Unobservable unmeasured variable

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... 

As an example, recall Fig. 3-6 where the j -th variable is nonredundant. As drawn, all the unmeasured variables are observable. Note that if we deleted the j -th measurement, the three unmeasured streams j , i, k forming a circuit would become unobservable. 

Generally, in a nonlinear system the solvability with respect to the unmeasured variables depends on the measured values and, as is frequently the case, the system is not solvable unless the measured values are adjusted by (nonlinear) reconciliation. In the above example however, a detailed analysis would show that the (iterative) reconciliation procedure would not converge for reasons of principle. Instead of speaking of observability/unobservability, one would rather say that the problem is not well-posed. See chapter 8 for more detailed discussion. 

The definition of an observable variable thus depends only on the matrix B in the partition (7.1.10), not perhaps on the fixed value x". An unmeasured variable that is not observable is called unobservable. From (7.1.18) follows... 

The l-th unmeasured variable is unobservable if and only if there exists some J-vector u such that... 

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. 

In the example in Section 5.5, we have shown that the whole system can be considered observable with the exception of certain special values of the measured variables. On the other hand, in the last example we have shown that an unmeasured variable is unobservable (not uniquely determined) with the exception of certain special measured values. Thus in the first case, a variable is almost always observable , in the second case almost always unobservable . 

That means that some set of L (= rankB) columns of B(z), not comprising the j-th column constitutes a basis of ImB(z) the L column vectors are thus linearly independent. By standard arguments, one concludes that the equality (8.5.45) holds true also in some neighbourhood of point z. We thus can state that the variable yj is not observable. It is not even locally observable, because the condition (8.5.43) is necessary even for local observability. The statement (8.5.45) thus disqualifies the j-th unmeasured variable we cannot expect that with an arbitrary measured (and adjusted) x, the value of yj will be determined. It can happen that the condition (8.5.43) is fulfilled at certain particular values of z, and even that the yj-value is uniquely determined by some it, see the example 4 in Section 8.1, Fig. 8-2. But such case is exceptional, due to some coincidence. It is left to the reader s taste, if he then will call the variable unobservable or perhaps observable at some x. 

Thus for well-posed problems (and analytic manifolds), the classification observable / not observable (unobservable) is almost complete . The remaining cases ( exceptional observability ) represent mere coincidence. Briefly, an unmeasured variable is... 

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 merit of the formula (9.2.15) consists in that it applies to any model (9.2.1) with any partition (9.2.2) we can have rankB = L < Af, in which case some unmeasured variables are unobservable thus not uniquely determined by the reconciled x. We shall now give other formulae restricted, however, to the... 

The problem of minimum as formulated above can be solved by sequential methods of nonlinear (in particular quadratic) programming. The idea of the sequential approach consists, most simply, in linearizing the equation g(z) = 0 at point z of the sequence and subjecting the linearized constraint equation to a minimum condition thus the next approximation is found, and so on. Some problems can arise when the whole unmeasured vector y is not observable (not uniquely determined) although the latter case is less frequent in practice, possibly it can happen that the values of some unmeasured variables are not required and admitted as unobservable (undetermined). In what follows we shall outline two methods that do not require the full observability of vector y. 

Given the topology of the process and the placement of the sensor, variable classification procedure aims to classify measured variables as redundant and non-redundant and the unmeasured variables as observable and unobservable. This is an... 

In practice we can meet with even more complicated situations - see Fig. 2-7d. Streams 1, 2, 4 and 5 are measured and redundant (one stream can be calculated from the others). Stream 6 is measured, but nonredundant. Streams 3 and 7 are unmeasured and observable. Streams 8 and 9 are unmeasured and unobservable. The general classification of balancing variables is presented in Fig. 2-8. Anyway, we can see that even in a relatively simple flowsheet with the... 

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