# Computation of a Reference Model for Robust Fault Detection and Isolation Residual Generation

Authors: Emmanuel Mazars; Imad M. Jaimoukha; Zhenhai Li
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

This paper considers matrix inequality procedures to address the
robust fault detection and isolation (FDI) problem for linear
time-invariant systems subject to disturbances, faults, and
polytopic or norm-bounded uncertainties. We propose a design
procedure for an FDI filter that aims to minimize a weighted
combination of the sensitivity of the residual signal to
disturbances and modeling errors, and the deviation of the faults
to residual dynamics from a fault to residual reference model, using theℋ∞-norm as a measure. A key step in our
procedure is the design of an optimal fault reference model. We
show that the optimal design requires the solution of a quadratic
matrix inequality (QMI) optimization problem. Since the solution
of the optimal problem is intractable, we propose a linearization
technique to derive a numerically tractable suboptimal design
procedure that requires the solution of a linear matrix inequality
(LMI) optimization. A jet engine example is employed to
demonstrate the effectiveness of the proposed approach.


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## Body

## 1. Introduction

In the past
decade, great attention has been devoted to the design of model-based fault
detection systems and their robustness [1, 2]. With the rapid development of
robust control theory and ℋ∞ optimization
control techniques, more and more methods have been presented to solve the robust FDI problem. The ℋ∞-filter is designed such that the ℋ∞-norm of the estimation error is minimized (see [3–5] and the references therein).
Some of the approaches used for this problem include frequency domain
approaches [6], left and right eigenvector assignment [7], structure parity
equation [8], and an unknown input observer with disturbances decoupled in the
state estimation error [9]. Recently developed LMI approaches offer numerically
attractive techniques [10–12].A reference residual model is an ideal solution for
robust FDI under the assumption that there are no disturbances or model
uncertainty. The idea is to design a filter for the uncertain system that
approximates the solution given by the reference model [13]. In [14], a new
performance index is proposed using such a reference residual model. Frisk and
Nielsen [15] give an algorithm to design a reference model and a robust FDI
filter that fits into the framework of standard robust ℋ∞-filtering
relying on established and efficient methods. However, their framework consists
in solving two optimization problems successively, which results in a
suboptimal solution.In this paper, we propose a performance index that
captures the requirements of fault isolation and disturbance rejection as well
as the design of the optimal reference model. The fault isolation performance
is measured by the size of the deviation of the fault to residual dynamics from
the reference dynamics model, while the disturbance rejection performance is
measured by the size of the input to residual and disturbance to residual
dynamics. In all cases, theℋ∞ norm is used as
a measure. The design of the optimal reference model is incorporated in the
robust FDI framework. We consider systems subject to norm-bounded or polytopic
uncertainties. For systems described by polytopic and unstructured norm-bounded
uncertainties, we derive an optimal FDI filter obtained as the solution of a
QMI optimization. For systems described by structured uncertainties, we derive
a suboptimal QMI-based solution. Since the solution of a QMI optimization is,
in general, intractable, we propose a linearization technique to derive a
suboptimal design procedure that requires the solution of a numerically
tractable LMI optimization. This note extends our work in [16] by proposing
algorithms for the design of suboptimal reference dynamics.The structure of the work is as follows. After
defining the notation, we review filter-based FDI techniques for residual
signal generation and give the problem formulation in Section2. Section 3
presents a matrix inequality formulation for the FDI problem, and gives the
solution and the design of the optimal reference model for both norm-bounded
and polytopic uncertainties in a form of QMI's. Section 4 gives a suboptimal
solution in both cases through the use of an algorithm that necessitates
solving LMI's. Finally, a numerical example is presented in Section 5, and
Section 6 summarizes our results.The notation we use is fairly standard. The set of
realn×m matrices is
denoted by ℛn×m. For A∈ℛn×m, we use the notation AT to denote
transpose. For A=AT∈ℛn×n, A≻0 (A≺0) denotes that A is positive
(negative) definite, that is, all the eigenvalues of A are greater
(less) than zero. The n×n identity matrix
is denoted as In and the n×m null matrix is
denoted as 0n,m with the
subscripts occasionally dropped if they can be inferred from context.Letℒ2 be the set of
square integrable functions. The ℒ2-norm of u∈ℒ2 is defined as ∥u∥2=∫0∞u(t)Tu(t)dt. A transfer matrix G(s)=D+C(sI−A)−1B will be denoted
as G(s)=s(A,B,C,D) or (1)G(s)=s[ABCD], and dependence on the variable s will be
suppressed. For a stable transfer matrix G, we define (2)‖G‖∞=sup⁡{‖Gu‖2/‖u‖2:0≠u∈L2},‖G‖−=inf⁡{‖Gu‖2/‖u‖2:0≠u∈L2}. In section 3, we use the following result.Lemma 1  (see [17]).
Letϕ(s)=R+BT(−sI−AT)−1C+CT(sI−A)−1B+BT(−sI−AT)−1Q(sI−A)−1B  with (A,B)
sign
controllable, RT=R, and  QT=Q. Then  ϕ(s)
has a spectral
factor G(s)∈ℛℒ∞m×p
(i.e.,  ϕ(s)=GT(−s)G(s)
) if and only
if there exists symmetric  P
that satisfies
the following linear matrix inequality:  (3)PA+ATP+QPB+C(PB+C)TR⪰0.


