The chapter starts by presenting some preliminaries on SA in Section 6.1. Then the state space realization, developed in this work, is described in Section 6.2. After that FDI on the centrifugal pump is considered. This is done by presenting the model of the pump in Section 6.3, followed by the results obtained using SA in Section 6.4. One of the re-sults of the SA is that the system can be splitted into two cascade connected subsystems, of which only the second is affected by the mechanical and hydraulic faults considered in this chapter. Therefore only an observer has to be considered for the first subsys-tem. This observer must observe the connecting variables between the two subsystems.
The design of this observer is described in Section 6.5. In Section 6.6 the design of the residual observers are considered. These observers are based on results obtained using SA on the second subsystem, and the realization theory developed in Section 6.2 of the chapter. Test results obtained on an industrial test-bench, which has been particularly developed for this purpose, are presented in Section 6.7. Finally, concluding remarks end the chapter.
c1 c2 c3
x1 x1 x2 x2 u
c4
(a) Graph representation.
K X
u x1 x˙1 x2 x˙2
c1 1 1
c2 1 1
c3 1 1
c4 1 1 1 1
(b) Table representation.
Figure 6.1: Structural representation of the connection between set of equations denoted constraintc1toc4, and the set of variablesZ={u, x1,x˙1, x2,x˙2}. HereK ={u}is the known variable, andX={x1,x˙1, x2,x˙2}is the set of unknown variables.
Assumption 6.1.1 (Blanke et al., 2003, Ass. 5.1) Any algebraic constraintc∈Cdefines a manifold of dimensionnc−1in the space of the variablesQ(c).
Assumption 6.1.2 (Blanke et al., 2003, Ass. 5.2) All the constraints inCare compati-ble.
Assumption 6.1.3 (Blanke et al., 2003, Ass. 5.3) All the constraints inCare indepen-dent.
The structural model ofS, defined in Definition 6.1.1, is a model describing the connec-tion between the variablesZand the constraints inCby a bi-partite graph. This graph is defined in the following definition.
Definition 6.1.2 (Structural model) (Blanke et al., 2003, Sec. 5.2) The structural model (or the structure) of the system (C,Z) is a bi-partite graph (C,Z,E) where E⊆C×Zis the set of edges defined by:
(ci, zj)∈Eif the variablezj∈Zappears in the constaintci∈C.
An example of such a bi-partite graph is shown in Fig. 6.1. Both a graphical and a table representation of the the graph are shown here. Using the bi-partite graphs defined in Definition 6.1.2 the structure of the model equations forming the set of constraintsCcan be analysed. Two important properties in this analysis are Reachability and Matching.
These are defined in the following definitions.
Definition 6.1.3 (Reachability)(Blanke et al., 2003, Def. 5.8) A variablez2is reach-able from a varireach-ablez1 if there exists an alternated chain fromz1 toz2 in the graph (C,Z,E). A variablez2 is reachable from a subsetX ⊆ Z\{z2}if the existsz1 ∈ X
such thatz2is reachable fromz1. A subset of variablesZ2is reachable from a subset of variablesZ1if any variable ofZ2is reachable fromZ1.
Definition 6.1.4 (Matching) (Blanke et al., 2003) A matchingM is a subset of the edgesEsuch that the endpoints of the edges have no common vertices, i.e.
∀ei, ej∈ M : ei= (a, b), ej = (α, β), thena6=α, b6=β.
Definition 6.1.5 (Complete matching) (Blanke et al., 2003) A matching is called com-plete with respect toCif|M|=|C|holds. A matching is called complete with respect toZif|M|=|Z|.
When there exist a complete matching with respect to the unknown variablesX ⊂ Z in the system, it means that, in almost all cases, these variables can be eliminated by rewriting the system equations. To ensure this property the notation of calculability is used. Calculability is defined in the following definition.
Definition 6.1.6 (Calculability) (Izadi-Zamanabadi and Staroswiecki, 2000) Let zi, i = 1,· · ·, p,· · ·, n be variables, which are related through a constraint ci, e.g.
ci(z1,· · · , zn). The variable zp is calculable if its value can be determined through the constraintci under the condition that the values of the other variables zj, j = 1,· · ·, n , j6=pare known.
When a matching of an unknown variable is done under the calculability constraint it is possible to calculate the given unknown variable in the space where the calculabil-ity condition is fulfilled. Unfortunately, this is not enough to state that a matching, where all the unknown variables are calculable, implies that all unknown variables can be eliminated. This fact will be considered later when the notation of causal matches is considered.
To connect the definitions given above to a system description as known from system theory, let a dynamic systemSbe described by the following set of equations,
S:
Cf : x˙d=fx(xd,xa,u) Cm : 0 =mx(xd,xa,u) Ch : y=hx(xd,xa,u) Cd : dxdtd = ˙xd
, (6.1)
where the set of variablesZ = xd∪x˙d∪xa∪u∪y. Zcan be decomposed into a set of unknown variablesX=xd∪x˙d∪xaand a set of known variablesK =u∪y.
