# A Method for Designing Fault Diagnosis Filters for LPV Polytopic Systems

Authors: Sylvain Grenaille; David Henry; Ali Zolghadri
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

The work presented in this paper focuses on the design of robust Fault Detection and Isolation (FDI) filters for dynamic systems characterized by LPV (Linear Parameter Varying) polytopic models. A sufficient condition is established to guarantee sensitivity performance of the residual signal vector to faults. Robustness constraints against model perturbations and disturbances are also taken into account in the design method. A key feature of the proposed method is that the residual structuring matrices are optimized as an integral part of the design, together with the dynamic part (i.e. the filter). The design problem is formulated as a convex optimization problem and solved using LMI (Linear Matrix Inequalities) techniques. The proposed method is illustrated on the secondary circuit of a Nuclear Power Plant.


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## Body

## 1. Introduction and Motivations

The issue of Fault Detection and Isolation (FDI) in dynamic
systems has been an active research area in the last two decades. Model-based
FDI techniques use mathematical models of the monitored process and extract
features from measured signals, to generate fault indicating signals, that is, the
residuals. LTI models have been widely used to solve the problem of FDI. Tools
are now available to enhance robustness against small parameter variations and
other disturbances (see, [1–3] for surveys). The resulting robust FDI
problem is generally formulated as a min-max optimization setting to maximize
fault sensitivity performance and at the same time, to minimize the influence
of unknowns inputs.More recently, some research works have appeared that consider Linear
Parameter Varying (LPV) modeling of the monitored system to take into account
wider and more rapid parameters variations. Such models can be used efficiently
to represent some nonlinear systems (see, e.g., [4, 5]). This motivates
some researchers from the FDI community to develop model-based methods using
LPV models (see [6–8] among others). The two commonly used
approaches are fault estimation methods where the fault indicating signal is an
estimate of the fault signal, and residual generation methods where the
residuals are synthesized to be robust against modeling errors and unknown
inputs, while being sensitive to the faults. In this context, a geometric
approach is proposed in [6] to design a LPV observer in a Luenberger form. A
procedure is derived to obtain the observer parameters via the construction of
a suitable family of invariant subspaces (parameter varying (C,A)-invariant
and un-observability subspaces). In [7], a multi-model approach is used to
solve the FDI problem for nonlinear systems. The nonlinear system is modeled using
polytopic models and a robust polytopic unknown input observer is then
synthesized by means of pole assignment. The method uses LMI optimization
techniques to synthesize the observer gain. The major limitation of this
approach is that sensitivity of the residual signal against faults can only be checked
a posteriori. More precisely, if the distribution matrices of the fault model
and the effects that faults could have on the decoupled state is not of full
column rank, then faults could go undetectable. To overcome this problem, a
solution is provided in [8] where the main idea is to build a fault estimate
using a LPV filter such that the worst-case gain (i.e., the
H∞
performances measure for LPV systems) from
disturbances and faults to the estimation error, is minimized.In this paper a different approach based on residual generation
is considered for LPV systems that can be modeled within a LPV polytopic
setting. Robustness against exogenous disturbances and sensitivity against
faults are considered in a framework similar to the well-knownH∞/H−  setting for LTI systems. The robustness objectives
are expressed in terms of a minimization problem using the
H∞
norm for LPV systems, and the sensitivity requirement
is formulated in terms of a maximization constraint using also the H-index for LPV systems. The main
difference between this problem and the standard
H∞
problem for LPV systems is that it involves the
residual structuring matrices that are unknown.The paper is organized as follows: In Section2, the general FDI
filter design problem and the corresponding solution are presented. In Section 3,
the proposed method is applied to real data set coming from the secondary
circuit of a nuclear power plant in France. Finally, some concluding remarks are made in a final section.Preliminaries
The Euclidean norm is always used for vectors and is written without
a subscript; for example∥x∥.
Similarly in the matrix case, the induced vector norm is used: ∥A∥=σ¯(A) where σ¯(A) denotes the maximum singular value of A. Signals, for example w(t) or w,
are assumed to be of bounded energy, and their norm is denoted by ∥w∥2,
that is, ∥w∥2=(∫−∞+∞∥w(t)∥2dt)1/2<∞. LTI models, for example, P(s) or simply P,
are assumed to be in RH∞,
real rational functions with ∥P∥∞=supωσ¯(P(jω))<∞. ∥P∥∞,
that is, the largest gain of P, is accompanied by the smallest gain of P, infωσ̲(P(jω)),
which may be equal to zero for some P (e.g., strictly proper systems), if
the frequency range of interest is infinite. This motivated [1, 9–12] to define the non-zero smallest gain of P, that is, the H-index, as the restriction of infωσ̲(P(jω)) to a finite frequency domain Ω,
that is, ∥P∥−=infω∈Ωσ̲(P(jω)).
In  [1, 9], an evaluation function ∥•∥e which is a restriction of the H2 signal norm to Ω,
is defined by the authors as ∥w∥e=∥w∥2,Ω=(∫ω1ω2∥w(jω)∥22dω)1/2.
Then, given P so that y=Pu, it follows that (1)∥y∥e2=12π∫ω1ω2∥P(jω)u(jω)∥22dω=12π∫ω1ω2∥P(jω)u(jω)∥u∥2∥22∥u∥22dω≥∥P∥−2∥u∥e2 and thus that
∥P∥−≤∥y∥e/∥u∥e. This motivates the introduction of an evaluation function, denoted ∥•∥sens, which
is defined according to: (2)∥P∥sens=inf∥u∥e≠0∥y∥e∥u∥e≥∥P∥−.
From (2), it follows that ∥P∥sens takes the sense of the smallest value of a
singular value of P(jω) evaluated on Ω. Then it follows that ∥P∥−≤∥P∥sens≤∥P∥∞.
The underlying LPV system is modeled by the following state
space representation(3)x⋅=A(θ(t))x+B(θ(t))uy=C(θ(t))x+D(θ(t))u
which is denoted in a compact form as
(4)M(θ)=(A(θ)B(θ)C(θ)D(θ)).x is the state vector, u is the input vector, y is the
output vector and θ(t) is a varying parameter vector. It is assumed
that all parameters θi(t), i=1,…,r are bounded, measurable (or estimated) in real
time and take their values in the domain Θ,
so that Θ is a convex polytope.
The LPV system (3) admits a (non-conservatism) polytopic model
if it is possible to determine a set of matrices Mi, i=1,…,N, constituting the vertices of a
polytope defined by (5)Co{β1,…,βN}={∑i=1NβiM(Πi),βi≥0,∑i=1Nβi=1}
and such that it corresponds to the image by M of the domain Θ:    (6){M(θ),θ∈Θ}≡Co{M(Π1),…,M(ΠN)}.
Then, βi,  i=1,…,N define barycentric coordinates of Θ and the following convex decomposition yields:  (7)θ(t)=β1Π1+⋯+βNΠN,βi≥0,∑i=1Nβi=1.
Referring to the LPV system (3), the worst-case RMS gain from u to   y which is known as the H∞-norm for LPV systems is defined by: (8)∥M(θ)∥∞=sup∀θ∈Θ∥u∥2≠0=∥y∥2∥u∥2.
Following the definition of the index ∥•∥sens of a LTI transfer given by (2), we will
introduce the following evaluation function, that will be useful in the following
to formulate fault sensitivity requirements for LPV fault detection schemes:
(9)∥M(θ)∥sens=inf∀θ∈Θ∥u∥e≠0=∥y∥e∥u∥e∥M(θ)∥sens is also a generalization of (2) to LPV case.