## 2. Problem Formulation

Consider a
linear time-invariant dynamic system subject to disturbances, modeling errors
and process, sensor and actuator faults modeled as(4)x˙(t)y(t)=ABBdBfCDDdDf︷Mx(t)u(t)d(t)f(t),  where x(t)∈ℛn, u(t)∈ℛnu, and y(t)∈ℛny are the process
state and input and output vectors, respectively, and where d(t)∈ℛnd and f(t)∈ℛnf are the
disturbance and fault vectors, respectively. Here, Bf∈ℛn×nf and Df∈ℛny×nf are the
component and instrument fault distribution matrices, respectively, while Bd∈ℛn×nd and Dd∈ℛny×nd are the
corresponding disturbance distribution matrices [18].We consider two types of uncertainties: norm-bounded
and polytopic uncertainties. In the case of norm-bounded uncertainties,(5)M∈{AoBoBdoBfoCoDoDdoDfo︷Mo..+FAFCΔHEAEBEdEf:Δ∈ΔAoBoBdoBfoCoDoDdoDfo︷Mo}=:MΔ, where ℳo represents the
nominal model, ΔH=Δ(I−HΔ)−1, where (6)Δ:={Δ=diag(δ1Iq1,…,δlIql,Δl+1,…,Δl+f):‖Δ‖≤1,δi∈R,Δi∈Rqi×qi}⊂RnΔ×nΔ,  and where FA, FC, EA, EB, Ed, Ef, and H are known and
constant matrices with appropriate dimensions. This linear fractional
representation of uncertainty, which is assumed to be well-posed over Δ (i.e., det(I−HΔ)≠0 for all Δ∈Δ), has great
generality and is widely used in control
theories.In the case of polytopic uncertainties,(7)M∈{∑i=1pξiAiBiBdiBfiCiDiDdiDfi︷Mi:∑i=1pξi=1,ξi≥0}=:Mξ, where ℳi, i=1,…,p, are known
constant matrices with appropriate dimensions.A residual signal in an FDI system should represent the
inconsistency between the system variables and the mathematical model. The
objective is to design an FDI filter of the form(8)x^˙(t)r(t)=AkBkuBkyCkDkuDkyx^(t)u(t)y(t),  where x^(t)∈ℛnk is the filter
state and r(t)∈ℛnf is the residual
signal. Figure 1 illustrates this filter in the robust residual generation
scheme.Figure 1
Filter-based robust FDI scheme.By defining an augmented statez(t)=[x(t)Tx^(t)T]T the residual
dynamics are given by (9)z˙(t)r(t)=A˜B˜fB˜dB˜uC˜D˜fD˜dD˜uz(t)f(t)d(t)u(t) or (10)TrfMTrdMTruM=s[A˜B˜fB˜dB˜uC˜D˜fD˜dD˜u]=[A0BfBdBBkyCAkBkyDfBkyDdBku+BkyDDkyCCkDkyDfDkyDdDku+DkyD],  where Trfℳ, Trdℳ, and Truℳ denote the
dynamics from faults, disturbances, and inputs to the residual, respectively.
Note that dependence on the uncertain data is indicated by a superscript ℳ.Ideally, the residual signal is required to be
sensitive only to faults. This corresponds toTrdℳ=0, Truℳ=0, and Trfℳ≠0. For fault isolation, it is further required that the
fault signature can be deduced from the residual. This corresponds to Trfℳ∈𝒮, where 𝒮 is a set of
transfer matrices with a special structure that depends on the nature of the
faults, for example, if the faults can occur simultaneously, 𝒮 is the set of
stable diagonal transfer matrices with nonzero diagonal entries. Unfortunately,
characterizing a general structured set 𝒮 is intractable,
and we will assume that we can define a subset (11)S={Tref=s(Aref,Bref,Cref,Dref):Aref∈SA⊆Rnref×nref,Bref∈SB⊆Rnref×nf,Cref∈SC⊆Rnf×nref,Dref∈SD⊆Rnf×nf}⊆S  such that
subsequent optimizations over the structured state-space data sets 𝒮A, 𝒮B, 𝒮C, and 𝒮D are tractable.
For example, if 𝒮 denotes the set
of all stable nf×nf diagonal
transfer matrices with nonzero diagonal entries, we may define 𝒮A as the set of
all nf×nf diagonal
matrices with negative entries, and 𝒮B, 𝒮C, and 𝒮D as the sets of
all nf×nf diagonal
matrices. An example of this simplification procedure is given in Section 5
below. We also replace the requirement of nonzero diagonal entries for 𝒮 by a condition
of the form ∥Tref∥−≥β>0 for all Tref∈𝒮.Due to the presence of disturbances and modelling
uncertainties, exact FDI is not possible. For robust FDI, we propose the
following, more realistic, problem formulation.Problem 1.
Assume that the system dynamics in (4) are
quadratically stable. For any  γ>0, find a stable fault reference dynamics Tref=s(Aref,Bref,Cref,Dref)∈𝒮     such that ∥Tref∥−≥1  (to ensure the
requirement of nonzero diagonal entries for 𝒮), and find a
stable filter of the form given in (8), if it exists, such that the residual
dynamics in (10) are quadratically stable and  (12)sup⁡M∈M‖TrfM−TrefTrdMTruM‖∞<γ,  where ℳ=ℳΔ for
norm-bounded uncertainties and ℳ=ℳξ for polytopic
uncertainties.Recall that quadratic stability for the dynamics in
(4) is equivalent to the existence of P=PT≻0 such that ATP+PA≺0 for all A (see [19] for
more details).A modified version of Problem1 uses a weighted cost
function, say, of the form (13)‖(TrfM−Tref)WfTrdMWdTruMWu‖∞, where Wf, Wd, and Wu are constant or
frequency-dependent weighting functions that can vary the emphasis between
fault detection (small ∥Trdℳ∥∞ and ∥Truℳ∥∞) and fault
isolation (small ∥Trfℳ−Tref∥∞). In the
sequel, we assume that any such weighting functions are absorbed in the system
data.Remark 1.
The objective is to find the smallestγ
for which (12)
is satisfied. Indeed, by minimizing γ, we will ensure ∥Trdℳ∥∞  (which measures
the disturbance rejection level), ∥Truℳ∥∞  (which measures
the effect of uncertainty), and ∥Trfℳ−Tref∥∞  (which measures
the deviation of the fault to residual dynamics from the reference dynamics,
and hence is a measure of fault isolation capability) to be small.A poorly chosen reference model can result in a
residual generator with poor robustness. Here, we incorporate its design into
our scheme so as to improve the robustness of the FDI filter.In some approaches, a common assumption is thatDf=0 and/or CBf has full rank
[20, 21]; in others, the assumption Df has full rank
and is widely used [22, 23]. Here, we do not impose any of these assumptions.
Note, however, that if Df does not have
full column rank, for example, if Df=0, then this will have an adverse effect on the minimum
values of γ since γ≥∥[Trfℳ−TrefTrdℳTruℳ]∥∞≥∥Trfℳ(∞)−Tref(∞)∥=∥DkyDf−Dref∥=∥Dref∥≥1. This would
therefore limit the overall performance of the filter, which is measured by the
value of γ.


