Parameter variations variables

The IIEC model was also used to study the importance of various design parameters. Variations in gas flow rates and channeling in the bed are not the important variables in a set of first-order kinetics. The location of the catalytic bed from the exhaust manifold is a very important variable when the bed is moved from the exhaust manifold location to a position below the passenger compartment, the CO emission averaged over the cycle rose from 0.14% to 0.29% while the maximum temperature encountered dropped from 1350 to 808°F. The other important variables discovered are the activation energy of the reactions, the density and heat... 

The parameter for variable batch time is defined by constraint (2.9). This gives the amount of time required to process a unit amount of a batch corresponding to a particular effective state in a corresponding unit operation. Constraint (2.10) denotes the minimum processing time for the effective state in the corresponding unit operation. This is, in essence, the minimum residence time of a batch within a unit operation. In constraints (2.10) and (2.11), v (j n is the percentage variation in processing time based on operational experience. 

Sampling point representativeness is a measure of accuracy and precision, with which sample data represent a characteristic of a parameter or parameter variation at the sampling point. Sampling point representativeness is greatly affected by inherent sample variability. To sustain the sampling point representativeness, we must use the sampling procedures that are appropriate for the nature of contaminant and that will best serve the intended use of the data. These field procedures should... 

To distinguish further between framework residues, which could show the usual mutational noise, and those positions which might be involved in antibody complementarity and thus show much greater variation, a parameter termed variability was defined by Wu and Kabat (1970) ... 

Thus the objective here is a generally applicable simulation of steady, two-dimensional, incompressible flow between rigid rolls with film splitting. The results reported are solutions of the full Navier-Stokes system including the physically required boundary conditions. The analysis is also extended to a shearthinning fluid. The solutions consist of velocity and pressure fields, free surface position and shape, and the sensitivities of these variables to parameter variations, valuable information not readily available from the conventional approach (10). The rate-of-strain, vorticity, and stress fields are also available from the solutions reported here although they are not portrayed. Moreover, the stability of the flow states represented by the solutions can also be found by additional finite element techniques (11), and the results of doing so will be reported in the future. 

While we have now derived from the minimization graph a good estimate of the steady-state gain from parameter variation to measured variable, our ultimate aim is to relate the variance of the parameter to the variance of the output of the distorted model. Recalling the first equality in equation (24.23), repeated below ... 

Let us make the usual assumption that the range of frequencies generated by the variable in controlling the Y vector to match the z vector may be represented by filtered white noise. By this condition we are assuming implicitly that the parameter varies randomly about a central value (a p,j) and that the parameter variation has a limited bandwidth. We envisage the companion model being excited as in Figure 24.3. 

Finding the parameter variations needed to match the behaviour of all recorded variables... 

The time-domain method presented here tackles head-on the problem of finding the parameter variations necessary to cause model matching for all k recorded variables. In some cases, the derivative of the measured state will itself be a state in the model, in which case the behaviour of two state variables may be found from a single record using the simple procedure of differentiation, which may be carried out to very good accuracy off-line. For example, suppose both position, X, and velocity, v = dxjdt, are state variables... 

As to FDI robust with regard to parameter uncertainties, an approach based on so-called uncertain bond graphs in linear fractional transformation form (LFT) has been reported in the literature [8-10] for time-continuous models. In an uncertain bond graph, bonds carry power variables uncertain with regard to parameter variations... 

An incremental bond graph can be constructed in a systematic manner from the original bond graph of a switched LTI system by replacing an element that is due to parameter variations by its incremental element model. Equations for variations of power variables can be automatically derived in the same way as they are derived from an initial bond graph with nominal parameters. 

In order to see how an incremental bond graph model for a bond graph element is obtained, a linear 1-port C-element with the nominal capacitance Cn is considered. In the following, an index n indicates a parameter or a variable of the non-faulty bond graph model with nominal parameters. In the case of a time constant parameter variation AC the linear constitutive equation of a 1-port C element in derivative causality takes the form... 

The power variable fc controlling the modulated source is an output variable of the original bond graph model. If fc has been obtained by measurements of the real system, then the contribution to the output of the incremental bond graph model of the C element may contain sensor noise. In any case, the outputs of the incremental bond graph of a bond graph element indicate a parameter variation. 

If measurement uncertainties can be assumed to be bounded, then application of the triangle inequality may yield thresholds for parameter variations of ARR residuals that are independent of measurement uncertainties. For instance, let z be the predicted part of an output variable z of the diagnostic bond graph that controls a modulated sink of the incremental bond and let Az < be the bounded measurement uncertainty. Furthermore, let Ar be the variation of an ARR residual r that depends on z and its derivative. Then... 

Section 5.3.1 has shown that parameter sensitivities of ARR residuals may be obtained from an incremental bond graph. The latter bond graph can be systematically developed from an initial bond graph with nominal parameters by replacing elements with parameters to be estimated by their incremental component model. Inputs into the incBG are variations of the parameters to be estimated multiplied by a power variable of the initial BG. Outputs may be parameter variations of ARR residuals. They are a weighted sum of the parameter variations and the weighting factors are just the residual sensitivity functions. 

As explained in Sect. 5.3.2, variations of ARR residuals due to parameter variations are obtained by summing increments of power variables at junctions of an incremental bond graph to which a virtual detector has been attached. Replacement of the switch model as well as the other bond graph elements in the DBG of the boost converter in Fig. 8.2 by their incremental model gives the incremental DBG depicted in Fig. 8.14. 

Incremental bond graphs present another approach. Similar to the LFT bond graph approach, they are obtained by replacing elements by their incremental model. The latter one is also a decomposition of a bond graph element into a nominal part and one that accounts for parameter variations. Opposed to LFT bond graphs, bonds of incremental bond graphs carry variations of the conjugated power variables due to parameter variations. 

In contrast to sensitivity pseudo-bond graphs, bonds of incremental bond graphs carry variations of power variables instead of their sensitivities with respect to a parameter. The idea is that a parameter variation A results in a perturbation of both power variables at the ports of an element due to the interaction of the element with the rest of the model [1], Hence, a power variable v t) (either an effort or a flow) has a nominal part v t) and a variation Av(t) due to a parameter change ... 

Let be the variable from the system bond graph controlling the modulated source (sink) representing the jth parameter variation A j in the incremental bond graph. Then, according to (4.14), the output of the modulated source is... 

Figure 6.13 The root mean square error (RMSE) as derived from the calibration within the teaching set (RMSEC) or from evaluating the independent validation set (RMSEP) as a function of the ratio between the number of samples used to teach the algorithm and the number of parameters (latent variables) 5, 10, or 20 samples out of the teaching set were used for teaching. The dashed line indicates that the ratio should be larger than 5 in order to obtain reasonable values for independent validation. For Nteach/Npara < 5 the RMSEC decreases due to overfitting, but the RMSEP reveals that in fact only random variations had been fitted, which were not found in the independent validation set.

The interactions among these variables are in part so complicated that the examples of individual parameter variations provided in the following cannot always be generalized, but instead reflect only certain tendencies. However, they allow appropriate selection of matrix and reinforcing materials, their configuration, and content that lead to acceptable results in processing. 

In this equation, r is a random number following the imiform distribution between 0 and 1 and Aj is a given mrmber representing the user-defined maximum perturbation allowed in the decision variable. In this context, it could control the decision parameters variation considering the constraints in the SC model. For the example in Figure 7.6, the aforementioned mutation is illustrated in Figure 7.7, in which Sq", S2 and S2 represent y/in Eq. (7.3). 

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