Fault signature matrix

ARRs from a bond graph representing a hybrid system model and the condensation of structural information in a system mode dependent stmctural fault signature matrix are presented. The chapter concludes by addressing off-line numerical computation of ARR residuals. To that end, a model of a system subject to faults is coupled to a bond graph of the healthy system by means of so-called residual sinks. This way, faults may be deliberately introduced without any risk and their effect, detection and isolation may be studied. 

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... 

Table 8.5 Structural fault signature matrix of the three phase rectifier with sensors Df ia, Df ib, Df ic and De Ud...

Structural analysis of ARRs also enables to decide whether a fault can be detected and moreover can be isolated. It is common to add two columns to the fault signature matrix holding information about whether a fault can be detected and moreover can be isolated. 

In case the signature matrix is not diagonal, Samantaray and Ghoshal use parameter estimation for isolation of simultaneous faults [11]. Parameters are estimated by least squares optimisation of residuals. In that approach, values for sensitivities of residuals with respect to parameters are needed. Beyond this optimisation problem, knowledge of how sensitive residuals are with respect to certain parameters helps assessing the information in a fault signature matrix. 

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. 

Table 7.3 Fault signature matrix of the two-tank system...

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. 

Fault f. will be located if for all non-zero elements of the signature matrix, the... 

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). 

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