Single fault hypothesis

Note that it depends on the number of sensors and where they are placed whether parametric faults can be isolated. Under the assumption of a single fault hypothesis and N given sensors, the maximum number of parameter faults that can be isolated is equal to 2 — 1. However, often, the number of sensors, N, is less than the number of component parameters, p, so that the FSM is not quadratic. Some component parameters may have the same component fault signature (in some modes) so that faults in these parameters cannot be isolated by inspection of the all-mode FSM. Clearly, additional sensors may improve the isolability of faults. But detectors can be placed in a model only for those variables that are accessible by real sensors in the real system. Even if quantities can be measured, cost considerations may suggest to limit the number of real sensors. For illustration, inspection of ARRs (4.6)-(4.8) yields the FSM in Table 4.1. 

In online FDI, the coherence vector is computed at every sampling step. If it is not a null vector, a fault is detected and an alarm is raised. Clearly, detectability is a necessary condition for a fault to be isolated. In order to simplify the task of isolating the fault, often a single fault hypothesis is adopted. It is assumed that more than one fault have not occurred simultaneously, that only one single fault may occur at a time. 

This also means that faults do not cancel each other in their effect on an ARR residual. Given a single fault hypothesis, the faulty component is identified by comparing the coherence vector against the rows of the FSM, i.e. the component fault signatures. If this comparison results in a match, the faulty component is isolated. However, there may be no match, or more than one match may be obtained. That is, the faulty component cannot be isolated. In the case of multiple simultaneous faults, FDI can be performed e.g. by means of parameter estimation as is discussed in Chap. 6. 

On the basis of a single fault hypothesis, fault isolation is performed by comparing the periodically updated coherence vector with the rows of the FSM. However, for a hybrid system model, the entries of a FSM are mode dependent. For a model with Ks switches, < 2 physical feasible switch state combinations, i.e. n/ system modes are to be considered. The FSM holding for all modes provides a specific FSM for each mode. To make sure that the coherence vector is compared with the component fault signatures in the right FSM, the current system mode of operation must be identified from measured system or process outputs. Figure4.7 depicts a flowchart of a bond graph model-based FDI process. 

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. 

In the following, two fault scenarios are studied. In both cases, the single fault hypothesis is adopted. Names of faulty quantities carry a tilde to distinguish them from names of their faultless counterparts. In figures, this is expressed by preceding names with the letter t . 

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. 

Let us consider the (72, 64) SEC-F7EC proposed in [1]. This UEC code has E+ = u eI" , i.e. it has full error correction in the strongly controlled area and single error correction in the weakly controlled area. Under different fault hypothesis, UEC alternatives are very limited, but FUEC codes allow a great flexibility. 

As far as the random process y(0 is concerned, a popular PSD model attributed to Kanai (1957) and Tajimi (1960) represents the ground acceleration as the absolute acceleration of a linear single-degree-of-freedom (sdof) system excited by a white noise. In this sense, the oscillator represents the terrain and the input white noise is the acceleratiOTi at the faulting point. Under this hypothesis the PSD of the process y(t) is given as ... 

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