Sensitivity Pseudo Bond Graphs

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. 

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 ... 

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