FAULT DETECTION AND IDENTIFICATION OF A SPECIAL CLASS

OF GENERALIZED STATE-SPACE SYSTEMS

by

Yaang (Ken) Zhao

B.Eng., Xi'an Jiaotong University, 1984

MBA, The University of Alberta, 1988

THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIEEMENTS FOR

THE DEGREE OF

MASTER OF APPLIED SCIENCE

in the School

of

Enginee~g Science

O Yaang (Ken) Zhao 1997

SIMON FRASER UNIVERSITY

May 1997

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Abstract

The focus of this thesis is on fault detection and identification of constrained mechanical

systems. This kind of system can not be exclusively described by dynamic equations because

the constraints represent algebraic relations among certain system variables. Such systems

which are partially dynamic and partiaiiy algebraic are caiied generalized state-space systems,

descnptor systems or singuiar systems. Constrained mechanicd systems are a special class of

descriptor systems because they lack infinte observability and complete controiiability, which

are desirable system properties for state estimator based Mt detection rnethods. This thesis

deals with the unique characteristics of constrained mechanical systems and presents a

systematic approach for fauIt detection and control of such systerns under uncertainties. In

this thesis, actuator faults are modeled as unknown inputs to the dynamic equations of

typically nonhem constrained mechanical system. Sensor W t s are added to the output

equations of the system. The noniïuiear system mode1 is first linearized about an operating

point. Then a coordinate transformation technique is used to convert the resultant linear

descriptor form representation of the system into two sub-systerns: a dynamic subsystem plus

an algebraic subsystem. Based on the dynamic subsystem representation, an unknown input

observer is designed to provide estirnates of displacements, velocities, constraint forces, and

sensor faults simultaneously. The estimates of sensor faults provide immediate means for

iv

sensor Fdult detection and identification. The estimates of displacements, velocities, and

constraint forces can be used in state feedback control of the system. Actuator fadt detection

and identification is accomplished by estimating actuator faults using a least square solution

technique which uses the estimation of the state vector of the system. This model-based

analytical redundancy approach offers many advantages. 1t can detect a wide varïety o f faults.

It generates not ody the magnitude but also the shape of the fa& and thus possesses the

capability of distinguishing between momentary fadts and persistent ones. Moreover, Its

mathematical simplicity and computational efficiency makes it a better candidate for

computer simuiation andlor reai-time implernentation. Simulation performed using a practical

system (an UMS-2 robot) mode1 indicates that the proposed approach is capable of detecting

and identGng multiple andor simultaneous actuator faults and sensor faults almost

immediately.

Dedication

To the fountain of civilization - applied science and the people that love me

or believe in me in my life

Acknowledgments

F i 1 would very much iike to thanlc Dr. Mehrdad S n a my supanrishg professor, for

guiding me through my program of study, finding potential thesis topics for me, and helping

me overcorne some of the difliculties encountered in this thesis. Es suggestions, comments,

and corrections with respect to this thesis are very much appreciated.

Secondly, 1 would iike to thank Nancy, my Iovhg wife, for her constant and strong support.

Thirdly, 1 would iike to thank Professor William Gruver and Professor Emeritus George

Bojadziev for serving on my thesis committee and spending their vaiuable tirne reviewing the

thesis. 1 would also iike to thank Dr. Andrew Rawicz, Chairman of the thesis examining

committee, for his t h e .

Fourthly, 1 would iike to thank Ms. Brigitte Rabold, Graduate Secretary of the School of

Enginee~g Science at SFU, for the excelient seMces she provided during my program of

study.

FiRhly, 1 would iike to thank my fiiend Yidong Zhang for advising me to use Microsoft Word

to wite the thesis and acting as a communication agent between me and SFü while I was

doing the last part of my thesis off campus.

Contents

Title Page

Approvd Page

Abstract

Açknowledgments

Dedication

Table of Contents

List of Figures

1 Introduction to Fault Detection and Identification

2 Constrained Mechanical Systems

2.1 Introduction

2.2 Description of Constrained Dynamic Systems

2.3 S pecial Form of Constrained Linear Mechanical systems

2.4 Normal Foms for Constrained Linear Mechanical systems

2.5 Regularity of Constrained Linear Mechanical systems

2.6 Controllability of Constrained Linear Mechanical systems

2.7 Observability of and Observer Existence Conditions

for Constrained Linea. Mechanid systems

2.8 Sumrnary

3 An Unknown Input Observer Based Fault Detection and Identification Method

3.2 Introduction

3 -2 S ystem Representation and its Observable Canonical Fom

3 -3 Assumptions for the Design of UIO's

3 -4 Unknown Input Observer(UIû) Design

3 -5 Necessary and SufEcient Conditions for the Existence of UIO's

3 -6 Problem Formulation for Fault Detection and Identification

3 -7 Actuator Fault Detection and Identification

3.8 Senssr Fault Detection and Identification

3.9 Summary

4 FDI Study of a Constrained Mechanical System - Approach and Simulation

4.1 htroduction

4.2 Approach and Simulation

4.5 Summary

5 Conclusions 94

Appendices

Appendk A : Proof of Regularity of the Onginal System

Appendk B : Proof of Infinite Unobservability of the Original System

Appendix C: Simulation Program Source Code

Figures

References

List of Figures

Figure 1 - simulation #1: sensor 1 failure and its estirnate

Figure 2 - simulation #1: actuator 1 faiiure and its estimate

Figure 3 - simulation #1: actuator 2 failure and its estimate

Figure 4 - simulation #2: sensor 1 failure and its estimate

Figure 5 - simulation #2: actuator 1 fdure and its e shate

Figure 6 - simulation #2: actuator 3 fdure and its estimate

Chapter 1 introduction to Fauit Detection and Identincation L

Chapter 1

Introduction to Fault Detection and Identification

Automatic systems have been wideiy employed in commerce and industry for many years.

Technologicai progress has made rnany of these systems more complex and sophisticated.

Exarnples of these dynamic systerns include commercial and rnilitary aircrafi, navigation

systems, space shuttle, nuclear reactors, chernicd reactors, robots, and many others. These

systems can consist of many working parts which may maifùnction or fail at any time.

Complete failure of these systems, especially those mission-critical ones, can resuIt in

unacceptable economical loss andor human casualty. The need for reliability and fault-

tolerance in these systems at reasonable cost prompted and in some cases fueled research in

fault detection, isolation, and accommodation. New developments in fadt diagnosis of

dynamic systems started to appear in the 1970's. Sorne basic theoretical and application

results were achieved in the 1980's and emly 1990's. Research in this cornplex, diverse, and

relatively new field continues today and is expected to continue welI into the distant firture.

For exarnple, several major aircraft manufacturers and car makers currently have some kinds

of then own R&D activities in this area. On-board fault detection or registration may becorne

a design critenon in some models of airplanes and automobiles.

A dynamic system, or a plant as it is commonly referred to, cm be divided into three types of

subsystems: actuators, main structure or process (which may consist of components), and

instrumentation/sensors. Let's take an aircraft flight control system as an example. The

actuators are the servornechanism that drive the control surfaces and engines which provide

the driving thmst. The autopilot controiier provides the actuators with the input or control

signds. The main structure is the airfiame with its cargo and appendages, dong with the

aerodynarnic forces exerted on the control surfaces. The instrumentation consists of several

sensors or transducers attached to the airfiame. The sensors provide signals proportionai to

the vital motions of the airftarne. These signals include airspeed, aititude, heading,

acceleration, attitude, rate of change of attitude, control surface deflection, enghe thmst,

Chapter 1 Introduction to Fauit Detection and Identification 2

etc.. Sensor signals are fed back to the autopilot which uses the feedback information and

referencdcommand inputs in its dynamic determination of new control signals. The actuaton

execute the new control signals dynamically and possibly affect the state of the system and

sensor measuements again. Such a system is called a closed-loop control system or a

feedback control systern in control system engineering.

Research in the field of fault diagnosis has led to the invention of some jargons. Three of the

most comrnody used ones are FDI (Fault Detection and Isolatiodidentincation), FDIA

(Fauit detection, Isolation/Identi£ication, and Accommodation), and IFD (Instrument Fault

Detection). Fault detection and identification means declaring the occurrence of fauits and

indicating which sensors, actuators, or cornponents are faulty. Fault accommodation refers to

the recor@uration of system signals or component actions in order to permit continued

operation of the system. Fault accommodation is an application-specific task and is not

addressed by most researchers. Another thing that has not been addressed by most

researchers in FDIA is the reliability of typically digital cornputers, which are usuaily used in

the implementation of FDI algorithms. Research in fault diagnosis have been focused on

sensor fault detection and to a lesser extent on actuator fault detection, dthough one research

work using least square parameter estimation methods has shown its capability of detecting

and localizing process faults or component failures. A typical fault monitoring scheme is

usually designed to detect and correct faults in one or two of the three subsystems. Early

proposed schemes were primarily concemed with sensor fault detection. Once detected,

sensor faults could usudly be corrected by electronic switching techniques and do not require

the reconfiguration of mechanical parts. On the other hand, actuator fault accommodation is

usually more dficult than re-directhg electricd signals. The compensation of faults in the

main structure is even less feasible and usually requires expert knowledge of the underiyhg

system. This is probably one of the most challenging aspects of any practical FDLA scheme.

The traditional approach to fault tolerance in dynarnic systems is hardware redundancy.

Typically three or four identical or similar hardware elements (actuators, measurement

sensors, process components, etc.) are distributed spatially around the system to provide

protection against localized damage. Multiple elements are used to perform a single task for

Chmter 1 Introduction to Fault Detection and Identification 3

which one element is d c i e n t if it was completely reiïable. For example, three or more

sensors could be instailed to measure the same output. The rneasurements ikom the sensors

codd be compared in a logic circuit for consistency. If the measurement fiom one sensor

deviates too much f h m the average of the measurements fiom the other sensors, then this

sensor is declared faulty. The underlying reasonable assumption is that the other sensors

remain within a smaii merence fiom each other. Additionally, the logic circuit gives some

ailowances for electronic noise, manufacturing tolerance, and monitoring errors inherent in

instruments. The hardware redundancy approach is generaily simple and straightforward to

apply . It is therefore widely used. It is essential in the control of aircraf?, space vehicles and in

certain safety-critical process plants that involve nuclear reactors or dangerous chemicals.

The major problems associated with hardware redundancy or physical redundancy are the

extra cost and software and, furthemore, the additional space required to accommodate the

redundant equipment andor the extra weight brought on by the redundant equipment. In

aircraft, for example, the additiond space could be used for more mission-oriented

equipment. The additional weight Iimits the pay-load for defensive equipment and, most

particularly, for fbel. Moreover, since redundant senson tend to have similar life

expectancies, it is likely that when one sensor fails the other will soon become fadty too.

New developments in FDIA have been prompted by the high cost of excess hardware and the

space and weight penalties associated with hardware redundancy since the early 1970's. The

avdability of reiiable and powerful cornputers also contrîbuted to the developments of new

approaches which elirninate some or all of the redundant hardware. These new approaches to

FDIA are based on fùnctional redundancy inherent in the systems. The fitndamental idea is

that entirely different measurements nom three (or more) dissimilar sensors are driven by the

same dynamic state of the system and are therefore fiuictionally related. These different

signals cm be used in a cornparison scheme more sophisticated than the simple majonty-vote

logic used in hardware redundancy approaches to detect and iden* sensor faults. These

newer schemes were initially cded inherent redundancy or fiinctional redundancy to

distinguish them fiom physicai or hardware redundancy. They are now better or alternatively

known as analytical redundancy or artificial redundancy. VirtualIy ail of the published

Chaoter 1 Introduca'on to Fa&! De-on and Identincation 4

research works in FDIA belong to this new cIass of approaches, dthough it bas been

recognized that both hardware redundancy and anaiyticai redundancy approaches can be and

in many cases should be employed together to advantage.

Analytical redundancy cm use and has used knowledge fiom severd academic discipiines.

These include but are not Iimited to control theory, statistics, and computer science. Specific

techniques employed in anaiyticai redundancy FDI approaches include state estimation,

parameter estimation, adaptive filtering, variable threshold Iogic, statisticd decision theory,

and combinatorial and logic operations. There are plenty books and papers on these

disciplines and subjects. For exarnple, the book of Swisher (1976) and the book of Chen

(1984) contain information on reduced-order and Mi-order observer design techniques which

can be used for state estimation. Other basic concepts in control theory such as state-space

rnodeling, state (variables), state controllability, output controlIabiiity, observability, state

feedback, output feedback, and stability are also covered in these books. The book of Dai

(1989) provides singular control system theory which is usefiil in dealing with generalized

state-space or descriptor systems among which are constrained dynamic systems.

AU of the aforementioned techniques can be implemented using high speed digitai computers

or electricai circuits. Kigh level system simutation or rnodeling languages such as MATLAB,

Simulink, or Ma- can be used in simulation of FDI schemes on dynamic systems. Lower

level langu~ges such as Assembly or C cm be used in experirnentation or real-time

application.

Analytical redundancy FDI approaches are essentially based on rnodeling dynamic systems in

one way or another. Either the dycarnic nature of the system is known to a reasonable degree

of precision or the physical parameters of the system can be determineci by some lcincis of on-

line identification techniques. Normaliy the R)I subsystem is constructed in parallel to the

monitored system. It can use both the input signals and the output signals of the monitored

system to generate signais within itseif These generated signais serve the same purpose as

the majority-vote signais used in hardware redundancy, Le., they cm be used in logic

Chapter 1 Introduction to Fadt Detection and Identification 5

processing or other kinds of sophisticated aigorithm to detect faults and identm faulty

efements.

To illustrate the basic notion of IFD scheme, assume that there are p sensors and one of the

them is known to be refiable. Also assume that an observer or state estimator can be

constructed using the measurement signal f?om this reliable sensor and the inputs to the

monitored plant. In uiis case the FDI subsystem can jenerate estimates of the measurement

signals of ail the other sensors. These estimates can then be compared with their actual

counterparts. Simple threshold logic can be applied to the difference signals to detea and

identQ sensor faults. In view of the noise in sensor signals and the inaccuracy in syaem

modehg and estimation, the thresholds shall be non-zero to prevent false alamis and yet

small enough to allow the FDI scheme remain sensitive to rnoderate faults. Obviously, there

is a compromise or balance between sensitivity to incipient (slowly developing or small) faults

and false alarm rate in this case as in many other cases. Incidentaily, this example is h o w n as

dedicated observer scheme (DOS), which was presented by Clark (1979). Many variations

and alterations of this simple idea are possible.

