof.

Remark 6.2 It is noticed in Theorem 6.1 that the derived sufficient conditions for the existence of the proposed SMOs include a linear matrix equality (6.30). The solvability of (6.30) can be converted into the problem of finding the minimum of a positive scalar γ satisfying the following inequality constraint: 

γ I P02 D2 T D2T P02 γI

 > 0.

(6.45)

However, this method can only make P02 D2 approximate to 0. In Theorem 6.2, we propose a new method to solve this problem. By employing the nonsingular transformation z = T0 h0 , we convert the problem of determining the value of F0 into the problem of determining the value of P0 . The obtained P0 and F0 can ensure the simultaneous satisfaction of (6.8) and (6.13). Theorem 6.2 Consider System (6.1) with Assumptions 6.1–6.4. Given a positive T scalar μ, if there exist matrices As1 , X, Y , Z, P1 = P1T > 0, P01 = P01 > 0 and T P¯ 03 = P¯ 03 > 0, and positive scalars α1 and α2 such that the following LMI feasibility problem has a solution: ⎡

⎤ X + X T + H1T H1 P1 A2 0 P1 E1 P1 T1 0 ⎢ AT2 P1 Π4 + H2T H2 C4T P¯ 03 P01 E2 0 P01 T2 ⎥ ⎢ ⎥ T T ⎢ ¯ 03 C4 0 P Y + Y + H H 0 0 0 ⎥ 3 3 ⎢ ⎥ < 0, ⎢ E2T P01 0 −μI 0 0 ⎥ E1T P1 ⎢ ⎥ ⎣ 0 0 0 −α1 I 0 ⎦ T1T P1 0 0 0 −α2 I 0 T2T P01 (6.46) where Π4 = AT4 P01 + P01 A4 + (α1 + α2 )Lf2 T −1 2 In−m + ZC4 − ZD2 D2+ C4 + (ZC4 )T − (ZD2 D2+ C4 )T , then the estimation error dynamics is asymptotically stable √ with the prescribed H∞ tracking performance rL2 ≤ μΔψL2 . Proof A solution for the unknown matrix P02 in (6.14) exists if and only if [19]  rank

D2 0

 = rank(D2 ).

(6.47)

Since the proof of (6.47) is straightforward, a solution to (6.14) exists and can be expressed as

6.3 SMOs-Based FE Scheme

125

P02 = Z(Ip−m − D2 D2+ ),

(6.48)

where Z ∈ R (n−m)×(p−m) is a design matrix and D2+ is the generalized inverse of D2 , i.e., D2+ = (D2T D2 )−1 D2T . Such a D2+ always exists because D is of full column rank. Substituting (6.48) into (6.31), and letting X = P1 AS1 and Y = P¯ 03 L, and then using Schur complement, the LMI (6.46) can be obtained immediately. After getting the value of P¯ 03 , F0 can be computed from F0 = D2T P¯ 03 . This completes the proof.  Remark 6.3 The effect of system uncertainties on the estimation errors is decided by the value of μ. The accuracy of the FE increases with smaller value of μ. The minimization of μ can be found by solving the following LMI optimization problem: minimize μ subject to X < 0, P1 > 0, P01 > 0, P¯ 03 > 0, and (6.46).

(6.49)

We have proved that the proposed observers are asymptotically stable with the prescribed H∞ performance in Theorem 6.1. The objective now is to determine the constant gain η1 in (6.22) and η2 in (6.23) such that the error systems can be directed to the sliding surface S which is defined as S = {(e1 , e2 , e3 )|e1 = 0, e3 = 0}

(6.50)

in finite time and maintain on it thereafter. Theorem 6.3 Given System (6.1) with Assumptions 6.1–6.4 and the proposed observers (6.20) and (6.21). Then the error dynamics (6.24) and (6.26) can be driven to the sliding surface given by (6.50) in finite time and remain on it if the LMI optimization problem formulated in (6.49) is solvable and the gains η1 and η2 satisfy η1 ≥ B1−T (A2 e2  + Lf T1 T −1 e2  + E1 ξ ) + η3 , C4 e2  + η4 η2 ≥ , D2T P¯ 03 e3 

(6.51) (6.52)

where η3 and η4 are two positive scalars. Proof Consider the Lyapunov functions V1 = eT1 P1 e1 and V3 = eT3 P¯ 03 e3 . The time derivative of V1 can be obtained as V˙ 1 = eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A2 e2 + 2eT1 P1 E1 Δψ

 ˆ t) + 2eT P1 B1 (fa − ν1 ). + 2eT1 P1 T1 f (T −1 h, t) − f (T −1 h, 1 It is easy to see that As1 T P1 + P1 As1 < 0 since As1 is a stable matrix by design. Then, from the Cauchy–Schwartz inequality and (6.22), we get

126

6 Simultaneous Estimation of Actuator and Sensor…

 ˆ t) V˙ 1 < 2eT1 P1 A2 e2 + 2eT1 P1 E1 Δψ + 2eT1 P1 T1 f (T −1 h, t) − f (T −1 h, + 2eT1 P1 B1 (fa − ν1 ) ≤ 2P1 e1 (A2 e2  + Lf T1 T −1 e2  + E1 ξ ) − 2η1 B1T P1 e1    ≤ 2B1T P1 e1  B1−T (A2 e2  + Lf T1 T −1 e2  + E1 ξ ) − η1 . (6.53) It follows from (6.51) that  1 V˙ 1 ≤ −2η3 B1T P1 e1  ≤ −2η3 B1  λmin (P1 )V12 . Similarly, it can be verified that if (6.52) is satisfied, then 1

V˙ 3 ≤ −2η4 P¯ 03 e0  ≤ −2η4 λmin (P¯ 03 )V32 . This shows that the reachability condition [20] is satisfied and an ideal sliding motion is achieved and maintained after some finite time. This completes the proof. 

6.3.2 Estimation of Faults Given the observers in the form of (6.20) and (6.21), the objective now is to simultaneously estimate actuator and sensor faults. From Theorem 6.3, we know that an ideal sliding motion (6.50) will take place after some finite time if the conditions (6.51) and (6.52) are satisfied. During the sliding motion, (6.24) becomes ˆ t)) + B1 (fa − ν1eq ) + E1 Δψ, 0 = A2 e2 + T1 (f (T −1 h, t) − f (T −1 h,

(6.54)

where ν1eq denotes the equivalent output error injection signal to maintain the sliding motion [1], and can be approximated to any degree of accuracy by replacing (6.22) with B1T P1 (C1−1 S1 y − hˆ 1 ) , (6.55) ν1 ≈ (ρa + η1 ) BT P1 (C −1 S1 y − hˆ 1 ) + δ1 1

1

where δ1 is a small positive scalar to reduce the chattering effect. Since B1 is invertible, (6.54) can be further rewritten as

 ˆ t)) + E1 Δψ . fa − ν1eq = −B1−1 A2 e2 + T1 (f (T −1 h, t) − f (T −1 h,

(6.56)

6.3 SMOs-Based FE Scheme

127

Computing the L2 norm of (6.56) yields ˆ t)) + E1 Δψ)L2 fa − ν1eq L2 = B1−1 (A2 e2 + T1 (f (T −1 h, t) − f (T −1 h, ≤ (σmax (B1−1 A2 ) + σmax (B1−1 T1 )Lf T −1 )e2 L2 + σmax (B1−1 E1 )ΔψL2 ≤ (σmax (B1−1 A2 ) + σmax (B1−1 T1 )Lf T −1 )eL2 + σmax (B1−1 E1 )ΔψL2 .

(6.57)

√ Since eL2 ≤ σmax (H −1 ) μΔψL2 , it can be obtained that fa − ν1eq L2 ≤

√

μ(σmax (B1−1 A2 ) + σmax (B1−1 T1 )Lf T −1 )σmax (H −1 )  (6.58) +σmax (B1−1 E1 ) ΔψL2 .

It follows that fa − ν1eq L2 √ = μβ1 + β2 , Δψ L2 ΔψL2 =0 sup

(6.59)

where β1 = (σmax (B1−1 A2 ) + σmax (B1−1 T1 )Lf T −1 )σmax (H −1 ) and β2 = σmax (B1−1 E1 ). √ Thus for a small ( μβ1 +β2 )ΔψL2 , the actuator faults fa can be approximated as fˆa (t) ≈ (ρa + η1 )

B1T P1 (C1−1 S1 y − hˆ 1 ) . BT P1 (C −1 S1 y − hˆ 1 ) + δ1 1

(6.60)

1

Similarly, we can get that fs − ν2eq L2 √ = μσmax (H −1 )σmax (D2+ A¯ 3 ), Δψ L2 ΔψL2 =0 sup

(6.61)

where ν2eq is the equivalent output error injection signal which can be approximated as ν2eq ≈ (ρs + η2 ) where δ2 is a small positive scalar.

D2T P¯ 03 (w3 − wˆ 3 ) , D2T P¯ 03 (w3 − wˆ 3 ) + δ2

(6.62)

128

6 Simultaneous Estimation of Actuator and Sensor…

Therefore for small estimated as

√ μσmax (H −1 )σmax (D2+ A¯ 3 )ΔψL2 , sensor faults fs can be

fˆs (t) ≈ (ρs + η2 )

D2T P¯ 03 (w3 T¯ D2 P03 (w3 −

− wˆ 3 ) . wˆ 3 ) + δ2

(6.63)

6.4 SMO- and AO-Based FE Scheme Based on the matching condition (6.13), the SMOs-based FE scheme is developed in Sect. 6.3. However, for many physical systems, the matching condition is truly a harsh condition and the solvability of Lyapunov equation together with the matching condition is not an easy job. Therefore, there is an incentive to develop an FE scheme when the matching condition is not satisfied, which is the main objective of this section.

6.4.1 Design of Observers When the matching condition is not satisfied, Systems (6.5) and (6.11) in the new coordinate z := [z1T z2T ]T = T0 h0 become 

−1 P02 w3 + T1 f (T −1 h, t) + B1 (u + fa ) + E1 Δψ h˙ 1 = A1 h1 + A2 z1 − A2 P01 w1 = C1 h1

⎧ −1 P02 D2 fs ⎨ z˙1 = A¯ 1 z1 + A¯ 2 z2 + A3 h1 + T2 f (T −1 h, t) + E2 Δψ + P01 ¯ ¯ z˙2 = A3 z1 + A4 z2 + D2 fs ⎩ w3 = z2

(6.64) (6.65)

where matrices A¯ 1 , A¯ 2 , A¯ 3 , and A¯ 4 have been defined in (6.12). For System (6.64), the same SMO as (6.20) is designed: ⎧ ˙ −1 ⎪ ˆ t) + B1 (u + ν) ⎨ hˆ 1 = A1 hˆ 1 + A2 zˆ1 − A2 P01 P02 w3 + T1 f (T −1 h, −1 s + (A1 − A1 )C1 (w1 − wˆ 1 ) ⎪ ⎩ wˆ 1 = C1 hˆ 1

(6.66)

where the output error injection input ν is defined by ⎧ ⎨(ρ + η) B1T P1 (C1−1 S1 y−hˆ 1 ) if C −1 S y − hˆ = 0 a 1 1 1 B1T P1 (C1−1 S1 y−hˆ 1 ) ν= ⎩0 otherwise where η is a positive scalar to be determined.

(6.67)

6.4 SMO- and AO-Based FE Scheme

129

For System (6.65), we construct the following AO: ⎧ ˆ t) + P−1 P02 D2 fˆs ⎪ z˙ˆ1 = A¯ 1 zˆ1 + A¯ 2 w3 + A3 C1−1 w1 + T2 f (T −1 h, ⎪ 01 ⎪ ⎨˙ zˆ2 = A¯ 3 zˆ1 + A¯ 4 zˆ2 + (A¯ 4 − L)(w3 − wˆ 3 ) + D2 fˆs wˆ 3 = zˆ2 ⎪ ⎪ ⎪ ⎩ ˙ˆ fs = Γ DT P¯ 03 (e3 + e˙ 3 )

(6.68)

2

where L ∈ R (p−m)×(p−m) is the traditional Luenberger observer gain, fˆs is the estimated value of the sensor fault, and Γ ∈ R q×q is a symmetric positive definite matrix representing the learning rate. Define the state and fault estimation errors as e1 = h1 − hˆ 1 , e2 = z1 − zˆ1 , e3 = z2 − zˆ2 , and ef = fs − fˆs , then the error dynamics after the occurrence of faults are given as

 ˆ t) + B1 (fa − ν) + E1 Δψ, (6.69) e˙ 1 = As1 e1 + A2 e2 + f1 (T −1 h, t) − f1 (T −1 h,

 ˆ t) + E2 Δψ + P−1 P02 D2 ef , e˙ 2 = A¯ 1 e1 + f2 (T −1 h, t) − f2 (T −1 h, (6.70) 01 e˙ 3 = A¯ 3 e2 + Le3 + D2 ef .

(6.71)

We now present Theorem 6.4 which gives the sufficient condition for the existence of the proposed observers (6.66) and (6.68) with the prescribed H∞ performance r2L2 ≤ μ1 Δψ2L2 + μ2 f˙s 2L2 , where μ1 and μ2 are two positive scalars and T  represents the weighted estimation error vector with H = r = H eT1 eT2 eT3 eTf diag(H1 , H2 , H3 , H4 ). Theorem 6.4 Consider System (6.1) with Assumptions 6.1–6.4. If there exist matriT T ces X, Y , P01 = P01 > 0, P02 , and P¯ 03 = P¯ 03 > 0, and positive scalars α1 , α2 , μ1 , and μ2 such that the following LMI optimization problem has a solution: minimize μ1 + μ2 subject to X < 0, P1 > 0, P01 > 0, P¯ 03 > 0 and ⎡ ⎤ Π5 + H1T H1 P1 A2 0 0 P1 E1 0 P1 T1 0 ⎢ Π6 + H2T H2 C4T P¯ 03 Π9 P01 E2 0 0 P01 T2 ⎥ AT2 P1 ⎢ ⎥ ⎢ ⎥ T T 0 P¯ 03 C4 Π7 + H3 H3 −Y D2 0 0 0 0 ⎥ ⎢ ⎢ ⎥ ⎢ 0 Π5T −D2T Y Π8 + H4T H4 0 Γ −1 0 0 ⎥ ⎢ ⎥ ⎢ ET P E2T P01 0 0 −μ1 I 0 0 0 ⎥ ⎢ ⎥ 1 1 ⎢ ⎥ 0 0 0 Γ −1 0 −μ2 I 0 0 ⎥ ⎢ ⎢ ⎥ ⎣ T1T P1 0 0 0 0 0 −α1 I 0 ⎦ 0 T2T P01 0 0 0 0 0 −α2 I < 0,

(6.72)

130

6 Simultaneous Estimation of Actuator and Sensor…

T where Π5 = X + X T , Π6 = AT4 P01 + P01 A4 + P02 C4 + C4T P02 + (α1 + α2 )Lf2 T −1 2 In−m , Π7 = Y + Y T , Π8 = −2D2T P¯ 03 D2 , and Π9 = P02 D2 − C4T P¯ 03 D2 , then the estimation error dynamics is asymptotically stable with the prescribed H∞ tracking performance r2L2 ≤ μ1 Δψ2L2 + μ2 f˙s 2L2 . Once the problem is solved, As1 and −1 L can be obtained from As1 = P1−1 X and L = P¯ 03 Y , respectively.

Proof Consider the Lyapunov function as V = V1 + V2 + V3 + V4 ,

(6.73)

where V1 = eT1 P1 e1 , V2 = eT2 P01 e2 , V3 = eT3 P¯ 03 e3 , and V4 = eTf Γ −1 ef . The time derivatives of V1 (t), V2 (t), V3 (t), and V4 (t) can be shown to be   1 V˙ 1 ≤ eT1 As1 T P1 + P1 As1 + P1 T1 T1T P1 e1 + 2eT1 P1 A2 e2 + 2eT1 P1 E1 Δψ α1

(6.74) + α1 Lf2 T −1 2 e2 2 ,   1 T + P01 T2 T2T P01 + α2 Lf2 T −1 2 I e2 V˙ 2 ≤ eT2 AT4 P01 + P01 A4 + P02 C4 + C4T P02 α2 + 2eT2 P01 E2 Δψ + 2eT2 P02 D2 ef , V˙ 3 =

eT3 (P¯ 03 L

+L

T

P¯ 03 )e3 + 2eT3 P¯ 03 C4 e2

(6.75) + 2eT3 P¯ 03 D2 ef ,

(6.76)

˙ V˙ 4 = 2eTf Γ −1 f˙s − 2eTf Γ −1 fˆs = 2eTf Γ −1 f˙s − 2eTf D2T P¯ 03 e3 − 2eTf D2T P¯ 03 (C4 e2 + Le3 + D2 ef ).

