HIGHER-ORDER SLIDING MODE TECHNIQUES FOR FAULT DIAGNOSIS · HIGHER-ORDER SLIDING MODE TECHNIQUES...

Ph.D. in Electronic and Computer EngineeringDept. of Electrical and Electronic Engineering

University of Cagliari

HIGHER-ORDER SLIDING MODETECHNIQUES FOR FAULT

DIAGNOSIS

Nicola Orani

Advisor: Elio UsaiCurriculum: ING-INF/04 Automatica

XXII CycleMarch 2010

Ph.D. in Electronic and Computer EngineeringDept. of Electrical and Electronic Engineering

University of Cagliari

HIGHER-ORDER SLIDING MODETECHNIQUES FOR FAULT

DIAGNOSIS

Nicola Orani

Advisor: Elio UsaiCurriculum: ING-INF/04 Automatica

XXII CycleMarch 2010

Dedicated to my family

Acknowledgments

Special thanks to my supervisor Elio Usai.

Also, i would like to thank Alessandro Pisano and Giorgio Bartolini for their indispens-able collaboration.

Finally, special thanks to my wife Valentina and to my children Federica and Gabriele,both born during my PhD studies, that supported, tolerated and motivated me duringthese three years. Tanks again.

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Contents

1 FDI-Generalities 31.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Fault diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Modelling of faulty systems . . . . . . . . . . . . . . . . . . . . . . . . 71.4 (Process) Model-Based Fault Detection Methods . . . . . . . . . . . . . . . . 111.5 Residual generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5.1 Parameter Estimation Techniques . . . . . . . . . . . . . . . . . . . . 151.5.2 Parity equations techniques . . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 State estimation and observer based techniques . . . . . . . . . . . . . . . . 221.6.1 Basic fault detection observer-based schemes . . . . . . . . . . . . . 241.6.2 Unknown Input Observer for Fault Detection . . . . . . . . . . . . . 261.6.3 FDI schemes based on UIO and output observers . . . . . . . . . . . 29

2 Structural fault detectability, isolability and identifiability 352.1 Structural fault detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Structural fault isolability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Structural fault identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Sliding Mode Observers 433.1 SMO for Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 SMO for linear systems partially driven by unknown inputs . . . . . . . . . 47

3.2.1 A classical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.2 HOSMO to weaken the 1-relative degree condition . . . . . . . . . . 53

3.3 Non-linear approaches to Sliding Mode Observers design . . . . . . . . . . 583.3.1 Systems in the companion form . . . . . . . . . . . . . . . . . . . . . 593.3.2 Triangular systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.3 Quasi-continuous HOSM observers . . . . . . . . . . . . . . . . . . . 633.3.4 Algebraic observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Sliding Modes for FDI 694.1 SMO for faults reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.1 SMO for reconstructing of the input fault signals . . . . . . . . . . . 734.1.2 SMO for reconstructing of the output fault signals . . . . . . . . . . 74

4.2 Filtration-free fault reconstruction via full order HOSMO . . . . . . . . . . . 75

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iv CONTENTS

4.2.1 Actuator faults reconstruction . . . . . . . . . . . . . . . . . . . . . . 814.3 Filtration-free fault reconstruction via reduced order HOSMO . . . . . . . . 82

4.3.1 Fault observer design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3.2 Actuator fault reconstruction . . . . . . . . . . . . . . . . . . . . . . . 844.3.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Discrete state reconstruction via HOSMO 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2.2 Comments on the considered class of systems . . . . . . . . . . . . . 91

5.3 Discrete state observer design . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.1 Observer input design . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.2 Discrete state reconstruction . . . . . . . . . . . . . . . . . . . . . . . 95

6 Application results 976.1 Three-tank system case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.1.1 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2.1 Fault reconstruction via HOSM . . . . . . . . . . . . . . . . . . . . . . 1026.2.2 Discrete state estimation via HOSM . . . . . . . . . . . . . . . . . . . 107

6.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.3.1 System identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.3.2 Fault reconstruction via HOSM . . . . . . . . . . . . . . . . . . . . . . 1146.3.3 Discrete state estimation via HOSM . . . . . . . . . . . . . . . . . . . 120

7 Concluding remarks 123

Bibliography 125

List of Figures

1.1 The fault topology in the controlled system . . . . . . . . . . . . . . . . . . . . . 81.2 The monitored system and fault topology . . . . . . . . . . . . . . . . . . . . . . 81.3 Time-dependency of faults : a)abrupt; b)incipient; c)intermittent . . . . . . . . 91.4 Linear input output model and faults . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Basic model faults: additive(a); multiplicative (b) . . . . . . . . . . . . . . . . . . 91.6 General scheme for analytical fault-detection and diagnosis method . . . . . . 121.7 Residual generation general structure . . . . . . . . . . . . . . . . . . . . . . . . 131.8 Residual generation via output estimator . . . . . . . . . . . . . . . . . . . . . . 141.9 Model structure for parameter estimation with equation error . . . . . . . . . . 161.10 Model structure for parameter estimation with output error . . . . . . . . . . . 171.11 Scheme for Output error via parity equation method . . . . . . . . . . . . . . . 181.12 The referred system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.13 Scheme for equation error via parity equation method . . . . . . . . . . . . . . 191.14 Parity equation method for a MIMO state-space model; Differentiator filter . . 201.15 Process and state observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.16 Multivariable process with disturbance v(t ) , w(t ) and faults fu(t ), fy (t ) . . . . 231.17 Process and output observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.18 The UIO Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.19 Bank of estimators for output residual generation (DOS) . . . . . . . . . . . . . 301.20 The GOS scheme for FDI of system inputs . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Utkin Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1 The state variable and the state observation error. . . . . . . . . . . . . . . . . . 864.2 The actual fault and the reconstructed signal fault. . . . . . . . . . . . . . . . . 87

6.1 Configuration of the considered three-tank system . . . . . . . . . . . . . . . . . 986.2 System input-output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3 Shape of tank 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.4 Shape of tank 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.5 Shape of tank 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.6 Fault topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.7 System inputs, outputs and disturbances . . . . . . . . . . . . . . . . . . . . . . 1026.8 Test 1. Faults and disturbance reconstruction performance . . . . . . . . . . . . 1056.9 Test 2. Reconstructed faults and disturbance in presence of noisy measures . . 1066.10 Test 1. δi (t ) vs. δi (t ). From top to bottom: i = 1,2,3. . . . . . . . . . . . . . . . . 109

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vi LIST OF FIGURES

6.11 Test 1. δi (t ) vs. δi (t ). From top to bottom: i = 1,2,3. . . . . . . . . . . . . . . . . 1096.12 Test 1. Actual σ(t ) and reconstructed σ(t ) . . . . . . . . . . . . . . . . . . . . . . 1106.13 Test 2. σ(t ) (solid) and σ(t ) (dashed) varying observer gain . . . . . . . . . . . . 1106.14 Test 3. Actual δi (t ) vs. δi (t ) discrete inputs . . . . . . . . . . . . . . . . . . . . . . 1116.15 Test 3. Actual σ(t ) vs. σ(t ) discrete states . . . . . . . . . . . . . . . . . . . . . . . 1116.16 The experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.17 Principle of parameters identification . . . . . . . . . . . . . . . . . . . . . . . . 1136.18 Test 1- Reference signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.19 Test 1- Measured signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.20 Test 2- Reference signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.21 Test 2- Measured signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.22 Test 1- Reconstructed faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.23 Test 2- Reconstructed faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.24 Actual δ1(t ) and non-thresholded reconstructed δ1(t ) discrete inputs . . . . . 1206.25 δ1(t ) (solid) and δ1(t ) (dashed) varying observer gain . . . . . . . . . . . . . . . 1216.26 δi (t ) vs δi (t ). From top to bottom: i = 1,2,3. . . . . . . . . . . . . . . . . . . . . . 1216.27 Actual σ(t ) and reconstructed σ(t ) discrete state . . . . . . . . . . . . . . . . . . 122

List of Tables

1.1 Fault Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.1 Parameters of the Three Tank System . . . . . . . . . . . . . . . . . . . . . . . . . 101

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Summary

The field of research inherent this thesis is that of "Sliding Modes (SM) techniques forModel Based Fault Diagnosis and Identification (FDI) for dynamic systems". In generalthis topic is of great importance and interest in industry and engineering, mainly for eco-nomic, security and reliability reasons, in recent decades, there was a significant devel-opment in science and a wide range of theoretical approaches have been formulated andare therefore available in literature. Many contributions to FDI are based on the principleof analytical redundancy, which is essentially based on using a model of the system un-der observation in order to estimate the evolution of the corresponding real variables. Ifa fault occurs it causes a discrepancy or "residual" between the actual behavior and themodel, which is then used to identify faults. A very important property that FDI systemsshould have is the rejection of false alarms that could occur in the presence of model mis-match or measurement noises and therefore discern between residues generated by faultsand those due to disturbances. To this purpose it is important to use "robust techniques"to overcome the problems due to measurement noises and uncertainties.

In general sliding modes theory has good robustness properties and it can be used todeal with both structural and unmatched uncertainties, so that the application of slidingmode techniques to robust FDI offers good potential.

First of all a careful analysis of the literature was made to assess the state of the artabout "model based Fault Diagnosis and Identification", with particular attention on avail-able Sliding Modes techniques for Model Based FDI. We proceeded first to examine thetechniques currently available in the literature and at the same time, deepen and improvesome methodologies SM Observer-Based for Nonlinear Uncertain Systems. It must behighlighted that dissimilar to initial approaches aimed to design SM observers like resid-ual generators, for which in the presence of a fault the sliding motion was destroyed, theunderlying philosophy of the mayor promising techniques available in literature providefor the design of SMOs in which the sliding motion is maintained even in the presenceof faults, which are then detected and identified by analyzing the so-called equivalentoutput injection. In this regard, it must be highlighted that, the major contributions ofmy research activities involving FDI concerns the use of HOSMO (Higher Order SlidingMode Observers). The use of continuous time output injection signals, in order to circum-vents some limitations resulting from the use of a conventional first order SM observers,provide for finite time reconstruction, theoretically exact, of additive actuator faults fornonlinear uncertain systems. Furthermore, based on the assumption that it is always pos-sible to associate a specific nonlinear dynamic to a system affected by specific faults, anHOSMO for the identification of the actual dynamic, and therefore of the fault signature,

1

2 LIST OF TABLES

for nonlinear uncertain switched system has been proposed.The studied techniques have been extensively verified by using a the three tank water

process experimental set up.It is worth to remark that the problems studied in the present thesis are related to the

topics of the ongoing FP7 project “PRODI - Power plants Robustification based On faultDetection and Isolation algorithms" which supported part of the work made during myPhD course.

Chapter 1

FDI-Generalities

1.1 Nomenclature

By going through the literature, one recognizes immediately that the terminology in thefield of the FDI (Fault Detection and Identification) is not consistent. This makes it diffi-cult to understand the goals of the contributions and to compare the different approaches.For example, which are the differences between faults, failures, malfunctioning and er-rors? Or the differences between fault (or failure) detection, isolation, identification anddiagnosis? Or supervision and monitoring? Or supervisory functions and supervisorycontrol? The SAFEPROCESS Technical Committee discussed this matter, and tried to findcommonly accepted definitions. Below, some definitions are suggested, based on the dis-cussions within the Committee.

States and Signals

Fault An unpermitted deviation of at least one characteristic property or parameter ofthe system from the acceptable, usual or standard condition.

Failure A permanent interruption of a system’s ability to perform a required function un-der specified operating conditions.

Malfunction An intermittent irregularity in the fulfillment of a system’s desired function.

Error A deviation between a measured or computed value of an output variable and itstrue or theoretically correct one.

Disturbance An unknown and uncontrolled input acting on a system.

Perturbation An input acting on a system, which results in a temporary departure fromthe current state.

Residual A fault indicator, based on a deviation between measurements and model-equation-based computations.

Symptom A change of an observable quantity from normal behavior.

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4 CHAPTER 1. FDI-GENERALITIES

Functions

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

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

Fault identification Determination of the size and time-variant behavior of a fault. Itfollows fault isolation.

Fault diagnosis Determination of the kind, size, location and time of detection of a fault.It follows fault detection. It includes fault detection and identification.

Monitoring A continuous real-time task to determining the conditions of a physical sys-tem, by recording information, recognizing and indicating anomalies in the behav-ior.

Supervision Monitoring a physical and taking appropriate actions to maintain the oper-ation in the case of fault.

Protection Means by which a potentially dangerous behavior of the system is suppressedif possible, or means by which the consequences of a dangerous behaviour areavoided.

Models

Quantitative model Use of static and dynamic relations among system variables and pa-rameters in order to describe a system’s behaviour in quantitative mathematicalterms.

Qualitative model Use of static and dynamic relations among system variables in orderto describe a system’s behaviour in qualitative terms such as causalities and IF-THEN rules.

Diagnostic model A set of static or dynamic relations which link specific input variables,the symptoms, to specific output variables, the faults.

Analytical redundancy Use of more (not necessarily identical) ways to determine a vari-able, where one way uses a mathematical process model in analytical form.

System properties

Reliability Ability of a system to perform a required function under stated conditions,within a given scope, during a given period of time. Measure: MTBF = Mean TimeBetween Failures.

Safety Ability of a system not to cause danger to persons or equipment or the environ-ment.

Availability Probability that a system or equipment will operate satisfactorily and effec-tively at any point of time. Measure: A=MTBF/(MTBF+MTTR) where MTTR = MeanTime To Repair.

1.2. INTRODUCTION 5

Dependability A form of availability that has the property of always being available whenrequired. It is the degree to which a system is operable and capable of performing itsrequired function at any randomly chosen time during its specified operating time.

Time dependency of faults

Abrupt fault Fault modeled as stepwise function. It represents bias in the monitored sig-nal.

Incipient fault Fault modeled by using ramp signals. It represents drift of the monitoredsignal.

Intermittent fault Combination of impulses with different amplitudes.

Fault terminology

Additive fault Influences a variable by an addition of the fault itself. They may represent,e.g., offsets of sensors.

Multiplicative fault Are represented by the product of a variable with the fault itself. Theycan appear as parameter changes within a process.

1.2 Introduction

For the improvement of reliability, safety and efficiency advanced methods of supervi-sion, fault-detection and fault diagnosis become increasingly important for many tech-nical processes. This holds especially for safety related to power plants and chemicalplants. In active FTC (Fault Tolerant Control), FDI plays a vital role to provide informa-tion on faults/failures in the system and to enable appropriate reconfiguration to takeplace. Therefore the main function of FDI is to detect a fault or failure and to find its lo-cation so that corrective action can be made to eliminate or minimize the effect on theoverall system performance. The classical approaches are limit or trend checking of somemeasurable output variables based on hardware or physical redundancy methods whichuse multiple sensors, actuators, components to measure and control a particular variable.Typically, a voting technique is applied to the hardware redundant system to decide if afault has occurred and its location among all the redundant system components. The ma-jor problems encountered with hardware redundancy are the extra equipment and main-tenance cost, as well as the additional space required to accommodate the equipment.Furthermore they do not give a deeper insight into the process behavior and usually donot allow for an effective prior fault diagnosis. In view of the conflict between reliabilityand the cost of adding more hardware, it is possible to use the dissimilar measured valuestogether to cross-compare each other, rather than replicating each hardware individually.This is the meaning of analytical or functional redundancy. In the analytical redundancyscheme, the resulting difference generated from the comparison of different variables iscalled a residual or symptom signal. In brief the residual should be zero when the systemis in normal operation and should be different from zero when a fault has occurred. Thisproperty of the residual is used to determine whether or not faults have occurred. Con-sistency checking in analytical redundancy is normally achieved through a comparison


between measured signals and their estimated values. The estimation is generated by amathematical model of the considered plant. The comparison is done using the resid-ual quantities which are computed as differences between the measured signals and thecorresponding signals generated by the mathematical model.

Classical approaches, consisting in limit or trend checking of some measurable outputvariables, do not give a deeper insight into the process since they are only able to reactafter a relatively large change of a feature occurs, i.e., after either a large sudden fault or along-lasting gradually increasing fault. In addition, an in-depth fault diagnosis is usuallynot possible. Therefore advanced methods of supervision and fault diagnosis are neededwhich satisfy the following requirements:

1. Early detection of small faults with abrupt or incipient time behavior;

2. Diagnosis of faults in the actuator, process components or sensors;

3. Detection of faults in closed loops;

4. Supervision of processes in transient states.

To this purpose, model-based methods of FDI were developed by using input and out-put signals together with dynamic process models. Various methods are proposed in theliterature.In the following section the main tasks of fault detection and fault diagnosis are intro-duced, some basic problems and methods in supervision, fault detection and fault diag-nosis are considered. Later model-based fault-detection methods are analyzed, whichallow a deep insight into the process behavior.

1.3 Fault diagnosis

The task of fault diagnosis consists of determining the type, size and location of the faultas well as its time of detection based on the observed analytical and heuristic symptoms.If no further knowledge on fault symptom causalities is available, classification methodscan be applied which allow for a mapping of symptom vectors into fault vectors. To thisend, methods like statistical and geometrical classification or neural nets and fuzzy clus-tering can be used. Note that geometrical analysis, whilst simple to implement, has afew drawbacks. The most serious is that, in the presence of noise, input variations andchange of operating point of the monitored process, false alarms are possible. If a-prioriknowledge of fault-symptom causalities is available, e.g. in the form of causal networks,diagnostic reasoning strategies can be applied. Forward and backward chaining, withBoolean algebra for binary facts and with approximate reasoning for probabilistic or pos-sibilistic facts, are examples. Finally a fault decision indicates the type, size and locationof the most possible fault, as well as its time of detection.

In the following we focalized our attention on the techniques based on the dynamicmathematical process model of the system under supervision, i.e. the model-based fault-detection methods are considered, which allow a deep insight the process behavior.

1.3. FAULT DIAGNOSIS 7

1.3.1 Modelling of faulty systems

The first step in FDI model-based approach consists of providing a mathematical descrip-tion of the system under investigation which shows all the possible faulty conditions,as well. The detailed scheme for FDI techniques here presented is depicted by Figure1.4. The main components are the Plant under investigation, the Actuators and Sensors,which can be further sub-divided as input and output sensors, and finally the controller.In the following, the system working conditions will be monitored by means of its inputu(t ) and output y(t ) measurements and signals from the controller uR which are sup-posed completely available for FDI purposes.

It is worth noting that also the behavior of any controller that drives the system canbe taken into consideration for FDI purpose. Concerning the occurrence of malfunctions,the location of faults and their modelling, the system under diagnosis can be separatedinto the following different parts which can be affected by faults:

• Actuators

• Process or system components

• Input sensors

• Output sensors

• Controller

Typical examples of such faults are: structural defects, such as cracks, ruptures, frac-tures, leaks, and loose parts; defects in the gears, and aging effects; faults in sensors, suchas scaling errors, hysteresis, drift, dead zones, shortcuts, and contact failures; abnormalparameter variations; external obstacles, such as collisions and clogging of outflows.

This fault scenario can be summarized by the Figure (1.1).Figure (1.1) also shows the situation where the controller can be affected by faults,

since the monitored process consists of a closed-loop system. However, because of tech-nological reasons (e.g., the control action is performed by a digital computer), when theactuator is considered as a part or a component of the whole controller device, the formercan be treated as subsystem where faults are likelier to occur whilst the latter remainsfree from faults. Under these assumptions, as depicted in Figure (1.2) when system isconsidered in view of fault location, since input and output measurements are supposedcompletely available for FDI purposes, the controller behavior in the design of a fault diag-nosis scheme can be neglected as well as the interconnection between the control systemand the process.

A fault is defined as an unpermitted deviation of at least one characteristic propertyof a variable from an acceptable behavior. Therefore, the fault is a state that may lead to amalfunction or failure of the system. The time dependency of faults can be distinguished,as shown in Fig (1.3), abrupt fault (stepwise), incipient fault (driftlike), intermittent fault.

Now lumped-parameter processes are considered, which operate in open loop. Thestatic behavior (steady states) is frequently expressed by a non-linear characteristic. Byconsidering small signal deviations around an operating point (Y00/U00) the input/outputbehavior of a SISO process can frequently be described by ordinary linear differentialequations


Figure 1.1: The fault topology in the controlled system

Figure 1.2: The monitored system and fault topology

y(t )+a1 y (1)(t )+·· ·+an y (n)(t ) = b0u(t )+b1u(1)(t )+·· ·+bmu(m)(t ) (1.1)

where y i represent the i-th time derivative and y(t ) = Y (t )−Y00;u(t ) =U (t )−U00. Thecorresponding transfer function becomes, through Laplace transformation:

GP (s) =Gyu(s) = y(s)

u(s)= b0 +b1s +·· ·+bm sm

1+a1s +·· ·+an sn(1.2)

In Fig. (1.4) has been denoted how different faults can affect the nominal system,where input signal faults fu and output signal faults fy , are additive faults and ∆ai ,∆bi

represent change in model parameters.Must be highlight that, regard to the process models, as we’ll see with more detail, the

faults can be further classified. Additive faults appear, e.g., as offsets of sensors, whereasmultiplicative faults are parameter changes within a process. According to Fig. (1.5) addi-

1.3. FAULT DIAGNOSIS 9

Figure 1.3: Time-dependency of faults : a)abrupt; b)incipient; c)intermittent

Figure 1.4: Linear input output model and faults

tive faults influence a variable Y by an addition of the fault f, and multiplicative faults bythe product of another variable U with f .

Figure 1.5: Basic model faults: additive(a); multiplicative (b)

Under the hypothesis of linearity, process dynamics can be described by the followingcontinuous time, time-invariant, linear dynamic system (LTI) in the state-space form

x = Ax(t )+Bu(t )y =C x(t )+Du(t )

(1.3)

where x ∈Rn , u ∈Rp , y ∈Rm and with the corresponding transfer function matrix

Gyu(s) = y(s)

u(s)=C (sI − A)−1B +D (1.4)

There exist a number of ways to model faults, among them the extension of model 1.4 to

y(s) =Gyu(s)u(s)+Gy f (s) f (s) (1.5)


is a widely used one, where f ∈Rq is an unknown vector that represents all possible faultsand will be zero in the fault-free case. In this work f is assumed to be deterministic. Sup-pose that a minimal state-space realization of 1.5 is given by

x = Ax(t )+Bu(t )+E f (t )y =C x(t )+Du(t )+F f (t )

(1.6)

with known matrices E ,F . Then we have

Gyu(s) = y(s)

u(s) =C (sI − A)−1B +D

Gy f (s) = y(s)f (s) =C (sI − A)−1E +F

(1.7)

It becomes evident that the fault distribution matrices E ,F indicate where a fault occurand its influence on the system components. A shown in figure Fig. (1.2) we divide thefault in three categories:

• sensor faults fs : these faults directly act on the process measurement

• actuator faults fa : these faults cause changes in the actuator response

• process faults fp : they are used to indicated malfunctions within the process

It’s straightforward to note that sensor faults can be modeled by setting F = I , i.e, y =C x(t )+Du(t )+ fS(t ), while actuators faults by setting E = B , F = D , i.e, x = Ax(t )+Bu(t )+B f A(t ); y =C x(t )+Du(t )+D f A(t ). For systems affected by actuator, sensor and processfaults, we define

f =

f A

fp

fS

, E = [B EP 0], F = [D FP I ] (1.8)

for some EP and FP and apply 1.6 to represent the system dynamics. Due to the wayhow they affect the system dynamic, the faults modeled in 1.6 are called additive faults.It’s very important to note that the occurrence of an additive fault will not affect the systemstability, independently from the fact that a feedback control loop is integrated into thesystem under observation or not. Typical additive faults met in practice are, for instance,sensors and actuators offsets, described by a constant, or drift in sensors. The formercan be described by a constant, while the latter by a ramp. In practice, malfunction inthe process or in the sensors and actuators often cause changes in the model parameters.They are called multiplicative faults and generally modeled in term of parameter change.They can be described by extending 1.6 to

x = (A+∆A)x(t )+ (B +∆B)u(t )y = (C +∆C )x(t )+ (D +∆D)u(t )

(1.9)

where ∆A,∆B ,∆C ,∆D represent the multiplicative faults in the plant, actuators andsensors, respectively. It is assumed that

∆A =∑l Ai=1 AiθAi ∆B =∑lB

i=1 BiθBi

∆C =∑lCi=1 CiθCi ∆D =∑lD

i=1 DiθDi

(1.10)

1.4. (PROCESS) MODEL-BASED FAULT DETECTION METHODS 11

where Ai ,Bi ,Ci ,Di are matrices of known and of appropriate dimensions, and θAi ,θBi ,θCi ,θDi are unknown time functions. Multiplicative faults are characterized by their pos-sible direct influence on the system stability. This fact is evident for the faults describedby ∆A. It must be highlighted that also multiplicative faults could be modeled as additivefaults, in such case the new fault vector is a function of the state and input variable of thesystem and thus will affect the system stability. In any case, the major focus of this workwill be on the diagnosis of additive faults.

Considering additive and multiplicative faults, and taking into account that environ-mental disturbance, uncertainty or a mismatch between the nominal plant and the dam-aged plant as well as measurement and process noises are often modeled as unknown in-put vectors, we’ll denote them by d ,η,ν and integrate them into input-output state spacemodels 1.9 and 1.6 as follows

x = Ax(t )+ Bu(t )+E f +Ed d +ηy = C x(t )+ Du(t )+F f +Fd d +ν (1.11)

where A = A+∆A, B = B+∆B ,C =C +∆C ,D = D+∆D , d ∈RKD represent a determinis-tic unknown input vector, and η ∈Rn ,ν ∈Rm represents a steady stochastic process whichis assumed to be, if no additional remarks is made, a white, normal distributed noise vec-tor with zero mean.

1.4 (Process) Model-Based Fault Detection Meth-ods

Different approaches to FDI using mathematical models have been developed in the last30 years, see, e.g., [41], [42],[43],[46] ,[63],[52],[58],[62],[61],[68],...According to the nomenclature explained in previous paragraph, model-based FDI can bedefined as the detection, isolation and identification of faults The task consists of the diag-nosis of faults in the processes, actuators and sensors by using the dependencies betweendifferent measurable signals, i.e., using the priori information on the process. These de-pendencies are expressed by mathematical process models.

Figure 1.6 shows the basic structure of model-based FDI. Based on measured inputsignals U and output signals Y , the detection methods generate residuals r, parameterestimates or state estimates x, which are called features. By comparison with the normalfeatures (nominal values), changes of features are detected, leading to analytical symp-toms s. In FDI model-based techniques "symptoms" and "residual" terms are treated assynonymous, and in the following we shall refer to residual term. In Fig. (1.6) two mainblocks can be recognized:

1. Residual generation:First this block generates feature signals (parameters, state estimation or directlyresidual if a parity equation method is used to detect the feature) using availableinputs and outputs from the monitored system. This feature of fault, in conjunctionof normal behavior, should indicate that a fault has occurred. The residual shouldnormally be zero or close to zero under no fault condition, whilst distinguishablydifferent from zero when a fault occurs.


