Extended error estimation

Error estimation. By indicating with Th and T2h two applications of the extended trapezoid rule with integration steps h and 2h, respectively, the error with step h is estimated ... 

An alternative technique consists of using as the error estimation the difference between the extrapolation obtained with the extended trapezoid formula and the one obtained with the extended central point formula. 

Predictive methods, albeit traditional ones, may be the only option available to obtain a first estimate of these properties. It should be noted that significant errors may be involved since extrapolating (or extending) satisfactory estimation procedures at the macroscale level may not always be reasonable. Notwithstanding these concerns, procedures to estimate some of the key physical and chemical properties in chemical kinetics given below are available in the literature. ... 

Because of the difficulty of obtaining satisfactory photometer records of electron diffraction photographs of gas molecules, we have adapted and extended the visual method to the calculation of radial distribution curves, by making use of the values of (4t sin d/2)/X obtained by the measurement of ring diameters (as in the usual visual method) in conjunction with visually estimated intensities of the rings, as described below. Various tests of the method indicate that the important interatomic distances can be determined in this way to within 1 or 2% (probable error). 

At a later stage, the basic model was extended to comprise several organic substrates. An example of the data fitting is provided by Figure 8.11, which shows a very good description of the data. The parameter estimation statistics (errors of the parameters and correlations of the parameters) were on an acceptable level. The model gave a logical description of aU the experimentally recorded phenomena. 

A second source of error may be in the detector. Detector linearity is an idealization useful over a certain concentration range. While UV detectors are usually linear from a few milliabsorbance units (MAU) to 1 or 2 absorbance units (AU), permitting quantitation in the parts per thousand level, many detectors are linear over only one or two decades of operation. One approach in extending the effective linear range of a detector is high-low injection.58 In this approach, an accurate dilution of a stock sample solution is prepared. The area of the major peak is estimated with the dilution, and the area of the minor peak is estimated with the concentrated stock. This method, of course, relies on linear recovery from the column. Another detector-related source of error that is a particular source of frustration in communicating... 

The next task is to seek a model for the observer. We stay with a single-input single-output system, but the concept can be extended to multiple outputs. The estimate should embody the dynamics of the plant (process). Thus one probable model, as shown in Fig. 9.4, is to assume that the state estimator has the same structure as the plant model, as in Eqs. (9-13) and (9-14), or Fig. 9.1. The estimator also has the identical plant matrices A and B. However, one major difference is the addition of the estimation error, y - y, in the computation of the estimated state x. 

Equations (8.11) and (8.12) are approximate expressions for propagating the estimate and the error covariance, and in the literature they are referred to as the extended Kalman filter (EKF) propagation equations (Jaswinski, 1970). Other methods for dealing with the same problem are discussed in Gelb (1974) and Anderson and Moore (1979). 

The solution of the minimization problem again simplifies to updating steps of a static Kalman filter. For the linear case, matrices A and C do not depend on x and the covariance matrix of error can be calculated in advance, without having actual measurements. When the problem is nonlinear, these matrices depend on the last available estimate of the state vector, and we have the extended Kalman filter. 

Only a few publications in the literature have dealt with this problem. Almasy and Mah (1984) presented a method for estimating the covariance matrix of measured errors by using the constraint residuals calculated from available process data. Darouach et al. (1989) and Keller et al. (1992) have extended this approach to deal with correlated measurements. Chen et al. (1997) extended the procedure further, developing a robust strategy for covariance estimation, which is insensitive to the presence of outliers in the data set. 

The length-distribution studies were recently extended to include an estimate of the radii of gyration of the protein in situ (Ramakrishnan et ai, 1981). The constraint used was that the proteins must have a radius of gyration greater than that of an anhydrous sphere. From their results Ramakrishnan to/. (1981) conclude that only SI and S4 show signs of an extended conformation in situ, whereas 12 proteins (S3, S5, S6, S7, S8, S9, SIO, Sll, S12, S14, S15, and S18) appear quite compact and globular. However, the experimental errors of these estimations are very large. 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

---