Horton, KirkGerritt. Fault Detection andModelIdenticationinLinear Dynamical
Systems. (Under the directionof Dr. Stephen La Vern Campbell.)
Lineardynamicalsystems,Ex 0
+Fx=f(t),inwhichE issingular,areuseful ina
widevariety ofapplications. Because ofthis widespreadapplicability,muchresearch
has been donerecently todevelop theory for the design of linear dynamicalsystems.
A key aspect of system design is fault detection and isolation (FDI). One avenue of
FDIisviathemulti-modelapproach,inwhichtheparametersofthenominal,unfailed
modelofthesystemareknown,aswellastheparametersofoneormorefaultmodels.
The design goalis to obtain an indicator for when a fault has occurred, and, when
morethan one type ispossible,whichtypeoffaultit is. Achoicethat must bemade
in the system design is how to modelnoise. One way isas a bounded energy signal.
This approach places very few restrictions on the types of noisy systems which can
be addressed, requiring nocomplex modelingrequirement.
This thesisappliesthe multi-modelapproachtoFDIinlineardynamicalsystems,
modeling noise as bounded energy signals. A complete algorithm is developed,
re-quiring very littleon-linecomputation,with whichnearly perfect faultdetection and
isolationoveranitehorizonisattained. Thealgorithmappliestechniquestoconvert
complexsystemrelationshipsintonecessaryandsuÆcientconditionsforthesolutions
isolation via the separating hyperplane. The algorithm is implemented and tested
ona suite of examplesin commercialoptimization software. The algorithmis shown
to have promise in nonlinear problems, time varying problems, and certain types of
LINEAR DYNAMICAL SYSTEMS
by
Kirk Gerritt Horton
a dissertation submitted to the graduate faculty of
north carolina state university
in partial fulfillment of the
requirements for the degree of
doctor of philosophy
operations research program
raleigh, north carolina
February 2001
approved by:
S. L. Campbell R. Smith
chair of advisory committee
K. Ito H. T. Tran
KirkGerritt Hortonwas born September 14,1963in Paterson, New Jersey. He grew
up attending public schools with his three sisters in West Milford, New Jersey. He
was the valedictorian of the West Milford HighSchool graduatingclass of 1981.
He received hisB. Engineeringdegreewith amajorinElectricalEngineering and
Computer Science from Stevens Institute of Technology in Hoboken, New Jersey in
1985. Shortly after graduation he received a commission in the United States Air
ForcefromAFROTC.From1985 to1993hewasapilot,yingthe T-37andtheT-38
trainer,thentheRF-4Creconnaissance/ghter, andthentheF-111Eghter/bomber.
He ew 19 missions over enemy territory in Iraq during Operation Desert Storm,
receivingthe AF DistinguishedFlying Cross forone of those missions. From 1993to
1995heattendedthe AirForceInstituteofTechnology,earningaM.S.inOperations
Analysis. From 1995to1997he pilotedtheF-117A StealthFighter,endingthat tour
asaninstructor in theaircraft. He arrived atN. C. State in1997to pursuea Ph. D.
inOperationsResearch. He iscurrently serving onactiveduty asa Major inthe Air
Force.
The author is marriedto the former Susan Elaine Pringels,of Sumter, S.C. and
As countless students have beforeme, I owe anincredibleamountof gratitude tomy
advisor, Dr. Stephen L. Campbell. An exceptional teacher, a meticulous researcher,
a prolic writer, an accomplished artist, and a natural conversationalist, he guided
me through the morass of graduate study with a rm but gentle hand. Without his
expertise I would not have been able tocomplete this project.
IamalsogratefultoDr.RalphSmith,Dr.HienT.Tran,Dr.KazufumiIto,andDr.
Ethelbert Chukwuforservingonmycommittee. Theirmathematicaland Operations
Research experience were invaluablein ensuring the accuracy of my research.
I would like to thank the United States Air Force, and in particular the faculty
and sta at the Air Force Institute of Technology for preparing and selecting mefor
the program that allowed me to attend North Carolina State University to pursue
my degree.
Finally, I must acknowledge my wife, Susan, and my daughters, Madisyn and
Ashlyn. Without their love and support none of this would have been possible, nor
wouldithavebeenworthdoing. Ipraythatallofourfather'sblessingswillbepoured
out on them as we travel around the world serving our country and spreading the
List of Tables vii
List of Figures ix
1 Introduction and Review of Prior Research 1
1.1 Linear DescriptorSystems . . . 2
1.1.1 Basic Theory . . . 2
1.1.2 NumericalSolutions . . . 7
1.2 Fault Detection and Isolation . . . 11
1.2.1 Basic Theory . . . 11
1.2.2 Feedback and Observer Design. . . 13
1.2.3 OptimalControl . . . 16
1.2.4 H 1 Control . . . 18
1.2.5 Prior Research . . . 19
1.2.6 Conclusion. . . 24
1.3 Outlineof Thesis . . . 24
1.4 Contributions of Thesis . . . 25
2 Fault Detection via the Detection Signal 26 2.1 The Problem- Findingthe Minimum Energy DetectionSignal . . . . 26
2.1.1 ProblemSetup . . . 27
2.1.2 Formulationas anOptimalControlProblem . . . 30
2.1.3 ProblemStatement . . . 37
2.2 Necessary Conditions . . . 37
2.2.1 Computing the Necessary Conditions . . . 38
2.2.2 Riccati Form of Necessary Conditions . . . 39
2.2.3 ProblemFormulation inTerms of the Necessary Conditions. . 49
2.2.4 SuÆcientConditions . . . 52
2.4.2 Unreduced Model . . . 61
2.4.3 ControlledSystems . . . 63
2.4.4 AlternativeCost Functions . . . 64
2.4.5 Knowledgeof InitialConditions . . . 65
2.4.6 Conclusion. . . 66
3 Model Identication via the Separating Hyperplane 68 3.1 The Problem- Determiningthe Origin of a Given Output . . . 68
3.1.1 ProblemSetup . . . 69
3.1.2 The Separating Hyperplane . . . 71
3.1.3 Approximatingthe SeparatingHyperplane . . . 72
3.1.4 ProblemStatement . . . 74
3.2 The ModelIdentication Algorithm . . . 76
3.3 Variations . . . 79
3.3.1 Multiple faultmodels . . . 79
3.3.2 AlternativeFormulations . . . 80
3.3.3 ControlledSystems . . . 81
3.3.4 AlternativeCost Functions . . . 81
3.3.5 Knowledgeof InitialConditions . . . 81
3.3.6 Conclusion. . . 82
4 Examples and Analysis of Results 83 4.1 The Complete Problemand Algorithm . . . 83
4.2 Introduction of Software . . . 87
4.2.1 SOCS Parameters . . . 88
4.2.2 Choosing aValue of . . . 90
4.3 Introduction of Examples. . . 90
4.4 One-DimensionalState Examples . . . 92
4.4.1 Primary One-DimensionalExample . . . 93
4.4.2 Other One-DimensionalExamples . . . 99
4.5 Two-Dimensional StateExamples . . . 103
4.5.1 Primary Two-DimensionalExample . . . 104
4.5.2 Other Two-Dimensional Examples. . . 108
4.5.3 Common Mode Two-Dimensional Example . . . 115
4.6 IndustrialExample . . . 118
4.7 Multiple Fault ModelExamples . . . 121
4.7.1 One-DimensionalExample . . . 121
5 Future Work and Conclusions 129
5.1 FutureWork . . . 129
5.1.1 The Half-Innite Interval . . . 130
5.1.2 Linear Time Varying Models . . . 131
5.1.3 NonlinearModels . . . 132
5.1.4 Independent NoiseBounds . . . 135
5.1.5 Sensitivity Issues . . . 135
5.2 Conclusions . . . 136
List of References 139 A Software Drivers 145 A.1 ModelReduction . . . 145
A.2 FortranCode Generation . . . 148
A.3 Optimizationvia the FDMI Algorithm . . . 149
A.4 Analysisand Presentation of Results . . . 163
A.4.1 Detection SignalPhase Processing. . . 163
4.1
and kbvk for Example 4.1: t
f
