Studies in Systems, Decision and Control 127

Ke ZhangBin JiangPeng ShiVincent Cocquempot

Observer-Based Fault Estimation Techniques

Studies in Systems, Decision and Control

Volume 127

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Ke Zhang • Bin Jiang • Peng ShiVincent Cocquempot

Observer-Based FaultEstimation Techniques

123

Ke ZhangCollege of Automation EngineeringNanjing University of Aeronauticsand Astronautics

NanjingChina

Bin JiangCollege of Automation EngineeringNanjing University of Aeronauticsand Astronautics

NanjingChina

Peng ShiSchool of Electrical and ElectronicEngineering

University of AdelaideAdelaide, SAAustralia

Vincent CocquempotUMR 9189, CRIStAL—Centre deRecherche en Informatique, Signal etAutomatique de Lille

CNRS, Université de Lille, Centrale LilleLilleFrance

ISSN 2198-4182 ISSN 2198-4190 (electronic)Studies in Systems, Decision and ControlISBN 978-3-319-67491-9 ISBN 978-3-319-67492-6 (eBook)https://doi.org/10.1007/978-3-319-67492-6

Library of Congress Control Number: 2017952028

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Preface

Many practical control systems are subject to possible malfunctions which maycause significant performance degradation and even instability of the system. Toimprove reliability, performance, and safety of dynamical systems, fault diagnosistechniques are receiving considerable attention both in research and applicationsand have been the subjects of intensive investigations. Fault detection, which acts asthe first step of the fault diagnosis, is a binary decision process determining whethera fault has occurred or not. Fault isolation is to determine the location of the faultycomponent, while fault estimation is to online identify the size of the occurred fault.Compared with the problems of fault detection and isolation, fault estimation ismore challenging. In this book, observer-fault estimation techniques are furtherinvestigated and new results related to fault estimation are presented.

In Chap. 1, the background of fault estimation is given and motivations of ourstudies are presented in detail.

In Chap. 2, the design of a multi-constrained full-order fault estimation observer(FFEO) with finite-frequency specifications is studied for continuous-time systems.By constructing an augmented system, a multi-constrained FFEO in finite-frequencydomain is proposed to achieve fault estimation. Meanwhile, the presented FFEO canavoid the overdesign problem generated by the entire frequency domain by thegeneralized Kalman–Yakubovich–Popov (KYP) lemma. Furthermore, by intro-ducing slack variables, improved results on FFEO design in different frequencydomains are obtained such that different Lyapunov matrices can be separatelydesigned for each constraint.

In Chap. 3, a framework of fault estimation observer design in finite-frequencydomain is proposed for discrete-time systems, including FFEO and reduced-orderfault estimation observer (RFEO). A FFEO in finite-frequency domain is designedto achieve fault estimation by using the generalized KYP lemma to reduce con-servatism generated by the entire frequency domain. Then, a RFEO is constructed,which results in a new fault estimator to realize fault estimation using current outputinformation. Furthermore, improved results on FFEO and RFEO design withfinite-frequency specifications are obtained.

v

In Chap. 4, the problem of fault estimation observer design with finite-frequencyspecifications is addressed for discrete-time Takagi-Sugeno fuzzy systems. Thenfuzzy unknown input observer-based fault estimation is investigated for discrete-time T-S fuzzy systems.

In Chap. 5, the issue of fault estimation observer design with finite-time con-vergence specification is studied for continuous-time dynamic systems subject toexternal disturbances. The unknown input observer is constructed to achieveaccurate estimation of the occurred fault and to guarantee robustness against thedisturbance. Then a pole placement-based fault estimation observer is constructedusing time-delay design such that the fault estimation error converges to zero infinite time. Meanwhile, the proposed fault estimator with finite-time convergencespecification does not contain discontinuous sign function.

In Chap. 6, a novel adjustable parameter-based fault estimation design isaddressed for continuous-time/discrete-time dynamic systems. First, a fault esti-mation observer with adjustable parameter is constructed to online identify the sizeof occurred faults. The fault estimation design not only possesses a wider appli-cation compared with adaptive observers, but also uses the current output infor-mation to enhance fault estimation performance. Then a multi-constrained approachis proposed to determine gain matrices of fault estimation observer. Moreover, faultestimation results with the slack-variable technique are obtained to further reducethe conservatism.

