Bond graph uncertain

As to FDI robust with regard to parameter uncertainties, an approach based on so-called uncertain bond graphs in linear fractional transformation form (LFT) has been reported in the literature [8-10] for time-continuous models. In an uncertain bond graph, bonds carry power variables uncertain with regard to parameter variations... 

For small switched LTI systems, variations of ARR residuals can be manually derived from an incremental bond graph by applying the principle of superposition. That is, only one bond graph element at a time is assumed to have an uncertain parameter. It is replaced by its incremental model. Detectors are replaced by a dual virtual detector for the variation of an ARR residual. Summing variations of flows or efforts, respectively, at these junctions and eliminating unknowns yields variations of residuals of ARRs as a weighted sum of the inputs supplied by those modulated sinks that represent parameter variations. The weighting factors in these sums are the sensitivities to be determined. 

As an example, the circuit with one switch in Fig. 4.1, is considered. To keep the illustration of the procedure short and simple it is assumed that only one parameter is uncertain. Accordingly, the incremental bond graph is obtained by replacing the element by its incremental model and by replacing the constant voltage source Se Vi by an effort source of value zero and by replacing detectors by dual virtual detectors for the variations of ARR residuals. 

Fig. 5.8 Incremental bond graph of the switched circuit in Fig. 4.1 in the case of an uncertain parameter R ...

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. 

Figure5.15 illustrates this situation assuming that measurement uncertainties are additive. A flow / = f + Af with a predicted part / and an uncertain part A/ due to measurement uncertainty is the output of a non-ideal sensor and an input into the diagnostic bond graph. The input / into the diagnostic bond graph results in an effort e = e + Ae that controls a modulated sink MSe where e denotes the predicted part and Ae the uncertain part. The output w = 5(e - - Ae )A0 of the modulated sink is an input into the incremental bond graph that is needed to compute the variation Ar of an ARR residual r.

The result is a threshold thr t) > thr t) that is independent of measurement uncertainties but that depends on the sampling rate of the measurement. In [14], Touati et al. have accounted for measurement uncertainties by adding modulated sources to an uncertain bond graph. 

Constant excitations to a system are represented by an effort or a flow source that provides an output of constant value. In the incremental bond graph these sources are replaced by sources of value zero. If a constant excitation, however, is to be considered uncertain, its source may be replaced in the incremental bond graph by a source modulated by the nominal value. For instance, let Se En represent a constant voltage or constant hydraulic pressure supply. If there is a relative uncertainty 8e = AE/E , then the constant effort source may be replaced in the incremental bond graph by an effort source MSe SsEn modulated by the nominal effort E obtained from the bond graph with nominal parameters. If the internal structure and the parameters of the device are known that provides the excitation and if possible disturbances acting on the device can be modelled, then an incremental bond graph model can be constructed that accounts for the uncertainty of the excitation. 

Chapter 3 introduces a special decomposition of bond graph elements in a part with nominal parameters and one with uncertain parameters. The resulting bond graph model of a bond graph element is called linear fractional transformation (LFT) model. In case of linear models, bond graphs with elements replaced by their LFT model enable the derivation of state space and output equations in LFT form as used for stability analysis and control law synthesis based on /r-analysis. 

Moreover, LFT bond graphs can also support robust fault detection and isolation (FDI) of systems with uncertain parameters. The decomposition of bond graph elements leads to a derivation of analytical redundancy relations (ARRs) composed of a nominal part representing their residuals and an uncertain part due to parameter... 

Keywords Bond graph Fault detection and isolation (EDI) Uncertain systems ... 

Diagnosis of uncertain systems has been the subject of several recent research works [1-6]. This interest is reflected by the fact that physical systems are complex and non-stationary and require more security and performance. The bond graph model in LFT form allows the generation of analytical redundancy relations (ARRs) composed of two completely separated parts a nominal part, which represents the residuals, and an uncertain part which serves for both the calculation of adaptive thresholds and sensitivity analysis. 

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