=1;10;20;100 . . . 95
4.2 Formulationcomparison of Example 4.1: t
f
=1;10;20 . . . 98
4.3
and kbvk for Examples 4.1-4.4: t
f
=1;10 . . . 101
4.4 Performance comparison of Example 4.5on various time intervals . . 107
4.5 Performance comparison of Examples 4.2-4.10: t
f
=1 . . . 114
4.6 Performance comparison of Examples 4.2-4.10: t
f
3.1 Outputsets under application of bv and Æbv,Æ >1. . . 73
3.2 Outputsets under full and reduced noise contributions . . . 75
4.1 Typicalvariation of a
(t) with . . . 91
4.2 bv, forExample 4.1: t
f
=1(left),t
f
=10(right) . . . 93
4.3 bv for Example4.1: t
f
=20(left),t
f
=100 (right) . . . 94
4.4 bv for Example4.1: t
f
=20;100 . . . 94
4.5
for Example 4.1asa function of t
f
. . . 95
4.6 Rescaledbv for Example 4.1: t
f
=1;10;20;100 . . . 96
4.7 y
(t) and a
(t) for Example4.1: =0:3;0:5;0:7;0:9 . . . 97
4.8 bv for Examples4.2 (left),4.3(center), 4.4(right): t
f
=1 . . . 101
4.9 bv for Examples4.1-4.4: t
f
=1 . . . 102
4.10 y
(t) and a
(t) for Example4.4: =0:3;0:5;0:7;0:9 . . . 103
4.11 bv for Example4.5: t
f
=1(left) and t
f
=20(right) . . . 105
4.12 bv for Example4.5: t
f
=1;2;4;6;8;10;20 . . . 105
4.13
forExample4.5asafunctionoft
f
(left)andcomparedwith
Exam-ple4.1(right) . . . 107
4.14 y
(t) and a
(t) for Example4.5: =0:3;0:5;0:7;0:9 . . . 108
4.15 bv for Examples4.5-4.10: t
f
=1(left), t
f
=10(right) . . . 112
4.16 bv for Examples4.2,4.3,4.4,4.9: t
f
=10 . . . 113
4.17 Normalizedbv for Examples 4.5-4.10: t
f
=10 . . . 113
4.18 y
(t) and a
(t) for Example4.10: =0:7 . . . 115
4.19 Componentsof bv for Example 4.11: t
f
=5 . . . 117
4.20 Componentsof y
(t) and a
(t) for Example 4.11: =0:7 . . . 117
4.21 Componentsof bv for Example 4.12: t
f
=1 . . . 120
4.22 Componentsof y
(t) and a
(t) for Example 4.12: =0:7 . . . 120
4.23 bv for Example 4.13 sequentialvs. simultaneous (left)and full interval
two-modelvs. simultaneous (right) . . . 123
4.24 y
(t) and a
(t) for Example4.13: sequentialsolve . . . 123
4.25 y
(t) and a
4.27 y
(t) and a
(t) for Example4.14: sequential . . . 126
4.28 y
(t) and a
Introduction and Review of Prior
Research
Models of dynamicalsystems that consistof aset of linear dierential and algebraic
equations (DAEs)
Ez 0
+Fz =f(t) (1.1)
inwhichthe(square)matrixEissingular,are calledlineardescriptorsystems. Many
systems throughouta widevarietyof applicationsare most easilydescribed aslinear
descriptorsystems. Variationalproblemssubjecttoconstraints,suchastheequations
of motion for a robotic arm, can often be written as descriptor systems. Network
modeling problems, as in electrical circuit design, are another example. The list
continues withmodelreductionproblems, singularperturbations,and discretizations
of partial dierential equations, just to name a few. (See [5, 8] for an in-depth
description of applications and examples.) Because of this wide spread applicability,
much research has been done recently involvinglinear DAEs.
A key aspect of system design in linear DAE modeled systems is fault detection
and isolation (FDI). One avenue of FDI is via the multi-model approach, in which
the parametersofoneormorefaultmodels. Thedesigngoal istoobtainanindicator
that tellsthe operator when a fault has occurred, and, when more than one type is
possible, which type of faultit is.
Another aspect of system design is the modeling of noise. One way to model
noiseisasaboundedenergysignal. Thisapproachplacesveryfewrestrictionsonthe
typesofnoisysystems whichcan beaddressed. Italsopresentsnocomplexmodeling
requirement,a very useful computationaltool of which we can take full advantage.
In this thesis we apply the multi-model approach to FDI in linear descriptor
systems, modeling noise as bounded energy signals. The combination appears to
be under-explored, in that very little research seems to exist that uses both the
multi-modelapproach and bounded energy noise. We develop a complete algorithm,
requiring very little on-line computation by an operator, with which nearly perfect
faultdetection and isolationover anite horizon isattained.
1.1 Linear Descriptor Systems
Webeginwithashortintroductiontodescriptorsystems,thebasictheoryandseveral
numericalmethodsused to obtainsolutions.
1.1.1 Basic Theory
As described above, DAEs occur in many applications. Models that consist of a set
of ordinary dierential equations (ODEs) often are rst written as DAEs. A DAE
is manipulated through dierentiation and substitution to convert it to ODE form.
Consider
x 0
in which a, b, c, and d are scalar constants. Equation (1.2) consists of a dierential
equation(1.2a)andanalgebraicconstraint(1.2b). TheJacobianof(1.2)withrespect
tox 0
;y 0
is
2
4 1 0
0 0 3
5
whichis singular. By dierentiating (1.2b) weobtain the full system
x 0
= ax+by (1.3a)
y = cx+d (1.3b)
y 0
= cx 0
: (1.3c)
Wecan substitute (1.3b)into (1.3a) toobtain the ODE
x 0
= (a+bc)x+bd (1.4a)
y 0
= cx 0
(1.4b)
the solution of which is easilyobtained.
Frequently,reasonsexistfornotattemptingtomanipulateasystemlike(1.2)into
explicit (ODE) form. First, physical problems initially modeled as DAEs contain
relationshipsbetween variablesofinterest. Changing toanexplicit modelmay result
in less meaningful variables, as well as a loss of the importance of the relationships
between those variables. In addition, sparsity is usually lost. Numerical methods
that rely on the sparsity of a DAE may not be suitable for solving the ODE that
is obtained by dierentiation and substitution. Finally, it may not be easy or even
possibletoconvertacomplexsystemintoODEform. Whenitispossible,itmightbe
easiertosolvetheDAEdirectlythan todothemathematicalmanipulationnecessary
It is due to these reasons, among others, that the base of research in DAEs has
continuouslygrown overthe lastseveral years. Attheheartof thetheoryare twokey
concepts, solvability and the uniform dierentiation index[5].
Definition 1.1. The system (1.1), where E and F are m m matrices, is
solvable on an interval if for every m-times dierentiable f(t), there is at least one
continuously dierentiablesolution to (1.1). Inaddition, solutions are denedon the
entire interval and are uniquely determined by their value at any t in the interval.
We willreturnto the necessary and suÆcient conditions for solvabilityof certain
typesof DAEs a bitlater.
Definition 1.2. The minimum number of times that all or part of (1.1) must
be dierentiatedwith respect tot in order todetermine z 0
as acontinuousfunction of
z;t is the index, , of DAE (1.1).
Example (1.2) is an index one DAE. Numerical methods are well developed for
index one DAEs. Higher index problems are notoriously more diÆcult to solve via
numerical methods. Fortunately, allof the DAEs we willdeal with in this thesis are
of index one.
Of the several special structural forms for DAEs found in the literature, two will
be of interest inthis research:
Linear Time Invariant DAE
Ez 0
+Fz =f(t) (1.5)
Linear Time Varying DAE
E(t)z 0
+F(t)z =f(t) (1.6)
Wemention three other typesfor completeness:
Linear inthe derivative,nonlinear DAE
Semi-Explicit(nonlinear) DAE
z 0
= f[z(t);u(t);t] (1.8a)
0 = g[z(t);u(t);t] (1.8b)
Fully Implicit(nonlinear) DAE
F(z 0
;z;t) =0 (1.9)
The extension of our algorithmtothese problems willbe leftto future research.
