Causal analytic structure

IX. Causality. The requirement of Causality, namely that the current situation can be influenced only by past and contemporaneous events, may be shown to impose a constraint on the analytic structure of the complex moduli in the complex (JD plane, and also on combinations of the moduli multiplying Green s functions in the solution of non-inertial boundary value problems. These quantities can have no singularities in the lower half-plane. In a restricted sense, this can be shown directly for certain combinations of complex moduli using properties of the individual complex moduli. 

We wish to take the inverse transform with respect to k of this relation and utilize the causal property of f k) mentioned at the end of Sect. 7.1. However, before we do this, it is necessary to examine the analytic structure of f k) in the complex k plane, in particular to see whether it has any singularities in the lower halfplane. We deduce from the discussion in Sect. A3.2, that if such singularities exist, it is necessary to take the inverse transform along contours off the real axis, below these singularities, to preserve the causal property. 

In Section 2.6 we discussed the nature of cause and effect, and in Section 7.2 we noted that empirical rules are obtained inductively from observations and experiences regarding real structures from three broad categories of situation, success, failure and the near-miss or narrowly avoided failure. Information about successful projects enables us to identify sufficient conditions for success failures tell us the necessary conditions for success. What we would like, of course, are the conditions which are necessary and sufficient but we will never be in that position, for we will never know that there are no unknown phenomena which could occur. Our technical knowledge enables us usually to describe, at least approximately, the technical causal chain in any success or failure. The variancy (Section 2.6) involves all the factors not accounted for in our analytical models, whether theoretical or physical, and it is from this that we learn our lessons for the future. In effect the engineer s judgement based upon his experience is the result of his synthesis of these factors in a teleological explanation. 

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. 

First, a series of criteria concerns the invertibility checking of a model. An approach based on different I/O causal paths (see Definition 6.8) and the system matrix determinant has been proposed in [40]. Here the approach based on disjoint I/O causal paths (see Definition 6.9) is presented [15, 24, 25]. It uses two structural criteria which, if not verified, enable the inversion process to be stopped early in the procedure. A third criterion is formulated at a behavioral level. This level is called behavioral in the sense that it requires analytical developments based on the constitutive and conservation laws in the bond graph representation. 

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