Figure 1.6: General scheme for analytical fault-detection and diagnosis method

2. Residual evaluation:This block examines symptoms for the likelihood of faults and a decision rule is thenapplied to determine if any faults have occurred. The residual evaluation block mayperform a simple threshold test (geometrical methods) on the instantaneous valuesor moving averages of the residuals. On the other hand, it may consist of statisti-cal methods, e.g., generalized likelihood ratio testing or sequential probability ratiotesting

Most contributions in the field of quantitative model-based FDI focus on the resid-ual generation problem, since the decision-making problem can be considered relativelystraightforward if residuals are well-designed.

The generation of residual (i.e. symptoms) is the main issue in model-based fault diag-nosis. A variety of methods are available in the literature for residual generation and thissection presents briefly some of the most common methods. The following basic processmodel-based fault detection schemes will be considered and summarized:

1. Observers-based approach (Output Observers, State Observers, estimators, filters);

2. Parity equations;

3. Identification and parameter estimation

An important aspect of these methods is the kind of fault to be detected. As noted inthe following, one can distinguish between additive faults, which influence the variablesof the process by a summation, and multiplicative faults, which are products of the pro-cess variables. The basic methods show different results, depending on these types offaults.

If only output signals y(t) can be measured, signal based methods can be applied, e.g.vibrations can be detected, which are related to rotating machinery or electrical circuits.Typical signal model-based methods of fault detection are:

1.5. RESIDUAL GENERATION 13

1. Bandpass Filters;

2. Spectral analysis (FFT);

3. Maximum-entropy estimation;

The characteristic quantities or features generated by fault detection methods showstochastic behavior with mean values and variances. Deviations from the normal behav-ior must then be detected by methods of change detection like:

1. Mean and variance estimation;

2. Likelihood-ratio test;

3. Run-sum test;

As previously observed each basic method offers different potentiality in term of capa-bility of fault detection. This is due to different aspects that in following shall be consid-ered.

1.5 Residual generation

The most frequently used FDI methods exploit the a priori knowledge of the characteris-tics of certain signals. As an example, the spectrum, the dynamic range of the signal andits variations may be checked. However, the necessity of a priori information concern-ing the monitored signals and the dependence of the signal characteristics on unknownworking conditions of the system under diagnosis are the main drawbacks of such a classof methods. The most significant contribution in modern model-based approaches is theintroduction of the symptom or residual signals, which depend on faults and are indepen-dent of system operating states. They represent the inconsistency between the actual sys-tem measurements and the corresponding signals of the mathematical model. The residualgenerator block can be interpreted as illustrated in Figure (1.7) [45].

Figure 1.7: Residual generation general structure


In the above structure, the auxiliary redundant signal z(t ) is generated by the function(possibly dynamical) W1(u(·), y(·)) and, together with the measurement y(t ), the symp-tom signal r (t ) is computed by means of W2(z(·), y(·)).

When a fault occurs in the plant, the residual r (t ) will be different from zero. The sim-plest residual generator is depicted in Fig (1.8) and it is obtained when the system W1 is amodel z(t ) = W1(u(·)) of the plant or it is an input-output description for the actual pro-cess obtained from system identification procedure (e.g., an Auto Regressive eXogenous(ARX) model). In the former case, the measurement y(t ) is not required in W1 becauseit is a system simulator. The signal z(t ) represents the simulated output and the residualis computed as r (t ) = y(t )− z(t ). Since it is an open-loop system, the process simulationmay become unstable.

Figure 1.8: Residual generation via output estimator

An extension to the model-based residual generation is to replace W1(u(·)) by W1(u(·),y(·)), i.e. an output estimator fed by both system input and output. In such a case, func-tion W1 generates an estimation of a linear function of the output W1(u(·), y(·)) = M y(t ),whilst function W2 can be defined as W2(z(·), y(·)) =W (z(t )−M y(t )), W being a weightingmatrix. Concluding, no matter which type of method is used, the residual generation pro-cess is nothing but a linear mapping whose inputs consist of process inputs and outputs.According to the definition, ideally, the residual signal r(t) has to be designed to becomezero for fault-free case and different from zero in case of failures. This means that

r (t ) = 0 if and only if f (t ) = 0 (1.12)

However, modelling errors and disturbances are inevitable. In general, both faultsand uncertainty affect the residual, and discrimination between these two effects is diffi-cult. Aspect inherent to the robustness of residuals has to be considerate. To overcomefalse alarm the simplest and most widely used way to fault detection, after generatingthe residual, is achieved by directly comparing residual signal r (t ) or a residual functionJ (r (t )) with a fixed threshold ε or a threshold function ε(t ) as follows

J (r (t )) ≤ ε(t ) f or f (t ) = 0J (r (t )) ≥ ε(t ) f or f (t ) 6= 0

(1.13)

where f (t ) is the general fault vector. If the residual exceeds the threshold, a fault maybe occurred. This test works well especially with fixed thresholds if the process operates


approximately in a steady state and it reacts after relatively large feature, i.e. after either alarge sudden or a long-lasting gradually increasing fault. Clearly the residual signal shouldbe near zero for the fault-free case, with consequent small thresholds to improve the sen-sitivity of fault detection, and should increase significantly when a fault appears in thesystem. On the other hand, adaptive thresholds ε(t ) can be exploited which depend onplant operating conditions, for example when ε(t ) is expressed as a function of plant in-puts.

The generation of symptoms is the main issue in model-based fault diagnosis. A varietyof methods are available in the literature for residual generation and this section presentsbriefly some of the most common methods. Most of the residual generation techniques arebased on both continuous and discrete system models [46], [63].

1.5.1 Parameter Estimation Techniques

In most practical cases, the process parameters are not known at all, or they are not knownexactly enough. Then, they can be determined by means of parameter estimation meth-ods, measuring input and output signals, u(t ) and y(t ), if the basic structure of the modelis known [63]. This approach is based on the assumption that the faults are reflected in thephysical system parameters and the basic idea is that the parameters of the actual processare estimated on-line using well-known parameter estimations methods. The results arethus compared with the parameters of the reference model, obtained initially under fault-free assumptions. Any discrepancy can indicate that a fault may have occurred. Now wecompare two different approaches for modelling the input-output behavior of the moni-tored system:

1. Equation Error Methods

2. Output Error Methods

Equation Error Methods

The process model written in following form


u(s)= b0 +b1s +·· ·+bm sm

1+a1s +·· ·+an sn(1.14)

can be rewritten in vector form as

y(t ) =ΨT (t )Θ (1.15)

where Θ= [a1 . . . an ,b1 . . .bm]T is the parameter vector and ΨT (t ) is

ΨT (t ) = [−y (1)(t ) . . .− y (n)(t )+u(t ) . . .+u(m)(t )] (1.16)

According to Figure (1.9), for parameter estimation, the equation error e(t ) is intro-duced

e(t ) = y(t )−ΨT (t )Θ (1.17)

or, assuming the transfer function of the process in the following form


u(s)= B(s)

A(s)(1.18)


the equation error, via La Place transform, becomes

e(t ) =L −1(B(s)u(s)− A(s)y(s)) (1.19)

in which A(s) and B(s) correspond to the estimate of A(s) an B(s). The least-square(LS) estimate

Θ= [ΨTΨ]−1ΨT y (1.20)

is obtained if the minimization of the sum of least-squares is computed

J (Θ) =Σt e2(t ) = eT ed J (s))

dΘ) = 0(1.21)

Figure 1.9: Model structure for parameter estimation with equation error

The equivalent procedure in z-domain is straightforward to obtain and will be omit-ted. The least-squares estimated can be also expressed in recursive form (RLS) [46] withrespect to the estimates at the instant k, with k = t

T0= 1,2,3, . . .

Θ(k +1) = Θ(k)+Υ(K )[Υ(K +1)−ΨT (K +1)Θ(k)] (1.22)

where Υ(K ) = 1

ΨT (K+1)P (k)Ψ(K+1)P (k)Ψ(k +1)

P (k +1) = [I −Υ(K )ΨT (K +1)]P (k)(1.23)

To improve estimates, filtering methods can be also exploited. In particular, when mea-surements are affected by noise, a Kalman filter can be used for the parameter estimation.

Output Error Methods

Instead of the equation error computed previously, the output error

e(t ) = y(t )− y(Θ, t ) (1.24)


where

y(Θ, s) = B(s)

A(s)u(s) (1.25)

can also be used as depicted in Fig.(1.10).

Figure 1.10: Model structure for parameter estimation with output error

Unfortunately, direct calculation of the parameter estimateΘ is not possible, becausee(t ) is non-linear in the parameters. Therefore, the problem of minimization of the sumof least-squares has to be done by numerical optimization methods. The computationaleffort is then much larger and on-line real-time application is in general impossible. How-ever, relatively precise parameter estimates may be obtained. If a fault within the processchanges one or several parameters by ∆Θ, the output signal changes for small deviationsaccording to

∆y(t ) =ΨT (t )∆Θ(t )+∆ΨT (t )Θ(t )+∆ΨT (t )∆Θ(t ) (1.26)

and the parameter estimator indicates a change ∆Θ(t ). Generally, the process param-eters Θ depend on physical process coefficients p (like stiffness, damping factor, resis-tance,....)

Θ(t ) = f (p) (1.27)

via non-linear algebraic equations. If the inversion of the relationship exists, changesp of the process coefficients can be calculated. These changes in the coefficients are inmany cases directly related to faults. Thus, although the knowledge of p facilitates thefault diagnosis problem, it is not necessary for fault detection only. Parameter estimationcan also be applied to non-linear static process models.

1.5.2 Parity equations techniques

The basic idea of the parity relations approach is to provide a proper check of the parity(consistency) of the measurements acquired from the monitored system. In the early de-velopment of fault diagnosis, the parity vector (relation) approach was applied to static or


parallel redundancy schemes which may be obtained directly from measurements (hard-ware redundancy) or from analytical relations (analytical redundancy). A survey of thesemethods can be found in [67],[47]. In the case of hardware redundancy, two methods canbe exploited to obtain redundant relations. The first requires the use of several sensorshaving identical or similar functions to measure the same variable. The second approachconsists of dissimilar sensors to measure different variables but with their outputs beingrelative to each other. Even if these techniques have been successfully applied for faultdiagnosis, the attention of this section is focused on analytical forms of redundancy. A

straightforward model-based method of fault detection is to take a model B(s)A(s)

and run it

in parallel to the process B(s)A(s) , thereby an output error vector e(t ) is obtained

e(t ) =L −1[(B(s)

A(s)− B(s)

A(s)

)u(s)

](1.28)

The methodology is depicted in Fig. 1.11

Figure 1.11: Scheme for Output error via parity equation method

It worth noting that the model parameter and structure of the monitored process haveto be known a priori. Let consider the situation depicted in Fig. 1.12

Figure 1.12: The referred system model

For the system in Fig. 1.12 under assumption of exact agreement between process andmodel, the output error assume the following form

e(t ) =L −1(B(s)

A(s)fu(s)+ fy (s)

)(1.29)


According to Fig. 1.13, another possibility is to generate a polynomial error

e(s) = A(s)y(s)− B(s)u(s) (1.30)

= B(s) fu(s)+ A(s) fy (s) (1.31)

Figure 1.13: Scheme for equation error via parity equation method

In both cases, different time responses are obtained for an additive input or outputfault. Moreover, the error vector e(s) corresponds to the output error of parameter esti-mation method computed by e(t ) = y(t )− y(θ, t ). On the other hand, e(t ) concerns theequation error, of the corresponding parameter estimation method, in the form e(t ) =y(t )−ΨT (t )Θ. The equations relative to e(s) and e(s) generate residuals, and can there-fore used to implement and design the residual generation system in order to meet faultdetection and isolation specifications, and are called parity equations [48]. under the as-sumptions of fault occurrence and of exact agreement between process and model:

GM (s) =GP (s) i .e.B(s)

A(s)= B(s)

A(s)(1.32)

Therefore in the following can be assume

e(t ) ≡ e(t ) ≡ r (t ) (1.33)

However, it must be highlighted that, within the parity equations, the model parame-ters are assumed to be known and constant, whereas the parameter estimations can varythe parameters of A(s) and B(s) in order to minimize the residuals. Moreover, for the am-plification of specific characteristics of the parity vector r (s) the residuals can be filteredaccording to matrix G f (s) to compute the filtered residual vector r f (s) with accordance of

r f (s) =G f (s)r (s) (1.34)

r f (s) can therefore be used to implement and design the residual generation system,in order to meet fault detection and isolation specification, as well. However, for SISOprocesses only one residual can be generated and it is obviously not easy to distinguishbetween different faults. On the other hand, more freedom in the design of parity equa-tions can be obtained, for SISO processes, when intermediate signals can be measured, or


for MIMO systems. As an extension of the parity equation method, the parity relation con-cept presented here can be generalized [49], [51], [50] and then extended to state-spacedescriptions, as shown in [58] for discrete-time models. The redundancy relations for acontinuous-time model are now specified mathematically as follow. Given the system

x = Ax(t )+Bu(t )y =C x(t )

(1.35)

by substituting the second of Equations in the first one and differential several times,the following system is obtained

y(t )y(t )y(t )

...

=

CC AC A2

...

x(t )+

0 0 0 · · ·C B 0 0 · · ·

C AB C B 0 · · ·...

......

. . .

u(t )u(t )u(t )

...

(1.36)

that can be rewrite in compact form as

Y f (t ) = T x(t )+QU f ( f ) (1.37)

In order to remove the non-measurable states x(t ), and to obtain a parity vector usefulfor FDI, equation above is multiplied by W, such that

W T = 0 (1.38)

This leads to residualsr (t ) =W Y f (t )−W QU f (t ) (1.39)

as shown in Fig. 1.14

Figure 1.14: Parity equation method for a MIMO state-space model; Differentiator filter

The filtered input and output vectors U f and Y f are obtained by digital state variablefilters for order n ≤ 3. The design of the matrix W gives some freedom to generate a struc-tured set of residuals. One possibility is to select the elements of W such that one mea-sured variable has no impact on a specific residual. Then, this residual remains small inthe case of an additive fault on this variable, and the other residuals increase [50], [52].


Finally, because of the previous results, it is clear that some correspondence exists be-tween parity relation and observer-based methods. This aspect was firstly pointed out in[54] and later was demonstrated by [66].

The problem was re-examined in detail by Chen and Patton [50] and the equivalenceunder different conditions and in different meanings was discussed. It was shown thatthe parity relation approach is equivalent to the use of a dead-beat observer. This im-plies that the parity relation scheme provides less design flexibility when compared withmethods which are based on observers without any restriction. More recently, a compar-ison between observer-based and parity space techniques was proposed [57] . Both themethods were first explored for SISO systems and then they were extended to the compari-son of MIMO systems. The comparison was performed using linear discrete-time models.In particular, considering MIMO systems described by estimated input-output discrete-time forms (e.g., ARX or Auto Regressive Moving Average eXogenous (ARMAX) models) ofparity equations leads to a representation in which parameters redundancy can not beavoided. To overcome this drawback Delmaire et al. proposed in [57] to use observersdesigned from identified canonical state-space forms [56]. Moreover, in the case of pa-rameters redundancy, multiple identification of some parameters may occur, leading toinconsistent estimations which might produce inconsistent FDI decisions [57]. This factstates again the FDI capabilities of the observer-based methods with respect to parity re-lation schemes.


1.6 State estimation and observer based techniques

The most common model-based approach makes use of observers [66].[59]to generate di-agnostic signals - residuals. In the framework of FDI, faults are detected by setting a (fixedor variable) threshold on each residual signal. A number of residuals can be designed,each having special sensitivity to individual faults occurring in different locations in thesystem. The subsequent analysis of each residual, once a threshold is exceeded, thenleads to fault isolation [66]. Therefore, the essential issue in model-based FDI is the gen-eration of residuals. Model-based FDI is built upon a number of idealized assumptions,one of which is that the mathematical model used is a faithful replica of the plant dynam-ics. This is, of course, not possible in practice, as an accurate and complete mathematicaldescription of a process is never available. Sometimes the mathematical structure of thedynamic system is not fully known. For other applications, the parameters of the systemmay not be fully known, or may only be known over a limited range of the plant’s opera-tion. There is therefore always a "model-reality mismatch" between the plant dynamicsand the model used for FDI. The basic idea behind the observer or filter-based techniquesis to reconstruct the state of the system by using either Luenberger observers (in a deter-ministic setting) or Kalman filters (in a stochastic noisy environment) and, by means ofthe reconstructed state, to compute an "expected" output which can be compared withthe measured output therefore building the so-called "output estimation error". Such out-put estimation error (or its weighted value) can be therefore used as residual. It is worthnoting that when an observer is exploited for FDI purpose, the estimation of the outputsis necessary, whilst the estimation of the state vector is usually not needed [52]. Moreover,the advantage of using the observer is the flexibility in the selection of its gains whichleads to a rich variety of FDI schemes [60]-[61]

In order to obtain the structure of a (generalized) observer, the continuous-time, time-invariant, linear dynamic model for the process under consideration in a state-space formis considered


(1.40)

Assuming that all matrices A, B and C are perfectly known, an observer is used to re-construct the system variables based on the measured inputs and outputs u(u) and y(t).

˙x(t ) = Ax(t )+Bu(t )+He(t )e(t ) = y(t )−C x(t )

(1.41)

The observer scheme described by previous equation is depicted in Fig. 1.15For the state estimation error ex(t ), it follows that

ex(t ) = x(t )− x(t )ex(t ) = (A−HC )ex(t )

(1.42)

The state error ex(t ) vanishes asymptotically

limt−→∞ex(t ) = 0 (1.43)

if the observer is stable, which can be achieved by proper design of the observer feedbackmatrix H. Let the process be influenced by disturbance and faults as depicted in Fig. 1.16

1.6. STATE ESTIMATION AND OBSERVER BASED TECHNIQUES 23

Figure 1.15: Process and state observer

Figure 1.16: Multivariable process with disturbance v(t ) , w(t ) and faults fu(t ), fy (t )

it is described by the following system

x(t ) = Ax(t )+Bu(t )+Qν(t )+L1 f (t )y(t ) =C x(t )+Rw(t )+L2 f (t )

(1.44)

where v(t ) and w(t ) represent the non-measurable disturbance vector at the inputand at the output respectively, and the entries of vector f (t ) = [ fu(t )T fy (t )T ] ∈ Rk corre-spond to specific faults acting on the system through fault distribution matrices L1 and L2,and they can represent actuator, process, input and output sensor additive faults. For thestate estimation error, under assumption that disturbance are neglected, the followingequation hold

ex(t ) = (A−HC )ex(t )+L1 f (t )−HL2 f (t ) (1.45)

and the output error becomes


e(t ) =Cex(t )+L2 f (t ) (1.46)

The vector f (t ) represent additive faults because they influence e(t ) and x(t ) by a sum-mation. When sudden and permanent faults f(t) occur, the state estimation error will de-viate from zero, ex(t ) as well as e(t ) show dynamic behavior which are different for L1 f (t )and L2 f (t ). Both ex(t ) and e(t ) can be taken as residuals. In particular, the residual e(t)is the basis for different fault detection methods based on output estimation. For thegeneration of residual with special properties, the design of the observer feedback matrixH is of interest [52]-[53]. Limiting conditions are the stability and the sensitivity againstdisturbances v(t) and w(t). If the signals are affected by noise, the Kalman filter must beused instead of classical observers. If faults appear as changes ∆A or ∆B or ∆C of theparameters, the process behavior becomes

x(t ) = (A+∆A)x(t )+ (B +∆B)u(t )y(t ) = (C +∆C )x(t )

(1.47)

while the state ex(t ) and the output estimation e(t ) errors

ex(t ) = (A−HC )ex(t )+ (∆A−H∆C )x(t )+∆Bu(t )y(t ) =Cex(t )+∆C x(t )

(1.48)

The changes ∆A, ∆B and ∆C are then multiplicative faults [62]-[63]. In this case, thechanges in the residuals depend on the parameter changes, as well as input and statevariable changes. Hence, the influence of parameter changes on the residuals is not asstraightforward as in the case of the additive faults f(t).

1.6.1 Basic fault detection observer-based schemes

The following observer-based fault detection schemes and configurations are briefly sum-marized and recalled [66],[62],[52],[63].

1. Dedicated observers for MIMO process

2. Fault detection filters for MIMO process

3. Output observers

Dedicated observers for MIMO process

• Observer excited by one output: one observer is driven by one sensor output. Theother outputs are reconstructed and compared with measured outputs y(t). Thisallows the detection of single output sensor faults.

• Bank of observers, excited by all outputs: several observers are designed for a definitefault signal and detected by hypothesis test.

• Bank of observers, excited by single outputs: several observers for single sensors out-puts are used. The estimated outputs are compared with the measured outputsy(t). This allows for the detection of multiple sensor fault (DOS, Dedicated ObserverScheme).


• Bank of observers, excited by all outputs and all input except one: as before, but eachobserver is excited by all inputs except one sensor input, which is supervised (GOS,Generalized Observer Scheme).

Fault detection filters for MIMO process

The feedback matrix H of the state observer is chosen so that the residual vectors causedby different fault sources have well distinct direction in the residual space. With such"directional" residual vectors, the fault isolation problem consists of determining whichof the known fault signature directions the residual vector lies the closest to. The orig-inal form of the "Failure detection filter" was proposed by Beard [64] and Jones [65] togenerate directional residual vectors. Many more straightforward methods have followed,including methods to achieve "robust fault detection filter" [61]. The fault (or failure) de-tection is a class of Luenberger observers with a specially designed feedback gain matrix.It allows output estimation errors having directional characteristics associated with someknown fault directions, to be obtained. These fault detection methods mostly require sev-eral measurable output signals and make use of internal analytical redundancy of multi-variable systems. Recently it was proposed to improve their robustness with respect toprocess parameter changes and unknown input signals v(t ) and w(t ) [50],[61]. This canbe reached, for example, through filtering the output error of the observer by r (t ) =W e(t )together with a special design of the observer feedback matrix H (see Fig. 1.15).

Output observers

Another possibility is the use of output observers (or UIO, see next section) in the recon-struction of the output signals, if the estimate of the state variable x(t ) is not of primaryinterest. In this context, it is worthy to mention the paper by Chen, Patton and Zhang[61] concerning the design of output observers for robust FDI using eigenstructure as-signment method. Through a linear transformation

z(t ) = T x(t ) (1.49)

the state-space representation of the observer becomes

˙z(t ) = F z(t )+ Ju(t )+G y(t ) (1.50)

and the residual is determinate by

r (t ) =Wz z(t )+Wy y(t ) (1.51)

The state estimation errore(t ) = z(t )−T x(t ) (1.52)

and the residuals r (t ) are then designed, such that they are independent of the processstates x(t ), the known input u(t ) and the unknown inputs v(t ) and w(t ), as depicted inFigure 1.17. In this way the residuals are dependent only on fault signals f (t ).


Figure 1.17: Process and output observer

1.6.2 Unknown Input Observer for Fault Detection

As the complexity of a dynamic system increases, the harder is the task of modeling thesystem and its disturbances, and one can speak of an "uncertain" system, for which thereis an uncertainty of knowledge of the system’s structure, its parameters and the distur-bance effect. There are therefore robustness problems in FDI with respect to modellingerrors and disturbances. The goal of robust FDI is to discriminate between the fault ef-fects and the effects of uncertain signals and perturbations. The robustness problem inFDI is thus defined as the maximization of the detectability of faults, together with theminimization of the effect of modelling errors and disturbances on the FDI procedure.The ultimate goal of robustness is to provide rapid and reliable detection and isolationof system faults when the plant under control is disturbed, and when the mathematicalmodel upon which the diagnosis is based cannot faithfully reproduce the full dynamicoperation of the plant. There are many approaches for residual generation [66]-[68]. Themost common one uses observers. The basic idea behind the observer or filter-basedapproach is to estimate the outputs of the system from the measurements (or a subsetof measurements) by using either Luenberger observer(s) in a deterministic setting orKalman filter(s) in a stochastic setting. Then, the (weighted) output estimation error (orinnovations in the stochastic case), is used as a residual. It is worth noting that for FDIpurposes, only the output estimation is required, the estimation of the state vector is notstrictly necessary. The most important task in model-based FDI is the generation of resid-uals which are independent of disturbances. The method is based on disturbance de-coupling principle. In this approach, uncertain factors in system modeling or identifica-tion are considered to act by means of an "unknown input", the disturbance, on a linearsystem model. The disturbance vector is unknown but its distribution matrix is usuallyassumed known. Based on the disturbance distribution matrix obtained by modelling oridentification procedure, the unknown input can be de-coupled from the residual. Theprinciple of the Unknown Input Observer (UIO) is to make the state (or output) estima-tion error decoupled from the unknown inputs. Since the residual is a weighted estima-tion error, it may be de-coupled from each disturbance. The first step in the disturbance


de-coupled residual generation consists of designing an UIO. Very important contribu-tions to this subject can be found in [52]. UIO or other disturbance de-coupling basedresidual generation approaches require that the unknown input distribution matrix mustbe known a priori. The actual unknown input itself does not need to be known. Whenuncertainties are caused by modelling errors, linearization errors, parameter variations,etc, such a disturbance de-coupling approach cannot be directly applied because the dis-tribution matrix E is normally unknown. To solve this problem, which is of paramountimportance in real industrial system applications, various techniques to identify the dis-turbance distribution matrix are available in literature. The problem of designing an ob-server for unknown inputs has been studied for nearly two decades and many approachesfor the design of both full-order and reduced-order UIO have been proposed (geometricand algebraic methods, singular value decomposition and matrix inversion techniques,linear transformation algorithms)[52]. It is worth noting that for fault detection purposealso the faults can be considered as an unknown input acting on the system. Differentlyfrom other type of disturbance acting on systems, for which residuals must be insensitive,residual signals have to be sensitive to faults themselves. In the following we will see a pro-cedure for fault detection using UIO scheme. In this paragraph, a full-order UIO structureis considered and a mathematical method for designing UIO is presented. The necessaryand sufficient conditions for this observer to exist are also recalled. These conditions areeasy to verify and the design procedure is easy to implement. Let consider a continuoustime, time-invariant, linear dynamic systems with an additive unknown disturbance termin the following form:

x = Ax(t )+Bu(t )+E d(t )y =C x(t )

(1.53)

Where, x(t ) ∈ Rn is the state vector, y(t ) ∈ Rm is the output vector, u(t ) ∈ Rr is the knowninput vector and d(t ) ∈ Rq the unknown input vector. A,B ,E ,C are known matrices withappropriate dimensions. It must be highlight that the unknown term E d(t ) can be usedto describe an additive disturbance, different kinds of modelling uncertainties (noise, un-modelled non-linear terms, time-varying dynamics, etc.) as well as fault terms. The un-known input term may also appear in the output equation, i.e.,

y(t ) =C x(t )+Ey d(t ) (1.54)

but this case is not considered because the term E d(t ) can be nullified by using a trans-formation of the output signal y(t ) [52]. Sometimes for systems described by Equation1.53, there is a term relating the control input u(t ) in the output equation, i.e.,

y(t ) =C x(t )+Du(t ) (1.55)

however, the term Du(t ) is omitted in the following since this does not affect the gen-erality of the discussion on the observer design.