The theory for (1.5) and (1.6) is fairly well understood. For (1.5), solvability is
expressed in terms of a matrix pencil. For square matrices E and F, and complex
parameter , E +F is called a matrix pencil. If its determinant is not identically
zero as a functionof , then the pencil E +F is said to be regular. Equation (1.5)
is solvable if and onlyif E+F isa regularpencil[5]. If(1.5) is solvable wecan let
z =Qw and premultiplyby P so that (1.5) becomes
PEQw 0
+PFQw =Pf(t)=g(t) (1.10)
where P,Q are nonsingular matricessuch that
PEQ= 2
4 I 0
0 N 3
5
; PFQ= 2
4 C 0
0 I 3
5
:
N isanilpotentmatrix the index of whichis the same asthe uniform dierentiation
index of DAE (1.5). The system is then decoupled, and can be writtenas
w 0
1 +Cw
1 =g
1
(t) (1.11a)
Nw 0
2 +w
2 =g
2
(t): (1.11b)
Equation(1.11a)isanODEforwhichasolutionexists forany initialvalue ofw
1 and
any continuous forcing function g
1
(t). The unique solution to(1.11b) is
w
2
=(ND+I) 1
g
2 (t)=
1
X
( 1) i
N i
g (i)
where is the index, or degree of nilpotency of N, and D is the dierentiation
operator. Notethat the initialvalues of w
2
are completelydetermined.
In the linear time-varying case, (1.6), a similar result holds. Whilethe nature of
the matrix pencil E(t)+F(t) is no longer an indicator of solvability, the form of
(1.11) isstill important inlinear time-varying DAEs.
Definition 1.3. The system (1.6) is in standard canonical form if it is in the
form
2
4
I 0
0 N(t) 3
5
z 0
+ 2
4
C(t) 0
0 I
3
5
z =f(t) (1.12)
where N is strictly lower(or upper) triangular.
If E(t), F(t) are real analytic, then (1.6) is solvable if and only if, after linear
time-varying coordinate changes, it can be written as (1.12). The problemexists in
the diÆculty of nding those coordinate changes that allow us to write the DAE in
standard canonical form.
For the other three cases mentioned above, (1.7)-(1.9), the theory is much newer
and alsomuchless understood. Forthe purpose of this thesis, it issuÆcient to note
that the concepts presented aboveserve as abasis for the development of the theory
forthesecases. Itshouldbenotedthatwhilethis newertheoryisbeyondthe scopeof
this thesis, the commonstarting point serves asa goodindicatorthat the algorithm
developed herein for linear time-invariant and linear time-varying DAEs may have
applications inthe more general case aswell.
While there are many more interesting and useful items in the theory of DAEs,
these few properties and denitions that we have mentioned willsuÆce for our
1.1.2 Numerical Solutions
While it is not our goal to present an exhaustive overview of the numerical
meth-ods that can be used to solve descriptor systems, we briey mention those methods
which will be used in later chapters. The discretization methods we will review are
those that are used by the commercial software in which we implement the FDI
al-gorithm developed in this thesis, namely the trapezoidal method, the Compressed
Hermite-Simpsonmethod, and the 4-stage implicit Runge-Kutta method. These
di-rect transcription discretizationswillbedescribed using the semi-explicitDAE form
(1.8). After discretization, several methods exist for solving the resulting nite
di-mensionalproblem. Ofthosemethods,onlythesparse quadraticprogram(SQP)will
be described here. While the software, Boeing's Sparse Optimal Control Software
(SOCS), which will be introduced in a later chapter, can solve DAEs via an
ana-lytic transformation, as well as Euler's and linear multistepmethods, these schemes
will not be used in this thesis, and thus will not be mentioned here. Many of the
approaches applied toDAEs are described in detailin [5,13].
For our discussion of discretizationand nite dimensional problemsolution,
con-sider a simpleoptimization problembased onthe semi-explicitDAE (1.8)
min
t
0 tt
f
J[x(t);u(t);t] (1.13a)
subject to
x 0
= f[x(t);u(t);t] (1.13b)
Discretization
In general, transcription discretization schemes start by dividing the time interval,
[t
0 ;t
f
],inton segments
t 0 <t 1 <t 2
<:::<t
n =t
f
where the points t
k
, k = 0;:::n, are referred to as mesh points. Let x
k
= x(t
k ) be
the value of astate variable ata mesh point. Likewise, denote the value of a control
at a mesh point as u
k
= u(t
k
). Let f
k = f(x k ;u k ;t k
) and g
k = g(x k ;u k ;t k
) be the
right-handsidesof(1.13b)and (1.13c),respectively. Finally,leth
k =t k t k 1 bethe
step size for k =1;:::;n.
Utilizing this notation, the trapezoidalmethodapproximates the state equations
(1.13b)and algebraicconstraints(1.13c) as
x k = x k 1 + h k 2 (f k +f k 1 ) (1.14a)
0 = g
k
: (1.14b)
In the Compressed Hermite-Simpson scheme we denote the value of the control at
the midpointof asegmentasu
k =u(t
k
) wheret
k = 1 2 (t k +t k 1
),fork =1;:::;n. The
discreteapproximationsfor this methodare given by
x k = x k 1 + h k 6 (f k +4f k +f k 1 ) (1.15a)
0 = g
k
(1.15b)
where
f
k
=f(x
k ;u;t
for k=1;:::;n. The 4-stageimplicit Runge-Kuttadiscretizationuses four
intermedi-ate, implicitsteps
c 1 = h k f(x k 1 ;u k 1 ;t k 1 ) (1.16a) c 2 = h k f(x k 1 + c 1 2 ;u k ;t k ) (1.16b) c 3 = h k f(x k 1 + c 2 2 ;u k ;t k ) (1.16c) c 4 = h k f(x k 1 +c 3 ;u k ;t k ): (1.16d)
The discrete approximations forthis method then become
x k = x k 1 + 1 6 (c 1 +2c 2 +2c 3 +c 4 ) (1.17a)
0 = g
k
(1.17b)
where u
k
is dened as before.
These methods have all been proven to converge for index one DAEs, and are
thusappropriateforour purpose [4,5]. Inevery case,the resultof the discretization,
when combined with the cost function, J, is a sparse nonlinear programming(NLP)
problem. Thevariablesoftheproblemarethe discretizedstates,x
k
,controls,u
k ,and
time, t
k
, fork =0;:::n.
Solving the Finite Dimensional Problem
One way to solve this NLP problem,and the approach used by SOCS, is via a SQP
approach [3]. Dropping subscripts for now, let w be the vector of state and control
variables,(x;u),andletF(w;t)betheconstraintset resultingfromthediscretization
of the DAE. That is, (1.14), (1.15), or (1.17), after shifting everything to the right
hand side, becomes
Note that F is a function of the state, control, and time variables atall time steps.
The SQP algorithmrequires an initialguess, w 0
, and forms a new iterate by adding
a scalarmultiple, , of the search direction, p. That is,
w 1
=w 0
+ p:
The search directionis found by solving aquadratic programming(QP) subproblem
dened at the current point. The QPsubproblem is dened as
minJ T
w p+
1
2 p
T
Hp
subject to
0=Gp
where J
w
is the gradient vector of the cost function, H is an approximation to the
Hessian matrix ofthe Lagrangian of the NLP (L=J T
F), and Gisthe Jacobian
matrix of gradients of the constraints F. The step length, , is computed such that
H remains positive denite. The QP subproblem can be solved via either a sparse
Schur-Complement method, when appropriate, or a null-space quadratic
program-ming algorithm when G and/or H are dense. Details about the latter can be found
in [3].
