Unmeasured variable

Chemical process data inherently contain some degree of error, and this error may be random or systematic. Thus, the application of data reconciliation techniques allows optimal adjustment of measurement values to satisfy material and energy constraints. It also makes possible the estimation of unmeasured variables. It should be emphasized that, in today s highly competitive world market, resolving even small errors can lead to significant improvements in plant performance and economy. This book attempts to provide a comprehensive statement, analysis, and discussion of the main issues that emerge in the treatment and reconciliation of plant data. 

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. 

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. 

Romagnoli and Stephanopoulos (1980) proposed an equation-oriented approach. Solvability of the nodal equations was examined and an output set assignment algorithm (Stadtherr et al., 1974) was employed to simultaneously classify measured and unmeasured variables. These ideas were modified to take into account special situations and a computer implementation (PLADAT) was done by Sanchez etal. (1992). 

In this chapter the classification of measurements and unmeasured variables of chemical processes is discussed. After the statement of the problem, variable categorization is posed in terms of a structural analysis of the flowsheet. Then graph-and matrix-based strategies are briefly described and discussed. Illustratives examples of application are included. 

An unmeasured variable, belonging to the subset u, is determinable if it can be evaluated from the available measurements using the balance equations. 

In the previous section we showed that process variables could be divided into vectors x and u, corresponding to measured and unmeasured variables, respectively. Accordingly, linear systems of balance equations can be represented in terms of compatible... 

By a natural extension of the concepts developed in the previous chapter (structural estimability), if the generic rank of the composite matrix (Aj A2) is not less than n (n number of unmeasured variables), then the system does not include structural singularities. Furthermore, if all the unmeasured nodes are determinable, then there are no isolated variables, which cannot be computed from the balance equations. 

Consequently, the following can be stated. The structural pair (Aj A2) is completely solvable with respect to the unmeasured variables, if the following two conditions are satisfied ... 

These two conditions stated for determinability correspond to those for the existence of an output set, given by Steward (1962). The first condition warrants that the number of equations is at least equal to the number of unmeasured variables, while the second condition of accessibility takes into account the existence of a subset of equations containing fewer variables than equations. We have shown that if either of the above two conditions is not satisfied, the structural pair (Aj A2) admits a decomposition analogous to that given in the previous section. Thus the same results are still valid when only the structural aspects are considered. A graphical interpretation of these two conditions is instructive. 

The authors (Meyer et al., 1993) introduced a variant method derived from Kretsovalis and Mah (1987) that allows chemical reactions and splitters to be treated. It leads to a decrease in the size of the data reconciliation problem as well as a partitioning of the equations for unmeasured variable classification. 

Romagnoli and Stephanopoulos (1980) proposed a classification procedure based on the application of an output set assignment algorithm to the occurrence submatrix of unmeasured variables, associated with linear or nonlinear model equations. An assigned unmeasured variable is classified as determinable, after checking that its calculation may be possible through the resolution of the corresponding equation or subset of equations. 

For linear plant models Crowe et al. (1983) used a projection matrix to obtain a reduced system of equations that allows the classification of measured variables. They identified the unmeasured variables by column reduction of the submatrix corresponding to these variables. 

The classification of unmeasured variables and measurements is accomplished by permuting rows and columns of the occurrence matrix corresponding to the Jacobian matrix of the model equations. 

The output set assignment is not unique however, this does not affect the result of the classification. As Steward (1962) has shown, if there is no structural singularity, the determinable unmeasured variables are always assigned independently of the obtained output set assignment. The classification of the unmeasured variables allows us to define the sequence of calculation for these variables. That is, expressions are obtained to solve them as functions of the measurements. The expressions are also used in the classification of the measured variables and in the formulation of the reconciliation equations. After the reconciliation procedure is applied to the measurements, these equations are used to find an estimate of the unmeasured determinable variables in terms of the reconciled measurements. 

After the classification of the unmeasured variables is completed, we need to classify the measured ones. First, the set of equations is divided into two groups ... 

Using a classification algorithm we can determine the measured variables that are overmeasured, that is, the measurements that may also be obtained from mathematical relationships using other measured variables. In certain cases we are not interested in all of them, but rather in some that for some reason (control, optimization, reliability) are required to be known with good accuracy. On the other hand, there are unmeasured variables that are also required and whose intervals are composed of over measured parameters. Then we can state the following problem Select the set of measured variables that are to be corrected in order to improve the accuracy of the required measured and unmeasured process variables. 

Consider a system that after the classification has all the unmeasured variables determinable. Suppose also that the system under study has some overmeasured variables. Then we want to select which of the overmeasured variables need not be measured, while preserving the condition of determinability for the unmeasured variables. That is, we want to minimize the number of measurements in such a way that all the unmeasured variables are determinable. This problem can be stated as... 

In some cases, we do not want all the variables to be determinable only those that are required. Consequently, we must identify which of the measurable variables have to be measured. Let p be the set of variables that for various reasons should be known correctly p may be composed of measured and unmeasured variables. Sometimes we are not interested in the whole system being determinable, so we want to select which of the process variables have to be measured to have complete determinability of the variables in set p. This problem can be stated as follows Select the necessary measurements for the subset of required variables to be determinable. 

In order to classify measurements and unmeasured variables for the process flowsheet in Fig. 8, the following tasks are performed ... 

Application of output set assignment algorithms to classify the unmeasured variables. For the process under study, the type and placement of instruments is such that all unmeasured variables are determinable. 

Formulation of expressions for the unmeasured variables in terms of the measured ones (see Appendix 3-B), using the sequence of calculations that is obtained as a by-product of the assignment procedure. 

Substitution of the determinable unmeasured variables in NA2 by the corresponding expressions in terms of measurements to obtain the set NA2. ... 

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... 

Furthermore, a variable classification strategy based on an output set assignment algorithm and the symbolic manipulation of process constraints is discussed. It manages any set of unmeasured variables and measurements, such as flowrates, compositions, temperatures, pure energy flows, specific enthalpies, and extents of reaction. Although it behaves successfully for any relationship between variables, it is well suited to nonlinear systems, which are the most common in process industries. 

A2 submatrix corresponding to unmeasured variables for linear model equations... 

A-X =f 7 Assigned equation for the estimation of unmeasured variable 7 it belongs to the output... 

Crowe et al. (1983) proposed an elegant strategy for decoupling measured variables from the linear constraint equations. This procedure allows both the reduction of the data reconciliation problem and the classification of process variables. It is based on the use of a projection matrix to eliminate unmeasured variables. Crowe later extended this methodology (Crowe, 1986, 1989) to bilinear systems. 

An equivalent decomposition can be performed using the Q-R orthogonal transformation (Sanchez and Romagnoli, 1996). Orthogonal factorizations were first used by Swartz (1989), in the context of successive linearization techniques, to eliminate the unmeasured variables from the constraint equations. 

The columns of P span the null space of A2, and thus the unmeasured variables are eliminated. 

Since the unmeasured variables do not appear in the remaining equations, the first reduced subproblem becomes the following problem. 

Estimate the unmeasured variables, u, by solving Eq. (4.17) where the components un ru were arbitrarily set. The uniqueness of u is related to the system estima-bility and will be discussed later. 

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