Definition 1. An observer is defined as an Unknown Input Observer for the systemdescribed by 1.53, if its state estimation error vector ex(t ) approaches zero asymptotically,regardless of the presence of the unknown input term. In other words an UIO is a robustobserver in which the state estimation errors become insensitive to disturbance.


The full-order UIO, for the system 1.53, has the following mathematical form [61]

z(t ) = F z(t )+T Bu(t )+K y(t )x(t ) = z(t )+H y(t )

(1.56)

Where z(t ) ∈ ℜn is the state of the UIO, x(t ) the estimated state vector x(t ) whilstF,T, H and K are matrices to be designed to achieve the unknown input de-coupling. Theobserver structure is depicted in Figure 1.18.

Figure 1.18: The UIO Structure

The state estimation error (ex(t ) = x(t )− x(t )) obtained by the UIO 1.56 applied to thesystem 1.53 is described by the equation:

ex(t ) = [A−HC A−K1C ]ex(t )− [F − (A−HC A−K1C )]z(t )−[K2 + (A−HC A−K1C )H ]y(t )−[T − (I −HC )]Bu(t )− (HC − I )E d(t )

(1.57)

where K = K1 +K2.If the following relations hold:

(HC − I )E = 0I −HC = TA−HC A−K1C = FF H =−K2

(1.58)

then the state estimation error becomes equal to:

ex(t ) = Fex(t ) (1.59)

This means that, if all the eigenvalues of F are stable, ex(t ) will approach zero asymp-totically, i.e., x(t ) → x(t ). Hence, according to the Definition 1, the observer described


by Equations 1.56 is an UIO for the system 1.53. The design of this UIO consists of solv-ing matrix equalities stated in eq. 1.58 and making all eigenvalues of the system matrix Fwith negative real part. A special solution for the matrix H under conditions 1.58 is givenin [52]:

H∗ = E(C E)+ (1.60)

where (·)+ represent the pseudo-inverse matrix notation.The following Theorem states the existence conditions for the UIO.

Theorem.1. Necessary and sufficient conditions for the existence of an UIO 1.56 for thesystem defined by Equation 1.53 are [52]:

1. r ank(C E) = r ank(E)

2. (A1,C ) is a detectable pair, where A1 = A−E(C E)+C A

It is worth noting that the number of independent row of the matrix C must not beless than the number of the independent columns of the matrix E to satisfy condition1 in Theorem 1. It means that the maximum number of disturbances which can be de-coupled cannot be larger than the number of the independent measurements. Moreover,without unknown inputs in the system, by setting T = I , H = 0 and E = 0, the observer1.56 will be a simple Luenberger observer. In this situation, condition 2 in Theorem 1clearly holds true and such condition is the detectability of pair (A,C ).

UIO design procedure

It can be seen how K1 is a free matrix of parameters in the design of an UIO. After K1 iscomputed, in order to stabilize the dynamic system matrix F, other parameter matricesin the UIO can be computed by the relation K = K1 +K2 and conditions 1.58. Some de-sign freedom left in the choice of K1 may be exploited to make the diagnostic residual hasdirectional characteristics. In this work, as we’ll see later, because the input-output linkof the Multiple-Input Multiple-Output (MIMO) system under investigation is obtainedby means of the identification of a collection of Multiple-Input Single-Output (MISO)models, this further degree of freedom will not be used in the residual design. Underthese assumptions, if the pair (A1,C ) is observable, in order to stabilize the system ma-trix F = A1 −K1C , the pole placement routine available in the Control System Toolbox forMATLAB can be used. If (A1,C ) is not observable, an observable canonical decompositionshould be applied to the pair. If (A1,C )is detectable, the matrix F can be stabilized.

1.6.3 FDI schemes based on UIO and output observers

Considering the system in 1.53 and explaining the dependency from fault signals

x = Ax(t )+Bu(t )+B fu(t )y =C x(t )+ fy (t )

(1.61)

The vectors faults fu(t ) = [ fu1 (t ), . . . . . . , fur (t )]T and fy (t ) = [ fy1 (t ) . . . . . . fym (t )]T repre-sent actuator and sensor faults respectively, and either assuming values different from


zero only in the presence of faults. Usually these signals are described by step and rampfunctions representing abrupt (bias) and incipient faults (drift).

u(t ) = u∗(t )+nu

y(t ) = y∗(t )+ny(1.62)

It is worth noting that actual input and output measurement can be corrupted withsignal noise, usually assumed as white, zero mean, uncorrelated Gaussian signal. Touniquely isolate a fault concerning one of the output sensors, fy (t ), under the hypothe-sis that inputs are fault-free ( fu(t ) = 0) a bank of classical dynamic Luenberger observers,in a deterministic setting or Kalman Filter (KF) in a stochastic setting, is used, accordingto Figure 1.19.

Figure 1.19: Bank of estimators for output residual generation (DOS)

This observer configuration represents the Dedicated Observer Scheme (DOS) [69].The number of these observers (estimators) is equal to the number m of system outputs,and each device is driven by a single output and all the inputs of the system. In this casea fault on the i − th output affects only the residual function of the output observer orfilter driven by the i − th output. To uniquely isolate a fault concerning one of the systeminputs, fui (t ), under the assumption that outputs are fault-free, ( fy (t ) = 0) a bank of UIOor UIKF is used (Fig. 1.20). Such a solution is known as the Generalized Observer Scheme(GOS) [66]. The number of these observers is equal to the number r of control inputs. Thei-th observer is driven by all but the i-th input and all outputs of the system and generatesa residual function which is sensitive to all but the i-th input fault fui (t ), i-th unknowninput. In this way the detection of single input measurement faults is possible, since afault on the i-th input affects all the residual functions except that of the device which isinsensitive to the i-th unknown input. In order to summarize the isolation capabilitiesof the schemes presented, the table below shows the "fault signatures" for the case of asingle fault in each input-output signal.

The residuals which are affected by the input and output faults are described by an en-try ’1’ in the corresponding table entry, while an entry ’0’ means that the input or output


Figure 1.20: The GOS scheme for FDI of system inputs

Table 1.1: Fault Signature

u1 u2 · · · ur y1 y2 · · · ym

rU IO1 0 1 · · · 1 1 1 · · · 1rU IO2 1 0 · · · 1 1 1 · · · 1

......

......

......

......

...rU IO3 1 1 · · · 0 1 1 · · · 1

rO1 1 1 · · · 1 1 0 · · · 0rO2 1 1 · · · 1 0 1 · · · 0

......

......

......

......

...rOm 1 1 · · · 1 0 0 · · · 1

fault does not a affect the corresponding residual. Note how multiple faults in the systemoutputs can be isolated since a fault on the i-th output signal affects only the residual func-tion rOi of the output observer driven by the i-th output, but all the UIO or UIKF residualfunctions rU IOi . On the other hand, multiple faults on the inputs channel cannot be iso-lated by means of this simplified technique since all the residual functions are sensitive tofaults regarding different inputs. However, to overcoming this limitations, dedicated UIOfor multiple fault input channel can be designed.

DOS

With reference to Figure 1.19, in order to diagnose a fault on the i − th system outputwhen the measurement noises are negligible (nu = 0,ny = 0) and fu(t ) = 0, the model ofthe i − th observer (i = 1, . . . ,m) has the form

xi (t ) = Ai xi (t )+B i u(t )+K i (yi (t )−C i xi (t )) (1.63)


where xi (t ) is the observer state vector and the triple (Ai ,B i ,C i ) is a minimal statespace representation (completely observable) of the link among the inputs of the processand its i − th output yi (t ). Such a triple can be obtained by means of a MISO identifiedmodel. The entries of K i must be designed in order to assign stable, and suitably chosen,eigenvalues to the matrix (Ai −K i C i ). In this situation and in the absence of faults, i.e.,fy (t ) = 0, it can be verified that for the i − th output residual ri (t ) the following relationholds

limt−→∞ri (t ) = lim

t−→∞(yi (t )−C i xi (t )) = 0 (1.64)

and the rate of convergence depends on the position of the eigenvalues of the (Ai −K i C i )matrix in the complex left half plane. In the presence of a fault (step or ramp signal) onthe i − th process output only the i − th output residual reaches a value different fromzero and this situation leads to a complete failure diagnosis.

GOS

With reference to the device for the FDI of the input channels, depicted in Figure 1.20, thestructure of the i−th UIO (i = 1, . . . ,r ) for residual generation [52], under the assumptions(nu = 0,ny = 0) and fy (t ) = 0, is the following

zi (t ) = (T i A−K i C )zi (t )+ J i u(t )+Si y(t )r i (t ) = L1

i xi (t )+L2i y(t )

(1.65)

where zi (t ) ∈Rn denotes the observer state vector, r i (t ) ∈Rm is the residual vector andF i , J i ,Si ,Li

1 and Li2 are matrices to be designed with appropriate dimensions. Let T i be a

linear transformation of the state x(t ) of the system and define the state estimation erroras

exi (t ) = zi (t )−T i x(t ) (1.66)

On the imposition (nu = 0,ny = 0) and fy (t ) = 0 it can be shown that the dynamics ofthe state estimation error become

exi (t ) = F i ex

i (t )+ (F i T i −T i A+Si C )x(t )+ (J i −T i B)u(t )−T i B fu(t ) (1.67)

whilst the residual vector is given by

r i (t ) = L1i ex

i (t )+ (L1i T i +L2

i C )x(t ) (1.68)

It can be seen that if the following holds

F i T i −T i A+Si C = 0J i = 0L1

i T i +L2i C = 0

(1.69)

Equation 1.67 and 1.68 becomes

ex

i (t ) = F i exi (t )+T i B fu(t )

r i (t ) = L1i ex

i (t )(1.70)


If the linear transformation T i is chosen as [70]

T i = In −Bi (C Bi )+C (1.71)

where Bi is the i − th column on B matrix and K i is selected such that F i = T i A −K i C isasymptotically stable, then, the solutions of Equation 1.69 are obtained as

F i = T i A+K i CSi = K i +F i Bi (C Bi )+ J i = T i BL1

i =−C L2i = [Im − (C Bi )(C Bi )+]

(1.72)

The selection of the Bi matrix in Equations 1.71 and 1.72 sets to zero the i − th col-umn of the J i matrix. That is, the estimation error and then the residual of the i − th UIObecome independent of the i − th system input. Under the hypothesis of observabilityof the system 1.61 and in the absence of input faults ( fu(t ) = 0), it can be seen that thei − th residual vector reaches zero as t approaches infinity and the rate of convergencedepends on the position of the eigenvalues of F i matrix. In the presence of a fault on thei − th input, the i − th residual reaches zero asymptotically while the residuals of the r −1remaining observers are sensitive to the fault signal. This situation leads to the possibilityof unique detection and isolation of all process input faults. The design of this UIO re-quires the knowledge of a minimal form model (A,B ,C ) for the system 1.61. Such a triplecan be computed by using a realization procedure from a MIMO identified model.

Residual Robustness

The model-based FDI uses a mathematical model for the system. As discussed in earliersections, the main and most challenging task of model-based FDI is the generation ofresiduals in which outputs and inputs of the system are processed to generate a fault in-dicator signal (residual). Ideally, this signal should be near to zero for the fault-free case,and should increase significantly when a fault appears in the system. The better the modelrepresents the system, the better will be the reliability and performance in FDI. However,modelling errors and disturbances are inevitable, and hence there is a need to develop ro-bust FDI algorithms. A robust FDI system is sensitive only to faults, even in the presenceof a model reality mismatch. To achieve robustness in FDI, the residual should be insen-sitive to uncertainty, whilst sensitive to faults, and therefore robust ([60], [66], [61]). Theresidual that has this property can then be used to detect and isolate faults reliably. In gen-eral, both faults and uncertainty affect the residual, and discrimination between these twoeffects is difficult. The effects of disturbances act as a source of false alarms which mustbe minimized. The ideal case is to make the residual itself become de-coupled from dis-turbances (robust residual generation). This is the principle of a robust residual generatorwhich can be achieved by minimizing the effect of disturbances on residuals. In particu-lar, we have to consider that the basis of fault diagnosis technique, under consideration,is the use of mathematical models. Hence, the model should have a certain accuracy. Inorder to make a diagnosis algorithm robust against modelling uncertainty, we should alsohave some knowledge about modelling uncertainty. Otherwise, if an FDI algorithm canbe made robust without a priori knowledge of the modelling, a model would clearly notbe required in the first place. The information of modelling uncertainty is normally rep-resented by assumptions on uncertainty. These assumptions should be easy to handle


by the robust design in a systematic manner, otherwise it does not provide assistance forrobust design. The disturbance representation of uncertainty can be handled by the un-known input observer or the eigenstructure assignment. However, this assumption is notrealistic, i.e., the distribution matrix cannot always be obtained directly in practice. In realsituation, we can obtain some descriptions about uncertainty, for example, parameters ofthe system are within a certain bound. However, these descriptions are not easy to handlein designing robust FDI algorithms.

Chapter 2

Structural fault detectability,isolability and identifiability

The concept of structural fault detectability, isolability and identifiability are introducedto describe the structural property of a system from the FDI point of view. Generallyspeaking one have to distinguish those properties from the performance of a generic FDIscheme, therefore such structural properties will be introduced without any reference tothe specified FDI scheme.Study on structural fault detectability, isolability and identifiability plays a central role inthe structural analysis for the construction of a technical process and for the design of anFDI scheme. In following, according to the notation in [42], we shall introduce the con-cepts of structural fault detectability, isolability and identifiability, study their checkingcriteria and illustrate the major results. In our analysis we’ll only consider additive faultsand, according to 1.6, we’ll refer to the following faulty MIMO system

x(t ) = Ax(t )+Bu(t )+E f (t )y(t ) =C x +Du(t )+F f (t )

(2.1)

where f is the fault vector f = [ f1, . . . , fq ]T .

2.1 Structural fault detectability

In the literature, one can find a number of definition of fault detectability, introducedunder different aspect. In order to give a definition valid to introduce also the conceptsof isolability and identifiability, we first specify our intension of introducing the conceptof structural fault detectability. First structural fault detectability should be understoodas a structural property of the system under consideration, which describes how a faultaffects the system behavior. It should be expressed independent of the system input vari-ables, disturbances as well as model uncertainties. Secondly structural fault detectabilityshould indicate if a fault would cause changes in the system output. Finally, the structuralfault detectability should be independent of the type and the size of the fault under con-sideration. Bearing these in mind, we adopt an intuitive definition of fault detectability

35

36 CHAPTER 2. STRUCTURAL FAULT DETECTABILITY, ISOLABILITY AND IDENTIFIABILITY

which says: A fault is detectable if its occurrence, independent of its size and type, wouldcause a change in the nominal behavior of the system outputs.

Definition 1 Given system 2.1, a fault fi is structurally detectable if for some u result

∂y

∂ fi

∣∣∣fi=0

d fi 6= 0 i = 1, ..., q (2.2)

Therefore a fault become detectable if its occurrence, independently of its size andtype, lead to change in the system outputs at least at some time instant and for somesystem input.

Theorem 1 Given system 2.1, a fault fi is structurally detectable iff

C (sI − A)−1Ei +Fi 6= 0 (2.3)

where Ei ,Fi denoting the i-th column of matrices E ,F respectively. It must be highlightthat an additive fault is structurally detectable as far as the transfer function from the faultto the system output is non zero. In the following the transfer matrix

G fi =C (sI − A)−1Ei +Fi (2.4)

will be called fault transfer matrix.In this work, the rank of a transfer matrix is understood as the so-called normal rank,

i.e. maximum rank, if no additional specification is given. It must be highlighted that thedetection of additive faults can be realized independently of the system inputs [42].

2.2 Structural fault isolability

We say that a group of faults are isolable if any simultaneous occurrence of these faultswould lead to a change in the system output. For the sake of simplicity, if we consider onlytwo different detectable faults fi , f j , i 6= j , we say that the faults are isolable if the changesin the system output caused by these two simultaneous faults are distinguishable.

Definition 2 Given system 2.1. The faults in a fault vector f are isolable, when

∂y

∂ f

∣∣∣f =0

d f 6= 0 (2.5)

In general case, we say that a group of faults are isolable if any simultaneous occur-rence of these faults would lead to a change in the system output. It must be highlightedthe similarity between the isolability of additive faults and the so-called input observabil-ity which is widely used for the purpose of input reconstruction. Consider the system

x(t ) = Ax(t )+E f (t )y(t ) =C x(t )+F f (t )

(2.6)

in which, without loss of generality, we consider the fault vector f as the system input.The input f(t) of system 2.6 is said observable if y(t ) = 0 for t > 0 implies f (t ) = 0 for t > 0provided that x(0) = 0 [44].

2.2. STRUCTURAL FAULT ISOLABILITY 37

Except the assumption of initial condition x(0) = 0, the physical meanings of the isolabil-ity of additive faults and input observability are equivalent.

With the aid of the concept of fault transfer matrices 2.4, we now derive existenceconditions for the structural fault isolability.

Theorem 2 Given system 2.1, than f (t ) with fault transfer matrix

G f (s) = [G f1 (s)...G fq (s)] (2.7)

is structurally isolable iff

r ank(G f (s)) =q∑

i=1r ank(G fi (s)) (2.8)

Due to the its straightforwardness, the proof given in [42] will be omitted.Corollary 1 Given the system 2.1 and assume that fi , i = 1, ..., l ≤ q, are additive faults.

Then, these l faults with fault matrix G f (s), are isolable iff

r ank(G f (s)) = l (2.9)

This result reveals that, to isolate l different faults, we need at least an l-dimensionalsubspace in the measurement space spanned by the fault transfer matrix. Consideringthat r ank(G f (s)) ≤ mi nm, l , where m is the number of sensors, we have the followingclaim which is very useful for the practical application.

Claim. The additive faults are isolable only if the number of the faults is not largerthan the number of the sensors.

The condition 2.9 can be equivalently expressed in terms of the matrices of the statespace description. Indeed, denoting the minimal state space realization of G f (s) by

G f (s) =C (sI − A)−1E +F (2.10)

the following corollary holdCorollary 2 Given the system 2.1 and assume that fi (i = 1, ..., l ≤ q ≤ m) are additive

faults. Then these l faults are isolable iff

r ank

[A− sI E

C F

]= n + l (2.11)

Noting that the relationship between fault isolability (for additive faults) and input ob-servability, as previously discussed consequently to Definition 2, can be further proved bythe rank condition in 2.11, since this is the condition for which the input f (t ) of system2.6 is observable with knowledge of initial condition x(0) [44] .

To the aim of find out alternative conditions for checking condition 2.9 or 2.11, difficultto use in practical context, now we’ll derive some sufficient conditions, on the assumptionthat m ≥ l , for the fault isolability. Let us consider a generic additive fault vector ξ =[ξ1,ξ2, . . . ,ξl ]T , acting on the system 2.6, to which correspond the fault transfer matrixG f (s) =C (sI − A)−1Eξ+Fξ. In the following, for the sake of simplicity, we’ll refer to matrix


Eξ and Fξ without subscripts. Initially, to simplify our study, we first consider F = 0. Itfollows from Cayley-Hamilton Theorem that

G f (s) =C (sI − A)−1E = 1

Φ(s)C (

n∑i=1

Si sn−1)E = 1

Φ(s)C (

n∑i=1

αi (s)Ai−1)E (2.12)

Φ(s) = det (sI − A) = sn +a1sn−1 +a2sn−2 +·· ·+an−1s +an

Si = Si−1 A+ai−1I , i = 2, · · · ,n; S1 = Iα1(s) = sn−1 +a1sn−2 +·· ·+an−1; · · · ;αn−1(s) = s +α1(s), αn(s) = 1

(2.13)

which can be rewritten into

C (sI − A)−1E = 1

Φ(s)[α1Iα2I · · ·αn I ]

C EC AE

...C An−1E

(2.14)

Thus

r ank

C EC AE

...C An−1E

= l (2.15)

builds a necessary condition for the fault isolability. Now if for some j ∈ 1, · · · ,n

r ank(C A j−1E) = l (2.16)

where l is the dimension of fault vector, then 2.14 can be rewritten into

C (sI − A)−1E = 1

Φ(s)[α j I +

n∑i=1,i 6= j

αi (s)Qi ]C A j−1E (2.17)

where Qi ∈ℜm×m (i = 1, · · · ,n, i 6= j ) are some matrices. Considering that

r ank(α j I +n∑

i=1,i 6= jαi (s)Qi ) = m ≥ l , r ank(C A j−1E) = l (2.18)

we finally have r ank(C (sI − A)−1E) = l .This proves the following theorem.

Theorem 3 Given C (sI − A)−1E as defined in 2.12 with m ≥ l and satisfying 2.15. Assumethat for some j ∈ 1, · · · ,n, r ank(C A j−1E) = l . Then

r ank(C (sI − A)−1E) = l (2.19)

In the framework of linear system theory, C Ai E , with i = 0,1, ..., are called Markov ma-trices. The theorem above provide us a sufficient condition for checking the isolability ofadditive faults by means of Markov matrices.

2.3. STRUCTURAL FAULT IDENTIFIABILITY 39

In a similar manner, like the proof of the previous theorem, we are able to prove thefollowing theorem that gives an alternative sufficient condition for the fault isolability.

Theorem 4 Given C (sI − A)−1E. Let Γi = C Si E, i = 1, · · · ,n and assume that for somej ∈ 1, · · · ,n

r ank(Γ j ) = l (2.20)

then the r ank(C (sI − A)−1E) = l .

The procedure to extend the founded condition to the more general case of system 2.1,for which F 6= 0, has been given in [42]. In particular it result that condition 2.15 can beequivalently written as

FC E

C AE...

C An−1E

= l (2.21)

while conditions 2.16 and 2.20 respectively as

r ank(C A j−1E) ≡

r ank(F ) = l , i f j = 1r ank(C A j−2E) = l , i f j ∈ 2, · · · ,n +1

(2.22)

and

r ank(Γ j ) = l , j ∈ 0, · · · ,n,Γ0 = F,

Γ j = [an I an−1I · · ·a1I I ]

0...0F

C E...

C A j−1E

, j ∈ 1, · · · ,n(2.23)

2.3 Structural fault identifiability

Roughly speaking, the concept of structural fault identifiability is understood as a char-acterization of system structure that is essential to reconstruct faults from the systemoutputs. From the mathematical point of view, fault identifiability characterize the map-ping from the system output to the faults under consideration. If this mapping is unique,then the faults are identifiable. Usually, we intend to express this mapping in terms ofthe model from the faults to the system output, then the structural fault identifiability isequivalent to the model invertibility. Motivated by this fact, the concept of structural faultidentifiability will be introduce in terms of fault transfer matrices.

Definition 3 Given system 2.1 and let


G f (s) = [G f1 (s)...G fl (s)] (2.24)

be the fault transfer matrix of fault vector f (t ) = [ f1(t ) · · · fl (t )]T . The vector f (t ) is calledstructurally identifiable if G f (s) is invertible and its inverse is stable and causal [42].

Note that the requirements on the stability and causality of the inverse of G f (s) is anexpression for the realizability of inverting G f (s). It’s evident that without these two re-quirements, the structural fault identifiability would be equivalent to the structural faultisolability. In another word, the structural fault isolability is a necessary condition for thefault to be identifiable.

In the work of Hou and Patton [44] has been considered the problem of input recon-struction from the viewpoint of input observability. In particular has been shown thatthe input observability is a necessary and sufficient condition for the existence of an es-timator for reconstructing inputs. To understand better the structural conditions behindthese concepts it’s important to introduce the concept of invariant zero, since it has alsoimportant implications for the arguments addressed in the following chapter.

Consider the system (2.1) and assume that the initial state is given by x(0). TakingLaplace transformation of the system representation yields

[A− sI E

C F

][X (s)f (s)

]=

[X (0−)Y (s)

](2.25)

The polynomial system matrix

P (s) =[

A− sI EC F

](2.26)

is sometimes referred to as Roosenbrock’s matrix. A necessary and sufficient conditionfor an "input"

f (t ) = f (0)ezt (2.27)

to yield rectilinear motion in the state space of the form

x(t ) = x(0)ezt (2.28)

such that the output of the system is identically zero for all time is that z, x(0) and f (0)satisfy

P (z)

[x(0)f (0)

]= 0 (2.29)

This result a set of complex frequencies z which are associated with specific directionsx(0) and f (0) in the state and input spaces for which the output of the system is zero.These elements are called invariant zeros.

Practically, in the case of square system, i.e. system with equal number of inputsand outputs, in order for equation (2.29) to have a nonzero solution for x(0) and u(0),det (P (z)) must be zero. In such cases invariant zeros are presents and the condition (2.11)is not true.

2.3. STRUCTURAL FAULT IDENTIFIABILITY 41

A similar discussion may also be extended in terms of input observability [44]. In fact,setting y(t ) = 0 in (2.25), we obtain

[A− sI E

C F

][X (s)f (s)

]=

[X (0−)

0

](2.30)

whatever x(0) and therefore x(t ) are, system (2.30) has a unique solution for f as f =0 iff matrix P (s) has no invariant zeros. However if the the Roosenbrock’s matrix P (s),related to the system (2.6), has invariant zeros ∈ C− it results that the "input" ( for ourcase the fault) is detectable, where the input f (t ) is said to be detectable if y(0) = 0 fort ≥ 0 implies f (t ) −→ 0 as t −→ 0. In these cases, as shown in the next chapter, the faultreconstruction problem can still be solved.

Chapter 3

Sliding Mode Observers

The concept of sliding mode control has been extended to the problem of state estima-tion by an observer, for linear systems, uncertain linear systems and nonlinear systems.Using the same design principle as for variable structure control, the observer trajectoriesare constrained to evolve after a finite time on a suitable sliding manifold by the use of aninjection signal designed according a SM control algorithm. Subsequently the sliding mo-tion provides an estimate, asymptotically or in finite time, of the system states. It worthnoting that the sliding manifold is usually given by the difference between the observerand the system outputs, therefore in such cases we refer to the control signal as outputinjection signal.

