The polynomial vector functionsfp2 andap2 are given by ap2(θp2) =
a2h2 a2t2
ah2at1ah1−at2a2h1+ 2a2h2at0−2ah2ah0at2
a1(θ) +a2(θ)
2ah2at2at0+ah2a2t1−2ah0a2t2−ah1at2at1
2ah2at2
,
where
a1(θ) =at2(a2h0at2+ah0ah1at1−at0a2h1)
a2(θ) =ah2(a2t0ah2+at0at1ah1−ah0a2t1−2ah0at2at0),
and
fp1(yp1) =
B2ω2r−2BTeωr+Te2 Y32
Bω3r−Teω2r ω4r Y3ωr2
−Y3Te+BY3ωr
.
ARR based on the set of constraintsCm3
The set of constraintsCm3 contains the following parameters and measurements de-scribed on vector form,
θp3 =¡
ah2 ah1 ah0
¢T
yp3 =¡
ωr Y3 Y4
¢T . The polynomial vector functionsfp3 andap3 are given by
ap3(θp3) =
−1
−at2
at1 at0
fp3(yp3) =
Y3
Y42 Y4ωr
ωr2
.
Moreover, the density of the liquid is not constant due to temperature changes and im-purities in the liquid. This density affects the value of the parameters in the hydraulic model of the pump given by (3.29). Therefore, parameter variations must be taken into account, for the developed algorithms to be robust and thereby usable in real applica-tions.
7.3.1 Robustness with Respect to Parameter Variations
The residuals are in this case calculated using functions on the formr=f(y)Ta(θ)as seen in Section 7.2.3. In these equationsr= 0when the parameter vectorθ=θ0and no fault has occured in the system. Hereθ0is the parameter vector used in the design of the residual generators. Robustness is now formulated as the problem of handling the residual generation problem whenθ∈Θ, whereΘis the set of possible parameter values includingθ0. This means thatr= 0is not necessary fulfilled in the no fault case anymore. However, in all real life systems it is assumed thatris bounded on the setΘ.
One way to treat the parameter variation problem, sketched above, is the use of a set-valued approach, as presented in for example (Tornill et al., 2000; Idrissi et al., 2001).
When using this approach a set of possible residual values is obtained. If0is not in this set it can be concluded that there is a fault in the system in spite of parameter variations.
This is formalized in the following definition,
Definition 7.3.1 The set of possible residual valuesRhas the following properties,
• If the system is not affected by faults i.e.e=0then0∈ R.
• If0∈ R/ then there are faults in the system i.e.e6=0 whereeis a given fault vector andR ⊂Ris a connected set.
Now letr =g(y,θ)be an ARR describing the behaviour of a given system, wherey contains the measurements andθcontains the parameters in the ARR. For this ARRr only equals0in the no fault case, if the structure and parameters of the system are known exactly. When, for such a system, parameter uncertainties are taken into account the set of possible residualsRdefined in Definition 7.3.1 is given by the following lemma, Lemma 7.3.1 If the set of possible parametersΘis a compact set, given by
Θ={θ|θi= [θi, θi], i= 1,· · ·, n}, (7.16) whereθ ∈Rn, and the residual functiong : Y ×Θ → Ris continuous onΘfor all y∈ Y ⊂Rm, then the set of possible residual values, in the no fault case, is defined by the maximum valuerand minimum valuerofR. These maximum and minimum values are given by
r= max
θ∈Θg(y,θ) r= min
θ∈Θg(y,θ).
This is a well known fact from mathematical analysis and the proof can be found in (Apostol, 1974). Using this lemma the boundaries ofRcan be calculated. Unfortunately it is not straightforward to calculate these maximum and minimum values. However, in the case treated in this work the functiong can be separated into a function of the parameters and a function of the measurements i.e.g(y,θ) =f(y)Ta(θ). When this is the case it is easy to see that one set of boundaries on the residual setRis given by
r≤rmax= Xk
i=1
maxθ∈Θ(ai(θ)fi(y))
r≥rmin= Xk
i=1
minθ∈Θ(ai(θ)fi(y)) .
From these expression it is seen that onlyaii= 1,· · ·, kis a function of the parameters θ. The maximum and minimum ofaican be calculated offline, leaving only a sign check offito be done online. This is expressed in the following lemma.
Lemma 7.3.2 Ifr = f(y)Ta(θ)wherea and f are two vector functions, which are continuous on respectivelyΘandY, then an upper (lower) boundary ofr(r) is given by
r≤rmax=X
i
ri,max r≤rmin=X
i
ri,min,
where
ri,max=
½ fi(y) maxθ∈Θ(ai(θ)) if fi(y)>0 fi(y) minθ∈Θ(ai(θ)) otherwise ri,min=
½ fi(y) minθ∈Θ(ai(θ)) if fi(y)>0 fi(y) maxθ∈Θ(ai(θ)) otherwise .