An algorithmbased onthe combinationof a directtranscription scheme and the
SQP approachbegins withadiscretizationand aninitialguess. The SQPproblemis
then solved via the QP subproblem iteration. After each QP subproblem is solved,
the current point is updated and the procedure is repeated. The subproblem
itera-tionterminateswhenapointisreachedwhichsatisesnecessary conditionsforalocal
minimum within a given set of tolerances. The solutionis then compared to that of
the previous discretizationiteration, orthe initialguess if it isthe rst iterate. The
succes-demonstratesquadraticconvergenceundertherightconditions[3]. Convergencerates
for the direct transcription schemes, when applied to index one descriptor systems,
are at least quadratic, and, under the right system coeÆcient conditions, often are
considerably better [5].
1.2 Fault Detection and Isolation
With this basic understanding of the theory of descriptor systems and numerical
methodsfortheirtreatment,wenowturnour attention tothevariousapproaches for
treating faults in those systems. We begin with basic controltheory, and then turn
to feedback, the linkbetween control theory and FDI. Following that is a discussion
of the elements of optimal control and H
1
control pertinent to our approach. We
conclude with a discussion of existing research into FDI in descriptor systems and
the methodsused.
1.2.1 Basic Theory
A descriptor system is one possible result of a system design problem. The problem
begins witha tasktobe accomplished,andthe design engineeris usuallygiven goals
orobjectivesthatdescribethedesiredperformancecharacteristicsofthesystemalong
withaset ofconstraintsby whichthe systemisbound. Thedevelopment ofasystem
whichaccomplishesthe objectiveswhilemeetingthe constraintsisthe system design
problem.
A particular type of system design problem is the control problem, in which the
goal is to generate certain outputs from the system or to maintain the state of the
system within certain bounds. For example, an engineer might be asked to design
essential elements of sucha controlproblemare
a mathematicalmodelof the system,
a desiredoutput,
a set of admissiblecontrols,
a performance measure.
Often, as stated above, a descriptor system is the natural product of the system
design problem. For the remainder of this thesis, we will restrict most of our study
to linear time-invariant systems (1.5). Comments extending our algorithmto linear
time-varyingsystems (1.6) are included in alater chapter.
Consider asystem based on the linear time invariantDAE (1.5)
x 0
= Ax+Bu (1.18a)
y = Cx (1.18b)
wherex,y,anduarethestate,output,andcontrolvectors,respectively,andthetime
intervalconsidered ist2[t
0 ;t
f
]. Systemsoften allowfornoise orunknown inputsby
adding aterm to each equation of (1.18)
x 0
= Ax+Bu+M (1.19a)
y = Cx+N (1.19b)
where is the noise or unknown input, and the matrices M and N are the weight
matricesfor the state and output noise, respectively.
Central to the study of system (1.18) are the concepts of controllability,
observ-ability,and stability [6].
Definition 1.4. Alinear system issaidtobe controllableat t
0
if itispossibleto
nd some input function u(t), dened over t 2[t
0 ;t
f
], which will transfer the initial
state x(t
0
) to the origin at some nite time t
1 2[t
0 ;t
f ]; t
1 >t
0
. If this istrue for all
Definition 1.5. A linear system is said to be observable at t
0 if x(t
0
) can be
determined from the output function y
[t
0 ;t
1 ]
for t
0 2 [t
0 ;t
f
] and t
0 t
1
, where t
1
is some nite time, t
1 2 [t
0 ;t
f
]. If this is true for all t
0
and x(t
0
), the system is
completelyobservable.
Since controllabilitydescribesthe abilityof the controltoaect the system state,
itinvolvesthematricesAandB. Likewise,sinceobservabilitydescribestheabilityof
theoutputtocharacterizethestate, itinvolvesthe matricesAand C. Simplystated,
the nth-order system (1.18) is controllable if and onlyif [sI A j B] has rankn for
allvalues of s. The same system isobservable if and only if [sI A T
jC T
] has rank
n for all values of s. Proofs of these characteristics can be found in [6], along with
the requirements forcontrollabilityand observabilityin more complex systems.
The concept of stability helps us deal with systems that may not be controllable
and/or observable. Stability is closely related to the eigenvalues of the A matrix.
Intuitively, a solutionto (1.18) isstable if we can stay close to the solutionby
start-ingclose enoughto it viathe initialcondition. A solution isasymptotically stable if,
by starting close enough, we converge to the solution. A system is stabilizable if all
unstable modes are controllable, and detectableif allunstable modes are observable.
Thusthesystemcan behandledeectivelyprovided alluncontrollableand
unobserv-able modes are stable. This situation can often be tolerated in a control system [6].
For the remainder of this thesis, we will assume that we are dealing only with the
controllableand/or observable modes of controlsystems.
1.2.2 Feedback and Observer Design
Thebridgebetweenbasiccontroltheoryandfaultdetectionistheconceptoffeedback.
In a feedback controlsystem, the control, u(t), is modiedby informationabout the
information to the controller, which adjusts the control based on the input from
the sensors. One of the fundamental goals of feedback compensator design is to
improve the performance of the system through eigenvalue placement. As stated
earlier, stability depends on the eigenvalues of the A matrix. By assigning desirable
values to eigenvalues, system stability can be enhanced. Forthe state feedback case,
the relation
u(t)=Fv(t) Kx(t) (1.20)
isused,wherethematrixKiscalledthefeedbackgainmatrix,andF thefeed-forward
matrix. Substituting into(1.18), we obtain
x 0
= (A BK)x+BFv (1.21a)
y = Cx: (1.21b)
Clearly, the eigenvalues of the A BK matrix now determine the stability of the
system. By careful construction of the feedback gain matrix K, the eigenvalues are
assignedthe desiredvalues. For the output feedback case, the relation
u(t)=Fv(t) Ky(t) (1.22)
isused, where the K and F matricesare asdened above. Substitutingthis relation
into (1.18),we obtain
x 0
=(A BKC)x+BFv: (1.23)
Now the eigenvalues of the A BKC matrix determine the stability of the system.
Unfortunately,duetothepresenceoftheCmatrixinthisexpression,outputfeedback
usually cannot place all of the eigenvalues of the system. This limitationis present
havenoimpactoncontrollability. Outputfeedback canimpactneithercontrollability
nor observability of a system [6].
Using feedback, the basic tool for many FDI approaches can be constructed: the
observer. For most systems the only information about the system state is through
the outputvector, which oftenprovidesonly partialinformation. Thus, output
feed-back is the only option, and not all system eigenvalues can be placed wheredesired.
To improve system stability in these cases, the most frequently used method is to
reconstruct information about the remaining elements of the state vector through
development of anobserver of the system. Consider the observer
^ x 0
=A^x+Bu+L(y Cx )^ (1.24)
where x^is the observer estimatefor the state vector. Note thaty is the outputfrom
the real system, (1.21), and Cx^ is the observer output. Subtracting (1.18a), and
letting e=x^ x be the observer error,we obtain
e 0
=(A LC)e:
Since L is arbitrary and (A;C) is observable, we can guarantee that observer error
goes to zero by selecting L so that A LC is stable. With this construction, state
feedback is possibleusing the observer estimateforthe state vector. Thus allsystem
eigenvaluescanbeplacedwheredesired,andcompletecontroloversystemstabilityis
possible. It should be noted that since the complete state vector is reconstructed by
theobserver, faultswhichsendthesystem intounpredictedorundesirablestatesmay
be detectable by such an observer simply by comparing the observer estimate with
those elements of the system state vector which are available. This fault detection
can be accomplishedwithout using the observer to aect any feedback compensator
1.2.3 Optimal Control
Later,whenwedevelopour algorithm,wewillworkwithacontrolsystemwhichacts
astheconstraintsinanoptimizationproblem. Thisoptimalcontrolstructureiskeyto
themulti-modelapproachtoFDI,whichwewilldiscussinSection1.2.5. Accordingly,
we briey review optimal control theory. While this area of study is vast, the only
topic whichwe willneedfor our discussion isthe state regulatorproblem,alsocalled
the linear quadratic regulator (LQR) problem. Consider the optimization problem
J(x;u)=min 1
2 x(t
f )
T
S
f x(t
f )+
1
2 Z
t
f
t
0 x
T
Qx+u T
R udt (1.25a)
subject to
x 0
=Ax+Bu (1.25b)
as well as some initial conditions at the beginningof the interval, where S
f
, Q, and
R are the weight matricesfor the terminalcost, the trajectory,and the control. It is
assumed that Q is positive semi-denite and R is positive denite. This is one form
ofthe LQRproblemanditisimportantforthreereasons. First,the theoryiselegant
and robust. Results are easyto understand and implement in numericalalgorithms.
