Kalman filter of a kinetics model

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 

Time Concentrations Wavelength j Absorp- tivities Absorb- ance Estimate of the concentration of A  

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