Component fault signature

Faults in a system component can be related to deviations of parameter values from those of the healthy component. For instance, if a hydraulic check valve that autonomously switches on and off is modelled as a non-ideal switch, then a stuck-open fault can be captured by a permanent change in its switch state. Thus, a matrix entry Sij that is non-zero for some system modes means that in these system modes, residual rj is structurally sensitive to faults in the tth component. A FSM thus relates discrepancies in components to changes in residuals. The columns of a FSM indicate the fault signatures of the ARRs of the residuals. Structurally independent ARRs, i.e. ARRs that cannot be algebraically constructed from other ARRs have a unique fault signature. The rows of a FSM are called component fault signatures. An all-mode FSM will be termed hybrid FSM (HFSM). 

Clearly, a fault in a component can be detected if in some system mode, at least one entry in the component fault signature does not vanish, i.e. at least one residual in one system mode is sensitive to it. If, moreover, the component fault signature differs from all others in that system mode, then the fault can be isolated. If a fault in the ith component can be detected in all modes, then this is captured by an entry equal to one in the ith row of an extra column with the heading Db added to the FSM. For a component fault signature that depends on switch states, the detectability of the fault is given by the logical OR of the switch states. 

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 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. 

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. 

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. 

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. 

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 least squares minimisation of ARR residuals only the m parameters contributing to the unstructured part of a FSM need to be considered. Residuals in that part structurally depend on more than one parametric fault and the unstructured part of a FSM can be further subdivided into subspaces (rows of the FSM) with common component fault signatures. Which parameters in which subspaces can be identified... 

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... 

The representation of a hybrid system model by means of a bond graph with system mode independent causalities has the advantage that a unique set of equations can be derived from the bond graph that holds for all system modes. Discrete switch state variables in these equations account for the system modes. In this chapter, this bond graph representation is used to derive analytical redundancy relations (ARRs) from the bond graph. The result of their numerical evaluation called residuals can serve as fault indicator. Analysis of the structure of ARRs reveals which system components, sensors, actuators or controllers contribute to a residual if faults in these devices happen. This information is usually expressed in a so-called structural fault signature matrix (FSM). As ARRs derived from the bond graph of a hybrid system model contain discrete switch state variables, the entries in a FSM are mode dependent. Moreover, the FSM is used to decide if a fault has occurred and whether it can unequivocally be attributed to a component. Finally, the chapter discusses the numerical computation of ARRs. 

The structural fault signature matrix considered so far indicates which component parameters are contained in which ARR. As faults in a component can be related to unwanted changes of parameter values, a structural FSM displays which ARRs are structurally sensitive to which faults. This matrix does not capture that variations in some of the parameters contained in an ARR may have only little effect on the residual of an ARR and may be overshadowed by the affect of other parameter variations in an ARR. As to hybrid models, the sensitivity of an ARR residual with... 

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. 

The fault signature matrix (FSM), S, is a binary structure which describes the participation of various components (physical devices, sensors, acmators, and controllers) in each ARR. This structure links the discrepancies in components to changes in the residuals. The elements of matrix S are determined from the following analysis ... 

The fault signature for the residual t of the considered example is given as ATi = [<2p. M, Cdb, P, P2] which may be written in terms of the components involved as 1 = [0p> Ti, Vb, 1, Pi]- Likewise, the fault signature for the residual f2 is K2 = [ Vb, T2, Vo, Pi, P2]- Because Ki K2, residuals r and T2 are said to be structurally independent. Because we have not considered controller ARRs, we will consider no fault in controllers. Moreover, we will consider that sensors in the system are robust so that faults in them can be ruled out. The fault signature matrix S is then derived as shown in Table 7.3. 

Note that for the two-tank system, all the faults can be monitored, i.e., detected, since there is at least one non-zero element in the signature of each component. However, faults in T2 and To are not isolatable because they have identical signatures. To improve the isolation ability of the supervision platform, more sensors should be added to the process. In this case, an output flow sensor will make all component faults isolatable. This information can be obtained from causal path analysis even before ARRs are derived [3,6]. 

From these eausal paths, the components involved in the residuals r and ra are obtained as Ki = [Qp, Ti, Vt, -Pi, P2] and K2 = [Vb, P2, Vo, Pi, P2], respectively. The fault signatures obtained from causal path analysis are identical to the ones obtained before (Table 7.3). Thus, sensor placement problem is reduced to a... 

The fault signature matrix excluding sensor faults is given in Table 7.4, where Ui and di (i = 1, 2) are binary and complementary to each other, i.e fl = di, and Rdi (i = 1,..., 4) represent residuals. The fault signature, monitorability, and isolatability of some components change with the operating modes of the system. Simulation results for this system with an adaptive threshold-based decision procedure are presented in the next section. 

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. 

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