The upper and lower boundaries, found using this lemma, are fast and easy to calculate online, as the hard part of the calculations can be done offline. Unfortunately, the found upper and lower boundaries are in general not very tight. However, if the residual ex-pressionr=g(y,θ)is affine with respect to the parametersθan exact solution exists.
Ifgis affine with respect to the parametersθit can be rewritten to be come,
r=g(y,θ) =G0(y) +Gl(y)θ. (7.17) For an expression of this form the upper and lower boundaries can be found on an interval setΘusing interval algebra (Boukhris et al., 1998). Using this approach the exact maximum and minimum values of (7.17) can be calculated, in the no fault case, using the following lemma.
Lemma 7.3.3 For the parameter affine system (7.17) the maximum and minimum values of the residual setRcan be calculated exactly, and are given by
r=G0(y) +Gl(y)θ0+|Gl(y)|(θ−θ0) r=G0(y) +Gl(y)θ0+|Gl(y)|(θ−θ0),
whereθ0=12(θ+θ).
This is a well-known fact from interval analysis (Boukhris et al., 1998; Moore, 1979).
In the following, Lemma 7.3.2 and 7.3.3 are used to calculate an upper and a lower boundary of the residual expressions found in the previous section.
All the residual expressions derived in Section 7.2.3 are on the form r=f(y)Ta(θ),
where bothaandf are vector functions with only polynomial nonlinearities inθand y. As this is a nonlinear expression, Lemma 7.3.2 could be used to find upper and lower boundaries of the residual set. However, these upper and lower boundaries are in general far from the real maximum and minimum values ofR, which is also the case in centrifugal pumps. Therefore, the linear dependency between the parameters is extracted using a first order Taylor Series expansion ofa(θ). This Taylor Series expansion is given by
a(θ) =a(θ0) +£∂a
∂θi(θ0)¤
(θ−θ0) +O(θ−θ0).
When this expression is used in the residual equation the residual is calculated by the sum of two termsr=rl+rO. These terms are given by
rl(y) =¡
f(y)Ta(θ0)¢ +¡
f(y)T£∂a
∂θi(θ0)¤¢θ˜ (7.18a) rO(θ,y) =f(y)T³
O(˜θ)´
, (7.18b)
where˜θ =θ−θ0, andθ0is chosen such thatθ0= 12(θ−θ). It is immediately seen that (7.18a) has the same structure as (7.17), meaning that Lemma 7.3.3 can be used to calculate the maximum and minimum values of r1 in the no fault case. This does however not form the boundaries onrbecause of the higher order termsO(θ−θ0). The dependency onrof these terms is described by (7.18b). This expression is a nonlinear expression on the form treated in Lemma 7.3.2, which therefore gives the upper and lower boundaries on this expression.
Using interval algebra it is easy to find the boundaies ofrfrom the boundaries of rlandrO. This is formalized in the following algorithm, which is used for robust fault detection.
1. At each new sampleycomputef(y),G0(y) =f(y)Ta(θ0)and Gl(y) =f(y)T£∂a
∂θi(θ0)¤ .
2. Compute the maximum and minimum values rl and rl using Lemma (7.3.3).
3. Compute the boundariesrO,minandrO,maxusing Lemma 7.3.2.
4. Compute the boundaries of the residual set using, r≥rmin=rl+rO,min
r≤rmax=rl+rO,max
5. Compute the decision signalDusing, D=
½ 0 ifrmin≤0≤rmax
1 otherwise
Remark 7.3.1 This algorithm can calculate the boundaries of the residual set of one single residual expression. However, parameter dependencies between residuals, when more than one residual is considered, are not taken into account. Therefore, the identi-fication of incipient faults can create problems in some cases.
Interpretation of the algorithm
The idea of the algorithm is to find the possible variation on the residual value, which can be caused by parameter divergence form the nominal parameter values. This is illustrated in Fig. 7.1.
+ yi
θ 1
θ 2
Θ
r rl
r rl
yi rmax
rmin
rO,max
rO,min
Figure 7.1: Illustration of the different boundary values in the proposed algorithm.
To the left of Fig. 7.1 the set of possible parameter valuesΘ is shown. At each measurement vector yi this set of parameter values is mapped into a set of possible
residual values. In the right-hand side figure of Fig. 7.1 this set is defined byr,r. Now considering the developed algorithm. The residual set defined byrl,rlin Fig. 7.1 is calculated in step 2, andrO,min,rO,max are calculated in step 3. These values sum up to a set defined byrmin,rmax, which is in general conservative compared to the real residual set defined byr,r. Ifr= 0is not in betweenr,ror alternative not in between rmin,rmaxit is guaranteed that a fault has happened in the system.