Functiondy-redundant R)I schemes may be further classified into at Ieast three subslasses

according to the techniques used in the schemes. The first sub-class of schemes uses state

estimation technique which is believed to be the most widely employed technique in all

analytical redundancy FDI schemes. This technique is suitable for systems for which a set of

differential equations (plus a few algebraic equations in the case of constrained dynamic

systems) can be fairly easily obtained by applying the physical or engineering laws governing

the motions of the system. Examples of such systems can Uiclude aircraft and robots. The

approach presented in this thesis fds into this sub-class. Typically the nonluiear mathematical

mode1 of the dynamic system in this sub-class is Iinearized and also converted hto a state-

space representation format. The analysis of the system and the design of state estimator or

observer based FDI subsystem is canied out in the realrn of linear system theory, or linear

singular system theory in the case of constrained dynamic systems. The second sub-class of

analyticai redundancy approaches uses parameter estimation techniques. A survey of the

schemes in this sub-class is presented in the paper of Isermann (1984). A thoroughly studied

Cha~ter 1 Inboduction to Fault Detection and Identincation 6

method in this sub-class is the so called least-squares parameter estimation technique. This

approach can provide on-line estbates of physical system parameters. Estuiiated parameters

associated with specific subsystems of the plant or process c m be used to detect and identm

faults in these subsystems or components. This method is considered to be particuIarly

important for process plants such as chernical processes and nuclear reactors. In these

process plants, parameter variations result from process faults can cause rapid parameter

estimate changes, even though the process itself typicaliy has a slow dynamic behavior. This

approach can detect and identifil both component faults and sensor faults. The thkd sub-class

of anaiytical redundancy based approaches uses the so caiied paramemc modehg technique.

A parametric mode1 is essentiaiiy an estimator of a process variable using other process

variables as inputs. Some simulation and actual experirnents using this approach have been

performed at several auclear power stations. Readers who are interested in this technique are

referred to the works of Kitamura (1980), Kitamura, et al. (1979), and Kitamura, et al.

(1985).

Sûll another major class of FDIA schemes use the knowledge-based methodology. These

knowiedge-based expert systems are designed using artificial intelligence (AI) techniques.

Expert systems are currently hding application to an increasuig repertory of human [Xe

domains, in the center of which lies the fault diagnosis and repair domain of technological

processes. Interested readers are referred to the survey paper of Pau (1986), the paper of

Tzafestas (1987), and the book of Tzafestas, et al. ( 1989).

Some of the criteria for evaluating the performance of an FDI scheme are: a) promptness of

detection, b) sensitivïty to incipient faults, c) fdse alarm rate, d) missed fault detections, and

e) incorrect fault identification. A discussion of each of these criteria is now given.

The basic hction of a FDI scheme is to register an alarm when an abnomal condition

develops in the system and to identiQ the abnomial component. Assuming that a fault is

detected successfuliy, the issue of promptness may be of vital importance. In certain

applications such as aerospace, a fadt that persists for a second may destroy the mission of

the operating system, ifnot the operating system itself

Chanter I Inaoduction to Fault Detection and Identification 7

In certain applications it may be more desirable to have reliable detection of minor faults at

the sacrifice of speed in detection time or promptness. In some systems a fault detection

scheme is intended to enhance maintenance operation by early detection of worn equipment.

In this case promptness of detection may be of secondary importance to sensitivity. In other

systems sensitivity and promptness may both be required. This lads to more complex

detection schemes, possibly require both hardware and analytical redundancy.

Faise dams are generaiiy indications of poor performance of FDI schemes. Even a low false

aiarm rate d u ~ g normal operation of the rnonitored plant may not be acceptable because it

can quickly lead to lack of confidence in the detection scheme. However, a FDI scheme that

has an acceptable false alarm rate might register a faise alarm when a plant undergoes an

unusual excursion, and this may be acceptable in some applications.

In other applications smail faults may be so serious that it is preferable to react to fdse

alarms, replacing unfailed components with spare parts, than to s a e r deteriorated

performance fiom an undetected, though smd, fadt. In these cases it is preferable to

minimize the number of missed detections at the expense of the creditability of detections.

Incorrect fault identification means that the system correctly registers that a fault has

occurred but incorrectly identifies the component that has failed. If the reconfiguration

system proceeds to compensate for the wrong fault, it could produce a consequence as

serious as a missed detection in some applications.

The compromises in detection system design among false d m rate, sensitivity to incipient

faults, and promptness of detection are difficdt to make because they require extensive

knowledge of the working environment and an explicit understanding of the important

performance criteria of the monitored system.

In deaiing with rnalfiinctions of fault detection schemes, especiaüy the problem of false

alarms, some researchers have developed FDI schemes that use variable or adapting fault

C h t e r 1 ~ u c t i o n to Fault Detection and Identincation 8

detection thresholds. Some of these techniques have demonstrated capability of reducing or

minimizing false alatms. ûther researchers have focused their attention on the problem of

robust fault detection by design. Most of these robust fadt detection schemes were designed

with the goal of m-g the sensitivity of the detector to actual sensor malfùnctions,

while discrimùiating between these faults and disturbance effects due to noise and uncertain

dynamics.

Robustness of a fault detection and identification scheme can be dehed as the degree to

which its performance is unafTected by conditions in the operating system which tums out to

be diierent fiom what they were assumed to be in the design of the scheme. Robustness

problems occur with respect to four features of the operating plant: a) parameter

uncertainties, b) unmodelled non-linearities or uncertain dynamics, c) disturbance and noise,

and d) fault types. A brief discussion of each of these issues is now given.

Parameters refer to physical characteristics such as properties of mass, moments of inertia,

electrical circuit parameters, heat transfer properties, etc. Many FDI schemes use state

estimation techniques which are based on mathematical modeling of the monitored systern.

The modeIs are often iinearized and simplifïed and result in linear and tirne-invariant (the

simplest class of dynarnic systerns) system representations. The inaccuracy of the mode1

depends on the uncertainty of the vaIurs of the parameters. if ail the parameters are known

with precision, then state estirnates can be very accurate and the FDI scheme may be

remarkably sensitive to incipient faults and immune to false alarms. However, parameter

values are ody known approximately in most applications, especiaiiy in systems that involve

fluid flows or heat tramfers. Therefore, state observers or estirnators have to be constructed

using only nominal values for uncertain parameters. This d l result in erroneous estimates.

The severity of the error depends on maneuvers of the system which can not easily be

determined. The algorithm or logic devices used for processing the redundant signals (e.g.

state estimates) may generate false alarms, or if they are protected against this, they may fail

to detect faults. This is the robust problem with respect to parameter uncertainty.

Chapter I introduction to Fault Detection and Identification 9

Nonlineanty is a naîural characteristic of alI practical systems. Strictly speaking, hear

dynamic systems don? exïst in the red world. One of the two major reasons that we study

linear systems and use linear system theory in analysis and design is that many of these

nonhear systems behave alrnost linearly within a narrow range of a nominai operating point.

The other reason is that iinear system theory is more established and easier to apply than

nonlinear system theory. A FDI scheme based on linear (hearked) models could be quite

satidactory as long as the plant does not operate outside the range used for Iinearization.

However, outside of this range nonlinearity may produce signals which are not modeled

accurately by the FDI scheme. These signals may then be interpreted as faults. This is the

robust problem with respect to unmodeled nonlinearity or uncertain dynamics.

Real world dynamic plants are always subject to disturbances. Disturbances are unintended

system inputs originating fiom the operating environment. For example, wind fluctuation is a

disturbance for certain systems. Disturbances are usually random signals. Furthemore,

sensors are uwaiiy subject to the influence of random signals which typically originate f h m a

different source. These random signals are cded noises. Most signal processing techniques

used in FDI schemes are based on the assurnption that the disturbances and noises are

stationary Gaussian processes and uncorrelated. If the raadorn signds are non-stationary,

non-ûaussian, or correlated in some way, then the performance of the FDI scheme d l be

worse than expected or even unsatisfactory. This is the robust problem with respect to

disturbances and noises.

Faults can take many forms such as a nonlinearity due to Wear or mion , excess noise, or a

stuck value at any level within its dynamic range. Some FDI schemes are designed to detect

only specific types of fdures. If a malfiinction or fault occurs and it is not in the repertoire,

then the FDI scheme cm not detect it. This is the robust problem with respect to fault types.

Some techniques have been developed by some researchers in the field of fault diagnosis to

deal with some of the aforementioned robustness problems. For example, unmodeled or

uncertain dynamics have been shown to act like a disturbance on a linear system in observer

or state estimator based FDI schemes. The robust fault detection problem becomes one of

Chauter f Introduction to Fault Detedon and Identification LO

disturbance-decoupihg by design. This type of approach is known as the Unknown Input

Observer Scheme @JIOS). Techniques used to deal with robustness problems with respect to

fault types include hypothesis-generation and hypo thesis-testing . The hypo thesis-generation

procedure is to build up a repertoire of known or hypothesized possible mafinctions or

faults in system components or instnunents (sensors). Interested readers are referred to

chapter 10 and chapter 1 1 of the book of Patton, Frank, and Clark (1989).

The most chailenging and usuaiiy missing part of research works in fadt diagnosis is testhg

or ushg the FDI scheme on a reai system or operating plant. Nomally the application of new

and developing FDI schemes to actual operating systems are prohibited because of expense

or safety. If and when one does get an opportunity or authorkation to test hidher FDI

scheme on a real world system, numerous practical and unforeseen dficulties wiil present

themselves. To overcome these challenges the designer of the FDI scheme must l e m to

understand the nature of the practical problems. This usually requires that hefshe follows

hisher work into a specinc engineering field which may or may not be familiar to himnier.

He/she has to either perfonn the implementation himseWherseifor work very closely with the

one who does the implementation. It is for this reason that most research works such as this

thesis end at the simulation stage.

The large scope and great divenity of unconstrained and constrained dynarnic systerns

prohibits a single research work to generate a general-purpose M t detection and

identification approach that is applicable to al l systems. In this thesis we focus our effort on a

special yet major sub-class of such systems - constrained mechanical systems. The

sigificance of studying this kind of system is threefold: a) There are many constrained

mechanical systems in the r d world. Some of them are used in industrial applications. b)

These systems are less studied than regular (unconstrained) dynamic systerns, especially in the

area of fault diagnosis. c) A systematic approach to detect and idente faults in these systems

has not been found but shodd be developed. The FDI scheme in this thesis relies purely on

analytical redundancy. It is model-based and uses only quantitative reasoning. Furthemore, It

falls into the sub-class of state estùnator based approaches.

Chapter 1 introduction to FauIt Detection and Identification L 1

This thesis consists of five chapters. Chapter 2 focuses on the description, mode- and

andysis of consbained mechanical systems. It shows that constrained mechanical systems are

a special ciass of generalized state-space systems, which are also known as descriptor systems

or singuiar system. Some properties which are speciai to constrained mechanical systerns are

discussed in this chapter. Linearization and nonsingular transformations are performed in this

chapter to yield a purely dynamic subsystem which becornes the foundation for filrther

analysis. The resuit is that all subsequent anaiyticai work can be carrieci out in the domain of

(regular) Iinear system theory rather than the domain of ünear suigular system theory.

Chapter 3 presents a design of an Unknown input Observer (UO) and shows how such an

UIO c m be used for fault detection and identitication in hear or iinearized dynamic systems.

Similarity transformations and a nonsingular transformation are used in this chapter to heip us

to divide and conquer the problem. Chapter 4 uses a practical constrained mechanical system

in demonstrating the appiicability of the proposed unknown input observer based fault

detection and identification approach. Two actuators and one sensor faults are detected and

identified in the simulations of a UMS-2 robot. Finally, chapter 5 summarizes the

advantages/contributions of the thesis and iists the limitations of the proposed scheme and the

opportunities for fùrther research on this subject by any interested persons in the future.

Chaoter 2 Constrained Mechanical Smems 12

Chapter 2

Constrained Mechaoieal Systems

2.1 Introduction

Dynarnic systems can be classified into unconstrained systerns and constrained systems.

Unconstrained continuous dynarnic systems c m be described by ordinary differentiai

equations of motion, wkch are easy to simulate. AU forces that do not work virtually are

eliminated from the formulation of unconstrained systerns. Exarnples of workiess forces

include contact forces in stidig-without-fiction, rolling-without-slipping, and the interna1

forces maintaining rigidity of a body. On the other hand, constrained dynarnic systems pose

sorne special problems. First of ail, they can no longer be described exclusiveIy by ordinq

differentid equations. Presence of constraint equations makes this type of systern more

dBicuIt to analyze and simulate. Additionally, because knowledge of constrahed forces is

crucial in some applications and such forces may not be rneasured diredy or indirectiy,

estimation of constrained forces poses another issue and challenge. Let us consider a robotic

manipulator (Mills & Goldenberg, 1989) performing a task on a rigid surEace as an example

of constrained dynamic systems. in the absence of a force sensor, the constrained forces

applied by the manipulator end-effector on the envkonment must be estimated for control

purposes so that i) neither the manipulator nor the rigid surface is damaged due to contact, ii)

contact is maintained during the task, and iii) the required forces are applied to successfully

complete the task. The study of constraùied dynarnic systerns has been going on since the

foundation of analytical dynamics. Understanding of analytical dynamics can be obtained

from the books of Meirovitch(l970), Goldstein(1 %O), Greenwood(l965), Neirnark(l972),

and Kane & Levinson(l985). The reader is referred to the last two of the above five books

for methods of d e r i h g equations of motion for constrained dynamic systerns. Basically, a

constrained dynarnic or mechanical system involves positions or disptacements, velocities,

forces, and constraints. Constraints involving only displacements or positions are caiied

geometrïc comaints. Constraints involvîng velocities and possibly displacements as w d are

cded velocity constraints. Geomemc constraints and velocity constraints that c m be

integrated into geomemc constraints are cded holonomic constraints. Velocity constraints

that can not be integrated into geometric constraints are cailed nonholonomic constraints.