(6.77)

Therefore, the derivative of V (t) can be obtained as   1 V˙ ≤ eT1 As1 T P1 + P1 As1 + P1 T1 T1T P1 e1 α1

T T T + e2 A4 P01 + P01 A4 + P02 C4 + C4T P02

 1 P01 T2 T2T P01 + (α1 + α2 )Lf2 T −1 2 I e2 + eT3 (P¯ 03 L + L T P¯ 03 )e3 α2 T T¯ − 2e D2 P03 D2 ef + 2eT1 P1 A2 e2 + 2eT1 P1 E1 Δψ + 2eT2 P01 E2 Δψ + 2eT2 P02 D2 ef +

f

+ 2eT3 P¯ 03 C4 e2 − 2eTf D2T P¯ 03 C4 e2 − 2eTf D2T P¯ 03 Le3 + 2eTf Γ −1 f˙s ⎤⎡ ⎤ ⎡ ⎤T ⎡ Π + 1 P T T T P P1 A2 0 0 5 e1 e1 α1 1 1 1 1 ⎥⎢ ⎥ 1 TP TP TP ⎢ e2 ⎥ ⎢ ¯ 03 e Π + P T T C Π A ⎢ ⎥ 2⎥ 1 6 01 2 01 9 2 2 4 ⎥ α2 =⎢ ⎥⎢ ⎣ e3 ⎦ ⎢ ⎣ ⎦ ⎣ 0 P¯ 03 C4 Π7 −L T P¯ 03 D2 ⎦ e3 ef ef 0 Π5T −D2T P¯ 03 L Π8    + 2eT1 P1 E1 Δψ

+ 2eT2 P01 E2 Δψ

W T −1 ˙ + 2ef Γ fs .

(6.78)

6.4 SMO- and AO-Based FE Scheme

131

When Δψ = 0 and f˙s = 0, we define V0 = V˙ + r T r − μ1 Δψ T Δψ − μ2 f˙sT f˙s .

(6.79)

Substituting (6.78) into the constraint (6.79) yields ⎤T ⎡ Π5 + H1T H1 + α11 P1 T1 T1T P1 P1 A2 e1 T T ⎢ e2 ⎥ ⎢ A2 P1 Π6 + H2 H2 + α12 P01 T2 T2T P01 ⎥ ⎢ ⎢ ⎢ e3 ⎥ ⎢ 0 P¯ 03 C4 ⎥ ⎢ V0 ≤ ⎢ ⎢ ef ⎥ ⎢ 0 Π9T ⎥ ⎢ ⎢ T ⎣ Δψ ⎦ ⎣ E1 P1 E2T P01 ˙fs 0 0 ⎤⎡ ⎤ 0 0 0 P1 E1 e1 ⎢ ⎥ C4T P¯ 03 Π9 P01 E2 0 ⎥ ⎥ ⎢ e2 ⎥ T T¯ ⎢ ⎥ Π7 + H3 H3 −L P03 D2 0 0 ⎥ ⎢ e3 ⎥ ⎥. (6.80) T¯ T ⎢ ⎥ −D2 P03 L Π8 + H4 H4 0 Γ −1 ⎥ ⎥ ⎢ ef ⎥ 0 0 −μ1 Ir 0 ⎦ ⎣ Δψ ⎦ −1 f˙s 0 −μ2 Iq 0 Γ ⎡

According to the Schur complement, the condition (6.72) indicates that V0 < 0 and therefore r2L2 ≤ μ1 Δψ2L2 +μ2 f˙s 2L2 when Δψ = 0 and f˙s = 0. Moreover, from (6.72), we can get that W < 0, thus V˙ (t) < 0 when Δψ = 0 and f˙s = 0, which implies that e → 0 as t → ∞. This completes the proof.  Theorem 6.5 Under the Assumptions 6.1–6.4, an ideal sliding motion will take place after some finite time on the hyperplane S = {(e1 , e2 , e3 )|e1 = 0} if η ≥ B1−T (A2 e2  + Lf T1 T −1 e2  + E1 ξ ) + η0 ,

(6.81)

where η0 is a positive scalar. Proof The proof is similar to that of Theorem 6.3 and thus omitted here.



6.4.2 Estimation of Faults √ Following the same procedure as given in Sect. 6.3.2, for small values of ( μβ1 + β2 )ΔψL2 and δ, the actuator faults can be estimated from fˆa (t) ≈ (ρa + η)

B1T P1 (C1−1 S1 y − hˆ 1 ) , BT P1 (C −1 S1 y − hˆ 1 ) + δ 1

(6.82)

1

where β1 = (σmax (B1−1 A2 ) + σmax (B1−1 )Lf1 T −1 )σmax (H −1 ) and β2 = σmax (B1−1 E1 ).

132

6 Simultaneous Estimation of Actuator and Sensor…

The sensor faults can be estimated as fˆs (t) ≈ Γ D2T P¯ 03 e3 (t) + Γ D2T P¯ 03



t

e3 (τ )dτ,

(6.83)

tf

where tf is the time instant when a sensor fault occurs.

6.5 Simulation Results To illustrate the effectiveness of the proposed FE schemes, we consider the following one-link flexible joint manipulator system [21], which can be described by the model of the form (6.1) where ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 1 0 0 θm 0 1 ⎢ −48.6 −1.25 48.6 0 ⎥ ⎢ ωm ⎥ ⎢ 21.6 ⎥ ⎢0⎥ ⎥ ⎢ ⎥, B = ⎢ ⎥ ⎢ ⎥ x=⎢ ⎣ θl ⎦ , A = ⎣ 0 ⎣ 0 ⎦, E = ⎣1⎦, 0 0 10 ⎦ ωl 1.95 0 −1.95 0 0 1 ⎤ ⎡ ⎡ ⎡ ⎤ ⎤ 0 1000 1 ⎥ ⎢ 0 ⎥. C = ⎣ 0 1 0 0 ⎦ , D = ⎣ 0 ⎦ , f (x, t) = ⎢ ⎦ ⎣ 0 0010 0 −0.333 sin(x3 ) ⎡

The terms associated with system uncertainties Δψ, actuator faults fa , and sensor faults fs have been added to demonstrate the effectiveness of the proposed methods. In the simulation, Δψ = 0.1 sin(t) and the faults are ⎧ t ≤ 5s ⎨0 5 < t < 15 s fa = 0.02(t − 5) ⎩ 0.2 t ≥ 15 s ⎧ t ≤ 10 s ⎨0 10 s < t < 20 s fs = 0.4 ⎩ 0 t ≥ 20 s Note that fa is an incipient fault and fs is an abrupt fault. Most common faults in electrical systems such as electronic short circuit, device saturation, data losses in the on-board computer, and gradual creation of bias in measurements can be modeled as these two types of signals.

6.5 Simulation Results

133

Introduce nonsingular transformation matrices T and S with ⎡

0 ⎢1 T =⎢ ⎣0 0

1 0 0 0

0 0 1 0

⎤ ⎡ ⎤ 0 010 0⎥ ⎥, S = ⎣1 0 0⎦, 0⎦ 001 1

such that in the new coordinate the system matrices become ⎤ ⎡ ⎤ 0 −1.2500 −48.6000 48.6000 1000 ⎥ ⎢ 1.0000 0 0 0 ⎥ , SCT −1 = ⎣ 0 1 0 0 ⎦ , TAT −1 = ⎢ ⎣ 0 0 10.0000 ⎦ 0 0010 1.9500 −1.9500 0 0 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 21.6000 0 0 ⎢ ⎥ ⎥ ⎢ 0 ⎥ , TE = ⎢ 1 ⎥ , SD = ⎣ 1 ⎦ . TB = ⎢ ⎣ ⎦ ⎣1⎦ 0 0 0 1 ⎡

For the first FE scheme, we select H = I6 . By solving the LMI optimization problem (6.49), we get the minimized feasible μ = 1e − 4, and P1 = 0.0111, X = −1.4909 ⎡ ⎡ ⎤ ⎤ 0.8130 −0.1109 −0.7004 0 −0.8262 P01 = ⎣ −0.1109 0.3138 −0.1936 ⎦ , Z = ⎣ 0 −2.2948 ⎦ , −0.7004 −0.1936 0.9025 0 −1.8018     0.5493 −0.0053 −1.4093 0 P¯ 03 = , Y= . −0.0053 0.7346 −0.0055 −1.3823 Then As1 , L, and F0 can be computed as 

As1

   −2.5661 −0.0180 = −134.2431, L = , F0 = 0.5493 −0.0053 . −0.0259 −1.8817

It can be verified that the matching condition D0T P0 = F0 C0 is satisfied. The MATLAB code used for computing above parameters is presented in the following file chapter6_scheme1lmi.m.

134

File Chapter6_scheme1lmi.m

6 Simultaneous Estimation of Actuator and Sensor…

6.5 Simulation Results

135

136

6 Simultaneous Estimation of Actuator and Sensor…

6.5 Simulation Results

137

For the second scheme, we select H = I7 and Γ = 10. By solving the LMI (6.72) in Theorem 6.4, we obtain the minimal μ1 and μ2 which equals to 1e − 4 and 1e − 2, respectively, and P1 = 0.0209, As1 = −114.7882, ⎤ ⎡ ⎤ 1.8304 −0.2191 −1.6092 0.0064 −2.1438 ⎥ ⎢ ⎢ ⎥ = ⎣ −0.2191 0.5723 −0.3412 ⎦ , P02 = ⎣ −0.0131 −3.6265 ⎦ , −1.6092 −0.3412 1.9628 −0.0505 −2.6181 ⎡

P01

! P¯ 03 =

2.2134 −0.0232

"

−0.0232 1.3308

! , L=

−0.9616 −0.0293 −0.0228 −1.5986

" .

The MATLAB code for getting these results is described in the file chapter6_ scheme2lmi.m. File Chapter6_scheme2lmi.m

138

6 Simultaneous Estimation of Actuator and Sensor…

6.5 Simulation Results

139

140

6 Simultaneous Estimation of Actuator and Sensor…

6.5 Simulation Results

141

The simulation is carried out by running the file chapter6.mdl and the plots are obtained from file chapter6_plot.m. In the simulation, we have selected η1 = 1.5, δ1 = 0.002 for both schemes and η2 = 1.6, δ2 = 0.001 for the SMO-based scheme. The initial state is chosen as x(0) = [0.01, 0, 0.1, 0.05]T , xˆ (0) = [0, 0, 0, 0]T , fˆa (0) = 0 and fˆs (0) = 0. The state and fault estimation results are provided in Figs. 6.1, 6.2 and 6.3. It can be seen from these figures that both the proposed schemes can estimate the faults as well as the system states with satisfactory accuracy despite the existence of the system uncertainty Δψ. Both methods have similar performance

x1

2 0 −2

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

x2

2 0 −2

x3

2 0 −2

x4

0.2 0

−0.2

Time (s) Fig. 6.1 System states x and their estimates xˆ (actual state: solid line; scheme 1: dashed line; scheme 2: dotted line)

142

6 Simultaneous Estimation of Actuator and Sensor… 0.4 Actual fa

0.35

Estimation by scheme1 Estimation by scheme2

0.3

Magnitude

0.25 0.2 0.15 0.1 0.05 0

−0.05 −0.1

0

5

10

15

20

25

Time (s) Fig. 6.2 Actuator fault fa and its estimate fˆa 0.6 Actual fs Estimation by scheme1 Estimation by scheme2

0.5

Magnitude

0.4 0.3 0.2 0.1 0

−0.1

0

5

10

15

20

25

Time (s) Fig. 6.3 Sensor fault fs and its estimate fˆs

in the actuator fault estimation, while the AO can estimate the sensor fault more accurately than the SMO. It is worth noting that we apply an integrator to transform sensor faults into the form of actuator faults when estimating sensor faults. The term t w3 = 0 w2 (τ )dτ actually acts as a low-pass filter and therefore can attenuate the effect of high-frequency disturbances which exist in the output signals. To demonstrate the measurement noise suppression property of the proposed schemes, we add

6.5 Simulation Results

143 Actual fs Estimation by scheme1 Estimation by scheme2

0.5

Magnitude

0.4 0.3 0.2 0.1 0 −0.1

0

5

10

15

20

25

Time (s) Fig. 6.4 Sensor fault fs and its estimate fˆs when considering measurement noise

noise v(t) to the measurement y(t). The considered noise v(t) is assumed to satisfy ⎡

⎤ 0.0001 0 0 E {v(t)} = 0, Cov{v(t)} = ⎣ 0 0.0001 0 ⎦ , 0 0 0.0001 where E {·} and Cov{·} denote the expectation and covariance of a signal, respectively. The simulation results show that the proposed schemes are very sensitive to the measurement noise and fail to estimate the actuator fault. However, they can still estimate the sensor fault satisfactorily. As is shown in Fig. 6.4, the effect of the measurement noise on the sensor fault estimation has been greatly reduced and the proposed AO exhibits a stronger robustness.

6.6 Conclusions In this chapters, two schemes for simultaneously estimating actuator and sensor faults for uncertain Lipschitz nonlinear systems are proposed. The method proposed in the first scheme uses two SMOs which can effectively estimate both the actuator and sensor faults provided certain matching condition is satisfied. In the second scheme, we relax the constraint of matching condition and employ an AO to estimate sensor faults. Moreover, H∞ filtering is integrated into both schemes to attenuate the effects of the system uncertainties on state and fault estimates. The simulation results show that both methods can successfully estimate actuator and sensor faults simultaneously

144

6 Simultaneous Estimation of Actuator and Sensor…

despite the existence of system uncertainties. When the outputs are contaminated by noises, sensor fault estimates can still be obtained.

References 1. Edwards C, Spurgeon SK, Patton RJ (2000) Sliding mode observers for fault detection and isolation. Automatica 36:541–553 2. Tan CP, Edwards C (2003) Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int J Robust Nonlinear Control 13:443–463 3. Tan CP, Edwards C (2002) Sliding mode observers for detection and reconstruction of sensor faults. Automatica 38:1815–1821 4. Yan XG, Edwards C (2007) Sensor fault detection and isolation for nonlinear systems based on a sliding mode observer. Int J Adapt Control Signal Process 21:657–673 5. Alwi H, Edwards C, Tan CP (2009) Sliding mode estimation schemes for incipient sensor faults. Automatica 45:1679–1685 6. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 7. Yan XG, Edwards C (2008) Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems. Int J Syst Sci 39(4):349–359 8. Raoufi R, Marquez HJ, Zinober ASI (2010) H∞ sliding mode observer for uncertain nonlinear Lipschitz systems with fault estimation synthesis. Int J Robust Nonlinear Control 20:1785– 1801 9. Liu M, Cao X, Shi P (2013) Fuzzy-model-based fault-tolerant design for nonlinear stochastic systems against simultaneous sensor and actuator faults. IEEE Trans Fuzzy Syst 21:789–799 10. Bejarano FJ, Figueroa M, Pacheco J, Rubio J (2012) Robust fault diagnosis of disturbed linear systems via a sliding mode high order differentiator. Int J Control 85:648–659 11. Lee DJ, Park YJ, Park YS (2012) Robust H∞ sliding mode descriptor observer for fault and output disturbance estimation of uncertain systems. IEEE Trans Autom Control 57:2928–2934 12. Zhang J, Swain AK, Nguang SK (2012) Detection and isolation of incipient sensor faults for a class of uncertain nonlinear systems. IET Control Theory Appl 6:1870–1880 13. Zhang J, Swain AK, Nguang SK (2013) Robust sensor fault estimation scheme for satellite attitude control systems. J Frankl Inst 350:2581–2604 14. Floquet T, Edwards C, Spurgeon SK (2007) On sliding mode observers for systems with unknown inputs. Int J Adapt Control Signal Process 21:638–656 ˙ SH, (2010) Sliding-mode observers for systems with unknown 15. Kalsi K, Lian J, Hui S, Zak inputs: a high-gain approach. Automatica 46:347–353 16. Zhu F (2012) State estimation and unknown input reconstruction via both reduced-order and high-order sliding mode observers. J Process Control 22:296–302 17. Tan CP, Edwards C (2001) An LMI approach for design sliding mode observers. Int J Control 74:1559–1568 18. Chen W, Saif M (2007) Observer-based strategies for actuator fault detection, isolation and estimation for certain class of uncertain nonlinear systems. IET Control Theory Appl 1:1672– 1680 19. Rao C, Mitra S (1971) Generalized inverse of matrices and its applications. Wiley, New York 20. Utkin VI (1992) Sliding modes in control optimization. Springer, Berlin 21. Zhang K, Jiang B, Shi P (2009) Fast fault estimation and accommodation for dynamical systems. IET Control Theory Appl 3:189–199

Chapter 7

Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

For the same faulty system that has been considered in Chap. 6, we develop an alternative scheme which is based on a sliding-mode observer (SMO) and an unknown-input observer (UIO), to simultaneously estimate actuator and sensor faults.