## 3. Matrix Inequality Formulation

In this
section, we derive a matrix inequality formulation of Problem1. The main
idea is to express (12) in terms of QMIs using the bounded real lemma and change
of variables techniques, and then to derive necessary and sufficient conditions
for solvability.The dynamics in (12) can be written as (14)TrfM−TrefTrdMTruM=s[AcMBcMCcMDcM]:=[Aref0Bref000A˜B˜fB˜dB˜u−CrefC˜D˜f−DrefD˜dD˜u]. It follows from
the bounded real lemma that there exists a stable filter of the form given in
(8) such that (12) is satisfied if and only if there exists Pc=PcT such that Pc≻0 and (15)(AcM)TPc+PcAcMPcBcM(CcM)T(BcM)TPc−γI(DcM)TCcMDcM−γI≺0  for all ℳ (see
[24, Theorem 3]). We deal separately with norm-bounded and polytopic uncertainties.


## 3.1. Solution with Norm-Bounded Uncertainties

For
norm-bounded uncertainties, we separate the terms involving modeling
uncertainties from the other terms as(16)(AcM,BcM,CcM,DcM)=(Aco+AcΔ,Bco+BcΔ,Cco+CcΔ,Dco+DcΔ), where (17)[AcoBcoCcoDco]=[Aref00Bref000Ao0BfoBdoBo0BkyCoAkBkyDfoBkyDdoBku+BkyDo−CrefDkyCoCkDkyDfo−DrefDkyDdoDku+DkyDo],(18)[AcΔBcΔCcΔDcΔ]=[Fr1Fr2]ΔHEr1Er2:=[0FABkyFCDkyFC]ΔH[0EA0EfEdEB].  Using this
separation, the inequality (15) can be rewritten as (19)(Aco)TPc+PcAcoPcBco(Cco)T(Bco)TPc−γI(Dco)TCcoDco−γI︷Tco+(AcΔ)TPc+PcAcΔPcBcΔ(CcΔ)T(BcΔ)TPc0(DcΔ)TCcΔDcΔ0︷TcΔ≺0.  A calculation
shows that TcΔ=FΔHE+ETΔHFT, where (20)F˜=(PcFr1)T0Fr2TT,E˜=Er1Er20.  The next result uses the fact that Δ∈Δ to remove
explicit dependence on Δ.Lemma 2  (see [25]).
LetΔ
be as described
in (6) and define the subspaces  (21)Σ={diag(S1,…,Sl,λ1Iql+1,…,λsIql+f):Si=SiT∈Rqi×qi,λj∈R},Γ={diag(G1,…,Gl,0ql+1,…,0ql+f):Gi=−GiT∈Rqi×qi}. Let R=RT, F, E, and H be matrices
with appropriate dimensions. We have det(I−HΔ)≠0 and R+FΔ(I−HΔ)−1E+ET(I−ΔTHT)−1ΔTFT≺0 for every Δ∈Δ if there exist S∈Σ and G∈Γ such that S≻0 and (22)R+ETSEF+ETSH+ETGFT+HTSE+GEHTSH+HTG+GH−S≺0.  If Δ is unstructured
(i.e., if  Δ={Δ∈ℛnΔ×nΔ:∥Δ∥≤1}
), then (22)
becomes (23)R+λETEF+λETHFT+λHTEλ(HTH−I)≺0  for some scalar λ≥0. In this case, condition (23) is both necessary and
sufficient.When the uncertainty set is unstructured, then Lemma2 implies that (24)(19)⟺Tco+F˜ΔHE˜+E˜TΔHTF˜T≺0⟺Tco+λE˜TE˜F˜+λE˜THF˜T+λHTE˜λ(HTH−I)≺0 for some λ≥0. Using a Schur complement argument shows that (19) is
equivalent to (25)T¯c:=(Aco)TPc+⋆PcBco(Cco)TPcFr1λEr1T⋆−γI(Dco)T0λEr2T⋆⋆−γIFr20⋆⋆⋆−λIλHT⋆⋆⋆⋆−λI≺0, where ⋆ denotes terms
readily inferred from symmetry. Next, we introduce a change of variables to
linearize the above matrix inequality [26]. Assume that nk=nref+n, that is, the filter order is equal to the order of
the system plus the order of the reference model. Let (26)Pc=YNNTY^,Pc−1=XMMTX^,Π1=IIMTZ0, where X,Y,X^,Y^∈ℛnk×nk are symmetric
matrices, and Z=X−1. Define T=diag(Π1,I). Then T¯c≺0 if and only if TTT¯cT≺0. From PcPc−1=I, we have the following calculations, where boldface
letters are used to indicate optimization variables:   (27)L11:=Π1TPcAcoΠ1=ZA¯+ZE1ArefE1TZA¯+ZE1ArefE1TYA¯+YE1ArefE1T+A^+B^yC¯YA¯+YE1ArefE1T+B^yC¯,(28)L12:=Π1TPcBco=ZB¯f+ZE1BrefZB¯dZB¯YB¯f+YE1Bref+B^yDfoYB¯d+B^yDdoYB¯+B^yDo+B^u,(29)L13:=(CcoΠ1)T=(D^yC¯−CrefE2+C^)T(D^yC¯−CrefE2)T,(30)L14:=Π1TPcFr1=ZF¯AYF¯A+B^yFC,(31)L15:=Π1TEr1T=E¯ATE¯AT,(32)L23T=D^yDfo−DrefD^yDdoD^yDo+D^u,(33)L25T=EfEdEB,where we have
defined (34)A¯E1B¯fB¯dB¯F¯A:=[00Inref00000Ao0BfoBdoBoFA],(35)C¯E2E¯A:=0CoInref00EA,(36)A^B^yB^uC^D^yD^u=NAkMTZNBkyNBkuCkMTZDkyDku. If M and N are invertible,
the variables
Ak, 	Ck, Bky, Bku, Dky, Dku can be replaced
by the new variables A^, B^y, B^u, C^, D^y, D^u without loss of
