Measurement equation

In this chapter we discuss the principles of the Kalman filter with reference to a few examples from analytical chemistry. The discussion is divided into three parts. First, recursive regression is applied to estimate the parameters of a measurement equation without considering a systems equation. In the second part a systems equation is introduced making it necessary to extend the recursive regression to a Kalman filter, and finally the adaptive Kalman filter is discussed. In the concluding section, the features of the Kalman filter are demonstrated on a few applications. 

Equation (41.11) represents the (deterministic) system equation which describes how the concentrations vary in time. In order to estimate the concentrations of the two compounds as a function of time during the reaction, the absorbance of the mixture is measured as a function of wavelength and time. Let us suppose that the pure spectra (absorptivities) of the compounds A and B are known and that at a time t the spectrometer is set at a wavelength giving the absorptivities h (0- The system and measurement equations can now be solved by the Kalman filter given in Table 41.10. By way of illustration we work out a simplified example of a reaction with a true reaction rate constant equal to A , = 0.1 min and an initial concentration a , (0) = 1. The concentrations are spectrophotometrically measured every 5 minutes and at the start of the reaction after 1 minute. Each time a new measurement is performed, the last estimate of the concentration A is updated. By substituting that concentration in the system equation xff) = JC (0)exp(-A i/) we obtain an update of the reaction rate k. With this new value the concentration of A is extrapolated to the point in time that a new measurement is made. The results for three cycles of the Kalman filter are given in Table 41.11 and in Fig. 41.7. The... 

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. 

For polydisperse samples, the weight average intrinsic viscosity [p])w is measured (Equation (8)) and the so-called viscosity averaged molar mass is obtained ... 

The activity size distributions were determined from the calculated penetration values in the diffusion batteries using the method outlined for aerosol size measurement (equation (6) for RnWL and equations (8) and (9) for 222Pb concentration). 

Recently, the 13C-NMR spectrum of l-mesityl-3,3-dimethylallenyl cation (51) generated from the propargyl alcohol was measured (equation 19)39. The cation exhibits strong shielding for the C+-atom (192 ppm). 

Here 4 is the target state vector at time index k and Wg contains two random variables which describe the unknown process error, which is assumed to be a Gaussian random variable with expectation zero and covariance matrix Q. In addition to the target dynamic model, a measurement equation is needed to implement the Kalman filter. This measurement equation maps the state vector t. to the measurement domain. In the next section different measurement equations are considered to handle various types of association strategies. 

The target state vector tk measured by the multilateration procedure can be considered directly as a target plot input of the association process. In this case, the input of the Kalman filter describes the same parameters that the internal state vector does. It is characteristic for the plot-to-track association procedure that the measurement equation contains directly the target state vector tk which is influenced by noise ftsk only ... 

Alternatively the radar sensor specific measured ranges and velocities ml can be used for a track update. In this case the tracking procedure can even be applied in the low target detection situation where the multilateration process cannot be applied. In the range-velocity-to-track association scheme the corresponding measurement equation is based on range and velocity calculations and has a nonlinear analytical structure,... 

The respective Kalman filter equations for the position correction and prediction steps can now be formulated based on equations (18) and (19), (20) or (21) accordingly for the different mentioned association schemes. Since the measurement equation is nonlinear in case of range-velocity-to-track or frequency-to-track association, the Extended Kalman filter is used for this particular application [16]. 

Gas absorption. See also Absorption ion exchange in, 14 423 measurement equation for, 24 456-457 two-film theory of, 26 154—158 Gas-air explosions, prevention of,... 

The right-hand side of Eq. (6.25) represents the covariance of the residuum when both blocks of measurement equations are used. The first term in the right-hand side is the covariance resulting from considering the information provided by the first set of measurements. Hence, the second term arises as a correction to the estimate, due to the incorporation of new measurements (new information) as it becomes available ... 

In this section, we first consider that, in addition to the measurement equations model (6.33), we have also the conditions defined by (6.35) imposed on the vector of process variables. Furthermore, we will assume, as a more general formulation, that a priori information enters into the estimation problem. 

Now, for the sequential treatment of constraint and measurements equations, consider the system partitioned into two groups as follows ... 

In the following discussion, one or several sensor failures are assumed, so a constant bias of magnitude mb is added to the measurement vector y. In the presence of a failure in the sensors, the measurement equation takes the form... 

Moran LJ, Kimhle JO, Mefferd RB Repetitive psychometric measures equating alternate forms. Psychol Rept. 14 335, 1964. 

In order to analyze the results from polymer characterization measurements, equation II is, normally, transformed into its "specific form" i.e. ... 

In order to assess steric and electronic effects on the bond enthalpies of reaction of phosphine and phosphite ligands to platinum(II), the enthalpies (AH) of reaction of these ligands L on reaction with [PtMe(PPhMe2)2(THF)]PF6 have been measured (equation 412). These authors measured the enthalpies of 34 ligands, and correlated the measured enthalpy with the ligand cone angles.1342,1343 Representative data are shown in Table 8. 

Our study of the kinetics of the reaction of carbon with steam (152) was conducted by the circulation flow method under atmospheric pressure in the temperature-range of 900-1000°C. Dilution with helium was employed to vary the sum of partial pressures of the reaction participants. The experiments were carried out with nonporous graphite of high purity (the content of admixtures did not exceed 10"5%). The roughness factor of graphite was found to be 2-2.5 (from electrochemical measurements). Equation (390) proved not to be obeyed quantitatively the results of the variation of PHl in a broad range by addition of H2 to the gas mixture at the inlet do not form a straight line in the plot 1/r11 vs. PH2 ... 

A 1 -cm path length quartz cuvette is most often used to make absorbance measurements. However, quartz cuvettes with shorter path lengths, 0.01 to 0.5 cm, are available (e.g., from Hellma Cells or Beckman) these shorter path length cuvettes allow higher concentrations of protein solutions to be measured. Equation B1.3.1 and Equation B 1.3.2 assume the cuvette has a path length of 1 cm when cuvettes of shorter path length are used, the correct value for b must be substituted in the equation. 

The value of q0 was determined from electro-osmosis measurements. Equations (64) and (65) were found to satisfy only at 0.01 N solution. On the other hand there was reasonable agreement with equation (69)... 

In this model (shown conceptually in Figure 3.22) a stagnant boundary layer exists on the fluid side of the crystal interface. Across this layer there exists a concentration gradient taken as the bulk fluid concentration (c) minus the interfacial concentration (c, ) in the fluid. Because the interfacial concentration (c,) is difficult to measure, Equations 3.12 and 3.13 are usually combined by eliminating a to obtain ... 

Calorimetric (DSC) measurements yield thermodynamic properties of duplex melting in these oligonucleotides independent of any assumptions concerning the model of melting, such as a cooperative all-or-none process versus a noncooperative, multiple-stage melting process. Comparison of calorimetric enthalpies with van t Hoff enthalpies obtained either from the manipulation of heat capacity curves outlined in equations (16.19) to (16.22), or from optical or NMR measurements [equations (16.14) to (16.17)] allows conclusions to be drawn concerning the size of the cooperative unit. If the two... 

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