Model-based fault diagnosis and fault-tolerant control for ... · Abstract The work presented in...

Model-based fault diagnosis and fault-tolerant control

for a nonlinear electro-hydraulic system

Modellbasierte Fehlerdiagnose und fehlertolerante Regelung

fur ein nichtlineares elektrohydraulisches System

Vom Fachbereich Elektrotechnik und Informationstechnikder Technischen Universitat Kaiserslauternzur Verleihung des akademischen Grades

Doktor der Ingenieurwissenschaften (Dr.-Ing.)genehmigte Dissertation von

M. Eng. Liang Chengeb. in Jiangsu, VR China

D 386

Tag der mundliche Prufung: 26.03.2010Dekan des Fachbereichs: Prof. Dipl.-Ing. Dr. G. Fohler1. Berichterstatter: Prof. Dr.-Ing. S. Liu2. Berichterstatter: Prof. Dr.-Ing. habil. L. LitzVorsitzender der Prufungskommission: Prof. Dr.-Ing. R. Zengerle

Acknowledgments

First and above all, I am really grateful to my advisor Prof. Steven Liu. This work wouldnot have been accomplished without his invaluable guidance. I am also thankful for hispermission of the four year scholarship. Without his help my stay in TU Kasierslauterncan not be such an enjoyable experience.

I thank Prof. Lothar Litz for reviewing my dissertation. I also want to thank Prof.Zengerle for hosting my Ph.D examination.

I thank all the faculty, staff and students of Institute of control systems, especially TimNagel, Daniel Gorges, Philipp Munch, Dr. Tuttas, Michal Izak, Nadine Stegmann, JensKroneis, Wei Wu, Peter Muller, Jianfei Wang, Qian Liu, Jutta Lenhardt, Swen Becker,Jie Zhao and Zheng Liu for their valuable help, assistance and suggestions.

Last but not least, I want to thank my family. I would not be where I am today withouttheir constant support and love.

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Abstract

The work presented in this thesis discusses the model-based fault diagnosis and fault-tolerant control with application to a nonlinear electro-hydraulic system. High per-formance control with guaranteed safety and reliability for electro-hydraulic systems isa challenging task due to the high nonlinearity and system uncertainties. This thesisdeveloped a diagnosis integrated fault-tolerant control (FTC) strategy for the electro-hydraulic system. In fault free case the nominal controller is in operation for achievingthe best performance. If the fault occurs, the controller will be automatically reconfig-ured based on the fault information provided by the diagnosis system. Fault diagnosisand reconfigurable controller are the key parts for the proposed methodology. The sys-tem and sensor faults both are studied in the thesis.

Fault diagnosis consists of fault detection and isolation (FDI). A model-base residualgenerating is realized by calculating the redundant information from the system modeland available signal. In this thesis differential-geometric approach is employed, whichgives a general formulation of FDI problem and is more compact and transparent amongvarious model-based approaches. The principle of residual construction with differential-geometric method is to find an unobservable distribution. It indicates the existence of asystem transformation, with which the unknown system disturbance can be decoupled.With the observability codistribution algorithm the local weak observability of trans-formed system is ensured. A Fault detection observer for the transformed system canbe constructed to generate the residual. This method can not isolated sensor faults. Inthe thesis the special decision making logic (DML) is designed based on the individualsignal analysis of the residuals to isolate the fault.

The reconfigurable controller is designed with the backstepping technique. Backsteppingmethod is a recursive Lyapunov-based approach and can deal with nonlinear systems.Some system variables are considered as “virtual controls” during the design procedure.Then the feedback control laws and the associate Lyapunov function can be constructedby following step-by-step routine. For the electro-hydraulic system adaptive backstep-ping controller is employed for compensate the impact of the unknown external load inthe fault free case. As soon as the fault is identified, the controller can be reconfiguredaccording to the new modeling of faulty system. The system fault is modeled as theuncertainty of system and can be tolerated by parameter adaption. The senor fault actsto the system via controller. It can be modeled as parameter uncertainty of controller.All parameters coupled with the faulty measurement are replaced by its approximation.After the reconfiguration the pre-specified control performance can be recovered.

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FDI integrated FTC based on backstepping technique is implemented successfully onthe electro-hydraulic testbed. The on-line robust FDI and controller reconfiguration canbe achieved. The tracking performance of the controlled system is guaranteed and theconsidered faults can be tolerated. But the problem of theoretical robustness analysisfor the time delay caused by the fault diagnosis is still open.

Abstract (German)

Die presentierte Dissertation behandelt die modellbasierte Fehlerdiagnose und fehler-tolerante Regelung mit Anwendung fur ein nichtlineares elektrohydraulisches Systems.Die hochperformante Regelung elektrohydraulischer Systeme mit garantierter Sicher-heit und Zuverlaessigkeit ist aufgrund starker Nichtlinearitaten, unbekannter außererStorungen und komplexen Stromungseigenschaften eine herausfordernde Aufgabe. Indieser Dissertation wird eine fehlertolerante Regelung mit integrierter Fehlerdiagnosefuer das untersuchte elektrohydraulische System entwickelt. Im fehlerfreien Fall wirdder Regler Regelung angewendet. Im Fehlerfall wird der Regler automatisch rekonfig-uriert basierend auf der Fehlerinformation, die aus der Fehlerdiagnose verfugbar sind.Die Fehlerdiagnose und der rekonfigurierbare Regler sind somit integrale Bestandteileder vorgeschlagenen Methodik.

Die Fehlerdiagnose umfasst die Fehlererkennung und Fehlerisolation (Fault Detectionand Isolation (FDI)). Eine modellbasierte Residuengenerierung wird realisiert durchBerechnung der redundanten Information aus dem Systemmodell und den verfugbarenMessgroßen. In dieser Dissertation wird ein differentialgeometrischer Ansatz verwen-det, der auf eine allgemeine Formulierung des FDI-Problems fuhrt sowie kompakter undtransparenter als andere bekannte modellbasierte Ansatze ist. Das zentrale Problem derResiduengenerierung basierend auf dem differentialgeometrischen Ansatz besteht in derBestimmung einer unbeobachtbaren Distribution. Diese zeigt die Existenz einer Sys-temtransformation an, durch die die unbekannten Systemstorungen entkoppelt werdenkoennen und der gesuchte Fehler isoliert werden kann. Mit dem Observability Codistri-bution Algorithm wird die lokale schwache Beobachtbarkeit des transformierten Systemssichergestellt. Ein Beobachter zur Fehlererkennung kann fur das transformierte Systementworfen werden, um das Residuum zu generieren. Eine Fehlerisolation ist mit dieserMethode nicht moglich. In dieser Dissertation wird eine Entscheidungslogik (DecisionMaking Logic (DMI)) basierte auf der individualen Signalanalyse des Residiuums en-twickelt.

Der rekonfigurierbare Regler wird mit dem Backstepping-Verfahren entworfen. DasBackstepping-Verfahren ist ein rekursiver Lyapunov-basierter Ansatz, der gut fur nicht-lineare Systeme geeignet ist. Das Regelgesetz und die zugehorige Lyapunov-Funktionkonnen durch folgendes schrittweise Verfahren konstruiert werden: Fur das elektrohy-draulische System wird ein adaptiver Backstepping-Regler angewendet, um den Einflussder unbekannten außeren Last im fehlerfreien Fall zu kompensieren. Sobald ein Fehlererkannt wird, wird der Regler rekonfiguriert basierend auf dem Modell des fehlerbe-

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hafteten Systems. Der Systemfehler wird als Parameterfehler des Systemes modeliertund durch eine weitere Paramteradaption toleriert. Der Sensorfehler kann durch Reglerdas System beeinflußen. Dann wird der Sensorfehler als Parameterfehler des Reglersbehandelt. Die Paraemter koppelt mit der fehlerhaften Messung wurde durch ihre Ap-proximation ersetzt. Nach der Rekonfiguration des Reglers war die Performaz wiederverbessert.

Die fehlertolerante Regelung mit integrierter Fehlerdiagnose basierend auf Backsteppingwurde erfolgreich an einem elektrohydraulischen Pruefstand mit kunstliche herbeige-fuehrten Fehlern implementiert. Eine echtzeitfahige robuste Fehlerdiagnose und Rekon-figuation des Reglers konnte erreicht werden. Die Performanz des geregelten Systemsbezuglich der Fuhrungsgeroße wurde garantiert und die betrachteten Fehler konnten to-leriert werden. Eine theoretische Robustheitsanalyse hinsichtlich der Totzeit, die durchdie Fehlerdiagnose verursacht wird, ist offen und sollte in weiteren Untersuchungen be-trachtet werden.

Contents

Acknowledgments I

Abstract III

Abstract(German) V

List of Figures XIII

Terminology XXI

1 Introduction 11.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives and structure of the dissertation . . . . . . . . . . . . . . . . 2

2 Description of the electro-hydraulic system 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Servo-pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Functional description . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Modeling of servo-pump . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Hydraulic cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 One-cylinder model . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Two-cylinder model . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 System identification and verification . . . . . . . . . . . . . . . . . . . . 132.4.1 Parameter identification for the pump . . . . . . . . . . . . . . . 132.4.2 Parameter identification for the cylinder . . . . . . . . . . . . . . 142.4.3 Verification of whole system . . . . . . . . . . . . . . . . . . . . . 16

2.5 Fault construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.1 Internal leakage in cylinder . . . . . . . . . . . . . . . . . . . . . . 192.5.2 Sensor faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Model-based fault detection and isolation 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.2 Residual generation . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Differential-geometric approach . . . . . . . . . . . . . . . . . . . . . . . 243.2.1 FDI for linear time-invariant systems . . . . . . . . . . . . . . . . 24

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VIII Contents

3.2.2 FDI for nonlinear input-affine systems . . . . . . . . . . . . . . . 293.3 FDI for the electro-hydraulic system . . . . . . . . . . . . . . . . . . . . 33

3.3.1 Disturbance decoupling . . . . . . . . . . . . . . . . . . . . . . . . 333.3.2 Fault detection observer . . . . . . . . . . . . . . . . . . . . . . . 373.3.3 Decision making . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.4 Sensor total failure checking . . . . . . . . . . . . . . . . . . . . . 41

3.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.1 Simulation settings . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Fault-tolerant control for the electro-hydraulic system 494.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Backstepping based FTC . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.2 Standard backstepping . . . . . . . . . . . . . . . . . . . . . . . . 544.2.3 Adaptive backstepping . . . . . . . . . . . . . . . . . . . . . . . . 564.2.4 Backstepping based FTC for electro-hydraulic system . . . . . . . 58

4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.1 Scenario 1: fault free . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.2 Scenario 2: with the internal leakage QLin . . . . . . . . . . . . . 674.3.3 Scenario 3: with the sensor offset ∆PB . . . . . . . . . . . . . . . 684.3.4 Scenario 4: with the sensor offset ∆PA . . . . . . . . . . . . . . . 694.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Experimental results 735.1 Practical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1 Hardware and software . . . . . . . . . . . . . . . . . . . . . . . . 735.1.2 Parameters and experiment setting . . . . . . . . . . . . . . . . . 74

5.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.1 Scenario 1: fault free . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.2 Scenario 2: with the internal leakage QLin . . . . . . . . . . . . . 795.2.3 Scenario 3: with the sensor offset ∆PB . . . . . . . . . . . . . . . 835.2.4 Scenario 4: with the sensor offset ∆PA . . . . . . . . . . . . . . . 865.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Summary and outlook 936.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Contents IX

7 Kurzfassung 95

Appendix 101

A Background of geometric theory 101A.1 Affected/unaffected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.2 Vector spaces and subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 101A.3 Dual spaces and annihilators . . . . . . . . . . . . . . . . . . . . . . . . . 102A.4 Factor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.5 Invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.6 Reachable subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.7 Unobservable subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.8 Distribution and codistribution . . . . . . . . . . . . . . . . . . . . . . . 104

B Technical data of the electro-hydraulic system 105

Bibliography 113

Lebenslauf 115

List of Tables

2.1 Common faults in electro-hydraulic systems . . . . . . . . . . . . . . . . 182.2 The designed faults of tesbed . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 The relation between the faults and the residuals . . . . . . . . . . . . . 393.2 The fault isolation table with rLin . . . . . . . . . . . . . . . . . . . . . . 40

5.1 The experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

B.1 Mannesmann-Rexroth A10VSO 28 DFE1 . . . . . . . . . . . . . . . . . . 105B.2 Mannesmann-Rexroth CD H3 MF3/200/140/500 . . . . . . . . . . . . . 105B.3 TR-Electronic CE-65-SSI . . . . . . . . . . . . . . . . . . . . . . . . . . . 106B.4 Mannesmann-Rexroth DBE 62-1X/315 . . . . . . . . . . . . . . . . . . . 106

XI

List of Figures

2.1 Block diagram of the electro-hydraulic testbed . . . . . . . . . . . . . . . 52.2 Cross section view of servo-pump [Rex97] . . . . . . . . . . . . . . . . . . 62.3 Block diagram of servo-pump [Rex97] . . . . . . . . . . . . . . . . . . . . 72.4 Closed control loop of servo pump . . . . . . . . . . . . . . . . . . . . . . 82.5 Nonlinear model of servo-pump . . . . . . . . . . . . . . . . . . . . . . . 92.6 Differential cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Load cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 Swashplate time response to PMRS input . . . . . . . . . . . . . . . . . 132.9 Swasplate frequency response to PMRS input . . . . . . . . . . . . . . . 142.10 Swasplate time response to step input . . . . . . . . . . . . . . . . . . . . 142.11 Swasplate time response with sinusoidal input . . . . . . . . . . . . . . . 142.12 Velocity-friction curve [JK04] . . . . . . . . . . . . . . . . . . . . . . . . 152.13 Verification of cylinder friction . . . . . . . . . . . . . . . . . . . . . . . . 152.14 Verification of volume flow of chamber B . . . . . . . . . . . . . . . . . . 162.15 Pressure in chamber A of working cylinder . . . . . . . . . . . . . . . . . 172.16 Pressure in chamber B of working cylinder . . . . . . . . . . . . . . . . . 172.17 Velocity of working cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 172.18 Position of working cylinder . . . . . . . . . . . . . . . . . . . . . . . . . 182.19 Artificial internal leakage of cylinder . . . . . . . . . . . . . . . . . . . . 192.20 Valve controlled internal leakage of testbed . . . . . . . . . . . . . . . . . 20

3.1 Fault progression over time [Zog02] . . . . . . . . . . . . . . . . . . . . . 213.2 Structure of model-based diagnosis [CP99] . . . . . . . . . . . . . . . . . 223.3 The principle of the designed DML . . . . . . . . . . . . . . . . . . . . . 403.4 Characteristic curves of proportional directional valve . . . . . . . . . . . 413.5 External load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 Estimation in fault free case . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 The residuals without faults . . . . . . . . . . . . . . . . . . . . . . . . . 433.8 the internal leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.9 Estimation with the internal leakage . . . . . . . . . . . . . . . . . . . . 443.10 The residuals with the internal leakage . . . . . . . . . . . . . . . . . . . 443.11 Estimation with fault ∆PB . . . . . . . . . . . . . . . . . . . . . . . . . 453.12 The residuals with fault ∆PB . . . . . . . . . . . . . . . . . . . . . . . . 453.13 Estimation with fault ∆PA . . . . . . . . . . . . . . . . . . . . . . . . . 463.14 The residuals with fault ∆PA . . . . . . . . . . . . . . . . . . . . . . . . 46

XIII

XIV List of Figures

4.1 The structure of fault-tolerant control [BKLS03] . . . . . . . . . . . . . . 494.2 The structure of backsteping based FTC for electro-hydraulic system . . 594.3 Tracking control (fault free) . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Estimation of Fload (fault free) . . . . . . . . . . . . . . . . . . . . . . . 664.5 Tracking control (with QLin) . . . . . . . . . . . . . . . . . . . . . . . . . 674.6 Estimation of Fload (with QLin ) . . . . . . . . . . . . . . . . . . . . . . . 674.7 Estimation of QLin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.8 Tracking control (with ∆PB) . . . . . . . . . . . . . . . . . . . . . . . . . 684.9 Tracking control (with ∆PA) . . . . . . . . . . . . . . . . . . . . . . . . . 694.10 Estimation of Fload (with ∆PA ) . . . . . . . . . . . . . . . . . . . . . . . 694.11 Tracking control (FTC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.12 Tracking control (adaptive backsteppping) (zoomed) . . . . . . . . . . . . 70

5.1 Structure of the software . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Estimation in fault free case . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 Estimation errors in fault free case . . . . . . . . . . . . . . . . . . . . . 755.4 Residuals in fault free case . . . . . . . . . . . . . . . . . . . . . . . . . . 765.5 The position trajectories in fault free case . . . . . . . . . . . . . . . . . 765.6 The velocity trajectories in fault free case . . . . . . . . . . . . . . . . . . 775.7 The external load force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.8 The position trajectories with PI controller (fault free) . . . . . . . . . . 785.9 The velocity trajectories with PI controller (fault free) . . . . . . . . . . 785.10 Estimation with QLin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.11 Estimation errors with QLin . . . . . . . . . . . . . . . . . . . . . . . . . 795.12 Residuals with QLin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.13 The decision making logic with QLin . . . . . . . . . . . . . . . . . . . . 805.14 The position trajectories with QLin . . . . . . . . . . . . . . . . . . . . . 815.15 The velocity trajectories with QLin . . . . . . . . . . . . . . . . . . . . . 815.16 Estimation of external load force with fault QLin . . . . . . . . . . . . . . 825.17 Estimation of QLin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.18 Estimation with ∆PB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.19 Estimation errors with ∆PB . . . . . . . . . . . . . . . . . . . . . . . . . 835.20 Residuals with ∆PB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.21 The decision making logic with ∆PB . . . . . . . . . . . . . . . . . . . . 845.22 The position trajectories with ∆PB . . . . . . . . . . . . . . . . . . . . . 855.23 The velocity trajectories with ∆PB . . . . . . . . . . . . . . . . . . . . . 855.24 Estimation with ∆PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.25 Estimation errors with ∆PA . . . . . . . . . . . . . . . . . . . . . . . . . 865.26 Residuals with ∆PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.27 The zoomed rPA

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.28 The decision making logic with ∆PA . . . . . . . . . . . . . . . . . . . . 885.29 The position trajectories with ∆PA . . . . . . . . . . . . . . . . . . . . . 885.30 The velocity trajectories with ∆PA . . . . . . . . . . . . . . . . . . . . . 895.31 The external load force with ∆PA . . . . . . . . . . . . . . . . . . . . . . 89

List of Figures XV

5.32 The position trajectories with total PA failure . . . . . . . . . . . . . . . 905.33 The velocity trajectories with total PA failure . . . . . . . . . . . . . . . 90

7.1 Das elektrohydraulische System . . . . . . . . . . . . . . . . . . . . . . . 967.2 Die Struktur der prasentierten Methde . . . . . . . . . . . . . . . . . . . 98

List of Symbols

A system matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A1 area of cylinder piston ring side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10A2 area of cylinder piston ring side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10As area of pump control cylinder piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9B input matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25C output matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25CH hydraulic capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9dv damping coefficient of valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9e1, e2 · · · ei error terms of backsteeping controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56eo error of observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37EH bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10EA bulk modulus in chamber A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11EB bulk modulus in chamber B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Ee effective bulk modulus of oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10E matrix of residual generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25f(x, t) nonautonomous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53F matrix of residual generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25FeL external load of pump control cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Ff friction of cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11FL external load of cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Fsf friction of pump control cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9G matrix of residual generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25H matrix of residual generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25Inf infimal element (minimal) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28jth fault mode index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25k integer set (1, . . . , k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25kair air/oil ration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10ksL leakage coefficient of pump control cylinder . . . . . . . . . . . . . . . . . . . . . . . . .9ksp pump gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9kp parameter of P controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9kpu pump angle-voltage gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9kvu spool position-voltage gain of valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9KQ pump flow coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8kqb1, kqb2 flow coefficient of chamber B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11kQv valve flow coefficient of servo pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

XVII

XVIII List of Figures

L1, L2 input vector for fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Lo observer gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37ms piston mass of pump control cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9mc moving mass of single cylinder model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11m moving mass of two cylinder model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12M matrix of residual generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25n motor speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8PA pressure in chamber A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10PB pressure in chamber B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10PsL pressure in control cylinder of pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Po atmospheric pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10o.c.a observability codistribution algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33q output dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25q integer set (1, . . . , q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Q fluid volume flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10QA volume flow of chamber A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11QB volume flow of chamber B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11QLex external leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10QLin internal leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Qp volume flow of pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Qv volume flow of valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8R real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25R

n n-dimensional real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25R+ positive real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53ri residual set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25S unobservability subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Tv time constant of valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9u input vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25uin input voltage of pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9uv input voltage of valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9vc velocity of cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11V Lyapunov function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56VH volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10V10 initial volume in chamber A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11V20 initial volume in chamber B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11Vgmax maximal displacement of pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8x state vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25xc stroke of cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11xr reference trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56z state variable of residual generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25yv valve spool stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8α swshplate angle of pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8β2, β3, . . . , βn stabilizing function of backstepping controller . . . . . . . . . . . . . . . . . . . . . 55ηv volumetric efficiency of pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

List of Figures XIX

ν fault signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25σ() spectrum of matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27ξ disturbance signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29ωv natural frequency of valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Ωj coding set j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Terminology

The definitions of terminology listed here stem from [IB97] and [BFK+00]. They aresuggested by the Safeprocess Technical Committee of IFAC.