## 2. FDI Filters for LPV Systems

### 2.1. Problem Setting

Consider the general FDI design problem for LPV systems
represented on Figure1.
G(θ) is a polytopic LPV model. K is a known controller. d∈Rqd represents exogenous disturbances and f∈Rqf represents the faults to be detected. F(θ) is the FDI LPV filter to be designed. z^∈Rqr is an estimation of z=Myy+Muu,
a subset of the measurements y∈Rm and the controlled inputs u∈Rp. My∈Rqr×m and Mu∈Rqr×p are the two residual structuring matrices to be designed.Figure 1
The general FDI filter design problem.It is assumed that the problem depicted on Figure1 is well
posed and thus the lower fractional transformation F1(G(θ),K) always exists.The FDI design problem we are interested in can then be formulated
as follows.Problem 1.
Assume that the faults f are detectable (the interested
reader can refer to [13] for a discussion on fault detectability).
The goal is to find the state space matricesAF(θ)∈RnF×nF, BF(θ)∈RnF×(m+p), CF(θ)∈Rqr×nF and DF(θ)∈Rqr×(m+p) of the (stable) polytopic LPV filter F(θ) and the residual structuring matrices My∈Rqr×m and Mu∈Rqr×p defining the residual vector
(10)r=z−z^=Myy+Muu−F(θ)(yu),r∈Rqr
such that the residual vector meets the following requirements:
(11)∥Trd(θ)∥∞<γ1(12)∥Trf(θ)∥sens>γ2
where
Trd and
Trf denote the looped transfers between d and r and f and r,
respectively. This problem can be represented by the block diagram
illustrated on Figure 2, where P(θ) is derived from  F1(G(θ),K) so that (yu)=P(θ)(df).
In this formulation, γ1 and γ2 are two positive constants referring
respectively to the robustness and sensitivity performances levels.Figure 2
General setup for FDI/LPV filter design problem.Equation (11) represents the worst-case robustness of the
residual to disturbances d for all θ∈Θ,
in the H∞-norm sense.
Under plant perturbation, the effect that the exogenous disturbances have on
the residuals, can greatly increase and the fault detection performance may
then be considerably degraded. A robust fault sensitivity specification is then
needed to maintain a detection performance level of the FDI unit. Here the sensitivity
measure (9) for LPV fault detection scheme is used to guarantee the worst-case
sensitivity of the residual to faults.Of course, the smallerγ1 and the bigger γ2
are the better the fault detection performance
will be.Remark 1.
In Problem1 formulation, it is assumed that the structuring matrices My and Mu do not depend on θ. If this assumption vanishes, it can be verified that the
following theoretical developments still yields. The only difference in such a
case is that, if we consider My(θ) and Mu(θ) in Problem 1, a set of structuring matrices  My(Πi) and Mu(Πi) for each vertex of the polytope Θ would be
obtained rather than constant matrices.


### 2.2. Design of the FDI Filter

In this section, a solution is provided to compute
simultaneouslyMy, Mu and F(θ) so that the requirements (11) and (12) are satisfied.
It is straightforward to verify that the major difficulty in this problem is
related to the fault sensitivity requirement (12) since (11) can be solved
using the techniques developed in the robust control community (see, e.g., [14] or [15]). To overcome this problem, a sufficient condition is
established in terms of a fictitious H∞ problem. It is then shown in the following that a solution to this fictitious
problem is a solution of the original one.


#### 2.2.1. Standard Setup for the Filter Design Problem

To proceed, let(13)P(θ)=(A(θ)Bd(θ)Bf(θ)C(θ)Dd(θ)Df(θ)).Using some algebra manipulations, the filter design problem illustrated on
Figure2, can be re-casted into the setup depicted in Figure 3, where P¯(θ,My,Mu) is deduced from P(θ),My and
Mu, according to:
(14)P¯(θ,My,Mu)=(A(θ)Bd(θ)Bf(θ)0n×qr(MyMu)C(θ)(MyMu)Dd(θ)(MyMu)Df(θ)−IqrC(θ)Dd(θ)Df(θ)0(m+p)×qr)
where  Iqr and 0i×j denote respectively the identity matrix of dimension
qr and the
null matrix of dimension  i×j. n is also the order of P¯(θ,My,Mu),
that is, A(θ)∈Rn×n.Figure 3
FDI/LPV filter design problem.Following the method proposed in [10, 11], the requirements
(11) and (12) are now expressed in terms of loop shapes, that is, of desired gain
responses for the appropriate closed-loop transfers. These shaping objectives
are then turned into uniform bounds by means of the shaping filters. Let
Wd and
Wf
denote the (dynamical) shaping filters associated
with (11) and (12) respectively, so that:
(15)∥Wd∥∞≤γ1(16)∥Wf∥−≥γ2∥Wd∥∞ and ∥Wf∥− denote respectively the H∞ and
H− norm of the LTI transfer
Wd
and
Wf
(see preliminaries).Assume thatWd
is
invertible (this can be done without loss of generality because it is always
possible to add zeros in
Wd
to make it invertible).
Wd
and
Wf
are also defined in
order to tune the gain responses for, respectively, Trd(θ) and Trf(θ).
Then it is straightforward to verify that the specification (11) yields if the
following constraint is satisfied:
(17)∥Trd˜(θ)∥∞<1.
In this formulation, d˜∈Rqd˜ is a fictitious signal generating d through
Wd
(see Figure 4(a) for easy reference) and Trd˜ denotes the looped transfer between r and d˜.Fictitious quadraticH∞
formulation for the filter design problem.
(a)(b)The following lemma allows the sensitivity constraint (12) to be
transformed into a fictitious
H∞
one.Lemma 1.
Consider an invertible transfer matrixWF such
that ∥Wf∥−=(γ2/λ)∥WF∥− and  ∥WF∥−>λ where λ=1+γ2.
Define the (fictitious) signal r˜∈Rqr such that r˜=r−WFf (see Figure 4(a)). Then a sufficient condition
for the specification (12) to hold, is:
(18)∥Tr˜f(θ)∥∞<1,
where
Tr˜f denote the looped transfer between r˜ and f.Proof.
Consider the signalr˜ introduced in Figure 4, that is,
(19)r˜=r−WFf,
where
WF
is define as in Lemma
1.
Then it can be verified
that the following relation yields:
(20)∥Trf(θ)∥sens≥∥WF∥−−∥Trf(θ)−WF∥∞
that can be re-written due to the
definition of
r˜
given by (19):
(21)∥Trf(θ)∥sens≥∥WF∥−−∥Tr˜f(θ)∥∞.
Now consider
the weighting function
WF
defined in Lemma
1. Since, WF
is supposed to be invertible, we get
(22)1∥WF∥−=∥WF+∥∞,Ω,
where
∥WF+∥∞,Ω=supω∈Ωσ¯(WF+(jω))⋅WF+
denotes the inverse of
WF
which always exists by assumption (see Lemma 1). Then, factorizing the right term
of (21) by
∥WF∥−
gives
(23)∥Trf(θ)∥sens≥(1−∥Tr˜f(θ)∥∞∥WF∥−)∥WF∥−
that
can be done since, by definition, ∥WF∥−≠0.
With (22), it then follows that:
(24)∥Trf(θ)∥sens≥(1−∥Tr˜f(θ)∥∞∥WF+∥∞,Ω)∥WF∥−.
Now, since by construction
∥WF∥−>λ,
it is straightforward to verify that the following relation yields:
(25)∥WF+∥∞,Ω<1λ.
Suppose
now that inequality (18) yields, that is,
∥Tr˜f(θ)∥∞<1.
From (25), it follows that
(26)∥Tr˜f(θ)∥∞∥WF+∥∞,Ω<1λ
and
with (24), we get
(27)∥Trf(θ)∥sens>λ−1λ∥WF∥−.
Thus, if
∥Wf∥−=(γ2/γ)∥WF∥−withλ=1+γ2, then (27) implies that
(28)∥Trf(θ)∥sens>∥Wf∥−
which terminates the proof.Following (17) and (18), the design problem can be
re-casted in a fictitious LPV H∞-framework
as depicted in Figure 4(a), where d˜ and r˜ are two fictitious signals, so that   d˜=Wdd and r˜=r−WFf.
Then, including γ1,λ,WF
and Wd−1 into the model P¯(θ,My,Mu) leads to the equivalent block diagram of Figure 4(b), where P˜(θ,My,Mu) is defined according to:
(29)(rr˜)=F1(P˜(θ,My,Mu),F(θ))(d˜f).
The residual generation problem can now be formulated in a
framework which looks like a standard
H∞
problem for LPV systems, by combining both requirements (17) and (18)
into a single
H∞
constraint.
Using Lemma 1, it can be verified that a sufficient condition for (17) and (18)
to hold, is (30)∥F1(P˜(θ,My,Mu),F(θ))∥∞<1.
As mentioned above, this equation seems to be similar to a standard
H∞
LPV problem.
In fact, this is not the case since the transfer P˜(θ,My,Mu) depends on
My
and
Mu
that are unknown.
In the following section, a procedure is given to overcome this problem.Remark 2.
It is clear that a key feature in the proposed formulation is the a priori choice
of the shaping filtersWd and Wf. From a practical point of
view, it is required that the residuals r
are as “big” and as “fast” as possible, when a fault occurs. Then, if the considered
faults manifest themselves in low frequencies, this leads to select Wf as a low pass
filter with the static gain and the cutting frequency, the highest possible.
With regards to the robustness objectives, it is required that the effects of
the disturbances on the residuals are as “small” as possible. This implies to
choose the gain of Wd as
“small” as possible in the frequency range where the energy content of the
disturbances is located. In other words, it is required a high attenuation
level of the disturbances on the residuals in the appropriate frequencies (see Section
3 where a practical case is presented). However, both sensitivity to
faults and robustness against disturbances might be not achieved in some cases.
Faults having similar frequency characteristics as those of disturbances might
go undetected. In such cases, the proposed formulation provides a framework to
find a good balance between fault sensitivity and robustness via the
construction of the shaping filters Wd and Wf. Finally, note
that the work reported in [16, 17] could be an interesting method to
select Wd and Wf.