Second, ithas stronggeometry. J(x;u)isactuallyaninnerproductnormwith useful
properties. Finally, there are strong physical correlations to this type of problem.
Energy is a quadraticform,as is power.
As with any optimization problem, the LQR problempossesses necessary
condi-tions for aminimum. Forthe problem (1.25), weconstruct the Hamiltonian
H(x;;u)= 1
2 (x
T
Qx+u T
R u)+ T
(Ax+Bu) (1.26)
must besatised by any extremum of the problem,are
H T
= x 0
(1.27a)
H T
x
=
0
(1.27b)
H T
u
= 0: (1.27c)
When applied to(1.26), we obtain
0
= Qx A
T
(1.28a)
x 0
= Ax+Bu (1.28b)
0 = R u+B T
: (1.28c)
Using (1.28c) and our assumption that R > 0, we can eliminate u from (1.28) to
obtain aset of dierentialequations in xand
x 0
= Ax BR
1
B T
(1.29a)
0
= Qx A
T
: (1.29b)
Whilethisformwillbeusefulinouralgorithm,itispossibletotakeanadditionalstep
and eliminate , resulting in a matrix Riccati dierential equation for the optimal
controlfeedback gain matrix. The derivation of the Riccatiequation willbe detailed
when we develop our algorithm in the next chapter. It should be noted that our
assumptions on the Q and R matrix, while not restrictive in an applicability sense,
guarantee that the extremum which satises the necessary conditions represents at
least a local minimum of the cost J(x;u). In fact, Q is often positive denite, and
in that case, the conditions for an extremum are necessary and suÆcient. Detailed
1.2.4 H
1
Control
H
1
control in the time domain is similar to optimal control. It takes advantage of
the linear quadratic (LQ) forminaddressing signicant uncertainties inthe energies
ofsystemnoises. Forboundedenergynoiseinputs,wherelittleornootherknowledge
isavailableaboutthe signal,the LQRformulationisanelegantworst-case approach.
Themodelgenerallytakestheformof(1.19),andallfunctionsareassumedtoexistin
the spaceof squareintegrablefunctions, denoted L 2
. WhileH
1
performancecriteria
vary, they all share the structure of the optimal control cost function, that is, they
are all inLQ form.
In this setting, ltering, smoothing, and compensator design are eÆciently
ac-complished. Nagpal and Khargonekar [27] apply a ltering and smoothing method
using an H
1
performance criterion on both the nite and half-innite intervals to
accomplish state estimation(ltering) and output smoothing. Tadmor [36] attempts
tond,inLQgame-theoreticterms,thecompensatorwhichprovidesthebest control
in response to the worst disturbance. Matrix Riccati equations provide solutions in
eachcase.
While the structure of our problem is very similar to the H
1
problem, several
key dierences willbecomeapparent. First,wewill solve adierentproblem. While
Tadmor [36] designs a worst-case compensator, and Nagpal and Khargonekar [27]
solve for the optimal lter and smoother inthe face of various initialconditions, we
will solve for the optimal fault detection signal. In addition, while both [27] and
[36] work in single model systems, we will work in a multi-model system. Finally,
while the noise present in our system is also L 2
, it is not the same kind of signal as
is commonly assumed in H
1
control. The impact of these dierences will become
1.2.5 Prior Research
In addition to the two approaches mentioned above, fault detection and isolationin
linearcontrolsystemshas been attemptedfrommanyangles. Tobegin, wenote that
there exist two basic types of approaches to FDI: passive and active. In the passive
approach,onlymonitoringofsystem performance isallowed. Nointeractionwith the
systemoccurs,eitherformaterialorsecurity reasons. Thesystem states(oroutputs)
aremeasuredand comparedto\normal"system behavior,generatingaresidual. The
residualiscomputed suchthat itis equalorcloseto zerowhen nofaultsare present,
and much dierent from zero when a fault occurs. The vast majority of research in
FDI using the passive approach appliesobservers to generate residuals.
Passive Methods
Nuninger et al. [32] use analytic redundancy in order to detect sensor and actuator
failuresorprocessdisturbances. Analyticredundancyattemptstogeneratearesidual
that might contain information about the faults. Two methods for generating the
residual are examined. First, direct residual generation is based onthe parity space
approach, using the input-output transfer function (the parity equation). Second,
indirect residual generation is based on output estimation, using an observer to do
state estimation rst. The authors apply the rst method to known input systems
and the second method to both known and unknown input systems. A drawback of
thisapproachisthatsomefaultsmayhavenoinuence onthe residualsgeneratedby
eithermethod,soperfect FDIisnot attained. ChenandSpeyer[15]alsouse analytic
redundancy, generating a residualvia anobserver that reconstructs the system state
vector. Their model has the target fault direction explicitly in the detection lter
Koenig et al.[24] present a comparative study of several design methodsfor
un-known input observers (UIO) used for FDI and Correction. Their goal is to design
anintegrated approachwhichcan detect, isolate,and correctalargevariety of faults
fora desiredsystemwith real-timecomputationconstraints. Methodscompared are:
failure isolation by using banks of observers (robust to some faults, but sensitive to
others, incombinationso that allfaults are detectable), failureisolationby observer
pole placement (to create an unknown input fault detection observer), and failure
correctionviageneralstructuredUIO(designoffullorderobservertoestimatestates
as well asunknown inputs). Chung and Speyer [17] develop a game theoretic
detec-tion lter, which is similar to the UIO, that attenuates disturbances, bounding all
signals except the faultto be detected, embeddingthe exogenous signals intoan
un-observable, invariantsubspace. The subspacestructure isused toreducethe orderof
the limitinglterby factoringtheinvariantsubspace out ofthe statespace, resulting
in alower order lter sensitive only to the fault tobe detected. The lter isapplied
tothe ight controlcharacteristics of the F-16XL and a simplerocket.
The parity relation approach to residual genertion is applied by Youssouf and
Kinnaert [40]. The method is based on the inverse of the map from both unknown
inputs and faults to the observable signals (measured inputs and outputs), using a
variablechangeinthe frequency domain. Toolsavailablefornonsingularsystems can
be used on the resulting map. The authors contend that FDI for singular systems
depended previously on state estimation, which put unnecessary requirements on
the plant, as there is no need to reconstruct the entire state vector to do residual
generation. WhereYoussoufandKinnaert[40]applytheirmethodtocontinuoustime
systems, Sauter et al. [35] do the same for the discrete case in mechanical systems,
though the theory and algorithm are completely dierent. They do state equation
of fromthe residual.
Chowdhury and Aravena [16] go in a slightly dierent direction. They apply a
modularmethodologytotheareaoffastfaultdetectionandclassicationindynamical
electricalpowersystems. Module I isthe generation of faultindicators in one of two
ways:
1. model-based, in which a residual is generated using either an accurate
mathe-matical orI/O model of the nominal system, or an I/O modelis built on-line,
which is very diÆcult,
2. model-free, in which detectable variables are measured and enhanced if
neces-sary by signal-processingtechniques.
The authorspresent amodel-free orthogonaldecomposition basedon multiratelter
bankstoproduceafaultindicator. ModuleIIisthemeasuringandtestingoffault
in-dicatorsviaeitherstatisticaltest orfeedforward neural-networktesting. Theauthors
explore the neural network aspect. Fault classication occurs in module III, another
neural network, the operation of whichdepends onthe existenceof asystem model.
