Complete fault diagnosis of uncertain polynomial systems
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Conference Paper: A setbased framework for coherent model invalidation and parameter estimation of discrete time nonlinear systems
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ABSTRACT: This work introduces a unified framework for model invalidation and parameter estimation for nonlinear systems. We consider a model given by implicit nonlinear difference equations that are polynomial in the variables. Experimental data is assumed to be available as possibly sparse, uncertain, but (set)bounded measurements. The derived approach is based on the reformulation of the invalidation and parameter/state estimation tasks into a setbased feasibility problem. Exploiting the polynomial structure of the considered model class, the resulting nonconvex feasibility problem is relaxed into a convex semidefinite one, for which infeasibility can be efficiently checked. The parameter/state estimation task is then reformulated as an outerbounding problem. In comparison to other methods, we check for feasibility of whole parameter/state regions. The practicability of the proposed approach is demonstrated with two simple biological example systems.Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on; 01/2010  SourceAvailable from: JeanBernard Lasserre[Show abstract] [Hide abstract]
ABSTRACT: We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control con straints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (lin ear matrix inequality)relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assump tions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.04/2007;  SourceAvailable from: Jan Hasenauer[Show abstract] [Hide abstract]
ABSTRACT: Analysis and safety considerations of chemical and biological processes frequently require an outer approximation of the set of all feasible steadystates. Nonlinearities, uncertain parameters, and discrete variables complicate the calculation of guaranteed outer bounds. In this paper, the problem of outerapproximating the region of feasible steadystates, for processes described by uncertain nonlinear differential algebraic equations including discrete variables and discrete changes in the dynamics, is adressed. The calculation of the outer bounding sets is based on a relaxed version of the corresponding feasibility problem. It uses the Lagrange dual problem to obtain certificates for regions in state space not containing steadystates. These infeasibility certificates can be computed efficiently by solving a semidefinite program, rendering the calculation of the outer bounding set computationally feasible. The derived method guarantees globally valid outer bounds for the steadystates of nonlinear processes described by differential equations. It allows to consider discrete variables, as well as switching system dynamics. The method is exemplified by the analysis of a simple chemical reactor showing parametric uncertainties and large variability due to the appearance of bifurcations characterising the ignition and extinction of a reaction.Proceedings of International Symposium on Advanced Control of Chemical Processes ADCHEM09 (2009).
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Complete Fault Diagnosis Of Uncertain
Polynomial Systems
Philipp Rumschinski∗Jan Richter∗∗Anton Savchenko∗
Steffen Borchers∗Jan Lunze∗∗∗Rolf Findeisen∗,1
∗Institute of Automation Engineering, OttovonGuericke Universit¨ at
Magdeburg, Germany
∗∗Siemens AG, Industry Sector, Gleiwitzer Str. 555, 90475
Nuremberg, Germany
∗∗∗Institute of Automation and Computer Control, RuhrUniversit¨ at
Bochum, Germany
Abstract: The increase in complexity in process control goes along with an increasing need for
complete and guaranteed fault diagnosis. In this contribution, we propose a setbased method for
complete fault diagnosis for polynomial systems. It is based on a reformulation of the diagnosis
problem as a nonlinear feasibility problem, which is subsequently relaxed into a semidefinite
program. This is done by exploiting the polynomial/rational structure of the discretetime model
equations. We assume the measurements of the output and the input to be available as uncertain,
but bounded convex sets. The applicability of the method is demonstrated considering a two
tank system subject to multiple faults.
1. INTRODUCTION
Fault diagnosis methods aim at deciding whether a fault
has occurred or not, given some measured information.
The result of the diagnosis is then either used for mon
itoring purposes only, or to inform a subsequent control
readjustment step. Introductions to the most common ap
proaches for fault diagnosis are provided by books Blanke
et al. [2006], Ding [2008], Gertler [1998], Isermann [2006].
In literature fault diagnosis is subdivided in methods rely
ing on the analysis of signals (signalbased) and methods
incorporating a model of the considered process (model
based). The latter methods are often founded on consis
tency tests. Here the measurement data is compared with
the ability of a system model to reproduce exactly those
measurements Blanke et al. [2006] or on consistency tests
based on identified system parameters Isermann [2006].