#### 2.2.2. Synthesis of the FDI Filter

In the following, a solution is derived in
terms of a SDP (Semi Definite Programming) problem. To proceed, let{Awd,Bwd,Cwd,Dwd} and {AwF,BwF,CwF,DwF} be the state-space representations of Wd−1 and
WF
respectively, and denote  nwd and nwF the associated order, that is, Awd∈Rnwd×nwd and AwF∈RnwF×nwF.
Using some linear algebra manipulations, it can be verified from (14) that the
matrices A˜(θ), B˜(θ), C˜(θ) and D˜(θ) of the state-space representation of P˜(θ,My,Mu) are definedaccording to:(31)A˜(θ)=(A(θ)Bd(θ)Cwd00Awd000AwF),A˜(θ)∈R(n+nwd+nwF)×(n+nwd+nwF)(32)B˜(θ)=(B˜1(θ)∣B˜2)=(Bd(θ)DwdBf(θ)0Bwd000BwF0),B˜(θ)∈R(n+nwd+nwF)×(qd˜+qf+qr)(33)C˜(θ)=(C˜1(θ)C˜2(θ))=((MyMu)C(θ)(MyMu)Dd(θ)Cwd0(MyMu)C(θ)(MyMu)Dd(θ)Cwd−CwFC(θ)Dd(θ)Cwd0),C˜(θ)∈R(2qr+m+p)×(n+nwd+nwF)(34)D˜(θ)=(D˜11(θ)D˜12D˜21(θ)D˜22)=((MyMu)Dd(θ)Dwd(MyMu)Df(θ)−Iqr(MyMu)Dd(θ)Dwd[(MyMu)Df(θ)−DwF]−IqrDd(θ)DwdDf(θ)0),D˜(θ)∈R(2qr+m+p)×(qd˜+qf+qr). Having in mind the definition of A˜(θ), B˜(θ), C˜(θ) and D˜(θ),
it can be noted that the
H∞
optimization
problem formulated by (30) is non convex since it involves simultaneously the
residual structuring matrices
My and
Mu
and
the filter state-space matrices AF(θ), BF(θ), CF(θ) and DF(θ).
A solution to this problem may consist in chosen heuristically
My
and
Mu. However, as it has been outlined in [10], there is no
guarantee to the optimal solution.The following lemma which is an adaptation of Proposition 1 in
[8] for our purpose, gives the solution to this problem. The proof is omitted
here as it can be found in [8].Lemma 2.
LetA˜(Πi), B˜(Πi), C˜(Πi), D˜(Πi)∀i=1,…,N be the evaluation of A˜(θ), B˜(θ), C˜(θ), D˜(θ) at each vertex Πi of the polytope Θ.
Assume that C˜2(θ) and D˜21(θ)  do not depend of  θ (see Remark 3 for a
discussion about this assumption) and let Ns=(C˜2D˜21)⊥.
Then, there exists a solution of (31) if and only if there exists γ<1 and My∈Rqr×m and Mu∈Rqr×p and two symmetric matrices R∈R(n+nwd+nwF)×(n+nwd+nwF)>0 and S∈R(n+nwd+nwF)×(n+nwd+nwF)>0 solving the following SDP problem involving
2N+1LMI constraints:
(35)minγs.t.(A˜(Πi)R+RA˜T(Πi)B˜1(Πi)B˜1T(Πi)−γI)<0,i=1,…,N
(36)(NS00I)T(A˜T(Πi)S+SA(Πi)SB˜1(Πi)C˜1T(Πi)B˜1T(Πi)S−γID˜11T(Πi)C˜1(Πi)D˜11(Πi)−γI)(NS00I)<0,i=1,…,N
(37)(RIIS)≥0.
Moreover, F(θ) is a full order filter if nF=n+nwd+nwF.
The state space realization of the LPV filter F(θ) is then computed using the barycentric coordinates of Θ given by (7), so that:
(38)(AF(θ)BF(θ)CF(θ)DF(θ))=∑i=1Nβi(AF(Πi)BF(Πi)CF(Πi)DF(Πi))AF(Πi), BF(Πi), CF(Πi) and DF(Πi)∀i=1,…,N are the state space matrices of the N LTI filters
F(Πi)∀i=1,…,N that are deduced from the unique solution
(R,S,My,Mu,γ) following the procedure described in [14].Remark 3.
As
it is outlined in Lemma2, it is required that C˜2 and D˜21 do not depend on θ. This assumption is also done for  NS to be computed. If, by construction, such an
assumption is not verified, the solution consists in post filtering (yTuT)Tby a LTI filter at a high cutting frequency.
This solution has already been proposed in [14].