In Chap. 7, the distributed fault estimation observer (DFEO) is studied based onH1 and H2 strategies for discrete-time multi-agent systems (MAS). For each agent,a fault estimation observer is designed using relative output estimation errors. Bydenoting global estimation error vectors, the global error dynamics is constructedfor MAS. Then the existence condition of the presented DFEO is further discussed.

In Chap. 8, under the directed communication topology, an adaptive observer-based DFEO is studied for MAS. First, a corresponding fault estimation observer isconstructed based on relative output estimation errors. To consider DFEO designfrom an overall perspective, the whole error dynamics is obtained by definingglobal error vectors. Then an adaptive technique-based DFEO design is proposedfor MAS with directed communication topologies.

In Chap. 9, an adjustable parameter-based DFEO is proposed for MAS withdirected communication topologies to improve the accuracy of fault estimation.Based on H1 and H2 with pole placement, multi-constrained design is given tocalculate gain matrices of DFEO.

In summary, conclusions are presented in Chap. 10.

Nanjing, China Ke ZhangMay 2017 Bin Jiang

Peng ShiVincent Cocquempot

vi Preface

Acknowledgements

This work is dedicated to our parents and

Jingping and Jinghang—Dr. Ke ZhangWen and Xinhao—Prof. Bin Jiang

Mei, Lisa and Michael—Prof. Peng ShiDelphine, Clément, Bastien, Nathan and Anaïs—Prof. Vincent Cocquempot

This work was supported in part by the National Natural Science Foundation ofChina (61490703, 61673207, 61533008, U1509217), Fundamental Research Fundsfor the Central Universities (NO. NE2014202), Qing Lan Project, PriorityAcademic Program Development of Jiangsu Higher Education Institutions, and theAustralian Research Council (DP170102644).

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Finite-Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Unknown Input Observer . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Finite-Time Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Adjustable Parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.5 Multi-agent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Book Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Fault Estimation of Continuous-Time Systemsin Finite-Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 FFEO Design in Finite-Frequency Domain . . . . . . . . . . . . 122.3.2 Fault Estimation with Less Conservatism . . . . . . . . . . . . . 16

2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Fault Estimation of Discrete-Time Systemsin Finite-Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 FFEO Design in Finite-Frequency Domain. . . . . . . . . . . . . . . . . . 293.4 RFEO Design in Finite-Frequency Domain . . . . . . . . . . . . . . . . . 393.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

ix

4 Fault Estimation of Fuzzy Systems in Finite-FrequencyDomain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Robust H1 Fault Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 FUIO-Based Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Fault Estimation with Finite-Time Convergence Specification . . . . . 875.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.1 Finite-Time Observer for Constant Faults . . . . . . . . . . . . . 885.3.2 Finite-Time Observer for Time-Varying Faults . . . . . . . . . 94

5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 AP-Based Fault Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3.1 Fault Estimation with AP . . . . . . . . . . . . . . . . . . . . . . . . . 1066.3.2 Fault Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.3 Fault Estimation with Less Conservatism . . . . . . . . . . . . . 114

6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 H1 and H2 Distributed Fault Estimation for MAS . . . . . . . . . . . . . . 1277.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.4 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.4.1 Existence Condition of DFEO. . . . . . . . . . . . . . . . . . . . . . 1317.4.2 H1 Performance Based Design. . . . . . . . . . . . . . . . . . . . . 1337.4.3 H2 Performance Based Design . . . . . . . . . . . . . . . . . . . . . 135

7.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

x Contents

8 Adaptive Technique-Based Distributed Fault Estimationfor MAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9 AP-Based Distributed Fault Estimation for MAS . . . . . . . . . . . . . . . 1579.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

9.3.1 DFEO with AP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589.3.2 H1 Performance Based Design. . . . . . . . . . . . . . . . . . . . . 1619.3.3 H2 Performance Based Design . . . . . . . . . . . . . . . . . . . . . 1649.3.4 Analysis and Comparison of DFEO with AP . . . . . . . . . . 165

9.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1669.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Contents xi