The emphasis is on model-free methods, those lesser explored and lesser restrictive
cases wheremodelsare notavailable,non-linearitiesprevent modelderivation,ortoo
manyuncertainties existinthesystem. Thesecases appear toholdthe mostpromise
for neural-network applications in faultdetection.
Hybrid Passive-Active Methods
Some research has been done using a hybrid of the passive/active approaches. The
passive approach is used to detect faults, then an active approach is used to
cor-rect or compensate for faults through feedback. Zhang and Jiang [43] investigate
to discrete-time stochastic vertical take-o and landing aircraft systems. A bank of
two-stage adaptiveKalmanlters is used forFDI, and statistical decisionsare made
forfaultdetection,diagnosisandactivationofcontrollerreconguration. Inarelated
paper[42], the same authors apply aninteracting multiple-modelbased approachto
the same type of control problem. A nite-state Markov chain is linked tothe same
Kalman lter bank for fault diagnosis. The decision from this process is used to
activate system reconguration via eigenstructure assignment.
Active Methods
The drawback inherent in the passive approach is that faults can be masked by the
applicationofthe control. Thus itispossiblethatafaultcouldgoundiscovereduntil
it is too late tocorrect it. In direct contrast, the active approach interacts with the
system on aperiodic basis, orat criticaltimes, todetect faults,thus eliminatingthe
possibility of the presence of undetected faults. The approach uses various types of
interaction with the system to detect faults. A test signal, which is constructed in
such a way that faults are highlighted, is fed into the system. Observation and/or
manipulationof the resultingoutput isused to makea decisionabout system faults.
Observers designed to aid in feedback, as well as various other types of feedback
compensators, are examples of the active approach.
Bennett et al. [2] apply speed dependent feedback (a stable time-varying linear
observer) todetectintermittent, shortdurationfaultsin bilineardynamicalsystems.
The AC drive system for an electric train is considered. These systems experience
abrupt disconnections which introduce severe transient errors and which are hard to
detectdue totheir shortdurations. The parityequation approachisnot preferredin
this casedue totheintermittentnatureof the faults. Bycombiningthe observerand
This case isan example of the applicationof atest signal as part of the feedback to
controlthe system and correctfaults.
The multi-modelapproachis well-suitedto thecase whereitis desirableto apply
atest signal independentof feedback control. The approachrelies onthe presence of
the system model
x 0
i
= A
i x
i +B
i v+M
i
i
(1.30a)
y = C
i x
i +N
i
i
(1.30b)
for i = 0;:::;m, where m is the number of faults expected from the system, and
v is the test signal. A dierent system model exists, with known parameters, for
each possible fault. It is assumed that any feedback control has been absorbed into
the A
i
matrix, thus eliminating the control u from the dierential equation. The
diÆculties inthisapproachlieindeterminingfromwhichmodelanoutputyderives,
as well as the computation of system parameters for each fault model. Nikoukhah
[28] presents the use of a test signal for active FDI in discrete-time linear systems
subject to inequality-bounded perturbations. Detectability is required, but when
present, guaranteed FDI is attained. The discrete time case lends itself to recursive
algorithms, and so recursion is used extensively by the author to develop the test
signal. After constructing a test signal that separates outputs into disjoint convex
sets, the author uses the separating hyperplane approach to determine which set a
certainoutputfallsinto,and thuswhetherafaulthasoccurred. Linear programming
is used to construct the separating hyperplane. Nikoukhah et al. [29] has the same
goal as [28], but goes about it completely dierently. Among the dierences, fault
isolationis accomplishedby aratio test, and optimal controltheory isapplied. This
paper is the inspiration for our current research, and thus uses some of the same
and algorithmic dierences. In addition, both [28] and [29] consider only two model
systems, whereas our approachcan handle problems with three or moremodels.
The multi-modelapproach is also useful with Kalmanltering. Keller et al. [21]
presents the multi-modelapproachfor faultdetection in stochastic systems with
un-known inputs. The method uses the two-stage Kalman lter with unknown inputs
and constant biases, the rst stage of which is bias-free (for faultdetection) and the
second stage is a bias lter (for fault isolation). The optimum state estimate is
ex-pressedastheoutputofthebias-freeltercorrectedwiththeoutputofthebiaslter.
Dierent fault types are detected using a bank of such lters. The two stages of the
lter reduce computationaltime associated with the presence of multiple faults.
1.2.6 Conclusion
As mentioned in the introduction to this chapter, the combination of the
multi-model approach and the bounded energy noise model seems to be under-explored.
The commonthread runningthrough most of the applications mentioned inthe last
section is the modeling of noise. [16, 21, 22, 23, 33, 37, 41, 42, 43] model noise as
sometypeofrandomvariable. Manyuse lteringorstatisticalteststomakethefault
decision, and thus do not model noise at all. Only [17, 27, 28, 29, 36] model noise
as bounded energy signals. As we shall see, the bounded energy noise modelis very
suited tothe multi-modelapproach, and the combinationas developed in this thesis
providesa powerfultool for faultdetection and isolation indescriptor systems.
1.3 Outline of Thesis
In the next chapter, we present the theory and algorithm for the fault detection
problem. Following that, Chapter 3 is the development of the algorithm for the
model identication phase. In Chapter 4, we state the full algorithm, then present
andanalyzeseveralexamples. Lastly,Chapter5istheconclusionandoutlineoffuture
research possibilities in this area. Software codes for the algorithm are in Appendix
A.
1.4 Contributions of Thesis
The research in this thesis will appear, or has already appeared in the following
publications:
S. L. Campbell, K. Horton, R. Nikoukhah, and F. Delebecque, Rapid
Model Selection and the Separability Index, in Proc. 4th IFAC
Sympo-sium on Fault Detection, Supervision and Safety for Technical Processes
(SAFEPROCESS 2000), Budapest, Hungary, June 2000, pp. 1187-1192.
R. Nikoukhah, F. Delebecque, S. L. Campbell, and K. Horton,
Multi-model Identication and the Separability Index, in Proc. 14th
Interna-tional Symposiumof the Mathematical Theory of Networks and Systems
2000, Perpignan, France, June 2000, CDROM.
R.Nikoukhah,S.L.Campbell,KirkHorton,andF.Delebecque,Auxiliary
signal design for robust multi-model identication,IEEE Transactions on
AutomaticControl, accepted subject to nal revision.
S. L. Campbell, Kirk Horton, R. Nikoukhah, and F. Delebecque,
Auxil-iary signal design for rapid multi-modelidentication using optimization,
Fault Detection via the Detection Signal
2.1 The Problem - Finding the Minimum Energy
Detection Signal
Asintroducedinthepreviouschapter,ourgoalistoattainnear-perfectfaultdetection
andmodelidenticationinlineardescriptorsystemsusingthemulti-modelapproach.
This approachallows the treatmentof the problemintwo steps. Inthis chapter, our
focus willbe onthe fault detection step of the problem, while the next chapter will
tackle the modelidenticationstep.
Multi-model fault detection and model identicationmeans that we have two or
morepossiblemodelsforasystem,andwedecidewhichonecorrespondstothesystem
based on measurements of the inputs and outputs of the system over a nite test
period, [0;t
f
]. Whileother possible test periods exist,we will restrictour discussion
tothe nite interval.
In order to exclude all but one model based on input-output measurements, the
input signal must have special properties. A signal with such properties is called a
proper detection signal. For the remainder of the present discussion we will assume
the fault model. This assumption is not restrictive in any way, and later we will
describe how the algorithmcan be extended to include the case inwhich more than
one fault modelispresent.