## 2.1. Problem Setting

Consider the general FDI design problem for LPV systems
represented on Figure1.
G(θ) is a polytopic LPV model. K is a known controller. d∈Rqd represents exogenous disturbances and f∈Rqf represents the faults to be detected. F(θ) is the FDI LPV filter to be designed. z^∈Rqr is an estimation of z=Myy+Muu,
a subset of the measurements y∈Rm and the controlled inputs u∈Rp. My∈Rqr×m and Mu∈Rqr×p are the two residual structuring matrices to be designed.Figure 1
The general FDI filter design problem.It is assumed that the problem depicted on Figure1 is well
posed and thus the lower fractional transformation F1(G(θ),K) always exists.The FDI design problem we are interested in can then be formulated
as follows.Problem 1.
Assume that the faults f are detectable (the interested
reader can refer to [13] for a discussion on fault detectability).
The goal is to find the state space matricesAF(θ)∈RnF×nF, BF(θ)∈RnF×(m+p), CF(θ)∈Rqr×nF and DF(θ)∈Rqr×(m+p) of the (stable) polytopic LPV filter F(θ) and the residual structuring matrices My∈Rqr×m and Mu∈Rqr×p defining the residual vector
(10)r=z−z^=Myy+Muu−F(θ)(yu),r∈Rqr
such that the residual vector meets the following requirements:
(11)∥Trd(θ)∥∞<γ1(12)∥Trf(θ)∥sens>γ2
where
Trd and
Trf denote the looped transfers between d and r and f and r,
respectively. This problem can be represented by the block diagram
illustrated on Figure 2, where P(θ) is derived from  F1(G(θ),K) so that (yu)=P(θ)(df).
In this formulation, γ1 and γ2 are two positive constants referring
respectively to the robustness and sensitivity performances levels.Figure 2
General setup for FDI/LPV filter design problem.Equation (11) represents the worst-case robustness of the
residual to disturbances d for all θ∈Θ,
in the H∞-norm sense.
Under plant perturbation, the effect that the exogenous disturbances have on
the residuals, can greatly increase and the fault detection performance may
then be considerably degraded. A robust fault sensitivity specification is then
needed to maintain a detection performance level of the FDI unit. Here the sensitivity
measure (9) for LPV fault detection scheme is used to guarantee the worst-case
sensitivity of the residual to faults.Of course, the smallerγ1 and the bigger γ2
are the better the fault detection performance
will be.Remark 1.
In Problem1 formulation, it is assumed that the structuring matrices My and Mu do not depend on θ. If this assumption vanishes, it can be verified that the
following theoretical developments still yields. The only difference in such a
case is that, if we consider My(θ) and Mu(θ) in Problem 1, a set of structuring matrices  My(Πi) and Mu(Πi) for each vertex of the polytope Θ would be
obtained rather than constant matrices.


## 2.2. Design of the FDI Filter

In this section, a solution is provided to compute
simultaneouslyMy, Mu and F(θ) so that the requirements (11) and (12) are satisfied.
It is straightforward to verify that the major difficulty in this problem is
related to the fault sensitivity requirement (12) since (11) can be solved
using the techniques developed in the robust control community (see, e.g., [14] or [15]). To overcome this problem, a sufficient condition is
established in terms of a fictitious H∞ problem. It is then shown in the following that a solution to this fictitious
problem is a solution of the original one.


### 2.2.1. Standard Setup for the Filter Design Problem

To proceed, let(13)P(θ)=(A(θ)Bd(θ)Bf(θ)C(θ)Dd(θ)Df(θ)).Using some algebra manipulations, the filter design problem illustrated on
Figure2, can be re-casted into the setup depicted in Figure 3, where P¯(θ,My,Mu) is deduced from P(θ),My and
Mu, according to:
(14)P¯(θ,My,Mu)=(A(θ)Bd(θ)Bf(θ)0n×qr(MyMu)C(θ)(MyMu)Dd(θ)(MyMu)Df(θ)−IqrC(θ)Dd(θ)Df(θ)0(m+p)×qr)
where  Iqr and 0i×j denote respectively the identity matrix of dimension
qr and the
null matrix of dimension  i×j. n is also the order of P¯(θ,My,Mu),
that is, A(θ)∈Rn×n.Figure 3
FDI/LPV filter design problem.Following the method proposed in [10, 11], the requirements
(11) and (12) are now expressed in terms of loop shapes, that is, of desired gain
responses for the appropriate closed-loop transfers. These shaping objectives
are then turned into uniform bounds by means of the shaping filters. Let
Wd and
Wf
denote the (dynamical) shaping filters associated
with (11) and (12) respectively, so that:
(15)∥Wd∥∞≤γ1(16)∥Wf∥−≥γ2∥Wd∥∞ and ∥Wf∥− denote respectively the H∞ and
H− norm of the LTI transfer
Wd
and
Wf
(see preliminaries).Assume thatWd
is
invertible (this can be done without loss of generality because it is always
possible to add zeros in
Wd
to make it invertible).
Wd
and
Wf
are also defined in
order to tune the gain responses for, respectively, Trd(θ) and Trf(θ).
Then it is straightforward to verify that the specification (11) yields if the
following constraint is satisfied:
(17)∥Trd˜(θ)∥∞<1.
In this formulation, d˜∈Rqd˜ is a fictitious signal generating d through
Wd
(see Figure 4(a) for easy reference) and Trd˜ denotes the looped transfer between r and d˜.Fictitious quadraticH∞
formulation for the filter design problem.
(a)(b)The following lemma allows the sensitivity constraint (12) to be
transformed into a fictitious
H∞
one.Lemma 1.
Consider an invertible transfer matrixWF such
that ∥Wf∥−=(γ2/λ)∥WF∥− and  ∥WF∥−>λ where λ=1+γ2.
Define the (fictitious) signal r˜∈Rqr such that r˜=r−WFf (see Figure 4(a)). Then a sufficient condition
for the specification (12) to hold, is:
(18)∥Tr˜f(θ)∥∞<1,
where
Tr˜f denote the looped transfer between r˜ and f.Proof.
Consider the signalr˜ introduced in Figure 4, that is,
(19)r˜=r−WFf,
where
WF
is define as in Lemma
1.
Then it can be verified
that the following relation yields:
(20)∥Trf(θ)∥sens≥∥WF∥−−∥Trf(θ)−WF∥∞
that can be re-written due to the
definition of
r˜
given by (19):
(21)∥Trf(θ)∥sens≥∥WF∥−−∥Tr˜f(θ)∥∞.
Now consider
the weighting function
WF
defined in Lemma
1. Since, WF
is supposed to be invertible, we get
(22)1∥WF∥−=∥WF+∥∞,Ω,
where
∥WF+∥∞,Ω=supω∈Ωσ¯(WF+(jω))⋅WF+
denotes the inverse of
WF
which always exists by assumption (see Lemma 1). Then, factorizing the right term
of (21) by
∥WF∥−
gives
(23)∥Trf(θ)∥sens≥(1−∥Tr˜f(θ)∥∞∥WF∥−)∥WF∥−
that
can be done since, by definition, ∥WF∥−≠0.
With (22), it then follows that:
(24)∥Trf(θ)∥sens≥(1−∥Tr˜f(θ)∥∞∥WF+∥∞,Ω)∥WF∥−.
Now, since by construction
∥WF∥−>λ,
it is straightforward to verify that the following relation yields:
(25)∥WF+∥∞,Ω<1λ.
Suppose
now that inequality (18) yields, that is,
∥Tr˜f(θ)∥∞<1.
From (25), it follows that
(26)∥Tr˜f(θ)∥∞∥WF+∥∞,Ω<1λ
and
with (24), we get
(27)∥Trf(θ)∥sens>λ−1λ∥WF∥−.
Thus, if
∥Wf∥−=(γ2/γ)∥WF∥−withλ=1+γ2, then (27) implies that
(28)∥Trf(θ)∥sens>∥Wf∥−
which terminates the proof.Following (17) and (18), the design problem can be
re-casted in a fictitious LPV H∞-framework
as depicted in Figure 4(a), where d˜ and r˜ are two fictitious signals, so that   d˜=Wdd and r˜=r−WFf.
Then, including γ1,λ,WF
and Wd−1 into the model P¯(θ,My,Mu) leads to the equivalent block diagram of Figure 4(b), where P˜(θ,My,Mu) is defined according to:
(29)(rr˜)=F1(P˜(θ,My,Mu),F(θ))(d˜f).
The residual generation problem can now be formulated in a
framework which looks like a standard
H∞
problem for LPV systems, by combining both requirements (17) and (18)
into a single
H∞
constraint.
Using Lemma 1, it can be verified that a sufficient condition for (17) and (18)
to hold, is (30)∥F1(P˜(θ,My,Mu),F(θ))∥∞<1.
As mentioned above, this equation seems to be similar to a standard
H∞
LPV problem.
In fact, this is not the case since the transfer P˜(θ,My,Mu) depends on
My
and
Mu
that are unknown.
In the following section, a procedure is given to overcome this problem.Remark 2.
It is clear that a key feature in the proposed formulation is the a priori choice
of the shaping filtersWd and Wf. From a practical point of
view, it is required that the residuals r
are as “big” and as “fast” as possible, when a fault occurs. Then, if the considered
faults manifest themselves in low frequencies, this leads to select Wf as a low pass
filter with the static gain and the cutting frequency, the highest possible.
With regards to the robustness objectives, it is required that the effects of
the disturbances on the residuals are as “small” as possible. This implies to
choose the gain of Wd as
“small” as possible in the frequency range where the energy content of the
disturbances is located. In other words, it is required a high attenuation
level of the disturbances on the residuals in the appropriate frequencies (see Section
3 where a practical case is presented). However, both sensitivity to
faults and robustness against disturbances might be not achieved in some cases.
Faults having similar frequency characteristics as those of disturbances might
go undetected. In such cases, the proposed formulation provides a framework to
find a good balance between fault sensitivity and robustness via the
construction of the shaping filters Wd and Wf. Finally, note
that the work reported in [16, 17] could be an interesting method to
select Wd and Wf.