Fault is an unpermitted deviation of at least one characteristic property or parameterof the system from the acceptable/usual/standard condition.

Failure Permanent interruption of a systems ability to perform a required function underspecified operating conditions.

Analytical redundancy Use of more than one not necessarily identical ways to determinea variable, where one way uses a mathematical process model in analytical form.

Hardware redundancy Use of more than one independent instrument to accomplish agiven function.

Fault diagnosis Determination of kind, size, location, and time of occurrence of a fault.Includes fault detection, isolation

Fault detection Determination of faults present in a system and time of detection.

Fault isolation Determination of kind, location, and time of detection of a fault. Followsfault detection.

Residual Fault information carrying signals, based on deviation between measurementsand model based computations.

Threshold Limit value of a residual’s deviation from zero, so if exceeded, a fault isdeclared as detected.

XXI

1 Introduction

1.1 Background and motivation

Hydraulic systems are widely used in today’s industries. Compared to electrical drivesthey can generate large forces or torques very fast with simple structures. Hydraulicsystems can broadly classified into two categories. The first is industrial hydraulics,which is typically applied in mechanical manufacture, like extrusion molding machines,pressers, rolling mills and other [Fin06]. The second is mobile hydraulics, which are thecore components of steering, braking, suspension and power-train system of the vehicles[KZH04] and also popular in the robot manipulators, ship rudders, airplane landing gearsand so on [Mun06].

Modern hydraulic systems are installed with electronic, for instance, sensors, servo-valves. Such systems are known as electro-hydraulic systems. The inputs are usuallyelectrical signals and the relevant system variables are measured and transformed intoelectrical signals too. Thus, the hydraulic components can be precisely controlled toachieve better performance. But the control of electro-hydraulic systems is a challengingtask due to inherent nonlinearities from complicated flow properties, friction in actuator,varying external load etc [BCR02]. Therefore, advanced control strategies are necessaryfor handling above problems [JK04].

Meanwhile, reliability and safety of the systems are most important issues of automatedsystems. Usually, a small fault can have a serious effect to control system, such asactuator malfunction or sensor offset, sometimes even crushes the system. If faults canbe detected and identified earlier, the collapse of whole system can be avoided. Thisrequirement motivates the research in fault diagnosis, which includes fault detection andisolation (FDI). If the redundant information is from the system model and availablesignal, it is labeled as model-based FDI, which is more attractive than conventionalhardware redundancy based diagnosis, because the former gives more information aboutmonitored system and improves the systems safety and reliability without additionalhardware cost [CP99]. Furthermore, if a controller is able to tolerate possible faultsautomatically, the control is known as fault-tolerant control (FTC). Usually the FTCneeds the fault information from FDI system. The controller can react to the fault andcan be automatically reconfigured.

High performance control [AL00], [TSL98], [CLCL08] and model-based diagnosis [AS03],[Mun06] [GY05] of electro-hydraulic systems both have attracted a lot of researchers.

1

2 1 Introduction

But there are few results, which combines the two aspect together. It is very meaningfulto apply the FDI integrated FTC to electro-hydraulic systems, which work usually incritical places, like in automobiles, aircrafts. Besides the improvement of safety andreliability, if the fault in electro-hydraulic system can be detected early, the maintenancecost can be cut down. The whole system will be more intelligent and efficient.

1.2 Objectives and structure of the dissertation

Motivated by above considerations the objective of this dissertation is to develop anapplicable model-based FDI integrated FTC scheme for the nonlinear electro-hydraulicsystem. The FDI system should be robust to the all uncertainties of the system but theconsidered faults. The high performance tracking control should be guaranteed even inthe presence of the fault. The main tasks of this thesis are:

• modeling of system

• construction of faults

• design of robust model-based FDI systems

• design of FTC system

• keeping stability in the critical period during which the fault is present but notidentified.

The rest of the dissertation is structured as follows.

In chapter 2, the model of the electro-hydraulic system is derived based on the physicalrelations among the system variables and experimental modeling technique. The derivedmodel is validated with the measurements. The system is subject to a varying externalload, which is model as an unknown disturbance to the system. The feature of systemmodel gives the instructions for choice of FDI and FTC methods. In this chapter, theconstruction of the fault is introduced too. Among the most common faults in electro-hydraulic systems a system fault and two sensor faults are designed and constructedphysically on the testbed. This chapter is the foundation of the rest of thesis.

In chapter 3, the modern model-based fault diagnosis methods are reviewed preliminarily.The differential-geometric approach is chosen for construction of FDI system because itgives a general formulation of FDI problem and is more compact and transparent thanother model-based approaches. This approach gives a systematic design procedure to finda distribution, with which the system can be transformed so that it is only affected by onefault and unaffected by other system faults or disturbance. The local weak observabilityof the transformed system is also guaranteed with the proposed construction method.Thus, a fault detection observer based on the subsystem can be designed for residualgeneration. The fault isolation is achieved by a decision making logic (DML), which is

1.2 Objectives and structure of the dissertation 3

based on the reaction of residuals to the fault.

In chapter 4, the design of reconfigurable controller for FTC scheme is conducted. TheState-of-the-Art of FTC is introduced at first. The backstepping control strategy isemployed to deal with the high nonlinearity. The feedback control laws and the associateLyapunov function can be constructed by following step-by-step routine. To compensatethe effect of varying external load of the system parameter an adaption law is integrated.Then the tracking error can asymptotically converge to zero in the fault free case. Thereconfigurations of controller in considered faulty cases are derived explicitly. Overall,the adaptive backstepping based reconfigurable controller and the FDI system proposedin chapter 3 construct the FTC scheme.

In chapter 5, the proposed FDI integrated FTC strategy is successfully implemented onthe electro-hydraulic system. All considered fault scenarios are tested. Detailed resultsabout FDI and FTC are presented.

In chapter 6, the summary and the advice to the work in future are given as the end ofthis dissertation.

2 Description of theelectro-hydraulic system

2.1 Introduction

!" # !"

Figure 2.1: Block diagram of the electro-hydraulic testbed

The researched testbed is built by company Mannesmann Rexroth1 and simulates theextruding machine, as shown in Fig. 2.1. It consists of two identical cylinders, whichare named as working cylinder and load cylinder respectively. The working cylinderrepresents the press stem of an extrusion machine. It’s fluid power is supplied by a vari-able displacement axial piston servo-pump. The load cylinder simulates the resistanceforce, which is generated when the material is pressed. It is connected with a fixeddisplacement pump so that load pressure can be built up quickly. With the equipmentof a proportional pressure relief valve an arbitrary load can be achieved. The direction

1Now the company is named as Bosch Rexroth after the merge in 2001.

5

6 2 Description of the electro-hydraulic system

change of fluid flow is realized through several 2-position directional valves. The oper-ation pressure of whole system is limited under 250 Bar [Wit97]. For simplicity it isassumed that the temperature of hydraulic circuit is constant and the pressure drop inpipeline is zero. The direct measured system variables are: swashplate angle, pressure inboth chambers of working cylinder, pressure in chamber A of load cylinder and cylinderstroke. Cylinder velocity and angular velocity of swashplate can be calculated based onthe respective position signals.

There is a Modular-4 card, in which signal measurements and control strategies areimplemented. The card is actually a independent computer system with a 486-CPU. Itcan be installed in PC’s ISA slot and works parallel to the PC. The real-time applicationcan be executed with a sampling rate up to 1k Hz. PC is used only for visualization,data saving and parameter setting.

Generally the hydraulic circuit of testbed can be divided in two main parts: servo-pumpand cylinders. In the following sections of this chapter the modeling of the cylindersand pump, the verification of the models and construction of faults will be introducedin sequence.

2.2 Servo-pump

2.2.1 Functional description

$ %&'%()*'+, - )./)/.+0/1'* 2'*2, 3 /44%,+ 56*017,. 8 5/1+./* 56*017,.9 ).,*/'7,7 %).01: ; %)//* )/%0+0/1 +.'157<5,. = '1:*, >)/%0+0/1? +.'1%7<5,. @ 2'*2, %/*,1/07A 2'*2, %)//* $B 2'*2, %).01:Figure 2.2: Cross section view of servo-pump [Rex97]

2.2 Servo-pump 7

The pump, which is usually driven by electrical machine, transforms the mechanicalenergy into hydraulic energy. The variable displacement pump means the displacement,which is amount of fluid pumped per revolution of the pump’s input shaft, can bevaried while the pump is running. The pump used here is actually a servo-pump andthe structure is shown in Fig. 2.2. It consists of an axial piston pump with built-onproportional valve including inductive position transducer for sensing the swivel angleand valve position, a pressure transducer, and an external amplifier card for control.

The pump’s pistons slide axially in a rotating cylinder block to give a pumping action tothe fluid. The piston shoes bear on a non-rotating swashplate. Variation of the swash-plate angle changes the travel distance of the pistons, which then changes the hydraulicdisplacement of the pump. The swashplate angle is determined by offset cylinder andcontrol cylinder. The stroke of control cylinder is controlled by the proportional valve.

Figure 2.3: Block diagram of servo-pump [Rex97]

The block diagram of servo-pump system is shown in Fig.2.3. If the pump is off, theswashplate angle is held on 100% because of the preload spring in offset cylinder side.With a rotating pump and de-energized proportional valve the big control cylinder isregulated to zero stroke as the valve spool is pushed to the right by the spring andtherefore the pump pressure is applied to the control piston via valve port A. The balance


between the pump pressure and the spring force is achieved between 8 to 12 bar. Increaseof the command value of swashplate angle makes the valve spool move from centerlposition to left side until the swashplate angle (stroke of the control cylinder) reachesthe setting value. The stroke of control cylinder reduces because of the connection ofport A to tank. Thus the swashplate angle will increase. A reduction of command valueof swashplate angle make the valve spool move from center to right. More fluid flowsinto the control cylinder leading to the increases of the stroke. Therefor the swahsplateangle is reduced.

Theoretically the volume flow rate can be calculated through the formula

Qp =Vgmax · n · tan(α) · ηv

1000 · tan(αmax)(2.1)

where Vgmax is the maximal displacement, α stands for swashplate angle and αmax isthe maximum of the angle, n is the rotation speed of the pump and ηv stands for thevolumetric efficiency [Fin06]. The swashplate angle varies usually in a small range, thenα ≈ tan(α). The Qp can be approximated as

Qp ≈ KQ · α, (2.2)

whereKQ is defined as pump flow coefficient, which is determined only by the mechanicalstructure of the pump. Hence the output flow of pump can be viewed as linear to theswashplate.

2.2.2 Modeling of servo-pump

The servo-pump in fact is a closed-loop control system. The swashplate angle α ischosen as the state variable of the pump model [MJ96] [Pra01]. Fig.2.4 shows the cascadstructure of the control loop. uin stands for the command value of the swashplate angle

α

rα

Figure 2.4: Closed control loop of servo pump

and the yv represents the stroke of valve spool. The angle control is the outer-loop. Theangle error is amplified by the P controller and then passed on to the inner control-loopas a command value. The valve spool stroke is regulated by the inner PD controller.Thus the stroke of the control cylinder can be adjusted.

For precise modeling it is necessary to introduce the model of the proportional valveand control cylinder. The fluid flow of proportional valve Qv usually is assumed to be

2.2 Servo-pump 9

linear to the spool stroke yv. The controlled valve can be approximated as a secondorder system [SS00] [Wat05]. In Laplace domain the relation of valve spool stroke yv toinput voltage uv can be written as

yv

uv

=kvuω

2v

s2 + 2dvωvs+ ω2v

, (2.3)

where dv stands for damping coefficient, ωv is the natural frequency and kvu representsthe spool position-voltage gain.

By integrating the cylinder model the block scheme of the servo-pump can be expandedas Fig.2.5. It should be pointed out that the structure is partly simplified but it is

vy kp

r 22

2

2 vvv

vvu

sds

k

!!

!

""kQv

Qv

#HC

1As

PsL

# #

Fsf

FeL

As

kQv flow coefficient of valve CH hydraulic capacity As piston area of control cylinder

PsL pressure in control cylinder ms mass of piston ksL leakage coefficient

Fsf friction of control cylinder FeL external load kp P controller

ksL

sm

1

Figure 2.5: Nonlinear model of servo-pump

still nonlinear because of the saturation of valve spool stroke, the friction in controlcylinder, non-constant parameter CH and so on. This is a fifth-order system. Some ofthe parameters and state variables are still unknown. Consequently the model in Fig.2.5is not appropriate for real-time application.

More simplifications should be made without losing much accuracy of modeling.

• The valve is simplified as a first-order system.

• The cylinder is approximated as an integrator by ignoring the friction, externalload and leakage2.

Thus the servo-pump can be described by a second-order system as following:

α

uin

=kpuksp

AsTvs2 + Ass+ ksp

, (2.4)

where kpu is the angle-voltage gain, uin represents the command value of the swashplateangle, ksp stands for the pump gain, which depends on the controller parameters andvalve flow coefficient and Tv is the valve time constant.

2A detailed cylinder model can be found in chapter 2.3.


2.3 Hydraulic cylinder

2.3.1 One-cylinder model

Cylinders are very common for linear actuation. The modeling of hydraulic cylinderscan be easily found in textbooks on hydraulics [Mer67] [MR03] [AGS06]. Thus only abrief introduction will be made and the background of hydrodynamics will be omitted.

QC QDPC PDAE AF

QGHI QGJKxLFG

FMFigure 2.6: Differential cylinder

As shown in Fig.2.6 is a hydraulic differential cylinder. PA and PB represent the pressurein the chamber A and B respectively. A1 and A2 denote the piston area and ring-sidearea. QLin and QLex stand for the internal leakage flow and external leakage flow. xc isthe cylinder displacement, Ff is the friction force and FL is external load. The pressuredynamics is derived according the rule of mass conservation

∑

Qin −∑

Qout = VH +VH

EH

P , (2.5)

where Q means liquid flow, VH is the volume and EH is the bulk modulus. The bulkmodulus indicates the compressibility of the fluid and can be formed as

EH = −VH

∂PH

∂VH

. (2.6)

For mineral oil the value is around 1.4 ∼ 1.6 × 104 Bar. The bulk modulus is affectedby the system pressure, entrained air, mechanical compliance and temperature. Forengineering applications only the effective bulk modulus, which is denoted as Ee, isutilized, which is usually expressed empirically. Considering the significant influence ofentrained-air the following approximation is employed [Wat05].

Ee =PPo

+ kair

kairEo

P+ P

Po

, (2.7)

where Po is the atmospheric pressure and constant parameter kair is the air/oil volumeratio.

2.3 Hydraulic cylinder 11

Now the pressure dynamics in both chambers can be derived as

PA =EA(PA)

V10 + A1xc

(−A1vc +QA −QLin) (2.8)

PB =EB(PB)

V20 − A2xc

(A2vc −QB +QLin −QLex). (2.9)

where V10 and V20 are the initial volume of chamber A and B respectively, vc standsfor the piston velocity, EA and EB can be calculated according to Eq.(2.7). QA is theflow rate of chamber A, which is equal to the pump output flow QA = KQα because ofthe pump controlled structure of the system. QB is the flow rate of chamber B. Thepressure in chamber B is low, therefore the effect of the directional valves can not beignored here. According to the characteristic curve of the directional valve QB can beapproximated by

QB =

√

kqb1(PB − Po) + kqb2 if kqb1(PB − Po) + kqb2 > 0;

0 if kqb1(PB − Po) + kqb2 ≤ 0.(2.10)

The parameters kqb1 and kqb2 can be identified through experiments.By ignoring theleakages and combining Newton’s second law the dynamic model of working cylindercan be described as:

xc = vc (2.11)

vc =1

mc

(PAA1 − PBA2 − Ff − FL) (2.12)

PA =EA(PA)

V10 + A1xc

(−A1vc +KQα) (2.13)

PB =EB(PB)

V20 − A2xc

(A2vc −QB), (2.14)

where xc is the stroke of the cylinder, vc is the velocity of the cylinder piston, mc meansthe mass of the piston, Ff and FL represent the friction and external load respectively.

2.3.2 Two-cylinder model

According to the cylinder model derived in last section the external load FL is generatedby the extrusion of two cylinder rods. This force can not be measured with the existinghardware. Now a two-cylinder model will be derived, in which the external load is sameas the pressure in load cylinder chamber A, which can be measured directly.

The load cylinder shown in Fig.2.7 is exactly same as the working cylinder. If theworking cylinder moves forwards, it moves backwards with the same velocity. Assumingthe friction in load cylinder is same as in working cylinder piston motion can be written


PAL

PBL

xc

FL

Ff

Figure 2.7: Load cylinder

as:

xc = vc (2.15)

vc =1

mc

(PBLA2 − PALA1 − Ff + FL), (2.16)

here PAL and PBL represent the pressure in chamber A and B respectively. The chamberB is in subpressure state and the pressure PBL is close to tank pressure, so PBL can beignored. Now add (2.16) to (2.12). The following equation can be derived.

vc =1

2mc

(PAA1 − PBA2 − 2Ff − PALA1) (2.17)

Define 2mc = m, 2Ff = Ffric and PALA1 = Fload, so the velocity dynamics is governedby

vc =1

m(PAA1 − PBA2 − Ffric − Fload). (2.18)

In this equation the external load is equal to the pressure in chamber A of load cylinder,which can be directly adjusted by the proportional pressure relief valve. The two-cylindermodel can be written as

xc = vc (2.19)

vc =1

m(PAA1 − PBA2 − Ffric − Fload) (2.20)

PA =EA(PA)

V10 + A1xc

(−A1vc +KQα) (2.21)

PB =EB(PB)

V20 − A2xc

(A2vc −QB), (2.22)

2.4 System identification and verification 13

2.4 System identification and verification

2.4.1 Parameter identification for the pump

Some parameters of the pump model should be identified by input-output data. Thepump model with the unknown parameters p1 and p2 is written in state-space format.

α = vα (2.23)

vα = −p1α− p2vα + p1kpuαr, (2.24)

where vα stands for the angular velocity of pump swashplate. It is important for systemidentification to design appropriate inputs [Lju99]. Here there are three kinds of inputs:

• Pseudo random multi-level (PRMS) signals

• Step signal

• Sinusoidal signal.

PRMS signal is recommended for system identification for exciting all amplitudes andfrequencies of the system. Step signal is used to test the time domain response to thesystem. Sinusoidal signal is additionally applied here because later the reference valuesfor control is sinusoidal-like curves.

The identification procedure is carried out with System Identification Toolbox of Mat-lab. The sampling time is chosen as Ts = 5ms, which is a compromise with respect tothe sampling theory and hardware characteristic [AW96]. By analyzing the test resultsresponding to the three inputs the final values of estimated parameters are p1 = 44414.0and p2 = 333.4. The system verifying results are shown in Fig. 2.8–2.11. If the fre-quency of input signal is under 100 rad/s and amplitude change is under 4v the modelestimations match well with measurements.

0 2 4 6 8 10 12−2

0

2

4

6

8

10

12

14

16

18PRMS Response

Time (sec.)

Sw

aspl

ate

Ang

le (

deg.

)

MeasurentModelfit: 69.56%

Figure 2.8: Swashplate time response to PMRS input


10−1

100

101

102

103

104

10−4

10−2

100

Am

plitu

de

Frequency Response

10−1

100

101

102

103

104

−400

−300

−200

−100

0

100

Pha

se (

deg.

)

Frequency (rad/s)

MeasurementModel

Figure 2.9: Swasplate frequency response to PMRS input

0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

Step Response

Time (sec.)

Sw

ashp

late

Ang

le (

deg.

)

MeasurementModel;fit: 95.94%

Figure 2.10: Swasplate time response to step input

5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

sinusoid input

Time (sec.)

Sw

ashp

late

Ang

le (

deg.

)

MeasurementModel;fit: 99.07%

Figure 2.11: Swasplate time response with sinusoidal input

2.4.2 Parameter identification for the cylinder

In the cylinder model the friction force Ffric, flow coefficients kqb1 and kqb2 should beidentified.


The friction in hydraulic cylinder consists of three parts [JK04] [Nis02]: viscous frictionfv, static friction fs and Coulomb friction fc. The velocity-friction curve usually has theform like Fig. 2.12. It is obvious that in the low velocity area the friction is strongly

−10 −8 −6 −4 −2 0 2 4 6 8 10−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

Velocity

Fric

tion

0

fc + f

s

fc

Figure 2.12: Velocity-friction curve [JK04]

nonlinear and hardly representable mathematically. Here only the viscous and Coulombfriction are considered. The rest can be treated as the uncertainty in external load.Then the friction model is written as

Ffric = Dvv + fc. (2.25)

Estimation of Dv and fc requires the value of friction. By making cylinder move withconstant velocity the friction is calculated through

Ffric = PAA1 − PBA2 − Fload

The optimal values of Dv = 2.718e3 N/mm and fc = 3.596e3 N are obtained by

0 2 4 6 8 10 12 14 160.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Velocity (mm/sec.)

Fric

tion

forc

e (N

)

Velocity vs Friction

MeasurmentModeling

Figure 2.13: Verification of cylinder friction


using polynomial fitting Technique. Fig. 2.13 shows the good linearity of friction. Themaximal error occurs at the smallest velocity because of the influence of static friction.