2.1.1 Problem Setup
The true model of the system isone of two models
x 0
i
= A
i x
i +B
i v+M
i
i
(2.1a)
y = C
i x
i +N
i
i
(2.1b)
for i = 0and 1, and for t 0, where x
i
, y, v, and
i
are the system states, output,
detection signal, and noise, respectively. The matrices A
i , B
i , C
i , M
i
, and N
i are
matrices of appropriate dimensions. We assume that v and
i
are in L 2
[0;t
f ]= L
2
,
forcingx
i
andytobeinL 2
aswell. WhileweassumefullrowrankoftheM
i
andN
i ,
and controllability/observabilityof the systemfor computationalreasons, thereisno
assumptionthatthedimensionsofthestateornoisevectorsofthe twomodelsare the
same. We alsoassumeno apriori informationabout the system beforet=0,and in
particularnoinformationaboutx
i
(0). Thus,unlikesomeexistingtheory,inparticular
[30], we have no weights on x
i
(0). (We willdiscuss the impact of information about
initialconditionsandthe subsequentpresenceofweightmatricesonx
i
(0)laterinthis
chapter.) In addition, we assume that any feedback control has been absorbed into
theA
i
matricesasdescribed inChapter1,orelseisnulledatt=0forthe durationof
the test period. Thus, the only commonelements of the two models are the output,
y, and the detection signal, v. Notethat (2.1) is alinear descriptor system since the
output y isknown.
Consider the detection signalv and letA 0
(v)bethe set ofpossibleoutputs
letA 1
(v)betheset ofoutputsifModel1,the faultmodel,isthecorrectmodel. Then
perfect modelidentication based onoutputmeasurementimplies that
A 0
(v)\A 1
(v)=;: (2.2)
This is achievable thanks to the bounded energy noise model. This noise model can
be expressed as
S
i (
i
)k
i k
2
= Z
t
f
0 j
i (t)j
2
dt<1; i=0;1 (2.3)
wherejjisthe(pointwise)Euclideannorm,andthuskkistheL 2
norm. In practice
one has bounds k
i k
2
<K. It is always possible to rescalethe M
i , N
i
toget K =1,
so weassume without lossof generality that K =1.
This expression for thenoise allows ustodistinguish between the two basic types
of detection signals.
Definition 2.1. Thedetection signalv isnotproperifthere existx
0 ,x
1 ,
0 ,
1 ,
and y satisfying (2.1) and (2.3). The detection signal v iscalled properotherwise.
Thus we say that the L 2
vector function v is a proper detection signal if its
application implies that we are always able to distinguish the two candidate models
based on observation y. That is, condition (2.2) is satised [30]. Notethat v =0 is
not proper since the zero solutionis always in the intersection of (2.2). In addition,
if v isproperthen cv isalsoproperfor c1,but if v is not proper then thereexists
an >0 such that cv isalso not properfor 0c1+. These factswill be useful
when we develop the optimization problemlater inthe chapter.
The conditions for the existence of proper detection signals are quite weak. For
their characterization, let
L
i (f)=
Z
t
0 e
A
i (t s)
be the solution of z 0
=A
i
z+f, z(0)=0. Then the solutionsto (2.1) are
x i = L i (B i v)+L
i (M
i
i )+e
A i t i (2.5a)
y = C
i L
i (B
i
v)+C
i L i (M i i )+C
i e A i t i +N i i (2.5b)
for i=0;1,where
i
isthe freeinitialcondition for x
i
. Thus the output set for each
model is the sum of three terms
y i = C i L i (B i
v) which is a vector depending linearly on the detection
signal, v, f(C i L i M i +N i ) i :k i
k<1g which is anopen convex set,
fC i e A i t i : i 2< n (orC n
)g which is anite dimensional subspace of L 2
.
Because of these facts, and noting that y
0
and y
1
are respectively the outputs of
Model 0 and Model 1 corresponding to zero noise and zero initialstate, we see that
the output sets A 0
(v) and A 1
(v) are translates by y
0
and y
1
of bounded open sets.
Since y
0 and y
1
depend linearly on v,either y
0 =y
1
forall v,or y
0 y
1
can be made
arbitrarily large with proper choice of v. So proper detection signals exist provided
the linear mapping of v to y
0
is distinct from the linear mapping of v to y
1
[30]. In
the time invariant case, this is equivalentto
C 0 (sI A 0 ) 1 B 0 C 1 (sI A 1 ) 1 B 1
6=0 (2.6)
for some s.
The amountof energy requiredfor adetection signaltobe properdetermines the
separability of the output sets A 0
(v) and A 1
(v).
Definition 2.2. Let V denote the set of proper detection signals v. Then,
= inf v2V kvk 2 1 2 (2.7)
Thus, (
) 2
isalowerbound onthe energyof properdetectionsignals. Also,the
inverse relationship between the separability index and the proper detection signal
energyindicatesthatsystemswithlowerenergyproperdetectionsignalshaveahigher
separability index. The separability index is zero if there are no proper detection
signals. Later, the algorithm we develop will compute
as the objective function
of an embedded optimal control problem. In Section 2.4.5 we describe an existing
algorithm that computes
[30]. Our approach has the advantage of being able to
address several problems that the algorithmin[30] cannot handle.
2.1.2 Formulation as an Optimal Control Problem
Before we describe the algorithm, however, the problem of nding the minimum
energy proper detection signal must be formulated as an optimal control problem.
First, note that for the detection signal v to be not proper, (2.1) must hold as well
as (2.3). We can rewrite(2.3) as
max Z
t
f
0 j
0 (t)j
2
dt; Z
t
f
0 j
1 (t)j
2
dt
<1: (2.8)
This expression can alsobe writtenas
max
01 Z
t
f
0 j
0 (t)j
2
+(1 )j
1 (t)j
2
dt <1: (2.9)
Thusweobtain a useful characterization of not proper detection signals[30]
Lemma 2.1. The detection signal v is not proper if and only if
inf max
01 Z
t
f
0 j
0 (t)j
2
+(1 )j
1 (t)j
2
dt<1 (2.10)
where the inmum is taken over (x
i ;
i
;y) in L 2
, subject to (2.1), i=0;1.
This characterizationisuseful becausethe algorithmwedevelop willcompute the
The next step in formulating the computation of the separability index as an
optimal control problem involves dimension reduction. By assumption the N
i are
bothfullrowrank. Thus, wecanperformaconstantorthogonalchangeofcoordinates
onthe N
i
(viaa QRdecomposition onN T
i
). As aresult we obtain
N i =[N i 0] (2.11) where N i
isinvertible, and
M i =[M i f M i ]: (2.12) Let i = 0 @ i e i 1 A
with the same decomposition as N
i
, and subtract (2.1b) for i = 1
from(2.1b) for i=0. Equation (2.1b) becomes
0=C
0 x 0 C 1 x 1 +N 0 0 N 1 1 : (2.13)
Nowwe cansolve foreither
i
anduse theresultingexpression toeliminate(2.13)by
substituting itinto(2.1a). Solving for
0
, we obtain
0 @ x 0 0 x 0 1 1 A = 2 4 A 0 M 0 N 1 0 C 0 M 0 N 1 0 C 1 0 A 1 3 5 0 @ x 0 x 1 1 A + 2 4 f M 0 M 0 N 1 0 N 1 0 0 M 1 f M 1 3 5 0 B B B @ e 0 1 e 1 1 C C C A + 2 4 B 0 B 1 3 5 v: (2.14)
Withtheobviouscorrespondences,thereducedsystem,nolongeradescriptorsystem,
can be written as
x 0
=Ax+Bv+M: (2.15)
and we desire to detect the fault in a short test period to prevent the instability of
the fault fromcreating problems forthe system.
The characterization of not proper, (2.10), forthe reduced system becomes
inf max
01
P(x;;) <1 (2.16)
where
P(x;;)= Z t f 0 (j N 1 0 C 0 x 0 +N 1 0 C 1 x 1 +N 1 0 N 1 1 j 2 +je 0 j 2 )+ (1 )(j 1 j 2 +je 1 j 2
)dt (2.17)
and the inmum is nowtaken over (x;) inL 2
, subject to(2.15).