### 2.2.2. Synthesis of the FDI Filter

In the following, a solution is derived in
terms of a SDP (Semi Definite Programming) problem. To proceed, let{Awd,Bwd,Cwd,Dwd} and {AwF,BwF,CwF,DwF} be the state-space representations of Wd−1 and
WF
respectively, and denote  nwd and nwF the associated order, that is, Awd∈Rnwd×nwd and AwF∈RnwF×nwF.
Using some linear algebra manipulations, it can be verified from (14) that the
matrices A˜(θ), B˜(θ), C˜(θ) and D˜(θ) of the state-space representation of P˜(θ,My,Mu) are definedaccording to:(31)A˜(θ)=(A(θ)Bd(θ)Cwd00Awd000AwF),A˜(θ)∈R(n+nwd+nwF)×(n+nwd+nwF)(32)B˜(θ)=(B˜1(θ)∣B˜2)=(Bd(θ)DwdBf(θ)0Bwd000BwF0),B˜(θ)∈R(n+nwd+nwF)×(qd˜+qf+qr)(33)C˜(θ)=(C˜1(θ)C˜2(θ))=((MyMu)C(θ)(MyMu)Dd(θ)Cwd0(MyMu)C(θ)(MyMu)Dd(θ)Cwd−CwFC(θ)Dd(θ)Cwd0),C˜(θ)∈R(2qr+m+p)×(n+nwd+nwF)(34)D˜(θ)=(D˜11(θ)D˜12D˜21(θ)D˜22)=((MyMu)Dd(θ)Dwd(MyMu)Df(θ)−Iqr(MyMu)Dd(θ)Dwd[(MyMu)Df(θ)−DwF]−IqrDd(θ)DwdDf(θ)0),D˜(θ)∈R(2qr+m+p)×(qd˜+qf+qr). Having in mind the definition of A˜(θ), B˜(θ), C˜(θ) and D˜(θ),
it can be noted that the
H∞
optimization
problem formulated by (30) is non convex since it involves simultaneously the
residual structuring matrices
My and
Mu
and
the filter state-space matrices AF(θ), BF(θ), CF(θ) and DF(θ).
A solution to this problem may consist in chosen heuristically
My
and
Mu. However, as it has been outlined in [10], there is no
guarantee to the optimal solution.The following lemma which is an adaptation of Proposition 1 in
[8] for our purpose, gives the solution to this problem. The proof is omitted
here as it can be found in [8].Lemma 2.
LetA˜(Πi), B˜(Πi), C˜(Πi), D˜(Πi)∀i=1,…,N be the evaluation of A˜(θ), B˜(θ), C˜(θ), D˜(θ) at each vertex Πi of the polytope Θ.
Assume that C˜2(θ) and D˜21(θ)  do not depend of  θ (see Remark 3 for a
discussion about this assumption) and let Ns=(C˜2D˜21)⊥.
Then, there exists a solution of (31) if and only if there exists γ<1 and My∈Rqr×m and Mu∈Rqr×p and two symmetric matrices R∈R(n+nwd+nwF)×(n+nwd+nwF)>0 and S∈R(n+nwd+nwF)×(n+nwd+nwF)>0 solving the following SDP problem involving
2N+1LMI constraints:
(35)minγs.t.(A˜(Πi)R+RA˜T(Πi)B˜1(Πi)B˜1T(Πi)−γI)<0,i=1,…,N
(36)(NS00I)T(A˜T(Πi)S+SA(Πi)SB˜1(Πi)C˜1T(Πi)B˜1T(Πi)S−γID˜11T(Πi)C˜1(Πi)D˜11(Πi)−γI)(NS00I)<0,i=1,…,N
(37)(RIIS)≥0.
Moreover, F(θ) is a full order filter if nF=n+nwd+nwF.
The state space realization of the LPV filter F(θ) is then computed using the barycentric coordinates of Θ given by (7), so that:
(38)(AF(θ)BF(θ)CF(θ)DF(θ))=∑i=1Nβi(AF(Πi)BF(Πi)CF(Πi)DF(Πi))AF(Πi), BF(Πi), CF(Πi) and DF(Πi)∀i=1,…,N are the state space matrices of the N LTI filters
F(Πi)∀i=1,…,N that are deduced from the unique solution
(R,S,My,Mu,γ) following the procedure described in [14].Remark 3.
As
it is outlined in Lemma2, it is required that C˜2 and D˜21 do not depend on θ. This assumption is also done for  NS to be computed. If, by construction, such an
assumption is not verified, the solution consists in post filtering (yTuT)Tby a LTI filter at a high cutting frequency.
This solution has already been proposed in [14].