Identification of QB is similar to friction. During the constant velocity motion of cylinderthe pressure dynamics in chamber B is almost zero, then the QB can be calculated by

QB = A2vc. (2.26)

With optimal values of kqb1 = 61.275 and kqb2 = −63.478 the verifying result is shown

1 1.5 2 2.5 3 3.5 4 4.5 52

4

6

8

10

12

14

16

PB

−Po (Bar)

QB

(L/

min

)

Figure 2.14: Verification of volume flow of chamber B

in Fig. 2.14. With bigger pressure difference the approximation error is smaller.

2.4.3 Verification of whole system

Applying the identified parameters the system model can be represented as following

xc = vc

vc =1

m(PAA1 − PBA2 −Dvvc − fc − Fload)

PB =EB(PB)

V20 − A2xc

(

A2vc −√

kqb1(PB − Po) + kqb2

)

PA =EA(PA)

V10 + A1xc

(−A1vc +KQα)

α = vα

vα = −p1α− p2vα + p1kpuuin.

(2.27)

The external load Fload has significant influence to the system dynamics. Unfortunately itcan not be precisely simulated. The proportional pressure relief valve can only determinea rough area of the pressure in load cylinder, because the pressure varies with velocitychange. For accurate verification of the whole system a look-up table based on measuredload pressure is applied for simulation.


The setting of system verification:

• The input voltage is sinusoidal-like;

• Pressure relief valve is set as 50 Bar;

• Parameter kair is assumed as 0.005.

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Time (sec.)

PA

(B

ar)

Pressure in Chamber A

MeasurementModel

Figure 2.15: Pressure in chamber A of working cylinder

0 5 10 15 20 25 30 35 401

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time (sec.)

PB

(B

ar)

Pressure in Chamber B

MeasurementModel

Figure 2.16: Pressure in chamber B of working cylinder

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

Time (sec.)

v c (m

m/s

ec.)

Cylinder Velocity

MeasurementModel

Figure 2.17: Velocity of working cylinder


0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

Time (sec.)

x c (m

m)

Cylinder Displacement

MeasurementModel

Figure 2.18: Position of working cylinder

Fig. 2.15 and 2.16 show the good match between the modeling and measurements. Themodeling error is mainly caused by the approximation of the effective bulk modulus,especially in low pressure situation. It is remarkable that the cylinder velocity of thesimulation model is also very noisy. This possibly results from the varied load.

2.5 Fault construction

Like all the technical devices the components of electro-hydraulic system can be faultydue to aging, wrong operation or sudden change of work environment, etc. The commonfaults in electro-hydraulic systems are briefly listed in table 2.1.

Components Faults Consequence

Pumpfailure in electrical machine, breakdown of whole system,

leakage insufficient output

Valve destroyed valve seat breakdown of whole system

Cylinder, Motor leakage degraded control performance

Sensoraging, degraded control performance,

cable break even system breakdown

Others polluted oil accelerated aging

Table 2.1: Common faults in electro-hydraulic systems

Not all listed faults can be studied within this research, because some faults are diffi-cult to be simulated, like oil pollution or impossible to be tolerated, such as a suddenbreakdown of electrical power. On the other hand it is not economical to simulate a

2.5 Fault construction 19

failure with high cost. As mentioned in literature [Mun06] leakage and senor faults oc-cur frequently and can be simulated by simple reconstruction in hardware. Thereforethese two kinds of faults are constructed on the testbed. In the range of this thesis theassumption is always held that only one fault is present for faulty system.

Designed faults Symbol

internal leakage in cylinder QLin

sensor fault of PA ∆PA

sensor fault of PB ∆PB

Table 2.2: The designed faults of tesbed

2.5.1 Internal leakage in cylinder

QN QOPN POAP AQxR

FSFTUVWXYY QSZ[

Figure 2.19: Artificial internal leakage of cylinder

The main cause of cylinder internal leakage is the wear of moving component, namelythe piston. The man-made internal leakage is constructed through the bypass pipelinebetween the chamber A and B of working cylinder, like Fig.2.19. The volume flow ofleakage is controlled by a proportional directional valve. The leakage volume flow willnot be measured directly by the flow meter for economy. The value can be obtainedaccording to the characteristic curves of the valve. The pictures of the constructedinternal leakage on real testbed are shown in Fig. 2.20.

2.5.2 Sensor faults

Sensor failures are very common in the electro-hydraulic systems. The electrical sig-nals from sensors are influenced by noise, offsets, etc. The noise usually come from the


Figure 2.20: Valve controlled internal leakage of testbed

electromagnetic compatibility (EMC) problem and consists of high frequency compo-nents. The noise can be easily filtered by anti-aliasing filters. Therefore sensor noisesare excluded in this dissertation.

Offsets of the sensors are caused by sensor aging, potential difference or similar reasons.The offsets in pressure sensors of chamber A and B of working cylinder are seen as sensorfaults and realized by addition of an amount after A/D conversion.

2.6 Summary

This chapter introduces the mathematical model of the electro-hydraulic system. Themodel is built based on the physical characteristic with some reasonable simplifications.System parameters are identified through experiments. Then the system model is verifiedwith the measurements. They have good consistency. The design of the faults is alsointroduced in this chapter. This chapter is the base of whole thesis.

3 Model-based fault detection andisolation

3.1 Introduction

3.1.1 Background

Engineering systems are always subject to unexpected changes, such as componentsmalfunction, change of working condition, etc. The guarantee of safety and reliabilityis extremely important for automated systems. For safety-critical systems, like aircraftor nuclear power plant, hardware redundancy is usually applied to make the systemsnormal running even if fault occurs. This solution can not be widely used because ofhigh cost of hardware. Furthermore the extra installed hardware can also introducefaults into the system. On the other hand the progression of a fault is usually gradualas shown in Fig. 3.1. If the fault can be detected and identified early, not only the totalbreakdown of the system can be avoided but also the maintenance cost can be reducedsignificantly. Hardware redundancy methods can not fulfill these requirements.

Figure 3.1: Fault progression over time [Zog02]

21

22 3 Model-based fault detection and isolation

Thus, since 1970s analytical redundancy methods have been intensive researched andinvestigated [Bea71]. Analytical redundancy means to calculate the required redun-dant information by using the system model and the available signal information. Thisapproach is also referred to as model-based FDI [Loo01]. The major advantage of model-based FDI is that no additional hardware is needed, except a powerful computer system.Although a great deal of real-time computations are necessary for on-line diagnosis, Itis not so arduous with today’s PCs and measurement cards.

Model-based FDI basically consists of two main procedures: residual generation anddecision making, as shown in Fig. 3.2. Residual generation is the key part of thediagnosis system. The residual is zero or almost zero if the system is fault free. Thesignal should distinguished deviate from zero if the fault occurs. A “good” residualsignal should be sensitive to one special fault but robust to the all system uncertainties(modeling error, disturbance, etc) and other faults. Then the fault can be detected andisolated. Decision making is based on the residual signals. It can be carried out bysimply setting thresholds or using statistical methods, like hypothesis test or generalizedlikelihood ratio (GLR) test, etc. Usually a compromise between false alarm occurrenceand fault detection sensitivity should be made to obtain satisfactory results.

SystemInput Output

Residual Generation

Decision Making

Fault Information

Figure 3.2: Structure of model-based diagnosis [CP99]

3.1.2 Residual generation

It is necessary to give a brief review of model-based residual generation approaches.Three major methods are introduced here.

1. Parity space approach is first presented by Chow and Willsky in 1984 [CW84].The parity equations are usually derived by system dynamical model, as state spaceor transfer functions. Residual signal can be generated by checking the consistency

3.1 Introduction 23

of the mathematical relations between the outputs and inputs in a time window(i.e. the parity check), during which the information of occurred faults are presentin measurement. Robustness to additive model uncertainty can also be realized byorthogonal parity equations. The same principle can be applied for fault isolationby treating the uninterested faults as unknown inputs [GS90], [MV88], [PC91].Further researches involved parity space based FDI can be found in [GCF+95]and [Ger97]. The main advantage of parity relation approach is its simplicity.The algorithm can be easily implemented with computers. If the system model isavailable, the parity relation always can be established. But it is difficult for thismethod to deal with nonlinear systems or noisy measurements. Using linearizedmodel can introduce so much modeling uncertainties that the fault diagnosis is notpossible.

2. Parameter estimation and identification approach works based on the ideathat faults affect the system physical parameters such as resistance, friction, vis-cosity, etc. The parameters of actual process are continuously estimated. Residualgeneration can be achieved by comparison estimated parameters to their nominalvalues. An substantial difference indicates the occurrence of one or more faults.Several researches in this area can be found in [Ise84], [IF91] and [Ise93]. Thebasic procedure for carrying out parameter estimation based FDI can be foundin [CP99]. This approach can be applied for both linear and nonlinear systemsand is very effective to parameter faults. An accurate parameter identificationrequires persistently exciting input signal, i.e. all modes of the system shouldbe excited during the identification procedure. For closed-loop control systemssuch a condition can not always be satisfied. Fault isolation could be difficult ifthe physical parameters do not uniquely correspond to model parameters. If thephysical parameters varies with the system dynamics, it will be difficult for thedecision making logic to distinguish them from faults. These restrictions limit thisapproach in application.

3. Observer based approach is the most active research area of model-based FDI.The idea is to reconstruct the states or outputs of the system from the mea-surements by using either observers for deterministic systems or Kalman-Filtersfor stochastic systems [Yan97]. The difference between the estimation and themeasurement is usually viewed as residual. Various design schemes have been pro-posed: in [BZH04], [Bes03] and [JV00] high-gain observers are applied for nonlin-ear systems; in [TE02] and [CS07] sliding-mode observers are employed for robustresidual generation; in [CH98] Kalman Filters are utilized for detecting faults. Fur-thermore many advanced methods can be integrated for improving the robustnessof diagnosis system, such as H∞ technique [PH97] [PMZ06], unknown input ob-servers method [CPZ96] [SO05]; differential-geometric algorithm [HKK00] [DI01];, [KS03]; adaptive techniques [WD96] [JSC04], etc. Fault isolation for observerbased approaches is more complicated. Different methods are employed, such asMulti-Model approach, unknown input decoupling approach and so on. The com-


mon of these methods is that a band of observers or filters are running parallelfor isolating a fault. It should be pointed out that adaptive observers differ fromother observer approaches. Adaptive observers combine observer and parameteridentification techniques [Din08]. The fault can be directly estimated by parameteradaption law. Residual generating can be archived according to the fault ampli-tude. It is very convenient to accomplish fault detection, isolation and estimationsimultaneously. But this approach also has more restrict constraints to the systemstructure [JSC04].

More details and advanced techniques can be found in recent published books [SFP02],[Ise06], [Wit07] and [Din08]. No general method can deal with all kinds of systems.Which approach should be employed depends on the system model.

3.2 Differential-geometric approach

Among the various model-based fault diagnosis approaches introduced in last sectionthe differential geometric method is of high interest. This approach provides a lot ofadvantages as it gives a general formulation of FDI problem and is more compact andtransparent than other methods [Loo01]. Massoumnia first proposed this method in[Mas86a] and [Mas86b] for linear systems. It has been proven that a necessary and suf-ficient condition for the linear fundamental problem of residual generation (FPRG) tobe solvable is the existence of an unobservability subspace leading to a quotient (observ-able) subsystem unaffected by all fault signals but one1. DePersis and Isidori extendedthis concept to nonlinear system by means of construction of the unobservability distri-bution, which can be viewed as the nonlinear analogue of the unobservability subspace[DI01].

In this section the differential-geometric approaches for linear and nonlinear systems willbe reviewed.

3.2.1 FDI for linear time-invariant systems

Consider a linear time-invariant system (LTI) as following:

x = Ax+Bu+k∑

i=1

Liνi (3.1)

y = Cx (3.2)

where x ∈ Rn is the state vector, u ∈ R

w represents the control input, νi ∈ R is thefault signal and y ∈ R

q is the output vector. A, B and C are the system matrices with

1Definition of affected/unaffected can be found in Appendix A

3.2 Differential-geometric approach 25

appropriate dimensions. Sensor faults model is omitted here. By augmenting systemdynamics (3.1) the sensor faults can be modeled as pseudo-actuator faults. The detailscan be found in [MVW89].

The general FDI filter problem (FDIFP) of LTI systems is formulated by [MVW89] asfollowing:

Considering the system (3.1) and (3.2), the FDIFP is to design an LTI dynamic residualgenerator that takes the known signals u and y (observables of the system) as inputs andgenerates a set of residual vectors ri, i ∈ q = 1, 2, . . . , q, with the following properties:

1. When no fault is present, all the residuals ri decay asymptotically to zero. Hence,the net transmission from u to the residuals is zero, and the modes observable fromthe residuals are asymptotically stable.

2. In the jth fault mode (i.e., when νi 6= 0), the residuals ri for i ∈ Ωj are nonzero,and the other residuals ra for a ∈ q − Ωj decay asymptotically to zero. Here thepre-specified family of coding sets Ωj ⊆ q, j ∈ k = 1, 2, . . . , k is chosen suchthat, by knowing which of the ri are (or decay to) zero and which are not, the faultνj can be uniquely identified.

The concept of coding set used in the above formulation contains a set of numbers thatrepresents a specific subset of the different residuals ri; i ∈ q, i.e. Ωj ⊆ q, j ∈ k; wherek denotes the number of faults and q the number of residuals. If the complete set ofresiduals defined by the coding set Ωj is affected by an occurring fault it can be said thatthe occurring fault is the jth fault. If q = k and Ωj = j, then the simplest coding setcan be obtained and the jth residual is only affected by the jth fault. More details canbe found in [Loo01] and [MVW89]. Massoumnia et al. have first proposed the definitionof the fundamental problem of residual generation (FPRG) for LTI system [MVW89].FPRG is in fact a restricted version of FDIFP. It considers only two-faults case. Theresidual signal is only sensitive to one fault and unaffected by another one.

Assume a LTI system with two fault signals and the vector L1 is monic, i.e. it has fullcolumn rank.

x = Ax+Bu+ L1ν1 + L2ν2

y = Cx(3.3)

The residual generator has the following form:

z = Fz − Ey +Gu

r = Mz −Hy +Ku(3.4)

where z ∈ Rq, r ∈ R

p and E, F , G, H, K, M are the matrices with appropriatedimensions. By combining the system and residual generator equations the extended


system and residual generator can be obtained.

x

z

=

A 0

−EC F

x

z

+

B L2

G 0

u

ν2

+

L1

0

ν1 (3.5)

r =(

−HC M)

x

z

+(

K 0)

u

ν2

(3.6)

The system (3.5) and (3.6) can be formulated as:

xe = Aexe +Beue + Leν1 (3.7)

r = Hexe +Keue (3.8)

where xe ∈ Rn+q, ue ∈ R

w+1 and Ae, Be, Le, He, and Ke are the matrices with appro-priate dimensions. Based on the equations (3.7) and (3.8) the definition of FPRG canbe proposed, which is same as the definition in [Loo01].

Definition 3.2.1. Fundamental problem of residual generation (FPRG) of thesystem (3.3) is to design a LTI dynamic residual generator by finding the appropriatematrices in (3.4) such that the following conditions are satisfied for system (3.7) and(3.8):

1. r is unaffected by ue.

2. The map from ν1 to r is input observable.

3. The observable modes of the pair (He, Ae) should be asymptotically stable.

With the condition 1 the residual is unaffected by uninteresting inputs (control input andanother fault signal), which prevents the false alarm of the diagnosis system and enablesthe isolation of fault ν1 from ν2. It also enhances the robustness of the residual generatorif signal ν2 is regarded as disturbance. The condition 2 assures the detectability of faultν1 with the designed residuals, which guarantees that the residual is affected by the faultsignal. The condition 3 makes the residual r asymptotically approach to zero if no faultis present.

The nature way to solve the problem is to view the ν1 as input to system and checkwhether the system for ν1 is input observable. It means that for a system with triple(C,A,B), B is monic and the image of B does not intersect the unobservable subspaceof (C,A). This approach can only ensure that the condition 2 and 3 are satisfied. Thefault isolation problem is not considered by this approach.

The solution of definition 3.2.1 can be derived in terms of differential-geometric methodas represented in [MVW89].


Theorem 3.2.1. The linear fundamental problem of residual generation (FPRG) indefinition 3.2.1, has a solution if and only if:

S∗ ∩ L1 = 0 with S∗ = InfS(L2) L1 = ImL1 L2 = ImL2 (3.9)

S(L2) stands for set of all (C,A) unobservability subspaces2 containing the subspace L2.Inf S(L2) can be understood as the minimal element of set S(L2). Im represents theimage of matrix. If condition (3.9) is satisfied, the dynamics of residual generator can bearbitrarily designed. Theorem 3.2.1 is the base of the differential-geometric methodology.Therefore, the proof is repeated here.

Proof. 3 (Sufficiency)The proof of sufficiency part is in fact a design procedure.

Assume D0 ∈ D(S∗), i.e. (A + D0C)S∗ ⊆ S∗. There is a canonical projection P :X → X /S∗. P is a system transformation matrix that the transformed system statesare unaffected by the another fault ν2. Calculate the matrix A0 = (A + D0C : X /S∗)and make A0P = P (A + D0C). Let the matrix H from system (3.4) be the solutionof KerHC = S∗ + KerC and matrix M is the solution of MP = HC, where Kerrepresents the null space. The pair (M,A0) is constructed as observable. Then thereexist a matrix D1 such that σ(F ) = Λ, where F = A0 + D1M and Λ is an arbitraryself-conjugate set and σ(F ) is the spectrum of F.

Let Pr denote the right inverse of P . Then let D = D0 + PrD1H, E = PD, G = PBand K = 0.

Define e = z − Px. So the following can be conducted

e = Fe− PL1ν1 (3.10)

r = Mz −Hy = Me (3.11)

It is obvious that residual vector r in Eq. (3.11) is only affected by the fault ν1. If thecondition (3.9) holds, the FPRG for LTI system has a solution.

(Necessity) If the FPRG for LTI system has a solution, the r in system (3.8) can not beaffected by ue in any case. Hence, the Ke = 0 and

〈A|Be〉 ⊆ Se (3.12)

where Be = ImBe and Se= 〈KerHe|A〉. Se is the unobservability subspace of (He, Ae).According to the proposition 1 in [MVW89] there exists a matrix Q that Q−1Se is a(C,A)-unobservability subspace.

2see Appendix A.73see Appendix A about the basic of differential geometry


Let S = Q−1Se, then Q−1Be ⊆ S. As L2 ⊆ Be and S(L2) denotes the set of all (C,A)unobservability subspaces containing the subspace L2, then

S ∈ S(L2). (3.13)

L1 is assumed to be monic so Le should be monic too. For condition 2 of definition 3.2.1to hold Le ∩ Se = 0. Thus

Q−1(Le ∩ Se) = Q−1Le ∩Q−1Se

= L1 ∩ S = 0. (3.14)

Now it is obvious that (3.13) and (3.14) hold only if (3.9) is true.

It is worth pointing out that there could be other subspaces S∗ ⊂ S, which meansthe dimension of residual generator can be further reduced. But it is more difficultto find a lower dimension residual generator because there is no good design routineto find the unobservability subspace except S∗. The above procedure gives a systemicand straightforward way to find the unobservability subspace. With this procedure thedetectability and isolatability of faults can be easily tested. If the fault is detectableand isolatable, then the residual generator with desired performance can be directlyconstructed based on this approach.

Although the solution (3.13) is derived from two faults case, it can be easily extendedto multi-faults case. The multi-faults case is defined as extension of the fundamentalproblem in residual generation (EFPRG).

Consider a system with faults and disturbances

x = Ax+Bu+ L1ν1 + Pξ

y = Cx(3.15)

where x ∈ Rn is the state vector, u ∈ R

w represents the control input, ν1 ∈ R is theinterested fault signal, y ∈ R

q is the output vector and ξ ∈ Rd denotes the disturbances

(including uninteresting faults). Vector L1 is monic. A, B, C, and P are the systemmatrices or vectors with appropriate dimensions.

Theorem 3.2.2. The extension of linear fundamental problem of residual generation(EFPRG), has a solution if and only if:

S∗ ∩ L1 = 0 with S∗ = InfS(P), L1 = ImL1 and P = ImP (3.16)

The proof of theorem 3.2.2 is similar to 3.2.1. The details can be found in [MVW89]. Ifthe solution of EFPRG can be found, then it is possible to detect and isolate the faulteven in simultaneous faults occurring situation.


3.2.2 FDI for nonlinear input-affine systems

In the preceding section the linear fundamental problem of residual generation (FPRG)for LTI system is formulated. The systematic design procedure of residual generatorbased on the ideal of [MVW89] is represented. But it is restricted to linear systems. Faultdetection and isolation for nonlinear systems is more difficult. Linearization around oneor more operation points can introduce a lot of modeling errors into the diagnosis system.After the decoupling of these uncertainties the residual generating is hardly possible. Itis necessary to extend geometric approach into nonlinear systems. The modeling uncer-tainties can be reduced massively by precise nonlinear description. Thus the robustnessof residual generator can be significantly improved. The limited measurements can bekept for generating residual signals.

Geometric approach based fault diagnosis for nonlinear systems is intensive researchedin [DI99], [DI00a] and [DI01]. They introduced the concept of unobservable distribution.The existence and regularity of this distribution indicated the solution of fault detectionand isolation for nonlinear systems. The principle of construction of residual generatoris to carry out a system transformation according to the unobservable distribution fordisturbance decoupling or fault isolation. The transformed system is only affected byone fault and fault detection observer is designed for the transformed system. In thefollowing of this section this method will be introduced in detail.