One of the major hdings of past studies is that dynamic or mechanicd systems with

constraints result in a description of differential-algebraic equations, Le., the natural or

original representation of constrained mechanical systems in terms of a number of dynamic

equations plus another number of constraint equations can be rewritten into a descrîptor form

(Shin and Kabamba, 1988). Descriptor systems are aiso called singular systems or generalized

state-space systems. For information on singular systems, the readers are referred to the book

of Dai(1988), the early work of Luenberger(I974 & 1978), the paper of Yip &

Sincovec(l981), and the survey of lewis(l986). The application of singular system theory to

constrauied mechanical systens has recently appeared as a new research topic. Generally,

linear time-invariant descriptor systems cm be described by the foUowing:

E Y (t) = Ax(t) + Bu(t)

y(t)=Cx(t)

where

We shall now present some definitions that wiii be usefùl in the remahder of this thesis.

Definition 2.1 - Matrk Pend

Let E and A be two matrices of appropriate dimensions with red values. A matrix pend is

then a polynornial matrix given by (SE-A). This pencil is reguiar if ISE - A( F O for a square

pencil, othenvise the pencil is singular.

Definition 2.2 - Normal Systerns

Dynarnic systems that can be described by only differential equations are called normal

systerns. An example of such a system described h state-space formulation is given by:

C h ~ t e r 2 Constrained Mechanical Svstems 14

x (t) = &(t) + Bu(t) (2.1.3)

~(t)'cx(t) (2.1.4)

where

Definition 2.3 N o d Forms for Constrained Linear Mechanical Systems

A normal fonn for a constrained linear mechanical system refers to the representation of the

system in the form of a normal(dynarnic) subsystem plus a set of aigebraic constraints. For

example, a normal form of the system deiïned in the last definition can be given by the

following :

x, (t) = A x, (t) f l? u(t) (2.1.5)

Y@) = C x, (0 (2.1.6)

5 (t) = &, (t) + i? u(t) (2.1.7)

where

Definition 2.4 Regularïty/Solvability

A descriptor system descnbed by equations (2.1.1) and (2.1.2) is regular or in other words

has a guaranteed existence and uniqueness of a solution if and only if the followhg matrix

pend is regular, i.e.,

I S E - A I 60,s EC (2.1.8)

Note a computationally attractive method for v e m g the system's regularity is provided by

Luenberger's shuffie algorithm, which cm be found in the book of Dai (1 989).

Definition 2.5 InfiniteLImpulse Observability

A descriptor system described by equations (2.1.1) and (2.1.2) is infinitely observable or

possess impulse observability if and only if

C h t e r 2 Consuaineci Mechanical Svstems 15

rank O E = n + rank E [U :] A more direct and more understandable defuiition of infinitdipulse observability is as

foiiows:

Systern (2.1.1)-(2.1.2) is infinitelimpulse observable if the impulsive behavior of x(t) at t=O-

can be uniqueIy determined fiorn y(t), t > O in the absence of input u(t).

Definition 2.6 FUiitdReachable Observability

A descriptor system described by equations (2.1 - 1) and (2.1.2) is bitely observable or

possess reachable observability if and only if'

A more direct and more understandable definition of finite/reachable observability is as

follows:

System (2.1.1)-(2.1.2) has fiaitelreachable observability if given any descriptor vector x(t),

t>O in the reachable set, it can be uniquely detennined through knowledge of the output y(t),

~ ( 0 , t] in the absence of input u(t).

Definition 2.7 Complete Controiiability (C-controllability)

A descriptor system described by equations (2.1.2) and (2.1.2) is C-controllable if and only if

rank[sE-A BI = n V S E C (2.1.11)

and

rankE BI = n (2-1.12)

A more direct and more understandable definition of C-controiiabity is as follows:

System (2.1.1)-(2.1.2) is completely controllabie (C-controllable) if there exists a control

input that can make one reach any state from any initial state in a 6nite t h e period.

Definition 2.8 Reachable Controllability (R-controllability)

A descriptor systern described by equations (2.1.1) and (2.1.2) is R-controllable if and only if

Chaoter 2 Constraîneci Mechanical Svstems 16

rank[sE-A BI = n V S E & (2-1.13)

A more direct and more understandable definition of R-controllabiiity is as follows:

System (2.1.1)-(2.1.2) is R-controiiabIe if there exists an admissible control that can make the

state of the system to go from any initial state to a point in the set of reachable States ( a

subspace of Rn ) .

The above dennitions are of value and will be used in the rest of the thesis. In section 2.2, a

natural mathematical description of constrained non-linear mechanicd systems is initially

given in the fonn of dynamic equations plus constraint equations. The non-iinear

representation is then iïnearized. In section 2.3, any possible redundancy in the constraints is

elirninated and the linearized mathematical model ts rewritten into a special form as weU as a

descriptor form. The special form is needed for deriving a normal form of the representation.

In section 2.4, a normal form of the linear mechanicd descriptor system is derived. In section

2.5 and 2.6, properties of iinear mechanical descnptor systems and their impacts on observer

design for such systems are discussed. Finaiiy, section 2.7 surnrnarizes this chapter and

explains the link between this chapter and subsequent chapters.

2.2 Description of Constrained Dynamic Systems

Constraints in dynamic systems can be classifted as scIeronomic constraints or rheonornic

constraints depending on whether the tirne variable t is explicitly contained in the constraints.

Systems with tirne-invariant constraints are called scleronomic systems. Systems with tirne-

varying constraints are caiied rheonomic systems. The most cornmon model for dynamic

systems with constraints is that of Lagrange's equations. Modeling of constrained dynamic

systems using Lagrange's equations can be found in the book of Goidstein (1 980). According

to Shin and Kabamba (1 988), Constrained dynamic systems can be modeled as:

where

q(i) E !Rn is the generalized coordinates vector

q(r) E !Rn is the velocity vector

q(t ) E Sn is the accelerations vector

M(q) E 'Rn'" is the symmetric positive definite inertia ma&

H(q,@ E !J? If is the force vector

6 : represents a set of holonomic constraints

g, : represents a set of iionholonomic constraints

A E '3 * is the Lagrange multiplier vector

T E Sn represents input forces acting as controls

r E Sm is an output vector

and

J' (g, 4 ) is called the Jacobian of constraint equations which is defined as

(a) In the case of only holonomic constraints represented by (2.2.2)

m 4 ) JT (q) = - @l

(b) In the case of only noaholonomic constraints represented by (2.2.3)

(c) In the case of both holonomic and nonholonomic constraints

represented by (2.2.2) and (2.2.3)

r&N4,~)1

Chaoter 2 Coosuained Mechanical Svsîems 18

Note that since g(Q represents generalired coordinates, its components are independent and

the co&t equations in (2.2.2) and (2.2.3) are linearly independent.

The process of lineuking the system represented by (2.2.1-(2.2.4) requires multivariable

Taylor series expansions involving only the fint order ternis. &en a nominal state

Then we have the following f h t order Taylor series expansions:

Chauter 2 Consbaineci Mechanical Svstems 19

and the Jacobian takes one of the following fonns dependhg on the types ofconstraints:

Substituting (2.2.5)-(2.2.11) Uito (2.2.1)-(2.2.4) and simplifyuig the resultant equations

redts in:

ME +Di +Kz = f + ~ ' R

The holonomic constraints in equation (2.2.16) and the nonholonomic constraints in equation

(2.2.17) can be represented in the following generalized fonn:

where

G = O and H = L in the case of only holonomic constraint

G = G and p= H in the case of only nonholonomic constraint

G = [a] ai ET= [:] in the case ofboUi types of constrainti

Therefore linear or iinearized constrained mechanical systems have the following form:

The above representation c m be rewritten in a iinear descriptor form:

Chapter 2 ConStrained Mechanical Svstems 2 1

2.3 Special Form of Constrained Linear Mechanical Systems

In this section we wili perform a nonsingular transformation on the generalized constraint

equation (2.2.19). nie motivation of this transformation is bea understood in the next

seaion(2.4). The process of this transformation yields a nonsingular(orthogonal)

transformation matrix T and a special form of the linear descriptor system representation.

Both will be used in deriving the nomal form of the descriptor system in the section 2.4.

We start with a matrix pend (A . G + H ) where

A is a cornplex variable in the complex plane or Laplace operator

c, H E !RPxn

is the number of holonomic plus nonholonomic constraints

First, let us define row compression matrix and column compression matrix for an arbitrary

singuiar matrix denoted by H. According to singular value decomposition theory in hear

algebra, Orthogonal matrices R and Q exkt such that

where

is a diagonal matrix filled with singuiar values of H

Then the foilowing equations can be established:

[el =

Thus R* can be used as a row compression ma& and Q can be used as a column

compression matrix.

Cha~ter 2 Constrained Mecbanicai Svstems 22

Now perfiirm the row compression of using an orthogonal rnatrix P , such that

then, we have

Perform fùrther column and row compression of 6 using orthogonal matrices P , and T , such that

where

H l is a nonshgular matrix

Thus, we have

where ' x ' indicates a usually nonzero matrix pend. Then, perform the same operations on - -

rubpencil 1 G , + , as on I G + H . Repeat the process until , in the resulting

subpencil A. G , +M is of full row rank. Hence, we have

Finally, perform the column compression of G , to get

( A G ,+Z,)T, =[Rkvl lGk

where

Cbavter 2 Constrained Mechanical Systems 23

and therefore the above equation can be rewritten as

where

and

T = T , ... T, is orthogonal

Now let us take Laplace transfomi on the constraints equation G i + H z = O

Let us fùrther define a new generaiized state vector

,g = T - ' = = T T

Taking Laplace transform on (2.3 -6) yields

Premultiply equation (2.3.4) by P and substituthg (2.3 -7) into it results

P ( l c + H ) T E ( A ) = O (2.3.8)

Chapter 2 Consuained Mechanical Svstems 24

Substituthg (2.3.1) into (2.3 -8) and partitionhg E (A ) results in

Equation (2.3.9) fùrther results in the foUowuig two equations:

%,I t,(A) + ( Â I + HIw2 ) Z2(A)+ [ X O - - - - - X] Z1(A) = O (2.3. L O)

Simphfjing the above two equations yi elds

' q ( n ) + ( A I + ) z?(A) = O

Taking inverse Laplace transformation of (2.3.12) results in

Equation (2.3.13) can be rewriten as

Substitut ing (2.3.5) into (2.3.14) yields

Then we obtain the following resdts

N i ( t ) + S z(t) = O

N T = [ 0 I O]

S T = [s, S2 O]

The above results can be summarized and stated as the following theorem:

Tbeorem 2.3.1 - Through the nonsinguiar transformation of matrk pends, constraint

equations (2.2.16) and (2.2.17) can be transformed into one of the following equivalent

forms:

(a) holonomic constraints:

F z = 0 7 F E !Ilq'" (2.3.21)

(b) nonholonomic constraint s :

N i + S z = O, N,S ~ ' 3 3 ~ ' "

(c) holonomic and nonholonomic constraints:

Nt+Sz=O and Fz=O

where

N, s E 9 1 ~ ~ ~ ~ FE%^^^^, q,, + qp = q

q is the number of independent constraints

~ I P is the number of independent holonomic constraints

q, is the number of independent nonholonomic constraints

The Jacobian J will be one of the following forms:

(a) For constraint equation (2.2.7): J=F

(b) For consaaint equation (2.2.8) J=N

Chapter 2 Constrained Mechanid Svstems 26

(c) For constraint equation (2.2.15) J = [FI (2.3.26)

Moreover, fiom the transformation which brings equation (2.2.7) and (2.2.8) into one of its

special forms (2.3.21), (2.3.22) or (2.3.23), an orthogonal matrix T, Le., T-1 = T ~ , can be

obtained such that

(a) for constraint equation (2.2.7): FT = [O I,] (2.3.27)

(b) for constrauit equation (2.2.8): NT= [O I,] , ST= [S, S- 1 (2.3.28)

(c) for constraint equation (2.2.15):

NT= [O 1, O], ST=[S, S O], FT= [O O 1 c ] (2.3 .29)

The above constraints can be denoted in a generalized form

N i + S Z = O (2.3.30)

where

3 = O , 3 = F in the case of only holonornic constraints

% = N , 5 = S in the case of only nonholonornic constraints

= [O], S = [FI in the case ofboth icincis of constraints

Thus, the specid fonn representation of linear mechanical system cm be written as

The above results wiii be used in deriving a dynamic subsystem(normd form) for the hear

mechanical system in the foilowing section.

C h t e r 2 Coostraùied Mechiinid Svstems 27

We now use an example to illustrate how the special form transformation is performed, Le.,

we will apply theorem 2.3.1 to a specifïc system- The example used here is a robg ring drive

which has one holonomic constraint and one non-holonomic constraint. This system was

found in the paper of Hou et-al. (1993). The

foIlowing form:

which corresponds to the form in equations (2.2.15) - (2.2.18).

The matrix pend ( A G + E ) for this particular systern would be

representation has the

Since the above matrix pend is already in row compressed form, there is no need to perform

row compression. Therefore we have the followhg:

The last equation (2.3-43) means that

R2 = [O 1 -1 O]

Now we need to find orthogonal matrices P and T l such that

P, RJ', =[O O O H , ]

It c m be verified that the following orthogonal matrices satisQ (2.3.45)

1[0 1 - 1 O]

Then, we have

Therefore we have

The transformation matrix T is determined as

Then we can calculate the foUowing matrices:

Finaiiy we obtain the special form representation of the linearized system as foiiows:

Y = [c, " O] [j]

The numericd representation of equations (2.3 -58) and (2.3.59) is given in the following:

This concludes our illustration of transfomùog a linear descriptor system to its special form.

1 0 0 0 0 0 0 0 0 0 - i l

0 1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 0

o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 ~ i 2

- .

i2

4 i4

z1 5,

f3

f4

i ,

Chmter 2 Constrained Mechanical Svstems 32

2.4 Normal Forms for Constrained Linear Mechanical Systems

In this section we will derive normal forms for linear mechanical systems with various

constraints. First Let us consider the most general case which involves both holonomic and

nonholomonic constraints. Let n be the number of descriptor variables, q be the total number

of independent holonomic and nonholonomic constraints, q, be the number of independent

nonholonomic constraints, and q, be the number of independent holonomic constraints.

