On the topological testability conjecture for analog fault diagnosis problems

Abstract

This paper gives a proof of a sharpened version of the conjecture in [1]. Let \bf{C}^{r} be r -dimensional complex vector space and let g \in {\bf C}^{r} be the vector of branch admittances of an analog network. A subset of {\bf C}^{r} is said to be ample if (i) its complement has Lebesgue measure zero, (ii) it is open, and (iii) it is dense. The sharpened version of the conjecture claims that the k -node fault testability condition [1] is satisfied on an ample subset of values of g , if, and only if, for any set X of inaccessible nodes, there are at least k + 1 nodes in X^{C} (complement of X ) each of which is connected with X via a branch. This is extremely powerful because the result depends only on the topology of a network and the condition can be checked by inspection. The proof justifies the fault location method developed in [1].