Consider an input-affine nonlinear system as modeled by equations of the form

x = f(x) + g(x)u+ l(x)ν + p(x)ξ

y = h(x)(3.17)

with state vector x ∈ Rn, control input u ∈ R

w, specific fault signal ν ∈ R, disturbancesignals (including other fault signals) ξ ∈ R

d and output y ∈ Rq. f(x), g(x) and p(x)

are smooth vector fields and h(x) is a smooth mapping with f(0) = 0 and h(0) = 0.Here only the actuator faults are considered. Similar to linear systems case the followingdefinition is formulated

Definition 3.2.2. (Local nonlinear fundamental problem of residual genera-tion (l-NLFPRG))[Loo01]: Considering the system 3.17 the l-NLFPRG is to find,if possible, a filter

z = f(z, y) + g(z, y)u

r = h(z, y).(3.18)

f(z, y), h(z, y) and g(z, y) are smooth too with f(0, 0) = 0 and h(z, y) = 0. z ∈ Rn

and r ∈ Rq. An augmented system can be constructed based on the systems (3.17) and

(3.18).

x

z

=

f(x)

f(z, h(x))

+

g(x)

g(z, y)

u+

l(x)

0

ν +

p(x)

0

ξ (3.19)


r = h(z, h(x)), (3.20)

which can be reformulated as

xe = f e(xe) + ge(xe)u+ le(xe)ν1 + pe(xe)ξ (3.21)

r = he(xe) (3.22)

and is defined on neighborhood U e of the origin (x; z) = (0; 0). The following propertieshold:

1. if ν = 0, then r is unaffected by u and ξ;

2. r is affected by ν;

3. if ν = 0,

limt→∞

‖r‖ = 0, ∀t > 0

for any initial condition (x0; z0) in a suitable set containing the origin (x; z) =(0; 0) and any set of admissible input.

Condition 1 ensures the robustness of the residual generator if the uncertainty of thesystem can be modeled as an additive disturbance and isolatability to other faults.Condition 2 relates the fault ν to residual vector so that the effect of the fault canbe reflected by residual signals. The condition 3 guarantees the stability of the residualgenerator. Here l indicates the fact that fault detection and isolation features are requiredto hold locally [AAE01].

Solution of the l-NLFPRG is derived in [DI99], [DI01] and [AAE01]. Assume the obser-vation distribution Oe of the system (3.21) and (3.22). The codistribution of Oe can bedefined as

ΩOe = span dR(xe), R ∈ Oe , (3.23)

where d stands for the differential.

Based on the proof in [Isi95] solution of the definition 3.2.2 can be desreibed by

span ge(xe), pe(xe) ⊂ (ΩOe)⊥ and le(xe) /∈ (ΩOe)⊥ (3.24)

It is not easy to find the subspace (ΩOe)⊥. An assumption to the system is inducted tosimplify the construction of (ΩOe)⊥.

The key problem is to find the distribution ΩOe . If xe = (x; z) = (0; 0) is a regular pointof ΩOe , then in the neighborhood of this point ΩOe can be described by the smallestcodistribution M e, which is invariant under f e, ge, pe and contains span dhe. In thiscase the formulation in (3.24) can be represented as

span ge(xe), pe(xe) ⊂ (M e)⊥ and le(xe) /∈ (M e)⊥. (3.25)


Now the problem is transformed to the construction of the codistribution M e, which ismore convenient in calculation. Then the solution of l -NLFPRG 4is [AAE01]:

Theorem 3.2.3. Let M be an involutive conditioned invariant distribution (unobserv-ability subspace) of system (3.17) such that

span p ⊂M ⊂ Kerd(ϕ h) and l /∈M (3.26)

for some surjection: ϕ : Rq → R

q, defined locally around y = 0 and with ϕ(0) = 0. Then,there exists a change of state coordinates z = Φ(x) and a change of output coordinatesy = Ψ(y), defined locally around x = 0 and, respectively, y = 0, such that, in the newcoordinates, the system (3.17) admits the normal form:

z1 = f1(z1, y2) + g1(z1, y2)u1 + l1(z1, z2)ν (3.27)

z2 = f1(z1, z2) + g2(z1, z2)u2 + l2(z1, z2)ν + p(z1, z2)ξ (3.28)

y1 = h1(z1) (3.29)

y2 = h2(z1, z2), (3.30)

with z1 ∈ Rn, n = n− dim(Q) = codim(Q), l1(z1, z2) 6= 0 locally around Φ(0) = 0.

The proof of the theorem 3.2.3 is omitted here. The details can be found in [DI01]. Whatis important is how to find the distribution M . If M is a generic conditioned invariantdistribution, it is not ensured that the subsystem (3.27) and (3.29) is observable. Theobservability of system (3.27) and (3.29) depends on the construction method of M .DePersis and Isidori [DI00b] have developed the observability codistribution algorithm,which guarantees the locally weak observability of subsystem (3.27) and (3.29) aftersystem transformation. This method will be introduced in the following.

3.2.2.1 Calculation of the involutive conditioned invariant distribution M(unobservability distribution)

The construction of distribution M is carried out in two steps. First, the distribution,which is involutive, conditioned invariant, containing the span p, is constructed. Next,based on the algorithm of first step the observability codistribution M⊥ can be obtained.Then the unobservability distribution is simply M = (M⊥)⊥.

Step 1 : Consider a nonlinear input-affine system

x = g0(x) +w∑

i=1

gi(x)ui

y = h(x)

(3.31)

4Because it is the “regular” (stronger) version of the fundamental problem of residual generation, it isalso defined as rl -NLFPRG in some literatures.


with x ∈ Rn, ui ∈ R, y ∈ R

q, in which g0(x), g1(x), . . . , gs(x) are smooth vector fieldsand h(x) is a smooth map. Recall the definition in [Isi95] that a distribution ∆ is saidto be conditioned invariant for system (3.31) if it satisfies

[gi,∆ ∩Ker dh] ⊂ ∆, for all i = 0, . . . ,m (3.32)

where [ ] is the Lie Bracket. Now let p1(x), . . . , pd(x) be the set of additional smoothvector fields and consider the distribution

P = span p1, . . . , pd . (3.33)

Define that P denotes the smallest involutive distribution that contains P (involutiveclosure of P ). Then consider the non-decreasing sequence of distributions

S0 = P (3.34)

Sk+1 = Sk +w∑

i=0

[

gi, Sk ∩Ker dh]

(3.35)

According to the proof in [DI00b] there exists a integer k∗ such that

Sk∗+1 = Sk∗ (3.36)

and Sk∗ is also denoted as ΓP∗ . Then ΓP

∗ is involutive, contains P and conditioned invari-ant. If there exists other distribution ∆ that is involutive, contains P and conditionedinvariant, ΓP

∗ satisfies ΓP∗ ⊂ ∆.

Now suppose that the distribution ΓP∗ is well-defined and nonsingular, so that its anni-

hilator (ΓP∗ )⊥ is locally spanned by exact differentials. Suppose also ΓP

∗ ∩ Ker dh isalso a smooth distribution. Then it can be said that (ΓP

∗ )⊥ is the maximal conditionedcodistribution which is locally spanned by exact differentials and contained in P⊥. Moredetails can be found in [DI00b].

Step 2 : There exists a fixed codistribution Θ of system (3.31) and the followingnon-decreasing sequence of codistribution:

M0 = Θ ∩ span dh (3.37)

Mk+1 = Θ ∩

(

s∑

i=0

LgiMk + span dh

)

, (3.38)

where LgiMk is the Lie Derivative of Mk. Suppose that all codistribution of this sequence

are nonsingular, so that there is an integer k∗ ≤ n−1 such that Mk = Mk∗ for all k > k∗,and set ξ∗ = Mk∗ . Then it is usually denoted as

ξ∗ = o.c.a.(Θ),

3.3 FDI for the electro-hydraulic system 33

where o.c.a is short for observability codistribution algorithm. The algorithm has theproperty that o.c.a.(Θ) = o.c.a.(o.c.a.(Θ)) and if Θ is conditioned invariant, so is thecodistribution ξ∗. A codistribution ξ is called observability codistribution if

Lgiξ ⊂ ξ + span dh ∀i = 0, 1, . . . , s (3.39)

Ω = o.c.a.(Ω). (3.40)

Furthermore if the codistribution Ω = ∆⊥, then ∆ is the unobservability distribution.

After substituting the (ΓP∗ )⊥, o.c.a.((ΓP

∗ )⊥) is the maximal observability codistributionwhich is locally spanned by exact differentials and contained in P⊥. The correspondingunobservability distribution M can be construed by

M = (o.c.a.((ΓP∗ )⊥))⊥ (3.41)

As a result of the maximality of o.c.a.((ΓP∗ )⊥), M is the smallest involutive conditioned

invariant unobservability distribution that contains P . The locally weak observabilityof the transformed system in fault free situation can be guaranteed, which is proved in[DI00b]. Therefor the M is most possible distribution to fulfill the condition in theorem3.2.3.

3.3 FDI for the electro-hydraulic system

Fault diagnosis for electro-hydraulic systems is a challenging task because of high nonlin-ear dynamics and unknown disturbance of the system. The nonlinear model is employedfor a precise modeling. The difficulty is to archive the robustness to the unknown distur-bance and keep the sensitivity to the considered faults simultaneously. Several researchesin this area have been presented: Gayaka and Yao [GY08] applied the adaptive robustapproaches to construct the fault detection observer for residual generation; An andSepehri [AS03] used extended Kalman Filter for detecting the pump pressure drop; Shiand his colleagues [SGLB05] employed linear observer and adaptive threshold for robustFDI of a nonlinear electro-hydraulic system.

What is different from the systems discussed in references is that our electro-hydraulictestbed has a varied external load, which has significant effect to system dynamics. Ifthe change of the external load is ignored, the estimations of the system states can notconverge to the actual values. So far there is no precise modeling of the varying externalload. Therefore, the only way to make the diagnosis system robust to the external loadis to decouple it from the mathematical model.

3.3.1 Disturbance decoupling

For the electro-hydraulic testbed the external load is viewed as unknown disturbance.The considered faults are internal leakage and sensor offset on pressure transducers of


chamber A and B. Firstly, only the internal leakage, which belongs to actuator faults,is studied. The sensor faults are researched later.

Based on the differential-geometric approach an unobservability distribution of the non-linear system should be constructed. Then system transformation in states and outputspace can be carried out. Finally an observable subsystem, which is affected only by thespecified fault, can be extracted for fault diagnosis. Design a fault detection observerfor the subsystem. Then the residual generation is obtained.

Recall the electro-hydraulic system with internal leakage:

xc = vc

vc =1


PB =EB(PB)

V20 − A2xc

(

A2vc −√

kqb1(PB − Po) + kqb2 +QLin

)

PA =EA(PA)

V10 + A1xc

(−A1vc −QLin +KQα)

α = vα


(3.42)

The state variables are cylinder position xc, cylinder velocity vc, the pressure PB, thepressure chamber PA, the swashplate angle α and the angular velocity vα. The bulkmodulus EA(PA) and EB(PB) are nonlinear functions of respective pressures as describedin Eq.(2.7). The physical meanings of other parameters can be found in chapter 2. Itis obvious that the pump dynamics has no intersection to cylinder dynamics exceptthe pressure change in chamber A. Neither the internal leakage nor the external loadcan affect the pump dynamics. It is nature to consider only the cylinder dynamics forresidual generator and treat the swashplate angle as control input of cylinder system.Then the new system for fault diagnosis is represented as:

xc = vc

vc =1

m(PAA1 − PBA2 −Dvvc − fc) −

1

mFload

PB =EB(PB)

V20 − A2xc

(

A2vc −√


)

+EB(PB)

V20 − A2xc

QLin

PA = −EA(PA)

V10 + A1xc

A1vc −EA(PA)

V10 + A1xc

QLin +EA(PA)

V10 + A1xc

KQα;

(3.43)

with the state vector x = xc vc PB PAT . All state variables are measurable, then

y = xc vc PB PAT . (3.44)

Now it should be tested whether the desired unobservability distribution does exist.The corresponding vectors are (For avoiding confusion the state variable x is omitted in


denotation.):

g0 =

vc

1m

(PAA1 − PBA2 −Dvvc − fc)

EB(PB)V20−A2xc

(

A2vc −√


)

− EA(PA)V10+A1xc

A1vc

g1 =

0

0

0

KQEA(PA)

V10+A1xc

l =

0

0

EB(PB)V20−A2xc

− EA(PA)V10+A1xc

p =

0

1/m

0

0

The terms EB(PB)V20−A2xc

, EA(PA)V10+A1xc

, KQ and m will never be zero because of the physicalconstraints. Therefore g1, l and p are all regular distributions.

Following the procedure (3.34) and (3.35) to compute ΓP∗ begins with:

S0 = P = P, with P = span p (3.45)

As P is one dimension subspace and constant, the involutive closure is P equal to P .Obviously, S0 = S0. All states are measurable, then

dh = I4 ⇒ Ker(dh) = 0 ⇒ S0 ∩Ker(dh) = 0

where I4 is the 4 × 4 identity matrix. Furthermore:

S1 = S0 +w∑

i=0

[gi, 0] ⇒ S1 = S0 + 0 = S0

⇒ k∗ = 0 ⇒ ΓP∗ = P

This leads to

(ΓP∗ )⊥ = P⊥ = span (1 0 0 0) , (0 0 1 0) , (0 0 0 1) .

Meanwhile

span dh = span (1 0 0 0) , (0 1 0 0) , (0 0 1 0) , (0 0 0 1) .

Now calculate the involutive conditioned invariant distribution M according to the ob-servability codistribution algorithm (o.c.a.).

M0 = P⊥ ∩ span dh = P⊥


Furthermore

M1 = M0 ∩

(

w∑

i=0

LgiM0 + span dh

)

= M0 ∩ (0 + span dh)

= M0.

This leads toM = (P⊥)⊥ = P (3.46)

Obviously, l /∈ P . Thus according to the theorem 3.2.3 there exists a subsystem, whichis unaffected by external load and affected by fault internal leakage.

Next step, a system transformation based on the observability distribution can be im-plemented. The procedure is derived in [DI00b]. It will be repeated here.

If n1 is the dimension of the observability distribution ∆ and q − n2 is the dimensionof the subspace ∆ ∩ span dh, then there exists a smooth surjection Ψ1 : R

q → Rq−n2

such that∆ ∩ span dh = span d(Ψ1 h) ,

where q is the dimension of the output space.

Furthermore two local diffeomorphisms Ψ(y) and Φ(x) can be found with the format

Ψ(y) =

y1

y2

=

Ψ1(y)

H2y

, Φ(x) =

z1

z2

z3

=

Φ1(x)

H2h(x)

Φ3(x)

,

where H2 is a n2 × p selection matrix, function Φ1 satisfies ∆ = span dΦ1 and Φ3 :R

n → Rn−n2−n1 is an arbitrary function.

For the cylinder system it can be obtained dim(∆) = dim(P⊥) = 3. It leads ton1 = 3 and n2 = 1. Define x = x1 x2 x3 x4

T = xc vc PB PAT and

y = y1 y2 , y3 , y4T = x1 x2 x3 x4

T Then it is nature to choose:

Ψ1 = y1 y3 y4 T ; H2 = 0 1 0 0 ; Φ1(x) = x1 x3 x4 T .

As n− n1 − n2 = 0, Φ3(x) does not exist. It leads to

z1 = y1 = x1 x3 x4 T z2 = y2 = x2

In the new coordinates the system is rewritten as

xc = vc

PB = −EB(PB)

V20 − A2xc

√

kqb1(PB − Po) + kqb2 +EB(PB)

V20 − A2xc

A2vc +EB(PB)

V20 − A2xc

QLin

PA = −EA(PA)

V10 + A1xc

A1vc −EA(PA)

V10 + A1xc

QLin +EA(PA)

V10 + A1xc

KQα.

(3.47)


y1 = xc PB PAT (3.48)

The transformed system has three orders. It is locally weakly observable in fault freecase. The cylinder velocity is regarded as input signal.

3.3.2 Fault detection observer

Nonlinear observer of system (3.47) and (3.48) should be designed for fault detection.The difficulty is that the new system dynamics is mainly described by the input signals.The nonlinear observer design methodology can not be applied due to the “abnormal”structure of system. Fortunately, all state variables are measurable. The system nonlin-earity can be compensated by output feedback. Thus, a linear Luenberger observer isconstructed for fault detection.

The structure of the observer is designed as:

˙x = Ax+ f(y) + g(y)u+ Lo(y − y)

y = Cx(3.49)

Define observer error eo = x− x and residual r = y − y = ceo. Then it can be obtained

eo = (A− LoC)eo

r = Ceo.(3.50)

Lo is the observer gain, which ensures that (A− LoC) is stable. Therefor, the residualr satisfies

limt→∞

‖r‖ = 0 ∀t > 0, (3.51)

if no fault occurs.

3.3.2.1 The internal leakage

The internal leakage can be viewed as actuator fault. If the fault occurs, the errordynamics of the observer is

eo = (A− LoC)eo + ELinfLin

r = Ceo,(3.52)

where

ELin =

0 0

1 0

0 1

, fLin =

EB(PB)V20−A2xc

QLin

− EA(PA)V10+A1xc

QLin

.

Then the residual vector is the response of linear filter with transfer function C(sI −(A−LoC))−1ELin to the input vector fLin. The elements of the fLin are always boundedand the transfer function is stable. Therefore, the boundedness of the residual can beguaranteed if the internal leakage occurs.


3.3.2.2 The sensor faults

It is always difficult to detect sensor faults by applying the observers with closed struc-ture. There is no systematic method to design an observer detecting sensor fault of non-linear systems. The detectability depends on the structures of systems and observers.Usually, the observers work so “well” that they track the measurements even if the mea-surements drift from the real values. Thus, the senor fault effect of the electro-hydraulicsystem should be studied in individual.

If there is an offset ∆PB in senor of pressure PB, then the outputs of system are

y1 = xc PB + ∆PB PAT

It can be obtained that the observer dynamics related to pressure estimation PB isaffected. Then,

ePB= −Lo2ePB

− Lo2∆PB + Π(∆PB)

rPB= ePB

+ ∆PB,(3.53)

where ePB= PB − PB and

Π(fB) =EB(PB)

V20 − A2xc

(

A2vc −√


)

−EB(PB + ∆PB)

V20 − A2xc

(

A2vc −√

kqb1(PB + ∆PB − Po) + kqb2

)

.

If the effect of Π(fB) is not considered, then the residual rPBis the response of the linear

filter ss+Lo2

to input ∆PB. It is obvious that if ∆PB is assumed to be a step signal

limt→∞

‖rPB‖ = 0 ∀t > 0.

It can be concluded that the sensor fault effect depends only on the term Π(∆PB).Theoretically, fault detection observer can sense the sensor fault, but the amplitude ofthe residual depends on not only the amount of offest but also the system structure.

The similar analysis can be carried out for the sensor of PA by defining ePA= PA − PA.

The same conclusion can be obtained.

3.3.3 Decision making

Residual signals are generated by the fault detection observer. But it can not be deter-mined yet if a fault is happened or which fault is happened. Usually a decision makinglogic (DML) is designed for residual evaluation and fault isolation. An intelligent DMLsometimes can also improve the robustness of the fault diagnosis system. In this thesisthe procedure is divided into two steps.


Step 1: Signal processing

The residual signals can not be sent to the DML directly due to the measurement noise,which can trigger false alarms. Simply increasing the range of thresholds can preventfrom the impact of noise but bring about the risk of missed detection of the fault. Analternative is filtering of the residuals. The original residuals are filtered by a low-passfilter before they are used for decision making. The part of fault information probablyis also filtered out, which can delay the FDI. In this thesis the later method is employed.

Based on the practical test it can be observed that the pressure signals are coupled withnoise. The position signal delivered by the absolute-encode is so precise and smooththat it can be viewed as noise free. Therefore, the estimation errors of PA and PB arefiltered firstly. Then the residuals of the cylinder systems are defined as

rxc(k) = xc(k) − xc(k), rPB

(k) = Gf1(z−1)ePB

(k) rPA(k) = Gf2

(z−1)ePA(k),

where Gf1(z−1) and Gf2

(z−1) are the discrete transfer functions of the filters. Here thek denotes the residual signal of kth sampling time.

Step 1: Decision making logic

According to the structure of fault detection observer the connection between faultsand residuals can be derived as shown in table (3.1), where 1 means affected and 0is unaffected. It is obvious that only the residuals rPA

and rPBare sensitive to the

XX

XX

XX

XX

XX

XXFault

Residualrxc

rPBrPA

QLin 0 1 1

∆PB 0 1 0

∆PA 0 0 1

Table 3.1: The relation between the faults and the residuals

faults. The residual rxcis not affected by the faults because the velocity is a input signal

for transformed system. As the internal leakage affects two residuals rPAand rPB

, theproduct of them rLin is defined as the special residual to the leakage. The advantage ofintroducing the rLin is that all useful information is concentrated on one signal, which cansimplify the decision making procedure. Tab. 3.1 shows how the fault can be identified,where O means the alarm is triggered and × means not.


PP

PP

PP

PP

PAlarm

FaultQLin ∆PB ∆PA

rPB × O ×

rPA × × O

rLin O × ×

Table 3.2: The fault isolation table with rLin

threshold

residual

t\] alarm (if t\] >= t _ ) time

Figure 3.3: The principle of the designed DML

Furthermore, timer is set for recording how long is the residual out of the range of thethreshold. If the time interval tre is bigger than the predefined value tpf . The fault alarmwill be activated. The principle is illustrated in the Fig. 3.3.

