Redundant measured variable

Finally, let us consider fi measured instead of /3. The new occurrence matrix is in Table 3. Now, we can assign all the equations by assigning Equations 1, 2, and 3 to /3, /s, and fi, respectively. In this case, all the unmeasured process variables are determinable from the available information however, there are no redundant measured variables. 

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... 

Context-dependent situations often lead to a large-scale input dimension. Because the required number of training examples increases with the number of measured variables or features, reducing the input dimensionality may improve system performance. In addition, decision discriminants will be less complex (because of fewer dimensions in the data) and more easily determined. The reduction in dimensionality can be most readily achieved by eliminating redundancy in the data so that only the most relevant features are used for mapping to a given set of labels. 

Filters are designed to remove unwanted information, but do not address the fact that processes involve few events monitored by many measurements. Many chemical processes are well instrumented and are capable of producing many process measurements. However, there are far fewer independent physical phenomena occurring than there are measured variables. This means that many of the process variables must be highly correlated because they are reflections of a limited number of physical events. Eliminating this redundancy in the measured variables decreases the contribution of noise and reduces the dimensionality of the data. Model robustness and predictive performance also require that the dimensionality of the data be reduced. 

Generally, a significant amount of the information contained in the experimentally measured variables is redundant (correlated with other variables) or nonrele-vant. PCA allows the transformation of this set of variables into a new set of noncorrelated variables, which is more easily interpretable, and underlines the... 

Similarly, some of the elements of vector x of measured variables can be classified into redundant and non-redundant measured process variables (Fig. 2). 

A measured process variable, belonging to subset x, is called redundant (overdetermined) if it can also be computed from the balance equations and the rest of the measured variables. 

In this case we assign Equation 2 to fs (determinable), leaving two unassigned equations, 1 and 3. Variables fo and fi are still not determinable, but now Equation 1 is not assigned and contains only measured variables thus, this is a redundant equation and the associated variables are also redundant or overmeasured. 

The unmeasured determinable variables in set NA2 are then substituted by their corresponding expressions as function of the measured variables and set NA2 is obtained. After this is accomplished, sets NA1 and NA2 contain only measured variables, which are then redundant. The corresponding equations constitute the set of constraints in the reconciliation problem. 

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. 

If A22 0, the system possesses unmeasured variables that cannot be determined from the available information (measurements and equations). In such cases the system is indeterminable and additional information is needed. This can be provided by additional balances that may be overlooked, or by making additional measurements (placing a measurement device to an unmeasured process variable). Also, from the classification strategy we can identify those equations that contain only measured variables, i.e., the redundant equations. Thus, we can define the reduced subsystem of equations... 

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. 

The outlined strategy has been applied to the subsystem of Example 4.4 in Chapter 4. The flow diagram, shown in Fig. 4 of Chapter 4, consists of 7 units interconnected by 15 streams. There are 8 measured flowrates and 7 unmeasured ones. The flowrate measurements with their variances are given in Table 3. In Chapter 4 we identified the subset of redundant equations. In this case it is constituted by one equation that contains the five redundant process variables. By applying the data reconciliation procedure to this reduced set of balances, we obtain the estimates of the measured variables, which are also presented in Table 3. 

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 ... 

Expressions for the unmeasured process variables, as functions of the measured ones, were also obtained. These were solved sequentially after the reconciliation of the redundant measurements was completed ... 

Thus, we have identified the subset of redundant equations containing only measured (redundant) process variables. Applying the data reconciliation procedure to this reduced set of balances we obtain for the estimate of the measured variables... 

There are 7 redundant equations containing all of the 25 measured variables. 

In view of the conflict between the reliability and the cost of adding more hardware, it is sensible to attempt to use the dissimilar measured values together to cross check each other, rather than replicating each hardware individually. This is the concept of analytical i.e. functional) redundancy which uses redundant analytical (or functional) relationships between various measured variables of the monitored process e.g., inputs/outputs, out-puts/outputs and inputs/inputs). Figure 3 illustrates the hardware and analytical redundancy concepts. 

Both the Parameter and Reconcile cases determine (calculate) the same set of parameters. However, these cases do not get the same values for each parameter. A Parameter case has an equal number of unknowns and equations, therefore is considered "square" in mathematical jargon. In the Parameter case, there is no objective function that drives or affects the solution. There are typically the same measurements, and typically many redundant measurements in both the Parameter and Reconcile case. In the Parameter case we determine, by engineering analysis beforehand (before commissioning an online system for instance) by looking at numerous data sets, which measurements are most reliable (consistent and accurate). We "believe" these, that is, we force the model and measurements to be exactly the same at the solution. Some of these measurements may have final control elements (valves) associated with them and others do not. The former are of FIC, TIC, PIC, AIC type whereas the latter are of FI, TI, PI, AI type. How is any model value forced to be exactly equal to the measured value The "offset" between plant and model value is forced to be zero. For normally independent variables such as plant feed rate, tower... 

The correlations between the original features of one set and the canonical variables of the second set are called inter-set loadings. They are also redundancy measures and demonstrate the possibility of describing the first data set with the features of the second set. The inter-set loadings characterize the overlapping of both sets or, in other words, the variance of the first set explained by the second set of features. 

These contributions represent the overall correlations with each canonical feature. Again we find factor patterns which are not very pronounced. The extraction and the redundancy measures are reported in Tab. 5-9. From the total values of the variance explained we see that both sets are well represented by their canonical variables. On the other hand the redundancy measure (90% or 72%) indicates that both feature sets may be of equal practical weight. 

In the last two decades, the researchers interest has been focused mainly on quantitative model-based methods, based on the concept of analytical or functional redundancy, which use a mathematical model of the process to obtain the estimates of a set of variables characterizing the behavior of the monitored system. The inconsistencies between estimated and measured variables provide a set of residuals,... 

Information can be defined as the scatter of points in a measurement space. Correlations between measurement variables decrease the scatter and subsequently the information content of the space [39] because the data points are restricted to a small region of the measurement space because of correlations among the measurement variables. If the measurement variables are highly correlated, the data points could even reside in a subspace. This is shown in Figure 9.2. Each row of the data matrix is an object, and each column is a measurement variable. Here x3 is perfectly correlated with x, and x2, since x3 (third column) equals x3 (first column) plus x2 (second column). Hence, the seven data points lie in a plane (or two-dimensional subspace), even though each point has three measurements associated with it. Because x3 is a redundant variable, it does not contribute any additional information, which is why the data points lie in two dimensions, not three dimensions. 

---