Parameter unmeasured

Equation (8-42) can be used in the FF calculation, assuming one knows the physical properties Cl and H. Of course, it is probable that the model will contain errors (e.g., unmeasured heat losses, incorrect Cl or H). Therefore, K can be designated as an adjustable parameter that can be timed. The use of aphysical model for FF control is desirable since it provides a physical basis for the control law and gives an a priori estimate of what the timing parameters are. Note that such a model could be nonlinear [e.g., in Eq. (8-42), F and T t. re multiplied]. 

More accurate information on k3 is obtainable if the equilibrium constant K is also determined at various crown ether concentrations, as shown by Nakazumi et al. (1981, 1983). The results with benzenediazonium tetrafluoroborate and 3- and 4-substituted derivatives demonstrate that k3 is not unmeasurably small, but that ky-values are generally 1-2% of k2 for complexation with 18-crown-6, 0.1-0.5% of k2 with 21-crown-7, and 2-10% of k2 with dicyclohexano-24-crown-8. A dual substituent parameter (DSP) analysis of A 3-values (Nakazumi et al., 1987) showed that the dediazoniation mechanism of the complexed diazonium ions does not differ appreciably from that of the free diazonium ions. 

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. 

Case (1). All unmeasured parameters are estimable and a unique solution for the unmeasured variables is possible using the rectified measured values and the balance equations. 

Parameters and unmeasured variables can be incorporated into this formulation (Alburquerque and Biegler, 1996). Let us consider that we have measured variables, x, and unmeasured variables, u, and also suppose that they are constrained by some physical model and are dependent on unknown parameters, 8. Then we can represent the equalities in Eq. (9.57) by the general form... 

An advanced control system has been implemented for efficient operation of the pyrolysis reactor. However, it faced problems due mainly to the difficulty of measuring the high coil and coil skin temperature reliably and consistently, because of regular drifting of the high-temperature sensors. Thus, there is a need for a data reconciliation package (DRP) to increase the level of confidence in key measured variables, to indicate the status of sensors, and to estimate the value of some unmeasured variables and parameters (Weiss et al., 1996). 

It should be noticed that a choice of s colunms of N other than N is also possible and that, whatever the choice, the existence of is necessary for rebuilding the unmeasured states. However, without loss of generality, it will be assumed that N = klg where k is an arbitrary, real and positive constant parameter. Other reason for this choice will be detailed below. 

As shown in Eq. (23), the heat released (Qr(k )), which cannot be measured, is needed in the GMC algorithm. Here, the EKF algorithm, as used in on-line optimization strategy, coupled with the simplified reactor model, given by [26], is also applied to estimate the heat released (Qr(k)). The reason of using the simplified model, not the exact model of the plant, is because if the exact model were used, too many uncertain/unknown parameters as well as too many unmeasurable states would be involved. That may lead to poor performance of the EKF. Hence, the simplified model with less uncertain/unknown parameters and unmeasurable... 

Mass models seek, by a variety of theoretical approaches, to reproduce the measured mass surface and to predict unmeasured masses beyond it Subsequent measurements of these predicted nuclear masses permit an assessment of the quality of the mass predictions from the various models Since the last comprehensive revision of the mass predictions (in the mid-to-late 1970 s) over 300 new masses have been reported Global analyses of these data have been performed by several numerical and graphical methods These have identified both the strengths and weaknesses of the models In some cases failures in individual models are distinctly apparent when the new mass data are plotted as functions of one or more selected physical parameters ... 

Figure 4.18 Evolution of the process composition variables for a 25% unmeasured increase in the inlet impurity levels y 0 occurring at t = 0, under plant-model parameter mismatch, (a) Product purity and (b) reactor impurity level.

The vector u is an n+m dimensional vector which can be partitioned into two vectors the n-dimensional vector x of measured parameters and the m-dimensional vector of unmeasured ones. Some of the unmeasured variables can be evaluated from the measurement of the others variables using the balance equations, and some not. Thus, the unmeasured parameters may be classified as "determinable" or "indeterminable". On the other hand, some of the elements of vector x of measured variables can be computed from the balances and the rest of the measured parameters. Such measured variables will be called "overdetermined". The rest of the elements of vector x will be called "just determined". Measurement of x is denoted by x, and the difference of any measured system parameter and its true value is called the "error" denoted by 6, i.e. 

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... 

Then for the unmeasured parameters we will have the following two cases ... 

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. 

After the classification of the unmeasured parameters is carried out, the following task is to classify the measured ones. The overmeasured parameters can be found by the following two step procedure. 

Subset E2 of not assigned equations which contain unmeasured but determinable parameters (i.e. they have been assigned as output to equations not included in set E) ... 

Subset E3 of equations not assigned which contain unmeasured and indeterminable parameters. 

Construct the occurrence matrix of the balance equations for the system, dividing the parameters in unmeasured and measured ones. 

Apply an algorithm to assign the unmeasured parameters, as output of the balance equations. (See appendix A)... 

If a given Du contain at least one external unmeasured parameter, then this disjoint subsystem does not add new parameters to M. Repeat for all D fs. 

Selection of the Necessary Measurements for the Required Parameters to be Determinables. It follows from the above development that the parameters in set Mi aure redundant and can be reconciled. Also the parameters in set M2 are not redundant and there is no other indirect way of obtaining their values. Hence this classification of parameters can serve to test a given design of measurement places. There is however another important task concerning this problem. We want ot present an algorithm to select the necessary measurement for a set of required parameters to be completely determinable. Let C be this set of parameters which for various reason should be known. C may be composed of measured and unmeasured parameters. The necessary measurements can be found using the following scheme ... 

Identify the "interval" of the required unmeasured parameters (N ) (see Identification algorithm). 

The set Ny of all the parameters found in 2) constitute the "interval" of required unmeasured parameters. 

All the equations in set E and E2 belong to set E4, after all the unmeasured determinables parameters have been replaced by their "intervals". 

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