In both cases the goal is to determine the set of models
consistent with the measurements (fault candidates). As
suming that for all faults a corresponding model is known
(closedworld assumption), a fault diagnosis algorithm is
said to be complete if the true fault is never excluded from
the set of fault candidates. Every complete consistency
based fault diagnosis method, starting from an initial fault
candidate set, seeks to iteratively exclude fault scenarios
that are inconsistent with the observations. If only one
fault remains, it is uniquely diagnosed. In general, it is
not possible to uniquely distinguish between all faults due
to some overlap in the inputoutput behavior of the corre
sponding models. However, it is clear that if two behaviors
belonging to these fault scenarios differ from another, then
there exists an inputsequence that permits distinction
between them (active fault diagnosis). This dependence is
intimately linked to the persistence of excitation condition
1Corresponding author rolf.findeisen@ovgu.de
encountered in system identification, which is however out
of the scope of this work.
Consistencybased approaches for fault diagnosis are avail
able for linear parameter varying systems Blesa et al.
[2007], for uncertain linear systems Combastel and Raka
[2009], Tornil et al. [2003], and for nonlinear systems sub
ject to biased uncertain measurements Planchon [2007].
Other approaches are based on residuals generated by
means of observers or Kalman filters and compared to a
threshold Theilliol et al. [2008], Videau et al. [2009], Zhang
et al. [2008]. Further fault diagnosis methods for nonlinear
systems are available in Aßfalg and Allg¨ ower [2006], Selmic
et al. [2009], Zhang et al. [2002].
In this work, we propose a setbased approach for fault
diagnosis for polynomial and rational systems in which we
directly aim to classify what fault situations are consistent
with the taken measurements. Our framework derives from
a parameter estimation and model invalidation approach
presented in Borchers et al. [2009], which is based on
formulating the regarded problem in terms of a nonlinear
feasibility problem. We extend this technique to fault
diagnosis, by reformulating the fault detection and fault
isolation problems in a similar way. Coupled with an
efficient semidefinite solution strategy of the feasibility
problem, we are able to provide conclusive proofs on
inconsistency of certain fault situations with respect to
the measurements. Under the assumption of a complete
description of the set of possible faults we can furthermore
isolate the corresponding fault candidates and guarantee
completeness of our method.
2. PROBLEM SETUP
In this contribution, we consider discretetime systems Mf
subject to a specific fault f ∈ F = {f0,f1,...,fnf}, where
f0is associated with the nominal (fault free) system. The
Proceedings of the 9th International Symposium on
Dynamics and Control of Process Systems (DYCOPS 2010),
Leuven, Belgium, July 57, 2010
Mayuresh Kothare, Moses Tade, Alain Vande Wouwer, Ilse Smets (Eds.)
MoMT4.5
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127
Page 2
behavior of these systems is described by polynomial or
rational difference equations of the form
Gf(xk+1,xk,wk,p) = 0,
Hf(yk,xk,wk,p) = 0.
(1)
(2)
Here xk ∈ Rnxdenotes the system states, p ∈ Rnpthe
model parameters and wk ∈ Rnw, yk ∈ Rnydenote the
measured input and output respectively.
For simplicity of presentation, we assume throughout the
paper that only a single fault affects the considered process
in the time horizon of interest and that for all faults a
corresponding model is known. Furthermore, we assume
the measurements to be unknownbutbounded and to be
given as convex sets such that measurement uncertainties
can be taken into account.
Definition 1. (Consistency). Consider a measurement Wk
of the applied input taken at timeindex k and a mea
surement Yk of the output of the considered process. A
model Mfis said to be consistent with the measurements
if wk∈ Wkand yk∈ Yk.
With Definition 1 we can state the following problems:
Problem 1. (Fault detection). A fault has occurred if the
model of the nominal case Mf0is inconsistent with the
measurements.
Problem 2. (Fault isolation). A fault f is a fault candi
date, if the model Mf is consistent with the measure
ments.
Note that consistency can online only be checked in a
necessary manner since only past measurements can be
taken into account, but not future ones.
3. FAULT DIAGNOSIS AS A FEASIBILITY
PROBLEM
In this section, we propose a reformulation of Problem 1
and Problem 2 as a nonlinear feasibility problem. There
fore, assume the following collections of measurements
Y = {Yk⊂ Rny,k ∈ T} and W = {Wk⊂ Rnw,k ∈ T} in
a certain time window T = {t0,...,te}. This time window
just specifies the time instances when a measurement was
taken. Furthermore, assume a candidate fault model Mf
to be given, as described in the previous section. We can
then gather all information in the following semialgebraic
equations
Ff(P) :
Gf(xk+1,xk,wk,p) = 0, k ∈ T,
Hf(yk,xk,wk,p) = 0,
p ∈ P,
xk∈ Xk,
wk∈ Wk,
yk∈ Yk,
k ∈ T,
k ∈ T,
k ∈ T,
k ∈ T,
(3)
where P,Xkdenote some given convex sets bounding the
parameters and the states, respectively. For instance such
bounds can be derived from the physical meaning of the
parameters or states (e.g. concentrations have to be non
negative), or from conservation principles. Note that these
bounds can be in general arbitrary large, but from a
practical perspective tighter bounds are preferable for the
proposed relaxation procedure.