The thirdstepinthetransformationtoanoptimalcontrolprobleminvolvesusing
the denition of the Euclidean norm to expand the integrand. After doing so and
combininglike terms,we can rewrite (2.17) as
P(x;;)= 1 2 Z t f 0 x T
Qx+x T
H+
T
R dt (2.18)
where
Q=2 2 4 C T 0 N T 0 N 1 0 C 0 C T 0 N T 0 N 1 0 C 1 C T 1 N T 0 N 1 0 C 0 C T 1 N T 0 N 1 0 C 1 3 5 (2.19)
H=4 2 4 0 C T 0 N T 0 N 1 0 N 1 0 0 C T 1 N T 0 N 1 0 N 1 0 3 5 (2.20)
R=2 2
6
6
6
4
I 0 0
0 (1 )I+N T 1 N T 0 N 1 0 N 1 0
0 0 (1 )I
Finally,lettingS
v
bethesetofL 2
functions(x;)satisfyingtheconstraints(2.15),
and dening
J
v
()= inf
(x;)2Sv
P(x;;) (2.22)
we callona useful result [30].
Theorem 2.1. Thefunction P has at least one saddlepoint (x s
; s
; s
)on S
v
[0;1]and
inf
(x;)2S
v max
01
P(x;;)= min
(x;)2S
v max
01
P(x;;)=
max
01 min
(x;)2Sv
P(x;;)=P(x s ; s ; s ): (2.23)
Proof (from [30]) Let (x
;
) be the solution of problem (2.22). Then S
i (
i ),
i=0;1, depend continuously on0< <1. Moreover, since
S
0 (
0
)=0; if =1; (2.24)
S
0 (
0
) iscontinuous for 2(0;1], and since
S
1 (
1
)=0; if =0; (2.25)
S
1 (
1
) iscontinuous for 2[0;1). Suppose
lim !1 S 1 ( 1
)>0: (2.26)
Then for some 0< s
<1, we must have
S 0 ( s 0 )=S
1 ( s 1 ): (2.27) Let (x s ; s
)=(x s ; s ). Then P(x s ; s
;)S
1 (
s
(holdingat equality because cancels out) and
P(x;; s
)S
1 (
s
1
); 8(x;)2S
v (2.29) because (x s ; s
) is the optimal solution of (2.22) for = s
. This implies that
(x s ; s ; s
) is a saddle point and the rest follows. Now suppose that (2.26) does not
hold so that
lim !1 S 1 ( 1
)=0: (2.30)
In that case S
0
and S
1
can be made arbitrarily small simultaneously. This implies
that J
v
()=0 forall which meansthat there exists (x s
; s
)suchthat (2.27) holds
with equality tozero. Then,clearly (2.28) holdsbecauseboth sides ofthe inequality
are zero. In addition, (2.29) holds for all s
2 [0;1] because the right hand side of
the inequality is zero and the left hand side cannot be negative. This implies that
(x s ; s ; s
) isa saddle point and the rest follows.
Note that the aboveproof in[30] is done withknowledgeof, and weight matrices
onthe initialstate, x
i
(0). In that case, the bounded energy noisemodelbecomes
S i (x i (0); i )x
i (0) T F i;0 x i (0)+ Z t f 0 j i (t)j 2
dt<1; i=0;1: (2.31)
Since each S
i
is the sum of positive semi-denite terms, letting one term go to zero
does not alter the proof.
This result allows us to interchange the order of the inf and the max in (2.16),
and replace inf with min. Thus
J
v
()= min
(x;)2Sv
P(x;;) (2.32)
and, the characterizationof not proper becomes
max J
v
Expandingthis resulttoitsfullyexplicit form,wesee thatadetection signalv isnot
properif and only if
max
01 min
1
2 Z
t
f
0 x
T
Qx+x T
H+
T
R dt<1 (2.34)
where the minis subject to
x 0
=Ax+Bv+M: (2.35)
Theinner minimization,the J
v
()problem,isastandardLQRoptimalcontrol
prob-lemwith an added cross term inthe objective function and the forcing function Bv
in the constraint. J
v
() iscalled the auxiliary cost function for the problem.
The auxiliary cost functionexhibits several useful qualities[12].
Lemma 2.2. For allv 2L 2
,for 0 1, J
v
()isdened andhas thefollowing
properties:
1. It is zero for =0 and =1,
2. It is quadratic in v, i.e., for allscalar c, J
cv
()=jcj 2
J
v (),
3. It is a continuous function of ,
4. If v is not proper, then J
v
() < 1 for all 0 1. Equivalently, J
v
() 1
for some implies v is proper,
5. It isa strictly concave function of if the set of proper detection signals isnot
empty, otherwise it isidentically zero.
Theproofisstraightforwardandreliesoncontinuityandlinearity. Itcanbefound
in [12]. With this result, we can state the original problem of nding a minimum
energy properdetection signal v as
minkvk subject to max J
v
Note that the cases = 0 and = 1 are excluded because J
v
(0)= J
v
(1) = 0, and
Lemma 2.2demonstrates continuity of J
v
() atthese points.
UsingthefactthatJ
v
()isquadraticinv,wearriveatthefollowingfundamental
result
Theorem 2.2. Let
J
()=sup
v6=0 J
v ()
R
t
f
0 jvj
2
dt
= sup
kvk=1 J
v
(): (2.37)
Then
(
) 2
= max
0<<1 J
() (2.38)
where
is the separability indexdened previously.
This theorem,whilesimilartoresultsin[29]and[30], hasaddedtechnical
diÆcul-ties due tothe presence of the innite dimensionalspace of the independent variable
and the unbounded nitedimensionalsubspace of the outputsets. Despite these
dif-ferences, the proof is an extensionof that in[29]. However, itis somewhat technical
and requires functional analysis and convergence theory for sequences. See [12] for
the complete proof.
Notethat the ease ofseparating the nominaland faultmodels ofa systemis
pro-portionalto the size of
. When
=0,the models are indistinguishable regardless
of the detection signal.
As a nal result before dening the optimization problems we will address, we
state a useful corollary toLemma 2.2.
Corollary 2.1. Adetection signal v isproper if andonly ifJ
v
()1forsome
0< <1.
Proof Lemma 2.2, part 4,shows that v isproperif J
v
()1 for some. Toshow
where J
v
() attains itsminimum. Clearly, the values producing a minimum ateach
endpoint are
1
=0 for =0, and
0
=0 for =1. Thus there willbe a value
where k
0
()k =k
1
()k. But then k
i
()k<1, which shows that v is not proper.
2.1.3 Problem Statement
We can now state the two versions of the problem solved by the rst half of the
algorithm. Version One, from (2.37-2.38) is:
(
) 2
= max
kvk=1
0<<1 J
v
(): (2.39)
VersionTwo, from(2.7) and (2.36) is:
(
) 2
= min
Jv()1
0<<1 Z
t
f
0 jvj
2
dt: (2.40)
These problems will be solved by rst calculating the necessary conditions for a
minimum of the inner problem which denes J
v
(), then numerically solving the
outer problemusing the previously computed necessary conditions as constraints.
2.2 Necessary Conditions
Aswithmanytypesofoptimizationproblems,theJ
v
()problempossessesconditions
that any extrema must satisfy in order to be an optimal solution. In Chapter 1 we
introduced the necessary conditions for an optimal solution to the standard LQR
inChapter1becauseofthepresenceofthecrosstermintheintegral,sointhissection
we develop the necessary conditions for the J
v
() problemexplicitly.
2.2.1 Computing the Necessary Conditions
Recall from(2.32) that
J
v
()=min 1
2 Z
t
f
0 x
T
Qx+x T
H+
T
R dt (2.41a)
subject to
x 0
=Ax+Bv+M: (2.41b)
The Hamiltonianfor system (2.41) is
b
H = 1
2 x
T
Qx+ 1
2 x
T
H+
1
2
T
R + T
(Ax+Bv+M): (2.42)
As described inChapter 1, the Euler equations for anextremum are
b
H T
= x 0
(2.43a)
b
H T
x
=
0
(2.43b)
b
H T
= 0: (2.43c)
These conditions appliedto (2.42)give
x 0
= Ax+Bv+M (2.44a)
0
= Qx
1
2
H A
T
(2.44b)
0 = R + 1
2 H
T
x+M T
: (2.44c)
whichis anindex one DAE in(x;;) since R >0.
