Fault candidates

If the number of fault candidates exceeds the number of sensors, structured residuals cannot be obtained. In that case, the FSM is non-square, rows, i.e. some component fault signatures are identical and not all faults can be isolated. More faults may be isolated by adding more detectors to the model if the real system permits to attach more real sensors. 

In online fault detection, ARR residuals are close to zero for a healthy system. Generally, they are not identical to zero for various reasons such as modelling uncertainties, uncertain parameters, noise, or numerical inaccuracies. For correct online fault detection it is important that true faults are reliably detected and false alarms are avoided. To that end, residuals are fed into a fault decision procedure. The result is a coherence vector. If this vector is a null vector, then the system is healthy, no fault has happened. If some of its entries are non-zero, then the coherence vector is compared with the rows of the structural FSM. Given a single fault hypothesis, the fault is isolated if there is a match with one row of the FSM. If there is more than one match then the detected fault cannot be isolated. Also, if the number of fault candidates exceeds the number of sensors, not all faults can be isolated. Isolation of multiple simultaneous faults by means of parameter estimation is considered in Chap. 6. 

Let m denote the number of component parameters. If the number m < m oi parametric fault candidates exceeds the number n of sensors then a set of structured ARRs in which each ARR is sensitive to only one parameter cannot be achieved. That is, the FSM is not diagonal. Some rows in the FSM will have the same component fault signature so that some faults cannot be isolated. This result cannot be improved with regard to a further isolation of faults if the real system does not permit to add more sensors. That is, faults cannot be structurally isolated. 

For these reasons, in the following, it is assumed that no parametric faults happen during system mode identification and that an initial system mode is known. System mode identification in the presence of faults is more difficult. In [4], Arogeti et al. present an advanced method for this more general case that categorises ARRs into different types and provides a refined set of fault candidates to the fault parameter estimation procedure. Multiple fault detection, isolation and identification for hybrid systems with no available information on the nature of faults (abrupt or incipient) and on system mode changes has been recently addressed in [5],... 

In the previous section, each discrete switch state has been taken into account by a row in the FSM. System mode changes can be viewed as faults in discrete switch states. This means that there are far more fault candidates than sensors. Accordingly, discrete switch states will share the same component fault signature so that a switch fault cannot be isolated. If multiple simultaneous faults have happened only in parameters that change continuously with time then parameter estimation can be used to isolate them. However, if there are discrete switch state faults among the multiple simultaneous faults then parameter estimation may result in meaningless real values for the discrete switch states. Therefore, in [6], Alavi and her co-authors propose to chose a combination of switch states, to insert them into the functional to... 

Assume that online monitoring and fault detection produce a coherence vector c = [1 0] in the time interval 0 < < 10 s when the circuit is in mode b = 0. A comparison of the coherence vector with the FSM in Table 9.2 reveals efficiency p and resistance Ri as possible fault candidates. Parameter estimation can identify p as a true fault. However, the incipient fault in R2 starting simultaneously with the fault in /3 at = 5 s cannot be detected. Residual ri is not sensitive to a fault in R2 and the term /(/ on + Riit)) in ARR2 and ARR3 is cancelled out by the switch state b... 

If component parameters in a system mode share a fault signature in the FSM then a fault can be detected but not isolated by simple inspection of the FSM. In case online fault detection provides a coherence vector that matches with more than one row in the FSM in a system mode, the result is a set of potential fault candidates. One way to identify multiple faults is to perform parameter estimation by Gauss-Newton least squares output error minimisation. Bond graph modelling can support this approach to multiple fault isolation by providing ARRs. Their residuals are used in the functional to be minimised. This has been discussed in Chap. 6. 

Thus the initial fault set detected from the thermal domain tree is Qp, Q, R, and P. However, the initial fault hypothesized from the hydraulic domain analysis and. The common hydraulic fault in both these sets is Qp and it too has same qualitative state. The final fault candidate list is Qj, Pj , and P. Since single fault hypothesis is considered, Qp should be the cause of the fault. 

The TCG can be traversed in both forward and backward directions from an observed or hypothesized fault. The backward propagation is used to construct a list of fault candidates (fault hypothesis), whereas the forward propagation derives predictions for posteriori behavior (temporal evolution) for hypothesized faults. 

We assume that the fault is abrupt. Then, for each fault candidate, we generate the qualitative trend (QT), i.e., change in magnitude, slope, and so on, of the output immediately after inception of the fault. The forward propagation of the TCG takes account of temporal delays encountered in the path. 

As an example, consider the two-tank system shown in Fig. 7.16 with a constant delivery pump. We assume that the flow through valve is proportional to the pressure difference. Four fault candidates are C C//, R Rv, C Ch2 and R Rv for which unknown input models are given in order in Figs. 7.26, 7.27, 7.28, and 7.29. 

For instance, consider the double fault case. In this case, the level sensor is lowly biased at 100 s (LS-BL). After 100 s, the reactor temperature is highly biased (TS-BH). The symptom of L(-) is detected at 105 s, CA(-) at 175 s, CB(+) at 200 s, T(+) at 205 s, and TR(-) at 215 s. The diagnosis procedure will be explained with the data of 215 s. The initial fault candidates set is LS-BL, RX-LK, TS-BL, TS-BH, FS-BH, CW-TCL) and TS-BL and FS-BH is removed because T(-) and FR(+) are not detected. Therefore, the initial fault candidates set is (LS-BL, RX-LK, CW-TCL, TS-BH. Because LS-BL among these faults has the biggest nES of 4, LS-BL is selected as first fault candidates. In the next step, the fault sets of CW-TCH, TS-BH is assigned to LS-BL. The final fault candidate is two double-fault of LS-BL, TS-BH, LS-BL, CW-TCL. In this example, though the diagnosis is later than the time of fault occurrence, it is accurate. The qualitative method failed to obtain the true solution because it is masked multiple-... 

To measure the diagnostic performance, three parameters are used. Accuracy is 1 if the diagnosis is accurate, that is, a true fault is involved in the final fault candidates set. Otherwise, accuracy is 0. Robustness is the number of wrong detected symptoms independent to the true faults. When accuracy is 1 and robustness is 0, resolution denotes the number of final fault candidates. 

In table 2, the former is average over the total results taken by the frequency of 5 s during the simulation period of 2000 s, and the latter is the worst result during all diagnosis periods. Table 2 shows the diagnosis result of the selected double-fault except a failed case and 24 cases of the worst resolution under 2. Case 20 of FV-BL and WP-BK shows the worst resolution of 8. In the case, the primary fault candidates of VT-BL, WP-BK and VR-BL, FS-BL, PUMP-EF, RP-BK from the detected residuals of FW(-) and FR(-) are obtained, respectively. Thus, 8 (2 X4) double-faults became the final fault candidates. 

Nica et al. [16] propose a method for combining debugging, testing and mutants to reduce the set of possible fault candidates. Contrary to our work, they use white-box methods by mutating the faulty code, they try to find mutants behaving correctly, while we mutate the correct model, trying to find mutants that show the same faulty behavior as the implementation. 

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