Recall that the goal of the fault detection problem is to
show that under the allowed variations in p the measure
ments are not reproducible by the nominal model Mf0.
We denote therefore as feasibility problem the problem of
checking whether Ff(P) admits a solution or not.
If the feasibility problem does not admit a solution, then
there exists no input for which the model Mfis consistent
with the measurements Y,W.
Problem 1 and Problem 2 are transfered to
Proposition 1. (Fault detection/Fault isolation). If Ff(P)
does admit a solution, the fault f is a fault candidate, i.e.
Mf is consistent with the measurements.
However, it is in general not possible to determine an
exact solution of the feasibility problem Ff(P), due to
the nonlinearities of the model equations. But we will
show in the next section that it is possible to address a
relaxed version instead of the original feasibility problem
for polynomial/rational systems to give conclusive answers
to the problems included in Proposition 1. Note that as a
consequence of the relaxation the fault candidates will be
determined by elimination of all other possibilities.
4. PROBLEM RELAXATION
As shown in Kuepfer et al. [2007], Borchers et al. [2009]
for polynomial/rational systems it is possible to relax
Ff(P) into a convex semidefinite program. The method
used is based on an image convexification described in
Lasserre [2001], Ramana [1994]. Semidefinite programs as
a generalization of linear programs can then be efficiently
solved via interior point methods, e.g. with Sturm [1999].
In literature several approaches for reformulating Ff(P)
are known, i.a. Lasserre [2001], Parrilo [2003]. For the
purpose of this work a quadratic reformulation is chosen,
as it leads to SDPs of moderate size. For the sake of
completeness, we present a short overview of the necessary
relaxation steps following Borchers et al. [2009].
As a first step the original feasibility problem Ff(P) is
rewritten as a quadratic feasibility problem (QP). There
fore, we introduce a vector ξ ∈ Rnξ, consisting of a minimal
basis of monomials of the model and output equations (1)
(2), in the form
ξ = (1, xi, pj, wl, ym, xipj, xiwl, ...)T,
where the indexes i,j,l,m correspond to the respective
number of states x, parameters p, inputs w and outputs y.
Equations (1) can be transformed to
Gf
i(xk+1,xk,p,w) = ξTQi
kξ = 0, (4)
in which Qi
i is again the number of states. Apparently the same is
possible for (2) whereas i takes values in {1,...,ny}. Note
that if the model equations (1)(2) contain higher order
terms (products of lower degree monomials), additional
equality constraints of the form (4) have to be introduced.
k∈ Rnξ×nξis a symmetric matrix and the index
For simplicity of notation we redefine the index i such that
it covers the number of states nx, the number of output
equations nyand the number of additional constraints nd
as i ∈ I = {1,...,nx+ ny+ nd}.
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128
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The bounds describing the subsets P,Xk,Wk,Ykappear
ing in Ff(P) can be described as linear constraints
Bξ ≥ 0.
Here B ∈ R2(nξ−1)×nξprovides explicit upper and lower
bounds on all components of ξ except the first one.
Then Ff(P) can be rewritten as
Such a quadratic decomposition can always be found for
a polynomial/rational system (1)(2), but QPf(P) is of
course still nonconvex. However, by introducing X = ξξT
and relaxing the rank(X) = 1 and tr(X) ≥ 1 condition
into the weaker constraint X ? 0, see e.g. Parrilo [2003],
we get the convex semidefinite program
QPf(P) :
find ξ ∈ Rnξ
subject toξTQi
ξ1= 1,
Bξ ≥ 0.
kξ = 0, i ∈ I,k ∈ T,
SDPf(P) :
find
subject to tr(Qi
X ∈ Rnξ×nξ
kX) = 0,
tr(eeTX) = 1,
BXe ≥ 0,
BXBT≥ 0,
X ? 0,
i ∈ I,k ∈ T,
where e = (1,0,...,0)T∈ Rnξ. The relaxation process
will increase in general the solution space of Ff(P) and
therefore a fault could be wrongly included in the fault
candidate set. However, the true fault will never be ex
cluded from the fault candidates. Note that the redundant
constraints BXBT≥ 0 were added to reduce this effect
Lasserre [2001].