## 2.2.1. Standard Setup for the Filter Design Problem

To proceed, let(13)P(θ)=(A(θ)Bd(θ)Bf(θ)C(θ)Dd(θ)Df(θ)).Using some algebra manipulations, the filter design problem illustrated on
Figure2, can be re-casted into the setup depicted in Figure 3, where P¯(θ,My,Mu) is deduced from P(θ),My and
Mu, according to:
(14)P¯(θ,My,Mu)=(A(θ)Bd(θ)Bf(θ)0n×qr(MyMu)C(θ)(MyMu)Dd(θ)(MyMu)Df(θ)−IqrC(θ)Dd(θ)Df(θ)0(m+p)×qr)
where  Iqr and 0i×j denote respectively the identity matrix of dimension
qr and the
null matrix of dimension  i×j. n is also the order of P¯(θ,My,Mu),
that is, A(θ)∈Rn×n.Figure 3
FDI/LPV filter design problem.Following the method proposed in [10, 11], the requirements
(11) and (12) are now expressed in terms of loop shapes, that is, of desired gain
responses for the appropriate closed-loop transfers. These shaping objectives
are then turned into uniform bounds by means of the shaping filters. Let
Wd and
Wf
denote the (dynamical) shaping filters associated
with (11) and (12) respectively, so that:
(15)∥Wd∥∞≤γ1(16)∥Wf∥−≥γ2∥Wd∥∞ and ∥Wf∥− denote respectively the H∞ and
H− norm of the LTI transfer
Wd
and
Wf
(see preliminaries).Assume thatWd
is
invertible (this can be done without loss of generality because it is always
possible to add zeros in
Wd
to make it invertible).
Wd
and
Wf
are also defined in
order to tune the gain responses for, respectively, Trd(θ) and Trf(θ).
Then it is straightforward to verify that the specification (11) yields if the
following constraint is satisfied:
(17)∥Trd˜(θ)∥∞<1.
In this formulation, d˜∈Rqd˜ is a fictitious signal generating d through
Wd
(see Figure 4(a) for easy reference) and Trd˜ denotes the looped transfer between r and d˜.Fictitious quadraticH∞
formulation for the filter design problem.
(a)(b)The following lemma allows the sensitivity constraint (12) to be
transformed into a fictitious
H∞
one.Lemma 1.
Consider an invertible transfer matrixWF such
that ∥Wf∥−=(γ2/λ)∥WF∥− and  ∥WF∥−>λ where λ=1+γ2.
Define the (fictitious) signal r˜∈Rqr such that r˜=r−WFf (see Figure 4(a)). Then a sufficient condition
for the specification (12) to hold, is:
(18)∥Tr˜f(θ)∥∞<1,
where
Tr˜f denote the looped transfer between r˜ and f.Proof.
Consider the signalr˜ introduced in Figure 4, that is,
(19)r˜=r−WFf,
where
WF
is define as in Lemma
1.
Then it can be verified
that the following relation yields:
(20)∥Trf(θ)∥sens≥∥WF∥−−∥Trf(θ)−WF∥∞
that can be re-written due to the
definition of
r˜
given by (19):
(21)∥Trf(θ)∥sens≥∥WF∥−−∥Tr˜f(θ)∥∞.
Now consider
the weighting function
WF
defined in Lemma
1. Since, WF
is supposed to be invertible, we get
(22)1∥WF∥−=∥WF+∥∞,Ω,
where
∥WF+∥∞,Ω=supω∈Ωσ¯(WF+(jω))⋅WF+
denotes the inverse of
WF
which always exists by assumption (see Lemma 1). Then, factorizing the right term
of (21) by
∥WF∥−
gives
(23)∥Trf(θ)∥sens≥(1−∥Tr˜f(θ)∥∞∥WF∥−)∥WF∥−
that
can be done since, by definition, ∥WF∥−≠0.
With (22), it then follows that:
(24)∥Trf(θ)∥sens≥(1−∥Tr˜f(θ)∥∞∥WF+∥∞,Ω)∥WF∥−.
Now, since by construction
∥WF∥−>λ,
it is straightforward to verify that the following relation yields:
(25)∥WF+∥∞,Ω<1λ.
Suppose
now that inequality (18) yields, that is,
∥Tr˜f(θ)∥∞<1.
From (25), it follows that
(26)∥Tr˜f(θ)∥∞∥WF+∥∞,Ω<1λ
and
with (24), we get
(27)∥Trf(θ)∥sens>λ−1λ∥WF∥−.
Thus, if
∥Wf∥−=(γ2/γ)∥WF∥−withλ=1+γ2, then (27) implies that
(28)∥Trf(θ)∥sens>∥Wf∥−
which terminates the proof.Following (17) and (18), the design problem can be
re-casted in a fictitious LPV H∞-framework
as depicted in Figure 4(a), where d˜ and r˜ are two fictitious signals, so that   d˜=Wdd and r˜=r−WFf.
Then, including γ1,λ,WF
and Wd−1 into the model P¯(θ,My,Mu) leads to the equivalent block diagram of Figure 4(b), where P˜(θ,My,Mu) is defined according to:
(29)(rr˜)=F1(P˜(θ,My,Mu),F(θ))(d˜f).
The residual generation problem can now be formulated in a
framework which looks like a standard
H∞
problem for LPV systems, by combining both requirements (17) and (18)
into a single
H∞
constraint.
Using Lemma 1, it can be verified that a sufficient condition for (17) and (18)
to hold, is (30)∥F1(P˜(θ,My,Mu),F(θ))∥∞<1.
As mentioned above, this equation seems to be similar to a standard
H∞
LPV problem.
In fact, this is not the case since the transfer P˜(θ,My,Mu) depends on
My
and
Mu
that are unknown.
In the following section, a procedure is given to overcome this problem.Remark 2.
It is clear that a key feature in the proposed formulation is the a priori choice
of the shaping filtersWd and Wf. From a practical point of
view, it is required that the residuals r
are as “big” and as “fast” as possible, when a fault occurs. Then, if the considered
faults manifest themselves in low frequencies, this leads to select Wf as a low pass
filter with the static gain and the cutting frequency, the highest possible.
With regards to the robustness objectives, it is required that the effects of
the disturbances on the residuals are as “small” as possible. This implies to
choose the gain of Wd as
“small” as possible in the frequency range where the energy content of the
disturbances is located. In other words, it is required a high attenuation
level of the disturbances on the residuals in the appropriate frequencies (see Section
3 where a practical case is presented). However, both sensitivity to
faults and robustness against disturbances might be not achieved in some cases.
Faults having similar frequency characteristics as those of disturbances might
go undetected. In such cases, the proposed formulation provides a framework to
find a good balance between fault sensitivity and robustness via the
construction of the shaping filters Wd and Wf. Finally, note
that the work reported in [16, 17] could be an interesting method to
select Wd and Wf.