Using the orthogonal transformation ma& T ( T T ' = T' T = 1 ) developed in the last

section enables us to do the following:

Partition the transformed generalized state vector defined in equation (2.3 - 5 ) as

Partition the transformed input vector as

where

Partition the transformed mass ma& M, the stiflhess matrix K, and the damping matrix D as

PR-multiplying both sides of the descriptor form representation of the last section by a

nonsingular matriv Q = diag(T ', T ' , 1 ) and nothg that z = T 5 results in

(2.4.7)

Substituthg (2.4.1) through (2.4.5) and the following results frorn the last section

into (2.4.7) redts in

where

Note that (2.4.8) can be expanded into as many equations as its number of rows. The seventh

and eighth rows of (2.4.8) offer the foliowing equations:

O = SI<, + S A +

O = r , These two equations result in the foUowing:

Substituting (2 -4.1 1)- (2.4.1 5) into (2.4.16) and rearranging tems results in the following:

- Ml, S, i , - MI, s2 5: + MII $1 = ( - K I I + ~ ~ &)CI

+(-K12+D,2S2)gt -q1k1 (2.4.17)

The fifih and sixth rows of (2.4.8) offer the following equations:

M II 5, +M& +M,?, =-ICI 5,-&5= -K,,5,

- D 21 i , - ~ , è 2 - 4 2, + h, + f, (2.4.18)

and

M 31 4 1 +M37 - i7 - + M33 8 3 = -K3151-& 62 -K;j 63

-4, il -03? ' $ 2 - ~ 3 3 4; + + f3 (2.4-19)

Substituting (2.4.12)- (2.4.15) into (2.4.18)-(2.4.19) and writuig the two equations in a

matrix form result in the following:

Cha~ter 2 Constraioed Mechanical Svstems 36

Substituting (2 -4.1 1) into (2.4.20) and re-arcanging tenns result in the followhg equation:

where

and

In view of equations (2.4.1 l), (2.4.17), and (2.4.21), equation (2.4.8) can be rewritten as

L O O O 0 0 0 -

O 1 cf.

O O O 0 0

O O O O O 0 0

-M12S, -MI,& 0 0 0 0

-MZsI -M,sI O MZI 0 0 0

O O O O 0 0 0

O O O O O 0 0

O O 0 L - q

-SI -st O O

O 0 O O

Chauter 2 Coastrained Mechanid Svstems 37

Equation (2.4.22) can be rewritten in the foiiowing fom by re-arranghg or re-grouping

variables in the generalized state vector:

- L q O O O 0 0 0 O I

q. O O 0 0 0

-M12Sl Ml, 0 0 0 0

O O O 0 0 0 0

O O O O 0 0 0

O O O O 0 0 0

--M21Sl - el 0 0 0 0

O O L q O 0 0 0

-31 -& 0 O 0 0 0

-Kll+D12Sl -Ki7+DI7& -Dl1 O 0 0 0

O O O 1 O 0 0 9~

O O O 0 1 O 0 9~

SI s 2 O O O I O 4.

-$, +D,s, -K, +a,& - O O 0 1,

Premultiplying both sides of (2.4.23) by the foiiowing nonsingular matrix

P =

results in

where

Then the Erst three rows of (2.4.25) offer the dynamic part of the normal form representation

as expressed by the following:

Expanding the 1 s t two rows of (2.4.23) results in:

F. 1

Simplification of equation (2.4.30) yieids the algebraic part of the normal form representation

as the following:

where

Chanter 2 Consuaïned Mechanical Svstems 39

- - E, =M2, A, + f i 1 S, S , +z ,,-D -SI (2.4.32)

- - - - EZ = M2, A2 +ME SI S2 +KI-DESZ (2.4.33)

- E3 = MIIA3 - M- S I +E2, (2.4.34)

Moreover, the output equation fkom (2.3 -33) of the last section

y = [c, C" O]

can be rewritten as

Substituthg 4, = O fiom equation (2.4.13) into the above equation results in

Further substitutingi, = [-S, -SI O] 5, from (2.4.1 1) into the above eguation (2.4.35) LI results in the output equation of the normal fom representation as the following:

Chapter 2 Constrained Mecbanid Svstems JO

The above development generated a normal form for a system with both holonornic and

nonholonomic constraints. This coordinate transformation procedure can be performed on

systerns with only holonmnic constraints to derive normal form representations for this kind

of system. It can also be used to derive normal forms for systems with only nonholonomic

constraints. The results of the derivation of n o d forms for the aforementioned three kinds

of constrained hear mechanical systems can be summarized as foliows:

Theorem 2.4.1 - The constrained mechanical systems of fonn (2.3 -32)-(2.3.33) can always

be transformed into one ofthe following forms:

(a) in the case of only holonornic constraints:

(b) in the case of only nonholonomic constraints:

(c) in the case of both holonomic and nonholonomic constraints:

Cha~ter 2 Constrained Mechanical Svstems JI

where 6,'s are defined in equation (2.4.1) and other matrices in the above normal forms are

defined as foiiows:

1) for case (a) and (b):

A denotes M, D, or K

2) for case (c):

T T A T =

2.5 Regularity of Constrained Linear Mechanical Systems

As defmed in definition 2.4 of section 2.1, regularity refers to the existence and uniqueness of

a solution. Obviously regulafity is an important property of the systems being studied in this

thesis. Working with the foliowing special fom representation of linearized constrained

mechanical systems

1, O o O M O 0 0 0

we can corne up with a theorem as folows:

Theorem 2.5.1 Constrained linear mechanicd systems as described by equations (2.5.1) and

(2.5 -2) are always regular, i.e.

Proof A proof using the s h d e algorithm (LuenbergerJ978) is provided in Appendu A of

this thesis.

Chapter 2 Constrained Mechanid Svstems 43

2.6 ControUability of Constrained Linear Mechanical Systems

As defined in section 2.1, the necessary and SuffiCient condition for reachable controllability is

one of the two conditions for cornplete controfiability. Consequently, complete controilability

is denniteiy a stronger condition than reachable controllability. A system that has reachable

controliability does not necessarily have complete controllability. A system that have

cornplete controllability must have reachable controilability. It can be s h o w that the stronger

controilability property is not possessed by constrained hear mechanical systems. Being a

special class of iinear descriptor systems, constrained linear mechanicd systems or linear

mechanical descriptor systems can have reachable controllability at best.

Theorem 2.6.1 Linear mechanical descriptor systems as descnbed by equations (2.5.1) and

(2.5 -2) are aiways not completely controllable or do not have C-controflability.

Proof The proof is easily obtained by showing that the second rank condition in the

definition of complete controllability is not satisfïed, Le., [E BI is not of &Il row rank.

First we need to rewrite (2.5.1) to show what B and E are

Now it is obvious that

rank [E BI = rank O M O I ;t number of rows in E or B [u , a :] The second rank condition is not satisfied and this completes the proof

Cha~ter 2 Constrainai Mechanical Svstems 44

2.7 Observabiiiîy of and Observer Existence Conditions for

Constrained Linear Mechanical Systems

Any observer-based fault detection and idenacation schemes inevitably relies on systern

observability conditions of one kind or another. 1 now present the foilowing theorem

regarding the infinite observability of linearized constrained mechanical systerns:

Theorem 2.7.1 Linear mechanical descriptor systems as described by equations (2.5.1) and

(2 -5 -2) are aiways infinitely unobservable or do not possess impulse observability.

Proof see Appendix B of this thesis.

Shin and Kabamba(l988) noticed that when constrained forces are not directly or indirectly

measurable, a constrained mechanical system is not iafinitely observable. The mathematical

proof in Appendix B confirms this physicd explanation.

Idhite observability is a desirable feature as far as unknown-input decoupled observer design

is concerned. Some generai results of conventional observers for descriptor systems with

unknown inputs are provided by Hou and Muiler(1992a). The existence conditions of the

unknown-input observer has the nature of the infinite observability and therefore can not be

met in constrained mechanical systems considered here. The design of unknown-input

observer must be based on weaker or alternative observability conditions. Designing

unknown-input observers is an essential part of any observer-based R)I schemes. An

unknown input observer (UIO) design method is presented in the next chapter.

Incidentdly, the observer existence condition for constrained hear mechanical systems

driven by totdy known inputs is very simple. Hou and et. al .(Hou and et. a1.,1993) pointed

out thaï finite observability of the system is a necessary and sufficient condition. They also

proved that finite observability of the descriptor system is equivalent to the observability of

the correspondhg conventional system in the minllnai coordinates. Since this thesis is most

Chanter 2 Constrained Mechanical Svstems 45

concemed with unknown input observer based FDI, their condition is not redy u W .

Necessary and SuffiCient conditions for the existence of unknown-input observers rernain to

be found in a subsequent chapter.

2.8 Summary

In this chapter, we enriched and extended the discussion and analysis of constrained dynamic

systerns and constrained mechanical systems by some previous researchers such as Hou, et al.

(1993). We recognized and classified constrained mechanical systems as a special kind of

singuiar or descriptor systems. Starting with the standard noniinear Lagrange equations

mode1 of constrained mechanical systerns, we fïrst obtained a lineaîzed mode1 of the system

by using the standard Taylor series expansion technique. We were able to rewrite the linear

mode1 in a generalized state-space format. Then we performed a nonsingular transformation

on the constraint equations and obtained a special descriptor fom representation. This

transformation process generated a nonsingular (orthogonal) mat+ which we subsequentiy

used in performing a coordinate transformation and deriving normal foms for mechanical

systerns with holonomic andor nonholonomic constraints. We used a numencal example in

demonstrating how one can perform the important transformation. The resultant dynamic

subsystem in the normal form of a constrained mechanical system leads our subsequent

studies in the following chapter to the domain of linear system theoiy. In the last few sections

of this chapter, we identified and discussed some rather special properties of constrained

mechanical systems such as their lack of infinite observability and complete controliability.

We pointed out that Iack of infinite observability restricted o u choices of approaches to

observer design and Bult detection and identification. By doing so we built a bridge between

this chapter and the next one, which presents the design of an observer that is capable of

estimating the state of a system driven by both known and unknown inputs.

Chauter 3 An Unknown input Observer Based Fault Detection and Identification Method 46

Chapter 3

An Unknown Input Observer Based Fault Detection and

Identification Method

3.1 Introduction

As discussed in chapter one, the need for safe and retiable operation of complex engineering

processes at reasonable cost has been promoting research and investigation into the problem

of fault detection, idenacation, and accommodation (FIXA). Among the various FDIA

techniques, there is a class of model-based approaches that are commonly referred to as the

analytical redundancy techniques. Detaiied survey of these methods couid be found in Willsky

(1976), Isermann (1984)- Merriif (1985), and Frank (1990). Since the introduction of

dedicated observer scheme @OS) by Clark (1978), more sophisticated approaches based on

it utilizing some detection fùnction or statisticai tests have been proposed.

One of the major difEiculties in the application of model-based techniques to practicai FDIA

cases is the problem of plant uncertainties or parameter variations. In such situations there

usudy exists a need for a robust FDIA methodology. A number of different approaches to

robust FDIA problems have been proposed. One such approach is a sensitivity discriminating

observer scheme proposed by Frank and Kelier (1980), Another approach deaüng with

uncertainties is the threshold selector method proposed by Ernami-Naeini, et aI. (1988).

Recently, there have been some studies in the area of FDIA based on the theory of unknown

input observers (UIO). A survey of the UIO-based approaches can be found in Frank (1990).

Severai somewhat dEerent UIO design methods have been proposed by Kudva (1980),

Kurek (1983), Wang et. al. (1975), Yang and Wilde (1988), Guan and Saif(1991), and Hou

Cha~ter 3 An Unknown b u t Observer Based Fault Detection and Identification Method 47

and Muller (1992). The UIû theory has been utiIized for actuator fault detection and isolation

by Viswanadharn and Srkhander (1987), and Park and Stein (1988).

U O design has been an active area of research in the past several years due to its widespread

applications. UIO's are primarily designed to accommodate unknown exogenous

disturbances in the dynamics of the plant. Conventional observers that reconstructs the state

vector under the assurnption that all inputs are known have been used in state feedback

control of various systems. This traditional approach of control neglected the presence of

certain uncertainties (such as inaccessible inputs and plant disturbances) and often is not

suflicientiy usefùl for fault detection and identification purpose. Because most uncertainties

and plant faults can be modeled as unknown inputs to the system, designing unknown input

observers (UIO) is of tremendous use for robust control, fadi detection, identification, and

accommodation (FDIA).

Basicdy, there are two types of UIO design methods. The first category of approaches

includes a number of attempts that assumed some a priori information about the

unmeasurable inputs to the system Specificaily, Johnson (1975) assumes a polynomid

approximation to these inputs, and in Meditch & Hostetter (1974), it is assumed that the

unknown inputs can be modeled as the response ofa known dynamic system represented by a

constant coefficient Werenûal equation. The other categov of UIO studies assumes no

knowledge of the inaccessible inputs. Among the more recent works are those of Yang &

Wilde (1988), Guan and Saif (1991), and Hou & Muller (1992). Yang & Wilde proposes a

fiil-order observer that is claimed to have sornewhat better rate of convergence than a

reduced-order observer. Although they claim that they use straightforward matrix

cdculations, thei. procedure involves singular value decomposition or Jordan fom

transformation. Their method also requires solving a system of linear equations that has more

unknowns than equations. In the work of Hou and Muller (1992), a reduced-order observer

and a minimal-order observer are derived via a technique of coordinate traosfomütion. The

derivation is rather mathematicaüy involved and hard to understand. In this chapter, we

propose a mathematically simple and computationally efficient unknown input observer

C C

(UIO) design method. This method is inspired by and owes its merits to the eariy work of

Guan and Saif (1991). In the foUowing few sections of this chapter, the CTTO design is

discussed and some modification of the approach of Guan and Saif (1991) is made to make

the UIO design more systematic. The first step of the procedure is fonnulating the probkm as

a iinear time-invariant system with unknown inputs. The second step is specwg the

assumptions used in UIO design. The third step involves performing a nonsingular

transformation on the partitioned system and actudy deriving an observer for one of the

three reduced order subsystems. It turns out that States of the other two subsystems have

direct algebraic relationships with the output of the systern which is assumed to be availabie

(measurable) for observer design purpose. Thus the combined state of the whole system can

be estimated using a single conventional observer. Furthemore, a necessary and mflicient

condition for the existence of an UIO is presented and proved in this chapter. This condition

can be expressed in terms of the matrices in the linear tirne-invariant representation of the

system for the convenience of checking if the condition is met. Finaily, methods for detecting

and ident-g actuator faults and sensor faults are presented in the iast part of this chapter.

