Incremental Bond Graphs for FDI

Let u denote the vector of inputs into the original bond graph model with nominal parameters, x its state vector composed of all inputs into storage elements in preferred derivative causality and let an index n indicate a dependency from nominal parameter values. The state space model derived from the original bond graph then reads 

Furthermore, let 0 denote the vector of aU component parameters. The state space model derived from the incremental bond graph then is of the form 

The matrices A and C are obtained deriving equations from the original bond graph with nominal parameters. Matrix B can be automatically set up from the incremental bond graph. The vector w denotes the outputs of the modulated sinks in the incremental bond graph that represent parameter variations. These outputs are of the form 

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As fault indicators should be sensitive to real faults but insensitive to parameter uncertainties, adaptive system mode dependent thresholds are needed for FDI in hybrid systems robust with regard to parameter uncertainties. Chapter 5 demonstrates that incremental bond graph can serve this purpose for switched LTI systems. To that end some basics of incremental bond graphs are recalled. It is shown how parameter sensitivities of ARRs and ARR thresholds can be obtained. A small example illustrates the approach. 

In contrast to an original bond graph, the bonds of an incremental bond graph carry variations of power variables. Outputs of interest with regard to FDI are variations of ARR residuals. For a switched LTI system, these variations are a weighted sum of the... 

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