7.1 Introduction In Chap. 6, we have developed two fault estimation (FE) schemes for uncertain Lipschitz nonlinear systems with simultaneous actuator and sensor faults based on SMO and AO. The essence behind these methods is to employ two cascade coordinate transformations to transform the system matrices to a special structure. However, the use of cascade transformations may bring complexities to the design of observers and therefore limits the application of the proposed methods. In this chapter, we develop a different FE method, which does not need transformations twice in the design. The proposed method in this chapter is inspired by the work of [1], where UIOs are designed to detect and isolate actuator faults. The method begins with an introduction of a system transformation such that the original system is transformed into two subsystems (subsystem-1 and 2), where subsystem-1 includes the effects of actuator faults but is free from sensor faults and subsystem-2 only has sensor faults. In contrast to the methods proposed in Chap. 6, where sensor faults in subsystem-2 are transformed into the form of actuator faults using an integral observer-based approach, in this chapter, we take sensor faults as auxiliary states and accordingly construct an augmented system with the original states and sensor faults as the new states. For the augmented system, a nonlinear UIO is developed, which has the ability to completely eliminate the derivatives of sensor faults from the error dynamics and simultaneously estimate the states of subsystem-2 and sensor faults. For actuator faults appearing in subsystem-1, an SMO is designed to estimate their values. The effects of system uncertainties on the estimation errors are minimized by integrating a prescribed H∞ disturbance attenuation level into the proposed observers. Sufficient conditions for © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_7

145

146

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

the stability of the observers have been derived and the procedure for computing their parameters is given in terms of LMIs. The rest of the chapter is organized as follows: Sect. 7.2 introduces the problem and some mathematical preliminaries required for designing the observers. In Sect. 7.3, the proposed SMO and UIO are presented. The sufficient conditions for the stability of the observers are derived. The design procedure is provided using LMI technique. It follows in Sect. 7.4 the estimation of faults using the proposed observers. The results of simulation are shown in Sect. 7.5 with conclusions in Sect. 7.6.

7.2 Problem Formulation Consider a nonlinear system described by 

x˙ (t) = Ax(t) + W f (x, t) + B(u(t) + fa (t)) + EΔψ(t) y(t) = Cx(t) + Dfs (t)

(7.1)

where x ∈ R n , u ∈ R m , and y ∈ R p denote respectively the vector of states, inputs, and outputs. fa ∈ R m represents the vector of actuator faults and fs ∈ R q represents the vector of sensor faults. Δψ represents the unknown bounded uncertainties and the nonlinear continuous term f (x, t) ∈ R n is assumed to be known. A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R p×q , E ∈ R n×r and W ∈ R n×j (p − m ≥ q) are known constant matrices with B and D being of full rank. Before stating the main results of this chapter, the following assumptions are made: Assumption 7.1 rank(CB) = rank(B). Assumption 7.2 The nonlinear function f (x, t) is Lipschitz about x uniformly, that is, ∀ x, xˆ ∈ R n , (7.2) f (x, t) − f (ˆx , t) ≤ Lf x − xˆ , where Lf is the Lipschitz constant. Assumption 7.3 The actuator fault fa satisfies fa  ≤ ρa . The sensor fault fs is differentiable after its occurrence (the continuity at the fault occurrence time is not required), and f˙s belongs to L2 [0, ∞). Remark 7.1 It is worth noting that the sensor faults considered in this chapter can be unbounded. Thus, it is clear that the Assumption 7.3 is more general compared with some of the previous papers, for example in [2–4] where sensor faults are assumed to be bounded by some known functions or constants.

7.2 Problem Formulation

147

Lemma 7.1 Under Assumption 7.1, there exist state and output transformations 

z z= 1 z2





     x1 w1 y =T , w= =S 1 , x2 w2 y2

(7.3)

where z1 ∈ R m and w1 ∈ R m , such that in the new coordinate, the system matrices become       B1 A1 A2 E1 −1 , TB = , , TE = TAT = A3 A4 0 E2       C1 0 W1 0 SCT −1 = , TW = , SD = , W2 0 C4 D2    S1 T1 n×n ∈R ,S= ∈ R p×p , T1 ∈ R m×n , S1 ∈ R m×p , z1 ∈ R m , where T = T2 S2 w1 ∈ R m , A1 ∈ R m×m , A4 ∈ R (n−m)×(n−m) , B1 ∈ R m×m , E1 ∈ R m×r , C1 ∈ R m×m , C4 ∈ R (p−m)×(n−m) , D2 ∈ R (p−m)×q and W1 ∈ R m×j . B1 , and C1 are invertible. Therefore, in the new coordinate, System (7.1) is converted into the following two subsystems: 

 

z˙1 = A1 z1 + A2 z2 + W1 f (T −1 z, t) + B1 (u + fa ) + E1 Δψ w1 = C1 z1

(7.4)

z˙2 = A3 z1 + A4 z2 + W2 f (T −1 z, t) + E2 Δψ w2 = C4 z2 + D2 fs

(7.5)

For Subsystem (7.5), if we take sensor faults fs as auxiliary states, then an augmented system can be constructed as 

z¯˙2 = A¯ 4 z¯2 + A¯ 3 z1 + W¯ 2 f (T −1 z, t) + E¯ 2 Δψ + E¯ f˙s w2 = C¯ 4 z¯2

where    z2 A4 0 ∈ R n+q−m , A¯ 4 = ∈ R (n+q−m)×(n+q−m) , fs 0 0     ¯A3 = A3 ∈ R (n+q−m)×m , W¯ 2 = W2 ∈ R (n+q−m)×j , 0 0     E2 0 ∈ R (n+q−m)×q , ∈ R (n+q−m)×r , E¯ = E¯ 2 = 0 Iq   C¯ 4 = C4 D2 ∈ R (p−m)×(n+q−m) . 

z¯2 =

(7.6)

148

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

Accordingly, System (7.4) can be rewritten as 

z˙1 = A1 z1 + A¯ 2 z¯2 + W1 f (T −1 z, t) + B1 (u + fa ) + E1 Δψ w1 = C1 z1

(7.7)

  where A¯ 2 = A2 0 .

7.3 Design of Observers For System (7.6), we construct the following SMO: ⎧ ˙ ¯ ˆ W1 f (T −1 zˆ , t) + B1 (u + ν) ⎪ ⎪ zˆ1 = A1 zˆ1 + A2 z¯s2 +−1 ⎨ +(A1 − A1 )C1 (w1 − wˆ 1 ) 1ˆ ⎪ + k W W T P C −1 (w1 − wˆ 1 ) ⎪ 2 1 1 1 1 1 ⎩ wˆ 1 = C1 zˆ1

(7.8)

where zˆ1 , zˆ¯2 , and wˆ 1 denote respectively the estimated z1 , z¯2 , and w1 . As1 ∈ R m×m is a stable matrix to be determined and P1 ∈ R m×m is the symmetric definite Lyapunov matrix for As1 . Note that zˆ is defined as zˆ := col(C1−1 S1 y, [In−m 0]zˆ¯2 ) and does not represent the state estimate vector col(ˆz1 , zˆ2 ). The discontinuous output error injection term ν is defined by ν=

 BT P (C −1 w −ˆz ) (ρa + η) B1T P1 (C1−1 w1 −ˆz1 ) if C1−1 w1 − zˆ1 = 0 1

1

1

1

1

0

(7.9)

otherwise

where η is a positive scalar to be determined. kˆ 1 satisfies the following adaptation law: ˙ kˆ 1 = lk1 W1T P1 (C1−1 w1 − zˆ1 )2 ,

(7.10)

where lk1 is a positive scalar. For System (7.7), we design the following UIO: 

h˙ = F0 h + M0 W¯ 2 f (T −1 zˆ , t) + L0 w2 + M0 A¯ 3 C1−1 w1 + 21 kˆ 2 M0 W¯ 2 H0 (w2 − wˆ 2 ) zˆ¯2 = h + N0 w2 (7.11)

where wˆ 2 = C¯ 4 z¯ˆ2 is the estimated w2 , h ∈ R n+q−m is a middle variable, F0 ∈ R (n+q−m)×(n+q−m) , M0 ∈ R (n+q−m)×(n+q−m) , L0 ∈ R (n+q−m)×(p−m) , N0 ∈ R (n+q−m)×(p−m) and H0 ∈ R j×(p−m) are matrices to be determined.

7.3 Design of Observers

149

kˆ 2 satisfies the following adaptation law: ˙ kˆ 2 = lk2 H0 (w2 − wˆ 2 )2 ,

(7.12)

where lk2 is a positive scalar. It follows from (7.11) that z˙ˆ¯2 = h˙ + N0 w˙ 2 = F0 zˆ¯2 + (L0 C¯ 4 + N0 C¯ 4 A¯ 4 − F0 N0 C¯ 4 )¯z2 + M0 W¯ 2 f (T −1 zˆ , t) + N0 C¯ 4 W¯ 2 f (T −1 z, t) 1 + (M0 + N0 C¯ 4 )A¯ 3 z1 + N0 C¯ 4 E¯ 2 Δψ + N0 C¯ 4 E¯ f˙s + kˆ 2 M0 W¯ 2 H0 C¯ 4 (w2 − wˆ 2 ). 2

(7.13) Define the state estimation errors as e1 = z1 − zˆ1 and e¯ 2 = z¯2 − zˆ¯2 . Their dynamics after the occurrence of faults can be obtained as

e˙ 1 = As1 e1 + A¯ 2 e¯ 2 + W1 f (T −1 z, t) − f (T −1 zˆ , t) + B1 (fa − ν) + E1 Δψ 1 − kˆ 1 W1 W1T P1 e1 , (7.14) 2 e˙¯ 2 = (A¯ 4 + F0 N0 C¯ 4 − L0 C¯ 4 − N0 C¯ 4 A¯ 4 )¯z2 − F0 zˆ¯2 + (In+q−m − N0 C¯ 4 )W¯ 2 f (T −1 z, t) − M0 W¯ 2 f (T −1 zˆ , t) + (In+q−m − N0 C¯ 4 )E¯ 2 Δψ + (In+q−m − N0 C¯ 4 )E¯ f˙s 1 − kˆ 2 M0 W¯ 2 H0 C¯ 4 e¯ 2 . 2

(7.15)

If matrices F0 , M0 , L0 , and N0 satisfy the following conditions: M0 = In+q−m − N0 C¯ 4 , F0 = M0 A¯ 4 + (F0 N0 − L0 )C¯ 4 = M0 A¯ 4 + K0 C¯ 4 , M0 E¯ = 0.

(7.16) (7.17) (7.18)

then, the error dynamics (7.15) can further be written as

e˙¯ 2 = F0 e¯ 2 + M0 W¯ 2 f (T −1 , t) − f (T −1 zˆ , t) + M0 E¯ 2 Δψ 1 − kˆ 2 M0 W¯ 2 H0 C¯ 4 e¯ 2 . 2

(7.19)

Define the controlled estimation error r as  r = He = H

e1 e¯ 2

 (7.20)

150

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

 where H is the prespecified weight matrix having the structure H =

 H1 0 with 0 H2

H1 ∈ R m×m and H2 ∈ R (n+q−m)×(n+q−m) . We now present the Theorem 7.1 which establishes the sufficient condition for the existence of the proposed observers satisfying:

1. The observer error dynamics (7.14) and (7.19) are asymptotically stable when Δψ = 0, namely, there is no system uncertainty; ∞ 2. Under zero initial condition, the H∞ performance index J = 0 (r T r − μΔψ T Δψ)dt < 0 holds for all nonzero Δψ ∈ L2 [0, ∞), where μ > 0 is a prescribed √ scalar and μ represents the attenuation level of disturbance. Theorem 7.1 Consider System (7.1) with Assumptions 7.1–7.3. Given a positive scalar μ and matrices F0 , L0 , M0 , and N0 which satisfy conditions (7.16)–(7.18). If there exist matrices P1 = P1T > 0, P2 = P2T > 0 and H0 such that: H0 C¯ 4 = W¯ 2T M0T P2 , ⎡ ⎤ Π1 + H1T H1 P1 A¯ 2 P1 E1 Λ := ⎣ A¯ T2 P1 Π2 + H2T H2 P2 M0 E¯ 2 ⎦ < 0, T E1 P1 −μIr E¯ 2T M0T P2

(7.21) (7.22)

where Pi1 = As1 T P1 +P1 As1 , Π2 = P2 F0 +F0T P2 +2In+q−m , then the estimation error dynamics are asymptotically stable with the prescribed H∞ tracking performance. Proof Consider the Lyapunov function as V = V1 + V2 + V3 + V4 ,

(7.23)

e2k1 /2, V4 = lk−1 e2k2 /2, ek1 = k1 − kˆ 1 , and where V1 = eT1 P1 e1 , V2 = e¯ T2 P2 e¯ 2 , V3 = lk−1 1 2 ek2 = k2 − kˆ 2 . k1 and k2 are two positive constants which can be determined from (7.33). The time derivative of V1 along the trajectories of error dynamics (7.14) can be shown to be

V˙1 = eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A¯ 2 e¯ 2 + 2eT1 P1 W1 f (T −1 z, t) − f (T −1 zˆ , t) + 2eT1 P1 E1 Δψ + 2eT1 P1 B1 (fa − ν) − kˆ 1 W1T P1 e1 2 . From the fact that zˆ := col(C1−1 S1 y, [I 0]zˆ¯2 ), it follows that   −1 f (T z, t) − f (T −1 zˆ , t) ≤ Lf T −1 ¯e2 .

(7.24)

7.3 Design of Observers

151

Since for any scalar α > 0, the inequality 2X T Y ≤ α1 X T X + αY T Y holds [5], then 1 V˙1 ≤ eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A¯ 2 e¯ 2 + 2eT1 P1 E1 Δψ + eT1 P1 W1 W1T P1 e1 α1

T

+ α1 f (T −1 z, t) − f (T −1 zˆ , t) f (T −1 z, t) − f (T −1 zˆ , t) + 2eT1 P1 B1 (fa − ν) − kˆ 1 W1T P1 e1 2 1 P1 W1 W1T P1 )e1 + 2eT1 P1 A¯ 2 e¯ 2 + 2eT1 P1 E1 Δψ α1 + α1 Lf2 T −1 2 ¯e2 2 + 2eT1 P1 B1 (fa − ν) − kˆ 1 W1T P1 e1 2 .

≤ eT1 (As1 T P1 + P1 As1 +

(7.25)

From (7.9), it is easy to show that eT1 P1 B1 (fa − ν) = eT1 P1 B1 fa − (ρa + η)

B1T P1 e1 2 B1T P1 e1 

≤ −ηB1T P1 e1  < 0.

(7.26)

Therefore 1 V˙1 ≤ eT1 (As1 T P1 + P1 As1 )e1 + eT1 P1 W1 W1T P1 e1 + α1 Lf2 T −1 2 ¯e2 2 + 2eT1 P1 A¯ 2 e¯ 2 α1 + 2eT P1 E1 Δψ − kˆ 1 W T P1 e1 2 . (7.27) 1

1

Let α1 = 1/Lf2 T −1 2 . We have V˙1 ≤ eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A¯ 2 e¯ 2 + (Lf2 T −1 2 − kˆ 1 )W1T P1 e1 2 + ¯e2 2 + 2eT1 P1 E1 Δψ.

(7.28)

With the satisfaction of (7.21), the derivatives of V2 with respect to time can be obtained as

V˙2 = e¯ T2 (P2 F0 + F0T P2 )¯e2 + 2¯eT2 P2 M0 W¯ 2 f (T −1 z, t) − f (T −1 zˆ , t) + 2¯eT2 P2 M0 E¯ 2 Δψ − kˆ 2 e¯ T2 P2 M0 W¯ 2 H0 C¯ 4 e¯ 2 1 ≤ e¯ T2 (P2 F0 + F0T P2 )¯e2 + e¯ T2 P2 M0 W¯ 2 W¯ 2T M0T P2 e¯ 2 + α2 Lf2 T −1 2 ¯e2 2 In+q−m α2 + 2¯eT2 P2 M0 E¯ 2 Δψ − kˆ 2 W¯ 2T M0T P2 e¯ 2 2 . (7.29)

152

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

Let α2 = 1/Lf2 T −1 2 . Then it follows that V˙2 ≤ e¯ T2 (P2 F0 + F0T P2 )¯e2 + (Lf2 T −1 2 − kˆ 2 )W¯ 2T M0T P2 e¯ 2 2 + ¯e2 2 + 2¯eT2 P2 M0 E¯ 2 Δψ. (7.30) Moreover, the time derivatives of V3 and V4 are

Set

V˙3 = −ek1 W1T P1 e1 2 , V˙4 = −ek2 W¯ 2T M0T P2 e¯ 2 2 .

(7.32)

k1 = k2 = Lf2 T −1 2 .

(7.33)

(7.31)

From (7.28), (7.30)–(7.32), the time derivative of V can be obtained as: V˙ = V˙1 + V˙2 + V˙3 + V˙4    T  Π1 P1 A¯ 2 e1 e + 2eT1 P1 E1 Δψ + 2¯eT2 P2 M0 E¯ 2 Δψ. ≤ 1 e¯ 2 e¯ 2 A¯ T2 P1 Π2   

(7.34)

W

If there exists a feasible solution to (7.21), then it is straightforward that W < 0, thus V˙ < 0 when Δψ = 0. This implies that e → 0 as t → ∞. Therefore the error dynamics are asymptotically stable when Δψ = 0. When Δψ = 0, to attain the robustness of the proposed observers to the disturbances Δψ in L2 sense, we define V0 = V˙ + r T r − μΔψ T Δψ.

(7.35)

If Condition (7.21) is satisfied, it follows that ⎡

⎤T ⎡ ⎤ e1 e1 V0 ≤ ⎣ e¯ 2 ⎦ Λ ⎣ e¯ 2 ⎦ < 0. Δψ Δψ

(7.36)

Then, under zero initial conditions, we get 

∞

2

r − μΔψ2 dt =

 

0

0

= 

∞

∞

2 r − μΔψ2 + V˙ dt −



∞

V˙ dt

0

2

r − μΔψ2 + V˙ dt − V (∞) + V (0)

0

≤ 0

T

V0 dt < 0,

(7.37)

7.3 Design of Observers

153

which implies that 

T



T

(r r)dt ≤ μ T

0

(Δψ T Δψ)dt,

(7.38)

0

namely, rL2 ≤

√

μΔψL2 .

