HFDM Structure, Terminologies and Basic Definitions

The proposed hierarchical decomposition is represented with a generic directed acyclic graph structure as shown in Figure 5.1 where the entire system under con- sideration is represented with a single node at the highest level that consists of subcomponents located at the lower levels. In this chapter, we identify this struc- ture as a hierarchical fault diagnosis model (HFDM). We denote the p-th component at level l in the hierarchy as Cl

Figure 5.1: Directed graph representation of the proposed hierarchical fault diagno- sis model (HFDM).

system, for l = 1, C1

p would correspond to a sensor or an actuator whereas for

l = L, C₁L would correspond to the “satellite formation component”. Let the set of all components located in two levels namely, l1 and l2, in the proposed hierar-

chical organization be denoted as Cl1

p and Cql2, respectively; where p = 1, 2, ..., Pl1;

q = 1, 2, ..., Ql2; l1, l2 ∈ {1, 2, ..., L}; with L denoting the total number of hierarchy

levels. Therefore, when l1 = l2 = l, two different components located at the same

level l are represented as Cl

p and Cql, assuming Pl = Ql as shown in Figure 5.1. For

any Cl

p, the sets of components that are parents of Cpl (as represented in Figure 5.1)

are denoted by pa(Cl p).

For fault diagnosis/isolation of physical systems, relationships among observed symptoms and faults are necessary. Symptoms are the manifestations of the faults in the available diagnostic signals (measured from the process or analytically calcu- lated). In other words, symptoms are obtained by monitoring some pre-identified diagnostic signals that are of one of the following types [19]: (1) residuals that are generated based on the system models, (2) binary or multi-valued signals that are resulting from residual classification/quantification, (3) statistical parameters or fea- tures that are describing signal properties, and (4) process states and/or variables (measured or calculated).

In this thesis, we utilize features that are extracted from process states and/or variables as diagnostic signals. Furthermore, a fault manifestation corresponding to a specific fault in this thesis is considered to be some pre-identified value(s) of a diagnostic signal that indicates the presence of that fault. The fault signature is represented in the form of rule(s) that identifies all the fault manifestations of interest for that specific fault. A fault signature model of a component refers to a set of rules that corresponds to all the faults that are to be identified in that component. Diagnostic signals, fault manifestations, and fault signatures are formally specified in Section 5.2.3.

The set of Pl components located at level l is denoted as Cl. The compo-

nents, manifestable faults, diagnostic signals, fault manifestations, and fault signa- ture models of the p-th component at level l are denoted by C_pl, F_pl, S_pl, M_pl, and F SM_pl, respectively; where p = 1, 2, ..., Pl. The fault signature model (F SMpl) of

Cl

p consists of fault signatures that correspond to both types of faults — the ones

that are originated at Cl

p and the ones that are originated at some lower level but

manifested at C_pl. It should be noted here that if pa(C_pl) 6= ∅, a fault f_kl ∈ Fl p and

its corresponding fault signature in F SM_pl may represent more than one fault that originates at some lower levels which will be formally stated in Proposition 5.2.2 in Section 5.2.4.

We distinguish the components within the proposed hierarchical decomposi- tion according to the following definitions:

Definition 5.2.1 (Independent Component). For any given fault severity level under consideration, a component Cll

p is independent of another component Cql2,

where l1 > l2, if a change in the diagnostic signal value(s) in Cql2 never manifests or

leads to a change in the diagnostic signal value(s) in Cl1

p due to the presence of any

fault fl3

k; where l2 ≥ l3.

under consideration, a component Cl1

p is dependent on another component Cql2,

where l1 > l2, if a change in the diagnostic signal value(s) in Cql2 always manifests

or leads to a change in the diagnostic signal value(s) in Cl1

p due to the presence of

some fault fl3

k; where l2 ≥ l3.

The set of all components on which Cl

p is dependent is denoted as DEP (Cpl).

Definitions 5.2.1 and 5.2.2 imply that we represent the dependencies among the different components within a particular level l at the next higher level l + 1. Note that a component C_pl is dependent on itself; consequently, C_pl ∈ DEP (Cl

p). The

following assumption regarding the component dependencies is now made explicit. Assumption 5.2.1 (Component Dependency). Component dependencies are not considered for diagnosis at a component Cl

p with pa(Cpl) = ∅.

Note that according to Definitions 5.2.1 and 5.2.2, dependencies of components and fault manifestations are directly related to each other. Furthermore, within an independent component Cl

p, it is useful to distinguish faults that are originated at

Cl

p and those that are originated at some lower level but manifested at Cpl. We

decompose an independent component C_pl into two virtual subcomponents, namely (1) independent subcomponent of Cl

p which is denoted by D(Cpl), and (2) dependent

subcomponent of Cl

p which is denoted by U (Cpl) as shown in Figure 5.2. The faults

that are originated at Cl

p correspond to D(Cpl) and those that are originated at

lower level(s) but manifested at C_pl correspond to U (C_pl). Once a component C_pl is decomposed into the subcomponents D(Cl

p) and U (Cpl), the subcomponents are

treated as independent and dependent components, respectively. The dependencies among components are represented by directed arcs as shown in Figures 5.1 and 5.2. To represent dependencies, where Cl1

p is dependent on Cql2, we add arcs from Cql2 to

Cl1

p and represent the arcs with an nK0 × n_K link matrix Ll2,l1

q,p where nK0 and n_K

are the number of manifestable faults in Cl2

Figure 5.2: Decomposition of a component into “independent subcomponent” and “dependent subcomponent”.

the Ll2,l1

q,p is denoted by Llq,p2,l1(k

0_{, k), where k}0 _{= 1, 2, ..., K}0 _{and k = 1, 2, ..., K. In this}

chapter, we assume that the dependencies are known with certainty. Consequently, given that a fault fl2

k0 ∈ C_ql2 manifests in C_pl1 and identified as f_kl1 ∈ C_pl1, then

Ll2,l1

q,p (k 0

, k) = 1. If two components are not dependent given the faults fl2

k0 and f_kl1,

then Ll2,l1

q,p (k

0_{, k) = 0.}

It follows from Definition 4.2.1 and the above discussion that the signature of an level l fault is not possible to be observed at lower levels, and the origin of an level l fault is in one of the “independent subcomponents” D(C_pl). It is important to note that in the graphical representation of Figure 5.2, pa(U ) 6= ∅ and pa(D) = ∅, and it is possible that a component Cl

p where l > 1 does not have any “dependent

subcomponent”. In practical sense this is possible due to lack/unavailability of an in-depth information about a subsystem or a component.

We propose to perform fault diagnosis at individual components in the hierar- chical fault diagnosis model (HFDM) which are represented by nodes in Figures 5.1 and 5.2. We now formally define a fault diagnosis module (FDM) that corresponds to a node or component C_pl as follows:

Definition 5.2.3 (FDM of a Component). A fault diagnosis module (FDM) corresponding to a component Cl

p is denoted by F DM (Cpl) = (Spl, Fid,pl , Rlp,K, Olp,

Al

p); where the diagnostic signals Spl are the inputs to the system, the identified

rules Rl

p,K that describe the relations among the K faults and their corresponding

symptoms, a set of operators Ol

p that are utilized to compute the rule activation

levels, and a set of assumptions Al

p that are made during the rule synthesis.

As described in Section 5.2.3, the rules Rl_p,K are obtained from the correspond- ing fault signature model F SMl

p.