Coherency vector

In order to isolate the faulty component the coherence vector is matched with the rows of the FSM, i.e with the component fault signatures. Given a hybrid system model, there is a FSM for each system mode. That is, in order to use the correct FSM for comparison, it is important to know in which mode the monitored system is at the present time point. Chapter shows that ARRs derived from a diagnostic bond graph can also be used for system mode identification. 

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. 

The diagram clearly indicates the fundamental role of generating ARRs. A DBG model with system inputs u t) and measurements y(t) from the real system can serve this purpose. In the decision procedure, ARR residuals are assessed and mapped onto a coherence vector. Faults that have happened are indicated by components of this... 

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. 

For illustration of parametric fault isolation by means of least squares ARR residual minimisation, the simple hybrid network in Fig. 4.1 shall be considered once again. The all-mode FSM (Table4.1) indicates that resistors Ri and R2 have the same component signature when the switch is on so that a parametric fault in one of the two resistors cannot be isolated by inspection of the coherence vector. 

Likewise, a subsequent mode change at f = 13 with the coherence vector c = [0 1] leads to mode [0 0 0] (mode 0) and if a final mode change takes place ext = U with the coherence vector c = [1 1] then mode [0 1 0] (mode 2) is identified. 

Time instant (s) Coherence vector Switch states [m3 m2 "ill Detected mode TauTt... 

In real-time FDI and in real-time mode identification, ARRs are evaluated with sampled data. Residuals are used by a decision procedure that provides an update of the coherence vector. A system mode change is indicated by a coherence vector... 

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. 

A fault is detected, when C 7 [0,0,..., 0], i.e., at least one element of the coherence vector is non-zero (alternatively, at least one residual exceeded its threshold). When a fault is detected, an alarm is raised. This non-null coherence vector is then matched with the fault signatures stored in the FSM to isolate the fault. 

The polarization characteristics of the incident field can also be described by the coherency and Stokes vectors. Although the ellipsometric parameters completely specify the polarization state of a monochromatic wave, they are difEcult to measure directly (with the exception of the intensity Eq). In contrast, the Stokes parameters are measurable quantities and are of greater usefulness in scattering problems. The coherency vector is defined as... 

As mentioned before, a scattering particle can change the state of polarization of the incident beam after it passes the particle. This phenomenon is called dichroism and is a consequence of the different values of attenuation rates for different polarization components of the incident light. A complete description of the extinction process requires the introduction of the so-called extinction matrix. In order to derive the expression of the extinction matrix we consider the case of the forward-scattering direction, = 6fc, and define the coherency vector of the total field E = Eg + E. by... 

---