Diagnoser Construction Procedure

2.4 DES Modeling and Fault Diagnosis Framework

2.4.3 Diagnoser Construction Procedure

event-based model as follows: τ₁ is fired from statex₁ to state x₂ due to the occurrence of “valve open” event (say, V E) at state x₁. So, τ₁ can defined as τ₁ =hx₁, V E, x₂i. In an event-based approach, a fault is generally represented using an unobservable event (say, event f). Therefore, its corresponding failure causing transition from a state x to another statex⁺, denoted as τf =hx, f, x⁺i also becomes unobservable. In event-based diagnosis approaches, fault diagnosis (detection and identification of the occurrence of a fault) is performed based on the observation of event sequences [103, 104]. Here, a system model G is said to be diagnosable for any fault event if its occurrence can be detected within a finite delay using the record of observed events. In a state-based ap- proach, the state set of the system can be partitioned according to the faulty status of the state [115]. Each state x is assigned a failure label defined by an unmeasurable status variable C ∈ S with its domain being {N, F₁, F₂, ..., F_k}, where F_i,1 ≤ i ≤ k, stand for permanent failure status and N stands for normal status. For example, let us consider the two states x1 and x5 of model G shown in Figure 2.5(b). Even though the measurable state variables (S₁, S₂, S₃) ofx₁andx₅ have same values, their unmeasurable status variable C has different values, that is, x₁(C) = N and x₅(C) = F. Therefore, the transition τ_F =hx₁, x₅i represents a failure causing transition and is unmeasurable.

In state-based approaches, fault diagnosis is performed based on the sequence of output measurements associated with the system states. The assumption on partitioning the state space of the system has two benefits [115]. First, this is particularly useful in cases where the failure might have occurred before the start of diagnosis. In these situations, a failure can be diagnosed by determining the faulty status of the states using the se- quence of measurements. Another benefit is that this framework simplifies the transition function of the diagnoser. Specifically, at each step, after receiving a new measurement, this approach only has to update the estimate of the system’s state as normal or faulty or uncertain, and thus it avoids label propagation as done in [103].

2.4 DES Modeling and Fault Diagnosis Framework

G as G-states, G-transitions and G-traces, respectively. Similarly, we use D-states, D- transitions and D-traces, respectively to represent the diagnoser states, transitions and traces.

The diagnoser is represented as a directed graph

G_diag =hZ, Ai (2.9)

whereZ is the set ofD-states, andAis the set ofD-transitions. EachD-statez ∈Z is a set ofGstates representing the uncertainty about the actual state and eachD-transition a ∈ A of the form hz_i, z_fi is a set of measurement equivalent transitions, representing the uncertainty about the occurrence of the actual measurable transition. The following definition is introduced to discuss the procedure for constructing the diagnoser from G.

Definition 2.4.6. Unmeasurable reach of a set of G-states: The unmeasurable reach of a setY ofG-states is the transitive closure (Kleene closure) of the unmeasurable successors of Y and is denoted as U^∗(Y), where the unmeasurable successor of a set Y of G-states is defined as U(Y) = S

x∈Y{x⁺|τ =hx, x⁺i ∈ =u}

Construction of the Diagnoser: The initial D-state (z₀) is obtained as U^∗(X₀).

Now, consider the construction of the transitions from a D-state z. Let =_mz denote the set of measurable G-transitions from the states x ∈ z. Let A_z be the set of all measurement equivalence classes of transitions obtained from =_mz. Corresponding to each of these classes, there is a transition a, emanating from z. For a transition a emanating from the D-state z, the successor state z⁺ via the transition a is computed in two steps: (i) first, a set z⁺_a is computed as the set {f inal(τ)|τ ∈ a}, (ii) the set z⁺ is then obtained as z⁺ =U^∗(z_a⁺). For each a ∈A, the initial and the final D-states are designated asinitial(a) andf inal(a), respectively. Therefore, from the above discussion, initial(a) = z and f inal(a) = z⁺. The set of the diagnoser transitions is augmented as A ← A∪ {a} and the set of states is augmented as Z ← Z∪ {z⁺}. It may also be noted that each D-state z 6= z₀ contains measurement equivalent states; z₀, however, may contain (initial) states, which are not necessarily measurement equivalent.

Diagnosability Analysis: Now, we introduce certain definitions and properties needed for diagnosability analysis on the diagnoser.

Definition 2.4.7. Embedding of G-traces in D-traces:. Given a D-trace γ = ha1, a2, ..., aki, a G-trace q, where P(q) = hτ₁, τ₂..., τ_ki, is said to be embedded in γ, if τ_i ∈ ai, 1 ≤ i ≤ k. The set of all G-traces embedded in a D-trace γ is represented as A_D(γ).

The fault label of any D-statez =hx₁, x₂, ...x_i, ...i is defined as z(C) = S

x∈zx(C).

Definition 2.4.8. Normal D-state:. AD-statez is callednormaland denoted as z_N, if z(C) ={N}; the set of all normal D-states is denoted as Z_N.

Definition 2.4.9. F_i D-state:. AD-statez is called anF_i D-state and denoted asz_Fi, if F_i ∈z(C). The set of all F_i D-states is denoted as Z_Fi.

Definition 2.4.10. F_i-certain D-state:. An F_i D-state z is called an F_i-certain D- state if z ⊆X_Fi.

Definition 2.4.11. F_i-uncertain D-state:. An F_i D-state which is not F_i-certain is called F_i-uncertain.

Property 1. If two traces q, y ∈ A_D(γ), where q is an F_i G-trace and y is a normal G-trace, then the D-states traversed by γ are F_i-uncertain.

Proof. The property also follows from diagnoser construction. As anyD-transitiona∈γ has a normal G-transition and a Fi G-transition (which are equivalent), so source and destination D-states of a are Fi-uncertain.

Therefore, an Fi-certain D-state contains only Fi G-states whereas an Fi-uncertain D-state contains both Fi G-states and normal G-states. So, a fault is diagnosed if the diagnoser reaches any F_i-certain D-state. Let consider a D-trace γ consisting of a sequence of F_i-uncertainD-states which is actually a composition of a normal as well as faulty G-trace (see Property 1 and Definition 2.4.7). If so, a fault cannot be diagnosed untilγ is exited.