Classification in practice

In practice, the adjustment problem as formulated in Subsection 8.5.2 (see (8.5.11)) is most frequently a reconciliation problem some measured values (vector ) are adjusted (reconciled) so as to make the model solvable. The classification of variables gives one an idea of what can be expected from the reconciliation. Thus, first, the degree of redundancy H informs us on the number of independent constraints (scalar equations) the adjusted value x has to obey, thus how many measured variables are redundant in the manner that having deleted their measurement, they will be still determined by the remaining measured values. In particular if // = 0 then all the / measured values are necessary (none is redundant). If it happens that H = I then the whole measurement is redundant because the constraints determine the I variables uniquely. Generally, not any H measured variables are determined by the other values, thus redundant-, some of them can be nonredundant thus not subject to the constraints (solvability conditions), hence their measurement cannot be deleted. Under frequent hypotheses adopted by the statistical model of measurement, the nonredundant values remain unadjusted by the reconciliation so they are also called nonadjustable. 

Further, if some vector x of adjusted measured values obeys the solvability condition then given certain L independent constraints (scalar equations) are imposed upon the J components of the unmeasured vector y. The number J-L determines the number of degrees of freedom for the unmeasured variables. It can happen that certain unmeasured variables are uniquely determined by the latter constraints (thus by x) they are called observable. In particular if L = / then there is no degree of freedom and all the y-variables are uniquely determined. If L 7 then at least some of the unmeasured variables remain undetermined they are called unobservable. 

The above intuitive concepts have a precise meaning when the model is linear. Then the variables can be classified a priori, and the partition of variables into measured and unmeasured (measurement placement) possibly modified. In any case (assuming 0 H I), the measured vector x can be adjusted by standard reconciliation. The situation is not that simple when the model is nonlinear, as the previous analysis has shown. A well-posed adjustment (reconciliation) problem allows one at least to expect that a reconciliation procedure will converge to some adjusted value x, if the measured x is not too bad . Then also an a priori classification of variables makes sense, based on the same ideas as in the linear case. 

The analysis of solvability and classification of variables is not so much a physical or technological problem, but rather a problem of computation relative to the mathematical model adopted. Observe that any model (even if regarded as rigorous ) represents a mathematical idealisation of the reality. If we put up with a linear approximation, we can make use of the methods described in Chapter 7. We thus assess a representative value Zq of the state variable (taken for instance from the design of the plant) and in a neighbourhood of point Zq, linearize the model according to (8.4.4). Then the model (8.5.8) reads 

Let now the model be nonlinear. Then the Jacobi matrix depends on the unknown vector z and the reconcilation consists of a number of steps, say of a sequence of approximations z if the sequence converges then the limit value, say z, represents a point on the solution manifold M, thus an estimate of the actual value of the state vector. So as to have an a priori idea of what can be expected, one can proceed as follows. 

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