Observable unmeasured variable

An observable unmeasured variable (yj) is uniquely determined by the measured variables values (obeying the condition of solvability), say (x)... 

We can also examine the confidence intervals for the estimates see Section 10.3, last paragraph. Consider for example the observable unmeasured variable m3. The (pseudo)variance a can be computed by (9.3.63) with (9.3.60) thus by... 

Steady-state process variables are related by mass and energy conservation laws. Although, for reasons of cost, convenience, or technical feasibility, not every variable is measured, some of them can be estimated using other measurements through balance calculations. Unmeasured variable estimation depends on the structure of the process flowsheet and on the instrument placement. Typically, there is an incomplete set of instruments thus, unmeasured variables are divided into determinable or estimable and indeterminable or inestimable. An unmeasured variable is determinable, or estimable, if its value can be calculated using measurements. Measurements are classified into redundant and nonredundant. A measurement is redundant if it remains determinable when the observation is deleted. 

The internal structure of an observer is based on the model of the considered system. Of course, the model can be extremely simple or reduced to a simple algebraic relationship binding available measurements. However, when the model is of the dynamical type, the value of a variable is no longer influenced uniquely by the inputs at the considered moment but also by the former values of the inputs as well as by other system variables. These phenomena are then described by differential equations. Since these models carry information on the interactions between the inputs and the state variables, they are used to estimate unmeasured variables from the readily available measurements. 

With regard to the interval observers, they will be described in detail further on. However, their use for the robust observation of unmeasured variables in the context of a biological WWTP is justified in the following section. 

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 whole vector of unmeasured variables is observable if and only if all the connected components of subgraph G° are trees (or also isolated nodes). 

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. 

More generally, let us consider the case where all the unknown (unmeasured) variables are observable. As shown in Subsection 3.3.1, the necessary and sufficient condition is that the connected components G° k=, —, K) of G [N, J ] are all trees. According to (A.l and 3), this is equivalent to the condition... 

If in particular all the unmeasured variables are observable and no measured variable is redundant, the system is called just determined see Section 3.5. A necessary and sufficient condition is that the subgraph G°[N,J°] is a tree thus a spanning tree of G[N,J]. This is a special case in Section 3.2 where thus G° is connected, the condition (3.6.4) is absent, and = 1 in... 

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. 

Also the analysis presented in Section 3.3 can be generalized. We shall use the same terminology. Irrespective of the way how the value of the subvector x has been actually fixed, we call its components measured, and those of y unmeasured. If z and z" are two vectors (7.1.9), the components of z resp. z" are y] and x resp. y- and x . Let us suppose that some j-th measured variable has the following property Whatever be two solutions z and z" of Eq.(7.1.1), if xl = jcj for ai h i then also Jt = jc". Then the i-th variable is called redundant its value is determined by the other measured values. Let us further suppose that some j-th unmeasured variable has the property Whatever be two solutions z and z" of Eq.(7.1.1) such that x = x", we have y] = y". Then the j-th variable is called observable its value is uniquely determined by the measured values, provided they make the equation solvable. 

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... 

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 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. 

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