Matrix Tools for General Observability Analysis in Traffic Networks

We deal with the problem of observability of a given subset
${\bf V}_{\bf 1}$
of flows in terms of another subset
${\bf V}_{\bf 2}$
, no matter which type of flows [link, origin–destination (OD), route, node, plate scanned, etc.] they contain or whether they are mixed types. Two problems are stated: The first consists of determining which subsets of flows in
${\bf V}_{\bf 1}$
can be calculated in terms of the observed flows
${\bf V}_{\bf 2}$
. The second consists of determining which subset of flows
${\bf V}_{\bf 2}$
needs to be observed to calculate a given subset
${\bf V}_{\bf 1}$
. A theorem providing necessary and sufficient conditions for observability is provided and used in the proposed methods to solve the two problems. Two theorems, one lemma, and one corollary provide the bases for optimizing the numerical procedures to solve these problems. Some examples of applications are used to illustrate the proposed methods.