This chapter present an overview of both linear and nonlinear sliding mode observerparadigms. Many of the concepts in this chapter are closely based on the book by Ed-wards and Spurgeon [16]

3.1 SMO for Linear Systems

We consider initially the linear system described by


(3.1)

where A ∈ Rnxn ,B ∈ Rnxm ,C ∈ Rpxn and p ≥ m. Assume that the matrices B and C are offull rank and the pair (A,C ) is observable. Since a sliding motion on the error output spaceis going to be enforced, it is convenient to introduce a coordinate transformation so thatthe outputs appear as the last p components of the states. One possibility is to considerthe non-singular transformation x → Tc x as

Tc =[

NcT

C

](3.2)

where Nc ∈Rnx(n−p) and the columns span the null space of C. This transformation isnon-singular, and with respect to this new coordinate system, the distribution matricesof the similar system are

43

44 CHAPTER 3. SLIDING MODE OBSERVERS

A = Tc ATc−1 =

[A11 A12

A21 A22

]; B = Tc B =

[B1

B2

]; C =C Tc

−1 = [0 Ip ] (3.3)

then the nominal system (3.1) can be rewritten as

x1(t ) = A11x1(t )+ A12 y(t )+B1u(t )y(t ) = A21x1(t )+ A22 y(t )+B2u(t )

(3.4)

where

Tc x =[

x1

y

] l n −pl p

(3.5)

The observer proposed by Utkin [1]-[2] has the form

˙x1(t ) = A11x1(t )+ A12 y(t )+B1u(t )+Lν˙y(t ) = A21x1(t )+ A22 y(t )+B2u(t )−ν (3.6)

where (x1, y) represent the state estimates for x1 and y , L ∈R(n−p)xp is a constant feedbackgain matrix and the discontinuous vector ν, of appropriate dimension, is define compo-nentwise by

νi = M sg n(yi − yi ) (3.7)

where M ∈ R+. If the errors between the estimates and the true states are written as e1 =x1 − x1 and ey = y − y , then from equations (3.4) and (3.6) the following error dynamicalsystem is obtained

e1(t ) = A11e1(t )+ A12ey (t )+Lνey (t ) = A21e1(t )+ A22ey (t )−ν (3.8)

that in compact form can be rewritten as follows

e = Ae(t )+Γν wher e Γ=[

L−Ip

](3.9)

Since the pair (A,C ) is observable, the pair (A11, A21) is also observable. As a conse-quence, L can be chosen to make the spectrum of A11 +L A21 lie in C−. Define a furtherchange of coordinates, dependent on L, by

T =[

In−p L0 Ip

](3.10)

it results

e =[

e1

ey

]= Te =

[e1(t )+Ley (t )

ey (t )

](3.11)

and system in compact form (3.9), with respect to the new coordinates, can be rewrite as

˙e = T Ae(t )+TΓν (3.12)

From (3.11), system (3.12) can be rewritten as

˙e1(t ) = A11e1(t )+ A12ey (t )ey (t ) = A21e1(t )+ A22ey (t )−ν (3.13)

3.1. SMO FOR LINEAR SYSTEMS 45

where A11 = A11 +L A21, A12 = A12 +L A22 − A11L and A22 = A22 − A21L.It follows from (3.13) that in the domain

Ω= (e1(t ),ey ) : ‖A21e1(t )‖+ 1

2λmax(A22 + AT

22)‖ey‖ < M −η (3.14)

where η< M is some small positive scalar, the reachability condition

eTy ey <−η‖ey‖ (3.15)

is satisfied. Consequently, an ideal sliding motion will take place on the surface

So = (e1,ey ) : ey = 0 (3.16)

It follows that after some finite time ts , for all subsequent time, ey = 0 and ey = 0 (in meanvalue, i.e., in the Filippov sense). Therefore from equation (3.13) result

˙e1(t ) = A11e1(t ) (3.17)

which, by choice of L, represents a stable system and so e1 → 0, i.e., e1 → 0 and conse-quently x1 → x1 asymptotically. Equation (3.17) presents the reduced order sliding modeerror dynamics.

ExampleConsider now the problem of designing a sliding mode observer for the system in (3.1)described by

A =[

0 1−2 0

]B =

[01

]C = [

1 1]

(3.18)

which is observable since (r ank([C ;C ∗ A]) = 2) and represent a simple harmonic oscilla-tor. For simplicity assume u = 0. Define a nonsingurar matrix

Tc =[

1 01 1

](3.19)

and the change of coordinates according to (3.10)-(3.5)

C =C Tc−1 = [0 1]; A = Tc ATc

−1 =[ −1 1−3 1

]; B = Tc B =

[01

](3.20)

The system is now in the form given in equation (3.4). An appropriate choice of gain inthe observer given in (3.6) is L = 0.5 which results in an error system governed by A11.The simulation results which follows were obtained setting the gain of the discontinuousoutput injection term M = 1 and the following initial conditions:[x1(0), y(0)] = [1 0], [x1(0), y(0)] = [0 0].


Figure 3.1: Utkin Observer

3.2. SMO FOR LINEAR SYSTEMS PARTIALLY DRIVEN BY UNKNOWN INPUTS 47

3.2 SMO for linear systems partially driven by un-known inputs

The problem of designing an observer for a multivariable linear system partially drivenby unknown inputs is of great interest. Such a problem arises in systems subject to dis-turbances or with inaccessible/unmeasurable inputs and in many applications such asfault detection and isolation, parameter identification and cryptography. This problemhas been studied extensively for the last two decades.

Consider initially the uncertain dynamical system

x = Ax(t )+Bu(t )+E d(t , y,u)y =C x(t )

(3.21)

where A ∈ Rnxn ,B ∈ Rnxm ,C ∈ Rpxn and E ∈ Rnxq with p ≥ q . Assume that the matrices B,C and D are full rank and the function ξ(t , y,u) is unknown but bounded, so that

‖d(t , y,u)‖ ≤ r1‖u‖+α(t , y) (3.22)

where r1 is a known scalar and α :R+×Rp →R+ is a known function.

3.2.1 A classical approach

The problem of estimating the states of the uncertain system given in (3.21) was approachedby W al cot t and Z ak (1987) [17]. Such a strategy, although intuitively appealing, necessi-tates the use of algebraic manipulation tools to effectively solve an associated constrainedLyapunov problem for systems of reasonable order. Edwards and Spurgeon (1998) [16]propose an observer strategy, similar in style to that in [17], which circumvents the useof symbolic manipulation and offers an explicit design algorithm. This approach will beoutlined here.

Let (A,E ,C ) represent the linear part of the uncertain system which represents thepropagation of the uncertainty through the output. Define an observer for the uncertainsystem (3.21) of the form

z(t ) = Az(t )+Bu(t )−GlCe(t )+Gnν (3.23)

where e=z-x and ν is discontinuous about the hyperplane

S0 = e ∈Rn : ey =Ce = 0 (3.24)

and Gl ,Gn ∈ Rn×p are gain matrices whose precise structure is to be determinate. Asdemonstrated in [16] the following proposition holds:

Proposition 1 A sliding mode observer of the form (3.23) which rejects the uncertaintyclass in (3.21) exist if and only if the nominal linear system, defined by the triple (A,E,C),satisfies:

• rank(CE)=q

• any invariant zeros of (A,E,C) must lie in C−


For a square system, where p = q , it should be noted that the above two conditionsfundamentally require the triple (A,E,C) to be relative degree one and minimum phase.Note also that these system theoretic conditions depend upon a specific selection of un-certainty channel. In the following a canonical form will be introduced, as presented in[16], since it’s useful to explore the pertinent characteristics of SMO for linear, uncertainsystems.

Lemma 1: Let the triple (A,E ,C ) represent a linear system with p > q and supposer ank(C D) = q. Then a change of coordinates x → T0x exist [16] so that the triple (A, E ,C )with respect to the new coordinates has the following structure:

(a) The system matrix can be written as

A = T0 AT0−1

A11 A12

A211

A212A22

(3.25)

where A11 ∈R(n−p)×(n−p), A211 ∈R(p−q)×(n−p) and when partitioned have the structure

A11 =[

A011 A0

120 A0

22

]and A211 = [0 A0

21] (3.26)

where A011 ∈Rr×r and A0

21 ∈R(p−q)×(n−p−r ) for some r ≥ 0 and the pair (A022, A0

21) is com-pletely observable. Furthermore, the eigenvalues of A0

11 are the invariant zeros of (A,E,C).

(b) The disturbance distribution matrix has the form

E = T0E =[

0E2

]and E2 =

[0

E2

] l p −ql q

(3.27)

where E2 ∈Rq×q is nonsingular.(c) The output distribution matrix has the form:

C =C T0−1 = [

0 T]

(3.28)

where T ∈Rpxp is orthogonal.

By referring to this canonical form, the necessary and sufficient existence conditionsfor the existence of the observer (3.23), that provides quadratic stability of the estimateerror system despite the presence of bounded matched uncertainty, can be obtained asformally proved in [16]. For completeness, an outline of the key arguments is presentedhere in order to emphasize the key structure of a SMO as in 3.23.

(proof of necessity)

Let Gl and Gn in (3.23) be appropriate gain matrices so that

A0 = A−GlC is stable (3.29)


and assume first that an ideal sliding mode insensitive to the uncertainty exist on thehyperplane in the error space given by S0 in (3.24). Defining the observation error in suchway e = z −x, and considering systems (3.21) and (3.23), the error system satisfies:

e(t ) = A0e(t )−E d(t , y,u)+Gnν (3.30)

Suppose at time Ts the observation error lie on hyperplane S0 = e ∈ Rn : Ce = 0, i.e.,an ideal sliding motion take place. Mathematically this can be expressed as Ce(t ) = 0 andC e(t ) = 0 for all t > Ts . Substituting for e(t ) from (3.30) gives

C e(t ) =C A0e(t )−C E d(t , y,u)+CGnν= 0 (3.31)

Definition 1: If det (CGn) 6= 0, the equivalent control associated with the nominal sys-tem (3.30), written as νeq , is defined to be the unique solution to the algebraic equation(3.31), namely

νeq = (CGn)−1(C E d(t , y,u)−C A0e(t )) (3.32)

It must be highlight that the equivalent control is directly dependent from the uncer-tain term, as expected. The ideal sliding motion is given by substituting the expression ofequivalent control (3.32) into equation (3.30) which result in a free motion, i.e., a motionindependent of the control action and given by

e(t ) = (I −Gn(CGn)−1C )A0e(t )− (I −Gn(CGn)−1C )E d(t , y,u) (3.33)

To be insensitive to the uncertainty it follows that (I −Gn(CGn)−1C )E = 0, or equivalently

E =Gn(CGn)−1C E (3.34)

Since by assumption, for original uncertain linear system in (3.21), r ank(E) = q , it fol-lows immediately that from equation (3.34) rank(CE)=q, further it means that we canassume, without loss of generality, that the system (A,E ,C ) is in canonical form given bythe change of coordinates stated in Lemma 1.

If the nonlinear gain matrix is partitioned so that

Gn =[

G1

G2

] l n −pl p

(3.35)

then, from (3.28) and (3.35), result CGn = TG2 and so det (G2) 6= 0. From equation(3.33) and using standard arguments, it follows that the poles of the (linear) reduced-ordermotion, given by e(t ) = (I −Gn(CGn)−1C )A0e(t ), are

λ((A0)11 −G1G−12 (A0)21) (3.36)

where (A0)11 and (A0)21 represent the top left and bottom left sub-block of the closed-loop matrix A0 partitioned in a compatible way to the canonical form. By definition ofthe matrix A0, given in (3.29), result: (A0)11 = A11 − (G1C )11, where (G1C )11 representsthe top left sub-block of the square matrix (G1C ). However, it’s straightforward to verifythat (G1C )11 = 0 for all Gl ∈ Rn×p and so (A0)11 = A11. Similarly it can be shown that(A0)21 = A21 and consequently

λ((A0)11 −G1G−12 (A0)21) =λ(A11 −G1G−1

2 A21) (3.37)


From equation (3.34) it follows that G1G−12 E2 = 0 which, after considering the structure

of E2, implies G1G−12 = [G 0], where G ∈ R(n−p)×(p−q) and therefore from the definition of

A21 it follows that A11 −G1G−12 A21 = A11 −G A211.

By construction the pair (A11, A211) is such that

zer os o f (A,E ,C ) =λ(A011) ⊂λ(A11 −G A211) f or al l G ∈R(n−p)×(n−p) (3.38)

And therefore for a stable sliding motion is necessary that any invariant zeros of (A,E ,C )must lie in C−.

(proof of sufficiency)

Conversely, let (A,E ,C ) represent the system and suppose r ank(C E) = q and any invari-ant zeros lie in C−. It’s assume that the system is already in the canonical form required tofacilitate sliding mode observer design where the matrix A0

11 is stable. As a consequencethere exists a matrix L ∈R(n−p)×(p−q) such that A11+L A211 is stable. Define a non-singulartransformation as

TL =[

In−p L0 T

](3.39)

whereL = [L 0(n−p)×q ] (3.40)

After changing coordinates with respect to TL , the new output distribution matrix be-comes

E =C T −1L = [0 Ip ] (3.41)

and from (3.27)-(3.40) result that

LE2 =[

L 0][

0E2

]= 0 (3.42)

and so the uncertainty distribution matrix is given by

E = TLE =[

LE2

T E2

]=

[0

T E2

](3.43)

Finally, if A = TL AT −1L , it can be shown by direct evaluation that

A11 = A11 +L A211 (3.44)

which is stable by choice of L.

The system triple (A,D ,C ) is now in the form

x1 = A11x1(t )+ A12 y(t )+B1u(t )y = A21x1(t )+ A22 y(t )+B2u(t )+E2d(t , y,u)

(3.45)

where x1 ∈R(n−p), y ∈Rp and the matrix A11 has stable eigenvalues.


Define the the corresponding observer by

˙x1 = A11x1(t )+ A12 y(t )+B1u(t )− A12ey (t )˙y = A21x1(t )+ A22 y(t )+B2u(t )− (A22 − AS

22)ey (t )+ν (3.46)

where AS22 is a stable design matrix and ey = y − y . Let P2 ∈ Rp×p be symmetric positive

Lyapunov matrix for AS22, then the discontinuous vector ν is defined by

ν=−ρ(t , y,u)‖E2‖P2ey

‖P2ey‖(3.47)

where the scalar function ρ :R+×Rp ×Rm →R+ satisfies

ρ(t , y,u) ≥ r1‖u‖+α(t , y)+γ0 (3.48)

and γ0 is a positive scalar. If the state estimation error e1 = x1−x1, then it’s straightforwardto show

e1(t ) = A11e1(t )ey (t ) = A21e1(t )+ AS

22ey (t )+ν−E2d(3.49)

Define Q1 ∈ R(n−p)×(n−p) and Q2 ∈ R(p×p) as symmetric positive definite design ma-trices and define P2 ∈ R(p×p) as the unique symmetric positive definite solution to theLyapunov equation

P2 AS22 + (AS

22)T P2 =−Q2 (3.50)

Using the computed value of P2 define

Q = AT21P2Q−1

2 P2 A21 +Q1 (3.51)

Note that Q = QT > 0 and let P1 ∈ R(n−p)×(n−p) be the unique symmetric positive definitesolution to the Lyapunov equation

P1 A11 + AT11P1 =−Q (3.52)

Taking the quadratic formV (e1,ey ) = eT

1 P1e1 +eTy P2ey (3.53)

as a candidate Lyapunov function, and considering the derivative along the system trajec-tory it result V (e1,ey ) < 0 [16] for (e1,ey ) 6= 0, i.e., the error system is quadratically stable.Therefore considering the hyperplane given in (3.24) then it result that An ideal Slidingmotion takes place on (3.24) in finite time. This finite time convergence property is a keyadvantage of sliding mode observer schemes. Many other observer paradigms guaranteeonly asymptotical properties.If x(t ) represents the state estimate for x and e = x − x, then the robust observer canconveniently be written as

˙x(t ) = Ax(t )+Bu(t )−GlCe(t )+Gnν (3.54)

where the linear gain

Gl = T −10

[A12

A22 − AS22

]. (3.55)


and the non-linear gain

Gn = ‖E2‖T −10

[0IP

](3.56)

and

ν=−ρ(t , y,u)P2Ce

‖P2Ce‖ (3.57)

A key development in this formulation of the sliding mode observer design frame-work is that there is no requirement for (A,C ) to be observable. This is straightforwardto demonstrated as a SMO can be developed for a system that is unobservable as long asthe nominal triple (A,E ,C ) has r ank(C E) = q and any invariant zeros of (A,E ,C ) lie inC−. In many sense, the problem of sliding mode observer design for systems which canbe assumed to possess a core linear triple is solved. Clearly, when transmission zeros arepresent in the representation, these will appear in the poles of the sliding mode dynamicsand only asymptotically stable error convergence will be possible [9].


3.2.2 HOSMO to weaken the 1-relative degree condition

The issue of broadening the class of systems so that the r ank(C E) = q and/or the stabilitycondition on the invariant zeros of (A,E ,C ) is of ongoing interest. Floquet et. al. [24] showthat the relative degree condition can be weakened if a classical sliding mode observeris combined with SM exact differentiators to essentially generate additional independentoutput signals from the available measurements. It’s proved that the transmission zerosof the nominal triple (A,E ,C ) still contribute to the dynamics of the error signal in thesliding mode and thus the condition that any invariant zeros of (A,E ,C ) must lie in C−still remains. We will refer to the linear time-invariant system subject to unknown inputsor disturbances in 3.21, where the output takes the following expression

y = [y1 . . . yp ]T =C x, yi =Ci x (3.58)

As for the previous case d(t , y,u) ∈ Rq stands for the bounded, unknown inputs. It isfurther supposed that q ≤ p. Basing on Proposition 1, if r ank(C E) = r ank(E) = q , i.e.observer matching condition hold, there exist a linear change of coordinates that puts theoriginal system into the canonical form given in 3.45 for which there exist an observer ofthe form

˙x(t ) = Ax(t )+Bu(t )+Gl (y −C x)+Gnν (3.59)

where Gl and Gn are design gains and ν is an injection signal which depends on theoutput estimation error in such a way that a sliding motion in the state estimation errorspace is induced in finite time, and therefore the state estimation error e = x− x is asymp-totically stable and independent of the unknown signal d during the sliding motion.Here the aim is to extend the previous result so that a sliding mode observer can be de-signed for (3.58) when the standard matching condition is not satisfied, more in particular,when rank(CE)<q. To this end, introduce the notation of relative degree µ j ∈ N+, 1 ≤ j ≤ pof the system with respect to the output y j , that is to say the number of times the outputy j must be differentiated in order to have the unknown input d explicitly appear. Thus,µ j is defined as follows:

C j Ak E = 0 f or al l k <µ j −1C j Aµ j−1E 6= 0

(3.60)

without loss of generality, it is assumed that µ1 ≤ . . . ≤ µp . The following assumptions aremade:

• the invariant zeros of A,E,C lie in C−

• there exists a full rank matrix

Ca =

C1...

C1 Aµα1−1

...Cp

...Cp Aµαp −1

(3.61)


where the integers 1 ≤µαi ≤µi are such that r ank(CaE) = r ank(E) = q and the µαi

are chosen such that∑P

i=1µαi is minimal.

Before describing the observer scheme proposed in [24] we introduce the followinglemma that in the cited article is also demonstrated.

Lemma 2: the invariant zeros of the triples A,E ,C and A,E ,Ca are identical

The previous lemma, on which practically the main idea of the method under investiga-tion stands, says that the invariant zeros of the triple A,E ,C and the newly created triplewith additional (derivative) outputs A,E ,Ca are identical. Consequently, if the originalsystem is minimum phase the new triple A,E ,Ca is both minimum phase and relativedegree one and hence a "classical" observer of the form given in (3.59) can be designedfor A,E ,Ca.

The observer design is based in two step.First a sliding mode observer is designed for a system described by the following triangu-lar observable form:

ξ1 = ξ2 +bT1 u

ξ2 = ξ3 +bT2 u

...ξl−1 = ξl +bT

l−1uξl = bl+1

Tθ+bTl u

yξ = ξ1

(3.62)

where ξ= [ξ1 · · ·ξl ]T ∈Rl , (l>1) is the state vector, yξ ∈R is the output, u ∈Rm is the knowninput vector and θ ∈ Rq stands for some unknown inputs. The vectors bi , with i = 1, . . . , l ,of appropriate dimension.Assume that the system is bounded input bounded state in finite time and that the signaloutput yξ, the unknown input θ and θ are bounded.Most of the sliding mode observer designs for (3.62) are based on a step-by-step pro-cedure using successive filtered values of the so-called equivalent output injections ob-tained from recursive first-order sliding mode observers (see e.g. [3]). However, the ap-proximation of the equivalent injections by low pass filters at each step will typically intro-duce delays that lead to inaccurate estimates or to instability for high-order systems. Toovercome this problem it’s possible to replace the discontinuous first-order sliding modeoutput injection by a continuous second-order sliding mode one. The observer is built asfollows:

˙ξ1 = ν(ξ1 − ξ1)+bT1 u

˙ξ2 = E1ν(ξ2 − ξ2)+bT2 u

...˙ξl−1 = El−2ν(ξl−1 − ξl−1)+bT

l−1u˙ξl = El−1ν(ξl − ξl )+bT

l uyξ = ξ1

(3.63)

where ξ1 := yξ and


ξ j := ν(ξ j−1 − ξ j−1), 2 ≤ j ≤ l (3.64)

where the continuous output error injection ν(·) is given by the so-called super twist-ing algorithm [14, 15]:

ν(s) =ϕ(s)+λs |s|12 si g n(s)

ϕ(s) =αs si g n(s)λs ,αs > 0

(3.65)

For i = 1, . . . , l −1, the scalar functions Ei are defined as

Ei = 1 i f |ξ j − ξ j | ≤ ε f or al l j ≤ i el se Ei = 0 (3.66)

Denoting ξ= ξ− ξ, the error dynamic is given by

˙ξ1 = ξ2 −ν(yξ− ξ1)˙ξ2 = ξ3 −E1ν(ξ2 − ξ2)...˙ξl−1 = ξl −El−2ν(ξl−1 − ξl−1)˙ξl = bl+1

Tθ−El−1ν(ξl − ξl )

(3.67)

As argued in [3] the sliding manifolds are reached one by one sequentially , from i = 1to i = l . At each step, a sub-dynamic of dimension one is obtained and consequently nopeaking phenomena appear. It can be verified that with a suitable choice of gains λs andαs , a sliding mode is attained in finite time on the manifold ξ1 = ξ2, . . . ,= ξL = 0 and thefollowing equivalent output injection is obtained

ν(ξl − ξl ) = bTl+1θ (3.68)

Note that the step-by-step observer achieves finite time recovery of the state components.

As second step, in order to estimate the state of system (3.58), the following SMO hasbeen proposed [24]

˙z(t ) = Az(t )+Bu(t )+Gl (ya −Ca z)+Gnν(ya −Ca z) (3.69)

where the auxiliary output ya is defined by

ya =

y1

ν(y1 − y11)

...

ν(yµα1−11 − y

µα1−11 )

...yp...

ν(yµαp −1p − y

µαp −1p )

(3.70)


where the constituent signals in (3.70) are given from the step-by-step observer:

y1i = ν(y1

i − y1i )+Ci Bu

y2i = E1ν(y2

i − y2i )+Ci ABu

...

yµαi −1

i = Eµαi −2ν(yµαi −1

i − yµαi −1

i )+Ci Aµαi −2

i Bu

(3.71)

for 1 ≤ i ≤ p, with

y1i := yi

y ji := ν(y j−1

i − y j−1i ), 2 ≤ j ≤µαi −1

(3.72)

where the injection operator ν(·) is defined by (3.65). The discontinuous output injectionνc in (3.69) could be defined by

νc (ya −Ca z) =−ρ P2(ya −Ca z)

‖P2(ya −Ca z)‖ (3.73)

where ρ is a positive constant larger than the upper bound of d . The definition of thesymmetric positive matrix P2 has been given in 3.50.

Let us define the state estimation error e = x−z and the augmented output estimationerror ey =Ca x − y , with

ey = [e11, . . . ,e

µαi −11 , . . . ,e1

p , . . . ,eµαp −1p ]T

y = [y11 , . . . , y

µαi −11 , . . . , y1

p , . . . , yµαp −1p ]T

(3.74)

where

e1i =Ci Ax −ν(yi − y1

i )e2

i =Ci A2x −E1ν(y2i − y2

i )...

eµαi −1

i =Ci Aµαi −1x −Eµαi −2ν(yµαi −1

i − yµαi −1

i )

(3.75)

for i ≤ i ≤ p. Thus, choosing suitable output injections ν, the following relations holdafter a finite time T:

ν(yi − y1i ) =Ci Ax

ν(y2i − y2

i ) =Ci A2x...

ν(yµαi −1

i − yαi−1i ) =Ci A2x

(3.76)

for i ≤ i ≤ p. This means that ya =Ca xThen it is straightforward to show that

e = Ae +Dw −Gl (ya −Ca z)−Gnνc (ya −Ca z) (3.77)

Thus, for all t > T , the error dynamics (3.77) are given by

e = (A−GlCa)e +E d −Gnνc (Cae) (3.78)

Since by construction r ank(CaE) = r ank(E) and by assumption the invariant zeros ofthe triple (A,E ,Ca) lie in the left half plane, the design methodology presented in §3.2.1


can be applied so that e = 0 is an asymptotically stable equilibrium point of (3.78) andthe dynamics are independent of d once a sliding motion on the sliding manifold e : s =Cae = 0 has been attained. In addition, the method here presented enables an estimationof the unknown inputs. Define (vc )eq as the equivalent output error injection required tomaintain the sliding motion in 3.78, during the sliding motion

s =Ca e =Ca(A−GlCa)e +CaE d −CaGn vc (Cae) = 0 (3.79)

Since e → 0 and using (3.78)CaGn(vc )eq →CaE d (3.80)

As CaE is full rank, an approximation d of d can be obtained from vc eq by

d = ((CaE)T CaE)−1(CaE)T CaGn vc eq (3.81)


3.3 Non-linear approaches to Sliding Mode Observersdesign

In this sections the problem of designing observers for state estimation of non-linear sys-tems using sliding mode will be addressed [9]. Early contribution in this area were devel-oped independently by Walcott and Zak ([10]-[11]) and Slotine et al. [12], where the latterteam considered a more extended class of system. Probably the next major breakthroughin the development of SMO for non-linear systems appears in the article by Drakunovand Utkin [2] where the equivalent injection concepts was introduced for observer design.Barbot et al. [3] developed a sliding mode observer for a non-linear system in a triangu-lar input form. Such systems were originally considered in the work of Drakunov andUtkin [2] and are important because it is possible to develop an observer without usinginput derivatives. It must be highlight that sliding mode observers, are usually designedunder the assumptions that the nominal uncertain system can be put into a triangular ob-servable form, where the uncertain term act only on the last dynamics. This assumptionis usually known as the observability matching condition for analogy to the well-knownmatching condition for a sliding mode controller to be insensitive to matched perturba-tions. In [25] is considered the case of a step-by-step SMO for autonomous nonlinearsystems with unknown inputs, based on the hierarchical application of the supertwistingalgorithm in such a way similar to that presented in §3.2.2. In such approach the systemunder observation is assumed already in triangular observable form. In this section willbe presented an approach to finite time HOSMO [31] that does not require the system tobe reduced to any normal form, which can be difficult to achieve in the presence of modeluncertainties.An important implications of the former observer is the possibility of reconstruct the un-certain term affecting the uncertain nonlinear system. Another method very useful whenthe problem of the input reconstruction is considered, is associated to the the concept ofalgebraic observability. The problem of the input reconstruction for non-linear time in-variant dynamic system, making use on the concept of algebraic observability and slidingmode differentiators, has been approached successfully in the work of Cannas et al. [36].