(7.39) 

This completes the proof.

Remark 7.2 Given a positive constant μ, the problem of finding matrices P1 = P1T , P2 = P2T and H0 to simultaneously satisfy (7.21) and (7.22) can be converted into the following LMI optimization problem: minimize γ subject to P1 > 0, P2 > 0, (7.22) and   γ Ij H0 C¯ − W¯ 2T M0T P2 > 0. (H0 C¯ − W¯ 2T M0T P2 )T γ In+q−m

(7.40)

Note that Theorem 7.1 states the stability condition of the proposed observers when matrices F0 , L0 , M0 , and N0 satisfying conditions (7.16)–(7.18) are provided. However, it does not suggest how to determine the value of these matrices. This problem will be addressed in the following theorem which outlines a constructive procedure for designing the proposed observers. Theorem 7.2 Consider System (7.1) with Assumptions 7.1–7.3. Given a positive constant μ, if there exist matrices P1 = P1T > 0, P2 = P2T > 0, X, Y , and U, and a positive scalar γ such that the following LMI optimization problem has a solution: minimize γ subject to P1 > 0, P2 > 0 and ⎡ ⎤ Π3 + H1T H1 P1 A¯ 2 P1 E1 ⎢ ⎥ A¯ T2 P1 Π4 + H2T H2 Π5 ⎦ < 0, ⎣ 

E1T P1

Π5T

−μIr

γ Ij W¯ 2T J1T P2 + W¯ 2T J2T X T − H0 C¯ ¯ T γ In+q−m (W¯ 2T J1T P2 + W¯ 2T J2T X T − H0 C)

(7.41)  > 0,

(7.42)

154

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

where Π3 = U + U T , Π4 = P2 J1 A¯ 4 + A¯ T4 J1T P2 + XJ2 A¯ 4 + A¯ T4 J2T X T + Y C¯ 4 + C¯ 4T Y T + 2In+q−m , and Π5 = P2 J1 E¯ 2 + XJ2 E¯ 2 , then the estimation error dynamics are asymptotically stable with the prescribed disturbance attenuation level μ. After getting Z = P2−1 X and K0 = P2−1 Y , the parameters of the proposed observers (7.8) and (7.11) can be obtained as As1 = P1−1 U, M0 = J1 + ZJ2 , N0 = J3 + ZJ4 , F0 = M0 A¯ 4 + K0 C¯ 4 , and L0 = F0 N0 − K0 . Proof The conditions (7.16) and (7.18) can be combined into a more compact form as 

 M0 N0 S1 = S2 ,

(7.43)

   In+q−m E¯ and S2 = In+q−m 0 . ¯ C4 0 It is easy to see that 

where S1 =



S rank 1 S2

 = rank(S1 ) = n + 2q − m.

Therefore, a set of solutions of unknown matrices [M0 N0 ] always exists [6], and can be represented as M0 = J1 + ZJ2 , N0 = J3 + ZJ4 ,

(7.44)

where Z ∈ R (n+q−m)×(n+p+q−2m) is a design matrix to be determined and 

    In+q−m  In+q−m + , J2 = In+p+q−2m − S1 S1 , J1 = 0 0       0 0 , J4 = In+p+q−2m − S1 S1+ , J3 = S2 S1+ Ip−m Ip−m S2 S1+

where S1+ is a generalized inverse of S1 , i.e., S1+ = (S1T S1 )−1 S1T . Substituting M0 = J1 + ZJ2 , N0 = J3 + ZJ4 into (7.22) and (7.40) and letting X = PZ and Y = PK0 , we can get the LMIs (7.41) and (7.42) directly. This completes the proof.  In the following theorem, we will show how to determine the value of η in (7.9) such that the error dynamics (7.14) can be driven to the sliding surface S which is defined as S = {(e1 , e¯ 2 )|e1 = 0} (7.45) in finite time and maintain on it thereafter. Theorem 7.3 Given System (7.1) with Assumptions 7.1–7.3 and the proposed observers (7.8) and (7.11). Then the error dynamics (7.14) can be driven to the

7.3 Design of Observers

155

sliding surface given by (7.45) in finite time and remain on it if the LMI optimization problem formulated in Theorem 7.2 is solvable and the gain η satisfies η ≥ B1−T (A¯ 2 ε + Lf W1 T −1 ε + E1 ξ ) + η1 ,

(7.46)

where η1 is a positive scalar and ε is the upper bound of e, i.e., e < ε. Proof Consider the Lyapunov function V1 = eT1 P1 e1 . The time derivative of V1 can be obtained as

V˙1 = eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A¯ 2 e¯ 2 + 2eT1 P1 W1 f (T −1 z, t) − f (T −1 zˆ , t) + 2eT1 P1 E1 Δψ + 2eT1 P1 B1 (fa − ν) − kˆ 1 W1T P1 e1 2

≤ eT1 (As1 T P1 + P1 As1 )e1 + 2eT1 P1 A¯ 2 e¯ 2 + 2eT1 P1 W1 f (T −1 z, t) − f (T −1 zˆ , t) + 2eT1 P1 E1 Δψ + 2eT1 P1 B1 (fa − ν). It is easy to see that As1 T P1 + P1 As1 < 0 since As1 is a stable matrix by design. Then, from the Cauchy–Schwartz inequality, (7.9) and the fact that B1 is invertible, we get

V˙1 < 2eT1 P1 A¯ 2 e¯ 2 + 2eT1 P1 E1 Δψ + 2eT1 P1 W1 f (T −1 z, t) − f (T −1 zˆ , t) + 2eT1 P1 B1 (fa − ν1 ) ≤ 2P1 e1 (A¯ 2 ¯e2  + Lf W1 T −1 ¯e2  + E1 ξ ) − 2ηB1T P1 e1 

(7.47) ≤ 2B1T P1 e1  B1−T (A¯ 2 ε + Lf W1 T −1 ε + E1 ξ ) − η1 . It follows from (7.46) that  1 V˙1 ≤ −2η1 B1T P1 e1  ≤ −2η1 B1  λmin (P1 )V12 . This shows that the reachability condition [7] is satisfied and an ideal sliding motion is achieved and maintained after some finite time. This completes the proof. 

7.4 Estimation of Faults In this section, we will show how to obtain simultaneous estimation of actuator and sensor faults using observers designed in Sect. 7.3. It is shown in Theorem 7.3 that an ideal sliding motion (7.45) will take place after some finite time if the condition

156

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

(7.46) is satisfied. During the sliding motion, (7.14) becomes 0 = A¯ 2 e¯ 2 + W1 (f (T −1 z, t) − f (T −1 zˆ , t)) + B1 (fa − νeq ) + E1 Δψ,

(7.48)

where νeq denotes the equivalent output error injection signal to maintain the sliding motion [8], and can be approximated to any degree of accuracy by replacing (7.9) with BT P1 (C −1 S1 y − zˆ1 ) , (7.49) ν ≈ (ρa + η) T 1 −11 B1 P1 (C1 S1 y − zˆ1 ) + δ where δ is a small positive scalar to reduce the chattering effect. Since B1 is invertible, (7.48) can be further rewritten as

fa − νeq = −B1−1 A¯ 2 e¯ 2 + W1 (f (T −1 z, t) − f (T −1 zˆ , t)) + E1 Δψ .

(7.50)

Computing the L2 norm of (7.50) yields

fa − νeq L2 = B1−1 A¯ 2 e¯ 2 + W1 (f (T −1 z, t) − f (T −1 zˆ , t)) + E1 Δψ L2 ≤ (σmax (B1−1 A¯ 2 ) + σmax (B1−1 W1 )Lf T −1 )¯e2 L2 + σmax (B1−1 E1 )ΔψL2 ≤ (σmax (B1−1 A¯ 2 ) + σmax (B1−1 W1 )Lf T −1 )¯eL2 + σmax (B1−1 E1 )ΔψL2 .

(7.51)

√ Since eL2 ≤ σmax (H −1 ) μΔψL2 , we get fa − νeq L2 ≤

√ μ(σmax (B1−1 A¯ 2 ) + σmax (B1−1 W1 )Lf T −1 )σmax (H −1 )

(7.52) + σmax (B1−1 E1 ) ΔψL2 .

It follows that sup

ΔψL2 =0

fa − νeq L2 √ = μβ1 + β2 , ΔψL2

(7.53)

where β1 = (σmax (B1−1 A¯ 2 ) + σmax (B1−1 W1 )Lf T −1 )σmax (H −1 ) and β2 = σmax (B1−1 E1 ). √ Thus for a small ( μβ1 +β2 )ΔψL2 , the actuator faults fa can be approximated as fˆa (t) ≈ (ρa + η)

B1T P1 (C1−1 S1 y − zˆ1 )

B1T P1 (C1−1 S1 y − zˆ1 ) + δ

.

(7.54)

7.4 Estimation of Faults

157

Using observer (7.11), the estimated state zˆ¯2 of the augmented state vector z¯2 can be obtained with the prescribed performance. Then sensor faults fs can be obtained from zˆ¯2 by   fˆs ≈ 0 Iq zˆ¯2 .

(7.55)

7.5 Simulation Results To illustrate the effectiveness of the proposed FE schemes, we consider the following system where ⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 1 0 0 0 1 0 ⎢ −48.6 −1.25 48.6 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , B = ⎢ 21.6 ⎥ , E = ⎢ 0 ⎥ , W = ⎢ −0.333 ⎥ , A=⎢ ⎣ 0 ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ 0 0 10 0 1 0 ⎦ 1.95 0 −1.95 0 0 1 0 ⎡ ⎡ ⎤ ⎤ 1000 1 C = ⎣ 0 1 0 0 ⎦ , D = ⎣ 0 ⎦ , f (x, t) = sin(x3 ). 0010 0 In the simulation, Δψ is assumed to be 0.1 sin(t) and faults are given as ⎧ ⎨0 fa = 0.1 sin(0.2π(t − 10)) ⎩ 0 ⎧ t ≤ 15 s ⎨0 15 s < t < 30 s fs = 0.1 ⎩ 0 t ≥ 30 s

t ≤ 10 s 10 < t < 30 s t ≥ 30 s

Introduce nonsingular transformation matrices T and S with ⎡

0 ⎢1 T =⎢ ⎣0 0

1 0 0 0

0 0 1 0

⎤ ⎡ ⎤ 0 010 0⎥ ⎥, S = ⎣1 0 0⎦. 0⎦ 001 1

Then the original system can be transformed into two subsystems in the form of (7.4) and (7.5).

158

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

Choose μ = 1e − 10. Solving the LMI optimization problem formulated in Theorem 7.2 gives the following solution: P1 = 0.0158, U = −1.8558, ⎡ ⎤ 1.0346 −0.0000 −0.8411 0.0299 ⎢ −0.0000 1.0645 0.0000 0.0000 ⎥ ⎥ P2 = ⎢ ⎣ −0.8411 0.0000 1.8788 0.8411 ⎦ , 0.0299 0.0000 0.8411 1.0346 ⎤ ⎡ −0.0000 −0.3272 1.0000 1.0000 0.0000 −0.0000 ⎢ 0.0000 −1.0645 1.0000 1.0000 0.0000 0.0000 ⎥ ⎥ X=⎢ ⎣ −0.0000 −0.3930 1.0000 1.0000 0.0000 −0.0000 ⎦ , 0.0000 0.3272 1.0000 1.0000 −0.0000 0 ⎡ ⎤ −0.3153 −2.9370 ⎢ 1.6021 −2.1659 ⎥ ⎥ Y =⎢ ⎣ −1.8485 3.6970 ⎦ . −1.7838 −0.0000 Then As1 , F0 , M0 , L0 , and N0 can be computed as As1 = −117.6568, ⎡ −1.3570 −0.6375 ⎢ 1.5050 −2.0346 F0 = ⎢ ⎣ 0.6339 0.6822 −0.6149 −2.1215 ⎡ 1.0000 −1.0000 ⎢ −0.0000 0.0000 M0 = ⎢ ⎣ 0.0000 −1.0000 −1.0000 1.0000

⎡ ⎤ ⎤ −10.0000 −1.3570 0 −10.0000 ⎢ ⎥ −0.0000 1.5050 ⎥ ⎥ , L0 = ⎢ −0.0000 −0.0000 ⎥ , ⎣ ⎦ −10.0000 −1.3161 0.0000 −10.0000 ⎦ 10.0000 −0.6149 0 10.0000 ⎡ ⎤ ⎤ 0 0 0.0000 1.0000 ⎢ 0.0000 1.0000 ⎥ 0 0⎥ ⎥ , N0 = ⎢ ⎥ ⎣ −0.0000 1.0000 ⎦ . 1.0000 0 ⎦ 0 0 1.0000 −1.0000

The process of computing above parameters is presented in the following file chapter7_lmi.m. File chapter7_lmi.m

7.5 Simulation Results

159

160

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

7.5 Simulation Results

161

162

7 Simultaneous Estimation of Actuator and Sensor Faults Using SMO and UIO

x1

5 0 −5

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

30

35

40

x2

2 0 −2

x3

5 0 −5

x4

0.2 0 −0.2

Time (s) Fig. 7.1 System states x and their estimates xˆ (actual state: solid line; estimated state: dashed line) 0.2 Actual fa Estimated fa

0.15

Magnitude

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2

0

5

10

15

20

Time (s) Fig. 7.2 Actuator fault fa and its estimate fˆa

25

30

35

40

7.5 Simulation Results

163

0.16 Actual fs

0.14

Estimated fs

Magnitude

0.12 0.1 0.08 0.06 0.04 0.02 0 −0.02

0

5

10

15

20

25

30

35

40

Time (s) Fig. 7.3 Sensor fault fs and its estimate fˆs

The simulation results are obtained by running the file chapter7.mdl and the plots are obtained from chapter7_plot.m. In the simulation, we have selected η = 9.9, δ = 0.03, and lk1 = lk2 = 1. The initial state is chosen as x(0) = [0.1, 0.2, 0, −0.1]T and xˆ (0) = [0, 0, 0, 0]T . The state and fault estimation results are shown in Figs. 7.1, 7.2 and 7.3. It can be seen that the proposed method can estimate the faults as well as the system states with satisfactory accuracy despite the existence of system uncertainties.

7.6 Conclusions In this chapter, we have developed a new scheme which consists of an SMO and an UIO to simultaneously estimate actuator and sensor faults for uncertain Lipschitz nonlinear systems. Sufficient conditions for the existence of the observers have been established and expressed as an LMI optimization problem. The simulation results have shown the effectiveness of the proposed observers on estimating both the sensor and actuator faults and the states.

References 1. Gao C, Zhao Q, Duan G (2013) Robust actuator fault diagnosis scheme for satellite attitude control systems. J Frankl Inst 350:2560–2580 http://dx.doi.org/10.1016/j.jfranklin.2013.02.021 2. Edwards C, Tan CP (2006) Sensor fault tolerant control using sliding-mode observers. Control Eng Pract 14(8):897–908

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3. Alwi H, Edwards C, Tan CP (2009) Sliding mode estimation schemes for incipient sensor faults. Automatica 45:1679–1685 4. Liu M, Shi P (2013) Sensor fault estimation and tolerant control for Itˆo stochastic systems with a descriptor sliding mode approach. Automatica 49:1242–1250 5. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 6. Rao C, Mitra S (1971) Generalized inverse of matrices and its applications. Wiley, New York 7. Utkin VI (1992) Sliding modes in control optimization. Springer, Berlin 8. Edwards C, Spurgeon SK, Patton RJ (2000) Sliding mode observers for fault detection and isolation. Automatica 36:541–553

Chapter 8

Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

We have discussed the issues of fault detection and isolation (FDI) and fault estimation (FE) for normal Lipschitz nonlinear systems in previous chapters. In this chapter, we will focus on descriptor systems which can be considered as a generalization of normal state-space systems, and develop an FE method for this type of systems.