Let tr1, tr2

and tr3be the recorded time intervals and tpf1

, tpf2and tpf3

be the predefinedvalues of rPB

, rPAand rLin respectively. Three boolean signals are defined for decision

making:

DMLPB=

1 tr1≥ tpf1

and DMLQLin= DMLQPA

= 0

0 otherwise(3.54)

DMLPA=

1 tr2≥ tpf2

and DMLQLin= DMLQPB

= 0

0 otherwise(3.55)

DMLQLin=

1 tr3≥ tpf3

and DMLQPA= DMLQPB

= 0

0 otherwise(3.56)

where 1 means alarm triggered and 0 means not. This DML is based on the assumptionthat the effect of the fault can be persistently reflected by the residual. It enable to

3.4 Simulation 41

isolate three faults with only two real residuals. The predefined values can be obtainedby simulations but should be adapted through experimental tests. Because the leakageaffects the rPA

and rPBboth, tr3

should be smaller than the others. Otherwise the DMLwill regard it as a sensor fault. It is assumed only one fault is present. Therefore, assoon as one fault is identified, then the alarm for other faults will never be triggered.

3.3.4 Sensor total failure checking

It could happen for the sensors that the total information is lost during the operation.Such failure should be identified as soon as possible. So it is necessary to make a totalfailure checking for some important measurements by every sampling time. The rule isvery simple: if the residual is extremely large, for example, ten times as the threshold,and the actual measured value is almost zero. Then it can be concluded that a totalfailure is happened. Such a failure has the highest priority and the alarm should betriggered immediately. In this thesis the total failure of PA is considered. Becausethe cylinder works under high pressure, a sudden information lost could cause seriousconsequence. How should the controller react to sensor total failure PA is discussed inchapter 4.

3.4 Simulation

3.4.1 Simulation settings

Figure 3.4: Characteristic curves of proportional directional valve


Simulation model of the electro-hydraulic system for Matlab/Simulink is built based onthe system description (3.42). External load is also imitated for testing the robustness ofthe fault generator. It is not exact same as the actual load pressure due to the modelingerror of the proportional pressure relief valve. The internal leakage flow is simulatedaccording to the data of characteristic curves of the proportional directional valve asshown in Fig. 3.4. The command value is set as 40%, whose curve possesses goodlinarity. Measurment noise of PA and PB signal is added for matching the real situation.The operation pressure is about 50 Bar. But for testing the robustness of the observerthe external load changes until the value is around 100 Bar as shown in Fig. 3.5.

0 5 10 15 20 25 30 350

20

40

60

80

100

120

Time (sec.)

PLo

ad (

Bar

)

external load

Figure 3.5: External load

3.4.2 Simulation results

3.4.2.1 Case 1: fault free

The first simulation is executed for fault free case. The results are shown in Fig. 3.6and Fig. 3.7.

3.4 Simulation 43

0 5 10 15 20 25 30 350

50

100

150

200

x c (m

m)

0 5 10 15 20 25 30 350

1

2

3

4

PB

(B

ar)

0 5 10 15 20 25 30 350

50

100

150

Time (sec.)

PA

(B

ar)

modelestimation

modelestimation

modelestimation

Figure 3.6: Estimation in fault free case

0 5 10 15 20 25 30 35−0.01

−0.005

0

0.005

0.01

0.015

0.02

r x c

0 5 10 15 20 25 30 35−0.02

−0.01

0

0.01

0.02

0.03

0.04

r PB

0 5 10 15 20 25 30 35−0.4

−0.2

0

0.2

0.4

Time (sec.)

r PA

0 5 10 15 20 25 30 35−10

−8

−6

−4

−2

0

2x 10

−3

Time (sec.)

r Lin

Figure 3.7: The residuals without faults

The estimation tracks the measurement very well even the external load varies a lot. Allthe residuals are near zero. The value of rPA

is much larger than other residuals due tothe big value of PA itself. It indicates that making selection of threshold should considerthe range of the variables.

3.4.2.2 Case 2: with the internal leakage QLin

Now the internal leakage is added in the simulation. The leakage flow is set as a ramp-likesignal and saturates on 5 (L/min) as shown in Fig. 3.8.


0 5 10 15 20 25 30 350

1

2

3

4

5

6

Time (sec.)Q

Lin

(L/m

in)

internal leakage

Figure 3.8: the internal leakage

0 5 10 15 20 25 30 350

50

100

150

200

x c (mm

)

0 5 10 15 20 25 30 350

2

4

6

P B (Bar

)

0 5 10 15 20 25 30 350

50

100

150

Time (sec.)

P A (Bar

)

modelestimation

modelestimation

modelestimation

Figure 3.9: Estimation with the internal leakage

0 5 10 15 20 25 30 35−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

r x c

0 5 10 15 20 25 30 35−8

−6

−4

−2

0

2

Time (sec.)

r PA

0 5 10 15 20 25 30 35−12

−10

−8

−6

−4

−2

0

2

Time (sec.)

r Lin

0 5 10 15 20 25 30 35−0.5

0

0.5

1

1.5

2

r PB

Figure 3.10: The residuals with the internal leakage

Residuals rPA, rPB

and rLin are affected by the leakage. These residuals deviates sig-nificantly deviated from the value range in fault free case. It can be said the residualscan indicate the fault occurrence quickly and clearly. The fault isolation is based on thedesigned DML.

3.4 Simulation 45

3.4.2.3 Case 3: with the fault ∆PB

The sensor offset is a setp-like signal and the value is 20% of the measurement.

0 5 10 15 20 25 30 350

50

100

150

200

x c (m

m)

0 5 10 15 20 25 30 350

2

4

6

PB

(B

ar)

0 5 10 15 20 25 30 350

50

100

150

Time (sec.)

PA

(B

ar)

modelingestimation

modelingestimation

modelingestimation

Figure 3.11: Estimation with fault ∆PB

0 5 10 15 20 25 30 35−0.01

−0.005

0

0.005

0.01

0.015

0.02

r x c

0 5 10 15 20 25 30 35−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

r PB

0 5 10 15 20 25 30 35−0.4

−0.2

0

0.2

0.4

Time (sec.)

r PA

0 5 10 15 20 25 30 35−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time (sec.)

r Lin

Figure 3.12: The residuals with fault ∆PB

The residual rPBdiverged obviously from zero after the sensor fault occurrence. The

other residuals kept in the near of zero.

3.4.2.4 Case 4: with the fault ∆PA

The sensor offset is also setp-like signal and the value is 20% of the measurement.


0 5 10 15 20 25 30 350

50

100

150

200

x c (m

m)

0 5 10 15 20 25 30 350

1

2

3

4

PB

(B

ar)

0 5 10 15 20 25 30 350

50

100

150

Time (sec.)

PA

(B

ar)

modelingestimation

modelingestimation

modelingestimation

Figure 3.13: Estimation with fault ∆PA

0 5 10 15 20 25 30 35−0.01

−0.005

0

0.005

0.01

0.015

0.02

r x c

0 5 10 15 20 25 30 35−0.02

−0.01

0

0.01

0.02

0.03

0.04r P

B

0 5 10 15 20 25 30 35−0.5

0

0.5

1

1.5

2

2.5

Time (sec.)

r PA

0 5 10 15 20 25 30 35−0.02

−0.01

0

0.01

0.02

0.03

0.04

Time (sec.)

r Lin

Figure 3.14: The residuals with fault ∆PA

The fault ∆PA can affect residual rPA. But the shape of residual is like a impulse, which

is far away from zero only at the beginning of fault occurrence. After the transient phase,the residual rPA

returned back to zero, although the fault is still present. Following thedefinition in [CP99] the sensor fault ∆PA is only detectable and not strong detectableby the rPA

.

3.4.3 Conclusion

The simulation results shows that the designed fault detection observer is robust to thetime-variant disturbance. The residuals are almost zero if the system is fault free. If

3.5 Summary 47

the fault occurs, the fault detection observer responses quickly and meanwhile keepsthe boundedness. But the disturbance decoupling takes the cost of one measurement(cylinder velocity), which is a input signal for transformed system. This leads to that theresidual rxc

is unaffected by the faults. The transformed system has a “bad” structure.Here “bad” means that the linear part of the system dynamics is zero, which results inthe non-strong detectability of the fault ∆PA. Therefore, the DML should be modifiedaccording to the residual response to ∆PA. Then, the DMLPA

in (3.55) is modified as

DMLPA=

1 tpf2≤ tr2

≤ tpf2(tpf2

> tpf2) and DMLQLin

= DMLPB= 0

0 otherwise

(3.57)where tr2

is the time interval, during which the rPAalways exceeds the threshold. The

lower limit prevents from the false alarm caused by noise or other stochastic disturbances.The upper limit isolates the sensor offset from the leakage. The residuals now can bedistinguished from each other, if different faults occur. Therefore, the discussed threefaults can be detected and isolated by the proposed approach.

In the simulations the values of thresholds and parameters for DML are not presented.Based on the simulation results an optimal setting of these parameters is always possible.More details can be found in chapter 5, which shows the results of experimental tests.

3.5 Summary

This chapter gives the detailed design procedure of fault diagnosis system for the electro-hydraulic system Firstly, the state of the art of model-based detection and isolation isintroduced. The differential-geometric approach is employed for robust fault detection.With differential-geometric method a system transformation can be carried out. Thetransformed system is robust to system disturbance (the varying external load) butaffected by the considered faults. Then the fault detection observer is designed for thetransformed system. Furthermore, the faults can be isolated by the designed decision-making logic (DML). The simulation shows good results of the proposed fault diagnosisstrategy. One problem is that after system transformation the measurement of positioncan not be handled as residual, because the velocity is an input of the transformedsystem.

4 Fault-tolerant control for theelectro-hydraulic system

4.1 Introduction

4.1.1 Background

Fault-tolerant control (FTC) namely is a kind of reconfigurable control, which can main-tain the control performance or at least, guarantee the stability of whole system in theevent of faults. FTC is different from the conventional control. It contains not only acontrol law but also the fault diagnosis subsystem, which helps the reconfiguration ofcontroller. The structure of FTC can be generally described as shown in Fig. 4.1. FTCsystem contains two layers: supervision and execution [BKLS03]. The supervision layermonitors the system behavior through the inputs, measurements and system model. Thefault diagnosis block will give all the information about faults. The results are used bycontroller redesign mechanism to make an appropriate decision. The execution layer issimilar to usual control loop. The feature is that the controller should be reconfigurable,which means the controller can react to the fault. With this architecture the reliabilityand safety of the system can be improved.

Supervision

ExecutionController System

df

Diagnosis System

u

Controller redesign

yy`-

f

Figure 4.1: The structure of fault-tolerant control [BKLS03]

49

50 4 Fault-tolerant control for the electro-hydraulic system

4.1.2 State-of-the-Art

FTC system has attracted more and more attention in industry and academia. Generallyspeaking FTC system can be divided into two classes: passive FTC and active FTC.The both are briefly reviewed below.

4.1.2.1 passive FTC

Passive FTC system is designed to achieve insensitivity to certain faults by means ofmaking the system robust to them. The faults are usually modeled as the system uncer-tainties so that robust controller can be designed to guarantee the desired performance.The control system take into account all possible faults in design stage. If the controllerexists, it is fixed. Basically the robust control approaches can be applied for passive FTC,for instance, robust H∞ approach in [ZR01], [NS03] and [CP01], quantitative feedbacktheory in [NS02], reliable control technique in [ZJ98], [Fer02].

The advantage of the passive FTC is that as long as the controller exists, the system canmaintain the desired performance in all considered fault scenarios. The design procedureis carried out off-line. It is not constrained to real-time computation problem in imple-mentation. Fault diagnosis system is not necessary. Therefore, the passive method isvery applicable for the systems, which have only limited computational resource. How-ever the design of passive FTC system is subject to all considered faults so that thedesigned controller is usually conservative. Usually the best control performance will belost due to the conservatism, although the system runs mostly in nominal state.

4.1.2.2 active FTC

In contrast, active FTC systems react to faults actively by reconfiguring control actionsso that the pre-specified performance of the entire system can be ensured. The reconfigu-ration mechanism is based on the latest fault information from a real-time fault diagnosissystem and executed on-line. Various active FTC methods have been developed. Someof them are introduced here as overview.

Pseudo inverseThe pseudo inverse is one of the most used approaches due to its computational simplicityand its ability to handle a very large class of system faults. The basic version of pseudoinverse is derived as follows. Consider a nominal system

xk+1 = Akxk +Bkuk

yk = ckxk.(4.1)

with state-feedback control law uk = Fnxk. Furthermore, the description of the faulty

4.1 Introduction 51

system (without sensor faults) is

xfk+1 = Af

kxk +Bfku

fk

yfk = cfkx

fk .

(4.2)

The reconfigured controller is designed with the same structure, i.e. ufk = Ffx

fk . Now the

problem is transformed to find the gain matrix Ff , which should minimize the distance

between the matrix (Ak +BkFn) and (Afk +Bf

kFf ). Namely,

Ff = arg minFf

∥

∥

∥(Ak +BkFn) − (Af

k +BfkFf )

∥

∥

∥

= B†(Ak +BkFn − Afk),

(4.3)

where B† is the pseudo-inverse of the matrix Bfk . The computation cost of this approach

is low. It can be easily implemented for real-time applications. One of the serious dis-advantages is that the optimal control law computed by (4.3) does not always stabilizethe closed-loop system. Simple examples are demonstrated in [GA91]. Therefore, themodified passive method is developed to overcome this problem by introducing systemconstraints, which complicates the optimal solution. Other disadvantages are that allsystem states should be measurable and sensor fault and system uncertainties are notconsidered. The recent research in this area can be found in [YB00] and [KV00].

Multiple ModelThe multiple model (MM) is based on a set of linear models Mi i = 1, 2, . . . N . Eachmodel represents a different faulty system. The key part of the multiple model approachis to design an on-line procedure, which determines the global control action througha (probabilistically) weighted combination of the different control actions that can betaken as described in [MS91] and [ZJ01]. The control inputs of each model are sometimesmixed, which is known as blending [MS97]. The mixing procedure is usually based ona bank of Kalman-Filters, in which each Kalman-Filter is designed for one of the localmodels Mi. Through the residuals of the Kalman-Filters the probabilities ςi of eachmodel in effect are computed. By weighting the control with probability factor the realinput is constructed as

u =N∑

i=1

ςiui,N∑

i=1

ςi = 1

The multiple model can be applied for nonlinear system or combined with differentcontrol strategies for achieving better performance, such as model predictive control,adaptive control, etc. Fault diagnosis system can also be easily integrated. The maindisadvantage is the high computation burden for real-time implementation. Also themodeling uncertainties may be neglected.

Model predictive control (MPC)The MPC is a kind of optimal control based on matching a prediction of the system


output to some desired reference trajectory. The optimization is executed at every timeinstant, which is the main difference from normal optimal control strategies. The MPCarchitecture allows fault-tolerance to be simply integrated by redefining the constraints torepresent certain faults such as actuator faults or changing the internal model or chang-ing the control objectives to reflect limitations due to the faulty components [AAE01].Therefore, as soon as the fault is diagnosed the controller reconfiguration is accomplishedautomatically. More details about MPC based FTC can be found in [KM99], [MJ03].

Generally speaking, the MPC approach is very suitable for FTC. But it can not handlethe modeling uncertainties and hardly can be applied for nonlinear systems.

Adaptive controlAdaptive methods based FTC has attracted many researchers due to its inherent self-reconfiguration capacity. The effect of the faults can be compensated by parameteradaption. Adaptive algorithms usually can not handle the sensor fault and the parame-ter estimation can not asymptotically converge to the actual value. Then an additionalFDI system is necessary to assist the reconfiguration of the controller. This structure isalso utilized by this thesis. Adaptive control can be applied to nonlinear system by com-bining nonlinear control techniques such as feedback linearization, backsteeping and etc[TaXTJ04]. More references on this topic can be found in [AZIGR91], [IS98] [KTT03].

More approaches for active FTC can be found in references [ZJ08], [Jia05], [BKLS03],[JMZ03]. The active FTC system is more intelligent than the passive FTC system. Thereconfiguration of the controller is carried out on-line according to the fault information.Hence, in fault free case the system possesses best performance. The serious problemof the active FTC is time delay caused by FDI. The time delay can not be avoided.Can system still be stable during the period that the fault occurs before it is identified?Therefore, the controller should be not only reconfigurable but also robust, at least fora short time. Thus, the performance of the active FTC also depends on the efficiencyof FDI system. Another problem is high computation burden, which limits the activeFTC strategies in practical applications.

4.2 Backstepping based FTC

The backstepping methodology is a recursive Lyapunov-based approach. By followinga step-by-step algorithm feedback control laws and associated Lyapunov functions canbe constructed systematically. A major advantage of backstepping is the flexibility toavoid cancellations of useful nonlinearities and achieve regulation and tracking proper-ties [KKK95]. The backstepping controller can be reconfigured by integrating parameteradaption or changing the internal model. The system or actuator faults can be modeledas unknown parameters of the system. Hence, these faults can be tolerated by param-eter adaption. The sensor faults are brought into the system by the controller and canbe modeled as uncertainties of the controller parameters. Hence, the system model for

4.2 Backstepping based FTC 53

controller design should be changed so that the effect of the sensor fault can be com-pensated or eliminated. An alternative for dealing with the sensor fault is to replace thefaulty measurement by the estimation of an observer, in which the faulty measurementis abandoned. In this case the observer-based control structure is turned out. For theelectro-hydraulic system it is impossible to construct such an observer due to the de-coupling of the external load. Therefore, in this thesis only the first method is underconsideration. The details of construction the backstpping controller and reconfigurationfor the fault accommodation are shown in following.

4.2.1 Preliminaries

Firstly, some basic knowledge about nonlinear systems is reviewed for completeness andconvenience [KKK95].

4.2.1.1 K and KL function

Definition 4.2.1. A continuous function γ : [0, a) → R+ is said to belong to class Kif it is strictly increasing and γ(0) = 0. It is said to belong to class K∞ if a = ∞ andγ(r) → ∞ as r → ∞.

Definition 4.2.2. A continuous function β : [0, a) × R+ → R+ is said to belong classKL if for each fixed s the mapping β(r, s) belong to class K with respect to r, and for eachfixed r the mapping β(r, s) is decreasing with respect to s and β(r, s) → 0 as s→ ∞. Itis said to belong to class KL∞.

4.2.1.2 Stability

Consider the nonautonomous system

x = f(x, t), x(t) ∈ Rn (4.4)

where x(t0) = x0 and f(0, t) = 0, ∀t ≥ t0. Then the stability properties of theequilibrium point xe = 0 are defined as follows.

Definition 4.2.3. The xe = 0 of system (4.4) is

uniformly stable , if there exists a class K function γ(·) and a positive constantc, independent of t0, such that

|x(t)| ≤ γ(|x(t)|), with t ≥ t0 ≥ 0 and |x(t)| < c; (4.5)

uniformly asymptotically stable, if there exists a class KL function β(·, ·) and apositive constant c, independent of t0, such that

|x(t)| ≤ β(|x(t0)| , t− t0), with t ≥ t0 ≥ 0 and |x(t)| < c; (4.6)


exponentially stable, if (4.6) is satisfied with

β(r, s) = kre−ρs, with k > 0 and ρ > 0. (4.7)

Then the LaSalle-Yoshizawa theorem can be introduced here.

Theorem 4.2.1. (LaSalle-Yoshizawa Theorem) Let x = 0 be an equilibrium pointof (4.4) and suppose f is locally Lipschiz in x uniformly in t. Let V : R

n × R+ → R+

be a continuously differentiable function such that

γ1(|x(t)|) ≤ V (x, t) ≤ γ2(|x(t)|) (4.8)

V =∂V

∂t+∂V

∂xf(x, t) ≤ −W (x) ≤ 0 (4.9)

where γ1 and γ2 are class K∞ functions and W is a continuous function. Then, allsolutions of (4.4) are globally uniformly bounded and satisfy

limt→∞

W (x(t)) = 0. (4.10)

If W (x) is positive definite, then the equilibrium is globally uniformly asymptoticallystable.

The proof [KKK95] is omitted here.

4.2.2 Standard backstepping

The idea of backstepping is to design a controller recursively by considering some of thestate variables as “virtual controls” and designing for them intermediate control laws.The derivation of standard backstepping can be illustrated by the following example.

Assume a second-order system

x1 = f1(x1) + g1(x1)x2

x2 = f2(x1, x2) + g2(x1, x2)u,(4.11)

where gi 6= 0 with i = 1, 2. The reference trajectory of x1 is denoted by xr and itssecond order derivatives are piecewise continuous and bounded. The control input ushould guarantee that

limt→∞

(x1 − xr) = 0. (4.12)

The design procedure is illustrated as follows.


Step 1: Introduce e1 = x1 − xr and e2 = x2 − β2, where β2 is a stabilizing function tobe designed. Let the Lyapunov function V1 = 1

2ρ1e

21 (ρ1 > 0), then the derivative is

V1 = ρ1e1e1

= ρ1e1 [f1(x1) + g1(x1)x2 − xr] . (4.13)

Substitute e2 = x2 − β2 into (4.13)

V1 = ρ1e1(f1(x1) + g1(x1)e2 + g1(x1)β2 − xr). (4.14)

Choose the stabilizing function as

β2 =1

g1(x1)[−f1(x1) + xr − k1e1] with k1 > 0. (4.15)

Then the derivative of V1 is

V1 = −ρ1k1e21 + ρ1g1(x1)e1e2. (4.16)

So far the V1 ≤ 0 is not ensured due to the term ρ1g1(x1)e1e2.