3.3. NON-LINEAR APPROACHES TO SLIDING MODE OBSERVERS DESIGN 59

3.3.1 Systems in the companion form

As in Slotine et al.[12], let us consider a non-linear system in companion form

x(n) = f (x, t ) (3.82)

where f (x, t ) is a non-linear, uncertain function of the system state and x1 is the singlemeasurement available. Define a corresponding sliding mode observer

˙x1 =−α1e1 + x2 −k1sg n(e1)˙x2 =−α2e1 + x3 −k1sg n(e1)· · ·˙xn =−αne1 + f −kn sg n(e1)

(3.83)

where e1 = x1 − x1, f is an estimate of f (x, t ) and the constant αi are chosen as for aclassical Luenbergher observer to ensure asymptotical error decay of a corresponding lin-earized system representation, where ki = 0. The corresponding error dynamics are givenby

e1 =−α1e1 +e2 −k1sg n(e1)e2 =−α2e1 +e3 −k1sg n(e1)· · ·en =−αne1 +∆ f −kn sg n(e1)

(3.84)

where ∆ f = f − f is assumed bounded and

k(n) ≥ |∆ f | (3.85)

The sliding condition (d/d t )(e1)2 < 0 is satisfied in the region

e2 ≤ k1 +α1e1 i f e1 > 0e2 ≥−k1 +α1e1 i f e1 < 0

(3.86)

From first equation of (3.84), when a sliding mode is attained on e1 = 0 it follows that inFilippov sense the following hold

e2 −k1sg n(e1) = 0

and thereforee2 = e3 − k2

k1e2

· · ·en =∆ f − kn

k1e2

The valuesαi are thus seen to only affect the dynamic performance prior to the reach-ing of this region, which is often called the sliding patch, and the dynamics on the patchare determined by ∣∣∣∣∣∣∣∣∣∣

λIn−1 −

−k2k1

1 0 · · · 0

−k3k1

0 1 · · · 0· · · · · · · · · · · · · · ·−kn

k10 0 · · · 1

∣∣∣∣∣∣∣∣∣∣= 0 (3.87)


Assuming kn is selected as a constant ratio with k1 and that the poles defining thedynamics on the patch are critically damped i.e. are real and equal to some constantvalue γ, then Slotine et al. [12] show that

|e(i )2 | ≤ (2γ)i k1 i = 0, . . . ,n −2 (3.88)

from which the precision of the state estimates can be determined. As well as defining theconcept of the sliding patch, the contributions of Slotine et al. [12] discussed the effectof measurement noise on sliding mode observer. It was demonstrated that, as would beexpected, the system does not attain a sliding mode in the presence of noise, but effec-tively remains within a region of the sliding patch which is determined by the bound onthe noise. Moreover, it was demonstrated that the average dynamics can be modified byselection of the ki which in turn can tailor the contribution of the noise to the state esti-mates.


3.3.2 Triangular systems

As in [3], let us consider a non-linear system in the triangular input form

ξ1 = ξ2 + g1(ξ1,u)ξ2 = ξ3 + g2(ξ1,ξ2,u)· · ·ξn−1 = ξn + gn−1(ξ1,ξ2, . . . ,ξn−1,u)ξn = fn(ξ1,ξ2, . . . ,ξn)+ gn(ξ1,ξ2, . . . ,ξn ,u)

(3.89)

where y = ξ1, the terms gi (·) are assumed known, gn(·,0) = 0 for i = 1, . . . ,n, and the systemis assumed bounded input bounded state (BIBS) in finite time.

As in Drakunov and Utkin [2], Barbot et al. [3] define the following SMO

˙ξ1 = ξ2 + g1(ξ1,u)+λ1si g n(ξ1 − ξ1)˙ξ2 = ξ3 + g2(ξ1, ξ2,u)+λ2si g n(ξ2 − ξ2)· · ·˙ξn−1 = ξn + gn−1(ξ1, ξ2, . . . , ξn−1,u)+λn−1si g n(ξn−1 − ξn−1)˙ξn = fn(ξ1, ξ2, . . . , ξn)+ gn(ξ1, ξ2, . . . , ξn ,u)+λn si g n(ξn − ξn)

(3.90)

whereξi = ξi +λi−1si g n(ξi−1 − ξi−1) (3.91)

for i = 2, . . . ,n−1 and the si g n(·) function is computed using filtered version of argumentand the anti-peaking methodology of Khalil [13] is employed. Effectively the observationerror information is not used in the implementation before the corresponding slidingmanifold is reached. The manifolds are reached sequentially one by one and ξi − ξi con-verges to zero if the error ξ j − ξ j , with j < i , have already converged to zero. The motiva-tion for the above, which is effectively a sequential consideration of a series of first-orderdynamics, is easily seen by forming the error dynamics for ei = ξi − ξi :

e1 = e2 −λ1si g n(ξ1 − ξ1)e2 = e3 + g2(ξ1,ξ2,u)− g2(ξ1, ξ2,u)−λ2si g n(ξ2 − ξ2)· · ·en−1 = en + gn−1(ξ1,ξ2, . . . ,ξn−1,u)− gn−1(ξ1, ξ2, . . . , ξn−1,u)−λn−1si g n(ξn−1 − ξn−1)en = fn(ξ1,ξ2, . . . ,ξn−1,u)− fn(ξ1, ξ2, . . . , ξn ,u)

+gn(ξ1,ξ2, . . . ,ξn ,u)− gn(ξ1, ξ2, . . . , ξn ,u)−λn si g n(ξn − ξn)(3.92)

It can be verified that for sufficiently large λ1, a sliding mode is attained on e1 = 0 infinite time and it follows that

e2 =λ1si g n(ξ1 − ξ1) (3.93)

which with 3.91 yields ξ2 = ξ2. The observation error dynamics become


e1 = 0e2 = e3 −λ2si g n(e2)· · ·en−1 = en + gn−1(ξ1,ξ2, . . . ,ξn−1,u)− gn−1(ξ1,ξ2, . . . , ξn−1,u)−λn−1si g n(ξn−1 − ξn−1)en = fn(ξ1,ξ2, . . . ,ξn−1,u)− fn(ξ1, ξ2, . . . , ξn ,u)

+gn(ξ1,ξ2, . . . ,ξn ,u)− gn(ξ1, ξ2, . . . , ξn ,u)−λn si g n(ξn − ξn)(3.94)

Proceeding as before it can be shown that for sufficiently large λ2 a sliding mode isthen attained on e2 in finite time and it follows that

e3 =λ2si g n(ξ2 − ξ2) (3.95)

which yields ξ3 = ξ3. Applying the same methodology up to the en dynamic produce

e1 = 0e2 = 0· · ·en−1 = 0en =−λn si g n(en)

(3.96)

and it follows trivially that a sliding mode is finally attained on en = 0 in finite time. Thisfinite time property of sliding mode observer is very attractive and has led to the develop-ment of a number of application specific sliding mode observers.


3.3.3 Quasi-continuous HOSM observers

The referred nonlinear uncertain system is the following [31]

x = f (x)+ f (x)y = h(x)

(3.97)

with the state vector x ∈ X ⊆ Rn and the scalar output y ∈ Y ⊆ R. The vector fieldsf (x) : X →Rn and h(x) : X →Y represent the known nominal part of the system dynam-ics, while f (x) : X →Rn is assumed to be uncertain.

The aim is that of designing a finite-time converging observer for the above system.The following observer structure has been considered [31]

˙x = f (x)+ g (x)uy = h(x)

(3.98)

with observed state vector x ∈ Rn and observed output variable y ∈ R. The design ofthe vector field g (·) and the corrective term u ∈ R are the main topics of the followingdiscussions. In particular, under a certain “ observability condition “ that is going to bespecified, we shall select the vector function g (·) in such a way that the observer outputhas full relative degree n with respect to the observer input u.

To the aim of introducing some important concepts let us introduce the operatord q(z) applied to a generic scalar function q with vector argument z defined on an openset Ω⊂Rn , q(z) :Rn →R.

We denote

d q(z) = ∂q(z)

∂z=

[∂q(z)

∂z1,∂q(z)

∂z2, . . . ,

∂q(z)

∂zn

]

Now, let us define the matrix

M(z) =

dh(z)dL f (z)h(z)

...dLn−2

f (z)h(z)

dLn−1f (z)h(z)

(3.99)

where L f (z)h(z) is the so-called Lie derivative of h(z) along f (z) and is defined as

L f (z)h(z) = ∂h(z)∂z f (z) and the kth derivative of h(z) along f (z) is defined as Lk

f (z)h(z) =∂Lk−1

f (z)h(z)

∂z f (z).The following assumption is assumed to holdAssumption 1 The matrix M(z) is nonsingular for every possible value of z.

The vector g (x) in (3.98) will be designed according to equation

M(x)g (x) = [0,0, . . . ,1]T (3.100)

from which, from assumption 1, descend

g (x) = M−1(x) · [0,0, . . . ,1]T (3.101)


i.e., g (x) is the last column of matrix M−1(x).In light of (3.101) the observer input/output dynamics is

d

d t

y˙y...

y (n−1)

=

L f (x)h(x)L2

f (x)h(x)...

L(n)f (x)h(x)

+

00...01

u (3.102)

which means that the observer output y has full relative degree n with respect to theobserver input u.

Let us define the n-dimensional output error vector ε, containing the output errorey = y − y ∈R and its first n −1 derivatives:

ε=

ε1

ε2...εn

=

ey

ey...

e(n−1)y

. (3.103)

and the state observation error

e = x −x =

e1

e2...

en

.

It must be highlight that the observer output error possesses the same relative degree nwith respect to u.

In [31] has been demonstrated that, under assumption 1, the observer (3.98), (3.101)can reconstruct the state of the nominal system (3.97) with f (x) = 0, i.e., the system isperfectly known, exactly and in finite time, provided that the observer input u is selectedin such a way that the vector ε is steered to zero in finite time. In other words it meansthat

ε= 0 ⇔ e = 0 (3.104)

It must be highlight that it is true for any uniformly observable system (see [32]), i.e.,for systems which are observable independently of the inputs.

Taking into account the uncertain term f (x), not modeled in the observer structure,an additional, necessary and sufficient, condition is needed to guaranty the preservationof implication 3.104. Such condition, referring to the system (3.97), consists in the follow-ing requirement

L f (x)h(x)L f (x)L f (x)h(x)

...L f (x)L

n−2f (x)h(x)

=

00...0

(3.105)


It means that the relative degree of the disturbance with respect the output, and there-fore also with respect the output error ey , must be equal to n.In order to design a robust state observer, for the system (3.97), of the form (3.98), (3.101),under the condition 3.105, a controller which stabilizes in finite time the observation errorε must be designed.

The observation output error dynamics takes the following Brunovsky canonical form

ε1 = ε2

ε2 = ε3

. . .εn = Φ(e, x)+u

(3.106)

whereΦ(e, x) = Ln

f (x)h(x)−Lnf (x−e)h(x −e)−L f (x−e)L

n−1f (x−e)h(x −e) (3.107)

for which the following assumption is meet

|Φ(e, x)| < Γ (3.108)

In [31], the “quasi-continuous “arbitrary-order sliding mode controller [33, 34], based onthe so-called “arbitrary-order" sliding-mode approach [33], was suggested in order to sta-bilize (3.106)-(3.108) in finite time. For completeness, the cited algorithm is reported infollowing.

Let i = 1, . . . ,n −1 and denote

ϕ0,n = ey , N0,n = |ey | (3.109)

Ψ0,n = ϕ0,n/N0,n = sign ey , (3.110)

ϕi ,n = e(i )y +βi N (n−i )/(n−i+1)

i−1,n Ψi−1,n , (3.111)

Ni ,n = |e(i )y |+βi N (n−i )/(n−i+1)

i−1,n , (3.112)

Ψi ,n = ϕi ,n/Ni ,n (3.113)

where β1, ..., βn−1 are positive numbers. The quasi-continuous n-sliding controller is

u =−αΨn−1,n(ey , ey , ..., e(n−1)y ). (3.114)

It was shown in [33, 34] that provided that the tuning parameters β1, ..., βn−1,α arechosen sufficiently large in the given order then the control law defined by (3.109)-(3.114)stabilizes the system (3.106)-(3.108) in finite time.

Note that control defined by (3.109)-(3.114) is globally bounded (|u| ≤ α) and contin-uous everywhere but the origin of the n-dimensional error space, from which the nameof the controller was derived. Following are reported example of second and third-orderquasi-continuous controllers:

u =−α ey+|ey |1/2 signey

|ey |+|ey |1/2 ,

u =−α ey+2(|ey |+|ey |2/3)−1/2(ey+|ey |2/3 signey )

|ey |+2(|ey |+|ey |2/3)1/2

(3.115)

The n-th order quasi-continuous controller 3.114 requires the availability of the suc-cessive derivatives of the output estimation error up to the order n −1. In order to recon-struct such derivatives exactly and in finite time, the well known Arbitrary-Order sliding-mode differentiator by A. Levant [35] can be used.


The n-th order differentiator can be expressed in the following non-recursive form

z0 = v0 = z1 −κ0|z0 −ey (t )| nn+1 si g n(z0 −ey (t )),

z1 = v1 = z2 −κ1|z1 − v0|n−1

n si g n(z1 − v0),. . .

zi = vi = zi −κi |zi − vi−1|n−i

n−i+1 si g n(zi − vi−1),. . .zn =−κn si g n(zn − vn−1)

(3.116)

for suitable positive constant coefficients κi to be chosen recursively large in the givenorder ([35]). Under the assumption that the n −1th derivative of ey is Lipshitz, i.e., a real

positive a constant C exists such that |e(n)y | ≤ C , the following equalities are true in the

absence of measurement noise after a finite time transient process:

|zi −e(i )y (t )| = 0 i = 0, ...,n (3.117)

Clearly, for the considered closed-loop system (3.106)-(3.108) the C constant existsand it is is overestimated by C = Γ+α. The separation and robustness results relevant tothe combined use of the above differentiator and any n-sliding homogenous controllerwere discussed in [35]. It was demonstrated by Levant [35] that non-idealities like mea-surement noise and finite frequency commutation cause a bounded error in the esti-mated derivatives and, as a result, a bounded loss of accuracy for the controller that usesthe "noisy" derivative estimates [33], [34], [35].

An important implications of the presented observer is the possibility of reconstructthe uncertain term f (x) of system 3.97. In the work of Davila et al. [31] a special instanceof the general uncertain dynamics f (x) represented by some bounded unknown inputterm v(x, t ) ∈ R premultiplied by a known distribution matrix D of appropriate dimen-sion, has been considered, and a technique aimed to reconstruct the exact unknown in-put v(x, t ) has been proposed.


3.3.4 Algebraic observers

Let us introduce some concepts about the algebraic observability and the input recon-struction by differential algebra [37]. Let we refer to the following non-linear system

x = f (x,u(t ))y = h(x)

(3.118)

where x is the n-dimensional state variable vector, u is a scalar input, y a scalar out-put, and f and h, are algebraic function vectors in x. We assume that the state is notdirectly available and only the scalar output is measured. The goal to be attained is thereconstruction of the system input u by using only the information contained in y .

An algebraic observer for the input u of system 3.118 is a polynomial of the form

P = p(u, y, y (1), . . . , y (m)). (3.119)

with m ≤ n. In such cases the input of system 3.118 can be reconstructed if a smoothfunction g exists such that u = g (y, y (1), . . . , y (m)).

The algebraic observability property can be easily tested within the differential alge-bra context by resorting to the concept of "Characteristic Set" associated to the dynamicequations. In order to define the Characteristic Set, we need to introduce some conceptsinherent to the Differential Algebra.

Remark 1If one chooses the ranking of the variables and their derivatives

u < u(1) < u(2) < ·· · < y < x1 < x2 < ·· · < y (1) < x1(1) < x2

(1) < ·· · < y (n) < xn(2) < x2

(n) < ·· · (3.120)

for a system of the form 3.118, the Characteristic Set exhibit n+1 differential polynomials,that is:

• an Input/Output (I/O) relation that is a differential polynomial in u, y and theirderivatives, and is denoted by k(u, y);

• n differential polynomials, triangular with respect to the state components, and de-noted by the n-dimensional vector K (u, x, y).

Property 1The input is algebraically observable if at least one of the following equivalent rela-

tions is verified:

1. derivatives of the input u do not appear in the Characteristic Set;

2. k(u, y) is of order n in the output y .

Proposition 1A necessary condition for the finite time global reconstruction of the input of system

(1) is that the I /O relation k(u, y) is an algebraic observer of the input of the type P =p(u, y, y (1), . . . , y (m))


It means that if the I/O relation is not an algebraic observer, it contains some deriva-tives of the input, hence the input is the solution of a differential equation whose initialconditions are unknown, therefore if no further information is available the input admitsinfinite solutions. Conversely, if the I/O relation is an algebraic observer, the number ofsolutions is finite.

Proposition 2The input function can be globally recovered in a finite time iff it appears in the I/O

relation with order zero and degree one.

Therefore, considering the reference dynamic system described by Eq. 3.118 and as-suming that the input is algebraically observable, there exists an algebraic I /O relationbetween the input u and the output y with its first m ≤ n derivatives, i.e.,

ξ(y, y (1), y (2), · · · , y (n),u) = 0 (3.121)

For a globally recoverable input, Eq. 3.121 can be written as

u = g (y, y (1), y (2), · · · , y (n)) (3.122)

Thus, in the case of global reconstruction, an m-order differentiator, realized for in-stance with the Arbitrary-Order sliding-mode differentiator by A. Levant [35] in 3.116, al-lows one to determine the input to the system uniquely by starting from the output mea-sures and independently of the initial conditions. Conversely, in the case of local recon-struction, the state of the system and the input signal are determined uniquely, providedthat the initial conditions of the reconstructor are close to the initial conditions of thesystem [24] and [36].

Chapter 4

Sliding Modes for FDI

It is well known that the core element of model-based fault detection in control systemsis the generation of residual signals which act as indicators of faults. The residual signalsare generated through comparison between measurement estimates and real measuredquantities. For the design of residual generators, various approaches have been discussedin the literature. In particular, the basic idea behind the use of the observer for fault detec-tion is to estimate the outputs of the system from the measurements by using some typeof observer, and then construct the residual by a properly weighted output estimate er-ror. The residual is then examined for the likelihood of faults by using a fixed or adaptivethreshold. When, in the observer-based approaches, a full order observer is used in resid-ual generator design, the main design procedure becomes an equivalent state feedbackcontrol problem because of the dual relation between the state feedback control and thefull order observer design. Based on this idea, some well-established approaches for statefeedback control can be readily applied to robust fault detection using full order unknowninput observers. Because of the existence of system complexities such as nonlinearities,disturbances, and uncertainties in a typical complex control system, fault diagnosis forsuch dynamical systems still pose a number of challenging problems. Amongst variousuncertainties, unknown inputs are one type of uncertainty that has received consider-able attention. To deal with the unknown inputs, robust approaches are often employed.Furthermore when the system under consideration is subject to unknown disturbanceor unknown inputs, to achieve effective fault detection, the effect of the disturbance hasto be de-coupled from the residual signal to avoid false alarms in detection. This prob-lem is known in the literature as robust fault detection or fault detection using unknowninput observers. Sliding mode techniques have good robustness and are completely in-sensitive to so-called matched uncertainty [16]-[7]. It has been shown that sliding modetechniques can also be used to deal with both structural and unmatched uncertainty [18],therefore the application of sliding mode techniques for robust FDI offers good potential.A sliding mode observer was used for FDI in the early nineties [21] where Sreedhar, Fer-nandez and Masada consider a model-based sliding mode observer approach although intheir design procedure it is assumed that the states of the system are available. A differentapproach is adopted by Hermans and Zarrop [22], who attempt to design an observer insuch a way that in the presence of a fault the sliding motion is destroyed. More recently

69

70 CHAPTER 4. SLIDING MODES FOR FDI

Edwards et al.(2000) [4] proposed an approach based on the concept of equivalent outputinjection in which the resulting reconstruction signal can approximate the actuator faultsto any required accuracy (this is called ’precise’ fault reconstruction). In [4] the authorsconsider the practical situation when the system states are not available. The observer isdesigned to maintain a sliding motion even in the presence of faults which are detectedand identified by analyzing the so-called equivalent output injection. The novelty lies inthe manipulation of the equivalent output injection signal to explicitly reconstruct faultsignals rather than detect the presence of a fault through a residual signal (This may be al-lied to the equivalent control signal which appears in the analysis of sliding mode). Faultreconstruction is a powerful alternative to the detection of a fault via the use of a resid-ual signal as long as the location of the fault effect on the system is known. The residualapproach is more suited to the combined problem of fault detection and fault isolation,when the structure of the fault influence on the system is not perfectly known. A bank ofdissimilar (but redundant) residual signals can then be used to infer the location of thefault in the system. On the other hand, the fault estimation approach is a direct way ofproviding fault information which, when compared with other fault estimation signals(from the same system), can be used to isolate all faults. The fault estimation methodalso provides a direct estimate of the size and severity of the fault, which can be impor-tant in many applications. Later, it was extended by Tan and Edwards (2002) [19] wheresensor faults were considered. However, uncertainty was not considered in these earlypapers. It is well-known that the observer-based approach is very dependent on the sys-tem model. In practice a precise and accurate model for a real system is often not avail-able due to unknown exogenous disturbances and/or time-varying parameters (compo-nent aging). Modelling uncertainty can cause false and missed alarms, which may makethe FDI system useless. Hence, it is very important to consider robustness when imple-menting FDI schemes. A FDI scheme for a class of linear systems with uncertainty wasproposed by Tan and Edwards (2003) [20] which focused on minimizing the L2 gain be-tween the uncertainty and the fault reconstruction signal by using linear matrix inequal-ities (LMI). A robust fault detection method for nonlinear systems with disturbances wasconsidered in Floquet et. al. 2004 [23] where strict geometric conditions are exploitedto the aim of design a residual generator when the faults cannot be decoupled from thedisturbance inputs. It should be emphasized that ’precise’ fault reconstruction is verychallenging for nonlinear systems especially in the presence of uncertainty. When uncer-tainty is considered, all the results concerning sliding mode observer-based fault recon-struction only provide an estimate of the fault signal. It is a valuable meaningful task toestablish an approach for fault reconstruction in nonlinear systems, or to find conditionsunder which ’precise’ fault reconstruction is feasible. Moreover, since FDI is required totake place on-line in real engineering systems, this requires the reconstruction fault sig-nal to be based only on the available measured information. Recently Yan and Edwards(2007) [5] extends previous result, Edwards et al. (2000) [4], for a class of nonlinear un-certain systems where the uncertainty is allowed to have a nonlinear bound. A sufficientcondition based on LMIs is presented for the existence and stability of a robust slidingmode observer. Then, fault estimation and fault reconstruction methods are presentedusing the equivalent output injection approach. It is shown that, under certain geometricconditions associated to the uncertainty structure matrix and the fault distribution ma-trix, ’precise’ fault reconstruction is available for a class of nonlinear systems by exploit-

71

ing the features of the sliding motion and the structure of the uncertainty. The proposedreconstruction signal converges to the fault with arbitrary accuracy even in the presenceof uncertainty. If the geometric condition does not hold, then a strategy is presented toestimate the fault signal, and the estimation error depends on the bounds on the uncer-tainty. The proposed fault estimation/reconstruction signals are only based on the avail-able plant input/output information and can be calculated on-line. It must be highlightthat the main limitation of the fault diagnosis schemes, that make use of conventionalfirst order SMOs, is that the relative degrees from the inputs and/or the unknown inputsto the outputs must be one. Furthermore for the fault reconstruction signals, obtainedby processing the so called "equivalent output error injection", the use of low pass filtersis needed and the observer guaranties the exact reconstruction of fault only in Filippovsense. Because many physical systems such as satellite control systems, and mechani-cal systems can not satisfy this condition, new fault diagnosis beyond using conventionalfirst order SMOs are needed. One promising strategy is to use the recently developed highorder sliding mode techniques such as high order sliding mode observers and differentia-tors. Based on high order sliding modes, arbitrary-order exact robust differentiators havebeen studied in the literature. The proposed differentiators can provide exact estimationfor the derivatives of a signal of any order if there is no measurement noise. When noiseis present, the estimation errors of the derivatives will be small if the magnitude of thenoise is small. These properties make high order sliding mode differentiators appealingin fault diagnosis. Although high order sliding mode observers and differentiators haveappealing properties that could be used in fault diagnosis, their great potential has notbeen well recognized in the fault diagnosis community and there are very few results inthis direction.


4.1 SMO for faults reconstruction

As previously discuss, fault reconstruction is a powerful alternative to the detection of afault via the use of a residual signal as long as the location of the fault effect on the systemis known. Based on the approach described in §3.2.1, where an observer for linear systemis designed to maintain a sliding motion even in the presence of matched unknown input,in this section will be presented a method [16] to detected and identified such unknowninputs by analyzing the so-called equivalent output injection. More precisely the manip-ulation of the equivalent output injection signal will be used to explicitly reconstruct faultsignals rather than detect the presence of a fault through a residual signal.

In this section we consider a nominal linear system subject to certain faults describedby

x = Ax(t )+Bu(t )+E fi (t )y =C x(t )+ fo(t )

(4.1)

where A ∈ Rnxn ,B ∈ Rnxm ,C ∈ Rpxn and E ∈ Rnxq with q ≤ p < n, and matrices B,C and Dare of full rank.It must be highlight that in 4.1 we denote with E the fault distribution matrix in order toapply directly the results presented in §3.2.1. The functions fi (t ) and fo(t ) are deemedto represent actuator and sensor faults, respectively, and are assumed to be bounded. It’sfurther assumed that the states of the system are unknown, i.e. p < n and only the signalsu(t ) and y(t ) are available. As one can see the fault signal fi (t ) takes the place of the distur-bance term d(t , y,u) of the uncertain dynamical system considered in 3.21, furthermore. The objective is to synthesize an observer to generate a state estimate x(t ) and outputestimate y =C x such that a sliding mode is established in which the output error

ey (t ) = y(t )− y(t ) (4.2)

is forced to zero in finite time. The particular observer structure described in section 3.2.1will be considered, namely

z(t ) = Az(t )+Bu(t )−Gl ey (t )+Gnν (4.3)

where Gl and Gn are defined in (3.55) and (3.56) respectively.The discontinuous output injection term ν, designed according the works [4]-[16] in

which a first order SM control law has been proposed, takes the following form

ν=−ρ(t , y,u)P2ey

‖P2ey‖(4.4)

where the upperbounding function ρ(t , y,u) and the symmetric positive design matrixP2 are introduced in 3.48 and 3.50, respectively. It will be shown that, provided a slidingmotion can be attained, estimates of fi (t ) and fo(t ) can be computed from approximatingthe so-called equivalent output injection signal required to maintain the sliding motion.