The methods are based on modeling actuator fauIts as unknown inputs to the dynamic

equations of the system and modeling sensor faults as unkriown inputs to an augmented

system of which sensor faults are part of the state. Both actuator fault detection method and

sensor fault detection method rely on state estimation which is accomplished via the unknown

input observer.

3.2 System Representation and its Observable Canonical Form

System description and modeling has been discussed in the previous chapter and will be

discussed fiirther in the last part of this chapter. In the next few sections we are only

concerned with deriving an UIO. For this purpose we assume that ail the necessary

iinearization and transformations have been performed and our system representation has

resulted in the sirnplest form of al1 dynamic systern representations, namely hear time-

invariant systems. These systems cm be assumed to be driven by partially unknown inputs

Chauter 3 An Unknown In~ut Observer Based Fault Detection and Identification Mechod 19

which may be used to represent p h t f d t s and parameter uncertainties. The state-space

formulation can be given as foiiows:

Without loss of generality, the concemed system cm be written in the following observable

canonical form:

where

A€%""", B E !Rnxq, CE%^'", DE %"'"

x E n: number of state variables

u E 9Iqx' , q: number of known inputs

d E %"", m: nwnber of unknown inputs

y E 9IPx', p: number of outputs

I is an identity matrix of order p x p

Rernark: If C is of full row rank, there always exists a similarity transformation that can b ~ g

the representation in (3.2.1) & (3.2.2) into its observable canonical form in (3.2.3) & (3.2.4).

Details of this procedure and the proof of this claim can be found in the book of Chen(I984).

3.3 Assumptions for the Design of UIO's

Three assumptions are made in the rest of this work. These assumptions have been used

implicitly or expiicitly in ail the works on UIO theory and design. As can be explained later,

they are not restrictive assumptions:

Chapter 3 An Unkuown Input Observer Based Fault Detection and Identification Method 50

Assumption 1

The measurement matrix C in (3.2.2) is assumed to be of £Ùii row rank, Le.

rank C = p

If the measurement matrix C is not of tùU rank, then there exists at Ieast one redundant

output. This redundancy wi be eliminated by redefining the output vector y and the

measurement matrk C such that the new outputs are hearly independent. Therefore, this is

not a restrictive assumption.

Assumptiou 2

The D matrVr in (3.2.3) is assumed to be of fÙii coIumn rank, Le.

rank D = m

If D is not of tùlI rank, it can always be decomposed as a product of two f u U rank matrices

via the following proposition:

Proposition 3.3.1 Any p x q matrix 4 whose rank is r can be decomposed as follows:

where

B is a px r fiiU rank matrix

C is an r x q fuii rank matrix

Proof See the proof of proposition 2 in Saif and Guan (1 993)

Thus, D can be decornposed as

Chapter 3 An Unknown Input ûbserver Baseci Fault Detection and IdentiEication Meîhod 5 1

D=DN (3 -3 -3)

where

5 has fill colurnn rank

2 has tùll row rank

and a fùll rank new D and a new d for (3.2.3) can be defined as

In the early work of Kudva et al. (1980), a necessary condition for the existence of any

unknown input observers for the system descnbed by (3.2.3) and (3.2.4) is proposed and is

subsequentiy used explititly or implicitly by many others. It c m be stated as follows:

Assumption 3

A necessary condition for the existence of a stable unknown input observer for the hear

dynamic system descnbed by (3 -2.3) and (3 -2.4) is that the number of linearly independent

outputs is greater than or equal to the number of unlaiown inputs, Le.,

Proof See theorem 1 in Saif and Guan(1993)

3.4 Unknown Input Observer(UI0) Design

First, we apply the partition technique developed by Saif and Guan (1993) to divide the

dynamic system in (3 -2.3)-(3.2.4) into three subsystems:

Chauter 3 An Unhown h ~ u t Observer Based Fault Detedion and IdenOfication Method 52

As show in the later part of this section, the UIO design procedure invoives using a

nonsingdar transformation matrix which contains the inverse of D, . In general, a simple

straigh~orward partition of the observable canonicai fonn represented by (3 -2.3) and (3.2.4)

does not necessarily result in an invertiile D, . A procedure is needed to deai with the lack of

invertibility of D, . It tums out that this can easily be accompiished by reordering state

variabIes in the observable canonical form representation.

%a3 [q],.. is of fidi column rank by assurnption 3 in the last section, it must have a

rnx rn subrnatrix whose determinant is nonzero. This submatrix is therefore invertible and can

be defhed as the new D , . The new D, just contains dierent rows of [4] than the oid

D, does. It can be obtained by switching the rows of [q] Switching the rows of

[:] Pm is equivalent

variables are contained

to reordering the last p state variables in (3.4.1). The first (n-p) state

in D , and therefore does not need reordering. Thus a nonsingular or

invertible D, can always be obtained by reordering state variables. What must be pointed

out is that reordering state variables also affects the output equation and thus the

measurement matrix C. As a matter of fact, reordering state variables result in column

exchanges in C. Since only the last p state variables are reordered, column switchings in CF=,

Chauter 3 An Unknown Inmt Observer Eiased Fauit Detection and Identfication Method 53

are limited to the last p columns. Because the last p colum~s of C,,, in the canonical form

(3.2.4) is an identity submatrix 1, which is absolutely nonsingular, the new submatrix

resulting from switching columns la 1, has to remain nonsingular. This inverthle subrnatrix

can be denoted as C, . Then D, can be assurned nonsingular whiie C is of the fom

It is worth noting that reordering state variables is equivdent to perforrning a sirnilarity

transformation on the observable form representation. The observable form representation

itself' can also be obtained by perforrning a similarity transformation on the original state-

space model.

Therefore, without loss of generality, the systern may be assumed to be of the foiiowing form:

where

AE'R"~", B E Xnaq, CE%^"", DE 'ifIn*"'

XEW"", n: number of state variables

U E % ~ ~ ' , q: number of known inputs

d E !RmXi , m: number of unknown inputs

p: number of outputs

C , is an invertible matrix of order p x p

The output equation (3.4.2) can be partitioned as:

Cba~ter 3 An Unicnown Innit Observer Based Fault Detection and Identification Method 54

The inverse of C , cm be defined as:

Then equation (3 -4.3) can be rewritten as:

The folowing matrix operator c m be defined:

Post-mdtiplying both sides of equation (3.4.1) wiîh the above operator results in:

Substituthg (3 -4.8) and (3.4.9) into the first two "rows" of (3.4.1 1) yields:

and

where

Chapter 3 An Unknom Input Observer Based Fault Detection and Identification Methai 55

where

and A,, 's are elernents of partioned A ma& of the following fom

and

where

and

Now ushg the partioned matrices in (3 -4.12) and (3 -4.1 3) wil: result in:

- XI =Al, x, + r (3 -4.16)

- Z =A2, XI (3.4.17)

r = (Al2 c,, +&C,~)~+D,D,-'C,~ y+-+ (3 -4.18)

- z=(C,, - D , D , - ' C ~ ~ ) ~ - ( & C ~ , +&Cpdy-B2U (3 -4.19)

Cha~ter 3 An Unknown In~ut Observer Based Fadt Detection and Identi6cation Method 56

According to observer theoxy(Chen, 1984), the state of the dynamic system represented by

(3.4.16) and (3.4.17) cm be estimated by a Luenberger observer. The dynamics of this

reduced-order observer is given by:

where M is the observer's gain. Substituthg for r and z into (3.4.20):

Equation (3.4.21) contains the derivative of the output which is not available for direct

measurement. This problem cm be dealt with by defining a new variable w as follows:

Rewriting (3 -4.2 1) in tems of the new variable w will result in:

where

F = (A,, - MA,,)

Chapter 3 An Unknown h ~ u t Observer Based Fauit Detection and Identification Method 57

L = (Ë(-M&) (3.4.26)

The foUowing theorern will conclude and summarize UIO design:

Theorern 3.4.1 If the pair (&, Â,, } is observable, the state of the dynamic EyRem

described by (3.4.1) and (3.4.2) can be estimated by using the UIO proposed in (3.4.23)-

(3.4.26). The estimate of the state is given by:

where

N = D,D,-~c,, - M(c,, -D,D,-'c,,)

In addition, aii the eigenvalues of F can be placed at any desired location. The proof of the

above theorem has been irnplicitiy given in the foregoing discussion.

3.5 Necessary and Sufficient Conditions for the Existence of UIO's

Given the iinear tirne-invariant dynamic system with partially unknown inputs as described in

the previous sections:

X=Ax+Bu+Dd (3.5.1)

y = Cx (3.5.2)

where

AEW"'", B EX^^^, C E (J iprn, D .E 'W1"

x E w"", n: number of state variables

u E Yiq*' , q: nurnber of known inputs

d E % m x ' , m: number of unknown inputs

y E Y i p , p: number of outputs

Chmter 3 An Unknown In~ut Observer Based Fault Detection and Identification Method 58

there exists a necessary and sufncient condition for the existence of unknown input observers.

Sime different researchers in this a r a use different methods to design different UIO's, there

exist quite a few seemingly different necessary and sufncient conditions. Mer close

examination of these competing conditions it is found that these conditions are virtuaily

equivaient to each other. The generdy accepted format of the condition can be stated in the

foilowing theorem:

Theorem 3.5.1 - A Necessary and Suflieient Condition for the Existence of UlO's

A necessary and SuffiCient condition for the existence of a . UIO for the systern descriied by

(3.5.1) and (3.5.2) is that

it has to be pointed out that the concemed system such as (3.5.1)-(3 5 2 ) has been assurned

to s a t i e the minimum necessary conditions expressed in Assumption 1, Assumption 2, and

Assumption 3 of section 3 -3.

Proof: The above theorern can be proved inciirectly by showing its equivdence to Theorems

3.4.1 which states that the necessary and sufficient condition as the obsenrabiiity of the pair

Define the foiiowing nonsingular rnatrix

where

Cha~ter 3 An Unkaown input Observer B a d Fauii Detection and Identification Method 59

(3 -5.6)

and

Note that C can be assurned to be of the fonn [O ~,]without loss of generality. The

matrices A, C and D can be partitioned as the following as they were partioned in section 3.4:

It is a known tact and a theorem in ma& theory that pre-multiplying a matrix with a

nonsingular matrix preserves its rank Now we can perfonn the following multiplication:

Chaoter 3 An Unlcnom hout Observer Based Fault Deteclion and Identikation Method 60

Now let us evaluate the upper two submatrices by defining

R e d that in equation (3 -4.13) of section 3.4 we have dehed

Thus equation (3 S. 17) can simplified as

w=

Now

Chmter 3 An Unknown b u t Observer Based Fault Detection and Identification Method 6 1

Then we have

SI,,-&, O O

-D~-' A ~ , O O

Therefore equation (3.5.22) can be rewritten as

Chaoter 3 An Unknown Input Observer B a d Fadt Detection and Identifkation Method 62

This shows the equivalency between the two conditions and concludes the proof This aiso

concludes our discussion of unknown input observers.

3.6 Problem Formulation for Fault Detection and Identification

In the previous sections, we outlined the design of an unlaiown input observer that can be

used to estimate the state of a dynamic system dnven by partiaily unknown inputs. In the

following sections we WU essentiaily re-present Saif and Guan's (1993) approach to fault

detection and identification using our modiied UIO as the state estimator. First we need to

establish a link between unknown input observers (UIO) and fault detection and identification

(FDI). The design of the UIO in the previous sections was based on the assumption that a

system mode1 is known with precision. In reaiity, however, parameter values may be known

only approximately or time-varying. There may be actuator f&Iures and/or sensor fdures

which affect the behavior of the system. Let us now consider the effects of actuator fauits,

sensor faults, and parameter uncertainties on system dynamics and outputs one at a the.

The only Somation commonly availabie about the faults is the location of their possible

appearance. No assumption cm be made about their mode, Le., their tirne evolution and size.

Suppose the nominal values of the parameters are known and our system can be iinearized.

The representation of the system can be written as:

C h a ~ 3 An Unknown Input Observer B a d Fault Detection and Identification Method 63

where

subscript , means nominal system model parameten

A. € !Rnnn , Bo €Wxq

Co E R P x " is the measurement matrix

x E % n X 1 is the state vector

u E w'" is the known or control input vector

y E W p x L is the output vector

Actuator faults act directly on system dynamics. They affect the dynamic equations of the

system model and these effects can be modeled using an actuator fault distribution matrix and

an actuator fault vector in the foilowing format:

where

Do E ! R " ~ * is actuator fault distribution mathv

do E 'I is actuator fault vector

Sensor faults can be modeled as additive bias components in the output equation through a

sensor fault distribution matrix.

y. = C o x o + E , e

w here

E, E W is sensor fadt distribution matrix

e EW' is sensor fault vector

Parameter uncertainties may be initidy modeled as deviations fiorn their nominal values:

Chamer 3 An Unknown haut Observer Based Fmit Detection and Identinotion Method 64

A=A, -t AA (3.6.5)

B=B, + A B (3 -6.6)

The uncertainty matrices AA and AB can be rewritten as

L u = I , AA,

AB = I,AB*

by giving the following defbitions:

Definition 3.6.1 The n by k uncertainty indicator matrix IR of any n by m matrix R is

defined as IR (r, , . . . ,r, ) , where k is the number of rows of R that contain unknown elements.

Thejth column ofthis matrix has zero enhies except for the a, th entiy which has a value of

one.

As an example, if A is a 4 by 4 rnatrix and there are uncertain elements in the kst and third

row, then

k=2, a, = 1 , a- = 3 , and I , (a i ,a2 )=I . , (1 ,3 ) =

Definition 3.6.2 The k by m uncertainty matrix AR, of any n by m matrix R is defined as

AR, = [?' 1 where M~ iir the r, rmv of AR

For example, in the above example of a 4 by 4 matrix A with uncertain elements in the 6rst

and third rows.

