ARR Residuals as Fault Indicators

An essential step in FDI is the evaluation of the time history of residuals serving as fault indicators. To that end, one approach is to establish ARRs from a model of a system (cf. Fig. 1.3). These relations are algebraic or dynamic constraints between known continuous variables, i.e. system inputs u and measured output variables y that include known model parameters O. ARRs derived from a bond graph of a hybrid system model also depend on discrete switch state variables. For a healthy system, the time evolution of ARRs should ideally be identical to zero in all system modes. In practice, residuals will be within certain small error bounds due to measurement noise, parameter uncertainties, and numerical inaccuracies. If, however, faults in some system components occur, then the values of some residuals will be outside given thresholds and can serve as fault indicators. The sfructure of ARRs expressed in the FSM will then indicate whether faults can be isolated. Let m t) denote the vector 

Borutzky, Bond Graph Model-based Fault Diagnosis of Hybrid Systems, 

4 Bond Graph Model-based Quantitative FDI in Hybrid Systems 

Clearly, chosen thresholds must not be too big in order to avoid that true faults are not detected, nor must they be too small in order to avoid the report of faults that did not happen. 

In order to get a clear indication for a fault by numerical evaluation of ARRs, these constraints should be insensitive to disturbances while sensitive to true faults. Moreover, analysis of their structure should allow to locate the cause of a fault. For a specific fault, only a subset of ARR residuals should be outside given thresholds. Ideally each residual is sensitive to only one fault while aU other residuals are insensitive to that same fault. In that case, ARRs are said to be structured. However, if the number of faults taken into account exceeds the number of sensors or if the latter are not appropriately located then a set of structured ARRs cannot be achieved. If a residual is affected by a set of faults that does not affect any other residual then this residual is called structurally independent. Such ARRs cannot be obtained by algebraic manipulation of other ARRs. The number of structurally independent ARRs that can be derived from a model equals the number of sensors used in the model [1]. 

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As a result, ARR residuals as fault indicators may be obtained by evaluating ARRs derived from a diagnostic bond graph with nominal parameters. In order to assess the effect of uncertain parameters on ARR residuals, parameter variations of ARR residuals may be derived from an incremental bond graph. Application of the triangle inequality then gives adaptive bounds for these variations. 

ARR residuals as fault indicators should be distinctly sensible to true faults and robust with regard to parameter uncertainties. That is, if parameters varies, the time evolution of ARR residuals should be within prescribed bounds. For real systems described by a hybrid model bounds should be adapted to system modes as the dynamic behaviour can be quite different in different system modes. 

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