generality. We can now rewrite (19) as (37)L11+L11TL12L13L14λL15⋆−γIL230λL25⋆⋆−γID^yFC0⋆⋆⋆−λIλHT⋆⋆⋆⋆−λI≺0,  which is
nonlinear in the variables. Note that the nonlinearities involve the terms Aref and Bref only. The
constraint Pc≻0 can be
expressed as an LMI as follows: (38)Pc≻0⟺Π1TPcΠ1≻0⟺ZZZY≻0.
The constraint ∥Tref∥−≥1 can be
expressed as a quadratic matrix inequality using the next lemma.Lemma 3.
LetTref  be as defined
above. Then ∥Tref∥−≥1  if and only if
there exists Pref=PrefT∈ℛnref×nref  such that  (39)Tro:=PrefAref+ArefTPref+CrefTCrefPrefBref+CrefTDrefBrefTPref+DrefTCrefDrefTDref−Inf⪰0.Proof.
Letϕ(s)=Tref~(s)Tref(s)−Inf. Then ϕ(s)≽0 if and only if ∥Tref∥−≥1. It is easy to show that ϕ(s) can be written
as follows: (40)ϕ(s)=DrefTDref−Inf+BrefT(−sInf−ArefT)−1CrefTDref+(CrefTDref)T(sInf−Aref)−1Bref+BrefT(−sInf−ArefT)−1CrefTCref(sInf−Aref)−1Bref. The result then
follows from Lemma 1.The next lemma summarizes our result so far by giving
necessary and sufficient conditions for the solution of Problem1, in the
case that the uncertainty set is unstructured, in the form of a QMI feasibility
problem.Lemma 4.
Assume thatΔ
is
unstructured. Then there exist a stable filter of the form given in (8) and a
stable fault reference dynamics model  Tref=s(Aref,Bref,Cref,Dref)∈𝒮, where 𝒮
is defined in
(11), such that ∥Tref∥−≥1, residual dynamics in (10) are quadratically stable,
and (12) is satisfied for ℳ=ℳΔ if and only if
there exist Aref∈𝒮A, Bref∈𝒮B, Cref∈𝒮C, Dref∈𝒮D,  A^, B^y, B^u, C^, D^y, D^u,λ, and symmetric matrices  Pref, Y, and Z
such that (37),
(38), and (39) are satisfied. If such variables exist, the filter dynamics are
obtained by solving (36) where M  and N
are chosen such
that NMT=I−YZ−1.Approximate solutions to these QMIs can be obtained by
using algorithms with guaranteed global convergence [27, 28], as well as
local numerical search algorithms that converge (without a guarantee) much
faster [29, 30]. A related discussion of the solution algorithms for QMIs can
also be found in [31]. In Section 4 below, we develop an alternative procedure
for the approximate solution of these QMIs.Remark 2.
In
the case thatTref  is preassigned
to a known value, (37) becomes linear and (39) becomes irrelevant, therefore,
the optimal solution is given in a form of an LMI optimization. This case has
been considered in [16].WhenΔ is structured,
we proceed as follows. By using (22) from Lemma 2, we get (41)(19)⟺Tco+F˜ΔHE˜+E˜TΔHTF˜T≺0⟸∃S∈Σ&G∈Γs.t.S≻0&TSG≺0, where (42)TSG=TcoF˜+E˜TGF˜T+GTE˜HTG+GTH−S+E˜THTSE˜H. Using a Schur
complement argument and the expression of E and F in (20), we
get (43)TSG≺0⟺T¯SG:=(Aco)TPc+⋆PcBco(Cco)TPcFr1+Er1TGEr1TS⋆−γI(Dco)TEr2TGEr2TS⋆⋆−γIFr20⋆⋆⋆HTG+GTH−SHTS⋆⋆⋆⋆−S≺0.As we did for unstructured uncertainties, we use the
same matrixT=diag(Π1,I) to allow to
change variables, it follows that T¯SG≺0 if and only if TSG:=TT¯SGTT≺0. We multiply TSG by K=diag(I,S) and KT from left and
right, respectively, to get TSG≺0 if and only
if (44)L11+L11TL12L13L14+L15GL15S⋆−γIL230L25S⋆⋆−γID^yFC0⋆⋆⋆HTG+GTH−SHTS⋆⋆⋆⋆−S≺0.  Therefore, when Δ is structured,
we have the following sufficient condition for solvability.Lemma 5.
Suppose thatΔ
has the
structure defined in (6). Then there exist a stable filter of the form given in
(8) and a stable fault reference dynamics model Tref=s(Aref,Bref,Cref,Dref)∈𝒮, where 𝒮
is defined in
(11), such that ∥Tref∥−≥1, the residual dynamics in (10) are quadratically
stable, and (12) is satisfied for ℳ=ℳΔ  if there exist Aref∈𝒮A, Bref∈𝒮B, Cref∈𝒮C,  Dref∈𝒮D,  A^, B^y, B^u, C^, D^y, D^u, S∈Σ,  G∈Γ, and symmetric matrices Pref,  Y, and  Z
such that (38),
(39), and (44) are satisfied.