Since we are only interested in proving infeasibility of
Ff(P), an efficient approach is to consider the Lagrangian
dual Lf of the semidefinite relaxation.
Lf(P) :
max ω
subject to
?
+BTλ1eT+ BTλ2B + λ3= 0,
k∈T
?
j∈I
νj
kQj
k+ ωeeT+ eλT
1B+
λ1≥ 0, λ2≥ 0, λ3? 0,
k, ω are the Lagrangian multipliers corresponding
to the equality constraints in the semi definite program,
and λ1 ∈ R2nξ−1,λ2 ∈ R(2nξ−1)×(2nξ−1), λ3 ∈ Rnξ×nξ
those corresponding to the remaining constraints.
Theorem 1. If the Lagrangian dual Lf(P) is unbounded,
then Mf is inconsistent with the measurements.
The Lagrangian weakduality property and the relaxation
process guarantee that if the Lagrangian dual is un
bounded, then Ff(P) does not admit a solution Waldherr
et al. [2008].
(5)
where νj
5. PARAMETER ESTIMATION
Recall that a way for proving inconsistency of a model Mf
is to verify that the Lagrangian dual Lf(P) is unbounded.
But since we allow uncertainties in the parameters as
well as in the measurements it is very likely for a fault
resulting in a slow change in the system dynamics, that the
corresponding model Mfcannot be excluded immediately.
In such a case it might be necessary to estimate the system
parameters from the measurements. The same is true for
a fault resulting in a slow drift in one of the parameters.
The goal is then to approximate the subset Pc ⊆ P
of consistent parameters. We denote this approximation
asˆPc. Therefore, a subregion Q ⊆ P is tested via the
Lagrangian dual whether a consistent parameterization
is contained or not. The subset Pc is approximated by
systematically exploring subregions of P and cutting out
those that lead to an unbounded Lf(P), i.e.
ˆPc:= P\
?
Q⊆P : Lf(Q)→∞
Q.(6)
A possible way of systematically investigating the param
eter space is using a recursive bisection algorithm.
Algorithm 1. (Q∗= Outer − approximate(Mf,Q)).
if Lf(Q) is unbounded
then return Q∗= ∅
else if volume(Q) ≤ precision threshold δ
then return Q∗= Q
else partition Q into Q1and Q2,
i.e. Q1∪ Q2= Q and Q1∩ Q2= ∅
Q?
Q?
returnQ∗= Q?
fi
fi
1:= Outer − approximate(Q1)
2:= Outer − approximate(Q2)
1∪ Q?
2
In Figure 1 the outcome of Algorithm 1 is depicted. The
quality of the outerapproximation is directly dependent
on the chosen precision threshold δ, whereas a decrease of
δ results of course in an increase of computational effort.
Fig. 1. Result of the outerapproximation algorithm for
a consistent parameter region Pc (dark gray area).
Light gray areas do not contain consistent parameter
izations.
Note that in the case when the applied solver is not well
tuned, e.g. the solution is not converging fast enough
and the number of allowed iterations is too low, it might
also be necessary to implement this algorithm for proving
inconsistency.
6. FAULT DIAGNOSIS ALGORITHM
In the previous section, we have shown, that the set of
parameters Pcleading to a consistent behavior of a model
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129
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Mfcan be approximated. In this section we want to show
how the parameter estimation algorithm can be extended
to a complete fault diagnosis algorithm. As a first step
we have to introduce a way of dividing the measurement
collections Y and W into subsequences. This derives from
Borchers et al. [2009], but is used here for formalizing the
fault diagnosis algorithm and not only for reducing the
computational complexity.
We split the collection of measurements Y and W into
smaller collections
S = {Sj⊆ Y,j = 1,...,nS}
with a corresponding shortened time window Tj ⊆ T as
depicted in Figure 2.
(7)
Fig. 2. Split collection of measurements.
The consistent parameters Pc can then be bounded by
intersecting the estimates obtained for each individual
subsequence, i.e.
