Classification methods and transformation of equations

The formal definitions enable us to classify the variables by graph algorithms. Basically, the classification algorithms consist in eliminating circuits containing unmeasured streams see for instance Mah (1990), 8-2-2 and 9-1-1. As an example, let us outline a procedure that performs the classification simultaneously with transforming the equations in a manner suitable for adjustment (reconciliation) of measured variables, and computation of the unmeasured observable ones. 

Given is the connected graph G[N, J] of the system and the partition of the streams (arcs) 7 e J into unmeasured (J ) and measured (J ). Define subgraph G° [N, J°]. Then 

On the other hand, as shown in the last two paragraphs of Section A.3, we thus can find the set S, of arcs separating G . Indeed, having decomposed 

Having completed the classification according to steps (a) and (b) above, we can make use of the additional information obtained in the described manner. First, the reduced graph G determines, having selected a reference node, the reduced incidence matrix A. It is the matrix occurring in (3.2.4)2, thus in the constraint equation for the measured vector m (in fact, only for the subvector m of redundant variables). The equation is employed for adjusting the given values if the components of m have been actually measured then for reconciliation by statistical methods. 

Interpreted in terms of balance equations, the figure represents the solutions of a subsystem (3.2.4), where the connected component G° drawn with incident fixed streams can look like 

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