Finding an Output Set

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. 

The reassignment phase continues until a column is reassigned to a row that has not been previously identified with a column. Then the reassigment phase stops and the assignment phase takes up again. 

We now consider how Steward s algorithm can help to ascertain whether or not the system of equations describing the process is determinate. It should be noted that if a system of equations having the same number of variables as equations incorporates a subset of equations that contains fewer variables than equations, a unique solution of the system equations is unlikely to exist. We have used the words is unlikely to rather than does not because there are some special classes of equations that specify more than one variable, and if such an equation is included in the system, the system may have a subset of equations with fewer variables than equations and still be determinate. For example, consider the system of Eq. (5)  

The second and third equations together contain only one variable and are not independent. The system, however, has a unique solution since the first equation specifies that both x and y, if real, must be zero. However, these 

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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... 

Various methods of carrying out the different phases of decomposition are compared in this section. Only one method of finding an output set has been reported in the literature, namely that of Steward (see Section III). 

C M IF THE PROGRAM CANNOT FIND AN OUTPUT SET. A LIST OF EQUATION c NUMBERS IS GIVEN WHICH COMPRISES THE SUBSET IN WHICH THE EQUATIONS C M ARE GREATER IN NUMBER THAN THE NUMBER OF VARIABLES APPEARING IN C ALL OF THE EQUATIONS OF THE SUBSET. THE PROBLEM MUST BE REVIEWED C M TO MAKE IT DETERMINATE. (IF THE NUMBER OF EQUATIONS IS LESS THAN C M THE NUMBER OF VARIABLES. THE PROGRAM WILL PROCESS THE EQUATIONS.)... 

C < THIS SUBROUTINE FINDS AN OUTPUT SET FOR ROWS AND COLUMNS FROM MIN... 

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). 

Ledet (LI) proposed a somewhat different algorithm for tearing based on tearing a variable from all the equations except one in a block. He evaluated the effectiveness of the tears in terms of finding the minimum number of tears to reduce the torn block to subsystems of single equations. In this method no initial output set in a block is chosen, and the tearing and assignment of an output set are carried out simultaneously. For example, consider the block of Eqs. (14), which must be solved simultaneously. We could tear x3 from equations /j and... 

If you cannot find what you need in an output data set from the statistical procedure, use ODS OUTPUT to send your statistics to a data set. To determine... 

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. 

The primary interest in the pole placement literature recently has been in finding an analytical solution for the feedback matrix so that the closed loop system has a set of prescribed eigenvalues. In this context pole placement is often regarded as a simpler alternative than optimal control or frequency response methods. For a single control (r=l), the pole placement problem yields an analytical solution for full state feedback (e.g., (38), (39)). The more difficult case of output feedback pole placement for MIMO systems has not yet been fully solved(40). 

The last step in a fiizzy logic system is defuzzification. As the name suggests, defuzzification is the opposite of fuzzification, which produces crisp output f for a fuzzy logic system from the aggregated output of fuzzy set B. A number of defuzzifiers have been developed the most popular is the centroid defuzzifier, which finds a vertical line and divides an aggregated set into two equal portions. Mathematically the center of gravity (COG) can be defined by ... 

Fuel cell-based power plants that have an output of up to 10 kW are under vigorous development as well, and they find ever wider practical uses. Table 24.3 shows the number of such plants produced every year from 2001 to 2010. Approximately half of the units produced in 2006 had a power of about 1 kW, and the other half had an output of 1.5-10 kW, the numbers being distributed evenly over this time interval. The overwhelming number (more than 50%) of these plants were produced and set up in Japan, with the United States taking the second place. Most of the low-power units were built with polymer electrolyte fuel cells. The fraction of solid oxide fuel cells has decreased gradually. 

These models offer a basic set of movements that far exceeds what a real face can achieve, so a research goal has been to find the natural patterns of movements which constitute an effective set of control parameters. A basic problem concerns where in the overall process the control should be manifested. While we may be able to build a realistic lip model that exhibits the correct degrees of freedom for the lips, the lips themselves are the outputs of other more-fundamental controls such as speaking, smiling and other actions governed by other facial movements. An attractive solution, then, is to use more-fundamental controls based on articulatory degrees of freedom. 

Step 5. After the simulation, you can find the report file in the CAOutput directory. At this point you may wish to rename the output file so that you can relate it to the specific settings employed. You may also wish to remove this file from the CAOutput subdirectory so that this directory does not become too crowded. If you wish to perform a statistical analysis of the simulation data, you can import the data into an appropriate statistical analysis program, such as Microsoft s EXCEL ... 

Besides ODS RTF, there are several other ways to create attractive RTF output from SAS. If you do an Internet search on RTF SAS macro, you will find SAS macros available that convert a rectangular data set of information into attractive proportional-font RTF files. 

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