Chuter 3 An Unlrnown Inuut Observer Based F a t Deteaion and Identitication Method 65

AA, = &jr h , t h 3 3 MN

Now the dynarnic equations (3.6.1) can be rewrîtten to incorporate parameter uncertainties:

x =(A, + I , AA,) x+@, + IB AB,)u, (3.6.9)

This can be fiirther rewntten as

X = A, X + Bo U, +IR (Ma X) + I B ( M b u0 )

By definhg

and

v = ['A 1.1

Parameter uncertainties can also be modeled as unknown inputs to a known system:

So far in this section we just considered the individual effécts of actuator faults, sensor faults,

and parameter uncertaulties one at a the. If any two or al1 of the three factors are present,

then we can just stick the relevant tenns into the dynarnic equations andlor output equations.

For example, in the simulation chapter we choose not to concern our selves with parameter

vanation of an UMS-2 robot and detect only two actuator Faults and one sensor fault rather

than component faults. In this case the mode1 of the system is of the foliowing fom:

Chapter 3 An Unknown imut Observer Baseci Fault Detection and Identification Method 66

3.7 Actuator Fault Detection and Identifcation

As discussed in the previous section, actuator faults can be modeled as unknown inputs to a

known system with known or nominal parameter values:

where

subscript denotes original or open-loop systems

matrices and vectors are of appropriate orders as defied in section 3.6

Actuator fault vector d, cm be estimated by using the following theorem:

Theorem 3.7.1 The unknown input in system (3.7.1)-(3.7.2) can be estimated by an

estimation technique of the foilowing fonn if Do is of full rank and T is a small enough

sarnpling interval.

where

and

Proof: Applying the formula(Rugh, 1993) of the complete solution to a forced hem and

continuous-time system in discrete form, the value of the state vector x(t) at t h e (k+l)T is:

where T is the sampling period in the time domain

D e m g S(k) as in (3.7.4) results in

It must be pointed out that the estimated state X(kT) rather than the reai state x(kT) is to be

used in evaluating Se) in computer simulations. This is because the me state x(kT) is

usually not available for measurement. It can ody be asyrnptotically estimated by an estirnator

such as the UIO designed in the early sections of this chapter.

Plotting each component of d, (k) against time index k would show if the corresponding

actuator has failed. This technique identifies not only the magnitude but dso the shape of

actuator faults.

3.8 Sensor Fault Detection and Identification

Since no knowledge can be assumed about the time histories of sensor fauit signais, it is

reasonable to mode1 them by a dynamic system driven by an unknown input signal. To do this

we fint present the following proposition:

Proposition 3.8.1 For any piecewise continuous vector function f E R' " , and a stable r x r

rnatrix A/, there wili always exist an input vector 5 such that

Chauter 3 An Unknm Inout Obsem Based Fault Detection and Identincalion Method 68

f = ~ ~ f + c (3.8.1)

Proof. the pmof is Unmediate simply by taking 5 = f - A, f -

Now we can assume that sensor faults have the followuig dynamics:

A, is a stable rx r matrix

u, is a rx 1 unknown input vector

e is a r x 1 sensor fault vector

Augmenting (3.6.14)-(3.6.15) with (3 -8.2) results in the following (n+r)th order dynamic

system:

Define

Chapter 3 An Unknown hmt Observer Based Fadt Detection and Identification Method 69

then (3.8.3)-(3 -8 -4) is in the standard fom of (3 -2.1)-(3 -2.2) of section 3 -2:

Therefore the state of the system in (3.8.5)-(3.8.6) can be estimated by ushg the UIO

designed in section 3.4 provided that all the necessary and suflicient conditions related to the

existence of an UIO are satisfied-

It is now clear that sensor fauits are part of the state of the augmented system. Therefore

monitoring the state estimates ( ) would provide an immediate means of the detection of [il sensor failures. The fdure detection logic is very simple. Any nonzero component of 2

would indicate a sensor fadure. It is also extremely easy to identa or isolate sensor failure(s)

by checkhg which component of ê is nonzero. For example, if only the second component

of ê is nonzero, then only sensor 2 has fded; if the fira two components of ê are nonzero,

then both sensor I and sensor 2 have faiied.

Plotting each component of P would indicate if the correspondhg sensor has failed. This

technique could identw not only the magnitude but also the shape of sensor fadts.

3.9 Summary

In the first five sections of this chapter we presented a modified unknown input observer

(UIO) capable of estimating the state of a linear dynamic system driven by both known and

unknown inputs. By performing a couple of similarity transformations and a nonsingular

Chaoter 3 An Unknown b u t Observer Based Fauit Detection and Identification Method 70

transformation, we were able to partition the system into three subsystems. One of these

subsystems was a dynamic systern driven by known inputs ody. The other two subsystems

are nothing but explicit algebraic relationships between the States of the subsystems and the

meamrable outputs. This made it possible to use a conventional Luenberger observer with

slight modifications to estimate the state of the transformed system. The estimate of the date

of the original systern cm be obtained by perfonning inverse transformations. It was possible

to state a similar necessary and sufticient condition to that of a conventionai observer for the

existence of a stable estimator and arbitrary pole placement capability. It was also shown and

proven that this necessary and sufficieut condition can be expressed in tenns of ofiginal

system matrices. This alternative expression of the necessary and &cient condition provides

a much easier way for checking whether the condition is satisfied before any transformations

are undertaken. In view of a couple of competing UIO design methods, it is feit that the

design and computational cornplexities involved in designhg UIO's is greatiy reduced in our

proposed approach. Our simulation program in Appendix C also shows that our UIO

aigorithm is quite easy to code.

In the last few sections of this chapter we used our modined UIO in fault detection and

identification of uncertain dynamic systerns. We were able to mode1 parameter uncertainties

as unknown inputs to a known system with nominal or assumed parameter values. We also

modeled aciuator fauits as unknown inputs to the dynamic equations of a known systern

because they act directly onto system dynamics. We dealt with sensor faults by modeling

them as additive biases to the output equations. We used a generalized inverse solution

technique in estimating the actuator fadt vector for the purpose of actuator fault detection

and identification. This technique c m also be used to estimate one or more parameter

variations in some systems. By modehg sensor faults as the state of a dynarnic system driven

by unknown input, we w m able to obtah an augmented system whose state vector contains

not ody the original state variables but also sensor fauit signals. We could thus obtaui the

estimates of sensor faults by extracting a sub-vector from the estirnate of the state vector of

the augrnented system. The estimates of sensor faults provides an immediate means of sensor

fault detection and identification (F'DI). In both actuator FDI and sensor FDI, we were able

Cha~ter 3 An Unknown input Observer Based Fadt Detection and Identification Method 71

to obtain not only the shape but also the magnitude of the fauits. nùs enables us to

distinguish between a momentary fauit that dears its self and a persistent one. It is recognized

that this UIO based FDI approach aüows us to detect and identify multiple a d o r even

simuitaneous actuator and sensor faults as weil as parameter variations so long as the total

number of faults and uncertainties to be detected and identified is less than the number of

available outputs.

Chauter 4 FDI S W of a Consminecl Mechaaical Svstern - Ammch and Simulation 72

Chapter 4 FDI Study of a Constrained Mechanical System - Approach and Simulation

4.1 Introduction

In chapter 2, constrained mechanical systerns were initidy mathematidy descnied by

nodiear equations with Lagrange mdtipliers. Linearization was performed using standard

Taylor series expansion. The pecuiiar structure and important properties of iinearized

constrained mechanical systems were anaiyzed and normal (dynamic) forms of the system

representations were derived. The resultant purely dynamic subsystem of the Iinear

mechanical descriptor system representation is in the form of linear time-invariant dynamic

equations. This aiiows us to shifi our analysis from the domain of iinear singular system

theory to the domain of linear systern theory. In chapter 3, an observer design method was

proposed for linear time-invariant dynamic systems driven by both lcnown and unknown

inputs and a FDI approach based on UIO theory was presented. We were able to mode1

actuator faults as unknown inputs and sensor fauits as additive biases to the outputs. In this

chapter, we combine the results of chapters 2 and 3 and use them in fault detection and

control of a UMS-2 robot manipulator system. The following is a drawing of this robot.

UMS-2 robot

This robot has three degrees of eeedom during unconstrained motion. However, we study it

in the context of motion with a holonomic constraint. A sketch of the robot manipulator

geometric workspace is given in the foiiowing:

Chaoter 4 FDI Studv of a Constrained Mechanical Svstem - Aonroach and Simulation 73

UMS-2 Robot Manipulator Task Geometry

This robot was found ù1 the paper of Milis & Goldenberg (1989). This paper used this robot

as a numerical example in force and position control of manipulators during constrained

motion tasks. It gave little information on the nature of the specific tasks performed by this

robot manipulator. For the purpose of simulation it is d c i e n t to know that the UMS-2

robot is assumed to be in contact with a &id fnctionless surface. Robots which are similar to

but not identicai with a WS-2 robot can be found in the book of Vukobratovic & Potkonjak

(1982). Some of these robot manipulators can perforrn tasks such as spraying powder dong a

prescribed trajectory.

4.2 Approach and Simulation

We deploy a systematic approach to fault detection and identification of the UMS-2 robot

manipuiator system. This approach may be outlined as foiiows:

1) Write original nonlinear mathematicai description of the system with actuator faults

appended to dynamic equations and sensor faults appended to output equations

2) Lineanze the nonlinear mode1 and rewrite it in a generaiiized state space format

3) Perform a nonsingular coordinate transformation and derive the nomai f o m

4) Perform sirnilarity transformations to bting the dynamic subsystem into its canonical form

Cha~ter 4 FDI Shi& of a Constrained Mechanical Svstem - Ap~roach and Simulation 74

4) Design an unknown input observer and a state-feedback cootroller for the dynamic

subsystem

5) Obtain necessary results and convert the results back into the original coordinates by

reversing transformations

In the simulations we will detect and identify two actuator faults and one sensor fault. The

dynamic equations of motion of this manipuiator in unconstrained form are given by:

The output equations chosen for the simulations are as foliows:

where

C h ~ t e r 4 FDI Studv of a Coristraioed Mechaaihaaical Svstem - Amroach and SimuIation 75

L J

e is an unknown signal representhg a sensor tàult or €dure

rd is output vector with appended sensor Mt

E =

The position vector p is gïven by

The constraint f ic t ion representing the robot end-effector being in contact wîth a ngid flat

surface is the following :

O

O

O

This holonomic constra.int equation can be rewritten as

is sensoc fadt distniution vector

We Iinearize this constrained mechanical system about a point at which the robot manipulator

is stationary but being in contact with a flat surface. The nominal dynamic parameters and the

nominal values of the generaiized state variables are given in the foflowhg table:

Cha~ter 4 FDI Studv of a Constrained Mechanical Svstem - AD~roacb and Shuiation 76

NOMINAL DYNAMIC AM) KINEMATIC PARAMETERS

Nominal

State

Nominal Dynarnic

Parameters

qio = 0.4363 radian Ij = 0.2 m

qZo = 0.3 m - = l k g

q30 = 0.3 m q =2kg

-1 kg-rn A0 =- - f i s'

The above dynarnic system with constraint but without sensor fault can be described by the

foilowing equations:

where

q =

Cha~ter 1 FDI Studv of a Coustrained Mechanical S-em - Awroach and Simulation 77

The Jacobian in this single hotonomic constraint system may be defined as

Now applying the linearization approach outlined in Section 2.2 of Chapter 2 results in:

Chaoter 4 FDI Studv of a Constrained Mechanical Svstem - Aooroach and Simulation 78

Chauter 4 FDI SN& of a Constrained Mecbanid Svstem - A ~ v m c b and Sbuiation 79

3

The Jacobian J of the system is of the fom: J = L = [02418 1 1.32891

The iiiearized descriptor form representation of the system is

Chauter 4 FDT Studv ofa Consuaineci Mechanid Svstem - Auoroach and Simulation 80

L

where

is the generalized coordinates in linearized form

is the derivative of generalized coordinates in linearized fonn

is the constrained force in linearized form

is the known input signal in hearized form

is the actuator fauIt or failure signal in iînearized fonn

is the output vector in linearized form

is the partition of the dispiacement vector

is the partition of the known input vector

is the partition of the unknown input vector

Now we can apply the nonsiigular transformation technique presented in section 2.3 of

chapter 2 to derive the normal form of the iinearized descriptor form representation:

It c m be verified that the following nonsingular transformation matrk

O 4.1356 O

(4.2.10) -0.7525

satisfies L T = [ 0 O 11

Chaoter 4 FDI Shidv of a Constrained Mechanical Svstem - Awroach and Simulation 8 1

where

Pre-multiplying both sides of (4.2.6) by ITT O T~ O O \ O and performing lengthy simplification

10 O GJ will yield the following normal form representation:

where

Chapter 4 R)I Snidv ofa Constrained Mechanid Svstem - A~proach and Simulation 82

The algebraic part of the system is described by the foliowing:

Evaluation of equai oas (4.2.12)-(4.2.13) result s in the following numencal fom:

Chauter 4 FDi Snidv of a Constcaïned Mechanial Svçtem - Awroach and Simulation 83

Evaluation of equations (4.2.12)-(4.2.13) r d t s in the foUowing numerical form:

The above representation can be rewritten in the following unaugmented open-loop form:

where

Chaoter 1 FDI Studv of a Consaainecf Mechanical Svstem - Approach and Simulation 84

(4.2.3 5)

The open-loop system is not stable because A, has eigenvalues 5 0.523 5 and f 0.6675i.