4.1. SMO FOR FAULTS RECONSTRUCTION 73

4.1.1 SMO for reconstructing of the input fault signals

Consider first the case when only actuator faults are presents, i.e., fo = 0. Assume that anobserver has been designed as in §3.2.1 and that a sliding motion has been establishedso that ey = ey = 0 (in mean value). Therefore, in appropriate coordinates, the equation(3.49) becomes

0 = A21e1(t )−E2 fi (t )+νeq (4.5)

where νeq is the equivalent control that represent the average behavior of the discontin-uous component ν and represents the effort necessary to maintain the motion on thesliding surface. From (3.49) - (3.53) it result that the error system is quadratically stableand therefore, after a finite time, it result that

νeq −→ E2 fi (t ) (4.6)

A commonly used approach to reconstruct the equivalent injection is by the use of alow pass filter [7]. Since r ank(E2) = q it follows from (4.6) that

fi (t ) ≈ (E T2 E2)−1E T

2 νeq (4.7)

The key point is that the signal on the right-hand side of the equation above can becomputed on-line and depends only on the output estimation error ey , thus the fault fi

can be approximated to any degree of accuracy.


4.1.2 SMO for reconstructing of the output fault signals

Now consider the case when only sensor faults are presents, i.e., fi = 0. In such situationsince y(t ) =C x(t )+ fo(t ) it follows that

ey (t ) =Ce(t )− fo(t ) (4.8)

therefore, assuming the referred system 4.1 in a the normal form of 3.45 and consideringthe particular observer structure given in 3.46, it follows a state estimation error of theform

e1(t ) = A11e1(t )+ A12 fo(t )ey (t ) = A21e1(t )+ AS

22ey (t )+ A22 fo(t )− fo(t )+ν (4.9)

Note that fo(t ) and fo(t ) appear as output disturbances and thus ρ(·) in equation (4.4)must be chosen to be sufficiently large to maintain sliding in the presence of these dis-turbances. Provided a sliding motion can be attained, making the assumption ey = 0, thefollowing holds

0 = A21e1(t )+ A22 fo(t )− fo(t )+νeq (4.10)

Thus for slowly varying faults, i.e. fo(t ) ' 0, provided the dynamics of the sliding mo-tion are sufficiently fast, from the first component of the error dynamic 4.9, and assuminge1 = 0, we obtain e1 = A−1

11 A12 fo(t ). Substituting such expression into the second compo-nent of the error dynamic and imposing ey = ey = 0, the following holds

νeq ≈−(A22 − A21 A−111 A12) fo(t ) (4.11)

As in the previous paragraph, the equivalent output injection νeq can be calculatedfrom (4.4) and consequently if (A22 − A21 A−1

11 A12) is non-singular, the fault signal can beobtained from equation (4.11).Note that from Schur expansion

det (A) = det (A11)det (A22 − A21 A−111 A12) (4.12)

and thus (A22 − A21 A−111 A12) is nonsingular if and only if det (A) 6= 0.

However even if (A22−A21 A−111 A12) is singular, inference can still be made about certain

fault channels depending on the precise nature of the rank deficiency.

4.2. FILTRATION-FREE FAULT RECONSTRUCTION VIA FULL ORDER HOSMO 75

4.2 Filtration-free fault reconstruction via full or-der HOSMO

Based on the same concepts, but applied in a nonlinear framework, in this section is pre-sented an observer for a class of nonlinear uncertain systems, which contains a correctiveterm designed on a second-order sliding mode control algorithm featuring global conver-gence properties. Such observer allow the reconstruction in finite time of the actuatorfaults performed directly from the continuous observer output injection signal withoutrequire any filtration.Let us consider the class of uncertain nonlinear systems ([5],[82]) described by

x = Ax +G(x,u)+EΨ(x,u, t )+D f (y,u, t ) (4.13)

y = C x (4.14)

where x ∈ Rn , u ∈ Rm and y ∈ Rp are the state, input and output vectors, respectively.A ∈ Rnxm , E ∈ Rnxr , D ∈ Rnxq and C ∈ Rpxn (q ≤ p < n) are known constant matrices withD and C having both full rank. The known nonlinear vector field G(x,u) is assumed to beLipschitz, the unknown nonlinear term Ψ(x,u, t ) ∈ Rr represent all model uncertaintiesand disturbance affecting the system, and the unknown functions f (y,u, t ) ∈ Rq repre-sents the actuators faults to be identified.

Some assumption are met about the uncertain terms affecting system (4.13)-(4.14).Let known function τ(t ),ξ(t ),τd (t ),ξd (t ) , possibly depending on the system inputs/outputs,exist such that

‖ f (y,u, t )‖ ≤ τ(t ), ‖Ψ(x,u, t )‖ ≤ ξ(t ) (4.15)

‖ f (y,u, t )‖ ≤ τd (t ), ‖Ψ(x,u, t )‖ ≤ ξd (t ) (4.16)

It is assumed that the known control input vector u(t ) contains smooth functions. Inparticular it is assumed to know a positive function Ud (y, t ) such that

‖u(t )‖ ≤ Ud (y, t ) (4.17)

The following assumption is also met:

r ank(C [E D ]) = r ank([E D ]) ≤ p (4.18)

Under assumption (4.18) it was shown [5] that it can be found a linear transformation,with matrix T0, such that, in the new coordinates system, the system equations become

x1 = A1x1 + A2x2 +G1(x,u) (4.19)

x2 = A3x1 + A4x2 +G2(x,u)+E2Ψ(x,u, t )+D2 f (y,u, t ) (4.20)

y = C2x2 (4.21)

where x = [x1, x2]T with x1 ∈Rn−p , G1(x,u) and G2(x,u) are known nonlinear vector fields.The triple (A, [E D],C ) assume therefore the following form

( [A1 A2

A3 A4

],

[0(n−p)×(r+q)

E2 D2

],

[0p×(n−p) C2

] )(4.22)


where A1 ∈R(n−p)×(n−p), and C2 ∈Rp×p is non singular.If, furthermore, r ank([E D ]) = q ≥ q , then the transformation matrix can be selected

in such a way that E2 and D2 have the following structure

E2 =[

0(p−q)×r

E22

],D2 =

[0(p−q)×q

D22

](4.23)

where E22 ∈Rq×r , furthermore the matrix D22 ∈Rq×q is of full rank (see [4] and [5]).

Under the assumption that the invariant zeros of the triple (4.22) lie in C−, there exista matrix L ∈R(n−p)×p of the form

L = [L1 0(n−p)× q

](4.24)

with L1 ∈R(n−p)×(p−q) such that A1 +L A3 is Hurwitz [5].

Let us consider system (4.19)-(4.21), and introduce a new coordinate transformationz = T x where

T =:

[I(n−p) L

0 Ip

](4.25)

and L is given by (4.24). Then, in the new coordinates the transformed dynamics become

z1 = M0z1 +M1z2 + [In−p L]G(T −1z,u) (4.26)

z2 = A3z1 +M2z2 +G2(T −1z,u)+E2Ψ(T −1z,u, t )+D2 f (y,u, t ) (4.27)

y = C2z2 (4.28)

where z = [z1 z2]T , with z1 ∈Rn−p and z2 ∈Rp , and

M0 = A1 +L A3

M1 = A2 +L A4 − (A1 +L A3)L

M2 = A4 − A3L

As said before, Matrix M0 is Hurwitz, it’ll have important consequences on the nextdevelopments.

For the system (4.26)-(4.28), consider a dynamical observer of the following form

˙z1 = M0z1 +M1C2−1 y + [In−p L]G(T −1z,u) (4.29)

˙z2 = A3z1 +M2z2 +G2(T −1z,u)−K (y −C2z)+w (4.30)

y = C2z2 (4.31)

where z = [z1 C2−1 y]T , y is the observer output, and w is an appropriate control law to be

designed.

Let e1 = z1 − z1 and ey = y − y then the error dynamics becomes

e1 = M0e1 + [In−p L](G(T −1z,u)−G(T −1z,u)) (4.32)

ey = C2 A3e1 +C2My ey +C2(G2(T −1z,u)−G2(T −1z,u))

+C2E2Ψ(T −1z,u, t )+C2D2 f (y,u, t )−C2w (4.33)


where My = (A4 − A3L)C2−1 +K and K ∈ Rp×p is a gain matrix selected to guarantee the

stability of the linear part of the state estimation error dynamics, i.e., in order to make thefollowing matrix an Hurwitz one

[M0 0

C2M2 C2My

](4.34)

Proposition 1: Consider system (4.32). If the following LMI is feasible

AT P T + P A+ 1

εP P T +εLG

2In−p +αP < 0 (4.35)

P := P [In−p L], A :=[

A1

A3

](4.36)

with P > 0, ε and α are positive constants, LG is the Lipschitz constant of G(x,u) withrespect to x, and L has the structure in (4.24), then e1 tends asymptotically to zero fulfillingthe following inequality during the transient

‖e1(t )‖ ≤ e1max (t ) ≡ M‖e1(0)‖exp−α t/2 (4.37)

where M :=√λmax(P )/λmi n(P ).

Proof: see [5].

Remark 1: The LMI (4.35) follows from a Lyapunov approach with candidate functionV = e1

T Pe1, and, in particular, by imposing V ≤−αV .

Proposition 1 explain that vector e1 tends to zero asymptotically. It means that in orderto reconstruct the system state implementing the equation (4.29) would be enough, but,in order to reconstruct the fault vector, additional dynamics are required.

Then, the main problem here is to design the observer control w in such a way thatvectors ey and ey tends exactly to zero in finite time. In [5] a solution was suggested basedon standard first order sliding mode control. Such an approach opens the way to achievethe fault reconstruction via using the equivalent control principle (i.e., via low pass filter-ing) and it is therefore an approximate method [7],[4],[5].

Here we propose a different approach based on second-order sliding modes that en-able us to reconstruct the fault without any filtration, therefore leading to an exact solu-tion.

Consider the following well-defined transformation:

e∗y =C2

−1ey (4.38)

and write the expression for the second derivative of e∗y

e∗y = A3e1 +MyC2ey

∗+Gy +E2Ψ+D2 f − w (4.39)

were Gy is compactly defined as follows

Gy = (G2(T −1z,u)−G2(T −1z,u)) (4.40)


Considering (4.32)-(4.33) together with the transformation (4.38) yields to rewrite (4.39)as

e∗y = (A3M0 +MyC2 A3)e1 + (MyC2)2e∗

y + ([In−p L]+MyC2)Gy

+MyC2E2Ψ+MyC2D2 f +Gy +E2Ψ+D2 f −MyC2w − w

= ϕ(e1,e∗y ,u, w, t )− w (4.41)

with implicit definition of the drift term ϕ(·).

The equation (4.41) defines p scalar subsystems having the following formγ1i = γ2i , i = 1..pγ2i =ϕ i (·, t )+ vi

(4.42)

where γ1i and ϕ i (·, t ) are the i-th entry, for i = 1..p, of vectors ey∗ and ϕ(·, t ), respectively,

and vi =−wi .The problem is to find a set of discontinuous controls laws vi stabilizing the uncertain

SISO systems (5.15) in finite time. To solve this problem the second-order sliding modecontrol approach appears to be particularly appropriate because of systems (5.15) haverelative degree two. The boundedness properties of the drift term ϕ i plays a crucial role.In the standard literature on 2-SMC it was often assumed the existence of a constant up-perbound Φ∗

i , known a-priori, such that |ϕi | ≤Φ∗i .

Here we refer to a recently proposed "Global" version [30, 8] of the suboptimal algo-rithm which can work under the more general assumption

|ϕi (·, t )| ≤Φi (t ) (4.43)

where Φi (t ) is a (possibly time-depending) function given in real time. The proposedsolution is sumarized in the following Theorem 1.

Theorem 1: Consider system (5.15), whose uncertain dynamics satisfies (4.43). Applythe control law

vi (t ) =

−[Φi (t )+χ]

sign(γ1i (t )−γ1i (0)

)

0 ≤ t ≤ tMi j

−[Φi (t )+χ]

sign (γ1i (tMi j ))tMi j < t ≤ tci j

[Φi (t )+Π+χ]

sign (γ1i (tMi j ))tci < t ≤ tMi , j+1

(4.44)

where

Π ≥ 1

3η2 (4.45)

χ and η are positive arbitrary constants tMi j ( j = 1,2, . . .) is the sequence of time instantsat which γ2i (t ) = 0, and tci j is the first time instant subsequent tMi j at which one of the


following relationships is verified

γ1i (tci j ) = 12γ1(tMi j )

γ2i (tci j ) = η√

|γ1i (tMi j )|(4.46)

where γ2i (t ), which represents an instantaneous upperbound of |γ2i (t )|, is defined as

γ2i (tMi j ) = 0 j = 1,2, . . .

γ2i (t ) =

2(Φi (t )+χ) tMi j ≤ t ≤ tci j

0 tci j < t < tMi , j+1

(4.47)

Then, the global finite-time attainment of conditions

γ1i = γ2i = 0 (4.48)

is provided.

Proof of Theorem 1: See [30, 8].

Remark 2: The parameter η, which is free to be chosen according to the desired tran-sient specifications, defines the prescribed upperbound for the modulus of γ2i during thetransient process. Note that the smaller η is, the slower the convergence process. Fur-thermore, it must be highlight that, the proposed algorithm requires the sequence of thevalues of γ1i at the time instants at which γ2i is zero. The corresponding time instantscan be detected, with an arbitrarily-small delay, by existing peak-detector devices. In pre-vious works [6], the sub-optimal algorithm has been shown to be robust against the ap-proximate detection of the singular points, and the same considerations still apply to theactual case.

Let Φi (t ) be positive functions fulfilling (4.43). Define

Φ(t ) = col (Φ1(t ),Φ2(t ), ...,Φp (t )) (4.49)

Let us derive an expression for the norm of the upperbounding vector Φ(t ) that isrequired for the synthesis of the control law. By (4.41) we can rewrite the drift termϕ(·, t ) =col (ϕ1(·, t ),ϕ2(·, t ), ...,ϕp (·, t )) as

ϕ(·, t ) = Me1 + (MyC2)2e∗y + ([In−p L]+MyC2)Gy ++ MyC2E2Ψ+MyC2D2 f +Gy +E2Ψ+D2 f −MyC2w

(4.50)

where M ≡ A3M0 +MyC2 A3. It is worth noting that whilst e∗y is known and measurable,

e1 is unknown, and, in order to evaluate explicitly an upperbound to (4.50), it must beoverestimated. To this end, the following proposition is useful.


From (4.24) and (4.25), it can be concluded that

‖x − x‖ ≡ ‖T −1z −T −1z‖ = ‖e1‖ (4.51)

in light of which it follows from (4.40) that

‖Gy‖ = ‖G2(T −1z,u)−G2(T −1z,u)‖ ≤ LG‖e1‖ (4.52)

Furthermore we are interested in finding an upperbound of ‖Gy‖, so expliciting thederivative of Gy :

Gy (x, x,u) = ∂G2

∂x(x,u)x − ∂G2

∂x(x,u) ˙x +

∂G2

∂u(x,u)u − ∂G2

∂u(x,u)u (4.53)

Adding and subtracting the term ∂G2∂x (x,u) ˙x, and making some manipulations, yields

the following inequality

‖Gy‖ ≤ LG2x‖Re1‖+LG2x‖Re1 ˙x‖+LG2u

‖Re1u‖ (4.54)

where R is an appropriate matrix, and LG2x ,LG2x,LG2u

represent the Lipschitz con-

stants with respect to x, uniformly for u, of G2, ∂G2∂x and ∂G2

∂u , respectively.Eq. (4.54) together with (4.32) implies that

‖Gy‖ ≤ ‖R‖LG2x (‖M0‖+LG‖[In−p L]‖)+‖R‖LG2x‖ ˙x‖+LG2u

K ‖u‖‖e1‖ (4.55)

that can be rewritten in compact form as

‖Gy‖ ≤ Gymax‖e1‖ (4.56)

Now we have all the ingredients to evaluate the upper bounding function (4.49). By(4.50), and considering the initial assumptions (4.15)− (4.17) together with (4.56), (4.52)and (4.37), it can be found a scalar function Φ∗(t ) such that Φ∗(t ) ≥ Φi (t ) ∀i which hasthe following structure

Φ∗(t ) = (‖M‖+Gymax +Γ1)e1max (t )+‖(MyC2)2‖ ‖ey∗‖

Γ2ξ(t )+Γ3τ(t )+‖E2‖ξd (t )+‖D2‖τd (t )

+‖MyC2‖‖w‖ (4.57)

where the positive constants Γi (i = 1,2,3) are defined as

Γ1 = ‖([In−p L]+MyC2)‖LG

Γ2 = ‖My‖ ‖C2‖ ‖E2‖Γ3 = ‖My‖ ‖C2‖ ‖D2‖ (4.58)

The upperbound defined in (4.57)-(4.58) is available in real time and can be used inall the control laws in Theorem 1 (in other words, the conservative approximationΦi (t ) =Φ∗(t ) could be made, for the simplicity sake, when implementing the proposed observer).


4.2.1 Actuator faults reconstruction

We address in this section the problem of reconstructing the vector f (y,u, t ) representingthe actuator faults. This procedure is conventionally approached by resorting to the equiv-alent control method [4]-[5] and, more precisely, by exploiting the possibility of achievingon-line an estimate of the equivalent control by means of low pass filtering.

In this contest the use of second order sliding mode control, that provides a continu-ous observer control signal w steering to zero both ey and ey in finite time, allows us toreconstruct the fault without filtration.

Assume that Im(E22)⋂

Im(D22) = 0, where the matrices E22 and D22 are those givenin (4.23). Then [5] there exist a nonsingular matrix W ∈Rq×q such that

W [E22 D22] =[

H1 00 H2

](4.59)

where H1 ∈R(q−q)×r and H2 ∈R(q×q). If (4.59) holds, the effect of the faults can be “separated"from the effect of the uncertainty, thus permitting the precise reconstruction of the faultvector.

From (4.33) and (4.38), imposing the conditions e∗y = e∗

y = 0, it yields that

A3e1 +Gy (z, z,u, t )+E2Ψ(T −1z,u, t )+D2 f (y,u, t )−w = 0 (4.60)

From the latter, taking into account (4.52), (4.37) and (4.23) it derives that

E22Ψ(T −1z,u, t )+ D22 f (y,u, t )−w q = 0 (4.61)

where w q denotes the last q elements of vector w . By (4.59),

W w q =W [E22 D22]

[Ψ(T −1z,u, t )

f (y,u, t )

]=

[H1Ψ(T −1z,u, t )

H2 f (y,u, t )

](4.62)

Let Wq denote the last q rows of W ; then it follows the following formula for recon-structing the fault vector

f = H2−1Wq w q (4.63)

Remark 3: If Im(E22)⋂

Im(D22) 6= 0 the exact reconstruction of the faults is no longerpossible and the application of the given fault reconstruction formula gives rise to an esti-mation error overestimated as follows [5] ‖ f (t )− f (y,u, t )‖ ≤ ‖D2

+E2‖ξ(x,u, t )+D2LG‖e1‖where D2

+ is the left pseudo-inverse of D2 (which exists since D2 is full column rank).

Simulative and experimental analysis of the proposed full order observer scheme iswidely given in chapter 6, "Application Results".


4.3 Filtration-free fault reconstruction via reducedorder HOSMO

In this section, by dispensing with the request of estimating the state vector, in otherwords by only aiming to the reconstruction of the faults, a reduced order observer willbe designed. Also for such case, as in previous section, the output injection of the pro-posed observer is built by means of a second-order sliding mode control algorithm, fea-turing global convergence properties, that permit the reconstruction of the actuator faultwithout requiring any filtration.

Let us consider the same class of uncertain nonlinear systems of the former case (4.13)−(4.14), except for the nonlinear vector G(·), for which, in this discussion, it may dependonly from the output vector y and the control inputu.

The same assumption (4.15)−(4.18) are meet, furthermore we assume that the systemis BIBS and that a controller, acting on the system, guarantees the following constraints

‖x1‖ ≤ Γ(‖y‖,‖u‖) (4.64)

where Γ : (R+×R+) →R+ is a positive scalar function.It worth noting that, under the rank condition (4.18) it can be found a linear transfor-

mation, such that the reference system in the new coordinates become

x1 = A1x1 + A2x2 +G1(y,u) (4.65)

x2 = A3x1 + A4x2 +G2(y,u)+E2Ψ(x,u, t )+D2 f (y,u, t ) (4.66)

y = C2x2 (4.67)

where x = [x1, x2]T with x1 ∈Rn−p , G1(y,u) and G2(y,u) are known nonlinear vector fields,and C2 ∈Rpxp is non singular.

As one can see, in (4.66) only the disturbance vector Ψ(x,u, t ) is dependent from thewhole state vector x.

4.3.1 Fault observer design

The aim is to design a dynamical system allowing us to reconstruct the actuator faults infinite time by dispensing with the request of estimating the whole state vector. Considersystem (4.65)-(4.67) and rewrite (4.66) as follows by embedding the state vector compo-nent x1 into the new, larger, “disturbance" vector ζ

x2 = A4x2 +G2(y,u)+Pζ(x,u, t )+D2 f (y,u, t ) (4.68)

y = C2x2 (4.69)

where

P = [A3 E2] P ∈Rp×(n−p+r ) (4.70)

ζ(x,u, t ) =[

x1

Ψ(x,u, t )

]ζ ∈R(n−p+r )×1 (4.71)

4.3. FILTRATION-FREE FAULT RECONSTRUCTION VIA REDUCED ORDER HOSMO 83

Consider the following reduced-order observer

˙x2 = A4x2 +G2(y,u)+w (4.72)

y = C2x2 (4.73)

where y is the observer output and w is an appropriate control law. The aim is to recon-struct the actuator fault vector f (y,u, t ) exactly and in finite time.

Let ey = y − y , and consider the following transformation:

e∗y =C2

−1ey ≡ x2 − x2 (4.74)

from (4.68), (4.72) and (4.74) it results

e∗y = Pζ(x,u, t )+D2 f (y,u, t )−w (4.75)

The second derivative of ey∗ is

e∗y = P ζ(x,u, t )+D2 f (y,u, t )− w (4.76)

= ϕ(·, t )− w (4.77)

with implicit definition of the drift term vector field ϕ(·).The problem of stabilizing ey

∗ and e∗y , exactly and in finite time, is formally equiva-

lent to that considered for the system 4.41 and therefore can be solved by using the sameapproach.

Let Φi (t ) be positive functions fulfilling (4.43). Define Φ∗(t ) as a positive scalar func-tion such that

Φ∗(t ) ≥Φi (t ) ∀i = 1,2, ..., p (4.78)

Let us derive an expression for Φ(t ) which is required for the synthesis of the controllaw. By (4.77) we can rewrite the drift term ϕ(·, t ) = col (ϕ1(·, t ),ϕ2(·, t ), ...,ϕp (·, t )) as

ϕ(·, t ) = P ζ(x,u, t )+D2 f (y,u, t ) (4.79)

First, from the expression for ζ given in (4.71), it follows that

ζ(x,u, t ) =[

x1

Ψ(x,u, t )

](4.80)

so that, from (4.16)‖ζ(x,u, t )‖ ≤ ‖x1‖+ξd (t ) (4.81)

Now considering the system equation (4.65), in conjunction with assumption (4.64), itfollows that

‖x1‖ ≤ ‖A1‖‖x1‖+‖A2‖‖x2‖+‖G1(y,u)‖≤ ‖A1‖Γ(‖y‖,‖u‖)+K (y,u) ≡ Γd (‖y‖,‖u‖)

(4.82)

with implicit definition of the positive scalar function Γd


Now we have all ingredients to evaluate the upper bounding functionΦ∗(t ), accordingto (4.78), as follows

Φ∗(t ) = ‖P‖ [Γd (‖y‖,‖u‖)+ξd (t )]+‖D2‖τd (t ) (4.83)

The upperbound defined in (4.83) can be used in all the control laws in Theorem 1 (inother words, the conservative approximation Φi (t ) = Φ∗(t ) could be made, for the sim-plicity sake, when implementing the proposed observer).

4.3.2 Actuator fault reconstruction

Now we address the problem of reconstructing the vector f (y,u, t ) representing the actu-ator faults. As for the full order observer case, §4.2.1, the use of second order sliding modecontrol, that provides a continuous observer control signal w steering to zero both ey andey exactly and in finite time, allows us to reconstruct the fault without filtration.

Assume that Im(P )⋂

Im(D2) = 0, then [5] there exist a nonsingular matrix W ∈Rp×p

such that

W [P D2] =[

H1 00 H2

](4.84)

where H2 ∈R(q×q).If (4.84) holds, the effect of the faults can be “separated" from the effect of the uncer-

tainty, thus permitting the precise reconstruction of the fault vector [5].From (4.75), imposing the condition e∗

y = e∗y = 0 , the following holds after a finite time

Pζ(x,u, t )+D2 f (y,u, t )−w = 0 (4.85)

Bringing w into the left hand side and multiplying both sides for W , it results

W w =W [P D2]

[ζ(x,u, t )f (y,u, t )

](4.86)

Let W2 denote the last q rows of W , then it follows the following formula for reconstructingthe fault vector

f = H2−1W2w (4.87)

Remark 4: It must be highlight that, unlike the approaches in [4]-[5] where ey = 0 istrue only in Filippov sense, here the condition ey = ey = 0 is attained in finite time, and,since the control signal w is already continuous, no filtering of discontinuous signal isnecessary anymore.


4.3.3 Simulation results

The effectiveness of the suggested observation and fault reconstruction scheme is testedon a simple example. Consider a system in the general form (4.13)-(4.14) with n = 4, m = 1,p = 3 and the following matrices and vector fields:

A =

−2 −1 0 −11 0 0 −1−4 −1 −2 02 1 0 −2

, C =

1 1 0 00 0 1 11 0 0 0

G(y,u) =

y12 +u

si n(y2)−3u−u0

As one can see the nonlinear vector G(·) depend only from the output vector. Intro-duce a linear transformation with matrix

T0 =

0 0 0 11 1 0 00 0 1 11 0 0 0

The matrices of the transformed dynamic system become

[A1 A2

A3 A4

]=

−2 1 0 1−2 −1 0 00 0 −2 −2−1 −1 0 −1

G(y,u) = col (G1,G2) =

0si n(y2)+ y2

2 −2u−u

y12 +u

D =[

0D2

]=

0−2−11

C2 =

1 0 00 1 00 0 1

E =[

0E2

]=

00.10.20

and Ψ(x,u, t ) = ‖x‖si n2x3. The P in (4.70) takes the following form

P =

−2 0.10 0.2−1 0


Let us design the observer according to (4.72)-(4.73). For the system under analysis, itresult that Im(P )

⋂Im(D2) = 0, so there exist a matrix W ∈ R3x3, as in (4.84), that allows

to decouple uncertainties from faults thereby permitting to reconstruct the fault signalsfrom w in accordance with the formula (4.87). A suitable choice for decoupling matrixconduce to

W2 =[

1 −0.5 −2]

H2 =−3.5

The reconstruction of fault is made via (4.87).

A fault signal, shown in the top plot of Figure 2, has been introduced in input channel.

Figure 4.1: The state variable and the state observation error.

