Graphs Elements

The bond graph method defines the structure and constitutive equations of the system. Standard bond graph elements are used to build a model of the structure of the system. Suitable computer programs are available to generate the governing equations, and alternative methods have also been developed for deriving equivalent block diagrams, which can represent nonlinear systems. 

The first element of each SE-vector, i.e. the 0-order molecular path count, corresponds to the count of the considered graph elements for example, the first entry in the TT-vector is the number of atoms in the molecule, in the OT-vector it is the number of oxygen atoms, and in the 2T-vector the number of double bonds. 

Several information indices, usually calculated as total information content and - mean information content, are based on partitioning graph elements or matrix elements in - equivalence classes according to two basic criteria ... 

Make the data stand out and use visually distinctive graph elements. 

Fig. 2.4 The ideal switch as an additional bond graph element...

That is, the characteristic of a storage element is approximated by a piecewise constant function other bond graph elements are static elements. A 2-port transformer, for instance, takes two piecewise constant inputs... 

Another approach that also allows to approximate each continuous time element of a bond graph by a DEVS model so that a DEVS simulation can be performed has been reported in [57, 58]. The task is to transform piecewise continuous input and output trajectories of a bond graph element into discrete event trajectories. To translate, e.g. the continuous time model of a C element in integral causality into a discret event model, the input trajectory of the flow f t) between two time instances U and tj is approximated by a linear function f(t) = a t + ao- The output trajectory of e(t) is a second order polynomial e(t) = b2t + bit = aQlC)t. This... 

Derivation of ARRs from a diagnostic bond graph then starts by summing power variables at all those junctions that have a BG sensor element in inverted causality attached to it. At first, these balances of power variables will contain unknown variables. They may be eliminated by following causal paths and by using constitutive equations of bond graph elements. The result may be a set of ARRs in closed symbolic form [cf. (4.2)] if nonlinear constitutive element equations permit necessary... 

An incremental bond graph of an LTI system is constructed from an initial bond graph with nominal parameters by replacing those bond graph elements by their incremental model for which a parameter variation has taken place. Clearly, sources that do not depend on a parameter become null sources. Furthermore, the incremental model of a l(0)-junction remains a l(0)-junction. As to the incremental models of storage elements, resistors, transformers and gyrators, it turns out that they differ from the respective element just by modulated sinks added to junctions. The sinks... 

In order to see how an incremental bond graph model for a bond graph element is obtained, a linear 1-port C-element with the nominal capacitance Cn is considered. In the following, an index n indicates a parameter or a variable of the non-faulty bond graph model with nominal parameters. In the case of a time constant parameter variation AC the linear constitutive equation of a 1-port C element in derivative causality takes the form... 

The power variable fc controlling the modulated source is an output variable of the original bond graph model. If fc has been obtained by measurements of the real system, then the contribution to the output of the incremental bond graph model of the C element may contain sensor noise. In any case, the outputs of the incremental bond graph of a bond graph element indicate a parameter variation. 

In the same manner, incremental models for the other bond graph elements may be obtained. As an example. Fig. 5.3 depicts the incremental bond graph model of a transformer TF ... 

Incremental Models of Nonlinear Bond Graph Elements... 

The incremental bond graph approach is not limited to linear 1 -port elements with one parameter [5]. The constitutive equation of the incremental model of a bond graph element can be easily obtained by taking the total differential of the constitutive equation of the bond graph element. 

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. 

In the same manner, sensitivity component models can be obtained for the other bond graph elements. As junctions do not depend on parameters they remain junctions in a sensitivity pseudo bond graph. Sources that provide a constant become sources of value zero. Sensitivity component models of other elements differ from their element only by additional sinks. As a result, a sensitivity pseudo bond graph is of the same structure as the behavioural system bond graph. Moreover, causalities of the latter one are retained. 

As explained in Sect. 5.3.2, variations of ARR residuals due to parameter variations are obtained by summing increments of power variables at junctions of an incremental bond graph to which a virtual detector has been attached. Replacement of the switch model as well as the other bond graph elements in the DBG of the boost converter in Fig. 8.2 by their incremental model gives the incremental DBG depicted in Fig. 8.14. 

Fig. B.5 Admissible causal patterns at bond graph element ports...

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... 

Incremental bond graphs present another approach. Similar to the LFT bond graph approach, they are obtained by replacing elements by their incremental model. The latter one is also a decomposition of a bond graph element into a nominal part and one that accounts for parameter variations. Opposed to LFT bond graphs, bonds of incremental bond graphs carry variations of the conjugated power variables due to parameter variations. 

Breedveld P.C. (1985) Multibond graph elements in physical systems theory. J. Franklin Inst. 319(1/2) 1-36. 

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