Causality inversion method

Given a causal bond graph of a hybrid model, the question is how ARRs containing only known system inputs, outputs, system parameters, and information about the system mode can be derived. This section considers three methods that have been reported in the literature. One approach is the so-called causality inversion method [2-4], It has been introduced for causal bond graphs of continuous time models but can also be applied to bond graphs of hybrid models [5, Chap. 7]. 

Causalities at bond graph sensor element ports (detector ports) are inverted. In online FDI, outputs of a real process or a real system are measured by real sensors. For a model of the system used for the determination of ARRs, these measurements are known inputs. 

Derivation of ARRs from a diagnostic bond graph then starts by summing power variables at all those junctions that have a BG sensor element in inverted causality attached to it. At first, these balances of power variables will contain unknown variables. They may be eliminated by following causal paths and by using constitutive equations of bond graph elements. The result may be a set of ARRs in closed symbolic form [cf. (4.2)] if nonlinear constitutive element equations permit necessary 

4 Bond Graph Model-based Quantitative FDI in Hybrid Systems 

Remark 4.2 This approach to the derivation of ARRs in closed form has been implemented in the software ModelBuilder [7]. Also, a module of the software Symbols [8] can automatically generate ARRs. If ARRs in closed form are not possible, i.e. unknowns cannot be eliminated then ARR residuals are to be computed numerically simultaneously with the model equations.  

---

For comparison with the causality inversion method, this procedure is used to generate again ARRs for the switched network in Fig. 4.1. ARRs are derived from the bond graph in Fig. 4.4. 

ARRs could also be derived from a bond graph in preferred integral causality and without causality inversion at sensor ports. The covering path method [3, 10, 11]... 

Hogan JW, Lancaster T. 2004. Instrumental variables and inverse probability weighting for causal inference from longitudinal observational studies. Slat. Methods Med. Res. 13 17-48. 

When forming a model, we wish to minimize the influence of random variations in the data and maximize the influence of the underlying causal effects. It is therefore important to use many (X, Y) pairs for training, as the standard error of a sample set is inversely proportional to the number of samples. Conversely, for estimation of the MSEP we need many (X, Y) test pairs, for exactly the same reasons. When forming a model, we usually have a limited number of (X, Y) pairs to use. There is therefore a trade-off between using samples for model formation and MSEP estimation. A number of methods have been suggested for addressing this problem and these will now be discussed. 

---