Pseudo bond graph

Sensitivities of the outputs of a model with respect to a parameter can be derived from a sensitivity bond graph [5-7]. Sensitivities of ARR residuals with respect to a parameter can be obtained from incremental bond graphs (Chap. 5), from sensitivity pseudo bond graphs [8] and from diagnostic sensitivity bond graphs [9]. 

In the following, first, sensitivity pseudo bond graphs are briefiy reviewed and are then used to obtain residual sensitivity functions needed for the previously presented least squares ARR residuals minimisation. The simple hybrid network in Fig.4.1 is used again for illustration of the approach. 

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. 

Deducing Residual Sensitivity Functions from a Sensitivity Pseudo Bond Graph... 

A sensitivity pseudo bond graph from which residual sensitivity functions for parameter estimation can be deduced is constructed by simply replacing those elements in a DBG by their sensitivity component model whose parameters are to be estimated. Equations for parameter sensitivities of ARR residuals can then be deduced from the SPBG in the same way as the equations of a state space model are deduced from a behavioural BG or equations for ARR variations from an incBG. 

We now have nine basic bond labels (M)C, (M)I, (M)Se, (M)Sf, (M)R(S), (M)TF, (M)GY, (X)0, and (X)l, which, for reasons of clarity can also be introduced bottom-up, in the sense that each is defined in the simplest form possible and where ports are power ports (as mentioned before, in case ports are not power ports, bond graphs are commonly addressed as pseudo-bond graphs with pseudo-bonds). This simplest form is the minimum number of ports and a minimum number of constitutive parameters, which results in the linear form. This results in one-port C, I, Se, Sf, and R, as well as the two-port TF, GY, and RS that are all characterized by one parameter and the 0- and 1-junctions, which are always linear, have no parameter at all. Junctions should have at least two ports to be power continuous, but that would limit the possible structures to chains only. This means that the simplest form of a junction required to be able to build arbitrary structures should have a minimum of three ports. In case the constitutive parameter of an element is replaced by a function of time, the modulated and switched versions of the basic elements, i.e., MC, MI, MSe, MSf, MR(S), MTF, MGY, XO, and XI, are obtained (cf. Table 1.3). 

D. Kamopp. (1990). State variables and pseudo bond graph for compressible thermo-fluid systems . Transaction of ASME, Journal of Dynamic Systems, Measurement and Control, 101(3), 201-204, September 1979. 

In contrast to sensitivity pseudo-bond graphs, bonds of incremental bond graphs carry variations of power variables instead of their sensitivities with respect to a parameter. The idea is that a parameter variation A results in a perturbation of both power variables at the ports of an element due to the interaction of the element with the rest of the model [1], Hence, a power variable v t) (either an effort or a flow) has a nominal part v t) and a variation Av(t) due to a parameter change ... 

A thermo-fluid system is shown in Fig. 7.1. The fluid is under-saturated. The pseudo-bond graph power variables e and /) for thermo-fluid systems are chosen as e = [en tI = [F T],/ = [/h /t] = h H where subscripts H and T are used to represent the hydraulic and thermal domains, respectively, and the state variables arising out of storage of mass and energy (enthalpy) are m and H. P and T, respectively, represent pressure and temperature. The pseudo-bond graph model of the system is shown in Fig. 7.2, where the CETF element [1] couples the hydraulic and thermal domains. 

The pseudo-bond graph model of the system, in the preferred derivative causality, is given in Fig. 7.17. The pressures and the mass flow rates have been considered as the generalized effort and flow variables, respectively. Performing the substitutions defined before, we obtain a model shown in Fig. 7.18, which is called a diagnostic bond graph (DBG) model [3,4],... 

---