Pc⊆
?
j=1...nS
ˆPj
c,(8)
where
subsequence j. A direct consequence is of course that a
model Mfcan only be consistent with the measurements if
for all subsequences Sja nonempty consistent parameter
setˆPj
Hence it is sufficient to prove that one subsequence leads
to the empty set. In the case that only one subsequence
is considered the detectability of a fault consequently
depends on the size of the regarded subsequence.
ˆPj
c denotes the result of Algorithm 1 for one
ccan be found.
If we now specify the starting point of a shortened time
window with k and the length of the timewindow with j,
the fault diagnosis is given by
Algorithm 2. (ˆ F =FaultDiagnosis(F,k,j)).
initializeˆ F = F
if Fault − Detection(Mf0,k,j) == false
thenˆ F =ˆ F \ f0
display a fault has occurred fi
for fi∈ˆ F
if Fault − Detection(Mfi,k,j) == false
thenˆ F =ˆ F \ fifi
end
returnˆ F
function consistent = Fault − Detection(Mf,k,j)
Q := Outer − approximate(Mf,P)
if Q == ∅ then return consistent = false fi
if Q ⊆ P then return consistent = true fi
Theorem 2. Algorithm 2 is a complete fault diagnosis
algorithm, since the true fault f∗is never excluded from
the initial fault set F, i.e. f∗∈ˆ F.
The completeness of Algorithm 2 results directly from
Theorem 1. If we consider an initial fault set F a fault
f will only be excluded if and only if Ff(P) is infeasible.
At the same time Mf might be considered as consistent
due to the relaxation, even though Ff(P) does not admit
a solution. In other words if we denote the best possible
diagnostic result as F∗then
F∗⊆ˆ F.
7. EXAMPLE
In this section we will show the applicability of our method
considering the simple twotank system as described in
Blanke et al. [2006] and depicted in Figure 3.
Fig. 3. Twotank system.
We only consider the case that H1,H2are measurable, be
cause, as demonstrated in Blanke et al. [2006], measuring
only one of the heights results in a loss of diagnosability.
7.1 System description
The system consists of two tanks connected by a valve, an
inflow qP, an outflow q2and a possible leakage qL. H1,H2
denote the measured waterlevels. the maximum allowed
height hmaxfor H1is reached qP will be set to zero. All
parameters are given in Table 1 and are taken from Blanke
et al. [2006]. We assume for reasons of simplicity in the
remainder of this work that under operating conditions
the fill level H1 will always be greater or equal to H2.
If one would want to incorporate the case that H1< H2
than one could apply a strategy similar to Hasenauer et al.
[2009] by adding some discrete switching conditions. A
mathematical description of the system is then given by
the following nonlinear differential equations
˙H1(t) =1
A(qP(t) − qL(t) − q12(t)),
˙H2(t) =1
A(q12(t) − q2(t)),
(9)
(10)
with
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130
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qp(t) =
?¯ qp, H1(t) ≤ hmax,
?
H1(t) − H2(t), V12is open,
0, H1(t) > hmax,
?
(11)
qL(t) =
cL
0,
H1(t), H1(t) > 0,
H1(t) ≤ 0,
(12)
q12(t) =
?
?
c12
0,
?
?
V12is closed,
(13)
q2(t) =
c2
0,
H2(t), H2(t) > 0,
H2(t) ≤ 0.
(14)
The equations (12)(14) contain nonpolynomial parts,
therefore, we extend the model with three additional states
and three additional constraints
∆H2(t) = H1(t) − H2(t),
H2
1(t) = H1(t)H1(t),
H2
2(t) = H2(t)H2(t).
This approach of approximating the nonlinearities might
not be suited for other nonlinearities (e.g. exponential
functions) or for other measurement setups. In such cases
stricter constraints have to be applied, e.g. enveloping
the nonlinearities by means of polynomial functions, for
further details see Hasenauer et al. [2009].
(15)
(16)
(17)
As our method requires the considered models to be
in discretetime, we apply Euler discretization to the
equations (12)(14) with a step size of 2 seconds.
Table 1. Nominal parameters
Parameter:
A
hmax
unom
c12
c2
cL
¯ qP
Value: Description:
Area of both tanks
Height of both tanks
Nominal pump velocity
Flow constant valve V12
Flow constant of the outflow
1.54 · 10−2m2
0.6m
1
6 · 10−4m5/2s−1
2 · 10−4m5/2s−1
2.6 · 10−4m5/2s−1Flow constant of the leakage
1.5 · 10−4m3s−1
Flow constant of pump
7.2 Scenario and Setup
We study the presented approach in a series of simulation
studies. To get a realistic setup the parameters are not
assumed to be known a priori, but are first estimated
following the algorithm proposed in Section 5. The con
sidered case is depicted in Figure 4, we performed it by
simulating the temporal evolution of the two states with
two slightly different initial conditions for the lower and
upper bound (H1(0) = 0.275m,H2(0) = 0.0375m, and
H1(0) = 0.325m,H2(0) = 0.0625m). We also added to the
bounds an additional absolute error of 1.2cm. The results
of the parameter estimation are given in Table 2.
