True bond graph

Each port in a true bond graph is characterized by four relevant objects effort, flow, power, and the constitutive relation between effort and flow that may contain an integration or differentiation with respect to time. In case of pseudo-bonds there are only three relevant objects, effort, flow, and constitutive relation, as the conjugation of effort and flow is not related to power. In case of linearity, the constitutive relation is characterized by just one parameter per port. It depends on the purpose of the model that is being represented by the bond graph which of these objects are independent and which are dependent with respect to a particular port. If the constitutive relation is a known and therefore independent object, either the effort... 

Abstract Incremental true bond graphs are used for a matrix-based determination of first-order parameter sensitivities of transfer functions, of residuals of analytical redundancy relations, and of the transfer matrix of the inverse model of a linear multiple-input-multiple-output system given that the latter exists. Existing software can be used for this approach for the derivation of equations from a bond graph and from its associated incremental bond graph and for building the necessary matrices in symbolic form. Parameter sensitivities of transfer functions are obtained by multiplication of matrix entries. Symbolic differentiation of transfer functions is not needed. The approach is illustrated by means of hand derivation of results for small well-known examples. 

Keywords Incremental true bond graphs Parameter sensitivities of transfer functions Linear inverse models Fault detection and isolation Parameter sensitivities of the residuals of analytical redundancy relations... 

Advantages of the incremental true bond graph-based approach presented in this chapter are that the matrices can be automatically set up in symbolic form from an original bond graph and its associated incremental bond graph by available software. Parameter sensitivities of transfer functions are then obtained by multiplication of matrix entries which can be performed by software in symbolic form. There is no need for symbolic differentiation of transfer functions. The purpose of determining sensitivities of transfer functions in symbolic form is that, in the design of a robust control, it may be useful to know how sensitive transfer functions are with respect to certain parameter uncertainties. 

The product Ae) Af) of the incremental power variables of a bond clearly has the physical dimension of power. This suggests to consider incremental bond graphs as true bond graphs, although the product Ae) Af) is only a part of the power change A due to a parameter change [1]. 

Fig. 4.12 Incremental true bond graph of the DC motor accounting for a variation in mechanical...

An incremental true bond graph approach to a matrix-based determination of parameter sensitivities of transfer functions of linear MIMO models and of residuals of ARRs in symbolic form has been presented. The approach has the following advantages ... 

Abstract Fuel cells are environmentally friendly futuristic power sources. They involve multiple energy domains and hence bond graph method is suitable for their modelling. A true bond graph model of a solid oxide fuel cell is presented in this chapter. This model is based on the concepts of network thermodynamics, in which the couplings between the various energy domains are represented in a unified manner. The simulations indicate that the model captures all the essential dynamics of the fuel cell and therefore is useful for control theoretic analysis. 

The true bond graph model of the SOFC system is given in Fig. 10.4. This model uses the four-port C-field (presented in Section 10.2.3) for representing the energy storage of the gases inside the anode and the cathode flow channels. It also uses the R-field representation discussed in Section 10.2.4 for modelling the convection at the inlet and the outlet of the SOFC channels. 

Representing a thermodynamic system in terms of true bond graph involves the concepts of network thermodynamics [9]. The true bond graph model of the SOFC, shown in Fig. 10.4, is constructed by using the concepts of network... 

In this section, the true bond graph model of the SOFC described in Section 10.2.5 is simulated to obtain the static characteristic curves and dynamic responses to a step change in the load current. In order to simulate the steady-state operation of the SOFC, the single port C-elements in the trae bond graph model have to be initialised with the values of generalised displacements (initial entropies in this case). Similarly, the two C-field elements have to be initialised with the values of the initial masses of the constituent gases and their entropies. 

The model is properly initialised and simulations are performed to obtain the static characteristics and dynamic responses of the SOFC. For obtaining the static characteristic curves of the SOFC, the FU and the OU have been interpreted in terms of the partial pressures of the gas species in the channels, for a given set of known and input parameters. The application of the tme bond graph model presented in this chapter for the optimisation of the operational efficiency of a SOFC system consisting of the cell, the after-burner and two pre-heaters under varying loads can be consulted in [12,14,16]. Readers may also refer to [12,15] for a control scheme to improve the dynamic performance of the SOFC using the true bond graph model presented in this chapter. 

In NaCl (18189), this principle would require all atoms to be identical. Clearly this symmetry is already broken by the constraint imposed by the chemical formula which requires half the atoms to be Na" " and half CP. However, all the Na" " ions are indistinguishable from each other, and the same is true for the CP ions. The bonds likewise, six for each formula unit, are also equivalent in the bond graph (Fig. 2.4). The crystal structure (Fig. 1.1) is then determined by applying the principle of maximum symmetry to the constraints imposed by three-dimensional space as described in Section 11.2.2.4. The crystal structure is thus uniquely determined by the principle of maximum symmetry and the chemical and spatial constraints. 

For a bond graph of a hybrid system model with system mode independent causalities, model equations derived from the bond graph contain discrete switch state variables and thus hold for all system modes. The same is true for ARRs. 

ARR residuals serving as fault indicators should be distinguishably sensitive to true faults but little sensitive to noise and parameter faults in order to avoid false alarms on the one hand side and to make sure that fault detection does not miss any faults. Therefore, appropriate thresholds for ARR residuals are to be set. As the dynamic behaviour of hybrid systems can be quite different in different modes, predefined bounds of constant value may not be suitable. In this book, the incremental bond graph approach [4] has been briefly recalled and applied to hybrid system models to deduce adaptive mode-dependent ARR residual thresholds that account for parameter uncertainties. 

In order to explain these results two alternatives immediately come to mind. The first, which is not preferred, would contend that the larger average Av(C=0) values for Bond A are caused by a larger degree of double-bond character of that bond in all three hydrocarbons. If this is true, apparently one must conclude that Bonds A and B are not equivalent even in o-xylene and that Bond A has about 7, 10 and 14 per cent more double-bond character than Bond B in indan, o-xylene, and tetrahydronaphthalene, respectively. These figures follow directly from our earlier-published graph relating Av(C=0) values to per cent double-bond character [lb]. This alternative appears to be unacceptable because certainly all aromatic bonds in o-xylene are equivalent. 

Fig. 2.19 First level graph sets for the three polymorphs of anthranrhc acid 2-V. The two molecules in the asymmetric unit of Form 1 are shown in their true crystallographic relationship to each other (see text). In (d) and (e) the third ammoniacal hydrogen, which does not participate in a hydrogen bond, is hidden hy the nitrogen to which it is attached, (a) Form II (b) Form ni (c) Form I, C(6) of A type molecnles (d) Form I, C(6) of B type molecules (e) Form I, intramolecular motifs S(6) for molecules A and B and the dimer D relationship connecting them. (Adapted from Bernstein et al. 1995, with permission.)...

A graph y fulfills R if J (y) = true. Otherwise y violates the constraint. A constraint R is called monotonic or consistent with augmentation (addition of further bonds) if the violation of a graph ytoR implies that every augmentation y of y violates R ... 

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