8.1 Introduction During the past several decades, various types of FE methods have been developed using several approaches, e.g., sliding-mode observer (SMO) [1–5], unknown input observer (UIO) [6, 7], adaptive observer (AO) [8, 9], and descriptor observers [10–12]. It should be noted that these results only focused on FE for normal state-space systems. Few efforts have been made to investigate FE for descriptor systems (also known as singular systems, generalized systems, and implicit systems). Descriptor systems can be considered as a generalization of state-space systems where algebraic relations resulting from the interaction of the process dynamics exist. This class of systems can be used to model many engineering systems. Applications can be found in aircraft systems, chemical systems, power systems, and economic systems. Therefore, it is of significance to design and implement fault diagnosis schemes for descriptor systems due to their extensive application in practical engineering. Research results on this topic are very limited. In [7], a proportionalintegral observer was designed for descriptor systems with faults and unknown inputs. In [13], an SMO-based method was employed to detect, isolate, and reconstruct faults. In [14], proportional multiple-integral and derivative (PMID) observer technique was presented to estimate fault signals and their derivatives. In [15] a model-

© Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_8

165

166

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

based fault estimation method was developed for discrete-time parameter-varying systems. In [16], the strong observability and detectability of a general class of linear system was investigated and unknown inputs were estimated. In the literature, most of the existing results considered the FE for linear descriptor systems. The study on nonlinear descriptor systems is more difficult and challenging since this class of systems not only has singular nature of the algebraic constraints, but also possesses nonlinearities. In [17], a descriptor estimator technique was presented to estimate output noises and sensor faults for systems with measurement output noises. In [18], a robust state-space observer was proposed to estimate actuator faults and their finite time derivatives. And in [19], an adaptive FE observer was designed to approximate the actuator fault by exploiting the online learning ability of radial basis function (RBF) neural networks. It is worth noting that the work in [17] focuses on sensor faults, while the results in [18, 19] are for actuator faults only. Compared with single-fault problems, multiple-fault ones are more complicated. Since in many practical systems, actuators and sensors may become faulty at the same time, it is desirable to consider actuator and sensor faults under one unified framework, which motivates the present study to develop a novel FE approach for nonlinear descriptor systems which are subject to actuator and sensor faults simultaneously. Inspired by the observer design in [12], we develop a robust H∞ adaptive descriptor observer to investigate the problem of simultaneously estimating actuator and sensor faults for Lipschitz nonlinear descriptor systems in this chapter. One distinct advantage of the proposed method is that the restrictive transformations, as in [17], are not needed and only original coefficient matrices are utilized in the observer design. By taking sensor faults as auxiliary states, an augmented descriptor system is constructed. For this system, an adaptive descriptor observer is developed such that the simultaneous estimation of the original system states, sensor and actuator faults can be obtained. Unlike in [12] where a high gain is chosen to reduce the effects of sensor faults on estimation errors, our method integrates a prescribed H∞ disturbance attenuation level into the observer to minimize the effects of system uncertainties and the derivative of actuator and sensor faults. Sufficient conditions for the existence of the proposed observer have been derived and given in terms of bilinear matrix inequalities (BMIs). The parameters of the observer are obtained by solving BMIs using the cone complementarity linearization algorithm [20]. The rest of the chapter is organized as follows: Sect. 8.2 introduces the problem and some mathematical preliminaries required for designing observers. In Sect. 8.3, the proposed observer is presented. The sufficient conditions of the stability of the observer are derived and the design procedure is provided using LMI. The results of simulation are shown in Sect. 8.4 with conclusions in Sect. 8.5.

8.2 Problem Formulation

167

8.2 Problem Formulation Consider a nonlinear descriptor system with actuator and sensor faults as E x˙ (t) = Ax(t) + f (x, t) + B(u(t) + fa (t)) + Δψ(t), y(t) = Cx(t) + Dfs (t).

(8.1)

where x ∈ R n is the state vector; u ∈ R m is the control input vector; y ∈ R p is the measurement output vector; fa ∈ R m and fs ∈ R q denote respectively the unknown vectors of actuator and sensor faults; Δψ(t) ∈ R n represents the system uncertainties and belongs to L2 [0, ∞); f (x, t) is the known nonlinear continuous term; E ∈ R n×n , A ∈ R n×n , B ∈ R n×m , C ∈ R p×n and D ∈ R p×q (p ≥ q) are known constant matrices; E may be rank deficient, i.e., rank(E) = l < n, and D is of full rank, i.e., rank(D) = q. When E = In , System (8.1) becomes a normal state-space systems. Remark 8.1 It is worth noting that the uncertainty Δψ considered in this chapter is unstructured and can represent various types of uncertainties such as high-frequency noises, slow-varying signals,and nonlinear functions of system states. This type of uncertainty is more general than the structured uncertainty which has been studied in the literature (e.g., [14, 21–23]). Before stating the main results of this chapter, the following assumptions are made. Assumption 8.1 The system (E, A, C) is observable, i.e.,  rank

E C



 = n, rank

 sE − A = n, ∀s ∈ C . C

Assumption 8.2 The nonlinear term f (x, t) is assumed to be known and Lipschitz about x uniformly, i.e., ∀ x, xˆ ∈ R n , f (x, t) − f (ˆx , t) ≤ Lf x − xˆ ,

(8.2)

where Lf is referred to as the Lipschitz constant and is independent of x and t. Assumption 8.3 The faults fa and fs are differentiable after their occurrence (note that the faults need not to be differentiable at the time of their occurrence), and both f˙a and f˙s belong to L2 [0, ∞). Remark 8.2 When E = In , Assumption 8.1 becomes the detectability definition for normal systems. Assumption 8.3 states that the continuity at the fault occurrence time is not required, which means that the faults can occur suddenly. The assumption also implies that the faults can be unbounded. Thus it can be seen that the considered

168

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

faults can represent a large class of faults, such as incipient faults, abrupt faults, and unbounded faults. Assumption 8.3 is more general compared with some of the previous papers, e.g., in [24], the fault is assumed to belong L2 [0, ∞); In [2, 22, 25], the fault is assumed to be bounded and the upper bound is known; The first order derivative of actuator fault is assumed to be zero in [26] and within a known bound in [9].

8.3 Design of Observer Taking the sensor faults fs as auxiliary states, an augmented descriptor system can be constructed as ¯ + f¯ (x, t) + B(u ¯ Δψ, ¯ z = Az ¯ + fa ) + G ¯ E˙ (8.3) ¯ y = Cz, where

    E 0 x ∈ R n+q , E¯ = ∈ R (n+q)×(n+q) , 0 Iq fs     B A0 ∈ R (n+q)×m, A¯ = ∈ R (n+q)×(n+q) , B¯ = 0 00     ¯ = In 0 ∈ R (n+q)×(n+q,) , C¯ = C D ∈ R p×(n+q) , G 0 Iq     f (x, t) ¯ = Δψ ∈ R n+q . f¯ (x, t) = ∈ R n+q , Δψ 0 f˙s z=

(8.4)

System (8.1) now becomes the augmented descriptor system in the form of (8.3) with both the original system state vector x and sensor fault vector fs being the components of the new state vector z. Therefore, if an effective observer can be designed for System (8.3), then an accurate estimation of x and fs can directly be obtained. In this section, a new H∞ adaptive descriptor observer will be developed for the augmented System (8.3) to generate the asymptotic estimation of z as well as fa . Remark 8.3 In [23, 27], both actuator and sensor faults are taken as auxiliary states, and an augmented descriptor system is then constructed. If an estimator can be designed for this system, then estimates of states, actuator and sensor faults can be obtained readily. Different from the methods in [23, 27], in our proposed scheme, we only augment sensor faults into the state vector and use adaptive algorithm to estimate actuator faults.

8.3 Design of Observer

169

It follows from Assumption 8.1 and the structure of E¯ and C¯ that 

E¯ rank ¯ C



⎡

⎤ E 0 = rank ⎣ 0 Iq ⎦ = n + q. CD

(8.5)

Therefore, there exists a matrix LD ∈ R (n+q)×p such that the matrix S¯ = E¯ + LD C¯ is nonsingular. Notice that when E = In , LD can be selected to be 0 to make S¯ nonsingular. Based on this discussion, we now propose the following observer for System (8.3): ¯ −1 C¯ T ey , ¯ z + B(u ¯ + fˆa ) + LP y + f¯ (ˆx , t) + α SP w˙ = (A¯ − LP C)ˆ ¯ −1 (w + LD y), zˆ = (E¯ + LD C) (8.6) ˙ ¯ −1 C¯ T ey , fˆa = Γ F1 ey + Γ F2 e˙ y + αΓ F2 CP ¯ z. yˆ = Cˆ where w ∈ R n+q is the middle variable, zˆ = [ˆx T , fˆsT ]T ∈ R n+q is the estimation of [x T , fsT ]T , fˆa is the estimated actuator fault, ey = y − yˆ is the output estimation error, Γ ∈ R m×m is a symmetric positive definite matrix which represents the learning rate, LP ∈ R (n+q)×p and LD ∈ R (n+q)×p are, respectively the proportional gain and derivative gain to be designed, matrices F1 ∈ R m×p , F2 ∈ R m×p , P ∈ R (n+q)×(n+q) > 0 and the scalar α > 0 are parameters to be determined. The proposed observer (8.6) can be rewritten as ¯ −1 C¯ T ey . ¯ z + f¯ (ˆx , t) + B(u ¯ + fˆa ) + LP y + LD y˙ + α SP S¯ z˙ˆ = (A¯ − LP C)ˆ

(8.7)

Adding LD y˙ to both sides of (8.3) yields ¯ z = (A¯ − LP C)z ¯ Δψ. ¯ + f¯ (x, t) + B(u ¯ + fa ) + LP y + LD y˙ + G ¯ S˙

(8.8)

Define the state estimation error as e = z − zˆ . Then subtracting (8.7) from (8.8), the error dynamics is obtained as ¯ −1 C¯ T Ce, ¯ e = (A¯ − LP C)e ¯ Δψ ¯ + [f¯ (x, t) − f¯ (ˆx , t)] + Be ¯ f +G ¯ ¯ − α SP S˙

(8.9)

where ef = fa − fˆa represents the actuator fault estimation error and has the dynamics as follows: ¯ − Γ F2 C¯ e˙ − αΓ F2 CP ¯ −1 C¯ T Ce. ¯ e˙ f = f˙a − Γ F1 Ce

(8.10)

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8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

Equations (8.9) and (8.10) can be represented in a more compact form as ⎛

⎞



     ⎜   ⎟  ¯ e˙ LP  ¯ ⎟ e S¯ 0 f (x, t) − f¯ (ˆx , t) ⎜ A¯ B¯ = − + C 0 ⎜ ⎟ Γ F1  ⎠ ef e˙ f 0 Γ F2 C¯ Im ⎝ 0 0



  ˜ 



 e˙˜

E˜

 +

A˜

K˜





C

e˜



f˜



¯ ¯ −1 C¯ T ¯ 0 Δψ SP G ¯ ˙fa −α Γ F2 CP ¯ −1 C¯ T Ce, 0 Im

  ˜ G

(8.11)

˜ Δψ

or equivalently ˜ e + f˜ + G ˜ Δψ ˜ −α E˜ e˙˜ = (A˜ − K˜ C)˜



 ¯ −1 C¯ T SP ¯ ¯ −1 C¯ T Ce. Γ F2 CP

Define the controlled estimation error r as   e , r=H ef

(8.12)

(8.13) 

 H1 0 where H is the prespecified weight matrix having the structure H = with 0 H2 (n+q)×(n+q) m×m H1 ∈ R and H2 ∈ R . The following theorem gives the existence condition of the proposed observer satisfying the following: ˜ = 0, 1. The observer error dynamics (8.12) is asymptotically stable when Δψ namely, there is no system uncertainty and fault, or the fault is time-invariant. ∞ 2. Under zero initial condition, the H∞ performance index J = 0 (r T r − ˜ ˜ ∈ L2 [0, ∞), where μ > 0 is ˜ T Δψ)dt < 0 holds for all nonzero Δψ μΔψ √ a prescribed scalar and μ represents the attenuation level of disturbance. Theorem 8.1 Consider System (8.1) with Assumptions 8.1–8.3. Given a positive scalar μ, if there exist matrices P = PT ∈ R (n+q)×(n+q) > 0, LP ∈ R (n+q)×p , F1 ∈ R m×p and F2 ∈ R m×p such that ⎡

⎤ ¯ Π1 + H1T H1 Π2 + C¯ T LPT S¯ −T C¯ T F2T PS¯ −1 G 0 ⎢ Π T + F C¯ S¯ −1 L C¯ ¯ Γ −1 ⎥ −F2 C¯ S¯ −1 G Π3 + H2T H2 ⎢ ⎥ 2 P Λ := ⎢ 2 ⎥ < 0, ¯ T S¯ −T C¯ T F2T ¯ T S¯ −T P ⎣ −G −μIn+q 0 ⎦ G 0 Γ −1 0 −μIm (8.14)

8.3 Design of Observer

171

where Π1 = PS¯ −1 A¯ + A¯ T S¯ −T P − PS¯ −1 LP C¯ − C¯ T LPT S¯ −T P + P, Π2 = PS¯ −1 B¯ − A¯ T S¯ −T C¯ T F2T − C¯ T F1T and Π3 = −F2 C¯ S¯ −1 B¯ − B¯ T S¯ −T C¯ T F2T + Γ −1 , then the error dynamics (8.12) is asymptotically stable with the prescribed H∞ performance. Proof Consider the Lyapunov function as ˜ e, V (˜e) = e˜ T P˜

(8.15)

 P 0 . Note that this quadratic form of P˜ has been used in many 0 Γ −1 previous papers such as [9, 28]. ˜ E, ˜ C˜ and e˜ , it is easy to see that E˜ is nonsingular and From the structure of P, where P˜ =



E˜ P˜ −1 C˜ T =



 ¯ −1 C¯ T SP ¯ ˜ ¯ −1 C¯ T , Ce = C e˜ . Γ F2 CP

(8.16)

Therefore, the error dynamics (8.12) can further be written as ˜ e + E˜ −1 f˜ + E˜ −1 G ˜ Δψ ˜ − α P˜ −1 C˜ T C˜ e˜ . e˙˜ = E˜ −1 (A˜ − K˜ C)˜

(8.17)

Then the time derivative of V (˜e) along the trajectories of error dynamics (8.17) can be shown to be ˜e V˙ (˜e) = e˜ T P˜ e˙˜ + e˙˜ T P˜ T ˜ ˜ −1 ˜ ˜ + (A˜ − K˜ C) ˜ T E˜ −T P]˜ ˜ e + 2˜eT P˜ E˜ −1 G ˜ Δψ ˜ = e˜ [PE (A − K˜ C) + 2˜eT P˜ E˜ −1 f˜ − 2αC˜ e˜ 2 .

(8.18)

We further assume that the nonlinear function f (x, t) satisfies the following Lipschitz constraint: f (x, t) − f (ˆx , t) ≤ U(x − xˆ ),

(8.19)

where U ∈ R n×n is a known constant matrix. Provided that rank[ U T C T ]T = rank(C), there exists a matrix Z = UC + such that U = ZC. Therefore, we have f (x, t) − f (ˆx , t) ≤ ZC(x − xˆ ) ≤ θ0 C(x − xˆ ) ≤ Lf x − xˆ ,

(8.20)

where θ0 is a positive scalar and can be chosen as θ0 = Lf /C. Since for any given vectors x, y and a positive definite matrix P, the inequality x T y + yT x ≤ x T Px + 2yT P−1 y holds [18], it follows that

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8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

˜ e + 2f˜ T E˜ −T P˜ E˜ −1 f˜ 2˜eT P˜ E˜ −1 f˜ ≤ e˜ T P˜ ˜ e + 2P˜ 1/2 E˜ −1 2 f˜ 2 ≤ e˜ T P˜ ˜ e + 2θ02 P˜ 1/2 E˜ −1 2 C(x − xˆ )2 ≤ e˜ T P˜ ˜ e + 2θ02 P˜ 1/2 E˜ −1 2 C˜ e˜ 2 . ≤ e˜ T P˜

(8.21)

If the scalar α in (8.6) is chosen to satisfy α ≥ Lf2 P˜ 1/2 E˜ −1 2 /C2 , then 2αC˜ e˜ 2 ≥ 2θ02 P˜ 1/2 E˜ −1 2 C˜ e˜ 2.

(8.22)

Substituting (8.21) and (8.22) into (8.18)yields ˜ + (A˜ − K˜ C) ˜ T E˜ −T P˜ + P]˜ ˜ e + 2˜eT P˜ E˜ −1 G ˜ Δψ. ˜ (8.23) V˙ (˜e) ≤ e˜ T [P˜ E˜ −1 (A˜ − K˜ C) ˜ C, ˜ E, ˜ P˜ and K, ˜ after some algebraic ˜ = 0, from the structure of A, When Δψ manipulation, we get V˙ (˜e) ≤



e ef

T 

  Π2 + C¯ T LPT S¯ −T C¯ T F2T Π1 e . ef Π2T + F2 C¯ S¯ −1 LP C¯ Π3 



(8.24)

W

If there exist feasible solutions to (8.14), then we can conclude that W < 0. Thus V˙ (˜e) < 0. This implies that e˜ → 0 as t → ∞. Therefore the error dynamics (8.12) ˜ = 0. is asymptotically stable when Δψ Now we will establish the H∞ performance for the proposed observer. Under zero initial conditions, we have  ∞ ˜ ˜ T Δψ)dt (r T r − μΔψ J= 0 ∞   ∞  ˜ + V˙ dt − ˜ T Δψ = r T r − μΔψ V˙ dt 0 0  ∞  ˜ + V˙ dt − V (∞) + V (0) ˜ T Δψ = r T r − μΔψ 0 ∞   ˜ + V˙ dt. ˜ T Δψ ≤ r T r − μΔψ

(8.25)

0

It follows from (8.23) that ˜ T Δψ ˜ + V˙ ≤ ϑ T Λϑ, r T r − μΔψ ˜ T ]T and Λ is defined in (8.14). where ϑ = [eT , eTf , Δψ

(8.26)

8.3 Design of Observer

173

If (8.14) holds, then J < 0, namely 

∞

˜ ˜ T Δψ)dt < 0. (r T r − μΔψ

(8.27)

Therefore the H∞ performance has been established. This completes the proof.