Hence state feedback is used to stabilize the open-loop system. The closed-loop system takes

the form:

k c = A 0 x , + B o u , + D o d o -Bo& X,

= A c x, +Bo uo +Do do (4.2.3 7)

Y, =Co Xe (4.2.3 8)

where

subscrïpt . denotes closed-loop and is an estimate of xe

By placing the poles of the closed-loop matrix

at arbitrarily chosen locations -5 i 4 i and -6 t 2 i, the state feedback gain mat* Kc and the

matrix Ac are computed by MATLAB as:

108.3 527 -0.820 1 262529 -02808

-46.4048 37.7784 -1 3.1420 1 1.3378 1

Chapter 4 FDI S a d v of a ConStraiaed Mecbanid Svstem - A ~ ~ m a c h and Simulation 85

As discussed in chapter 5, sensor fadt estimation is accommodated by adding a term Ee in

the output equations of aii previous representations, Le.,

The augmented open-loop system is of the followîng fom:

where

A, is a stability matrix(a negative constant in one dimensional case)

u, is a sensor fadt input vector(a scalar hction in the above case)

Then last represent ation (4.2.4 1)-(4.2.42) can be written as

Chapter 4 FDI Stodv of a Constrained Mecbanid Svstem - Ammach and Simulation 86

Once again state feedback is used to stabilize the above systern. The augrnented closed-loop

dynamic systern representation is

- E, -A, x, +B+, + D a d a

Y = C ~ X,

where

A , = A a - B a K ,

Let us now determine K, and A, by performing the followhg analysis:

Assuming K, is of the form [ Kc K, ] where K, is the state feedback gain ma& used in

the unaugmented case and K, is an unknom submatrix to be determined.

where

A, is the unaugmented closed-loop matrix defhed previously

Note that

det(A, ) = det(A, ) x det( A# )

poles of A, are poles of Ac plus pole(s) of A,

Chatlter 4 FDI Studv of a Constrained Mechanical Svstem - A~~roach and Simulation 87

Therefore no rnatter what value K, takes the eigenvalues of A, remain the same. in this

simulation we use the following arbitrary values:

and A, = -5

then the resultant A, computed by MATLAB using the aforementioned expression is

with poles or eigenvalues at

The estimation of the state vector of a system that has actuator faults and/or sensor fkults

relies on the evaiuation of an unknown input observer(üI0) outhed in chapter 3. In the

MATLAB simulation program, the foiiowing were done or obtained:

1, Necessary and sufncient conditions for the existence of an UIO are verifled numericdy

2, Two similarity transformations are performed in bring the augmented cIosed-loop hear

Chapter 4 FDI Studv of a Constrained Mechanical SvStem - Ammach and Simulation 88

dynamic system to its special canonical form and partitionhg it into three subsysterns

3, The output vector Y, of the unaugmented closed-loop system is obtauied by doing a

hear dynamic system simulation using the lsIm command in MATLAB

4, The output vector Y, of the augmented closed-loop system is calnilateci ushg equation

where

.=j:/ is sensor fault distribution ma& (vector)

e = u, = 0.5*u(t-3) is the assumed form of sensor 1 failure

5, The observer equation is: w = -6 w + [O -1.0054 0.0330 0.0050] Y, + [O O] u,

6, The state vector of the augmentecl ciosed-Ioop system is estimated using the equation:

7, The estimate of the state vector of the unaugmented closed-loop system Ir is extracted

8, The estirnate of sensor fault ê is also extracted fiom Ca using equation X,, =

Once the estimate of the state vector of the unaugmented closed-loop system gr is available,

transformed actuator fadt vector d, can be estirnated using the least square solution

technique presented in chapter 3, i-e.,

where

and

& is the estirnate of transformed actuator fault & A

& is the estimate of transformed actuator fault d2 n

Cc is the estimate of the state vector of the unaugrnented closed-

loop system

Then estimates of the original actuator faults can be obtained by reversing the transformation

defined by equation (4.2.24), Le.,

(4.2.5 Ib)

Note that the aigebraic equation (4.2.26) cm not be used to estimate transformed actuator

fault & because the constrained force A. in this equation is also unknown. This means that

4 in equation (4.2.5 1) can not be detennined or evaluated. The variables that we do have A

estimates for are just 4 and dz . To obtain the estimates of d,, d,, and d, fkom the

e s t h t e s of d, and d2 is equivalent to solving a system of two hear equations with three

unknowns as specined by (4.2.51). A solution can ody be obtained by assuming one of the

Cha~ter 4 FDI Studv of a Constrained Mechanical Svstem - Amroach and SimuMion 90

estimates of d l , d?, and d, is zero or known. This essentiaily requires that one of the

actuators is fadtiess. In this particuiar system it does not matter which actuator is assumed to

be healthy. As long as one of the three actuators c m be assumed faultiess, the other two

actuator fadts can be uaiquely detected and identified by solving a system of two hear

equations with two unknowns. For example, suppose the 3-rd actuator is fdtless, Le., d, =

O = (î,, then d, and d2 can be determined by solving two equations. g3 can be obtained via

the transformation and used in (4.2.26) to generate an estimate of the constraint force A .

In this thesis we perfomed two simulations to estimate aU three actuator faults. The first

simulation estimates actuator 1 and actuator 2 fadts based on the assumption that actuator 3

is highly reliable and faultless. The second simulation estimates actuator 1 and actuator 3

faults based on the assumption that actuator 2 is highly reliable and faultless.

Although the unlaiown inputs representing two actuator failures in the simulations can be of

any form, we have to specify a specific fùnction for each one of them for the purpose of

estimation. In this simulation we just happen to use step fiindon as a fom of possible

failures. The sofi actuator failures are assumed to be of the following form:

(a) For simulation number 1

d, = O. 5 *u(t-3)

d, = 0.4*u(t-3)

d3 = O.O*u(t-3)

(b) For simulation number 2

dl = 0.5 *u(t-3)

d2 = 0.O1u(t-3)

d, = 0.4*~(t-3)

The known control inputs are assumed to be of the foUowing form for both simulations:

f , = Ul = 7* u(t)

f, =u, = 8* u(t)

Chauter 4 FDI S W of a Constrained Mechanical Svstem - Auuroach and Simulation 91

f, = u, = 9* u(t)

a v e n the above control inputs f, , f,, & f, and actuator fadt inputs 4 , d2, & d3 , we cari

use equations (4.2.23)-(4.2.24) and equations (4.2.32)-(4.2.33) to obtain the known inputs

and unknowu inputs used for simulating the normal form (4.2.37):

(a) for simulation number 1

(b) for simulation number 2

Then the original actuator f d t estimates are obtained by performing reverse transformation

on the estimates generated by the simulations. The reverse transformation equation (4.2.5 1 b)

reduces to the foliowing foms for the following cases:

(a) Cor simulation number 1

@) for simulation number 2

where

[$ ] was obtained in the simulations using equation (4.2.50)

Once the estimates of the original actuator fàults are obtained, the estirnates and their

corresponding fault signais are plotted for easy Mt detection and identification. In

simulation number 1, 4 (the estimate of d, ) and d, itseif are plotted against tirne to show the

transient and arymptotic behavior of actuator 1 fault. &(the estimate of d 2 ) and d, itseif

are plotted against tirne to show the transient and asymptotic behavior of actuator 2 huit.

Each of the two plots shows that the estimate has a big spike iaitially, another spike at the 3-

rd second, and then quickIy settles down to the asymptotic value. The fkst spike is due to the

transient behavior. The second spike indicates that the actuator had a fault at tirne t = 3

seconds. The asymptotic behavior confirms the stabiiity of the observer and the correctness

of the theoreticai work. Similarly in simulation number 2, actuator 1 and actuator 3 faults are

detected and identified using two sirnilar plots. The combination of simulation number 1 and

simulation number 2 detects and identifies ali three(3) actuator faults.

It can be seen fiom equation (4.2.57) that sensor fadt is part of the state of the augmented

system. Sensor fault estimate can be obtained fiom the estimate of the state vector of the

augrnented system. Sensor fàuit estirnate provides an immediate means for sensor huit

detection and identiiïcation.

The plot of sensor fauIt estunate ê and the original assumed sensor fadt bc t ion shows a

sensor fault at time t = 3 seconds. The objective of sensor fault detection is achieved.

Chauter 4 FDI Studv of a Constrained Mechanid Svstem - Ap~roach and Simulation 93

Tii this particular simulation case, we performed two simulations each of which is based on

the assumption that one of the three actuaton is highly reliable(faultless). The observer

equations for these two simulations are essentially the same. Consequently, we codd say that

we used only one observer (but two simulations). In fact, it can be seen fiom Appendix C

that the first part of the source codes of the two simulations are identical. The merence only

exists in the Iast part of the program. This is the reason that two simulations were written in

one source code program.

The plots of actuator and seosor faults and their corresponding estimates against t h e are

shown in the figures of this thesis.

4.3 Summary

In conclusion, this chapter bas illustrated a systematic or at least a procedural approach to

fauit detection and identification of a major subclass of generalized state-space systems. By

performing severai nonsingular and similarity transformations and using an unknown input

observer we were able to convert a problem of fault detection and control of a hearized

constrained mechanical systems to a problem of fault detection and control of a hear tirne-

invariant dynamic system with partially unknown inputs. The methodology appears to be

mathematicaiiy elegant yet simple. The procedure or aigorithm is quite straightforward and

fairly easy to code or Unplement. As long as the necessary and wflicient conditions of the

existence of an unknown input observer is met and the system is stabilizable, our proposed

approach cm detect and ident* multiple and/or simultaneous actuator faults and sensor

fauit (s) almost immediat ely .

Cha~ter 5 Conclusions 94

Chapter 5

Conclusions

In this thesis au approach for the controi, fault detection and identification of constrained

mechanical systems is presented. The major advantages of this state estimator or observer

based analytical redundancy approach and the major contributions of this thesis can be

summarized as the foilowing:

(1) It is a systernatic approach for fault detection and identification of a special class of

descriptor systems that is neither infinitely observable nor completely controilable.

(2) It can detect and isolate multiple andlor sirnultaneous actuator and sensor faults

aimost immediately. The promptness of detection can be adjusted through changing

the eigenvalues of the closed-loop A matrix and the eigenvalues of the observer.

(3) It is capable of distinguishing momentary fdures fiom persistent fdures. This

capability exists because the FDI scheme can estimate not only the magnitude but also

the shape of the faults during t ie entire time period in which the faults last.

(4) It cari detect almost aii kinds of faults. This is because that the scheme assumes no

apriori knowiedge about the nature or the mode of the fdures

(5) It uses only a single observer instead of a bank of estimators.

(6) It is mathematically simple yet elegant, computationally straightforward and efficient,

and relatively easy for computer simulation andor red time implementation.

(7) A technique for numericdy testing the necessary and wllicient condition under which

an unknown input observer exists is found and used. Note tbat the following condition

cm not be possibly numerically tested because s takes an infinite nurnber of values. My

experiencehypothesis is that testing s at al the eigenvalues of A and zero is sufficient.

(8) A rnodified unlaiown input observer whose equations are different f?om those contained

Cha~ter 5 Conclusions 95

in a previous research work is derived.

(9) Simulations are performed using the model of a practical system - a UMS-2 robot.

(10)A modified coordinate transformation technique using a nonsingular but not orthogonal

transformation matmE is developed for any mechanical system that has otdy one

holonomic constrauit. The coordinate transformation technique using an orthogonal

transformation matrk which was presented in a previous research work is not applicable

to the speciai case of a single constraint. Norrnai fom of the linearized descriptor system

model of a single constraint system (such as a UMS-2 robot) is derived in this thesis and

can be shown to be different eom normal forms of systems with multiple constraints.

The Limitations of the proposed approach and the aspects of the topic that could be further

researched by somebody else in the fiture cm be summarized as foilows:

(1) The maximum number of actuator failures and sensor failures that can be detected and

identified by the approach is I i t e d to the number of measuable outputs.

(2) Not ail actuator failures can be detected and identified ifthey al1 faiI at the sarne time.

(3) The approach requires that the considered systern behaves almost linearly within an

operating range, Le., IineariLation of the system can be justified.

(4) A mathematicai proof is not available for the experimentally correct numerical testing

technique (hypothesis) with respect to the necessary and sufficient condition for the

existence of an unknown input observer

On one hand, Our proposed approach does not need infinite observability or complete

controliability. On the other hand, for an unknown input observer to exist, at least one fairly

strong (necessaxy and sufficient) condition has to be net. The capability of our observer

based anaiytical redundancy approach primariiy depends on the number of available outputs.

The larger the number of independent outputs, the more faults we can potentiaiiy detect and

ident*. in the situations where a stable unknown input observer with pole placement

capability does exist, our proposed approach can be very simple yet powefil.

Atmendices 96

Appendix A

Proof of Regularity of Constrained Linear Mechanical Systems

The proof uses Luenberger's shuffie algorithm (Luenberger 1978). A presentation of this

algorithm and a numerical example can be found in the book of Dai (1989). BasicaIly the

algorithm invoives a series of shuffling and row operations of the matrix combination

[E A ] . Shufniog means the interchanging of a row of the lefl half of the combined matrix

with that of the right half of the çombined matrix. A row operation involves multiplying one

row of the combined matrix with another rnatrix and addisubtract the product to/fiorn

another row. If the left haif' of the combined matrix cm be made nonshguiar by performing a

series of altemating s h d i n g and row operations, then the system is said to be regular by

Luenberger. Here, we present only the proof for the case of hdonomic constraints because

the simufation system used in chapter 4 has oniy one holonomic constraint. O u proof here is

sirnilar to the proof of the more generai case of cornbined holonomic and nonholonomic

constraints, which cm be found in Schimidt and Muller (1991).

The linear mechanical descriptor systern describeci by equations (2.2.23) reduces to the

following fom in the case of holonomic consb'aints:

where

Multiplying row 1 by F and subtracting the product nom row 3 results in

rowoperation 1- O M O -K -D F~ [D : : : :F :] 1, O O O 1, O M O -K -D F* 1 O - F O O O 0 1

Multiplying row 2 byEW-' and add the product to row 3 results in

rn O O O 1,

row operation 2 3 O M O -K -LI O O O -FiW1K -FM-'D FU-'Fr

Addiig FM-'D * row 1 and FM-' * row 2 * M-'K to row 3 results in

row operation 3

- -

Since the mass matrix M is positive dennite, so is M-' . Then given any non-zero arbitrq

vector x and its transpose xT we have

Therefore FM-' F' is positive definite by dennition and E, is nonsingular. The system is

hence reguiar by Luenberger's theorem.

The augrnented system descnied in chapter 2

is in the descriptor form

It is infinitely observable if and only if

rank O E = number of rows or columns of A + rank(E) [: rl Theorem 1 The rank of a matrix will not change after the pre- or post- multiplication of a

non-singular matru<.

Appendices 99

Proof Let M be an arbitrary nonsinguiar matrix of order m by rn

N be an arbitrary ma& of order m by p and is of rank n

P be an arbitrary nonsîngular matrix of order p by p

and assurning ns rns p without Ioss of generaiity

then using a theorem in ma& theory, we have

This proves that the rank of a rnatrix is not changed by the premultiplication of a nonsingular

matrix. Applying the same theorem in a similar manner will prove that

and

Therefore the theorem is valid.

