All-mode structural fault signature matrix

ARRs derived from a bond graph of a hybrid system model contain discrete switch state variables. Depending on their values, i.e. depending on the system mode, some parts of the ARRs are disabled. An inspection of ARRs derived from a bond graph of a hybrid system model thus shows which component parameters affect which residuals in which system mode. This information may be expressed in an all-mode 

Let u t) denote the known inputs into a system, y(t) known measured outputs, m(r) the vector of all discrete switch states at time t and the vector of component parameters. Each ARR, j = l.N, may then be expressed by means of function 

4 Bond Graph Model-based Quantitative FDI in Hybrid Systems 

If the inputs into the system, the measured outputs and the discrete switch states may be considered faultless, gj () becomes a function of the component parameter vector, their rows may be removed from the FSM and the FSM can be considered a structural parameter sensitivity matrix for ARR residuals. 

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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