Acronyms

AP Adjustable parameterCAFE Conventional adaptive fault estimationDFEO Distributed fault estimation observerFAFE Fast adaptive fault estimationFFEO Full-order fault estimation observerFUIO Fuzzy unknown input observerKYP Kalman–Yakubovich–PopovLMIs Linear matrix inequalitiesMAS Multi-agent systemsRFEO Reduced-order fault estimation observerUIO Unknown input observer

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Chapter 1Introduction

1.1 Background

1.1.1 Fault Diagnosis

The rapid increase of productivity requirements in modern control systems leads tomore challenging operating conditions. A fault is deemed to occur, and is defined asan unpermitted deviation of at least one characteristic property (feature) of the systemfrom the acceptable, usual, standard condition. The failure of actuators, sensors andother components can result in significant performancedegradation, severe damageofthephysical systemsor evendisaster. Fault diagnosis algorithmsand their applicationshave received considerable attention and been the topics of intensive investigationsover the past several decades.Many significant results have been achieved, and can befound in several excellent works [2, 6, 15, 26, 46, 48, 76, 94, 140].

Model-based fault diagnosis techniques have been widely recognized as powerfulapproaches and successfully applied to many practical systems. The main idea offault diagnosis is to formulate a residual signal to detect faults presented in a systemand to construct a scheme to determine the location and amplitude of a fault. Residualgeneration, using observers or filters, have been widely used, where the differencebetween the system and the observer outputs is processed to form the so-calledresiduals.

1.1.2 Fault Estimation

In general, fault diagnosis contains three steps: fault detection, fault isolation andfault estimation, which is illustrated in Fig. 1.1. Fault detection always acts as thefirst step of the fault diagnosis process, which is a binary decision to confirmwhetheran unexpected fault has occurred or not. Timely detection can avoid the developmentof more serious faults. Then, fault isolation is usually applied to determine the exact

© Springer International Publishing AG 2018K. Zhang et al., Observer-Based Fault Estimation Techniques, Studies in Systems,Decision and Control 127, https://doi.org/10.1007/978-3-319-67492-6_1

1

2 1 Introduction

Fig. 1.1 Fault diagnosisFault

DetectionFault

IsolationFault

Estimation

location of the detected fault [42, 98, 120]. Finally, fault estimation aims to identifythe magnitude of the faults in order to be used in a fault accommodation procedure,which is an important part of fault tolerant control [38, 126]. Compared with faultdetection and isolation, fault estimation, as a bridge between fault diagnosis andtolerant control, is more challenging and fruitful results were obtained during thepast two decades [61, 62, 91, 115, 130].

1.2 Motivations

1.2.1 Finite-Frequency Domain

As we all know, H∞ methods are used in control theory to synthesize controllersachieving stabilization with guaranteed performance and to attenuate the externaldisturbance with the purpose of minimizing the impact of the disturbance on systemperformance. The standard H∞ norm is the maximum singular value of the functionover that space. Note that this can be interpreted as a maximum gain in any directionand at any frequency [151]. However, once external noises/disturbances belong toa certain finite-frequency range, the conventional H∞ approach may lead to muchconservatism because of the overdesign. Meanwhile, frequency domain inequalitiescannot be solved easily and will lead to inconvenience for numerical calculation.Reference [41] not only considered the H∞ design properties in finite-frequencydomains, but also provided exact linear matrix inequalities characterizations basedon the generalized Kalman-Yakubovich-Popov (KYP) lemma. On the basis of [41],analysis anddesignoffinite frequencyhavebeen ahot topic and received considerableattention [8, 14, 30, 57, 99, 116].

However, most addressed filtering problems, stability analysis and feedback sta-bilization, and very few results are related to the issue of fault diagnosis in the finitefrequency domain. The issue of fault detection in the finite-frequency domain hasbeen addressed in [107, 109, 117, 118]. Fault estimation is utilized to online deter-mine the size of the fault. Recently, literatures [138, 139] considered a fault estimationobserver design, but these results were based on the design of the entire-frequencydomain and thus the conservatism may be introduced. Therefore, it is necessary todevelop a new method such that the conservatism generated by the entire frequencycan be reduced.

