Recursive multicomponent analysis

At this point we introduce the formal notation, which is commonly used in literature, and which is further used throughout this chapter. In the new notation we replace the parameter vector b in the calibration example by a vector x, which is called the state vector. In the multicomponent kinetic system the state vector x contains the concentrations of the compounds in the reaction mixture at a given time. Thus x is the vector which is estimated by the filter. The response of the measurement device, e.g., the absorbance at a given wavelength, is denoted by z. The absorbtivities at a given wavelength which relate the measured absorbance to the concentrations of the compounds in the mixture, or the design matrix in the calibration experiment (x in eq. (41.3)) are denoted by h.  

Kalman filter algorithm equations for time-invariant system states 

Step 2. Update of the Kalman gain vector k(l) and variance-covariance matrix P(l) 

These steps are summarized in Tables 41.5 and 41.6. The concentration estimates should be compared with the true values 0.1 and 0.2 respectively. For design B the results listed in Table 41.7 are obtained. From both designs a number of interesting conclusions follow. 

Calculated gain vector, variance covariance matrix and estimated concentrations (see Table 41.5 for the starting conditions) 

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Fig. 41.5. Multicomponent analysis (aniline (jC ), azobenzene (xi), nitrobenzene (xi) and azoxy-benzene (X4)) by recursive estimation (a) forward run of the monochromator (b) backward run (k indicates the sequence number of the estimates solid lines are the concentration estimates dotted lines are the measurements z).

P.C. Thijssen, L.J.P. Vogels, H.C. Smit and G. Kateman, Optimal selection of wavelengths in spectrophotometric multicomponent analysis using recursive least squares. Z. Anal. Chem., 320(1985) 531-540. 

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