2.4 DES Modeling and Fault Diagnosis Framework
2.4.2 Fault Modeling
Each statexis assigned a failure label defined by an unmeasurable status variableC ∈S with its domain = {N, F1, F2, ..., Fk}, where Fi,1 ≤ i ≤k, stand for permanent failure status and N stands for normal status. Therefore, a state in the model may either denote a normally operating status of the system or a faulty status. The set of all normal G-states is denoted as XN, where xN ∈ XN represents a normal G-state. We assumek types of fault in the system. A state is denoted asxFi if it is a faulty state due to the fault of type i. For a normal G-state xN, xN(C) = {N}. Similarly, for a failure state (or synonymously, an Fi G-state) xFi,xFi(C) ={Fi}. The set of all states x such
that Fi ∈ x(C) is denoted as XFi. We use the words fault and failure interchangeably because here, the fault or failure of a system mean the same. Simultaneous occurrence of more than one fault is not considered here.
In fault modeling, a G-transition τ = hx, x+i is called a normal G-transition if x, x+ ∈ XN. The set of all normal G-transitions is denoted as =N. A G-transition τ =hx, x+iis called an Fi G-transition ifx, x+ ∈XFi. The set of allFi G-transitions is denoted as=Fi. Similarly, a G-trace q is called normalG-trace if all transitions inq are normalG-transitions. If all transitions in aG-traceqareFi G-transitions thenqis called Fi G-trace. A failure causing transition τFi = hx, x+i, where x(C) 6= x+(C) indicates the first occurrence of some failureFi. Since failures are assumed to bepermanent, there is no transition from any xFi to any xN.
A DES model G is said to be diagnosable if it is always possible to determine the faulty status of the states after the occurrence of a fault, using the sequence of mea- surements. Let Ψ(XFi) ={q|q ∈Lf(G) and f inal(q)∈XFi and q ends in a measurable transition}.
Definition 2.4.5. Fi-Diagnosability: A DES model G is said to be Fi-diagnosable under a measurement limitation for fault Fi if the following holds
(∃n ∈N)[∀q ∈Ψ(XFi)](∀r∈Lf(G)/q)[|r| ≥n⇒D] (2.8) where the condition D is ∀u∈P−1[P(qr)], f inal(u)∈XFi.
This definition is taken from the literature, [103, 104]. It means: let q be any finite prefix of a trace ofG that ends in an Fi-state and let rbe any sufficiently long continu- ation of q. Condition D then requires that every sequence of transitions, measurement equivalent with qr, shall end into an Fi-state. This implies that, along every continua- tion r ofq, one can detect the occurrence of faultFi within a finite delay, specifically in at most n transitions of the system after q.
Example: Consider a benchmark system consisting of a pump, a valve and a controller (Figure 2.5(a)). Assume that the system is equipped with a valve flow sensor and let
2.4 DES Modeling and Fault Diagnosis Framework
its outputs be no flow and flow. The state-based DES model of this pump-valve system, defined as G=hX, S,=, X0i is shown in Figure 2.5(b).
Pump
Valve Controller
Flow sensor (a)
x1
S1=0,S2=0, S3=0
N
x2
S1=1,S2=0, S3=0
N
x3
S1=1,S2=1, S3=1
N
x4
S1=1,S2=0, S3=0
N
x5
S1=0,S2=0, S3=0
F
x6 S1=1,S2=0,
S3=0 F
x7
S1=1,S
2=1, S3=0
F
x8 S1=1,S
2=0, S3=0
F 1
F
2 3
4
5 7
8 6
(b)
Figure 2.5: (a) Pump-valve system; (b) its state-based DES model.
Now, we illustrate various notions corresponding to this state-based DES modeling formalism in detail. In a state-based approach, the state set of the system is partitioned according to the failure status of the state [115]. We assume a “Stuck Closed” failure of the valve (say, fault F) in the system. The pump and controller are assumed to be fault-free. The occurrence of the fault F in G is represented through an unmeasurable transition τF. Therefore, the state set X of G is partitioned as X = XN ∪XF, where XN = {x1, x2, x3, x4} and XF = {x5, x6, x7, x8}. X0 = {x1}. S = {s1, s2, s3, C} is the set of state variables associated with each statexofG. Here, the state variabless1, s2, s3 denote status of the valve (open (1)/closed (0)), status of the pump (on (1)/off (0)) and readings of the flow sensor (flow (1)/no flow (0)), respectively, andC is the failure label with its domain being {N, F}. Sm ={s1, s2, s3} and Su = {C}. All transitions except τF are measurable. Here, transitions τ1 and τ5 are measurement equivalent, i.e., τ1Eτ5
due to x1Ex5∧x2Ex6 (see, Definitions 2.4.1, 2.4.4).
Note: There has traditionally been two distinct streams of works related to fault diag- nosis based on whether the DES modeling methodology is event-based or state-based.
The transition τ1 of model G shown in Figure 2.5(b) can be equivalently defined in an
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, ..., Fk}, where Fi,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 τF =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].