0

˜ = [Δψ T , f˙sT , f˙aT ]T on the Remark 8.4 The effect of the uncertainty term Δψ estimation error e˜ is decided by the value of μ. The accuracy of state and fault estimates increases with decrease in value of μ. Therefore, the robustness of the proposed observer can be enhanced by minimizing μ. The minimum value of μ can be obtained by solving the following optimization problem: minimize μ subject to P > 0 and (8.14). This optimization problem seeks two objectives. The first one is to find proper matrices P, LP , F1 and F2 such that the proposed observer is asymptotically stable; while the second objective is to boost the robustness of the observer against uncer˜ by minimizing the L2 gain between the controlled estimation error r tainties Δψ ˜ and Δψ. Remark 8.5 The proposed method is inspired by the PID observer design in [12], but it is not a trivial extension. We now illustrate the following discussion to show the differences between these two methods. It can be found that the singular nature of the algebraic constraints, Lipschitz nonlinearity and uncertainties have not been taken into account in [12], and only constant actuator fault has been considered. The dynamics of actuator fault estimation considered here has the following form: ˙ ¯ fˆa = KI Ce,

(8.28)

where KI is the integral gain. Compared with our proposed observer in (8.6), the derivative of ey is added as an additional term such that the rapidity and accuracy of fault estimation can be further enhanced. Moreover, the error dynamics in [12] is given by ⎡

⎤ −L1 ˜ e + ⎣ I + CL1 ⎦ L2−1 Dfs , e˙˜ = E˜ −1 (A˜ − K˜ C)˜ 0

(8.29)

      LP L1 E¯ + L¯ D C¯ 0 ˜ ˜ ,K = and LD = . In the approach of [12], LD where A = KI L2 0 Im was chosen as L1 = 0 and L2 = diag(β1 , . . . , βp ) with β1 , . . . , βp being reasonable large positive scalars to reduce the effect of sensor fault fs . Instead, in this chapter,

174

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

a prescribed H∞ disturbance attenuation level is integrated into the observer to minimize the effect of the derivative of fs . Hence, our method is more robust and practical to be implemented than high gain observers introduced in [12, 14], which requires a higher computation cost. Note that (8.14) is a nonlinear matrix inequality, which can not be solved directly using the LMI toolbox of MATLAB. Therefore we need to make a further development to give modified conditions in LMI forms such that P, LP , F1 , and F2 can be computed. Before doing so, we recall the following lemma. Lemma 8.1 (See [28, 29]) Given matrices Q = QT , F, M, and N of appropriate dimensions, then Q + MFN + N T F T M T < 0

(8.30)

for all F satisfying F T F ≤ I, if and only if there exists some scalar ε > 0 such that Q + εMM T + ε−1 N T N < 0.

(8.31)

Theorem 8.2 Consider System (8.1) with Assumptions 8.1–8.2. If there exist matrices P = PT ∈ R (n+q)×(n+q) > 0, P¯ = P¯ T ∈ R (n+q)×(n+q) > 0, Y ∈ R (n+q)×p , F1 ∈ R m×p and F2 ∈ R m×p , and positive scalars ε and ε¯ , such that the following optimization problem has a solution: minimize μ subject to   In+q P¯ ≥ 0, (8.32) P¯ In+q ⎤ ⎡ ¯ M1 + H1T H1 M2 PS¯ −1 G 0 0 C¯ T Y T ⎢ ¯ Γ −1 F2 C¯ M2T M3 + H2T H2 −F2 C¯ S¯ −1 G 0 ⎥ ⎥ ⎢ ⎥ ⎢ ¯ T S¯ −T C¯ T F2T −μIn+q ¯ T S¯ −T P −G ⎥ ⎢ G 0 0 0 ⎥ < 0, (8.33) ⎢ ⎢ −1 0 Γ 0 −μIm 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ 0 C¯ T F2T 0 0 −εIn+q 0 ⎦ Y C¯ 0 0 0 0 −¯ε In+q ¯ PP = In+q , εε¯ = 1, (8.34) where M1 = PS¯ −1 A¯ + A¯ T S¯ −T P − Y C¯ − C¯ T Y T + P, M2 = PS¯ −1 B¯ − A¯ T S¯ −T C¯ T F2T − C¯ T F1T and M3 = −F2 C¯ S¯ −1 B¯ − B¯ T S¯ −T C¯ T F2T + Γ −1 , then for a given positive scalar α, the observer error dynamics (8.12) is asymptotically stable with the prescribed ¯ −1 Y . H∞ performance. The observer gain LP can then be computed from LP = SP

8.3 Design of Observer

175

Proof Let Y = PS¯ −1 LP , then (8.14) can be rewritten as ⎡

⎤ ¯ M1 + H1T H1 M2 PS¯ −1 G 0 ⎢ ¯ Γ −1 ⎥ M2T M3 + H2T H2 −F2 C¯ S¯ −1 G ⎢ ⎥ T T −T T −T T ⎣ G ¯ S¯ C¯ F2 −μIn+q ¯ S¯ P −G 0 ⎦ 0 −μIm 0 Γ −1 ⎡ T T⎤ ⎤ ⎡ 0 C¯ Y ⎢ 0 ⎥ −1    ⎢ F2 C¯ ⎥ −1  ⎥ ⎥ +⎢ 0 C¯ T F2T 0 0 + ⎢ Y C¯ 0 0 0 < 0. ⎣ 0 ⎦P ⎣ 0 ⎦P 0 0

(8.35)

It follows from Lemma 8.1 that for matrix P satisfying P−2 ≤ In+q ,

(8.36)

the inequality (8.35) holds if and only if there exists some ε > 0 such that ⎤ ¯ M2 PS¯ −1 G 0 M1 + H1T H1 ⎢ ¯ Γ −1 ⎥ M2T M3 + H2T H2 −F2 C¯ S¯ −1 G ⎥ ⎢ T T −T T −T T ⎣ G ¯ S¯ C¯ F2 −μIn+q ¯ S¯ P −G 0. ⎦ 0 −μIm 0 Γ −1 ⎡ ⎡ T T⎤ ⎤ 0 C¯ Y ⎢ ¯ ⎥ ⎢ 0 ⎥   −1 ⎢ F2 C ⎥ ⎥ ¯ ¯T T +ε⎢ ⎣ 0 ⎦ Y C 0 0 0 + ε ⎣ 0 ⎦ 0 C F2 0 0 < 0. 0 0 ⎡

(8.37)

Using Schur complement, (8.36) and (8.37) can further be written as 

 In+q P−1 ≥ 0, (8.38) P−1 In+q ⎡ ⎤ ¯ M1 + H1T H1 M2 PS¯ −1 G 0 0 C¯ T Y T ⎢ ⎥ ¯ Γ −1 F2 C¯ M2T M3 + H2T H2 −F2 C¯ S¯ −1 G 0 ⎢ ⎥ T T −T T −T T ⎢ G ⎥ ¯ S¯ C¯ F2 −μIn+q ¯ S¯ P −G 0 0 0 ⎢ ⎥ < 0. −1 ⎢ ⎥ 0 −μI 0 0 0 Γ m ⎢ ⎥ ⎣ ⎦ 0 C¯ T F2T 0 0 −εIn+q 0 Y C¯ 0 0 0 0 −ε−1 In+q (8.39) Letting P¯ = P−1 and ε¯ = ε−1 , (8.32)–(8.34) can immediately be obtained. This completes the proof.  Remark 8.6 The scalar α is chosen so that the condition α ≥ Lf2 P˜ 1/2 E˜ −1 2 /C2 is satisfied. If the value of Lf is unknown, we can select arbitrarily large α such that the above inequality holds. As seen in the Theorems 8.1 and 8.2, the introduction

176

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

of this positive scalar eliminates Lf from the LMI formulation (8.33), thus makes the feasibility of the LMI be independent of the knowledge of Lf . This is one of the distinct advantages of the proposed observer over some of the existing methods introduced in [9, 10, 22, 30]. If the value of Lf is unknown or exceeds the admissible value, those methods often fail to find feasible LMI solutions. Using the cone complementary linearization algorithm, the nonconvex problem formulated by (8.32)–(8.34) can be converted into the following nonlinear minimization problem subject to LMIs: minimize trace(PP¯ + εε¯ In+q + μIn+q ) subject to (8.32)–(8.33) and     P I εI ≥ 0, ≥ 0. I ε¯ I P¯

(8.40)

To solve this optimization problem, the following iterative algorithm is being used: Step 1: Set i = 0 and solve (8.32), (8.33) and (8.40) to obtain the initial solutions ¯ Y , F1 , F2 , ε, ε¯ , μ)0 (P, P, ¯ i + εε¯ i I + Step 2: Solve the LMI minimization problem: Minimize trace(PP¯ i + PP i ε¯ ε I + μI) subject to (8.32)–(8.33) and (8.40). The obtained solutions are denoted ¯ Y , F1 , F2 , ε, ε¯ , μ)i+1 . as (P, P, Step 3: If the obtained solutions satisfy (8.36) and (8.37), then compute the pro¯ i+1 )−1 Y i+1 and EXIT. Otherwise, set i = i + 1 and portional gain from LP = S(P return to Step 2. If the above optimization problem is solvable, then the proposed observer in the form of (8.6) can be constructed and asymptotic estimates of states and faults can be obtained. The estimates of system states x and sensor faults fs can be obtained directly from the state estimate zˆ of System (8.3). Specifically, the estimates of states and sensor faults can be obtained as xˆ = [In 0n×q ]ˆz and fˆs = [0q×n Iq ]ˆz, respectively. Moreover, the estimates of actuator faults can be calculated from (8.6) as fˆa (t) = Γ



t

¯ −1 C¯ T ey + F2 e˙ y )dτ, (F1 ey + αF2 CP

(8.41)

tf

where tf is the time when an actuator fault occurs. Remark 8.7 The proposed method is also valid for simultaneous actuator and sensor faults estimation for a normal system, that is when E = In . We can choose LD = 0 for this case. The effectiveness will be shown in the following section.

8.4 Simulation Results

177

8.4 Simulation Results 8.4.1 Example of FE for Descriptor Systems To illustrate the effectiveness of the proposed observer for estimating faults of nonlinear descriptor systems, we consider the following plant: 

00 01



x˙ 1 x˙ 2

       1 x1 0 0.1 (u + fa ) + + + 0 x2 0.1347 sin(x2 ) 0.2      0 2 1 x1 + y= f, x2 1 s −1 2 (8.42)





=

1 0 1 −3



where d = 0.2 sin(20t) represents the disturbance. The input u = 4. The faults are in the form as follows: fs = 0,  0.2(t − 3) + 0.1 sin[ω(t − 3)] + 0.5e−t fa = 0

t ≥ 3s t < 3s

where ω represents the angular frequency. The effectiveness of the proposed method in estimating the actuator faults is demonstrated by considering different frequencies. Note that the system uncertainty Δψ considered in this example is structured and has the following decomposition: Δψ = Bd d,

(8.43)

where Bd ∈ R n×r is called the structural matrix which isusedto characterize the 0.1 . In order to apply structure of the uncertainty Δψ. In this simulation, Bd = 0.2 the proposed observer to systems with this special type of uncertainty, we only need ¯ in (8.4) to be to modify the matrix G   ¯ = Bd 0 ∈ R (n+q)×(q+r). G 0 Iq Selecting ⎡

⎡ ⎤ ⎤ 0.1 0 100 LD = ⎣ 0 0.1 ⎦ , H1 = ⎣ 0 1 0 ⎦ , H2 = 1, Γ = 100 5 10 001

178

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

and solving the optimization problem described by (8.32)–(8.33) and (8.40) gives μ = 0.2770,   F1 = 1.0865 −0.6920 ,   F2 = 0.1228 −0.0500 , ⎡ ⎤ 1.4342 −1.6312 0.4653 P = ⎣ −1.6312 12.7010 0.8428 ⎦ , 0.4653 0.8428 2.7035 ⎡ ⎤ 0.9087 0.1298 −0.1969 P¯ = ⎣ 0.1298 0.0989 −0.0532 ⎦ , −0.1969 −0.0532 0.4204 ⎡ ⎤ 5.4590 −0.9702 Y = ⎣ 2.3023 1.9405 ⎦ , 0.8544 8.3690 ε = 0.0756, ε¯ = 13.2202. The detailed process of computing above parameters is shown in the following file example_8.1.m. Example_8.1.m

8.4 Simulation Results

179

180

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

8.4 Simulation Results

181

182

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

8.4 Simulation Results

183

¯ −1 Y , the proportional It can be verified that PP¯ = I3 and εε¯ = 1. By using LP = SP gain LP can be computed as ⎡

⎤ 1.1074 −0.4934 LP = ⎣ 0.4760 0.1335 ⎦ 13.0522 30.1885

(8.44)

184

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

The simulation results are obtained by running the file example_8.1.mdl and the plots are obtained from example_8.1_plot.m. In using the simulation, we have selected the initial state as x2 (0) = −0.5, w(0) = [0, 0, 0]T and fˆa (0) = 0, and input as u(t) = 4. It can be computed that Lf2 P˜ 1/2 E˜ −1 2 /C2 = 1.0305. Thus we choose the parameter α as α = 2. It is worth noting that if the value of Lf is unknown, we can still select any arbitrarily large value for α such that α ≥ Lf2 P˜ 1/2 E˜ −1 2 /C2 can be guaranteed. The performance of the proposed method in simultaneously estimating actuator and sensor faults is compared with the method developed in [23]. When ω = 30 rad/s, the results of state and fault estimates are shown in Figs. 8.1, 8.2, 8.3, and 8.4. It can be seen that both methods can estimate states successfully and the proposed method estimates actuator and sensor faults more accurately.

−3.8 Actual x 1

−4

Estimate by proposed method Estimate by [142]

Magnitude

−4.2 −4.4 −4.6 −4.8 −5 −5.2 −5.4 −5.6

0

2

4

6

8

10

Time (s)

Fig. 8.1 x1 and its estimate xˆ 1 of Example 8.1

0 Actual x2

−0.2

Estimate by proposed method Estimate by [142]

−0.4

Magnitude

−0.6 −0.8 −1 −1.2 −1.4 −1.6 −1.8 −2

0

2

4

Time (s)

Fig. 8.2 x2 and its estimate xˆ 2 of Example 8.1

6

8

10

8.4 Simulation Results

185

1 0.9 0.8

Magnitude

0.7 0.6 0.5 0.4 0.3 0.2

Actual f a Estimate by proposed method

0.1

Estimate by [142]

0

4

4.5

5

5.5

6

6.5

7

7.5

8

Time (s)

Fig. 8.3 Actuator fault fa with ω = 30 rad/s and its estimate fˆa of Example 8.1

0.1 Actual fs

0.08

Estimate by proposed method Estimate by [142]

0.06

Magnitude

0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1

0

2

4

6

8

10

Time (s)

Fig. 8.4 Sensor fault fs and its estimate fˆs of Example 8.1

With the increase of the angular frequency of the fault signal, the accuracy of actuator fault estimate obtained by [23] decreases, while the proposed method remains almost the same. This can be seen from Figs. 8.5 and 8.6, where the performances of both methods on estimating signals with ω = 10 rad/s and ω = 50 rad/s are compared, respectively. The root mean square errors (RMSE) of signals with different frequencies are calculated in Table 8.1, which clearly exhibits that the performance of the proposed method is better when estimating actuator faults with high frequencies.