Table 2. Achieved parameter bounds
Parameter: Lower bound: Upper bound:
c12
5 · 10−4
c2
1 · 10−4
¯ qP
0.5 · 10−4
In the following, we consider four different scenarios con
cerning the measurements. For this reason, let us consider
the measurement collection Y∗:= {Yk = (Hk
k ≤ 300}, with each measurement providing information
on both states. If we split the measurement collection,
7 · 10−4
3 · 10−4
2.5 · 10−4
1,Hk
2),0 ≤
Fig. 4. Measurements taken of the two states from the
faultless model. The red lines give the upper and lower
bounds on the measurements of H1 and the dashed
blue lines the bounds on the measurements of H2.
following (7), into subsequences Sj
with ? ∈ {1,2,4,9}, we can investigate how many time
steps after a fault f has occurred the fault can be de
tected/isolated. Two different fault scenarios are consid
ered: First (f1), the valve V12 gets stuck in the closed
position or the flow through it is obstructed suddenly at
timestep k = 150 (Figure 5) and second (f2) the leakage
qLoccurs at timestep k = 50 (Figure 6).
= {Yj,...,Yj+?}
Fig. 5. Fault f1occurs on time step 150.
Fig. 6. Fault f2occurs on time step 150.
7.3 Simulation results
Table 3 shows the number of timesteps until a fault is
detected and isolated. The number of considered mea
surements is apparently deciding the time necessary for
detecting/isolating the fault. An interesting observation
is that if only 2 measurements are considered at once, a
detection of the second fault is not possible before the
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131
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Table 3. Necessary time steps
Fault:
Number of timesteps k:
? = 1? = 2
92

Description of faults:
? = 4
1
2
? = 9
1
1
f1
f2
Valve V12is closed
Tank 1 is leaking
new steadystate is reached. This implies that one has to
carefully choose the amount of considered measurements.
Also, as noted in Blanke et al. [2006], the detection of f2
is more difficult then the detection of f1. This seems to be
a result of the less drastic change in the output measure
ments. In addition fault f1can still be detected when even
larger errors in the measurements are assumed (results not
shown). One can conclude that if the measurements would
not allow a certain precision, i.e. the error is (very) large,
a detection/isolation is not possible.
8. CONCLUSIONS AND OUTLOOK
We have studied in this contribution fault diagnosis for
a quite general class of process control models. Based on
an existing setbased parameter estimation, we proposed
a solution method to the fault detection and isolation
problems that is complete under the closedworld assump
tion. The method furthermore provides conclusive results
even if the measurements and the model parameters admit
uncertainties. We demonstrated for the wellknown two
tank example, that our approach is capable of determining
which of the considered fault situations are exhibited by
the plant.
For the considered class of uncertain polynomial/rational
systems we were able to show that the fault detec
tion/isolation tasks can be reformulated as a nonconvex
feasibility problem. Additionally, we have shown that it
is sufficient to address a relaxed convex version of this
feasibility problem and still achieve conclusive results.
With the help of this socalled semidefinite program we
could derive an efficient algorithm for fault diagnosis. This
algorithm is complete since the true fault is never excluded
from the set of fault candidates. Furthermore, we proposed
a method for reducing the computational complexity.
In practice, even with the proposed reduction technique,
the number of resulting problems might be too large for
very complex processes, especially if the direct diagnos
ability of the faults cannot be guaranteed. A combination
of the method with a state prediction scheme could then
be used to limit the number of fault models which has to
be addressed simultaneously. For instance, if more than
one fault model is consistent with the measurements a
investigation of the reachable state sets for all models
could help discarding models as soon as the next measure
ment arrives and thus reducing immediately the number
of possible fault situations. Such a prediction could also
be used for finding a specific input sequence that allows
to discriminate fault alternatives (active diagnosis). Both
extensions will be subject of future work. Furthermore,
it might be possible to extend the proposed framework to
continuoustime models as shown in Lasserre et al. [2008].
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