1.2 Motivations 3

1.2.2 Unknown Input Observer

In practical situations, there almost are unknown inputs in control systems, whichleads to system performance degradation, such as process noises, external distur-bances, etc. How to effectively deal with unknown inputs of practical systems is aninteresting and attractive topic. During the past three decades, special attention hasbeen focused on the design of unknown input observer (UIO) [5, 51, 52, 121]. AnUIO can realize the state estimation for dynamic systems subject to unknown inputsand one of the most significant features resorts to the unknown input decouplingprinciple, so the design of UIO for uncertain control systems subject to external dis-turbances has been extensively studied in both theory and application such as craneset-up, lateral vehicle dynamics, chemical process, et al. [18, 66, 127]. An UIO hasalso provided a useful method to achieve fault diagnosis with robustness againstunknown inputs, in which the residual is designed to be insensitive to unknowninputs. For UIO-based fault detection and isolation, many contributions have beenproposed in [9, 17, 49, 67] and applied to machine infinite bus systems, ship models,etc.

Different from fault detection and isolation, fault estimation is used to onlinedetermine fault’s size and magnitude, aimed at providing accurate fault informationto active fault-tolerant control. In [113], an approach for robust fault estimationand reconstruction for a class of nonlinear systems with uncertainties was proposedbased on a sliding mode observer and simulations of a single-link flexible jointrobot system were used to verify the effectiveness. For a class of nonlinear systems,an exact observer design for nonlinear locally detectable systems with unknowninputs was proposed based on higher-order sliding-mode observers and a satellitemodel was taken as a simulation model [27]. In [54], an UIO-based fault estimationstrategy was proposed by using a coordinate transformation, but the derivative of theoutput was required. In practice, it is not an easy task to obtain signal derivativesbecause of the presence of noises. Therefore, how to realize UIO-based robust faultestimation for control systems with unknown inputs is challenging and motivates ourstudy.

1.2.3 Finite-Time Convergence

Compared with asymptotic convergence and bounded stability, the problem of finite-time convergence is more attracting because many practical systems require severetime response constraints. In [21], a state estimator was proposed to estimate the statevalue in finite time. Based on a time delay, two full-order observers were combinedin one equation to estimate the unknown state in finite time and the convergence timecan be chosen freely. Furthermore, the finite-time convergent observer was extendedto a class of nonlinear systems [68] and functional observer-based linear systems[79]. In [53], for continuous-time systems subject to noises and uncertainties, a state

4 1 Introduction

estimator was designed to estimate the unknown state in finite time. In [54], a finite-time state observer design was proposed to achieve fault estimation. However, thefinite-time design was used to estimate the unknown state, and the fault estimatorneeded the derivative of the output, which amplifies the noise influence. However,few attention is paid to the finite-time design for the problem of fault estimation.

For the issue of fault estimation, many important results have been obtained, suchas popular adaptive observer and sliding mode observer-based methods [19, 44,69, 114, 131, 146]. Adaptive observers can achieve asymptotic estimation for con-stant faults and bounded estimation for time-varying faults. Sliding mode observersproduct bounded estimation for both constant and time-varying faults when the dis-continuous sign function is replaced by a smooth approximation. However, thesepopular design methods do not consider the accurate estimation within finite timefor the occurred fault. Estimating the fault in finite time is a meaningful researchsubject and motivates our study.

1.2.4 Adjustable Parameter

Many important results have been obtained for fault estimation in continuous-timesystems. Adaptive observer-based fault estimation techniques have attracted con-siderable attention in [82, 132, 146]. However, the adaptive observer-based designneeds a strictly positive real condition to be satisfied, which is also the case in thetraditional sliding mode observer-based design. To enlarge the application scope ofadaptive observer, an auxiliary output technique-based sliding mode observers wereproposed to deal with this situation that matching conditionwas not satisfied in [119].And an augmented fault estimation design was proposed in [136] through robust H∞observer design, in the case where there is no invariant zero at the origin. The faultestimator is an integral term. Correspondingly, the augmented fault estimation designwas extended to discrete-time systems [136], for the case where there is no invari-ant zero at one. The fault estimator uses the output information of the previous timeinstant. In [105], a fault estimator was proposed using the high-order time derivativesof the output, but thismethod amplifies the effect of noises on the fault estimation per-formance. In [152], a parity space-based fault estimator was formulated as finding aminimum of a quadratic form, which provides an unified solution to the fault estima-tion problem with different design criteria. In addition, there are model uncertaintiesin practical systems, such as modeling errors, noises and unknown disturbances,so it is impossible to obtain the exact mathematical model. Therefore, the robusttechnique is needed to be considered for fault diagnosis and fault-tolerant control[100, 108, 110]. For the problem of robust fault estimation, fault estimation errorsare insensitive to these system uncertainties, aimed at enhancing accuracy of faultestimation. Based on the above analysis, how to further enhance fault estimationperformance of augmented fault estimation observer design is an interesting issueand motivates our study.