Step 2: Now augment the Lyapunov function as V2 = V1+12ρ2e

22(ρ2 > 0). The derivative

of V2 is

V2 = V1 + ρ2e2e2

= −ρ1k1e21 + ρ1g1(x1)e1e2 + ρ2e2

[

f2(x1, x2) + g2(x1, x2)u− β2

]

, (4.17)

where u is chosen to ensure V2 ≤ 0 that leads to

u =1

g2(x1, x2)

[

−f2(x1, x2) −ρ1

ρ2

g1(x1)e1 + β2 − k2e2

]

with k2 > 0, (4.18)

where

β2 =1

g21

[

g1

(

−f1 − k1e1 + xr

)

− g1 (−f1 − k1e1 + xr)]

.


V2 = −ρ1k1e21 − ρ2k2e

22. (4.19)

Now it can be obtained

V2 ≤ 0 and V2 = 0 only by e1 = e2 = 0. (4.20)

Because gi 6= 0 and xr are second order differentiable, the designed u can be obtained andis bounded. By applying the LaSalle-Yoshizawa theorem, the global uniform asymptoticstability of e1 and e2 is guaranteed. It follows that

limt→∞

ei = 0. (4.21)


Then the desired tracking performance in (4.12) can be achieved.

This recursive design procedure can be easily extended to n-th order system. Assumethe nonlinear system has the structure

x1 = f1(x1) + g1(x1)x2

...

xn−1 = fn−1(x1, · · · , xn−1) + gn−1(x1, · · · , xn−1)xn

xn = fn(x1, · · · , xn) + gn(x1, · · · , xn)u

(4.22)

with gi 6= 0 (i = 1, 2, · · · , n). In addition, system with such a form is known as strict-feedback system. The reference trajectory xr is nth order differentiable. Define

ei = xi − βi, (4.23)

where β1 = xr and βi(i > 1) are the virtual control signals, which are determined in therecursive procedure.

The Lyapunov function is

V =1

2

n∑

i=1

ρiei2 with ρi > 0, i = 1, 2 · · ·n. (4.24)

The control input based on backstepping approach can be recursively derived as

u =1

gn

(−fn −ρn−1

ρn

gn−1en−1 − knen + βn) (ki > 0) (4.25)

Then the derivative of the Lyapunov function (4.24) is

V = −n∑

i=1

ρikiei2. (4.26)

Thus, the tracking error is asymptotic stable at the origin according to LaSalle-Yoshizawatheorem.

4.2.3 Adaptive backstepping

Adaptive techniques can be easily inserted in backstepping design procedure. Assume asecond order system with an unknown, constant parameter. The parameter appears inthe system linearly. The system is described as

x1 = f1(x1)θ1 + g1(x1)x2

x2 = f2(x1, x2) + g2(x1, x2)u(4.27)


with gi 6= 0 (i = 1, 2) and second order differentiable reference trajectory xr. θ1 standsfor the unknown parameter. The control input u should ensure that the tracking errorx1 − xr converges to zero. Carry out the similar procedure as the illustration example(4.11).

Step 1: Introduce e1 = x1−xr, e2 = x2−β2 and eθ1= θ1− θ1, where θ1 is the estimation

of θ1. Let the Lyapunov function V1 = 12ρ1e

21, then the derivative is

V1 = ρ1e1e1

= ρ1e1 [θ1f1(x1) + g1(x1)x2 − xr] (4.28)

Choose the stabilizing function as

β2 =1

g1(x1)[−θ1f1(x1) + xr − k1e1] with k1 > 0. (4.29)

But the parameter θ1 is unknown, it is substituted by its estimation θ1. Then

β2 =1

g1(x1)

[

−θ1f1(x1) + xr − k1e1

]

. (4.30)

That leads to

V1 = −ρ1k1e21 + ρ1g1(x1)e1e2 + ρ1f1(x1)e1eθ1

. (4.31)

Step 2: Now augment the Lyapunov function as V2 = V1 + 12ρ2e

22 + 1

2ρθ1e2θ1

. Thederivative of V2 is

V2 = V1 + ρ2e2e2 + ρθ1eθ1θ1

= −ρ1k1e21 + ρ1g1(x1)e1e2 + ρ1f1(x1)e1eθ1

+ρ2e2

[

f2(x1, x2) + g2(x1, x2)u− β2

]

+ ρθ1eθ1θ1. (4.32)

Choose the control input as

u =1

g2(x1, x2)

[

−f2(x1, x2) −ρ1

ρ2

g1(x1)e1 + β2 − k2e2

]

with k2 > 0, (4.33)

Then the derivative of Lyapunov function V2 is

V2 = −ρ1k1e21 − ρ2k2e

22 + ρ1f1(x1)e1eθ1

+ ρθ1eθ1eθ1

(4.34)

To guarantee a non-positive V2 for arbitrary θ1, the updating law of θ1 is designed to

make the sum of last two terms in (4.34) is zero. Because θ1 is constant, then eθ1= −

˙θ1.


Thus, the adaption law of θ1 is

˙θ1 =

ρ1

ρθ1

e1f1(x1) (4.35)

That leads to V2 ≤ 0 and V2 = 0 only if e1 = e2 = 0. The tracking error is asymptoticstable on origin even with the unknown parameter in system. The estimation error eθ1

is bounded based on the LaSalle-Yoshizawa.

This procedure can be generalized to the parametric strict-feedback system, which hasthe following form

x1 = f(x1) + g1(x1)x2 + ΦT1 (x1)θ

x2 = f(x1, x2) + g2(x1, x2)x3 + ΦT2 (x1, x2)θ

...

xn−1 = f(x1, · · · , xn−1) + gn−1(x1, · · · , xn−1)xn + ΦTn−1(x1, · · · , xn−1)θ

xn = fn(x1, · · · , xn) + ΦTn (x1, · · · , xn)θ + gn(x1, · · · , xn)u

(4.36)

where θ is the unknown parameter vector, which is constant or changes slowly. gi 6= 0and is known with i = 1, 2, · · · , n. The reference xr is nth order differentiable. Thenthe Lyapunov function can always be constructed as

Va =1

2

n∑

i=1

ρiei2 +

1

2eT

θ Γeθ with ρi > 0, i = 1, 2 · · ·n. (4.37)

where ei = xi − βi, β1 = xr and βi(i > 1) are the stabilizing functions, which aredetermined by backstepping procedure are coupled with the estimation of parameters.Γ is a positive definite matrix. Then there exists a control input u to ensure that

Va = −n∑

i=1

ρikiei2 (ki > 0), (4.38)

which implies that the tracking error will asymptotically converge to zero. The estima-tion error eθ is bounded. The proof is also based on the LaSalle-Yoshizawa theorem. Itis noted that eθ is only bounded and but not asymptotically approaches to zero.

4.2.4 Backstepping based FTC for electro-hydraulic system

The backstepping method is applied to realize the position tracking for the electro-hydraulic system, which is subject to the varied external load Fload and the consideredfaults QLin, ∆PA and ∆PB. Fload and QLin can be viewed as parameter uncertainties ofthe system. So the adaptive backstepping should be employed to tolerate these uncer-tainties. The sensor faults ∆PA and ∆PB affect the system through the backstepping


controller itself. Some parameters of system model are dependent on the measurementsof PA and PB, such as EA(PA) and EB(PB). Then the system model must be adaptedfor reconfiguration of the backstpping controller.

The structure of backstepping based FTC for the electro-hydraulic system is shown inFig. 4.2. In fault free case the adaptive backstepping is utilized for compensation of thevariation of Fload. If the fault is present, the controller is reconfigured based on a newmodel. Which model should be adapted is dependent on the up-to -date information ofthe fault from FDI system. The new model usually is not so precise as the model innominal state. It may cause performance depredation.

adap. backsteppingabcde fghij kl mreference

values

Electro-hydraulic System

y

nopqrFDI System rs

adap. backstepping uty

uu Qvsuy

wxyz| ~

Figure 4.2: The structure of backsteping based FTC for electro-hydraulic system

4.2.4.1 System transformation

It is necessary for the system to be formulated into strict-feedback or parametric-strict-feedback structure for the implementation of backstepping approach. Recall the modelof the electro-hydraulic system

xc = vc

vc =1


PB =EB(PB)

V20 − A2xc

(

A2vc −√


)

PA =EA(PA)

V10 + A1xc

(−A1vc −QLin +KQα)

α = vα


(4.39)


Assume a new state variable FHY = PAA1 −PBA2. Then the original system (4.39) canbe rewritten as

xc = vc

vc =1

m(FHY −Dvvc − fc − Fload)

FHY =EA(PA)A1

V10 + A1xc

(−A1vc −QLin +KQα) (4.40)

−EB(PB)A2

V20 − A2xc

(

A2vc −√


)

α = vα


It is obvious that new system possesses the strict-feedback structure. For simplicity thedynamics of FHY is given by

FHY = f(xc, vc) + g(xc)α, (4.41)

where EA(PA), EB(PB) and√

kqb1(PB − Po) + kqb2 are viewed as known parameters.According to the actual parameters of the testbed, g(xc) 6= 0 can be obtained.

4.2.4.2 Reference trajectories

The goal of control is to realize precise tracking position with good transient performance.With the consideration of hardware constraints the reference trajectory of position isdesigned as

xr = 5t− 40sin(0.125t) + x0, (mm) (4.42)

where x0 is the initial position. It is obvious that this trajectory is always differentiable.The corresponding reference velocity is

vr = 5 − 5cos(0.125t) (mm/sec.). (4.43)

If the cylinder moves along the reference trajectory, it can reach the final position verygently.

4.2.4.3 Controller for fault free case

In this section the position tracking controller for fault free case is derived.

Step 1: Define

e1 = xc − xr, e2 = vc − β2, e3 = FHY − β3

e4 = α− β4, e5 = vα − β5,(4.44)


with βi (i = 2, 3, 4, 5) are the stabilizing functions. Then the Lyapunov function in thisstep is

V1 =1

2ρ1e

21 (ρ1 > 0). (4.45)


V1 = ρ1e1e1

= ρ1e1 (β2 + e2 − xr) .

(4.46)

Choose β2 = −k1e1 + xr, with k1 > 0. This follows that

V1 = −ρ1k1e21 + ρ1e1e2. (4.47)

Step 2: Define the Lyapunov function of this step

V2 = V1 +1

2ρ2e

22 (ρ2 > 0). (4.48)

The derivative of V2 is

V2 = −ρ1k1e21 + ρ1e1e2 + ρ2e2e2

= −ρ1k1e21 + ρ1e1e2 + ρ2e2

(

−Dv

mvc +

1

m(β3 + e3) −

1

mFload − β2

)

Assume the Fload is known. β3 is chosen as

β3 = Dvvc + fc + Fload −ρ1

ρ2

me1 −mk2e2 +mβ2 (k2 > 0) (4.49)

This leads to

V2 = −ρ1k1e21 − ρ2k2e

22 +

ρ2

me2e3. (4.50)

The solution (4.49) is based on the constraint that the external load Fload is known. Nowthis constraint is released by employing parameter adaption.

Define θ1 = Fload and the estimation error eθ1= θ1− θ1. Augment the Lyapunov function

V2 as

V2a = V2 +1

2ρθ1e2θ1

ρθ1> 0. (4.51)

The derivative of V2a is

V2a = −ρ1k1e21 + ρ1e1e2 + ρθ1

eθ1eθ1

+ρ2e2

(

−Dv

mvc +

1

m(β3 + e3) −

1

m(eθ1

+ θ1) − β2

)

(4.52)


Then the β3 is rewritten as

β3 = Dvvc + fc + θ1 −ρ1

ρ2

me1 −mk2e2 +mβ2 (4.53)

with the adaption law of θ1

˙θ1 = −

ρ2

mρθ1

e2 (4.54)


V3 = V2 +1

2ρ3e

23 (ρ3 > 0). (4.55)

The derivative is

V3 = −ρ1k1e21 − ρ2k2e

22 +

ρ2

me2e3 + ρ3e3 [f(xc, vc) + g(xc)(β4 + e4)] . (4.56)

Because g(xc) 6= 0, the virtual control signal β4 can be designed as

β4 =1

g(xc)

(

−f(xc, vc) −ρ2

ρ3me2 − k3e3 + β3

)

(k3 > 0). (4.57)

This leads to

V3 = −ρ1k1e21 − ρ2k2e

22 − ρ3k3e

23 + ρ3g(xc)e3e4. (4.58)


V4 = V3 +1

2ρ4e

24 (ρ4 > 0). (4.59)

Then the derivative is

V4 = −ρ1k1e21 − ρ2k2e

22 − ρ3k3e

23 + ρ3g(xc)e3e4

+ρ4e4

(

β5 + e5 − β4

)

. (4.60)

Choose β5 as

β5 = −ρ3

ρ4

g(xc)e3 − k4e4 + β4. (k4 > 0) (4.61)

This leads to

V4 = −ρ1k1e21 − ρ2k2e

22 − ρ3k3e

23 − ρ4k4e

24 + ρ4e4e5. (4.62)


V5 = V4 +1

2ρ5e

25 (ρ5 > 0). (4.63)


The derivative is

V5 = −ρ1k1e21 − ρ2k2e

22 − ρ3k3e

23 − ρ4k4e

24 + ρ4e4e5

+ρ5e5(−p1α− p2vα + p1kpuuin − β5). (4.64)

Then the stabilizing control input is

uin =1

p1kpu

(

p1α+ p2vα −ρ4

ρ5

e4 − k5e5 + β5

)

(k5 > 0) (4.65)

This leads to

V5 = −ρ1k1e21 − ρ2k2e

22 − ρ3k3e

23 − ρ4k4e

24 − ρ5k5e

25. (4.66)

With designed control input (4.65) the position tracking error will asymptotically con-verge to zero.

It is noted that for deriving the final control input the derivative of the stabilizing func-tion should be available. It can be calculated analytically. However, this is a fifth ordersystem. The calculation is very cumbersome and is not suitable for real-time implementa-tion. Therefore, the numeric differential is applied. Sometimes the calculated stabilizingfunction is so large that the value is far away from the limits of the corresponding statevariable. Thus, the values of the stabilizing functions are restricted according to thephysical constraints.

4.2.4.4 The case with leakage fault QLin

If the internal leakage occurs, the model is different from nominal model. It can bedescribed as

FHY = f(xc, vc) + ψ(xc)θ2 + g(xc)α, (4.67)

where θ2 denotes the leakage QLin in model (3.43). θ2 is an unknown constant andappears linearly in equation (4.67). Thus, the parameter adaption can be utilized tocompensate the effect of the leakage.

The new controller design procedure is similar to the one in fault free case. The param-eter adaption law is inserted in V3. Define the estimation error eθ2

= θ2 − θ2. The newLyapunov function is

V3a = V3 +1

2ρθ2e2θ2

(ρθ2> 0). (4.68)

Then the derivative is

V3a = −ρ1k1e21 − ρ2k2e

22 +

ρ2

me2e3 + ρθ2

eθ2eθ2

+ρ3e3

[

f(xc, vc) + ψ(xc)(θ2 + eθ2) + g(xc)(β4 + e4)

]

. (4.69)


The stabilizing function is chosen as

β4 =1

g(xc)

(

−f(xc, vc) − ψ(xc)θ2 −ρ2

ρ3me2 − k3e3 + β3

)

(k3 > 0), (4.70)

with the adaption law of θ2

˙θ2 =

ρ3ψ(xc)

ρθ2g(xc)

e3. (4.71)

This leads to

V3a = V3 = −ρ1k1e21 − ρ2k2e

22 − ρ3k3e

23 + ρ3g(xc)e3e4. (4.72)

The rest of the design is the same as the no fault case. Then the asymptotic convergenceof the position tracking error can be guaranteed even with the internal leakage in system.

4.2.4.5 The case with the sensor fault ∆PB

For eliminating the effect of ∆PB the system model should be adapted. The simplestmethod is to assume the dynamics of PB is zero at every sampling time. That leads toQB = A2vc. Then the PB can be approximated as

PBapp=

(A2Vc)2−kqb2

kqb1+ P0 if (A2Vc)

2 − kqb2 ≥ 0,

P0 if (A2Vc)2 − kqb2 < 0,

(4.73)

where the parameters have the same meaning as in equations (2.9) and (2.10). Fur-thermore, FHY = PAA1 + PBapp

A2 and FHY = PAA1. This approximation can degradethe performance of adaptive backstepping. But the value of pressure PB is very smallcompared to the pressure PA. Therefore, this impact can be ignored. The rest of designis same as the no fault case.

4.2.4.6 The case with the sensor fault ∆PA

The Sensor fault ∆PA can bring more serious consequence than ∆PB, especially if thesystem is running under high pressure. Fortunately, the reconfiguration can be easilyaccomplished thanking to the structure of adaptive backstepping. The ∆PA affects twoparameters of the backstepping controller: error term e3 and the bulk modulus EA(PA).Firstly, it can be proved that the sensor offset on error e3 can be compensated by theadaption law of Fload (θ1). Let FHY = FHY + ∆FHY , where ∆FHY = ∆PAA1. Design anew stabilizing function β3a, which satisfies that e3 = FHY − β3a. It leads to

FHY = β3a + e3 − ∆FHY . (4.74)

4.3 Simulation 65

Recall the derivative of V2a (4.52)

V2a = −ρ1k1e21 + ρ1e1e2 + ρθ1

eθ1eθ1

+ρ2e2

(

−Dv

mvc +

1

mFHY −

1

mθ1 − β2

)

. (4.75)

Substitute (4.74) into V2a. Then

V2a = −ρ1k1e21 + ρ1e1e2 + ρθ1

eθ1eθ1

+ρ2e2

(

−Dv

mvc +

1

m(β3a + e3) −

1

m(θ1 + ∆FHY ) − β2

)

. (4.76)

∆FHY is a step-like signal and its derivative can be viewed as zero. So if define θ1a =θ1 + ∆FHY then it can be estimated by the same adaption law as θ1. Also β3a can beobtained as

β3a = Dvvc + fc + θ1 + ∆FHY −ρ1

ρ2

me1 −mk2e2 +mβ2 (4.77)

It leads to FHY − β3a ≈ FHY − β3 = e3. It can be concluded that the error term e3 isalmost unaffected by ∆PA.

The second affected parameter EA(PA) by the fault approximated by a constant. Thecalculation of EA(PA) can not be accurate, when the value of PA is wrong. Furthermore,if the pressure is high enough, then the bulk modulus can be viewed as a constant.

Thus, the reconfiguration for ∆PA is accomplished by setting the bulk modulus EA(PA)as constant.

4.3 Simulation

The backstepping based FTC for all discussed fault scenarios is simulated in this section.Actually the fault diagnosis system developed in chapter 3 has been integrated in theFTC scheme. But the detailed results about fault diagnosis are omitted here for avoidingpresentation of similar contents as chapter 3. In this section the control performance isthe highlight and is presented in details. Average errors of position and velocity denotedby exc

and evcare introduced for evaluating the control performance.


4.3.1 Scenario 1: fault free

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(m

m)

xref

xc

0 10 20 30 40 50 60−0.2

0

0.2

0.4

0.6

Time (sec.)

Pos

ition

Err

or (

mm

)

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(m

m/s

ec.)

vref

vc

0 10 20 30 40 50 60−1

−0.5

0

0.5

Time (sec.)V

eloc

ity E

rror

(m

m/s

ec.)

Figure 4.3: Tracking control (fault free)

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Time (sec.)

Extern

al load

(N)

settingestimation

Figure 4.4: Estimation of Fload (fault free)

Fig. 4.3 shows excellent tracking performances of proposed backstepping controller in thefault free case. The small “peak” in the velocity trajectory is caused by the variation ofload force. The position error converges to zero very soon due to the parameter adaptionof the load force. The estimation of Fload by adaptive controller has a bounded error withthe actual value. The average errors are exc

= 0.3086 (mm) and evc= 0.078 (mm/sec.).

4.3 Simulation 67

4.3.2 Scenario 2: with the internal leakage QLin

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(m

m)

xref

xc

0 10 20 30 40 50 60−0.2

0

0.2

0.4

0.6

Time (sec.)

Pos

ition

Err

or (

mm

)

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(m

m/s

ec.)

vref

vc

0 10 20 30 40 50 60−1

−0.5

0

0.5

Time (sec.)

Vel

ocity

Err

or (

mm

/sec

.)

Figure 4.5: Tracking control (with QLin)

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Time (sec.)

Exter

nal loa

d (N)

settingestimation

Figure 4.6: Estimation of Fload (with QLin )


0 10 20 30 40 50 600

1

2

3

4

5

6

Time (sec.)

Interna

l leaka

ge (L/m

in)

settingestimation

Figure 4.7: Estimation of QLin

The internal leakage QLin is activated at t = 12 second. It is detected and isolated inabout 5 seconds. It can be seen from Fig. 4.6 the control performance is almost same asin the fault free case. The fault QLin is tolerated by the reconfigured adaptive controller.The estimation errors of Fload and QLin are bounded but not converges to zero. Theaverage errors are exc

= 0.315 (mm) and evc= 0.078 (mm/sec).

4.3.3 Scenario 3: with the sensor offset ∆PB

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(mm

)

xref

xc

0 10 20 30 40 50 60−0.2

0

0.2

0.4

0.6

Time (sec.)