The initial conditions of the state variables are x1(0) = x2(0) = x3(0) = x4(0) = 1.0, whilethe initial conditions of the observed states are x2(0) = x3(0) = x4(0) = 0.0. Fig.4.1 showsthe actual state variables, and the state observation errors. Fig.4.2 shows the actual andreconstructed fault signal and the reconstruction error. The finite time convergence ofthe estimate, and the high accuracy of the estimate, are both apparent from the analysisof Fig.4.2.


Figure 4.2: The actual fault and the reconstructed signal fault.

Chapter 5

Discrete state reconstruction viaHOSMO

5.1 Introduction

As explained in Chapter 1 the knowledge of a model for the system under investigation,the better possible accurate, is the main point in model based FDI systems, either fortechniques based on residual generation or not. In particular it is well-known that theobserver-based approaches are very dependent on the system model since modelling un-certainty can cause false and missed alarms, which may make the FDI system useless.It must be highlighted that many process, in real applications, show nonlinear dynamicbehavior, especially if wide areas of operation are involved during the functioning.

It’s a common practice, in several engineering fields, to refer to non-linear processesthat operate at different regimes as distinct models each associated to an admissible op-erating condition.

Based on these assumptions we focalize our attention into those cases in which thesystem under diagnosis is characterized by different nonlinear dynamics, varying accord-ing to an unknown logic. In such a case the use of a prefixed FDI scheme based on aunique model could led to a residual strongly dependent from model mismatches. Assum-ing indeed to know the active nonlinear model and its parameter, and also to be able ofdesigning an appropriate FDI scheme for each single nonlinear model, the overall schemewill be obviously more accurate and the false alarm rate will strongly decrease. Further-more, it could be possible to associate a specific model for the faulty system. In this case,the detection of the actual dynamics leads to fault detection, directly. In this respect a cru-cial issue in model based FDI is the capability to identifying the actual nonlinear dynamicof systems that could be described appropriately by a switched system. A switched sys-tem has a discrete dynamics represented by a finite state machine that evolves accordingto the occurrence of discrete events. To each discrete state (or "mode") a continuous dy-namics is associated. In the last decade the control community has devoted a great dealof attention to the study of switched systems [71],[72],[73].

Therefore, a problem of great interest is the reconstruction of the discrete-state throughthe observation of measurable system outputs. The techniques developed in this frame-work can be applied to several problems where the discrete events are not observable.

89

90 CHAPTER 5. DISCRETE STATE RECONSTRUCTION VIA HOSMO

In the framework of discrete event systems, several approaches have been proposed toestimate the discrete state [74],[75]. In a more general hybrid context, the discrete stateestimation has been discussed in [76],[77].

In this chapter we investigate the problem of the discrete state reconstruction forswitched systems building on the idea that a general class of switched systems can bemodeled by nonlinear systems with an affine boolean input representing the system dis-crete state.

Objective of the present work [81] is to reconstruct such a boolean input despite boundeduncertainties affecting the system dynamics. To this aim we propose a 2-SM based tech-nique, that exhibit remarkable properties of robustness against uncertainties and distur-bances [7], for reconstructing the unmeasurable quantities affecting the system.

5.2. PROBLEM FORMULATION 91

5.2 Problem formulation

In this paper we examine the class of switched systems that can be represented in theform:

x(t ) =G(x,u, t )+D(x,u, t )δ(σ(t ))+ε(x, t ) (5.1)

where x(t ) ∈ X ⊂ Rn is the continuous state, u(t ) ∈ U ⊂ Rp is the input to the system,G(x,u, t ) ∈Rn and D(x,u, t ) ∈Rn×L are known vector fields, and ε(x, t ) ∈Rn is an uncertainterm representing model mismatches and/or external disturbances.

The piecewise-constant integer function σ(t ) ∈ 0, . . . ,k − 1 represents the unknowndiscrete state of system (5.1). Vector δ(σ(t )) ∈ 0,1L contains boolean elements. It mapsthe discrete state σ(t ) into an L-dimensional boolean vector which “encodes" the actualdiscrete state.

Model (5.1) can represent switched systems with, at most, k ≤ 2L different dynamics.The problem tackled in this paper is the reconstruction of the discrete state σ(t ).

5.2.1 Assumptions

We now specify the assumptions which are met about the considered class of systems (5.1).The continuous state x(t ) is supposed to be fully measurable, and G(x,u, t ), D(x,u, t ) aresupposed to be known.

The dimension L of vector δ(t ) must not exceed the dimension of the continuous state:

L ≤ n (5.2)

The boolean vector δ(t ) is not available due to the uncertainty in the discrete state.The discrete state σ(t ) can be uniquely recovered from the boolean vector δ(t ), and vicev-ersa. Let the time evolutions of the continuous state x and exogenous input u variablesbe a-priori confined in the compact domainsX and U.

Assumption with respect the smoothness and norm-bounded are made with respectthe full-rank matrix D(x,u, t ) and the unmeasurable, state dependent, uncertainty termε(x, t )

‖D(x,u, t )‖ ≤ D0 ‖D(x,u, t )‖ ≤ D1 (5.3)

‖[DT (x,u, t )D(x,u, t )]−1DT (x,u, t )‖ ≥ D2 (5.4)

‖ε(x, t )‖ ≤ ε0 ‖ε(x, t )‖ ≤ ε1 (5.5)

Note that no boundedness or smoothing requirements are met on the vector fieldG(x,u, t ).

5.2.2 Comments on the considered class of systems

We now show that the considered class of switched systems (5.1) is general enough torepresent the following continuous-time switched nonlinear dynamics

x(t ) =Gσ(t )(x,u, t )+ε(x, t ) (5.6)


whereσ(t ) ∈ 0, . . . ,k−1 is the discrete state. A simple systematic procedure to put system(5.6) into the form (5.1) is now given. Define the matrices D(x,u, t ) and G(x,u, t ) as follows(omitting the dependence of its entries from x, u and t )

D(x,u, t ) = [G1 −G0 G2 −G0 · · · Gk−1 −G0

]

G(x,u, t ) =G0(x,u, t )(5.7)

and letδ(σ(t )) = [δ1(σ(t )),δ2(σ(t )), ...,δL(σ(t ))]T (5.8)

where

δ j (σ(t )) =

1 if j =σ(t ) j = 1,2, ...,L0 otherwise

(5.9)

One can readily verify that system (5.6) is equivalent to (5.1),(5.7)-(5.9), whose maincharacteristics is that of being affine in the boolean vector δ(σ(t )) defining the currentmode of operation.

5.3 Discrete state observer design

The proposed discrete state estimator (which assumes the knowledge of the continuoussystem state) takes the following form:

z = G(x,u, t )+w(t ) (5.10)

where z represents the observer state and w is an observer input to be designed. Lete = x − z be the observer error variable. Then, from (5.1) and (5.10), the observer errordynamics is

e = D(x,u, t )δ(t )+ε(x, t )−w(t ) (5.11)

5.3.1 Observer input design

Our objective is to design an observer control vector w = [w1, w2, . . . , wn]T guaranteeingthe finite-time convergence to zero of e and e.

A second-order sliding modes based approach, that enables us to reconstruct the dis-crete state, will be proposed. Must be highlight that, dissimilar to other works [5] withina distinct framework related to a fault diagnosis, such an approach, theoretically exact,exhibits a solution converging in finite time.Consider the second time derivative of the error variable e

e = D(x,u, t )δ(t )+Dd

d tδ(t )− ε(x, t )− w(t ) (5.12)

which can be rewritten in compact form as follows:

e =ϕ(x,u, t )− w(t ) (5.13)

The uncertain “drift term" ϕ(·) = [ϕ1(·),ϕ2(·), ...,ϕn(·)]T takes the following form

5.3. DISCRETE STATE OBSERVER DESIGN 93

ϕ(x,u, t ) = D(x,u, t )δ(t )+Dd

d tδ(t )− ε(x, t ) (5.14)

which depend from δ(t ) that must be reconstructed.Denote v(t ) = [v1, v2, ..., vn]T ≡−w(t ), yi ,1 = ei and yi ,2 = ei , where ei and ei represent

the i -th entry of vectors e and e. Then it is possible to rewrite system (5.13) in terms of nde-coupled single input subsystems having the following form

yi ,1 = yi ,2, i = 1,2, ..,nyi ,2 =ϕ i (x,u, t )+ vi

(5.15)

The problem is to find a set of control inputs vi stabilizing the uncertain SISO sys-tems (5.15) in finite time. To solve this problem, the second-order sliding mode controlapproach [15], [29] appears to be particularly appropriate because of systems (5.15) haverelative degree two with respect to the inputs vi ’s which are treated as auxiliary controlvariables. The control task is complicated by the two issues: (a) variables yi ,2 (i = 1,2, ...,n)are not measurable, and (b) the drift terms ϕ i (x,u, t ) are uncertain.

Let the mode switching sequence of the hybrid dynamics have a dwell time td . Thismeans that ti+1 − ti ≥ td , for i ≥ 0. The main idea is to use the discontinuous controller in[26]. Under the condition that a constant upperboundΦi to the drift term magnitude canbe computed

|ϕ i (x,u, t )| ≤Φi ∀t (5.16)

such controller is able to stabilize the uncertain SISO systems (5.15) in a finite time t∗ <<td starting from any initial conditions (yi ,1(0), yi ,2(0)) norm bounded by arbitrarily largeconstants.

Denote as ti (i = 1,2, ...) switching instants at which the active dynamics is commut-ing. The discrete state σ(t ), and then also the vector δ(t ), are piecewise constant duringthe time intervals Ti = (ti , ti+1) between two adjacent mode switchings. The effect of theimpulsive term d

d t δ(t ) at the switching instants ti is a jump in the (e, e) state trajectoriesof system (5.13), and in particular, from (5.12), it result ‖e‖ = ‖D(x,u, t )‖‖δ(t )‖ ≤ D0

pn,

since the norm of the n-dimensional boolean vector δwill never exceed the valuep

n. So,considering the single i-th decoupled subsystem (5.15), at the first switching instant t1 thepoint (yi ,1(t1), yi ,2(t1)) will leave the origin according to (0, yi ,2(t+1 )) with

‖yi ,2(t+1 )‖ ≤ D0 (5.17)

After a new transient whose duration can be made less than t∗ the system will be steeredback to the origin. Thus, at any time t ∈ [t1 + t∗, t2) and any i = 1, ..,n, the conditionsyi ,1(t ) = yi ,2(t ) = 0, i.e., e(t ) = e(t ) = 0 will be satisfied. The reasoning is repeated over allthe successive switching intervals. The key point is the capability of the robust controllerpresented in [26], that will be specified in the sequel, of steering to zero the SISO systems(5.15)arbitrarily fast during the time interval between two adjacent mode switchings.

Along any interval Ti ≡ (ti−1, ti ), (i = 1,2, ...), t0 ≡ 0, the drift term (5.14) can be upperbounded in the form

‖ϕ(x,u, t )‖ ≤ Φ≡ D1p

n + ε1, t ∈ (ti−1, ti ) (5.18)

The existence of such a constant upper bound allows for designing a robust controllerfeaturing the desired finite time convergence properties. Next theorem outlines the main


result by introducing the so-called “ Suboptimal“ second-order sliding mode control al-gorithm [26, 27], together with the tuning rules that allow to obtain an arbitrarily fast con-vergence, which is a basic requirement of the present problem.

Theorem 1: Consider system (5.15) and the control law

vi (t ) =−VM si g n(yi ,1(t )− 1

2 yi ,1(ξi , j ))

ξi , j ≤ t < ξi , j+1

j = 1,2, ...(5.19)

where ξi ,0 = t0 ≡ 0, ξi , j is the sequence of time instants at which yi ,2(t ) = 0, and

VM = ΓΦ (5.20)

Denote the sequence of the switching instants as th , h = 1,2, .... Then there is Γ∗ such that,for any Γ≥ Γ∗, the following conditions are provided.

yi ,1(t ) = yi ,2(t ) = 0, t ∈ (th + t∗, th+1) (5.21)

where i = 1,2, ...,n and with t∗ being arbitrarily small.

Proof of Theorem 1. The proof exploits the basic convergence properties of the subop-timal second-order sliding mode control algorithm [27]. Let us consider the single i − thdecoupled subsystem (5.15) and denote with t1 the first switching instant. At t = t1, whenthe yi ,1 and yi ,2 variables are assumed both zero, on the basis of (5.17), we can infer thatyi ,2 undergoes a jump that, for the worst case, lead the system states (yi ,1(t+1 ), yi ,2(t+1 ))

in the point (0,D0). Starting from here, after a the time tM1 ≤ D0

Φ(Γ−1), the extremal point

( D02

2Φ(Γ−1),0) is reached . The suboptimal control strategy, with the additional constraint

Γ> 2, causes the generation of a sequence of states, with coordinates (yi ,1M1,0), featuringthe following contraction property:

|yi ,1M1+ j | ≤α|yi ,1M1|, j = 1,2, ....., α= 1

Γ−1∈ [0,1) (5.22)

Such sequence converge to the origin at finite time t∗ that can be made arbitrarilysmall opportunely tuning the gain parameter Γ. By imposing the following condition t∗ <td , descend

D0

Φ(Γ−1)+

2ΓΦ

√D0

2

2Φ(Γ−1)

(Γ−1)√Φ(Γ+1)

1

1−√

1Γ−1

< td (5.23)

From (5.23) it follows the rule for selecting Γ in such a way that the convergence timet∗ fulfills the inequality t∗ < td . An admissible interval for Γ exist due to the fact that theleft and side, i.e. t∗, converge to zero for Γ−→∞. On the basis of previous observations,denoting with Γ∗ the value of Γ such that t∗ = td , choosing the tuning parameter Γ ∈]Γ∗,∞), the the condition t∗ < td is guarantees.

¤

5.3. DISCRETE STATE OBSERVER DESIGN 95

5.3.2 Discrete state reconstruction

It was shown in the previous Section that there is t∗ such that, in every “inter-switching"interval Ti ≡ (ti−1, ti ) the next conditions hold

e = e = 0, t ∈ [ti−1 + t∗, ti ) (5.24)

From the definition (5.11) of e, its zeroing implies that

D(x,u, t )δ(t )+ε(x, t )−w(t ) = 0 (5.25)

Notice that the observer input w(t ) is obtained integrating the discontinuous signalv(t ), whose sign switches at very high (theoretically infinite) frequency (Zeno behaviour),then w(t ) is a continuous signal.

By neglecting the uncertainty ε(x, t ) in (5.25) it yields naturally the following recon-struction formula that defines a non-thresholded estimate of the boolean vector δ.

δ(t ) = [DT (x,u, t )D(x,u, t )]−1DT (x,u, t )w(t ) (5.26)

The non-thresholded estimate is not robust against the uncertainty ε(x, t ). By (5.25),the estimation error δ(t )−δ(t ) will be such that

‖δ(t )−δ(t )‖ ≤ ‖[DT (x,u, t )D(x,u, t )]−1DT (x,u, t )‖‖ε‖(5.27)

It can be fruitfully exploited the boolean nature of the vector δ(t ) by introducing athresholding that rounds the value of δ(t ) to the closest integer value between 0 and 1. Ityields the thresholded estimate δ(t ) defined according to

δi (t ) =

1 δi (t ) > 0.50 δi (t ) ≤ 0.5

(5.28)

where δi (t ) and δi (t ) are the i-th entries of vectors δ(t ) and δ(t ) respectively. Thethresholded estimate results to be robust against any error (δi (t )−δi (t )) less, in magni-tude, than 0.5. Thus it can be explicitly given a bound to the maximal tolerated magnitudefor the uncertainty term.

From the requirement that ‖δ(t )−δ(t )‖ ≤ 0.5 it yields by (5.27), (5.4), (5.5) the followingmaximal acceptable bound for the norm of the uncertainty term

‖ε(x, t )‖ ≤ ε0 ≤0.5

D2(5.29)

The fulfillment of (5.29) guarantees the insensitivity of the estimate δ against the un-certainty, namely the condition

δ(t ) = δ(t ), t ∈ [ti−1 + t∗, ti ), i = 1,2, ... (5.30)

Lemma 1 Under the condition that the norm of the uncertain term ε(x, t ) fulfills therestriction (5.29), the proposed estimation procedure given by (5.26), (5.28) provides for


the exact reconstruction of the boolean vector δ in the time intervals t ∈ [ti−1 + t∗, ti ),according to (5.30)

Remark 1 The requirement of providing the observer convergence within the arbitrar-ily small transient time t∗ << td would correspond, in the linear context, to locate theeigenvalues of the error dynamics far away from the origin. Generally, this strongly dete-riorates the robustness against the measurement noise of the resulting linear “high gain"observer. It can be argued, due to the analysis made in [28, 29], that the magnificationof the noise in the considered 2-SMC observer could be less severe than in the linear ob-server counterpart. This topic will be addressed in more detail in next research activities.

Chapter 6

Application results

6.1 Three-tank system case study

As one of popular experimental systems in control laboratories, the three-tank water pro-cess is regarded as a valuable setup for investigating, theoretically and experimentally,nonlinear multivariable feedback control as well as fault diagnosis schemes ([38],[39],[40]).

The multi-tank system that we shall consider, see in Fig.6.1, is composed of three tanksof different shape disposed vertically, so that the potential energy of water allows the wa-ter to flow from one tank to another in a lower position. A variable-velocity pump thatsupplies the upper tank1, and three automatic (electrical) regulation valves RV1,RV2 andRV3 allow for regulating the outflow from each tank. A reflux tank is present below thelower tank3. Furthermore, three manual valves MV1, MV2 and MV3 are also present be-sides the electrically controlled ones.

The three tanks are equipped with piezo-resistive pressure transducers (P Z1,P Z2,P Z3)which permit to measure the water levels H1, H2 and H3. As depicted in figure 6.2, weselect, as the modifiable control inputs, the water inflow to the tank 1 (provided by thevariable-speed DC pump) and the opening of the valves RV1, RV2 and RV3.

The input variables U (t ),U1(t ), U2(t ) and U3(t ) (whose exact meaning shall be ex-plained later on) control the DC pump and the regulation valves RV1, RV2 and RV3 re-spectively.

Concerning the FDI aspects, we’ll investigate the effectiveness of the FDI proposedscheme, discussed in Chapter 4, against additive faults fp , f1, f2 and f3 acting on thepump and the three regulation valves RV1, RV2 and RV3 respectively.

6.1.1 Mathematical model

The flow balance equations leads to the following system model

V1 = q −C1p

H1 (6.1)

V2 =C1p

H1 −C2p

H2 (6.2)

V3 =C2p

H2 −C3p

H3 (6.3)

97

98 CHAPTER 6. APPLICATION RESULTS

Figure 6.1: Configuration of the considered three-tank system

Figure 6.2: System input-output

where V1,V2,V3 represent the actual volume of water contained in the three tanks, q isthe in-flow to the upper tank provided by the pump, H1, H2, H3 are the nonnegative waterlevels, and C1,C2,C3 are the outflow coefficients of the valves that can be adjusted.

The time derivatives of the actual volumes of water depend on the time derivatives ofthe water levels inside the tanks according to the simple relationships

Vi =βi (Hi )Hi , i = 1,2,3 (6.4)

where βi , i = 1,2,3 represent the cross sectional area of tank i at the level height Hi .

6.1. THREE-TANK SYSTEM CASE STUDY 99

Figure 6.3: Shape of tank 1



An expression for cross-sectional areas is now given in explicit form. Taking into ac-count the shapes of the tanks in Fig. 6.3- 6.5, which are derived from the experimentalsetup provided by Inteco [78], the following relationships hold, where a,b,c,w,R,Hmax areconstant parameters reported in the Table 1.

β1 = aw (6.5)

β2(H2) = cw +bw(H2/Hmax) (6.6)

β3(H3) = w√

R2 − (R −H3)2 (6.7)

Clearly, all the above functions are strictly positive. It yields the simple model


H1 = (q −C1

√H1)β−1

1 (6.8)

H2 = (C1

√H1 −C2

√H2)β−1

2 (H2) (6.9)

H3 = (C2

√H2 −C3

√H3)β−1

3 (H3) (6.10)

The coefficients C1, C2, C3 actually depends on the valves opening. Obviously whenthe generic valve RVi is in fully closed position, the corresponding coefficient Ci is equalto zero. Vice versa if the valve is in fully opened condition Ci = C∗

i . Therefore seemsappropriate to represent such outflow coefficients by the notation Ci = Ci (t ) = C∗

i Ui (t ),where Ui (t ) ∈ [0,1] is time-varying and represents the relative actual valve opening. Weassume that U (t ), U1(t ), U2(t ) and U3(t ), the input references for the system actuators,are user-modifiable and therefore known. We also denote the adjustable water inflowgenerated by the DC pump as

q =C∗U (t ) U (t ) ∈ [0,1] (6.11)

It results the following model

H1 = (C∗U (t )−C1∗U1(t )

√H1)β−1

1 (6.12)

H2 = (C1∗U1(t )

√H1 −C2

∗U2(t )√

H2)β−12 (H2) (6.13)

H3 = (C2∗U2(t )

√H2 −C3

∗U3(t )√

H3)β−12 (H2) (6.14)

In the following additive actuator faults will be induced both manually, making use ofthe manual values MV1, MV2, MV3 depicted in Fig. 6.1, or numerically by additive termscorrupting the adjustable reference signals according to Fig.6.6.

Figure 6.6: Fault topology

6.2. SIMULATION RESULTS 101

6.2 Simulation results

The effectiveness of the suggested fault reconstruction scheme and of the discrete stateidentification is studied preliminarily by making some simulative analysis.

The parameter values used for simulating tests have been evaluated through a proce-dure of system identification. The dimensional parameters a, w,b,c,R have been directlymeasured and an identification procedure, described later, has been applied to evaluateappropriate values for C∗, C1

∗, C2∗,C3

∗. The obtained values are

Table 6.1: Parameters of the Three Tank System

Parameter Value Unit

C∗ 1.296∗10−4 m3

s

C1∗ 6.1∗10−5 m3

s

C2∗ 6.5∗10−5 m3

s

C3∗ 6.4∗10−5 m3

sa 0.035 mw 0.035 mb 0.348 mc 0.1 mR 0.365 m

Hi−max 0.35 m


6.2.1 Fault reconstruction via HOSM

In this paragraph we address the problem of detecting and exactly identifying certainfaults and disturbances acting on a three tank system [80]. For this purpose we exploit a ro-bust model-based technique based on 2-order SMO, and more precisely, on the manipula-tion of the equivalent output injection signal to explicitly reconstruct fault signals. Basedon the particular structure of the system under investigation, the method presented in§4.2, for a class of nonlinear uncertain systems, seems particularly suitable. It is straight-forward noting that the proposed scheme enables us to reconstruct certain additive faultsand external disturbances without any filtering, therefore leading to a solution convergingin finite time and theoretically exact.

As depicted in figure 6.7, we select, as the modifiable control inputs, the water inflowto the Tank 1 (provided by the variable-speed DC pump) and the opening of the valves RV1

and RV2. Considered faults are associated to a malfunctioning in the control valves RV1

and RV2. The outflow through the valve RV3 is regarded as a disturbance term. This term,to be treated as an external disturbance, will be reconstructed by the proposed observer.Concerning the FDI aspects, it is our aim to detect and reconstruct additive faults f1 and f2

on the regulation valves RV1 and RV2. Pumping systems are often redundant in practice,so that there is a limited possibility of pump fault.

Figure 6.7: System inputs, outputs and disturbances

In the following we denote, according to Eq. (6.3), the overall outflow from the lowertank 3 in compact form as

ψ(·) =C3

√H3 (6.15)

we also denote the adjustable water inflow generated by the DC pump as

q(t ) =C∗U (t ) U (t ) ∈ [0,1] (6.16)


According to (6.12)− (6.14), the following model results

H1 = (C∗U (t )−C1∗U1(t )

√H1)β−1

1 (6.17)

H2 = (C1∗U1(t )

√H1 −C2

∗U2(t )√

H2)β−12 (H2) (6.18)

H3 = (C2∗U2(t )

√H2 −ψ(·))β−1

3 (H3) (6.19)

As previously said, we assume that actuator faults (such as permanent or intermittentbiases, or gain degradation) and component faults (such as leakage in the tanks and clogsin the pipes) can occur in the valves RV1 and RV2.

Our aim is to realize an Observer-Based FDI scheme guaranteeing the precise recon-struction of the (possibly simultaneous) additive actuator faults f1 and f2 that can be oc-cur in control valves RV1 and RV2, and of the disturbance term ψ(t ). It worth noting thatan estimate of the flow disturbance ψ(t ) would be very useful for level control purpose.

The actuator fault signals can be modeled by additive terms corrupting the adjustablevalve signals U1(t ) and U2(t ). Thus, by including such possible fault signals in the modelone can rewrite system (6.17)-(6.19) according the general form (4.13)-(4.14).

x = G(x,u)+D(x) f (x,u, t )+E(x)Ψ(x, t ) (6.20)

y = C x (6.21)

where x = [H1, H2, H3]T , u = [U ,U1,U2]T , C = I3 being the identity matrix, and G(x,u) =G∗(x)u with

G∗(x) =

C∗β1

−C1∗pH1β1

0

0 C1∗pH1

β2(H2) −C2∗pH2

β2(H2)

0 0 C2∗pH2

β3(H3)

(6.22)

D(x) =

−C1

∗pH1β1

0C1

∗pH1β2(H2) −C2

∗pH2β2(H2)

0 C2∗pH2

β3(H3)

(6.23)

E(x) =

00

− 1β3(H3)

, f (x,u, t ) = [ f1, f2]T (6.24)

Ψ(x, t ) =ψ(·) (6.25)

The nonlinear vector fields G∗(x), D(x), E(x) are assumed to be known, while vec-tors f (x,u, t ) and Ψ(x, t ) represent the considered actuators faults and the external dis-turbance, respectively.

As one can see, the measurements actually represent the complete state vector. Nev-ertheless, the design of an appropriate observer is necessary to identify and reconstructsignal faults and disturbance term. Assuming that an appropriate closed-loop controlsystem has been designed, capable of guaranteeing that the tanks never become empties(i .e., Hi > 0, i = 1,2,3), the follows hold

Hi ≡ xi 6= 0; f or i = 1,2,3 ⇒ no lost of full rank for E(x) and D(x) (6.26)


Observing the structure of (6.20) - (6.25), the system can be reviewed as

y =C [G(x,u)+D(x) f (x,u, y)+E(x)Ψ(x, t )] (6.27)

or equivalently in the following compact form

y =C G(x,u)+ [D(x) E(x)]τ(x,u, t ) (6.28)

with implicit definition of the uncertainty term and disturbance distribution matrix

τ(x,u, t ) =[

f (x,u, t )Ψ(x, t )

]; [D(x) E(x)] =

−C1

∗pH1β1

0 0C1

∗pH1β2(H2) −C2

∗pH2β2(H2) 0

0 C2∗pH2

β3(H3) −1/β3(H3)

(6.29)

Under the full column-rank condition (6.26) for the matrices E(x) and D(x), the fol-lowing assumption is globally met:

r ank(C [E(x) D(x) ]) = r ank([E(x) D(x) ]) ≤ p ≡ 3 (6.30)

Therefore, from (6.27)-(6.30), we can infer that the propagation of the overall uncer-tainty vector term τ(x,u, t ) through the output is governed by a transfer function charac-terized by a relative degree one, i.e., all the uncertain element of τ(x,u, t ) acting on thefirst derivative of the correspondent signal output. Furthermore, in view of applying themethod presented in §4.2, we should verify that all invariant zeros of the triple (A,[E D], C)lie in C−. The check, from which follows that no invariant zeros are present, is trivial andwill be omitted.