186

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems 1 0.9

Magnitude

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Actual fa Estimate by proposed method

0.1

Estimate by [142]

0

4

4.5

5

5.5

6

6.5

7

7.5

8

Time (s)

Fig. 8.5 Actuator fault fa with ω = 10 rad/s and its estimate fˆa of Example 8.1 1

0.9

Magnitude

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Actual fa Estimate by proposed method

0.1

Estimate by [142]

0

4

4.5

5

5.5

6

6.5

7

7.5

8

Time (s)

Fig. 8.6 Actuator fault fa with ω = 50 rad/s and its estimate fˆa of Example 8.1 Table 8.1 RMSE of actuator fault estimates ω (rad/s) Proposed method 10 20 30 40 50

0.0296 0.0312 0.0315 0.0329 0.0347

Method of [23] 0.0579 0.0677 0.0706 0.0734 0.0752

8.4 Simulation Results

187

8.4.2 Example of FE for Normal Systems In this example, we further test the effectiveness of the proposed observer for normal state-space systems. Here, we consider the single link flexible joint robot arm system with the following state-space form: ⎡

⎤ ⎡ x˙ 1 0 1 0 ⎢ x˙ 2 ⎥ ⎢ −48.6 −1.25 48.6 ⎢ ⎥=⎢ ⎣ x˙ 3 ⎦ ⎣ 0 0 0 x˙ 4 19.5 0 −19.5

⎤ ⎤⎡ ⎤ ⎡ ⎤ ⎡ x1 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0⎥ ⎥ ⎥ ⎢ x2 ⎥ + ⎢ 21.6 ⎥ (u + fa ) + ⎢ ⎦ ⎣ 0 1 ⎦ ⎣ x3 ⎦ ⎣ 0 ⎦ −3.33 sin(x3 ) x4 0 0

+ Δψ ⎡ ⎤ ⎡ ⎤ ⎤ x ⎡ 1 1 0 0 0 ⎢ 1⎥ x2 ⎥ ⎣ ⎦ 0 + fs , y = ⎣0 1 0 0⎦⎢ ⎣ x3 ⎦ 0 0011 x4



(8.45)

In this simulation, the vector of uncertainties Δψ is assumed to be Δψ = T 0 0 0 0.2 sin(10t) and faults are in the following forms: ⎧ t < 20 s ⎨0 20 s ≤ t < 30 s fa = sin(0.5t) + 0.2 cos(5t) ⎩ 0 t ≥ 30 s ⎧ 0 t < 15 s ⎪ ⎪ ⎨ 0.1(t − 15) 15 ≤ t < 20 s fs = 0.5 − 0.1(t − 20) 20 ≤ t < 25 s ⎪ ⎪ ⎩ 0 t ≥ 25 s Choosing ⎡

0 ⎢0 ⎢ LD = ⎢ ⎢0 ⎣0 0

0 0 0 0 0

⎤ ⎡ 0 00 ⎢0 0 0⎥ ⎥ ⎢ ⎢ 0⎥ ⎥ , H1 = ⎢ 0 0 ⎦ ⎣0 0 0 0 00

0 0 0 0 0

0 0 0 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ , H2 = 1, Γ = 100 0⎦ 1

and applying Theorem 8.2 gives the following parameters:

188

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

μ = 6.1267,   F1 = 10.3906 15.7547 −3.4573 ,   F2 = 0.0186 0.1768 0.1067 , ⎡ 18.2879 0.6298 −16.8765 −2.0048 ⎢ 0.6298 1.0567 −0.3982 −0.0213 ⎢ P=⎢ ⎢ −16.8765 −0.3982 25.7767 4.3800 ⎣ −2.0048 −0.0213 4.3800 1.9443 4.0048 0.2534 −0.2660 0.5937 ⎤ ⎡ −1.6335 −3.7078 4.9673 ⎢ −0.9541 26.7926 0.0193 ⎥ ⎥ ⎢ ⎥ LP = ⎢ ⎢ −0.1682 −0.3860 1.1232 ⎥ . ⎣ −4.5219 −7.7388 14.3440 ⎦ 7.3939 5.5092 −9.4568

⎤ 4.0048 0.2534 ⎥ ⎥ −0.2660 ⎥ ⎥, 0.5937 ⎦ 3.5324

The process of computing above parameters is shown in the following file example_8.2.m. Example_8.2.m

8.4 Simulation Results

189

190

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

8.4 Simulation Results

191

192

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

8.4 Simulation Results

193

The simulation results are obtained from example_8.2.mdl and the plots are obtained from example_8.2_plot.m. In the simulation, we have selected the initial state as x(0) = [0.15, 0.2, 0, −0.3]T , w(0) = [0, 0, 0, 0, 0]T and fˆa (0) = 0, and input as u(t) = 2 sin(2π t). It can be computed that Lf2 P˜ 1/2 E˜ −1 2 /C2 = 336.2547. Thus we choose the parameter α as α = 1000. Figures 8.7, 8.8, 8.9, and 8.10 show the trajectories of the actual states and their estimates. It can be seen from the figures that the proposed observer can estimate the states accurately, before and after the occurrence of any fault. The results of fault estimation are illustrated in Figs. 8.11 and 8.12. These figures clearly demonstrate that the proposed observer is able to estimate both actuator and sensor faults successfully, irrespective of uncertainties in the system. It should be noted that fa changes abruptly at time instants 20 s and 30 s, which indicates that the proposed observer has the ability to track abrupt faults. For the objective of comparison, the methods proposed in [10, 23, 25] are applied for this single link flexible joint robot arm system. However, it is found that for this particular system, there does not exist any solution to the LMIs proposed in [10, 23], which shows that the fault estimation methods in these two papers are not applicable for the considered system. The results of sliding-mode descriptor

194

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems 10 0

Magnitude

−10 −20 −30 −40

Actual x1 Estimated x1

−50

0

5

10

15

20

25

30

35

40

25

30

35

40

Time (s)

Fig. 8.7 x1 and its estimate xˆ 1 of Example 8.2 15 10

Magnitude

5 0 −5 −10

Actual x 2 Estimated x 2

−15

0

5

10

15

20

Time (s)

Fig. 8.8 x2 and its estimate xˆ 2 of Example 8.2

observer developed in [25] are similar to our method, and thus omitted in the figures. However, coordinate transformations are utilized in [25], which makes the observer design more complicated than our method.

8.5 Conclusion In this chapter, we consider a nonlinear descriptor system and explore a novel FE scheme for this type of systems. Specifically, sensor faults are taken as auxiliary states and the original system is accordingly transformed into an augmented descriptor

8.5 Conclusion

195 10

Magnitude

0 −10 −20 −30 −40

Actual x 3 Estimated x3

−50

0

5

10

15

20

25

30

35

40

25

30

35

40

Time (s)

Fig. 8.9 x3 and its estimate xˆ 3 of Example 8.2 15 10

Magnitude

5 0 −5 −10

Actual x4 Estimated x4

−15

0

5

10

15

20

Time (s)

Fig. 8.10 x4 and its estimate xˆ 4 of Example 8.2

system. An H∞ adaptive descriptor observer is therefore developed for this system. Sufficient conditions for the stability of the proposed observer have been established by using Lyapunov stability theory and expressed as a BMI optimization problem such that the H∞ gain between estimation errors and uncertainties can be minimized. The simulation results demonstrate that the proposed observer is effective both for descriptor and normal systems.

196

8 Simultaneous Estimation of Actuator and Sensor Faults for Descriptor Systems

Fig. 8.11 Actuator fault fa and its estimate fˆa of Example 8.2

1.5

Magnitude

1 0.5 0 −0.5 −1

Actual f a Estimated fa

−1.5

0

5

10

15

20

25

30

35

40

Time (s)

Fig. 8.12 Sensor fault fs and its estimate fˆs of Example 8.2

0.6 Actual fs Estimated fs

0.5

Magnitude

0.4 0.3 0.2 0.1 0 −0.1

0

5

10

15

20

25

30

35

40

Time (s)

References 1. Tan CP, Edwards C (2003) Sliding mode observers for robust detection and reconstruction of actuator and sensor faults. Int J Robust Nonlinear Control 13:443–463 2. Chen W, Saif M (2007) Observer-based strategies for actuator fault detection, isolation and estimation for certain class of uncertain nonlinear systems. IET Control Theory Appl 1:1672– 1680 3. Liu M, Shi P (2013) Sensor fault estimation and tolerant control for Itˆo stochastic systems with a descriptor sliding mode approach. Automatica 49:1242–1250 4. Zhang J, Swain AK, Nguang SK (2011) Reconstruction of actuator fault for a class of nonlinear systems using sliding mode observer. In: Proceedings of American control conference, pp 1370–1375 5. Liu M, Shi P, Zhang L, Zhao X (2011) Fault-tolerant control for nonlinear Markovian jump systems via proportional and derivative sliding mode observer technique. IEEE Trans Circuits Syst.-I: Regul Papers 58:2755–2764 6. Aldeen M, Sharma R (2008) Estimation of states, faults and unknown disturbances in nonlinear systems. Int J Control 81(8):1195–1201

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7. Koenig D (2005) Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation. IEEE Trans Autom Control 50:212–217 8. Xu A, Zhang Q (2004) Nonlinear system fault diagnosis based on adaptive estimation. Automatica 40:1181–1193 9. Jiang B, Staroswiecki M, Cocquempot V (2006) Fault accommodation for nonlinear dynamic systems. IEEE Trans Autom Control 51:1578–1583 10. Gao ZW, Ding SX (2007) Sensor fault reconstruction and sensor compensation for a class of nonlinear state-space systems via a descriptor system approach. IET Control Theory Appl 1:578–585 11. Ha QP, Trinh H (2004) State and input simultaneous estimation for a class of nonlinear systems. Automatica 40:1779–1785 12. Gao ZW, Wang H (2006) Descriptor observer approaches for multivariable systems with measurement noises and application in fault detection and diagnosis. Syst Control Lett 55:304–313 13. Yeu TK, Kim HS, Kawaji S (2005) Fault detection, isolation and reconstruction for descriptor systems. Asian J Control 7:356–367 14. Gao ZW, Ding SX (2007) Fault estimation and fault-tolerant control for descriptor systems via proportional, multiple-integral and derivative observer design. IET Control Theory Appl 1:1208–1218 15. Astorga-Zaragoza CM, Theilliol D, Ponsart JC, Rodrigues M (2012) Fault diagnosis for a class of descriptor linear parameter-varying systems. Int J Adapt Control Signal Process 26:208–223 16. Bejarano FJ, Floquet T, Perruquetti W, Zheng G (2013) Observability and detectability of singular linear systems with unknown inputs. Automatica 49:793–800 17. Gao ZW, Ho DWC (2006) State/noise estimatior for descriptor systems with application to sensor fault diagnosis. IEEE Trans Signal Process 54:1316–1326 18. Gao ZW, Ding SX (2007) Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems. Automatica 43:912–920 19. Wang Z, Shen Y, Zhang X (2012) Actuator fault estimation for a class of nonlinear descriptor systems. Int J Syst Sci 43:1492–1503 20. Ghaoui LE, Oustry F, Aitrami M (1997) A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans Autom Control 42(8):1171–1176 21. Yan XG, Edwards C (2008) Robust sliding mode observer-based actuator fault detection and isolation for a class of nonlinear systems. Int J Syst Sci 39(4):349–359 22. Yan XG, Edwards C (2007) Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43:1605–1614 23. Gao ZW, Ding SX (2008) Fault reconstruction for Lipschitz nonlinear descriptor systems via linear matrix inequality approach. Circuits Syst Signal Process 27:295–308 24. Nguang SK, Shi P, Ding S (2006) Delay-dependent fault estimation for uncertain time-delay nonlinear systems: an LMI approach. Int J Robust Nonlinear Control 16:913–933 25. Lee DJ, Park YJ, Park YS (2012) Robust H∞ sliding mode descriptor observer for fault and output disturbance estimation of uncertain systems. IEEE Trans Autom Control 57:2928–2934 26. Wang H, Daley S (1996) Actuator fault diagnosis: an adaptive observer-based technique. IEEE Trans Autom Control 41:1073–1078 27. Liu M, Cao X, Shi P (2013) Fuzzy-model-based fault-tolerant design for nonlinear stochastic systems against simultaneous sensor and actuator faults. IEEE Trans Fuzzy Syst 21:789–799 28. Zhang K, Jiang B, Chen W (2009) An improved adaptive fault estimation design for polytopic LPV systems with application to helicopter models. In: Proceedings of the 7th Asian control conference 29. Xie LH (1996) Output feedback H∞ control of systems with parameter uncertainty. Int J Control 63(4):741–750 30. Raoufi R, Marquez HJ, Zinober ASI (2010) H∞ sliding mode observer for uncertain nonlinear Lipschitz systems with fault estimation synthesis. Int J Robust Nonlinear Control 20:1785– 1801

Chapter 9

Conclusions and Future Work

9.1 Conclusions This book proposes novel observer-based methods to diagnose faults for a class of nonlinear systems. The nonlinear systems are assumed to be Lipschitz about the state uniformly and contaminated by modeling discrepancies and external disturbances, which are lumped as additive system uncertainties. Detection, isolation, and estimation of actuator and sensor faults are three main topics which have been studied. In the present research, methods based on sliding-mode observer (SMO), adaptive observer (AO), unknown-input observer (UIO), and descriptor observer have been developed. Several situations including when the Lipschitz constant is known or unknown, the matching condition is satisfied or unsatisfied, uncertainties are structured or unstructured have been discussed. Based on the Lyapunov method, the sufficient conditions for the existence of various fault estimators are derived in terms of linear matrix inequalities which can be solved using MATLAB. The effectiveness of the proposed methodologies have been verified by simulations of practical systems such as singlelink joint robot system, seventh-order aircraft system and satellite attitude control system, and numerical examples. The source codes for generating the simulation results are included in the book. Initially, actuator fault detection and isolation (FDI) has been studied. These results are then extended to sensor FDI by converting sensor faults into the form of actuator faults using integral observer-based approach. Since FDI only indicates when and where a fault occurs, but can not provide more comprehensive information of the fault such as the magnitude and duration, we further investigate faults and study the topic of fault estimation (FE) in the following chapters. In Chaps. 4 and 5 it is assumed that only actuator faults or sensor faults occur in the system at one time. However, in many practical systems, both actuators and sensors are simultaneously prone to faults. This motivates us to consider actuator and sensor faults in the same framework. In Chap. 6, two schemes based on SMOs and AOs are proposed. However, the proposed schemes employ two cascade coordinate © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4_9

199

200

9 Conclusions and Future Work

transformations to transform the system matrices to have special structures, which may bring complexities to the design of the observers. Therefore, we develop a different FE method in Chap. 7 to solve the same FE problem. Chapters 2–7 all focus on normal Lipschitz nonlinear systems; FE for descriptor systems is studied in Chap. 8. The main contributions of this study are as follows: • The problem of detection, isolation, and estimation of actuator faults and sensor faults is thoroughly investigated and reported in this book. Several methods based on coordinate transformations, SMOs, AOs, UIOs, and descriptor system approach have been developed. • Incipient faults are almost unnoticeable during their initial stage and their effects to residuals are most likely to be concealed by system uncertainties. Therefore the diagnosis of incipient fault is difficult. In this book, the diagnosis of incipient faults, which occur in sensors and actuators, is studied. • Sufficient conditions for the existence and stability of the proposed fault estimators are derived and expressed in LMIs. However, the feasibility of LMIs depends on the value of Lipschitz constant. When this constant is too large, the LMI solver may not provide a feasible solution. In order to solve this problem, adaptation laws are integrated into the design of observers, which makes the fault diagnosis schemes more applicable for the situations where either the Lipschitz constant is large or unknown. In summary, the book provides an integrated approach for the design of observers for detecting, isolating, and estimating actuator and sensor faults. It represents a valuable and meaningful contribution to the development of an LMI-based fault diagnosis.

9.2 Future Work The robust observer-based fault diagnosis has been studied extensively over the last two decades. However, this research area still remains open. The following are some of the directions that could be pursued for the future research: • The book assumes that the system uncertainties are structured. It will be an interesting problem when the uncertainties are unstructured. Moreover, most of the existing works on robust FDI simply treat modeling uncertainties as additive unknown terms. This is questionable because the model uncertainty describes internal property of the dynamic systems. The future work will extend the results of the proposed schemes in this book to systems with parametric uncertainties and other types of uncertainties. • Networked control systems (NCSs) have advantageous over traditional systems in many aspects such as efficiency, practicality, energy consumption, and installation. However, one of the major problems of NCSs is the channel time delay and quantization error due to the limited communication capacity. The network-induced

9.2 Future Work

•

•

• •

201

delay, including sensor-to-controller delay and controller-to-actuator delay, will deteriorate the system performance as well as stability. Therefore, it is desirable to develop fault diagnosis schemes for networked control systems and for nonlinear systems with fixed or varying time delay in the states, outputs. The schemes proposed in this book are believed to have the potential to be extended to such systems. In this book, only fault detection, isolation, and estimation were studied. However, successful fault diagnosis is not the ultimate goal for real applications. Faulttolerant control (FTC) is needed to preserve the stability and reliability of the system when it is subject to a set of possible faults. The existing strategies of fault compensation control are based on adding an additional control input to the original control input in order to reduce or compensate the effects of faults, so that the controlled system can still continue to operate according to its original specifications. The additional input signal can be obtained from fault estimation and therefore the fault estimation schemes proposed in this book forms the foundation for FTC systems. As engineering plants grow in size and complexity, and the popularity of distributed systems, FDI for large-scale nonlinear systems becomes increasingly important. In general, a fault that occurs in one subsystem will not only affect the behavior of this system, but it will also affect the behavior of the neighboring subsystems. It is believed that the results of this book can be extended to FDI for large-scale nonlinear systems by taking interactions between subsystems into account. This book assumes that the output equation is linear. More complicated systems with both state dynamics and output dynamics being nonlinear can be studied in the future. In this book, we only focus on the fault diagnosis for Lipschitz nonlinear systems. More general nonlinear systems which could be represented by Takagi–Sugeno fuzzy models [1] or polynomial systems [2] should be studied in the future.

References 1. Chae S, Nguang SK (2013) SOS based robust fuzzy dynamic output feedback control of nonlinear networked control systems. IEEE Trans Cybern 44:1204–1213 2. Saat S, Nguang SK (2015) Nonlinear H∞ output feedback control with integrator for polynomial discrete time systems. Int J Robust Nonlinear Control 25:1051–1065

Appendix A

Solving Linear Matrix Inequality (LMI) Problems

In this section, we present a brief introduction about linear matrix inequalities which have been used extensively to solve the FDI problems described in this book.