Since the mess matrix M in the thesis is positive definite( see Hou et. al., 1993, second h e

from the top right corner on page 612), it has an inverse M I . Post-multiplying the ma&

O E by a nonsingular diagonal ma& containhg M' in the following form preserves the ["] rank:

Performing elementary row operations on the above matrix yields:

L.H.S. = rank

The right hand side(R.H.S.) of equation (1) is evaluated as :

RH-S. = (n + n + q + e) + rank

Obviously L.H.S. of equation b. 1 # RH.S. of equation b. 1, the necessary and wfficient

condition for infinite observability as expressed by equation b. 1 does not hold. Therefore the

system is infinitely unobservable.

Appendix C

Simulation Program Source Code

% SIMULATION PROGRAM SOURCE CODE

YO Simulation of a UMS-2 Robot in MATLAB

% this simulation detects and identifies 2 actuator failures % and 1 sensor failure

Na = 2 % Na: number of actuator fdwes Ne = 1 % Ne: number of sensor fdure

% specify matrices used in iinearized descriptor system mode1

J = [0.2418 1 1.32891 % Jacobian J = L in the holonomic case

% nonsingular coordinate transformation begias

T=[O 4.13560 I -1 1

-0.7525 O O] % transformation matrk

% ve@ that LT = [O O 11

% nonsingular coordinate transformation ends

% perform controliability test on the normal form representation

%specfi measurement matrix used in original system representation

% obtain output matrices used in normal form representation

C l = Cp*inv(T')*[eye(n-q);LI IL123

CS = Cv*inv(T')*[eye(n-q);L 1 1 L 123

C=[C1 C3]

% spec% actuator fault distrifiution matrix

% perform observability test

ObservMatrk = obsv(4[C1 C33)

ObservMatrivRank = rank(0bservMatrVr)

% test observer existence conditions

Appendices 105

% Redefine the above matrices as open-loop matrices % use subscript O to denote open-loop

CoDo = &*Do

% Stabilize the open-loop system using state feedback technique

Pc = [-6+2i;-6-2i;-5+4i;-5-4i] % choose closed-loop poles

Kc = place(Ao,Bo,Pc)% state feedback gain matrix

eigenvalue = zeros(n, 1);

Eig-VdAc = eig(Ac)

% Augment the system to accomodate sensor failure

% define open-loop system matrices Aao, Bao, Cao, Dao, Kao

Aao = [Ao zeros(4,Ne);zeros(Ne,4) -51 % set additional eigenvalue at -5

Bao = P o ; zeros(Ne,(n-q))]

% speciQ sensor failure distribution vector

Appendices 106

Cao = [Co Ef]

Dao = [Do zeros(4,l); zeros(l,2) eye( 1 )]

CaoDao = Cao*Dao

% check observability of augmented open-loop system

Observability-Augm = obsv(Aao,Cao)

Obsew-Augm-Rank = rank(ObservabilitytYAu~)

% check rank conditions

Rank-Dao = rank(Da0)

Rank-CaoDao = rank(Cao*Dao)

% Obtain augmented closed-loop dynamic mode1

Karb = ones(2,l) % feedback gain matrix used to stabilize augmented system % It can be proved that this matrix can be chosen arbitrarily

Kao = WC Karb]

Aac = Aao - Bao*Kao

Eig-Val-Aac = zeros(5,I);

Eig-Val-Aac = eig(Aac)

%Redefine system order using closed-loop augmented representation

n = 5 % number of state variables

p = 4 % number of outputs

rn = 3 % number of combined actuator failures and sensor fdure

% Check augmented closed-hop system observer existence condition

Ranks-wrt-Eig-Vd = zero@, 1);

Observability-Test = [-AacJao;Cao,zeros@,m)]

for i = l:n

Ranks-wrt-Eig-Vai(i) = tank([EiggValalAac(i)*eye(n)-Aac,Dao;Cao,zeros(pym)]);

end

RankRankwrtwrtZero-Eig_Val = rank([-Aac,Dao;Cao,zeros@,m)])

Ranks-wrtEig-Val

% check rank conditions

Rank-Dao = rankoao)

Rank-CaoDao = rank(Cao*Dao)

% check observability of augmenteci cfosed-loop system

% Observability-Augm = obsv(Aac,Cao)

% Observ-Augrn-Rank = rank(0bservability - Augm)

% [-Aac,Dao;Cao,zeros@,m)]

% rank([-Aac,Dao;Cao,zeros(p,rn)])

% proceed to obtaining reduced order observer

Q = k o s(n-p, PI, eye(n-pll

P=[Q;Cao] % transformation matrix to b ~ g C to [O,i] f o m

Pinv=inv(P)

% First transformation is now taking place

As=P*Aac*Pinv

Bs=P*Bao

% Convert Cs=[O Ip] to Cn=[O Cp] to deal with the invertibility of Ds3

% The new transformation matrix Pn is chosen such that Da3 is % invertible. This is accomplished by switching row 2 and % row 3 in Ds

% the new representation are defked by An, Bq Cn, & Dn. % n denotes new

Dn = Pn*Ds

% Obtain the Cp in Cn=[O Cp] and Cp 1 & Cp2 in x2=Cp 1 *y & xj=CpZ*y

B2=Bn(n-p+ 1 : n-m, :)

B3=Bn(n-m+ lm,:)

D 1 =Dn( 1 :n-p, :)

D2=Dn(n-p+ 1 :n-m, :)

D3=Dn(n-m+l :n,:)

D3 inv=inv(D3)

Al bar=Al -D 1 *D3inv*A3

A2ba~A2-D2*D3 inv*A3

B I bar=B 1-D 1 *D3Ïnv*B3

B2ba~B2-D2*D3inv*B3

Al 1 bar=Al bar(:, l :n-p)

A1 2bar=Al bar(:,n-p+l :nom)

A 1 3 bar=A 1 bar(:, nom+ 1 :n)

A2 1 bar=A2bar(:, 1 :n-p)

A22bar=.42bar(:,n-p+1 mm)

A23 bar=A2bar(:,n-m+l :n)

% obtain observer in the form of (3 S. 1.19)-(3 51-24) of thesis

Pole - observer = -6 % choose observer pole at -6

M = place(A 11 bar,A2 l bar,Pole-O bserver) % observer gain mat*

F = Al 1 bar-M*A2l bar

E 1 = (Al l bar-M*A2 1 bar)*@ 1 *D3UivZCp2-M*(Cp l -D2*D3invSCp2));

E2 = ((Al 2bar*Cp l+Al3 bar*Cp2)-M*(A22bar*Cp l+AUbar*Cp2));

E = E1+E2 % E is too long to be typed in one row

%contiming expression in 2nd row would have resulted in E=Et

L = B I bar-M*B2bar

N = D 1 *D3inv*Cp2-M*(Cp l-D2*D3inv*Cp2)

% PREPARE FOR LINEAR DYNAMIC SYSTEM SIMLTLATION

% DEFINE SAMPLiNG PERIOD

Ts = [0:0,1: 19.91'; % sample taken at O. 1 sec. interval % for 20 seconds

% spec* applied generalized forces or known inputs

u 1 = 7*ones(200,1);

u2 = 8*ones(200,1);

u3 = 9*ones(200,1);

Tt = T' % transpose of nonsingular transformation mat~ix

U = (Tt(l:2,:)*[u l';uS';u3'])'; % known input vector

% STARTING SlMüL,A'ITON #I

% specify arbitrary actuator failures and sensor fdure % for the sake of simulation

dl = [zeros(30,1); 0.5*ones(170,1)]; % actuator #1 fdure

ci2 = [zeros(30,1); 0.4*ones(170, l)]; % actuator #2 fdure

d3 = [zeros(30,1); O.O*ones(l70, l)]; % actuator #3 faultless

Tsub = [0.0 1.0 4.1356 -1.01

d = (Tsub*[dl';d2'])'; % unknown input matrix

fl = [zeros(30,1); 0.5*ones(170,1)]; % sensor # l fdure

Appendices 111

% start simulation ofcontinuous tirne state-space mode1 of % the unaugmented closed-loop system

XcO =[0;0;0;0]% arùitrary initial condition of the state vector

WC, Xc] = Isim(~c, @o,Do],Co,zeroIp,Na+Na), CCl. dl ,Ts,XcO);

% Get output vector for the augmented closed-loop system

Yac = Yc + (EPf)';

% Start simulation of reduced order observer

WO=O % arbitrary initial condition of reduced order observer

W = zeros(1,200);

~observer,Wj=lsim(F, [E,L],O,zeros( l,6), a , UJ,Ts, WO);

% Estimate state vector of twice transformed representation Xn

Yn = Yac; % output doesn't change during transformation

Xn = ([ i ;O;O;O;O] *Ur + m;Cp 1 ;Cp2] *Ynt)';

% estimate state vector of once transformed representation Xs

Xs = (inv(Pn)*(?h)')';

% estimate state vector of augmented closed-loop system Xac

Xac = (inv(P)*(Xs)')';

% estimate state vector of un-augmented closed-loop system Xc

Xc = Xac(:, l ml);

% estimate sensor fault failure

% plotting sensor fault and its estimate against tirne

figure( 1

plot(Ts, fl ,'g-',T~,fl~eStimate,~r-~,Ts~fl~'g-'~Ts,fl-e~ate,'y~')

title('Figure 1 - simulation #1: sensor 1 failure and its estimate')

xlabel(%ne(s)')

print figure1 -dps

% obtain actuator fdure estimates using least-square approach

v = zeros(Na, 199); % specify the dimension for % unknom inputs estimates

for k = LI99

S i = Ac*Uiv(expm(Ac*O -1 )-eye(n- 1));

S(: , k)=S 1 * (Xc(k+l , :)'-exprn(Ac*O. l)*Xc(k, :)');

v(:,k) = inv(Dot*Do)*Do'*(SC,k)-Bo*U(k,:)');

end

d2-b-stirnate = v(2, :)';

T-b = Tt(l:2,1:2) % upper left sub matrix of Tt

d lestirnate = d-estimate(1, :)'; % estimate of actuator 1 fault

d2estimate = d-estimate(2, :)'; % estimate of actuator 2 fault

% Plotting actuator fdures and their estimates agauist t h e

dlt = d1(1:199,1);

d2t = d2(1: 199,I);

title('Figure 2 - simulation #1: actuator 1 fdure and its estimate')

title(Tigure 3 - simulation #1: actuator 2 fdure and its estimate')

% STARTING SIMüLATION #2

% specify arbitrary actuator fdures and sensor fàilure % for the sake of simulation

d 1 = [zeros(30,1); 0.5 *ones(170,1)]; % actuator #l fdure

d2 = [zeros(30,1); O.O*ones(l70,1)]; % actuator #2 faultless

d3 = [zeros(30,1); 0.4*ones(170, l)]; % actuator #3 failure

Tsub = [O -0.7525 4.1356 O ]

d = (Tsub* Cd 1 ';d3'])'; % unknown input rnatrix

f l = [zeros(3 0,l); 0.5 *ones(170, l)]; % sensor #1 failure

f=f l ;

% start simulation of continuous time state-space mode1 of % the unaugnented closed-loop system

XcO =[0;0;0;0]% arbitrary Wal condition of the state vector

[Yc, Xc] = Isim(Ac,[Bo~o],Co,zeros@,Na+Na),~,d],Ts,XcO);

% Get output vector for the augmented closed-loop system

Yac = Yc + (EPf)';

% Start simulation of reduced order observer

WO = O % arbitrary initial condition of reduced order observer

W = zeros(1.200);

~observer,W]=Isim(F,[E,L],07zeros(l,6),[Yac, u,Ts7 WO);

% Estimate state vector of twice transformed representation Xn

Yn = Yac; % output doesn't change during transformation

Xn = ([ 1 ;O;O;O;O] *W + [N;Cp 1 ;Cp2]*Yn1)';

% estimate state vector of once transformed representation Xs

Xs = (iv(Pn)*(Xn)')';

% estimate state vector of augmented closed-loop system Xac

Xac = (inv(P)*(Xs)')';

% estimate state vector of un-augmented closed-loop system Xc

% estimate sensor fadt fdure

flestimate = Xac(:,n);

% plotting sensor fauk and its estimate against tirne

figure(4)

plot(Ts,fl ,'g-',Ts,fl-estimate,'r-',Ts,fl,'g-',Ts,fi-estimate,'y.')

title('Figure 4 - simulation #2: sensor 1 fdure and its estimate')

xlabel('time(s)')

% obtain actuator failure estimates using least-square approach

v = zeros(Na, 199); % speci@ the dimension for % unknown inputs estimates

for k = Ir199

end

d lbar-estimate = ~(1,:)';

d2bar-estimate = v(2, :)';

Tt-sub = Tt(l:2,1:2) % upper lefl sub matrix of Tt

d-estimate = inv(Tsub)*[dl-bararehate';&-bacestimate'];

d 1 estimate = d-estunate(1, :)'; % estimate of actuator 1 fault

d3 estimate = d-estUnate(2, :)'; % estimate of actuator 3 fault

d 1 -difFerential= d 1 (1 : 199) - d 1-estimate;

d3-dEerential= d3(2 : 199) - d3-estimate;

% Plotting actuator failures and thek estimates against t h e

T l = [0:0.1: 19.81';

dlt = d1(1:199,1);

d3t = d3(I: 199,l);

figure(5

piot(T1,dl t,'g-',T1,dl-e~timate,'r-'~Tl,dl~g-',Tl,dI-esthate,'y.')

title(Tigure 5 - simulation #2: actuator 1 fdure and its estimate')

xlabel('time(s)')

print figure5 -dps

fiwe(6)

plot(T1 ,d3t,'g-',Tl,d3estllnate,'r',Tl,d3t,'g-',T I,&-estirnate,'y.')

title(l3gure 6 - simulation #2: actuator 3 fdure and its estimate')

xlabel('time(s)')

print figure6 -dps

% END OF MATLAB SOURCE CODE

Figures 117

Figure 1 - simulation #1: sensor 1 failure and its estimata I I I I I I 1 I I

Figures 118

Figure 2 - simulation #l : actuator 1 failure and its estimate I l I I I I ! I I

Figures 11s

Figure 3 - simulation #1: actuator 2 failure and its estimate I I I I I I l l I

Figures 120

Figure 4 - simulation #2: sensor 1 failure and its estimate I I I I I 1 1 1 I

Figures 122

Figure 6 - simulation #2: actuator 3 failure and its estimate I I I 1 I I I I I

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