## 3.2. Solution with Polytopic Uncertainties

In this
section, we derive necessary and sufficient conditions for solvability of the
robust FDI problem for a system subject to polytopic uncertainties, in the form
of LMIs. Now,(45)[AcMBcMCcMDcM]∈{∑i=1pξi[AciBciCciDci],∑i=1pξi=1,ξi≥0}, where (Aci,Bci,Cci,Dci) are as defined
in (17), but with superscript (o) replaced by (i). We assume that the polytopic system is quadratically
stable. Recall that (12) is satisfied if and only if (15) is satisfied for all ℳ. Now, (46)(15)⟺∑i=1pξi(Aci)TPc+PcAciPcBci(Cci)T(Bci)TPc−γI(Dci)TCciDci−γI︷Tpoli≺0,∀ξis.t.∑i=1pξi=1,ξi⪰0,⟺Tpoli≺0,i=1,…,p.Noting that the change of variable defined in (36) is
independent of ℳi, we can therefore use it in this scheme as well.
Letting T=diag(Π1,I), it follows that (47)Tpoli≺0⟺T¯poli:=TTpoliTT=L11i+(L11i)TL12iL13i⋆−γIL23i⋆⋆−γI≺0, where the (ℒjki)Ts are as
defined in (27)–(33) and (34)-(35), but with the nominal model data ℳo replaced by ℳi. Therefore, for polytopic uncertainties, we can
derive necessary and sufficient conditions for solvability of Problem 1 in
the form of a QMI feasibility problem as follows.Lemma 6.
Letℳ=ℳξ. Then there exist a stable filter of the form given
in (8) and a stable fault reference dynamics model  Tref=s(Aref,Bref,Cref,Dref)∈𝒮, where 𝒮
is defined in
(11), such that ∥Tref∥−≥1, the residual dynamics in (10) are quadratically
stable, and (12) is satisfied for ℳ=ℳΔ  if and only
there exist Aref∈𝒮A, Bref∈𝒮B,  Cref∈𝒮C, Dref∈𝒮D,  A^,  B^y,  B^u, C^,  D^y,  D^u,  S∈Σ,  G∈Γ, and symmetric matrices  Pref,  Y, and  Z  such that (38),
(39), and  T¯poli≺0
are satisfied
for i=1,…,p.


