Output set assignment

In this chapter, the mathematical formulation of the variable classification problem is stated and some structural properties are discussed in terms of graphical techniques. Different strategies are available for carrying out process-variable classification. Both graph-oriented approaches and matrix-based techniques are briefly analyzed in the context of their usefulness for performing variable categorization. The use of output set assignment procedures for variable classification is described and illustrated. 

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). 

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. 

A more detailed description of an update strategy based on the use of output set assignments will be presented in the next main section. 

To classify the variables, one must first establish what information each equation is to supply, that is, to obtain an output set assignment for the balance equations. 

The output set assignment assigns to any unmeasured process variable one equation, or to two or more variables the same number of equations. This is equivalent to transforming the original undirected graph to a directed one. 

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. 

When the balance equations are formulated around individual units only, it is possible that the classification by output set assignment may not be satisfactory. Some variables classified as indeterminable may actually be determinable if we consider additional balances around groups of units. An erroneous measurement classification is also possible. The problem is in the system of equations used in the classification rather than in the assignment method. The most common problem arises because of the presence of parallel streams between two units. 

According to the output set assignment approach, flowrates fa, fa, and fa are indeterminable from this set of equations. However, if one of these equations is substituted by a balance around units 1 and 2, the result is different. In this case the set of balances is given by... 

A similar situation arises for measurement categorization when fa and fa are measured. Although fa and fa constitute an output set assignment for the individual... 

By obtaining the output set assignment on the previous set of balances we can classify the unmeasured process variables as determinable or nondeterminable. The results are given in Table 5 this classification is coincident with those from other works cited in the literature. 

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. 

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. 

Where, x and y are the measured and unmeasured parameters respectively, and A and A2 are compatibles matrices. The topology of the balance equations is represented by the structure of these matrices. In order to classify the parameters one must first establish what information each equation is to supply, that is, to obtain the output set assignment for the balance equations. With the output set assignment we assign to any unmeasured... 

The output set assignment is not unique but whether an unmeasured parameter is determinable or not does not depend on it. 

As Steward (13) has proved, all the unmeasured variables will be assigned to every possible output set assignment if they are determinables, i.e. if there is no structural singularity. This result is general and does not depend on the functional form of the balance equations. Consequently, except for isolated numerical singularities the determinability of an unmeasured parameter is specified from whether it can be assigned in an output set assignment or not. 

First Step. From the output set assignment we obtain equations that are not assigned. This set E of not assigned equations is composed of three subsets of equations. 

This simple stochastic criterion is now imbeded in a generalized structural search. The set of linearized balance equaticns is represented by a directed bipartite graph. We develop an output set assignment with the output from equation i and... 

The variable influence path leading to x, is shown in Fig. 14. As a consequence, x, can only be influenced by variables that appear on the path causality is established by the input variables. This occurs because the output set assignment given to the set of relationships in the structural matrbc is unique, as it is made evident by rearranging the tabulation order of relationships and variables comprising the structural matrbc ... 

Unique output set assignment occurs when the structural incidence matrix is triangular, such as the above. As a consequence, the variable-influence path leading to the top-level output variable (e.g., x,) is also unique and can be affected only by inputs that occur along the path. 

Alternatively output set assignments, and consequently variable-influence paths, result from the block structure established by variables Xj, x, and x. Thus, variable x, can take its value from any one of the three relationships, since there is no way that x, can be uniquely and unambiguously assigned as an effect to any one of these three relationships. 

To break the loop of cause-and-effect relationships, we make the unqualified assumption that Op2, the variable denoting the human operator, obtains its value from relationship (17) i.e., the operator responds to the temperature in the reactor (T ) and not vice versa. This assumption is then catalogued, as Assumption-1, so that it may be retracted at a later stage, as required. Such retraction allows the causality that emanates from the set of process equations to be modified. Once we have eliminated Eq. (17) and the column corresponding to variable Op, we can repeat the steps of the output set assignment (see Section IV.B), and find that F, obtains its value from Eq. (15). The set of input and output assignments, made so far, establish the causality between... 

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