1.2 Motivations 5

A fault estimation observer designwith adjustable parameter (AP) for continuous-time/discrete-time dynamic systems will be considered to enhance fault estimationperformance by adding an adjustment term. The proposed method has the sameapplication range as the augmented fault estimation design, but is more flexible inthe fault estimator design compared with the augmented fault estimation design.

1.2.5 Multi-Agent Systems

Over the past two decades, distributed control of multi-agent systems (MAS) hasattractedmuch attention frommany scientific communities because of its wide appli-cation in various fields, such that consensus, formation control of vehicles, distributedcontrol of multiple robotics, etc. [3, 16, 37, 39, 55, 56, 63, 74, 80, 81, 90, 128].Designing distributed protocols based on the relative information guarantees that thestates of all agents reach an agreement, known as the consensus problem. All agentsneed to interact with each other and eventually reach an agreement.

In past few years, fault diagnosis of MAS has attracted attention and become avery hot topic [78]. Compared with the centralized architecture, the fault diagnosisstudy of MAS is more complex because of the information exchanges among allagents. The fault occurred in a certain agent would be propagated to other onesthrough the communication graph and affects fault-free agents’ behavior, whichresults in performance degradation or even catastrophic accidents for the wholeMAS. Therefore, the issue of fault diagnosis is very critical for MAS to enhance thesystem safety. Meanwhile, there are different types of agents modeled by first-order,second-order or general linear dynamics.Comparedwithfirst-order and second-orderdynamics, the general linear dynamics is more representative and more preciselydescribes the control system. In [103], the problem of distributed fault detection andisolation for large-scale interconnected systemswith respect to different fault modelswas studied. In [148], an adaptive neural network-based distributed fault detectionand isolation approachwas discussed for a class of interconnected uncertain nonlinearsystems. In [70, 71], the design and analysis of actuator fault detection and isolationfilters for a network of unmanned vehicles was investigated. [47] considered thedistributed fault detection for a class of MAS with networked-induced delays andpacket dropouts. The problem of distributed fault detection and isolation for a classof second-order discrete-time MAS was studied by using an optimal robust observerapproach [87, 89]. In [103], the problem of distributed fault detection and isolationin large networked systems with uncertain system models was discussed.

But most of these works only dealt with fault detection and isolation, and few ofthese results addressed the problem of online fault estimation, which is a challengingissue. For a class of MAS, a robust fault estimation method based on sliding modeobservers was proposed for a collection of agents, but only undirected graphs wereconsidered [69]. In [25], a consensus-tracking based distributed fault estimation anddistributed fault tolerant control problem for a multi-agent system were proposed,but the studied plant was a special class of power systems.

6 1 Introduction

1.3 Book Outline

In this book, inspired by the previous work, our objective is to analysis and developobserver-fault estimation techniques for dynamic systems. The rest of this book isorganized as follows:

In Chap.2, the design of a multi-constrained full-order fault estimation observer(FFEO) with finite frequency specifications is studied for continuous-time systems.By constructing an augmented system, a multi-constrained FFEO in finite-frequencydomain is proposed to achieve fault estimation, which are given in terms of linearmatrix inequalities (LMIs).Meanwhile, the presented FFEOcan avoid the overdesignproblem generated by the entire frequency domain by the generalized KYP lemma.Furthermore, by introducing slack variables, improved results on FFEO design indifferent frequency domains are obtained such that different Lyapunov matrices canbe separately designed for each constraint.