## 4. Robust FDI Filter Design Using LMIS

In this
section, we give a suboptimal solution to Problem1. An optimal solution
would necessitate the solution of a quadratic matrix inequality and is in
general intractable. Here, we propose a linearization procedure to derive an
upper bound on the optimal solution that requires solving an LMI optimization
problem.The following general result demonstrates that if we
have one feasible solution to a QMI optimization, then we can construct an LMI
optimization problem whose solution gives an upper bound on the QMI
problem.Lemma 7.
LetQo,Ro∈ℛm×n  and  S=ST∈ℛm×m  be given and
let  Eo=Qo−Ro  and  Fo=Qo+Ro. For  Q,R∈ℛm×n, define  (48)ℳ(Q,R)=S+QRT+RQT,N(Q,R)=S+EoEoT−(Q−R)EoT−Eo(Q−R)TRQRT−I0QT0−I,O(Q,R)=S−FoFoT+(Q+R)FoT+Fo(Q+R)TRQRTI0QT0I. Then (49)N(Q,R)≺0⟹ℳ(Q,R)≺0,(50)ℳ(Qo,Ro)≺0⟹N(Qo,Ro)≺0,(51)O(Q,R)≻0⟹ℳ(Q,R)≻0,(52)ℳ(Qo,Ro)≻0⟹O(Qo,Ro)≻0.Proof.
Let𝒥(Q,R)=S+QQT+RRT+EoEoT−(Q−R)EoT−Eo(Q−R)T. Then by using a Schur complement argument, we
get (53)N(Q,R)≺0⟺J(Q,R)≺0.  Let ξ(Q,R)=(Q−R−Eo)(Q−R−Eo)T. Now, (54)S+ξ(Q,R)=S+(Q−R)(Q−R)T−(Q−R)EoT−Eo(Q−R)T+EoEoT, and it follows
that (55)S+QRT+RQT=S+QQT+RRT+EoEoT−(Q−R)EoT−Eo(Q−R)T−ξ(Q,R). That is, (56)ℳ(Q,R)=J(Q,R)−ξ(Q,R). Using (53) and
noting that ξ(Q,R)≽0 and ξ(Qo,Ro)=0, we have (57)N(Q,R)≺0⟹J(Q,R)≺0⟹ℳ(Q,R)≺0,ℳ(Qo,Ro)≺0⟹J(Qo,Ro)≺0⟹N(Qo,Ro)≺0. A similar proof
can be used to derive (51) and (52).In order to simplify the subsequent analysis, we adopt
the convention that variables appended with a subscript “x” denote
feasible values of the variables for the QMIs (37) and (39).In (37), the only nonlinear terms are ZE1ArefE1T, YIE1ArefE1T, ZE1Bref, and YE1Bref. We  denote the matrix in (37) by Trfw, and set Srfw to be the
matrix Trfw, with the nonlinear terms removed.LetZx=ZxT∈ℛnk×nk, Yx=YxT∈ℛnk×nk, Arefx∈ℛnref×nref, and Brefx∈ℛnref×nf be given. We
use Lemma 7 to derive an LMI formulation. We can write Trfw as (58)Trfw=Srfw+QcRcT+RcQcT, where (59)Qc=ZE100YE1000000000000,Rc=E1ArefT0E1ArefTE1ArefTBrefTBrefT0000000000⋅LetQcx denote the
value of Qc when Z and Y are replaced by Zx and Yx, respectively, let Rcx denote the
value of Rc when Aref and Bref are replaced by Arefx and Brefx, respectively, and define Ex=Qcx−Rcx. Using (49) from Lemma 7, we get (60)Trfwlin≺0⟹Trfw≺0,  where (61)Trfwlin=Srfw+ExExT−(Qc−Rc)ExT−Ex(Qc−Rc)TRcQcRcT−I0QcT0−I.To linearize the matrix inequality in (39), we need
Lemma 7 and the following lemma, whose proof is similar to that of Lemma 7
and is therefore dropped.Lemma 8.
LetUo∈ℛm×n  and  S=ST∈ℛm×m. For  U∈ℛm×n, define  (62)ℳ(U)=S+UUT=ℳ(U)T,N(U)=S+UUoT+UoUT−UoUoT=N(U)T. Then (63)N(U)≻0⟹ℳ(U)≻0,ℳ(Uo)≻0⟹N(Uo)≻0.LetPrefx=PrefxT∈ℛnref×nref, Crefx∈ℛnf×nref, and Drefx∈ℛnf×nf be given. Using
(51) from Lemmas 7 and 8, it is easy to get (64)Trolin⪰0⟹Tro⪰0,
where (65)Trolin=V⋆⋆⋆V′V′′⋆⋆Pref0I⋆ArefBref0I,
where
V={(Pref+ArefT)(Prefx+Arefx)+⋆−CrefxTCrefx+CrefTCrefx+⋆−(Prefx+ArefxT)(Prefx+Arefx)}, V′={DrefTCrefx+DrefxTCref−BrefxT(Prefx+Arefx)−DrefxTCrefx+BrefT(Prefx+Arefx)+BrefxT(Pref+Aref)}, V′′={DrefTDrefx+DrefxTDref−BrefxTBrefx−DrefxTDrefx−I+BrefTBrefx+⋆}. The next lemma
summarizes the results of this section by giving a linearized formulation of
the optimization problem defined in Lemma 4 using (60)  and (64).Lemma 9.
Assume thatΔ
is
unstructured. Let  Zx=ZxT∈ℛnk×nk,  Yx=YxT∈ℛnk×nk,  Prefx=PrefxT∈ℛnref×nref,  Arefx∈𝒮A,  Brefx∈𝒮B, Crefx∈𝒮C, and Drefx∈𝒮D
be given. Then
there exist a stable filter of the form given in (8) and a stable fault
reference dynamics model  Tref=s(Aref,Bref,Cref,Dref)∈𝒮, where  𝒮
is defined in
(11), such that  ∥Tref∥−≥1, residual dynamics in (10) are quadratically stable,
