Nonredundant measured variable

Classification of the measured variables included in NA1 and NA2 as redundant. The other measurements are categorised as nonredundant. Measured variable classification results for this example are in Table 7. 

From the classification it was found that, for this specific problem, there are 10 redundant and 6 nonredundant measured variables, and all the unmeasured process variables are determinable. Symbolic manipulation of the equations allowed us to obtain the three redundant equations used in the reconciliation problem ... 

Let us further consider a nonredundant measured variable Xj. Let again 2 e iW (8.5.31). We have a curve on parametrized as... 

A nonredundant measured variable is not affected by the solvability condition it can be freely varied in some interval leaving the model equations solvable, if the remaining fixed measured variables obey the condition of solvability. The necessary condition reads, according to (8.5.39)... 

For measurement adjustment, a constrained optimization problem with model equations as constraints is resolved at a fixed interval. In this context, variable classification is applied to reduce the set of constraints, by eliminating the unmeasured variables and the nonredundant measurements. The dimensional reduction of the set of constraints allows an easier and quicker mathematical resolution of the problem. 

A measured process variable, belonging to subset x, is called nonredundant (just-measured) if it cannot be computed from the balance equations and the rest of the measured variables. 

Remark 4. As indicated by Crowe et al. (1983), measured variable classification is performed by examining the matrix associated with the reconciliation equations. The zero columns of G or Gx correspond to variables that do not participate in the reconciliation, so they are nonredundant. The remaining columns correspond to redundant measurements. 

Chapters 7 and 8 are devoted to the problems of solvability. We call a set of equations solvable when there exists some vector of solutions, not necessarily unique. In Chapter 3, we have shown that the set of mass balance equations is always solvable if no variable has been fixed a priori. With redundant measured variables, the equations need not be (and usually are not) solvable, unless the fixed variables have been adjusted. Then certain unknown (unmeasured) variables are uniquely determined (observable), other still not (unobservable variables). Certain measured variables can be nonredundant they... 

The j-th measured variable is nonredundant if and only if arc j is deleted by the graph reduction (merging the nodes connected by unmeasured streams). 

The structure of the graph G along with the partition (3.6.1) allows one to classify all the variables (mass flowrates) with respect to the solvability see Section 3.3. According to the standard terminology, the variables mj, i J are called measured, the remaining (j J°) unmeasured. This classification will be used henceforth any a priori fixed variable of a model will be called measured , else unmeasured . The classification of the measured variables follows immediately from the partition (3.6.3) if i e J then m is redundant, else (i G J ) nonredundant. The nonredundant variables are also characterized by the property that i g J if and only if arc i closes a circuit in G with certain unmeasured streams (/ g J°). The nonredundant variables are unaffected by the solvability condition. 

The definition of a redundant variable thus depends only on the matrix (B, A), not on the value x. A measured variable that is not redundant is called nonredundant. If it is the h-ih. variable then the condition reads a = 0 where a is the h-th column vector of A . If in particular L = M, we have no additional condition, thus no redundant variable, and all the measured variables are nonredundant. Summarizing... 

More generally, if > 0 then the choice of x = x is not arbitrary, and if L < y then the solution in y is not unique. The classification of variables enables one to decide which of the variables x-, (components of x) is, perhaps, still arbitrary thus must be determined a priori so as to determine a unique solution (a nonredundant variable ), and which of the variables (components of y) is, having satisfied the solvability condition, perhaps still uniquely determined by the given x (an observable variable ). It will be shown later (in Chapter 8) that generally (for a nonlinear system), such a verbal classification is somewhat vague. For a linear system, it can be precisely formulated and the classification based on the partition C = (B, A) only, not on the particular choice of x = X . We call a measured variable redundant if its value is uniquely determined by the other neasured variables and the solvability condition, else nonredundant. There are H (redundant) variables X at most whose values are simultaneously determined by the other measured variables values the number H is called the degree of redundancy. We further call an unmeasured variable y observable if it is uniquely determined by x obeying the condition of solvability, else unobservable. The classification criteria ate (7.1.17) and (7.1.18), from where also (7.1.19). 

The classification criterion (7.1.17) requites a matrix projection to obtain some matrix A (7.4.3), but is independent of the particular choice of the transformation (matrix) L whatever be such L A = A = (a), —, aj), if a 0 resp. a] = 0 then X is redundant resp. nonredundant, and if = 0 then all the measured variables are nonredundant by definition. The criterion can also be formulated directly as independent of the transformation (projection) of (B, A), according to (7.1.27) If the original A = (a , Uj) then Xj is nonredundant if the column vector Uj is a linear combination of the columns of matrix B, else redundant. The classification criterion (7.1.18) with (7.1.19) depends on matrix B only If the j-th component of any /-vector u obeying Bu = 0 (thus u KerB) equals zero then y is observable, and if there exists some u KerB such that its j-th component u O then yj is unobservable. An equivalent formulation of the criterion is (7.1.28) The y-th unmeasured variable y is unobservable if the y-th column vector bj of B is a linear combination of the other /-I columns, if not then yj is observable. Thus clearly, if a measured variable x-, is redundant resp. nonredundant then including the variable in the list of the unmeasured ones ( deleting the t-th measurement ), it becomes observable resp. unobservable and vice versa, for an unmeasured variable yj added to the measured ones. 

It is not difficult to show that if the condition (8.1.12) is fulfilled then y and rankB = 2 in the whole admissible region, in particular on iM. In the linearized system, we thus obtain the degree of redundancy equal to 1. On the other hand, so long as > 0, we obtain again the condition (8.1.14) and this condition satisfied, we have m >0 and m > 0. Further, because the two columns of B are linearly independent and because on as shown above, with the third column of (8.5.3) the rank of the matrix equals also 2, the m3-column of the Jacobi matrix is a linear combination of the columns of the new matrix B (8.5.6). According to (7.1.27), at points of Mthe. measured variable m3 can be qualified as nonredundant, in accord with the tentative qualification before formula (8.1.16). But observe that at points zi M, the matrix (8.5.3) is of rank 3, thus in the linearized system, again by (7.1.27), m3 will be qualified as redundant. If... 

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. 

If the admissible region of concentrations is (by the technology) limited by y < a and > b (a < b) then the problem is well-posed in our special example, all the measured variables are nonredundant and the minimized adjustments equal zero. 

Of course also some measured variables, z g Jf can be nonredundant in addition). In (11.1.18) we can have n Nj, >1, thus at least two nodes with accumulation (inventories) have been merged for example... 

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