The constaintsCof system (6.1) are given by the setC = Cf ∪Cm∪Ch∪Cd. The structural model of this system is shown in Table 6.1. HereG, Fi, Mi, Hi are boolean matrices describing the connections between the variablesZand the constaintsC. Iis the identity matrix andX is a diagonal matrix denoting thatxdcan only be calculated formx˙dup to an unknown but constant offset.
Table 6.1: Incident matrix of a structural graph. This structural graph is a graph of a minimal over-constrained system, which can be a subsystem of a larger system.
yT uT x˙Td xTd xTa fx 0 G I F1 F2
mx 0 M3 0 M1 M2
hx I H3 0 H1 H2
dx 0 0 I X 0
The structural model presented in Table 6.1 of the systemS can have, but does not always have, one of following three properties,
Definition 6.1.7 (Over-constrained graph)(Blanke et al., 2003, Def. 5.9) A graph (C,Z,E)is called over-constained if there is a complete matching on the variablesZ but not on the constaintsC.
Definition 6.1.8 (Just-constrained graph)(Blanke et al., 2003, Def. 5.10) A graph (C,Z,E)is called just-constained if there is a complete matching on the variablesZ and on the constaintsC.
Definition 6.1.9 (Under-constrained graph)(Blanke et al., 2003, Def. 5.11) A graph (C,Z,E)is called under-constained if there is a complete matching on the constaintsC but not on the variablesZ.
In the cases where the systemSfails to conform to any of the three above properties, it can be proven that there exists a unique decomposition of Sinto three subsystems (Blanke et al., 2003),
S+= (C+,Z+) S0 = (C0,Z+∪Z0) S−= (C−,Z+∪Z0∪Z−),
whereC = C−∪C0∪C+ andZ =Z−∪Z0∪Z+. For this decomposition the sub-systems(C+,Z+),(C0,Z0), and (C−,Z−)are over-constrained, just-constrained, and under-constrained respectively. On the over-constrained and just-constrained subsys-tems the notion of causality is important, as it is used to state the conditions for structural observability. Causality is defined in the following definition.
Definition 6.1.10 (Causality) (Blanke et al., 2003) A subsystem is called causal if there exists an alternating chain fromki ∈ Ktoxj ∈ Xfor all reachablexj ∈ Xand this chain is composed of only calculable matchings.
A special and very important over-constrained subsystem is defined in the following definition.
Definition 6.1.11 (Minimal over-constrained subsystem) (Izadi-Zamanabadi, 2001, Def. 4) A minimal over-constrained subsystem, Smin = (Cmin,Zmin) is an over-constrained subsystem with the following property:
|Cmin|= 1 +|Xmin| (6.2) where Xmin ⊆ Zmin are the unknown variables contained in the set of constraints Cmin. Additionally,Cmin⊆CandZmin⊆Z.
Such a minimal over-constrained subsystem contains information enough to derived ex-actly one residual. Therefore, if a set of minimal over-constrained subsystems is identi-fied in a system, and the matchings, which define each of these subsystems, are causal, a set of residual generators can be derived. When the residual generators are derived by eliminating all unknown variables in the subsystem they are called Analytical Re-dundant Relations (ARR). If each of these residual generators are sensitive to different subsets of the faults affecting the system this can be used for fault identification. This is formalized in the following two theorems.
Theorem 6.1.1 (Structural observability) (Blanke et al., 2003, The. 5.2) A necessary and sufficient condition for system (6.1) to be structural observable is that, under deriva-tive causality
1. all the unknown variablesXare reachable from the known ones, 2. the over-constrained and the just-constrained subsystems are causal, 3. the under-constrained subsystem is empty.
For systems which are structural observable as stated in Theorem 6.1.1, and where the over-constraint subsystem is non empty, it is always possible to identify a number of minimal over-constraint subsystems, as defined in Definition 6.1.11.
Let one of the constraintsϕ∈Cbe corrupted by a fault. Let the unknown variables containted in the constraint be given byXϕ=Q(ϕ). Then the following theorem states the conditions for the fault corrupting the constraintϕto be monitorable or detectable.
Theorem 6.1.2 (Monitorability) (Blanke et al., 2003, The. 5.3) Two equivalent neces-sary conditions for a faultϕto be monitorable are:
(i) Xϕ is structural observable - according to Theorem 6.1.1 - in the system (C\{ϕ},Z),
(ii) ϕ belongs to the structurally observable over-constrained part of the system (C,Z).
Algorithms exist, which can decompose a system S into all possible minimal over-constraint subsystems. For each of these minimal over-over-constraint subsystems the con-nectionnc =nx+ 1holds, wherenc =|C|is the number of constraints andnx=|X|is the number of unknown variables in these constraints (Zamanabadi, 2001; Izadi-Zamanabadi and Staroswiecki, 2000). It is possible to derive an Analytical Redundant Relation (ARR) for each of these subsystems. Each of these ARR can be used to gen-erate one residual, which is sensitive to a subset of the faults in the systemS (Blanke et al., 2003). The connection between the subsystem used in the derivation of a given ARR and the faults is described by Theorem 6.1.2.