Pos

ition

Err

or (m

m)

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(mm

/sec

.)

vref

vc

0 10 20 30 40 50 60−1

−0.5

0

0.5

Time (sec.)

Vel

ocity

Err

or (m

m/s

ec.)

Figure 4.8: Tracking control (with ∆PB)

4.3 Simulation 69

The sensor fault ∆PB is set as 20% of the meassument. It is added in measument att = 12 second. Although the effect of ∆PB is not signficant to the system, it is detectedin 0.2 seconds. The tracking error is similar to the faul free case with exc

= 0.310 (mm)and evc

= 0.076 (mm/sec). The estimaton of Fload is also alomst same as in the faultfree case, so the figure is not present here.

4.3.4 Scenario 4: with the sensor offset ∆PA

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(mm

)

xref

xc

0 10 20 30 40 50 60−0.2

0

0.2

0.4

0.6

Time (sec.)

Pos

ition

Err

or (m

m)

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(mm

/sec

.)

vref

vc

0 10 20 30 40 50 60−2

−1

0

1

2

3

Time (sec.)

Vel

ocity

Err

or (m

m/s

ec.)

Figure 4.9: Tracking control (with ∆PA)

0 10 20 30 40 50 600

1

2

3

4

5x 10

5

Time (sec.)

Extern

al load

(N)

settingestimation

Figure 4.10: Estimation of Fload (with ∆PA )


In this case the sensor offset ∆PA is also set as 20% and added to the measurementat t = 12 second. The fault is detected in almost 0.1 second. It can be observed inFig. 4.10 the effect of ∆PA is viewed as the uncertainty of the load by the adptivecontroller. Thus control performanc is not so affected by the fault. The average errorsare exc

= 0.317 (mm) and evc= 0.095 (mm/sec).

The extreme situation is simulated to illustrate the robustness of the FTC strategy. Themeasurement value of PA is set as zero during the simulation.

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Posit

ion

(mm

)

xref

xc

0 10 20 30 40 50 60−1

−0.5

0

0.5

Time (sec.)

Posit

ion

Erro

r (m

m)

0 10 20 30 40 50 600

5

10

15

Time (sec.)

Velo

city

(mm

/sec

.)

vref

vc

0 10 20 30 40 50 60−10

−5

0

5

Time (sec.)

Velo

city

Erro

r (m

m/s

ec.)

Figure 4.11: Tracking control (FTC)

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Posit

ion

(mm

)

xref

xc

11.5 12 12.5 13 13.5 14 14.5−2.5

−2

−1.5

−1

−0.5

0

Time (sec.)

Posit

ion

Erro

r (m

m)

10 11 12 13 14 15 16

0

2

4

6

8

10

12

Time (sec.)

Velo

city

(mm

/sec

.)

vref

vc

11.5 12 12.5 13 13.5 14 14.5

−8

−6

−4

−2

0

2

4

Time (sec.)

Velo

city

Erro

r (m

m/s

ec.)

Figure 4.12: Tracking control (adaptive backsteppping) (zoomed)

The results in Fig. 4.12 are from the simulation with normal adaptive backstpping con-

4.4 Summary 71

troller. The position tracking error can be still limited in a small range. But the systembegins to oscillate due to the information lost of PA. The proposed FTC performs excel-lent with the aid of FDI system. The control performance has only a slight degradationwith the results exc

= 0.386 (mm) and evc= 0.114 (mm/sec).

4.3.5 Conclusion

The simulation study shows wonderful performance of the developed fault diagnosisintegrated FTC scheme. The tracking of position and velocity is realized. By employ-ing adaptive backstpping technique the controller can easily eliminate the effect of thevariation of external load. The faults can be accommodated by applying pre-designedreconfiguration laws. The control performance is almost same as the nominal state eventhe system is faulty. The parameter estimation of the adaptive controller has boundederror and can not asymptotically converge to the actual value. It indicates that adaptivecontroller can not be applied as a residual generator here. An additional FDI systemis really necessary for application. In summary it can be concluded that the proposedcontrol scheme has realized the precision control with guaranteed safety and reliabilityfor the electro-hydraulic system.

4.4 Summary

In this chapter the fault-tolerant control (FTC) scheme is designed. The focus is tofind a controller that can perform the position tracking and tolerate the possible faultsautomatically. Backstepping technology is applied to design the controller. It can han-dle the nonlinearity and unknown parameter of the system. The asymptotic stabilityis guaranteed in the fault free case. The considered faults can be accommodated byparameter adaption and model modification. The control performance has only a slightdegradation. According to the simulation results it can be concluded that the proposedFTC can achieve the desired control performance even in the presence of the fault.

5 Experimental results

5.1 Practical issues

5.1.1 Hardware and software

The discussed method of fault diagnosis and FTC is implemented on the practical electro-hydraulic testbed. The key hardware of the controller is the a real-time measurementcard MODULAR-4 installed in an ISA slot of a PC. The measurement card has a 486CPU and various I/O units and is delivered with its own real-time multitasking operatingsystem in ROM [Mod]. The card works parallel with the computer. The algorithms arewritten in C and complied as executable files. These files are downloaded into the card’smemory as tasks. According to the function of individual task the priority is determinedimplicitly. The real-time property is guaranteed by the Timer-Interrupt.

The PC is served as a graphic user interface (GUI) for visualization, parameter settingand data storage. These function is realized through the NI software LabWindows/CVI.The computer can access to the card through the drivers. For example, as soon as thebuffer area of card is full, the compute will read all saved data. Then the buffer areawill be refreshed with the new data. All data can be saved in computer. Hence, theproblem of limited memory capacity in card can be well solved. The general descriptionof software structure in computer and measurement card is shown in Fig.5.1.

Figure 5.1: Structure of the software

73

74 5 Experimental results

5.1.2 Parameters and experiment setting

The sampling time is chosen as Ts = 5ms. The system parameters are same as those insystem identification. The maximum of load pressure is 120 Bar.

The Controller parameters are chosen as (for standard units)

k1 = 200, k2 = 2000, k3 = 4, k4 = 100, k5 = 100,

ρ1 = 1, ρ2 = 1, ρ3 = 1, ρ4 = 100, ρ5 = 10,1

ρθ1

= 6e7, ρθ2= 230.

The faults are set as following:

• QLin ≈ 5 (L/min), if t ≥ trv. Here trv denotes the rising time of the commandvalue, which is a ramp-like signal with the saturation of 5 (L/min).

• ∆PB is 20% of the measurement and acts to measurement as step signal.

• ∆PA is 20% of the measurement and acts to measuremtn as step signal.

The setting of the DML is chosen as follows (with the time unit (sec.)).

DMLPB=

1 tr1≥ 0.25 and DMLQLin

= DMLQPA= 0

0 otherwise(5.1)

DMLPA=

1 0.05 ≤ tr2≤ 0.25 and DMLQLin

= DMLQPB= 0

0 otherwise(5.2)

DMLQLin=

1 tr3≥ 0.10 and DMLQPA

= DMLQPB= 0

0 otherwise(5.3)

5.2 Experiments 75

5.2 Experiments

5.2.1 Scenario 1: fault free

0 10 20 30 40 50 600

100

200

300

Time (sec.)

x c (mm

)

measurementestimation

0 10 20 30 40 50 600

50

100

150

Time (sec.)

PA

(Bar

)

0 10 20 30 40 50 600

2

4

6

Time (sec.)

PB

(Bar

)

Figure 5.2: Estimation in fault free case

0 10 20 30 40 50 600

2

4

Time (sec.)

r x c(mm)

0 10 20 30 40 50 60−0.5

0

0.5

Time (sec.)

e P B(Bar

)

0 10 20 30 40 50 60−5

0

5

Time (sec.)

e P A (Bar

)

Figure 5.3: Estimation errors in fault free case


0 10 20 30 40 50 600

1

2

3

4

5

6

Time (sec.)

r x c

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Time (sec.)

r PA

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

Time (sec.)

r PB

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

Time (sec.)

r Lin

Figure 5.4: Residuals in fault free case

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(m

m)

xref

xact

0 10 20 30 40 50 60−2

−1

0

1

2

Time (sec.)

Pos

ition

Err

or (

mm

)

Figure 5.5: The position trajectories in fault free case

5.2 Experiments 77

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(m

m/s

ec.)

vref

vact

0 10 20 30 40 50 60−2

−1

0

1

2

Time (sec.)

Vel

ocity

Err

or (

mm

/sec

.)

Figure 5.6: The velocity trajectories in fault free case

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5x 10

5

Time (sec.)

Exter

nal lo

ad (N

)


Figure 5.7: The external load force

There is no fault but the varying external load in system. The fault detection observerpossesses good convergence. The residuals are unaffected by the unknown external loadforce and vary always under the threshold. No false alarm is trigged. The initial errorshave significant effect to the estimation. The estimation of pressure PB is worse thanothers due to the modeling error. Adaptive backstepping also archives excellent trackingperformance. The “peaks” in the position error trajectory shown in Fig. 5.5 are causedby load variation. The average errors are exc

= 0.379 (mm) and evc= 0.237 (mm/sec.).


The estimation of external load is bounded but had a poor accuracy as shown in Fig.5.7.

For comparison the results with a PI controller are shown in following.

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(mm

)

xref

xact

0 10 20 30 40 50 60−4

−2

0

2

4

Time (sec.)

Pos

ition

Err

or (m

m)

Figure 5.8: The position trajectories with PI controller (fault free)

0 10 20 30 40 50 600

5

10

15

Time (sec.)

Vel

ocity

(mm

/sec

.)

vref

vact

0 10 20 30 40 50 60−4

−2

0

2

4

Time (sec.)

Vel

ocity

Err

or (m

m/s

ec.)

Figure 5.9: The velocity trajectories with PI controller (fault free)

It shows that the velocity signal in Fig. 5.9 has began to oscillate due to the variation

5.2 Experiments 79

of the external load. The average errors are exc= 1.88 (mm) and evc

= 1.27 (mm/sec.).The conclusion can be drawn that backstepping controller performs better than theconventional PI controller in the fault free case.

5.2.2 Scenario 2: with the internal leakage QLin

0 10 20 30 40 50 600

100

200

300

Time (sec.)

x c (mm

)


0 10 20 30 40 50 600

50

100

150

Time (sec.)

P A (Bar

)

0 10 20 30 40 50 600

5

10

Time (sec.)

P B (Bar

)

Figure 5.10: Estimation with QLin

0 10 20 30 40 50 600

1

2

3

Time (sec.)

r x c(mm

)

0 10 20 30 40 50 60−1

0

1

2

Time (sec.)

e P B(Bar

)

0 10 20 30 40 50 60−5

0

5

10

Time (sec.)

e P A(Bar

)

Figure 5.11: Estimation errors with QLin


0 10 20 30 40 50 600

1

2

3

4

5

6

Time (sec.)

r x c

0 10 20 30 40 50 600

0.5

1

1.5

2

Time (sec.)

r PA

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Time (sec.)

r PB

0 10 20 30 40 50 600

0.5

1

1.5

Time (sec.)

r Lin

Figure 5.12: Residuals with QLin

0 10 20 30 40 50 60−1

0

1

Time (sec.)

DM

LP

A

0 10 20 30 40 50 60−1

0

1

Time (sec.)

DM

LP

B

0 10 20 30 40 50 600

1

2

Time (sec.)

DM

LQ

Lin

Figure 5.13: The decision making logic with QLin

5.2 Experiments 81

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(m

m)

xref

xact

0 10 20 30 40 50 60−3

−2

−1

0

1

Time (sec.)

Pos

ition

Err

or (

mm

)

Figure 5.14: The position trajectories with QLin

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(m

m/s

ec.)

vref

vact

0 10 20 30 40 50 60−2

−1

0

1

2

Time (sec.)

Vel

ocity

Err

or (

mm

/sec

.)

Figure 5.15: The velocity trajectories with QLin


0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

Time (sec.)

Ext

erna

l loa

d (N

)


Figure 5.16: Estimation of external load force with fault QLin

0 10 20 30 40 50 600

1

2

3

4

5

6

Time (sec.)

Inter

nal le

akag

e (L/m

in)

settingestimation

Figure 5.17: Estimation of QLin

The leakage QLin is introduced into the system at t ≈ 10 second. The fault is identifiedabout 10 seconds later. The long time delay is caused by fault detection. The residualrLin can not reflect the fault until the value of leakage flow is settled. Furthermore,rLin has the maximum when the velocity is fastest. The DMLLin makes a right choiceand the second parameter adaption is activated. The position error quickly converges tozero with the reconfigured controller. The estimations of external load and leakage arebounded but not accurate. In spite of the inaccuracy good control performance is stillachieved with the average errors exc

= 0.408 (mm) and evc= 0.237 (mm/sec.).

5.2 Experiments 83

5.2.3 Scenario 3: with the sensor offset ∆PB

0 10 20 30 40 50 600

100

200

300

Time (sec.)

x c (mm

)


0 10 20 30 40 50 600

50

100

150

Time (sec.)

PA

(Bar

)

0 10 20 30 40 50 600

2

4

6

Time (sec.)

PB

(Bar

)

Figure 5.18: Estimation with ∆PB

0 10 20 30 40 50 600

2

4

Time (sec.)

r x c(mm

)

0 10 20 30 40 50 60−0.5

0

0.5

1

Time (sec.)

e PB

(Bar

)

0 10 20 30 40 50 60−5

0

5

Time (sec.)

e PA

(Bar

)

Figure 5.19: Estimation errors with ∆PB


0 10 20 30 40 50 600

1

2

3

4

5

6

Time (sec.)

r x c

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Time (sec.)

r PA

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

Time (sec.)

r PB

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

Time (sec.)

r Lin

Figure 5.20: Residuals with ∆PB

0 10 20 30 40 50 60−1

0

1

Time (sec.)

DM

LP

A

0 10 20 30 40 50 600

1

2

Time (sec.)

DM

LP

B

0 10 20 30 40 50 60−1

0

1

Time (sec.)

DM

LQ

Lin

Figure 5.21: The decision making logic with ∆PB

5.2 Experiments 85

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(mm

)

xref

xact

0 10 20 30 40 50 60−3

−2

−1

0

1

2

Time (sec.)

Pos

ition

Err

or (m

m)

Figure 5.22: The position trajectories with ∆PB

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(mm

/sec

.)

vref

vact

0 10 20 30 40 50 60−3

−2

−1

0

1

2

Time (sec.)

Vel

ocity

Err

or (m

m/s

ec.)

Figure 5.23: The velocity trajectories with ∆PB

The fault is added at t ≈ 9 second and is detected about 14 seconds later. The fault isnot detected until the value of PB becomes large. Then the effect of the sensor offsetappeared distinctly. The controller reconfiguration is activated as soon as the fault isidentified. It can be obtained through the error trajectory of position that the fault∆PB has less impact on system than the load variation. Although the dynamics of PB is


ignored in controller redesign procedure, the performance degradation is also very smallwith the average errors exc

= 0.433 (mm) and evc= 0.280 (mm/sec.). The result of

estimation of Fload is similar as in scenario 1 and is omitted here.

5.2.4 Scenario 4: with the sensor offset ∆PA

0 10 20 30 40 50 600

100

200

300

Time (sec.)

x c (mm

)


0 10 20 30 40 50 600

50

100

Time (sec.)

P A (Bar

)

0 10 20 30 40 50 600

2

4

6

Time (sec.)

P B (Bar

)

Figure 5.24: Estimation with ∆PA

0 10 20 30 40 50 600

1

2

3

Time (sec.)

r x c(mm

)

0 10 20 30 40 50 60−0.5

0

0.5

Time (sec.)

e P B(Bar

)

0 10 20 30 40 50 60−10

0

10

20

Time (sec.)

e P A(Bar

)

Figure 5.25: Estimation errors with ∆PA

5.2 Experiments 87

0 10 20 30 40 50 600

1

2

3

4

5

6

Time (sec.)

r x c

0 10 20 30 40 50 600

0.5

1

1.5

Time (sec.)

r PA

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

Time (sec.)

r PB

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

Time (sec.)

r Lin

Figure 5.26: Residuals with ∆PA

16.5 16.6 16.7 16.8 16.9

0.5

1

1.5

Time (sec.)

r P A

Figure 5.27: The zoomed rPA


0 10 20 30 40 50 600

1

2

Time (sec.)

DM

LP

A

0 10 20 30 40 50 60−1

0

1

Time (sec.)

DM

LP

B

0 10 20 30 40 50 60−1

0

1

Time (sec.)

DM

LQ

Lin

Figure 5.28: The decision making logic with ∆PA

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Pos

ition

(m

m)

xref

xact

0 10 20 30 40 50 60−2

−1

0

1

2

Time (sec.)

Pos

ition

Err

or (

mm

)

Figure 5.29: The position trajectories with ∆PA

5.2 Experiments 89

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (sec.)

Vel

ocity

(m

m/s

ec.)

vref

vact

0 10 20 30 40 50 60−2

−1

0

1

2

Time (sec.)

Vel

ocity

Err

or (

mm

/sec

.)

Figure 5.30: The velocity trajectories with ∆PA

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5x 10

5

Time (sec.)

Exter

nal lo

ad (N

)


Figure 5.31: The external load force with ∆PA

The sensor offset ∆PA is added at t ≈ 16.63 second and is detected in 0.2 second. Theoffset only caused a impulse-like impact on the residual rPA

. Then the residual signallay under the threshold. Nonetheless, the fault is exactly identified by the DMLPA

as shown in Fig. 5.28. Although the residual is over the threshold because of theinitial error at the begging, the DML make a right decision. The DML activated thefault accommodation algorithm. The parameter estimation of Fload is much bigger thanactual load. The sensor offset is viewed as load variation by the controller. The average


errors increases slightly compared to the fault free case with the values exc= 0.486 (mm)

and evc= 0.265 (mm/sec.).

The FTC is also tested in extreme situation, namely the sensor of PA has a total failure.

0 10 20 30 40 50 600

50

100

150

200

250

300

Time (sec.)

Posit

ion

(mm

)

xref

xact

0 10 20 30 40 50 60−2

−1.5

−1

−0.5

0

0.5

Time (sec.)

Posit

ion

Erro

r (m

m)

Figure 5.32: The position trajectories with total PA failure

0 10 20 30 40 50 600

5

10

15

Time (sec.)

Velo

city

(mm

/sec

.)

vref

vact

0 10 20 30 40 50 60−6

−4

−2

0

2

4

Time (sec.)

Velo

city

Erro

r (m

m/s

ec.)

Figure 5.33: The velocity trajectories with total PA failure

The tracking performance of position is still ensured. But the system has a smalloscillation in high speed area. The average errors are exc

= 0.601 (mm) and evc=

0.35 (mm/sec.). It is should be pointed out that detection and isolation the total failureis simply realized by checking the value of rPA as explained in chapter 3. The timerestrict is released in such situation.

5.2 Experiments 91

5.2.5 Conclusion

The all experimental results are summarized in the following table, where tf denotesthe fault occurrence time, td represents the detection time and PA× means senor of PA

has a total failure. Although the external load varied a lot. The robust fault detection

ControllerPI FTC FTC FTC FTC FTC

(no fault) (no fault) (QLin) (∆PB) (∆PA) (PA×)

exc(mm) 1.88 0.379 0.409 0.433 0.486 0.601

evc(mm/sec.) 1.27 0.265 0.268 0.283 0.265 0.35

tf (sec.) — — 10 9 16.63 21.29

td (sec.) — — 20 23 16.8 21.3

Table 5.1: The experimental results

and tracking control are realized by the presented FDI integrated FTC. The designedthresholds and the DML ensures zero false alarm in fault free case and correct faultidentification, if the considered fault occurs. The residual rxc

is unaffected by the allconsidered faults, which is in conformity with the analysis and simulations. The fault∆PA can be detected by the ad hoc signal checking of the residual. But it has higherrisk of missed detection or false alarm than the others. Backstepping based controllersshow good tracking performance and robustness. With the parameter adaption of theunknown external load the best results is obtained in fault free case. The control perfor-mance degrades just slightly, when the faults are present. Unfortunately, the estimationsof parameters are only bounded and have large deviations with true values. Althoughthe FDI for the QLin and ∆PB takes long time, the robustness of the controller can stillbe kept during the critical period. A theoretical robustness analysis of the system inthis time interval is already out of the range of this thesis.

6 Summary and outlook

6.1 Summary

This thesis proposed a model-based methodology of the fault diagnosis integrated fault-tolerant control (FTC) for a nonlinear electro-hydraulic system, whose main componentsinclude a servo-pump and hydraulic cylinders. The advantage of such a control scheme isthat the pre-specified control performance can be maintained or has only an acceptabledegradation with the presence of the considered fault. Meanwhile, the best controlperformance can be obtained in fault free case. The design procedure was divided intwo parts: 1) fault detection and isolation 2) design of a reconfigurable controller. Theconsidered faults were internal leakage QLin, sensor offsets ∆PA and ∆PB.

Firstly, the mathematical model of the electro-hydraulic testbed, which is the founda-tion of diagnosis and control, was derived based on the physical relations among thestates variables. The unknown parameters, such as parameters of transfer function ofthe servo pump or friction coefficient of the cylinder, were identified with the help ofpowerful toolbox of Matlab/Simulink. The established model was validated with themeasurements. A good consistency was achieved.