and (12) is satisfied for  ℳ=ℳΔ  if there exist  Aref∈𝒮A,  Bref∈𝒮B,  Cref∈𝒮C,  Dref∈𝒮D,  A^,  B^y,  B^u,  C^,  D^y,  D^u, λ, and symmetric matrices  Pref,  Y, and  Z  such that  (66)Trfwlin≺0,(67)Trolin⪰0,(68)ZZZY≻0.Remark 3.
This scheme
can also be applied to Lemmas5 and 6 to obtain a suboptimal solution
involving linear matrix inequalities.Next, we need to choose the initial parameters (Zx, Yx, Prefx, Arefx, Brefx, Crefx, and Drefx) to reduce γ. The idea is to derive an algorithm where at each
iteration, we solve the optimization problem given in Lemma 9, using the
solution of this problem at the previous iteration, for initial parameters. The
algorithm will use initial values Zxinit, Yxinit, Prefxinit, Arefxinit, Brefxinit, Crefxinit, and Drefxinit, which must guarantee that the LMI constraints (66)
and (67) will be feasible.From the above discussion, an algorithm for choosing
the initial parameters can be listed as follows.Algorithm 1.
(1) Set initial
valuesArefxinit, Brefxinit, Crefxinit, and Drefxinit such that Trefinit=s(Arefxinit,Brefxinit,Crefxinit,Drefxinit) satisfies ∥Trefinit∥−≥1 and Trefinit∈𝒮.
(2) Find the
solutionsZ, Y, and Pref of the
optimization derived from Lemmas 3 and 4, which is linear since the matrix inequalities (37), (38), and (39) become linear when Tref:=Trefinit is fixed. (The
matrices Z, Y, and Pref always exist
since the cost function in (12) is bounded because Δ is bounded).
(3)   SetZxinit:=Z, Yxinit:=Y, and Prefxinit:=Pref.
(4)   Start loop.
(5)   Since the
initial values are feasible for (37), (38), and (39), the LMIs (66) to (68) have feasible solutions from (50) and (63) in Lemmas 7 and 8. Compute solutions (Z, Y, etc.) of (66) to (68) to minimize γ.
(6)  RenameZxinit=Z, Yxinit=Y, Prefxinit=Pref, Arefxinit=Aref, Brefxinit=Bref, Crefxinit=Cref, and Drefxinit=Dref, and go to Step 5.
(7) End loop.Algorithm1 is convergent, possibly to a local minimum, in the sense that the quantity γ is nonincreasing
after each iteration. This can be easily shown using (50) and (52) from Lemma 7, and (63)
from Lemma 8.Remark 4.
Lemmas7, 8, and Algorithm 1 can also be applied to other problems involving
QMIs and bilinear matrix inequalities (BMIs). The procedure potentially gives an improvement and seems
to work well in practice.Remark 5.
In
the case that we choose a diagonal structure forTref, we may use  (69)Arefinit=−Inref,Brefinit=0nref×nf,Crefinit=0nf×nref,Drefinit=Inf, as initial
values. This will ensure that Tref is stable and ∥Tref∥−=1. We can solve the LMI optimization problem given in
Lemma 4. This will give Zxinit, Yxinit, Prefxinit, and so forth. These initial values are not unique
and can be chosen using other criteria.Remark 6.
A
more systematic technique for generating valid initial values is as follows:
first, generate anyBrefinit,  Crefinit,  Drefinit, and a stable  Arefinit
with the
structure chosen, then, compute  α=∥Trefinit∥−. If  α>0, redefine the matrices as  Arefinit=Arefinit/α,  Brefinit=Brefinit/α,  Crefinit=Crefinit/α, and  Drefinit=Drefinit/α, which fulfill that the conditions  Trefinit  are stable and  ∥Trefinit∥−=1.Remark 7.
The requirement forTref  to be diagonal
is to ensure fault isolability in the case that all faults may occur
simultaneously. If we ignore disturbances and uncertainty, and assuming perfect
fault isolation, then r=Treff  so that fault  fi  only affects
residual  ri. If none of the faults can occur simultaneously, then
we need only to impose the constraint that  Tref  is upper (or
lower) triangular. While it is not difficult to modify our analysis under these
conditions, we only consider the case when  Tref  is diagonal so
that all faults may occur simultaneously since our contribution is focussed on
reducing the effect of disturbances and uncertainties under the most stringent
fault scenarios.


