Causal path disjoint

Consider the simple linear electrical network depicted in Fig. 4.19. It can be viewed as an electrical analogue of the coupled hydraulic tank system considered in Section 4.4.1. A bond graph of the direct model with the two inputs I(t) and E(t) and the two outputs e and /2 appears in Fig. 4.20. There is one set of two disjoint input-output causal paths... 

While the power line is an acausal concept, i.e., it does not require any organization of the model equations, the causal path needs a causality assignment in the bond graph representation. Its definition is firsf recalled. Then the length and the order of a causal path are introduced, and finally, botii different and disjoint causal paths are defined. The latter concepfs, as for tiie power line, will be used in fhe invertibilify criteria. The concept of different causal paths will also be used to characterize the structure at infinity of a model from its bond graph representation. 

Definition 6.9 (Disjoint causal path) In a causal or (bicausal) bond graph representation, two causal paths are said to be disjoint only if they have no variable in common [34]. This translates into a graphical disjunction of these two causal paths in the bond graph representation. 

First, a series of criteria concerns the invertibility checking of a model. An approach based on different I/O causal paths (see Definition 6.8) and the system matrix determinant has been proposed in [40]. Here the approach based on disjoint I/O causal paths (see Definition 6.9) is presented [15, 24, 25]. It uses two structural criteria which, if not verified, enable the inversion process to be stopped early in the procedure. A third criterion is formulated at a behavioral level. This level is called behavioral in the sense that it requires analytical developments based on the constitutive and conservation laws in the bond graph representation. 

Criterion 2 (Causal) In the bond graph representation in preferential integral causality, ifno set of disjoint I/O causal paths (Definition 6.9) exists then the model is not invertible. 

Then the preferential integral causality is assigned to give the causal bond graph representation in Fig. 6.13. The causal structural analysis results in four I/O causal paths, each of length 1, and two sets of I/O disjoint causal paths of length 2 which is minimal. The second criterion is also verified and Fig. 6.13 displays the set associated with that of Fig. 6.12 power lines. 

Finally, the bicausal bond graph representation is obtained in Fig. 6.14. It has been obtained, on the one hand, from the set of the disjoint I/O causal paths of minimal length equal to 2 and, on the other hand, from the set of the associated disjoint... 

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