Based on the system model the differential-geometric method was employed for robustfault detection. The solution of fault detection and isolation in terms of differential-geometry is to find an unobservable distribution, which indicates that a system transfor-mation can be carried out for disturbance decoupling or isolation of other faults (sensorfault is not included). Also the transformed system is locally weakly observable. There-fore, it can be applied for decoupling the unknown external load. The construction ofthe unobservable distribution is realized by establishing the observability codistribution,which is the annihilator of the unobservable distribution. After such a transformation thenew system is totally robust to external load and takes variable velocity as an input. Thefault detection observer based on the new system model could detect all pre-specifiedfaults. The fault isolation was realized through a special decision making logic. Thesimulation study showed good results of fault diagnosis. The residuals were independentof the unknown disturbance. Every considered fault could be detected and isolated.The only problem was that the fault diagnosis would cause time delay for the controllerreconfiguration.

The backstepping based controller was applied for position tracking. The feedback con-trol law and associated Lyapunov function is designed by recursive approach. The system

93

94 6 Summary and outlook

states can be viewed as “virtual controls”. In fault free case the parameter adaption wasinserted for the unknown external load. Then the tracking error asymptotically con-verges to zero. As soon as the fault was identified, the backstepping controller can bereconstructed by parameter or model adaption. The fault QLin was handled as unknownparameter and could be tolerated by parameter adaption. Accommodation of sensorfault ∆PA and ∆PB was based on the model adaption. The faulty measurement wasreplaced by the approximation. Modeling of the faulty system was also simplified. ∆PA

affected adaptive controller more significantly than ∆PB. So the reconfigured controllerdropped the parameter adaption for unknown external load for preventing from possibleoscillation. In addition, numerical differential was applied for simplifying the calculationof derivatives of stabilizing functions. The saturations for the stabilizing functions wereintroduced according to the physical constraints of system states so that the values oferror terms of backstpping controller were limited in a reasonable range.

The proposed fault diagnosis system and backstepping based controller formed a faultdiagnosis integrated fault-tolerant control structure. It was successfully implemented inthe electro-hydraulic testbed. The on-line robust fault diagnosis and controller recon-figuration were realized. In fault free case the best control performance was obtained.The control performance had only a slight degradation. But the parameter estimationby adaptive controller was only bounded and had a poor accuracy. This also indicatesthat the parameter estimation of the adaptive can not be used as residual generator.

In conclusion the proposed methodology can perform excellent for precise control of theelectro-hydraulic testbed with guaranteed safety and reliability. The system variablesare monitored by the fault detection observers. It can be extend to control other electro-hydraulic systems, such as valve controlled hydraulic system.

6.2 Outlook

In future’s work it is necessary to analyze the robustness of the system during thecritical period. The adaption laws of the backstepping controller should be improved tomake a more precise estimation of parameters. Other observers should be tested, likeKalman-Filter, to enhance the sensitivity to the faults. The estimation of the amplitudeof the fault should be investigated to archive a complete monitoring for the system.More possible faults of the electro-hydraulic system should be studied and simulated,for example, the leakage in servo-valve or pulsation of the pump.

7 Kurzfassung

Die Grundlagen und Motivation

Hydraulische Antriebe kommen in vielfaltigen Anwendungen in der Stationar- und Mo-bilhydraulik zum Einsatz. Beispiele sind Pressen im Werkzeugmaschinenbau, Antrieb-ssysteme der Fahrzeuge oder Roboter. Im Vergleich zum elektrischen Antrieb ist diehohe Kraftdichte der großte Vorteil der Hydraulik. Um die Anforderungen der Au-tomatisierung zu erfullen, werden elektrische Komponenten in die hydraulischen Gerateintegriert. Solche Systeme werden elektrohydraulische Systeme genannt. Unter Verwen-dung elektrischer Mess- und Stellsignale konnen die elektrohydraulischen Antriebssys-teme geregelt und ein hoch praziser Antrieb erzielt werden. Elektrohydraulische Systemeweisen jedoch aufgrund komplexer Stromungseigenschaften, Reibungen in Aktuatorenund wechselnder Last ein inharent nichtlineares Verhalten auf. Um die gewunschte Per-formanz zu erzielen, sind moderne nichtlineare Regelungsmethoden notig.

Sicherheit und Zuverlassigkeit sind die wichtigsten Anforderungen an automatisierte Sys-teme. Schon ein kleiner Fehler kann in manchen Fallen den kompletten Systemausfallnach sich ziehen. Durch eine fruhzeitige Detektion des Fehlers, konnen entsprechendeMaßnahmen zur Kompensation des Fehlers ergriffen und der Systemausfall vermiedenwerden. Das Gebiet der Fehlerdiagnose war und ist deshalb Gegenstand vieler Forschung-svorhaben. Wird die redundante Information aus einem Systemmodell und den verfug-baren Messgroßen gewonnen, wird von der modellbasierten Fehlerdiagnose gesprochen.Sie umfasst Fehlerdetektion und Fehlerisolation (Fault detection and fault isolation(FDI)). Verglichen mit dem Einsatz redundanter Hardware ist die modellbasierte Fehler-diagnose vorteilhafter, da sie mehr Systeminformation liefert und weniger Hardwarebenotigt.

Eine Regelung, die auch nach dem Auftreten eines Fehlers ihre Funktion erfullt, heißtfehlertolerante Regelung (Fault tolerant control (FTC)). Die fehlertolerante Regelungbasiert auf einer Fehlerdiagnose. Der Regler reagiert auf die Fehler und wird ausgehendvon der Fehlerinformation automatisch rekonfiguiert. Auf diese Weise wird die Regelgutenicht wesentlich verschlechtert.

Die Anwendung der Fehlerdiagnose integrierte fehlertolerante Regelung auf die elek-trohydraulischen Systeme ist sehr bedeutend. Außer der Verbesserung der Sicherheitund Zuverlassigkeit werden die Wartungskosten auch wegen der Fehlerfruherkennungsinken. Und zwar das ganze System funktioniert intelligenter und effizienter. Die

95

96 7 Kurzfassung

Hauptschwierigkeit zu diesem Thema besteht in folgendes:

• hohe Nichtlinearitat und Unsicherheit des elektrohydraulischen Systems

• Robustheit des Fehlerdiagnosesystems

• prazise Regelung

• Rekonfiguration des Reglers im Fehlerfall

• echtzeitig Rechnen

Das Ziel der Arbeit

Motiviert von der obigen Betrachtung wird in dieser Arbeit eine anwendbare Strategiefur die Integration der modellbasierten Fehlerdiagnose in die fehlertolerante Regelungelektrohydraulischer Systeme entwickelt. Das Fehlerdiagnosesystem soll die Fehler trotzSystemunsicherheit detektieren und isolieren, die Regelung auch im Fehlerfall prazisefunktionieren.

Zusammenfassung der einzelnen Kapitel

¡¢ £ ¡¢

Figure 7.1: Das elektrohydraulische System

In Kapitel 2 wird das Modell der untersuchten elektrohydraulischen Anlage hergeleitet.

97

Der hydraulische Kreis, wie in Bild 7.1 gezeigt, besteht hauptsachlich aus einer Ser-vopumpe und zwei Zylindern, die jeweils als Arbeitszylinder und Lastzylinder benanntwerden.

Die Position des Arbeitszylinders soll geregelt werden. Die Modellierung basiert auf denphysikalischen Beziehungen der Systemzustandsvariablen und wird durch Vernachlassi-gen der unwichtigen Dynamik und unwichtiger Parameter vereinfacht. Parameteridenti-fikationen werden mit Hilfe einer leistungsfahigen Matlab/Simulink Toolbox durchge-fuhrt. Die auf das System wirkende wechselnde Last wird als unbekannte Storungmodelliert. Hierdurch wird deutlich, welche Ansatze der fehlertoleranten Regelung undFehlerdiagnose fur das System anwendbar sind.

Die kunstliche Realisierung der Fehler wird vorgestellt: eine innere Leckage als System-fehler und zwei Drucksensorfehler werden als haufig auftretende Fehler ausgewahlt. DasKapitel bildet die Basis fur die weitere Arbeit.

In Kapitel 3 wird zunachst ein Uberblick zur modelbasierten Fehlerdiagnose gegeben.Der differentialgeometrische Ansatz, der auf eine allgemeine Formulierung des FDI-Problems fuhrt sowie kompakter und transparenter als andere bekannte modellbasierteAnsatze ist, wird ausfuhrlich vorgestellt. Dieser Ansatz zeigt einen systematischen En-twurfprozess fur die Bestimmung einer Distribution, die als nicht beobachtbare Distribu-tion (unobservable distribution) definiert wird. Diese Distribution zeigt, wie der Einflussder wechselnden Last auf das fehlerbeeinflusste System mit Hilfe einer Systemtransfor-mation entkoppelt werden kann. Die Auswirkung des Fehlers wird hierdurch jedochnicht beeinflusst. Weiterhin ist das transformierte System im fehlerfreien Fall noch lokalschwach beobachtbar.

Der Fehlerdetektionsbeobachter wird basierend auf dem transformierten Systemmodelentworfen. Auf diese Weise sind die generierten Residuen unabhangig von der unbekan-nten Storung. Der Nachteil des Gebrauchs einer Systemstransformation liegt darin, dasseine Ausgangsgroße als Eingangsgroße fur das transformierte System behandelt werdenmuss. Im elektrohydraulischen System wird die Geschwindigkeit des Zylinders als Ein-gangsgroße betrachtet. Hierdurch ist das Positionssignal unabhangig vom betrachtetenFehler und dem Fehlerdiagnosesystem stehen nur zwei Residuen fur drei Fehler zur Ver-fugung. Um die Fehlerisolation zu ermoglichen, wird eine spezielle Entscheidungslogik(Decision Making Logic (DML)) entworfen. Die Entscheidungslogik zur Identifizierungder Fehler basiert auf einer Signalanalyse des einzelnen Residuums .

In Kaptitel 4 ist der Entwurf des rekonfigurierbaren Reglers fur FTC ausgefuhrt. ZumEliminieren des nichtlinearen Systemverhaltens wird ein Backstepping-Verfahren ver-wendet. Das Backstepping-Verfahren ist ein rekursiver Lyapunov-basierter Ansatz, derdas Regelgesetz und die zugehorige Lyapunov-Funktion schrittweise konstruiert. Furdas elektrohydraulische System wird ein adaptiver Backstepping-Regler verwendet, umso den Einfluss der unbekannten außeren Last im fehlerfreien Fall zu kompensieren. Eswird bewiesen, dass die Regelabweichung asymptotisch gegen Null konvergiert und derParameterschatzungsfehler im fehlerfreien System begrenzt ist. Um die Berechnung des

98 7 Kurzfassung

Reglers zu vereinfachen wird eine numerische Differentiation gefolgt von einem Tiefpass-Filter verwendet.

Das Auftreten des Systemfehlers, wird es als Parameterunsicherheit des Systems mod-elliert und durch weitere Parameteradaption kompensiert. Die Behandlung des Sensor-fehlers ist von der Systemstruktur abhangig. Der Sensorfehler wird durch den Reglerdas System beeinflusst. Normalerweise wird die fehlerhafte Messung ignoriert und durcheine Schatzung ersetzt. Fur das betrachte System ist es nicht moglich, die Beobacht-barkeit ohne Messung sicher zu stellen. In der Arbeit wird daher ein spezialer Ansatzentwickelt, der den Einfluss des Sensorfehlers auf den Regler kompensiert. Dies fuhrtdazu, dass nur ein vereinfachtes Modell verfugbar ist, wodurch der Sensorfehler nichtauf das System wirkt. Welche Konfiguration aktiviert werden soll, ist von der aktuellenFehlerinformation abhangig. Mit dem in Kapitel 3 prasentierten Fehlerdiagnosesystemwird eine fehlertolerante Regelungsstruktur fur das nichtlineare System aufgebaut. DieStruktur wird in Bild 7.2 gezeigt.

Figure 7.2: Die Struktur der prasentierten Methde

In Kapitel 5 wird das prasentierte Verfahren am realen Prufstand erprobt. Die prasen-tierte FDI mit integrierter FTC Strategie ermoglicht eine sehr gute Performanz mitgarantierter Sicherheit und Zuverlassigkeit. Die betrachten Fehler werden detektiert undrichtig isoliert. Die Regelgute im Fehlerfall zeigt keine wesentliche Verschlechterung. Einkritisches Problem ist die lange Fehlerdetektionszeit, wahrend der der Regler jedoch dasSystem stabilisiert. In weiteren Untersuchungen sollte daher eine theoretische Robus-theitsanalyse durchgefuhrt werden.

99

Ausblick

Die in dieser Arbeit entwickelte Methodik realisiert Fehlerdiagnose und fehlertoleranteRegelung am elektrohydraulischen System in Echtzeit. Sie bietet das Potential auf an-dere elektrohydraulische, z.B. ventilgeregelte Systeme erweitert zu werden. In weit-ergehenden Untersuchungen sollte die Fehlerdetektionsmethode verbessert werden, umso die Detektionszeit zu reduzieren. Die Robustheit wahrend der Fehlerdiagnosephasesollte analysiert werden. Außerdem sollten weitere Fehler, wie Leckagen in Ventilen oderPulsation der Pumpe simuliert und einbezogen werden.

A Background of geometric theory

Background of geometric theory is detailed described in [Isi95] and [Won85]. Some ofthem are repeated here.

A.1 Affected/unaffected

Affected/unaffected The jth output yj(t, x0, u1, . . . , uk) corresponding to an initial

condition x0 and to the set of input functions u1, . . . , uk is unaffected by (or invariantunder) the ith input ui if, for any initial condition x0 and any collection of admissibleinput functions u1, . . . , ui−1, ui+1, . . . , uk, there holds:

yj(t, x0, u1, . . . , ui−1, ua, ui+1, . . . , uk) = yj(t, x

0, u1, . . . , ui−1, ua, ui+1, . . . , uk)

for all t ≥ 0 and any pair ua, ub. The jth output yj(t, x0, u1, . . . , uk is said to be affected

by the ui if it is not unaffected by the ui.

A.2 Vector spaces and subspaces

Vector (linear) spaces consists of an additive group, of elements vectors, together withan underlying filed of scalars. Vector spaces are denoted by script capitals, such as Xand Y . Their elements are written as x and y. The dimension of the vector space X isdenoted by dim(X ).

A subspace M of the vector space X is a subset of X that is a vector space underthe operations of vector addition and scalar multiplication inherited from X : namely,M ⊂ X and for all x1, x2 ∈ M and c1, c2 ∈ R then c1x1 + c2x2 ∈ M. If the twosubspaces M,N ⊂ X , then M + N ⊂ X and M ∩ N ⊂ X . Here M + N can beunderstood as the smallest subspace containing both M and N , while M ∩ N is thelargest subspace contained in both M and N . M and N are linearly independent ifM ∩ N = 0. If dim(M) = d1 and dim(X ) = d2, then the difference of d2 and d1 isdefined as the codimension of M in X and can be denoted as:

codim(M) = dim(X ) − dim(M).

101

102 A Background of geometric theory

Let C : X → Y be a linear map. X is the domain of C and Y is the codomain. The nullspace (or kernel) of C is the subspace

KerC = x : x ∈ X & Cx = 0 ⊂ X ,

while the image (or range) of C is the subspace

ImC = Cx : x ∈ X ⊂ Y .

The map C : X → Y is an epimorphism (or epic) if ImC = Y . If C is epic there is aright inverse Cr : Y → X , such that CCr = IY .

The map C : X → Y is a monomorphism (or monic) if KerC = 0. If C is monic, thereis a left inverse Cl : Y → X , such that ClC = IX .

If C is both epic and monic, C is an isomorphism and this can happen only if dim(X ) =dim(Y). In this case Cl = Cr = C−1.

A.3 Dual spaces and annihilators

Let X be linear vector space over field F. The set of all linear functionals x′ : X → F isdenoted by X ′. X ′ becomes a linear space with the following definitions

(x′1 + x′2)x = x′1x+ x′2x; x′i ∈ X ′, x ∈ X

(ax′1)x = a(x′1x); x′1 ∈ X ′, x ∈ X , a ∈ F.

Then X ′ is called the dual space of X in F. The element of X ′ is named as covector.

The annihilator of M ⊂ X is usually denoted by M⊥ and is defined as

M⊥ = x′;x′M = 0, x′ ∈ X ′.

Therefor M⊥ ⊂ X ′.

A.4 Factor space

Let M ⊂ X . Call vectors x, y ∈ X equivalent mod M if x − y ∈ M.Tthe factor space(or quotient space) X /M is defined as the set of all equivalence classes:

x := y : y ∈ X , y − x ∈ M, x ∈ X .

For vector spaces the following equation is always true if the factor space exists.

dim(X /M) = dim(X ) − dim(M)

A.5 Invariant subspaces 103

A.5 Invariant subspaces

Let A : X → X and let M ⊂ X have the property AM ⊆ M is said to be A−invariant.Namely, an invariant subspace of this system is a linear subspace M ⊂ X with theproperty that every trajectory starting in M will stay in M in the future. Certainly Mitself, and the subspace 0, are trivially invariant subspaces.

For example, there is a linear system

x = Ax. (A.1)

where x ∈ Rn, A ∈ R

n×n. The solution of the differential equation (A.1) x starts in Mand ends in M. The subspace M ⊂ X is A− invariant.

A.6 Reachable subspace

Further if there is a linear control system

x = Ax+Bu (A.2)

y = Cx (A.3)

If x ∈ X ⊂ Rn, B = ImB, the reachable subspace is:

〈A|B〉 = B + AB + · · · + An−1B

〈A|B〉 also means minimal A-invariant subspace containing B.

A subspace R is called a reachability subspace if there are matrices F and G such that

〈(A+BF )|ImBG〉 .

A.7 Unobservable subspace

The concept of unobservable subspace can be viewed as dual to reachable subspace andis defined as

〈KerC|A〉 = Ker(C) ∩Ker(CA) ∩ · · · ∩Ker(CAn−1)

=n⋂

i=1

Ker(

CAi−1)

.

It also means maximal A-invariant subspace contained in KerC.

104 A Background of geometric theory

The system (A.2) and (A.3) is observable if and only if

n⋂

i=1

Ker(

CAi−1)

= 0

A subspace S is called unobservability space, if there exist a matrices L and H such thatS is the set of unobservable states of the system

x = (A+ LC)x (A.4)

y = HCx. (A.5)

A.8 Distribution and codistribution

A distribution on Let f1, f2, · · · , fk be smooth vector fields of open set D ⊂ Rn and

∆(x) = span f1(x), f2(x), · · · , fk(x)

be the k-dimensional subspace spanned by the vectors f1(x), f2(x), · · · , fk(x) at anyfixed point x ∈ D. The collection of all vector spaces ∆(x) for x ∈ D is called adistribution and is noted as

∆ = span f1, f2, · · · , fk .

The dimension of the distribition is

dim(∆) = rang [f1(x), f2(x), · · · , fk(x)] (A.6)

If ∆ defined on D ⊂ Rn is said to be nonsingular if there is a integer d such that

dim(∆) = d for ∀x ∈ D. A distribution dim(∆) is said to be involutive if two vectorfields f(·) and g(·) exist such that if ∀x ∈ D, f ∈ ∆, g ∈ ∆ then [f, g] ∈ ∆.

The codistribtuion is the dual of distribution. If the elements of Rn are column vectors,

then denote the dual space as Rn∗, whose elements are row vectors. Just as a vector

field on Rn is a smooth assignment of column vector f(x) to each point x ∈ R

n. Definea covector field to be a smooth assignment of covector ω(x) to each point x ∈ R

n. Thenthe definition of codistriubiton is

Ω = span ω1, ω2, · · · , ωk .

The annihilator of a distribution is its codistribution.

B Technical data of theelectro-hydraulic system

Servo-pump

rotation speed 1000 1/min

volume flow 28 L/min (max.)

power 11 kW

swashplate angle 17.7 (max.)

control cylinder stroke 17.72 mm (max.)

Table B.1: Mannesmann-Rexroth A10VSO 28 DFE1

Hydraulic cylinder

piston ø 200 mm

piston rod ø 140 mm

area ratio 1.96

piston rod length 1176 mm

piston rod stroke 500 mm (max.)

moving mass 166 kg

Table B.2: Mannesmann-Rexroth CD H3 MF3/200/140/500

105

106 B Technical data of the electro-hydraulic system

Absolute encoder

encoder capacity 24 bit (max.)

steps / revolution 12 bit (4096)

number of revolution 12 bit (4096)

resolution 20 µm

Table B.3: TR-Electronic CE-65-SSI

Proportional pressure relief valve

volume flow 30 L/min (max.)

operation pressure 315 Bar (max.)

Table B.4: Mannesmann-Rexroth DBE 62-1X/315

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Lebenslauf

Personliche Daten

Name: Liang Chen

Geburtsdatum: 24.08.1979

Geburtsort: Jiangsu, VR China

Familienstand: verheiratet

Staatsangehorigkeit: chinesisch

Studium

09/1998–05/2005 Masterstudium an University of Shanghai for Science and Technol-ogy, Shanghai, VR ChinaFachrichtung: Elektrotechnik/Automatisierungstechnik

01/2006–03/2010 Promotion am Lehrstuhl fur Regelungssysteme,TU Kaiserslautern, Kaiserslautern

Berufspraxis

07/2005–09/2005 Projektingenieur bei Kapolnek Engineering Technology (Shanghai)Co., Ltd.

01/2006–12/2009 wissenschaftlicher Mitarbeiter am Lehrstuhl fur Regelungssysteme,TU Kaiserslautern

Kaiserslautern, June 14, 2010

115

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