Making the assumption the non linear term G(x,u) is Lipschitz with respect to x uni-formly for u (that belong to an admissible control set) and that the unknown nonlinearterm τ(x,u, t ) and its first derivative is norm bounded, with known bounds , the assump-tions stated in §4.2 and §4.3 are met, so the proposed observer, making use of a 2SMbased output injection control low, can be used to reconstruct in finite time the vectorsignal τ(x,u, t ).

In the following it will be assumed that the system in (6.20)− (6.21) is under feedbackcontrol and the signals u(t ) are (smooth) functions of the states x(t ).In the absence of faults it is assumed that the controller has been well designed so thatx(t ) is close to its required operating point. If a fault occurs it is assumed that x(t ) lies ina bounded compact set for at least a finite time t f > 0, starting from the onset of the fault,which allows time for detection to take place.

The proposed observer has the following form:

˙x = G(x, t )+ν (6.31)

y = C x (6.32)

From (6.28) and (6.31-6.32), and considering that C = I , descend the following obser-vation error dynamics:

ey = y − ˙y = x − ˙x = [G(x, t )−G(x, t )]+ [D(x) E(x)]τ(x,u, t )−ν (6.33)


where ν is the continuous control vector whose realization should be capable of guaran-teeing the required global and finite-time convergence to zero of ey and ey .The approach based on second-order sliding modes, presented in §4.2, has been used inorder to enable us to reconstruct the fault in finite time without any filtration, thereforeleading to an exact solution.

Two simulative tests has been performed with sampling time Ts = 0.001s. In the firsttest (Test 1) noise-free measurements are used. In the second test (Test 2), a band-limitedadditive white noise is taken into account. Fault and disturbance signals of different shapeand magnitude have been considered. Note that a fault of magnitude of 0.5 represent anerror which is the 50% of the overall valve opening run.

In the figures (6.8) and (6.9) the results of the two simulation tests are shown. A precisereconstruction of the faults and disturbances is achieved in both tests.

Figure 6.8: Test 1. Faults and disturbance reconstruction performance


Figure 6.9: Test 2. Reconstructed faults and disturbance in presence of noisy measures


6.2.2 Discrete state estimation via HOSM

Let us show that the three-tank water process can be modeled as a switched affine system[81] according to the general formulation in Eq. 5.1.

The vertical multi-tank system that we shall consider has three on-off valves V 1sw ,V 2sw , V 3sw that determine whether an outflow from each Tank exists or not. The on-offstate of the three valves define the 8 possible operating mode of the considered system.

Referring to the schematic representation in Fig. 6.1, signal q(t ), the inflow to theupper tank, represent a measurable input to the system, the binary signals U1,U2 and U3

are the unknown states of the on-off valves, and H1, H2 and H3 are the water levels whichrepresent the continuous state of the three-tank system. It is our aim to present a scheme,based on the approach described in chapter 5, for reconstructing the states of the threeon-off valves by assuming the knowledge of the water levels and of the input inflow q(t )to the upper tank.

Let us consider the system formulation in (6.17-6.19) where the three boolean vari-ables U1(t ),U2(t ),U3(t ) represent the status of the three valves. Due to the on-off behaviorof the valves, the Ci (t ) coefficients (i = 1,2,3) can assume two values only according to

Ci (t ) =

0 when valve V isw is OFFC∗

i when valve V isw is ON(6.34)

Collecting into a boolean vector the discrete states of the on-off valves as follows

δ(t ) = [U1(t ),U2(t ),U3(t )]T ∈ 0,13 (6.35)

it is straightforward to put the model (6.17)-(6.19) in the form (5.1) with x = [H1, H2, H3]T ,u = q(t ) and

G(x,u, t ) =

q(t )β1

00

(6.36)

D(x,u, t ) =

−C1

∗pH1β1

0 0C1

∗pH1β2(H2) −C2

∗pH2β2(H2) 0

0 C2∗pH2

β3(H3) −C3∗pH3

β3(H3)

(6.37)

According to the notation introduced in chapter 5, it must be highlighted that in ourcase a particular instance for (6.35) represents one of the possible k = 8 discrete statesσ(t ) in which the three-tank system could be found.

In the derived three tank system the dimension L of vector δ(t ) is L = 3 which does notexceed the dimension n = 3 of the continuous state, as required in assumption (5.2).

The assumptions (5.3) on the matrix D(x,u, t ) are trivially satisfied since the waterlevels H1(t ), H2(t ), H3(t ) remain strictly positive during the observation process. Further-more the assumption (5.4) requires that the square matrix D(x,u, t ) is nonsingular. Since

detD(x,u, t ) =− C1∗C2

∗C3∗

β1(H1)β2(H2)β3(H3)

√H1H2H3 (6.38)

again the assumption (5.4) is fulfilled if none of the water levels become zero during theobservation process. Assuming that an appropriate closed-loop supervisory system has


been designed, capable of guaranteeing that Hi (t ) ≥ H∗i > 0, i = 1,2,3, the proposed ob-

server can provide for the reconstruction of the binary signal vector δ(t ).An additive error term ε(x, t ) may take into account possible discrepancies between

the actual and nominal system model as well as possible external disturbances. It hasbeen stated, in the Lemma 1 of §5.3.2, that the discrete state can be still reconstructedexactly provided that the norm of ε(x, t ) is sufficiently small.

It is worth noting that the discrete state σ(t ) ∈ 0,1, · · · ,7 can be reconstructed fromthe thresholded estimates δ1(t ), δ2(t ), δ3(t ) by means of the following expression

σ(t ) = δ1(t ) ·22 + δ2(t ) ·21 + δ3(t ) ·20 (6.39)

The effectiveness of the suggested discrete state observer, presented in §5.3, is nowstudied by means of some simulative analysis conducted on the three tank model (6.12)-(6.14).

The inflow input q(t ) =C∗U (t ) and the binary state δ(t ) have been selected in such away that the tanks never become empty, that would cause the loss of observability for thesystem.

The parameter values used in the simulations, evaluated by means of an identificationprocedure, are reported in the Table 1.

Euler integration method with the fixed sampling time Ts = 0.001s has been used. Adisturbance vector with elements of the form

εi (x, t ) = 0.1(|H1(t )|+ |H2(t )|+ |H3(t )|)si n(ωt ), i = 1,2,3 (6.40)

is considered, and a band-limited additive white noise is added to the level measurementsH1, H2, H3. The binary signal inputs U1(t ),U2(t ),U3(t ) defining the discrete state of thesystem have been selected as shown in the plot of the next figure 6.10 (the same profilefor all the simulation tests has been used). It can be noted that a dwell time of 0.5s hasbeen used.

In the first test (Test 1), the disturbance vector ε(x, t ) and the measurement noise arenot included. The plots in the figure 6.10 show the actual δi (t ) values together with thenon-thresholded reconstructed ones δi (t ). Figure 6.11 makes the same comparison byconsidering the thresholded reconstructed values δi (t ). Figure 6.12 shows the actual andreconstructed discrete states σ(t ) and σ(t ). It can be seen that the suggested methodprovides a prompt identification of the active mode.

In "Test 2" it is shown that by increasing the VM observer parameter it can be achievedan arbitrarily fast identification of the current mode after the mode switchings. To this endthree different values of VM have been considered, and a zoom across some switchinginstant is made in the Fig. 6.13. The differences in the transient duration confirm theexpected performance.

In the last test (Test 3), disturbances and measurement noise are considered. Figure6.14 shows that signals δ1i (t ) are corrupted by the noise as compared with the Test 1. But,since the resulting errors are less than 0.5, the successive thresholding removes the errorsand recovers the correct discrete state estimates according to Lemma 1 (see figure 6.15)


0 2 4 6 8 10-0.5

0

0.5

1

1.5

0 2 4 6 8 10-0.5

0

0.5

1

1.5

0 2 4 6 8 10-0.5

0

0.5

1

1.5

Time (sec)

Actual and reconstructed (non-thresholded) binary states

Figure 6.10: Test 1. δi (t ) vs. δi (t ). From top to bottom: i = 1,2,3.

0 2 4 6 8 10-0.5

0

0.5

1

1.5

0 2 4 6 8 10-0.5

0

0.5

1

1.5

0 2 4 6 8 10-0.5

0

0.5

1

1.5

Time (sec)

Actual and reconstructed (thresholded) bynary states

Figure 6.11: Test 1. δi (t ) vs. δi (t ). From top to bottom: i = 1,2,3.


0 2 4 6 8 10

0

1

2

3

4

5

6

7

8Actual and reconstructed discrete state

Time (sec)

Figure 6.12: Test 1. Actual σ(t ) and reconstructed σ(t )

0 0.5 1 1.5

0

1

2

3

4

Time (sec)

VM

=1.0

VM

=0.1

VM

=10

Actual and reconstructed discrete state

Figure 6.13: Test 2. σ(t ) (solid) and σ(t ) (dashed) varying observer gain


0 2 4 6 8 10-0.5

0

0.5

1

1.5Actual and Reconstructed (non thresholded) binary states

0 2 4 6 8 10-0.5

0

0.5

1

1.5

0 2 4 6 8 10-0.5

0

0.5

1

1.5

Time (sec)

Figure 6.14: Test 3. Actual δi (t ) vs. δi (t ) discrete inputs

0 2 4 6 8 10

0

1

2

3

4

5

6

7

8Actual and reconstructed discrete states

Time (sec)

Figure 6.15: Test 3. Actual σ(t ) vs. σ(t ) discrete states


6.3 Experimental results

Experimental results using the three-tank laboratory-size setup apparatus by Inteco [78]are presented and commented in this section. A picture of the experimental setup isshown in the figure 6.16.

Figure 6.16: The experimental setup

The multi-tank system is interfaced with an external PC-based data acquisition andcontrol system. The development of, both, the controller and observer systems is madein the MATLAB/Simulink environment, and the associated executable code is automati-cally generated by the RTW/RTWI rapid prototyping environment. The water level in thetanks are measured with piezo resistive pressure transducer and acquired by a DAC mul-tipurpose I/O board. There are four control signals sent out from the DAC board to themulti-tank system: the three valve control signals and the DC pump control signal.

The sampling time is set to 0.01 s. The closed-loop control system is made up of antiwind-up PI controllers. The identification of the plant, sensor and actuator parameters,is carried out in order to minimize the discrepancies between the real process and itsmathematical model.

6.3.1 System identification

A parameter identification procedure has been carried out for:

• i. Level sensors characteristic curves.

• ii. Dc pump actuator characteristic curves.

• iii. Proportional valve actuator characteristic curves.

6.3. EXPERIMENTAL RESULTS 113

• iv. Parameter identification of the mathematical model.

Points i, ii and iii are basically carried out with inflow and outflow experiment for eachtank, that led us to properly calibrate the piezo-resistive transducers and to determine, forthe pump and the automatic valves, look-up tables that should be used in real-time exper-iments. However, it must be highlighted that, controlling the actuators only with on/offvalues, i.e. U (t ) and Ui (t ) ∈ 0,1 as explained later, identification procedures, concerningDC pump and proportional valve actuator characteristic curves, are not necessary. Fur-thermore, the piezo-resistive water level transducers offer characteristic curves almostlinear, therefore only a bias calibration is necessary.

The parameters C∗, C∗1 , C∗

2 , and C∗3 of the mathematical model (6.12)-(6.14), were

identified experimentally. For each tank the outflow experiment has been performed, thedata are collected and the characteristic curves has been fitted to the data Fig.(6.17).

Figure 6.17: Principle of parameters identification

For this purpose FMINS procedure from MATLAB Optimization toolbox was appliedfor the objective function given as

J =3∑

k=1wk

N∑i=1

(Hk (i )−H mk (i ))2 (6.41)

where Hk (i ) are the measurements for k-tank at i-time point H mk (i ) are the simulation

results for k-tank and wk is a weighting coefficient for k-tankThe overall obtained values are reported in the Table 1.


6.3.2 Fault reconstruction via HOSM

Two different tests has been conducted:

Test 1 involving faults f1, f2 and f3

Test 2 involving faults fp , f2 and f3

Figure 6.18: Test 1- Reference signals

The signal references and outputs concerning the Test 1, during which faults f1 and f2

and f3 have been generated, are depicted in Fig. 6.18 and 6.19 respectively. In particular,according to equations 6.12-6.14, Fig. 6.18 shows the reference signals U ,U1,U2,U3, i.e.vector u∗ in Fig.6.6, and the fault signals fp , f1, f2, f3.

It worth noting that, according to Fig. 6.6, the experimental tests have been conductedproviding inputs U + fp , U1 + f1, U2 + f2 and U3 + f3 to the respective actuators.

As we can see f1, acting between 20 and 30 second and 70 and 80, represents a stuck inposition of full opened of the respective valve in correspondence to the instant in whichthe command reference to the valve switch from 1 to 0. Equivalently the reasoning holdfor f3. The fault f2, acting from 60 to 80 seconds, with its negative value represent a stuckof the respective valve in position of full closed, despite the command reference is 1.

In Fig. 6.19 the measured water levels are shown.The signal references and outputs concerning the Test 2, during which faults fp and f2

and f3 have been generated, are depicted in Fig. 6.20 and 6.21 respectively. As we can seevalve 1 is not affected by faults. As for Test 1, all the additive faults could represent a stuckof the pump/valve, in particular for the considered case, faults fp and f3 represents stuckin position of full open for the pump and the valve 3. Additive fault f2, that acting nega-tively respect to the reference signal for the valve 3, correspond to a stuck in completelyclosed position for the valve.

It must be highlight that the particular time-occurrence of the faults, that contempo-rary occurs, makes more difficult the issue of FDI.


Figure 6.19: Test 1- Measured signals

Figure 6.20: Test 2- Reference signals

Our aim is to realize an Observer-Based FDI scheme guaranteeing the precise recon-struction of the (possibly simultaneous) additive actuator faults triplet f1, f2 and f3, or thetriplet fp , f2 and f3. First to proceed showing the followed FDI methodology, it must behighlight that, for our system, the number of faults that we can diagnose simultaneouslyare maximum three, since there are only three measured variables. Furthermore sincefaults fp and f1, acting on the pump and on the valve 1 respectively, are booth presenton the first differential equation of the system model (6.12), it’s straightforward to un-derstand that the fault isolability is not permitted if such faults act on the system contem-porarily. This is the reason why only the formers triplets of faults are considered. However,


Figure 6.21: Test 2- Measured signals

it must be highlight that, this is not always true, especially for those faults acting on thesystem with different frequencies behavior that could allow the faults to be identified.

The actuators fault signals can be modeled by additive terms corrupting the adjustablesignals U (t ), U1(t )−U3(t ). Thus, by including all such possible fault signals in the model,the system (6.12)-(6.14) can be rewrite in the general form

x = G(x,u)+D(x) f (x,u, t )+Ψ(x, t ) (6.42)

y = C x (6.43)

where x = [H1, H2, H3]T , u = [U ,U1,U2,U3]T , C = I3 being the identity matrix, and G(x,u) =G∗(x)u(t ) with

G∗(x) =

C∗β1

−C1∗pH1β1

0 0

0 C1∗pH1

β2(H2) −C2∗pH2

β2(H2) 0

0 0 C2∗pH2

β3(H3) −C3∗pH3

β3(H3)

(6.44)

D(x) =

C∗β1(H1) −C1

∗pH1β1(H1) 0 0

0 C1∗pH1

β2(H2) −C2∗pH2

β2(H2) 0

0 0 C2∗pH2

β3(H3) −C3∗pH3

β3(H3)

(6.45)

Ψ(x, t ) =Ψ1(x, t )Ψ2(x, t )Ψ3(x, t )

, f (x,u, t ) = [ fp , f1, f2, f2]T (6.46)

The nonlinear vector fields G∗(x) and D(x) are assumed to be known, while vectorsf (x,u, t ) and Ψ(x, t ) represent the considered actuators faults and the external distur-bance, respectively.


As one can see, the measurements actually represent the complete state vector. Nev-ertheless, the design of an appropriate observer is necessary to identify and reconstructsignal faults and disturbance term.

The proposed observer assume the following form [80]:

˙x = G(x, t )+ν (6.47)

y = C x (6.48)

where x is the estimated vector state and ν is an appropriate control law to be de-signed. It is worth noting that in 6.47 we used G(x,u), instead of G(x,u), although inthe actual case the full state vector is directly accessible for measurement. This choiceis made in order to attenuate the effects of the measurement noise in the piezo-resistivetransducers. It has to to designed an observer control vector ν capable of guaranteeingthe required global and finite-time convergence to zero of e and ey .

Here the output injection ν will be designed according a second order sliding modesalgorithm, like SuperTwisting or Sub-Obtimal controllers, in order to obtain a continuouscontrol signal allowing to reconstruct the faults without any filtration.

From (6.42)-(6.43) and (6.47)-(6.48) descend the following observation error dynam-ics:

ey = y − ˙y = x − ˙x = [G(x, t )−G(x, t )]+D(x) f (x,u, t )+Ψ(x, t )−ν (6.49)

where ν is the continuous control vector, designed according a 2-SM control algorithm,whose realization is capable of guaranteeing the global and finite-time convergence tozero of ey and ey ([83]).Therefore if we assume that ey is guaranteed in finite time, by neglecting the uncertainterm Ψ(x, t ), we obtain

D(x) f (x,u, t )−ν= 0 (6.50)

Assuming that an appropriate control system guarantees that the tanks never becomeempties, i.e. Hi > 0(i = 1,2,3), and considering that at maximum three simultaneous faultcan be reconstructed since only three measures are available, the faults reconstructionformulas can be obtained from

fTest1(x,u, t ) = D1(x)−1ν (6.51)

fTest2(x,u, t ) = D2(x)−1ν (6.52)

where matrices D1 and D2 are obtained removing from matrix D the first and the sec-ond column, respectively.


The reconstruction formulas take the following expressions (6.53)-(6.56)

fp = ν1β1

C∗ (6.53)

f1 = −ν1β1

C∗1

(6.54)

f2 = −ν1β1 +ν2β2

C∗2p

H1(6.55)

f3 = −ν3β3 + f2C∗2p

H2

C∗3p

H3(6.56)

The following figures (6.22) and (6.22) shown the results obtained with the methoddescribed in previous section. As one can see the proposed FDI scheme performs goodresults for the considered case, even in presence of the following issues:

1. Presence of measurement noise

2. Presence of model mismatches, unmodelled nonlinearities

3. Sampling period is not so small

4. Strongly coupled system equations

5. Abrupt fault signals

6. Simultaneous faults (worst case between 70 and 80 seconds of test 1)

7. FDI scheme applied in real time.

Figure 6.22: Test 1- Reconstructed faults


Figure 6.23: Test 2- Reconstructed faults


6.3.3 Discrete state estimation via HOSM

According to the corresponding simulative section §6.2, now we’ll present some experi-mental result for the case of discrete state estimation. In Fig 6.24 it is shown a generictrend of the binary state δ1(t ) of valve V 1sw and the corresponding reconstructed oneδ1(t ). It must be highlighted that, after an initial transient time, the reconstructed signalmatches quite precisely the signal δ1(t ).

0 10 20 30 40 50−0.5

0

0.5

1

1.5Actual and reconstructed (non thresolded) first binary state

Time (sec)

Figure 6.24: Actual δ1(t ) and non-thresholded reconstructed δ1(t ) discrete inputs

Figure 6.25 shows how, after the thresholding procedure (5.28), apart from a tran-sient time depending on the magnitude VM of corresponding injection signal, the currentmode can be precisely reconstructed. In particular, as shown in Fig. 6.25, two differentvalues for VM have been used, and as expected, increasing VM the transient time can bereduced.

It must be highlighted that differently from what stated in the theoretical analysis, thetransient time cannot be made arbitrarily small. This is due to several reasons, like dis-cretization ad noise effect. Furthermore sensors and actuators are unavoidably affectedby unmodelled dynamic effects.

The plots in Fig. 6.26 show the actual δi (t ) values together with the thresholded recon-structed ones δi (t ).

Figure 6.27 shows the actual and reconstructed discrete states σ(t ) and σ(t ). It can beseen that the suggested method provides for a prompt identification of the active mode.


10 15 20 25 30−0.5

0

0.5

1

1.5Actual and reconstracted (thresolded) first binary state

Time (sec)

VM

=10

VM

=20

Figure 6.25: δ1(t ) (solid) and δ1(t ) (dashed) varying observer gain

0 20 40 60 80 100−0.5

0

0.5

1

1.5Actual and Reconstructed (thresolded) binary states

0 20 40 60 80 100−0.5

0

0.5

1

1.5

0 20 40 60 80 100−0.5

0

0.5

1

1.5

Time (sec)

Figure 6.26: δi (t ) vs δi (t ). From top to bottom: i = 1,2,3.


0 20 40 60 80 1000

1

2

3

4

5

6

7

8Actual and Reconstructed discrete states

Figure 6.27: Actual σ(t ) and reconstructed σ(t ) discrete state

Chapter 7

Concluding remarks

The first part of the thesis, Chapter 1 and 2, has been devoted to introduce some indis-pensable concepts inherent the model based FDI. The main methods for residual genera-tion and fault detection have been described with particular attention to observer-basedtechniques, well known in literature, and for schemes based on UIOs (Unknown Input Ob-servers). Furthermore, the concepts of structural fault detectability, isolability and identi-fiability were reviewed to describe the structural property of a system from the FDI pointof view. Chapter 3 provides an overview of various SM observer approaches for linearand nonlinear uncertain systems. Particular attention has been given to the structuralproperties for a system to be observable with a classical SMO and some techniques tobroadening the class of those systems. In order to be useful for FDI purpose, the chapterfocuses on those observation techniques that enable the estimation of the unknown in-put uncertainties acting on the system. Furthermore, the chapter gives an explanation ofsome important paradigms like the step-by-step observers and HOSM differentiators.

In Chapter 4 two HOSMOs for the additive actuator faults reconstruction in nonlinearsystems affected by simultaneous faults and disturbance signals were presented. The firstscheme, involving a full order observer, under certain condition and on the basis of a coor-dinate transformation aimed to put the system into a canonical form, makes use of a cor-rective output injection term, based on a 2-SMC algorithm. It allows for the precise faultreconstruction by means of the continuous corrective term and therefore avoiding anyfiltering requirement despite of the uncertain term, under certain conditions. Further-more, by dispensing with the request of estimating the state vector, in other words by onlyaiming to the reconstruction of the faults, a reduced order observer was designed. Alsofor such case, as for the full order case, the output injection of the proposed observer isbuilt by means of a second-order sliding mode control algorithm, featuring global conver-gence properties, that permits the reconstruction of the actuator fault without requiringany filtration.

In Chapter 5, we focalized our attention into those cases in which the system under di-agnosis is characterized by different nonlinear dynamics. In fact the identification of thecurrent dynamic can be fruitfully used to cope with the FDI problems for many reasons,first of all the possibility of associate a specific dynamic to a system affected by specifiedfaults. In this case, the detection of the actual dynamics leads to fault detection, directly.

123

124 CHAPTER 7. CONCLUDING REMARKS

On the basis of these considerations, since the problem can be described as that of thecurrent discrete state identification for switched systems evolving according to the occur-rence of discrete events, we investigated the problem of the discrete state reconstructionfor switched systems. The base idea is that a general class of switched systems can bemodeled by nonlinear systems with an affine Boolean input representing the system dis-crete state. With the aim to reconstruct such a binary input, despite bounded uncertain-ties affecting the system dynamics and making use of the intrinsic robustness of Booleansignals, we proposed an observer with a corrective output injection term, based on a 2-SMC algorithm, that exhibit remarkable properties of robustness against uncertaintiesand disturbances for reconstructing the unmeasurable quantities affecting the system. InChapter 6 the three tank water process system has been introduced to the aim of pro-vide a benchmark system respect to which the methods of FDI presented in Cap.4 and 5could be tested and evaluated. Two schemes for the simultaneous reconstruction of actu-ator faults and for the discrete state reconstruction have been applied to the real process.The proposed observers, which makes use of a second order sliding mode controller algo-rithm, permits the direct reconstruction of the faults and of the discrete states, from thecontinuous observer output injection signal, avoiding the delays and intrinsic errors as-sociated to any filtration process. Experimental results made on the laboratory size setupconfirm the effectiveness of the proposed schemes despite the presence of measurementnoise, unmodelled nonlinear phenomena and not very small sampling period.

Among possible lines for future investigations it appears interesting to embed the sug-gested FDI approaches into novel, HOSM based, fault tolerant control systems for nonlin-ear uncertain processes.

More significant and practically relevant industrial applications of the developed ap-proaches are devised in the framework of the previously mentioned PRODI project.

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List of Publications Related tothe Thesis

Published papers

Journal papers

[79] G. Bartolini , N. Orani, A. Pisano, E. Punta , E. Usai, “A combined first/second order sliding-mode technique in the control of a jet propelled vehicle“, International Journal of Robust andNonlinear Control, 18:4-5, pp. 570 - 585, 2007. [cited at p. -]

[80] N. Orani, A.Pisano, E. Usai, “Fault Detection and Reconstruction for a Three-Tank Sys-tem via high-order sliding-mode Observer“, Advances in Nonlinear Observation andIdentification for Dynamic Systems. Journal of the Franklin Institute, published on linedoi:10.1016/j.jfranklin.2009.11.010. [cited at p. 102, 117]

Conference papers

[81] N. Orani, A.Pisano, M. Franceschelli, A. Giua and E. Usai . “Robust reconstruction of thediscrete state for a class of nonlinear uncertain switched systems“, ADHS09: 3rd IFAC Con-ference on Analysis and Design of Hybrid Systems. Zaragoza, Spain September 16-18, 2009[cited at p. 90, 107]

[82] N. Orani, A.Pisano, E. Usai. “Exact reconstruction of actuator faults by reduced-order slidingmode observer“. IEEE Multi-conference on Systems and Control. Saint Petersburg, Russia, July8-10, 2009. [cited at p. 75]

[83] N. Orani, A.Pisano, E. Usai. “Fault Detection and Reconstruction for a Three-Tank Systemvia high-order sliding-mode observer“. IEEE Multi-conference on Systems and Control. SaintPetersburg, Russia, July 8-10, 2009. [cited at p. 117]

Submitted papers

[84] N. Orani, A.Pisano, M. Franceschelli, A. Giua and E. Usai. “Robust reconstruction of the dis-crete state for a class of nonlinear uncertain switched systems“. Special issue of the IFAC af-filiated Journal Nonlinear Analysis: Hybrid Systems, published by Elsevier [cited at p. -]

131

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