Modeling error

Does not introduce instability in the closed-loop response Sensitive to process/model error... 

Other Considerations in Feedforward Control The tuning of feedforward and feedback control systems can be performed independently. In analyzing the block diagram in Fig. 8-32, note that Gy is chosen to cancel out the effects of the disturbance Us) as long as there are no model errors. For the feedback loop, therefore, the effects of L. s) can also be ignored, which for the sei vo case is ... 

It can be shown [4] that the innovations of a correct filter model applied on data with Gaussian noise follows a Gaussian distribution with a mean value equal to zero and a standard deviation equal to the experimental error. A model error means that the design vector h in the measurement equation is not adequate. If, for instance, in the calibration example the model was quadratic, should be [1 c(j) c(j) ] instead of [1 c(j)]. In the MCA example h (/) is wrong if the absorptivities of some absorbing species are not included. Any error in the design vector appears by a non-zero mean for the innovation [4]. One also expects the sequence of the innovation to be random and uncorrelated. This can be checked by an investigation of the autocorrelation function (see Section 20.3) of the innovation. 

The above example illustrates the self adaptive capacity of the Kalman filter. The large interferences introduced at the wavelengths 26 and 28 10 cm have not really influenced the end result. At wavelengths 26 and 28 10 cm , the innovation is large due to the interfered. At 30 10 cm the innovation is high because the concentration estimates obtained in the foregoing step are poor. However, the observation at 30 10 cm is unaffected by which the concentration estimates are restored within the true value. In contrast, the OLS estimates obtained for the above example are inaccurate (j , = 0.148 and JCj = 0.217) demonstrating the sensitivity of OLS for model errors. 

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary. In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package SIMUSOLV. Beeause of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

The error-free likelihood gain, V,( /i Z2) gives the probability distribution for the structure factor amplitude as calculated from the random scatterer model (and from the model error estimates for any known substructure). To collect values of the likelihood gain from all values of R around Rohs, A, is weighted with P(R) ... 

Output errors can be especially insidious since the natural tendency of most model users is to accept the observed data values as the "truth" upon which the adequacy and ability of the model will be judged. Model users should develop a healthy, informed scepticism of the observed data, especially when major, unexplained differences between observed and simulated values exist. The FAT workshop described earlier concluded that rt is clearly inappropriate to allocate all differences between predicted and observed values as model errors measurement errors in field data collection programs can be substantial and must be considered. 

One can identify two major categories of uncertainty in EIA data (scientific) uncertainty inherited in input data (e.g., incomplete or irrelevant baseline information, project characteristics, the misidentification of sources of impacts, as well as secondary, and cumulative impacts) and in impact prediction based on these data (lack of scientific evidence on the nature of affected objects and impacts, the misidentification of source-pathway-receptor relationships, model errors, misuse of proxy data from the analogous contexts) and decision (societal) uncertainty resulting from, e.g., inadequate scoping of impacts, imperfection of impact evaluation (e.g., insufficient provisions for public participation), human factor in formal decision-making (e.g., subjectivity, bias, any kind of pressure on a decision-maker), lack of strategic plans and policies and possible implications of nearby developments (Demidova, 2002). 

The functional relationships that characterize the real process behavior are never known exactly. A conventional way to account for the inaccuracies generated by approximations is to introduce additive noise, which in some sense reflects the expected degree of modeling errors, that is,... 

The vector w is an additive error introduced to account for inaccuracies generated by approximations and reflects the expected degree of modeling errors. 

Table 20.1 summarises the model errors from the validation trials and shows that the model is successful in predicting the steady-state condition of the plant. Errors in waste brine strength and temperature must be compared with the total change across the cell which is about 13% for brine strength and 40°C for temperature. This is because the plant is a waste brine process changes in brine temperature and strength are much smaller for a resaturation process. 

If the model is used then this situation can be improved considerably. In Fig. 20.14 some of the most important cell operating conditions are taken into account when inferring the gap from k-factor. This means that the spread of possible k-factor values that can be associated with that gap is reduced and the alarm point can be reduced from KA to KA and yet still guarantee that the minimum gap constraint will not be violated. Notice that despite the use of the model there is still a range of gaps which could be prevalent when the alarm triggers. This spread is due to modelling errors and variables not used by the model. 

NRTL-SAC has been demonstrated through the case study on Cimetidine as a valuable aid to solubility data assessment and targeted solvent selection for crystallization process design. The average model error is typically 0.5 Ln (x) [1] and is sufficient as a solvent screening tool. Methods that can deliver greater accuracy would increase the value and utility of these techniques. It is impressive in the case of Cimetidine that the NRTL-SAC correlation is capable of reasonable accuracy and predictive capability on the basis of just 2 fitted parameters. Further work to extend the solvent database and optimize the descriptive parameters will be beneficial, and are planned by the developers. 

For example, a) in (radioactivity) counting experiments a non-Poisson random error component, equal in magnitude (variance) to the Poisson component, will not be detected until there are 46 degrees of freedom ( ), and b) it was necessary for a minor component in a mixed Y-ray spectrum to exceed its detection limit by -50 , before its absence was detected by lack-of-fit (x, model error) (7). 

The minimum singular value is a measure of the invertibility of the system and therefore represents a measure of the potential problems of the system under feedback control. The condition number reflects the sensitivity of the system under uncertainties in process parameters and modelling errors. These parameters provide a qualitative assessment of the dynamic properties of a... 

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