## 2.2.2. Synthesis of the FDI Filter

In the following, a solution is derived in
terms of a SDP (Semi Definite Programming) problem. To proceed, let{Awd,Bwd,Cwd,Dwd} and {AwF,BwF,CwF,DwF} be the state-space representations of Wd−1 and
WF
respectively, and denote  nwd and nwF the associated order, that is, Awd∈Rnwd×nwd and AwF∈RnwF×nwF.
Using some linear algebra manipulations, it can be verified from (14) that the
matrices A˜(θ), B˜(θ), C˜(θ) and D˜(θ) of the state-space representation of P˜(θ,My,Mu) are definedaccording to:(31)A˜(θ)=(A(θ)Bd(θ)Cwd00Awd000AwF),A˜(θ)∈R(n+nwd+nwF)×(n+nwd+nwF)(32)B˜(θ)=(B˜1(θ)∣B˜2)=(Bd(θ)DwdBf(θ)0Bwd000BwF0),B˜(θ)∈R(n+nwd+nwF)×(qd˜+qf+qr)(33)C˜(θ)=(C˜1(θ)C˜2(θ))=((MyMu)C(θ)(MyMu)Dd(θ)Cwd0(MyMu)C(θ)(MyMu)Dd(θ)Cwd−CwFC(θ)Dd(θ)Cwd0),C˜(θ)∈R(2qr+m+p)×(n+nwd+nwF)(34)D˜(θ)=(D˜11(θ)D˜12D˜21(θ)D˜22)=((MyMu)Dd(θ)Dwd(MyMu)Df(θ)−Iqr(MyMu)Dd(θ)Dwd[(MyMu)Df(θ)−DwF]−IqrDd(θ)DwdDf(θ)0),D˜(θ)∈R(2qr+m+p)×(qd˜+qf+qr). Having in mind the definition of A˜(θ), B˜(θ), C˜(θ) and D˜(θ),
it can be noted that the
H∞
optimization
problem formulated by (30) is non convex since it involves simultaneously the
residual structuring matrices
My and
Mu
and
the filter state-space matrices AF(θ), BF(θ), CF(θ) and DF(θ).
A solution to this problem may consist in chosen heuristically
My
and
Mu. However, as it has been outlined in [10], there is no
guarantee to the optimal solution.The following lemma which is an adaptation of Proposition 1 in
[8] for our purpose, gives the solution to this problem. The proof is omitted
here as it can be found in [8].Lemma 2.
LetA˜(Πi), B˜(Πi), C˜(Πi), D˜(Πi)∀i=1,…,N be the evaluation of A˜(θ), B˜(θ), C˜(θ), D˜(θ) at each vertex Πi of the polytope Θ.
Assume that C˜2(θ) and D˜21(θ)  do not depend of  θ (see Remark 3 for a
discussion about this assumption) and let Ns=(C˜2D˜21)⊥.
Then, there exists a solution of (31) if and only if there exists γ<1 and My∈Rqr×m and Mu∈Rqr×p and two symmetric matrices R∈R(n+nwd+nwF)×(n+nwd+nwF)>0 and S∈R(n+nwd+nwF)×(n+nwd+nwF)>0 solving the following SDP problem involving
2N+1LMI constraints:
(35)minγs.t.(A˜(Πi)R+RA˜T(Πi)B˜1(Πi)B˜1T(Πi)−γI)<0,i=1,…,N
(36)(NS00I)T(A˜T(Πi)S+SA(Πi)SB˜1(Πi)C˜1T(Πi)B˜1T(Πi)S−γID˜11T(Πi)C˜1(Πi)D˜11(Πi)−γI)(NS00I)<0,i=1,…,N
(37)(RIIS)≥0.
Moreover, F(θ) is a full order filter if nF=n+nwd+nwF.
The state space realization of the LPV filter F(θ) is then computed using the barycentric coordinates of Θ given by (7), so that:
(38)(AF(θ)BF(θ)CF(θ)DF(θ))=∑i=1Nβi(AF(Πi)BF(Πi)CF(Πi)DF(Πi))AF(Πi), BF(Πi), CF(Πi) and DF(Πi)∀i=1,…,N are the state space matrices of the N LTI filters
F(Πi)∀i=1,…,N that are deduced from the unique solution
(R,S,My,Mu,γ) following the procedure described in [14].Remark 3.
As
it is outlined in Lemma2, it is required that C˜2 and D˜21 do not depend on θ. This assumption is also done for  NS to be computed. If, by construction, such an
assumption is not verified, the solution consists in post filtering (yTuT)Tby a LTI filter at a high cutting frequency.
This solution has already been proposed in [14].


