Power line disjoint

Definition 6.3 (Disjoint power line) Two power lines are said to be disjoint only if there is no power in common [34]. 

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. 

Criterion 1 (Acausal) In the acausal bond graph representation, ifno set of disjoint I/O power lines (Definition 6.3) exists then the model is not invertible. 

Along each power line chosen at step I propagate bicausality from the double source to the double detector and propagate causality through the junction structure taking into account the causality constraints ofO- and I-junctions, TF-and GY-elements. If at this step causal conflicts or non-solvable causal loops appear, repeat the previous steps with another set of disjoint I/O power lines. If none of them solves the problem of causal conflicts or non-solvable causal loop appearance then the model is not invertible (Criterion 3) and the procedure stops. 

The application of phase 1 requires the acausal bond graph representation given in Fig. 6.12. The acausal structural analysis gives eight I/O power lines between the suspension forces Fi,Ft) and the heave and pitch velocities (z, Combining these I/O power lines, two sets of disjoint power lines exist thus the first criterion is verified. Figure 6.12 shows one of them. 

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. 

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