Analysis bond graph structure

Moreover, this strategy has the advantage to provide the designer with a guide in his design process. In fact, if he assesses his model as sufficiently faithful to the studied system, the results of the analysis levels (bond graph structure, behavioral... 

The theoretical investigation of these reactions requires quantum-mechanical methods, in particular, the study of chemical bonds in initial, final, and intermediate compounds, as well as the consideration of nuclear motions. Yet frequently the important information can be obtained without analysis of electronic structure of molecules and investigation of actual motion of nuclei, only resorting to the graph theory and employing the group-theoretical conceptions. Here we should define two terms permutation isomers and permutation isomerism reaction. 

Bond Graph-based Analysis of Structural Observability... 

Sueur, C., Dauphin-Tanguy, G. (1991). Bond graph approach for structural analysis of MIMO linear systems. Journal Franklin Institute, 328(1), 55-70. 

The representation of a hybrid system model by means of a bond graph with system mode independent causalities has the advantage that a unique set of equations can be derived from the bond graph that holds for all system modes. Discrete switch state variables in these equations account for the system modes. In this chapter, this bond graph representation is used to derive analytical redundancy relations (ARRs) from the bond graph. The result of their numerical evaluation called residuals can serve as fault indicator. Analysis of the structure of ARRs reveals which system components, sensors, actuators or controllers contribute to a residual if faults in these devices happen. This information is usually expressed in a so-called structural fault signature matrix (FSM). As ARRs derived from the bond graph of a hybrid system model contain discrete switch state variables, the entries in a FSM are mode dependent. Moreover, the FSM is used to decide if a fault has occurred and whether it can unequivocally be attributed to a component. Finally, the chapter discusses the numerical computation of ARRs. 

Figure 4.21 shows a bond graph representation of the inverse model. Contrary to the bond graph of the direct model, the bond graph of the inverse model has one energy store in differential causality (C Ci) as to be expected from the structural analysis of the direct bond graph. Hence, the order of the inverse model is 1. 

This section presents the theoretical material required for the methodology concepts and the proof of its effectiveness. A very brief review of model inversion is first recalled. Then the definitions of relative orders, orders of zeros at infinity, and essential orders are presented. These notions are also reviewed in the bond graph language for defining structural analysis in this framework. In particular the concepts of power lines and causal paths are defined. They will be used for checking the structural criteria of invertibility and differentiability. 

One of the main characteristics of an inverse model is the presence of the output derivatives in the equations (vector y (t)). This will be discussed in more detail in the following sections. In particular, in structural analysis, the notion of essential orders enables the necessary minimal number of output time differentiations to be anticipated before the constmction of the inverse model. This notion will be translated into the bond graph language. 

The previous concepts of structural analysis are now reviewed in the context of the bond graph language. First, the notions attached to power lines and causal paths are defined and then used for determining the relative orders, the orders of the zeros at infinity, and the essential orders. All the following definitions are given for the bond graph representation of an LTI system (A, B, C, D). 

Bicausality is the extension of causality for obtaining the inverse model equations directly from a bond graph representation. The way bicausality is assigned in a bond graph depends on the results of the invertibility criteria and of the structural analysis in terms of I/O power line (see Definition 6.2) sets and I/O causal path (see Definition 6.5) sets. 

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. 

The study of chemical reactions requires the definition of simple concepts associated with the properties ofthe system. Topological approaches of bonding, based on the analysis of the gradient field of well-defined local functions, evaluated from any quantum mechanical method are close to chemists intuition and experience and provide method-independent techniques [4-7]. In this work, we have used the concepts developed in the Bonding Evolution Theory [8] (BET, see Appendix B), applied to the Electron Localization Function (ELF, see Appendix A) [9]. This method has been applied successfully to proton transfer mechanism [10,11] as well as isomerization reaction [12]. The latter approach focuses on the evolution of chemical properties by assuming an isomorphism between chemical structures and the molecular graph defined in Appendix C. 

The mathematical theory of topology is the basis of other approaches to understanding inorganic structure. As mentioned in Section 1.4 above, a topological analysis of the electron density in a crystal allows one to define both atoms and the paths that link them, and any description of structure that links pairs of atoms by bonds or bond paths gives rise to a network which can profitably be studied using graph theory. 

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