In Chap.3, a framework of fault estimation observer design in finite-frequencydomain is proposed for discrete-time systems, including FFEO and reduced-orderfault estimation observer (RFEO). Under themulticonstrained idea, a FFEO in finite-frequency domain is designed to achieve fault estimation by using the generalizedKYP lemma to reduce conservatism generated by the entire frequency domain. Then,a RFEO is constructed, which results in a new fault estimator to realize fault estima-tion using current output information.

Chapter 4 firstly addresses the problem of fault estimation observer design withfinite-frequency specifications for discrete-time Takagi-Sugeno (T-S) fuzzy systems.Then the problem of fuzzy unknown input observer (FUIO)-based fault estimationis investigated for discrete-time T-S fuzzy systems.

Chapter 5 studies the problem of fault estimation observer design with finite-timeconvergence specification for continuous-time dynamic systems subject to exter-nal disturbances. First, the UIO is constructed to achieve accurate estimation ofthe occurred fault and to guarantee robustness against the disturbance. Then a poleplacement-based fault estimation observer is constructed using time-delay designsuch that the fault estimation error converges to zero in finite time. Unlike conven-tional literatures, the proposed fault estimator with finite-time convergence specifi-cation doesn’t contain discontinuous sign function.

Chapter 6 addresses an AP-based multi-objective fault estimation design forcontinuous-time/discrete-time dynamic systems. First, a fault estimation observerwith AP is constructed to on-line identify the size of occurred faults. The faultestimation design not only possesses a wider application compared with adaptiveobservers, but also uses the current output information to enhance fault estimationperformance.

In Chap.7, the distributed fault estimation observer (DFEO) is studied based onH∞ and H2 strategies for discrete-time MAS. For each agent, a fault estimationobserver is designed using relative output estimation errors. By denoting globalestimation error vectors, the global error dynamics is constructed for MAS. Theexistence condition of the presented DFEO is further discussed.

1.3 Book Outline 7

In Chap.8, under the directed communication topology, an adaptive observer-based DFEO is studied for MAS. Firstly, a fault estimation observer is constructedbased on their relative output estimation errors. To consider DFEO design from anoverall perspective, the whole error dynamics is obtained by defining global errorvectors. Then an adaptive technique-based DFEO design is proposed for MAS withdirected communication topology.

In Chap.9, an AP-based DFEO is proposed for MAS with the directed commu-nication topology to improve the accuracy of fault estimation.

Conclusions of this book are presented in Chap.10.

Chapter 2Fault Estimation of Continuous-TimeSystems in Finite-Frequency Domain

2.1 Introduction

In this chapter, inspired by the previous work, our objective is to provide a gen-eral robust fault estimation observer scheme with finite-frequency specifications forcontinuous-time systems. Main contributions of this chapter are twofold: (1) Basedon the generalized KYP lemma, a multi-constrained FFEO with finite-frequencyspecifications is proposed to achieve fault estimation, aimed at decreasing the con-servatism that results from the entire frequency domain; (2) By using the projectionlemma and introducing auxiliary slack variables, we obtain the improved results,which not only design different Lyapunov matrix for each constraint, but also areconvenient to calculate FFEO parameters for different frequency domains.

The rest of this chapter is organized as follows. The systemdescription is presentedin Sect. 2.2. In Sect. 2.3, based on the generalized KYP lemma, a multi-constrainedFFEO designwith finite-frequency specifications is proposed to avoid the overdesignproblem generated by the entire frequency domain, and improved FFEO results arefurther obtained by introducing slack variables. Simulation results are presentedin Sect. 2.4 to show the effectiveness of the proposed approach, followed by someconcluding remarks in Sect. 2.5.

2.2 System Description

Consider the following continuous-time system:

{x(t) = Ax(t) + Bu(t) + E f (t) + D1d(t)y(t) = Cx(t) + D2d(t)

(2.1)

where x(t) ∈ Rn is the state, u(t) ∈ R

m is the input, y(t) ∈ Rp is the output,

d(t) ∈ Rd is the disturbance and noise which belongs to L2[0,+∞) and f (t) ∈ R

r

© Springer International Publishing AG 2018K. Zhang et al., Observer-Based Fault Estimation Techniques, Studies in Systems,Decision and Control 127, https://doi.org/10.1007/978-3-319-67492-6_2

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