A.1

Introduction to LMI

LMIs are matrix inequalities which are linear or affine in a set of matrix variables. They are essentially convex constraints and therefore many optimization problems with convex objective functions and LMI constraints can easily be solved efficiently using many existing software. This method has been very popular among control engineers in recent years. This is because a wide variety of control problems can be formulated as LMI problems. An LMI has the following form: F(x) = F0 + x1 F1 + · · · + xn Fn = F0 +

n 

xi Fi > 0

(A.1)

i=1

where x ∈ R m is the vector of decision variables and F0 , F1 , . . . , Fn are given constant symmetric real matrices, i.e., Fi = FiT , i = 0, . . . , m. The inequality symbol in the equation means F(x) is positive definite, i.e., uT F(x)u > 0 for all nonzero u ∈ R n . This matrix inequality is linear in the variables xi. As an example, the Lyapunov inequality AT P + PA < 0

(A.2)

where A ∈ R n×n is given and X = X T is the decision variable that can be expressed in the form of LMI (A.1) as follows: Let P1 , P2 , . . . , Pm be a basis for the symmetric n × n matrices (m = n(n + 1)/2), then take F0 = 0 and Fi = −AT Pi − Pi A. © Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4

203

204

A.1.1

Appendix A: Solving Linear Matrix Inequality (LMI) Problems

Tricks Used in LMIs

Although many problems in control can be formulated as LMI problems, some of these problems result in nonlinear matrix inequalities. There are certain tricks which can be used to transform these nonlinear inequalities into suitable LMI forms. Some of the tricks which are often used in control are described here with suitable examples. 1. Change of variables By defining new variables, it is sometimes possible to linearize nonlinear matrix inequalities. Example A.1 Synthesis of state feedback controller The objective is to determine a matrix F ∈ R m×n such that all the eigenvalues of the matrix A + BF ∈ R n×n lie in the open left-half of the complex plane. Using Lyapunov theory, it can be shown that this is equivalent to find a matrix F and a positive definite matrix P ∈ R n×n such that the following inequality holds: (A + BF)T P + P(A + BF) < 0,

(A.3)

AT P + PA + F T BT P + PBF < 0.

(A.4)

or

Note that the terms with products of F and P are nonlinear or bilinear. Let us multiply either side of the above equation by Q = P−1 . This gives QAT + AQ + QF T BT + BFQ < 0.

(A.5)

This is a new matrix inequality in the variables Q > 0 and F. But it is still nonlinear. Let us define a second new variable L = FQ. This gives QAT + AQ + L T BT + BL < 0.

(A.6)

This gives an LMI feasibility problem in the new variables Q > 0 and L ∈ R m×n . After solving this LMI, the feedback matrix F and Lyapunov variable P can be recovered from F = LQ−1 and P = Q−1 . This shows that by making a change of variables, we can obtain an LMI from a nonlinear matrix inequality. 2. The Schur Complement Schur’s formula is used in transforming nonlinear inequalities of convex type into LMI. This says that the LMI 

 Q(x) S(x) < 0, S(x)T R(x)

(A.7)

where Q(x) = Q(x)T , R(x) = R(x)T and S(x) depends affinely on x, is equivalent to R(x) < 0, Q(x) − S(x)R(x)−1 S(x)T < 0.

(A.8)

Appendix A: Solving Linear Matrix Inequality (LMI) Problems

205

In other words, the set of nonlinear inequalities (A.8) can be transformed into the LMI (A.7). Example A.2 Consider the following matrix inequality: AT P + PA + PBR−1 BT P + Q < 0,

(A.9)

where P = PT > 0 and R > 0, is equivalent to 

 AT P + PA + Q PB < 0, −R BT P

(A.10)

3. The S-Procedure This procedure is adopted when we want to combine several quadratic inequalities into one single inequality. In many problems of control engineering we would like to make sure that a single quadratic function of x ∈ R m is such that F0 (x) ≤ 0, F0 (x) := x T A0 x + 2b0 x + c0 ,

(A.11)

whenever certain other quadratic functions are positive semi-definite, i.e., Fi (x) ≥ 0, Fi (x) := x T Ai x + 2b0 x + c0 , i ∈ (1, 2, . . . , q).

(A.12)

Example To illustrate the S-procedure, consider the case i = 1 for simplicity. We need to guarantee that F0 (x) ≤ 0 for all x such that F1 (x) ≥ 0. If there exists a positive (or zero) scalar τ such that Faug (x) := F0 (x) + τ F1 (x) ≤ 0∀x, s.t F1 (x) ≥ 0,

(A.13)

then our goal is achieved. This is because Faug (x) ≤ 0 implies that F0 (x) ≤ 0 if τ F1 (x) ≥ 0, since F0 (x) ≤ Faug (x) if F1 (x) ≥ 0, By extending this to q number of inequality constraints gives the following: F0 (x) ≤ 0 whenever Fi (x) ≥ 0 holds if F0 (x) +

q 

τi Fi (x) ≤ 0, τi ≥ 0. (A.14)

i=1

A.1.2

Solving LMI Using MATLAB Toolbox

The LMI toolbox of MATLAB provides a set of useful functions to solve LMIs. Some of these functions are discussed here with sample codes. Step-1: Initialization At the beginning, initialize the LMI description with the command setlmis([]). Note that this function does not take any parameter.

206

Appendix A: Solving Linear Matrix Inequality (LMI) Problems

Step-2: Defining the Decision Variables Next, it is necessary to define the decision variables, i.e., the unknown variables of the LMI problem. As an example, consider the LMI C T XC < 0, where C is a constant matrix and X is the matrix of decision variables. It is defined using the function lmivar which has the following syntax. X = lmivar(type, structure). This command allows us to define several forms of decision matrices such as symmetrical matrices, rectangular matrices or matrices of other type. Depending on the selected matrix type, the structure contains different information. Thus, first we define the type and then define the structure which depends on the type. • If type = 1, this implies that the matrix X is square and symmetrical. The structure element (i, 1) specifies the dimension of the ith-block, while structure element (i, 2) specifies the type of the ith-block (1 for full, 0 for scalar and −1 for zero block). • If type = 2, matrix X is rectangular of size m × n as specified in structure = [m, n]. • If type = 3, matrix X is of other type. Step-3: Define the LMIs one by one The syntax of the command is

This is done with the command lmiterm.

lmiterm(termID, A, B, flag). The lmiterm takes 3 or 4 arguments. The first argument termID is a 1 × 4 vector. The first element of this vector indicates which LMI is defined. The second and third entry in this vector gives the position of the term being defined. And the fourth entry indicates which LMI decision variable is involved. It can be 0, X or X depending on whether the term is constant, of type AXB or AX T B. The second and third arguments of lmiterm function are the left and right multiplier of the decision matrix. If the flag is set to ‘s’, it permits to specify with a single command that the given term and its symmetrical appears in the LMI. Example A.3 Consider the following set of LMIs: 

 CX T C T + BT YA XF , Y FT X DXDT > 0.

(A.15) (A.16)

Here we have two decision variables X and Y , and two LMIs. Let Y be a full symmetric matrix of dimension 5 and X has 5 blocks and the dimensions of the various block matrices are 5, 4, 3, 1 and 2.

Appendix A: Solving Linear Matrix Inequality (LMI) Problems

First, we define the variables X and Y using lmivar function as follows:

Then, we define the LMIs using lmiterm function as follows:

Lastly, we create an LMI object using the following command.

A.2

Examples of Applying LMIs to Solve Various Control Problems

Example A.4 Determine the stability of linear time-invariant systems Consider the linear time-invariant system x˙ = Ax. The system is stable provided the following inequalities are satisfied: P > 0, AT P + PA < 0. These two inequalities can be combined into a single LMI as

207

208

Appendix A: Solving Linear Matrix Inequality (LMI) Problems



 AT P + PA 0 < 0. 0 −P

 0 1 is given The MATLAB code for solving this stability problem for A = −2 −3 as follows. 

Programme example_A.4.m

After running this programme we get tmin = −2.615451 and P =



 65.9992 12.8946 . 12.8946 15.1836

Example A.5 The LQR Problem: Solution of Riccati Equation. Consider a system represented by the following linear continuous-time state equations: x˙ = Ax + Bu, x(0) = x0 , where x ∈ R n is the state vector, u ∈ R m is the input vector, A and B are known matrices of appropriate dimensions. The objective is to determine the control input u which minimizes the following performance index:  J=

∞

(x T Qx + uT Ru)dt,

0

where Q ∈ R n×n is a real symmetric positive semi-definite matrix and R ∈ R p×p is a real positive definite matrix.

Appendix A: Solving Linear Matrix Inequality (LMI) Problems

209

The optimal control input which minimizes J is given by u(t) = R−1 BT Px(t) = Kx(t), K = R−1 BT P,

(A.17)

where the matrix P is obtained by solving the following Riccati equation: AT P + PA + PBR−1 BT P + Q < 0, P > 0, R > 0. Note that the Riccati equation, in contrast to Lyapunov equations, is a nonlinear equation in P. This is because the quadratic term PBR−1 BT P appears in the inequality. Using the Schur Complement, we can represent this inequality as LMIs as follows: P > 0, Q > 0, R > 0 and  T  A P + PA + Q PB < 0. −R BT P The MATLAB code for computing the gain K is given as follows: Programme example_A.5.m

(A.18)

210

Appendix A: Solving Linear Matrix Inequality (LMI) Problems

After solving the LMIs, the control gain K is obtained as K = [ 0.2454 0.2210 ]. Example A.6 H∞ controller using full state feedback for the system x˙ = Ax + B1 w + B2 u, z = C1 x + D11 w + D12 u. The controller parameters can be obtained by solving the following LMIs: Y >0 ⎤ ⎡ T T YA + AY + Z T B2T + B2 Z B1 YC1T + Z T D12 ⎥ ⎢ T ⎥ 0,Y < 0]; In addition to these standard constraints, YALMIP also supports definition of integrity constraints, second-order cone constraints, and sum of squares constraints.

B.3

Solving Optimization Problems

Once all variables and constraints have been defined, the optimization problem using the command solvesdp can be solved. For example, a linear program min cT x subject to Ax 0, AT P + PA < 0. These two inequalities can be combined into a single LMI as:  T  A P + PA 0 < 0. 0 −P   0 1 is given The MATLAB code for solving this stability problem for A = −2 −3 as follows. Example_B.1.m

Example B.2 Solving Linear Quadratic Regulator Problem The LQR can be designed by solving the following Riccati equation: AT P + PA + PBR−1 BT P + Q < 0, where P, Q, R are symmetric positive definite matrices.

216

Appendix B: YALMIP Toolbox: A Short Tutorial

This inequality can be transformed into the LMIs as follows: P > 0, Q > 0, R > 0 and

AT P + PA + Q PB < 0. −R BT P  The MATLAB code for solving this LQR problem for A =   1 is given as follows. 1 Example_B.2.m

 −2 1 and B = 1 −1

Appendix B: YALMIP Toolbox: A Short Tutorial

217

The controller gain K is obtained as K = [0.2072 0.2182]. Example B.3 H∞ State Feedback Control Consider the discrete-time time-invariant linear system x(k + 1) = Ax(k) + Bw w(k) + Bu u(k), z(k) = Cz x(k) + Dzw w(k) + Dzu u(k), y(k) = Cy x(k) + Dyw w(k),

(B.1)

where the state vector x ∈ Rn and all other matrices and vectors have appropriate dimensions. Assume that the state vector x(k) is available for feedback and the state information is not corrupted by the input w(k). These assumptions can be incorporated into System (B.1) by assuming Cy = I and Dyw = 0. The objective is to find a linear state feedback control law u = Kx. Following theorem gives the basis of the design. Theorem ([2]) There exists a controller in the form of u(t) = Kx(t) such that the inequality Hwx (s)2∞ < γ holds if, and only if, the following LMI hold, where matrices X and L and symmetric Matrix P are the variables. ⎡

P AX + Bu L ⎢ (.)T X + X T − P ⎢ M=⎢ T (.)T ⎣ (.) T (.) (.)T

⎤ Bw 0 T ⎥ 0 X T CzT + L T Dzu ⎥ ⎥ > 0. T I Dzw ⎦ (.)T γI

(B.2)

In this example we have assumed γ = 1. The MATLAB code for this design are givenin the  state feedback     following for 0.01 0.1 0.01 0.5 the system (B.1) with A = , Bw = , Bu = , Cy = [1 1], 0.01 0.2 0.01 1.0 Cz = [0.1 0], Dyw = 2, Dzw = 0.1 and Dzu = 0 are given below. After computing X, L and P, the gain matrix K is calculated from K = LX −1 . Example_B.3.m

218

Appendix B: YALMIP Toolbox: A Short Tutorial

Solution X = [1.3217 0.0001; 0.0009 1.3358], L = [−0.0170 − 0.2671], Po = [1.3131 0.0010; 0.0010 1.3407] and controller K = [−0.0128 − 0.2000]. Example B.4 H∞ Output Feedback Control Consider System (B.1) and the following full-order linear dynamic controller xc (k + 1) = Ac xc (k) + Bc y(k), u(k) = Cc xc (k) + Dc y(k). The following theorem is the basis of this design.

(B.3)

Appendix B: YALMIP Toolbox: A Short Tutorial

219

Theorem ([2]) There exists a controller in the form (B.3) such that the inequality Hwx (s)2∞ < γ holds if, and only if, the following LMI holds, where the matrices X, L, Y , F, Q, R, S, J and the symmetric matrices P and H are the variables. ⎡

P

⎢ ⎢ (.)T ⎢ ⎢ ⎢ (.)T ⎢ M=⎢ ⎢ T ⎢ (.) ⎢ ⎢ T ⎢ (.) ⎣ (.)T

J

AX + Bu L

H

Q

A + Bu RCy Bw + Bu RDyw YA + FCy

YBw + FDyw

(.)T X + X T − P I + S T − J

0

(.)T

(.)T

Y + YT − H

0

(.)T

(.)T

(.)T

I

(.)T

(.)T

(.)T

(.)T

0

⎤

⎥ ⎥ ⎥ ⎥ T ⎥ X T CzT + L T Dzu ⎥ ⎥>0 T T T T ⎥ Cz + Cy R Dzu ⎥ ⎥ T T T ⎥ ⎥ Dzw + Dyw RT Dzu ⎦ γI (B.4) 0

Controller parameters are calculated from 

Ac Bc Cc Dc



 =

V −1 −V −1 YBu 0 I



Q − YAX F L R



 0 U −1 , −Cy XU −1 I

(B.5)

where UV = S − YX. In the example, we have assumed γ = 1. The following code gives the H∞ -output feedback controller design for System (B.1). Example_B.4.m

220

Appendix B: YALMIP Toolbox: A Short Tutorial

Appendix B: YALMIP Toolbox: A Short Tutorial

221

Solution X = [1.66170 − 0.0000; 0.0067 1.6798] Y = [1.6751 − 0.0068; −0.0322 1.6829] P = [1.6499 0.0066; 0.0066 1.6861] H = [1.6736 0.0001; 0.0001 1.6736] J = [0.0075 0.0120; 0.0120 0.0210] S = [−0.9755 0.0120; 0.0120 − 0.9790] Q = 1.0e − 03 × [0.6293 0.0038; 0.8481 0.0027] F = [−0.0253; −0.1027] R = −0.0557 L = [−0.0208 − 0.3360] U = [1 0; 0 1] V = [−3.7590 0.0233; 0.0542 − 3.8058] Dc = −0.0557 Bc = [−0.0056; 0.0025] Cc = [0.0722 − 0.2423] Ac = [−0.0064 − 0.0093; 0.0027 0.0042]. References 1. Lofberg J (2004) YALMIP: a tool box for modelling and optimization in matlab. In: Computer aided control systems design: 2004 IEEE international symposium, pp 284–289 2. De Oliveira MC, Geromel JC, Bernussou J (2002) Extended h2 and h∞ norm characterizations and controller parametrizations for discrete time systems. Int J Control 75(9):666–679

Index

A Actuator faults, 1, 11 Adaptation law, 92, 148 Adaptive observer, 5, 88, 116

B Bilinear matrix inequalities, 166

C Cone complementarity linearization algorithm, 166 Coordinate transformations, 14, 36, 60, 89, 117, 147

D Dedicated observer scheme, 12 Descriptor observer, 166 Descriptor systems, 165 Discontinuous output error injection term, 15 Disturbance attenuation, 62, 145, 150, 166

F Fault detection, 2, 11 Fault detection and isolation, 2, 35 Fault diagnosis, 2 Fault estimation, 2, 57, 87, 115, 145, 165 Fault isolation, 2, 12 Fault-tolerant control, 2

G Generalized observer scheme, 12 Gronwall–Bellman inequality, 18

I Integral observer, 58, 88

L Linear matrix inequality, 203 Lipschitz constant, 13, 88, 146 Lipschitz nonlinear systems, 6 Luenberger observer, 15, 39 Lyapunov function, 16, 40, 62, 92, 122, 150, 171

M Matching condition, 59, 115

O Observer-based fault diagnosis, 4

P Popov–Belevitch–Hautus test, 38

R Reachability condition, 19

© Springer International Publishing Switzerland 2016 J. Zhang et al., Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB , Advances in Industrial Control, DOI 10.1007/978-3-319-32324-4

223

224 S Schur complement, 18, 41, 65, 204 Sensor faults, 1, 35 Sliding-mode observer, 5, 11, 35, 88, 115, 145 Sliding-mode surface, 19 T Triangle inequality, 18

Index U Unknown-input observer, 5, 145

Y YALMIP toolbox, 213