Output set

Output set of fully integrated programs and elements and the processes that support them, developed and documented strategy for installation determined... 

The Boltzman Machine generalizes the Hopfield model in two ways (1) like the simple stochastic variant discussed above, it t(>o substitutes a stochastic update rule for Hopfield s deterministic dynamics, and (2) it separates the neurons in the net into sets of visible and hidden units. Figure 10.8 shows a Boltzman Machine in which the visible neurons have been further subdividetl into input and output sets. 

Pseudo-Code Implementation The Boltzman Machine Learning Algorithm proceeds in two phases (1) a positive, or learning, ph2 se and (2) a negative, or unlearning, phtise. It is summarized below in pseudo-code. It is assumed that the visible neurons are further subdivided into input and output sets as shown schematically in figure 10.8. 

In general for a set of nonlinear equations the necessary condition for determinancy is that there exists at least one admissible set of output variables for the equations (C7, S4). We can think of an output variable as that variable for which a given equation is solved either by an iteration process or by an elimination process. The set of all such assigned pairs of variables and equations is called the output set. Clearly an admissible output set must satisfy the conditions (i) each equation has exactly one output variable, (ii) each variable appears as the output variable of exactly one equation, (iii) each output variable must occur in the assigned equations in such a manner that it can be solved for uniquely. Such an output set (circled entities) is illustrated in terms of an occurrence matrix in Fig. 4. Algorithms for finding output sets have been published by Steward (S4) and Gupta et al. (G8). 

A defuzzifier is the opposite of a fuzzifier it maps the output sets into crisp numbers, which is essential if a fuzzy logic system is to be used to control an... 

Blog Entry 4 The best way to do this is to put in a low dropout linear regulator on the output. Set the Boost regulator output for 12.5-12.7V, and then use an LDO to drop your voltage to 12 V. 

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). 

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. 

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. 

Correspond to the Maximal Loops in the Adjacency Matrix and are Invariant of the Output Set. 200... 

We first describe in Section II how the information flow takes place in process models, give a compact method of representation of the system of equations, and point out the correspondence between a system of equations and a linear diagraph. In Section III, methods for finding an output set... 

In decomposing a system of equations, it is necessary to analyze the information flow among the equations concerning the values of the system variables (S3). In order to determine the direction of information flow in the system equations, one must first establish what information each equation is to supply, that is, the identity of the variable whose value is to be obtained from the equation. Further, the system equations together must supply all of the information about the system (the values of all of the variables). The variable for which an equation is to be solved is called its output variable and the set of all of the variables assigned to the equations as output variables is called an output set (S3, H2). Thus the information that one equation can... 

Once an output set has been established, the direction of information flow is fixed in the system of equations, and they can be represented either by a linear diagraph or its associated Boolean adjacency matrix. For our purposes it is more convenient to work with the Boolean adjacency matrix, which can be obtained directly from the occurrence matrix and output set as follows. First, assign numbers to the equations that correspond to the rows, and numbers to the variables that correspond to the columns of the occurrence matrix as in Fig. 5. Then pick an output set by the methods described in Section III. For the first equation, and the number of the column containing its output variable. The information flow transmitted by the variable designated by the column number goes to all other equations that have nonzero... 

The concept of an output set was introduced in Section II. In this section we discuss how to find an output set, and how the procedure that is presented can help in ascertaining system determinacy. 

The algorithm that has turned out to be the most effective in practice for finding an output set is that proposed by Steward (S3). Steward proved that application of the algorithm to a system of equations will lead to one of two outcomes. The first is that an output set will be found. The other is that a subset of equations is found that contains fewer variables than equations, in which case Steward has shown that no output set exists for the system of equations. 

Steward s algorithm, thus, can help locate errors in formulating a model of a process by identifying the set of equations that contains improperly specified design variables or parameters. The fortran program listed in Appendix A prints out the equation numbers in the subset that contains fewer variables than equations when an output set cannot be found. 

Neither equation can be solved independently because both equations contain the same variables. In terms of information flow, regardless of the output set chosen, the first equation must feed information to the second equation and the second to the first, constituting a loop of information flow. For any number of equations, as long as each equation feeds information to the next equation in sequence, and the last equation feeds information to the first equation, the whole system of equations has to be solved simultaneously. If a set of equations comprises part of a larger set of equations, which themselves form a larger loop of information flow, the subset must be solved together with the bigger system. Thus, any set of equations in which each equation is included in a maximal loop of information flow must be solved simultaneously. 

Previously we have termed the largest loop of information flow a maximal loop, and indicated that it is not tied into other loops, by definition. One might wonder, because the choice of an output set is not unique, whether the maximal loops of information flow in one adjacency matrix will differ from those of another adjacency matrix, i.e., one formed from a different output set. It is shown in the following paragraphs that the maximal loops will be the same and therefore any output set will suffice for accomplishing the partitioning. 

Consider the relationship between output sets. Steward (S3) has shown that all of the output sets of a system of equations can be generated from one output set and the loops of information flow for that set. He showed that the only way another output set could be obtained would be to reassign the output variable of each equation in a loop to the equation in that loop which it feeds. For example, consider the following set of three equations ... 

Suppose X is assigned as the output variable of fu x2 as the output variable of f2, and x3 as the output variable of /3. With this output set, /j feeds information to f2, f2 feeds information to /3, and /3 feeds information to fx constituting a loop of information flow. 

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