## 3. Application: The Secondary Circuit of a NPP

To illustrate the benefits of the proposed approach,
experimental results obtained from the secondary circuit of a Nuclear Power
Plant (NPP) are provided in this section.In a NPP, the secondary circuit erosion can occur in the
steam generators, releasing radio nuclides into the secondary coolant. This
problem is now well understood and has been the subject of some studies initiated
by EDF (Eléctricité De France) for its pressurized water reactors (PWR) to
overcome and master the steam generator corrosion problems. The main
degradation process is to be controlled by careful optimization of the
secondary water chemistry. There is a need to ensure that the optimum secondary
chemistry regime is selected and maintained. So, the process of erosion—corrosion of steel piping and other components is of critical importance during
operation of a NPP. Feed waterpH is
adjusted by hydrazine, so that the
pH is maintained within the limits specified by the nuclear authority (norm
ISO-14253-1).The NPP under consideration is a900MW pressurized water
reactors (PWR), located in France.
During the winter 2002, and thanks to a mandatory period of maintenance
operations, it has been possible to measure and record a set of experimental
data on the secondary circuit. The aim was to optimally control hydrazine and pH through an adaptive LQG control
scheme. The designed controller has been successfully tested under real
operational conditions [18].The dynamics of hydrazine and ammoniac concentrations behavior in
the secondary circuit of the NPP can be expressed as follows:(39)Vddt[N2H4]=−Qext(t)⋅[N2H4]+u(t−τ)Vddt[NH3]=43Qext(t)([N2H4]−[O2])−β⋅[NH3]V is the circuit's
water volume,
Qext
the
water extraction flow rate of the condenser and β is a parameter depending of the NPP operating conditions. [N2H4],[NH3] and [O2] represent hydrazine,
ammoniac and oxygen concentrations respectively. u is the flow rates of the pumps used to inject hydrazine in the
circuit. Moreover, the system present a time delay (τ=560s) corresponding to the chemical reaction after
the introduction of hydrazine in the circuit. It is assumed that the pH is measured, that is,
(40)pH=14+12log(Kb[NH3]+KM0[M0])+npH,
where npH
denotes the measurement noise, [Mo]
also denotes the morpholine concentration, Kb
and
KMo
are the basicity
constants of the ammoniac and morpholine respectively.Taken into account the dynamical equations (39), it follows that
(41)x⋅(t)=(−βV4Qext(t)3V0−Qext(t)V)x(t)+(01V)u(t−τ)+(−4Qext(t)3V0)[O2](t)y(t)=(1001)x(t)+(nNH3(t)nN2H4(t)),
where x=([NH3][N2H4])Trepresents the state vector. nN2H4represents the measurement noise of the
hydrazine sensor. nNH3 is the image of the measurement noise of the
pH
sensor via the relation (40). Then,
since (40) is static we will consider that an ammoniac concentration measure is
available through the
pH
sensor. Characteristics
of nNH3 are then deduced from the following equation,
which is the inverse of (40):
(42)[NH3]+nNH3=102[pH−14−npH]−KMo[Mo]Kb.Figure5 presents the behavior of the time varying parameter Qext(t) during a period of 3 days. We assume here that Qext(t) is not affected by the considered faults. As
it can be seen on Figure 5, Qext(t) varies between
Qmin
and
Qmax
with:
(43)Qmin=878l/sQmax=1097l/s.Figure 5
The time varying parameterQext(t).The considered faults are hydrazine sensor faults andpH
sensor faults. Note that monitoring
of
pH
sensors is a key feature for
well functioning the overall system.The relation between the ammoniac measurement and thepH
measurement is only algebraic (see (41)),
then the
pH
sensor fault is directly transmitted
to the ammoniac measurement. Therefore, we can consider an ammoniac sensor
fault in place of a
pH
sensor fault.
Consequently, the following state space representation derived from (42) models
the failing behavior of the secondary circuit (the notations are chosen to be
consistent with those used in Section 2):
(44)G(θ):{x˙=(−βV4θ(t)3V00−θ(t)V2τV00−2V)x+(0−1V2)u+(−4θ(t)3V00)[O2]y=(100010)x+(00)u+(nNH3(t)nN2H4(t))d1+(1001)(nNH3(t)nN2H4(t))fNH3 represent ammoniac sensor faults and  fN2H4 hydrazine sensor faults. In this model, the
time delay due to actuators is approximated using a first order Pade
approximation. The model (44) corresponds to the model described in Figure 1,
where 878≤θ(t)=Qext(t)≤1097.
It follows that the considered polytope Θ={θ:878≤θ≤1097} becomes a simple segment since dim(θ)=1.For the FDI purpose, two filtersF1(θ) and F2(θ) are computed such that the two residuals r1(t) and r2(t) satisfy the following requirements:(i)
r1(t) is sensitive to
pH
sensor faults through the (fictitious) ammoniac sensor and
robust to hydrazine sensor faults.(ii)
r2(t) is sensitive to hydrazine sensor faults and
robust to ammoniac sensor faults.This method allows to isolate sensor faults uniquely. Therefore,
we consider a different model for each filter to be synthesized. For the design
ofF1(θ),
the following model is used
(45)G1(θ):{x˙=(−βV4θ(t)3V00−θ(t)V2τV00−2V)x+(0−1V2)u+(−4θ(t)3V00000000000)d1+(000)fNH3y=(100010)x+(00)u+(01000011)d1+(10)fNH3,
where the disturbances vector d1 includes the oxygen concentration, the measurement
noises and the hydrazine sensor fault, that is, d1=[[O2]nNH3nN2H4fN2H4]T.
For the design of F2(θ),
the following model is retained
(46)G2(θ):{x˙=(−βV4θ(t)3V00−θ(t)V2τV00−2V)x+(0−1V2)u+(4θ(t)3V00000000000)d2+(000)fN2H4y=(100010)x+(00)u+(01010010)d2+(11)fN2H4,
where the augmented disturbances vector takes into account the
ammoniac sensor faults;
d2=[[O2]nNH3nN2H4fNH3]T.According to the methodology developed in the Section2, two
polytopic models P1(θ) and  P2(θ) are built as illustrated on Figure 2. To save
place and for a better understanding, the different steps are only detailed for F1(θ).
Here, because the system is placed in an open-loop control law, it follows from G1(θ) that (for clarity the index “1” is ignored from now):
(47)P(θ):{x˙=(−βV4θ(t)3V00−θ(t)V2τV00−2V)x+(0−4θ(t)3V000−1V000020000)d+(000)fNH3(yu)=(100010000)x+(001000001110000)d+(100)fNH3,
where d=[u[O2]nNH3nN2H4fN2H4]T. Following the developments in Section 2, the fault detector design problem
turns out to be the design of (F(θ),My,Mu) satisfying the following objectives:
(48)∥Trd(θ)∥∞<γ1∥Trf(θ)∥sens>γ2
Figure 6 gives an illustration of this problem.Figure 6
FilterF(θ(t)) synthesized scheme.Finally, following the method describes in Section2, the
problem is recasted into the setup depicted in Figure 3 where the model P¯(θ,My,Mu) is defined according to:
(49)P¯(θ,My,Mu)=(−βV4θ(t)3V00−4θ(t)3V000000−θ(t)V2τV1V00000000−2τ2000000My1My20Mu0My1My2My2My1−1100001001001000011000001000000)My1
and
My2 are the two components of the structuring
matrix My.As it is outlined in Remark2 an important step in the
proposed method is the choice of the shaping filters Wd and Wf.
Similarly to the developments presented in Section 2,
Wd
is refereed to the robustness objectives against   d and
Wf
to the sensitivity requirements against fNH3.
Here, due to the definition of d, it
is natural to choose
Wd
such as:
(50)Wd=diag(Wu,W[O2],WnNH3,WnN2H4,W[N2H4]).
The weighting functions Wu, W[O2], WnNH3, WnN2H4, W[N2H4] and  Wf
allow to manage separately the robustness
objectives against u, [O2], nNH3, nN2H4, fN2H4 and fNH3 respectively. The interested reader can refer
to [10] or [19] if necessary. These weighting functions are deduced from an off
line spectral analysis procedure of available measurements according to:
(51)Wu=γu(1+1,7.103s)2(1+103s)2W[O2]=γ[O2]1+1,5.104s1+102sWnNH3=γnNH3s+2.10−2s+1.10−5WnN2H4=γnN2H4s+2.10−2s+1.10−5W[N2H4]=γ[N2H4]1+103s1+10−3sWf=γ211+103s.
The parameters γu, γ[O2], γnNH3, γnN2H4, γ[N2H4] and γ2 allow to manage the gain of the weighting
functions separately. They are optimized by performing an iterative refinement.
Remember that the goal is to minimize the effects of disturbances on the
residual r(t) and maximize the effects of faults on r(t).
The numerical values of them have been fixed to
(52)γu=0.025,γ[O2]=10−4,γnNH3=γnN2H4=0.1,γ[N2H4]=2.10−5,γ2=2.
The method described in Section 2.2.2 is then used to synthesize
the filter F(θ),
and the structuring matrices
My
and
Mu. For the SDP
optimization problem computation, the SDPT3 solver is used.To analyze the computed solution, the principal gainsσ¯(Tdrk(jω)) and σ̲(TfNH3r(jω)) of the closed loop transfers Tdrk(jω) and TfNH3r(jω) are plotted versus the objectives Wdk and Wf for some θ∈Θ (see Figure 7). The notation “k” is
introduced to outline that the analysis is performed with respect to each
component of  d. As it can be seen on the figures, for each synthesis,
σ¯(Tdrk(jω))<σ¯(Wdk(jω))∀ω and σ̲(TfNH3r(jω))>σ̲(Wf(jω))∀ω∈Ω≈[0;10−4[rd/s
for all considered values of θ(t).
This indicates that the requirements (48) are satisfied for the considered
values of θ and by virtue of Lemma 2, we know that it
still yields for all values of θ.Behavior of the principal
gains of the closed loop transfersTdrk(jω) and TfNH3r(jω) versus the shaping objective filters (in dB).
(a)(b)(c)(d)(e)Simulation Results
The FDI unit is implemented within the simulator of the  secondary circuit. For simulating faults, a variation of ten percent of sensors
measurements betweent=80
hours and
t=85
hours for the
pH
sensor and between t=120 hours and t=125 hours for the
hydrazine sensor is made. Figures 8, 9 and 10 illustrate the behavior of the
residual signals r1(t) and
r2(t) in both fault free and faulty situations for
the aforementioned period of 3 days. As expected, it can be seen from figures
that r1(t) is only sensitive to
pH
sensor faults and r2(t) is only sensitive  to hydrazine sensor faults.
Finally, a sequential Wald decision test is also implemented
within the simulator to make a final decision about the faults. The
probabilities of non-detection and false alarms have been fixed to 0.1%. The
results are presented in Figures8, 9 and 10. As it can be seen, all faults are
successfully detected and isolated.Behavior ofri(t), i=1,2 and the decision test-fault-free situation.
(a)(b)Behavior ofri(t),  i=1,2 and the decision test-pH sensor fault (80 h–85 h).
(a)(b)Behavior ofri(t), i=1,2 and the decision test-hydrazine sensor fault (120 h–125 h).
(a)(b)


## 4. Conclusion

The problem which is addressed in this paper is that of
designing FDI filters for dynamic systems that can be described by LPV
polytopic models. The method can be seen as a generalization of the well knownH∞/H−  setting for LTI systems. The H∞ norm for LPV systems is used to formulate the
robustness specifications and a new index, that is, the
Hsens
index, which is deduced from the H-norm for LTI
systems, is introduced for fault sensitivity specifications. As a result,
various design goals and trades-off can be formulated and managed in a
systematic way by means of some high level design parameters formulated in
terms of dynamic weighting functions. A key feature of the proposed technique
is that the remaining control and measurement canals are optimally merged to
build the fault indicating signals. The resulting static matrices are also
optimized via LMI together with the dynamic FDI filter. The proposed technique is
also appropriate for fault diagnosis in nonlinear systems which can be
approximated efficiently by LPV models to cover a wider range of operating, and
to cope with rapid parameter variations. The method has been successfully
applied to experimental data set coming from the secondary circuit of a nuclear
power plant in France.



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*Source: 231697-2007-12-25.xml*