## 5. Numerical Example

In order to
illustrate the efficiency of Algorithm1 and the importance of the choice of Tref, we consider a jet engine state-space model [32]
supplied by NASA Glenn Research Center given as (70)x˙(t)=(Ao+FAΔHEA)x(t)+(Bo+FAΔHEB)u(t)+Bdd(t)+Bff(t),y(t)=(Co+FCΔHEA)x(t)+(Do+FCΔHEB)u(t)+Ddd(t)+Dff(t), where (71)Ao=−0.9835−0.0110−0.0039−0.0004−0.9858−0.002600.0002−0.9891,Bo=0.00800.2397−0.03830.00680.15650.02480.0003−0.00030.0003,Co=0.23830.48710.13900−.00080.00040.00002−0.000040,Do=0.4171−4.49200.48750.0008−0.00500.000300.0005−0.0021. The system is subject to three disturbances and three
potential faults. Here, the setup is given by (72)Bd=0.10000.10000.01,Bf=0.00800.2397−0.03830.00680.15650.02480.0003−0.00030.0003,Dd=0.1×I3,Df=−0.02050.62170.81150.2789−1.75060.63631.05830.69731.3101.Since no uncertainty parameters were given in this
example, we assume an unstructured norm-bounded uncertainty, with matricesFA, FC, EA, EB, and H randomly
generated as (73)H=−0.319−0.0800.142−0.2880.1380.2580.1140.1630.133,EA=−0.060−0.008−0.135−0.0150.1540.047−0.044−0.061−0.090,EB=0.1340.6300.4510.2070.3710.0440.6070.5750.027,FC=−0.0550.066−0.012−0.085−0.085−0.007−0.025−0.1200.049,FA=0.148−0.129−0.0840.114−0.0070.050−0.068−0.0330.149.A square and diagonal structure forTref is necessary to
get fault isolation in the case that the faults occur at the same time.Remark 8.
IfAref,  Bref, and  Cref  are chosen diagonal,
then  (74)Tref(s)=Cref(sI−Aref)−1Bref+Dref=CrefBref(sI−Aref)−1+Dref. The terms in Bref can be
incorporated in Cref. It follows that we can set Bref=I. Therefore, the nonlinearity in (37) comes only from
bilinear terms ZE1ArefE1T and YE1ArefE1T.To initialize Algorithm1, Trefinit is generated
following Remark 6 as (75)Arefinit=−1.60000−1.44000−0.57,Brefinit=I3,Crefinit=0.710001.290000.67,Drefinit=1.400001.690001.81. Then, by
following the first two steps of Algorithm 1, we get the optimal γ as (76)γ=0.6273 and the values
given below for Z, Y, and Prefx, which will be used to initialize the main loop of
the algorithm as follows. (77)Zref=0.5010−0.43850.03090.6192−0.43130.4861−0.43850.81840.0844−0.3629−0.6083−0.81050.03090.08440.06510.0986−0.3323−0.13970.6192−0.36290.09861.4722−1.27160.9082−0.4313−0.6083−0.3323−1.27165.7926−1.53370.4861−0.8105−0.13970.9082−1.53372.5593,Yref=5.005919.13112.75180.18603.06070.701619.1311165.323312.5435−13.288632.4525−15.35952.751812.54357.7249−4.82090.6873−3.44390.1860−13.2886−4.82097.4800−5.35954.93293.060732.45250.6873−5.359522.7920−4.15750.7016−15.3595−3.44394.9329−4.15758.6769,Prefx=−0.04921.4322−0.14441.4322−63.12468.1628−0.14448.1628−1.2084.Taking into  account Remark8, we implemented
Algorithm 1 in Matlab to minimize γ. Table 1
shows the evolution of the
optimization following Algorithm 1.Table 1Iterations0125102050100γ0.62730.62480.57910.54760.52530.47390.46410.4641Algorithm1 can clearly improve the result in a few
iterations; γ is reduced by 33% compared to the
one obtained with a fixed Tref. This shows that the choice of Tref is essential in
our FDI scheme. We get (78)Aref=−18.1665000−9.2781000−6.0914,Bref=I3,Cref=0.0007000−0.39590000.0572,Dref=1.24380004.37680001.8216.Remark 9.
In
our numerical experimentation, other choices forTrefinit
have been used;
however, all converged to the same solution but with different numbers of
iterations.In order to show that our filter is robust against
disturbances and model uncertainties, we introduce a randomly generatedΔ given by (79)Δ=0.4058−0.0534−0.3600−0.4097−0.54660.4822−0.00670.0877−0.2743, as well as
three disturbances. Simulated through MATLAB and SIMULINK, these disturbances
are three-band-limited white noises with mean 0 and standard
deviation 2. Faults f1 and f2 are both
simulated by a unit positive jump connected from the 14th second. Fault f3, simulated by a soft bias (slope =0.2), is connected
from the 20th second. Figure 2 gives the residual responses before the algorithm, where each residual is
dedicated to a particular fault, while Figure 3 gives the optimized residual
response using our algorithm.Figure 2
Time response of the residual before the optimization.Figure 3
Time response of the residual after the optimization.The linesopt1, opt2, and opt3 represent the
optimal trajectories that each residual must follow(80)opt1(s)opt2(s)opt3(s)=Treff1(s)f2(s)f3(s).In both figures, each fault can be
distinguished from the others and the disturbances; however, in Figure3, the faults can be
more easily distinguished and each residual follows its optimal trajectory
(green line) with more accuracy. Furthermore, the disturbances are more
attenuated compared to Figure 2, and the jumps that indicate faults are clearer
in Figure 3 since their amplitudes are bigger and therefore allow a better
fault detection using thresholds [33, 34]. The isolation performance is clearly
effective as each fault produces a deviation of its residual only, without
modifying the trajectory of the others. This example illustrates that the
designed filter satisfies the performance requirement of FDI which is
sufficiently robust against disturbances and modeling errors. Figure 3 also
justifies the efficiency of Algorithm 1 to improve the design of the reference
model and therefore the overall performance of our filter.


## 6. Conclusion

This paper has
addressed the problem of fault detection and isolation for linear
time-invariant systems subject to faults, disturbances, and model
uncertainties. We proposed a performance index that captures the FDI
requirements. Through QMI formulations, we gave the design of an optimal filter
for polytopic or unstructured norm-bounded uncertainties, and a suboptimal
filter for structured norm-bounded uncertainties. Suboptimality in the latter
case is inherited from the bounded real lemma, which gives only sufficient LMI
conditions for structured uncertainties (see Lemma2). By allowing the reference model Tref to be variable
in our formulation, we get its optimal design, which can be used in other
schemes dedicated to fault isolation. The optimal design of this reference
model is initially given in a form of a QMI optimization, then a suboptimal
solution is obtained by using a linearization procedure which derives an upper
bound on the optimal solution of the QMI formulation that requires the solution
of an LMI optimization. Note, however, that we have no indication concerning
the deviation of our design from the optimal filter. We have also illustrated
the effectiveness of our algorithm using a jet-engine example.



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*Source: 790893-2008-02-03.xml*